Bet Sizing, Kelly & Ruin

3.6.1 Same edge, different fates

A positive edge is necessary but not enough. How much of your wealth you stake on each bet decides whether the edge compounds into wealth or grinds you down.

Take the simplest bet with an edge: a coin that lands your way 55% of the time, paying even money. Stake $1 and you win $1 or lose it, so the expected value is $0.10 per dollar staked. Now hand the coin to 10,000 simulated bettors for 1,000 flips each. Every bettor stakes a fixed fraction of whatever they have at the time, and the only thing that differs between groups is that fraction. The lab uses the same flips for every group, so luck is identical:

• staking 2.5% of wealth, the median bettor turns $1 into $8.92;
• staking 10%, the median bettor turns $1 into $150;
• staking 25%, the median bettor turns $1 into $0.0012, and 81% of bettors end with less than they started.

Same coin, same edge, same luck. Deciding how much to put on each bet is called bet sizing, and in this example it is the difference between a hundredfold gain and losing almost everything. You met the reason in Session 3.1: when each bet stakes what the last one left you, wealth multiplies, and what almost every bettor experiences is the average log return per bet, not the average return. That average log return is the growth rate, $g = \E[\ln(W_{t+1}/W_t)]$: after $n$ bets typical wealth is about $W_0 e^{ng}$, because the law of large numbers applies to the sum of the logs. Plot $g$ against the fraction staked and you get the picture this whole session hangs on (fig 3.6.1).

Read it from left to right. Stake nothing and your wealth stands still. Stake a little and you grow slowly but safely. Growth peaks at one fraction, $f^\star$, the Kelly criterion's answer. Past the peak you get less growth for more risk. Past roughly $2f^\star$ the growth rate turns negative: a bet you win on average makes almost every bettor poorer over time, just as the all-in coin of Session 3.1 did.

3.6.2 The Kelly bet, derived

Maximise the expected log of wealth and the optimal bet falls out in two lines. It turns out to be the edge divided by the odds.

Stake a fraction $f$ of wealth on a bet that pays $b$ to 1 with probability $p$ and loses the stake with probability $q = 1 - p$. A win multiplies wealth by $1 + bf$, a loss by $1 - f$. The growth rate per bet is the probability-weighted average of the log of those multipliers. Set its derivative to zero:

The numerator $bp - q$ is the expected profit per dollar staked, the edge; the denominator is the odds. So Kelly stakes edge over odds. For the coin, $f^\star$ = 0.55 − 0.45 = 10% of wealth, and the growth rate peaks at 0.50% per bet, enough to multiply typical wealth by about 150 over 1,000 bets. The second derivative is negative everywhere, so this is a maximum, and there is only one. A bet with no edge gets $f^\star \le 0$: don't bet.

The same two lines work for any bet with two outcomes. If a win adds a fraction $u$ of the stake and a loss removes a fraction $d$, then $f^\star = p/d - q/u$. Take the +50%/−40% coin of Session 3.1, which, staked in full, has a growth rate of −0.0527: a typical change of −5.1% a round. Its Kelly fraction is $0.5/0.4 - 0.5/0.5$ = 25%, and staking a quarter of wealth each round turns the same coin into a winner: a growth rate of 0.0062, or about +0.6% a round for the typical player.

Maximising $\E[\ln W]$ is what an investor with log utility would do: someone who values each extra dollar in inverse proportion to how many they already have, so that doubling their money is worth exactly as much as halving it costs. John Kelly, a Bell Labs scientist, reached the same rule from information theory: a gambler who receives tips over a noisy private wire should bet so as to maximise the growth rate of capital, and the best achievable rate equals the rate at which the wire transmits information (Kelly 1956). Breiman (1961) proved the two properties that made it famous: over a long run of bets, the Kelly bettor's wealth pulls ahead of any essentially different strategy's, almost surely, and it reaches a distant wealth goal fastest on average.

