TEDAS portfolios: parameters optimisation and out-of - Hu

Transcription

TEDAS - Tail Event Driven ASset Allocation:
equity and mutual funds’ markets
Wolfgang Karl Härdle
Sergey Nasekin
Alla Petukhina
Ladislaus von Bortkiewicz Chair of Statistics
C.A.S.E. – Center for Applied Statistics
and Economics
Humboldt–Universität zu Berlin
http://lvb.wiwi.hu-berlin.de
http://case.hu-berlin.de
Motivation
1-1
Strategies comparison: hedge funds’ indices
Comparison of Strategies
9
8
Cumulative Return
7
6
5
4
3
2
1
0
2006
2007
2008
2009
2010
2011
2012
2013
2014
Time
Figure 1: Strategies’ cumulative returns’ comparison: TEDAS Basic,
S&P500, TEDAS Naïve, RR
1.6
Cumulative Return
1.5
TEDAS - Tail Event Driven Asset allocation
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-2
⇢ Härdle et al. (2014)
I TEDAS applied to hedge funds’ indices performs better than
benchmark models
⇢ Limitation of using hedge indices as portfolio assets
⇢ Application of TEDAS approach to global mutual funds’ data
and German stock market
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-3
Core & Satellites
Mutual funds, SDAX, MDAX and TecDAX constituents
⇢ diversification - reduction of the portfolio risk
⇢ construction - a more diverse universe of assets
⇢ allocation - a higher risk-adjusted return.
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-4
⇢ Comparison of the TEDAS with more benchmark strategies:
I 60/40 portfolio
I Risk Parity (equal risk portfolio contribution)
I Mean-Variance strategy
⇢ TEDAS parameters optimisation
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-5
Tail Risk
−0.25
−0.2
−0.15
−0.1
−0.05
0
S&P Returns
0.05
0.1
0.15
0.2
Figure 2: Estimated density of S&P 500 returns
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-6
The TLND challenge
⇢ Tail dependence
⇢ Large universe: p > n
⇢ Non normality
⇢ Dynamics
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Motivation
1-7
TEDAS Objectives
⇢ Hedge tail events
I Quantile regression
I Variable selection in high dimensions
⇢ Improve Asset Allocation
L
N
T
D
I Higher-order moments’ optimization
I Modelling of moments’ dynamics
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Outline
1. Motivation
X
2. TEDAS framework
3. Data
4. Empirical Results
5. Discussion: choice of ⌧ -spine
6. Conclusions
TEDAS framework
2-1
Tail Events
⇢ Y 2 Rn core log-returns; X 2 Rn⇥p satellites’ log-returns,
p>n
⇢
def
q⌧ (x) = FY |x1 (⌧ ) = x > (⌧ ) = arg minp EY |X =x ⇢⌧ {Y
2R
⇢⌧ (u) = u{⌧
X > },
I(u < 0)}
⇢ L1 penalty n kˆ
! > k1 to nullify "excessive" coefficients;
and !
ˆ controlling penalization; constraining  0 yields
ALQR Details
âdapt = arg min
⌧, n
p
2R
n
X
⇢⌧ (Yi
i=1
Xi> ) +
!
n kˆ
>
n
(1)
k1
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
TEDAS framework
2-2
TEDAS Step 1
Initial wealth W0 = $1, t = l, . . . , n; l = 120 length of the moving
window
⇢ Portfolio constituents’ selection
1. determine core asset return Yt , set ⌧t = Fbn (Yt )
⌧j=1,...,5 = (0.05, 0.15, 0.25, 0.35, 0.50)
2. ALQR for ˆ⌧t , n using the observations X 2 Rt
Y 2 Rt l+1,...,t
l+1,...,t⇥p
,
3. Select ⌧j,t according to the right-side q̂⌧j,t in: rt  q̂⌧1,t or
q̂⌧1,t < rt  q̂⌧j,t
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
TEDAS framework
2-3
TEDAS Step 1
TEDAS framework
TEDAS Example
TEDAS framework
2-6
TEDAS Example
1. Suppose
= 86 (Feb.
2007), W86 =
framework
1. Suppose t = 86 (Feb. TEDAS
2007),
W86 =t $1.125,
accumulated
W86 =
Example
wealth W86 = $1.429, TEDAS
Y161wealth
= 1.36%
< 0$1.429, Y161 = 1.3
𝒀
= −𝟎. 𝟎𝟏 → 𝝉
𝟏𝟐𝟎
Suppose
t = 86 (Feb.
2007), W86 = $1.125,
b=n𝟏𝟐𝟎
�𝒏 (𝒀161
2.1. Fwealth
(Y
0.35, so
estimate ˆ0.35
2. Fbn (Y161 ) = 0.35, so estimate
𝑭
𝟎. 𝟒𝟓
𝟏𝟐𝟎 ))==
W86 = $1.429, Y161 = 1.36% <
0
choose assets with
60⇥163
60 y
bn (Y161 )on
3.
ALQR
X
2
R
,
Y
2
R
2.
