# Numeracy #6: Kelly Criterion

Imagine we are given the opportunity to bet on a fair coin flip. If heads, we get 2 dollars for every dollar bet. If tails, we lose our bet. We can easily work out the expected value of a 1 dollar bet.

EV = 0.5(\$2) – 10.5(\$1)
EV = \$0.50

On average, we can expect to make 50 cents per bet. How could we possibly lose? Now, imagine that we have a bankroll (investable capital) of 10 dollars. We take our 10 dollars, bet it all on heads, and lose it all. We broke the first rule of gambling, don’t get wiped out.

The solution seems obvious: never bet your whole bankroll. But how much should we bet? 9 dollars, 7 dollars, 4 dollars, 0 dollars? This is where the Kelly Criterion comes into play. The Kelly Criterion, or Kelly Optimization Model, is a formula used for determining the optimal size of a series of bets. It other words, it tell us how much of our bankroll to bet given a chance of winning and a payout.

### History

John Kelly developed the Kelly Criterion while working on the problem of long distance telephone noise at AT&T’s Bell Laboratory. He published his findings in 1956. Soon after, the gambling community realized the potential application of this work to horse racing. Today, the Kelly Criterion has become a part of mainstream investment theory.

If you are still interested in the history of the Kelly Criterion, William Poundstone wrote the definitive history on the subject.

### Math

To derive the Kelly Criterion we need a basic understanding of calculus; specifically how to maximize a function. The equation we are trying to maximize is:

avg rate of return = (p)(log(1+bx))+(1-p)(log(1-x))

where
p = probability of winning
b = fraction of bankroll bet
x = payout for win

After taking the derivative, setting it equal to 0, and solving for b, we get:

b = (p)(x)-(1-p) / x

The top of the fraction is simply our expected net winnings from a 1 dollar bet. We divide this by the payout ratio to determine the percent of our bankroll to bet. Another way of looking at this is:

b = expected net winnings / net winnings if you win

Conceptually, this is saying that when the probability of winning (expected net winnings) is disproportional to the payout (net winnings if you win) we should bet more. Simple enough.

### Examples

Let’s start with that coin flipping example. Recall that we have a 50% chance of winning a payout of 2 to 1. Plugging this into the Kelly Criterion we get:

bet_size = (0.5)(2)-(1-0.5) / 2
bet_size = 0.25

The Kelly Criterion tells us that if we want to maximize our compounded rate or return, we should bet 25% of our bankroll on every coin flip.

What if the payout is even money?

bet_size = (0.5)(1)-(1-0.5) / 1
bet_size = 0

In this case we shouldn’t even be betting. How about an example where we are almost sure to win? Take the credit default swaps sold by banks in the run up to the 2008 financial crisis. For simplicity, lets assume that these swaps had a 99% chance of returning 2% per year (in reality the premiums were much lower). The other 1% of the time, the seller would lose the full value of the underlying security. What does the Kelly Criterion tell us about risk management in this situation?

bet_size = (0.99)(.02)-(1-.99) / .02
bet_size = 0.49

Nearly half of our bankroll should be in these investments. Cool right? Well not really. As we know, these investments blew up. This leads us to three important lessons.

### Draw backs of the Kelly Criterion

Drawback #1: Gamblers tend to overestimate their odds of winning. This can be easily seen in that CDS example. The real chance of losing money was far greater than 1%.

Drawback #2: They Kelly Criterion can mask extreme volatility. In the case of the first coin flip example, it would only take 4 tails in a row to completely wipe us out.

Drawback #3: It only works in situations where a large number of repeatable bets can be made.

For all of these reasons, many people argue that we should always bet less than the Kelly formula suggests. We always want to position ourselves so that we can survive the once in a generation 10 tails scenario. It is also important that we never put to much confidence in our models. As Box and Draper pointed out in their book Empirical Model-Building and Response Surfaces:

Essentially, all models are wrong, but some are useful.

### As it relates to investing

In investing, we don’t automatically lose our whole bet. We tend to gain x% when we are right and lose y% when we are wrong. The Kelly Criterion can be adapted to account for this:

bet_size = p – ((1-p) / (x/y))
bet_size = p – (pl / w)

where
p = probability of winning
x = amount gained for winning
y = amount lost for losing
pl = probability of losing
w = win/loss ratio

If we invest in a market index like the S&P500, we might have a 95% chance of a 7% return every year and a 5% chance of a 50% loss. The Kelly Criterion would tell us:

bet_size = (0.95) – (0.05)/((0.07)/(0.50))
bet_size = 0.59

The same drawbacks that were mentioned above apply here. A good rule of thumb is that if you bet half the Kelly amount, you get about three-quarters of the return with half the volatility. For this reason, many people chose to use half Kelly when sizing their bets. This allows them to sleep better at night.

### Conclusion

All of investing is a game of odds. The Kelly Criterion provides us with a framework for capitalizing on those odds. It’s not perfect, but it’s probably better than nothing. Charlie Munger put it best:

It’s not given to human beings to have such talent that they can just know everything about everything all the time. But it is given to human beings who work hard at it – who look and sift the world for a mispriced bet – that they can occasionally find one. And the wise ones bet heavily when the world offers them that opportunity. They bet big when they have the odds. And the rest of the time they don’t. It’s just that simple.