# The Term Structure of Currency Risk Premia

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The Term Structure of Currency Risk Premia

The Term Structure of Currency Risk Premia Hanno Lustig Andreas Stathopoulos Adrien Verdelhan UCLA and NBER USC MIT and NBER June 2013∗ Abstract The returns to the currency carry trade are much smaller at longer maturities. Countries with a high local term premium have an offsetting low or negative currency risk premium. The local term premium in bond markets compensates investors for the risk associated with temporary innovations to the pricing kernel. In the limiting case in which the permanent shocks are fully shared across countries and exchange rates are only driven by temporary innovations, the currency exposure completely hedges the interest rate exposure of the foreign bond portfolio, and the term premium in dollars is identical across countries. The empirical evidence suggests that there is more cross-country sharing of permanent shocks to the pricing kernel than of temporary shocks. That accounts for the downward sloping term structure of currency risk premia. ∗ Please do not quote. Incomplete Lustig: UCLA Anderson School of Management, 110 Westwood Plaza, Suite C4.21, Los Angeles, CA 90095 ([email protected]). Stathopoulos: USC Marshall School of Business, 3670 Trousdale Parkway, Hoffman Hall 711, Los Angeles, CA 90089 ([email protected]). Verdelhan: MIT Sloan School of Management, 77 Massachusetts Avenue E62-621, Cambridge, MA 02139 ([email protected]). 1 Introduction There is no point in chasing high long term bond yields around the world, at least not without hedging currency risk. The returns to the currency carry trade are much smaller at longer maturities. Figure 1 plots the cumulative log returns on an investment strategy that goes long in T-bills of high interest rate currencies and short in T-bills of low interest rate currencies against the returns on 10-year government bond portfolios for the same currencies. The returns on the bond investment strategy are much smaller. Between 1950 and 2012, the average spread in dollar returns between the high and low interest rate portfolios for T-bills is 4.36%, but only a 40 basis points for the 10-year bond portfolios. Countries with a high local term premium have an offsetting low or negative currency risk premium. The portfolio of high interest rate currencies yields a currency risk premium of 2.88% and a local term premium of -51 bps per annum, while the portfolio of low interest rate currencies yields a currency risk premium of -148 bps and a term premium of 198 bps. The dollar term premium, which is the sum of the local currency term premium and the currency risk premium, only differs by 40 bps. No arbitrage implies that foreign currency risk premia are high when there is less risk in those foreign countries’ pricing kernels than at home. If most of this variation in risk concerns temporary shocks to the pricing kernel, then these countries would also have lower term premia in foreign bond markets, because the local term premium in bond markets compensates investors for the risk associated with temporary innovations to the pricing kernel (Alvarez and Jermann (2005) and Hansen and Scheinkman (2009a)). In the limiting case in which the permanent shocks are fully shared across countries and exchange rates are only driven by temporary innovations, the currency exposure completely hedges the interest rate exposure of the foreign bond portfolio, because the exchange rate completely offsets the effect of ‘unshared’ temporary foreign shocks on the foreign bond portfolio. We refer to this as uncovered bond return parity. In this case, the term premium in dollars is identical across countries. Long-run uncovered bond parity is a better fit in the cross-section than in the time series. 1 Figure 1: The Carry Trade Premium and Term Premium 3 HML on Bonds HML on Currencies 2.5 2 1.5 1 0.5 0 −0.5 −1 1943 1957 1971 1984 1998 2012 2026 Cumulative log returns on high-minus-low in Currencies (sorting on monthly T-bill returns at t into 5 portfolios) and high-minus-low in 10-year Bonds (same sorting). Monthly data. 1950.1-2012.12. While we reject long-run uncovered bond return parity in the time series, we do find a secular increase in the sensitivity of foreign long-term bond returns to U.S. bond returns over time, our measure of risk sharing of permanent shocks in international financial markets. After 1991, a 100 basis points increase in U.S. long-term bond returns increases foreign bond returns in dollars by an average of 56 basis points. The exchange rate exposure accounts for a 1/3 to a quarter of this effect: The dollar appreciates on average against a basket of foreign currencies when the U.S. bond returns are lower than average, and vice-versa, except during flight-to-liquidity episodes. While permanent innovations are not completely shared across countries, the empirical evidence suggests that there is substantially more cross-country sharing of permanent shocks to the pricing kernel than of temporary shocks. That accounts for the downward sloping term structure of currency risk premia. An important question in international finance is the extent to which countries leave opportunities for risk sharing unexploited. As pointed out by Brandt, Cochrane, and Santa-Clara (2006), the combination of relatively smooth exchange rates (10% per annum) and much more 2 volatile stochastic discount factors (50% per annum) implies that state prices are highly correlated across countries (at least 0.98). 1 This paper sheds some light on the nature of international risk sharing by decomposing the pricing kernel of each country into a permanent component and a transitory component. Alvarez and Jermann (2005), Hansen and Scheinkman (2009a) and Hansen, Heaton, and Li (2008) have explored the implications of that decomposition for asset prices. From the relative size of the equity premium (large) and the term premium (small), Alvarez and Jermann (2005) infer that almost all the variation in stochastic discount factors arises from permanent fluctuations. By bringing information from the term structure of currency risk premia to bear, we learn that the shocks driving exchange rates and currency risk premia are much less persistent. The bulk of the persistent shocks to the pricing kernel may have been effectively traded away in international financial markets. This result is relevant to economists. The welfare gains from removing all aggregate consumption uncertainty are large, but almost exclusively because of the low frequency component in consumption, not the business cycle component Alvarez and Jermann (2004)). While international risk sharing gains may not have been fully exploited, they may be smaller than commonly assumed. Our paper makes contact with the the vast literature on UIP (Uncovered Interest Rate Parity) and the currency carry trade. We derive general conditions under which long-run UIP follows from no-arbitrage: if all permanent shocks to the pricing kernel are common, then foreign and domestic yield spreads in dollars on long maturity bonds will be equalized, regardless of the properties of the pricing kernel. Chinn and Meredith (2004) have documented some time-series evidence that supports UIP at longer holding periods. In closely related work, Koijen, Moskowitz, Pedersen, and Vrugt (2012) and Wu (2012) examine the currency-hedged returns on ‘carry’ portfolios of international bonds, sorted by a proxy for the carry on long-term bonds, but they do not examine the interaction between currency and term risk premia, the topic of our paper. We focus on portfolios sorted by interest 1 Colacito and Croce (2011) argue that only the persistent component of consumption growth is highly correlated across countries. Our finding provide model-free evidence in support of the view that the bulk of permanent shocks are shared across countries 3 rates, as well as yield spreads. Ang and Chen (2010) show that yield curve variables also forecast currency excess returns, but they do not examine the returns on foreign bond portfolios. Finally, Dahlquist and Hasseltoft (2013) study international bond risk premia in an affine asset pricing model and find evidence for local and global risk factors. Jotikasthira, Le, and Lundblad (2012) report similar findings. Asymmetric exposure to global or common innovations to the pricing kernel are key to understanding the global currency carry trade premium (Lustig, Roussanov, and Verdelhan (2011)). They identify innovations in the volatility of global equity markets as candidate shocks, while Menkhoff, Sarno, Schmeling, and Schrimpf (2012) propose the volatility in global currency markets instead. If these global shocks are temporary, and the permanent shocks are completely shared between countries, then there should be no carry trade premium for portfolios of bonds with long maturities. The downward sloping term structure of currency risk premia lend support to the view that the shocks driving exchange rates and currency returns are much less persistent than the bulk of the innovations to the pricing kernel. The rest of the paper is organized as follows. Section 2 derives the no-arbitrage restrictions imposed on currency and term risk premia. Section 3 describes the data and section 4 explores the correlation and volatility of foreign bond returns at various maturities. Section 5 documents a strong negative relation between currency risk premia and local currency term risk premia in the data. Section 6 directly tests uncovered bond return parity in the time-series and in the cross-section. 