F
=
0.35,
so
estimate
TEDAS
framework
ˆ ,0.35 < 0
>
ˆchoose
assets
with
( 0.77,
1.12, 0.41) , La
0.25 =
TEDAS
Example
ˆ ,0.3560< 0
3. ALQR on X 2 R60⇥163 , YNorth
2 R America
yields Macro, Emerging M
ˆ with
�for
ALQR
Suppose
t=
86American
W86 = $1.12
> ,𝜷
,0.35
ALQR
for
𝑿(Feb. 2007),
ˆ0.25 = ( 0.77, 1.12,1. 0.41)
𝟎.𝟒𝟓,𝝀
Latin
Arbitrage,
Futures
with
XW
and
Y
𝒀 from
the
and
wealth
= from
$1.429, Y161 = 1.36% < 0
86 moving
North America Macro, Emerging
Markets
window
the moving
windowCTA/Managed
4. FbTEDAS
CF-CVaR
optimization De
2.
0.35, so estimate
Futures
n (Y161 ) =
60⇥163
60
3. choose
ALQR on
X 2with
R
with
Choose
assets
Y 2R
yields
>; X
b87 = (0
b86 = (0.22, 0.16, ,0.62)
w
=<(Details
1.12, >0.41)> , Latin Amer
4. TEDAS CF-CVaR optimization
yields
ˆˆ0.25
�
0 0.77,(1
,0.35 =
𝜷
b86
𝟎.𝟒𝟓,𝝀
W
W
+w
Xb87 ) =Markets
$1.438
87 < 𝟎
86 Macro,
Emerging
b86 = (0.22, 0.16, 0.62)> ; North
w
Xb87 =America
(0.38%,
0.45%,
0.76%)> , CT
Futures
Y
3. ALQR
2 R60⇥163 , Y 2 R60 yields
87 =on X1.53%
>
b
b86 X87 ) =
$1.438 (return of 0.62%) >while
87 = W86 (1 + w
Window'length=120'
ˆ0.25
Window length l W
= 60
= (CF-CVaR
0.77, 1.12,
0.41) , Latin Am
4. TEDAS
optimization Details yields
Y87 = 1.53%
North America Macro, >Emerging
Markets
b
TEDAS
Event 0.16,
Driven0.62)
Asset allocation
b86- Tail
w
= (0.22,
; X87 = (0.38%, 0.4
TEDAS - Tail Event Driven Asset
allocation
>X
Futures
b87 )Asset
TEDAS
- Tail
allocation
b86
W87
= WEvent
w
= $1.438
(return of
86 (1 + Driven
allocation
Y87 = CF-CVaR
1.53%
4. TEDAS
optimization Details yie
1.6
Cumulative Return
1.5
1.6
TEDAS - Tail Event Driven Asset
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
1.4
0.9
March
1.3
May
July
September
Time
1.2
1.1
1
0.9
March
May
July
September
Time
b86 = (0.22, 0.16, 0.62)> ; Xb87 = (0.38%, 0
w
TEDAS framework
2-4
TEDAS Step 2
⇢ Portfolio selection
bt 2 Rk
1. apply TEDAS Gestalt to Xj , obtain w
2. determine the realized portfolio wealth for t + 1,
def
bt> Xbt+1 )
Xbt+1 = (Xt+1,1 , . . . , Xt+1,k )> : Wt+1 = Wt (1 + w
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
TEDAS framework
2-5
Rebalancing of portfolio:
⇢ one of inequalities in step 3 holds
I sell the core portfolio and buy satellites (step 4) with
estimated weights (step 5)
I stay "in cash" if there are no adversely moving satellites (step
4)
⇢ no one of inequalities holds: invest in the core portfolio
⇢ period (t+1), if no one of inequalities (step 3) holds, we return
to the core portfolio
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
TEDAS framework
2-6
TEDAS Example
1. Suppose t = 161 (May. 2011), accumulated wealth
W161 = $2.301, Y161 = 1.36% < 0
2. Fbn (Y161 ) = 0.35, so estimate ALQR for ˆ
,0.35
3. ALQR on X 2 R120⇥583 , Y 2 R120 yields
ˆ0.35 = ( 1.12, 0.41)> , Blackrock Eurofund Class I, Pimco
Funds Long Term United U.S. States Government Institutional
Shares
b161 = (0.014, 0.026)> ;
4. TEDAS CF-CVaR optimization w
> X
b161 ) = $2361
b161
Xb162 = (0, 1)> , W162 = W161 (1 + w
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
TEDAS framework
2-7
TEDAS Gestalten
TEDAS gestalt
TEDAS Naïve
Dynamics modelling
NO
Weights optimization
Equal weights
TEDAS Hybrid
NO
Mean-variance optimization
of weights Details
DCC volatility
CF-VaR optimization
TEDAS Basic
Details
Details
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-1
Small and mid caps German stocks
⇢ MDAX
I 50 medium-sized German public limited companies and foreign
companies primarily active in Germany from traditional sectors
I Ranks after the DAX30 based on market capitalisation and
stock exchange turnover
⇢ SDAX
I The selection index for smaller companies from traditional
sectors
I 50 stocks from the Prime Standard
⇢ TecDAX
I Comprises the 30 largest technology stocks below the DAX
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-2
Size premium
⇢ Banz (1981) and Reinganum (1981): the US small cap stocks
outperformed large-cap stocks (in 1936-1975)
⇢ Fama, French (1992, 1993): a size premium of 0.27% per
month in the US over the period 1963-1991
⇢ Results are robust:
I for stock price momentum by Jegadeesh , Titman (1993) and
Carhart (1997)
I for liquidity by Pastor, Stambaugh (2003) and Ibbotson, Hu
(2011)
I for industry factors, high leverage, low liquidity by Menchero et
al. (2008)
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-3
Why small and mid cap stocks?