The lab checks the formula against the 10,000 bettors of §3.6.1, asserting that each simulated growth rate lies within four Monte Carlo standard errors of $g(f)$:

def growth_rate_binary(f, p: float, b: float):
    """Expected log growth per bet when staking a fraction f of wealth on a p-to-win, b-to-1 bet."""
    return p * np.log1p(b * f) + (1 - p) * np.log1p(-f)


def kelly_binary(p: float, b: float) -> float:
    """Kelly fraction for a bet paying b-to-1 with win probability p: f* = p − q/b."""
    return p - (1 - p) / b

wins = rng_bin.random((N_BETS, N_BETTORS)) < P_WIN     # one row per bet, one column per bettor
for f in FRACTIONS:
    log_step = np.where(wins, np.log1p(ODDS * f), np.log1p(-f))
    log_wealth = log_step.cumsum(axis=0)
    growth = log_wealth[-1] / N_BETS                      # each bettor's average log growth per bet
    theory = growth_rate_binary(f, P_WIN, ODDS)
    se = growth.std(ddof=1) / np.sqrt(N_BETTORS)
    assert abs(growth.mean() - theory) < 4 * se, f"f={f}: simulated growth rate off by more than 4 SE"

Growth returns to zero at a stake of 19.87%, just under $2f^\star$; at exactly $2f^\star$ it is −0.014% per bet. For a two-outcome bet "twice Kelly gives zero growth" is a very good approximation, and in the continuous case of §3.6.4 it is exact.

3.6.3 Over-betting costs more than under-betting

Near its peak the growth curve is almost symmetric. The risk around it is not, and that asymmetry is why nobody sensible bets more than Kelly.

When the bets are small relative to wealth (or when you resize continuously, §3.6.4), the growth curve becomes an exact parabola. Measure the stake as a multiple $c$ of Kelly, $f = c f^\star$:

The first formula says half Kelly and one-and-a-half Kelly grow at the same rate, three-quarters of the maximum. The second says the bigger bet carries three times the volatility for that same growth. The third, from Thorp (2006, §7.4), is the one that stings: the chance of ever seeing your wealth halve is one in two at full Kelly, one in eight at half Kelly and about four in five at one-and-a-half Kelly. The exponent depends only on $c$, not on how good the edge is: a better edge makes everything happen faster, but not less often.

The lab's two-outcome coin shows the same shape, slightly tilted against over-betting because $\ln(1 - f)$ plunges faster than $\ln(1 + bf)$ rises: half Kelly keeps 74.9% of the peak growth and one-and-a-half Kelly only 74.6%.

3.6.4 Continuous Kelly is growth-optimal leverage

For a position you can resize at any time, Kelly is not new. It is the growth-optimal leverage you met in Session 2.3, now read as a bet size.

Put a fraction $f$ of wealth into an asset with expected return $\mu$ and volatility $\sigma$, keep the rest in cash at $r_f$, and rebalance continuously. The growth rate is the formula Session 2.3 wrote for leverage $L$, with $L$ renamed $f$:

Continuous Kelly is that session's $L^\star$. For its asset ($\mu$ = 8%, $\sigma$ = 20%, $r_f$ = 3%) it is 1.25: full Kelly means 1.25 dollars of exposure per dollar of capital. The last formula is sobering: with a Sharpe ratio $\SR$ of 0.25, the best possible excess growth rate over cash is only 3.125% a year, bought with 25.0% volatility. Half Kelly's excess growth rate is 2.34% at 12.5% volatility, the "75% of the growth for half the volatility" that the same session showed for leverage. And at $c = 2$ the excess growth is exactly zero: 2.5× leverage compounds like cash.

The lab re-checks the formula with daily rather than continuous rebalancing: 4,000 paths of 10 simulated years each, the same random shocks at every multiple of Kelly, and each mean growth rate asserted within four standard errors of theory.

Kelly and mean-variance

Look again at $g(f)$: expected excess return minus half the variance. That is the mean-variance objective $f(\mu - r_f) - \tfrac{\gamma}{2}f^2\sigma^2$ with risk aversion $\gamma = 1$. Merton (1969) showed that an investor with constant relative risk aversion $\gamma$ holds $(\mu - r_f)/(\gamma\sigma^2)$ in this setting, so log utility is the case $\gamma = 1$, and betting a fraction $c$ of Kelly is what a more risk-averse investor with $\gamma = 1/c$ would choose: half Kelly is $\gamma = 2$. The two frameworks agree when prices follow geometric Brownian motion and you rebalance continuously. They part ways when either fails:

3.6.5 You never know your edge

Kelly assumes you know the edge exactly. You never do, and the growth curve punishes a noisy estimate in a way that has a clean formula: the correct response is to bet a fraction of Kelly.

Session 2.3 and Session 1.4 showed that mean returns are the hardest thing in finance to estimate: after $Y$ years the standard error of an estimated mean is $\sigma/\sqrt{Y}$. Volatility is far easier, because daily data pin it down. The lab makes this concrete with 20,000 simulated researchers. Each sees 5 years of daily prices from the asset of §3.6.4, estimates $\mu$, sizes at Kelly on that estimate, and then trades for the next 10 years. After 5 years the standard error of $\hat\mu$ is 8.9% against a true excess return of 5%; for $\hat\sigma$ it is 0.4%, so the lab treats $\sigma$ as known. Nine in ten estimates of $\mu$ fall between −6.7% and 22.6%, and the middle 90% of the Kelly fractions built on them run from −2.4 to 4.9. 29% of researchers go short an asset that truly rises, and 21% lever it more than threefold.

