Common Risk Factors in Currency Markets

That UIP fails in the cross-section as well as the time series, and that a single global risk factor – measured by the return on high-minus-low interest rate currency portfolios – explains the cross-section of average currency excess returns. The paper’s own framing: “we show that currency risk premia are determined by their exposure to a single, global risk factor, and that interest rates measure currency exposure to this factor… After accounting for the covariance with this risk factor, there are no significant anomalous or unexplained excess returns in currency markets” (Introduction, pp. 2-3). The distinction from the earlier literature is precise. The time-series tradition, from Hansen and Hodrick (1980) and Fama (1984), implies “that currency investors need to know what ‘higher than usual’ means for a specific currency,” and Bansal and Dahlquist (2000) conclude “that country-specific attributes are critical to understanding the cross-sectional variation in currency risk premia” (p. 4). Sorting on current forward discounts instead shows that “currently high interest rate currencies depreciate 5.9 % per annum less than the interest rate differential, while currently low interest rate currencies appreciate 2.9 % per annum less than the interest rate differential,” so “investors earn more simply by holding bonds from currencies with interest rates that are currently high” (p. 4). The authors then draw the inference that constrains characteristic-based stories: “currency-specific attributes other than the interest rate cannot be the only explanation, because these currencies switch portfolios when their interest rates change, and these switches are frequent” (p. 4).

Six portfolios sorted monthly on the forward discount, from 37 currencies over end-1983 to early 2008, with currencies switching portfolios roughly every three months (Section 1.1, pp. 6-8). Log excess return on buying foreign currency forward and selling spot after one month is the forward discount minus the change in the spot rate; under covered interest parity the forward discount equals the interest differential. The sample starts with 9 countries in 1983 and ends with 26 in 2008, with a maximum of 34 attained during the sample, the decrease reflecting the launch of the euro; Turkey and the United Arab Emirates are excluded “because their forward rates appear disconnected from their spot rates,” while dollar-pegged currencies are kept “because forward contracts were easily accessible to investors” (p. 7). Turnover is substantial and asymmetric: “the average frequency is 29.32 percent, implying that currencies switch portfolios roughly every three months,” breaking down as 19.9 percent for the first portfolio, 33.8, 40.7, 43.4, 42.0 for the middle four, and 13.4 for the sixth – so “there is more persistence in the composition of the corner portfolios” (p. 8). The Japanese yen illustrates the dynamics: it starts in the fourth portfolio, falls to the first as Japanese rates decline in the late eighties, briefly returns to the sixth in the early nineties, and stays in the first thereafter. For net returns the authors assume investors “short all the foreign currencies in the first portfolio and go long in all the other foreign currencies” (p. 8).

Yes. The net spread between the extreme portfolios is 483 basis points per year with a Sharpe ratio of 0.54, comparable to US equities (Section 1.2, pp. 8-9). Gross of costs, the first portfolio’s average discount of -390 basis points against average appreciation of about 100 basis points gives a log excess return of -290 basis points; the sixth portfolio’s discount of 778 basis points against depreciation of 188 gives +590 (p. 9). Net of bid-ask spreads the first portfolio’s return drops to -170 basis points (Sharpe ratio -0.21, reported as minus the actual return since the investor is short) and the sixth to 314 basis points (Sharpe ratio 0.34); the high-minus-low spread is 483 basis points with a Sharpe ratio of 0.54 (p. 9). The authors flag that their net estimates are conservative – “since we rebalance portfolios monthly, and transaction costs are incurred each month” – and give the equity benchmark: “over the same sample, the (annualized) Fama-French monthly excess return on the US stock market is 7.11 percent, and the equity Sharpe ratio is 0.48. Note that this equity return does not reflect any transaction cost” (p. 9). On the developed subsample the long-short Sharpe ratio is 0.39, and the authors conclude “there is no evidence that time-varying bid-ask spreads can account for the failure of UIP in these data or that currency excess returns are small in developed countries, as suggested by Burnside et al. (2006)” (pp. 9-10).

