Dollar Factor Beta-Sorted Portfolios

If the conditional dollar factor model is an appropriate model, the expected excess return of a portfolio should increase monotonically in the portfolio’s expected conditional dollar factor exposure, entailing that a high minus low conditional dollar factor portfolio delivers a positive expected excess return. In practice, identifying such a risk-return relation relies critically on accurate measurements of dollar factor exposures. The slow-moving nature of the rolling window betas may not be very informative of the realized risk exposures over the course of, say, the next month, and especially not if there are rapid changes in the factors that drive exchange rates. Naturally, we may then ask if the ex-ante nature of option-implied betas, and their ability to instantaneously incorporate new information, make them better at anticipating future returns than historical betas.

As a first step in the comparative analysis of the betas, I construct beta-sorted portfolios using both methodologies. I follow the portfolio construction procedure of Verdelhan (2017).

Specifically, each month, for each type of beta separately, I allocate the currencies into three equal-weighted portfolios from low to high based on their dollar factor betas. I then construct three portfoliosP₁, P₂, andP₃ which are long the respective beta-sorted portfolios whenever the average foreign discount is negative (i.e., average foreign interest rate is larger than the U.S. dollar interest rate) and short otherwise. In other words, the portfolios are constructed based on their exposure to the conditional dollar factor. Table 2.4 shows mean excess returns, standard deviations, and Sharpe ratios for each of the portfolios at horizons of 1-12 months.

The brackets below the mean excess returns are t-statistics based on Newey and West (1987), with the automatic lag selection of Newey and West (1994). In the construction

of the Q-beta-sorted portfolios, I use options with the same time to expiry as the holding period of the forward contracts. The P-beta used for portfolio construction is calculated on the basis of overlapping daily rolling window regressions with a length of 252, i.e., the same beta is used for each holding period. Both shorter (126 days) and longer (504 days) rolling windows produce similar results, therefore I only report results for the 252-day P-betas.

Table 2.4 reveals a clear pattern: at any holding period, the mean excess portfolio returns increase in the ex-ante Q-beta, while a more dispersed pattern is to be found whenP-betas are used for portfolio construction. E.g., for the 1-month holding period, a high minus low (HML) factor based on Q-betas, which buys portfolio P3 and shorts portfolio P1, gives a significant (t-statistic of 2.58) mean annualized excess return of 3.35 percent (Sharpe ratio 0.41), while the HML factor based on P-betas has an insignificant (t-statistic of 0.57) mean excess return of 0.95 percent (Sharpe ratio 0.11). The mean excess returns to the Q-beta HML factors are positive at longer horizons as well (albeit only significant at the 2-month holding period), and larger than for the correspondingP-beta HML factors.

Figure 2.5 illustrates the cumulative returns to monthly rebalanced HML dollar factors, for both types of betas, along with the annualized 1-month Q-volatility and the 252-day rolling volatility of the dollar factor. We see that implementing an HML dollar strategy based on Q-betas, rather than P-betas, gives larger returns throughout the sample period.

Interestingly, while the dollar carry trade performs poorly from 2010-2016 (Figure 2.1), the HML dollar strategy continues to deliver high positive excess returns. In this period, the AFD is negative, therefore, the dollar carry trade is short the U.S. dollar and long the equal-weighted basket of foreign currencies. Thus, the dollar carry trade is exposed to an upwards shift in the level of the U.S. dollar relative to all foreign currencies, which in fact occurred over this period. The HML dollar factor, on the other hand, is immune to level shifts in the U.S. dollar, since the long and short side of the portfolio are affected equally.

Notably, there is no obvious link between the volatility of the dollar factor and the returns to the HML dollar strategy. For instance, the HML factor does not crash during the financial turmoil in 2008, as the HML carry trade (Jurek, 2014; Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a). This highlights that the HML factor and the HML carry trade appear to be driven by different risk factors (in this sample, their correlation is ∼19%).

Table 2.5 shows the mean excess returns of each beta-sorted portfolio decomposed into

a spot and interest rate component. Interestingly, at any holding period, the larger mean excess returns on the Q-beta HML factors compared to the P-beta HML factors stem entirely from the spot component. For example, at the 1-month holding period, the spot components are 2.35% and −0.18%, and the interest rate components are 1.00% and 1.12%, for theQ-beta and P-beta HML factors, respectively. Thus, for the Q-beta HML factor the largest proportion of the excess return is due to spot changes, which is in contrast to the HML carry trade, where the return is primarily driven by the interest rate differential (see Table 2.1). For both types of beta, the annualized interest rate components are virtually the same for the HML factor portfolios across different holding periods, whereas the spot components decrease, i.e., theQ-beta predictability of currency spot changes is confined to shorter horizons.

One potential explanation for why theQ-betas are better at explaining the cross-section of currency returns, relative to the P-betas, is that they are better at predicting realized volatility and to a lesser extent because they more accurately forecast correlations with the dollar factor. For instance, Jorion (1995) and Busch, Christensen, and Nielsen (2011) provide evidence that implied volatilities from currency options are better predictors of realized currency volatility than historical volatility measures.

I examine if this is the case by constructing portfolios on the basis of betas which are built from a mixture of Q and P-moments. Specifically, I follow the method suggested by French, Groth, and Kolari (1983), in which betas are constructed from historical correlations and option-based variances. Supposedly, if the Q-correlations are good predictors of real-ized correlations, this beta method will be less successful at identifying high and low-beta currencies ex-ante. In the same spirit, I also construct mixed betas based onQ-correlations in conjunction withP-variances.

Following the exact same procedure as previously, portfolios are constructed based on both types of mixed betas. Table 2.6 shows the results. The HML factor constructed from betas combining P-correlations and Q-variances gives a lower mean excess return, at any horizon, compared to the Q-beta HML factor. For instance, at the 1-month horizon, the mean annualized excess return is 2.53% (t-statistic 1.53), compared to 3.35% when Q-correlations are used (Table 2.4). Furthermore, the Sharpe ratio declines to 0.30 (11 percentage points), suggesting that the Q-correlations are more effective at constructing

betas for cross-sectional analysis than P-correlations.

Using betas built from Q-correlations and P-variances to construct portfolios reaffirms that Q-correlations are useful for computing betas. Using this mixed dollar factor beta, the HML factors have larger mean excess returns and Sharpe ratios relative to the corresponding P-beta HML factors at all horizons. There is a monotonic relation between ex-ante portfolio betas and ex-post portfolio excess returns when sorting on the mixed beta, albeit the HML factor excess returns are insignificant.

Among all types of betas, the pure Q-betas perform best for portfolio construction, which corroborates that bothQ-correlations and Q-variances contain useful information for the computation of betas.