How do you maximize the geometric mean?

November 29, 2017

The Kelly formula gives a gambler a simple way to calculate the best bet given a fair game with favorable odds. The best bet is not always the biggest. The best bet allows for the greatest compound growth of the gambler’s bankroll while eliminating the chance of financial ruin.

The simplest way to use the Kelly formula is to divide the “edge” by the odds. The edge is the expected value of all possible outcomes. The odds are the ratio of payoff to bet. To use the formula:

  • Consider all the possible outcomes and estimate the probabilities of each;
  • Estimate the (positive and negative) payoffs of each outcome;
  • Sum the products of each probability times its payoff;
  • Divide the total by the odds of the best case outcome;
  • Note that for a bet to be worthwhile, the edge must be greater than zero.

If the geometric mean is the one that investors should focus on, then how do you maximize it? In the 1950s, John Kelly derived the Kelly criterion. Designed for gamblers, the catch is that the formula only works with known probabilities with known payoffs.

The Kelly formula is:

f = p - ( \frac{q}{(\frac{b}{a})}), where
  • f is the percentage of the bankroll that should be bet;
  • p is the chance of a winning outcome;
  • q is the chance of a losing outcome;
  • b are the possible winnings; and
  • a is the possible loss. ( b / a are the “odds” in a game of chance.)

Betting 100% of the bankroll (and being infinitely successful) would result in the greatest long-term growth rate, but it’s too high risk. Unless the outcome is known, a Kelly bettor would never bet 100% of the roll. Bets are always adjusted according to the relative probabilities of winning and losing and the expected value of each bet.

Tossing coins

For example, in an “even money” coin toss (where p = 50%, q = 50%, and b / a = 1), we have a 50/50 chance of winning, and we’ll win $1 for every $1 we bet. Sounds good, right? Who wouldn’t want to double their money? But the Kelly formula says that we should bet nothing.

f = p - ( \frac{q}{(\frac{b}{a})}) = 50\% - \frac{50\%}{(\frac{\$1}{\$1})} = 50\% - 50\% = 0\%

With each bet, we could double our money or lose our bet. Sounds good, right? Not really. Over a long enough timeline, the expected value of the game is $0.

\$1(50\%) - \$1(50\%) = \$0

Tossing coins with bad gamblers

But if the odds change to 2:1 (we win $2 for every $1 bet), then we should bet 25% of our bankroll:

f = p - ( \frac{q}{(\frac{b}{a})}) = 50\% - \frac{50\%}{(\frac{\$2}{\$1})} = 50\% - 25\% = 25\%

If you have odds that could triple your bet, you’ll walk away with three dollars for every winning bet of one dollar. If you can continue playing a game with the same odds, then betting 25% of your bankroll each time will maximize its long-term growth.

This makes sense, because there is a positive expected value:

\$2(50\%) - \$1(50\%) = \$0.50

Cheating at tossing coins

Or if it were still an even money game, but we were playing with a rigged coin (i.e. p = 60%; q = 40%), then we should bet 20% of our bankroll:

f = p - ( \frac{q}{(\frac{b}{a})}) = 60\% - \frac{40\%}{(\frac{\$1}{\$1})} = 60\% - 40\% = 20\%

This also makes sense, because the expected value is still positive:

\$1(60\%) - \$1(40\%) = \$0.20

The Kelly Shortcut

If you look at the above equations, you’ll notice that the expected value is a product of f times the odds. You can reverse the equation to get a shortcut version of the Kelly formula. And because it doesn’t make sense to make bets that aren’t in your favor, the shortcut version is often described as edge / odds:

f = \frac{+ExpectedValue}{Odds} = \frac{Edge}{Odds}

You can see this in action using the first coin toss example.

f = \frac{Edge}{Odds} = \frac{1(50\%) - 1(50\%)}{(\frac{\$1}{\$1})} = 0\%
  • To find your edge, consider all the possible outcomes, their probabilities, and their payoffs. For every dollar in an even money coin toss, you have a 50% chance of winning 1x your bet, and a 50% chance of losing 1x your bet.
  • Next, sum the products of each outcome times its probability. If the sum is positive, then you have an edge. If the sum is 0 or negative–as in an even money coin toss–then you don’t have an edge. In that case, the odds will determine whether or not you should bet.
  • To find the portion of the bankroll that you should bet, divide your edge by the odds of the best case outcome.