The Kelly Criterion is a formula developed by Bell Labs' scientist John L. Kelly in 1956. It optimizes the amount to bet on an event with known odds in such a way as to maximize the expected profit over the long term. Therefore, it has long been a favorite tool of blackjack players and other professional gamblers. Unfortunately, it is less applicable to investing since explicit odds are not as well known, although investors such as Warren Buffett, Bill Gross, and Edward O. Thorp (who incidentally was also a blackjack legend) are purported to have used it effectively.
Since markets are so unpredictable, certainly at least in the short term, it is not practical to use the actual Kelly Formula. Therefore, approximations or variants of it have to be applied instead. The original formula is f = (bp-q)/b, where f is the fraction of your bankroll you should bet on an event with odds of b, win probability of p, and probability of loss of q, or 1-p. When trying to apply this to investing, it immediately becomes apparent that there's no exact way of defining the probability of winning or losing a bet on a stock, since the outcome falls on a spectrum of possibilities rather than a distinct all or nothing result.
Therefore, I thought the Kelly Criterion could instead be used as a way to size how much you invest on market corrections or pullbacks. If you're like me, you get nervous when the market has done nothing but go up continuously lately. The tendency is to raise cash and wait for a better opportunity to redeploy it when the market sells off. However, it is often difficult to determine how much to invest on declines, not wanting to invest it all at once lest it continue going down. Several methods are usually proposed or used, such as investing in tiers and putting in a quarter of your money with each decline of say ten percent, or a pyramid variation of this where you put an eighth in, then a quarter, then half for each level down.
It stuck me that the Kelly Criterion could provide a more scientific approach to these sensible but imprecise heuristics, or rules of thumb. If we consider that the market trends up over the long term, we could conservatively say that you have at least a 50 percent chance of "winning" any particular bet on the market. Also, since the long term trend is upwards, we would think that the market would pass its previous peaks at some point after a downturn. Armed with these simplifications, we can reduce the Kelly Formula to f = (.5b - .5)/b, where b is the "odds" we are getting by assuming the market will return to its previous high after we make the bet.
For example, if the market is down 10 percent, we have an expected return of 10 percent if it were just to return to the previous level at some point in the future. So on a 10 percent decline, we might venture to bet f = (.5*1.1 - .5)/1.1 = .045, or 4.5 percent of our portfolio. With a 25 percent pullback, it would be f= (.5*1.25 - .5)/1.25 = 10 percent. This might seem like a timid amount to invest into such a large pullback, but it would be a continuous stream of money put to work at lower levels as the market declines, in amounts dictated by the Kelly Criterion to maximize our long term growth rate.
To see if this method has any merit, we can run some simulations and model the market as a random walk that meanders along with an equal but random possibility of either going up or down in 5 percent increments, which you could consider to be the expected weekly or monthly performance of the overall market. Then we will execute each of three strategies and see which provides the optimal return.
The first would be the simple method of investing a fixed amount corresponding to the amount of each decline, say 10 percent of your cash for each 5 percent pullback. The second method would be the pyramid approach, investing 10 percent, then 20, then 40 percent and so on for each 5 percent selloff. Finally, we would examine the Kelly Method, with bet size determined by the formula according to the magnitude of the pullback.
We then can run a simple Matlab simulation with the following assumptions:
1. The market will proceed in a random walk with either 5% increases or declines over each time period
2. We will use a predetermined amount of cash to purchase shares of a market proxy according to each of the three methods outlined above
3. We will only purchase additional shares after market declines if the market is below the baseline peak value we started with
The results for a run are as follows:
This does not look like a very promising result, as the Kelly Method is trailing not only the other two but also the overall market as well! However, if we increase the length of the simulation the Kelly method begins to dominate:
The Kelly Strategy's power mostly comes from the fact that it prevents you from going "all-in" too early and having no more to invest when the market decreases further. While it may lag other strategies during bull market rallies, it weathers the storm better and continues to put money to work at market lows, allowing it to outpace any other method over a long enough timeframe.
Therefore, it would seem ideal for investing with very long term time horizons, such as a retirement account like a 401k. If you diligently invest your cash in a market ETF like the S&P 500 (SPY) during downturns according to the formula, this method should outperform any others in increasing the ending value of your holdings.