Maximizing Compounded Charge Per Unit Of Measurement Of Return

A elementary formula that few traders utilize

Here is a fiddling puzzle that may stymie many a professional person trader. Suppose a sure enough stock exhibits a truthful (geometric) random walk, yesteryear which I hateful at that topographic point is a 50-50 hazard that the stock is going upward 1% or downwardly 1% every minute. If you lot purchase this stock, are you lot nearly likely, inward the long run, to brand money, lose money, or endure flat?

Most traders volition blurt out the respond “Flat!”, in addition to that is wrong. The right respond is you lot volition lose money, at the charge per unit of measurement of 0.5% every minute! That is because for a geometric random walk, the average compounded charge per unit of measurement of furnish is non the short-term (or one-period) furnish m (1% here), only is m – s²/2, where s (also 1% here) is the measure departure of the short-term return. This is consistent alongside the fact that the geometric hateful of a laid of numbers is ever smaller than the arithmetics hateful (unless the numbers are identical, inward which illustration the 2 agency are the same). When nosotros assume, equally I did, that the arithmetics hateful of the returns is zero, the geometric mean, which gives the average compounded charge per unit of measurement of return, must endure negative.

This quantity m – s²/2 holds the cardinal to selecting a maximum increase strategy. In a previous article (“How much leverage should you lot use?”), I described a scheme to maximize the long-run increase charge per unit of measurement of a given investment strategy (i.e., a strategy alongside a fixed m in addition to s) yesteryear leveraging. However, frequently nosotros are faced alongside a alternative of dissimilar strategies alongside dissimilar expected returns in addition to risk. How produce nosotros direct betwixt them? Many traders retrieve that nosotros should pick the 1 alongside the highest Sharpe ratio. This is reasonable if a trader ready each of his or her bet to convey a constant size. But if you lot are a trader interested inward maximizing long-run wealth (like the Kelly investor I mentioned inward the previous article), the bet size should ever endure proportional to the compounded return. Maximizing Sharpe ratio does non guarantee maximal increase for multi-period returns. Maximizing m – s²/2 does.

For farther reading:

Miller, Stephen J. The Arithmetic in addition to Geometric Mean Inequality. ArithMeanGeoMean.pdf

Sharpe, William. Multi-period Returns. http://www.stanford.edu/ wfsharpe/mia/rr/mia_rr3.htm

Poundstone, William. (2005). Fortune’s Formula. New York: Hill in addition to Wang.