# Currency Hedged Excess Return

In the famous article "Global portfolio optimisation" of Black and Litterman, the authors defined the excess return on currency-hedged assets as the following :

$$E_t = 100 \frac{P_{t+1}X_t}{P_tX_{t+1}} + 100\frac{X_{t+1} - F_t^{t+1}}{X_t}(1+R_t) - R_t$$

where $$E_t$$ the currency-hedged excess return of an asset, $$P_t$$ the price of the asset in the foreign currency, $$X_t$$ the exchange rate in units of foreign currency per US dollar, $$R_t$$ the domestic short rate and $$F_t^{t+1}$$ is the one-period forward exchange rate at time $$t$$.

My question is : I do not understand the $$(1+R_t)$$ in the second element at the right side of the equation, does someone have an explanation ?

Thank you for your help !

• You hedge with one period forward contracts. How many contracts do you need? If you have 1 million USD equivalent in foreign assets now you sell forward contracts that will pay you 1*(1+R) million usd vs foreign currency in one period, where R is the USD one period interest rate. The second term in your equationis the return on these forward contracts. – noob2 Jul 23 '19 at 19:25
• Why forward contracts will "pay me" something ? I do agree that you take current and foreign domestic rate to do price a forex forward, but why, this will affect the return on an ex post perspective ? – JeanGuillaume Jul 25 '19 at 7:48
• In a forward you agree to pay one currency and receive another, or if you prefer you are long one currency and short another. Specifically to hedge foreign assets you are short the foreign currency and long ("receive") the domestic currency. But the hedge is never perfect and there will be a hedging error ex post. The formula you give captures this. – noob2 Jul 25 '19 at 8:28
• "The hedge is never perfect" because you do not know, at the beginning of the period, how much your foreign assets will be worth at the end of the period. So you cannot arrange a perfect hedge, unlike the case of a known future cash flow (ex: a customer promising to pay you X Swiss francs in 3 months). – noob2 Jul 25 '19 at 8:55
• I agree with what you wrote. But there is still a misunderstanding for me. May be let us start from the beginning. Let us suppose, I want to invest in the foreign asset P. To do so, I need $\frac{P_t}{X_t}\$ $and I will get back, after one period$\frac{P_{t+1}}{X_{t+1}}\ , multiplied by 100, it gives me in percentage my gross return without hedging. Now, I want to hedge the currency risk : So I will sell a forward for time $t+1$ to be as you said, short on the foreign and long on the domestic to compensate the first position. My payoff will be $(X_{t+1} - F^{t+1}{t})*N$. – JeanGuillaume Jul 25 '19 at 9:32

I think you have the answer in the comment you made. I will again explain with the inverse exchange rate S, and let me represent the forward price of this exchange by f. And let me represent the first time by 0 and the second by t, no more multi-period as in the previous answer! Now the unhedged asset value at next step will be:

$$P_t S_t$$

We want exposure to asset but not the exchange rate, so let’s hedge with N forward contracts:

$$P_t S_t+N\left(f-S_t\right)$$

If we can set N equal to $$P_t$$ then we have a perfect hedge but this value is unknown. So you can set N equal to $$P_0$$ or its forward value $$P_0 \left(1+r_f\right)$$. Let’s go with the second as you suggested.

$$P_t S_t+P_0\left(1+r_f\right) \left(f-S_t\right)$$

Now divide through by the initial value $$P_0 S_0$$, We get two terms:

$$\frac{P_t S_t}{P_0 S_0}+\frac{1}{S_0}\left(1+r_f\right) \left(f-S_t\right)$$

Now we know that:

$$f=S_0\frac{1+r_d}{1+r_f}$$.

Which means we can write the previous expression as follows:

$$\frac{P_t S_t}{P_0 S_0}+\frac{1+r_d}{f}\left(f-S_t\right)$$

Now just invert the exchange rate and the forward price to the format in the question: Substitute 1/X for S and 1/F for f.

$$\frac{P_t X_0}{P_0 X_t}+\left(1+r_d\right) F\left(\frac{1}{F}-\frac{1}{X_t}\right)$$

Combining the terms and cancelling the F, you get very close to their formula:

$$\frac{P_t X_0}{P_0 X_t}+ \frac{X_t-F}{X_t} \left(1+r_d\right)$$

Now, since we want an Excess Return, we subtract the dollar discount rate to get their full formula. Notice however this difference: they are dividing the second term by $$X_{0}$$ though instead of $$X_t$$.

• Concerning the difference at the end, it might be a typo. Thanks a lot for your explanation and your patience ! – JeanGuillaume Aug 30 '19 at 13:46

I have seen something similar in a multi-period context, so I will have a go though I don't see the detailed calculation that Black and Litterman might have used.

Let me first invert the exchange rate and call it S, which now represents the price of one unit of other currencies in dollars (e.g., one pound equal S dollars). This is just to simplify the calculation.

The value of the unhedged portfolio at any time t will be $$P_t S_t$$. Let's first assume that we are investing for one period. We will be receiving $$P_{t+1}$$ units of say Pounds at time t+1, the dollar value of which will fluctuate, so to hedge we buy dollars forward at say $$F_{t,t+1}$$, which represents the forward price of one pound in dollars. But we need to hedge, say $$P_t$$ pounds, so we buy P contracts instead of one. Our hedged portfolio value at time $$t+1$$ would be:

$$V_{t+1}=P_{t+1}S_{t+1}+P_t \left(F_{t,t+1}-S_{t+1} \right)$$

Now lets consider multi-period and assume we keep rolling the forward hedge - we keep hedging one period ahead. The value of the portfolio after, say T period, would be:

$$V_T=P_{T}S_{T}+\sum_{t=0}^{T-1}{P_{t} \left(F_{t,t+1}-S_{t+1} \right)}$$

We would have funded the asset, but the foreign exchange hedge will be making or losing money, so there are a few alternative assumptions one can make - e.g., 1) the investment in the asset is reduced or increased by the amount of loss/gain on the currency hedge, 2) the interest rates are so low, and the net position will be small, so it won't make a difference, 3) we should account for the time value of money. I will go with the third and add the charge/reward for the balance in the hedge account:

$$V_T=P_{T}S_{T}+\sum_{t=0}^{T-1}{P_{t} \left(F_{t,t+1}-S_{t+1} \right)\left( 1+R \right)^{T-t-1}}$$

If you divide the above by the initial value of the portfolio, you shall get the formula in the form you have written. Notice I have inverted X so that is why you will see some differences.

So i can explain the 1+R in multi period settings but I am not sure if this is what the writers intended. Another potential explanation would be margining - collateralised trade or mark-to-market.

• Thank you for your well-documented answer. I completely understand your point. However when we take your multi-period formula with $T=1$, we do not get the (1+R)… When you say " But we need to hedge, say $P_t$ " . could the answer be here ? We could say that we expect the risky asset "in average" to earn the risk-free rate … But in this case, I would take the foreign risk free rate and not the domestic one. – JeanGuillaume Jul 30 '19 at 11:08