How Inverting an Asset Changes Its Log-Return Correlation

Summary

The document derives the correlation between the log returns of an asset’s reciprocal price and another asset. Since taking the logarithm of the reciprocal negates the original log price, the transformed log return is the negative of the original log return.

Using the definition of correlation and the scaling rules for covariance and variance, negating one variable reverses the sign of its correlation with the other variable. Thus, if the original log-return correlation is rho, the reciprocal-price log-return correlation is minus rho. The argument is an algebraic identity; it assumes the relevant variances are defined and nonzero, and offers no empirical market data or broader trading strategy.

Key ideas

• The log of a reciprocal price equals the negative of the original log price.
• Negating one variable reverses the sign of its correlation with another variable.
• The result applies directly to log returns when one asset price is replaced by its reciprocal.

Tags

# correlation of 1/X


# correlation of 1/X












My question is the following. If the correlation between the log-returns of X and Y is rho, what would be the correlation between the log returns of 1/X and Y ?

Thanks for your answers.

## Answer by Dave (score 6)

https://quant.stackexchange.com/a/73354

A few identities will be helpful to remember.

$$\log(1/X)=\log(X^{-1})=-\log(X)$$

$$ cor(X,Y)=\dfrac{cov(X,Y)}{\sqrt{var(X)var(Y)}} $$

$$ var(X)=cov(X,X)\\ $$

$$ cov(aX,bY)=ab\times cov(X,Y)\\ var(aX)=a^2var(X) $$

(This last one means that $var(-X)=var(X)$.

Combining these:

$$ cor(-\log(X),\log(Y))\\ =-corr(\log(X),\log(Y)\\ =-\rho $$

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