2 The Term Premium and the Currency Risk Premium We use Λt to denote the nominal pricing kernel, or the marginal value of a dollar delivered at t in some state of the world $; the nominal stochastic discount factor (SDF) is the growth rate of the pricing kernel (Mt+1 = Λt+1 /Λt ). The price of a zero-coupon bond with maturity k periods into the future is given by Vt [1t+k ] = Et 4 Λt+k Λt (2.1) We define the one-period return on a zero-coupon bond with maturity k as: Rt+1,1 [1t+k ] = We use rxkt+1 to denote the log excess returns log ht [k] = Et Vt+1 [1t+k ] Vt [1t+k ] (2.2) Rt+1,1 [1t+k ] Rt+1,1 [1t+1 ] . Rt+1,1 [1t+k ] log Rt+1,1 [1t+1 ] We define the term premium as: Let us define the yield spread at long maturities: yt [k] = log Vt [1t+1 ] ! (2.3) Vt [1t+k ]1/k If the limits of yt [k] and ht [k] are well defined and the unconditional expectations of holding returns are independent of calendar time, then, according to results derived by Alvarez and Jermann (2005), the average term premium equals the average yield difference: lim E(ht [k]) = lim E(yt [k]). k→∞ k→∞ Hence, the average term premium is the average yield spread at longer maturities. 2.1 Pricing Kernel without Permanent Innovations We start by considering a pricing kernel that is not subject to permanent innovations. Definition 1 The pricing kernel has no permanent innovations if and only if (Alvarez and Jermann (2005)): lim Et log k→∞ Et+1 [Λt+k ] = 0. Et [Λt+k ] Under regularity conditions, this condition is equivalent to limk→∞ 5 Et+1 [Λt+k ] Et [Λt+k ] = 1. Term Premium on Domestic Bonds If there are no permanent innovations to the pric- ing kernel, then the return on the bond with longest maturity equals the inverse of the SDF: limk→∞ Rt+1,1 [1t+k ] = Λt /Λt+1 . High marginal utility growth translates into higher yields on long maturity bonds and low long bond returns, and vice-versa. As a result, this long bond commands the largest possible risk premium in an economy without permanent innovations to the pricing kernel. Proposition 2.1 If the pricing kernel has no permanent innovations, then the term premium is the largest risk premium in the economy (Alvarez and Jermann (2005)). Rt+1,1 [1t+k ] Rt+1 ≥ Et log ht [∞] = lim Et log k→∞ Rt+1,1 [1t+1 ] Rt+1,1 [1t+1 ] for any Rt+1 . Example 2.1 Consider a log-normal model of a pricing kernel: log Λt = ∞ X αi t−i + (t) log β (2.4) i=0 with ∼ N (0, σ 2 ), α0 = 1. It is easy to check that the condition outlined in definition 1 is satisfied provided that limk→∞ αk2 = 0. Proposition 2.2 In example 2.1, the term premium equals one half of the variance (Alvarez and Jermann (2005)). lim k→∞ Et rxkt+1 = lim Et k→∞ Term Premium on Foreign Bonds Rt+1,1 [1t+k ] log = (1/2)σ 2 . Rt+1,1 [1t+1 ] What is the term premium for foreign bonds? We use St to denote the spot exchange rate in pounds per dollar. Similarly, we use Ft to denote the 6 one-period forward exchange rate. The log local currency return on a foreign bond position in excess of the domestic risk-free rate can be restated as the sum of the log excess return in local currency plus the return on a long position in forward currency: ∗ [1t+k ] St Rt+1,1 = rxk,∗ log t+1 + (ft − st ) − ∆st+1 , St+1 Rt+1,1 [1t+1 ] where we have used covered interest rate parity: ∗ Rt+1,1 [1t+1 ] Ft = St Rt+1,1 [1t+1 ] The last two components represent the log excess return on a long position in foreign currency, given by the forward discount minus the rate of depreciation. By taking expectations, we see that the total term premium in dollars consists of a foreign bond risk premium Et [rxk,∗ t+1 ] plus a currency risk premium (ft − st ) − Et ∆st+1 . Proposition 2.3 In example 2.1, the foreign term premium in dollars is identical to the domestic term premium. h∗t [∞] + (ft − st ) − Et [∆st+1 ] = (1/2)σ 2 = ht [∞] In a log-normal model, the currency risk premium equals (1/2) σ ∗,2 − σ 2 . (see Bansal (1997), Bekaert (1996) and Backus, Foresi, and Telmer (2001)). Currencies with a high local currency term premium (high σ 2 ) also have an offsetting negative currency risk premium, while those with a small term premium have a larger currency risk premium. Hence, U.S. investors get the same dollar premium on foreign as on domestic bonds. There is no point in chasing high term premia around the world, at least not in economies with only temporary innovations to the pricing kernel. Currencies with the highest local term premium also have the lowest (i.e. most negative) currency risk premium. This result is equivalent to uncovered interest rate parity for very long holding periods. 7 Corollary 2.1 In example 2.1, the average foreign yield spread in dollars is identical to the domestic yield spread (Long-run UIP): yt∗ [∞] Non-normalities + (ft − st ) − E[ lim (1/k) k X k→∞ ∆st+j ] = (1/2)σ 2 = yt [∞] j=1 This result does not require log-normality. Gavazzoni, Sambalaibat, and Telmer (2012) convincingly argue that higher moments are critical for understanding currency returns. 2 We obtain the same result. Proposition 2.4 If the pricing kernels do not have permanent innovations, the foreign term premium in dollars equal the domestic term premium. h∗t [∞] + (ft − st ) − Et [∆st+1 ] = ht [∞]. The proof is straightforward. In general the foreign currency risk premium is equal to the difference in entropy (see Backus, Foresi, and Telmer (2001)): (ft − st ) − Et [∆st+1 ] = Lt Λt+1 Λt − Lt Λ∗t+1 Λ∗t (2.5) Also in the absence of permanent innovations, the term premium is equal to the entropy of the pricing kernel. The result follows. Actually, it turns out that we can prove a much stronger result. Not only are the risk premia identical. The returns on the foreign bond position are the same to those on the domestic bond position, because the foreign bond position automatically hedges the currency risk exposure. 2 More recently, Brunnermeier, Nagel, and Pedersen (2008) show that risk reversals increase with interest rates. Jurek (2008) provides a comprehensive empirical investigation of hedged carry trade strategies. Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2009) estimate a no-arbitrage model with disaster risk using a cross-section of currency options. Chernov, Graveline, and Zviadadze (2011) study jump risk at high frequencies. 8 We consider the multiplicative one-period excess return of the foreign k-maturity bond over the domestic k-maturity bond, with both returns in domestic currency: ∗ [1t+k ] St Rt+1,1 St+1 Rt+1,1 [1t+k ] Proposition 2.5 If the domestic and foreign pricing kernels have no permanent innovations, then the one-period returns on the foreign longest maturity bonds in domestic currency are identical to the domestic ones Long-run Uncovered Bond Return Parity): ∗ [1t+k ] St Rt+1,1 = 1, k→∞ St+1 Rt+1,1 [1t+k ] lim in all states. In this class of economies, the returns on long-term bonds expressed in domestic currency are equalized. We refer to this uncovered long-run bond return parity. For large maturity k, the spread in log bond returns expressed in domestic currency equals the rate of depreciation: k lim (rx∗,k t+1 − rxt+1 ) ≈ − ((ft − st ) − ∆st+1 ) . k→∞ In countries which experience higher marginal utility growth, the domestic currency appreciates but that is exactly offset by the capital loss on the bond. The foreign bond position automatically hedges the currency exposure. 2.2 Pricing Kernel with Permanent Innovations Following Alvarez and Jermann (2005), Hansen, Heaton, and Li (2008), and Hansen and Scheinkman (2009b), we decompose the pricing kernel into a transitory and a permanent component: Λt = ΛPt ΛTt . 9 (2.6) Definition 2 The transitory component, ΛTt , is defined as δ t+k , k→∞ Vt [1t+k ] ΛTt = lim (2.7) where the constant δ is chosen to satisfy the following regularity condition: 0 < lim k→∞ Vt [1t+k ] < ∞. δk (2.8) The permanent component, ΛPt is a martingale. To see why, note that ΛPt = lim k→∞ Vt [1t+k ] Λt , δ t+k (2.9) This expression is a martingale. That follows directly from the Euler equation for the zero coupon bond with maturity k. The one-period growth rate of transitory SDF components is given by ΛTt+1 Vt [1t+k ] = lim δ T k→∞ Vt+1 [1t+k ] Λt The infinite maturity bond return is given by Rt+1,1 [1t+∞ ] = lim Rt,1 [1t+k ] = k→∞ ΛTt ΛTt+1 We can decompose exchange rate changes into a permanent component and a transitory component, defined below: St+1 = St P Λ∗P t+1 Λt Λ∗P ΛPt+1 t ! T Λ∗T t+1 Λt T Λ∗T t Λt+1 ! = P ST St+1 t+1 StP StT Therefore, we can think of exchange rate changes as capturing differences in both the transitory and the permanent component of the two countries’ stochastic discount factors. We can use returns on long bonds to extract the permanent component of exchange rates. Example 2.2 We consider a log-normal model of the pricing kernel( Alvarez and Jermann 10 (2005)): 1 log ΛPt+1 = − σP2 + log ΛPt + εPt+1 , 2 ∞ X log ΛTt+1 = log β t+1 + αi εTt+1−i , i=0 where α is a square summable sequence, and εP and εT are i.i.d. normal variables with mean zero and covariance σT P . A similar decomposition applies to the foreign stochastic discount factor, where a ? denotes a foreign variable: 1 ?2 ?P ?P log Λ?P t+1 = − σP + log Λt + εt+1 , 2 ∞ X ?t+1 log Λ?T = log β + αi? ε?T t+1 t+1−i . i=0 Proposition 2.6 In example 2.2, the term premium is given by the following expression: ht [∞] = (1/2)σT2 + σT P . Corollary 2.2 In example 2.2, the foreign term premium in dollars is identical to the domestic term premium. h∗t [∞] + (ft − st ) − Et [∆st+1 ] = (1/2)(σ 2 − σP2,∗ ). Provided that σP2,∗ = σP2 , the foreign term premium in dollars equals the domestic term premium: h∗t [∞] + (ft − st ) − Et [∆st+1 ] = (1/2)σT2 + σT P High local currency term premia coincide with low currency risk premia and vice-versa. In the symmetric case, dollar term premia are identical across currencies. 