Inferring Currency-Rate Covariance from Triangular Variances

Summary

The document explains how covariance between two currency-rate returns can be inferred when a third rate is related to them through a currency triangle. In log terms, the cross rate is represented as the sum or difference of the two component rates, depending on quote conventions. Applying the variance identity for a sum or difference gives the variance of the cross rate as the two component variances plus or minus twice their covariance. Rearranging this identity yields the covariance formula.

This is an algebraic relationship, not the usual sample-covariance calculation from paired observations. The sign depends on how the exchange rates are quoted and combined, so the currency convention must be checked before applying the formula. The discussion does not provide data or a numerical example, and its direct application assumes the rates are expressed consistently and the variances refer to compatible log-return measures.

Key ideas

• Log exchange rates in a currency triangle combine through addition or subtraction, depending on quote conventions.
• The variance of a sum or difference includes the two individual variances and a covariance term.
• Rearranging the variance identity allows covariance to be inferred from three compatible variances.
• The sign in the formula depends on the direction in which the rates are quoted and combined.
• This implied covariance relationship differs from estimating covariance directly from paired observations.

Tags

# Calculating covariance from three variances


# Calculating covariance from three variances












I have been asked to look to refactor some code.

There is a line shown below:

$\text{implied covariance} = -\frac{(\text{var}_1 - \text{var}_2 - \text{var}_3)} {2}$,

where $\text{var}_1$ is the implied variance of AUDUSD, $\text{var}_2$ is the implied variance of USDCAD and $\text{var}_3$ is the implied variance AUDCAD

I understand that this is a calculation of covariance between AUDCAD.

However I don't understand the $\text{var}_1 - \text{var}_2 - \text{var}_3$ line. I thought the covariance between two variables was the variance of the two variables multiplied together divided by $n-1$.

## Answer by Kermittfrog (score 3, accepted)

https://quant.stackexchange.com/a/51820

I think what you are effectively looking at is

$$\ \begin{align} \log(S_{AUDCAD})&=\log(S_{AUDUSD})\pm\log(S_{USDCAD})\\ \Rightarrow z&=x\pm y \end{align} $$ Thus,

$$ \sigma_z^2=\mathrm{E}\left(\left(x\pm y\right)^2\right)- [\mathrm{E}(x\pm y)]^2 =\sigma_x^2+\sigma_y^2\pm 2\sigma_{xy} $$ Hence, $$ \tag{1} \sigma_{xy}=\frac{\sigma_z^2-\sigma_x^2-\sigma_y^2}{\pm 2} $$

Does that work for you?

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