# Average Variance, Average Correlation and Currency Returns

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5 numeraire that is based on the US dollar, the euro, the UK pound and the Japanese yen. We further demonstrate that implied volatility indices, such as the VIX for the equities market and the VXY for the FX market, are insignificant predictors of future carry returns, and hence cannot replicate the predictive information in average variance and average correlation. Finally, the predictive quantile regression framework allows us to compute a robust measure of conditional skewness, which is predominantly positive at the beginning of the sample and predominantly negative at the end of the sample. The remainder of the paper is organized as follows. In the next section we provide an overview of the literature on the intertemporal tradeoff between risk and return, and motivate the empirical predictions we examine in this paper. In Section 3 we describe the FX data set and define the measures for risk and return on the carry trade. Section 4 presents the predictive quantile regressions and discusses estimation issues. In Section 5, we report the empirical results, followed by robustness checks and further analysis in Section 6. Section 7 discusses the augmented carry trade strategies and, finally, Section 8 concludes. 2 The Intertemporal Tradeoff between FX Risk and Return: A Brief Review and Testable Implications Since the contribution of Merton (1973, 1980), there is a class of asset pricing models suggesting that there is an intertemporal tradeoff between risk and return. These models hold for any risky asset in any market and hence can be applied not only to equities but also to the FX market. For the carry trade, the intertemporal risk-return tradeoff may be expressed as follows: r C,t+1 = µ + κσ 2 t + ε t+1, (1) σ 2 t = ϕ 0 + ϕ 1 AV t + ϕ 2 AC t, (2) where r C,t+1 is the return to the carry trade portfolio from time t to t+1; σ 2 t is the conditional variance of the returns to the FX market portfolio, also known as FX market variance; AV t is 4

7 average variance and average correlation, as shown by Pollet and Wilson (2010) for equity returns. This decomposition is an aspect of our analysis that is critical for distinguishing the effect of systematic and idiosyncratic risk on future carry trade returns as well as for delivering robust and statistically significant results in predicting future carry trade returns. We begin with a brief discussion of the relation between average variance and future returns. Goyal and Santa-Clara (2003) show that although the equally weighted market variance only reflects systematic risk, average variance captures both systematic and idiosyncratic risk and, as a result, average variance is a powerful predictor of future equity returns. This implies that idiosyncratic risk matters for equity returns and our analysis investigates whether this is also the case for FX excess returns. 5 We now turn to the relation between average correlation and future returns. Standard finance theory suggests that lower correlations generally lead to improved diversification and better portfolio performance. This is also the case for carry trade strategies, as shown by Burnside, Eichenbaum and Rebelo (2008), who demonstrate that the gains of diversifying the carry trade across many currencies are large, raising the Sharpe ratio by over 50 percent. More to the point, Pollet and Wilson (2010) show that the average correlation of stocks is significantly positively related to future stock returns. In the context of the ICAPM, they argue that since the return on aggregate wealth is not directly observable (e.g., Roll, 1977), changes in aggregate risk may reveal themselves through changes in the correlation between observable stock returns. Hence an increase in aggregate risk can be related to higher average correlation and to higher future stock returns. In short, the first testable hypothesis of our empirical analysis is: H1: FX risk measures based on average variance and average correlation are more powerful predictors than market variance for future FX excess returns. The intertemporal risk-return model of Equations (1) (2) can be applied to the full conditional distribution of returns. As shown by Cenesizoglu and Timmermann (2010), the 5 There is a number of theoretical models and economic arguments that justify why a market-wide measure of idiosyncratic risk may matter for returns. For example, variants of the CAPM where investors hold undiversified portfolios for either rational (tax or transactions costs) or irrational reasons predict an intertemporal relation between returns and idiosyncratic risk (see Goyal and Santa-Clara, 2003, for a discussion of this literature). This relation also obtains in a cross-sectional context in models with incomplete risk sharing (e.g., Sarkissian, 2003). 6

