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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.

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    $\begingroup$ you know about $\log(1/X)=-\log X$ ? $\endgroup$
    – Kurt G.
    Sep 29, 2022 at 12:10

1 Answer 1

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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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