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
where:
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+0.2*5)^(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.
In Part II, I'll list some additional reading for the Kelly Criterion.
i want detail of formulas
Posted by: junai | June 06, 2007 at 03:17 PM