11 Proposition 2.7 In general, the foreign term premium in dollars equal the domestic term premium plus the difference in the entropy of the permanent component of the pricing kernel. h∗t [∞] + (ft − st ) − Et [∆st+1 ] − ht [∞] = Lt ΛPt+1 ΛPt ! − Lt ΛP,∗ t+1 ΛP,∗ t ! . In order to deliver a currency risk premium at longer maturities, entropy differences in the permanent component of the pricing kernel are required. At short maturities, the currency risk premium is determined by the entropy difference of the entire pricing kernel (see equation 2.5). Since carry trade returns are base-currency-invariant, heterogeneity in the exposure of the pricing kernel to a global component of the pricing kernel is required to explain the carry trade premium (Lustig, Roussanov, and Verdelhan (2011)). To deliver a carry trade premium at longer maturities, we would need heterogeneous exposure to a global permanent component. Certainty, the permanent component of the pricing kernel is important. Proposition 2.8 There is a lower bound on the volatility of the permanent component of the pricing kernel (Alvarez and Jermann (2005)): Lt ( ΛPt+1 ) ≥ Et (log Rt+1 ) − Et (log Rt+1,1 [1t+∞ ]) . ΛPt Given the size of the equity premium relative to the term premium, Alvarez and Jermann (2005) conclude that the permanent component of the pricing kernel is large and accounts for most of the risk. Lots of persistence is needed to deliver a low term premium and a high equity premium. 2.3 Measure of Risk Sharing The valuation of long-maturity bonds in bond markets encodes information about the nature of shocks that drive changes in exchange rates in currency markets. Using the prices of longmaturity bonds in two countries, we can decompose the changes in the bilateral spot exchange 12 rate into two parts: a part that captures cross-country differences in the transitory components of the pricing kernel and a part that encodes differences in the permanent components of the the pricing kernel. We assume that the transitory components of the domestic and foreign stochastic discount factors are bounded from below and above: 0< Λ∗T ΛTt+1 t+1 < ∞ and 0 < <∞ T Λt Λ∗T t We consider the one-period multiplicative excess return of the foreign k-maturity bond over the domestic k-maturity bond, where both returns are expressed in domestic currency terms: RXt+1,1 [1k ] ≡ ∗ [1k ] St+1 Rt+1,1 St Rt+1,1 [1k ] As maturity k approaches infinity, we can apply the boundedness condition above and show that differences in the returns of infinite maturity bonds allow us to trace how well countries share risk that arises from permanent innovations in their marginal utility: Proposition 2.9 In two economies with complete markets, the multiplicative excess return on the longest maturity foreign bonds in domestic currency measures the permanent component of exchange rates. RXt+1,1 [1∞ ] ≡ lim RXt+1,1 [1k ] = k→∞ P P St+1 Λ∗P t+1 Λt = Λ∗P ΛPt+1 StP t Corollary 2.3 If the domestic and foreign pricing kernels have common permanent innovations, ΛPt+1 Λ∗P t+1 = ΛPt Λ∗P t for all states, then the one-period returns on the foreign longest maturity bonds in domestic currency are identical to the domestic ones: RXt+1,1 [1∞ ] = 1 for all states. We recover uncovered long-bond return parity. In this polar case, most of the innovations to 13 the pricing kernel are highly persistent, but the shocks that drive exchange rates are not, simply because the persistent shocks are shared more efficiently across countries. Brandt, Cochrane, and Santa-Clara (2006) show that the combination of relatively smooth exchange rates and much more volatile stochastic discount factors implies that state prices are very highly correlated across countries. A 10% volatility in exchange rate changes and a volatility of marginal utility growth rates of 50% implies a correlation of at least 0.98. We can derive a tighter bound on the covariance of the permanent component across different countries. Proposition 2.10 The cross-country covariance of the SDF permanent components is bounded below by: covt ΛPt+1 ΛP ? , log log t+1 ΛPt ? ΛPt ! = 1 [V art 2 ΛP ? log t+1 ΛPt ? ! + V art ΛP log t+1 ΛPt ! − V art (log RXt,1 [1∞ ])] ? ? ≥ Et log Rt+1 − Et log Rt+1,1 [1t+∞ ] + Et (log Rt+1 ) − Et (log Rt+1,1 [1t+∞ ]) − 3 1 V art (log RXt+1,1 [1∞ ]) . 2 Data We use two different panels of panels: a smaller panel of countries consisting of zero coupon prices for the whole yield curve and a larger panel consisting of bond returns for a 10-year bond index. Small Panel First, we construct a small panel of countries with zero coupon bond return data. We use end-of-the-month data for the riskless zero-coupon yield curves, as proxied by the government debt zero-coupon yield curve, of 10 currencies: the US dollar, the German mark (the euro from 1999 onwards), the UK pound, the Japanese yen, the Canadian dollar, the Australian dollar, the Swiss franc, the New Zealand dollar, the Swedish krona and the Norwegian krone. 14 The sample starts in November 1971 and ends in September 2012, but we have full data only for the US dollar; for the rest of the currencies, the sample period is given in Table 1. From November 1971 to May 2009, we use the data in Wright (2011). From June 2009 to September 2011, we source the data from the Bank of International Settlements (for the US, German, Canadian, Swiss and Swedish sovereign debt yield curves) and the Bank of England (for the UK sovereign debt yield curve). For each currency, continuously-compounded yields are available at maturities from 3 months to 120 months (10 years), in 3-month increments. We also collect end-of-the-month data on spot exchange rates against the US dollar from MSCI (available through Datastream) for the same set of countries. Large Panel We also construct a larger panel. We collect data from Global Financial Data for a much larger panel of developed countries and a larger panel that includes all countries. The dataset includes a 10-year Government Bond Total Return Index for each of these countries in dollars and local currency and a T-bill Total Return index. We will use the 10-year bond returns as a proxy for the bonds with the longest maturity. We will check the robustness of our results. The entire sample of countries includes Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Israel, Italy, Japan, Malaysia, Mexico, Netherlands, New Zealand, Norway, Pakistan, Philippines, Poland, Portugal, Singapore, South Africa, Spain, Sweden, Switzerland, Taiwan, Thailand, United Kingdom and the United States. The sample of developed countries includes Australia Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, United Kingdom, and the United States. 4 International Bond Return Correlation and Volatility We use bond return data from the small panel of 10 developed countries to show that longmaturity bond returns expressed in US dollar terms are much more correlated across countries than the dollar returns of short-maturity bonds. We then formally test the long bond par- 15 ity condition and show that unshared permanent innovations do contribute to exchange rate variation. 4.1 Correlation and Maturity of Foreign Bond Returns If international risk sharing is mostly due to countries sharing their permanent pricing kernel fluctuations, we would expect to see that holding period returns on zero-coupon bonds, once converted to a common currency (the US dollar, in particular), become increasingly similar as bond maturities approach infinity. To determine whether this hypothesis has merit, we calculate the correlation coefficient between one-period nominal USD returns on foreign bonds and corresponding returns on US bonds for bonds of maturity ranging from 1 year to 10 years. To determine whether the patterns in correlation coefficients arise as a result of exchange rate properties, we also calculate the correlation coefficient between one-period nominal local currency returns on foreign bonds and corresponding returns on US bonds. The results for both USD and local currency returns, regarding overlapping 3-month holding periods, are presented in Figure 2. As we can see, all 9 foreign currency yield curves exhibit the conjectured pattern: correlation coefficients for USD returns start from negative or zero values and increase monotonically with bond maturity, tending towards 1 for long-term bonds. Interestingly, this striking monotonicity is not observed if we focus on local currency returns. The local currency 3-month return correlations do not exhibit any discernible pattern with maturity, implying that the convergence of USD return correlations towards 1 results from exchange rate changes which partially offset differences in local currency bond returns. 4.2 Volatility and Maturity of Foreign Bond Returns To further explore our intuition that USD bond returns of different countries become increasingly similar as bond maturity increases, we also calculate the ratio of foreign to domestic USD bond 16 Figure 2: The Maturity structure of Bond Return Correlations 10 1 0.5 0 2 4 6 8 Maturity (in years) NZD 10 1 0.5 0 2 4 6 8 Maturity (in years) 10 0 2 4 6 8 Maturity (in years) AUD 10 1 0.5 0 2 4 6 8 Maturity (in years) SEK 10 1 0.5 0 2 4 6 8 Maturity (in years) 10 Correlation coefficient 4 6 8 Maturity (in years) CAD 0.5 Correlation coefficient 2 JPY 1 Correlation coefficient 0 Correlation coefficient Local USD 0.5 Correlation coefficient GBP Correlation coefficient Correlation coefficient Correlation coefficient Correlation coefficient NOK 1 1 0.5 0 2 4 6 8 Maturity (in years) CHF 10 2 4 6 8 Maturity (in years) NOK 10 2 4 6 8 Maturity (in years) 10 1 0.5 0 1 0.5 0 Panel B: Developed Countries Notes: Correlation with U.S. bond returns. Sample: country-dependent. Monthly data. Holding period is 3-months. returns; the k-year maturity volatility ratio is given by V olRk = ∗ + ∆st+1 σ rt+1,k σ (rt+1,k ) For comparison, we also calculate the corresponding volatility ratio for local currency returns, given by V olRkL = ∗ σ rt+1,k σ (rt+1,k ) and report both V olRk and V olRkL for k = 1, 2, ...10 years in Figure 3 for 3-month returns. The pattern is unambiguous: the unconditional volatility of the USD 3-month foreign returns is much higher than that of the corresponding volatility of US bond returns for small maturities, but the volatility ratio falls sharply for higher maturities and is close to 1 for 10-year bonds. 