Plug-in full Kelly is a disaster: its expected growth rate, −3.88% a year, is below the 3% that cash pays, even though the asset is genuinely good. The reason has a formula. The estimated Kelly fraction $\hat f = (\hat\mu - r_f)/\sigma^2$ is right on average, with variance $1/(\sigma^2 Y)$, and because the growth curve is a parabola, its errors cost growth in proportion to that variance. Sizing at a multiple $k$ of the estimate gives

At $k = 1$ the estimation error costs $1/(2Y)$ of growth a year, 10 percentage points with five years of data, whatever the asset. The best multiplier $k^\star$ depends only on $t$, the t-statistic your data can be expected to give the edge (Session 1.4): here $t$ = 0.56 and $k^\star$ = 0.24. Kan & Zhou (2007) derived the optimal scaling for mean-variance portfolios built on estimated means and covariances; applied to one asset and five years of daily data, their formula gives 0.237 against our 0.238. Baker & McHale (2013) show that shrinking the plug-in Kelly bet improves out-of-sample growth in betting too. It is a shrinkage estimator in disguise, and Session 5.3 — Bayesian Thinking: Priors, Shrinkage & Updating gets there properly, by shrinking $\hat\mu$ toward a prior before you size.

Two caveats make the real answer smaller still. The table assumes your estimate is merely noisy; a backtest chosen as the best of many is also biased upward, which Session 6.7 measures. And it assumes the edge does not change, which Session 1.4 warned it will. Even the benchmark researcher who is told the true $\mu$ did worse than cash over ten years 35% of the time.

3.6.6 Ruin, halving and the drawdown cap

Growth rates describe the long run. You have to survive the short run to get there, so the last input to a bet size is how deep a fall you can live with.

Risk of ruin is the probability that wealth ever reaches a level from which you cannot, or will not, carry on: zero, a margin call, or the loss at which you or your investors stop the strategy. How you size decides which kind of ruin you face. Someone who stakes a fixed dollar amount can be wiped out: with $n$ stakes' worth of bankroll on the lab's 55/45 coin, the classical gambler's-ruin result (Feller 1968, ch. XIV) is about $(q/p)^n$. A bettor who stakes a fraction of current wealth can never hit zero, because each loss is smaller than the last, but can fall a long way. The lab plays both from a $100 bankroll until it reaches $1,000 or is gone, with 20,000 players per rule:

Risk of ruin with fixed stakes falls exponentially with the bankroll measured in stakes: 30 stakes would take it to 0.24%. Proportional sizing removes ruin at zero in theory. In practice minimum trade sizes, costs and overnight gaps (Session 3.2) bring it back, which is one more reason to stay below Kelly.

How likely is a 50% loss?

For a continuously rebalanced position, the chance of touching a given level within a horizon is the barrier problem of Session 3.5: wealth at $c$ times Kelly, measured against cash (i.e. $W_t / (W_0 e^{r_f t})$, so cash's own compounding is netted out), is a geometric Brownian motion with drift $cf^\star(\mu - r_f)$ and volatility $cf^\star\sigma$. Every probability below is that relative-to-cash figure, not the chance of nominal wealth halving — cash compounding for 10 years is itself a real cushion. The lab feeds the relative process into quantnb.stats.p_touch (with its correction for checking only once a day) and compares it with 20,000 simulated 10-year paths of the same asset:

The last column is the one that hurts. A formula that starts from today's wealth understates the pain, because a drawdown is measured from the running peak, and peaks keep rising. At full Kelly on this asset, 61% of paths suffered a drawdown of −50% or worse relative to cash within ten years (the distribution of the maximum drawdown is the subject of Session 9.8 — Drawdowns, Copulas, Tail Risk & Stress Tests). Few people, and fewer investors, sit through that.

Turning a pain limit into a cap

Invert Thorp's formula and a drawdown tolerance becomes a maximum bet. If you will accept at most a probability $\delta$ of ever falling to a fraction $x$ of today's wealth, bet no more than

times Kelly (all three percentages below still relative to cash, as above). "At most a 10% chance of ever halving" allows 0.46 of Kelly; 5% allows 0.38; "at most a 20% chance of ever losing 20%" allows 0.24. Busseti, Ryu & Boyd (2016) solve the general version, maximising growth subject to a drawdown-probability limit as a convex program, the kind Session 3.4 taught you to state.

