Analysis of Bond Risk Premia: Extensions to Macro

Transcription

Analysis of Bond Risk Premia:
Extensions to Macro-Finance and Multi-Currency Models
DISSERTATION
of the University of St. Gallen,
School of Management,
Economics, Law, Social Sciences
and International Affairs
to obtain the title of
Doctor of Philosophy in Management
submitted by
Lukas Wäger
from
Ernetschwil (St. Gallen)
Approved on the application of
Prof. Paul Söderlind, PhD
and
Prof. Dr. Karl Frauendorfer
Dissertation no. 4052
ADAG Copy AG, Zürich 2012
The University of St. Gallen, School of Management, Economics, Law, Social Sciences and International Affairs hereby consents to the printing of the present dissertation, without hereby expressing any opinion on the views herein expressed.
St. Gallen, May 11, 2012
The President:
Prof. Dr. Thomas Bieger
Acknowledgements
First and foremost, I would like to express my sincere gratitude to my supervisor,
Prof. Paul Söderlind, PhD, for his great support of my doctoral studies and the
many valuable comments and discussions. Moreover, I would like to thank my cosupervisor, Prof. Dr. Karl Frauendorfer, for his support and encouragement in an
early stage of my research.
Furthermore, I am indebted to Dominik Boos for numerous interesting and fruitful discussions about interest rate models and for proofreading my final manuscript.
Additional thanks go to all my friends and colleagues from work, university and private live for supporting me and my studies during all these years. Finally, I would
like to express my highest gratitude to my parents, my brother, my sister and last but
not least Christiane for their ongoing support.
Zürich, May 2012
Lukas Wäger
iii
iv
Summary
The focus of this thesis is on bond return predictability and providing an empirical
and economic understanding of bond risk premia. The thesis consists of an empirical
analysis of time-varying bond risk premia along three major branches of the current
term structure literature, namely yields-only, macro-finance and multi-currency term
structure models. All these models belong to the well-known class of affine models
introduced by Ang and Piazzesi (2003), whereas the latter two embed unspanned
factors. Unspanned factors are state variables that have an effect on bond risk premia
but do not span the cross-section of yields, as recently introduced by Duffee (2011),
Joslin, Priebsch and Singleton (2011) and Boos (2011).
The section concerning yields-only models contributes by providing evidence
of three priced risk premia of bonds in the US market, extending the analysis of
Cochrane and Piazzesi (2005) and Boos (2011). The section concerning macrofinance models adds to the new branch of models with unspanned macro factors and
extends existing research by analyzing the effects of unspanned macro factors on
risk premia beyond the level risk premium and extending into a broader and longer
data set of macroeconomic variables. The section concerning multi-currency models firstly introduces unspanned factors into international models by taking mutually
unspanned latent yield curve factors of domestic and foreign countries as state variables. The information in foreign yield curves is found to be partly unspanned by the
domestic yield curve and improves bond return predictability beyond local models.
v
vi
Contents
1
Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Field of Research . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Contribution and Results . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Yields-Only Models . . . . . . . . . . . . . . . . . . . . .
5
1.3.2
Macro-Finance Models . . . . . . . . . . . . . . . . . . . .
6
1.3.3
Multi-Currency Models . . . . . . . . . . . . . . . . . . .
8
Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
I
Yields-Only Models
13
2
Yield Regressions
15
2.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Yield Curve Data . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
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viii
CONTENTS
2.3
2.4
2.5
3
Level Risk Premium . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.1
Extended Time Series . . . . . . . . . . . . . . . . . . . .
20
2.3.2
Extended Cross-Section . . . . . . . . . . . . . . . . . . .
21
2.3.3
Rolling Window and Subsample Analysis . . . . . . . . . .
26
2.3.4
Quarterly Forecasting Horizon . . . . . . . . . . . . . . . .
29
2.3.5
Robustness Checks with a Different Data Set . . . . . . . .
29
Slope and Curvature Risk Premia . . . . . . . . . . . . . . . . . . .
30
2.4.1
Slope and Curvature Portfolios . . . . . . . . . . . . . . . .
31
2.4.2
Forecasting Slope and Curvature . . . . . . . . . . . . . . .
33
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Affine Term Structure Models
37
3.1
Why an Affine Model? . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.3
Valuation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.3.1
Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.3.2
Model Expectations . . . . . . . . . . . . . . . . . . . . .
42
Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . .
43
3.4.1
Self-Consistent Models . . . . . . . . . . . . . . . . . . . .
44
3.4.2
Forwards as State Variables . . . . . . . . . . . . . . . . .
45
3.4.3
Restricting the Number of Forecasting Factors . . . . . . .
47
3.4
CONTENTS
3.5
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.6
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.6.1
Quarterly Models . . . . . . . . . . . . . . . . . . . . . . .
51
3.6.2
Models Estimated on Different Frequencies . . . . . . . . .
58
3.6.3
Comparison to Regression Forecasts . . . . . . . . . . . . .
59
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.7
II
4
Macro-Finance Models
63
Macro Regressions
65
4.1
Macroeconomic data . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.2
Forecasting Bond Returns with Macro Data . . . . . . . . . . . . .
70
4.2.1
Unrestricted Forecasts . . . . . . . . . . . . . . . . . . . .
71
4.2.2
Principal Components of Groups . . . . . . . . . . . . . . .
71
Unspanned Macro Factors . . . . . . . . . . . . . . . . . . . . . .
73
4.3.1
Rolling Window and Subsample Analysis . . . . . . . . . .
76
4.4
Spanned Macro Factors . . . . . . . . . . . . . . . . . . . . . . . .
78
4.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
4.3
5
ix
Macro-Finance Term Structure Models
83
5.1
84
Macro-Finance Models . . . . . . . . . . . . . . . . . . . . . . . .
x
CONTENTS
5.1.1
5.2
5.3
5.4
III
6
Unspanned Factors . . . . . . . . . . . . . . . . . . . . . .
85
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5.2.1
Parametrization . . . . . . . . . . . . . . . . . . . . . . . .
87
5.2.2
Price of Unspanned Risks . . . . . . . . . . . . . . . . . .
89
5.2.3
Restricting the Number of Priced Risk Factors . . . . . . .
90
5.2.4
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.3.1
Model Estimates . . . . . . . . . . . . . . . . . . . . . . .
92
5.3.2
Comparison to Regression Forecasts . . . . . . . . . . . . .
95
5.3.3
Comparison to Yields-Only Models . . . . . . . . . . . . .
97
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Multi-Currency Models
International Regressions
103
105
6.1
International Yield Curve Data . . . . . . . . . . . . . . . . . . . . 106
6.2
Individual Forecasting Regressions . . . . . . . . . . . . . . . . . . 107
6.3
6.2.1
Unconditional risk premia . . . . . . . . . . . . . . . . . . 108
6.2.2
Conditional risk premia
. . . . . . . . . . . . . . . . . . . 108
International Forecasting Regressions . . . . . . . . . . . . . . . . 111
6.3.1
International Yield Curve Factors . . . . . . . . . . . . . . 114
CONTENTS
6.3.2
6.4
7
International Risk Premia . . . . . . . . . . . . . . . . . . 116
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Multi-Currency Term Structure Models
7.1
7.2
7.3
8
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127
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.1.1
General Two-Country Model . . . . . . . . . . . . . . . . . 130
7.1.2
Foreign Yields as Unspanned Factors . . . . . . . . . . . . 134
7.1.3
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.1.4
Restrictions on Risk Premia . . . . . . . . . . . . . . . . . 138
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2.1
Single-Country Models . . . . . . . . . . . . . . . . . . . . 141
7.2.2
Two-Country Models . . . . . . . . . . . . . . . . . . . . . 145
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Conclusion
161
Appendices
165
A Yield Curve Data
167
A.1 Term Structure Estimation Methods . . . . . . . . . . . . . . . . . 168
A.2 Unsmoothed Fama-Bliss Method . . . . . . . . . . . . . . . . . . . 169
A.3 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
xii
CONTENTS
B Estimation of Term Structure Models
175
B.1 State Space Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2 Backing-out State Variables from Yields . . . . . . . . . . . . . . . 177
B.3 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.4 Specification Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C Valuation Formulas
181
C.1 Bond Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.2 Yields and Forward Rates . . . . . . . . . . . . . . . . . . . . . . . 182
C.3 Forecast of Prices, Yields and Forward Rates . . . . . . . . . . . . 184
C.4 Expected Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
C.5 Multi-Period Expected Returns . . . . . . . . . . . . . . . . . . . . 185
D Yields-Only Models
187
D.1 Invariant Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
D.2 Other Choices for Self-Consistent Models . . . . . . . . . . . . . . 188
D.2.1 Bond Prices as State Variables . . . . . . . . . . . . . . . . 188
D.2.2 Yields as State Variables . . . . . . . . . . . . . . . . . . . 189
D.3 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 190
D.4 Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 192
D.5 Forecasting Power of Additional Models . . . . . . . . . . . . . . . 192
CONTENTS
xiii
E Macro-Finance Models
199
E.1 Yield Curve Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
E.2 Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 200
F Multi-Currency Models
203
F.1
Exchange Rate Return Forecasts . . . . . . . . . . . . . . . . . . . 203
F.2
Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . 204
F.3
Additional Tables and Figures . . . . . . . . . . . . . . . . . . . . 205
Bibliography
233
xiv
CONTENTS
List of Figures
2.1
Yield curve based on CBD data set . . . . . . . . . . . . . . . . . .
17
2.2
Loadings of forecasting regressions . . . . . . . . . . . . . . . . .
23
2.3
R2 of 20 year rolling forecasting regressions . . . . . . . . . . . . .
27
2.4
Loadings of different subsamples . . . . . . . . . . . . . . . . . . .
28
2.5
PCA of yields and excess returns . . . . . . . . . . . . . . . . . . .
31
3.1
Forecasted vs. realized returns of the A(4, 3) model . . . . . . . . .
56
3.2
Model forecasts vs. regression forecasts . . . . . . . . . . . . . . .
57
4.1
R2 of rolling window analysis
. . . . . . . . . . . . . . . . . . . .
79
5.1
Forecasted vs. realized returns of the B(4, 2, 3) model . . . . . . . .
96
5.2
Macro vs. yields-only model forecasts . . . . . . . . . . . . . . . . 100
6.1
Loadings of restricted regressions across countries . . . . . . . . . . 110
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LIST OF FIGURES
7.1
Comparison of expected yield curves in the conundrum period . . . 140
7.2
Comparison of risk premia across countries . . . . . . . . . . . . . 144
7.3
Decomposition of risk premia into domestic and foreign parts . . . . 153
7.4
International bond level risk premia . . . . . . . . . . . . . . . . . 156
7.5
A simple trading rule application . . . . . . . . . . . . . . . . . . . 158
D.1 Standard deviation of measurement errors . . . . . . . . . . . . . . 193
F.1
Forecasted vs. realized foreign exchange returns of the C(3, 6) model 209
F.2
Comparison of the C(3, 6) and CnoFX (3, 6) foreign exchange forecasts 210
F.3
Forecast of US bond returns in the USD-EUR model . . . . . . . . 215
F.4
Forecast of German bond returns in the USD-EUR model . . . . . . 216
F.5
Forecast of US bond returns in the USD-GBP model . . . . . . . . 217
F.6
Forecast of UK bond returns in the USD-GBP model . . . . . . . . 218
F.7
Forecast of German bond returns in the EUR-GBP model . . . . . . 219
F.8
Forecast of UK bond returns in the EUR-GBP model . . . . . . . . 220
List of Tables
2.1
Sample statistics for US bond data sets . . . . . . . . . . . . . . . .
19
2.2
R2 of forecasting regressions . . . . . . . . . . . . . . . . . . . . .
21
2.3
T-statistics for forecasting regressions . . . . . . . . . . . . . . . .
24
2.4
R2 of different instrument subsets . . . . . . . . . . . . . . . . . . .
25
2.5
R2 of principal components as factors . . . . . . . . . . . . . . . .
25
2.6
R2 of different subsamples . . . . . . . . . . . . . . . . . . . . . .
28
2.7
R2 of forecasting level, slope and curvature portfolios . . . . . . . .
34
3.1
Forecasting power of the quarterly three-factor models . . . . . . .
52
3.2
Forecasting power of the quarterly four-factor models . . . . . . . .
53
3.3
Forecasting power of the quarterly five-factor models . . . . . . . .
54
3.4
Likelihood ratio tests of the quarterly models . . . . . . . . . . . .
55
3.5
Yield curve fit of the dominant models . . . . . . . . . . . . . . . .
58
3.6
Forecasting power of the A(4, 3) model at different frequencies . . .
58
xvii
xviii
LIST OF TABLES
3.7
Model forecasts vs. regression forecasts . . . . . . . . . . . . . . .
59
3.8
Correlation of model and regression forecasts . . . . . . . . . . . .
60
4.1
Macro forecasts in comparison to forward rates . . . . . . . . . . .
72
4.2
Macro forecast t-statistics . . . . . . . . . . . . . . . . . . . . . . .
72
4.3
R2 of combined macro-finance forecasting regressions . . . . . . .
74
4.4
R2 and t-statistics for macro factors regressed onto yield factors . . .
75
4.5
R2 of forecasting regressions with unspanned macro factors . . . . .
76
4.6
T-statistics for forecasting regressions with unspanned macro factors
76
4.7
R2 of various subsamples . . . . . . . . . . . . . . . . . . . . . . .
77
4.8
R2 and t-statistics for yields regressed onto macro variables . . . . .
80
5.1
Forecasting power of the quarterly five-factor models . . . . . . . .
93
5.2
Forecasting power of the quarterly six-factor models . . . . . . . .
94
5.3
Likelihood ratio tests of the quarterly models . . . . . . . . . . . .
94
5.4
Model vs. regression forecasts . . . . . . . . . . . . . . . . . . . .
97
5.5
Forecasting power of the restricted macro models . . . . . . . . . .
98
5.6
Likelihood ratio tests of the quarterly BA models . . . . . . . . . . .
98
5.7
Correlation of macro and yields-only model forecasts . . . . . . . .
99
5.8
Macro part of model-implied risk premia . . . . . . . . . . . . . . . 101
6.1
Sample statistics for bond data sets . . . . . . . . . . . . . . . . . . 109
LIST OF TABLES
xix
6.2
R2 of single-country forecasting regressions . . . . . . . . . . . . . 112
6.3
T-statistics for single-country restricted forecasting regressions . . . 113
6.4
Explained variance of principal components . . . . . . . . . . . . . 115
6.5
Correlation of local and international principal components . . . . . 116
6.6
Correlation of bond excess returns . . . . . . . . . . . . . . . . . . 117
6.7
R2 of unrestricted international forecasting regressions . . . . . . . 119
6.8
R2 of restricted international forecasting regressions . . . . . . . . . 120
6.9
R2 of international forecasting regressions with principal components as instruments . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.10 Spanning of the foreign yield curve factors by domestic factors . . . 122
6.11 R2 of unspanned international forecasting regressions . . . . . . . . 123
7.1
Forecasting power of the quarterly three-factor single-country models 141
7.2
Likelihood ratio tests of the quarterly single-country models . . . . 142
7.3
Correlation of risk premia across countries . . . . . . . . . . . . . . 145
7.4
Forecasting power of the quarterly two-country C models . . . . . . 147
7.5
Likelihood ratio tests of the quarterly two-country C models . . . . 148
7.6
Correlation of risk premia in the USD-EUR model . . . . . . . . . 149
7.7
Correlation of risk premia in the USD-GBP model . . . . . . . . . . 150
7.8
Correlation of risk premia in the EUR-GBP model . . . . . . . . . . 150
7.9
Variance decompositions of bond risk premia . . . . . . . . . . . . 152
xx
LIST OF TABLES
7.10 Likelihood ratio tests of the CnoFX and CL models . . . . . . . . . . 154
7.11 Uncovered interest rate parity tests . . . . . . . . . . . . . . . . . . 155
D.1 Parameter estimates of the A(4, 3) model . . . . . . . . . . . . . . . 191
D.2 Forecasting power of the semi-annual models . . . . . . . . . . . . 194
D.3 Forecasting power of the annual models . . . . . . . . . . . . . . . 195
D.4 Likelihood ratio tests of the semi-annual models . . . . . . . . . . . 196
D.5 Likelihood ratio tests of the annual models . . . . . . . . . . . . . . 197
E.1 Yield curve fit of the dominant models . . . . . . . . . . . . . . . . 199
E.2 Parameter estimates of the B(4, 2, 3) model . . . . . . . . . . . . . 201
F.1
Forecasting power of the quarterly four-factor single-country models 205
F.2
Likelihood ratio tests of the quarterly single-country models . . . . 206
F.3
T-statistics for international forecasting regressions . . . . . . . . . 207
F.4
Comparison of USD-EUR and EUR-USD two-country models . . . 208
F.5
Forecasting power of the quarterly two-country CnoFX models . . . 211
F.6
Likelihood ratio tests of the quarterly two-country CnoFX models . . 212
F.7
Forecasting power of the restricted quarterly two-country CL models 213
F.8
Likelihood ratio tests of the restricted quarterly two-country CL models214
F.9
Parameter estimates of the USD-EUR C(3, 4) model . . . . . . . . 221
F.10 Parameter estimates of the USD-GBP C(3, 4) model . . . . . . . . . 222
LIST OF TABLES
xxi
F.11 Parameter estimates of the EUR-GBP C(3, 3) model . . . . . . . . . 223
xxii
LIST OF TABLES
Chapter 1
Introduction
1.1
Motivation
Term structure modeling is a broad field in finance. Reasons to be concerned about
yield curves range from debt and monetary policy to forecasting, hedging and pricing of all kinds of fixed income instruments and derivatives thereof. The focus of
this thesis is on forecasting bond returns and providing an empirical and economic
understanding of bond risk premia. Bond forecasting is of major interest for any
practitioner dealing with portfolio allocation and management of fixed income securities since risk premia on these instruments are time-varying. An understanding of
bond risk premia is thus crucial in the process of capital allocation for any investor.
This thesis provides a profound empirical analysis of bond risk premia and bond return forecasting and improves the understanding of the conditioning information and
structure of these bond risk premia considering several aspects beside the standard
yields-only case by extending to macro-finance and multi-currency term structure
models. The contribution of this thesis is threefold in that it covers three distinctive
fields of the huge term structure literature. Before summing up the contribution and
1
2
CHAPTER 1. INTRODUCTION
results of this thesis, the according fields of research that lay the foundation for the
advancement of this thesis are presented in the following section.
1.2
Field of Research
Today’s term structure models are predominantly structured as state space systems
with time series dynamics on one hand and cross-section restrictions on the other.
This was first been introduced by the single-factor models of Vasicek (1977) and
Cox, Ingersoll and Ross (1985) with continuous-time Gaussian and square-root diffusions respectively. Multivariate generalizations from Duffie and Kan (1996) and
the classification of Dai and Singleton (2000) set the standard to the class of affine
term structure models. Finally, Duffee (2002) introduces flexible time-varying prices
of risk that enables these models to plausibly forecast bond returns instead of only
fitting the actual yield curve in the risk-neutral measure with a constant price of risk.
The discrete-time analog of these term structure models were first proposed by Ang
and Piazzesi (2003), and is also the model class that is used in this thesis. The general form of these models allows for all kind of state variables ranging from yield
curve factors and other economic variables to statistical factors. The existing literature focuses mainly on historic yield curve information to forecast bond excess
returns. An early contribution is Fama and Bliss (1987), who forecast bond excess
returns with forward rates. Forward rates contain information about future yields
and/or excess returns on bonds. Since empirical research shows that the expectations hypothesis1 of the term structure only holds on average and for long maturity
bonds (Cochrane 2001, p. 426), it seems logical to test forward rates as a predictor
for bond excess returns. Cochrane and Piazzesi (2005) show that a tent-shaped linear
combination of forward rates (hereinafter referred to as the CP factor) predicts bond
excess returns with remarkably high R2 up to 44%. Boos (2011) provides further insights into bond forecasting with forward rates and finds a second forecasting factor
1 Campbell
dence.
and Shiller (1991) provide an overview of the expectations hypothesis and empirical evi-
1.2. FIELD OF RESEARCH
3
besides that in Cochrane and Piazzesi (2005). This is not obvious in the regression
framework of Cochrane and Piazzesi (2005) because the magnitude of the shocks
of this second factor is much smaller compared to the CP factor and thus vanishes
within the noise. By building duration-neutral portfolios to remove the predictability
of the CP factor, which is essentially forecasting level shocks, the second factor is
made visible and can be attributed to slope shocks.
A growing field of research in bond forecasting is about combining term structure
models with macroeconomic data. From a macroeconomic perspective the short rate
is a monetary policy instrument and an important variable in growth models. The
Taylor rule (Taylor 1993) and similar models attempt to predict the short term interest rate by changes in inflation and the output gap. Ang, Dong and Piazzesi (2007)
estimate a term structure model with an augmented Taylor rule and find that macroeconomic variables account for over half of the variation in time-varying excess bond
returns. There is a growing body of literature that combines macroeconomic and
finance models of bond yields, indicating the importance of macroeconomic variables in bond return predictability. A forecasting analysis of bond risk premia with
macroeconomic variables in particular is provided by Kim and Moon (2005) and
Ludvigson and Ng (2009). Both of these studies conclude that bond returns can be
predicted using macroeconomic data. Macro-finance term structure models are a
relatively new branch of research combining the term structure models of finance literature with macroeconomic variables or any models thereof, based on the findings
that macroeconomic variables predict bond returns. Ang and Piazzesi (2003) simply
extend the state space of an affine term structure model with macroeconomic variables. Mönch (2005) follows a similar approach by extending the macroeconomic
state variables with a factor-augmented VAR model to incorporate many different
time series. The specification of the VAR model decides how the macro and yield
factors are allowed to interact. Fischer (2010) proposes a model with two separate VARs for latent yield curve and macroeconomic factors. Models that allow for
bidirectional feedback are proposed by Rudebusch, Swanson and Wu (2006) and
Diebold, Rudebusch and Boragan (2006). Dewachter, Lyrio and Maes (2006) es-
4
timate a macro-finance model with the Kalman filter in a one-step procedure and
Dewachter and Iana (2009) extend the macro-finance model with additional finance
factors as liquidity measures. Some of the macroeconomic information that forecasts
bond returns is embedded in the yield curve and thus captured by the yield curve
factors. However, a significant part of the macroeconomic information that forecasts
bond risk premia is not spanned by the yield curve. This means that macroeconomic
variables have further predictive power in addition to the yield curve factors. None of
the above-mentioned macro-finance models capture the unspanned part of macroeconomic variables. Models that systematically cover these unspanned macroeconomic
factors in term structure models are provided by Duffee (2011), Joslin, Priebsch and
Singleton (2011) and Boos (2011).
Yield curves of open economies that interact with each other are obviously linked.
International term structure models have been introduced by Backus, Foresi and
Telmer (2001) among others. These multi-currency term structure models are often set up as a two-country model based on separating local and international state
variables. The literature on international term structure models often assumes common international yield curve factors as in Dewachter and Maes (2001). Pérignon,
Smith and Villa (2007) conclude that one international factor is enough to explain
the common movements in the yield curve, with the remaining variance explained
by local factors. The international factor is found to account for the common level of
all yield curves. Egorov, Li and Ng (2011) expand this specification to two common
factors and one local factor each. Modugno and Nikolaou (2007) estimate an international Nelson-Siegel model with the level, slope and curvature of the domestic
and foreign yield curve as state variables. Their international model outperforms the
domestic alternatives. Diebold, Li and Yue (2008) take this marginally further and
estimate an international Nelson-Siegel term structure model to explain the crosssection of yields with hierarchical dependence of country-specific level, slope and
curvature factors on a global variation thereof. They find that global level and slope
factors explain a significant fraction of the country-specific factors. Another approach to forecasting domestic risk premia with international yield curves is pre-
1.3. CONTRIBUTION AND RESULTS
5
sented by Dahlquist and Hasseltoft (2011). They add international factors to the
local level, slope and curvature factors in a single-country term structure model.
1.3
Contribution and Results
The major contributions and important results of this thesis are summed up in the
following. These are structured along the three areas of research introduced in the
previous section. The three sections are divided into yields-only, macro-finance and
multi-currency term structure models.
1.3.1
Yields-Only Models
The yields-only model part is about bond return predictability with yield curve factors only. In a simple regression framework I review and extend the analysis of
Cochrane and Piazzesi (2005) and Boos (2011) and contribute in several ways. Firstly,
I find that not only level and slope shocks, but also curvature risk are priced. This
seems reasonable since butterfly positions, which are standard trades that are neutralized against level and slope changes of the yield curve, are a common instrument in
trading swap markets. Priced slope and level risk premia have been stated in previous
studies, but not so the priced curvature risk premium. Slope and curvature risk premia are forecasted with R2 in the same size as the level risk premium, but the factors
to forecast these risks are not entirely the same. In addition to the level, slope and curvature of the yield curve, which forecast the level risk premium, slope and curvature
risk premia require the fourth principal component of yields as a forecasting factor.
Secondly, the data set is extended in the time series up to June 2009. This includes
the bursting of the US housing bubble that peaked in 2006 and the resulting subprime mortgage crisis with the bankruptcy of Lehman Brothers in September 2008.
By extending the time series up to June 2009, predictability measured in R2 drops
to 0.25, which is a huge decrease compared to the R2 of 0.35 reported in Cochrane
6
and Piazzesi (2005). Thirdly, rolling window and subsample analysis shows that the
predictability is not persistent over time. Most of the full sample predictability is
lost when excluding the high inflation period of the 1970’s and early 1980’s. In fact,
predictability of bond returns is boosted by the Great Inflation period and the R2 of
the full sample are overestimated. Besides, this is a strong indication of structural
breaks in the data.
The regression analysis is then taken to an affine model setup with the goal of
evaluating the optimal number of priced risk premia and driving state variables in
a yields-only term structure model. Compared against the simple regression framework, the affine term structure model includes no arbitrage assumptions or crossequation restrictions that improve the stability and plausibility of the estimation. The
number of priced risk premia and the number of state variables is shown to coincide
with the regression analysis. The dominant model prices three risk premia based on
four factors that drive the state dynamics. These three priced risk premia are reflected
in the pricing of level, slope and curvature portfolios. Both number of risk premia
and the number of state variables contradicts the existing literature, which proposes
one or two priced risk premia and mostly lower dimensional models.
1.3.2
The section concerning macro-finance models investigates the predictability of bond
returns with several macroeconomic variables. These are organized into groups according to their approximation of real activity, inflation and money supply. Macro
factors, especially real activity and inflation, are shown to predict bond returns, slope
and curvature portfolios. This predictability is persistent beside the yield curve factors. Since this macroeconomic information is not spanned by the yield curve, it is
known as unspanned macroeconomic information. Unspanned macro factors significantly forecast the level and slope risk premia, whereas the curvature risk premium
is spanned by the latent yield curve factors alone. This implies that the yield curve
does not contain all available information to forecast bond returns. In a regression
7
framework, the predictive power of unspanned macro factors is analyzed. This work
is most closely related to the study of Joslin, Priebsch and Singleton (2011), but differs in several important points. The analysis is extended along the time dimension
in both directions to increase the number of observations to periods beyond the low
inflation and decreasing yield sample. Additional subsample and rolling window
analysis reveal that the predictability is not stable throughout history. Furthermore,
macro data is covered by a broader set of variables, sorted into three distinct groups
according to their approximation of real activity, inflation and money supply. Finally,
not only is the level risk premium of bonds accounted for, but also the slope and curvature risk premia, which have been shown to be priced in yields-only models.
Accordingly, this thesis introduces an affine term structure model with the desired feature of including unspanned macro factors. General macro-finance models
assume that the yield curve is spanned by all the state variables jointly. In contrast,
empirical evidence suggests that the yield curve is sufficiently described by three
latent yield curve factors, often referred to as level, slope and curvature, but that
expectations about future yields are not fully spanned by these factors. Macroeconomic factors are those factors that are partially unspanned by the yield curve, but
help to predict bond excess returns. In this thesis the additional macro state variables
are restricted to affect the cross-section of yields, but allowed to influence bond risk
premia. This allows the addition of additional factors to term structure models that
do not span the yield curve but improve estimation of bond risk premia. Standard
term structure models without this restriction do not allow the addition of state variables that are unspanned by the yield curve by construction. The only other sources
that systematically cover this unspanned specification in term structure models are
Duffee (2011), Joslin, Priebsch and Singleton (2011) and Boos (2011). This study
contributes by extending the empirical analysis in the time axis and cross-section
of macroeconomic variables. Furthermore, the analysis is not restricted to the level
risk premium as the studies of Joslin, Priebsch and Singleton (2011) and Duffee
(2011) but extends to slope and curvature risk premia. The introduced model is
based on the class of Ang and Piazzesi (2003) models with the unspanned restriction
8
and parametrization applied in the spirit of Boos (2011) but with extensions to multiple macro variables. These macro-finance models with unspanned macro factors
improve predictability of expected returns beyond that of a yields-only model. The
dominant model exhibits four latent yield curve factors and two macro state variables
with three priced risk premia overall. This is the same number of priced risk premia
as in the dominant yields-only model. The main result is that unspanned macroeconomic information is significant for forecasting bond risk premia; this is in line with
the other studies of term structure models with unspanned factors. Furthermore, it is
shown that real activity and inflation span the level, slope and curvature risk premia
and that these risk premia react differently to macroeconomic shocks. The level risk
premia is negatively correlated with real activity. This implies that high growth is
associated with a low expected level risk premium. The slope risk premium is positively correlated with inflation, implying that high inflation is associated with a low
expected slope risk premium. The curvature risk premium also is spanned by inflation, but in comparison to the slope risk premium it tends to capture the unspanned
part.
1.3.3
This part extends the US yields-only regressions and models in two ways. On the
one hand, local models of the British and German yield curves are explored and
compared to the US case. This is simply an application of the forecasting regressions
and yields-only models to different empirical data. This has been explored before2
but not with the assessment of slope and curvature risk premia. Predictability of
bond excess returns with domestic yield curve factors as instruments is persistent to
countries beside the US, but with somewhat lower R2 ; this is especially true of the
UK.
On the other hand, international relations between countries are investigated in
a regression and term structure model framework. Principal component analysis of
2 see
Kessler and Scherer (2009) and Sekkel (2011)
9
a single country’s term structure empirically determines that three principal components almost fully span the yield curve.3 Less known is the joint principal component analysis of several countries. This decomposition shows that the number of
factors that sufficiently describe the local yield curves is not reduced compared to an
individual domestic setting. This implies that the yield curve shows significant independent movement across countries and is not captured by common international
factors. The same is true for bond risk premia. The local risk premia cannot be captured by a common international factor. Whereas the level risk premium still shows
a correlation of around 0.7 across countries, the slope and curvature risk premia do
not. Thus the yield curve is best spanned by its domestic yield curve factors. Risk
premia are instead driven by both domestic and foreign factors. The inclusion of
foreign yields as instruments in the forecasting regression alongside the standard domestic yields improves the predictability of domestic bond excess returns across all
the countries under consideration for the level, slope and curvature risk premia. R2
in the restricted regressions are boosted beyond 0.4 for the UK and beyond 0.5 for
Germany and the US. The unspanned parts of foreign yield curve factors dominate
and significantly forecast bond returns in addition to standard yield curve factors
with an R2 of up to 0.3. Unspanned foreign yield factors are important in forecasting
the domestic yield curve in the US, UK and German markets. International yield
curves exhibit information not spanned by the domestic yield curve but spanned by
domestic risk premia. This section serves mainly to explore the characteristics of unspanned international yield curves as a forecasting instrument for bond returns in the
domestic country. This is investigated by introducing an international term structure
model with the property that each country’s yield curve factors are mutually unspanned in the risk-neutral measure but affect one another’s the forecasting ability in
the physical distribution. The implementation of unspanned factors into international
term structure models has not yet been systematically explored to the author’s best
knowledge.4 Empirical analysis reveals that the number of priced bond risk premia
3 Litterman
4 Boos
and Scheinkman (1991)
and Wäger (2011) explore the same kind of models as a combination of results from Boos
(2011) and this thesis. Mirkov (2011) applies such an unspanned two-country model to investigate the
10
is not reduced in a multi-currency model compared to those of their single-country
alternatives. Therefore, a common international factor is not supported. However,
the unspanned state variables do affect and improve the predictability of the model
without affecting the yield curve. This is the case since some information that is
important in forming investor’s expectations but with opposite effect on the current
yield curve is netted out and only recovered by additional unspanned factors. Since
unspanned factors in one country are spanned by the yield curve of the other country
and vice versa, all the state variables are spanned by the model in the risk-neutral and
physical distribution and thus the price of risk is defined for any state variable. These
multi-currency models with comovement of domestic and foreign state variables outperform the local and single-country models in terms of predictability and likelihood
ratios. As a result, risk premia depend on international yield curve information,
which is not spanned by the local yield curve. Additionally, the international model
provides a model-implied forecast of foreign exchange returns. The specification of
the foreign exchange risk premium is shown to have no influence on the results of
the bond risk premia estimation. A better specification of the foreign exchange risk
premium should therefore be unable to improve bond risk premia predictability. Additionally, the estimated foreign exchange risk premium deviates from the uncovered
interest rate parity and is thus in line with the actual data and empirical research.
1.4
Structure of the Thesis
The thesis consists of three sections, which separate the contribution to yields-only,
macro-finance and multi-currency models. These sections are then split into a chapter pursuing a simple regression framework and a chapter based on an arbitrage-free
state space model approach. The chapters with the simple regression framework
provide a fundamental understanding of the relevant relations and illustrate some
stylized facts inherent in the empirical data sets. The state space models in the subsequent chapters are then structured along these insights and additional conclusions
influence of monetary policy decisions across countries.
1.4. STRUCTURE OF THE THESIS
11
are drawn in the consistent and more formal environment of these empirical model
estimates.
Chapter 2 starts with an extensive treatment of stylized facts of bond return forecasting, which are revised and extended beyond existing literature. Chapter 3 introduces a class of affine term structure models that serves as a reference throughout
the thesis. In addition, a self-consistent parametrization for yields-only term structure models, which identifies the model and allows for quick estimation is provided.
An empirical investigation of these models and their predictability of bond excess
returns is then presented.
The section concerning macro-finance models starts with an analysis of the forecasting performance of macroeconomic factors in Chapter 4. The finding that these
macroeconomic factors are not fully spanned by the yield curve leads to a term structure model specification that adds macroeconomic variables as unspanned factors
introduced in Chapter 5. Following the introduction of this model is an empirical
analysis of risk premia driven by these macroeconomic factors.
Chapter 6 extends the yields-only risk premia analysis to other countries and
links them by adding foreign yield curve factors as explanatory variables. In Chapter 7 these findings are embedded in a multi-currency term structure model, which
allows pricing of all bonds in two countries. The subsequent empirical analysis of
these models provides insights into international bond risk premia.
12
I
Yields-Only Models
13
Chapter 2
Yield Regressions
This chapter is about US government bond return predictability in a simple linear regression framework. Using a single forecasting factor, Cochrane and Piazzesi (2005)
forecast bond excess returns with a linear combination of forward rates with an R2
of up to 0.44. Their forecasting analysis is extended by Boos (2011), who concludes
that slope shocks are priced in addition to the level shocks. Cochrane and Piazzesi
(2005) miss this risk premium because they only evaluate forecasts of single bonds
and not portfolios thereof. I review these analyses and contribute in several ways.
Firstly, I find that curvature risk is priced in addition to the level and slope shocks.
This seems reasonable since butterfly positions, which are standard trades that are
neutralized against level and slope changes of the yield curve, are a common instrument in trading swap markets. Secondly, the data set is extended in the time series
up to June 2009. This includes the bursting of the US housing bubble that peaked
in 2006 and the resulting subprime mortgage crisis with the bankruptcy of Lehman
Brothers in September 2008. By extending the time series up to June 2009, predictability (measured in R2 ) drops to 0.25, which is a huge decrease compared with
the R2 of 0.35 reported in Cochrane and Piazzesi (2005). Thirdly, rolling window
and subsample analysis show that the predictability is not persistent over time. Most
15
16
CHAPTER 2. YIELD REGRESSIONS
of the full sample predictability is lost when excluding the high inflation period of
the 1970’s and early 1980’s. This is a strong indication of structural breaks in the
data.
2.1
Notation
To enhance readability and comparability the notation chosen is the same as in
Cochrane and Piazzesi (2005). The log bond price is given as
(n)
pt
≡ log price of n-year discount bond at time t.
(2.1)
Parentheses distinguish maturity from exponentiation in the superscript. The log
yield is
1 (n)
≡ − pt .
(2.2)
n
The log forward rate at time t for loans between time t + n − 1 and t + n is defined as
(n)
yt
(n)
ft
(n−1)
≡ pt
(n)
− pt .
(2.3)
The log holding period return from buying a n-year bond at time t and selling it as
an n − 1-year bond at time t + 1 is denoted by
(n)
(n−1)
(n)
rt+1 ≡ pt+1 − pt
(2.4)
and the excess log return by
(n)
(n)
(1)
rxt+1 ≡ rt+1 − yt .
2.2
(2.5)
Yield Curve Data
Throughout this thesis US government bond data considered as default-free is used.
Since term structure data is not observed itself but extracted from traded bond prices,
2.2. YIELD CURVE DATA
17
the method of term structure estimation is crucial to preserve important features for
forecasting. The CBD1 data set is used throughout this thesis and constructed from
several different data sources.2 The CBD data set ranges from 1952:06 to 2009:06
with monthly frequency and maturities up to ten years.3 This data set is plotted in
Figure 2.1. The data set provided by Fama and Bliss (1987) and used by other studies
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
10
0.02
0
1943
5
1957
1971
1984
1998
2012
0
Figure 2.1: Yield curve based on CBD data set
Surface plot of yields from the CBD data set ranging from 1946:12 - 2009:6 with maturities up to ten
years.
including that of Cochrane and Piazzesi (2005) is taken as a reference to compare
the results to these previous studies and is henceforth called FB. However, the crosssection of this data set is limited to annually spacing maturities up to five years
only. The data set constructed by Gürkaynak, Sack and Wright (2006), henceforth
called GSW, is estimated using a parametric method, which restricts the yield curve
1 Combined
2 The
Bond Data
CBD data set starts at 1952:06 with the data set provided in McCulloch and Kwon (1993). From
1970:01 to 2000:12 the data set of Diebold and Li (2006) is used. The extension up to 2009:06 is achieved
by the author itself using bond data from CRSP and Datastream. Further information on the data set and
estimation methodology is given in Appendix A.
3 Extending the data set further back into history is possible but not advisable since the interest rate data
before 1952:03 was not free-floating. According to Brennan, Wang and Xia (2004) the Federal Treasury
Accord that reasserted the independence of the Fed from the Treasury was adopted in March 1952.
18
information to the number of factors used in the parametrization; this is certainly not
helpful in analyzing the number of risk premia as conducted in this chapter. It is
only used to conduct a robustness check of the forecasting regressions. Any further
information on these data sets, term structure estimation methodologies and their
suitability for forecasting is deferred to Appendix A.
2.3
Level Risk Premium
This section extends the bond return forecasting regressions of Cochrane and Piazzesi (2005) and provides a thorough analysis of the level bond risk premium. Starting with the results of Cochrane and Piazzesi (2005) based on the FB data set, the
analysis is extended in various ways. Firstly, the data set is enlarged in both dimensions: cross-section and time series. In the cross-section dimension, bonds with
maturities beyond five years are included. In the time series, the extension sheds
light on how bond predictability is affected by more recent data, which was shaped
by the subprime crisis. Secondly, a breakdown analysis along the time series dimension with rolling window and subsample regressions reveals that bond forecasting
is not persistent across time and the in-sample prediction probably is overestimated.
Thirdly, a comparison with a quarterly forecasting horizon and additional robustness
checks with the GSW data completes the level risk premium analysis.
First, a short look at unconditional risk premia. Table 2.1 shows the full sample
estimates of mean and standard deviation including the unconditional Sharpe ratio
(Sharpe 1966). Bond excess returns are all positive across the sample, indicating that
there is an unconditional positive risk premium on bonds. Unconditional risk premia
in bond markets should result in a positive long-term expected return that compensates the risks taken. The Sharpe ratios range between 0.03 and 0.16 with a mean
of around 0.11. Over an observation period of more than 50 years, these values are
not significantly different from zero. There could be another reason for this. Either
there really is no priced unconditional risk premium, or the size of the premium is
2.3. LEVEL RISK PREMIUM
19
so insignificant that the small sample (less than 60 independent observations) cannot
satisfactorily estimate this premium. Due to the high persistence of yields4 , the estimation would need considerably more observations to achieve significant results.
However, since all the Sharpe ratios are positive and of similar order, this result is
still an indication of the existence of an insignificant unconditional risk premium.
Table 2.1: Sample statistics for US bond data sets
r(2)
r(3)
r(4)
r(5)
µ
6.1%
6.4%
6.5%
6.5%
σ
4.0%
5.0%
6.1%
7.0%
0.09
0.13
0.13
0.11
r(6)
r(7)
r(8)
r(9)
r(10)
Fama-Bliss 1952:06 - 2006:12
SR
Combined Bond Data 1952:06 - 2006:12
µ
6.2%
6.4%
6.6%
6.5%
6.8%
6.6%
6.7%
6.6%
6.1%
σ
3.9%
5.0%
6.0%
7.0%
8.1%
8.8%
10.2%
11.0%
11.7%
0.11
0.13
0.13
0.11
0.12
0.09
0.09
0.07
0.03
SR
All figures based on annualized discrete returns. µ is the return, σ the standard deviation and SR the
Sharpe ratio (ratio of excess returns to volatility).
The next step is to focus on conditional risk premia. These conditional risk premia are expected to primarily vary with the forward rates extracted from the yield
curve. This forecasting exercise is conducted in a linear OLS regression framework
(n)
that regresses one year excess returns of an n-year bond rxt+1 on a set of forward
rates ft :
(n)
rxt+1 = αn + ft βn + εt,n .
(2.6)
This forecasting regression is estimated using one-year excess return data with monthly
overlapping. The overlapping provides more observations (12 times more) but does
not increase the number of independent observations. Nevertheless, the inclusion of
intra-year data points should improve regression results by giving them more stability and robustness. This is common practice because the available data sets still
seem to be too small compared to the long interest rate cycles observed. The test
4 Yields
are very close to a unit root when modeled as an autoregressive process, which is a clear
indication of high persistence.
20
statistics need to be adjusted for the overlapping period. This is achieved using the
heteroscedasticity and autocorrelation-consistent Newey-West standard errors with
18 lags (Newey and West 1987).
The first line of Table 2.2 reports the R2 of the forecasting regression based on
the two- to five-year excess returns of the FB data set with one- to five-year forwards
as instruments. These are the same results as in Cochrane and Piazzesi (2005). The
in-sample R2 of these regressions are particularly high with values up to 0.37. The
corresponding loadings of Equation (2.6) show a tent-shaped forecasting factor as
plotted in Figure 2.2, which implies a single forecasting factor for excess bond returns at first sight. By restricting the forecasting regression (2.6) to the average of
excess returns rx = n1 ∑n rx(n) , only the level of excess returns is forecasted. This is
the same as forecasting the first principal component of excess returns. Since this
preserves the tent-shaped forecasting factor (not reported), Cochrane and Piazzesi
(2005) assume that there is only one forecasting factor with the implication that only
level risks are priced. This hypothesis is contradicted by including longer maturity
bonds as shown subsequently.
2.3.1
Extended Time Series
The results in Cochrane and Piazzesi (2005) are based on the time window spanning
from 1964:01 to 2003:12. Extending this data set in both directions to range from
1952:06 until 2006:12 results in heavily reduced R2 as shown in the second line of
Table 2.2. Reconducting this analysis with the CBD data set gives the same result
and is even more pronounced by extending the data set up to 2009:06.5 This further
extension includes the recent financial crisis, which heavily influenced bond markets
with the subprime mortgage crisis peaking in the bankruptcy of Lehman Brothers.
5 The
CBD data set would as well reach further back in history but this is not advised since the interest
rate data before 1952:03 was not free floating. According to Brennan, Wang and Xia (2004) the Federal
Treasury Accord that reasserted the independence of the Fed from the Treasury was adopted in March
1952. Data before 1952:06 is thus omitted.
21
The drop in predictability amounts to 0.10 in terms of R2 which is a reduction of
more than 25%. This reduced predictability supposes that the behavior of bond risk
premia changes over time and that the historic data set holds some structural breaks.
Further investigation based on subsample and rolling window analysis strengthen
this finding in Section 2.3.3.
Table 2.2: R2 of forecasting regressions
rx(2)
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
rx(9)
rx(10)
Fama-Bliss 1964:01 - 2003:12
Unrestricted
0.32
0.34
0.37
0.35
Fama-Bliss 1952:06 - 2006:12
Unrestricted
0.23
0.24
0.27
0.25
Unrestricted
0.34
0.38
0.41
0.39
0.43
0.42
0.41
0.42
0.41
Restricted
0.31
0.35
0.36
0.36
0.38
0.37
0.37
0.37
0.37
Unrestricted
0.31
0.35
0.37
0.36
0.39
0.38
0.36
0.37
0.36
Restricted
0.28
0.32
0.32
0.32
0.34
0.33
0.32
0.32
0.33
Unrestricted
0.24
0.26
0.28
0.27
0.30
0.30
0.29
0.31
0.31
Restricted
0.21
0.24
0.25
0.24
0.26
0.26
0.26
0.27
0.27
2.3.2
Extended Cross-Section
The FB data set and thus the analysis of Cochrane and Piazzesi (2005) is limited
in the cross-section to maturities up to five years. The CBD data set extends the
analysis to maturities of up to ten years, both in the instruments and the forecasted
bond returns dimension. Predictability is only marginally increased by adding yields
with longer maturities.6 This is most likely to be simply an effect of increasing the
set of instruments that is prone to improve the in-sample fit of the regression just
6 Reconducting
the regression analysis with the CBD data set but with limited instruments of up to
5-year forward rates delivers very similar results (not reported). This is obvious because the major part of
both data sets, the FB and CBD data sets, are constructed using the same method.
22
by matching some measurement errors. The estimated loadings of Equation (2.6)
do not reproduce the tent-shaped forecasting factor but show highly oscillating factor loadings in a zig-zag pattern as seen in the middle plot of Figure 2.2. This is a
strong indication for fitting measurement errors and arises due to the fact that about
60 independent observations are fitted with ten highly correlated instruments. The
combination of a rather small sample and many highly correlated regressors leads to
overfitting measurement errors or other noise specific to that sample and produces a
pattern similar to a multicollinearity problem. It seems that ten instruments are too
many to explain the underlying risk premia dynamics. The number of instruments is
thus reduced back down to five by selecting evenly spread forward rates. Forecasting with the reduced number of instruments is called the restricted regression and
the instruments included are the 1-, 3-, 5-, 8- and 10-year forward rates, compared
to the unrestricted regression that includes all the available instruments. The R2 of
the regressions are provided in Table 2.2 and the corresponding t-statistics in Table
2.3. The R2 of the restricted models are only slightly lower than in the unrestricted
case: an indication that few or even no information is lost by dropping some instruments. In the limited FB sample and the restricted CBD sample, the t-statistics show
high significance for most of the instrument loadings. In contrast, the unrestricted
CBD sample only produces selected significant instrument loadings caused by the
aforementioned dependency problems. The loadings of the restricted regressions are
shown in the right hand side plot of Figure 2.2 and produce a nice structure similar to the tent shape produced with the limited FB sample shown in the left hand
side plot of Figure 2.2, but with some divergence in forecasting the long maturity
bond returns. This divergence of the loadings is crucial because this pattern can no
longer be captured by a single linear combination of the instruments, as highlighted
by Boos (2011). This is an indication for at least one additional forecasting factor
and thus a risk premium in addition to the level risk, contrary to only one forecasting
factor as stated in Cochrane and Piazzesi (2005). At first sight, the divergence looks
minute and thus negligible. However, because the loadings on the ten-year forward
have different signs, this is not the case. Although the loadings are not significantly
different from zero (see Table 2.3), this effect is due to the fact that the information
3
2
23
8
12
6
10
8
4
1
6
2
4
0
0
2
−2
−1
0
−4
−2
−3
−2
−6
1
2
3
4
5
−8
−4
1
2
3
4
5
6
7
8
9
10
−6
1
3
5
8
10
Figure 2.2: Loadings of forecasting regressions
The left-hand figure shows the loadings of the forecasting regression with FB data for the range 1952:06 2006:12. The middle and right-hand figures show the same with CBD data for unrestricted and restricted
regression, respectively. The ranking of the absolute loadings corresponds to the ranking of the bond
maturity. The intercept is omitted.
at the long end is too similar. The loadings on the eight- and ten-year forwards are
individually significant and by excluding one of them in the restricted regression, the
other significantly shows the same pattern of divergent loadings with opposite sign
(not reported) as shown by Boos (2011). The predictability of additional risk premia
other than the level risk is covered in Section 2.4 below.
The selection of the five forward rates out of the ten available instruments obviously comes with a selection bias. This issue is addressed by examining all possible
combinations of a subset consisting of five instruments.7 The summary statistics in
Table 2.4 of these R2 show quite a high dispersion. The mean R2 of all these subsets
should provide a more robust result. The R2 of the most obvious choice of evenly
spread factors as applied here is beyond the mean. This could be caused either by
this selection bias or could also and more likely stem from not including neighboring
forwards and thus provide a more sound information base for the whole yield curve.
A breakdown of the level risk premium information into the well-known yield
7 The
number of five instruments is also arbitrary, but has been chosen to match the number of factors
used in Cochrane and Piazzesi (2005).
24
Table 2.3: T-statistics for forecasting regressions
rx(2)
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
rx(9)
rx(10)
Unrestricted, Fama-Bliss 1952:06 - 2006:12
f (1)
-5.03
-4.90
-5.10
-5.10
f (2)
2.55
1.58
1.82
1.99
f (3)
1.55
2.83
2.32
2.05
f (4)
1.14
0.77
2.38
1.81
f (5)
-2.23
-2.75
-2.98
-2.12
f (1)
-5.00
-5.26
-5.73
-6.03
-6.18
-6.33
-6.06
-6.13
-5.95
f (2)
0.86
0.03
0.13
0.28
0.45
0.64
0.69
1.04
1.10
f (3)
2.94
4.36
3.79
3.51
3.36
3.24
3.13
2.88
2.95
Unrestricted, Combined Bond Data 1952:06 - 2006:12
f (4)
2.62
2.41
3.70
3.44
3.02
2.41
2.25
2.43
2.34
f (5)
-1.87
-2.19
-2.31
-1.16
-1.56
-1.98
-2.20
-2.35
-2.54
f (6)
1.01
0.91
0.73
0.64
1.83
1.70
1.42
1.28
0.99
f (7)
-0.80
-0.72
-0.73
-0.93
-0.99
0.37
0.04
-0.18
-0.66
f (8)
-5.40
-5.70
-5.66
-5.70
-5.73
-5.64
-4.31
-4.46
-4.61
f (9)
-1.01
-0.81
-0.72
-0.65
-0.73
-0.69
-0.48
0.49
0.62
f (10)
-1.16
-0.99
-0.93
-0.79
-0.65
-0.59
-0.49
-0.39
0.74
f (1)
-5.25
-6.40
-6.58
-6.78
-7.14
-7.27
-6.87
-6.86
f (3)
7.23
8.41
8.50
7.76
7.65
7.45
6.78
6.79
6.86
f (5)
-1.81
-2.25
-2.23
-1.23
-1.32
-1.07
-1.51
-1.48
-1.93
f (8)
-4.03
-4.47
-3.79
-4.23
-3.94
-4.10
-3.02
-2.84
-3.09
f (10)
-0.84
-0.72
-0.35
-0.37
-0.24
0.28
0.17
0.12
1.43
Restricted, Combined Bond Data 1952:06 - 2006:12
T-statistics are Newey-West-adjusted with 18 lags
-6.88
25
Table 2.4: R2 of different instrument subsets
rx(2)
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
rx(9)
rx(10)
Mean
0.18
0.19
0.20
0.19
0.22
0.20
0.21
0.21
0.20
Std
0.05
0.07
0.07
0.07
0.08
0.08
0.08
0.07
0.08
Min
0.08
0.07
0.07
0.07
0.07
0.06
0.07
0.07
0.06
Max
0.29
0.33
0.35
0.35
0.37
0.36
0.35
0.36
0.35
Summary statistics of all different possible instrument combinations of five instruments out of the ten
$ forwards 10
5 . Std is the abbreviation for standard deviation.
curve factors (level, slope and curvature), which are termed due to the shape of their
loadings, is provided in Table 2.5. The loadings of these factors are shown in the
left-hand side plot of Figure 2.5. These first three principal components of the yield
curve explain more than 99.9% of the variation in yields, of which the level accounts
already to 98%. In terms of explaining the level risk premium, these three factors
already capture about 70% of the unrestricted forecasting regressions, as shown in
Table 2.5. The most information comes from the slope and curvature factors, the
level only provides information about short maturity bonds. The remaining information is captured by the seventh and eighth principal components, which are the
only significant ones besides the first three (not reported). Finally, the number of
significant principal components coincides with the selection of five instruments in
the restricted regressions above.
Table 2.5: R2 of principal components as factors
rx(2)
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
rx(9)
rx(10)
Level
0.07
0.05
0.04
0.04
0.04
0.03
0.04
0.04
0.02
Slope
0.05
0.07
0.09
0.10
0.13
0.13
0.15
0.14
0.13
Curvature
0.12
0.13
0.12
0.12
0.11
0.09
0.08
0.07
0.07
First three
0.25
0.25
0.27
0.27
0.29
0.25
0.27
0.26
0.23
All yields
0.31
0.35
0.37
0.36
0.39
0.38
0.36
0.37
0.36
R2 of forecasting regressions with principal components of yields as factors. The data is taken from the
CBD data 1952:06 - 2006:12.
26
2.3.3
Rolling Window and Subsample Analysis
Since the previous analysis of extending the time series showed that the regressions
are very sensitive to the sample size, a subsample and rolling window analysis further
investigates the persistence of forecasting ability over time. Since the already small
sample is further reduced, the following results should be interpreted with appropriate care. Nevertheless, this analysis shows that the forecasting ability of the level
bond risk premium is not persistent across time and corresponding R2 vary much.
The rolling window regressions are conducted for 20 years of rolling data, i.e. 240
data points with 20 independent observations. The average R2 is about 0.34 with the
highest observation being 0.54 and the lowest being 0.06 (plotted in Figure 2.3). R2
fluctuation is quite high and indicates unstable forecasting results over time. Low R2
are observed in the beginning and towards the end of the sample. The low R2 in the
beginning could be influenced, in particular, by the lower quality standards of the
data at the time, as is also noted in Cochrane and Piazzesi (2005). Towards the end
of the sample there is another drop in forecasting ability in terms of R2 , which does
not come with an obvious explanation. These regressions are heavily influenced by
special situations and outliers because there are only twenty independent observations in the rolling samples. Nevertheless, the last years of the sample were difficult
years for bond return predictability. The bursting of the US housing bubble and the
subprime crisis might have caused a lot of unexpected variation in bond returns. Although the average R2 lies in a plausible range compared to the full sample results,
the persistence is not given. It is in the most recent years of the sample, when literature on bond return predictability had already been published8 and thus publicly
available, that the R2 deteriorate.
In the following the results of a subsample analysis are provided. The subsamples
are intentionally chosen to punctuate an obvious bias in the US bond data history.
The sample is divided in two ways around the high inflation period of the 1970’s and
the peak of interest rates in 1981. Firstly, this sample is split into the increasing yield
8 see
Cochrane and Piazzesi (2005)
27
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1971
1976
1982
1987
1993
1998
2004
2009
Figure 2.3: R2 of 20 year rolling forecasting regressions
The figure shows the range of R2 for restricted forecasting regressions based on a rolling 20 year window
with a minimum and maximum line (and the mean thereof in between) along the cross-section of all
forecasted maturities.
period before August 1981 and the decreasing yield period thereafter. Secondly, the
high inflation period of the 1970’s is excluded from the sample and compared against
the remaining periods with lower inflation. The Great Inflation is broadly defined as
the period from the beginning of 1971 to the end of 1985.
Table 2.6 shows slightly higher R2 for the decreasing yield period than the increasing yield period. But in fact R2 are stable over these two subsamples, and the
loadings shown in Figure 2.4 imply no obvious difference between these subsamples.
However, as we look at the second pair of subsamples separating the Great Inflation
of the 1970’s from the rest, a different picture evolves. The high inflation period
reveals much higher R2 (around 0.48) than the low inflation period (around 0.12).
The loadings in Figure 2.4 display different loading structures for the two subsamples. The tent shape is not recovered in the low inflation sample, but only appears
in the high inflation subsample and finally in the full sample. This strengthens the
assumption that the high inflation period is boosting R2 in the full sample. The high
inflation period could be seen as a regime different from the remaining history with
28
Table 2.6: R2 of different subsamples
rx(2)
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
rx(9)
rx(10)
Period I
0.29
0.31
0.29
0.28
0.28
0.28
0.28
0.27
0.27
Period II
0.33
0.32
0.32
0.31
0.33
0.34
0.35
0.35
0.38
Period III
0.12
0.12
0.12
0.11
0.13
0.13
0.14
0.14
0.12
Period IV
0.45
0.50
0.48
0.47
0.48
0.48
0.47
0.48
0.47
The table shows the R2 of restricted forecasting regressions of different subsamples of the CBD data set.
Period I is the increasing yield subsample 1946:12 - 1981:08. Period II is the decreasing yield subsample
1981:09 - 2009:06. Period III is the low inflation subsample 1946:12 - 1970:12 and 1986:01 - 2009:06.
Period IV is the high inflation subsample 1971:01 - 1985:12.
its own structure in bond return forecasting. The concern about this data sample is
that either the interest rate history includes some major structural changes or interest
rates display very long cycles, meaning our sample only contains one major reversal
from increasing to decreasing rates or one high inflation period. Future data will
shed more light on that issue.
6
10
2
15
8
4
1
10
6
2
0
4
5
2
0
−1
0
−2
0
−2
−2
−4
1
3
5
8
10
−4
1
3
5
8
10
−3
1
3
5
8
10
−5
1
3
5
8
10
Figure 2.4: Loadings of different subsamples
The figure shows the loadings of the restricted forecasting regressions of different subsamples of the
CBD data set. The first plot from the left shows the increasing yields subsample 1946:12 - 1981:08, the
second the decreasing yields subsample 1981:09 - 2009:06, the next the low inflation subsample 1946:12
- 1970:12 and 1986:01 - 2009:06 and the last the high inflation subsample 1971:01 - 1985:12.
2.3.4
29
Quarterly Forecasting Horizon
The following analysis looks at quarterly forecasting regressions in order to focus on
the persistence of forecasting bond returns at a lower frequency. The advantage of a
lower frequency is the increased number of independent observations. However, the
signal-to-noise ratio is reduced because although noise-like measurement errors are
of the same size, returns are obviously reduced in size since they are not accumulated
over a year but only over a quarter. Forecasting power is thus expected to be lower
in terms of R2 and more sensitive to outliers and measurement errors. The analysis
needs quarterly bond yield data up to full maturity. This analysis is conducted with
the CBD data set in the range of 1952:06 - 2006:12 only up to maturities of two
years, since historical data is not available at this frequency for higher maturities.
Additionally, the analysis is done with the CRSP data set that is constructed by the
author and provides any requested frequency up to a maturity of ten years. This
CRSP data set covers the range of 1958:12 - 2006:12. The forecasting instruments
are the restricted set of forwards already used in the above analysis. The tent shape
of the forecasting factor is preserved in quarterly frequency for both data sets (not
reported). As expected, R2 are somewhat lower, with values between 0.10 and 0.12
for the CBD data set and 0.11 and 0.14 for the CRSP data set.
2.3.5
Robustness Checks with a Different Data Set
Although forecasting exercises above have already been conducted with two different data sets (FB and CBD), they are both very similar because the same methodology to retrieve the yields from observed bond prices is applied. The GSW data set is
thus used to check for the robustness of the results. This data set is constructed using
the Svensson (1994, 1995) parametric method, which is of limited use in forecasting
since some patterns are smoothed away by restricting the number of factors. Regressions are conducted with the restricted set of instruments for the period 1971:08 2009:01 where the GSW data is available for up to ten year maturity. The loadings
30
do not recover the tent-shaped pattern but display a zig-zag form that roots in the
parametric construction of the data set (not reported). Forecasting results are similar
but slightly lower, both in terms of R2 and t-statistics. The instrument loadings are
still significant and the R2 range is between 0.20 and 0.27. The CBD data set, on the
other hand, provides R2 for this period of between 0.24 and 0.32. This indicates that
the forecasting result is also persistent across different types of data sets and thus not
tied to measurement errors or any other data-set-specific patterns.
2.4
Slope and Curvature Risk Premia
Restricting the model to one forecasting factor, built as a linear combination of the
forward rates as in Cochrane and Piazzesi (2005), still shows the tent-shaped structure in factor loadings with slightly lower, but comparable, R2 . As pointed out in the
previous section, the divergence of loadings on the long end disappears because this
divergence is only possible with more than one forecasting factor.
Contrary to Cochrane and Piazzesi (2005) the level risk premium is not the only
priced risk factor in bond returns. The dominance of the level risk compared to slope
and curvature risk is huge. The level risk, which is the first principal component
of bond excess returns, explains 98.0% of the variance. The next two components,
slope and curvature, explain only 1.4% and 0.2%, respectively. These principal components of bond excess returns are plotted on the right-hand side of Figure 2.5. This
domination of the level risk is the reason that the forecasting pattern of additional factors such as slope and curvature risk is not visible in the given forecasting framework
and vanishes in the measurement errors. These risk factors are not to be confused
with the principal components of yields known as level, slope and curvature, as described by Litterman and Scheinkman (1991) and Knez, Litterman and Scheinkman
(1994) and plotted on the left-hand side of Figure 2.5. Principal components of
yields are referred to as instruments in the forecasting regression, which correspond
to moves in the yield curve. Principal components of returns are portfolios reacting
2.4. SLOPE AND CURVATURE RISK PREMIA
31
to isolated shocks such as shifts, turns and bends, which are called level, slope and
curvature risks, respectively. The level risk is long in all bonds and thus like an average across maturities. This level risk premium has been assessed in the previous
section. This section shows that in addition to the level risk, additional risk premia
PCA of yields
PCA of excess returns
Level
Slope
Curvature
0.2
0.15
0.15
0.1
0.05
0.1
0
0.05
−0.05
0
−0.1
−0.05
−0.15
Level
Slope
Curvature
−0.2
−0.1
−0.25
2
4
6
8
10
2
4
6
8
10
Figure 2.5: PCA of yields and excess returns
The figure shows the first three loadings of the eigenvector of a principal component analysis of yields on
the left-hand side and excess returns on the right-hand side. The data is taken from the CBD data set for
the period of 1952:06 - 2009:06.
for slope and curvature risks are also priced. In swap markets, standard strategies
that correspond to these risks exist. Exposure to slope risk is called a duration or
delta-neutral position, meaning that there is no exposure to level risk. Similarly, isolated exposure to curvature risk is called a butterfly position. Since these are standard
strategies with high trading volume in interest rate markets, the existence of priced
risk premia to compensate risks in these positions is likely.
2.4.1
Slope and Curvature Portfolios
In order to analyze these additional risks, they are isolated from bond returns by
constructing portfolios of single bonds. Separating these risks is important since
32
they differ greatly in variance and would vanish behind the most volatile risk in the
measurement errors if not isolated. The portfolios are built in two ways. Firstly,
the first three principal components of excess returns, which react to level, slope
and curvature risks are taken as described above.9 The orthogonality property of
eigenvectors ensures that these portfolios react to isolated risks. The loadings of
the principal components are shown in the right-hand plot of Figure 2.5. The level
portfolio represents 98.0% of variation of a standard long-only bond portfolio and is
thus expected to react the same as the average of the individual bonds in the previous section. Secondly, slope and curvature portfolios are constructed by weighting
according to the duration of the bonds.10 Slope and curvature portfolios are thus
set duration- and slope-neutral, respectively. The duration of a zero bond equals its
maturity and thus the excess return of a duration neutral portfolio srxm,n is built of
two bonds with distinct maturities of m and n years by
(m)
(n)
m,n
= wd rxt+1 − (1 − wd ) rxt+1
srxt+1
n
wd =
.
m+n
(2.7a)
(2.7b)
Duration- and slope-neutral portfolios crxm,n,o , which have exposure to curvature risk
are constructed by the difference of two slope portfolios, or at least three different
single bonds with maturities of m, n and o years as
m,n,o
m,n
n,o
crxt+1
= ws srxt+1
− (1 − ws ) srxt+1
(2.8a)
o−m
o o+m
o−m
m−n
o o+m
− m m+n
(2.8b)
ws =
.
This method allows the construction of many different slope and curvature portfolios
with adjacent maturities along the curve. However, the results are only shown for a
representative portfolio spanning the whole curve, which therefore also reacts to the
2,10
and for the
turns and bends of the entire curve. For the slope portfolio this is srxt+1
2,5,10
.
curvature portfolio crxt+1
9 The
10 The
gross exposure is arbitrarily set to 100%.
gross exposure is set to 100%.
2.4. SLOPE AND CURVATURE RISK PREMIA
33
Sample statistics of slope and curvature portfolios yield Sharpe ratios of up to
0.32. However, they are consistent neither in size nor sign and also sensitive to the
sample selection. A statement about unconditional risk premia is thus not evident.
Since these premia are rather small in size compared to measurement errors, which
do not reduce in order of magnitude by constructing long-short portfolios of single
bonds, the signal-to-noise ratio deteriorates. Additionally, conditional risk premia,
as analyzed in the following paragraph, could overlay unconditional long-term risk
premia.
2.4.2
Forecasting Slope and Curvature
Conditional risk premia on slope and curvature risks are analyzed in the same manner
as the level before by forecasting regressions with forwards as instruments. Table 2.7
shows the R2 of regressing the aforementioned portfolios onto the unrestricted and
restricted set of forward rates. As expected, the level portfolio is forecasted with
about the same R2 as the average of all individual single bonds in Table 2.2. This
provides empirical support for the fact that forecasting single bonds and long-only
portfolios is simply forecasting the level risk premium; all the other risk premia are
missed. The slope and curvature portfolios are predicted with similarly high R2 , both
in the principal component and duration- or slope-neutral construction method. This
is a strong indication for time-varying priced risk premia for both slope and curvature
risks. This result extends the study of Boos (2011) and states that in addition to
a curvature risk premium, level and slope risks are also priced. For the restricted
set of forwards, the R2 drops substantially, but the selection of the restricted set
is optimized to forecast the level and not slope or curvature risk. The breakdown
of forecasting power along principal components of the yield curve indicates what
features of the yield curve forecast level, slope and curvature risk premia, and is
shown in the lower half of Table 2.7. The level of the yield curve has almost no
forecasting power on any of the portfolios and is only significant in forecasting level
portfolios. The slope and curvature of the yield curve provide the main contribution
34
in forecasting the level risk premia. In contrast, slope and curvature risk premia are,
to a large extent, forecasted by an additional instrument, which is the fourth principal
component of yields. However, this additional instrument has no forecasting power
on the level risk premium. Overall, at least four individual instruments are needed
to forecast level, slope and curvature risk premia. Yields as instruments are not able
to capture any significant forecasting pattern in risks beyond the first three principal
components of bond returns; an analysis of them is thus omitted.
Table 2.7: R2 of forecasting level, slope and curvature portfolios
Portfolios
Level
Slope
Curvature
srx2,10
crx2,5,10
Forwards as instruments
Unrestricted
0.30
0.22
0.44
0.27
0.25
Restricted
0.27
0.11
0.20
0.22
0.18
Level
0.03
0.00
0.02
0.02
0.00
Slope
0.12
0.03
0.05
0.10
0.10
Curvature
0.06
0.03
0.01
0.00
0.06
PC4
0.02
0.11
0.22
0.12
0.00
First four
0.23
0.18
0.30
0.24
0.15
All yields
0.30
0.22
0.44
0.27
0.25
Principle components of yields as instruments
The table shows the R2 of forecasting regressions based on the CBD data set 1952:06 - 2009:06. The
portfolios’ level, slope and curvature are based on principal components of excess returns, the srx and crx
are slope and curvature portfolios, respectively, constructed as described in the text.
2.5
Conclusion
This chapter has analyzed risk premia of bonds in a linear regression framework
and extended existing studies in several ways. The inclusion of more recent data
proposes that the forecasting ability is not persistent with such high values of R2 as
posted in earlier studies about bond risk premia. A subsample analysis even presumes that the predictability of bonds is boosted by the Great Inflation period and
2.5. CONCLUSION
35
thus R2 of the whole sample are overestimated. In addition to the analysis of the level
risk premium, further risk premia are investigated and slope and curvature risks are
shown to be compensated. A priced slope risk premium has been stated in previous
studies, but not so the priced curvature risk premium. Although slope and curvature
risk premia are forecasted with R2 at the same size as the level risk premium, the
factors to forecast these risks are not entirely the same. As well as the level, slope
and curvature of the yield curve, which forecast the level risk premium, slope and
curvature risk premia require the fourth principal component of yields as a forecasting factor. Finally, there is no clear evidence for the existence of unconditional risk
premia on bonds for level, slope or curvature risks.
36
Chapter 3
Affine Term Structure Models
In this chapter, bond return predictability is analyzed in an affine model setting. The
goal is to evaluate the number of priced risk premia and the number of driving state
variables needed to span them in a yields-only term structure model. Compared
against the simple regression framework of the last chapter, the affine term structure model includes no arbitrage assumptions or cross-equation restrictions, which
improve the stability and plausibility of the estimation. Bringing together the time
series and cross-section dimension applies additional structure to the risk premia process. Furthermore, the signal-to-noise ratio is improved by specifying measurement
errors explicitly and filtering the state variables. The model is built as an affine term
structure model with a flexible specification of time-varying risk premia and latent
state variables, which are backed-out of the yield curve, a yields-only model. The
number of priced risk premia and the number of state variables is shown to coincide
with the empirical analysis in Chapter 2. The dominant model prices three risk premia based on four factors that drive the state dynamics. The term structure model
results are finally compared to the regression results.
The chapter is organized as follows. The next section argues the advantages of
37
38
CHAPTER 3. AFFINE TERM STRUCTURE MODELS
an affine model over the simple regression framework. Section 3.2 defines the model
and Section 3.3 derives the appropriate pricing equations. Section 3.4 deals with
parametrization and identification issues and Section 3.5 continues with the applied
estimation procedure. Finally, an analysis of empirical results is presented in Section
3.6, followed by the conclusion.
3.1
Why an Affine Model?
The main difference between forecasting regressions and affine yield curve models is
that the model approach considers not only the time series, but also the cross-section
dimension. The additional cross-equation restrictions arise because of imposing the
no-arbitrage property. The advantage of the regression framework is fast, easy and
consistent estimation with ordinary least squares. In comparison, estimating an affine
model is demanding and needs lots of computation time. However, the model approach with cross-equation restrictions implied by no arbitrage has many advantages
(Piazzesi 2003). Firstly, the no-arbitrage property makes the yield equation consistent in the time series and the cross-section. Secondly, affine models allow the
separation of risk premia from expectations about future yields. According to the
expectation hypothesis, expected bond excess returns are zero. However, evidence
suggests that expected bond returns are time-varying. Cross-equation restrictions are
necessary to model and test for risk premia and their behavior. Thirdly, the number of
variables does not have to be equal to the number of yields. These lower-dimensional
systems capture the common driving forces of yields in a few factors. Overidentified
systems result in more efficient estimates and this is particularly helpful for estimating a large number of parameters; even more so in yield curve models where the state
variables are highly correlated. The additional structure that follows from a consistent model setting instead of the simple regression framework helps to improve the
stability and plausibility of the forecasts.
The model class applied in this thesis has been defined by Ang and Piazzesi
3.2. THE MODEL
39
(2003) and is a discrete-time homoscedastic affine class of models with Gaussian
shocks. This type of model is also used in the studies of Cochrane and Piazzesi
(2005, 2008). The choice of an affine model is motivated by tractability and closedform solutions. A Gaussian vector autoregressive (VAR) process in a discrete-time
framework eases estimation, since a maximum likelihood estimation (MLE) is applicable with closed-form density and no discretization method is needed. Choosing discrete-time is not to the detriment of generality and even adds flexibility and
tractability compared to continuous-time analogues as stated in Joslin, Singleton and
Zhu (2011) and Dai, Le and Singleton (2010). All these affine Gaussian models have
a continuous-time analog with comparable properties.
3.2
The Model
This section defines the yield curve model of the class of Ang and Piazzesi (2003).
The base case is a single currency model with interest rates as the only state variables:
a so called yields-only model. The following model definition, however, is general
and valid for any kind of state variable whether latent, statistical or macroeconomic.
The yields-only restriction that interest rates are the state variables is not imposed
until the parametrization in Section 3.4.
The state equation defines the dynamics of the state variables under the physical distribution (P-measure). The state variables Xt follow a vector autoregressive
process (VAR)
Xt+1 = µ + ΦXt + εt+1
(3.1)
with Gaussian iid shocks εt+1 with zero mean and covariance matrix Σ.
The pricing kernel Mt+1 , together with the specification of the price of risk λt ,
40
completes the definition of the model:
1 ⊤
⊤
Mt+1 = exp(−δ0 − δ⊤
1 Xt − λt Σλt − λt εt+1 )
2
λt = λ0 + λ1 Xt .
(3.2a)
(3.2b)
The derivation of the affine pricing kernel is standard math and given in Ang and
Piazzesi (2003). The pricing kernel can be thought of as a stochastic discount factor
(SDF), which discounts any future payoff with a risk and time discount. The time
(1)
discount is reflected by the risk-free rate yt
= δ0 + δ⊤
1 Xt in the first part of the pric-
ing kernel (3.2a). The second term 21 λt⊤ Σλt is a correction for the Jensen’s inequality,
which arises because of the transformation from simple to log returns. The final term
λt⊤ εt+1 is the risk discount and the only source of stochastics in the pricing kernel.
The risk discount is driven by the innovations to the state variables εt+1 times the
price of risk λt . The price of risk is time-varying and affine in the state variables as
defined in Equation (3.2b).
The transformation of the state equation (3.1) into the risk-neutral measure (Qmeasure) is achieved by subtracting the risk compensation Σλt given as the covariance of state variable innovations times the price of risk. The risk-neutral VAR then
reads
Xt+1 = µ∗ + Φ∗ Xt + εt+1
(3.3)
with the risk-neutral parameters µ∗ and Φ∗
3.3
µ∗ = µ − Σλ0
(3.4a)
Φ∗ = Φ − Σλ1 .
(3.4b)
Valuation Formulas
Valuation formulas express the price or return of bonds in terms of the state variables
and thus link observed time series with the state variables and express the model
expectations of future returns. There is a major difference in the valuation formulas
3.3. VALUATION FORMULAS
41
of prices, yields and forward rates, which are contemporaneously observed and those
of future expectations of any of these. Contemporaneously observed values are only
dependent on the risk-neutral distribution alone, whereas expectations thereof are
dependent on the physical distribution parameters as well.
3.3.1
Cross-Section
The bond pricing equations are recursively derived from the basic model. The log
(n)
bond price pt
at time t with maturity n is given iteratively as a function of the
(n−1)
expected log bond price pt+1
kernel
(n)
pt
one period later, discounted by the expected pricing
h
i
(n−1)
= log Et Mt+1 exp pt+1
.
(3.5)
By iterating Equation (3.5) up to the maturity n of the bond and the assumption1 that
(0)
pt
= 0 leads to
(n)
= log Et [Mt+1 . . . Mt+n ] .
(3.6)
= log Et [Mt+1 ] = −δ0 − δ⊤
1 Xt ,
(3.7)
pt
For the one period bond this is
(1)
pt
which is non-stochastic because it only depends on Xt , which is known at time t.
(1)
This also gives the one period yield yt
(1)
yt
from Equation (2.2) as
= δ0 + δ⊤
1 Xt ,
(3.8)
which is the short rate of the model and thus also deterministic.
The calculation of further log bond prices is achieved by inserting the pricing
kernel Mt+1 into Equation (3.6). The log bond prices are affine in the state variables
(n)
pt
1 The
= An + B⊤
n Xt
(3.9)
price of a risk-free zero bond at maturity by definition equals the nominal value of that bond.
All bonds can be normalized to a nominal value of one without loss of generality; the log bond price at
(0)
maturity pt
is then zero.
42
with coefficients An and Bn following a recursive scheme depending only on the
parameters δ0 , δ1 , µ∗ , Φ∗ and Σ. The yields and forward rates implied by the model
are also affine in the state variables. Derivations of all relevant valuation formulas
are provided in Appendix C.
3.3.2
Model Expectations
The above formulas for prices, yields and forwards span the cross-section at time
t only. They are thus deterministic and depend only on the risk-neutral parameters. Model expectations of prices, yields and forwards are derived by iterating
forward along the time series dimension and taking expectations. Expected k-step(n)
ahead prices Et [pt+k ] are affine in the expected future state variables Et [Xt+k ] with
the known coefficients An and Bn from (3.9)
h
i
(n)
Et pt+k = An + B⊤
n Et [Xt+k ] .
(3.10)
The formula is solved by iterating the time series of the expected state variables back
to Xt along the VAR of Equation (3.1) with
Et [Xt+k ] = (I − Φk )(I − Φ)−1 µ + Φk Xt .
(3.11)
The equations for return forecasts are derived based on the difference of expected
(n)
bond prices. The 1-step-ahead expected return Et [rt+1 ] is derived from the definition
of return in (2.4) by taking expectations
i
i
h
h
(n)
(n−1)
(n)
Et rt+1 = Et pt+1 − pt .
(3.12)
(1)
Expected excess returns are retrieved by subtracting the risk-free rate yt = δ0 +δ1 Xt
from the expected returns. All these expected values depend on the physical and
risk-neutral parameters of the model. Further valuation formulas are provided in
Appendix C.
3.4. PARAMETER IDENTIFICATION
3.4
43
Parameter Identification
Identification of state space systems with latent state variables is difficult. Latent
variable models are well known to often suffer from identification problems (CollinDufresne, Goldstein and Jones 2008). Piazzesi (2003) states that the invertibility
condition on the information matrix, which should provide evidence of parameter
identification, is not useful within affine term structure models, especially those using numerical gradient methods, because they are imprecise. Theoretical conditions
of identification in the class of affine term structure models in continuous-time are
developed by Dai and Singleton (2000). They reach identification by applying invariant transformations; these reduce the number of free parameters by eliminating
unidentified parameters. The theoretical conditions for identification are derived
in a representation with a diagonal diffusion matrix. Later research by Cheridito, Filipovic and Kimmel (2006) show, by extending the representation to nondiagonalizable diffusion models, that the conditions of Dai and Singleton (2000) only
provide locally identified models and that the representation does not have maximum
flexibility. In addition, the approach of Dai and Singleton (2000) struggles with the
fact that latent state variable models often lead to only locally identifiable models
(Collin-Dufresne, Goldstein and Jones 2008).
A different approach, which circumvents the problem of latent state variables is
to rotate them into a set of state variables that are observable.2 The state variables
are thus tied to this observable factors and the number of free parameters is reduced
to an identifiable set. This approach has been introduced by Duffie and Kan (1996)
by rotating into yields. Collin-Dufresne, Goldstein and Jones (2008) extended this
methodology as they transform the system by rotation into observable state variables
with unique economic interpretation, in specifically the spot rate and its risk-neutral
drifts. This specific representation also circumvents the problem that it is often analytically difficult to rotate into observable variables because the functions have no
2 A formal description of invariant transformations of affine term structure models is given in appendix
D.1.
44
closed-form solution. Boos (2011) applies a similar parametrization in discrete-time
with the risk-free rate and its risk-neutral expectations as state variables. A similar
approach in discrete-time is given in Cochrane and Piazzesi (2006). They specify
the model as self-consistent, meaning that the state variables that go into the model
are rotated into any set of variables that come out of the model, such as bond prices,
yields or forward rates. This self-consistent model representation has the same desired properties as the parametrization proposed by Boos (2011). Joslin, Singleton
and Zhu (2011) achieve a similar transformation of the state variables into arbitrary
observable economic variables. All these methods rotate the state variables into
economic variables and therefore model parameters are identified by tying them to
observable state variables.
3.4.1
Self-Consistent Models
Self-consistency is a property of models that has proven to be successful in several
ways. Firstly, a general solution to the identification problem of term structure models is given as stated above. Secondly, empirical estimation is improved. Estimation
of standard term structure models has been computationally extensive and suffered
from problems of slow convergence speed to the optimum and a likelihood function
with multiple local optima. A detailed description of these challenges can be found
in Duffee and Stanton (2008) and Duffee (2009). The self-consistent parametrization improves estimation speed and accuracy as reported in Boos (2011) and Joslin,
Singleton and Zhu (2011).
The parametrization that ensures self-consistency is easily derived by imposing
that the state variables that go into the model are the same as some set of variables
that come out of the model. Therefore a multitude of self-consistent parametrizations possible, one for any set of state variables. Obvious choices for state variables
in term structure models are log bond prices, yields and forward rates. In the following, the parametrization with forward rates as state variables is derived. Different
parametrizations with log bond prices and yields as state variables are deferred to
45
Appendix D.2.
3.4.2
Forwards as State Variables
To obtain the parametrization, specific choices of Φ∗ , µ∗ , δ0 and δ1 are derived while
setting the forward rates as state variables. This choice is arbitrary but not restrictive
since affine rotation into forward rates is possible without loss of generality. The
(k)
state variables Xt are defined as the first K forward rates ft
(k)
Xt,k = ft ,
(3.13)
where K is the number of state variables and k = 1 . . . K. Pricing forwards with this
set of state variables yields along Equation (C.10)
i
h
(K) ⊤
(1)
(n)
.
ft = Anf + Bnf ⊤ ft · · · ft
f
(3.14)
f
This pricing equation is solved for the parameters An and Bn for n = 1 . . . K. Obvif
f
ously An = 0 and Bn = en for n = 1 . . . k, where en is a vector of zeros with length K
and a one in the n-th entry. Combining with equations (C.11) and (C.12) and solving
for n = 1 gives
δ0 = 0