17 In contrast with the observed pattern for V olRk , the local currency volatility ratio V olRkL is virtually flat with maturity, implying that the convergence in USD return bond volatility is due to the properties of the nominal exchange rate. Of course, even if exchange rates followed a random walk and exchange rate innovations are uncorrelated with returns, we could still observe this pattern, simply the exchange rates account for a smaller share of overall return volatility at longer maturities. However, we will show that exchange rates actually hedge interest rate risk. Figure 3: The Maturity structure of Bond Return Volatility NOK GBP 5 2 4 6 8 Maturity (in years) CAD 2 4 6 8 Maturity (in years) NZD 2 4 6 8 Maturity (in years) SEK 4 6 8 Maturity (in years) 10 10 2 4 6 8 Maturity (in years) NOK 10 2 4 6 8 Maturity (in years) 10 10 5 0 4 6 8 Maturity (in years) CHF 5 0 10 Volatility ratio Volatility ratio 2 2 10 10 5 5 0 10 5 0 10 10 Volatility ratio 4 6 8 Maturity (in years) AUD Volatility ratio 5 0 2 10 Volatility ratio Volatility ratio 10 0 5 0 10 10 Volatility ratio Local USD 0 JPY 10 Volatility ratio Volatility ratio 10 2 4 6 8 Maturity (in years) 10 5 0 Notes: Volatility of Foreign and U.S. bond returns. Sample: country-dependent. Monthly data. Holding period is 3-months. Our results are robust to an increase of the holding period. Specifically, in unreported results, 6-month and 12-month returns produce the same patterns: for both holding periods and for virtually all currencies, there is an almost monotonic relationship between correlation coefficients of USD returns and bond maturity. Furthermore, 6-month and 12-month local currency return correlations are not sensitive to maturity, the USD return volatility ratio is very high for short maturities, but quickly converges towards 1, and the local currency return 18 volatility ratio is flat with maturity. In sum, the behavior of USD bond returns and local currency bond returns differs markedly as bond maturity changes. While USD bond returns become more correlated and roughly equally volatile across countries as the maturity increases, the behavior of local currency returns do not appear to change when bond maturity changes. 5 The Maturity Structure of Currency Carry Trade Returns To test the predictions of the theory, we sort currencies into portfolios based on variables that predicts bond and currency returns. Interest rates predict currency returns. Hence, from the perspective of the theory, sorting by interest rates is equivalent to sorting by the total entropy of the foreign pricing kernel relative to the U.S. 5.1 Sorting Currencies by Interest Rates Following the work of Lustig and Verdelhan (2005), we start by sorting currencies into 5 portfolios based on the interest rate differences. We use the one-month return on the GFD Treasury-bills Total Return index that is realized at t − 1 to sort currencies into portfolios at time t.3 Then, we compute monthly returns between t and t + 1. The portfolios are rebalanced each month. In the large sample of countries, we have 5 to 6 currencies in each portfolio. Table 1 reports the annualized moments of log returns. The top panel uses the entire sample. The first three rows report moments of currency excess returns rxf x = (f − s) − ∆s. As we expected (see Lustig and Verdelhan (2005)) for detailed analysis, average excess returns increase from the first portfolio to the last portfolio. The average excess return on the first portfolio is -73 basis points per annum. The average excess return on the last portfolio is 269 basis points. The spread between the first and the last portfolio is 341 basis points per annum. The volatility of these returns increases only slightly from the first to the last portfolio. As a result, the Sharpe ratio (annualized) increases from -0.12 to 0.34 on the last portfolio. The Sharpe ratio on a long 3 We are being conservative by sorting on the T-bill return at t − 1. The Tbill return at t is largely known at t − 1. 19 position in the last portfolio and a short position in the first portfolio is 0.54 per annum. The results for the post-Bretton-Woods sample in the bottom panel are very similar. Hence, the currency carry trade is profitable at the short end of the maturity spectrum. Table 1: Interest-Rate Sorted Portfolios: All Countries 1 2 3 4 Panel A: 1950-2012 1.85 1.85 6.98 7.19 0.27 0.26 5 5-1 2.69 7.89 0.34 3.41 6.28 0.54 rxf x Mean Std SR -0.73 6.24 -0.12 0.60 7.07 0.09 rx? Mean Std SR 2.49 3.43 0.73 1.45 4.71 0.31 1.78 3.80 0.47 1.76 3.94 0.45 0.45 5.34 0.08 rx$ Mean Std SR 1.76 7.52 0.23 2.06 8.94 0.23 3.63 8.22 0.44 3.61 8.24 0.44 3.14 9.02 0.35 ∆s f −s r? − rU S Mean Mean Mean 0.79 -1.52 -0.54 0.43 0.18 0.05 rxf x Mean Std SR -0.50 7.53 -0.07 0.85 8.64 0.10 rx? Mean Std SR 3.14 4.06 0.77 2.22 5.63 0.39 2.06 4.49 0.46 2.55 4.48 0.57 0.78 5.26 0.15 rx$ Mean Std SR 2.63 9.04 0.29 3.07 10.85 0.28 4.43 9.79 0.45 4.57 9.88 0.46 4.27 10.73 0.40 ∆s f −s r? − rU S Mean Mean Mean 1.30 -1.80 -1.23 0.61 0.24 -0.11 0.91 1.46 0.95 -1.23 3.25 3.23 -3.86 7.35 5.57 0.61 -0.87 1.24 2.72 1.44 2.89 Panel B: 1971-2012 2.37 2.02 8.33 8.75 0.28 0.23 1.38 8.03 0.17 -2.95 5.64 4.58 3.49 8.89 0.39 3.99 6.66 0.60 1.64 9.44 0.17 Portfolios of currencies sorted at t − 1 by returns on T-bills realized at end of t − 1. Annualized monthly returns realized at t on 10-year Bond Index and T-bills. As we know from Bekaert (1996), Bansal (1997) and Backus, Foresi, and Telmer (2001), The currency risk premia reflect differences in the entropy of the domestic and the foreign pricing kernels: (ft − st ) − Et [∆st+1 ] = Lt 20 Λt+1 Λt − Lt Λ∗t+1 Λ∗t . Table 2: Interest-Rate Sorted Portfolios: Developed Countries 1 2 3 4 Panel A: 1950-2012 1.64 2.35 7.71 7.65 0.21 0.31 5 5-1 2.94 8.18 0.36 3.01 5.88 0.51 rf x Mean Std SR -0.06 7.22 -0.01 0.97 7.75 0.12 rx Mean Std SR 2.04 3.75 0.55 1.80 5.68 0.32 1.61 4.62 0.35 0.96 4.35 0.22 0.39 5.30 0.07 rx$ Mean Std SR 1.98 8.75 0.23 2.76 9.91 0.28 3.24 9.39 0.35 3.30 8.97 0.37 3.34 9.36 0.36 ∆s f −s r − rU S Mean Mean Mean 1.22 -1.28 -0.75 0.56 0.41 0.69 rf x Mean Std SR 0.28 8.80 0.03 1.32 9.47 0.14 rx Mean Std SR 2.58 4.44 0.58 2.65 6.82 0.39 1.89 5.39 0.35 1.40 4.88 0.29 1.18 6.10 0.19 rx$ Mean Std SR 2.86 10.60 0.27 3.97 12.03 0.33 3.80 11.34 0.33 4.25 10.39 0.41 4.70 11.02 0.43 ∆s f −s r − rU S Mean Mean Mean 1.75 -1.47 -1.45 0.77 0.55 0.64 0.38 1.52 0.85 -0.03 2.88 1.72 -2.17 5.70 4.31 0.31 -0.21 1.33 2.56 1.43 2.00 Panel B: 1971-2012 1.90 2.84 9.41 8.97 0.20 0.32 1.36 8.01 0.17 -1.93 4.88 3.76 3.52 9.74 0.36 3.24 6.82 0.48 1.84 9.27 0.20 Portfolios of currencies sorted at t − 1 by returns on T-bills realized at end of t − 1. Annualized monthly returns realized at t on 10-year Bond Index and T-bills. 21 High interest rate currencies have low entropy and low interest rate currencies have high entropy. This follows directly from no-arbitrage. In a log-normal world, entropy is just one half of the variance. In that case, high interest rate currencies have low variance of the pricing kernel, while low interest rate currencies have high variance of the pricing kernel. Hence, sorting by interest rates (from low to high) seems equivalent to sorting by pricing kernel entropy (from high to low). The next three rows report the excess return rx? on 10-year bond positions in each of these currencies. To be clear, these returns are reported in local currency. There is a strong decreasing pattern in local currency bond risk premia. The average excess return on the first portfolio is 249 basis points per annum. These excess returns decrease monotonically to 45 basis points on the last portfolio. Hence, there is a 294 basis points spread per annum between the first and the last portfolio. The Sharpe ratio on the first portfolio is 0.73. Hence, there is a very strong negative correlation between local currency bond risk premia and currency risk premia. Low interest rate currencies tend to produce high local currency bond risk premia, while high interest rate currencies tend to produce low local currency bond risk premia. In the absence of arbitrage, we know that the foreign term premium in local currency is given by: ?,P ? h?t (∞) = lim Et rxk,? t+1 = Lt (mt+1 ) − Lt (mt+1 ). k→∞ (5.1) Hence, the decreasing term premia are consistent with the decreasing entropy Lt (m?t+1 ) from the low interest rate portfolio 1 to the high interest rate portfolio 5 that we had inferred from the foreign currency risk premia. These are apparently not offset by equivalent increases in the entropy of the entropy of the permanent component of the foreign pricing kernel. In a log-normal world, the term premium is determined by (1/2) of the variance of the temporary component of the pricing kernel plus a covariance term. Hence, if the currency risk premia are driven (to seem extent) by the variance of the temporary component, that would explain why term premia are high for low interest rate currencies, with high currency risk premia. The monotonically decreasing pattern in term risk premia is direct evidence in favor of a risk-based explanation of foreign currency returns. Bond markets agree with currency markets 22 that