⇢ Strong absolute returns
⇢ Diversification benefits (Eun, Huang, Lai (2006))
⇢ High risk-adjusted returns
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-4
Strong absolute returns
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
2007
2008
2009
2010
2011
2012
2013
2014
Figure 3: Cumulative index performance: MSCI World Large Cap, MSCI
World Mid Cap, MSCI World Small Cap, MSCI World Small and Mid Cap
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-5
Diversification
0.04
0.03
0.02
Return
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Time
Figure 4: DAX and Hamborner REIT AG daily returns in 2013122020140831
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-6
German equity
I 125 stocks - SDAX (48),
MDAX (47) and TecDAX
(50) as on 20140801
I DAX index
2
1.8
Cumulative Return
⇢ Frankfurt Stock Exchange
(Xetra), weekly data
1.6
1.4
1.2
1
DAX
MDAX
0.8
SDAX
TecDAX
0.6
Feb11
Sep11
Apr12
Oct12
May13
Time
Nov13
Jun14
Dec14
⇢ Span: 20121221 - 20141127
(100 trading weeks)
⇢ Source: Datastream
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Jul15
Data
3-7
Mutual Fund
⇢ Open-End: buy and sell the shares, meet the demand for
customers
⇢ Unit Investment Trust: exchange-traded fund (ETF), Fixed/
unmanaged Portfolio
⇢ Closed-End: fixed number of shares, not redeemable by the
fund, buy and sell on the exchange
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-8
Mutual fund flowchart
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-9
Why Mutual Funds?
⇢ Importance of MF
I $30 trillion worldwide, 15 trillion in
U.S in 2013
I 88% investment companies managed
asset by holding MF
⇢ Big data: 76 200 MFs worldwide in
2013
⇢ Diversification
Structure of U.S.
Mutual funds, by asset
classes
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-10
Dynamics of Mutual funds investment
TNA, bln US$
Number of
Mutual funds
80,000
70,000
60000
50,000
1998 2000 2002 2004 2006 2008 2010 2012 2014
40,000
30,000
20,000
10,000
0
1998 2000 2002 2004 2006 2008 2010 2012 2014
Year
Figure 5: Worldwide Mutual Funds: total number and TNA
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-11
Mutual Funds
I Core asset (Y ): S&P500
I Satellite assets (X ): 583
Mutual funds
⇢ Span: 19980101 - 20131231
⇢ Source: Datastream
Sample Mutual Funds' Cumulative Gain
9
8
7
Stock Investment Value
⇢ Monthly data
6
5
4
3
2
1
0
1997
2000
2002
2005
2007
2010
2012
2015
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Data
3-12
Benchmark Strategies
1. RR: dynamic risk-return optimization
Details
2. ERC: Risk-parity portfolio (equal risk contribution)
3. 60/40 portfolio
Details
Details
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-1
Return
TEDAS approach:German stocks’ results
1.5
1.5
1.5
1.4
1.4
1.4
1.3
1.3
1.3
1.2
1.2
1.2
1.1
1.1
1.1
1
1
1
0.9
0.9
0.9
0.8
Jan14
Apr14
Aug14
Dec14
0.8
Jan14
Apr14
Aug14
Dec14
0.8
Jan14
Apr14
Aug14
Dec14
TEDAS Naïve, TEDAS Hybrid, DAX30
TEDAS_strategies
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-2
Return
1.6
1.6
1.6
1.4
1.4
1.4
1.2
1.2
1.2
1
1
1
0.8
Jan14
Apr14
Aug14
0.8
Dec14 Jan14
Apr14
Aug14
0.8
Dec14 Jan14
Apr14
Aug14
Dec14
Figure 7: Strategies’ cumulative returns’ comparison (with transaction
costs 1% of portfolio value): TEDAS Hybrid, 60/40, ERC, RR
TEDAS_strategies
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-3
Strategies’ performance: German stocks
Strategy
TEDAS Basic
TEDAS Naïve
TEDAS Hybrid
RR
ERC
60/40
DAX30
Cumulative
return
144%
143%
143%
108%
129%
121%
103%
Sharpe
ratio
0.3792
0.3184
0.3079
0.0687
-0.0693
0.0306
0.0210
Maximum
drawdown
0.1069
0.0564
0.1068
0.0934
0.1792
0.0718
0.1264
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-4
τ = 0.05
τ = 0.15
10
10
8
8
6
6
4
4
2
0
2
0
2
4
6
8
0
10
0
2
4
τ = 0.25
10
10
8
8
6
6
4
4
2
2
0
0
2
4
6
8
10
6
8
10
τ = 0.35
6
8
10
0
0
2
4
Figure 8: Frequency of the number of selected variables for 4 different ⌧
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-5
TEDAS approach:German stocks results
τ = 0.05
τ = 0.15
15
15
10
10
5
5
0
0
0
50
100
150
0
τ = 0.25
50
100
150
τ = 0.35
15
15
10
10
5
5
0
0
0
50
100
150
0
50
100
150
Figure 9: The frequency of stocks
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-6