From a principal component analysis of the six portfolio returns: a level factor and a slope factor, which the authors then replace with two traded returns (Section 2.1, pp. 10-11). Two components explain more than 80 percent of the variation in the six portfolio returns. The first explains 70 percent of common variation “and can be interpreted as a level factor, since all portfolios load equally on it”; the second, “responsible for over 12 percent of common variation, can be interpreted as a slope factor, since portfolio loadings increase monotonically across portfolios” (p. 10). The traded substitutes are the average currency excess return RX and the portfolio-6-minus-portfolio-1 return HML_FX, with correlations of 0.99 and 0.94 to the respective principal components; both are computed net of bid-ask spreads (pp. 10-11). Their economic reading: HML_FX “is the portfolio return of a US investor engaged in the usual currency carry trade,” and RX “is the average portfolio return of a US investor who buys all foreign currencies available in the forward market” (p. 11).

The fit is good and the restriction holds almost exactly: the estimated price of HML_FX risk is 546 basis points against a factor mean of 537 (Section 2.2, pp. 12-13). “The market price of HML_FX risk is 546 basis points per annum. This means that an asset with a beta of one earns a risk premium of 5.46 percent per annum.” Because the factors are traded returns, no arbitrage requires the risk price to equal the average excess return – “the Euler equation applies to the risk factor itself, which clearly has a regression coefficient of one on itself” – and “the estimated risk price is only 9 basis points removed from the point estimate implied by linear factor pricing.” The GMM standard error is 234 basis points and the Fama-MacBeth standard error 183, so “in both cases, the risk price is more than two standard errors from zero, and thus highly statistically significant” (p. 12). Overall, “the RMSE is around 95 basis points and the adjusted R2 is 69 percent. The null that the pricing errors are zero cannot be rejected, regardless of the estimation procedure” (p. 12). The authors note the plotted fit uses OLS betas times sample factor means rather than estimated risk prices, since “the latter would imply an even better fit by construction” (p. 12).

Pricing the level of average returns, not the cross-section. Its estimated risk price is 135 basis points against a factor sample mean of 136, and “all the portfolios have a beta close to one with respect to this second factor. As a result, the second factor explains none of the cross-sectional variation in portfolio returns, and the standard errors on the risk price estimates are large: for example, the GMM standard error is 168 basis points” (p. 12). Dropping it raises the RMSE from 95 to 168 basis points while leaving the adjusted R-squared at 76 percent, so “the dollar factor does not explain any of the cross-sectional variation in returns, but it is crucial to get the average returns right.” The authors also note a mechanical caution: “including a constant and the dollar risk factor is a problem, because the dollar factor acts like a constant in the cross-sectional regression” (p. 12).

Essentially no. One portfolio has a marginally significant alpha; the joint test does not reject. “The fourth portfolio has a large alpha of 162 basis points per annum, significant at the 10 percent level but not statistically significant at the 5 percent level. The other alpha estimates are much smaller and not significantly different from zero. The null that the alphas are jointly zero cannot be rejected at the 5 or 10 % significance level” (Section 2.2, p. 13). The betas do the work: HML_FX betas “increase monotonically from -.39 for the first portfolio to .61 for the last currency portfolio, and they are estimated very precisely,” with the first three significantly negative and the last two significantly positive (p. 13). The authors check the source of the covariance, noting that the conditional covariance between log currency returns and the carry factor depends only on spot rate changes, and report that “these conditional betas are almost identical to the unconditional ones” (p. 13).

By re-sorting: portfolios formed on estimated HML_FX betas reproduce the pattern in forward discounts and returns (Section 2.3, pp. 14-15). For each date, each currency’s excess return is regressed on HML_FX over a 36-month rolling window ending in the previous period – “note that it only uses information available at date t” – and currencies are sorted into six groups on those betas. Average forward discounts “increase monotonically from portfolio 1 to portfolio 6. Thus, sorts based on forward discounts and sorts based on betas are clearly related, which implies that the forward discounts convey information about riskiness of individual currencies.” Average log excess returns are also monotonically increasing, so “currencies that covary more with our risk factor - and are thus riskier - provide higher excess returns,” and post-formation betas vary monotonically from -0.31 to 0.38. The finding survives a shorter estimation window: “when we estimate betas using a 12-month rolling window, we also obtain a 300 basis point spread between the first and the last portfolio” (p. 15). Bid-ask spreads are not netted out in this exercise “because it is not obvious a priori when the investor wants to go long or short” (p. 14).