10 to substantial gains over the standard carry trade. The risk measures employed in our analysis have been the focus of recent intertemporal as well as cross-sectional studies of the equity market. The intertemporal role of average variance is examined by Goyal and Santa-Clara (2003), as discussed earlier, and also by Bali, Cakici, Yan and Zhang (2005). These studies show that average variance reflects both systematic and idiosyncratic risk and can be significantly positively related to future equity returns. In the cross-section of equity returns, the negative price of risk associated with market variance is examined by Ang, Hodrick, Xing and Zhang (2006, 2009). They find that stock portfolios with high sensitivities to innovations in aggregate volatility have low average returns. Similarly, Krishnan, Petkova and Ritchken (2009) find a negative price of risk for equity correlations. Finally, Chen and Petkova (2010) examine the cross-sectional role of average variance and average correlation. They find that for portfolios sorted by size and idiosyncratic volatility, average variance has a negative price of risk, whereas average correlation is not priced. 3 Measures of Return and Risk for the Carry Trade This section describes the FX data set and defines our measures for: (i) the excess return to the carry trade for individual currencies, (ii) the excess return to the carry trade for a portfolio of currencies, and (iii) three measures of risk: market variance, average variance and average correlation. 3.1 FX Data We use a cross-section of US dollar nominal spot and forward exchange rates by collecting data on 33 currencies relative to the US dollar: Australia, Austria, Belgium, Canada, Czech Republic, Denmark, Euro area, Finland, France, Germany, Greece, Hong Kong, Hungary, India, Ireland, Italy, Japan, Mexico, Netherlands, New Zealand, Norway, Philippines, Poland, Portugal, Saudi Arabia, Singapore, South Africa, South Korea, Spain, Sweden, Switzerland, Taiwan and United Kingdom. The sample period runs from January 1976 to February Note that the number of exchange rates for which there are available data varies over time; 9

11 at the beginning of the sample we have data for 15 exchange rates, whereas at the end we have data for 22. The data are collected by WM/Reuters and Barclays and are available on Thomson Financial Datastream. The exchange rates are listed in Table The Carry Trade for Individual Currencies An investor can implement a carry trade strategy for either individual currencies or, more commonly, a portfolio of currencies. In practice, the carry trade strategy for individual currencies can be implemented in one of two equivalent ways. First, the investor may buy a forward contract now for exchanging the domestic currency into foreign currency in the future. She may then convert the proceeds of the forward contract into the domestic currency at the future spot exchange rate. The excess return to this currency trading strategy for a one-period horizon is defined as: r j,t+1 = s j,t+1 f j,t, (5) for j = {1,..., N t }, where N t is the number of exchange rates at time t, s j,t+1 is the log of the nominal spot exchange rate defined as the domestic price of foreign currency j at time t + 1, and f j,t is the log of the one-period forward exchange rate j at time t, which is the rate agreed at time t for an exchange of currencies at t + 1. Note that an increase in s j,t+1 implies a depreciation of the domestic currency, namely the US dollar. Second, the investor may buy a foreign bond while at the same time selling a domestic bond. The foreign bond yields a riskless return in the foreign currency but a risky return in the domestic currency of the investor. Hence the investor who buys the foreign bond is exposed to FX risk. In this strategy, the investor will earn an excess return that is equal to: r j,t+1 = i j,t i t + s j,t+1 s j,t, (6) where i j,t and i t are the one-period foreign and domestic nominal interest rates respectively. 7 Note that our data includes no more than 33 currencies to avoid having exchange rate series with short samples and non-floating regimes. 10