Selected Stocks
Table 1: The selected German Stocks for ⌧ = 0.05
Top 5 influential Stocks
Sartorius Aktiengesellschaft
Frequency
12
Index
TecDAX
XING AG
8
TecDAX
Surteco SE
7
SDAX
Kabel Deutschland Holding AG
7
MDAX
Biotest AG
6
MDAX
Industry
Provision of laboratory and process
technologies and equipment
Online business communication services
Household Goods & Home Construction
Cable-based telecommunication services
Producing biological medications
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-7
- b in each window, ⌧ = 0.05
b in application; ⌧ = 0.05
Selected Stocks
1.6
1.5
Cumulative Return
Figure 10: Different
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-8
Return
TEDAS approach:Mutual Funds results
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
2007
2009
2011
2013
2015
0
2007
2009
2011
2013
2015
0
2007
2009
2011
2013
2015
TEDAS Naïve, TEDAS Hybrid, S&P500
TEDAS_strategies
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-9
Return
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
2007
2009
2011
2013
2015
0
2007
2009
2011
2013
2015
0
2007
2009
2011
2013
2015
Figure 12: Strategies’ cumulative returns’ comparison (with transaction
costs 1% of portfolio value): TEDAS Hybrid, 60/40, ERC, RR
TEDAS_strategies
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-10
Strategies’ performance: Mutual funds
Strategy
TEDAS Basic
TEDAS Naïve
TEDAS Hybrid
RR
ERC
60/40
S&P500
Cumulative
return
421%
454%
433%
116%
129%
121%
113%
Sharpe
ratio
0.6393
0.6974
0.6740
0.0214
0.0487
0.0252
0.0132
Maximum
drawdown
0.0855
0.0583
0.0276
0.4772
0.4899
0.3473
0.5037
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Empirical Results
4-11
τ = 0.05
τ = 0.15
8
5
4
6
3
4
2
2
0
1
1
2
3
4
5
0
1
2
τ = 0.25
3
4
5
4
5
τ = 0.35
10
8
8
6
6
4
4
2
2
0
1
2
3
4
5
0
1
2
3
Figure 13: Frequency of the number of selected variables for 4 different ⌧
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Cumulative Return
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0.9
March
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Time
September
Empirical Results
4-12
τ = 0.05
τ = 0.15
5
8
4
6
3
4
2
2
1
0
0
0
200
400
600
0
τ = 0.25
200
400
600
τ = 0.35
15
8
6
10
4
5
2
0
0
0
200
400
600
0
200
400
600
Figure 14: The frequency of mutual funds
1.6
Cumulative Return
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1
0.9
March
May
July
Time
September
Empirical Results
4-13
Selected Mutual Funds
Table 2: The selected Mutual Funds for ⌧ = 0.05
Top 5 influential Stocks
Blackrock Eurofund Class I
Pimco Funds Long Term United
States Government Institutional
Shares
Prudential International Value
Fund Class Z
Artisan International Fund Investor Shares
American Century 2OTH Century International Growth Investor Class
Frequency
12
8
Market
U.S.
U.S.
4
U.S.
3
U.S.
1
U.S.
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Choice of ⌧ -spine
5-1
How to choose optimal ⌧ -spine?
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
5-2
Generation of different ⌧ -spines
Tau = 0.5*Xn
0.5
0.45
0.4
0.35
Tau
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
X
Figure 15: Generation of 10 sets of ⌧ -spines
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
5-3
TEDAS Basic with different ⌧ -spines
Figure 16: Cumulative return for TEDAS Basic with various ⌧ -spines
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
5-4
TEDAS Basic with different ⌧ -spines
Cumulative return
2
1.8
1.6
1.4
1.2
1
Jan14
May14
Sep14
Jan15
Time
Figure 17: TEDAS Basic cumulative returns’ for ⌧ -spines:
⌧j=1,...,5 = (0, 0.002, 0.0233, 0.1311, 0.5),
⌧j=1,...,5 = (0.05, 0.15, 0.25, 0.35, 0.5),
⌧j=1,...,50 = (0.01, 0.02, 0.03 . . . 0.49, 0.5)
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
5-5
What is the best ⌧ -spine?
Monte Carlo simulations
⇢ Yi =q̂⌧i ⌧j=1,2,3 = (0.05, 0.15, 0.35), n = 100,
Yt ⇠ ALD(µ, , ⌧ ); Details
⇢ Xi ⇠ N(0, ⌦), n = 100 for every ⌧ , p = 150,
= ( 5, q2, 1, 3, 1, 0.5, 0, ..., 0), "i ⇠ N(0, 2 );
p
înit k0 log(n _ p)(log n)0.1/2 , !
ˆ j = 1/| înit | ^ n;
n = 0.25 k
j
înit ;
j
⇢ ⌦i,j = 0.5|i
j| ,
= 0.1, 0.5, 1 (three levels of noise);
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
5-6
What is the best ⌧ -spine?