Yes, with one partial exception for Japan. Reconstructing both factors in local currency for UK, Japanese and Swiss investors and using local-currency returns as test assets, “the correlation of HML_FX across different base currencies is above .95 in all cases. In fact, without the bid-ask spreads, HML_FX is identical across base currencies.” The estimated risk prices are 5.54 percent in the UK, 5.50 percent in Japan and 5.79 percent in Switzerland, all “less than 70 basis points removed from the sample mean of the factor” and “statistically different from zero in all three cases”; the two factors explain between 47 and 71 percent of the variation, with mean squared pricing errors of 95, 116 and 81 basis points, and “the null that the underlying pricing errors are zero cannot be rejected except for Japan, for which the p-values are smaller than 10 percent” (Section 2.4, pp. 15-16). The authors also revisit the sample of Burnside et al. (2008), who “build 5 currency portfolios and argue that these currency excess returns bear no relation to their riskiness,” and report that in that data the returns are explained by the two factors, with alphas “smaller than 60 basis points per annum,” while the high-minus-low return is 6.3 percent per annum gross of spreads (p. 16). Further checks – foreign-investor perspective, subsamples starting 1983 and 1995, and the longer Treasury-bill-based sample of Lustig and Verdelhan (2007) – are reported to confirm the results (p. 16).

Because it discriminates between a risk explanation and a characteristic explanation. The logic is stated in the introduction: “a characteristic-based explanation would imply that our risk factor has no explanatory power for currency portfolios not constructed by sorting on interest rates” (p. 3). Section 6 reports that “the carry risk factor can account for at least 50 % of the cross-sectional variation in momentum-driven currency returns, even though the momentum portfolios” are formed on past returns. The two strategies are shown to be genuinely distinct: high-momentum currencies do tend to have higher interest rates, but “the spread between the lowest and the highest momentum portfolio is less than 300 basis points on average, much smaller than the” carry spread, and “the return correlations between corresponding (i.e. high/high or low/low) carry and momentum strategies are small and sometimes” negative. The momentum strategy earns 9.32 percent before transaction costs, but “high momentum currencies tend to have larger bid/ask spreads,” and a three-factor model including a momentum factor achieves an adjusted R-squared of 0.83 with a 70 basis point RMSE on the twelve test assets.

A common SDF factor must exist, and low interest rate currencies must load more heavily on it precisely when the global price of risk is high (Section 3, pp. 3-4, and the Proposition in Section 3.2). The model has many countries and an exponentially-affine SDF composed of a country-specific and a common risk factor. “First, we need a common risk factor because it is the only source of cross-sectional variation in currency risk premia. Second, we need low interest rate currencies to be more exposed to the common risk factor in times when the price of common risk is high, i.e. in bad times” (p. 3). The model then justifies the empirical construction: “by sorting currencies into portfolios and constructing HML_FX, we measure the common innovation to the SDF,” while “the dollar risk factor RX measures the home-country-specific innovation to the SDF.” The authors present this as the paper’s theoretical contribution to the portfolio method – “we provide a theoretical foundation for building currency portfolios: by doing so, we recover the two factors that drive the pricing kernel” (p. 3). The closed-form proposition makes the carry premium proportional to the loading spread times the global state variable, so that “when this spread doubles, the carry trade risk premium doubles. However, the spread itself also increases when the global Sharpe ratio is high. As a result, the carry trade risk premium increases non-linearly when global risk increases.” The knife-edge case is explicit: “if there is no spread, i.e. if low and high interest rate currencies share the same loadings on the common risk factor, then HML_FX cannot be a risk factor, because the global component does not affect exchange rates” (pp. 3-4).

