In a previous post, we showed that **unhedged currency exposure adds unrewarded risk** **to our investment**, hurting risk-adjusted-performance. This risk should either be neutralized through passive hedging; or mitigated and turned into profit with an active overlay, the latter being what ETS has been doing for the last 20 years.

Now, let’s say we don’t want to get involved in currency matters and we decide to remain local investors. Even then, how can we guarantee that the assets we are invested in are not themselves exposed to currency risk? For instance, we may buy equities quoted in the local currency, but if those companies’ cash flows depend on the value of foreign currencies, the price of our stocks will vary according to these currencies’ fluctuations. In summary, even as purely local investors, we may be exposed to embedded currency risk.

Note that in our example, **we didn’t say these cash flows were in foreign currency but dependant on foreign currency value.** This makes it hard to identify embedded currency risk through a simple balance sheet analysis. As a result, statistical analysis of equity return is often a better alternative as Neil Record points out in his book Currency Overlay [1]. That same book gives a small introduction to such statistical analysis, which we will use as a basis for our discussion.

## Basic Statistical Analysis

At this point, the simplest metric that immediately comes to our mind is the correlation between equity and foreign currency returns. Neil Record takes as example BP, a UK oil and gas (which are priced in dollars in international markets) multinational company which comprises sizeable US businesses (e.g. Amoco). Using 5-year rolling correlation of monthly returns we obtain the following:

Our choice of the period is far from innocent as we will see later, but it also roughly coincides with that used in the referred book (which is 1988-2002). Anyway, we can see a consistent (although modest) negative correlation which would correspond with a long USD position. In other words, this result suggests that a British BP investor is inadvertently exposed to USDGBP.

As a side note, generally, it’s** important not to make hurried interpretations based solely on statistical analysis**. In this case, we are encouraged to make conclusions because results are consistent with our previous conceptual understanding of the problem.

Back to the topic, as Neil Record points out, correlation identifies the amount of variance in the BP share price returns that can be explained with GBPUSD returns. Nonetheless, it doesn’t give information about the precise percentage of embedded exposure. Thus he proposes a different approach based on volatility.

Since currency adds extra variance, a passive overlay that completely offsets embedded currency returns should result in minimal volatility. This is to say, **the hedge ratio of the passive overlay that minimizes volatility equals the percentage of embedded currency exposure**. We can plot variance against hedge ratio to gain a better intuition on how this works:

According to the plot above, the USD content in BP shares is slightly above 40%. Hedging this exposure would result in a modest 2.5% decrease in volatility. Neil Record’s study over the 1988-2002 period yields very similar results. A corollary of this is that a US investor should only hedge 60% of her exposure, as a 100% hedge would leave her 40% short GBP.

## Linking correlation and volatility minimization

So far we have described correlation and volatility minimization as two different approaches, while actually they are intimately related, as we will show in this section. We will call \(r_a\) the return of our asset, \(r_h\) the return of our currency hedge and \(hr\) the hedge ratio. Thus we obtain the following volatility for our investment:

$$ Var(r_a + hr · r_h) = Var(r_a) + hr^2 Var(r_h) + 2hr Corr(r_a, r_h) \sqrt{Var(r_a)Var(r_h)} $$

Therefore, volatility reduction will be given by:

$$ Var(r_a + hr · r_h) – Var(r_a) = hr^2 Var(r_h) + 2hr Corr(r_a, r_h) \sqrt{Var(r_a)Var(r_h)} $$

The quadratic form of the equation makes sense given the shape of the earlier plot. We want to find the value of \(hr\) which maximizes volatility reduction. Thus, we set the derivative with respect to \(hr\) to zero.

$$2hr_{min} Var(r_h) + 2Corr(r_a, r_h) \sqrt{Var(r_a)Var(r_h)} = 0 $$

$$ hr_{min} = -Corr(r_a, r_h) \sqrt{\frac{Var(r_a)}{Var(r_h)}} $$

Hence correlation determines the sign of currency exposure while the ratio of variances also affects its magnitud. Rearranging terms we can also express the equation in the following form:

$$ hr_{min} = -\frac{Cov(r_a, r_h)}{Var(r_h)} $$

Which means that \(hr_{min}\) is also the slope coefficient of the linear regression: \(r_a = -\beta · r_h + \alpha\).

Back to volatility reduction, if we substitute \(hr_{min}\) in our previous equation and do a little bit of algebra we obtain:

$$ Var \left(r_a + hr · r_h \right) – Var(r_a) = -Corr(r_a, r_h)^2 · Var(r_a) $$

Which according to our premises should be roughly equal to the initial embedded currency risk. As we can see, correlation is squared, meaning that low values will result in negligible risks.

## 2008 and beyond…

So far we have worked with pre-2008 data, to consider the same market environment Neil Record used in his calculations. However, if we widen our scope we will see how the financial crisis becomes a turning point.

Now, the fact that correlation turns positive after the crisis doesn’t necessarily invalidate our previous discussion. There are still good reasons both theoretical and empirical (20 years of consistent data) to believe that BP had an embedded long dollar exposure over the considered period.

It’s equally plausible that the change in correlation was caused by contingent events (e.g. the fact that GBP depreciated strongly against USD at the same time stocks were falling worldwide at the beginning of the crisis) or by a radical change in embedded currency exposure due to different market conditions. It’s interesting to note that correlation seems to be coming back to its pre-crisis values in recent years, although it’s still too soon to be certain.

In any case, this example serves to illustrate **the importance of working with dynamic strategies that can adapt to varying market environments.**

## Conclusion

We have introduced the concept of embedded currency risk and discussed a very basic method that attempts to quantify it. Also, we have seen the need to measure this risk dynamically in order to adapt to changing market conditions. We will leave it here for today but if you feel like trying more advanced approaches, factor analysis might be an interesting place to start.

## References

[1] N. Record, *Currency Overlay*. Wiley Finance, First edition (2003)