What are the mean and variance of the ratio of two lognormal variables?

Ratios of two normally distributed variables (e.g X/Y) have no moments (e.g. means and variances) because Y can have zero values. However, lognormal variables have no zero values. How can I calculate the mean and variance of the ratio of two lognormal variables?

Answer

Note that log(X/Y)=log(X)log(Y). Since X and Y are lognormally distributed, log(X) and log(Y) are Normally distributed.

I’ll assume that log(X) and log(Y) have means μX and μY, variances σ2X and σ2Y, and covariance σXY (equal to zero if X and Y are independent) and are jointly normally distributed. The difference Z is then normally distributed with mean μZ=μXμY and variance σ2Z=σ2X+σ2Y2σXY.

To get back to X/Y, note that X/Y=expZ, showing that X/Y is itself lognormally distributed with parameters μZ and σ2Z. The relationship between the mean and variance of a lognormal variate and the mean and variance of the corresponding normal variate is:

E(X/Y)=EeZ=exp{μZ+12σ2Z}

Var(X/Y)=Var(eZ)=exp{2μZ+2σ2Z}exp{2μZ+σ2Z}.

This can be rather easily derived by considering the moment-generating function of the normal distribution with mean μZ and variance σ2Z.

Attribution
Source : Link , Question Author : KuJ , Answer Author : cardinal

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