A requirement none of them was built to satisfy: a common heteroscedastic SDF component with heterogeneous loadings. The authors situate three fully-specified models – Verdelhan (2005) with habit preferences, Bansal and Shaliastovich (2007) with long-run risk, and Farhi and Gabaix (2007) with disaster risk – and note the two features they share: “a persistent variable drives the volatility of the log stochastic discount factor, and this variable comoves negatively with the country’s risk-free interest rate,” the latter being a necessary condition established by Backus et al. (2001) for log-normal models to reproduce the forward premium puzzle (p. 5). Their own addition: “To reproduce our finding that a single global risk factor explains the cross-section of currency returns, the SDF in these models needs to have a common heteroscedastic component, and the SDF in low interest rate currencies needs to load more on the common component. This heterogeneity is critical for replicating our empirical findings; we show that heterogeneity in the loadings on the country-specific factor cannot explain the cross-sectional variation in currency returns, even though it can generate negative UIP slope coefficients” (p. 5). They also report that HML_FX “is strongly related to macroeconomic risk; it has a US consumption growth beta between 1 and 1.5” (p. 5).

That the average forward discount predicts portfolio returns better than portfolio-specific discounts, and that expected returns are counter-cyclical (Section 4, pp. 21-22, and Introduction pp. 3-4). Portfolio-specific forward discounts “account for between 1.8 percent and 6.4 percent of the monthly variation in excess returns on these currency portfolios,” with the slope coefficient rising from 108 basis points for the first portfolio to 357 for the fourth before falling back to 72 for the sixth, so “deviations from UIP are highest for currencies with medium to high forward discounts” (p. 22). The authors treat inference carefully, reporting Newey-West standard errors with Andrews (1991) lag selection, Hansen-Hodrick standard errors with one lag, and bootstrapped small-sample errors, noting that “forward rates are strongly autocorrelated” and citing Bekaert, Hodrick and Marshall (1997) on small-sample performance (p. 22). The comparative claim is that “the average forward discount rate is a better predictor of portfolio returns than the forward discounts of individual currency portfolios,” echoing Cochrane and Piazzesi (2005) for bonds (p. 3). And the cyclicality: “expected excess returns on portfolios with medium to high interest rates co-move negatively with the US business cycle as measured by industrial production, payroll or help wanted indices, and they co-move positively with the term and default premia as well as the option-implied volatility index VIX,” with “US industrial production growth [having] predictive power for currency excess returns even when controlling for forward discounts” (pp. 3-4).

It matches seven targeted moments with seven parameters, reproducing both the carry premium and the failure of the CAPM in currency returns (Section 5, pp. 29-30). The calibration is monthly, targeting annualized moments, focused on developed countries over 1983-2008, and proceeds in two stages – real side first, then nominal SDFs matched to inflation moments. Four parameters govern the countries’ SDFs and three the state variables; the seven targets are “the mean, standard deviation and autocorrelation of real risk-free rates, the average conditional variance of changes in real exchange rates, the mean and standard deviation of the maximal conditional Sharpe ratio and the UIP slope coefficient,” with moments generated by “drawing 10,000 observations from a model with 40 currencies.” The simulated economy produces an annualised real risk-free rate with mean 1.2 percent, standard deviation 0.2 percent and autocorrelation 0.7, a real exchange rate standard deviation of about 10 percent, and a UIP regression coefficient “around -1, roughly consistent with our data.” Crucially the procedure is sequential: the authors “start by calibrating a completely symmetric version of the model, and then… introduce enough heterogeneity in the SDF loadings on the global shock across countries to match the carry trade risk premium” – that is, the loading heterogeneity is fitted to the target rather than derived.

It claims to have shown that currency excess returns are compensation for exposure to aggregate risk, and explicitly leaves the economic source of that risk, and of the loading heterogeneity, open. The conclusion states the findings as “consistent with the notion that carry trade profits are compensation for systematic risk” – a calibrated statement, not a claim of proof – and that the calibrated model replicates the empirical findings “provided that low interest rate currencies are more exposed to global risk in bad times, when the price of global risk is high” (Section 7, p. 35). The open question is named: “Identifying the economic mechanism that drives the relationship between macroeconomic risk and asset prices is therefore key to understanding the dynamics of currency markets.” The authors offer two candidate sources for the heterogeneity without testing either: it “can be driven by the differences in preferences (risk aversion) across investors in different countries or by the cross-sectional variation in the goods market integration” (p. 35).