there is more temporary risk in the pricing kernel of low interest rate currencies. Hence, temporary shocks to the pricing kernel play a major role as drivers of currency risk premia. If all of the shocks driving currency risk premia were permanent, then there would be no relation between currency risk premia and term premia. A natural question is whether U.S. investors can ‘combine’ the currency risk premium and the bond risk premium. To compute the dollar bond excess returns rx$ , we simply add the currency excess returns rxf x = (f − s) − ∆s and the local currency bond returns rx? . The results are reported in the next three rows. The decline in the local currency bond risk premia partly offsets the increase in currency risk premia. As a result, the average excess return on the last portfolio is only 138 basis points per annum higher than the returns on the first portfolio. The SR on a long-short position in bonds of the last and the first portfolio is only 0.17. U.S. investors cannot simply combine the currency carry trade with a yield carry trade, because these risk premia roughly offset each other. Interest rates are great predictors of currency excess returns and local currency bond excess returns, but not dollar excess returns. To get long-term carry trade returns, we need differences in the quantity of permanent risk, as can be verified from: h∗t [∞] + (ft − st ) − Et [∆st+1 ] − ht [∞] = Lt ΛPt+1 ΛPt ! − Lt ΛP,∗ t+1 ΛP,∗ t ! . The data do not seem to lend support to these differences in permanent risk. These results are essentially unchanged in the post-Bretton-Woods sample. The Sharpe ratio on the currency carry trade is 0.60, achieved by going long int he last portfolio and short in the first portfolio. However, there is a strong decreasing pattern in local currency bond risk premia, from 314 basis points per annum in the first portfolio to 78 basis points in the last portfolio. As a result, there is essentially no discernible pattern in dollar bond risk premia. Table 2 excludes non-developed countries and performs the same spring exercise. These moments look similar. The Sharpe ratio on the carry trade strategy is lower (0.51). We see a monotonically increasing pattern in local currency bond risk premia. The spread in returns is 243 basis points, but this spread shrinks to 136 basis points in dollars, because of the offsetting effects of the currency risk premia. 23 Sorting by Contemporaneous T-bill returns Since the construction of the total return index by Global Financial Data assumes the Tbill-price does not change, we could also use the return realized at t to sort currencies into portfolios of currencies at t − 1, because the return at t would be known at t − 1. The results are reported in Table 3. The annualized spread in currency risk premia between the first and the last portfolio is even larger: 436 bps per annum. The returns increase from minus 148 bps on the first portfolio to 288 bps per annum on the last portfolio. The SR on the long-short strategy is 0.68. The spread between the first and the last portfolio in local currency term premia is 397 bps per annum. Hence, the term spread is almost the same order of magnitude as the currency risk premium. As a result, the dollar term risk premium spread is only 40 bps per annum. Hence, the results on these portfolios sorted by returns at t are even starker. The term premia almost completely offset the currency risk premia. We replicate the same portfolio-building exercise on the subsample of developed countries, partly to guard against the possibility of credit risk contaminating our findings. These are reported in 4. Figure 6 and 7 depict the currency risk premia and the local currency bond risk premia. In both samples, there is a strong negative relation between these risk premia. Low interest rate currencies tend to have high entropy of the pricing kernel. This also leads to higher bond risk premia, as one would expect. 5.2 Sorting Currencies by Slope of the Yield Curve We also sorted currencies into portfolio by the slope of the yield curve in each country. Recently, Ang and Chen (2010) have documented that the slope of the yield curve adds additional foresting power for currency excess returns. We use the yield on the 10-year government bonds at t − 1 minus the T-bill rate at t − 1 to sort currencies into portfolios at t − 1. Then we compute returns at t. Table 5 reports the annualized moments of log returns on these portfolios. The slope of the yield curve, a measure of the term premium, is determined largely by the entropy of the temporary component of the pricing kernel (see equation 5.1). As this increases, 24 Table 3: Interest-Rate Sorted Portfolios: All Countries 1 2 3 4 Panel A: 1950-2012 1.85 2.45 7.09 7.02 0.26 0.35 5 5-1 2.88 7.37 0.39 4.36 6.40 0.68 rf x Mean Std SR -1.48 6.89 -0.21 0.56 7.06 0.08 rx Mean Std SR 3.46 4.37 0.79 1.90 4.46 0.43 1.46 4.06 0.36 1.33 3.86 0.34 -0.51 4.59 -0.11 rx$ Mean Std SR 1.98 7.50 0.26 2.46 8.84 0.28 3.31 8.41 0.39 3.78 8.13 0.47 2.38 9.13 0.26 ∆s f −s r? − rU S Mean Mean Mean 0.47 -1.95 0.00 0.41 0.15 0.55 rf x Mean Std SR -1.05 7.59 -0.14 0.76 8.64 0.09 r Mean Std SR 3.85 4.04 0.95 2.76 5.33 0.52 1.83 4.70 0.39 2.04 4.50 0.45 -0.05 5.29 -0.01 r$ Mean Std SR 2.80 9.01 0.31 3.52 10.74 0.33 4.36 9.98 0.44 4.74 9.80 0.48 3.22 10.83 0.30 ∆s f −s r? − rU S Mean Mean Mean 0.86 -1.90 -0.62 0.55 0.21 0.41 1.10 1.44 0.70 -0.54 3.24 2.71 -4.29 7.57 4.96 0.63 -0.25 1.22 2.70 1.17 2.52 Panel B: 1971-2012 2.54 2.70 8.47 8.56 0.30 0.32 0.40 8.19 0.05 -3.27 6.15 4.14 3.28 8.83 0.37 4.33 6.77 0.64 0.43 9.62 0.04 Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthly returns realized at t on 10-year Bond Index and T-bills. 25 Table 4: Interest-Rate Sorted Portfolios: Developed Countries 1 2 3 4 Panel A: 1950-2012 1.77 2.47 7.71 7.66 0.23 0.32 5 5-1 2.92 8.20 0.36 3.33 5.83 0.57 rxf x Mean Std SR -0.41 7.17 -0.06 1.03 7.79 0.13 rx? Mean Std SR 2.95 3.89 0.76 2.29 5.34 0.43 0.80 4.78 0.17 0.97 4.43 0.22 -0.77 5.39 -0.14 rx$ Mean Std SR 2.54 8.76 0.29 3.32 9.87 0.34 2.57 9.39 0.27 3.44 9.03 0.38 2.15 9.60 0.22 ∆s f −s r? − rU S Mean Mean Mean 0.92 -1.33 0.11 0.65 0.38 1.16 rxf x Mean Std SR -0.20 8.74 -0.02 1.38 9.53 0.14 rx? Mean Std SR 3.68 4.62 0.80 3.49 6.31 0.55 0.68 5.68 0.12 1.33 4.98 0.27 -0.13 6.16 -0.02 rx$ Mean Std SR 3.48 10.63 0.33 4.87 11.93 0.41 2.79 11.36 0.25 4.34 10.53 0.41 3.42 11.26 0.30 ∆s f −s r? − rU S Mean Mean Mean 1.33 -1.53 -0.42 0.86 0.51 1.44 0.60 1.51 -0.37 0.11 2.91 1.67 -2.29 5.83 3.14 0.45 -0.10 1.32 2.58 0.61 2.03 Panel B: 1971-2012 2.11 3.01 9.38 9.04 0.23 0.33 -0.39 8.16 -0.05 -2.06 4.98 2.70 3.54 9.72 0.36 3.74 6.74 0.55 -0.06 9.44 -0.01 Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. Annualized monthly returns realized at t on 10-year Bond Index and T-bills. 26 the local term premium increases as well. However, the dollar term premium only compensates investors for the relative entropy of the permanent component of the U.S. and the foreign pricing kernel. In the extreme case in which all permanent shocks are common, the dollar term premium equals the U.S. term premium. The first three rows repots the moments of the currency excess returns. These decline from an average of 232 bps per annum on the first portfolio to -96 bps per annum on the fifth portfolio. A long-short position delivers an excess return of -327 bps per annum and a Sharpe ratio of 0.45. This confirms the findings of Ang and Chen (2010). The slope of the yield curve predicts currency excess returns. These findings confirm that the entropy of the temporary component plays a large role in currency risk premia. The next three rows report the local currency bond returns. As expected, the highest slope portfolios produce large bond excess returns of 5.11 percent per annum, compared to -179 basis points per annum on the first portfolio. Hence, a long-short position produces a spread of 690 basis points per annnum. The next three rows report dollar returns. In dollars, this 690 spread is reduced to 363 basis points, because of the partly offsetting pattern in currency risk premia. What is driving these results? The high slope currencies tend to be low interest rate currencies, while the low slope currencies tend to be the high interest rate currencies, as is apparent from the last four rows in each the top panel. The first portfolio has an average slope of -86 bps and an interest rate difference of 386 bps relative to the U.S., while the last portfolio has a slope of 379 bps, and a negative interest rate difference of -54 bps per annum. These findings confirm that currency risk premia are driven to a large extent by temporary shocks to the pricing kernel. We also sorted currencies into portfolio at t − 1 based on the yield at t − 1 minus the T-bill returns that is realized at t. These results are reported in Table 7 and Table 8. We observe the same negative correlation between currency and term risk premia. The spread in local term premia is 953 basis points when we include all countries. Of course, a large portion of this spread is due to credit risk, because we’re sorting by the slope of the yield curve, provided that the term structure of credit risk premia is upward sloping. This spread gets reduced by 338 bps per 27 Table 5: Slope-sorted Portfolios: Developed Countries 1 2 3 4 Panel A: 1950-2012 1.82 0.50 7.28 7.53 0.25 0.07 5 5-1 -0.96 7.89 -0.12 -3.27 7.20 -0.45 rxf x Mean Std SR 2.32 7.03 0.33 1.51 7.04 0.22 rx? Mean Std SR -1.79 3.76 -0.48 1.15 3.64 0.32 1.86 4.27 0.44 2.66 4.84 0.55 5.11 6.95 0.74 rx$ Mean Std SR 0.52 8.18 0.06 2.66 7.86 0.34 3.67 8.74 0.42 3.16 9.18 0.34 4.16 8.49 0.49 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.86 -1.54 3.86 0.56 0.74 -0.21 1.73 1.36 rxf x Mean Std SR 2.57 8.55 0.30 1.89 8.57 0.22 rx? Mean Std SR -1.79 4.38 -0.41 2.12 4.23 0.50 2.31 5.04 0.46 3.41 5.67 0.60 5.28 5.56 0.95 rx$ Mean Std SR 0.78 9.86 0.08 4.01 9.46 0.42 4.68 10.52 0.45 4.15 10.84 0.38 4.99 10.06 0.50 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -1.19 -2.06 4.63 0.28 0.64 -0.15 2.04 1.59 1.31 1.00 1.37 1.12 1.91 0.17 0.57 1.41 3.31 -0.50 0.21 2.92 1.36 1.97 0.70 0.08 1.11 0.42 1.46 1.58 Panel B: 1971-2012 2.38 0.74 8.82 8.91 0.27 0.08 3.63 7.61 0.48 3.79 -0.41 -0.54 3.06 -0.29 7.23 -0.04 -2.86 6.00 -0.48 4.21 8.92 0.47 Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year Bond Index and T-bills. 