14 Zhang (2005), among others, this measure of market variance accounts for the autocorrelation in daily returns Average Variance and Average Correlation Our second set of risk measures relies on the Pollet and Wilson (2010) decomposition of MV into the product of two terms, the cross-sectional average variance (AV) and the cross-sectional average correlation (AC), as follows: MV t+1 = AV t+1 AC t+1. (9) The decomposition would be exact if all exchange rates had equal individual variances, but is actually approximate given that exchange rates display unequal variances. Thus, the validity of the decomposition is very much an empirical matter. Pollet and Wilson (2010) use this decomposition for a large number of stocks and find that the approximation works very well. As we show later, this approximation works remarkably well also for exchange rates. We can assess the empirical validity of the decomposition by estimating the following regression: MV t+1 = α + β (AV t+1 AC t+1 ) + u t+1, (10) where E [u t+1 AV t+1 AC t+1 ] = 0. The coefficient β may not be equal to one because exchange rates do not have the same individual variance and there may be measurement error in MV t+1, AV t+1 and AC t+1. However, the R 2 of this regression will give us a good indication of how well the decomposition works empirically. 11 This is similar to the heteroskedasticity and autocorrelation consistent (HAC) measure of Bandi and Perron (2008), which uses linearly decreasing Bartlett weights on the realized autocovariances. Our empirical results remain practically identical when using the HAC market variance, and hence we use the simpler specification of Equation (8) for the rest of the analysis. 13

15 We estimate AV and AC as follows: AV t+1 = 1 N t V j,t+1, (11) N t AC t+1 = j=1 1 N t (N t 1) N t N t i=1 C ij,t+1, (12) where V j,t+1 is the realized variance of the excess return to exchange rate j at time t + 1 computed as j i D t D t V j,t+1 = rj,t+d/d 2 t + 2 r j,t+d/dt r j,t+(d 1)/Dt, (13) d=1 and C ij,t+1 is the realized correlation between the excess returns of exchange rates i and j at time t + 1 computed as d=2 C ij,t+1 = V ij,t+1 = V ij,t+1 Vi,t+1 Vj,t+1, (14) D t d=1 D t r i,t+d/dt r j,t+d/dt + 2 r i,t+d/dt r j,t+(d 1)/Dt. (15) d=2 Note that we do not demean returns in calculating variances. This allows us to avoid estimating mean returns and has very little impact on calculating variances (see, e.g., French, Schwert and Stambaugh, 1987). 3.6 Systematic and Idiosyncratic Risk Define V j,d as the variance of the excess return to exchange rate j on day d. In this section only, for notational simplicity we suppress the monthly index t. Then, V j,d is a measure of total risk that contains both systematic and idiosyncratic components. Following Goyal and Santa-Clara (2003), we can decompose these two parts of total risk as follows. Suppose that the excess return r j,d is driven by a common factor µ d and an idiosyncratic zero-mean shock ε j,d that is specific to exchange rate j. For simplicity, further assume that the factor loading for each exchange rate is equal to one, the common and idiosyncratic factors are uncorrelated, and ignore the serial correlation adjustment in Equation (13). Then, the data generating 14

16 process for daily returns is: r j,d = µ d + ε j,d, (16) and the return to the FX market portfolio for day d in a given month t is: r M,d = 1 N t N t j=1 r j,d = µ d + 1 N t where the second term becomes negligible for large N t. It is straightforward to show that in a given month t: N t j=1 ε j,d, (17) MV = AV = D t d=1 D t d=1 ( D t rm,d 2 = µ 2 d + 2 N t 1 µ N d ε j,d + t [ d=1 1 N t rj,d 2 N t j=1 j=1 N t N t j=1 ] [ D t = µ 2 d + 2 N t µ N d ε j,d + 1 N t t N t d=1 j=1 ) 2 ε j,d, (18) j=1 ε 2 j,d ]. (19) The first two terms of MV and AV are identical and capture the systematic component of total risk as they depend on the common factor. The third term that depends exclusively on the idiosyncratic component is different for MV and AV. For a large cross-section of exchange rates, this term is negligible for MV, and hence MV does not reflect any idiosyncratic risk. For AV, however, the third term is not negligible and captures the idiosyncratic component of total risk. To get a better idea of the relative size of the systematic and idiosyncratic components, consider the following example based on the descriptive statistics of Table 2. In annualized terms, the expected monthly variances are: E [MV ] 10 3 = 5 = Systematic } {{ } 5 + Idiosyncratic } {{ } 0, (20) E [AV ] 10 3 = 10 = Systematic } {{ } 5 + Idiosyncratic } {{ } 5. (21) Therefore, half of the risk captured by AV in FX is systematic and the other half is idiosyn- 15

