Geometric Means, Kelly Criterion, etc.
We continuously have in-depth discussions surrounding the purchase or sale of a given security, but there exists another equally important problem for the investor — how to allocate capital between competing qualifying ideas. It is not good enough to simply say, “This is an undervalued business with a high margin of safety.” How do you decide how much capital to commit to such an idea? The task is not simple because each choice will have slightly different risk vs. return characteristics. While we build a portfolio one security at a time, we must use a rational system to allocate our limited capital in the most efficient way — that is to allocate it with the highest chance of outsized returns while minimizing real risk (business risk).
The goal of the investor, therefore, is to maximize the geometric mean of possible outcomes, as opposed to his arithmetic mean. The goal is such because the investor must always have all of his capital invested. The geometric mean is the nth root of the product of n numbers. The arithmetic mean is the sum of n numbers divided by n. For example, for the list of numbers {0.5, 2, 3, 1, 0.2}, the geometric mean is the 5th root of (0.5 x 2 x 3 x 1 x 0.2) or 0.90. For the same list, the arithmetic mean is (0.5 + 2 + 3 + 1 + 0.2) / 5 or 1.34. Maximizing the arithmetic mean is appropriate when you are playing games once, but in investing you are by nature playing games (investment opportunities) repeatedly. Therefore, the investor needs to pick the games that will maximize the long-run result of playing the games over and over again. The appropriate goal for the investor, then, is to maximize the geometric mean of his portfolio. The following example will explain this concept more clearly [1].
Let’s look at an example of a Wheel of Fortune with twenty-three spaces marked “$100” and one space marked “Bankrupt.” While the arithmetic mean of this wheel is $2,300 / 24, or $96, the geometric mean is $0 due to the presence of the “Bankrupt” space. While you might be very happy to play this game with part of your money, if you play this game with all of your money, you will go broke with 100% certainty in the long run. If you look at another Wheel of Fortune with twenty $50 spaces and four $10 spaces, the arithmetic mean will be $43 and the geometric mean would be $38. While the latter wheel will underperform the first wheel on 96% of spins, it is the only rational choice for one’s entire portfolio.
A special case formula for maximizing the geometric mean was developed by Bell Labs’ scientist John L. Kelly, Jr. He showed that the rational gambler should put a percentage of his bankroll in a given game based on the formula Edge/Odds. This formula has been dubbed the “Kelly Criterion.” Edge is calculated at the expected profit of playing the game. Odds is the public odds of playing the game. Suppose you have the opportunity to get paid 2-1 Odds on a coin-flip. The Edge for this game would be 3 for heads plus 0 for tails divided by 2 for the number of outcomes minus 1 for the initial investment, or 0.5. The Odds of the game are 2, so you should bet 0.5 / 2 or 25% of your bankroll (portfolio) on each coin flip in order to maximize your geometric mean and the long-run value of your portfolio. In investing, one can obviously not know the Edge and Odds exactly, but understanding this concept is critical. Intuitively, as the expected profit (Edge) of the investment choice rises, the investor should put more of his portfolio to work in this investment. But note also that given a constant expected value, as the Odds rise (in other words as the degree of certainty in the idea falls), the percentage of the portfolio deployed should fall as well. So if the investor has two ideas with the same expected return, but one is in a highly-leveraged financial company and one is in a very stable consumer products company, the investor should allocate substantially more money to the consumer products company because the Odds will be smaller. When we talk about risk, we are always talking about business risk NOT volatility. So the mathematics of maximizing the geometric mean proves that you make much bigger bets on ideas where the range of possible outcomes is smaller even if the expected value of such options are slightly lower. It is, in short, the logic behind concentration of capital when certainty is high.
A discussion of portfolio allocation is not complete without considering the nature of leverage (borrowed money). When the investor uses leverage, he will almost certainly raise the arithmetic mean of his portfolio returns, but he will be adding a bankruptcy space to his wheel. As we have shown above, regardless of the arithmetic mean, this one bankruptcy space will make the geometric mean (long-term outcome) zero. Even if there are thousands of spaces on the wheel and only one bankruptcy space, the long term result of running a portfolio by simply trying to maximize the arithmetic mean with leverage will be losing all your money – whether it happens in one year, 10 years, 100 years, or 1,000 years. Our goal at Cook & Bynum will continue to be to maximize our long-run returns without the permanent loss of capital, and that goal will continue to preclude our using leverage.
The Kelly Criterion shows that a key to success in investing is to maximize the difference between the actual odds of an investment opportunity and the odds that the market is giving you. In other words, the greater the discount between the current price of a stock and its true underlying intrinsic value the better. Our job is to find places where we have superior insights to the other market participants. The other key is to accurately estimate how big our information advantage is. If we underestimate the informational advantage, we will underperform as an investor. If we overestimate the advantage, we will go broke.
[1] The ideas in this section are simplified and synthesized from the following:
Kelly, J. L., Jr. (1956). “A New Interpretation of the Information Rate.” Bell System Technical Journal, 917-26.
http://www.bjmath.com/bjmath/thorp/paper.htm
Poundstone, William (2005). Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. New York, NY: Hill and Wang.