28 Table 6: Slope-sorted Portfolios: Developed Countries 1 2 3 4 Panel A: 1950-2012 1.04 0.41 7.87 7.76 0.13 0.05 5 5-1 0.28 7.00 0.04 -2.39 4.93 -0.49 rxf x Mean Std SR 2.68 7.48 0.36 2.37 7.35 0.32 rx? Mean Std SR -1.08 4.21 -0.26 1.51 4.15 0.36 1.69 4.56 0.37 2.26 5.33 0.42 2.96 7.31 0.40 rx$ Mean Std SR 1.59 8.52 0.19 3.88 8.43 0.46 2.73 9.21 0.30 2.67 9.92 0.27 3.24 11.06 0.29 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.58 -0.79 3.47 0.87 0.78 0.71 1.65 1.65 rxf x Mean Std SR 3.09 9.08 0.34 3.07 8.95 0.34 rx? Mean Std SR -0.74 4.87 -0.15 2.64 4.81 0.55 2.29 5.33 0.43 2.82 6.31 0.45 3.15 8.69 0.36 rx$ Mean Std SR 2.35 10.22 0.23 5.71 10.12 0.56 3.61 10.97 0.33 3.43 11.79 0.29 3.44 13.33 0.26 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.79 -0.88 3.97 0.66 0.72 1.21 1.86 1.94 1.27 0.08 1.24 0.97 1.86 -0.24 0.85 1.11 3.18 0.06 0.23 0.81 1.31 1.90 -0.04 -0.22 1.08 0.63 1.26 1.38 Panel B: 1971-2012 1.32 0.61 9.46 9.22 0.14 0.07 1.65 9.51 0.17 3.14 0.14 0.14 1.59 0.29 8.53 0.03 -2.80 5.91 -0.47 1.09 11.29 0.10 Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1. Monthly returns at t on 10-year Bond Index and T-bills. 29 annum when we convert the returns into dollars. When we look at developed countries only, the spread in local term premia of 727 bps is reduced to 413 basis points, because the spread in currency risk premia is 388 bps per annum. This spread in currency risk premia increases to 388 bps per annum in the post-Bretton-Woods sample. Table 7: Slope-sorted Portfolios: All Countries 1 2 3 4 Panel A: 1950-2012 1.39 0.57 7.18 7.53 0.19 0.08 5 5-1 -0.99 8.00 -0.12 -3.38 7.31 -0.46 rxf x Mean Stdev SR 2.39 6.97 0.34 1.95 6.98 0.28 rx? Mean Stdev SR -2.95 3.75 -0.79 0.32 3.83 0.08 1.59 4.14 0.38 3.62 4.72 0.77 6.58 6.97 0.94 rx$ Mean Stdev SR -0.55 8.14 -0.07 2.27 8.06 0.28 2.98 8.40 0.35 4.18 9.22 0.45 5.59 8.60 0.65 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.96 -1.57 3.97 -0.49 0.72 0.20 1.75 0.56 1.38 0.31 1.08 1.16 2.00 0.17 0.40 2.50 3.89 -0.36 -0.63 4.44 Panel B: 1971-2012 1.78 1.01 8.68 8.92 0.21 0.11 rxf x Mean Stdev SR 2.70 8.45 0.32 2.45 8.52 0.29 rx? Mean Stdev SR -3.11 4.40 -0.71 0.89 4.39 0.20 1.79 4.90 0.37 4.59 5.54 0.83 7.22 5.63 1.28 rx$ Mean Stdev SR -0.42 9.79 -0.04 3.33 9.70 0.34 3.58 10.10 0.35 5.59 10.89 0.51 6.74 10.23 0.66 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -1.33 -2.08 4.77 -0.90 0.63 0.38 2.07 0.39 1.32 0.45 1.33 0.56 1.94 0.46 0.55 2.57 3.43 -0.59 0.10 4.76 -0.48 7.44 -0.07 6.14 8.06 0.76 -3.18 6.16 -0.52 7.16 9.46 0.76 Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1 (defined as yield at t − 1 minus return on T-bill at t). Monthly returns at t on 10-year Bond Index and T-bills. 30 Table 8: Slope-sorted Portfolios: Developed Countries 1 2 3 4 Panel A: 1950-2012 0.93 0.59 7.84 7.69 0.12 0.08 5 5-1 -0.17 7.08 -0.02 -3.14 5.02 -0.63 rxf x Mean Std SR 2.97 7.46 0.40 2.41 7.38 0.33 rx? Mean Std SR -2.39 4.16 -0.58 0.64 4.28 0.15 1.14 4.57 0.25 3.22 5.24 0.61 4.88 7.28 0.67 rx$ Mean Std SR 0.57 8.52 0.07 3.04 8.56 0.36 2.07 9.10 0.23 3.81 9.83 0.39 4.71 11.10 0.42 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.68 -0.59 3.56 -0.35 0.77 0.74 1.67 0.79 rxf x Mean Std SR 3.54 9.02 0.39 2.97 9.02 0.33 rx? Mean Std SR -2.28 4.86 -0.47 1.54 4.81 0.32 1.15 5.43 0.21 4.25 6.07 0.70 5.68 8.74 0.65 rx$ Mean Std SR 1.26 10.19 0.12 4.51 10.24 0.44 2.28 10.88 0.21 5.25 11.63 0.45 5.33 13.45 0.40 y 10 − y 1 ∆s f −s r − rU S Mean Mean Mean Mean -0.91 -0.54 4.08 -0.76 0.71 1.08 1.89 0.87 1.29 -0.16 1.30 -0.12 1.89 0.20 0.80 2.49 3.29 -0.42 0.08 3.19 1.33 1.93 -0.18 0.00 1.11 0.59 0.74 2.30 Panel B: 1971-2012 1.13 0.99 9.40 9.15 0.12 0.11 4.13 9.64 0.43 3.22 -0.21 0.04 3.41 -0.34 8.62 -0.04 -3.88 5.99 -0.65 4.07 11.54 0.35 Portfolios of currencies sorted at t − 1 by slope of yield curve at t − 1 (defined as yield at t − 1 minus return on T-bill at t). Monthly returns at t on 10-year Bond Index and T-bills. 31 6 Testing Uncovered Bond Return Parity This section directly tests the Uncovered Bond Return Parity Condition. Uncovered bond return parity should hold for long bonds provided that countries share the permanent component. If the permanent component of the pricing kernel is common, then exchange rate exactly hedge the foreign interest rate risks in long foreign bond position, because exchange rates respond only to temporary innovations to the pricing kernels. These are the innovations driving long-term bond prices and yields. 6.1 Testing Uncovered Bond Return Parity in the Cross-section The last three rows in Table 3 decompose the results for the portfolios sorted by returns at t − 1. The currency excess return equals the interest rate difference minus the rate of depreciation (f − s) − ∆s. The rate at which the high interest rate currencies depreciate (327 bps per annum) is not high enough to offset the interest rate difference 615 bps. Similarly, the rate at which the low interest rate currencies appreciate (47 bps per annum) is not high enough to offset the low interest rates (minus 195 bps). UIP fails in the cross-section. However, the bond return differences (in local currency) are closer to being offset by the rate of depreciation. The bond return spread is 414 bps per annum for the last portfolio, compared to an annual depreciation rate of 327 bps, while the spread on the first portfolio is 0 bps, compared to depreciation of 47 bps. Figure 8 plots the rate of depreciation against the interest rate (bond return) differences with the U.S. The vertical distance from the 45-degree line is an indication of how far we are from UIP or long-run UBRP. Especially for portfolio 1 and portfolio 5, UBRP is a much better fit for the data. The currency exposure hedges the interest rate exposure in the bond position. High returns are off-set by higher depreciations. As a result, foreign bond portfolios are almost hedged against foreign interest rate risk. 6.2 Testing Uncovered Bond Return Parity in the Time-Series Alternatively, we could check whether bond return parity holds in the time series. To the extent the 10-year bond is a reasonable proxy for the infinite-maturity bond, uncovered long-bond 32 parity implies that the unconditional USD 10-year bond returns are not statistically different. To determine whether exchange rate changes completely eliminate differences in countries’ permanent SDF component, we test the long-bond return parity condition by regressing nominal USD holding period returns on 10-year foreign bonds on corresponding USD returns on 10-year US bonds: $ US rt+1,10 + ∆st+1 = α + βrt+1,10 + t+1 , where small letters denote the log of their capital letter counterpart. Uncovered long-bond parity implies α = 0 and β = 1. We run the same regression for the local currency bond returns (in logs) and the change in the exchange rates r$ , r? and ∆s on the U.S. bond return rU S . The sum of the local currency and the FX beta equal the total dollar bond return beta. Large Panel of Countries Table 9 reports the results for the entire sample in Panel A. Panel B and C report the results for the post-Bretton-Woods sample and for 1991-2012. We report the regression coefficients for the log local currency returns, the log exchange rate changes and for the log dollar returns on foreign bonds. The sum of the local currency coefficient (first two rows) and the exchange rate coefficient produces the dollar return coefficient in the last two rows. First, the average sloe coefficient for dollar returns is increasing over time for most of the countries in the sample. For the whole sample, the average is 0.38. This number increases to 0.43 in the post-Bretton-Woods sample and to 0.56 in the sample that starts in 1991. More than 50% of the permanent shocks are shared with the U.S. For some countries, the number is closer to 75%. The exchange rates account for up to 1/3 of this coefficient. When dollar returns are higher than average, the dollar tends to depreciate relative to other currencies. When dollar returns are lower than average, the dollar tends to appreciate relative to other currencies. Hence, exchange rates actively enforce long-run uncovered bond return parity. Interestingly, the AUD, the NZD, the NOK and to some extent the CAD are the main exceptions. We find negative slope coefficients in these currencies. These are positive carry currencies (with on average high 33 interest rates) of countries that are commodity exporters. To learn more about the time-variation in those coefficients, we use an equal-weighted portfolio of all currencies. We run a regression of average returns on U.S. bond returns. Figure 10 plots the 60-month rolling window of the regression coefficients for the basket of developed currencies. There are large increases in the dollar beta after the demise of the Bretton-Woods regime, mostly driven increases in the exchange rate betas, as well around the early 90s. In addition, there is a secular increase in the local return betas over the entire sample. There is clear evidence that the currency exposure hedges the interest rate exposure of the foreign bond position. When U.S. bond returns are higher than usual, the dollar depreciates on average, relative to all foreign currencies. There are two main exceptions: the LTCM crisis in 1998 and the recent financial crisis. During these episodes, the dollar appreciated even though U.S. bond returns were higher than usual. Small Panel of Countries Using the zero coupon bond returns, we test the long-bond return parity condition by regressing nominal USD holding period returns on 10-year foreign bonds on corresponding USD returns on 10-year US bonds: ∗ rt+1,10 + ∆st+1 = α + βrt+1,10 + t+1 , where small letters denote the log of their capital letter counterpart. Uncovered long-bond parity implies α = 0 and β = 1. We test the two hypotheses separately, as well as jointly, and present the results for 3-month, 6-month and 12-month holding period returns in Table 2. We report both Newey-West standard errors (with 12 lags) and bootstrap standard errors. A mixed pattern emerges: we can mostly reject the null hypothesis of long-bond return parity of US and foreign bonds for Germany, the UK, Canada, Australia and Switzerland, while we are unable to reject the parity condition between US and Japapese, New Zealand, Swedish and Norwegian bonds. Overall, permanent exchange rate components appear to non-trivially contribute to nominal exchange rate variation. 