1
$\begingroup$

I found in a paper the relation between the CAGR and the arithmetic average of returns to be

$$g \sim \mu - \frac{\sigma^2}{2}$$

where g is the geometric average, $\mu$ the arithmetic average and $ \sigma^2$ the variance of the returns. I cannot find any formal derivation of this relationship.

$\endgroup$
-1
$\begingroup$

It's a special case of the AM-GM inequality, assuming that market returns follow a lognormal distribution.

Consider the simple example of a stock that has a 50% probability of rising and falling 10% every period.

Its arithmetic average is obviously 0: (50% * +10%) + (50% * -10%) = 0

Its geometric average is (1+10%)^0.5*(1-10%)^0.5 -1 = -0.5%

Or a more extreme example the other way. A stock that doubles and halves with equal probability. It's geometric average is obviously 0. In the long-run, there's a doubling for every halving, and vice versa. But the arithmetic average is (50% * +100%) + (50% * -50%) = +25%.

For the full continuous lognormal distribution rather than my discrete examples above, the half variance formula above is derived from calculating its first moment. http://mathworld.wolfram.com/LogNormalDistribution.html

|improve this answer|||||
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.