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The proper approach to hedging currency risk for international assets continues to be a topic of substantial debate. Empirical evidence suggests that fully hedged benchmarks were substantially more efficient than unhedged benchmarks in terms of reward-to-risk across a broad cross-section of currencies and market regimes. Despite this, many consultants and plan sponsors continue to choose unhedged benchmarks. Our suspicion is that because of the severe time constraints faced by today’s plan sponsors and investment managers, the approach to currency is often one of benign neglect. Unfortunately, the benefits of such an approach are random at best, and a more rigorous framework is needed to approach the currency issue in a logical manner.

Generalised Sharpe ratio maximisation within currency management
Our generic approach is based on the assumption that, given an existing core investment, any additional asset (including hedges) must be viewed in terms of whether it improves the return to risk ratio** of the portfolio. Let us consider our primary asset to be an unhedged international asset portfolio, including local and currency effects, and the candidate second asset to be a 1-month forward sale of each of the currencies in the benchmark. We can then derive an exact formula for the unleveraged hedge ratio as candidate asset which maximises the Sharpe ratio. Figure 1 illustrates the corresponding optimal hedge ratio as a function of the expected hedge return and its correlation with the core equity portfolio. This assumes further that an international equity index of large cap stocks has an expected return and standard deviation of 17% per annum and a Sharpe ratio of 1.0; the currency hedge has an expected standard deviation of 9%.
Specifically, Figure 1 shows that even if a hedge is expensive, it can be justified in terms of Sharpe ratio as long as it has a more negative correlation (offsets the core risk more perfectly) and a less perfect hedge can be justified if it costs less.
Note in Figure 1 that even if a “hedge” programme has zero correlation with the unhedged underlying stocks, it should be included in some proportion if it has a positive expected return. This has interesting implications if an active currency overlay programme is treated as the candidate asset. Recent empirical studies have shown that active currency management has generated alpha in the past. Assuming an annualised outperformance by an overlay manager would have been approximately 175 basis points implies that the overlay programme should be used for approximately 36.75% of the portfolio in figure 1. Of course, the point here is not to suggest that this particular ratio is some sort of magic number, but to indicate that the debate on active currency management can be viewed within this same general analytical framework. Figure 1 indicates that even if a currency hedge programme has no expected return whatsoever, then as long as the programme has a negative correlation with the unhedged portfolio of less than –52.95%, then a fully-hedged benchmark will produce the highest Sharpe ratio.
We are able to deconstruct this important correlation coefficient further, and demonstrate two interesting points. First, the more volatile the currency component, the higher the prudent hedge ratio, a confirmation of conventional wisdom. Second, assuming a constant interest rate differential enables us to show that the effectiveness of the currency hedge is directly variable with the relative volatility of the local asset and currency returns. So for example, applying the conventional wisdom amongst international equity managers that the correlation between local and currency returns is low or zero produces Figure 2.
So holding all else constant, the more volatile the core assets become, the less urgent it is to hedge currency exposures. Though this validates the instinct of many international equity managers, only those managing extremely volatile sectors can make the assumption that currency volatility is negligible. Empirically, the volatility of currency is quite high relative to most other sectors.

Empirical Evidence: Were unhedged equity benchmarks the best approach?
In order to avoid drift and regime specific effects, we tested the formulae developed in the last two sections for several countries and equity benchmarks over various periods as follows:
q US dollar (USD) based MSCI EAFE benchmark
q British pound (GBP) based MSCI EAFE less UK plus US benchmark
q Japanese yen (JPY) based MSCI EAFE less Japan plus US benchmark
q Deutschemark (DEM) based MSCI EAFE less Euroland plus US benchmark
q 1973 – 2000 Full period data post-Bretton Woods
q 1980 – 1990 Japanese asset bubble, Plaza accord, German reunification
q 1990 – 2000 Japanese recession, end of Soviet Union, EMU and Mexican crises
q 1995 – 2000 NASDAQ explosion, internet, USD outperformance, birth of EUR
Portfolio rebalancing was assumed to take place at each monthly measurement. Results are annualised. Full statistics are provided in Tables 1 to 4. For the sake of completeness we also indicate the optimal hedge ratios using a conservative assumption of annualised transaction costs of 0.40%.
Fully currency hedged benchmarks exhibited less volatility than fully unhedged benchmarks in all cases except one – in the last five years for the USD, hedging has added a bit more volatility. The impact of hedging on returns is less clear. For the USD investor currency hedging had a negative return in three out of four periods; for the GBP investor it consistently added return; for the JPY investor it was 50/50; for the DEM investor currency hedging always cost money. In terms of Sharpe ratio, currency hedging offered improvement in all cases except for the USD in the 1990s as a whole. In fact in many cases, the optimal hedge ratio h* was not at either extreme. This again demonstrates the importance of understanding the relationship between the core unhedged asset and its currency hedge. Note also that even our high estimate of transactions costs has little impact on the optimal hedge ratio.
In 14 of 16 scenarios, the correlation between the underlying international index unhedged returns and the forward currency hedge returns was less than -50%. It is worthwhile to note, however, that the magnitude of correlation for a USD investor in a basket of EAFE equities has been steadily becoming less negative. Using a less than –50% correlation in our theoretical formula with the actual returns from the unhedged indices and the associated currency hedges would have produced optimal hedge ratios at or very close to 100%.
Empirical studies show that the conventional stance taken by many plan sponsors and managers to leave international assets unhedged has not been particularly efficient ex-post. There is a solid mathematical foundation for this result which shows that even if a hedge costs money it can be justified in terms of Sharpe ratio if it is sufficiently effective. Defining the effectiveness of the hedge as the degree of negative correlation with the core international investment, the hedge ratio can be seen as a function of the ratio between the volatility of the asset to currency. Studies of the relative volatility of currency and equity indicate that currency can be very volatile over all but the very long-term. This implies that ex-ante benchmark currency hedge ratios should be very high. A further implication is that currency overlay programmes (whether internal or external) can be an effective addition to an international asset portfolio as long as there is confidence that they can, in fact, add returns.
Finally, though they may find our argument for higher currency hedge ratios compelling, some managers may be reluctant to add hedges that have an ex-ante cost (negative interest rate differential). This constraint leads to a hedging programme known as the ‘Differential Forward’ strategy – hedge only when there is no ‘cost’. This simple intuitive approach has been incredibly robust historically. Academic studies have shown this approach to have attractive statistical properties as well, though application within a traditional mean-variance framework is problematic. Nevertheless, this represents a viable ‘hybrid’ alternative to traditional passive benchmark hedge programmes.
Bapi Maitra and Emmanuel Acar1 are with Citibank FX Engineering, in London
deeptesh.maitra@citi.com, emmanuel.acar@citi.com
* We gratefully acknowledge the valuable feedback received from Bruno Crastes at Indocam.
Further information including precise mathematical derivations and references are available from the authors.
** Though technically inaccurate, for simplicity we will refer herein to the return-to-standard deviation ratio as the Sharpe ratio.

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