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### Kelly Criterion

Kelly Criterion [Kelly 1956] permits the investor to maximise return on investment, geometrically. The theory was developed by John L Kelly (pictured left) out of the information theory work of Claude Shannon. The Kelly Criterion uses the analogy of a crooked gambler who has control of a private wire (this theory is from the 1950s hence the old terminology) to their local bookmaker.

The wire provides the bookmaker with results. Having control of the wire gives the gambler time to learn the result of a horse race and delay the result getting to the bookmaker long enough to place the winning bet. Obviously, if the wire is perfect you would bet your house on the result but what if the wire had an intermittent fault, how much do you then wager?

The answer is that you bet the size of your edge. If your wire gives you a 100% edge over the bookmaker then you bet 100% of of your bankroll but if you only have a 5% edge then you only bet 5%. No matter what the edge, so long as it is positive, your bankroll will grow geometrically. If your edge is negative then simply lay instead of backing the proposition.

The fraction to bet is given by the equation

where f is the fraction of your bankroll to bet, b is the net money returned on a £1 bet eg. if the odds are 2.1 on Betfair then the return is 1.1, p is the probability of winning and q is the probability of losing (1 - p).

For example, a fair bookie offers 2.0 on either tossing heads or tails. However, the coin is biased to heads with probability 0.6. You must therefore bet ((1 * 0.6) - 0.4) /1 = 0.2 (20% of your bankroll) on heads. Some people prefer to use a fractional Kelly bet as a full Kelly can make your bankroll fluctuate considerably. A Half Kelly, in the previous example, would have you betting 10% rather than 20%.

Model Kelly Criterion in a spreadsheet

You can model geometric growth with Kelly Criterion using a spreadsheet.

Click on the image above to see the detail.

In cell B6 we see the equation for the size of Kelly bet as a fraction of the total bankroll. The figures above are fixed values for predicted probabilities and return. In cells B5 and B7 we see variations of the Kelly Criterion; f/2 is a fractional Kelly bet of one half of the recommended amount. 2f is double the recommended amount.

Betting above the recommended Kelly Criterion will result in ruin. Betting a fractional amount of the Kelly Criterion may be wise if your system is inaccurate or open to fluctuations. A long losing streak will ruin a bankroll, even if using Kelly Criterion, so always consider optimising a fractional Kelly bet.

In the sample run we see a starting bankroll of 100, a wager given B6 multiplied by B10, a probability given by the random function in the range 0 to 1 and remainder being the bankroll updated with the win or the loss. As you can see, the bankroll takes time to ramp up but ramp up it does. Of course, in reality such ramping up will run into market capacity problems where your bet size is too big to be accepted.

Reference

Kelly, J., (1956), "A New Interpretation of the Information Rate", Bell System Technical Journal 1, pp. 917–926.