Lesson 6 of 7
Kelly, and the fraction used
The growth-optimal bet size is computable and almost nobody uses it. Work the arithmetic, then see what happens at half and at double the optimum.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
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In one minute
- Betting to maximise expected wealth means betting everything, which leads to ruin with probability one if repeated.
- Kelly's answer is to maximise the exponential rate of growth of capital instead, which gives a fraction of capital rather than all of it.
- Half the optimal fraction keeps about three quarters of the growth rate, and twice it turns a winning edge into a losing one.
There is an exact answer to how much of your capital to put behind an edge, and it has been available since 1956. Almost nobody uses it at full size, and the arithmetic explains why better than any argument about temperament.
Growth rate per bet at different fractions of capital
- 5 percent of capital0.88%
- 10 percent, half the optimum1.50%
- 20 percent, the optimum2.01%
- 30 percent1.48%
- 40 percent, twice the optimum−0.25%
What Kelly actually argued
Kelly's 1956 paper sets out the problem plainly. A gambler could bet his entire capital each time, and that would maximise the expected value of his capital. His words on what follows are the point: this would be little comfort, since when the number of bets was large he would probably be broke, and in fact would be broke with probability one if he continued indefinitely.
So he defines a different objective, the exponential rate of growth of capital, and maximises that instead. For an even-money bet the answer is the probability of winning less the probability of losing.
Working the numbers
An edge of 60 to 40, at even money
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The optimal fraction is 0.6 less 0.4, which is 0.2. Bet a fifth of capital each time.
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The growth rate at that fraction is 0.6 times the logarithm of 1.2 plus 0.4 times the logarithm of 0.8, which is 0.0201 per bet.
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At half the fraction, 10 percent, the growth rate is 0.0150 per bet, which is 74.7 percent of the maximum for half the exposure.
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At double the fraction, 40 percent, the growth rate is minus 0.0025. The same favourable bet now loses money over time.
The three reasons to halve it
The inputs are estimated
- The formula needs the probability and the payoff, and an investor has neither.
- A 60 percent edge that is really 55 percent moves the optimum from 0.2 to 0.1, so full Kelly on the wrong estimate is double Kelly in reality.
- Half Kelly is what full Kelly becomes when you allow for being wrong about the edge.
The penalty is asymmetric
- Underbetting costs a little growth. Overbetting destroys it: at twice the optimum the growth rate goes negative.
- The curve is flat near the top and falls away sharply on the right.
- There is almost nothing to gain from the last few points of size and a great deal to lose.
All three point the same way, and none of them is about courage.
The third reason is the path. Full Kelly produces falls that are unbearable in practice: the fraction that maximises long-run growth says nothing about how deep the interim declines get, and a holder who abandons the policy in a drawdown has not implemented it at all.
The same shape without the formula
The reason this is in a value investing academy is that the shape survives the loss of the inputs. Three statements hold whether or not anyone can compute a probability for a holding.
An edge justifies a position, and a larger edge justifies a larger one. There is a size beyond which more conviction should not buy more capital, because the cost of being wrong about the edge grows faster than the benefit of being right about it. And the penalty curve is lopsided, so when in doubt the error to make is the small position rather than the large one.
None of that needs a number. All of it falls out of the arithmetic above.
Why it still matters to a value investor
Nobody is computing Kelly fractions for a holding in a listed company, and this lesson does not suggest starting. What the arithmetic supplies is a shape.
There is an optimum, it is finite, it is well below everything, and going past it converts an edge into a loss while looking like conviction. That is worth knowing when the temptation is to put a third of an account into the idea that currently seems obvious.
Check yourself
What does betting to maximise expected wealth imply?
- Betting the entire capital each time, which leads to ruin with probability one if repeated indefinitely
- Betting a fixed fraction of capital
- Never betting at all
Show the answer
Answer: A.
Kelly makes exactly this point: maximising expected capital means betting everything, and the gambler who does so goes broke with probability one over enough repetitions.At an even-money bet won with probability 0.6, what is the growth-optimal fraction?
- 0.6
- 0.2
- 0.5
Show the answer
Answer: B.
For an even-money bet the fraction is the probability of winning less the probability of losing: 0.6 minus 0.4.What happens at twice the optimal fraction in the example?
- The growth rate doubles
- The growth rate is unchanged
- The growth rate turns negative, so a favourable bet loses money over time
Show the answer
Answer: C.
At 40 percent the growth rate is minus 0.0025 per bet. The penalty for overbetting is far larger than the penalty for underbetting.Sources
- Kelly, "A New Interpretation of Information Rate", Bell System Technical Journal, manuscript received 21 March 1956: the exponential rate of growth of capital, the observation that betting everything maximises expected capital and leads to ruin with probability one, and the optimal fraction for the even-money case.
- The growth rates at each fraction are computed for this lesson from the formula shown in the figure's source line.
What this leaves out
- The 60 to 40 even-money bet is invented. A holding in a listed company has neither a known probability nor a known payoff, which is the main reason the rule is not applied literally.
- The growth rates are per bet, in logarithmic units, for a repeated identical independent bet. Investments are none of those things.
- The rule maximises long-run growth and says nothing about the depth of interim declines.
- Nothing here is advice about any security or about how much to put in one.