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Sizing a Currency Forward to Hedge an Invested Foreign Currency

Article Quant Q&A · Author: Vishaal Sudarsan
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Summary

The document clarifies a currency forward hedge when foreign currency grows through investment before settlement. If one pound is invested at the pound risk-free rate, the amount held at maturity is greater than the initial pound. A forward position sized only for the initial amount would therefore leave part of the terminal currency exposure unconverted. The hedge must cover the full grown balance.

The explanation treats each forward contract as covering one unit of currency, then generalizes to a position of any size: multiply the terminal foreign-currency amount by the forward exchange rate to obtain the corresponding domestic-currency proceeds. This resolves why the strategy’s terminal value includes both the investment growth factor and the forward rate. The example is a no-arbitrage bookkeeping explanation under the stated risk-free investment setup; it does not discuss transaction costs, collateral, basis, or differences between textbook contract units and dealer trade sizing.

Key ideas

  • An invested foreign-currency balance grows at its own risk-free rate until forward settlement.
  • A hedge must cover the terminal foreign-currency amount rather than only the amount initially invested.
  • The domestic proceeds equal the hedged terminal currency balance multiplied by the forward exchange rate.
  • Forward contract counts depend on the currency units represented by each contract.

Tags

ForexArbitrageRisk management
Full text
# Compute Forward Exchange Rates using Risk Free Rates


# Compute Forward Exchange Rates using Risk Free Rates












In the following image :

- I am not able to understand how, the final value of strategy B can be equal to $e^{r_{GBP}T}F(0,T)$

- According to me it should be just $F(0,T)$

- My reasoning is that when you received 1 pound and invested it, you’ll get $e^{r_{GBP}T}$ pounds in return. But we use that to cover our short position in the forward contract entered at time 0, so at the end we’re only left with the forward price $F(0,T)$ paid to us at time T.

Please correct me where I am going wrong.

Source of image : Mathematical Finance : An Introduction to Financial Engineering by Marek Capinski and Tomasz Zastawniak

## Answer by nbbo2 (score 1, accepted)

https://quant.stackexchange.com/a/53726

What is slightly confusing here is where they mention that you should short $X$ forward contracts. Implicitly they are assuming that each "contract" refers to one GBP and you buy or sell as many contracts as you need. (In practice you would simply call the dealer and tell them the size of the GBP position you have in mind, no need to mention "contracts").

Now the analysis. The 1 Pound mentioned in B. will have grown to $1 \cdot e^{r_{GBP}T}$ Pounds at time T. So you must today sell forward not 1 forward contract but $e^{r_{GBP}T}$ of them, so you are able to convert at time T the full amount of GBP that you have and not just the initial 1 GBP, into USD. This justifies the RHS of the equation shown.

Another way to look at it (without mentioning contracts) is that if $F(0,T)$ is the forward rate then $X \cdot F(0,T)$ is the result of converting $X$ GBP into USD at this rate. Here $X$ is $1 \cdot e^{r_{GBP}T}$.

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.