Sizing FX Futures Hedges for Foreign-Currency Equity Holdings
Summary
The document considers an investor who borrows euros to buy US stocks and wants to offset the resulting EUR/USD exposure with currency futures. It explains why matching the initial euro borrowing amount may not maintain a precise hedge: the value of the stock investment in dollars is uncertain at the hedge’s maturity. The appropriate hedge notionals depend on the future dollar value that will ultimately be converted, which cannot be known in advance.
A hedge based on the expected future stock value can leave an over- or under-hedged position, with the mismatch tied to the stock investment’s uncertainty. The answer suggests adjusting the hedge over time as the investment evolves, which may reduce that mismatch. It also notes that correlation between the asset’s dollar value changes and EUR/USD movements can affect expected P&L; absent correlation, the hedge contribution is expected to average to zero, though exposure variance remains. No hedge ratio, contract specifications, or empirical test is provided, so implementation requires further modeling.
Key ideas
- The initial euro borrowing amount may not equal the right hedge notional at maturity.
- The future dollar value of an equity investment is uncertain, creating residual currency exposure.
- Hedging an expected future value can leave a mismatch whose size depends on the investment’s variability.
- Adjusting the currency hedge as the investment changes may reduce over- or under-hedging.
- Correlation between asset value changes and the exchange rate can influence hedge P&L.
Tags
Full text
# How much to hedge if borrow in EUR to buy USD assets?
# 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.
## Answer by Attack68 (score 1)
https://quant.stackexchange.com/a/50707
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.