δ1 = 


1
0
..
.
0
(3.15a)




.


(3.15b)
Solving for the remaining n = 2 . . . K gives the risk-neutral VAR parameter Φ∗ in
companion form as

0
 .
 .
 .
Φ∗ = 
 0

c1
1
0
..
.
0
c2
···





1 

cK
(3.16)
46
and µ∗ as





∗
µ =



1
2 Σ1,1
1
2 Σ2,2 + Σ2,1
..
.
1
2 ΣK,K
+ ΣK,K−1 + . . . + ΣK,1
cK+1





.



(3.17)
c1 . . . cK+1 are free parameters of the model. The covariance matrix of the VAR is
by definition positive definite and parameterized by a Cholesky factorization. This
adds thus another K(K + 1)/2 free parameters. Together with a maximally flexible
specification of the price of risk λt the number of parameters of a model with K state
variables is
3 2 5
K + K + 1.
2
2
A model with three state variables therefore has 22 free parameters. For a model
with four state variables the number of free parameters increases to 35, in one with
five state variables to 51.
This parametrization is very similar to the one in Boos (2011), which takes the
risk-free rate and its risk-neutral expectations as state variables
(1)
Xt,1 = ft
(3.18a)
Xt,k = EQ [Xt,k−1 ],
(3.18b)
where EQ denotes the expectation under the risk-neutral measure. This parametrization is identical to that with the first k forward rates as state variables, with the exception of a small difference in the constant VAR term µ∗ . This constant is thus defined
as





µ∗ = 



0
0
..
.
0
cK+1





.