⇢ for înit estimator ˆ⌧, ˆ from the model (2) is used, where ˆ is
chosen according to the BIC criterion
(
)
n
X
log(n) b
def
BIC n ,⌧ = log n 1 ·
⇢⌧ (Yi Xi> ˆ⌧ ) +
· df( n )
2n
i=1
⇢ Apply one of TEDAS modification with different ⌧ -spines
⇢ Choose that ⌧ -spine, which gives the highest wealth
Wi =
d
X
w j xi , ⌧ ,
j=1
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Conclusions
6-1
beginframe
⇢ TEDAS solves TLND challenge
⇢ TEDAS approach performs better than traditional benchmark
strategies
⇢ TEDAS outperforms for
I different regions (global and Germany),
I various assets
I alternative time periods (daily, weekly and monthly),
I big data and small data
⇢ Results for 3 gestalts of TEDAS are robust
1.6
1.5
Cumulative Return
⇢ Discussion:
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
I How to choose optimal ⌧ -spine?
Conclusions
TEDAS
6-3
- Tail Event Driven ASset Allocation:
equity and mutual funds’ markets
Wolfgang Karl Härdle
Sergey Nasekin
1.6
Cumulative Return
Alla Petukhina
1.8
1.4
1.2
1
0.8
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Time
Ladislaus von Bortkiewicz Chair of Statistics
C.A.S.E. – Center for Applied Statistics
and Economics
Humboldt–Universität zu Berlin
http://lvb.wiwi.hu-berlin.de
http://case.hu-berlin.de
1.6
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1.2
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1
0.9
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May
July
Time
September
Technical details
7-1
Notation
def
⇢ q̂⌧ = Fbn 1 (⌧ ), with
def
Fbn (Yt ) =
Z
n
1X
fˆn (u) du =
H
n
1
Yt
i=1
P
def
where fˆnR(Yt ) = (1/nh) ni=1 K {(Yt
x
H(x) = 1 K (u)du, K (·) = '(·);
Silverman (1986) rule-of-thumb:
h = 1.06sn
⇢ ˆ⌧,
1/5
✓
Yt
Yi
h
◆
(2)
,
Yi )/h},
, s sample standard deviation of Y
are the estimated non-zero ALQR coefficients
n
Back to "TEDAS Step 1"
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-2
Lasso Shrinkage
Linear model: Y = X + "; Y 2 Rn , X 2 Rn⇥p ,
i.i.d., independent of {Xi ; i = 1, ..., n}
2 Rp , {"i }ni=1
The optimization problem for the lasso estimator:
ˆlasso = arg min f ( )
p
2R
subject to
g( )
(3)
0
where
1
(y X )> (y
2
g ( ) = t k k1
f( )=
where t is the size constraint on k k1
X )
Back to "Tail Events"
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-3
Lasso Duality
If (1) is convex programming problem, then the Lagrangian is
L( , ) = f ( )
g ( ).
and the primal-dual relationship is
minimize sup L( , )
0
|
{z
}
primal
maximize inf L( , )
0
|
{z
}
dual
Then the dual function L⇤ ( ) = inf L( , ) is
1
L⇤ ( ) = y > y
2
with (y
1 ˆ> > ˆ
X X
2
X ˆ)> X ˆ/k ˆk1 =
t
(y
X ˆ)> X ˆ
k ˆk1
1.6
Cumulative Return
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1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-4
Paths of Lasso Coefficients
figures/lassop.pdf
Figure 18: Lasso shrinkage of coefficients in the hedge funds dataset example (6 covariates were chosen for illustration); each curve represents a coefficient as a function of the scaled parameter ŝ = t/k k1 ; the dashed line
represents the model selected by the BIC information criterion (ŝ = 3.7)
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-5
8
9
10
11
12
Example of Lasso Geometry
βOLS
4
−4
−3
−2
−1
0
1
2
3
β2
5
6
7
●
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
9
10
11
12
β1
Figure 19: Contour plot of the residual sum of squares objective function centered at the OLS estimate bols = (6, 7) and the constraint region
P
| j|  t
MVAlassocontour.xpl
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-6
Quantile Regression
The loss ⇢⌧ (u) = u{⌧
def
Fy |x1 (⌧ ) =
I(u < 0)} gives the (conditional) quantiles
q⌧ (x).
Minimize
ˆ⌧ = arg min
p
2R
n
X
⇢⌧ (Yi
Xi> ).
