# The Kelly Criterion

Position sizing is a key component of successful investing. Determining the sizing of positions can differentiate between a long term winning and a long term losing strategy.

A classic formulation for position sizing is called the Kelly Criterion developed by John L Kelly at Bell Labs (http://en.wikipedia.org/wiki/Kelly_criterion).  Kelly’s formulation is based on maximizing the expect long-run return over a series of bets.

Assume for instance, that you bet on a series of coin flips. There are two outcomes, heads or tails, and we assume that on any coin flip there is a 50% chance of getting a head and a 50% chance of getting a tail.  We also assume that for every dollar that you wager, you receive either two dollars if the coin is a head or you lose one dollar if the coin is a tail.  If you start with a bankroll of \$100 and invest 10% of your bankroll on each coin flip, you might have the following series of outcomes:

1. bankroll=\$100, bet \$10, coin is a head, win \$20
2. bankroll=\$120, bet \$12, coin is a head, win \$24
3. bankroll=\$144, bet \$14.40, coin is a tail, lose \$14.40
4. bankroll=\$129.60, bet \$12.96, coin is a tail, lose \$12.96
5. bankroll=\$116.64 …

So, by investing 10% of your bankroll on each flip your bankroll has grown from \$100 to \$116.64 after four coin flips.

If, however, you had invested 25% of your bankroll on the same set of coin flips, your money would have grown to \$126.5625:

1. bankroll=\$100, bet \$25, coin is a head, win \$50
2. bankroll=\$150, bet \$37.50, coin is a head, win \$75
3. bankroll=\$225, bet \$56.25, coin is a tail, lose \$56.25
4. bankroll=\$168.75, bet \$42.1875, coin is a tail, lose \$42.1875
5. bankroll=\$126.5625 …

The percentage of your bankroll you commit on each coin flip affects your long term returns. So, how does one determine the allocation to best size your bets?

To determine the sizing that maximizes long run returns, we generalize the coin flip game to one in which you have two outcomes: for each dollar you bet, outcome-1 (heads) occurs with probability p and pays you b dollars and outcome-2 (tails) occurs with probability q and you lose one dollar.  (The coin flip example above has parameters  p = q= 1/2 and b = 2.) As in the coin flip example, you can play this wager as many times as you want to and you wager as much or as little as you want to on each bet. Setting aside your desire to avoid wide swings in your net worth over time, one can determine the fixed percentage of your bankroll for each wager that will maximize your long term profitability.  The derivation in the box to the left shows the mathematical details.

To maximize your long run return the percentage of your bankroll to wager on each return is

S = (bp – q) / b

For the initial coin flip example,  the maximum gain occurs when S = 0.25, so any allocation other than a 25% of your bankroll on each flip will decrease your long run return.

For this example, S, the optimal percent that you wager, increases as your chances of winning increase and as b increases. This makes intuitive sense since the greater your advantage on each wager, the greater the amount that you should be willing to risk on each bet.

However, what may be less than intuitive is that, for any combination of p, q, and b, there is a upper limit on the amount that you should wager for even the most attractive bets. As an extreme case, we consider what happens if we wager 100% of our bankroll on each bet when there is some chance that we can receive outcome-2 (tails). In this case, we could receive very large returns for a period of time, but a single occurrence of outcome-2 will reduce your bankroll to zero. At this point, you will have nothing to wager, and as a result your long run return will be zero. At the other extreme, if you never make a wager, S = zero, you will never grow your bankroll. So, there is an intermediate wager percentage that grows your portfolio at the fastest rate.

In some cases, the wager is so unattractive that bp < q. In this case, no amount of sizing or strategizing will make you money in the long run.

The plot below shows that many of the bets with low winning probabilities are unattractive and are colored red indicating no wager is appropriate.  By the contrast, the most attractive combinations are colored blue or purple on the upper right corner. Here there is a high probably of success along with better than even odds, b. As indicated by the red areas in the lower-right quadrant, when the odds, b, are below 1, the value of b has a large effect on the optimal sizing. In those situations only very high success rates can made those wagers attractive.

wager.Setting aside your desire to avoid wide swings in your net worth over time, one can determine the fixed percentage of your bankroll for each wager that will maximize your long term profitability in the long run.  The derivation in the box to the left shows the mathematical details.
1. Валютчик says: