Recap
The Kelly Criterion tells you how much of your capital you should risk on a single bet, e.g. on a horse in a race. Kelly, in his original paper, contrived a rather odd situation where you actually have an “edge,” i.e. the expected payoff of your bet is positive. The original paper has been plundered by various investment bank quants to inform their investing style: to be frank, it seems to relate to investing more than betting at a race track. In fact, the original paper was mainly about optimising information flow through a noisy channel. Kelly was a colleague of Claude Shannon, a superstar for those of us who are interested in great scientists and mathematicians, and the originator of the concept of entropy in the context of communication theory.
Kelly’s criterion, as applied to investing, is equivalent to making the assumption of a logarithmic utility function. Roughly speaking, the utility an investor derives from a given amount of wealth is proportional to the logarithm of the amount of wealth he has. Logarithms have some pleasing properties: if your wealth grows exponentially, the utility you derive from your growing wealth grows linearly. In other words, both Jeff Bezos and a homeless person derive the same satisfaction from finding out that their spending power has gone up by five percent, even though it takes many orders of magnitude more dollars for Bezos than for the hobo. This, of course, is the justification for progressive taxation and re-distributive policies but this post is about economics, not policy.
Although the log function has these pleasing properties, it’s far from obvious why it should be preferred to other convex functions. I am not the first person to think about this: the anonymous author of this blog has though about this and written up his thoughts about the very topic. He seems to conclude that the choice is arbitrary.
Although Kelly’s paper seems to be terribly important, I don’t know anyone who actually uses it as a guideline for action. The problem is that the “bet size” is determined by the size of one’s edge: the extent to which the bookie’s odds are mis-priced. This is (obviously) impossible to measure, or even estimate with any confidence. In fact, there is a huge body of academic literature which supports the view that in large, liquid markets the edge one has is zero, which leads to the familiar “modern” portfolio theory conclusion that betting on a particular stock is a mugs game and that you can’t beat ‘em so you’d better join ‘em and invest in the whole market, or index.
Utility is such a central concept in economics, but one that is not directly measured. It explains so much, but remains, like the luminiferous ether, unobservable, its existence inferred from observable phenomena.
Although the Kelly Criterion does not directly inform my investment decisions, it does shape my attitude to risk. I don’t side with Harry Markowitz, who argued that risk is simply a matter of a variance of a future distribution of returns. For me, risk is about losing capital, being unable to invest tomorrow. This, not some sigma value of VaR measure seems to be what should be minimized; this is often done by buying some insurance — maybe a put or two. This doesn’t eliminate risk, but reduces it to an uninsurable element, which Frank Knight argued was the the sort of risk that will be rewarded with profit. Let’s hope he was right.
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