34 35 0.30 0.36 0.47 0.65 0.47 (0.05) (0.05) (0.05) (0.04) (0.05) 0.19 0.21 0.17 0.15 0.23 0.21 -0.06 0.07 (0.05) (0.06) (0.04) (0.04) (0.05) (0.05) (0.05) (0.04) 0.69 0.05 0.55 0.07 0.41 0.07 0.53 0.06 0.61 0.07 0.19 0.12 0.27 0.11 (0.06) (0.06) (0.07) (0.06) Monthly Returns. Regression of log return on bonds in local currency rlocal , log change in the exchange rate and the log return in dollars on the log return on U.S. bonds in dollars. OLS standard errors. r$ 0.66 0.65 0.64 0.50 0.71 0.64 0.71 0.71 0.72 0.56 0.57 0.75 0.44 0.24 0.43 0.50 0.44 0.52 0.35 s.e. (0.10) (0.09) (0.09) (0.07) (0.08) (0.10) (0.09) (0.09) (0.11) (0.12) (0.10) (0.08) (0.10) (0.09) (0.23) (0.10) (0.10) (0.09) (0.08) −∆s -0.06 0.23 0.23 -0.09 0.22 0.17 0.22 0.23 0.20 0.18 0.34 0.23 -0.01 0.04 0.23 0.26 0.13 0.27 0.04 s.e. (0.10) (0.09) (0.09) (0.07) (0.09) (0.10) (0.09) (0.09) (0.09) (0.09) (0.09) (0.09) (0.10) (0.09) (0.09) (0.09) (0.10) (0.09) (0.08) 0.55 0.40 0.51 0.69 0.07 0.19 0.37 0.28 0.43 0.29 0.08 0.07 0.08 0.07 0.08 0.06 0.07 0.06 0.07 0.06 Panel C: 1991-2012 rlocal 0.71 0.42 0.41 0.59 0.49 0.46 0.48 0.47 0.52 0.38 0.23 0.52 0.45 0.20 0.20 0.24 0.31 0.25 0.31 s.e. (0.04) (0.04) (0.04) (0.03) (0.04) (0.05) (0.04) (0.04) (0.07) (0.06) (0.04) (0.03) (0.05) (0.03) (0.22) (0.04) (0.03) (0.03) (0.02) r$ 0.33 0.42 0.52 s.e. (0.07) (0.06) (0.07) −∆s -0.04 0.23 0.22 0.01 0.21 0.15 0.21 0.23 0.19 0.18 0.27 0.24 -0.08 0.08 s.e. (0.06) (0.06) (0.06) (0.04) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.07) (0.06) 0.17 0.16 0.15 0.18 (0.04) (0.03) (0.02) (0.03) 0.32 0.25 0.37 0.27 (0.05) (0.05) (0.05) (0.05) 0.17 0.10 0.23 0.10 (0.05) (0.05) (0.05) (0.04) IRP ITL JPY NLG NZD NOK PTE ESP SEK CHF GBP Panel A: 1950-2012 0.28 0.34 0.30 0.19 0.20 0.40 0.14 0.11 0.16 0.15 0.14 0.17 (0.03) (0.02) (0.04) (0.03) (0.03) (0.02) (0.04) (0.02) (0.03) (0.02) (0.01) (0.02) FF DEK 0.47 0.56 0.47 0.34 0.44 0.61 0.08 0.18 (0.05) (0.06) (0.07) (0.06) (0.06) (0.05) (0.06) (0.05) Panel B: 1971-2012 rlocal 0.37 0.19 0.29 0.68 0.34 0.26 0.32 0.39 0.36 0.22 0.24 0.45 0.16 0.11 s.e. (0.04) (0.03) (0.03) (0.03) (0.04) (0.04) (0.03) (0.03) (0.05) (0.04) (0.04) (0.03) (0.05) (0.03) r$ −∆s -0.03 0.19 0.19 0.01 0.18 s.e. (0.05) (0.05) (0.05) (0.03) (0.04) rlocal 0.33 0.16 0.27 0.63 0.28 s.e. (0.03) (0.03) (0.02) (0.02) (0.03) AUD ATS BEL CAD DKK FIM Table 9: Bond Return Parity: Large Panel ∗ rt+1,10 = α + βrt+1,10 + t+1 , Currency Portfolio Betas Finally, we computed the same regression coefficients for each interest-rate sorted portfolio. These results are reported in Table 11. The top panel looks at developed currencies. There are interesting differences in the slope coefficient across these portfolios. The dollar slope coefficient declines from 46 (51) to 29 (32)% over the entire (PostBretton-Woods) sample. This due to a decline in the local currency betas from 33% (36%) to 20%(22%) and a decline in the exchange rate betas from 13% (15%) to 9% (9%). In the bottom panel, we see ben larger differences between portfolios. The dollar slope coefficient declines from 32 (34) to 13 (13)% over the entire (Post-Bretton-Woods) sample. This due to a decline in the local currency betas from 24% (25%) to 13%(13%) and a decline in the exchange rate betas from 8% (9%) to 0% (0%). As a result, it does look like there is more sharing of permanent innovations between the U.S. and lower interest rate countries than with higher interest rate countries. 7 Conclusion The term structure of currency risk premia is downward sloping. That implies that the shocks not shared in international financial markets are much less persistent than the overall shocks driving pricing kernels. This model-free evidence supports the mechanism proposed by Colacito and Croce (2011) to explain the Brandt-Cochrane-SantaClara puzzle of low exchange rate volatility, high Sharpe ratios and low correlation of consumption growth. 36 Table 10: Bond Return Parity: Small Panel DEM α NW s.e. BS s.e. β NW s.e. BS s.e. Wald 0.01∗∗ (0.01) (0.00) 0.63∗∗∗ (0.08) (0.07) 25.17∗∗∗ α NW s.e. BS s.e. β NW s.e. BS s.e. Wald 0.02∗∗ (0.01) (0.01) 0.66∗∗∗ (0.12) (0.09) 8.67∗∗ α NW s.e. BS s.e. β NW s.e. BS s.e. Wald 0.04∗ (0.02) (0.01) 0.69∗ (0.16) (0.11) 4.05 GBP JPY CAD AUD CHF 3-month holding period USD returns 0.01∗ 0.01 0.01∗∗ 0.02∗∗ 0.00 (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.00) (0.01) (0.00) 0.58∗∗∗ 0.69∗∗ 0.71∗∗∗ 0.57∗∗∗ 0.61∗∗∗ (0.08) (0.12) (0.11) (0.13) (0.10) (0.09) (0.14) (0.09) (0.13) (0.08) 27.13∗∗∗ 6.11∗∗ 8.87∗∗ 11.60∗∗∗ 16.65∗∗∗ 6-month holding period USD returns 0.02∗ 0.01 0.02∗∗ 0.04∗∗ 0.01 (0.01) (0.02) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) ∗∗∗ ∗∗∗ ∗∗∗ 0.58 0.76 0.65 0.48 0.61∗∗∗ (0.10) (0.19) (0.13) (0.16) (0.13) (0.10) (0.14) (0.09) (0.14) (0.10) 16.22∗∗∗ 1.71 8.08∗∗ 12.04∗∗∗ 9.52∗∗∗ 12-month holding period USD returns ∗∗ 0.05 0.03 0.05∗∗ 0.06∗∗ 0.02 (0.02) (0.04) (0.02) (0.03) (0.03) (0.02) (0.02) (0.01) (0.02) (0.02) 0.49∗∗∗ 0.73 0.63∗∗ 0.59∗∗ 0.54∗∗ (0.13) (0.29) (0.15) (0.21) (0.21) (0.10) (0.18) (0.09) (0.15) (0.15) 15.64∗∗∗ 0.89 6.96∗∗ 5.47∗ 5.62∗ NZD SEK NOK 0.01 (0.01) (0.01) 0.84 (0.11) (0.10) 2.45 0.01∗ (0.01) (0.01) 0.60∗∗∗ (0.10) (0.11) 17.42∗∗∗ 0.01 (0.01) (0.01) 0.48∗∗∗ (0.17) (0.13) 11.29∗∗∗ 0.01 (0.01) (0.01) 0.80 (0.14) (0.12) 3.07 0.02 (0.01) (0.01) 0.68∗∗ (0.15) (0.14) 5.08∗ 0.01 (0.01) (0.01) 0.65∗ (0.21) (0.18) 4.08 0.03 (0.03) (0.02) 0.83 (0.20) (0.15) 1.50 0.03 (0.03) (0.02) 0.80 (0.22) (0.16) 1.52 0.01 (0.03) (0.02) 0.91 (0.34) (0.26) 0.21 We regress the USD holding period return of a foreign 10-year bond on the corresponding holding period return of the US bond and report the constant and the slope coefficient. Standard errors are reported in the parentheses; we first report the Newey-West standard error (with 12 lags) and then the bootstrap standard error. For the latter, we apply block bootstrapping using 1,000 bootstrap samples, with block length equal to 3. We test the null hypotheses of constant equal to zero and slope coefficient equal to one both individually and jointly for each currency; Wald is the Wald test statistic for the joint hypothesis test. One, two and three asterisks denote rejection of the null hypothesis at the 10, 5 and 1 percent level of significance, respectively. 37 Table 11: Bond Return Parity: Currency Portfolios 1 rlocal 2 3 4 1950-2012 5 1 3 4 1971-2012 5 Panel A: Developed 0.33 0.30 0.28 0.24 0.20 0.36 0.32 0.30 0.26 0.22 (0.02) (0.03) (0.02) (0.02) (0.03) (0.02) (0.03) (0.03) (0.03) (0.03) rf x 0.13 0.16 0.17 0.11 0.09 (0.04) (0.04) (0.04) (0.04) (0.04) r$ 0.46 0.47 0.44 0.36 0.29 0.51 (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) Panel B: All 0.24 0.28 0.28 0.23 0.13 0.25 (0.01) (0.02) (0.02) (0.02) (0.02) (0.02) rlocal 2 0.15 0.19 0.19 0.13 0.09 (0.05) (0.05) (0.05) (0.05) (0.05) 0.51 0.49 0.40 0.32 (0.06) (0.06) (0.05) (0.06) 0.31 0.31 0.24 0.13 (0.03) (0.02) (0.02) (0.03) rf x 0.08 0.15 0.16 0.07 0.00 (0.03) (0.03) (0.03) (0.03) (0.04) 0.09 0.17 0.17 0.09 0.00 (0.04) (0.05) (0.05) (0.05) (0.05) r$ 0.32 0.43 0.44 0.31 0.13 (0.04) (0.04) (0.04) (0.04) (0.04) 0.34 0.49 0.48 0.33 0.13 (0.05) (0.05) (0.05) (0.05) (0.06) Monthly Returns. Regression of log return on bonds in local currency rlocal , log change in the exchange rate and the log return in dollars on the log return on U.S. bonds in dollars. OLS standard errors. Portfolios of currencies sorted at t − 1 by monthly returns on T-bills realized at end of t. 38 Figure 4: Sorts by Interest Rates: Whole Sample Panel A: All Countries Currency Risk Premium 4 2 0 −2 1 2 3 4 5 4 5 4 5 4 5 4 5 4 5 Local Term Risk Premium 4 2 0 −2 1 2 3 Dollar Term Risk Premium 4 3 2 1 0 1 2 3 Panel B: Developed Countries Currency Risk Premium 3 2 1 0 −1 1 2 3 Local Term Risk Premium 3 2 1 0 −1 1 2 3 Dollar Term Risk Premium 4 3 2 1 0 1 2 3 Notes: Annualized Monthly Returns. Sample: 1950-2012. Portfolios of Currencies sorted by T-bill Interest Rates. The top panel shows all countries. The bottom panel shows developed countries. 39 Figure 5: Sorts by Interest Rates: Post-Bretton-Woods Panel A: All Countries Currency Risk Premium 6 4 2 0 −2 1 2 3 4 5 4 5 4 5 4 5 4 5 4 5 Local Term Risk Premium 6 4 2 0 1 2 3 Dollar Term Risk Premium 8 6 4 2 0 1 2 3 Panel B: Developed Countries Currency Risk Premium 4 2 0 −2 1 2 3 Local Term Risk Premium 4 2 0 −2 1 2 3 Dollar Term Risk Premium 6 4 2 0 1 2 3 Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill Interest Rates. The top panel shows all countries. The bottom panel shows developed countries. 