Calculating the Return on Investment in Currency X When Investing in Currency Y
Abstract
This post presents a method for calculating the return on an investment denominated in a base currency X, when the actual investment is made in a foreign currency Y. The approach considers both the performance of the asset in Y and the movement of the exchange rate between X and Y. A general formula is derived and explained in detail, along with a simplified approximation for practical use. The analysis also accommodates trade costs, taxes, and inflation. Several scenarios are discussed to illustrate how currency appreciation or depreciation affects investment returns. A numerical example is provided, alongside R and Python implementations of the calculation.
Return on Investment in X from Investments in Y
To calculate the return on an investment denominated in currency X, when the investment is made in currency Y, we use the following formula (with derivation provided):
where:
- \(\mathrm{V}[\mathrm{X}]\) – starting value of the investment in X
- \(\mathrm{V}^{\prime}[\mathrm{X}]\) – final value of the investment in X
- \(\mathrm{r}_{\mathrm{X}}\) – return on investment in currency X
- \(\mathrm{r}_{\mathrm{Y}}\) – return on investment in currency Y
- \(\mathrm{e}_{\mathrm{YX}}\) – exchange rate of Y in terms of X (units of X per unit of Y)
- \(\mathrm{e}_{\mathrm{YX}}^{\prime}\) – future exchange rate after a given period
- \(\mathrm{R}_{\mathrm{YX}}\) – relative change in the exchange rate \(\mathrm{e}_{\mathrm{YX}}\)
Trade costs, taxes, and inflation can be included when calculating \(\mathrm{r}_{\mathrm{Y}}\). Exchange costs can be taken into account when determining \(\mathrm{e}_{\mathrm{YX}}^{\prime}\).
We know that:
So:
Rule of Thumb
One can write the discussed return as:
It leads to the following remarks.
Firstly, \(\mathrm{R}_{\mathrm{XY}}\approx 0\), then:
\[\mathrm{r}_{\mathrm{X}}\approx \mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}.\]So, if exchange rate of X changes slightly or negligibly, then the return on investment is just about the difference between \(\mathrm{r}_{\mathrm{Y}}\) and \(\mathrm{R}_{\mathrm{XY}}\).
Further, if \(\mathrm{R}_{\mathrm{XY}}\mathrm{ }>0\) (appreciation of X against Y), and \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}>0\), one can say that:
\[\mathrm{r}_{\mathrm{X}} < \mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}.\]The profit on the investment in X is slightly less than \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}\).
Lastly, if \(\mathrm{R}_{\mathrm{XY}}\mathrm{ }>0\) and \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}<0\), one can say that:
\[\mathrm{r}_{\mathrm{X}} > \mathrm{r}_{\mathrm{Y}} - \mathrm{R}_{\mathrm{XY}}.\]In this case, the investment loss does not exceed \(\left| \mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}} \right|\).
A similar analysis can be carried out for a depreciation of X against Y (\(\mathrm{R}_{\mathrm{XY}}\mathrm{ }<0\)).
Numerical example
Let: \(\mathrm{r}_{\mathrm{Y}}\mathrm{ }=\mathrm{ }8\%\), \(\mathrm{e}_{\mathrm{XY}}\mathrm{ }=\mathrm{ }4.00\), \(\mathrm{e}^{\prime}_{\mathrm{XY}}\mathrm{ }=\mathrm{ }4.10\).
Then:
or, as \(\mathrm{R}_{\mathrm{XY}}=0.025\mathrm{ }>0\) and \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}=0.055>0\), one can say approximately, using the upper bound, that \(\mathrm{r}_{\mathrm{X}}\) doesn’t exceed \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}=0.055\).
Repeating calculation in R and Python
R:
>> r_X <- function(r_Y, e_XY, e_XY_prime) {
+ R_XY <- (e_XY_prime / e_XY) - 1
+ r_X <- (r_Y - R_XY) / (1 + R_XY)
+ r_X
+ }
>
> r_X(0.08, 4, 4.1)
[1] 0.05365854
Python:
>>> def r_X(r_Y, e_XY, e_XY_prime):
... R_XY = (e_XY_prime / e_XY) - 1
... r_X_value = (r_Y - R_XY) / (1 + R_XY)
... return r_X_value
...
>>> r_X(0.08, 4, 4.1)
0.05365853658536584
Conclusion
When investing in foreign assets, evaluating returns in the investor’s domestic currency requires combining asset performance with exchange rate changes. The derived formula
captures this relationship precisely. The rule-of-thumb approximation is useful for intuitive reasoning, particularly when exchange rate changes are small or one knows the signs of \(\mathrm{R}_{\mathrm{XY}}\) and \(\mathrm{r}_{\mathrm{Y}}-\mathrm{R}_{\mathrm{XY}}\). The direction and magnitude of currency appreciation or depreciation can either amplify or erode investment gains. Importantly, if the return in Y is strong enough, it may outweigh a currency loss, resulting in a net gain. Conversely, a moderate loss in Y may be offset by favorable exchange rate movements. This dual-dependence highlights the importance of exchange rate exposure in cross-currency investing.
To measure portfolio risk accounting for currency exposure, try my free tool: VaRCalc — Multi-Method Value at Risk Calculator.
For a statistical inference framework for evaluating trading strategies incorporating FX returns, see my textbook: Scientific Backtesting.
To explore synthetic return simulation workflows based on GARCH models, see: Synthetic Data in Inferential Backtesting.
Download the Code Scripts
Currency Adjusted Roi Scripts (R + Python)