(3.19)
47
The difference to the parametrization in Equation (3.17) is due to Jensen’s inequality
terms. Henceforth this parametrization is used for estimating the models. Other
choices of self-consistent parametrizations are given in Appendix D.2.
3.4.3
Restricting the Number of Forecasting Factors
Independent of the parametrization, the model allows the number of forecasting factors to be restricted. In the model, returns are driven by the price of risk λt multiplied
by the shocks to the state variables εt+1 . Hence, expected returns are driven by Σλt ,
whereas Σλ0 contributes to the constant part and Σλ1 to the time-varying part. Restrictions to reduce the number of factors that forecast expected returns therefore
affect the parameters of the market price of risk λ0 and λ1 .
By setting both λ0 and λ1 to zero, the model is no longer able to produce risk
premia. The pricing kernel (3.2a) becomes non-stochastic and only incorporates the
time discount part based on risk-free interest rates. This is the case since the only
source of variation in the pricing kernel is driven by the price of risk multiplied by the
shocks to the state variables. The volatility of the state variables is constant over time
and therefore provides no additional variation. As a consequence, the physical (Pmeasure) and risk-neutral (Q-measure) distribution are equal, as are the parameters
of the corresponding VAR models, (3.3) and (3.1), respectively. With λ0 free and
only λ1 constrained to zero, the model produces risk premia that are constant over
time. Returns are therefore still not predictable in the model, since the time-varying
risk premia are governed by the parameter λ1 . In this case, the model is equal to
a multi-factor version of the Vasicek (1977) model. If both λ0 and λ1 are free, the
model is fully flexible and all state variables are allowed to drive time-varying risk
premia and thus determine the predictability pattern of the model. Restricting the
rank of the matrix λ1 controls the number of state variables driving the predictability
pattern. The rank of λ1 corresponds to the number of individual factors driving
expected returns and thus predictability. For the maximally flexible model, the rank
L of matrix λ1 is equal to the number of state variables K, and for the model with
48
constant risk premia that has no time-varying predictability pattern, L is set to zero.
For any L between zero and K the model displays time-varying risk premia driven by
a reduced rank factor model. Henceforth the notation for this class of term structure
models is A(K, L) with K as the number of state variables and L as the rank of the
matrix λ1 .
An A(K, L < K) model implies that not all state variables feed into the model
forecasts. This is economically reasonable since the model is not only conducting
forecasts in the time series dimension, but also pricing current bonds in the crosssection dimension. It is thus expected that some of the state variables will be necessary to explain the cross-section, and others are included for forecasting only. By
estimating the full A(K, L = K) model, the differentiation of the state variables into
forecasting, cross-section or both is not forced by restrictions. In the A(K, L < K)
models the differentiation is preselected by restricting the number of state variables
that drive the time-varying price of risk. Obviously, all reduced rank A(K, L < K)
models are nested in the full A(K, L = K) model.
3.5
Estimation
Parameter estimation is achieved using maximum likelihood estimation (MLE), whereas
all the yields are supposed to be observed with measurement errors. The resulting
Kalman filter algorithm introduced by Kalman (1960) is the optimal procedure to
do so. The Kalman filter is the same as simple MLE or OLS when state variables
are observed without measurement errors, as shown by Boos (2011) and Joslin, Singleton and Zhu (2011).3 In the case of yields affected by normally distributed measurement errors, Kalman filtering is the efficient estimation method and solves the
overidentification problem in the cross-section by extracting the optimal latent state
3A
comparison of MLE to Kalman filtering in terms of finite sample properties is provided by Duffee
and Stanton (2004). They find that Kalman filtering is an efficient method to estimate term structure
models especially for rather complex specifications.
3.5. ESTIMATION
49
variables, thus improving the accuracy of the state variables when estimated from
a broad cross-section. The reason is that the explicit specification of measurement
errors helps to reveal underlying common factors in a similar manner as a principal
component analysis.
The model in state space form provides the required equations. The transition
equation is equal to the dynamics of the state variables in the VAR (3.1)
Xt+1 = µ + ΦXt + εt+1
(3.20)
and the measurement equation links the observable data with the state variables in
the corresponding valuation formula. In the case of yields this is the pricing equation
(n)
for yields (C.7) with added measurement errors ηt
(n)
yt
(n)
= Ayn + By⊤
n Xt + ηt .
(3.21)
Generally, measurement errors ηt are assumed to be iid normally distributed and
have a diagonal covariance matrix to reduce the number of parameters. In this model
measurement errors are even assumed to be of the same size across maturities in
terms of volatility. The standard deviation of the measurement errors ηt is ση and
estimated jointly with the other parameters. This specification only adds another
parameter to the estimation problem and the reduction in the dimension of the measurement error specification does not seem to alter the results significantly. Throughout this thesis the Kalman filter optimization routines of Boos (2011)4 are used. A
complete description of the applied estimation procedure is provided in Appendix B.
Based on the estimation procedure, several model specifications are compared using
likelihood ratio tests. This specification test is easily applied to models estimated by
MLE and is shortly reviewed in Appendix B.4.
4 Thanks
to Dominik Boos for kindly providing me with the Matlab routines.
50
3.6
Empirical Results
This section discusses the empirical properties of the estimated models. The estimation is based on the CBD data set described in Section 2.2. Here the full data set
ranging from 1952:06 - 2009:06 is used. Model estimation is conducted at different frequencies ranging from quarterly to semi-annually and annually. The selection
of the frequency is a choice between a reasonable signal-to-noise ratio and a great
amount of independent observations. Quarterly data is a good balance and is treated
as the base case in this empirical analysis. In addition, semi-annual and annual data
is used to supplement the empirical results, although the number of observations in
both time series and the cross-section is reduced. Models estimated on monthly frequency are also investigated but dropped in this analysis due to three reasons. Firstly,
the results indicate a too low signal-to-noise ratio to reasonably calibrate these models to forecasting. Secondly, the evaluation based on one year forecasting power for
a monthly model might be too far in the future; thus calibrating the model to different features arising in monthly returns, which might be different to annual returns.
Thirdly, the CBD data set does not provide monthly spaced maturities on the short
end to fit the model in the parametrization applied in this thesis. The results from
the monthly models are thus not directly comparable to the other models estimated
in this section and not reported here.
All the models are estimated with three, four and five factors. The principal
component analysis shows that three factors are enough, but necessary to explain the
current yield curve accurate in the cross-section. The forecasting regression, however, revealed that information not included in the first three principal components is
necessary to forecast bond returns. Models with three, four and five factors are thus
estimated and compared in their forecasting power. In addition, all the models are
estimated with all possible rank restrictions on time-varying risk premia as described
in Section 3.4.3 in order to assess the number of priced risk premia. These models
are compared in terms of likelihood ratio specification tests as defined in Appendix
3.6. EMPIRICAL RESULTS
51
B.4 and in terms of the coefficient of determination R2 defined as follows
R2 = 1 −
var[rxt − Et−1 [rxt ]]
.
var[rxt ]
(3.22)
The forecasting power in terms of R2 is always evaluated for one year returns, since
historical data for bonds with longer maturity is only available at a yearly frequency.
The level forecasting power is analyzed at the short and long horizon with rx(2)
and rx(10) , respectively. For the slope and curvature, the same portfolios as in the
regression analysis5 are used, namely srx2,10 and crx2,5,10 .
3.6.1
Quarterly Models
The following models are estimated based on a short rate of three months with the
CBD data set and henceforth are called quarterly models. The number of independent observations in the time series of the selected data set is 229. The quarterly
models are set as the reference and thus analyzed in more detail than estimates based
on different frequencies.
Three-Factor Models
The estimation results for the quarterly three-factor models are shown in Table 3.1.
The A(3, 0) model is not able to produce time-varying risk premia and thus has no
forecasting ability. The A(3, 1) model with one priced risk premium already forecasts
one year excess returns on bonds with maturities of two and ten years with an R2 of
0.15 and 0.19, respectively. The forecasting power of slope and curvature portfolios
is visible, but lower with R2 of 0.05 and 0.13, respectively. The R2 for slope and
curvature portfolios is further increased with an additional priced risk factor in model
A(3, 2), but stays the same for the level. The full A(3, 3) model does not increase
forecasting power any further. The dominant model in terms of forecasting power
5 see
Section 2.4
52
is therefore A(3, 2). In terms of likelihoods, the A(3, 3) model does not significantly
increase the log likelihood compared to model A(3, 2). The likelihood ratio tests for
all combinations among three-factor models are given in the upper part of Table 3.4.
The specification tests indicate that the A(3, 2) model is the dominant model at a
significance level of 1%.
Table 3.1: Forecasting power of the quarterly three-factor models
log likelihood
A(3, 0)
20034.06
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
14
0.00
0.00
0.00
0.00
A(3, 1)
20044.94
19
0.15
0.19
0.05
0.13
A(3, 2)
20052.31
22
0.14
0.19
0.13
0.15
A(3, 3)
20052.46
23
0.14
0.19
0.13
0.15
The table shows the R2 of a one year forecast based on the quarterly three-factor models, the number of
parameters (“nvar”) and the estimated maximum log likelihood.
Four-Factor Models
The estimation results of the quarterly four-factor models are shown in Table 3.2.
The A(4, 0) and A(4, 1) models show no forecasting power. This is obvious in the
case of A(4, 0), but not for the A(4, 1), since the parametrization allows for one priced
risk premium. In terms of likelihoods the A(4, 1) model clearly dominates the A(4, 0)
model, as seen in the mid part of Table 3.4. The A(4, 2) model shows some forecasting power for level, slope and curvature, which is further increased by the models
A(4, 3) and A(4, 4). The dominant model in terms of forecasting power and likelihood ratios is A(4, 3), as stated in tables 3.2 and 3.4. The corresponding R2 are
comparable in size to the dominant three-factor model A(3, 2), only for the slope
portfolio the A(4, 3) model is much higher with 0.18 compared to 0.13. The number
of priced risk premia in the dominant model increases to three while the three-factor
dominant model shows only two priced risk premia. The increase in the number of
risk premia corresponds by adding a fourth state variable to the model, which enables
the model to separate another risk premium.
53
Table 3.2: Forecasting power of the quarterly four-factor models
log likelihood
A(4, 0)
20584.07
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
20
0.00
0.00
0.00
0.00
A(4, 1)
20603.65
27
0.00
-0.03
0.01
0.01
A(4, 2)
20618.64
32
0.06
0.15
0.15
0.12
A(4, 3)
20624.55
35
0.15
0.18
0.18
0.16
A(4, 4)
20624.66
36
0.15
0.18
0.17
0.16
The table shows the R2 of a one year forecast based on the quarterly four-factor models, the number of
Five-Factor Models
The estimation results of the quarterly five-factor models are shown in Table 3.3.
Again the A(5, 0) model shows no forecasting power, as expected. Like the fourfactor case, the A(5, 1) model also has limited forecasting abilities close to zero.
The A(5, 2) model shows some forecasting power, mainly for level and curvature.
Forecasting ability is further increased with more than two priced risk premia in the
A(5, 3), A(5, 4) and A(5, 5) models, which all show comparable performance. The
A(5, 3) model is thus the dominant model with level, slope and curvature forecasting
between around 0.15 and 0.17. In terms of the specification test the A(5, 3) model
is also dominant, based on the likelihood ratio tests in Table 3.4. The forecasting
power R2 of the dominant five-factor model corresponds to the values found for the
four-factor models. Adding the fifth state variable does not increase the number of
priced risk premia in the dominant A(5, 3) model in comparison to the four-factor
case.
Model Selection
A comparison of the dominant risk premia specifications among the three- to fivefactor models delivers the following result. The four- and five-factor models (both
with three risk premia) outperform the three-factor model (with only two risk premia)
54
Table 3.3: Forecasting power of the quarterly five-factor models
log likelihood
A(5, 0)
20705.18
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
27
0.00
0.00
0.00
0.00
A(5, 1)
20742.35
36
0.03
0.06
0.05
0.08
A(5, 2)
20750.80
43
0.14
0.14
0.07
0.12
A(5, 3)
20771.88
48
0.17
0.17
0.15
0.17
A(5, 4)
20774.28
51
0.15
0.17
0.15
0.17
A(5, 5)
20775.46
52
0.15
0.18
0.18
0.18
The table shows the R2 of a one year forecast based on the quarterly five-factor models, the number of
in terms of R2 , as already shown above. Likelihood ratio tests between these models
also indicate that the four- and five-factor models clearly outperform the three-factor
model with a difference in log likelihoods of 572 and 720, respectively. As a result,
the three-factor model is clearly dominated by the other two.
The dominance of the five-factor model over the four-factor model is less clear.
Although the likelihood ratio test strongly supports the dominance of the five-factor
model with a log likelihood increase of 147, the forecasting performance in R2 (as
noted above) is in equal range and provides no support for the dominance of the
five-factor model with additional 13 parameters. The fact that the additional state
variable in the five-factor model does not reveal any further priced risk premia for
the dominant specification is also unsupportive. The reasons for the dominance of
the five-factor model are therefore explored in more detail. Since the additional state
variable is not manifested in another risk premium, the A(5, 3) model might just
better fit the actual term structure and not improve the forecasting ability. The fit
of the actual term structure is evaluated as the root mean squared error (RMSE) and
absolute mean error (MAE) provided in Table 3.5.
The improvement from the A(3, 2) to the A(4, 3) model is much greater than
from the A(4, 3) to the A(5, 3) model, and the correlation of yields in the four- and
five-factor models with equal maturity is 1. The dominance of the A(5, 3) model is
thus likely to be based on a better fit of the actual term structure and not an improve-
55
Table 3.4: Likelihood ratio tests of the quarterly models
A(3, 0)
A(3, 1)
A(3, 1)
21.8
(15.1)
A(3, 2)
36.5
(20.1)
14.7
(11.3)
A(3, 3)
36.8
(21.7)
15.0
(13.3)
A(4, 0)
A(4, 1)
A(3, 2)
0.3
(6.6)
A(4, 2)
A(4, 1)
39.2
(18.5)
A(4, 2)
69.1
(26.2)
30.0
(15.1)
A(4, 3)
81.0
(30.6)
41.8
(20.1)
11.8
(11.3)
A(4, 4)
81.2
(32.0)
42.0
(21.7)
12.0
(13.3)
A(5, 0)
A(5, 1)
A(5, 2)
A(4, 3)
0.2
(6.6)
A(5, 3)
A(5, 1)
74.3
(21.7)
A(5, 2)
91.2
(32.0)
16.9
(18.5)
A(5, 3)
133.4
(38.9)
59.1
(26.2)
42.2
(15.1)
A(5, 4)
138.2
(43.0)
63.9
(30.6)
47.0
(20.1)
4.8
(11.3)
A(5, 5)
140.6
(44.3)
66.2
(32.0)
49.3
(21.7)
7.2
(13.3)
A(5, 4)
2.4
(6.6)
The table shows the likelihood ratios comparing unrestricted and constrained models. The unrestricted
model is indicated on the left, the constrained (and nested) model on top. The corresponding test statistic
with a significance level of 1% is given in parenthesis. If the likelihood ratio exceeds the test statistic, the
unrestricted model fits the data significantly better than the constrained model in terms of likelihood.
ment in forecasting ability. Since the objective here is forecasting, the parsimonious
A(4, 3) model is preferred. Furthermore, the improvement of the yield curve fit is
negligible, particularly compared to the variance of the expected excess returns. Figure 3.1 shows the model forecast for level, slope and curvature in comparison with
the realized returns for the dominant A(4, 3) model.
Considering the dominant model A(4, 3)6 , the number of priced risk premia in
the US yield curve is three with four state variables driving the model. This analysis
thus provides evidence for a model of four state variables contradicting some of the
affine term structure literature that typically suggests three state variables e.g. Dai,
Singleton and Yang (2004). Reducing independent forecasting factors to three leaves
6 Model
parameter estimates are provided in the appendix D.3 followed by an inspection of measure-
ment errors in D.4.
56
Level
0.4
0.2
forecast
realized
0
−0.2
−0.4
1950
1960
1970
1980
1990
2000
2010
1990
2000
2010
1990
2000
2010
Slope
0.04
0.02
0
−0.02
−0.04
1950
1960
1970
1980
Curvature
0.04
0.02
0
−0.02
1950
1960
1970
1980
Figure 3.1: Forecasted vs. realized returns of the A(4, 3) model
The plots show one year forecasts for the quarterly A(4, 3) model compared to one year realized returns
(plotted with quarterly overlapping). Level is the long bond rx(10) ; slope and curvature are the portfolios
srx2,10 and crx2,5,10 , respectively.
57
Level
0.4
0.2
0
Kalman filter
OLS regression
−0.2
−0.4
1950
1960
1970
1980
1990
2000
2010
1990
2000
2010
1990
2000
2010
Slope
0.04
0.02
0
−0.02
1950
1960
1970
1980
Curvature
0.02
0.01
0
−0.01
1950
1960
1970
1980
Figure 3.2: Model forecasts vs. regression forecasts
The plots show one year forecasts for the quarterly A(4, 3) model compared to one year forecasts of the
restricted OLS regressions. Level is the long bond rx(10) ; slope and curvature are the portfolios srx2,10
and crx2,5,10 , respectively.
58
Table 3.5: Yield curve fit of the dominant models
A(3, 2)
A(4, 3)
A(5, 3)
MAE
0.058%
0.041%
0.037%
RMSE
0.088%
0.069%
0.066%
The table shows the mean absolute error (MAE) and root mean squared error (RMSE) of the given models
for the fit of the actual yield curve measured in annual yields from one to ten years.
one state variable or driving factor solely for risk-neutral yield curve movements.
3.6.2
Models Estimated on Different Frequencies
The same models as above are also estimated based on semi-annual and annual frequency. Observations are thus reduced in half and quarters, respectively, which turns
parameter estimation into more problematic terrain. The number of independent time
series observations for the annual frequency is only 57. Results from these models
should therefore be handled with care. Concisely, they point in the same direction
as models estimated on a quarterly frequency when considering the likelihood ratio
tests.7 The A(4, 3) and A(5, 3) models are dominant in both additional frequencies.
Table 3.6: Forecasting power of the A(4, 3) model at different frequencies
rx(2)
rx(10)
srx2,10
crx2,5,10
Quarterly Model A(4, 3)
0.15
0.18
0.18
0.16
Semi-annual model A(4, 3)
0.19
0.23
0.23
0.21
Annual Model A(4, 3)
0.22
0.22
0.51
0.24
The table shows the R2 of a one year forecast based on the A(4, 3) model estimated in the indicated
frequency.
Table 3.6 compares the forecasting power of the A(4, 3) model in all the estimated frequencies. The R2 improve slightly on the semi-annual and further on the
annual estimation frequency. The models estimated on these other frequencies sup7 Detailed
forecasting performance and likelihood ratio test statistics are given in Appendix D.5.
59
port the findings of three priced risk premia and the four state variables. These are
two reasons why the higher R2 might be due to a better in-sample fit of the lower
frequency models. Firstly, there are fewer observations to fit the model to. Secondly,
the R2 is evaluated at a forecasting period of one year, which might be better matched
by a model with annual frequency. However, the forecasting ability for level, slope
and curvature portfolios is persistent across frequencies.
3.6.3
Comparison to Regression Forecasts
Table 3.7 compares the Kalman filtered model forecasts with the OLS regression
forecasts of the previous chapter. The R2 of the model are lower than those of the
regression forecast for the level. For slope and curvature portfolios the model is superior to restricted regression forecasts. However, unrestricted regressions still show
higher R2 . As already noted in the regression section, the unrestricted regression
suffers from an in-sample fitting of measurement errors while the restricted regression suffers from selection bias. Correcting the selection bias by taking the mean R2
of all possible combination of instruments reduces the R2 by a third. The R2 of the
regressions are, with this correction, in the same range as the ones from the model.
Since the Kalman filter does not suffer from selection bias (as the state variables are
latent8 and measurement errors are modeled explicitly) the model forecasts deliver a
more precise measure of the extent to which bond returns are predictable.
Table 3.7: Model forecasts vs. regression forecasts
rx(2)
rx(10)
srx2,10
crx2,5,10
Kalman filtering A(4, 3) model
0.15
0.18
0.18
0.16
Unrestricted OLS regression
0.24
0.31
0.27
0.25
Restricted OLS regression
0.21
0.27
0.14
0.07
The table compares the R2 of the Kalman filtered model forecasts to the unrestricted and restricted OLS
regression forecasts of the previous chapter. For estimation, the CBD data set from 1952:06 - 2009:06 is
used to make the regression forecasts comparable to the model forecasts.
8 For
a detailed treatment of this topic consult Boos (2011).
60
Figure 3.2 plots the model forecasts for level, slope and curvature portfolios
against the restricted regression forecasts. Visual inspection of the plots show some
peaks that are generally more extreme in the regressed than the filtered forecasts.
This is because the filtering algorithm detects higher measurement errors in these
extreme return observations and, unlike the regression, therefore does not attempt to
fit these values exactly. Apart from this the forecasts of the both estimation methods
appear similar. The correlation matrix of the forecasted return series is displayed in
Table 3.8. The correlation between the two methods is quite high with 0.67, 0.81 and
0.72 for level, slope and curvature risk premia, respectively.
Table 3.8: Correlation of model and regression forecasts
Model
rx(10)
srx2,10
Regression
crx2,5,10
rx(10)
srx2,10
rx(10)
1.00
srx2,10
-0.61
crx2,5,10
-0.85
0.49
1.00
rx(10)
0.67
-0.36
-0.74
1.00
srx2,10
-0.40
0.81
0.42
-0.40
1.00
crx2,5,10
-0.63
0.64
0.72
-0.68
0.60
Model
Regression
crx2,5,10
1.00
1.00
The table displays the correlation matrix of model and regression forecasts. Model forecasts are the
Kalman filtered model forecasts of the A(4, 3) model, regression forecasts are those of the unrestricted
OLS regressions.
3.7
Conclusion
This chapter has analyzed bond return predictability in a term structure model setting. Predictability has been shown to be persistent in the state space model setting
across several frequencies. The analysis implies four state variables and three priced
risk premia based on the dominant A(4, 3) model. This coincides with the results of
the regression analysis of Chapter 2. These three priced risk premia are reflected in
the pricing of level, slope and curvature portfolios. Both the number of risk premia
3.7. CONCLUSION
61
and the number of state variables contradict the existing literature, which proposes
one or two priced risk premia and mostly lower dimensional models. The corresponding forecasting power, measured in R2 , is lower than that of the regression
forecasts of the previous chapter, but this is due to a favorable reduction in selection bias by specific handling of the measurement errors. Level, slope and curvature
portfolios are predicted with an R2 of around 0.18.
62
II
63
Chapter 4
Macro Regressions
The yield curve does not contain all available information to forecast bond returns.
There is additional information in macroeconomic variables that forecasts bond excess returns. Since this macroeconomic information is not spanned by the yield
curve, it is known as unspanned macroeconomic information. Unspanned information has been recently addressed by Duffee (2011), Joslin, Priebsch and Singleton
(2011) and Boos (2011). This chapter explores the predictive power of unspanned
macro factors and is, in terms of the empirical results, closest to the the study of
Joslin, Priebsch and Singleton (2011), but differs in several important points. The
analysis is extended along the time dimension in both directions to increase the number of observations to periods beyond the low inflation decreasing yield sample. Additional subsample and rolling window analysis reveal that predictability is not stable
throughout history. Furthermore, macro data is covered by a broader set of variables,
organized into three groups according to their approximation of real activity, inflation
and money supply. Finally, not only is the level risk premium of bonds accounted
for, but also the slope and curvature risk premia, which have been shown to be priced
in chapters 2 and 3.
65
66
CHAPTER 4. MACRO REGRESSIONS
The following regression analysis of bond excess returns demonstrates that macroe-
conomic variables forecast level, slope and curvature risks of bond portfolios. Monetary economics also deals with the prediction of monetary policy actions based on
macroeconomic information: the Taylor rule (Taylor 1993) and similar models attempt to predict the short term interest rate by changes in inflation and output gap.
Ang, Dong and Piazzesi (2007) estimate a term structure model with an augmented
Taylor rule and find that macroeconomic variables account for over half of the variation in time-varying excess bond returns.1 There is a growing body of literature
that combines macroeconomic and finance models of bond yields2 , indicating the
importance of macroeconomic variables in bond return predictability. A forecasting
analysis specifically focusing on bond risk premia with macroeconomic variables
is provided by Kim and Moon (2005) and Ludvigson and Ng (2009). Both conclude that bond returns can be predicted using macroeconomic data. Some of the
macroeconomic information that forecasts bond returns is embedded in the yield
curve and thus also captured by the yield curve factors. However, a significant part
of the macroeconomic information that forecasts bond risk premia is unspanned by
the yield curve. This means that macroeconomic variables have further predictive
power in addition to the yield curve factors. The following analysis separates the
unspanned macroeconomic information from the observed variables and evaluates
its predictive power.
In addition to the fact that unspanned macro factors forecast bond returns, there is
a second reason to add macroeconomic variables to forecasting regressions and term
structure models. Estimation of term structure models is generally difficult due to the
a serious small sample problem that arises because of the highly persistent nature of
interest rates (Kim and Orphanides 2005). In addition, the yield state variables in
term structure models have the same kind of measurement errors as the bond returns
1 Söderlind, Söderström and Vredin (2005) provide evidence against the predictability of yield changes
using Taylor rules on the short end. However, in forecasting bond excess returns, macroeconomic variables might still prove successful.
2 Diebold, Piazzesi and Rudebusch (2005) provide a Q&A about combining finance and macroeconomics in term structure models.
4.1. MACROECONOMIC DATA
67
because all the data is constructed from the same source. Additional information
from a different source could help to overcome both problems. Macroeconomic
variables could do that job; they are related to yields3 , but do not show the same
measurement error patterns because they are gathered from different sources.
This chapter analyzes bond return predictability in a simple linear regression
framework, comparing the results to the regressions in Chapter 2, and is organized
as follows. Section 4.1 introduces the macroeconomic data used. Section 4.2 analyzes the macro forecasting regressions and compares them with the forward rate
regressions, while Section 4.3 combines both macro and yield curve factors and explores the unspanned part of macro variables. Section 4.4 investigates the spanned
part of macro factors and provides an economic interpretation of the yield curve
factors level, slope and curvature. Section 4.5 concludes.
4.1
Macroeconomic data
Macroeconomic time series are chosen in consideration of existing macro-finance
studies.4 Empirical work on macro-finance models of term structure preferably include very similar choices of macroeconomic variables. Ang and Piazzesi (2003) include inflation and real activity measured as the principal components of two groups
of variables. Price change measures include CPI and PPI of finished goods as well
as spot market commodity prices. Economic growth measures include Help Wanted
Advertising in Newspapers (HELP), unemployment, the growth rate of employment
and the growth rate of industrial production. Gürkaynak, Sack and Swanson (2005)
3 The
relation of macroeconomic variables to yields is explored later in this chapter. Dewachter and
Lyrio (2006) give a macroeconomic interpretation of level, slope and curvature as long-run inflation expectation, business cycle conditions and monetary policy factor, respectively.
4 The selection of macroeconomic time series obviously comes with a selection bias that possibly
derogates the results of the analysis. The main selection criteria applied are of practical reasons. The
macroeconomic time series are chosen based on their availability. Data must be available on monthly
frequency and cover the same time period of 57 years as the bond yields from 1952:06 until 2009:06. All
other choices are purely arbitrary in consideration of existing empirical literature as cited in the text.
68
add further variables such as consumer confidence, ISM manufacturing PMI composite index (NAPM) and new home sales. Diebold, Piazzesi and Rudebusch (2005)
and Dewachter and Lyrio (2006) as well as Evans and Marshall (2001) separate a
third group of factors, namely monetary policy instruments. This factor is approximated in Kim and Moon (2005) using money macro variables as M1, M2 and M3
including ratios and growth rates thereof. They find that these money variables help
considerably in predicting bond returns.
Following the above literature the macro variables in this thesis are selected according to the three groups of inflation, real activity and money supply factors. The
real activity factors indicate or forecast the state of the economy and are also called
business cycle indicators. These are:
• Industrial Production (IP5 ) from FRED6 - reports the industry’s output in
changes in the volume of goods produced. It does not take the price of goods
into account and therefore is not distorted by inflation. It is thus a purer measure of output and corresponds more closely to real GDP.
• Industrial Production Index of Materials (IPMAT) from FRED - the same as
IP but for materials only. This means that all final products are excluded.
• Civilian Labor Force Unemployment (UNEMPLOY) from FRED - measures
business conditions very timely and is one of the most closely watched economic activity indicators. It is thus an indicator of the financial health of consumers and of future spending.
• Consumer Credit Outstanding (CONCRED7 ) from Datastream - tracks monthly
changes in consumer installment debt.
• ISM Manufacturing PMI Composite Index (NAPM) from FRED - a monthly
survey distributed by the Institute for Supply Management (ISM) to production
5 The
FRED mnemonic is INDPRO.
is the Federal Research Economic Data of the Federal Reserve Bank of St.
6 FRED
(http://research.stlouisfed.org/fred2/).
7 The Datastream mnemonic is USCRDCONB.
Louis
4.1. MACROECONOMIC DATA
69
managers with a focus on manufacturing. The survey contains questions about
new orders, manufacturing production, employment, supplier deliveries and
inventories.
• New privately-owned housing units started (HSTARTS) from PhilFed8 - reports the number of new homes being built. Many experts view it as one of the
most reliable leading indicators of economic activity.
Inflation factors measure the price change in the economy with the following indices:
• Consumer Price Index of all Items (CPI9 ) from FRED - one of several inflation
measures, including all items for all urban consumers.
• Core Consumer Price Index (CPIcore10 ) from FRED - subtracts food and energy from the CPI to get rid of the highly volatile short term price movements
in these commodities.
• Producer Price Index of Finished Goods (PPI11 ) from FRED - measures the
inflation as well, but does not include services. Whereas the CPI consists of
more than half of services like housing and medical care.
Money supply factors record the money supply in different measures with:
• M1 Money Stock (M112 ) from FRED - measures the money supply as currency in circulation and demand deposits.
• Currency Component of M1 (CCM113 ) from FRED - measures the money
supply as the currency in circulation, one part of M1 money stock.
8 PhilFed
is the Federal Reserve Bank of Philadelphia.
http://www.philadelphiafed.org/research-and-data/.
9 The FRED mnemonic is CPIAUCSL.
10 The FRED mnemonic is CPILFESL.
11 The FRED mnemonic is PPIFGS.
12 The FRED mnemonic is M1SL.
13 The FRED mnemonic is CURRSL.
Research and data site available at
70
• US treasury securities outstanding (PUBDEBT14 ) from Datastream - a measure of public debt.
All these variables are annual log growth rates, except for NAPM and HSTARTS,
which are log levels, and cover the same period as the bond data introduced in Section 2.2 from 1952:06 until 2009:06 with monthly frequency. Descriptions of the
data series are taken from Baumohl (2005).
4.2
Forecasting Bond Returns with Macro Data
This section explores the annual forecasting regressions with macro factors and compares them to the yield curve factor forecast regressions of Chapter 2. Kim and Moon
(2005) present a similar regression analysis as an extension to Cochrane and Piazzesi
(2005). Ludvigson and Ng (2009) analyze the predictability of bond excess returns
with a huge number of macro variables. Both studies conclude that macroeconomic
data is a significant and robust predictor of bond returns. The following analysis
approves the significant forecasting power of macro variables, but the forecasting
power of the longer sample examined here is approximately a third lower than in the
studies of Kim and Moon (2005) and Ludvigson and Ng (2009). The reduced predictability appears to stem from the additional observations in the most recent years,
where macroeconomic variables lost some of their predictive power.15 Nevertheless,
macroeconomic information forecasts excess returns on bond portfolios and predicts
level, slope and curvature risk premia.
The analysis is conducted for two set of instruments, once with all the introduced
macro variables of Section 4.1 and once with the first principal components (PCs)
of the three groups of real activity, inflation and money supply only. The principal
components reduce the number of forecasting factors to one of each group by taking
14 The
Datastream mnemonic is USSECMNSA.
analysis is provided in Subsection 4.3.1; Table 4.7 is especially relevant in this case.
15 Subsample
4.2. FORECASTING BOND RETURNS WITH MACRO DATA
71
weighted averages of the individual series.16
4.2.1
Unrestricted Forecasts
Table 4.1 plots the R2 of annual forecasting regressions. Short, medium and long
maturity bonds as well as slope and curvature portfolios are regressed onto different
sets of instruments. The first line of Table 4.1 includes all twelve macroeconomic
variables from Section 4.1 as instruments. These are compared to the yields-only
instrument sets of unrestricted and restricted forwards,17 shown in the bottom part
of Figure 4.1. A comparison of macro and yield instruments demonstrates that the
former do better for the level, especially on the short end. Slope and curvature portfolios are forecasted using macro variables but with lower R2 than when using yield
factors.
4.2.2
Principal Components of Groups
In order to break down forecasting power into the contributions of each macro group
and to reduce the number of instruments, the first principal components of each
group of macro variables is taken as an instrument. The midsection of Table 4.1
presents these results. With the first principal components of each group of macro
factors (i.e. three instruments in total), R2 decrease to between 0.10 and 0.18. In
the case of level portfolios the contribution to these R2 is dominated by the real
activity factor, but inflation also adds its share, particularly for the long maturity
16 All
the principal components load positively onto the individual series, except onto UNEMPLOY,
which is in line with expectations since UNEMPLOY is negatively correlated to business cycles. The
interpretation of the principal component series is thus the same as a weighted average of the individual
series. The share of variance explained by the first principal component is 66%, 91% and 75% for the real
activity, inflation and money supply groups, respectively.
17 The instrument set of unrestricted and restricted forwards includes all annual forward rates from one
to ten years and only the 1-, 3-, 5-, 8- and 10-year forward rates, respectively, according to the forecasting
regressions in Chapter 2.
72
Table 4.1: Macro forecasts in comparison to forward rates
All macro factors
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
0.34
0.29
0.26
0.17
0.16
PCs of each group
0.18
0.15
0.12
0.12
0.10
Real activity PC
0.10
0.04
0.01
0.09
0.00
Inflation PC
0.01
0.03
0.05
0.07
0.05
Money supply PC
0.01
0.01
0.00
0.00
0.01
Unrestricted forwards
0.24
0.27
0.31
0.27
0.25
Restricted forwards
0.21
0.24
0.27
0.22
0.18
The table displays the R2 of annual forecast regressions over the period 1952:06 - 2009:06. Restricted
forwards means that the set of instruments is restricted to the 1-, 3-, 5-, 8- and 10-year forwards instead
of all forwards in the unrestricted forwards regression.
bonds. Slope portfolios are forecasted by both real activity and inflation, whereas
curvature portfolios are forecasted solely by inflation. The money supply factor does
not explain any of the level, slope and curvature portfolio returns.
Table 4.2: Macro forecast t-statistics
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
Multivariate regression with the PCs of each group as instruments
Real activity PC
-3.89
-3.32
-2.60
Inflation PC
-2.04
-2.36
-2.49
2.11
2.49
1.96
1.89
1.43
-0.06
-2.04
0.12
Money supply PC
-2.95
1.60
Univariate regressions
Real activity PC
-2.38
-1.66
-1.06
-3.92
Inflation PC
-0.67
-1.21
-1.69
4.15
1.76
0.60
0.63
0.39
0.21
-0.96
Money supply PC
The table reports the t-statistics of the regressions in Table 4.1. Standard deviations are Newey-Westadjusted with 18 lags. Intercept statistics omitted.
Table 4.2 reports the t-statistics from these regressions separately (in the upper
panel for the multivariate regressions, and in the lower panel for each univariate regression). The real activity and inflation principal components are significant. The
money supply factor has only slight and no significance in the multivariate and uni-
4.3. UNSPANNED MACRO FACTORS
73
variate regressions, respectively. These figures support the assumption that real activity forecasts level and slope risks while inflation forecasts all three risk premia.
The sign of the loadings is negative for the level risk premium, which implies that
high real activity and high inflation correspond with a low level risk premium. For
the slope risk premium the sign of real activity is negative, but that of inflation positive. In the case of curvature risk premium it is positive for inflation while real
activity is insignificant. The low significance of the money supply factor in combination with R2 close to zero (shown in Table 4.1) is strong indication that the money
supply factor does not forecast bond returns and therefore is excluded from the set
of macro instruments in any further analysis.
4.3
Unspanned Macro Factors
Macro variables, particularly the real activity and inflation factors, have shown significant forecasting power in forecasting level, slope and curvature risks. However,
some of the macroeconomic information is already contained in the yield curve factors. This section examines the persistence of predictability of the macroeconomic
information unspanned by the yield curve. Yields are market prices of bonds and,
with some limitations such as monetary policy on the short end, are freely determined by the forces of supply and demand in an open market. All available information relevant for building investors’ expectations about future yields and bond returns
should be priced into the actual yield curve if markets are perfect. The predictability
of bonds with information unspanned by the yield curve is either a sign of imperfect
markets or that the yield curve is not able to capture all the relevant information.
The economic interpretation of the existence of such unspanned factors is provided
in Section 5.1.1 in the more structured environment of the macro-finance term structure model of Chapter 5. The basic argument is that information that has opposite
effect on the expectations of bond returns and future yields is netted out in the actual
yield curve and therefore no longer visible even though this information is correctly
priced. Unspanned factors are thus able to predict bond excess returns beyond the
74
extent to which yield curve factors are.
Table 4.3: R2 of combined macro-finance forecasting regressions
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
All macro factors
0.47
0.46
0.48
0.38
0.27
PCs of each group
0.36
0.35
0.39
0.35
0.21
PCs of real activity and inflation
0.36
0.34
0.36
0.30
0.21
The table displays R2 of regressions with combined macro and yield instruments. The macro instruments
are indicated; yield instruments are the restricted forward rates (1, 3, 5, 8 and 10 years).
Unspanned macroeconomic variables forecast bond excess returns if macroeconomic variables significantly improve the yields-only forecast. Table 4.3 shows the
R2 of regression forecasts with a combined set of yields and macro factors. The
yield part is always covered by the restricted forward rates, whereas the macro part
is given by different sets of macro variables. The first line represents the R2 of regressions, which include all twelve macro variables and yields as instruments. Comparing these R2 with the yields-only case on the bottom line in Table 4.1 leads to
the assumption that macro variables contain information that is unspanned by yields.
The R2 have approximately doubled for the level, while the slope and curvature portfolios have also increased greatly, but not quite to the same extent. Similar but less
pronounced results are produced by the inclusion of the first principal components
of each macro group, whereas including the money supply factor does not increase
forecasting power. The increase in R2 is still between 0.10 and 0.15, except in the
case of the curvature portfolio where the increase is only approximately 0.03. The
Newey-West-adjusted t-statistics report significance for the two PCs real activity and
inflation in the level and slope forecasting, but not for curvature (not reported). An
F-test allows the joint significance of multiple parameter estimates to be tested. The
joint testing of the real activity and inflation PCs affirms their significance on a probability level of 5% for the level (and as much as 1% for the short level) and slope
portfolios, whereas curvature is not significant. These results of the F-test coincide
with those of the t-test and confirm that some of the macroeconomic information is
not spanned by the yield curve. Additional support for these findings is provided
75
by a contemporaneous regression of these macroeconomic factors onto the standard
yield curve factors. The corresponding t-statistics and R2 are reported in Table 4.4.
The real activity principal component is completely unspanned by the first four yield
curve principal components since none of the instruments is significant and the R2
is close to zero. The inflation principal component is spanned by about half of the
yield curve, indicated by an R2 of 0.53. The part of inflation that is spanned by yields
is explained by the first three yield curve factors; the fourth principal component of
yields (PC4) is not significant in spanning any of the macroeconomic information.
Table 4.4: R2 and t-statistics for macro factors regressed onto yield factors
Level
Real activity PC
Inflation PC
Slope
Curvature
PC4
R2
-0.51
0.65
0.04
0.25
0.01
7.16
3.79
2.85
0.53
0.53
The table reports t-statistics and R2 of regressing the macro factors (PCs) onto the first four yield curve
principal components. T-statistics are Newey-West-adjusted with a lag of 18. Intercept statistics omitted.
The part of the macroeconomic factors that is unspanned by the yield curve is just
the residuals of the contemporaneous regressions in Table 4.4. The R2 of forecasting
bond returns with these residuals is provided in Table 4.5 with the corresponding
t-statistics in Table 4.6. Both unspanned macro factors (the residuals of the real
activity and the inflation factors) forecast bond returns of level and slope portfolios
with R2 around 0.10, while the loadings are predominantly significant. The sign of
the loadings of the unspanned macro factors correspond with the observable macro
factors and thus the economic interpretation of the influence of these factors is along
the same lines. High unspanned real activity corresponds with low level and slope
risk premia and high unspanned inflation corresponds with high slope and curvature
risk premia.
In summary, these regressions strongly confirm that macroeconomic variables
as measures of real activity and inflation are to some extent unspanned by the yield
curve and forecast bond excess returns in level and slope portfolios along with yield
