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# Investment: The Kelly Criterion

The Kelly Criterion dates from the 1956 paper where John Kelly sketched the idea that by maximising the expected logarithm of final wealth in every period, one would have good final wealth. In 1961, Leo Breiman proved that the Kelly strategy would give more wealth than any other “sufficiently different” strategy in the long run. Indeed, the ratio of the log investor’s wealth to the “other strategy” is wider and wider as time increases without limit. Later, Thorsten Hens and Klaus Shenk-Hoppe proved that if there was one Kelly log better and a finite number of other betters all competing for the same wealth, then the Kelly better would, in the end, not only get the most wealth but would get all of it.

Sports betters and investment professionals (including some of the greatest hedge fund managers of all time) have all used such strategies. An understanding of the mathematics of gambling risks versus rewards helps these investors.

Flawed applications
In the short run, Kelly wagering is quite risky, since the log function is the most risky utility function that one would ever want to use. This is measured by the Arrow-Pratt risk aversion index, which for the utility function u(w) of wealth is: -u”(w)/u’(w)=1/w and is essentially zero for even moderate sized wealth as Figure 1 illustrates. Observe that for the half-Kelly strategy (50% in cash, 50% in the Kelly strategy), final wealth is usually lower but the wealth path is much smoother. To win, you must have a strategy with positive expectation, as the Kelly and half- Kelly strategies have. Betting on the favourite turns \$2500 into \$480 and is a losing, negative expected value wager.

The Kelly, half- Kelly and other fractional Kelly wagers are well-known to experts in the field but were little-known to the general public until the publication of William Poundstone’s book, Fortune’s Formula. Indeed, Motley Fool, Morningstar and other investment advisers recommend the strategy. Unfortunately, most applications like the discussion in Poundstone are flawed. One needs to use multivariate non-linear programming to compute optimal policies and there are dangers, as well as benefits, from these strategies.

A simulation Donald Hausch and I carried out in 1986 illustrates the medium-term properties of Kelly and half-Kelly strategies. You have five possible investments, each with a 14% advantage. So, a £1 investment returns £1.14 on average. The wagers are odds of 1:1 with a 57% chance of getting £2 and a 43% chance of losing. Then there are 2-1, 3-1, 4-1 and 5-1 wagers, as shown in Table 1.

You make 700 independent bets starting with £1000. Table 2 shows the results from 1000 simulations. The full Kelly strategy multiplies initial wealth by over 100 times, 16.6% of the time and by over 50 times, 30.2% of the time. This is huge growth but the cost is that there is only an 87% chance of being ahead, such as ending up with more than £1000. Half-Kelly loses the great growth but is ahead 95.4% of the time. Hence, there is a growth-security trade-off. However, even with the half-Kelly strategy, you are not assured to win in the end. Indeed, the minimum pay-off is £18 for the Kelly from £1000 and £145 for the half-Kelly. How can this be? You have 700 independent bets, all with a 14% advantage. Unfortunately, a bad scenario can occur and lead to losses. The Kelly strategy is not a sure thing but, if used carefully, is very valuable.

William Ziemba is the alumni professor of financial modelling and stochastic optimisation (Emeritus) at the University of British Columbia and a visiting professor at the Mathematical Institute, University of Oxford and the University of Reading.