Question 1

What is the Kelly criterion?

Accepted Answer

The Kelly criterion is a rule for how much of your bankroll to put on a bet that pays better than its chance of winning requires, so that your capital grows as fast as possible in the long run. J. L. Kelly Jr. of Bell Labs derived it in 1956 in “A New Interpretation of Information Rate” (Bell System Technical Journal 35) for a gambler who receives advance information over a noisy communication channel; he showed that the best growth rate equals Shannon's rate of transmission of information. Edward O. Thorp met Kelly's paper through Claude Shannon in 1960 and introduced it to gamblers in the first edition of Beat the Dealer (1962). The idea is to maximize the expected logarithm of your wealth. Staking everything each time maximizes the expected win but makes ruin almost certain, and staking almost nothing almost never grows the bankroll; the Kelly fraction lies between the two.

Question 2

How do I calculate the Kelly bet?

Accepted Answer

If a bet pays b units for each unit staked (the net odds) and wins with probability p, with q = 1 − p, the Kelly fraction is f* = (b·p − q) ÷ b = p − q ÷ b (Thorp 2006, section 2). Convert the odds you are quoted to net odds first: decimal odds minus 1 (2.00 is b = 1), American +150 is 1.5 and −110 is 100 ÷ 110 = 0.909, fractional 3/2 is 1.5. A bet is worth making only when the expected profit b·p − q is above zero, which means p > 1 ÷ (1 + b); otherwise f* is zero. Example, with the coin from Thorp's article: it pays even money (b = 1) and wins 53% of the time, so f* = 0.53 − 0.47 = 6% — on a $10,000 bankroll the first stake is $600, and the next stake is 6% of whatever the bankroll is then. A second example: a 40% chance at decimal odds of 3.00 (b = 2) gives f* = (2 × 0.40 − 0.60) ÷ 2 = 10%. For even-money bets the formula reduces to f* = p − q, the edge itself. The win probability is your own estimate and the calculator cannot know it; every number that follows depends on it.

Question 3

How fast does the bankroll grow, and how many bets does it take to double?

Accepted Answer

When you bet the same fraction f each time, the logarithm of the bankroll grows on average by g(f) = p·ln(1 + b·f) + q·ln(1 − f) per bet. For the coin example at f* = 6%, g = 0.001801, about 0.18% a bet. Since ln 2 ÷ g is the average number of bets needed to double (Thorp's continuous approximation, section 3.4, n = ln C ÷ g for a goal of C times the starting capital), the bankroll doubles in about 385 bets; Thorp's example 2.1 prints the same figures (f* = 0.06, g = 0.001801, about 385 bets). Kelly's 1956 even-money case gives a second way to see it: the best growth is ln 2 + p·ln p + q·ln q, which for p = 60% is 0.020136 per bet — the same as g at f* = 20% (0.020136). These are averages of the logarithm; real sequences of wins and losses scatter widely around them, and at the Kelly fraction large temporary losses are normal.

Question 4

Why is betting more than the Kelly fraction so dangerous?

Accepted Answer

The growth rate g(f) rises up to f* and then falls, and beyond a certain share it turns negative: the bankroll then shrinks in the long run even though every single bet has a positive edge. Thorp's Theorem 1 states that when g(f) is below zero the capital tends to zero almost surely. In the coin example the zero-growth point is 11.97% of the bankroll; at twice the Kelly fraction (12%) growth is already −0.0017% a bet. “Twice Kelly loses” is a rule of thumb rather than a law: it is exact when p = 50% (and in the continuous investing model with no interest), but with a 60% win chance at even money growth stops at 38.94%, below twice Kelly (40.00%, where growth is −0.2447% a bet), while for the 40% chance at b = 2 it stops at 20.39%, above twice Kelly (20.00%, where growth is still 0.0703% a bet). The calculator therefore shows the exact zero-growth point rather than assuming 2f*. (Thorp prints the coin example's zero-growth point as .11973; solving g(f) = 0 directly gives 0.119712.) Thorp's case study (section 8.2) states the practical lesson: because overbetting is much more harmful than underbetting, a fraction of Kelly is prudent to the extent the inputs are uncertain. And if f reaches 100%, one loss wipes out the bankroll.

Question 5

What are half Kelly and quarter Kelly?

Accepted Answer

Fractional Kelly means staking a fixed share of the Kelly amount. In the coin example, half Kelly (3%) grows the log of the bankroll by 0.1350% a bet, 75.0% of full Kelly's 0.1801%, and quarter Kelly (1.5%) grows it by 0.0788% a bet (43.7%). In his continuous-time analysis with no interest (section 7.4), Thorp finds that half Kelly keeps 3/4 of the growth rate, raises the chance of doubling the bankroll before halving it from 2/3 to 8/9, and cuts the chance of ever falling to half the starting capital from 1/2 to 1/8. The reason to bet a fraction is that your probability or return estimate is uncertain: Thorp shows that if the true edge is half of what you assumed, betting full Kelly on the assumed edge gives zero growth, and he suggests choosing a fraction between one half and the full Kelly amount, which protects against zero growth at a cost of likely no more than 25% of the growth.

Question 6

What is the Kelly formula for investing?

Accepted Answer

For an asset with expected return μ, volatility σ and a risk-free rate r at which you can both borrow and lend, Thorp (section 7.1) gives the Kelly share of capital to hold in the asset as f* = (μ − r) ÷ σ², with growth g = r + (μ − r)² ÷ (2σ²). Take example inputs of μ = 8%, r = 2% and σ = 20% a year, which are not forecasts: f* = 6% ÷ 4% = 150%, which means borrowing half of your capital; growth is 6.50% a year and the volatility of that growth is 30%. Half Kelly (75% in the asset) grows at 5.375% a year, which is 75% of the growth above the risk-free rate, because the extra growth at a fraction c of Kelly is c·(2 − c) of the best. The best growth also equals r + S²÷2, where S is the Sharpe ratio (here (8 − 2) ÷ 20 = 0.30, so 2% + 4.5% = 6.5%), which is the link Thorp draws to the Sharpe ratio — see the Sharpe ratio calculator. The model assumes lognormal prices with constant μ and σ, continuous rebalancing without costs or taxes, and borrowing at the risk-free rate; none of that holds exactly, and μ is the weakest input. Thorp warns that forecast excess returns in the stock market are more likely too high than too low.

Question 7

What are the limits of the Kelly criterion?

Accepted Answer

First, it needs a probability (or an expected return) you do not actually know. A quoted price is not that probability: bookmakers' odds include a margin, so the implied probabilities of all outcomes add up to more than 100%, and the true chance has to come from your own analysis. Second, the formula is for repeated, independent bets of the same kind with the whole bankroll at risk each time; several simultaneous or correlated bets need a joint calculation, which this calculator does not do. Third, Thorp assumes capital can be divided without limit; real bets have a minimum size, and costs, taxes and the possibility that odds change are not included. Fourth, even at the Kelly fraction the path is rough: in his continuous model with no interest the chance of ever being down 50% from the starting capital is 1/2 at full Kelly. This page is educational, not betting or investment advice, and it cannot tell you whether any bet or investment is a good idea.

Kelly criterion calculator