curve information. Because the macroeconomic information is not spanned by the
76
Table 4.5: R2 of forecasting regressions with unspanned macro factors
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
0.02
Unspanned macro factors
0.16
0.10
0.09
0.09
Real activity residual
0.08
0.03
0.01
0.09
0.00
Inflation residual
0.03
0.04
0.06
0.03
0.02
The table reports the R2 of forecasting regressions with the residual of a regression of macro factors (PCs)
onto the first four yield curve principal components.
yield curve but is important for enhancing bond return predictability, it needs to
be included as additional forecasting variables in forecasting regressions or term
structure models.
Table 4.6: T-statistics for forecasting regressions with unspanned macro factors
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
Multivariate regression
-3.29
-2.55
-1.95
-3.24
0.45
Inflation residual
-2.11
-2.22
-2.56
0.93
1.50
Univariate regressions
-2.03
-1.35
-0.76
-3.65
-0.14
Inflation residual
-1.15
-1.47
-2.00
1.79
1.33
The table reports t-statistics for forecasting regressions with the residual of a regression of macro factors
(PCs) onto the first four yield curve principal components. T-statistics are Newey-West-adjusted with a
lag of 18. Intercept statistics omitted.
4.3.1
Rolling Window and Subsample Analysis
For the subsample analysis, the full sample from 1952:06 - 2009:06 is split in two
ways into subperiods. Firstly, the sample is split in approximately August 1981 in
order to separate it into an increasing and a decreasing yield period. Secondly, the
high inflation period from 1971 to 1985 is removed and compared against the low
inflation period before and after. The results are presented in Table 4.7. The lines
with macro factors contain the R2 of the regressions with the PCs of each group
77
as instruments as presented in Section 4.2. The lines with forwards as instruments
repeat the restricted forward regressions from Section 2.3.3. The lines with combined instruments merge the macro regressions with the forward regressions of the
respective previous lines.
Macro regressions perform much better in the increasing yield period than in
the following decreasing yield period in terms of R2 , whereas for the yields-only
regressions there is no obvious difference between these two periods. However, the
combined regressions perform reasonably well in both periods. The difference between the high and low inflation periods are striking for macro variables and yields:
both have much higher R2 for the high inflation phase. The macro forecasts are in
line with the observation of eroding predictability in recent years for yields-only factors, as stated in Section 2.3.3. Caution applies because the high inflation sample is
rather short with only fifteen years.
Table 4.7: R2 of various subsamples
rx(2)
rx(5)
rx(10)
srx2,10
crx2,5,10
Subsample 1: increasing yields 1952:06 - 1981:08
Macro
0.14
0.17
0.23
0.26
0.14
Forwards
0.30
0.29
0.26
0.37
0.22
0.35
0.33
0.32
0.56
0.29
Combined
Subsample 2: decreasing yields 1981:09 - 2009:06
Macro
0.18
0.10
0.04
0.08
0.05
Forwards
0.28
0.28
0.36
0.28
0.16
Combined
0.46
0.41
0.47
0.39
0.19
Subsample 3: high inflation 1971:01 - 1985:12
Macro
0.37
0.39
0.37
0.34
0.26
Forwards
0.40
0.43
0.43
0.31
0.28
0.56
0.60
0.63
0.47
0.37
Combined
Subsample 4: low inflation 1952:06 - 1970:12 and 1986:01-2009:06
Macro
0.11
0.07
0.06
0.10
0.05
Forwards
0.17
0.21
0.26
0.20
0.20
Combined
0.27
0.27
0.35
0.40
0.21
The table reports the R2 of the various subsamples.
78
A rolling window analysis sheds some more light on when macro regressions
performed well in the past. Figure 4.1 shows the R2 of twenty year rolling window
regressions estimated monthly, whereas the R2 is plotted at the end of each twenty
year period. The period up to 1962 and after 2002 shows low R2 in level forecasting,
with both yield and macro factors. In forecasting curvature portfolios with macro information, this is even more severe, with around zero R2 at both ends of the sample.
As already noticed in Section 2.3.3, the forecasting power of yields erode continuously in the last years. The same is true of the macro factors, with the exception
of slope portfolio forecasts, where a sharp peak is observed. However, combined
macro-finance forecasting regressions show a surprisingly steady high R2 along the
rolling windows compared to their individual series.
4.4
Spanned Macro Factors
The part of macroeconomic factors that is spanned by yields obviously does not enhance the predictability of bond returns in addition to yield curve factors. However,
it does deliver an understanding of how the spanned macroeconomic information is
related to the yield curve factors and thus provides an economic understanding of
the latent yield curve factors level, slope and curvature, as introduced by Litterman
and Scheinkman (1991). The macroeconomic variables of real activity, inflation and
money supply are able to explain a large share of bond yield movements, as shown
by Dewachter and Lyrio (2006) and Diebold, Rudebusch and Boragan (2006), which
investigate between spanned macroeconomic information and common yield curve
factors.
The yield curve factors are built as the first four principal components of the
yield curve. These are regressed onto the macroeconomic variables with the results
given in Table 4.8. The level and slope of the yield curve are explained fairly well
through macro variables with an R2 of 0.62 and 0.54, respectively. Curvature is not
well matched by macroeconomic factors, as indicated by a low R2 of only 0.08. The
4.4. SPANNED MACRO FACTORS
79
Forecasting Level
0.8
0.6
0.4
0.2
0
1970
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Forecasting Slope
0.8
0.6
0.4
0.2
0
1970
1975
1980
1985
1990
1995
Forecasting Curvature
0.8
0.6
0.4
0.2
0
1970
1975
1980
1985
Forwards
1990
Macro
1995
Combined
Figure 4.1: R2 of rolling window analysis
R2 of twenty year rolling window analysis. Level is the mean R2 of the bond excess returns series rx(2) ,
rx(5) and rx(10) . Slope and curvature are the corresponding R2 of the portfolios srx2,10 and crx2,5,10 ,
respectively. Forwards indicate the restricted forward regressions, macro indicates the regressions with
the first principal components of each group (PCs) and combined indicates both.
80
Table 4.8: R2 and t-statistics for yields regressed onto macro variables
T-statistics
Level
Slope
Curvature
PC4
Real activity PC
-0.48
4.70
0.80
0.52
Inflation PC
4.81
7.48
1.84
0.57
Money supply PC
5.11
-8.44
-0.19
-0.54
R2
0.62
0.54
0.08
0.01
T-statistics are Newey-West-adjusted with 18 lags. Intercept statistics omitted.
fourth principal component of yields (PC4) and any further principal components
show neither any R2 higher than 0.05 nor any significant loadings by regressing onto
macroeconomic factors. The loadings of the regressors allow the yield curve components to be attributed to the macro factors. The level is explained by the inflation and
money supply principal components and slope of all three together. Explaining curvature by macro factors only produces one barely significant loading for the inflation
factor. These findings are similar to the results in Evans and Marshall (2001). They
conclude that level and slope are attributed to output growth, inflation and monetary shocks, but not so curvature and further principal components. Dewachter and
Lyrio (2006) find a high correlation of the level factor with their long-run inflation
expectation and of the slope factor with business cycle conditions, whereas the curvature factor expresses a monetary policy factor. The above analysis thus supports
the dependence of the level and slope on inflation and real activity, respectively.
4.5
Conclusion
This chapter has examined bond return predictability of macroeconomic variables,
organized into three groups according to their approximation of real activity, inflation
and money supply measures. Macro factors, especially real activity and inflation,
predict excess bond returns, slope and curvature portfolios, but in this long sample
R2 are lower compared to existing analyses. Subsection and rolling window analysis shows that the predictive power of unspanned macro factors is boosted by the
4.5. CONCLUSION
81
high inflation period, in a similar way to the yield curve factors. The predictability of
bond returns is persistent when using macro factors in addition to yield curve factors.
These macro factors thus capture information unspanned by the yield curve. Unspanned macro factors significantly forecast the level and slope risk premia, whereas
the curvature risk premium is spanned by the latent yield curve factors alone. However, this implies that bond excess returns are not fully spanned by yields and that
macroeconomic information is crucial in forecasting. The spanned part of macroeconomic information allows a decomposition of the common yield curve factors and
allocates the level and slope to inflation and real activity, whereas no explanation
could be found for curvature in terms of these macroeconomic variables.
82
Chapter 5
Macro-Finance Term Structure
Models
The previous chapter has identified solid improvement in the forecasting power of
bond excess returns by adding macroeconomic factors that are unspanned by the
actual yield curve. This chapter introduces an affine term structure model with the
desired feature of including unspanned macro factors. These unspanned macro factors are additional state variables, which only affect excess returns but not the actual yield curve. Building a term structure model with macroeconomic factors in
the unspanned specification is the obvious consequence of the forecasting properties
explored in Chapter 4. Macro-finance models link the term structure of yields to
macroeconomic variables in addition to common latent yield curve factors in a term
structure model setting. These models are prominent in recent literature.1 However,
none of these models capture the unspanned part of macroeconomic models. The
only other sources that systematically cover this unspanned specification in term
structure models are Duffee (2011), Joslin, Priebsch and Singleton (2011) and Boos
1 See
Section 5.1 for an overview of the recent literature on macro-finance models.
83
84
CHAPTER 5. MACRO-FINANCE TERM STRUCTURE MODELS
(2011). This study contributes by extending the empirical analysis in the time axis
and cross-section of macroeconomic variables and, as in the previous chapters, the
analysis is not restricted to level risk premium (as in the studies of Joslin, Priebsch and Singleton (2011) and Duffee (2011)), but extends to slope and curvature risk
premia. The introduced model is based on the class of Ang and Piazzesi (2003) models with the unspanned restriction and parametrization applied in the spirit of Boos
(2011), but with extensions to multiple macro variables. The unspanned macrofinance models are thoroughly compared and benchmarked to the standard yieldsonly models of Chapter 3.
The chapter is organized as follows. Section 5.1 provides an overview of the relatively new macro-finance term structure model literature as well as extensive treatment of the unspanned feature and its economic reasoning. Section 5.2 introduces a
model that is of the same affine class as the self-consistent term structure model of
Chapter 3, but with a different parametrization and restrictions that ensure the unspanned state variable feature. Section 5.3 then delivers empirical results including
rank restriction tests on the number of risk premia and compares them to the simple
regression forecasts of Chapter 4 and the self-consistent yields-only term structure
models of Chapter 3. The last section concludes.
5.1
Macro-finance term structure models are a relatively new branch of research combining the term structure models of finance literature with macroeconomic variables or
any models thereof, based on the finding that macroeconomic variables significantly
help to predict bond excess returns.
Macro-finance models can be grouped into structural and reduced form models.
Structural form models are motivated by macroeconomic theory and impose macroeconomic structure such as a New Keynesian model. These models are then linked
to standard finance term structure models with either no, unidirectional or even bidi-
5.1. MACRO-FINANCE MODELS
85
rectional feedback between the macroeconomic and latent yield curve variables. An
early combination of macroeconomic variables with interest rates has been popular in monetary policy with the Taylor rule (Taylor 1993) and similar models of the
short rate. Gürkaynak, Sack and Swanson (2005) link a New Keynesian model to a
simple interest rate rule through the monetary policy rate. Ang, Dong and Piazzesi
(2007) present a Taylor rule linked to a full affine term structure model instead of
only the short rate. Rudebusch and Wu (2008) link the macro model to an affine
term structure model and allow for bidirectional feedback between yield curve and
macro variables.
Reduced form models only impose some statistical dynamics, such as a VAR
process, and then model the latent yield curve factors together with the macroeconomic factors in one system, whereas the macroeconomic factors could be latent or
observed, depending on the specification. Ang and Piazzesi (2003) simply extend the
state space of an affine term structure model with macroeconomic variables. Mönch
(2005) follows a similar approach by extending the macroeconomic state variables
with a factor-augmented VAR model to incorporate many different time series. The
specification of the VAR model decides how the macro and yield factors are allowed
to interact. Fischer (2010) proposes a model with two separate VARs for latent yield
curve and macroeconomic factors. Models that allow for bidirectional feedback are
proposed by Rudebusch, Swanson and Wu (2006) and Diebold, Rudebusch and Boragan (2006). Dewachter, Lyrio and Maes (2006) estimate a macro-finance model
with the Kalman filter in a one-step procedure, and Dewachter and Iana (2009) extend the macro-finance model with additional finance factors as liquidity measures.
5.1.1
Unspanned Factors
All of the above-mentioned models assume that the yield curve is spanned by all the
state variables. However, empirical evidence suggests that while the yield curve is
sufficiently described by three latent yield curve factors (often referred to as level,
slope and curvature), expectations about future yields are not fully spanned by these
86
factors. Macroeconomic factors are factors that are partially unspanned by the yield
curve but help to predict bond excess returns. In a term structure model, these additional state variables are restricted to affect the cross-section of yields, but allow
an influence upon bond risk premia. Technically, this is a restriction on the model
where macroeconomic variables only affect yield state variables in the physical measure and not in the risk-neutral measure.2 Economically, this restriction allows the
addition of further factors to a term structure model that do not span the yield curve,
but improve estimation of bond risk premia. By definition, standard term structure
models without this restriction do not allow the addition of state variables that are
unspanned by the yield curve. Since all the state variables are generally backed-out
of the observed yield curve data, the model ensures that all factors are spanned by
yields.
Are there factors that affect bond risk premia but not the actual yield curve? A
particular example is provided by Joslin, Priebsch and Singleton (2011): they show
that in an unspanned macro-finance model, roughly the same yield curves could imply different expectations about future yield curves, solely based on the different
characteristics of the macroeconomic variables. These variables thus provide information that is not visible in the yield curve, since it is hidden by two opposing effects
that net themselves out. The information that cancels out in the yield curve is the unspanned part of the macroeconomic factors. The remaining part that is not canceled
out affects the actual yield curve and is thus spanned. As Duffee (2011) highlights,
unspanned information therefore always has an opposite effect on bond risk premia
and expected future yields in order to have no effect on the actual yield curve. Since
the unspanned information is not contained in the yield curve, this information cannot be reproduced by the cross-section of bond data. Other data unspanned by the
yield curve, e.g. macroeconomic variables, is needed in order to embed this information into a term structure model.
Unspanned factors in macro-finance term structure models are a recent finding
in finance literature. Duffee (2011) introduced unspanned hidden factors and Joslin,
2 This
is explained in more detail in Section 5.2.
5.2. MODEL
87
Priebsch and Singleton (2011) then explicitly applied these unspanned factors to
observed macroeconomic variables, i.e. the inflation rate and industrial production
growth. Boos (2011) extends a term structure model of the Ang and Piazzesi (2003)
class with unspanned macro factors and provides an example with survey data on
expected inflation to filter an unspanned factor. Based on this initial research some
related work emerged. Wright (2009) looked at unspanned macro factors in several countries besides the US. Li, Niu and Zeng (2011) investigated the difference
between unspanned and traditional macro-finance models and Barillas (2011) researched the optimal bond portfolio choice given unspanned macro models.
5.2
Model
The macro-finance model with unspanned macro risks is a special case of the model
outlined in Section 3.2. All the valuation formulas for yields and bond returns
apply here as well. The difference to the model outlined in Chapter 3 lies in the
parametrization and the unspanned restriction implied therein. Whereas the yieldsonly model has been parameterized as a self-consistent model, this is obviously not
possible with unspanned factors. The following paragraph derives the parametrization applicable in the case of unspanned macro variables.
5.2.1
Parametrization
Parametrization is derived by appending the self-consistent parametrization of Section 3.4 with additional state variables that drive the unspanned macroeconomic
risks. The state variables X are thus composed of ky yield-related state variables
y
Xt and km macro-related state variables Xtm as
Xt =
"
y
Xt
Xtm
#
.
(5.1)
88
The dynamics of the state variables are described by the VAR as given in Equation
(3.1) of dimension ky + km under the physical measure (P-measure) with parameters
µ, Φ and Σ. The specification of these parameters is maximally flexible, allowing for
bidirectional interaction between the yield and macro state variables. In contrast, the
dynamics in the risk-neutral measure (Q-measure), which follow the VAR defined
in (3.3) constrained such that the macro state variables do not affect the latent yield
curve factors; this is the restriction that enables unspanned factors. The risk-neutral
parameters µ∗ and Φ∗ are parameterized as follows:
#
"
µ∗y
∗
µ =
µ∗m
and
∗
Φ =
"
Φ∗11
0
Φ∗21
Φ∗22
#
(5.2)
.
(5.3)
µ∗y is a vector of length ky and defined the same way as in the yields-only model
parametrization in Section 3.4, i.e. all zeros except the last entry, which is a free
parameter. The vector µ∗m is of length km and consists of free parameters only. The
matrix Φ∗11 is a companion matrix of dimension ky × ky , equal to the parametrization
in the yields-only model of Section 3.4. Φ∗22 with dimension km × km controls the
dynamics among the macro state variables and consists basically of free parameters.
The upper-right and lower-left part of Φ∗ govern the interrelations from macro to
yield factors and vice versa. The upper-right part is thus restricted to zero so as to
not allow any influence of the macro state variables on the latent yield factors. The
reverse feedback is basically allowed and thus Φ∗21 consists only of free parameters.
Due to identification issues the parameters of Φ∗21 and Φ∗22 are further restricted. This
is explained in detail in the estimation in Section 5.2.4.
This specification allows the inclusion of factors in the VAR that are not spanned
by the cross-section but help to forecast both expected excess bond returns and expected yields. This is easily verified3 by looking at the valuation formulas for yields
3 An
empirical verification that these kinds of models only affects the predictability of the model and
not the fit of the cross-section is provided in Appendix E.1.
5.2. MODEL
89
and bond excess returns provided in Appendix C. The valuation formula for yields
only depends on the risk-neutral parameters µ∗ and Φ∗ and the parameters δ0 and
δ1 . The latter two parameters define the short-rate and are thus fixed to δ0 = 0 and
δ1 = [1 0 . . . 0]⊤ , as explained in the parametrization of the yields-only model in Secy
tion 3.4. Inspecting the yield equation (C.7), especially the part of Bn in Equation
(C.9), which is driving the link to the state variables Xt , it becomes obvious that only
the first ky state variables enter the yield equation if the upper-right ky × km part of
Φ∗ is equal to zero. However, the equation for expected yields (C.17) or expected
bond returns (C.22) also depend on the parameters of the physical dynamics µ and Φ
and therefore on all the state variables since the physical parameters are unrestricted.
5.2.2
Price of Unspanned Risks
Macro risks are not spanned by bonds in the risk-neutral measure and the price of
risk cannot be inferred by them. Without any additional assets that span the macro
factors in the risk-neutral measure beside bonds, the risk-neutral dynamics of the
macro state variables cannot be determined. An asset that spans inflation risk would
be inflation-protected securities like TIPS. However, the time series history is too
short to use them in this model setting. Assets that span the risk of real activity are
more difficult to find; equity indices might help here. Inspired by Boos (2011), the
risk-neutral dynamics of the macro factors are set equal to the physical dynamics.
This is achieved by restricting the last km elements of Σλ0 and the last km rows of
Σλ1 to zero because the parameters of the risk-neutral and the physical dynamics
are linked according to equations (3.4a) and (3.4b). This restriction implies that
the risk compensation, i.e. the variance of the shocks times its price of risk Σλt , is
constrained to zero for the unspanned macro variables. The matrix λ1 , which governs
the time-varying risk premia, is then singular with rank ky and its last km rows are
a linear combination of the others. The macro state variables still affect bond risk
premia through the upper-right ky × km part of λ1 , which is nonzero.
90
5.2.3
Restricting the Number of Priced Risk Factors
As with yields-only models, rank restrictions on λ1 are imposed to test for the number of priced risk premia. The methodology and interpretation is analogous to Section 3.4.3. Since the rank of λ1 of the macro-finance model is already reduced to
ky by the parametrization, rank restrictions are imposed beginning with ky − 1 until
zero. The macro-finance model with rank L of λ1 is labeled as B(ky , km , L) with ky
as the number of latent yield state variables and km as the number of macro state
variables.
An additional restriction on the model is tested by restraining the upper-right ky ×
km part of Σλ1 to zero. The macro state variables then do not have any influence on
bond risk premia and similar results to the yields-only case are expected. Since this
model is a B(ky , km , L) model restricted to an A(ky , L) model, it is called BA (ky , km , L).
Similar restrictions are possible by setting only one column of the upper-right ky ×km
part of the Σλ1 matrix to zero, or all columns except one. This allows testing for the
influence of single macro state variables on the model performance.
5.2.4 Estimation
The models are estimated by the Kalman filter as outlined in Appendix B. The transition equation is given by the dynamics of all state variables, which now include
the latent yield curve factors and the macroeconomic variables. The measurement
equation for yields is given in (B.1); that for the macro variables is analogous and
given as
zt = Am + B⊤
m Xt + υt
(5.4)
where zt are the observable macro variables and the measurement error υt is assumed
to be iid normally distributed with a diagonal covariance matrix. The standard deviation of the measurement errors συ is estimated jointly with the other model parameters. The macro state variables in Xt are filtered from the observed series with the
Kalman filter. They are thus similar to the residual of a regression of the observed
91
macro variables onto all the latent yield curve state variables. With all parameters
Am , Bm , µ and Φ free, the model obviously faces an identification problem because
only one time series is available in the cross-section to estimate the parameters. This
is solved by setting the diagonal elements of the lower km × km part of matrix Bm
to one and restricting the dynamics of the macro state variables to a simple onedimensional AR(1) process. This means that the lower-left km × ky part of Φ (as well
as of Φ∗ because of the restriction on Σλ1 ) is all zeros and the lower-right km × km
part of Φ is zero in its off-diagonal elements. This means that the dynamics of the
macro state variables are not influenced by any other state variables. The influence
of these other state variables is taken care of in Am and Bm . The macro state variables
are therefore rotated to a series that captures only the movement that it does not share
in common with interest rates and other macro variables. This specification ensures
identification by continuing to provide the maximally flexible model.4
5.3 Empirical Results
This section presents the empirical properties of the estimated models. The estimation is based on the same CBD data set as used for the yields-only models, described in Section 2.2 with the full data range from 1952:06 - 2009:06. The number
of macroeconomic state variables is set to two, based on the regression analysis in
Chapter 4. The corresponding observable macroeconomic time series for filtering the
macroeconomic risks are the first principal components of each of the two groups of
real activity and inflation measures. Model estimation is conducted at quarterly frequency, on the one hand because it is a reasonable trade-off between a good signal4 Estimation
of term structure models is well-known to be demanding with poor convergence. The
specific parameter identification and the Kalman filter algorithm with the analytical gradient dramatically
improves convergence speed. Multiple different starting values, which converge to the same parameter
sets and likelihood values affirm that convergence is reached at the optimum. Parameter identification
and further information about the difficulty of estimating term structure models is introduced in Section
3.4. The applied estimation procedure is described in detail in Appendix B. Further information about the
applied algorithm is provided in Boos (2011).
92
to-noise ratio and a high amount of independent observations, while on the other it
allows for direct comparison to the quarterly yields-only models. The models are
estimated with five and six factors, two of which are accounted to macro factors and
three or four to latent yield curve factors. These models are directly comparable to
the three- and four-factor models A(3, L) and A(4, L) from Chapter 3.5
The models are estimated with several restrictions as described in Section 5.2.3.
These are rank restrictions on the time-varying risk premia to assess the number
of priced risks and restrictions on the influence of macroeconomic state variables
to compare them against the yields-only model specification. All these models are
compared in terms of likelihood ratio specification tests and R2 .6 The forecasting
power in terms of R2 is always evaluated for one-year returns, since historical data
for bonds with longer maturities is only available at a yearly frequency. Predictability
of the level risk premium is analyzed at the short and long horizon by rx(2) and rx(10) ,
respectively. For the analysis of slope and curvature risks the same portfolios as in
the regression analysis in Section 2.4 are used, i.e. srx2,10 and crx2,5,10 .
5.3.1 Model Estimates
In the following the quarterly five- and six-factor macro-finance models are discussed
and compared against each other in order to select the dominant model; this includes
an inspection of the filtered macro state variables.
5 The
analogs to the five-factor A(5, L) models are not estimated due to the large number of free pa-
rameters in comparison with the number of individual observations, which makes estimation less robust.
The A(5, L) model analog in the macro-finance specification is the seven-factor B(5, 2, 5) model with 95
free parameters. This is a large number of free parameters to estimate using only a cross-section of 17
yields and two macro factors and a time series of 229 data points.
6 Likelihood ratio specification tests are defined in Appendix B.4 and the R2 calculation is given with
Equation (3.22).
93
Five-Factor Models
The estimation results of the five-factor B(3, 2, L) models are shown in Table 5.1.
The B(3, 2, 0) model is not able to produce time-varying risk premia and thus has no
forecasting ability. The B(3, 2, 1) model forecasts one-year excess returns on bonds
with a maturity of two and ten years with R2 of 0.28 and 0.12, respectively. The
forecasting power of the slope and curvature is close to zero. The predictability
of the slope and curvature portfolios is only present in the B(3, 2, 2) and B(3, 2, 3)
models, with R2 of 0.20 and 0.15, respectively. Forecasting of long bonds is also
increased up to an R2 of 0.21 for both models. The B(3, 2, 3) model is not able
to increase predictability beyond the B(3, 2, 2) model. In terms of likelihoods, the
B(3, 2, 2) model dominates the others with a significance level of 1%. The likelihood
ratio tests for all combinations among these five-factor models are given in the upper
part of Table 5.3.
Table 5.1: Forecasting power of the quarterly five-factor models
log likelihood
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
B(3, 2, 0)
20931.3
39
0.00
0.00
0.00
0.00
B(3, 2, 1)
20953.8
46
0.28
0.12
0.04
0.02
B(3, 2, 2)
20969.0
51
0.29
0.21
0.20
0.15
B(3, 2, 3)
20972.2
54
0.30
0.21
0.20
0.15
The table shows the R2 of a one-year forecast based on the quarterly five-factor models, the number of
Six-Factor Models
The estimation results of the quarterly six-factor B(4, 2, L) models are shown in Table 5.2. According to the restrictions, the B(4, 2, 0) model shows no forecasting
power. Model B(4, 2, 1) predicts short bonds and slope portfolios with R2 of 0.14
and 0.09, respectively. The B(4, 2, 2) model further improves the forecasting ability,
which peaks in model B(4, 2, 3) for level forecasts with R2 of 0.34 and 0.30. The
94
B(4, 2, 3) model is the dominant model in terms of R2 and likelihood ratio tests, as
provided in the lower part of Table 5.3. Model B(4, 2, 4) is not able to increase the
forecasting power overall, in fact there is a decrease in level but an increase in slope
predictability. The number of priced risk premia in the dominant model increases
to three compared to the dominant five-factor model, which shows only two priced
risk premia. The increase in the number of risk premia corresponds by adding a
sixth state variable to the model, which enables the model to separate another risk
premium.
Table 5.2: Forecasting power of the quarterly six-factor models
log likelihood
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
B(4, 2, 0)
21402.7
49
0.00
0.00
0.00
0.00
B(4, 2, 1)
21428.0
58
0.14
0.03
0.09
0.00
B(4, 2, 2)
21483.9
65
0.20
0.10
0.07
0.06
B(4, 2, 3)
21534.7
70
0.34
0.30
0.24
0.18
B(4, 2, 4)
21539.9
73
0.25
0.24
0.27
0.18
The table shows the R2 of a one-year forecast based on the quarterly six-factor models, the number of
Table 5.3: Likelihood ratio tests of the quarterly models
B(3, 2, 0)
B(3, 2, 1)
B(3, 2, 1)
45.0
(18.5)
B(3, 2, 2)
75.5
(26.2)
30.5
(15.1)
B(3, 2, 3)
81.9
(30.6)
36.9
(20.1)
B(4, 2, 0)
B(4, 2, 1)
50.6
(21.7)
B(4, 2, 1)
B(3, 2, 2)
6.4
(11.3)
B(4, 2, 2)
B(4, 2, 2)
162.5
(32.0)
111.9
(18.5)
B(4, 2, 3)
263.9
(38.9)
213.4
(26.2)
101.5
(15.1)
B(4, 2, 4)
274.5
(43.0)
223.9
(30.6)
112.0
(20.1)
B(4, 2, 3)
10.6
(11.3)
with a significance level of 1% is given in parentheses. If the likelihood ratio exceeds the test statistic,
the unrestricted model is able to fit the data significantly better than the constrained model in terms of
likelihood.
95
Comparing the five- and six-factor models, the six-factor model with three priced
risk premia B(4, 2, 3) clearly dominates the five-factor model B(3, 2, 2), both in terms
of likelihoods and R2 . The dominant macro-finance model thus is the B(4, 2, 3)7
model, a similar result to that reached with the yields-only models in Chapter 3.
Model forecasts for level, slope and curvature portfolios compared to the realized
returns are shown in Figure 5.1.
Filtered Macro Series
Since the macro state variables are filtered from the observed macro data, the filtered
series are interesting to look at. The presented figures correspond with the dominant
B(4, 2, 3) model. Both macro state variables correlate slightly negatively with -0.37.
The correlation of these variables to yields is around zero, with slightly higher values
around 0.3 for the second macro factor. The first factor is filtered from the real
activity principal component and shows a high correlation with the observed series
(0.97). The second factor, which is filtered from inflation measures, shows a lower
correlation of 0.3 with its observed factor. Since the observed inflation time series
correlates with yields at around 0.6, the filtered series contains some part of the signal
that is less correlated with yields.
5.3.2 Comparison to Regression Forecasts
Table 5.4 compares the model forecasts with the regression forecasts of the previous
chapter. The first line of the combined macro and yield regressions in Table 5.4
reproduces the R2 of forecasting with all twelve macroeconomic variables together.
Compared to the relevant regressions that include only the PCs of the real activity and
the inflation group, the dominant B(4, 2, 3) model matches predictability very closely
in terms of R2 . The model is in general not able to improve upon the predictability
of regressions, since the OLS regression maximizes the R2 of the linear relation
7 Model
parameter estimates are provided in Appendix E.2.
96
Level
0.4
0.2
0
−0.2
−0.4
1950
forecast
realized
1960
1970
1980
1990
2000
2010
1990
2000
2010
1990
2000
2010
Slope
0.04
0.02
0
−0.02
−0.04
1950
1960
1970
1980
Curvature
0.04
0.02
0
−0.02
1950
1960
1970
1980
Figure 5.1: Forecasted vs. realized returns of the B(4, 2, 3) model
The plots show the one-year forecasts of the quarterly B(4, 2, 3) model compared to one-year realized returns (plotted with quarterly overlapping). “Level” is the long bond rx(10) , while “slope” and “curvature”
are the portfolios srx2,10 and crx2,5,10 , respectively.
97
between the given data. However, as the additional structure of the macro-finance
model does not reduce the predictability (measured in R2 ), it is evidence that the
specification of the model produces the desired results and captures the forecasting
features of the simple regression setting of Chapter 4.
Table 5.4: Model vs. regression forecasts
B(4, 2, 3) model
rx(2)
rx(10)
srx2,10
crx2,5,10
0.34
0.30
0.24
0.18
Combined macro and yield regressions
All macro factors
0.47
0.48
0.38
0.27
PCs of each group
0.36
0.39
0.35
0.21
PCs of real activity and inflation
0.36
0.36
0.30
0.21
The table compares the R2 of the dominant model forecasts to the OLS regression forecasts of Table 4.3
of the previous chapter.
5.3.3
Comparison to Yields-Only Models
This section compares the macro-finance models to the yields-only models of Chapter 3 and the restricted models within the macro-finance parametrization, which prevent the macro state variables from having any influence on forecasting. The restricted models have the advantage that they are nested in the general macro-finance
models and thus allow specification tests. The standard B models are compared with
the restricted BA models, which do not allow for any influence of the macro state
variables and should therefore be analogous to the yields-only A models. Table 5.5
compares the forecasting power of all these models in terms of R2 . The BA and A
models are in fact almost identical in forecasting power, but are significantly lower
than the macro-finance B models. Table 5.6 provides likelihood ratio tests and concludes that the B models significantly outperform the BA models. The unspanned
macroeconomic information therefore increases predictability beyond the standard
yields-only models. Further restricted models are used to test for the individual influence of each of the two macro state variables. The BRA and BINF models only
98
Table 5.5: Forecasting power of the restricted macro models
log likelihood
B(3, 2, 3)
20972.2
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
54
0.30
0.21
0.20
0.15
BRA (3, 2, 3)
20967.0
51
0.29
0.23
0.20
0.16
BINF (3, 2, 3)
20970.4
51
0.31
0.24
0.21
0.16
BA (3, 2, 3)
20956.7
48
0.15
0.19
0.13
0.16
A(3, 3)
20052.5
23
0.14
0.19
0.13
0.15
B(4, 2, 4)
21539.9
73
0.25
0.24
0.27
0.18
BRA (4, 2, 4)
21473.4
69
0.27
0.21
0.23
0.17
BINF (4, 2, 4)
21475.9
69
0.28
0.22
0.23
0.17
BA (4, 2, 4)
21463.7
65
0.16
0.19
0.16
0.17
A(4, 4)
20624.7
36
0.15
0.18
0.17
0.16
The table shows the R2 of a one-year forecast based on the quarterly four-factor models, the number of
allow for the influence of one macro state variable: real activity and inflation, respectively. Both these models outperform the yields-only BA models in terms of
likelihood and R2 as shown in tables 5.5 and 5.6. The time series of expected returns
on the level, slope and curvature portfolios of the dominant macro finance model
B(4, 2, 3) compared to the dominant yields-only model A(4, 3) are plotted in Figure
5.2. The difference between the two series is attributable solely to the macro state
variables. The correlation of these time series are provided in Table 5.7. These risk
Table 5.6: Likelihood ratio tests of the quarterly BA models
B(3, 2, 3)
BA (3, 2, 3)
30.9
(16.8)
B(4, 2, 4)
BA (4, 2, 4)
152.4
(20.1)
BRA (3, 2, 3)
20.5
(11.3)
BRA (4, 2, 4)
19.3
(13.3)
BINF (3, 2, 3)
27.3
(11.3)
BINF (4, 2, 4)
24.4
(13.3)
model is indicated on top, the constrained (and nested) model on the left. The corresponding test statistic
likelihood.
99
premia of the macro-finance and yields-only models correlate still relatively high
with values of 0.82, 0.95 and 0.86 for level, slope and curvature risk premia, respectively.
Table 5.7: Correlation of macro and yields-only model forecasts
B(4, 2, 3)
rx(10)
B(4, 2, 3)
A(4, 3)
srx2,10
A(4, 3)
crx2,5,10
rx(10)
srx2,10
rx(10)
1.00
srx2,10
-0.52
crx2,5,10
-0.94
0.52
1.00
rx(10)
0.82
-0.55
-0.71
1.00
srx2,10
-0.57
0.95
0.56
-0.61
1.00
crx2,5,10
-0.80
0.47
0.86
-0.85
0.49
crx2,5,10
1.00
1.00
The table displays the correlation matrix of the macro-finance and yields-only model forecasts.
A more profound comparison of the model-implied risk premia is provided by
regressing the risk premia of the yields-only and macro-finance models onto the
observed macroeconomic time series. The corresponding t-statistics and R2 are provided in Table 5.8. The same regressions are applied also to the difference of the
macro-finance and yields-only risk premia, which is in fact the additional risk premia explained by unspanned macro factors alone. The macro factors explain the risk
premia of the macro-finance model significantly; in contrast to the risk premia of the
yields-only model, where only the inflation factor significantly influences the slope
risk premium. The inflation information is not fully unspanned by the yield curve,
thus the part of inflation that is spanned by the yield curve is obviously already incorporated in the yields-only model risk premia. For the difference of the two model
risk premia, the results are even stronger. The sign of the coefficients, which is evident from the sign of the t-statistics, is different for the three risk premia and thus
separates them in a macroeconomic perspective. These three risk premia can be differentiated by their reactions to real activity and inflation. In explaining the level
risk premium, the signs of the regression coefficients for the real activity and inflation factor are both negative, implying that high growth and inflation accompanies a
low expected level risk premium. The corresponding signs for the slope and curva-
100
Level
0.2
0.1
0
−0.1
−0.2
1950
macro−finance
yields−only
1960
1970
1980
1990
2000
2010
1990
2000
2010
1990
2000
2010
Slope
0.02
0.01
0
−0.01
1950
1960
1970
1980
Curvature
0.01
0.005
0
−0.005
−0.01
1950
1960
1970
1980
Figure 5.2: Macro vs. yields-only model forecasts
The plots show the one-year forecasts of the quarterly B(4, 2, 3) model compared to one-year forecasts
of the quarterly A(4, 3) model. “Level” is the long bond rx(10) , while “slope” and “curvature” are the
portfolios srx2,10 and crx2,5,10 , respectively.
101
Table 5.8: Macro part of model-implied risk premia
R2
T-statistics
A(4, 3)
B(4, 2, 3)
Difference
0.26
-4.22
-6.05
-0.83
-2.21
-2.99
A(4, 3)
B(4, 2, 3)
Difference
0.00
0.12
0.34
0.01
0.06
0.08
Level risk premium
Real activity PC
Inflation PC
Slope risk premium
Real activity PC
0.36
-1.45
-6.30
0.00
0.05
0.49
Inflation PC
3.03
2.59
1.29
0.09
0.08
0.02
Curvature risk premium
Real activity PC
Inflation PC
-0.67
3.66
5.46
0.00
0.10
0.36
2.30
3.35
3.32
0.13
0.17
0.09
The table reports the t-statistics (Newey-West-adjusted with 6 lags) and R2 of regressing the risk premia
of the yields-only and macro-finance models as well as their difference onto the macroeconomic time
series, all univariate in quarterly frequency.
ture risk premia are different. The expected slope risk premium is low when inflation
is high and growth is low, whereas a low expected curvature risk premium accompanies low inflation and low growth. Intuitively, low growth accompanies high risk
premia, which is the case for the level and slope risk premia. Low inflation would
be expected to accompany decreased risk premia. This is the case in the slope and
curvature risk premia. The influence of inflation on the level risk premium and of
real activity on the curvature risk premium might be an effect of separating these two
risk premia. The opposite signs for level and curvature risk premia are in accordance
with their relatively high negative correlation, as shown in Table 5.7. The difference
between the slope and curvature risk premia is that the former accounts mainly for
spanned inflation and the latter for unspanned inflation. This is apparent from the
fact that the part of inflation of the slope risk premium is also captured by the yieldsonly model and not improved by the macro-finance model. However, the part of
inflation captured in the curvature risk premia is improved by adding the unspanned
macroeconomic factors in the macro-finance specification. This is illustrated by the
significance of the coefficient for the difference of the risk premia across the two
models and the increased R2 . According to the R2 , real activity factor increases pre-
102
dictability for all three risk premia compared to the yields-only models. The inflation
factor only increases predictability for the level and curvature risk premia since the
part of the inflation factor that forecasts the slope risk premium is spanned by the
yield curve and thus already accounted for in the yields-only model.
5.4
Conclusion
This chapter has extended the yields-only term structure models to macroeconomic
state variables, while including the feature that these macroeconomic factors are unspanned by the yield curve. Although this leaves the current yield curve unaffected
by shocks to the unspanned factors, these factors influence expected bond risk premia. This macro-finance model improves predictability of expected returns beyond
that of a yields-only model with latent yield curve factors only. The dominant model
is the B(4, 2, 3) model with four latent yield curve factors and two macro state variables with a total of three priced risk premia. This is the same number of priced risk
premia as the dominant yields-only model from Chapter 3. The predictability of the
macro-finance term structure model is consistent with the results of the regression
forecasts, including the same macro and yield variables in Chapter 4. The major
result is that unspanned macroeconomic information is significant when forecasting
bond risk premia; this is in line with other studies of term structure models with unspanned factors.8 Furthermore, in addition to the level risk premium, the slope and
curvature risk premia are also forecasted by unspanned macroeconomic information.
These risk premia beyond the level risk premium react differently to macroeconomic
shocks. The expected curvature risk premium is positively correlated with inflation
and real activity, whereas the expected slope risk premium is negatively correlated
with real activity but positively with inflation.
8 These
are the studies of Duffee (2011), Boos (2011) and Joslin, Priebsch and Singleton (2011).
III
103
Chapter 6
International Regressions
This chapter extends the forecasting regressions of Chapter 2 to an international setting. Firstly, the forecasting pattern of bond markets in countries other than the US
is explored individually and compared among each other. This is simply an application of the forecasting regressions in the yields-only case of Chapter 2 to different empirical data. This has been explored before1 , but not with the assessment of
slope and curvature risk premia. Predictability of bond excess returns with domestic
yield curve factors as instruments is persistent to countries other than the US, but
with somewhat lower R2 , especially the UK. Secondly, bond return forecasting is
analyzed in an international setting. A decomposition of international yield curves
shows that the number of factors that sufficiently describe the local yield curves is
not reduced compared to an individual domestic setting. This implies that the yield
curves show significant independent variation across countries and are not captured
by common international factors. The same is true for bond risk premia. The local
risk premia cannot be captured by a common international factor. Whereas the level
risk premium still shows a correlation of around 0.7 across countries, the slope and
1 see
Kessler and Scherer (2009) and Sekkel (2011)
105
106
CHAPTER 6. INTERNATIONAL REGRESSIONS
curvature risk premia do not. Furthermore, the unspanned information in foreign
yield curves predicts domestic bond excess returns. International yield curves exhibit information not spanned by the domestic yield curves but spanned by domestic
risk premia. Exploring the characteristics of unspanned international yield curves
as a forecasting instrument for bond returns in the domestic countries is the main
contribution of this chapter to the understanding of international bond risk premia.
