When investing in several currencies, one can face currency risks. The below example illustrates the concept. The numbers have been chosen to make the example easily understandable. The code can be found in the notebook.

Let’s assume a portfolio is composed of 50% Peugeot (EUR) and 50% Roche (CHF). The reference currency is EUR, meaning that the performance is reported in EUR.

Let’s create a synthetical stock for Peugeot (EUR).

Similarly, we create a synthetical stock for Roche (CHF).

We simulate EURCHF with 2 situations:

• case 1: EUR is appreciating against CHF

• case 2: EUR is depreciating against CHF

To report the performance in EUR, we need to convert the CHF stock in EUR. Hence, when the EUR is depreciating (CHF is appreciating), we earn more thanks to the CHF stock increase. Consequently, the performance of our portfolio is higher when the CHF is appreciating.

Hedging

Holding assets in a foreign currency is thus a good strategy if we think such currency will remain stable or appreciate over time. If an investor wants only to enjoy the stock appreciation and has no view on the foreign currency, it may be good to consider FX hedging. One possibility is to purchase a FX forward so that at maturity we convert the Roche stock in EUR using a fixed rate. That way, if the CHF decreases (resp. increases), we don’t earn less (resp. more). As stated below, currency hedging is about removing the uncertainty regarding the future spot exchange rate.

Interest Rate Parity (IRP) refers to a theoretical condition (when NFL holds) in which the relationship between interest rates and the spot and forward currency values of two countries are in equilibrium.

E.g. say local and foreign currencies are trading at par (exchange rate = 1). However, the interest rate is lower in foreign currency. It would make sense to borrow some money in foreign currency, convert it, and then invest it in local. In doing so, one could remove uncertainty in entering a forward contract to convert back the amount later. If IRP holds, the forward rate would eradicate all the profit from the transaction.

We define:

• $A_F$: the amount we invest in foreign currency

• $S_0$: the spot rate i.e. the current exchange rate

• $F$: the forward rate i.e. the future spot rate that we will use to convert the amount back

• $r_{A,D}$: the return of the investment in domestic currency

• $r_{A,F}$: the return of the investment in foreign currency

• $r_D$: the interest rate in the domestic country

• $r_F$: the interest rate in the foreign country

$r_{A,D} = \frac{\text{price of the asset (in domestic currency) at time 1}}{\text{price of the asset (in domestic currency) at time 0}}-1= \frac{A_F(1+r_{A,F})F}{A_FS_0}-1$

Interest Rate Parity states that:

\[F = S_0 \frac{1+r_D}{1+r_F}\]

Using this formula, we have:

\[\begin{align*} r_{A,D} &= \frac{(1+r_{A,F})}{S_0}S_0 \frac{1+r_D}{1+r_F}-1 \\ \ & = (1+r_{A,F})\frac{1+r_D}{1+r_F}-1 \end{align*}\]

Which leads to:

$r_{A,D}(1+r_F) = (1+r_{A,F}) (1+r_D) - (1+r_F)$

$r_{A,D} + \color{red}{r_{A,D} r_F} = 1 + r_D + r_{A,F} + \color{red}{r_{A,F}r_D} - 1 - r_F$

We can simplify the terms in red because they are very close to zero (money market returns are usually very small). Finally:

\[r_{A,D} \approx r_{A,F} + r_D - r_F\]

This result shows that money market hedging involves entering a long position in domestic currency and a short position in foreign currency. $r_D - r_F$ can be seen as the interest rate differential.

Important: as stated in this paper, currency hedging is about removing the uncertainty regarding the future spot exchange rate. It does not mean that the investor earns the local return of foreign assets; it means that he earns the local return plus an interest rate differential.