i=1
Re-write:
minimize
(⇠,⇣)2R2n
+
n
⌧ 1>
n ⇠ + (1
⌧ )1>
n ⇣|X + ⇠
with ⇠, ⇣ are vectors of "slack" variables
⇣=Y
o
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-7
Non-Positive (NP) Lasso-Penalized QR
The lasso-penalized QR problem with an additional non-positivity
constraint takes the following form:
minimize
(⇠,⇣,⌘, ˜)2R2n+p
⇥Rp
+
subject to
⌧ 1>
n ⇠ + (1
>
⌧ )1>
n ⇣ + 1n ⌘
⇠
⇣ = Y + X ˜,
⇠
0,
⇣
0,
˜,
⌘
˜,
⌘
˜
(4)
0,
˜ def
=
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-8
Solution
Transform into matrix (Ip is p ⇥ p identity matrix; Ep⇥n =
minimize
c >x
✓
◆
In
0
):
subject to Ax = b, Bx  0
where A =
In
In 0 X
, b = Y, x =
0
1
⌧ 1n
B
B (1 ⌧ )1n C
B
C, B = B
c=B
B
@
A
1p
@
01p
0
Ep⇥n
0
0
0
0
⇠ ⇣ ⌘
0
Ep⇥n
0
0
0
0
0
Ip
Ip
0
>
0
0
Ip
Ip
Ip
,
1
C
C
C
C
A
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-9
Solution - Continued
The previous problem may be reformulated into standard form
minimize
c >x
subject to Cx = d,
x + s = u, x
0, s
0
and the dual problem is:
maximize
d >y
subject to C > y
u>w
w + z = c, z
0, w
0
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-10
Solution - Continued
The KKT conditions for this linear program are
8
9
Cx d
>
>
>
>
>
>
>
>
x +s u
<
=
>
C y w +z c
F (x, y , z, s, w ) =
= 0,
>
>
>
>
x
z
>
>
>
>
:
;
s w
with y 0, z
variables.
0 dual slacks, s
0 primal slacks, w
0 dual
This can be solved by a primal-dual path following algorithm based
on the Newton method
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-11
Adaptive Lasso Procedure
Lasso estimates ˆ can be inconsistent (Zou, 2006) in some
scenarios.
Lasso soft-threshold function gives biased results
figures/thresh-eps-converted-to.pdf
Figure 20: Threshold functions for simple and adaptive Lasso
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-12
Adaptive Lasso Procedure
The adaptive Lasso (Zou, 2006) yields a sparser solution and is less
biased.
L1 - penalty replaced by a re-weighted version; !
ˆ j = 1/| ˆjinit | ,
= 1, înit is from (3)
The adaptive lasso estimates are given by:
âdapt = arg min
p
2R
n
X
(Yi
i=1
Xi> )2 + kˆ
! > k1
init
adapt
(Bühlmann, van de Geer, 2011): ˆj = 0, then ˆj
=0
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-13
Simple and Adaptive Lasso Penalized QR
Simple lasso-penalized QR optimization problem is:
ˆ⌧, = arg min
p
2R
n
X
⇢⌧ (Yi
i=1
Xi> ) + k k1
(5)
Adaptive lasso-penalized QR model uses the re-weighted penalty:
âdapt = arg min
⌧,
p
2R
n
X
⇢⌧ (Yi
i=1
Xi> ) + kˆ
! > k1
(6)
Adaptive lasso-penalized QR procedure can ensure oracle properties
for the estimator Details
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-14
Algorithm for Adaptive Lasso Penalized QR
The optimization for the adaptive lasso quantile regression can be
re-formulated as a lasso problem:
⇢ the covariates are rescaled: X̃ = (X1
⇢ the lasso problem (5) is solved:
înit , . . . , Xp
1
ˆpinit );
n
ˆ˜ = arg min X ⇢ (Y
⌧
i
⌧,
p
2R
i=1
X̃i> ) + k k1
adapt
˜⌧,
⇢ the coefficients are re-weighted as ˆ⌧,
= ˆ
înit
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-15
Oracle Properties of an Estimator
An estimator has oracle properties if (Zheng et al., 2013):
⇢ it selects the correct model with probability converging to 1;
⇢ the model estimates are consistent with an appropriate
convergence rate;
⇢ estimates are asymptotically normal with the same asymptotic
variance as that knowing the true model
Back to "Simple and Adaptive Lasso Penalized QR"
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-16
Oracle Properties for Adaptive Lasso QR
In the linear model, let Y = X + " = X 1 1 + X 2 2 + ", where
X = (X 1 , X 2 ), X 1 2 Rn⇥q , X 2 2 Rn⇥(p q) ; q1 are true nonzero
coefficients, p2 q = 0 are noise coefficients; q = k k0 .
p
p
Also assume that q/ n ! 0 and /{ q log(n _ p)} ! 1 and
certain regularity conditions are satisfied Details
Back
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-17
Oracle Properties for Adaptive Lasso QR
Then the adaptive L1 QR estimator has the oracle properties
(Zheng et al., 2013):
1. Variable selection consistency:
P(
2
= 0)
1
6 exp
⇢
log(n _ p)
4
ˆk = Op (
2. Estimation consistency: k
def
.