17 cratic, whereas all of the risk reflected in MV is systematic. Similarly, the standard deviations are: ST D [MV ] 10 3 = 2, (22) ST D [AV ] 10 3 = 3. (23) As a result, the t-ratio of mean divided by standard deviation is 2.5 for MV and 3.3 for AV. In other words, AV is measured more precisely than MV, which can possibly make AV a better predictor of FX excess returns. 4 Predictive Regressions Our empirical analysis begins with ordinary least squares (OLS) estimation of two predictive regressions for a one-month ahead horizon. The first predictive regression provides a simple way for assessing the intertemporal risk-return tradeoff in FX as follows: r C,t+1 = α + βmv t + ε t+1, (24) where r C,t+1 is the return to the carry trade portfolio from time t to t + 1, and MV t is the market variance from time t 1 to t. This regression will capture whether, on average, the carry trade has low or negative returns in times of high market variance. The second predictive regression assesses the risk-return tradeoff implied by the variance decomposition of Pollet and Wilson (2010): r C,t+1 = α + β 1 AV t + β 2 AC t + ε t+1, (25) where AV t and AC t are the average variance and average correlation from time t 1 to t. For notational simplicity, we use the same symbol α for the constants in the two regressions. The second regression separates the effect of AV and AC in order to determine whether the decomposition provides a more precise signal on future carry returns. 16

18 The simple OLS regressions focus on the effect of the risk measures on the conditional mean of future carry returns. We go further by also estimating two predictive quantile regressions, which are designed to capture the conditional effect of either MV or AV and AC on the full distribution of future carry trade returns. It is possible, for example, that average variance is a poor predictor of the conditional mean return but predicts well one or both tails of the return distribution. After all, higher variance implies a change in the tails of the distribution and here we investigate whether this is true in a predictive framework. Using quantile regressions provides a natural way of assessing the effect of higher risk on different parts of the distribution of future carry returns. It is also an effective way of dealing with outliers. For example, the median is a quantile of particular importance that allows for direct comparison to the OLS regression that focuses on the conditional mean. It is well known that outliers may have a much larger effect on the mean of a distribution than the median. Hence the quantile regressions can provide more robust results than OLS regressions even for the middle of the distribution. In our analysis, we focus on deciles of the distribution of future carry returns. The first predictive quantile regression estimates the conditional quantile function: Q rc,t+1 (τ MV t ) = α (τ) + β (τ) MV t, (26) where τ is the quantile of the cumulative distribution function of one-month ahead carry returns. 12 The second predictive quantile regression yields estimates of the conditional quantile function: Q rc,t+1 (τ AV t, AC t ) = α (τ) + β 1 (τ) AV t + β 2 (τ) AC t. (27) The standard error of the quantile regression parameters is estimated using a moving block bootstrap (MBB) that provides inference robust to heteroskedasticity and autocorrelation of 12 We obtain estimates of the quantile regression coefficients {α (τ), β (τ)} by solving the minimization problem {α (τ), β (τ)} = arg mine [ρ τ (r C,t+1 α (τ) β (τ) MV t )], using the asymmetric loss function α,β ρ τ (r C,t+1 ) = r C,t+1 (τ I (r C,t+1 < 0)). We formulate the optimization problem as a linear program and solve it by implementing the interior point method of Portnoy and Koenker (1997). See also Koenker (2005, Chapter 6). 17

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