# An intuition behind currency risk

### Juan Martínez

#### 18/07/2019

Although we find currency risk particularly interesting, it is not often the case with many investors for whom it is no more than a necessary inconvenience. As such, they tend to neglect it, accepting undesirable non-remunerated risks and missing potential opportunities. To prevent this, in this post we will try to provide an understanding of the mechanics of currency risk. For the sake of simplicity, we will consider the case of foreign equities investment, although results can be easily generalized.

## A foreign investment example

Foreign investment is a very popular method of reducing a portfolio’s risk. Indeed, there is little sense in having a highly diversified portfolio across assets classes if they all depend on the performance of a single economy. As sensible as it is, foreign investment also entails certain risks, specially when securities are denominated in a currency other than ours as we will see below.

Let’s imagine we are a European investor seeking exposure to the Japanese equity market. In order to purchase stocks we will need yen, which we can buy in exchange for euros at the $$EURJPY$$ exchange rate. With an investment amount of $$M$$ euros, we will obtain $$M*EURJPY_0$$ yen which we will use to buy stocks. At the end of our investment, we will do the opposite process, selling our equities for $$M*EURJPY_0*(1+r_e)$$ yen (where $$r_e$$ is the return experienced by our stocks) and exchanging them for $$\frac{M*EURJPY_0*(1+r_e)}{EURJPY_f}$$ euros. As a result, the total return of our investment would be:

$$R=\frac{EURJPY_0*(1+r_e)}{EURJPY_f}-1$$

Now, note that the $$\frac{EURJPY_0}{EURJPY_f}$$ is intimately linked with the return of the exchange rate. Indeed, for convenience we can consider the spot on the opposite direction $$JPYEUR=\frac{1}{EURJPY}$$ which reflects the price of a yen in euros. This way, the return on the spot is:

$$r_s=\frac{JPYEUR_f}{JPYEUR_o}-1=\frac{EURJPY_0}{EURJPY_f}-1$$

With this in mind, we can take our total return expression and substitute the term $$\frac{EURJPY_0}{EURJPY_f}$$ by $$1+r_s$$ obtaining the following:

$$R=(1+r_s)(1+r_e)-1=r_s+r_e+r_s*r_e$$

## Leverage out of the blue!

If we work with daily returns, both $$r_e$$ and $$r_s$$ will be in the order of magnitude of $$10^{-2}$$. Therefore, $$r_e*r_s$$ will be in the order of magnitude of $$10^{-4}$$ which is basically negligible. As a side note, this does not hold for yearly returns but it is still a good enough approximation. Anyway, the expression for our investment returns is now:

$$R=r_e+r_s$$

Note that this means we will experience 100% of the spot returns plus 100% of our stocks returns and therefore our market exposure is 200%. In other words, we have unconsciously entered a leveraged investment! This idea of leverage provides a great intuition for understanding the risks involved with currency exchange.

It is not uncommon to hear that being exposed to the currency spot is another source of diversification, implying that this accordingly brings risk reduction. The reason this is unlikely to hold is that such diversification is based on the premise of leverage, so the risk mitigation provided by the former is frequently overwhelmed by the latter. We can better understand this by looking at the resulting volatility (measured through variance) as follows: $$Var(r_e+r_s)=Var(r_e)+Var(r_s)+2Corr(r_e,r_s)\sqrt{Var(r_e)Var(r_s)}$$

From this expression, we can obtain the necessary conditions under which the spot exposure reduces risk.

$$Var(r_e+r_s)\leq Var(r_e)\Rightarrow Corr(r_e,r_s)\leq-0.5\sqrt{\frac{Var(r_s)}{Var(r_e)}}$$

In summary, for diversification to offset leverage we need a negative correlation between the spot and our investment. As we can see with the examples below, this is rather unlikely; only in the SMI there is a relatively consistent risk reduction.

## Risk in exchange for nothing

So it is clear that leverage is leading to a risk increase; however, this is not necessarily a problem. Indeed, leverage is often employed by investors who are willing to accept greater risks in exchange for a higher expected return, as long as that expected rate of return surpasses the cost of leveraging (cost of funding). Generally speaking and specially among developed countries, this is not the case with currency spots, which do not show consistent movements in one direction in the long run. An intuitive explanation is that the exchange rate is bounded by the economic interdependencies between the involved economies; as a result, it can neither grow nor drop indefinitely.

On the other hand, other asset classes, such as equities, generate consistent profits under normal economic conditions. Although they also experience downturns, these are less significant so we can expect a positive return in the long run. Below we illustrate this contrast between equities and exchange rates with a few examples.

## Wrapping up

In conclusion, the concept of leverage helps to gain intuition on the nature of currency risk. Since exposure to this risk doesn’t imply an increase in the portfolio’s rate of return, the foreign investor is accepting risk in a non-efficient way. In future posts, we will discuss how such an investor can get rid of this risk or try to use it to their advantage. As you can imagine, this hardly ever comes for free. See you then!

This post was written in collaboration with Jorge Sánchez.