p
q/n)
3. Asymptotic normality: uq2 = ↵T ⌃11 ↵, 8↵ 2 Rq , k↵k < 1,
⇢
(1 ⌧ )⌧
1/2
1 T 1
1 L
ˆ
n uq ↵ (
) ! N 0, 2 ⇤
f ( )
where
⇤
is the ⌧ th quantile and f is the pdf of "
Back
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-18
Risk-Return Asset Allocation
Log returns Xt 2 Rp :
minp
wt 2R
def
2
P,t (wt ) =
wt> ⌃t wt
s.t. µP,t (wt ) = rT ,
(7)
wt> 1p = 1,
wi,t
def
where rT "target" return, ⌃t =
modeled with a GARCH model
0
t 1 {(Xt
Details
µ)(Xt
µ)> }, ⌃t is
Back to "Benchmark Strategies"
Return to "TEDAS Gestalten"
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-19
Dynamic Conditional Correlations Model
Assume: rt |Ft
1
⇠ N(0, Dt Rt Dt ), "t = Dt 1 rt ,
Dt2 = diag(!i ) + diag(↵i )
Qt = S
(11>
A
Rt = {diag(Qt )}
1
rt
>
1 rt 1
B) + A
{Pt
Qt {diag(Qt )}
where rt 2 Rp , Dt = diag (
def
it )
+ diag( i )
1
Dt2 1 ,
>
1 "t 1 "t 1 P t 1 }
+B
Qt
1,
2 Rp⇥p , "t 2 Rp standardized
def
returns with "it = rit it 1 , 1 vector of ones; Pt 1 = {diag(Qt )}1/2 ,
!i , ↵i , i , A, B coefficients, Hadamard (elementwise) product
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-20
The DCC Model - Continued
⇢ correlation targeting: S = (1/T )
PT
>
t=1 "t "t
⇢ Q0 = " 0 " >
0 positive definite, each subsequent Qt also positive
definite
⇢ consistent but inefficient estimates: the log-likelihood function
o
1 Xn
1
n log(2⇡) + 2 log |Dt | + log |Rt | + ">
R
"
t ,
t t
2
T
L(✓, ) =
t=1
where ✓ parameters in D and
parameters in R Back
additional correlation
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-21
The DCC Model - Continued
Re-write:
L(✓, ) = LV (✓) + LC (✓, ),
with volatility part LV (✓) and correlation part LC (✓, ),
o
1 Xn
n log(2⇡) + log |Dt |2 + rt> Dt 2 rt
2
t=1
T
d ⇢
r2
1 XX
log(2⇡) + log( it2 ) + it2 ,
2
it
T
LV (✓) =
=
t=1 i=1
1 Xn
1
log |Rt | + ">
t R t "t
2
T
LC (✓, ) =
t=1
o
">
"
t t .
Back
1.6
Cumulative Return
1.5
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1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-22
Cornish-Fisher VaR Optimization
Log returns Xt 2 Rp :
minimize
w 2Rd
Wt { q↵ (wt ) ·
p (wt )}
subject to wt> µ = µp , wt> 1 = 1, wt,i 0
Q
def
here Wt = W0 · tj=11 wt> j (1 + Xt j ), w̃ , W0 initial wealth,
def
2
p (w ) =
wt> ⌃t wt ,
Sp (w )
Kp (w )
Sp (w )2
+(z↵3 3z↵ )
(2z↵3 5z↵ )
,
6
24
36
here Sp (w ) skewness, Kp (w ) kurtosis, z↵ is N(0, 1) ↵-quantile If
Sp (w ), Kp (w ) zero, then obtain Markowitz allocation
def
q↵ (w ) = z↵ +(z↵2 1)
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Cumulative Return
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July
Time
September
Technical details
7-23
Risk Parity (Equal risk contribution)
Let (w ) =
p
w > ⌃w . Euler decomposition:
def
(w ) =
n
X
i (w )
=
i=1
n
X
i=1
wi
(w )
@wi
(w )
(w )
where @w
is the marginal risk contribution and i (w ) = wi @w
i
i
the risk contribution of i-th asset. The idea of ERC strategy is to
find risk balanced porfolio, such that:
i (w )
=
j (w )
i.e. the risk contribution is the same for all assets of the portfolio
Return to "Benchmark Strategies"
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-24
60/40 allocation strategy
60/40 portfolio allocation strategy implies the investing of 60% of
the portfolio value in stocks (often via a broad index such as
S&P500) and 40% in government or other high-quality bonds, with
regular rebalancing to keep proportions steady.
Return to "Benchmark Strategies"
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-25
Portfolio Skewness and Kurtosis
Skewness SP and excess kurtosis KP are given by moment
expressions
1
SP (w ) = 3
(m3 3m2 m1 + 2m13 )
(w
)
P
1
KP (w ) = 4
(m4 4m3 m1 + 6m2 m12 + 3m14 ) 3
(w
)
P
where portfolio non-central moments also depend on w :
def
m1 = µP (w ) = w > µ
m2 =
m3 =
2
2
P + m1
d X
d X
d
X
wi wj wk Sijk
i=1 j=1 k=1
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-26
Portfolio Skewness and Kurtosis - Continued
m4 =
d X
d X
d X
d
X
wi wj wk wl Kijkl ,
i=1 j=1 k=1 l=1
2
P (w ) =
E(ri ⇥ rj
w > ⌃w
where
and Sijk = E(ri ⇥ rj ⇥ rk ),
Kijkl =
⇥ rk ⇥ rl ) can be computed via sample averages
from returns data.
Sijk , Kijkl determine the d-dimensional portfolio co-skewness and
co-kurtosis tensors
def
S = {Sijk }i,j,k=1,...,d 2 Rd⇥d⇥d
def
K = {Kijkl }i,j,k,l=1,...,d 2 Rd⇥d⇥d⇥d .