6.1
International Yield Curve Data
The empirical analysis of this and the subsequent chapter is expanded to yield curves
in currencies other than the USD, specifically the euro (EUR) and the British pound
(GBP). These yield curves stem from open economies with flexible exchange rates
and a developed and fairly liquid government bond market on a comparable credit
risk level. These countries are also linked through their capital markets and foreign
trade activity.
The EUR data set is constructed by the author based on bond return data for
German government bonds available in Datastream. There is no public data set for
German bond yields available with quarterly maturities or even higher frequency.
The yield curve generation algorithm applied is the Fama-Bliss method described
in Appendix A. The final data set ranges from 1976:12 - 2009:06. For the United
Kingdom (UK) there is a data set available with quarterly spaced maturities that are
fitted based on the method proposed in Anderson and Sleath (2001) provided by the
Bank of England. Unfortunately, short yields up to one-year maturity are only spuriously available before 1997. Since bond return data before 1997 is also sparse in
Datastream, the author’s routine to estimate Fama-Bliss yield curves is only applicable in the range from 1997:12 - 2009:06. The yield curve in the previous period from
1976:12 - 1997:11 is taken from the Bank of England with an approximation of the
short yields.2 For the US yield curve, the CBD data set, as described in Section 2.2
2 Missing
three month yields are derived from the corresponding currency forwards, the exchange rate
6.2. INDIVIDUAL FORECASTING REGRESSIONS
107
is used, as in the previous chapters. However, the time window is adjusted to match
those of the EUR and GBP data sets ranging from 1976:12 - 2009:06.
Henceforth, the aforementioned data sets are at times abbreviated according to
their domestic currency code, USD for the CBD data set, GBP for the British data
set and EUR for the German data set. The regression results are based on these
data sets with yields and bond returns up to a maturity of ten years and 391 monthly
observations.
6.2
Individual Forecasting Regressions
This section applies the forecasting regressions of Chapter 2 to the British and German bond data and compares the results across these markets. Other studies extending the bond risk premia analysis of Cochrane and Piazzesi (2005) to data sets other
than the US are Kessler and Scherer (2009) and Sekkel (2011). Both these studies
apply the forward regressions to several countries individually and build trading rules
based on these. They find that predictability is persistent in all bond markets of the
countries under consideration. The section contributes by extending the sample in the
time dimension and evaluating the slope and curvature risk premia in addition to the
level risk premium. The following paragraph concisely presents the sample statistics
and sums up the findings of unconditional risk premia across these countries. The
paragraph thereafter applies a conditional risk premia analysis by forecasting level,
slope and curvature portfolios.
and the US interest rate. Since three month currency forward rates of the exchange rate between USD and
EUR are available, the quarterly interest rate can be derived implicitly according to the pricing equation
of a forward currency contract.
108
6.2.1
Unconditional risk premia
The sample statistics are provided in Table 6.1 and give an indication of the unconditional risk premia. Since the sample might only capture a single or even a fraction
of an interest rate cycle3 and unconditional risk premia are very sensitive to the selected time frame, the analysis might be distorted. Excess returns on level portfolios
obviously profit from the downward trend in this particular sample. However, the
unconditional risk premia for the level of Germany and the UK are comparable in
size to those in the US. The Sharpe ratios for the UK are marginally lower in the
range of 0.11 to 0.21 and those for Germany marginally higher than the US with
values between 0.14 and 0.31. The unconditional Sharpe ratios of the US level portfolios are in the range of 0.14 to 0.24, which is marginally higher than in the full
sample starting in 1952 (shown in Table 2.1). However, the unconditional slope and
curvature risk premia are close to zero for all these countries.
6.2.2
Conditional risk premia
Forecasting the level of US bond excess returns with yields delivers a tent-shaped
pattern of loadings. This pattern is not preserved when all forecasting instruments,
i.e. all forward rates (known as the unrestricted regression), are included in the
regression because of measurement errors and highly correlated instruments, as discussed in Chapter 2. Reducing the number of instruments (known as the restricted
regression) somewhat recovers this pattern, but leaves some indication that the forecasting factor cannot just be a linear combination of the forward rates. The loadings
of the restricted forecasting regressions of the three countries are shown in Figure
6.1. The patterns of the loadings differ across the countries and reveal, along the line
of the argument in Chapter 2, the existence of additional risk premia in addition to the
level risk premium. This is most obvious for long maturity forward rates in the GBP
3 Interest
mean.
rate cycles appear to be very long lasting since interest rates show very slow reversion to the
r(4)
r(5)
r(6)
0.16
6.3%
0.11
5.8%
2.9%
0.14
7.3%
4.3%
0.16
Sharpe ratio
µ
σ
Sharpe ratio
µ
σ
Sharpe ratio
0.18
7.9%
9.9%
0.20
9.0%
10.3%
0.26
4.7%
6.6%
0.25
5.5%
6.8%
0.30
6.5%
7.4%
Germany 1976:12 - 2009:06
0.17
6.6%
9.6%
0.24
6.7%
8.3%
0.22
7.8%
8.5%
0.24
9.2%
9.0%
0.21
10.1%
8.9%
0.31
7.1%
7.7%
0.21
9.9%
10.6%
r(7)
0.22
11.7%
9.3%
0.29
7.6%
7.7%
0.21
10.8%
10.9%
r(8)
0.19
12.7%
9.2%
0.24
8.1%
7.4%
0.21
11.7%
11.1%
r(9)
0.14
13.8%
8.6%
0.18
8.9%
7.0%
0.21
12.6%
11.3%
r(10)
0.07
3.8%
6.9%
0.03
2.6%
5.5%
-0.02
3.5%
8.3%
sr2,10
-0.03
3.6%
6.5%
-0.06
2.4%
5.2%
-0.02
3.3%
8.3%
cr2,5,10
All figures based on annualized discrete returns. µ is the return, σ the standard deviation and SR the Sharpe ratio (ratio of excess returns to volatility).
0.21
5.5%
7.9%
United States 1976:12 - 2009:06
0.24
3.9%
5.2%
4.0%
σ
9.3%
8.9%
United Kingdom 1976:12 - 2009:06
r(3)
µ
r(2)
Table 6.1: Sample statistics for bond data sets
6.2. INDIVIDUAL FORECASTING REGRESSIONS
109
110
GBP
EUR
USD
8
4
8
6
3
6
2
4
4
1
2
2
0
0
−2
−4
0
−1
−2
−2
1
3
5
8
10
−3
1
3
5
8
10
−4
1
3
5
8
10
Figure 6.1: Loadings of restricted regressions across countries
The figure shows the loadings of the restricted forecasting regressions (with only the 1-, 3-, 5-, 8- and
10-year forward rates as instruments) of the GBP, EUR and USD data set. Intercept omitted.
data set, but also to a lesser extent in the other two. Table 6.2 provides the R2 of the
unrestricted and restricted forecasting regressions for all single yields and the slope
and curvature portfolios. The level forecasting pattern is similar for German and US
bonds with unrestricted R2 of around 0.3. British bonds fall back on a much lower R2
of between 0.07 and 0.16. The restricted regressions reduce R2 only slightly across
all data sets, implying that the reduction to only five forward rates is not restrictive
to level forecasting. Yields are also able to forecast slope and curvature portfolios
with high R2 across the data sets. Since the restricted regressions for the slope and
curvature portfolios only contain the same maturity forwards as the bonds that are
used to construct the corresponding portfolios, this is more restrictive than the level
forecasting regressions. The slope and curvature restricted regressions only allow
for two and three different forecasting factors, respectively. Therefore, R2 erode by
these restrictions but still display predictability of these portfolios. Table 6.3 provides the t-statistics corresponding with the restricted regressions. In general, they
affirm highly significant loadings in forecasting slope and curvature portfolios. The
same is true for the level forecasting, except for the British data set, where the parameters are not statistically significant and thus result in low forecasting power in terms
of R2 . The sometimes insignificant coefficients reported by the t-statistics could be
6.3. INTERNATIONAL FORECASTING REGRESSIONS
111
due to the multicollinearity problem of highly correlated instruments.4 Finally, the
predictability of slope and curvature risk premia beyond the level risk premium is
persistent in countries other than the US. The regressions demonstrate that slope and
curvature risks are priced in all these markets.
Table 6.2 additionally provides a breakdown of the forecasting power into the
first three principal components and the rest, which is the sum of any further principal components. This breakdown shows barely any forecasting power stemming
from the level factor. This factor only spans the yield curve. In contrast, the slope
and curvature factors account for the main share of predictability in most cases. Further principal components explain the remaining predictability. In the US the fourth
principal component added much to the predictability, but this is not persistent to the
other countries, where the forecasting power of the fourth factor is very low in favor
of higher order principal components (not reported).
As a result, forecasting bond excess returns in the UK and Germany in general
shows a comparable pattern to in the US. The major forecasting properties are very
similar and therefore the findings of the results of the extensive treatment of individual yields-only single-country regressions in Chapter 2 carry over to these other
countries. The works of Kessler and Scherer (2009) and Sekkel (2011), both of
which extend the forecasting regressions to an even broader set of countries, reach
the same conclusion (that this model performs well across all these data sets), although they only evaluate the level risk premium.
6.3
International Forecasting Regressions
This section takes the domestic forecasting regressions to an international setting by
adding foreign yield curve factors as forecasting variables. The yield curves of open
4 An
F-test would provide a joint test of multiple parameters but is unfortunately not applicable in a
sample with overlapping data. Restricting the sample to non-overlapping data also reduces the sample
size to 32, which is too small to result in significant parameter estimates.
rx(3)
rx(4)
rx(5)
rx(6)
0.09
0.08
0.01
0.00
0.07
0.02
0.25
0.21
0.01
0.02
0.14
0.09
0.26
0.24
0.03
0.07
0.06
0.10
Unrestricted
Restricted
Level
Slope
Curvature
Further PCs
Unrestricted
Restricted
Level
Slope
Curvature
Further PCs
Unrestricted
Restricted
Level
Slope
Curvature
Further PCs
0.03
0.11
0.00
0.02
0.15
0.16
0.04
0.10
0.00
0.03
0.14
0.15
0.03
0.08
0.00
0.03
0.12
0.13
0.10
0.14
0.05
0.01
0.25
0.30
0.13
0.11
0.06
0.01
0.24
0.29
0.14
0.10
0.07
0.01
0.25
0.32
0.13
0.07
0.08
0.01
0.28
0.30
0.14
0.08
0.10
0.01
0.29
0.32
0.12
0.07
0.10
0.01
0.29
0.31
0.14
0.08
0.12
0.01
0.31
0.35
United States 1976:12 - 2009:06
0.13
0.14
0.03
0.01
0.25
0.30
Germany 1976:12 - 2009:06
0.03
0.10
0.00
0.01
0.12
0.14
rx(2)
0.17
0.06
0.12
0.00
0.32
0.35
0.12
0.09
0.09
0.01
0.27
0.30
0.03
0.06
0.00
0.03
0.10
0.11
rx(7)
0.14
0.07
0.14
0.01
0.32
0.35
0.11
0.07
0.10
0.00
0.24
0.28
0.02
0.04
0.01
0.03
0.08
0.09
rx(8)
Table 6.2: R2 of single-country forecasting regressions
0.16
0.06
0.14
0.01
0.33
0.37
0.13
0.05
0.11
0.00
0.25
0.29
0.02
0.02
0.02
0.02
0.07
0.08
rx(9)
0.19
0.05
0.13
0.00
0.35
0.38
0.18
0.05
0.08
0.00
0.27
0.31
0.01
0.01
0.03
0.02
0.06
0.07
rx(10)
0.22
0.00
0.08
0.03
0.12
0.33
0.11
0.03
0.07
0.01
0.14
0.22
0.02
0.04
0.03
0.02
0.06
0.11
srx2,10
0.10
0.10
0.07
0.01
0.05
0.28
0.23
0.07
0.05
0.00
0.15
0.34
0.15
0.23
0.03
0.05
0.29
0.41
crx2,5,10
112
-0.67
-0.37
-0.18
-2.71
f (5)
f (8)
f (10)
f (1)
-0.73
-3.00
1.40
-4.20
f (5)
f (8)
f (10)
f (1)
-2.59
-3.29
-0.02
f (5)
f (8)
f (10)
T-statistics are Newey-West-adjusted with 18 lags.
5.89
f (3)
f (2)
4.50
f (3)
f (2)
2.26
-1.49
f (3)
f (2)
f (1)
rx(2)
rx(4)
rx(5)
rx(6)
-0.57
-0.70
-0.24
2.84
-1.59
-0.62
-0.79
0.27
2.35
-1.53
-0.72
-0.57
0.28
2.12
-1.44
1.35
-2.52
-0.41
4.20
-3.12
1.64
-2.91
0.95
3.33
-3.05
1.86
-2.79
0.57
3.39
-3.20
0.02
-4.00
-3.05
6.61
-4.90
0.43
-3.81
-3.14
7.03
-5.30
0.39
-4.26
-2.16
6.38
-5.40
0.47
-4.42
-2.42
6.85
-5.82
United States 1976:12 - 2009:06
1.32
-2.76
-0.72
4.52
-2.88
Germany 1976:12 - 2009:06
-0.45
-0.56
-0.48
2.77
-1.58
rx(3)
1.00
-5.11
-2.19
6.84
-5.99
2.05
-2.72
0.30
3.57
-3.41
-0.71
-0.41
0.37
1.78
-1.36
rx(7)
0.87
-4.15
-2.56
6.99
-6.21
2.34
-1.85
0.10
3.16
-3.34
-0.77
-0.12
0.32
1.53
-1.31
rx(8)
0.67
-4.06
-2.44
7.25
-6.40
2.50
-1.76
0.01
3.34
-3.46
-0.67
0.05
0.27
1.25
-1.28
rx(9)
2.08
-4.48
-2.82
7.25
-6.69
3.64
-2.28
-0.31
3.29
-3.42
-0.23
-0.07
0.26
1.00
-1.21
rx(10)
Table 6.3: T-statistics for single-country restricted forecasting regressions
-3.96
5.13
-3.53
2.82
-2.04
1.35
srx2,10
0.17
-1.94
2.14
0.50
-4.78
3.02
4.75
-1.97
-0.68
crx2,5,10
113
114
economies that interact with each other are obviously linked. Before analyzing the
international forecasting regressions, the yield curves and bond risk premia are investigated in respect of common factors. Contrary to the existing literature, which
features a common international factor, the following analysis does not support a reduction of forecasting factors by common international factors, either in yield curves
or bond risk premia.
6.3.1
International Yield Curve Factors
According to Litterman and Scheinkman (1991), principal component analysis of a
single country’s term structure empirically determines that three principal components almost fully span the yield curve. Less known is the joint principal component
analysis of several countries. Investigation of these principal components is driven
by the question of whether there are any common yield curve factors across countries and how many factors are needed to jointly explain the yield curve of several
countries. Pérignon, Smith and Villa (2007) conclude that one international factor
is enough to explain the common variation in the yield curves, with the remaining
variance explained by local factors. The international factor is found to account for
the common level of all yield curves. Diebold, Li and Yue (2008) estimate an international Nelson-Siegel term structure model to explain the cross-section of yields
with hierarchical dependence of country-specific level, slope and curvature factors
on a global variation thereof. They find that global level and slope factors explain
a significant fraction of the country-specific factors. The following analysis does
not support common international factors but provides evidence that domestic yield
curves are best spanned by their local factors. Common international factors are
not able to reduce the number of total factors needed to explain the cross-section of
yields with similar precision.
Table 6.4 reports the explained variance of the first nine principal components of
each individual and joint term structure from these three countries. The high share
of explained variance captured by the first factor of the joint principal component
115
Table 6.4: Explained variance of principal components
GBP
EUR
USD
GBP-EUR
GBP-USD
EUR-USD
All
PC1
98.22%
97.22%
98.39%
93.75%
93.77%
91.74%
91.45%
PC2
1.64%
2.36%
1.48%
4.29%
4.56%
6.51%
4.20%
PC3
0.12%
0.24%
0.06%
1.32%
1.08%
1.12%
2.60%
PC4
0.02%
0.06%
0.02%
0.44%
0.47%
0.42%
0.98%
PC5
0.00%
0.04%
0.02%
0.08%
0.06%
0.07%
0.39%
PC6
0.00%
0.04%
0.01%
0.05%
0.03%
0.04%
0.22%
PC7
0.00%
0.02%
0.01%
0.02%
0.01%
0.02%
0.05%
PC8
0.00%
0.01%
0.01%
0.01%
0.01%
0.01%
0.03%
PC9
0.00%
0.01%
0.00%
0.01%
0.01%
0.01%
0.02%
This table reports the explained variance of the first nine principal components of yields for the indicated
data sets.
analysis confirms that there is a great deal of common variation in international yield
curves. However, the number of factors to reach 99.9% explained variance is not
obviously reduced. The number of factors is three in single countries (except for
EUR), between five and six in the two-country case, and eight for all three countries.
This indicates at most a reduction of one factor in an international setting in order to
still explain the cross-section of yields fairly well. This becomes clearer by looking
at the correlations of these principal components. Table 6.5 shows the correlations
of the individual countries with the joint principal components of all the countries
together. The first international principal component accounts for the level across all
countries. However the difference in the level of these three countries demands another two factors (PC2 and PC3) to account for this country-sepcific variation. Thus
the common slope across these countries does not appear until the fourth principal
component (PC4) and also requires two further factors to account for the differences
across these countries. The curvature factor has no commonality across countries
and is individually accounted for in the international setting. This pattern appears
in the same form in all three two-country cases (not reported). This implies that a
reduction in the number of factors is not feasible by building international common
factors, since some of the yield curve features will be lost in comparison to the indi-
116
Table 6.5: Correlation of local and international principal components
GBP
PC1
PC2
EUR
PC3
PC1
PC2
USD
PC3
PC1
PC2
PC3
PC1
0.98
-0.01
0.03
0.92
-0.16
0.00
0.97
-0.06
-0.09
PC2
0.14
0.05
-0.11
0.23
0.25
-0.15
-0.25
-0.25
-0.33
PC3
0.14
0.02
-0.09
-0.32
-0.30
-0.12
-0.04
0.20
-0.01
PC4
0.00
0.90
0.01
0.00
0.54
0.25
0.00
0.69
0.03
PC5
0.00
0.44
-0.01
-0.01
-0.29
0.44
0.00
-0.60
0.28
PC6
0.00
0.05
0.18
0.02
-0.67
0.03
-0.01
0.26
0.22
PC7
0.00
-0.01
0.97
0.00
0.03
0.08
0.00
-0.01
0.14
PC8
0.00
0.02
0.03
0.00
-0.01
-0.83
0.00
-0.02
-0.05
PC9
0.00
0.01
0.06
0.00
-0.02
0.03
0.00
0.00
-0.85
This table shows the correlation of the principal components of the individual countries (on the top) with
the joint principal components of all countries together (on the left). Values higher than 0.5 in absolute
terms are highlighted in bold.
vidual single-country models. This leads to the conclusion that the individual yield
curves are reasonably well spanned by their first three principal components (level,
slope and curvature), even in an international setting. International yield curves do
not provide enough common ground to reduce the number of factors by forming
international principal components.
6.3.2
International Risk Premia
The same is true for the risk premia across countries. Level, slope and curvature risk
premia based on the principal components of bond excess returns show the same
pattern as the yield curve factors above (not reported), although they share some
common ground. Table 6.6 provides the correlations between realized yearly bond
excess returns for the ten-year bond, a slope and a curvature portfolio. The correlations of the level risk premia across countries is given by values of 0.73, 0.61 and
0.70. These positive correlations indicate some common variation in the level risk
premia across countries, but also shows that a large fraction is still driven by indi-
117
vidual variation. The level risk premia are positively correlated across countries and
show some commonality, but still they are to a minor part individual. The slope and
curvature risk premia are also positively correlated, but on a fairly moderate level
with correlations below 0.5. This is evidence that risk premia across countries are
linked to some extent but that they also show individual variation. The foreign yield
curve factors, which explain foreign risk premia, should thus be able to improve
domestic forecasts.
Table 6.6: Correlation of bond excess returns
GBP
rx
GBP
EUR
USD
srx
EUR
crx
rx
srx
USD
crx
rx
srx
rx
1.00
srx
-0.51
1.00
crx
-0.67
0.33
1.00
rx
0.73
-0.37
-0.52
1.00
srx
-0.27
0.49
0.20
-0.49
crx
-0.47
0.12
0.36
-0.62
0.28
1.00
rx
0.61
-0.38
-0.60
0.70
-0.25
-0.39
1.00
srx
-0.28
0.37
0.23
-0.35
0.44
0.21
-0.55
1.00
crx
-0.43
0.25
0.44
-0.51
0.16
0.37
-0.77
0.47
crx
1.00
1.00
The table plots the correlation of one-year bond excess returns (all in local currency) of the level rx
(rx(10) ), slope srx (srx2,10 ) and curvature crx portfolio (crx2,5,10 ) for the three countries. Correlations of
the same type of portfolio across countries are highlighted in bold.
Forecasting Regressions with Foreign Yields
Tables 6.7 and 6.8 report the R2 of unrestricted and restricted forecasting regressions,
respectively. The first line (indicated by single-country) shows the resulting R2 by
including domestic yield curve factors only. This line is thus the equivalent of the
simple forecasting regressions from Section 6.2.2 above. The following lines include
the yield curve factors of the indicated countries alongside the domestic ones. The
increase in R2 by including foreign yield factors is sizable and takes place for all
combinations of countries, in both unrestricted and restricted regressions. By in-
118
cluding the yield curve factors of both foreign countries, the R2 again increase in
comparison to the inclusion of only one foreign country’s factors.
In more detail, the R2 of the UK level forecasts increase from a range between
0.07 and 0.16 to a range between 0.22 and 0.34 by adding the German yield curve
factors. Adding US yields results in a similar, but slightly greater, increase. By
adding both of them together, the R2 jump to values between 0.43 and 0.53. The
same pattern is observed for the slope portfolio. The curvature portfolio does not
show that great an increase, most likely due to the already high values in the domestic
regression with an R2 of 0.41. Nevertheless, this number still increases to 0.62.
Restricted regressions5 show the same pattern as unrestricted regressions, but on a
slightly lower level with R2 reduced by around 0.1 for the level forecasts and around
0.3 for the slope and curvature forecasts.
The German level forecasts increase from R2 of around 0.28 to 0.52 by adding
UK yields, and to around 0.45 by adding US yields. Adding all the yield curves
as factors, R2 peaks in a value of 0.64. The increase of predictability in slope and
curvature portfolios is analogous from 0.22 and 0.34 to 0.49 and 0.62, respectively,
by adding the other countries’ yield curves as factors. These results persist in the restricted regressions, except for the slope forecast, which does not improve by adding
foreign yields.
In the US data set the R2 of level forecasts also increase by adding foreign yields.
The increase ranges from R2 around 0.32 with domestic yields to around 0.52 by
adding either UK or German yields, up to around 0.62 by adding both together.
The predictability of slope and curvature portfolios is raised from 0.33 and 0.28 up
to 0.53 and 0.46, respectively, by adding foreign yield curves. R2 in the restricted
5 The
restricted regressions use a reduced set of instruments, which is the same as in all the previous
regression analyses in this thesis. The level of bond excess returns is forecasted with the five forward rates
of 1-, 3-, 5-, 8- and 10-year maturity. The slope and curvature portfolios are forecasted with the forward
rates that have the same maturity as the corresponding bond portfolio is constructed of. Explicitly, these
are the 2- and 10-year forwards for the srx2,10 portfolio, and for the crx2,5,10 the 2-, 5- and 10-year
forwards.
rx(3)
rx(4)
rx(5)
rx(6)
0.50
0.62
EMU
All
0.14
0.16
0.54
0.36
0.34
0.15
0.52
0.36
0.32
0.13
0.50
0.34
0.29
0.30
0.63
0.48
0.55
0.29
0.64
0.49
0.56
0.32
0.64
0.50
0.56
0.63
0.52
0.50
0.30
0.62
0.53
0.50
0.32
0.60
0.50
0.49
0.31
0.62
0.53
0.52
0.35
United States 1976:12 - 2009:06
0.62
0.46
0.54
0.30
Germany 1976:12 - 2009:06
0.54
0.35
0.34
0.61
0.52
0.51
0.35
0.62
0.49
0.54
0.30
0.48
0.32
0.26
0.11
rx(7)
0.61
0.51
0.52
0.35
0.60
0.47
0.51
0.28
0.46
0.31
0.24
0.09
rx(8)
0.63
0.53
0.54
0.37
0.58
0.46
0.51
0.29
0.44
0.29
0.23
0.08
rx(9)
0.63
0.54
0.54
0.38
0.59
0.48
0.51
0.31
0.43
0.29
0.22
0.07
rx(10)
0.53
0.48
0.42
0.33
0.40
0.31
0.36
0.22
0.49
0.30
0.33
0.11
srx2,10
0.46
0.40
0.38
0.28
0.69
0.54
0.61
0.34
0.62
0.53
0.49
0.41
crx2,5,10
are added alongside the domestic factors, each in local currency.
Single-country means that only the domestic forward rates enter the regression. In all the other specifications the respective forward rates of the country
0.26
0.47
0.57
All
UK
0.40
US
Single-country
0.25
0.53
All
0.48
0.31
US
UK
0.33
EMU
Single-country
0.09
Single-country
rx(2)
Table 6.7: R2 of unrestricted international forecasting regressions
119
rx(3)
rx(4)
rx(5)
rx(6)
0.28
0.23
0.43
0.21
0.41
0.38
0.48
0.24
0.43
0.44
0.56
EMU
US
All
Single-country
UK
US
All
Single-country
UK
EMU
All
0.12
0.15
0.44
0.29
0.29
0.14
0.41
0.29
0.27
0.12
0.39
0.27
0.25
0.25
0.54
0.44
0.46
0.24
0.55
0.45
0.47
0.25
0.54
0.46
0.46
0.57
0.46
0.46
0.28
0.55
0.47
0.45
0.29
0.53
0.45
0.44
0.29
0.54
0.46
0.46
0.31
United States 1976:12 - 2009:06
0.54
0.43
0.47
0.25
Germany 1976:12 - 2009:06
0.44
0.27
0.29
0.53
0.46
0.45
0.32
0.56
0.47
0.47
0.27
0.37
0.26
0.22
0.10
rx(7)
0.53
0.45
0.46
0.32
0.53
0.45
0.44
0.24
0.35
0.24
0.20
0.08
rx(8)
0.52
0.45
0.45
0.33
0.51
0.42
0.44
0.25
0.34
0.23
0.18
0.07
rx(9)
0.53
0.46
0.47
0.35
0.51
0.44
0.45
0.27
0.33
0.23
0.18
0.06
rx(10)
0.20
0.20
0.14
0.12
0.15
0.15
0.14
0.14
0.16
0.11
0.10
0.06
srx2,10
0.17
0.06
0.15
0.05
0.32
0.28
0.29
0.15
0.36
0.33
0.32
0.29
crx2,5,10
are added alongside the domestic factors, each in local currency.
Single-country means that only the domestic forward rates enter the regression. In all the other specifications the respective forward rates of the country
0.08
Single-country
rx(2)
Table 6.8: R2 of restricted international forecasting regressions
120
rx(3)
rx(4)
rx(5)
rx(6)
0.12
0.14
0.34
0.22
0.26
0.13
0.31
0.21
0.24
0.11
0.28
0.19
0.22
0.50
0.35
0.47
0.19
0.48
0.34
0.45
0.17
0.45
0.33
0.43
0.18
0.45
0.33
0.33
0.16
0.44
0.33
0.34
0.18
0.43
0.32
0.35
0.18
0.43
0.32
0.37
0.20
United States 1976:12 - 2009:06
0.48
0.32
0.45
0.17
Germany 1976:12 - 2009:06
0.35
0.20
0.27
0.39
0.27
0.34
0.17
0.45
0.33
0.42
0.18
0.25
0.17
0.19
0.09
rx(7)
0.42
0.30
0.37
0.21
0.43
0.31
0.41
0.16
0.23
0.16
0.17
0.07
rx(8)
0.39
0.28
0.36
0.20
0.41
0.28
0.39
0.16
0.21
0.15
0.15
0.06
rx(9)
0.37
0.25
0.34
0.18
0.40
0.26
0.38
0.13
0.20
0.14
0.14
0.06
rx(10)
0.27
0.21
0.17
0.12
0.16
0.12
0.15
0.10
0.29
0.16
0.26
0.09
srx2,10
0.34
0.29
0.26
0.17
0.36
0.30
0.25
0.12
0.39
0.37
0.36
0.32
crx2,5,10
alongside the domestic factors, all in local currency.
means that only the domestic instruments enter the regression. In all the other specifications the respective instruments of the foreign countries are added
The instruments are the first three principal components of the corresponding local yield curve, known as level, slope and curvature. Single-country
0.47
0.44
All
0.34
0.29
US
All
0.42
UK
EMU
0.16
Single-country
0.16
0.35
All
0.33
0.16
US
UK
0.25
EMU
Single-country
0.08
Single-country
rx(2)
Table 6.9: R2 of international forecasting regressions with principal components as instruments
121
122
regressions follow the same pattern with only slightly reduced values.
The number of highly correlated forecasting instruments in these international
regressions is fairly large and thus the estimates are prone to overfit some measurement errors or outliers. To check for the validity of these results the regressions are
rerun with only the first three principal components of the corresponding local yield
curves as instruments. This reduces the number of instruments to six and nine in the
two- and three-country regressions, respectively. The results are reported in Table
6.9 and the R2 are only slightly reduced compared to the restricted regressions of
Table 6.8 discussed above. The problem of overfitting due to the high number of instruments is thus of some concern, but does not alter the drawn conclusions since the
results are persistent in this setting with reduced instruments. The appendix provides
t-statistics in Table F.3, which report the significance of adding any of the international factors individually.6 These tests affirm high significance of the international
factors in forecasting domestic bond returns.
Forecasting Regressions with Unspanned Foreign Yields
Table 6.10: Spanning of the foreign yield curve factors by domestic factors
UK
EMU
US
Level
Slope
Curv.
Level
Slope
Curv.
Level
Slope
Curv.
UK
1.00
1.00
1.00
0.77
0.14
0.26
0.81
0.13
0.08
EMU
0.80
0.39
0.02
1.00
1.00
1.00
0.73
0.11
0.22
US
0.84
0.13
0.03
0.75
0.16
0.12
1.00
1.00
1.00
Spanned by
The table plots the R2 of contemporaneous regressions of the foreign yield curve factors (level, slope and
curvature (curv.)) onto those of the domestic yield curve. Dependent variables are therefore indicated on
top and independent on the left.
6 This
is done because an F-test of the joint significance of all international factors is not valid in
a sample with overlapping data. Using only non-overlapping data would lead to a sharp reduction in
the sample size to only 32 observations, which makes it difficult to reach significance in the parameter
estimates. Therefore t-tests with overlapping data and heteroskedasticity and autocorrelation consistent
(HAC) estimates of the covariance (Newey and West 1987) are applied instead.
0.09
0.28
0.25
0.12
0.27
0.18
0.18
0.31
US residuals
Both residuals
UK residuals
US residuals
Both residuals
UK residuals
EMU residuals
Both residuals
0.29
0.18
0.18
0.30
0.15
0.28
0.23
0.08
0.15
rx(3)
0.26
0.16
0.16
0.30
0.15
0.28
0.20
0.08
0.13
rx(4)
rx(6)
rx(7)
rx(8)
rx(9)
0.25
0.14
0.17
0.30
0.16
0.27
0.18
0.08
0.11
0.10
0.16
0.08
0.10
0.16
0.09
0.09
0.15
0.09
0.27
0.15
0.24
0.27
0.15
0.24
0.25
0.12
0.24
0.23
0.12
0.16
0.22
0.10
0.17
0.21
0.09
0.17
0.19
0.08
0.15
United States 1976:12 - 2009:06
0.28
0.15
0.25
Germany 1976:12 - 2009:06
0.17
0.08
0.11
rx(5)
0.18
0.07
0.16
0.26
0.12
0.24
0.14
0.08
0.08
rx(10)
0.15
0.09
0.05
0.07
0.02
0.05
0.20
0.07
0.17
srx2,10
0.16
0.11
0.08
0.25
0.18
0.14
0.07
0.05
0.04
crx2,5,10
The instruments are the residuals of the level, slope and curvature factors of the foreign country regressed onto those of the domestic country.
0.18
EMU residuals
rx(2)
Table 6.11: R2 of unspanned international forecasting regressions
123
124
As foreign yield curve factors are able to forecast domestic bond returns, these
foreign yield curve factors need to be unspanned by the domestic yield curve, at least
to some extent. The unspanned information, which is not contained by the domestic yield curve, improves predictability beyond an individual single-country model.
Table 6.10 provides the R2 of contemporaneously regressing the foreign yield curve
factors level, slope and curvature onto the domestic ones. These R2 indicate the fraction of variance that is spanned by domestic yield curve factors. The unspanned part,
which is simply the residual of the regressions, accounts for the remaining variation. Apart from the level factor, the unspanned part clearly dominates. Forecasting
regressions with only the unspanned foreign level, slope and curvature factors are
provided in Table 6.11. These provide evidence that bond risk premia are explained
by foreign yield curve factors with information beyond that already embodied in the
domestic yield curves. This is not limited to the level risk premium, but also true for
the slope and curvature risk premia. T-statistics for the individual parameters show
high significance for at least one, but usually several, of the dependent variables for
each forecasted bond return (not reported). In short, unspanned foreign yield curve
factors explain up to 31% of domestic bond returns and are important factors in forecasting bond returns in addition to the standard domestic yield curve factors.
6.4
Conclusion
This chapter has provided four main insights into predictability of bond excess returns in an international setting. Firstly, the predictability of bond excess returns with
yields is persistent beyond the US to other countries, specifically UK and Germany.
German forecasting regressions show similar predictability in terms of R2 as in the
US, while those for the UK are somewhat lower. Secondly, even though the first
principal component captures a lot of common ground of international yield curves,
the international setting does not allow the reduction of the number of yield curve
factors if the three individual level, slope and curvature features of every country are
preserved. The yield curve is therefore best spanned by domestic yield curve factors.
6.4. CONCLUSION
125
Thirdly, the inclusion of foreign yields as instruments in the forecasting regression
alongside the standard domestic yields improves the predictability of domestic bond
excess returns across all countries under consideration and along the level, slope and
curvature risk premia. The R2 in the restricted regressions are boosted beyond 0.4
for the UK and beyond 0.5 for Germany and the US. Fourthly, the unspanned parts
of foreign yield curve factors dominate and significantly forecast bond returns in addition to the standard yield curve factors with R2 of up to 0.3. Unspanned foreign
yield curve factors are important in forecasting the domestic yield curve in the US,
UK and German markets.
126
Chapter 7
Multi-Currency Term
Structure Models
This chapter builds an international term structure model with the property that yield
curve factors of each country are mutually unspanned in the risk-neutral measure but
affect the forecasting ability of each other. This is in line with the stylized facts found
in the regression analysis in Chapter 6: Firstly, the cross-section of yields is best described by its domestic yield curve factors only, without any influence from foreign
yield curves. Secondly, and in contrast, although bond returns depend on domestic
and foreign yield curve factors, these do not sum up in any common international
factors. The model introduced here is specified to match precisely these empirical
characteristics. Technically, this model extends the yields-only model of Chapter 3
by including the feature of unspanned state variables, as introduced in Chapter 5.
To the best of the author’s knowledge, the implementation of unspanned factors into
international term structure models has not yet been systematically explored.1 The
1 Boos
and Wäger (2011) explore the same kind of models as a combination of results from Boos
(2011) and this thesis. Mirkov (2011) applies such an unspanned two-country model to investigate the
127
128
CHAPTER 7. MULTI-CURRENCY TERM STRUCTURE MODELS
specification of the international model with mutually unspanned factors implicitly
assumes that there are individual local yield curve factors as opposed to common
international factors. This chapter therefore also provides evidence against international common factors as it finds no common bond risk premia in the sense that the
number of individual risk premia is not reduced compared to the single-country models. The unspanned factor feature is implemented analogous to the unspanned macro
factors from Chapter 5, but with an important difference. The factors are mutually
unspanned and thus restrict interaction in both directions. This in turn ensures that
state variables unspanned by one country are spanned by the other country’s yield
curve. The price of risk is then defined for any state variable, since the physical
and risk-neutral distributions are separated by the data. The international model also
provides a model-implied forecast of foreign exchange returns. The specification
of the foreign exchange risk premium is shown to have no influence on the results
of the bond risk premia estimation. A better specification of the foreign exchange
risk premium should therefore not be able to improve bond risk premia predictability. Furthermore, the estimated foreign exchange risk premium deviates from the
uncovered interest rate parity and is thus in line with the actual data and empirical
research.
International term structure models have been introduced by Backus, Foresi and
Telmer (2001) among others. These multi-currency term structure models are often
established as a two-country model based on separating local and international state
variables. The specification with only one local factor each and one international factor as in Dewachter and Maes (2001) does not match the data as described in Chapter
6. Egorov, Li and Ng (2011) expand this specification to two common factors and
one local factor each, but this still cannot capture the risk-neutral features of level,
slope and curvature in both countries. Modugno and Nikolaou (2007) estimate an international Nelson-Siegel model with the level, slope and curvature of the domestic
and foreign yield curve as state variables. Their model differs from that introduced
here in the specification of the influence of the foreign state variables to the domestic
influence of monetary policy decisions across countries.
7.1. MODEL
129
yield curve, because the restriction that domestic yields are unspanned by international yields is not included. Nevertheless, their international model outperforms
the domestic models. Another approach to forecasting domestic risk premia with
international yield curves is presented by Dahlquist and Hasseltoft (2011). They add
international factors to the local level, slope and curvature factors in a single-country
term structure model and are therefore closer to the unspanned feature implied in the
model presented here than most of the international two-country models cited above.
This chapter is organized as follows. Section 7.1 introduces the multi-currency
term structure model in a general two-country case and the parametrization to implement foreign yield curve state variables as unspanned factors, followed by a detailed description of the adjustments necessary for the estimation routine to fit multicurrency models. Section 7.2 provides empirical results for the estimated models and
compares their forecasting performance across countries as well as against yieldsonly models and simple regressions. Specification tests on the number of forecasting
factors are also provided. Finally, Section 7.3 concludes.
7.1
Model
Firstly, this section derives a very general two-country model, which is based on
combining two yields-only single-country models of the sort presented in Section
3.2. This is achieved by including the exchange rate to model foreign bonds in the
domestic currency. This model is a homoscedastic Gaussian affine term structure
model with time-varying risk premia, just as all the other models presented in this
thesis, but allows domestic and foreign risk premia and their interactions to be analyzed.
Secondly, this general specification of a multi-currency yield curve model is further restricted in the interaction of the state variables. The risk-neutral dynamics of
the state variables are assumed to be mutually independent for both countries. This
ensures that the yield curve of one country is not spanned by the yield curve of an-
130
other country and vice versa. However, dependencies across risk premia still allow
for full interaction of all state variables in the physical dynamics.
7.1.1
General Two-Country Model
The general two-country model is derived by joining all the domestic and foreign
state variables in one VAR that describes the dynamics. This results in a very general
international yield curve model, which prices domestic and foreign bonds, as shown
by Wäger (2007). It is basically a combination of two individual single-country
models. The state variables, which consist of the domestic and foreign factors, follow
a joint 2K-dimensional VAR with the form
x
.
xt+1 = µx + Φx xt + εt+1
(7.1)
The exchange rate S between the two countries is modeled depending on the total yield curve factors. Inci and Lu (2004) state that a model including both term
structures of the two countries fits the exchange rate between them very closely. Presumably, there are other factors driving exchange rates besides the interest rate dynamics. However, since forecasting exchange rates is not the purpose of this thesis,
this specification seems appropriate and is in favor of a more parsimonious model.
In this straightforward implementation the foreign exchange risk premium depends
on the total yield curve state variables. The innovations to the exchange rate S are
thus exponentially affine in the state variables with Gaussian iid shocks εs
s
log(St+1 ) − log(St ) = As + Bx,⊤
s xt + εt+1 .
(7.2)
Adding the exchange rate shock from Equation (7.2) to the state variable dynamics
(7.1) leads to an augmented VAR
Xt+1 = µ + ΦXt + εt+1
(7.3)
with Xt and the shocks εt+1 as
Xt =
xt
εts
!
, εt+1 =
x
εt+1
s
εt+1
!
7.1. MODEL
131
and the parameters µ and Φ
µ=
µx
0
!
, Φ=
Φx
0
0
0
!
.
This augmented VAR has dimension 2K + 1 but only 2K factors, since the foreign
exchange rate is not a state variable. The dynamics of the exchange rate dependent
on Xt now read
⊤
log(St+1 ) − log(St ) = As + B⊤
s Xt + e2K+1 εt+1
with
Bs =
Bxs
0
(7.4)
!
and e2K+1 stands for a vector of zeros with a one at the denoted element 2K + 1.
The definition of the pricing kernel completes the model. The pricing kernel follows the same structure as in the single-country model from Section 3.2, but with
individual parameter sets for pricing in the domestic and foreign currency. The parameters of the foreign pricing kernel are indicated with a tilde. The domestic pricing
kernel is then given as
1 ⊤
⊤
Mt+1 = exp(−δ0 − δ⊤
2
λt = λ0 + λ1 Xt
(7.5a)
(7.5b)
and the foreign pricing kernel with the same structure reads as follows
1e ⊤ e e ⊤
et+1 = exp(−e
M
δ0 − e
δ⊤
2
eλt = eλ0 + eλ1 Xt .
(7.6a)
(7.6b)
The relation between the domestic and foreign pricing kernels is easily derived from
the bond pricing formulas combined with an absence of arbitrage argument2 and
2 Imposing
m∗ .
absence of arbitrage ensures the existence and positiveness of the pricing kernels m and
For Equation (7.8) to hold in any case, the additional assumption of complete markets that ensures
132
equals the change in the exchange rate
St+1
Mt+1
=
.
e
St
Mt+1
(7.7)
A derivation of this relation is given in e.g. Backus, Foresi and Telmer (2001) and
Wäger (2007). Using lower-case characters for logs this reads
et+1 = mt+1 − (st+1 − st ).
m
(7.8)
Plugging in the definitions of the domestic and foreign pricing kernels (7.5) and (7.6)
and exchange rate dynamics (7.4) into equation (7.8) results in
1 ⊤
⊤
⊤
⊤
et+1 = −δ0 − δ⊤
m
1 Xt − λt Σλt − λt εt+1 − As − Bs Xt − e2K+1 εt+1
2
(7.9)
and yields the relation of the parameters of the foreign model to the domestic model
by a comparison of coefficients with equation (7.6) as
separated into
eλt = λt + e2K+1
(7.10)
eλ0 = λ0 + e2K+1
(7.11a)
eλ1 = λ1 .
(7.11b)
These equations imply that the constant vector µ∗ , analogous as defined in (3.4a), of
the risk-neutral VAR (3.3) depends on the currency. The difference is equivalent to
the last column of Σ. However, Φ∗ is not currency-dependent since the conversion
of Φ into the risk-neutral measure is given by (3.4b). The difference in the price of
risk λt of the two countries is constant and equals the variance of the exchange rate.
This could be explained as a constant risk compensation depending on the reference
currency of the investor. This term is always positive in the foreign pricing kernel.
uniqueness of the pricing kernels needs to be in place. In incomplete markets these pricing kernels are not
unique, but can be chosen to satisfy Equation (7.8). The model thus provides a solution without the strong
assumption of complete markets. A detailed discussion, derivation and proof of this theorem is provided
in Backus, Foresi and Telmer (2001).
7.1. MODEL
133
Finally, to solve for the remaining parameters, Equation (7.10) is replaced in the
foreign pricing kernel (7.6)
1e ⊤ e
⊤
e
et+1 = − δ0 − δ⊤
(7.12a)
m
1 Xt − λt Σλt + e2K+1 Σλt
2
1
Σe2K+1 − eλt⊤ εt+1 − As − B⊤
− e⊤
s Xt ,
2 2K+1
which yields the relations of the parameters e
δ0 to δ0 and e
δ1 to δ1 that are as follows
1
e
e
Σe2K+1 + As − e⊤
δ0 = δ0 + e⊤
2K+1 Σλ0
2 2K+1
⊤
⊤
⊤
e
e
δ⊤
1 = δ1 + Bs − e2K+1 Σλ1 .
(7.13a)
(7.13b)
These equations specify the parameters for As and Bs of the exchange rate equation (7.4). With the variance of the exchange rate dynamics σ2s = e⊤
2K+1 Σe2K+1 and
e
e
replacing λ0 and λ1 by (7.11), these are:
1
As = e
δ0 − δ0 + σ2s + e⊤
2K+1 Σλ0
2
⊤
⊤
e⊤
B⊤
s = δ1 − δ1 + e2K+1 Σλ1 .
(7.14a)
(7.14b)
In this general specification of an international yield curve model, exchange rate dynamics are driven by interest rate differentials and a time-varying foreign exchange
risk premium. This is obvious from inspecting equations (7.14a) and (7.14b), which
drive exchange rate dynamics. The equations can be decomposed into three parts.
(1)
(1)
The first is the short rate differential of the two countries short rates yt and yet ,
which are given as δ0 + δ⊤ Xt and e
δ0 + e
δ⊤ Xt , respectively. The second is a constant
1
1
Jensen’s inequality term of half the size of the exchange rate variance. The third is a
time-varying compensation for risk given as the variance-covariance matrix Σ times
the market price of risk λ, which thus is allowed to depend on all the risk factors in
the model.
Additionally, Equation (7.14b) for Bs together with the definition of Bs in Equation (7.4) derived from Bxs , implies the restriction that the last row of λ1 is all zeros.
This is obvious because the augmented exchange rate is not a state variable and thus
does not compensate any risk. In a 2K factor model with augmented exchange rate
shocks λ1 has rank 2K.
134
7.1.2
Foreign Yields as Unspanned Factors
So far the above derived model is a very general form of a two-country model. The
innovation in the model presented in this thesis lies in the restriction that the domestic
and foreign state variables are modeled as mutually unspanned factors. Unspanned
factors reflect state variables that are important in forecasting bond returns, but do
not span the current yield curve. In the two-country case this means that the riskneutral distributions of both the domestic and foreign bonds are only dependent on
their corresponding local yield curve factors. However, the distribution of bonds in
the physical measure is dependent on all the factors in the model, whether domestic
or foreign. The risk premia and thus the return forecasting ability are driven by all
the state variables.
Technically, the mutually unspanned yield curve factors are implemented by restricting bidirectional dynamics in the risk-neutral VAR. This results in a block diagonal structure of Φ∗ because the unspanned factors are not dependent on each other
in the risk-neutral measure Q. The parametrization applied on the local yield curve
factors is an extension of the single-country case. The first K factors are the domestic
yield curve factors. The last K factors are the foreign yield curve factors. Both sets
of factors are modeled by the self-consistent parametrization derived in Section 3.4,
whereas these two sets are mutually unspanned, analogous to the unspanned macrofinance models derived in Section 5.2. Combining the standard parametrization and
the block diagonal form that produces unspanned factors, yields the risk-neutral parameters


