Measuring FX Exposure Contributions to Portfolio Variance

Summary

The note considers how foreign exchange exposure affects the risk of a portfolio holding foreign assets. For an unhedged position, the return can be represented as the hedged portfolio return plus a scaled currency return. Its variance includes the portfolio variance, the currency variance scaled by the square of exposure, and a covariance term that may raise or lower total risk.

Differentiating this variance with respect to the exposure fraction gives a marginal change of twice the covariance plus twice the exposure fraction times currency variance. A separate response suggests freezing exchange rates and recomputing portfolio volatility to gauge an FX effect, while another illustrates the return-variance decomposition for a foreign equity. The material is a compact variance-based explanation, not a full decomposition of portfolio standard deviation across multiple simultaneous currency exposures; signs and magnitudes depend on covariance and exposure.

Key ideas

• Unhedged portfolio returns can be modeled as a hedged return plus a scaled currency return.
• Currency exposure changes variance through both currency variance and covariance with the underlying portfolio.
• The covariance contribution can be positive or negative.
• Differentiating variance with respect to the exposure fraction yields its marginal variance contribution.
• The formula isolates one exposure and does not fully specify attribution across multiple currencies.

Tags

# How to compute the foreign exchange volatility within a portfolio


# How to compute the foreign exchange volatility within a portfolio












Suppose I have a portfolio of 5 assets. Assets 1 and 2 have foreign exchange exposures and therefore foreign exchange volatility. How can I calculate the marginal contribution to the total portfolio volatility from the individual foreign exchange exposures?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/25323

How about letting the FX rates remain fixed, and recalculate the portfolio volatility. That seems very obvious - am i missing something?

## Answer by Alex C (score 0)

https://quant.stackexchange.com/a/25332

If a USD based investor owns shares of Toyota Motor in Japan, the variance of USD based returns is approximately equal to the variance of Toyota in yen, plus the variance of USDJPY plus twice the covariance between Toyota and the exchange rate. The last term could be positive or negative; if I had to guess for a big exporter like Toyota it is probably slightly negative.

## Answer by user20429 (score 0)

https://quant.stackexchange.com/a/25599

If $R$ and $r$ are the return on the portfolio after currency hedging and on the currency, if I write $V(\cdot)$ for variance, and a fraction of $t$ of the portfolio is exposed to currency risk, then the return of the unhedged portfolio is $R+tr$. Then: $$V(R+tr) = V(R) + 2t\mathrm{Cov}(R,r) + t^2 V(r)$$ so the marginal contribution (derivative with respect to t) is $$\frac {\text d} {\text d t}V(R+tr) = 2\mathrm{Cov}(R,r) + 2tV^2(r).$$

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