40 Figure 6: Sorts by Slope: Whole Sample Panel A: All Countries Currency Risk Premium 3 2 1 0 −1 1 2 3 4 5 4 5 4 5 4 5 4 5 4 5 Local Term Risk Premium 6 4 2 0 −2 1 2 3 Dollar Term Risk Premium 6 4 2 0 1 2 3 Panel B: Developed Countries Currency Risk Premium 3 2 1 0 1 2 3 Local Term Risk Premium 4 2 0 −2 1 2 3 Dollar Term Risk Premium 4 3 2 1 0 1 2 3 Notes: Annualized Monthly Returns. Sample: 1950-2012. Portfolios of Currencies sorted by T-bill Interest Rates. The top panel shows all countries. The bottom panel shows developed countries. 41 Figure 7: Sorts by Slope: Post-Bretton-Woods Panel A: All Countries Currency Risk Premium 3 2 1 0 −1 1 2 3 4 5 4 5 4 5 4 5 4 5 4 5 Local Term Risk Premium 6 4 2 0 −2 1 2 3 Dollar Term Risk Premium 6 4 2 0 1 2 3 Panel B: Developed Countries Currency Risk Premium 4 3 2 1 0 1 2 3 Local Term Risk Premium 4 2 0 −2 1 2 3 Dollar Term Risk Premium 6 4 2 0 1 2 3 Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill Interest Rates. The top panel shows all countries. The bottom panel shows developed countries. 42 Figure 8: UIP and Uncovered Bond Return Parity: All Countries 8 Bond Return Spread Interest Rate Spread 45−degree 7 6 Rate of Depreciation 5 4 3 2 1 0 −1 −2 −2 −1 0 1 2 3 4 Spread with U.S. 5 6 7 8 Notes: Annualized Monthly Returns. Sample: 1950-2012. Portfolios of Currencies sorted by T-bill returns at t. 43 Figure 9: UIP and Uncovered Bond Return Parity: All Countries 8 Bond Return Spread Interest Rate Spread 45−degree 7 6 Rate of Depreciation 5 4 3 2 1 0 −1 −2 −2 −1 0 1 2 3 4 Spread with U.S. 5 6 7 8 Notes: Annualized Monthly Returns. Sample: 1971-2012. Portfolios of Currencies sorted by T-bill returns at t. 44 Figure 10: Foreign Bond Return Betas: Developed Countries Average Betas Dollar Local FX 0.8 0.6 0.4 0.2 0 −0.2 1957 1971 1984 1998 2012 Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for the equal-weighted average of log bond returns in local currency, the log change in the exchange rate and the log dollar bond returns. Developed Countries. 45 Figure 11: Foreign Bond Return Betas: All Countries Average Betas Dollar Local FX 0.8 0.6 0.4 0.2 0 −0.2 1957 1971 1984 1998 2012 Notes: Sample: 1950-2012. 60-month rolling window estimation of beta with respect to US bond returns for the equal-weighted average of log bond returns in local currency, the log change in the exchange rate and the log dollar bond returns. All countries. 46 References Alvarez, F., and U. J. Jermann (2004): “Using Asset Prices to Measure the Cost of Business Cycles,” Journal of Policitical Economy, 112(6), 1123–1256. (2005): “Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth,” Econometrica ,, 73(6), 1977–2016. Ang, A., and J. S. Chen (2010): “Yield Curve Predictors of Foreign Exchange Returns,” Working Paper Columbia University. Backus, D., S. Foresi, and C. Telmer (2001): “Affine Models of Currency Pricing: Accounting for the Forward Premium Anomaly,” Journal of Finance, 56, 279–304. Bansal, R. (1997): “An Exploration of the Forward Premium Puzzle in Currency Markets,” Review of Financial Studies, 10, 369–403. Bekaert, G. (1996): “The Time Variation of Risk and Return in Foreign Exchange Markets: A General Equilibrium Perspective,” The Review of Financial Studies, 9(2), 427–470. Brandt, M. W., J. Cochrane, and P. Santa-Clara (2006): “International Risk-Sharing is Better Than You Think, or Exchange Rates are Too Smooth,” Journal of Monetary Economics, 53(4), 671–698. Brunnermeier, M. K., S. Nagel, and L. H. Pedersen (2008): NBER Macroeconomics Annual chap. Carry Trades and Currency Crashes. University of Chicago Press. Chernov, M., J. Graveline, and I. Zviadadze (2011): “Sources of Risk in Currency Returns,” Working Paper London School of Economics. Chinn, M. D., and G. Meredith (2004): “Monetary Policy and Long-Horizon Uncovered Interest Parity,” IMF Staff Papers, 51(3), 409–430. Colacito, R., and M. M. Croce (2011): “Risks for the Long Run and the Real Exchange Rate,” Journal of Political Economy, 119(1), 153–181. 47 Dahlquist, M., and H. Hasseltoft (2013): “International Bond Risk Premia,” Journal of International Economics, 90, 12–32. Farhi, E., S. P. Fraiberger, X. Gabaix, R. Ranciere, and A. Verdelhan (2009): “Crash Risk in Currency Markets,” Working Paper MIT Sloan. Gavazzoni, F., B. Sambalaibat, and C. Telmer (2012): “Currency Risk and Pricing Kernel Volatility,” Working Paper, Carnegie Mellon University. Hansen, L. P., J. C. Heaton, and N. Li (2008): “Consumption Strikes Back? Measuring LongRun Risk,” Journal of Political Economy, 166(2), 260–302. Hansen, L. P., and J. A. Scheinkman (2009a): “Long-Term Risk: An Operator Approach,” Econometrica, 77(1), 177–234. Hansen, L. P., and J. A. Scheinkman (2009b): “Long-Term Risk: An Operator Approach,” Econometrica, 77, 177 – 234. Jotikasthira, C., A. Le, and C. Lundblad (2012): “Why Do Term Structures in Different Currencies Comove?,” Working Paper University of North Carolina. Jurek, J. W. (2008): “Crash-neutral Currency Carry Trades,” Working Paper Princeton University. Koijen, R., T. J. Moskowitz, L. H. Pedersen, and E. B. Vrugt (2012): “Carry,” Working Paper Chicago Booth. Lustig, H., N. Roussanov, and A. Verdelhan (2011): “Common Risk Factors in Currency Markets,” Review of Financial Studies, 24 (11), 3731–3777. Lustig, H., and A. Verdelhan (2005): “The Cross-Section of Foreign Currency Risk Premia and Consumption Growth Risk,” American Economic Review, 97((1)), 98–117. Menkhoff, L., L. Sarno, M. Schmeling, and A. Schrimpf (2012): “Carry Trades and Global Foreign Exchange Rate Volatility,” Journal of Finance, 67 (2), 681–718. 48 Wu, L. J. (2012): “Global Yield Curve Risk Factors in International Bond and Currency Portfolios,” Working Paper UCLA. 49 A Proofs • Proof of Proposition 2.2. Proof. We use Mt+1 to denote the stochastic discount factor. To see why, consider the investor’s Euler equation: 0 = Et [log Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] + (1/2) lim V art [log Mt+1 + log Rt+1,1 [1t+k ]]. k→∞ k→∞ The last term drops out because we know that limk→∞ Rt+1,1 [1t+k ] = Λt /Λt+1 . Also, note that log Et [Mt+1 ] = Et [log Mt+1 ] + (1/2)V art [log Mt+1 ]. Hence, we obtain: 0 = log Et [Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] − (1/2)V art [log Mt+1 ]. k→∞ It follows directly that the term premium equals one half of the variance: Rt+1,1 [1t+k ] = (1/2)σ 2 . lim Et rxkt+1 = lim Et log k→∞ k→∞ Rt+1,1 [1t+1 ] • Proof of Proposition 2.3. Proof. In the log-normal model, the foreign term premium –the bond risk premium on ∗,2 the longest maturity bond limk→∞ Et [rxk,∗ t+1 ] – equals (1/2)σ . In a log-normal model, ∗,2 2 the currency risk premium equals (1/2) σ − σ . (Backus, Foresi, and Telmer (2001)). To see why, consider the investor’s Euler equation: 0 = Et [log Mt+1 ] + Et [log Rt+1 ] + (1/2)V art [log Mt+1 + (ft − st ) − ∆st+1 ]. We use the complete markets expression for the change in the spot rates: ∗ ∆st+1 = log Mt+1 − log Mt+1 . The result follows immediately. • Proof of Corollary 2.1: Proof. Using the equivalence between long-term yield spreads and expected excess returns, we know that: yt∗ [∞] + (ft − st ) − E[∆st+1 ] = (1/2)σ 2 = yt [∞] Note that E[ lim (1/k) k→∞ k X ∆st+j ] = lim (1/k) k→∞ j=1 k X E[∆st+j ] j=1 by the Lebesgue dominated convergence theorem. Also, note that E[∆st+j ] = E[∆st+1 ] 50 by assumption (because we assumed that expected holding period returns do not depend on calendar time). That delivers the result. • Proof of Proposition 2.4. Proof. First, note that the foreign currency risk premium is equal to: ∗ Λt+1 Λt+1 (ft − st ) − Et [∆st+1 ] = Lt − Lt Λt Λ∗t See Backus, Foresi, and Telmer (2001). Also, note that: ! ΛPt+1 Rt+1,1 [1t+∞ ] Λt+1 = Lt Lt + E log t Λt Rt+1,1 [1t+1 ] ΛPt This result is established in Alvarez and Jermann (2005). The final result follows immediately. • Proof of Proposition 2.7: Proof. First, note that the foreign currency risk premium is equal to: ∗ Λt+1 Λt+1 (ft − st ) − Et [∆st+1 ] = Lt − Lt Λt Λ∗t See Backus, Foresi, and Telmer (2001). Also, note that: ! ΛPt+1 Rt+1,1 [1t+∞ ] Λt+1 Lt = Lt + Et log Λt Rt+1,1 [1t+1 ] ΛPt This result is established in Alvarez and Jermann (2005). The final result follows immediately. • Proof of Proposition 2.5: Proof. This result follows directly from the definition of a pricing kernel without permanent innovations and from the no-arbitrage expression for the spot exchange rates when markets are complete: M ∗ /M ∗ St = t+1 t St+1 Λt+1 /Λt • Proof of Proposition 2.6: Proof. We use Mt+1 to denote the stochastic discount factor. To see why, consider the investor’s Euler equation: P T 0 = Et [log Mt+1 ]+ lim Et [log Rt+1,1 [1t+k ]]+(1/2) lim V art [log Mt+1 +log Mt+1 +log Rt+1,1 [1t+k ]]. k→∞ k→∞ P ] because we know that lim The last term simplifies to V art [log Mt+1 k→∞ Rt+1,1 [1t+k ] = T T Λt /Λt+1 . Also, note that log Et [Mt+1 ] = Et [log Mt+1 ] + (1/2)V art [log Mt+1 ]. 51 Hence, we obtain: P 0 = log Et [Mt+1 ] + lim Et [log Rt+1,1 [1t+k ]] − (1/2)V art [log Mt+1 ] + (1/2)V art [log Mt+1 ]. k→∞ It follows directly that the term premium equals one half of the variance: Rt+1,1 [1t+k ] = (1/2)σT2 + σT P . lim Et rxkt+1 = lim Et log k→∞ k→∞ Rt+1,1 [1t+1 ] • Proof of Corollary 2.2: Proof. The result follows immediately from Proposition 2.6. • Proof of Proposition 2.10: Proof. The proof is immediate: the ratio of foreign to domestic holding period returns for infinite maturity bonds (once converted to the same currency) measures the ratio of the domestic and permanent components of the SDF. Theorem 2.10 refines the main result in Brandt, Cochrane, and Santa-Clara (2006): the permanent components of the SDFs must be highly correlated across countries. 52