Back
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-27
Regularity Conditions for Adaptive Lasso QR
A1 Sampling and smoothness: 8x in the support of Xi , 8y 2 R,
0
fYi |Xi (y |x), f 2 C k (R), |fYi |Xi (y |x)| < f , |fYi |Xi (y |x)| < f 0 ; 9f ,
such that fYi |Xi (x > ⌧ |x) > f > 0
A2 Restricted identifiability and nonlinearity: let 2 Rp ,
T ⇢ {0, 1, ..., p}, T such that Tj = j if j 2 T , Tj = 0 if
j 62 T ; T = {0, 1, ..., s}, T ( , m) ⇢ {0, 1, ..., p}\T , then
9m 0, c 0 such that
inf
T E(X
>
i Xi )
2A, 6=0 k T [T ( ,m)
def
k2
> 0,
where A = { 2 Rp : k
T c k1
3f 3/2
8f
0
 ck
inf
2A,
E[|XiT |2 ]3/2
> 0,
6=0 E[|X T |3 ]
i
T k1 , k T c k0
 n}
Back
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-28
Regularity Conditions - Continued
A3 Growth rate of covariates:
q 3 {log(n _ p)}2+⌘
! 0, ⌘ > 0
n
A4 Moments of covariates: Cramér condition
E[|xij |k ]  0.5Cm M k
for some constants Cm , M, 8k
2
k!
2, j = 1, ..., p
A5 Well-separated regression coefficients: 9b0 > 0, such that
8j  q, | ˆj | > b0
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-29
Proof of the CF-CVaR Expansion 1
Define the Cornish-Fisher expansion:
q1
↵
def
= z1
2
↵ + (z1 ↵
def
def
1)s + (z13
↵
3z1
↵ )k
(2z13
↵
5z1
↵ )s
2
,
where s = S/6, k = K /24, S and K are skewness and excess
kurtosis, respectively; z1
↵
def
1 (1
=
↵).
Re-write:
q1
where a0 =
↵
= a0 + a1 z1
s, a1 = 1
↵
+ a2 z12
↵
+ a3 z13
3k + 5s 2 , a2 = s, a3 = k
(8)
↵,
2s 2
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-30
Define the conditional Value-at-Risk (CVaR) or expected shortfall (ES):
Z 1
1
def
CVaR↵ =
VaRq d q,
1 ↵ ↵
def
where VaRq =
1
(↵)
p
T
Observe:
CVaR↵ =
=
=
1
1
1
1
p
p
↵
T
↵
T
↵
Z
1
↵
Z 1
Z
1
1
(↵)
p
Tdq
(9)
(↵)d q
(10)
u'(u)d u,
(11)
↵
z↵
1
where (11) follows from the change of variable: u = zq =
1
(q)
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-31
Substitute (8) into (11):
CVaR↵ =
p
1
T
↵
Z
z↵
a0 + a1 z + a2 z 2 + a3 z 3 '(z)d z
1
= a0 A0 + a1 A1 + a2 A2 + a3 A3 ,
A0 =
A1 =
A2 =
A3 =
p
1
p
T
↵
Z
p
z↵
'(z)d z =
1
T,
p
z↵
T
'(z↵ ),
↵
p Z z
✓
◆
↵
p
'(z↵ )z↵
T
z 2 '(z)d z =
T
+1 ,
1 ↵
1 ↵
1
p Z z
p
↵
T
T 2
z 3 '(z)d z =
(z↵ + 2)'(z↵ ).
1 ↵
1 ↵
1
1
T
↵
Z
z'(z)d z =
1
1
Collecting terms and simplifying gives the desired result.
Return to "Conditional VaR Optimization"
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-32
density
0.00
0.10
0.20
0.30
Asymetric Laplace Distribution
−4
−2
0
2
4
X
Figure 21: Standart ALD: ⌧ = 0.3,⌧ = 0.5,⌧ = 0.7, ⌧ = 0.1
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
Technical details
7-33
Quantile regression using ALD
⇢ Yu & Moyeed(2001)
Yi ⇠ ALD(µ, , ⌧ ), if its pdf is given by
n
f (y |µ, , ⌧ ) = ⌧ (1 ⌧ ) exp ⇢⌧ (y
µ)
o
where µ is location, - scale and ⌧ -skewness parameters, and
loss function ⇢⌧ (u) = u{⌧ I(u < 0)}
⇢ Sanches et. al (2013)
yi = xi>
Re-write:
here ✓ =
⌧
+ "i , i = 1, . . . , n
Yi |Ui = ui ⇠ N(xi ⌧ + ✓ui , p⌧2 ui )
Ui ⇠ Exp( ), i = 1, . . . , n
1 2⌧
⌧ (1 ⌧ )
and p⌧2 =
2
⌧ (1 ⌧ )
Back to "Choice of ⌧ -spine"
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September
References
8-1
References
W. K. Härdle, S. Nasekin, D. K. C. Lee , K. F. Phoon
TEDAS - Tail Event Driven Asset Allocation
SFB 649 Discussion Paper 2014-032: Jun, 2014
L. B. Sanchez V. H. Lachos, F. V. Labra
Likelihood Based Inference for Quantile Regression Using the
Asymmetric Laplace Distribution
Discussion Paper, Universidade Estadual de Campinas: Sep,
2013
1.6
Cumulative Return
1.5
1.4
1.3
1.2
1.1
1
0.9
March
May
July
Time
September

TEDAS portfolios: parameters optimisation and out-of - Hu

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Discussion Paper No. 9245 - Forschungsinstitut zur Zukunft der Arbeit