Maximizing compounded rate of return

Here is a little puzzle that may stymie many a professional trader. Suppose a
certain stock exhibits a true (geometric) random walk, by which I mean there is
a 50-50 chance that the stock is going up 1% or down 1% every minute. If you buy
this stock, are you most likely, in the long run, to make money, lose money, or
be flat?

Most traders will blurt out the answer “Flat!”, and that is wrong. The correct
answer is you will lose money, at the rate of -0.005% every minute! That is because
for a geometric random walk, the average compounded rate of return is not the
short-term (or one-period) return m (0% here), but is m — s2/2, where s is the standard deviation of the short-term return. This is
consistent with the fact that the geometric mean of a set of numbers is always
smaller than the arithmetic mean (unless the numbers are identical, in which
case the two means are the same). When we assume, as I did, that the arithmetic
mean of the returns is zero, the geometric mean, which gives the average
compounded rate of return, must be negative.

This quantity m — s2/2 holds the key to selecting a maximum growth
strategy. In a previous article (“How
much leverage should you use?”), I described a scheme to maximize the
long-run growth rate of a given investment strategy (i.e., a strategy with a
fixed m and s) by leveraging. However, often we are faced with a
choice of different strategies with different expected returns and risk. How do
we choose between them? Many traders think that we should pick the one with the
highest Sharpe ratio. This is reasonable if a trader fixes each of his or her bets
to have a constant size. But if you are a trader interested in maximizing
long-run wealth (like the Kelly investor I mentioned in the previous article),
the bet size should always be proportional to the compounded return. Maximizing
Sharpe ratio does not guarantee maximal growth for multi-period returns.
Maximizing m — s2/2 does.

For further reading:

Miller, Stephen J. The Arithmetic and Geometric Mean Inequality.

ArithMeanGeoMean.pdf

Sharpe, William. Multi-period Returns.

https://www.stanford.edu/~wfsharpe/mia/rr/mia_rr3.htm