Φ∗ = 

and
µ∗ =
h
C1
0
0
0
C2
0
c1 . . . c2K
0
0 . . . 0 µ∗K




0 . . . 0 µ∗2K
(7.15)
cs
i⊤
.
(7.16)
The matrices C1 and C2 are companion matrices, which arise in the self-consistent
specification of term structure models, as do the free parameters µ∗K and µ∗2K for the
7.1. MODEL
135
domestic and foreign yield factors, respectively. According to the self-consistent
parametrization, the parameters δ0 and e
δ0 are both fixed to zero and δ1 and e
δ1 are
vectors of zeros with a one at the first local state variable, analogous to the specifi-
cation in (3.15). The off-diagonal blocks of zeros in Φ∗ implement the unspanned
feature by restraining interaction of the two sets of state variables in the risk-neutral
measure Q. The parameters c1 . . . c2K and cs correspond with the negative risk compensation of the exchange rate dynamics and are implicitly defined by the definition
of the VAR in the physical measure P (7.3) combined with the transformation into
the risk-neutral parameters given in (3.4b) and (3.4a), which results in
c1 . . . c2K = −e⊤
2K+1 Σλ1
cs = −e⊤
2K+1 Σλ0 .
(7.17a)
(7.17b)
Economically, the reasoning for unspanned factors is similar to the unspanned
macro model from Chapter 5. Some information that is important in forming investor’s expectations but with opposite effects on the current yield curves is netted
out. Chapter 6 provides empirical evidence that much of the information in the foreign yield curves is unspanned by the domestic yield curves and that this unspanned
part predicts domestic bond excess returns. This model differs from the specification
of the unspanned factors in the macro-finance model from Chapter 5 in an important point. Since the model is mutually unspanned, the matrix Φ∗ , which governs
the risk-neutral dynamics is block diagonal compared to the macro-finance model
that only restrains movements from the macroeconomic variables to the yield curve
and not vice versa. The beauty of the block diagonal structure of the multi-currency
model is that the state variables are unspanned factors in one country but fully span
the yield curve in the other country. This implies that the price of risk is defined for
any state variable.
136
7.1.3
Estimation
The model is estimated in state space form with the Kalman filter algorithm as applied in the other models in this thesis. The detailed description of the estimation
algorithm is given in Appendix B. Adaptions to this procedure due to the international setting are described in the following. The measurement equation extends to
(n)
(n)
(n)
the observed yields of both countries yt and yet by adding measurement errors ηt
et(n) for any observed maturity n. This yields the two sets of equations
and η
(n)
yt
(n)
yet
(n)
= Ayn + By,⊤
n Xt + ηt
=
eyn + Bey,⊤
A
n Xt
et(n) .
+η
(7.18a)
(7.18b)
Again, in order to reduce the number of parameters, the measurement errors are
assumed to be uncorrelated and the standard deviation of the measurement errors ση
is the same across all maturities in both countries. In other words, the covariance
e is a scaled identity matrix.
matrix of the measurement errors η and η
Observed exchange rate returns st − st−1 are implemented in the same way. The
measurement equation is equal to the definition of the foreign exchange rate given in
Equation (7.4) as
s
st − st−1 = As + B⊤
s Xt−1 + εt .
(7.19)
Unfortunately, observed exchange rate returns at time t are a function of the state
variables at time t −1. In contrast, contemporaneously observed yields are a function
of the state variables at time t, easily seen in the measurement equations above.
In order to perform Kalman filter estimation, these equations need to be merged
into a joint system of measurement equations. A convenient way to do this is to
augment the vector of state variables Xt with one lag. The following implementation
is borrowed from Boos (2011) and adapted to this specific case. The augmented state
vector Wt is then given as
Wt =
xt
xt−1
!
(7.20)
7.1. MODEL
137
and the augmented VAR thus is given as
!
µ
Φ
Wt+1 =
+
0
1
0
0
!
Wt +
εt+1
0
!
.
(7.21)
The exchange rate shock εts is not necessarily uncorrelated with the shocks to the
state variables εtx . Since the Kalman filter requires the measurement errors to be uncorrelated with the innovations of the transition equation, the linear dependence of
the exchange rate shocks to these of the state variables needs to be handled explicitly. The exchange rate shocks can be decomposed as linearly dependent of the state
variable shocks with
εts = C⊤ εtx + ηts
(7.22)
with a Gaussian iid distributed error ηts uncorrelated to the innovations of the state
variables. Written as a function of all the shocks εt this reads
εts = Cs⊤ εt + ηts
(7.23)
where Cs⊤ = [C⊤ 0]. By replacing εt with the rearranged VAR of Equation (7.3) gives
the expression
εts = Cs⊤ Xt −Cs⊤ µ −Cs⊤ ΦXt−1 + ηts ,
(7.24)
which decomposes the exchange rate shocks. Inserted into Equation (7.4) this reads
s
⊤
(7.25)
st − st−1 = As −Cs⊤ µ +Cs⊤ Xt + B⊤
s −Cs Φ Xt−1 + ηt .
Putting Equation (7.25) together with the measurement equations for domestic and
foreign yields (7.18) results in the following system of measurement equations
 
 




yt
Ay
By
0
ηt
 
 




=
 +  Bey

 Wt +  η

ey
yet
A
0
 
 


