The Kelly Criterion, Part I, Basic Kelly Math
As a supplement to Part I, Part II, and Part III of the reviews of William Poundstone’s book, Fortune’s Formula, I thought I’d summarize the actual Kelly Formula and some “Kelly Math” here. As Poundstone describes, the fraction of your bankroll you should wager on any given bet in a series of bets to maximize your expected growth of capital is:
f = Kelly bankroll fraction to bet = edge / odds
This incredibly simple equation is the result of some very complicated math given in Kelly’s original 1956 paper here. Edge and odds are further defined as:
edge = average expected winnings-losses = (Wo - L) odds = o
W = Actual probability of winning L = Actual probability of losing = 1 - W
The above equation is simpler to work with than other forms, but it mixes and matches odds and probabilities, which can lead to mistakes. Remember that odds are “chances against / chances for” whereas probability is “chances for / total chances.” Thus, probability (PR) can be written as:
PR = 1 / (1 + o)
Example. To use Poundstone’s book example, say the posted odds on a horse are 5:1, but you have a good tip that the horse has a 1 in 3 chance of winning or 2:1 odds. In this case W=1/3, L=2/3, and o=5:1 or 5. Thus f=(5/3 - 2/3)/5 = 0.2. A Kelly bettor would bet 20% of their bankroll to maximize capital growth. Betting more than this amount (overbetting) reduces returns and increases risk (drawdowns); betting less than this amount (underbetting) reduces returns, but decreases risk (drawdowns).
Expected Profit. The average expected profit in a given bet (P) can be computed as:
P = expected winnings - losses = (f * W * o) - (f * L)
Thus, for the above example, P = (0.20 * (1/3) * 5) - (0.20 * 2/3) = 0.33 - 0.13 or 20% - not a bad result for one bet. Note that this is not the same as the expected growth of capital. Expected Kelly growth (KG) is given by:
KG = ((1+f*o)^W) * ((1-f)^L)
Again, for the horse example, KG=((1+0.25)^(1/3))((1-0.2)^(2/3)) = 1.086 or 8.6%. After betting on 20 similar horse races, you might expect to have about 5 times your money. Unfortunately, due to the volatility of the Kelly system, you may end up with 1/5 your money easier than you might think.
Drawdown. Before salivating over your potential profits, any Kelly bettor should consider what will happen when you encounter a losing streak. The fraction of your bankroll remaining (frln) after any “n” losing bets is:
frln = (1-f)^n
Again using the above example, just five losing bets in a row will turn $1 into $0.33, a loss of 67%. The higher “f” is, the more dangerous Kelly can be to your psyche. The chance of losing “n” bets in a row (Ln) is (assuming you estimate “W” correctly):
Ln = L^n
For the horse example, you have a 13% chance of losing 67% of your bankroll. This is what many find unsettling about the Kelly Criterion - the inevitable losing streaks and their associated wicked drawdowns. Personally, I think the bigger hazard in the real world with the Kelly system is the potential to overestimate “W,” which results in a triple whammy according to the above equations: lower expected profits coupled with more frequent and more severe drawdowns. Be very wary of overestimating W!
Fractional Kelly. Poundstone and experienced gamblers discuss using a fractional Kelly system to limit the inevitable drawdowns (at the expense of lower returns). A popular recommendation is “1/2 Kelly” where you take the Kelly result and divide it in half. For the horserace example, expected profit per bet would drop in half from 20% to 10%, but your expected Kelly growth would drop only about 22%, and max drawdown after 5 straight losses would be reduced from 67% to 41% - better, but not exactly comforting for the risk averse.
The Kelly Criterion, Part II, Additional reading
Below are some references to consider regarding the Kelly Criterion / Kelly Formula. Be sure to also read the series on the book Fortune’s Formula and also Part I of this post.
A detailed review of the book.
The original paper by John Kelly.
A much simpler derivation of the Kelly Criterion.
How to use the Kelly Criterion to manage money based on your stock trade history. More thoughts on Kelly and trading here and here.
A 1997 paper by Ed Thorp, a smart man who basically got rich using the Kelly Criterion. An earlier 1984 paper is here.
A Kelly Criterion online calculator.
Some criticism of the Kelly Criterion, Samuelson style.
An article on how to apply the Kelly Criterion for more complicated situations (bets with multiple outcomes).
A bunch of links for advanced topics regarding the Kelly Criterion related to Blackjack (but applicable to other arenas as well).
Use of Kelly Criterion to judge betting market efficiency.
Update: Morningstar discusses optimizing a portfolio with the Kelly Formula.
Update II: The Adaptive Market Hypothesis and the Kelly Formula.
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