The Kelly formula and its derivations are employed in money management systems to define the optimum amount that should be invested and risked in a series of repeated trades that have a positive expectation. The aim of the formula is to maximise the long-term growth rate of funds based on the probability and size of the potential gain as well as the size of the potential losses.
The formula was originally developed in 1956 by John Kelly, a researcher for the telecommunications company AT&T, to find a solution to the problem of random interference (noise) on long-distance telephone lines. The Kelly formula ultimately helped to reduce information loss and increase data flow in telephone communication.
In his paper ‘A new interpretation of information rate’, Kelly compared information theory to the problem of a gambler who has inside information on a horse race. Subsequently, the formula was used by sports bettors and investors to determine the optimum percentage of a maximum amount that should be invested and risked in a bet or investment.
The formula requires the estimated percentage of wins (W%) and the win/loss ratio dividing the average size of wins (W) by the average size of losses (L).
\(\mathrm{Optimum \:risk \:in\%}=\mathrm{W\%\:–\:}\frac{1-w\%}{\frac{w}{L}}\)
For example, if the probability of winning is 70%, the average win is 60 and the average loss is 30, the optimum amount that should be risked is 55%.
\( 0.7\:–\: \frac{1\:–\:0.7}{\frac{60}{30}}=\mathrm{0.7\:–\:}\frac{0.3}{2}=\mathrm{0.55}\)
This means that in this scenario, risking 55% of total funds would provide the maximum rate of return. In essence, the formula states that the higher the risk (ie the lower the winning probability), the lower the percentage of total funds that should be invested and vice versa. While investors instinctively know this, the Kelly formula provides a calculation of the amount that should be risked.
Problems
There is however a major problem with the Kelly formula. In our example, risking 55% of funds in a series of trades that have a positive expectation of 70%, could still result in a sequence of losing trades that will diminish funds significantly before they can increase. While the formula will guarantee that not all funds can be lost in a series of trades, risking the amount that maximises the return can still lead to very volatile results.
For practical purposes, the Kelly formula is therefore often lowered by 20 – 50% in order to reduce volatility. For the same reason it is suggested never to risk more than 25% of total funds.
Another problem with the Kelly formula is that its assumptions are not directly applicable to the investment world. The formula considers only two possible outcomes, total win and total loss, whereas a typical investment can yield a range of returns. It is also very difficult to determine the exact probability of making a positive investment.
However, the formula can still be put to use in a money management system. The probability of winning can, for example, be calculated as the number of investments returning a positive yield divided by the number of total investments. The win/loss ratio can be calculated as the average size of positive returns divided by the average size of negative returns. When these numbers are put into the Kelly formula, the resulting percentage can give a good indication of the size of future positions to take in similar trades in order to maximise total returns.