 et  . (7.26)
st − st−1
As −Cs⊤ µ
Cs⊤
⊤
B⊤
s −Cs Φ
ηts
The errors ηts are now linearly independent of the shocks to the state variables as
required by the Kalman filter. Estimation is now achieved simply by applying maximum likelihood estimation (MLE) to the Kalman filter routine on the state space
described by the measurement equation (7.26) and the transition equation (7.3).3
3 Estimation
of term structure models is well-known to be demanding with poor convergence. The
138
7.1.4
Restrictions on Risk Premia
The risk premia in this model can be restricted in several dimensions. As before,
rank restrictions on the matrix λ1 will restrict the overall number of parameters in
the model. Since the model now includes domestic as well as foreign bond risk
premia with a foreign exchange risk premium on top, the overall restriction is not
really meaningful for analyzing bond risk premia. The restriction on the foreign exchange risk premium is separated from that on bond risk premia. In the standard
specification, although all the bond risk premia are constrained by the rank restriction, the foreign exchange risk premium is excluded and freely depends on all the
state variables. This model specification is called C(K, L) with K as the number of
yield factors in each country and L as the number of bond risk premia. Specification
CL (K, L) constrains the bond risk premia to be affected by local factors only. The
restriction is implemented by reducing the physical dynamics parameter Φ to block
diagonal form. The off-diagonal blocks are zero and therefore restrain any influence
of foreign factors on domestic risk premia and vice versa. The rank restriction on
bond risk premia is split in half, meaning that in each country the number of risk
premia is constrained to L/2. The foreign exchange risk premium is treated as in the
standard C(K, L) model. Another specification CnoFX (K, L) differs from C(K, L) by
constraining the time-varying foreign exchange risk premium to zero. This is implemented by restricting the exchange rate parameter Bs of Equation (7.14b) to e
δ1 − δ 1 .
This specification eliminates any predictability of exchange rate returns.
specific parameter identification and the Kalman filter algorithm with the analytical gradient dramatically
improve convergence speed. Multiple different starting values, which converge to the same parameter sets
and likelihood values affirm that convergence is reached at the optimum. The parameter identification and
further information about the difficulty of estimating term structure models is introduced in Section 3.4.
The applied estimation procedure is described in detail in Appendix B. Further information about the
applied algorithm is provided in Boos (2011).
7.2
139
Empirical Results
This section provides empirical estimates of the single and multi-currency models
and compares them against each other.4 The information provided by the foreign
state variables in the multi-currency model is shown to improve the forecasting ability compared to local models. As an example, the conundrum period of the US
yield curve is taken. This period features continuous hikes of the target rate starting
around mid-2004 until around end of 2006, but the yields of long maturity bonds
did not increase such that the yield curve was inverted at the end of that period.
The decrease of long maturity yields compared to the high increase of 4.25% at the
short end has not been expected and is not well matched by standard term structure
models, thus its name.5 Comparing the one-year yield curve forecasts of the singleand multi-currency models as of September 2005 with the effective yield curve as
of September 2006 shows that the single-currency model does not reproduce the
inverted yield curve, whereas the multi-currency model produces an inverted yield
curve and is even closer to realization one year later, as shown in Figure 7.1. The
forecasted excess returns of a ten-year bond in that period from the multi-currency
model is -0.42%, wheras that of the single-currency model is 0.53% and thus further
away from the -2.12% realized.
The remainder of this section introduces the model estimates and an empirical
analysis of the implied risk premia of the multi-currency models compared to their
single-currency benchmark. The main result is that the risk premia differ across
countries although foreign state variables improve the models significantly compared
to the single-country alternatives. The latter is a sign of improved international integration of bond markets, but the individual variation in risk premia is shown to still
dominate. The models are all estimated on quarterly frequency and in the range of
1976:12 - 2009:06. This are 131 observations in the time series and 17 yields in the
4 Due
to the large number of models estimated and used for comparison in this section, only the most
important results are reported with tables and figures. Additional tables and figures are provided in Appendix F and cross-references are given in the text when appropriate.
5 A treatment of this period is given by Backus and Wright (2007) and Cochrane and Piazzesi (2008).
140
0.05
0.048
0.046
0.044
0.042
0.04
Multi−currency model
Single−currency model
Realized
0.038
0.036
1
2
3
4
5
6
7
8
9
10
Figure 7.1: Comparison of expected yield curves in the conundrum period
The figure shows the expected one-year-ahead US yield curve from the two models as of September 2005
compared with the realized yield curve one year later.
cross-section of each country and the observed foreign exchange rate.6 Firstly, the
single-country model estimates are presented. This is an evaluation on a different
time frame for the US model and allows the single-country results from the US to
be compared with the German and UK models. Secondly, the two-country estimates
are reported and compared against both the single-country and the regression results.
The two-country models are estimated in the six-factor case, that is three factors for
each country which is the extension of the single-country three-factor models.7
6 Foreign
7 The
exchange rates are all taken from Datastream.
shorter time series dimension (the time series of the EUR and UK data does not reach as far
back as the US data) and the sharp increase in free parameters in the model prevents a robust parameter
estimation in dimensions above the C(3, L) models by going from the single- to the two-country case. The
two-country analog of the single-country four-factor case already is an eight-factor model C(4, 8) with 137
free parameters. This is even slightly more than the number of independent time series observations and
therefore parameter estimates would most likely suffer the curse of dimensionality problem.
7.2.1
141
Single-Country Models
Estimates of the single-country yields-only models from Chapter 3 are presented in
the following. These will serve as a benchmark in evaluating the two-country models
but also deliver an extension to countries other than the US compared to the previous
analyses. Table 7.1 reports the R2 of the three-factor UK, German and US models.
Table 7.1: Forecasting power of the quarterly three-factor single-country models
log likelihood
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
A(3, 0)
11050.5
14
0.00
0.00
0.00
0.00
A(3, 1)
11057.5
19
0.00
0.01
0.00
0.07
A(3, 2)
11062.6
22
0.02
0.07
0.03
0.15
A(3, 3)
11062.7
23
0.02
0.07
0.03
0.15
Germany 1976:12 - 2009:06
A(3, 0)
11282.8
14
0.00
0.00
0.00
0.00
A(3, 1)
11291.0
19
0.16
0.09
-0.02
0.03
A(3, 2)
11294.1
22
0.16
0.13
0.06
0.05
A(3, 3)
11294.1
23
0.16
0.13
0.06
0.05
A(3, 0)
11269.7
14
0.00
0.00
0.00
0.00
A(3, 1)
11275.7
19
0.13
0.14
0.02
0.04
A(3, 2)
11281.7
22
0.13
0.19
0.15
0.15
A(3, 3)
11281.8
23
0.13
0.19
0.15
0.15
United States 1976:12 - 2009:06
The table shows the R2 of a one-year forecast based on the quarterly three-factor models, the number of
A comparison of the forecasting power of the US model in Table 7.1 with the
same model but on a longer time series (as presented in Table 3.1 of Chapter 3)
shows very similar R2 values. The forecasting power seems not to be affected by
this cut in the time series. Judging by the forecasting ability, the A(3, 2) model is the
dominant case. The corresponding likelihood ratio test statistics are given in Table
7.2. These confirm the dominance of the A(3, 2) model in the US, as is the case in
the longer data sample of Chapter 3.
142
Table 7.2: Likelihood ratio tests of the quarterly single-country models
A(3, 0)
A(3, 1)
A(3, 1)
13.9
(15.1)
A(3, 2)
24.2
(20.1)
10.3
(11.3)
A(3, 3)
24.4
(21.7)
10.5
(13.3)
A(3, 2)
0.2
(6.6)
Germany 1976:12 - 2009:06
A(3, 0)
A(3, 1)
A(3, 1)
16.4
(15.1)
A(3, 2)
22.6
(20.1)
6.2
(11.3)
A(3, 3)
22.6
(21.7)
6.2
(13.3)
A(3, 2)
0.0
(6.6)
United States 1976:12 - 2009:06
A(3, 0)
A(3, 1)
A(3, 1)
11.9
(15.1)
A(3, 2)
24.0
(20.1)
12.1
(11.3)
A(3, 3)
24.1
(21.7)
12.2
(13.3)
A(3, 2)
0.1
(6.6)
likelihood.
The forecasting ability of the German models is in similar range to the US model
for the level, but slightly lower for slope and curvature portfolios. The R2 also indicate the dominance of the A(3, 2) model, but this is not confirmed by the likelihood
ratio statistics in Table 7.2. The statistics indicate that only one risk premium is
priced and that further risk premia do not enhance the model. Significance of the
second priced risk premia is reached at a p-value of 5%, which could be seen as
weakly significant.
The UK model forecasts are rather poor in comparison to the other two countries.
Level and slope portfolio forecasts are very low, as shown in Table 7.1. Only the
curvature portfolio is forecasted with an R2 comparable to the other models. The
A(3, 2) model is the dominant model in terms of R2 , since models with fewer risk
143
premia than two show barely any forecasting ability. The specification test in Table
7.2 shows a similar pattern to the German case with the A(3, 2) model only weakly
significant, with a p-value of 2%.
Since these single-country three-factor models serve as a benchmark for the twocountry case, tables with estimation results for the four-factor models are deferred
into Appendix F (tables F.1 and F.2). As expected, adding another state variable
the predictability pattern improves, but not by much as indicated by the R2 values
reported in Table F.1. Noteworthy are the results from the specification tests reported
in Table F.2. The number of priced risk premia is the same across the three countries
and equals two. Adding a fourth state variable therefore does not increase the number
of risk premia. This result is contrary (in the US case) to the longer sample results
from Chapter 3, where three risk premia are priced. It is not obvious why there
is only two priced risk premia in this shorter sample compared to three priced risk
premia in the full sample. Either the third priced risk premium is only priced in
the extended period of the full sample, or the third and smallest risk premium is not
statistically significant due to the short sample. However, because of these results,
it is appropriate to evaluate the two-country models based on the three-factor singlecountry case extension only, in the following section.
Comparison of Risk Premia Across Countries
The extension to different countries also allows risk premia to be compared against
each other. The A(3, 3) model-implied risk premia for level, slope and curvature portfolios are plotted in Figure 7.2 for the three countries UK (GBP), Germany (EUR)
and US (USD). The corresponding correlation matrix is given in Table 7.3, while
cross-correlations between the same type of risk premia are highlighted in bold. The
correlations of risk premia across these different countries are relatively low but positive, ranging from 0.07 to 0.53. The correlations of the level risk premia are generally
higher than those of slope and curvature risk premia, as has already been noticed in
the principal component analysis of international bond returns in the previous chapter
144
Level
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
1995
2000
2005
2010
2000
2005
2010
Slope
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
1990
Curvature
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
1990
USD
1995
EUR
GBP
Figure 7.2: Comparison of risk premia across countries
The plots show the one-year forecasts of excess returns (all in local currency) of the level rx (rx(10) ),
slope srx (srx2,10 ) and curvature crx portfolio (crx2,5,10 ) for the three countries (plotted with quarterly
overlapping).
145
Table 7.3: Correlation of risk premia across countries
USD
rx
USD
EUR
GBP
srx
EUR
crx
rx
srx
GBP
crx
rx
srx
rx
1.00
srx
-0.82
crx
-0.89
0.99
1.00
rx
0.53
-0.35
-0.41
1.00
srx
-0.25
0.22
0.22
-0.39
crx
-0.52
0.33
0.39
-0.99
0.23
1.00
rx
0.52
-0.25
-0.30
0.41
-0.44
-0.36
1.00
srx
-0.49
0.07
0.15
-0.42
0.28
0.41
-0.87
1.00
crx
-0.43
0.00
0.09
-0.40
0.06
0.42
-0.69
0.93
crx
1.00
1.00
1.00
(rx(10) ), slope srx (srx2,10 ) and curvature crx portfolio (crx2,5,10 ) for the three countries. Correlations of
the same types of portfolio across countries are highlighted in bold.
(see Table 6.6). These low correlations imply that there is considerable uncommon
variation in international risk premia and thus support the international model set-up
with domestic and foreign factors instead of common international factors.
The correlations of the different risk premia within each model show high correlations of the curvature risk premia with one of the previous factors, either level or
slope risk premia. This is obvious since the dominant specification of these threefactor models includes only two priced risk premia. Therefore, one of these three
risk premia is not priced and is thus modeled as linearly dependent on the other two.
7.2.2
Two-Country Models
This section provides the estimation results of the two-country models and evaluates
the number of priced bond risk premia. As these models allow the foreign exchange
risk premium to be estimated, this premium and its interaction with bond risk premia
are explored. In addition, a comparison to the single-country case and the simple
146
regression forecasts is provided. The estimated two-country models are labeled as
the USD-EUR, the USD-GBP and the EUR-GBP models, while the first three-letter
code indicates the domestic and the second the foreign country.8 The selection of the
domestic country is arbitrary and does not influence the estimation results because
these models are perfectly symmetric. An USD-EUR model thus results in the same
estimates as an EUR-USD model.9
Number of Priced Bond Risk Factors
The forecasting power of the standard two-country models is given in Table 7.4.
Throughout all countries, the forecasting ability shows a great improvement on the
single-country models. Level forecasts of German and US bond excess returns show
a high R2 of between 0.25 and 0.37, and for the UK between 0.08 and 0.17 These
values are approximately doubled compared to single-country models. According
to the specification of the single-country models where two priced risk premia are
dominant,10 four priced risk premia would be expected in the two-country models if
risk premia are individual across countries.
Looking at the specification tests in Table 7.5 shows exactly this pattern for the
USD-EUR model. The C(3, 4) specification is the dominant model, which implies
that the priced risk premia are significantly different across countries and that this
is vital in order to explain local expected returns.11 For the USD-GBP model, the
C(3, 4) specification is significant at a p-value of 4%. By looking at the forecasting
ability of the model in Table 7.4 this selection seems appropriate since the R2 of
forecasts of UK bond excess returns jump from zeros in the C(3, 3) specification to
8 The
9 An
three letter codes USD, EUR and GBP indicate the US, German and UK models, respectively.
example is given in table F.4 in the appendix, where likelihood values and forecasting power is
compared of two symmetric models and found to be equal. In fact the corresponding parameters are equal
as well (not reported).
10 In the US model this is clearly the case, but the UK and German models show only weak significance
of the specification with two priced risk premia as shown in table 7.2.
11 The parameter estimates of the dominant model of each country are provided in the appendix in the
tables F.9, F.10 and F.11 for the USD-EUR, USD-GBP and EUR-GBP models, respectively.
82
85
22651.9
22656.4
22665.5
22667.5
22667.5
C(3, 2)
C(3, 3)
C(3, 4)
C(3, 5)
C(3, 6)
70
77
82
85
22423.8
22432.2
22438.4
22440.4
22440.9
C(3, 2)
C(3, 3)
C(3, 4)
C(3, 5)
C(3, 6)
70
77
82
85
86
22492.6
22500.6
22502.0
22502.7
22504.4
C(3, 2)
C(3, 3)
C(3, 4)
C(3, 5)
C(3, 6)
0.37
0.35
0.35
0.36
0.25
0.23
0.00
0.28
0.27
0.25
0.23
0.09
0.04
0.00
0.27
0.28
0.28
0.28
0.27
0.18
0.00
rx(2)
0.35
0.31
0.30
0.30
0.23
0.19
0.00
0.32
0.32
0.29
0.19
0.08
0.06
0.00
0.26
0.26
0.27
0.27
0.25
0.21
0.00
rx(10)
0.08
0.07
0.07
0.07
-0.04
-0.09
0.00
EUR
0.18
0.17
0.16
0.04
0.07
-0.02
0.00
USD
0.19
0.18
0.18
0.19
0.04
0.05
0.00
USD
srx2,10
0.11
0.08
0.08
0.10
0.07
0.08
0.00
0.22
0.23
0.19
0.13
0.08
0.06
0.00
0.21
0.21
0.25
0.24
0.23
0.23
0.00
crx2,5,10
0.17
0.17
0.17
0.15
-0.01
-0.01
0.00
0.07
0.06
0.08
0.02
0.02
0.00
0.00
0.29
0.29
0.29
0.28
0.22
0.17
0.00
rx(2)
0.11
0.10
0.09
0.07
0.03
-0.05
0.00
0.11
0.11
0.11
0.00
-0.01
-0.04
0.00
0.23
0.23
0.26
0.21
0.10
0.07
0.00
rx(10)
0.19
0.18
0.17
0.18
0.11
-0.08
0.00
GBP
0.09
0.09
0.04
0.00
0.02
-0.01
0.00
GBP
0.04
0.04
0.08
0.06
-0.01
-0.02
0.00
EUR
srx2,10
0.14
0.09
0.07
0.09
0.09
0.00
0.00
0.15
0.14
0.16
0.12
0.04
0.05
0.00
0.18
0.18
0.18
0.05
0.02
0.02
0.00
crx2,5,10
0.16
0.15
0.16
0.16
0.16
0.16
0.14
0.14
0.14
0.14
0.14
0.14
0.10
0.10
0.11
0.11
0.12
0.10
0.10
0.07
0.10
FX
likelihood.
The table shows the R2 of a one-year forecast based on the quarterly models, the number of parameters (“nvar”) and the estimated maximum log
61
22482.3
C(3, 1)
50
22459.0
C(3, 0)
EUR-GBP model
61
22411.8
86
50
22388.1
C(3, 1)
86
70
C(3, 0)
USD-GBP model
77
22641.8
61
22623.1
C(3, 1)
50
nvar
C(3, 0)
USD-EUR model
log likelihood
Table 7.4: Forecasting power of the quarterly two-country C models
147
83.2
86.0
87.4
90.7
C(3, 3)
C(3, 4)
C(3, 5)
C(3, 6)
(24.7)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
(24.7)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
(24.7)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
C(3, 1)
(21.7)
(44.3)
(43.0)
(38.9)
(32.0)
(21.7)
(44.3)
(43.0)
(38.9)
(32.0)
44.3
41.0
39.5
36.8
20.6
(44.3)
(43.0)
(38.9)
(32.0)
(21.7)
EUR-GBP
58.1
57.3
53.2
40.9
24.1
USD-GBP
51.5
51.4
47.4
29.1
20.3
USD-EUR
23.6
20.3
18.9
16.1
34.0
33.2
29.1
16.8
31.2
31.1
27.2
8.9
(32.0)
(30.6)
(26.2)
(18.5)
(32.0)
(30.6)
(26.2)
(18.5)
(32.0)
(30.6)
(26.2)
(18.5)
C(3, 2)
7.5
4.2
2.8
17.2
16.4
12.3
22.4
22.3
18.3
(21.7)
(20.1)
(15.1)
(21.7)
(20.1)
(15.1)
(21.7)
(20.1)
(15.1)
C(3, 3)
4.7
1.4
4.9
4.1
4.1
4.0
(13.3)
(11.3)
(13.3)
(11.3)
(13.3)
(11.3)
C(3, 4)
3.3
0.8
0.1
(6.6)
(6.6)
(6.6)
C(3, 5)
test statistic, the unrestricted model is able to fit the data significantly better than the constrained model in terms of likelihood.
(and nested) model on top. The corresponding test statistic with a significance level of 1% is given in parentheses. If the likelihood ratio exceeds the
The table shows the likelihood ratios comparing unrestricted and constrained models. The unrestricted model is indicated on the left, the constrained
46.5
105.5
C(3, 6)
67.1
104.6
C(3, 5)
C(3, 1)
100.5
C(3, 4)
C(3, 2)
88.2
C(3, 3)
89.0
C(3, 6)
47.3
88.9
C(3, 5)
71.4
84.9
C(3, 4)
C(3, 1)
66.6
C(3, 3)
C(3, 2)
37.5
57.7
C(3, 1)
C(3, 2)
C(3, 0)
Table 7.5: Likelihood ratio tests of the quarterly two-country C models
148
149
positive values in the C(3, 4) specification and stay at this same level even if more
risk premia are allowed in the model. In the EUR-GBP model the C(3, 3) specification is dominant at a p-value of 2.5%. This is a reduction of priced risk premia
in comparison to the specification of the single-country models. This is supported
by the R2 of the model predictions as they do not improve with more than three
priced risk premia. This reduction in the number of risk premia could be for one of
two reasons: either the number of risk premia in the EUR-GBP model is actually
reduced in comparison to the individual single-country models, or the international
comovement of domestic and foreign risk premia renders one of the local risk premia insignificant. Both reasons imply that in the EUR-GBP two-country model, the
number of risk premia available in forecasting is reduced by one compared to the
single-country models.
Table 7.6: Correlation of risk premia in the USD-EUR model
USD C(3, 6)
rx
USD C(3, 6)
EUR C(3, 6)
USD A(3, 3)
EUR A(3, 3)
rx
1.00
srx
EUR C(3, 6)
crx
rx
srx
crx
srx
-0.58
crx
-0.91
0.67
1.00
rx
0.74
-0.55
-0.74
1.00
srx
0.19
0.05
-0.30
-0.29
1.00
crx
-0.58
0.29
0.56
-0.93
0.27
1.00
rx
0.92
-0.73
-0.81
0.70
-0.08
-0.44
0.49
1.00
srx
-0.74
0.91
0.87
-0.75
0.04
crx
-0.81
0.89
0.88
-0.76
0.06
0.50
rx
0.57
-0.37
-0.33
0.74
-0.34
-0.78
srx
0.07
0.31
-0.20
-0.30
0.91
0.18
crx
-0.62
0.33
0.39
-0.73
0.20
0.79
(rx(10) ), slope srx (srx2,10 ) and curvature crx portfolio (crx2,5,10 ).
150
Table 7.7: Correlation of risk premia in the USD-GBP model
USD C(3, 6)
rx
USD C(3, 6)
GBP C(3, 6)
USD A(3, 3)
GBP A(3, 3)
rx
1.00
srx
GBP C(3, 6)
crx
rx
srx
crx
srx
-0.68
crx
-0.92
0.82
1.00
rx
0.45
-0.65
-0.47
1.00
srx
-0.45
0.25
0.28
-0.76
1.00
crx
-0.19
0.27
0.21
-0.79
0.78
1.00
rx
0.82
-0.70
-0.75
0.79
-0.71
-0.41
srx
-0.69
0.88
0.81
-0.66
0.25
0.15
crx
-0.75
0.87
0.83
-0.70
0.39
0.21
rx
0.15
-0.30
0.03
0.71
-0.66
-0.51
srx
-0.22
0.14
0.02
-0.73
0.83
0.77
crx
-0.32
0.18
0.15
-0.66
0.83
0.85
1.00
Table 7.8: Correlation of risk premia in the EUR-GBP model
EUR C(3, 6)
rx
EUR C(3, 6)
GBP C(3, 6)
EUR A(3, 3)
GBP A(3, 3)
rx
1.00
srx
GBP C(3, 6)
crx
rx
srx
crx
srx
-0.34
crx
-0.94
0.20
1.00
rx
0.49
-0.01
-0.71
1.00
srx
-0.14
0.18
0.42
-0.80
crx
0.03
-0.09
0.21
-0.78
0.73
1.00
rx
0.59
-0.36
-0.77
0.82
-0.86
-0.54
0.02
1.00
1.00
srx
-0.23
0.80
0.19
-0.24
0.40
crx
-0.59
0.23
0.79
-0.83
0.84
0.57
rx
0.20
-0.04
-0.32
0.77
-0.55
-0.65
srx
-0.02
0.04
0.19
-0.77
0.62
0.91
crx
0.08
-0.01
0.09
-0.68
0.59
0.96
151
Decomposing Risk Premia
Further investigation of the comovement of risk premia in these models is achieved
by looking at their correlations. Tables 7.6, 7.7 and 7.8 report the correlations of the
risk premia implied by the USD-EUR, USD-GBP and EUR-GBP models, respectively. The upper part of the tables shows the correlations of the model-implied risk
premia. The lower part adds the correlations of the model-implied risk premia to
the risk premia of the corresponding single-country models. The correlations of the
local risk premia within a country are reduced in size compared to the single-country
models. This is the case since foreign information helps to span the three portfolio
risks given in the level, slope and curvature portfolios with more than two risk premia
in total. The high correlations of curvature risk to either slope or level risk is thus
reduced in the international models. Correlations between the countries’ level, slope
and curvature factors are low, mostly below 0.5. The risk premia across countries
therefore have a lot of individual variation not connected across countries. This is
even the case for the EUR-GBP model, where the number of risk premia is reduced
in the dominant specification compared to that in the single-country models. This is
contrary to the finding that domestic bond risk premia converge and are driven by
some global risk factors, as stated by e.g. Dahlquist and Hasseltoft (2011).
Even though country-specific risk premia do not converge to any set of global
risk premia, these local risk premia are nevertheless influenced by the information in
foreign yield curves. This is the case since foreign yield curve information improves
the model beyond the single-currency alternative and thus to some extent captures a
common part of both countries’ yield curves. Figure 7.3 decomposes the US level
risk premia into the parts from domestic and foreign state variables. The variation of
the foreign yield curve adds a rather large fraction to the expected returns. This foreign part of expected returns improves the multi-currency model beyond its singlecurrency analogs. Further information is recovered by a variance decomposition of
the ten-year maturity bond forecasts one year ahead. This decomposition separates
152
the forecasting variance into domestic and foreign factors.12 The share of explained
variance stemming from the domestic and foreign state variables is reported in Table 7.9 and accounts to about one fourth in average. This confirms that the foreign
information is important in explaining domestic risk premia. However, the influence
of the domestic factors still dominates.
Table 7.9: Variance decompositions of bond risk premia
USD-EUR
USD-GBP
EUR-GBP
USD
EUR
USD
GBP
EUR
GBP
Share of domestic state variables
77.1%
99.5%
75.7%
70.1%
49.9%
64.2%
Share of foreign state variables
22.9%
0.5%
24.3%
29.9%
50.1%
35.8%
The table shows the variance decomposition of forecast errors into the contributions of domestic and
foreign state variables for the annual forecasts of ten-year bond excess returns xr(10) . The exact calculation
of the variance decomposition is deferred to Appendix F.2.
Foreign Exchange Forecasts
These multi-currency models provide a foreign exchange forecast based on the foreign exchange risk premium in the model. In the C(K, L) models the foreign exchange risk premium is dependent on all the state variables, both domestic and foreign, independent of the rank restriction on bond risk premia. As already noted, the
model is not built to deliver a good foreign exchange forecast. The point here is to
examine the influence of the foreign exchange risk premium on the bond risk premia.
The R2 of annual foreign exchange forecasts are provided in the last column of Table
7.4 and are between approximately 0.10 and 0.15. The calculation of multi-period
foreign exchange return forecasts is provided in Appendix F.1 and a plot of these
foreign exchange return forecasts against the realized values is presented in Figure
F.1.
12 The
variance decomposition is sensitive to the ordering of variables because the orthogonalization
of the state variable shocks needs to assign the correlated variation of the state variables to one of them.
Here these correlated effects are assigned to the domestic state variables. The detailed computation of the
variance decomposition is provided in Appendix F.2.
153
Level risk premium
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Domestic part
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
Foreign part
0.15
0.1
0.05
0
−0.05
1975
1980
1985
1990
1995
Figure 7.3: Decomposition of risk premia into domestic and foreign parts
The figure shows the annual US level bond risk premium for ten-year maturity (rx(10) ) and its decomposition into the contributions of domestic and foreign state variables.
154
Table 7.10: Likelihood ratio tests of the CnoFX and CL models
CnoFX (3, 4)
CnoFX (3, 6)
CL (3, 4)
CL (3, 6)
USD-EUR
C(3, 4)
6.8
(16.8)
C(3, 6)
40.2
6.7
(32.0)
(16.8)
43.7
(34.8)
48.9
(34.8)
38.4
(34.8)
USD-GBP
C(3, 4)
10.0
(16.8)
C(3, 6)
44.8
10.1
(32.0)
(16.8)
EUR-GBP
C(3, 4)
14.9
(16.8)
C(3, 6)
33.7
15.6
(16.8)
(32.0)
likelihood.
The CnoFX (K, L) models are nested in the C(K, L) models and restrict the timevarying foreign exchange risk premium to zero. The variation in the foreign exchange forecast is thus solely based on the constant risk premium and the short
rate differential. The comparison of the foreign exchange rate forecasts from the
CnoFX (3, 6) and C(3, 6) model is plotted in Figure F.2. The predictability of foreign
exchange rate returns with the CnoFX (K, L) models is constrained to zero, but bond
risk premia in the domestic and foreign countries are not affected by this restriction.
The predictability of bond excess returns measured in R2 stays the same. The corresponding R2 and specification tests are provided in tables F.5 and F.6. In addition,
the free specification of the C(K, L) model does not improve the model beyond the
constrained CnoFX (K, L) model in terms of likelihood ratio tests. The test statistics
for the full C(3, 6) model and the dominant C(3, 4) model are reported in Table 7.10.
In conclusion, the foreign exchange risk premium specification does not influence
bond risk premia in these kind of models.
The uncovered interest rate parity (UIRP) relates the interest rate differential of
155
Table 7.11: Uncovered interest rate parity tests
Model expectations
Realized returns
β
β=0
β=1
R2
β
β=0
β=1
R2
USD-EUR model
-0.18
-3.91
-25.99
0.11
-0.20
-0.93
-5.66
0.01
USD-GBP model
0.22
4.34
-15.37
0.13
0.21
0.88
-3.30
0.01
EUR-GBP model
0.19
4.46
-18.95
0.14
0.20
1.17
-4.83
0.01
The table provides the results of regressing the quarterly foreign exchange model expectations and realized
returns onto the interest rate differential for the three two-country models indicated on the left. β is the
estimated regression coefficient on the interest rate differential; intercept statistics are omitted. β = 0 and
β = 1 provide the t-statistics for the corresponding null hypothesis H0 : β = 0 and H0 : β = 1, respectively.
the domestic and foreign short rates to the expected foreign exchange returns. The
UIRP assumes that the interest rate differential equals the appreciation or depreciation of the foreign exchange rate, which only holds if there is no foreign exchange
risk premium. The deviation of the UIRP is broadly covered by the literature.13
Table 7.11 provides a test of the UIRP with the model expectations of foreign exchange returns compared to the realized foreign exchange returns. The regressions
are specified such that the UIRP holds if the regression coefficient β is equal to one.
The corresponding t-statistics decisively reject the UIRP for both model expectations
and for realized returns. In the USD-EUR model, the coefficient is even significantly
negative for model expectations, whereas for the other two models the coefficient
is still positive. In the realized return setting, the coefficients are not significantly
different from zero and thus no statement about the sign is evident.
By their very nature, multi-currency models also induce bond risk premia for foreign investors. This is the premium a foreign investor receives by investing in bonds
denominated in a currency different from his reference currency. Without hedging
the exchange rate dynamics, the foreign investor’s risk premium decomposes into the
local bond risk premium and the foreign exchange risk premium. Figure 7.4 plots the
13 See
Fama (1984), Bekaert and Hodrick (1993) and Bekaert, Wei and Xing (2007). Engel (1996) pro-
vides an overview of the UIRP puzzle literature and Christiansen, Ranaldo and Söderlind (2010) present
an explanation of the UIRP puzzle.
156
Bond risk premia
0.4
0.2
0
Foreign investor
Domestic investor
−0.2
−0.4
1975
1980
1985
1990
1995
2000
2005
2010
2005
2010
Foreign exchange risk premia
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
Figure 7.4: International bond level risk premia
The figure shows the annual German level bond risk premium for ten-year maturity (rx(10) ) for an US
investor and its decomposition into the local bond risk premium and the foreign exchange risk premium.
bond risk premium of a ten-year German bond for a domestic (German) and foreign
(US) investor. The dependence on the foreign exchange risk premium is empirically
shown to increase the volatility of the position for a foreign investor compared to a
domestic one. The bottom plot of Figure 7.4 shows the foreign risk premium individually. The upward sloping trend in the second half of the time series accounts for the
devaluation of the US dollar against the euro. In summary, multi-currency models
provide a consistent framework, which serves any application where investments in
foreign bonds are considered alongside or in comparison to domestic bonds.
157
Comparison to Local Models
The advantage of modeling the two yield curves, domestic and foreign, in one framework is that the comovement of domestic and foreign risk premia is allowed. This
improves the forecasting ability of the model compared to the single-country case by
comparison of R2 . A formal test is provided in the following. Restricting the standard
C(K, L) model to only local state variables so that there is no comovement of domestic and foreign factors is represented in the CL (K, L) model. This model is nested
in the C(K, L) model but behaves like two single-country models. The comovement
of domestic and foreign state variables is restricted by forcing the upper-right and
lower-left blocks of the physical dynamics parameter Φ to zero. This is equivalent to constraining the same blocks of the risk compensation matrix Σλ1 to zero.
The predictability and the likelihood ratio statistics of these models are reported in
the appendix in tables F.7 and F.8, respectively. The specification tests imply the
same result as the single-country models: the CL (3, 4) models are dominant with
two priced risk factors in each country. The models are estimated with only an even
number of free risk premia L since the number of risk premia is divided equally in
the domestic and foreign parts of the model. Specifically, the CL (3, 6) model allows
three risk premia to be priced in each country. Since this model is nested in the
corresponding C(K, L) model, it allows likelihood ratio specification tests to be performed if the influence of the unspanned international factors is significant. These
tests are reported in Table 7.10 and demonstrate the dominance of the multi-currency
models C(K, L) over the local models CL (K, L). The comovement of state variables
across the two countries improves these models beyond the single-country models,
including in terms of predictability (measured in R2 ).14
Figure 7.5 compares the cumulative returns of a trading strategy applied on the
model-based risk premia forecasts. The strategy holds a long (short) position in the
US ten-year maturity bond for a quarter if the expected excess return of the model is
positive (negative). Although this strategy is applied based on in-sample estimated
14 R2
values of the CL (K, L) models are reported in the appendix in Table F.5.
158
1
10
Multi−currency model
Single−currency model
0
10
1980
1985
1990
1995
2000
2005
Figure 7.5: A simple trading rule application
The figure shows the cumulative excess return on a logarithmic scale of a trading strategy that goes
long/short in a US ten-year maturity bond (rx(10) ) according to the sign of the model implied expected
return for a quarterly holding period.
parameters, it indicates where on the time axis the additional foreign yield curve
information in the multi-currency model improves the level risk premium forecasts
beyond the single-country model. Interestingly, the foreign information helps in
forecasting bond returns in the second part of the sample, beginning around 1994
with further improvement towards the end of the sample in 2009. This is evidence
that foreign yield curve information has been increasingly important in the recent
data. This steadily increased influence could be explained with further integration
of capital markets between these countries. As a result, foreign yield curve information has become increasingly important as an information base for monetary policy
decisions.15
Comparison to Regression Forecasts
The two-country multi-currency model forecasts of the dominant models reported
in Table 7.4 are compared with the two-country forecasting regressions provided
in Table 6.9 of Chapter 6. The latter are the forecasting regressions, which only
include the first three principal components of each country’s yield curve as instru15 Mirkov
(2011) investigates the effect in the opposite direction: the impact of policy decisions onto
foreign yield curves.
7.3. CONCLUSION
159
ments.16 In this setting the number of instruments of the regressions corresponds
with the number of state variables in the multi-currency models. The two-country
model forecasts measured in R2 are only slightly lower than their regression alternatives. Only the UK forecasts are considerably lower than its regression analogs. The
model results are assumed to be more stable because of the additional restrictions
and structure applied and because the Kalman filtering algorithm reduces noise and
measurement errors. Since the OLS regression by definition maximizes the R2 of
the data, these findings support the multi-currency model being correctly specified
in order to incorporate the empirical facts discovered by the regressions in Chapter
6.
7.3
Conclusion
This chapter has introduced an international multi-currency term structure model,
which is fully symmetric and implements the domestic and foreign state variables as
mutually unspanned factors. Unspanned factors are able to improve bond risk premia estimation without affecting the yield curve. Since the unspanned factors in one
country are spanned by the yield curve of the other country and vice versa, all the
state variables are spanned by the model in the risk-neutral and physical distribution.
The price of risk is thus defined for any state variable. Empirical analysis reveals
that the number of priced bond risk premia is not reduced in a multi-currency model
compared to that in their single-country alternatives. An international common factor is therefore not supported. This has been anticipated by the low correlations of
risk premia across countries. However, the unspanned state variables do affect and
improve the predictability of the model. Domestic and foreign yield curve factors are
important in forecasting bond returns, while domestic yield curve factors still dominate the foreign ones in terms of explained variation. The multi-currency models
with comovement of domestic and foreign state variables outperform the local and
16 The
regressions using the principal components as instruments reduce the number of factors and thus
also the overfitting bias, which is anticipated due to the highly correlated instruments.
160
single-country models. Risk premia depend on international yield curve information
that is not spanned by the local yield curve. Vice versa, local yield curves do not
provide all the important information in forming bond return expectations. Furthermore, the foreign exchange risk premium is only poorly forecasted with yield curve
factors alone and the specification of the foreign exchange risk premium does not
influence the forecasting pattern of the bond risk premia in this model setting. An
improved foreign exchange risk premium forecast would therefore not alter the bond
risk premia predictability in the model.
The predictability of bond risk premia in single-country yields-only models, as
empirically analyzed for the US market in Chapter 3, is persistent in the German and
UK markets. The R2 of German bond return forecasting are comparable to the US
case in size, whereas these values drop sharply for the UK. Contrary to the results in
Chapter 3, the smaller sample size in the US case reduces the number of priced risk
premia to two.
Chapter 8
Conclusion
Overall, this thesis has advanced in three different fields of the actual term structure literature: yields-only, macro-finance and multi-currency models. In addition
to the theoretical aspects of model specification, several results of a broad empirical
analysis of bond risk premia are presented, which lead to an in-depth understanding of these bond risk premia. The main conclusions of this thesis are threefold.
Firstly, and contrary to most recent term structure literature, which finds only one or
two risk premia to be priced, this thesis has identified three linear independent and
priced risk premia. These risk premia are spanned by the yield curve and illustrated
by its predictability for level, slope and curvature portfolios. Secondly, bond risk
premia are shown to compensate for macroeconomic risks that are not spanned by
the yield curve. This means that contrary to the standard belief in term structure
modeling, the yield curve does not incorporate all relevant information that investors
consider when forming future expectations about bond yields. The cross-section of
the yield curve is still the only driver of risk-neutral pricing of bonds. However,
investors’ risk compensation also depends on information not contained in yields.
The unspanned macro-finance term structure model that is introduced is able to filter the unspanned part of macroeconomic information out of some macroeconomic
161
162
CHAPTER 8. CONCLUSION
time series and therefore captures these effects in bond risk premia. Thirdly, a new
approach to international term structure models shows that there is low support for
common international state variables, since the cross-section of yields is fully captured by latent local yield curve factors and risk premia show such high levels of
individual variation across countries that the number of risk premia is not reduced
in an international two-country framework. Foreign risk premia or state variables,
however, improve the bond return predictability in the domestic currency. Again,
although the foreign yield curve is not fully spanned by domestic yields, they have
an influence on investors’ future expectations of bond yields. The tools and models
given in this thesis allow bond risk premia (including these effects from unspanned
macroeconomic information and international yield curves) to be estimated.
The following ideas present some potential for future research along these kinds
of models. The macro-finance model specification could be improved by not only
filtering from principal components of some grouped data, but instead from a broad
range of individual and observable macroeconomic time series and letting the Kalman
filter choose how and what is filtered into the macroeconomic state variables. In the
same way, the model can easily be extended to additional information that does not
span the yield curve, but might be important in setting investors’ risk compensation,
such as different financial time series, e.g. a global equity risk premium. Additional
assets could be added to the model to properly identify these additional sources of
risk. In the case of macro time series, inflation linked securities could span and
price the inflation risk. Another approach could lead to adding investors’ expectations by including survey data, such as surveys about expected inflation1 or expected
interest rates2 , which serve as additional unspanned factors. An interesting but computationally more demanding task would be to merge both the macro-finance and
multi-currency models to include both effects at once. This could lead to further
improvements in bond risk premia forecasts and maybe even increase predictability
of the foreign exchange risk premium, which is quite low within yields-only mod1 Boos
(2011) proposes a model with expected inflation based on survey data as unspanned factor.
2 Kim and Orphanides (2005) propose a term structure model estimated on the basis of survey forecasts
of a short-term interest rate in addition to the standard yield data.
163
els. According to the purchasing power parity, the foreign exchange rate should
depend on the difference between the inflation rates of the two countries in the long
run. Adding macro factors to the international model should thus also have an effect
on long run exchange risk premium predictability.3 The same two-country multicurrency model could be used to model both the real and nominal yield curves of
one country in one model, with the unspanned feature applied as in the two-country
case. The price level can be thought of as an exchange rate between the nominal and
real yield curves as in Panigirtzoglou (2001).
3 Dong
(2006) concludes that macroeconomic data drives the foreign exchange risk premium.
164
CHAPTER 8. CONCLUSION
Appendices
165
Appendix A
Yield Curve Data
Throughout this thesis US government bond data, which is considered as default free
is used. Some different data sets are readily available. Common to all of them is that
the data is given as yields (or equivalently as zero bond prices or discount rates) with
fixed maturities. Firstly, yields are not directly observable in reality. These yields are
deducted by some technique from traded bonds. In the case of zero bonds, the conversion to yields is straightforward, but with coupon bonds and most traded bonds are
coupon bonds, it becomes more difficult. All future coupons up to expiration need to
be priced with an individual discount rate matching the coupon dates. In this case the
discount or yield curve serves as an input as well as an output. Secondly, yields and
zero bond prices are not observed on all maturities because there is a limited number
of traded bonds. The yield to a given maturity not directly observed in the market
must be approximated through other nearby bonds by means of some interpolation
method. Thirdly, bond prices are affected by measurement errors. Special situations
in which the price impact is not obviously captured by pricing methods distort the
resulting yield curve. This includes, but is not limited to, liquidity premia, hedging
demand and demand for deliverability into futures contracts. These are also some
true measurement errors because of bid-ask spreads, which do not quote the exact
167
168
APPENDIX A. YIELD CURVE DATA
mid-price and non-synchronous trading that arises due to the low trading activity in
some bonds.
A.1
Term Structure Estimation Methods
A variety of methods for extracting the yield curve out of coupon bond prices have
been proposed to handle these problems. Most of the term structure estimation methods used today are based either on the piecewise polynomial cubic spline method of
McCulloch (1971, 1975) or the parametric method of Nelson and Siegel (1987),
which fits a functional form with three terms for level, slope and curvature to the
data. These methods differ in the number of parameters and flexibility to fit the
curve. The spline-based methods fit a number of individual segments to the data
connected on an arbitrary number of knots. At these knots the splines need to follow some restrictions to smooth the curve; in most cases derivatives are set equal.
This method allows considerable flexibility in reproducing any shapes of the term
structure, but demands a high number of parameters to be estimated. The parametric
method fits a predefined functional form to the data. In contrast to the spline-based
method, the number of parameters is much lower and the yield curve much smoother,
because the entire term structure depends on the same function and parameters. The
drawback of this method is that the complexity of the term structure shapes is reduced dramatically. If the yield curve is deduced by a parametric function with three
parameters, the term structure model based on this data is obviously limited to this
number of state variables and it is not reasonable to retrieve more state variables
from the data to explain the cross-section. A further disadvantage of this method is
insufficient stability, as shown by Anderson and Sleath (2001). Changing a single
data point affects the entire term structure, whereas with spline-based methods the
impact is largely captured in one segment only. Some available data sets (mostly
from central banks) use the method proposed by Svensson (1994, 1995), which is
an extension to the Nelson-Siegel method that adds a fourth term (another curvature
term) to the functional form of the original model. Other data sets are constructed
A.2. UNSMOOTHED FAMA-BLISS METHOD
169
using smoothed spline methods as described in Fisher, Nychka and Zervos (1994)
and Waggoner (1997), an extension to the regression splines of McCulloch.
A.2
Unsmoothed Fama-Bliss Method
All the aforementioned methods describe the curve entirely or in parts using a set of
parameters. A different approach is provided by Fama and Bliss (1987) (known as
the unsmoothed Fama-Bliss method), which builds the yield curve iteratively. Bliss
(1996) compares different term structure estimation methods and concludes that the
unsmoothed Fama-Bliss method provides the best results overall. Cochrane and Piazzesi (2008) point out that data not smoothed across maturities as against smoothed
data preserves features important to forecasting bond excess returns. They conclude
that smoothing with the Svensson method removes some measurement errors along
with the forecasting signal. Dai, Singleton and Yang (2004) present a comprehensive
analysis of the implications of different term structure estimation methods in respect
of bond return predictability.1 The unsmoothed Fama-Bliss yield curves are very
close to equal to the smoothed yield curves in terms of linear dependence with R2
of 0.998 and higher. Nevertheless, the unsmoothed curve looks much choppier and
the additional predictability stems from principal components beyond the first three
(commonly known as level, slope and curvature). Predictability using the first three
principal components as instruments is similar across smoothed and unsmoothed
data sets.
There are two publicly available data sets2 deducted with the unsmoothed FamaBliss method, but both have their limitations either in time or maturity dimension. I
thus constructed data sets based on the unsmoothed Fama-Bliss method on my own.3
1 A comparison of the Svensson and spline methods for data sets other than the US is provided in Sekkel
(2011) (Canadian and British yield curves). The findings are in line with those in the US. Predictability is
lower for yields smoothed with the Svensson than with the spline method.
2 The description of these data sets follows in Section A.3.
3 Another benefit of constructing a partly overlapping data set in the US is the possibility of comparing
170
The method was first described in Fama and Bliss (1987) with additional comments
in Bliss (1996). A more technical reference is given with the monthly treasury US
database guide4 from the Center for Research in Security Prices (CRSP)5 , which
hosts the Fama-Bliss discount bonds file.
This method builds the yield curve iteratively by extracting forward rates from
bond prices. The extraction of forward rates starts at the bond with the shortest
maturity and continues with each successive maturity bond. Knowing the bond price
including accrued interest (the dirty price) the routine solves for the implied daily
forward rates with respect to the already constructed part of the forward curve and
the assumption that the forward rates between successive maturities of the available
bonds are constant. Coupons on bonds are discounted with the already constructed
forward curve matching the coupon payment date and are then subtracted from the
bond price. Coupons to be received in the time period where the forward curve
is not yet constructed are included in the bond price together with the principal.
A simple example should make it more obvious. In each step the routine solves
for the implied daily forward rates of one bond with maturity T with respect to the
already calculated forwards up to maturity t (t < T ) and the assumption that the daily
forward rates between t and T are all equal. Coupons of the bond to be received
prior to t are discounted with the available forward rates and subtracted from the
bond price. Coupons to be received between t and T together with the principal due
at T discounted with the daily forward rates, that are solved for, equate the price
of the bond. This is the basic routine that generates a step function for forward
rates and a smoother, but still serrated, discount rate curve that prices any included
bond without error by construction. The problem is that mispriced bonds or bonds
with other measurement errors force the forward curve into an oscillating pattern.
The major challenge of this method is to filter these bonds. This is achieved using
the yield curves to the existing data sets and thus verifying the term structure estimation routine. In
countries other than the US publicly available unsmoothed Fama-Bliss data sets do not yet exist and the
possibilities to double-check are therefore limited.
4 http://www.crsp.chicagogsb.edu/documentation/pdfs/monthly_treasury.pdf
5 A research center at the University of Chicago Graduate School of Business.
A.2. UNSMOOTHED FAMA-BLISS METHOD
171
a four-step filtering procedure. First of all, any bond with option features such as
callable and flower bonds are excluded anyway. Further bonds with special liquidity
situations, such as shortly issued bonds (on-the-run bonds pay a liquidity premium)
and bonds close to maturity (bonds with one year to maturity and bills with one
month to maturity are often illiquid), are not admissible for this specific date. The
remaining set of admissible bonds is then filtered in a four-step procedure as follows:
1. The first filter is based on yields to maturity. A bond is included if the difference of its yield to the average of the three preceding or subsequent bonds is
below 0.2% or the bond’s yield is between the two averages.
2. The second filter cleans up major yield reversals in the term structure by deleting the bonds that cause the reversal. A reversal is a sequence of changes in
the yield curve of more than 0.2% and of the opposite sign.
3. The third filter reviews the bonds excluded in steps one and two for possible
inclusion, based on the newly constructed yield curve. The inclusion criteria
are similar to the first filter, but are based on the yield curve including the bond
under inspection. A bond is included again if the difference of the yield to the
average of the three preceding or subsequent bonds is below 0.2% or if the
yield is between the two averages.
4. The fourth filter again cleans up major yield reversals in the same way as step
two, but with the bonds of step three included.
Additional refinements and how to deal with multiple issues of the same maturity are
not covered in the above description. However, these are not essential in understanding the main principle of this method.6
6 Additional
information can be obtained from the author directly upon request.
172
A.3
Data Sets
Publicly available data sets based on the aforementioned unsmoothed Fama-Bliss
estimation method comprise the data set provided with the original Fama and Bliss
(1987) paper (henceforth FB), and that provided with the Diebold and Li (2006)
paper (henceforth DL). The FB data set is available through the CRSP as the FamaBliss discount bonds file. It is based on monthly US treasury data from the CRSP,
which includes bond data back to 1925. Due to insufficient bonds, the FB data set
starts in 1952 and is annually update by the CRSP. The file includes monthly data
for annual maturities ranging from one to five years. The DL data set is basically an
extension of the FB data set in terms of maturities. It is based on the same CRSP
US treasury data and the same filtering programs. The data set is available as a
supplement to Diebold and Li (2006).7 The maturities are extended, ranging from
one month up to ten years, with quarterly spacing up to two years, annually thereafter. Because of the broader maturity dimension, the time dimension is limited to
a range with more bonds available, especially on the long end. The data set thus
ranges from 1970 to 2000 without any updates so far. The data set constructed along
with this thesis also uses the unsmoothed Fama-Bliss method, but based on routines
reproduced by the author. The underlying US bond data is taken from the CRSP and
Datastream8 . One data set was created solely based on the CRSP bond data in order
to test the routine by comparing the generated yield curves to the DL data set. As a
result, the two data sets are nearly identical, which is hard evidence of correct implementation of the estimation method. With the help of the Datastream bond data,
the yield curve is extended to a time window of monthly data from 1980 to 2009
providing any maturity up to ten years.
McCulloch and Kwon (1993) provide an alternative data set (henceforth McK),
derived using the McCulloch (1971, 1975) cubic spline estimation method. The McK
data set ranges from 1947 to 1991, with detailed maturities ranging from one month
7 http://www.ssc.upenn.edu/∼fdiebold/papers/paper49/FBFITTED.txt
8 Thomson
Reuters Datastream, a database vendor with proprietary bond data.
A.3. DATA SETS
173
up to 40 years on a monthly frequency. However, the very long maturities (over
20 years) are largely omitted due to the absence of corresponding bonds. The yield
curve before 1985 is calculated by relying heavily upon callable bonds, but without
any price adjustment for the embedded option.
The most current data set has been constructed by Gürkaynak, Sack and Wright
(2006) (henceforth GSW) and made available by the Federal Reserve Board9 providing periodical updates. The GSW yield curve is estimated using the Svensson
(1994, 1995) method with daily frequency and annual maturities from one to 30
years. The data set starts in 1961. The first part up to 1987 is based on the daily US
treasury data provided by CRSP. The second part up to the present rests upon data
collected by the Federal Reserve Bank of New York (FRBNY).
The choice of a data set is a trade off between a long time horizon that is favorable
in estimating risk premia, a broad cross-section across maturities and the estimation
method applied. The data set used throughout this thesis (henceforth CBD10 ) combines some of the aforementioned data sets in order to minimize the drawbacks of
each used individually. The DL data set basis is taken from 1970 to 2000 because
it is estimated with the Fama-Bliss method and provides a broad cross-section. The
extension up to 2009 is achieved using the data set constructed by the author as described above. The data set is extended back to 1947 using the McK data, which is
constructed using the cubic spline method. The FB and GSW data sets have not been
taken into consideration. In the case of FB this is because the data set is very limited in the cross-section in both frequency (only annually) and maturities on the long
end. Compared to quarterly frequency, the annual frequency reduces independent
observations by a factor of four and the missing long maturities omit some features
of the long end of the curve. However, the FB data set serves as a reference for comparing some of the analysis with the results shown in Cochrane and Piazzesi (2005)
and Fama and Bliss (1987). The omittance of the GSW data set is because of the
parametric method, which smooths the resulting data excessively. Still this data set
9 http://www.federalreserve.gov/econresdata/researchdata/feds200628.xls
10 Combined
Bond Data
174
might be useful and suitable for robustness checks.
Appendix B
Estimation of Term Structure
Models
Term structure models explain the yield curve based on a small set of state variables.
Since the cross-section of yields is much larger than the number of state variables,
the model faces stochastic singularity. This feature arises because the variancecovariance matrix of the yields is singular if the number of yields is greater than
the number of state variables (Piazzesi 2003). Stochastic singularity is a problem
since the model is easily rejected. The standard solution to resolve this singularity
problem is to add measurement error to the yield equation. Another problem is that
the state variables Xt are not observable, either because the aforementioned measurement errors exist or the state variables are purely statistical factors without clear
economic meaning. There are two prominent ways of solving this unobservability
associated with the estimation method applied thereafter. One is known as backingout state variables from yields, the other is known as Kalman filtering. For both
methods it is useful to write the affine model in state space form; this is conducted
next before describing the two estimation methods.
175
176
APPENDIX B. ESTIMATION OF TERM STRUCTURE MODELS
B.1 State Space Form
A state space system consists of two equations.1 The measurement equation links
observable data such as yields with the state variables. Normally distributed measurement errors ηt are added to break stochastic singularity.
Generally, measurement errors ηt are assumed to be iid normally distributed and
have a diagonal covariance matrix to reduce the number of parameters. In this model,
measurement errors are even assumed to be of the same size across maturities in
terms of volatility. The standard deviation of the measurement errors ηt is ση and
estimated jointly with the other parameters. This specification only adds one additional parameter to the estimation problem and the reduction in the dimension of the
measurement error specification does not appear to alter the results significantly.
In the case of yields, this is the pricing equation for yields (C.7) with added
measurement errors
(n)
yt
(n)
= Ayn + By⊤
n Xt + ηt .
(B.1)
The transition equation describes the dynamics of the state variables and is equal to
the VAR (3.1)
Xt+1 = µ + ΦXt + εt+1 .
(B.2)
In estimation the transition equation (B.2) equals the time series part and the measurement equation (B.1) the cross-section part. Parameter estimation in yield curve
models is often achieved using a (quasi-) maximum likelihood estimation (MLE)
approach, e.g. Duffee (2002). In the following, two approaches are presented, one
with backing-out state variables from yields and the other with filtering state variables from yields.
1 An
extensive treatment of state space models is given in Harvey (1981).
B.2. BACKING-OUT STATE VARIABLES FROM YIELDS
B.2
177
Backing-out State Variables from Yields
Backing-out the state variables from the yields by inverting the measurement equation (B.1) is only possible by relaxing the measurement assumption. As many yields
as there are state variables are assumed to be observed without measurement errors.
Based on these yields the state variables are retrieved and allow estimation of the
parameters with a combined approach of time series and cross-sectional MLE.2
Alternatively, the time series part can be estimated individually by simple ordinary least squares (OLS) on the transition equation (B.2), which leads to parameter
estimates of µ, Φ and Σ. In a second step, the cross-section estimation is performed
with an MLE on the measurement equation (B.1) by holding the parameters of the
time series estimation constant and thus estimating the remaining parameters λ0 ,
λ1 , δ0 and δ1 . To separate the time series from the cross-sectional estimation the
model must be self-consistent or at least the state variables must be economically
observable variables; both of these are fulfilled under the parametrization given in
this thesis. The advantage of this procedure is faster estimation in terms of computation time. However, it has the drawback that the interaction between the parameters
of the time series and cross-section is prevented.
B.3
Kalman Filtering
The approach of backing-out state variables from yields is unsatisfactory in two
ways. Firstly, the assumption of no measurement errors for some yields is not plausible and secondly, the selection of the yields observed without measurement errors is
arbitrary and therefore comes with a selection bias. The Kalman filter is an algorithm
for estimating a state space system with unobserved state variables and was first in2 A hands-on introduction to MLE with backing out the state variables from yields is given by Söderlind
(2007) including pure time series and cross-sectional MLE.
178
troduced by Kalman (1960).3 This estimation method thus allows state variables to
be filtered from yields without the no measurement assumption. The algorithm is a
recursive prediction-correction procedure, which is able to deal with fully unobservable state variables.
The recursive algorithm of the Kalman filter is directly applicable to the state
space system with the measurement equation (B.1) and transition equation (B.2),
and iteratively performs the following two steps:
1. The transition equation delivers a forecast X̂t+1|t of Xt+1 based on all available
information up to time t
X̂t+1|t = µ + ΦX̂t
(B.3)
The forecast is simply the expectation from the time series model applied recursively to the updated states from the previous step. In the first step the
forecast is based on an initial guess X0 .
2. As soon as the cross-section at time t + 1 is observed, the prediction of the
latent state variables X̂t+1|t is updated (or corrected) to X̂t+1 based on this new
cross-section information using the measurement equation.
ŷt+1|t
}|
{
z
−1
X̂t+1 = X̂t+1|t + Pt+1|t Bn Ft+1
X̂
−
A
(yt+1 − B⊤
n)
t+1|t
n
|
{z
}|
{z
}
Kalman gain
Ft+1 = B⊤
n Pt+1|t Bn + Ση
(B.4a)
measurement errors
(B.4b)
with the prediction error in the state variables Xt − X̂t|t−1 ∼ N(0, Pt|t−1 ) and
Ση as the covariance matrix of the measurement errors η. The prediction error
covariance matrix Pt|t−1 is iteratively estimated and based on an initial guess
P0 in the first iteration.
The updating step can be interpreted as a correction of the state variable based
on the cross-section information available and weighted by the Kalman gain.
3 A comprehensive treatment of the Kalman filter applied to financial time series is provided by Harvey
(1989).
B.3. KALMAN FILTERING
179
Kalman filter is the same as simple MLE or OLS when state variables are observed
without measurement errors, as shown by Boos (2011) and Joslin, Singleton and Zhu
(2011).4 In the case of yields affected by normally distributed measurement errors,
Kalman filtering is the efficient estimation method and solves the overidentification
problem in the cross-section by extracting the optimal latent state variables, whereby
it improves the accuracy of the state variables when estimated from a broad crosssection. The reason is that the explicit specification of measurement errors helps to
reveal the underlying common factors similar to a principal component analysis.
Numerical gradient methods converge very slowly in Kalman filter estimation
with many parameters, especially if the autoregressive state dynamics are close to the
unit root (Piazzesi 2003). There are two crucial ways to tackle this problem. Firstly,
analytical rather than numerical gradients make optimization more accurate, faster
and more reliable. The analytical gradients of the likelihood function of the Kalman
filter applied to affine term structure models have been derived by Boos (2011), who
provides also a detailed treatment of the exact algorithm. Throughout this thesis
the Kalman filter optimization routines of Boos (2011) are used. Secondly, feasible
initial values are crucial to increase convergence speed and to avoid local optima.
Since with observable state variables the OLS and MLE estimation is equal to the
Kalman filter, this will provide a good set of initial values. To increase the accuracy
of the OLS estimates in the time series regression, the standard parametrization is
rotated into the first K principal components, with K as the number of state variables.
The invariant rotation theory is given in Appendix D.1.
4 A comparison of MLE and Kalman filtering in terms of finite sample properties is provided by Duffee
and Stanton (2004). They find that Kalman filtering is an efficient method for estimating term structure
models, especially for rather complex specifications.
180
B.4 Specification Tests
Finally, the specification test used in this thesis is briefly introduced. To compare the
estimated models according to their specification of the number of factors and the
number of individual priced risk premia the likelihood ratio test is used. Econometric
textbooks, e.g. Verbeek (2004), provide the wide range of specification tests that are
available. The likelihood ratio test is chosen due to its simplicity and the fact that
all the likelihood values of the estimates are readily available. The likelihood ratio
test LR is simply computed based on the likelihood of the unrestricted model Lu and
constrained model Lc as
LR = 2 (log Lu − log Lc ) ,
(B.5)
which follows a χ2 distribution with degrees of freedom equal to the number of free
parameters under the null hypothesis (Verbeek 2004).
Appendix C
Valuation Formulas
C.1 Bond Prices
Bond pricing equations are recursively derived from the basic model. Extensive
derivation of all formulas is given in other studies, e.g. Cochrane and Piazzesi (2006,
(n)
2008). The log bond price pt
at time t with maturity n is given iteratively as a
(n−1)
function of the expected log bond price pt+1
one period later, discounted by the
expected pricing kernel
(n)
pt
i
h
(n−1)
.
= log Et Mt+1 exp pt+1
(C.1)
(0)
Iterating Equation C.1 up to the maturity n of the bond and assuming1 that pt
=0
leads to
(n)
pt
1 The
= log Et [Mt+1 . . . Mt+n ] .
(C.2)
price of a risk-free zero bond at maturity by definition equals the nominal value of that bond.
Without loss of generality all bonds can be normalized to a nominal value of one; the log bond price at
(0)
maturity pt
is then zero.
181
182
APPENDIX C. VALUATION FORMULAS
For the one-period bond this is
(1)
pt
= log Et [Mt+1 ] = −δ0 − δ⊤
1 Xt ,
(C.3)
which is non-stochastic because it only depends on Xt , which is known at time t.
(1)
This also gives the one-period yield yt
(1)
yt
from Equation (2.2) as
= δ0 + δ⊤
1 Xt ,
(C.4)
which is the short rate of the model and thus also deterministic.
Calculating further log bond prices is achieved by inserting the pricing kernel
Mt+1 into Equation (C.2). The log bond prices are affine in the state variables
(n)
pt
= An + B⊤
n Xt
(C.5)
and the coefficients An and Bn follow the recursion
A0 = 0
(C.6a)
B0 = 0
(C.6b)
1 ⊤
∗
An = −δ0 + An−1 + B⊤
n−1 µ + Bn−1 ΣBn−1
2
⊤
⊤
∗
⊤
∗n
∗ −1
B⊤
n = −δ1 + Bn−1 Φ = −δ1 (I − Φ )(I − Φ )
(C.6c)
(C.6d)
with the parameters µ∗ and Φ∗ from the risk-neutral VAR in Equation (3.3). The
algebra is provided in detail in other studies, e.g. Cochrane and Piazzesi (2006).
C.2 Yields and Forward Rates
The equation for yields is affine in the state variables along the lines of the bond
price formula
(n)
yt
= Ayn + By⊤
n Xt
(C.7)
C.2. YIELDS AND FORWARD RATES
y
183
y
with coefficients An and Bn easily derived according to the definition of yields in
Equation (2.2) and the coefficients of the bond price formula (C.6) as
An
n
Bn
y
Bn = − .
n
Ayn = −
(C.8a)
(C.8b)
Plugging in the coefficients An and Bn of (C.6) yields
y
A0 = 0
(C.9a)
y
B0
(C.9b)
=0
n y⊤
δ0
y
y⊤
y
− An−1 − Bn−1 µ∗ − Bn−1 ΣBn−1
n
2
δ⊤
y⊤
1
By⊤
− Bn−1 Φ∗ .
n =
n
Ayn =
(C.9c)
(C.9d)
The same is true of forward rates, which are also affine in the state variables
(n)
ft
f
= Anf + Bnf ⊤ Xt
(C.10)
f
and the coefficients An and Bn are derived from the definition of forward rates in
Equation (2.3) in combination with the coefficients of the bond price formula (C.6)
as
Anf = An−1 − An
(C.11a)
Bnf
(C.11b)
= Bn−1 − Bn .
Again, plugging in the coefficients from (C.6) yields
f
(C.12a)
f
(C.12b)
A0 = 0
B0 = 0
1 ⊤
∗
Anf = δ0 − B⊤
n−1 µ − Bn−1 ΣBn−1
2
f⊤
f⊤
∗
⊤ ∗n−1
Bn = Bn−1 Φ = δ1 Φ
.
(C.12c)
(C.12d)
184
C.3
Forecast of Prices, Yields and Forward Rates
The above formulas for prices, yields and forwards span the cross-section at time t
only. They are thus deterministic as a function of the current state variables. The
model is, of course, also able to produce future prices, yields and forwards by iterating forward along the time series dimension and taking expectations. Expected
(n)
k-step-ahead prices Et [pt+k ] are affine in the expected future state variables Et [Xt+k ]
with the already known coefficients An and Bn from (C.6)
i
h
(n)
Et pt+k = An + B⊤
n Et [Xt+k ] .
(C.13)
The formula is solved by iterating the time series of the expected state variables back
to Xt along the VAR of Equation (3.1) with
Et [Xt+k ] = (I − Φk )(I − Φ)−1 µ + Φk Xt .
The coefficients Ak,n and Bk,n of the affine price formula
h
i
(n)
Et pt+k = Ak,n + B⊤
k,n Xt
(C.14)
(C.15)
are then given as
k
−1
Ak,n = An + B⊤
n (I − Φ )(I − Φ) µ
(C.16a)
B⊤
k,n
(C.16b)
=
k
B⊤
nΦ .
The coefficients for expected k-step-ahead yields and forward rates are derived in the
same way. For yields the formula and corresponding coefficients are
h
i
(n)
y
y⊤
Et yt+k = Ak,n + Bk,n Xt
y
(C.17a)
k
−1
Ak,n = Ayn + By⊤
n (I − Φ )(I − Φ) µ
(C.17b)
y⊤
Bk,n
(C.17c)
k
= By⊤
n Φ
and for forward rates they are
i
h
(n)
f
f⊤
Et ft+k = Ak,n + Bk,n Xt
f
Ak,n = Anf + Bnf ⊤ (I − Φk )(I − Φ)−1 µ
f⊤
Bk,n = Bnf ⊤ Φk .
(C.18a)
(C.18b)
(C.18c)
C.4. EXPECTED RETURNS
C.4
185
Expected Returns
The equations for return forecasts are derived from the difference of price forecasts
given above in Equation (C.15). Since there are a multitude of different returns
to calculate along the time series dimension, the general formula is given in C.5
(n)
below. Following the formula for 1-step-ahead expected returns Et [rt+1 ] is provided.
The expected return is derived from the definition of the return in (2.4) by taking
expectations
h
i
h
i
(n)
(n−1)
(n)
Et rt+1 = Et pt+1 − pt .
(C.19)
Plugging in from (C.15) and (C.5) yields
i
h
(n)
⊤
⊤
Et rt+1 = An−1 + B⊤
n−1 µ + Bn−1 ΦXt − An − Bn Xt .
(C.20)
Returns are affine in today’s state variables Xt
i
h
(n)
Et rt+1 = Arn + Br⊤
n Xt
(C.21)
with coefficients
Arn = An−1 − An + B⊤
n−1 µ
1 ⊤
= δ0 − B⊤
n−1 Σλ0 − Bn−1 ΣBn−1
2
r⊤
⊤
⊤
Bn = Bn−1 Φ − Bn
(C.22a)
⊤
= δ⊤
1 + Bn−1 Σλ1 .
(C.22b)
(1)
Expected excess returns are retrieved by subtracting the risk-free rate yt = δ0 +δ1 Xt
from the expected returns.
C.5 Multi-Period Expected Returns
In the following the general expected return formula, which leads to the expected
q-period k-step-ahead returns is given. To achieve this, the definition of the return
186
from 2.4 needs to be adjusted. The log holding period return from buying an n-period
bond at time t + k − q and selling it as an n − q period bond at time t + k is denoted
by
(n)
(n−q)
(n)
rt+k−q→t+k = pt+k − pt+k−q .
(C.23)
Therefore q is the holding period and k is the time when the returns are realized
relative to today’s time t. For expected returns the definition reads
h
i
h
i
h
i
(n)
(n−q)
(n)
Et rt+k−q→t+k = Et pt+k − Et pt+k−q .
(C.24)
Expected returns are affine in the state variables Xt
i
h
(n)
Et rt+k−q→t+k = Ark,q,n + Br⊤
k,q,n Xt
(C.25)
and the coefficients Ark,q,n and Brk,q,n are given as
k
⊤
k−q
Ark,q,n = An−q − An + (B⊤
))(I − Φ)−1 µ
n−q (I − Φ ) − Bn (I − Φ
Br⊤
k,q,n
k−q
q
⊤
= (B⊤
n−q Φ − Bn )Φ
(C.26a)
(C.26b)
The 1-period returns derived in the text are simply a special case of this general
formula with q = k = 1. Expected excess returns are derived by subtracting the riskfree yield.
i
h
h
i
(n)
(q)
Et rt+k−q→t+k = Ark,q,n + Br⊤
X
−
E
y
t
t
k,q,n
t+k−q
rx⊤
= Arx
k,q,n + Bk,q,n Xt
(C.27a)
(C.27b)
rx
The coefficients Arx
k,q,n and Bk,q,n are given as
Arx
k,q,n = An−q − An
k−q
y⊤
k
⊤
))(I − Φ)−1 µ
+ (B⊤
n−q (I − Φ ) − (Bn + Bq )(I − Φ
⊤
q
⊤
y⊤
k−q
Brx⊤
k,q,n = (Bn−q Φ − Bn − Bq )Φ
(C.28a)
(C.28b)
Appendix D
Yields-Only Models
D.1 Invariant Rotation
The affine transformation of a state space Xt of an affine term structure model into
its transformation Xtτ is
Xtτ = AXt + B
(D.1)
where A is a quadratic matrix and B a vector, both matching the dimension of the state
space. Obviously, the transformation is invariant with A as an identity matrix and B
is equal to zero. The reverse transformation is given with Aτ = A−1 and Bτ = −A−1 B.
187
188
APPENDIX D. YIELDS-ONLY MODELS
The corresponding transformations of the parameters are as follows
δτ0
=
−1
δ0 − δ⊤
1A B
(D.2)
δτ1
=
A−1 δ1
(D.3)
τ
Φ
=
AΦA
µτ
=
Aµ + B − AΦA−1 B
(D.4)
=
AΦ A
(D.6)
µ∗τ
=
Aµ∗ + B − AΦ∗ A−1 B
(D.7)
λτ0
= (A−1 )⊤ λ0 − (A−1 )⊤ λ1 A−1 B
λτ1
∗ −1
(D.5)
∗τ
Φ
D.2
−1
−1 ⊤
= (A ) λ1 A
−1
(D.8)
(D.9)
Other Choices for Self-Consistent Models
Here two other obvious choices for self-consistent models are derived with parameter
restrictions: one for log bond prices and one for yields.
D.2.1
Bond Prices as State Variables
The state variables are defined as the first K log bond prices
(k)
Xt,k = pt
(D.10)
where K is the number of state variables and k = 1 . . . K. The restrictions on the
parameters Φ∗ , µ∗ , δ0 and δ1 are deducted by solving the log bond price equation
(n)
pt
= An + B⊤
n pt for n = 1 . . . K. Obviously, this gives An = 0 and Bn = en . Thus for
D.2. OTHER CHOICES FOR SELF-CONSISTENT MODELS
189
n = 1 we get
δ0 = 0

(D.11a)

−1


 0 


δ1 =  . 
 .. 


0
(D.11b)
and for n = 2 . . . K the following restriction:
⊤
⊤
∗
e⊤
n = e1 + en−1 Φ
(D.12a)
1 ⊤
∗
0 = e⊤
n µ + en Σen
2
(D.12b)
Solving for Φ∗ and µ∗ gives

−1
 .
 .
 .
Φ∗ = 
 −1

c1

···
..
.
1
0
c2
Σ11
 .

1  ..
µ∗ = − 
2 Σ
 KK
cK+1
D.2.2
···

0





1 

cK
(D.13a)






(D.13b)
Yields as State Variables
The state variables are defined as the first K yields
(k)
Xt,k = yt
(D.14)
where K is the number of state variables and k = 1 . . . K. The restrictions on the
(n)
parameters Φ∗ , µ∗ , δ0 and δ1 are deducted by solving the yield equation yt = − Ann −
190
B⊤
n
n yt
for n = 1 . . . K. Obviously this gives An = 0 and Bn = −nen . Thus for n = 1 we
get
δ0 = 0




δ1 = 


(D.15a)

1






0
..
.
0
and for n = 2 . . . K the following restriction
(D.15b)
⊤
⊤
∗
ne⊤
n = e1 + (n − 1)en−1 Φ
n
en µ∗ = e⊤
Σen
2 n
(D.16a)
(D.16b)
Solving for Φ∗ and µ∗ gives

−1 2
 .
 .
 .
Φ∗ = 
 −1 0
 K
c1 c2

Σ11

..

