# How much to hedge if borrow in EUR to buy USD assets?

Suppose an investor borrows EUR1m to buy USD stocks. He wants to hedge away the currency risk through EURUSD futures. He should go long EURUSD to hedge this risk. The question is how much of EURUSD futures should he buy to hedge this risk accurately? The intuitive answer is to long EUR1m worth of EURUSD. Are there other considerations that have been missed? Thank you.

• One potential subtlety: if the USD stocks contain multinational businesses some of whose earnings are generated in Euro, it might mean the best hedge is to buy less than 1:1 EURUSD. You would have to test this theory by looking at the data. – dm63 Jan 13 '20 at 5:06
• You're right but I prefer to keep things simple. I only want to hedge away the currency risk. It's hard to quantify how much the effect you mentioned contribute to lessen currency risk. – curious Jan 13 '20 at 5:22
• In practice 1 million EUR is probably good enuf. In theory since you are using futures of say 3 month maturity you have to estimate what the assets will be worth in USD 3 months from now. With bonds, you can use the bonds' yield for your estimate. With stocks I have seen people apply the USD risk free rate to the stock portfolio to estimate its value 3 months from now; clearly that is an approximation. There is a post on quant.stackchange that discusses this. – noob2 Jan 13 '20 at 13:10

If you purchase a Stock today in USD you will model that it has some value in USD in the future,

$$S_{t, usd} = S_{0, usd} + W_{t, usd}$$

where $$W_{t, usd}$$ is some random motion, possibly with drift, such that $$E[W_{t, usd}] = \mu$$.

Ideally you would exchange $$S_{t, usd}$$ at maturity, so this is the amount of notional that should be translated to futures hedges. But, you do not have certainty in its value. For a bond or fixed income investment you would have greater certainty.

Suppose you naturally hedged $$E[S_{t,usd}] = S_{0,usd} + \mu$$, then the amount that you might be over/under hedged at maturity is:

$$S_{t,usd} - E[S_{t, usd}] = W_{t, usd} - \mu$$

The variance in the amount over/under hedged is $$Var(W_{t,usd})$$.

You might consider dynamic hedging: i.e. adjusting the hedge on a daily basis adjusting for the information of daily evolution in $$W$$. You could then model the variance of the under/over hedged element which would presumably be lower.

If there is a correlation between EUR/USD and $$W$$ then you will have an expected PnL effect (whose size will be dependent on underlying volatility) but if the two are uncorrelated the PnL attributed to FX hedging would be expected to be zero, whilst it would have greater variance for increased volatility.