1
.
µ∗ = 
2  KΣ
KK

cK+1
D.3
···
..
.
0
K+1
K
···







cK







(D.17a)
(D.17b)
Parameter Estimates
This section provides the parameter estimates of the dominant A(4, 3) model. The
parameters δ0 and δ1 are not estimated but predefined by the parametrization. The
estimated parameters are given in Table D.1, whereas the physical VAR parameter
µ and Φ are implicitly estimated, as they are a combination of the risk-neutral VAR
D.3. PARAMETER ESTIMATES
191
parameters µ∗ and Φ∗ and the price of risk parameters λ0 and λ1 . The covariance
matrix of the VAR shocks is estimated in its Cholesky decomposition to satisfy the
symmetry and positive semi-definiteness conditions.
The eigenvalues of Φ and Φ∗ indicate the stability of the corresponding VAR. For
the estimated parameters, all the eigenvalues lie inside the unit circle with maximum
values of 0.978 and 0.996 for Φ and Φ∗ , respectively.
The standard deviations of the VAR shocks are between 0.22% and 0.25%; the
correlation between them is high, with values between 0.82 and 0.99, as expected of
forward rates.
Table D.1: Parameter estimates of the A(4, 3) model
µ∗⊤
0
0
0
2.63E-07
Φ∗
0
1
0
0
0
0
1
0
0
0
0
1
-0.31
1.75
-3.57
3.12
-6.03E-05
3.38E-04
4.11E-04
3.74E-04
-0.68
6.50
-9.30
4.43
-0.30
3.73
-5.21
2.75
-0.36
3.85
-6.06
3.56
-0.83
6.13
-10.19
5.88
λ⊤
0
-1.68E+04
7.67E+04
-1.11E+05
5.16E+04
λ1
-8.42E+07
3.57E+08
-4.97E+08
2.24E+08
3.90E+08
-1.65E+09
2.30E+09
-1.03E+09
-5.68E+08
2.40E+09
-3.34E+09
1.51E+09
2.64E+08
-1.11E+09
1.55E+09
-6.98E+08
6.05E-06
5.58E-06
5.01E-06
4.47E-06
5.58E-06
5.96E-06
5.67E-06
5.17E-06
5.01E-06
5.67E-06
5.53E-06
5.13E-06
4.47E-06
5.17E-06
5.13E-06
4.83E-06
µ⊤
Φ
Σ
ση
7.40E-04
192
D.4
Measurement Errors
Measurement errors η are significantly autocorrelated of first order. Autocorrelation
in the errors is a serious problem since it could be a sign of omitted state variables or
incorrect functional form assumptions. Measurement errors of affine term structure
models are typically highly autocorrelated (Piazzesi 2003). The measurement errors
do not appear to result from interpolation methods, or other data construction methods because traded yields such as swap yields are still highly autocorrelated (Duffie
and Singleton 1997).
Figure D.1 shows the standard deviation of the measurement errors in the time
series and cross-section. There is a large reduction in measurement errors from the
three-factor to the four-factor model. The further reduction of the five-factor model is
relatively small (not reported). In the time series there is a phase of higher measurement errors in the seventies and eighties, and again in recent history starting around
2008. In the cross-section the reduction from the three- to the four-factor model is
slightly higher for the short maturities, but there is no clear pattern.
D.5
Forecasting Power of Additional Models
This section provides the forecasting power of additional semi-annual (Table D.2)
and annual models (Table D.3) discussed in the text as well as the corresponding
likelihood ratio tests (tables D.4 and D.5) on the number of risk premia.
D.5. FORECASTING POWER OF ADDITIONAL MODELS
−4
14
193
Standard deviation in the cross−section
x 10
model A(4,3)
model A(3,2)
12
10
8
6
4
0.25 0.5 0.75 1
−3
3
1.25 1.5 1.75 2
2.5 3
4
5
6
7
8
9
10
Standard deviation in the time series
x 10
model A(4,3)
model A(3,2)
2.5
2
1.5
1
0.5
0
1950
1960
1970
1980
1990
2000
Figure D.1: Standard deviation of measurement errors
2010
194
Table D.2: Forecasting power of the semi-annual models
log likelihood
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
A(3, 0)
7626.48
14
0.00
0.00
0.00
0.00
A(3, 1)
7635.94
19
0.17
0.23
0.06
0.19
A(3, 2)
7638.21
22
0.19
0.23
0.10
0.20
A(3, 3)
7638.21
23
0.19
0.23
0.10
0.20
A(4, 0)
7714.65
20
0.00
0.00
0.00
0.00
A(4, 1)
7728.65
27
0.13
0.23
0.12
0.19
A(4, 2)
7737.35
32
0.16
0.23
0.19
0.21
A(4, 3)
7741.09
35
0.19
0.23
0.23
0.21
A(4, 4)
7741.10
36
0.19
0.23
0.23
0.21
A(5, 0)
7733.39
27
0.00
0.00
0.00
0.00
A(5, 1)
7763.17
36
0.01
-0.01
0.03
0.06
A(5, 2)
7776.98
43
0.10
0.24
0.48
0.18
A(5, 3)
7785.77
48
0.18
0.29
0.46
0.20
A(5, 4)
7786.67
51
0.19
0.29
0.47
0.22
A(5, 5)
7786.67
52
0.19
0.30
0.48
0.22
The table shows the R2 of a one-year forecast based on the semi-annual models, the number of parameters
(“nvar”) and the estimated maximum log likelihood.
195
Table D.3: Forecasting power of the annual models
log likelihood
nvar
rx(2)
rx(10)
srx2,10
crx2,5,10
A(3, 0)
2832.62
14
0.00
0.00
0.00
0.00
A(3, 1)
2840.04
19
0.19
0.18
0.00
0.22
A(3, 2)
2842.48
22
0.21
0.18
0.07
0.22
A(3, 3)
2842.48
23
0.21
0.18
0.07
0.22
A(4, 0)
2838.75
20
0.00
0.00
0.00
0.00
A(4, 1)
2855.62
27
0.16
0.17
0.02
0.21
A(4, 2)
2879.59
32
0.18
0.18
0.51
0.22
A(4, 3)
2882.01
35
0.22
0.22
0.51
0.24
A(4, 4)
2882.01
36
0.22
0.22
0.51
0.24
A(5, 0)
2840.62
27
0.00
0.00
0.00
0.00
A(5, 1)
2901.68
36
-0.01
0.13
0.38
0.01
A(5, 2)
2924.10
43
0.25
0.44
0.48
0.31
A(5, 3)
2927.12
48
0.31
0.48
0.50
0.34
A(5, 4)
2927.52
51
0.31
0.49
0.50
0.34
A(5, 5)
2929.48
52
0.33
0.33
0.50
0.24
The table shows the R2 of a one-year forecast based on the annual models, the number of parameters
(“nvar”) and the estimated maximum log likelihood.
196
Table D.4: Likelihood ratio tests of the semi-annual models
A(3, 0)
A(3, 1)
A(3, 1)
18.9
(15.1)
A(3, 2)
23.5
(20.1)
4.5
(11.3)
A(3, 3)
23.5
(21.7)
4.5
(13.3)
A(4, 0)
A(4, 1)
A(3, 2)
0.0
(6.6)
A(4, 2)
A(4, 1)
28.0
(18.5)
A(4, 2)
45.4
(26.2)
17.4
(15.1)
A(4, 3)
52.9
(30.6)
24.9
(20.1)
7.5
(11.3)
A(4, 4)
52.9
(32.0)
24.9
(21.7)
7.5
(13.3)
A(5, 0)
A(5, 1)
59.6
A(5, 1)
A(5, 2)
A(4, 3)
0.0
(6.6)
A(5, 3)
A(5, 4)
(21.7)
A(5, 2)
87.2
(32.0)
27.6
(18.5)
A(5, 3)
104.8
(38.9)
45.2
(26.2)
17.6
(15.1)
A(5, 4)
106.6
(43.0)
47.0
(30.6)
19.4
(20.1)
1.8
(11.3)
A(5, 5)
106.6
(44.3)
47.0
(32.0)
19.4
(21.7)
1.8
(13.3)
0.0
(6.6)
likelihood.
197
Table D.5: Likelihood ratio tests of the annual models
A(3, 0)
A(3, 1)
A(3, 1)
14.8
(15.1)
A(3, 2)
19.7
(20.1)
4.9
(11.3)
A(3, 3)
19.7
(21.7)
4.9
(13.3)
A(4, 0)
A(4, 1)
A(3, 2)
0.0
(6.6)
A(4, 2)
A(4, 1)
33.8
(18.5)
A(4, 2)
81.7
(26.2)
47.9
(15.1)
A(4, 3)
86.5
(30.6)
52.8
(20.1)
4.8
(11.3)
A(4, 4)
86.5
(32.0)
52.8
(21.7)
4.8
(13.3)
A(5, 0)
A(5, 1)
A(5, 2)
A(4, 3)
0.0
(6.6)
A(5, 3)
A(5, 1)
122.1
(21.7)
A(5, 2)
167.0
(32.0)
44.8
(18.5)
A(5, 3)
173.0
(38.9)
50.9
(26.2)
6.0
(15.1)
A(5, 4)
173.8
(43.0)
51.7
(30.6)
6.8
(20.1)
0.8
(11.3)
A(5, 5)
177.7
(44.3)
55.6
(32.0)
10.8
(21.7)
4.7
(13.3)
A(5, 4)
3.9
(6.6)
likelihood.
198
Appendix E
E.1
Yield Curve Fit
Since the feature of the unspanned macro model is to only affect expected returns
and not the cross-section of actual yields, the fit of the yield curve should be equal in
the macro-finance and yields-only models. The fit of the term structure is evaluated
as the root mean squared error (RMSE) and absolute mean error (MAE) provided in
Table E.1. This actually confirms that the fit is not improved by including unspanned
Table E.1: Yield curve fit of the dominant models
B(3, 2, 2)
B(4, 2, 3)
A(3, 2)
A(4, 3)
MAE
0.057%
0.041%
0.058%
0.041%
RMSE
0.088%
0.069%
0.088%
0.069%
The table shows the mean absolute error (MAE) and root mean squared error (RMSE) of the given models
for the fit of the actual yield curve measured in annual yields form one to ten years.
macro factors and that the chosen parametrization really only improves expected
returns. This is an empirical check to ascertain whether the specification of the
199
200
APPENDIX E. MACRO-FINANCE MODELS
unspanned model is reacting according to the presumptions.
E.2 Parameter Estimates
This section provides the parameter estimates of the dominant B(4, 2, 3) model. The
parameters δ0 and δ1 are not estimated but predefined by the parametrization. The
estimated parameters are given in Table E.2, whereas the physical VAR parameter
µ and Φ are implicitly estimated, as they are a combination of the risk-neutral VAR
parameters µ∗ and Φ∗ and the price of risk parameters λ0 and λ1 . The covariance
matrix of the VAR shocks is estimated in its Cholesky decomposition to satisfy the
symmetry and positive semi-definiteness conditions.
The eigenvalues of Φ and Φ∗ indicate the stability of the corresponding VAR. For
the estimated parameters, all the eigenvalues lie inside the unit circle with maximum
values of 0.967 and 0.997 for Φ and Φ∗ , respectively.
E.2. PARAMETER ESTIMATES
201
Table E.2: Parameter estimates of the B(4, 2, 3) model
µ∗⊤
0
0
0
2.88E-07
-0.52
-0.59
Φ∗
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
-0.30
1.72
-3.53
3.11
0
0
0
0
0
0
0.87
0
0
0
0
0
0
0.14
µ⊤
-0.36
-0.32
-0.28
-0.26
-0.52
-0.59
Φ
-1.86
10.59
-13.32
5.55
1.69E-03
-0.53
-1.81
9.32
-11.50
4.97
1.44E-03
-0.48
-1.69
8.71
-11.45
5.42
1.25E-03
-0.43
-1.83
9.62
-13.77
6.97
1.09E-03
-0.38
0
0
0
0
0.87
0
0
0
0
0
0
0.14
-12.84
-20.04
1
27.14
A⊤
m
29.20
0.70
B⊤
m
-16.73
46.37
-25.93
122.18
-169.11
73.91
0.01
1.00
λ⊤
0
5.94E+08
-2.64E+09
3.20E+09
-1.27E+09
1.60E+06
2.49E+08
λ1
Σ
2.83E+09
-1.50E+10
2.02E+10
-7.99E+09
-2.62E+06
8.88E+08
-1.26E+10
6.64E+10
-8.95E+10
3.55E+10
1.16E+07
-3.94E+09
1.51E+10
-8.00E+10
1.08E+11
-4.27E+10
-1.41E+07
4.78E+09
-5.99E+09
3.17E+10
-4.28E+10
1.69E+10
5.60E+06
-1.91E+09
7.83E+06
-4.12E+07
5.56E+07
-2.21E+07
-7.09E+03
2.39E+06
1.22E+09
-6.41E+09
8.65E+09
-3.44E+09
-1.10E+06
3.72E+08
4.98E-06
4.64E-06
4.18E-06
3.73E-06
-7.17E-05
3.23E-06
4.64E-06
5.15E-06
4.95E-06
4.53E-06
-7.83E-05
3.71E-06
4.18E-06
4.95E-06
4.89E-06
4.56E-06
-6.38E-05
3.45E-06
3.73E-06
4.53E-06
4.56E-06
4.31E-06
-4.50E-05
2.99E-06
-7.17E-05
-7.83E-05
-6.38E-05
-4.50E-05
2.05E-02
-2.02E-04
3.23E-06
3.71E-06
3.45E-06
2.99E-06
-2.02E-04
3.93E-06
ση
7.41E-04
σ⊤
υ
4.61E-06
1.13E-02
202
APPENDIX E. MACRO-FINANCE MODELS
Appendix F
F.1
Exchange Rate Return Forecasts
Exchange rate return forecasts from the multi-currency model described in Chapter
7 are derived based on the definition of the exchange rate dynamics in Equation (7.4)
as
Et [st+1 ] − st = As + B⊤
s Xt .
(F.1)
Future expected returns in are analogously
Et [st+1+k ] − st+k = As + B⊤
s Et [Xt+k ] ,
(F.2)
whereas Et [Xt+k ] is iterated backwards in t to obtain a term that is a function of Xt
by using the state dynamics of (7.3), which yields
Et [st+1+k ] − st+k =
As + B⊤
s
203
k
Φ −1
k
µ
+ Φ Xt .
Φ−1
(F.3)
204
APPENDIX F. MULTI-CURRENCY MODELS
Multi-period expected returns are simply the sum of these individual returns:
i
k Φ −1
i
+
Φ
X
(F.4a)
µ
Et [st+1+k ] − st = ∑ As + B⊤
t
s
Φ−1
i=1
Φ(Φk − 1)
kµ
Φk − 1
⊤
= kAs + Bs µ
−
+
Xt .
(F.4b)
(Φ − 1)2
Φ−1 Φ−1
F.2
Variance Decomposition
The forecast variance of any instrument linear in the state variables Xt is decomposed
as follows, according to the variance decomposition given in Hamilton (1994). The
k-period-ahead forecast error of the VAR is given as
Xt+k − Xbt+k|t =
k−1
∑ Φi εt+k−i
(F.5)
i=0
and easily extended to any instrument that is affine in these state variables by multiplying by the B, which controls the dependence of this instrument with the state
variables. The calculation of these B for the different instruments is provided by the
valuation formulas in Appendix C. The mean squared error MSE of the forecast is
then calculated as
k−1
MSE =
∑ B⊤ Φi ΣΦi⊤ B,
(F.6)
i=0
which can be further decomposed along the K state variables into
K k−1
MSE =
∑ ∑ B⊤ ΦiC jC⊤j Φi⊤ B
(F.7)
j=1 i=0
with P as the upper-triangular matrix of a Cholesky decomposition of Σ, such that
PP⊤ = Σ and Pj indicates the jth column in the matrix P. The relative contribution
of each of the K state variables is then given for the jth state variable as
k−1
i⊤
MSE −1 ∑ B⊤ ΦiC jC⊤
j Φ B.
i=0
(F.8)
F.3. ADDITIONAL TABLES AND FIGURES
F.3
205
Additional Tables and Figures
Table F.1: Forecasting power of the quarterly four-factor single-country models
log likelihood
rx(2)
nvar
rx(10)
srx2,10
crx2,5,10
0.00
0.00
0.00
0.00
A(4, 0)
11645.7
20
A(4, 1)
11674.7
27
0.04
-0.11
-0.16
0.22
A(4, 2)
11696.9
32
0.03
0.04
0.09
0.22
A(4, 3)
11699.1
35
0.04
0.04
0.10
0.22
A(4, 4)
11699.7
36
0.04
0.07
0.12
0.25
A(4, 0)
11416.4
20
0.00
0.00
0.00
0.00
A(4, 1)
11430.6
27
0.16
0.14
-0.02
0.06
Germany 1976:12 - 2009:06
A(4, 2)
11437.1
32
0.15
0.11
0.05
0.11
A(4, 3)
11441.0
35
0.16
0.11
0.09
0.16
A(4, 4)
11441.0
36
0.16
0.11
0.09
0.16
A(4, 0)
11510.6
20
0.00
0.00
0.00
0.00
A(4, 1)
11524.5
27
-0.01
0.01
0.03
0.01
A(4, 2)
11532.6
32
0.06
0.15
0.17
0.12
United States 1976:12 - 2009:06
A(4, 3)
11536.4
35
0.12
0.18
0.18
0.19
A(4, 4)
11536.7
36
0.12
0.18
0.17
0.19
The table shows the R2 of a one-year forecast based on the quarterly four-factor models, the number of
206
Table F.2: Likelihood ratio tests of the quarterly single-country models
A(4, 0)
A(4, 1)
A(4, 2)
A(4, 1)
58.1
(18.5)
A(4, 2)
102.5
(26.2)
44.4
(15.1)
A(4, 3)
106.8
(30.6)
48.6
(20.1)
4.2
(11.3)
A(4, 4)
108.0
(32.0)
49.9
(21.7)
5.5
(13.3)
A(4, 3)
1.3
(6.6)
Germany 1976:12 - 2009:06
A(4, 0)
A(4, 1)
A(4, 2)
A(4, 1)
28.4
(18.5)
A(4, 2)
41.6
(26.2)
13.2
(15.1)
A(4, 3)
49.3
(30.6)
20.8
(20.1)
7.7
(11.3)
A(4, 4)
49.3
(32.0)
20.8
(21.7)
7.7
(13.3)
A(4, 3)
0.0
(6.6)
United States 1976:12 - 2009:06
A(4, 0)
A(4, 1)
A(4, 2)
A(4, 1)
27.9
(18.5)
A(4, 2)
44.0
(26.2)
16.1
(15.1)
A(4, 3)
51.5
(30.6)
23.6
(20.1)
7.6
(11.3)
A(4, 4)
52.2
(32.0)
24.3
(21.7)
8.2
(13.3)
A(4, 3)
0.7
(6.6)
likelihood.
rx(3)
rx(4)
rx(5)
rx(6)
rx(7)
rx(8)
-0.04
US curvature
3.38
-0.01
-2.27
-1.13
2.02
1.43
3.16
0.26
-2.46
-0.96
2.08
1.03
2.96
0.38
-2.68
-0.86
2.16
0.71
2.79
0.62
-2.79
-0.74
2.25
0.46
2.59
0.83
-2.88
-0.67
2.28
0.19
2.40
1.07
-2.95
-0.62
2.27
-0.01
-3.34
0.72
US slope
US curvature
-3.83
0.54
-3.33
-1.83
1.27
-3.02
-3.80
0.52
-3.19
-2.08
1.31
-2.87
-3.95
0.58
-2.77
-2.41
1.60
-2.54
-3.78
0.41
-2.91
-2.31
1.56
-2.45
-3.83
0.53
-2.95
-2.37
1.56
-2.32
-4.02
0.44
-2.79
-2.58
1.59
-2.16
2.46
1.96
1.87
1.97
3.99
UK slope
UK curvature
EMU level
EMU slope
EMU curvature
3.94
2.03
1.66
2.34
2.71
-1.96
3.80
1.96
1.44
2.84
2.61
-2.02
3.72
1.78
1.09
3.04
2.48
-2.29
3.50
1.61
0.89
3.36
2.58
-2.22
3.01
1.39
0.73
3.70
2.29
-2.47
3.03
1.27
0.56
3.75
2.19
-2.69
2.70
1.14
0.41
3.28
2.05
-2.71
0.42
-2.61
-2.33
1.51
-2.07
-4.05
1.18
-2.98
-0.60
2.23
-0.17
2.20
rx(9)
2.51
0.98
0.26
3.65
2.07
-2.72
0.24
-2.76
-2.23
1.62
-1.82
-4.41
1.47
-2.85
-0.55
2.12
-0.34
2.03
rx(10)
1.00
1.92
2.91
-2.66
0.70
0.65
1.18
0.29
1.05
-0.57
-0.94
1.20
-2.67
0.61
-1.85
-0.80
3.34
2.05
srx2,10
-2.73
-1.86
0.02
-3.36
-1.58
1.84
-0.31
1.37
3.75
-1.32
1.67
2.50
0.12
2.88
0.17
-1.61
0.19
-1.64
crx2,5,10
The reported t-statistics always belong to the fourth factor, which is indicated on the left. T-statistics are Newey-West-adjusted with a lag of 18.
table refers to a multivariate OLS regression with four explanatory variables, which are the three domestic factors and the factor indicated on the left.
This table displays the t-statistics for the international factor added to the set of instruments of three domestic principal components. Every line of this
-1.91
UK level
First three principal components, United States 1976:12 - 2009:06
1.11
-1.46
US level
UK slope
UK curvature
-3.56
-3.18
UK level
First three principal components, Germany 1976:12 - 2009:06
-1.23
-2.01
EMU curvature
US slope
1.99
EMU slope
US level
3.42
1.86
EMU level
First three principal components, United Kingdom 1976:12 - 2009:06
rx(2)
Table F.3: T-statistics for international forecasting regressions
207
22667.4
22662.1
22661.6
22664.2
22664.2
C(3, 4)
C(3, 6)
C(3, 6)
CnoFX (3, 4)
CnoFX (3, 4)
CnoFX (3, 6)
CnoFX (3, 6)
EUR-USD
USD-EUR
EUR-USD
USD-EUR
EUR-USD
USD-EUR
EUR-USD
0.25
0.26
0.27
0.28
0.28
0.27
0.28
0.28
rx(2)
0.22
0.23
0.25
0.26
0.26
0.26
0.27
0.27
rx(10)
0.18
0.18
0.19
0.19
0.19
0.19
0.19
0.18
srx2,10
0.17
0.18
0.23
0.23
0.22
0.21
0.25
0.25
crx2,5,10
Forecast of US bonds
0.28
0.29
0.29
0.29
0.29
0.29
0.29
0.29
rx(2)
0.22
0.22
0.25
0.25
0.23
0.23
0.26
0.26
rx(10)
0.04
0.04
0.07
0.08
0.04
0.04
0.07
0.08
srx2,10
of the model
0.00
0.00
0.01
0.01
0.11
0.11
0.11
0.12
FX
on the domestic or foreign bond returns.
forecasts are in line and indicate that the models are in fact symmetric and that the specification of the exchange rate does not have a different influence
R2
0.18
0.18
0.18
0.18
0.18
0.18
0.18
0.18
crx2,5,10
Forecast of German bonds
This table displays empirical estimates of the USD-EUR and EUR-USD models in several specifications. Log likelihood values and
22667.5
22665.1
22665.5
C(3, 4)
USD-EUR
log likelihood
Model
Currencies
Table F.4: Comparison of USD-EUR and EUR-USD two-country models
208
209
USD−EUR
0.4
0.2
forecast
realized
0
−0.2
−0.4
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
USD−GBP
0.4
0.2
0
−0.2
−0.4
1975
1980
1985
1990
1995
EUR−GBP
0.4
0.2
0
−0.2
−0.4
1975
1980
1985
1990
1995
Figure F.1: Forecasted vs. realized foreign exchange returns of the C(3, 6) model
The plots show the one-year forecasts of the exchange rate returns of the quarterly C(3, 6) model compared
to one-year realized returns (plotted with quarterly overlapping).
210
USD−EUR
0.2
0.1
0
C(3,6)
CnoFX(3,6)
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
USD−GBP
0.1
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
EUR−GBP
0.1
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
Figure F.2: Comparison of the C(3, 6) and CnoFX (3, 6) foreign exchange forecasts
The plots show the one-year forecasts of the exchange rate returns of the quarterly C(3, 6) model compared
to the CnoFX (3, 6) model (plotted with quarterly overlapping).
22636.9
22648.7
22652.9
22662.1
22662.9
22664.2
CnoFX (3, 1)
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
22405.1
22419.1
22423.9
22433.4
22434.9
22435.8
CnoFX (3, 1)
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
22474.7
22485.0
22492.9
22494.6
22495.0
22496.6
CnoFX (3, 1)
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
80
79
76
71
64
55
44
80
79
76
71
64
55
44
80
79
76
71
64
55
44
nvar
0.35
0.34
0.32
0.33
0.21
0.20
0.00
0.27
0.27
0.25
0.15
0.07
0.03
0.00
0.26
0.28
0.28
0.29
0.28
0.05
0.00
rx(2)
0.33
0.29
0.27
0.27
0.20
0.16
0.00
0.31
0.31
0.28
0.14
0.06
0.06
0.00
0.23
0.24
0.26
0.25
0.23
0.05
0.00
rx(10)
0.08
0.07
0.06
0.07
-0.04
-0.11
0.00
EUR
0.18
0.18
0.16
0.08
0.07
-0.03
0.00
USD
0.18
0.19
0.19
0.18
0.02
0.00
0.00
USD
srx2,10
0.09
0.07
0.06
0.08
0.06
0.08
0.00
0.22
0.22
0.19
0.08
0.07
0.05
0.00
0.18
0.19
0.23
0.22
0.21
0.09
0.00
crx2,5,10
0.18
0.18
0.18
0.16
-0.02
-0.01
0.00
0.07
0.06
0.07
0.05
0.03
0.00
0.00
0.29
0.29
0.29
0.29
0.22
0.13
0.00
rx(2)
0.12
0.11
0.10
0.08
0.05
-0.05
0.00
0.11
0.12
0.11
0.06
-0.01
-0.03
0.00
0.22
0.25
0.25
0.24
0.12
0.05
0.00
rx(10)
0.19
0.18
0.17
0.17
0.11
-0.08
0.00
GBP
0.10
0.09
0.06
0.05
0.02
0.00
0.00
GBP
0.04
0.06
0.08
0.06
0.01
-0.06
0.00
EUR
srx2,10
0.15
0.09
0.08
0.08
0.12
0.01
0.00
0.14
0.16
0.16
0.08
0.05
0.05
0.00
0.18
0.19
0.18
0.08
0.03
0.01
0.00
crx2,5,10
-0.15
-0.15
-0.15
-0.15
-0.15
-0.14
-0.14
-0.03
-0.03
-0.03
-0.03
-0.03
-0.06
-0.06
0.00
0.00
0.01
0.00
0.01
0.01
0.01
FX
likelihood.
22452.6
CnoFX (3, 0)
EUR-GBP model
22381.0
CnoFX (3, 0)
USD-GBP model
22620.4
CnoFX (3, 0)
USD-EUR model
log likelihood
Table F.5: Forecasting power of the quarterly two-country CnoFX models
211
56.7
64.9
83.4
85.1
87.6
48.2
76.2
85.8
104.7
107.8
109.5
44.1
64.6
80.6
83.9
84.8
87.9
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
CnoFX (3, 1)
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
CnoFX (3, 1)
CnoFX (3, 2)
CnoFX (3, 3)
CnoFX (3, 4)
CnoFX (3, 5)
CnoFX (3, 6)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
(24.7)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
(24.7)
(58.6)
(57.3)
(53.5)
(47.0)
(37.6)
(24.7)
(44.3)
(43.0)
(38.9)
(32.0)
(21.7)
(44.3)
(43.0)
(38.9)
(32.0)
(21.7)
43.8
40.8
39.8
36.5
20.6
(44.3)
(43.0)
(38.9)
(32.0)
(21.7)
EUR-GBP
61.3
59.6
56.5
37.6
28.0
USD-GBP
54.6
52.0
50.4
31.9
23.7
USD-EUR
CnoFX (3, 1)
23.2
20.2
19.3
15.9
33.3
31.6
28.5
9.6
30.9
28.4
26.7
8.2
(32.0)
(30.6)
(26.2)
(18.5)
(32.0)
(30.6)
(26.2)
(18.5)
(32.0)
(30.6)
(26.2)
(18.5)
CnoFX (3, 2)
7.3
4.3
3.3
23.7
22.0
18.9
22.7
20.2
18.5
(21.7)
(20.1)
(15.1)
(21.7)
(20.1)
(15.1)
(21.7)
(20.1)
(15.1)
CnoFX (3, 3)
4.0
0.9
4.8
3.1
4.2
1.7
(13.3)
(11.3)
(13.3)
(11.3)
(13.3)
(11.3)
CnoFX (3, 4)
3.1
1.8
2.5
(6.6)
(6.6)
(6.6)
CnoFX (3, 5)
test statistic, the unrestricted model is able to fit the data significantly better than the constrained model in terms of likelihood.
(and nested) model on top. The corresponding test statistic with a significance level of 1% is given in parentheses. If the likelihood ratio exceeds the
The table shows the likelihood ratios comparing unrestricted and constrained models. The unrestricted model is indicated on the left, the constrained
33.0
CnoFX (3, 1)
CnoFX (3, 0)
Table F.6: Likelihood ratio tests of the quarterly two-country CnoFX models
212
22636.3
22645.4
22645.7
CL (3, 2)
CL (3, 4)
CL (3, 6)
22404.5
22416.0
22416.4
CL (3, 2)
CL (3, 4)
CL (3, 6)
22476.9
22485.2
22485.2
CL (3, 2)
CL (3, 4)
CL (3, 6)
68
66
60
50
68
66
60
50
68
66
60
50
nvar
0.15
0.15
0.15
0.00
0.10
0.11
0.10
0.00
0.06
0.06
-0.01
0.00
rx(2)
0.11
0.11
0.11
0.00
0.15
0.15
0.18
0.00
0.10
0.10
-0.03
0.00
rx(10)
0.02
0.02
-0.04
0.00
EUR
0.12
0.13
0.12
0.00
USD
0.12
0.12
0.05
0.00
USD
srx2,10
0.05
0.05
0.05
0.00
0.12
0.12
0.13
0.00
0.09
0.09
0.02
0.00
crx2,5,10
0.01
0.00
-0.03
0.00
-0.04
-0.06
0.00
0.00
0.13
0.13
0.11
0.00
rx(2)
0.03
0.02
-0.03
0.00
0.02
0.02
0.01
0.00
0.07
0.06
0.03
0.00
rx(10)
0.03
0.03
0.00
0.00
GBP
0.02
0.03
-0.01
0.00
GBP
0.01
0.01
-0.07
0.00
EUR
srx2,10
0.17
0.17
0.07
0.00
0.11
0.12
0.15
0.00
0.05
0.04
0.05
0.00
crx2,5,10
0.16
0.16
0.16
0.13
0.09
0.12
0.06
0.04
0.12
0.12
0.09
0.09
FX
likelihood.
22459.1
CL (3, 0)
EUR-GBP model
22386.9
CL (3, 0)
USD-GBP model
22622.9
CL (3, 0)
USD-EUR model
log likelihood
Table F.7: Forecasting power of the restricted quarterly two-country CL models
213
214
Table F.8: Likelihood ratio tests of the restricted quarterly two-country CL models
CL (3, 0)
CL (3, 2)
CL (3, 4)
USD-EUR
CL (3, 2)
26.9
(23.2)
CL (3, 4)
45.0
(32.0)
18.2
(16.8)
CL (3, 6)
45.7
(34.8)
18.8
(20.1)
0.6
(9.2)
0.8
(9.2)
0.1
(9.2)
USD-GBP
CL (3, 2)
35.2
(23.2)
CL (3, 4)
58.3
(32.0)
23.1
(16.8)
CL (3, 6)
59.1
(34.8)
23.9
(20.1)
EUR-GBP
CL (3, 2)
35.6
(23.2)
CL (3, 4)
52.1
(32.0)
16.5
(16.8)
CL (3, 6)
52.2
(34.8)
16.7
(20.1)
likelihood.
215
Level − USD
0.2
C(3,6)
A(3,3)
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − USD
0.015
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
1990
1995
Curvature − USD
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
1990
1995
Figure F.3: Forecast of US bond returns in the USD-EUR model
The plots show the one-year forecasts of the exchange rate returns of the two-country C(3, 6) model compared to the single-country A(3, 3) model for level rx(10) , slope srx2,10 and curvature crx2,5,10 portfolios
(plotted with quarterly overlapping).
216
Level − EUR
C(3,6)
A(3,3)
0.1
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − EUR
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
−3
5
1990
1995
Curvature − EUR
x 10
0
−5
−10
1975
1980
1985
1990
1995
Figure F.4: Forecast of German bond returns in the USD-EUR model
217
Level − USD
0.3
C(3,6)
A(3,3)
0.2
0.1
0
−0.1
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − USD
0.02
0.01
0
−0.01
1975
1980
1985
1990
1995
Curvature − USD
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
1990
1995
Figure F.5: Forecast of US bond returns in the USD-GBP model
218
Level − GBP
0.1
C(3,6)
A(3,3)
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − GBP
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
−3
4
1990
1995
Curvature − GBP
x 10
2
0
−2
−4
−6
−8
1975
1980
1985
1990
1995
Figure F.6: Forecast of UK bond returns in the USD-GBP model
219
Level − EUR
C(3,6)
A(3,3)
0.1
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − EUR
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
−3
5
1990
1995
Curvature − EUR
x 10
0
−5
−10
1975
1980
1985
1990
1995
Figure F.7: Forecast of German bond returns in the EUR-GBP model
220
Level − GBP
C(3,6)
A(3,3)
0.1
0.05
0
−0.05
−0.1
1975
1980
1985
1990
1995
2000
2005
2010
2000
2005
2010
2000
2005
2010
Slope − GBP
0.01
0.005
0
−0.005
−0.01
1975
1980
1985
−3
4
1990
1995
Curvature − GBP
x 10
2
0
−2
−4
−6
1975
1980
1985
1990
1995
Figure F.8: Forecast of UK bond returns in the EUR-GBP model
221
Table F.9: Parameter estimates of the USD-EUR C(3, 4) model
µ∗⊤
0
0
1.56E-05
0
0
3.52E-06
Φ∗
0
1
0
0
0
0
0
0
1
0
0
0
0.55
-2.05
2.49
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0.81
-2.61
2.79
1.17E-03
8.02E-04
5.75E-04
-7.09E-06
2.13E-04
3.99E-04
0.19
0.95
-0.19
1.20
-0.84
-0.43
0.24
0.06
0.69
3.23
-5.28
1.99
0.80
-1.86
2.07
4.33
-7.78
3.38
0.10
0.67
-0.74
3.81
-6.56
3.73
0.18
0.36
-0.50
2.46
-4.42
2.90
0.25
0.10
-0.30
2.09
-4.27
3.10
λ⊤
0
-1.39E+03
4.95E+03
-3.76E+03
1.27E+04
-3.13E+04
1.90E+04
λ1
-1.70E+06
8.19E+06
-6.89E+06
2.03E+07
-4.63E+07
2.66E+07
4.43E+06
-2.24E+07
1.89E+07
-6.11E+07
1.39E+08
-7.94E+07
-2.84E+06
1.48E+07
-1.26E+07
4.41E+07
-9.98E+07
5.67E+07
-5.92E+04
-1.39E+07
1.46E+07
-2.15E+08
4.54E+08
-2.40E+08
-9.44E+05
3.57E+07
-3.65E+07
4.98E+08
-1.06E+09
5.62E+08
1.14E+06
-2.20E+07
2.20E+07
-2.87E+08
6.12E+08
-3.27E+08
9.45E-06
8.54E-06
7.75E-06
2.12E-06
2.26E-06
2.37E-06
8.54E-06
8.23E-06
7.81E-06
1.99E-06
2.17E-06
2.31E-06
7.75E-06
7.81E-06
7.62E-06
1.85E-06
2.05E-06
2.21E-06
2.12E-06
1.99E-06
1.85E-06
1.94E-06
1.86E-06
1.77E-06
2.26E-06
2.17E-06
2.05E-06
1.86E-06
1.87E-06
1.86E-06
2.37E-06
2.31E-06
2.21E-06
1.77E-06
1.86E-06
1.91E-06
µ⊤
Φ
Σ
ση
1.05E-03
ση s
5.04E-02
222
Table F.10: Parameter estimates of the USD-GBP C(3, 4) model
µ∗⊤
0
0
1.51E-05
0
0
4.14E-07
Φ∗
0
1
0
0
0
0
0
0
1
0
0
0
0.55
-2.05
2.49
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0.62
-2.22
2.59
-5.45E-04
-5.34E-04
-4.70E-04
-3.23E-04
3.23E-05
2.66E-04
-0.34
2.87
-1.79
1.56
-4.48
3.12
0.10
1.02
-0.31
0.92
-3.02
2.26
0.92
-1.56
1.51
0.54
-2.14
1.72
-0.20
1.18
-0.96
-0.72
2.91
-1.16
0.17
0.04
-0.13
-0.07
0.48
0.53
0.40
-0.69
0.39
0.99
-2.68
2.59
λ⊤
0
1.98E+03
-5.12E+03
3.12E+03
2.04E+02
-1.46E+03
1.33E+03
λ1
-1.28E+06
5.23E+06
-4.05E+06
4.20E+06
-8.21E+06
4.02E+06
2.30E+06
-1.21E+07
9.85E+06
-8.87E+06
1.67E+07
-7.79E+06
-1.05E+06
7.31E+06
-6.28E+06
4.87E+06
-9.09E+06
4.20E+06
1.91E+06
4.59E+06
-7.01E+06
-3.62E+06
3.89E+06
5.27E+05
-5.62E+06
-8.23E+06
1.50E+07
7.03E+06
-5.20E+06
-3.61E+06
3.81E+06
3.58E+06
-8.02E+06
-3.57E+06
1.72E+06
2.83E+06
8.94E-06
8.12E-06
7.39E-06
2.66E-06
2.90E-06
3.01E-06
8.12E-06
7.90E-06
7.53E-06
2.57E-06
2.92E-06
3.10E-06
7.39E-06
7.53E-06
7.38E-06
2.44E-06
2.87E-06
3.08E-06
2.66E-06
2.57E-06
2.44E-06
5.34E-06
4.89E-06
4.51E-06
2.90E-06
2.92E-06
2.87E-06
4.89E-06
4.99E-06
4.96E-06
3.01E-06
3.10E-06
3.08E-06
4.51E-06
4.96E-06
5.16E-06
µ⊤
Φ
Σ
ση
1.07E-03
σηs
5.11E-02
223
Table F.11: Parameter estimates of the EUR-GBP C(3, 3) model
µ∗⊤
0
0
3.44E-06
0
0
2.25E-07
Φ∗
0
1
0
0
0
0
0
0
1
0
0
0
0.82
-2.61
2.79
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0.63
-2.23
2.60
1.53E-04
1.99E-04
2.35E-04
2.63E-04
1.45E-04
8.23E-05
1.10
0.11
-0.35
-1.15
2.98
-1.74
0.95
-0.43
0.35
-1.30
3.35
-1.96
µ⊤
Φ
1.58
-2.56
1.85
-1.41
3.60
-2.11
-1.86
5.55
-3.89
-1.41
4.48
-1.92
0.85
-0.57
-0.40
-0.98
2.52
-0.46
2.52
-4.36
1.77
-0.07
-0.31
1.42
λ⊤
0
-1.26E+04
2.73E+04
-1.46E+04
1.40E+04
-3.48E+04
2.11E+04
λ1
-1.32E+08
2.62E+08
-1.32E+08
1.07E+07
-3.34E+07
2.45E+07
2.95E+08
-5.89E+08
2.97E+08
-2.18E+07
6.96E+07
-5.16E+07
-1.66E+08
3.34E+08
-1.69E+08
1.02E+07
-3.41E+07
2.60E+07
-3.46E+07
9.34E+07
-5.89E+07
-2.23E+07
5.39E+07
-3.19E+07
7.77E+07
-2.13E+08
1.36E+08
5.42E+07
-1.31E+08
7.80E+07
-4.34E+07
1.21E+08
-7.75E+07
-3.23E+07
7.86E+07
-4.68E+07
2.03E-06
1.92E-06
1.80E-06
1.14E-06
1.28E-06
1.34E-06
1.92E-06
1.89E-06
1.85E-06
1.25E-06
1.43E-06
1.51E-06
1.80E-06
1.85E-06
1.88E-06
1.33E-06
1.54E-06
1.64E-06
1.14E-06
1.25E-06
1.33E-06
5.20E-06
4.71E-06
4.29E-06
1.28E-06
1.43E-06
1.54E-06
4.71E-06
4.76E-06
4.71E-06
1.34E-06
1.51E-06
1.64E-06
4.29E-06
4.71E-06
4.89E-06
Σ
ση
1.11E-03
ση s
3.84E-02
224
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Curriculum Vitae
Lukas Wäger, born on the 2nd of December 1981 in Uznach, Switzerland
Education
2007 - 2012
Doctoral Studies in Finance,
2005 - 2007
Studies in Banking and Finance (M.A. HSG),
University of St. Gallen
2002 - 2005
Studies in Business Administration (B.A. HSG),
1997 - 2001
Matura Typus C, Kantonsschule Wattwil (SG)
Work Experience
Since 2012
Senior Quant Engineer,
2006 - 2012
Quantitative Analyst,
swissQuant Group, Zürich
Kraus Partner Investment Solutions, Zürich
2005 - 2006
Internship in Fixed Income Derivatives,
UBS Investment Bank, Opfikon (ZH)
2001 - 2002
Internship in Product Structuring,
RMF Capital Markets, Pfäffikon (SZ)

Analysis of Bond Risk Premia: Extensions to Macro

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