3.6.7 From Kelly to a position, and the debate

Kelly is one step in a six-step sizing workflow. The steps around it decide whether it helps you or hurts you, and whether to use it at all is still argued about.

Run the asset of §3.6.4 through it, supposing your five-year estimate happened to land on the truth. Step 3 gives $f^\star$ = 1.25. Step 4 shrinks it by $k^\star$ = 0.24 to 0.30. Step 5, a limit of a 10% chance of ever halving relative to cash, caps it at 0.58. Step 6 takes the smaller: hold 30% of capital in the asset and the rest in cash. That is far from the 1.25× leverage the formula alone suggested, and it is what the evidence supports. Turning a fraction of capital into contracts, lots and shares is position sizing, the subject of Session 9.5.

Kelly in practice, and its critics

Edward Thorp used Kelly sizing for blackjack and sports betting, and from 1969 to allocate assets in an investment partnership (Thorp 2006). His advice is a practitioner's. When the edge is an estimate, assume the true edge is smaller and bet less than the estimated Kelly amount (his §7.3). And, from his own experience and that of many blackjack players and teams, "most people strongly prefer the increased safety and psychological comfort of 'half Kelly' (or some nearby value), in exchange for giving up 1/4 of their growth rate" (his §7.4). The economist Paul Samuelson objected from the other side, to the objective itself, and the argument about why anyone should maximise a logarithm has never quite ended:

Strip out the philosophy and the practical ground is shared. Nobody defends betting more than Kelly: it is beaten on growth and risk at once (§3.6.3). Fractional Kelly trades growth for safety along a smooth curve, so the fraction is a preference, and MacLean, Thorp & Ziemba (2010) lay out both sides of that trade. And estimation error pushes every honest answer below full Kelly (§3.6.5). Whether you call the objective "log utility" or "time-average growth", the number you bet ends up smaller than the formula's.

3.6.8 Try it

Open the companion lab and change the inputs. Predict each answer before you run it.

3.6.9 Glossary · Check yourself · Go deeper

Check yourself

📚 Go deeper

1. Kelly, J. L., Jr. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal 35(4), 917–926. doi:10.1002/j.1538-7305.1956.tb03809.x
2. Thorp, E. O. (2006). "The Kelly Criterion in Blackjack Sports Betting, and the Stock Market." In Zenios, S. A. & Ziemba, W. T. (eds.), Handbook of Asset and Liability Management, Vol. 1, ch. 9, 385–428. North-Holland/Elsevier. The fractional-Kelly case and the halving formula are in §7.3–7.4. doi:10.1016/S1872-0978(06)01009-X
3. MacLean, L. C., Thorp, E. O. & Ziemba, W. T. (2010). "Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria." Quantitative Finance 10(7), 681–687. doi:10.1080/14697688.2010.506108
4. Samuelson, P. A. (1979). "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long." Journal of Banking & Finance 3(4), 305–307. The critique, in words of one syllable. doi:10.1016/0378-4266(79)90023-2
5. Kan, R. & Zhou, G. (2007). "Optimal Portfolio Choice with Parameter Uncertainty." Journal of Financial and Quantitative Analysis 42(3), 621–656. doi:10.1017/S0022109000004129
6. Busseti, E., Ryu, E. K. & Boyd, S. (2016). "Risk-Constrained Kelly Gambling." The Journal of Investing 25(3), 118–134. doi:10.3905/joi.2016.25.3.118 · arXiv:1603.06183
7. Poundstone, W. (2005). Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Hill and Wang. The story of Kelly, Shannon, Thorp and Samuelson, told for a general reader. ISBN 978-0-8090-4637-9

Sources for the numbers

1. Breiman, L. (1961). "Optimal Gambling Systems for Favorable Games." In Neyman, J. (ed.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 65–78. University of California Press. https://projecteuclid.org/proceedings/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Fourth-Berkeley-Symposium-on-Mathematical-Statistics-and/Chapter/Optimal-Gambling-Systems-for-Favorable-Games/bsmsp/1200512159
2. Samuelson, P. A. (1971). "The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling." Proceedings of the National Academy of Sciences 68(10), 2493–2496. doi:10.1073/pnas.68.10.2493
3. Merton, R. C. (1969). "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case." Review of Economics and Statistics 51(3), 247–257. doi:10.2307/1926560
4. Baker, R. D. & McHale, I. G. (2013). "Optimal Betting Under Parameter Uncertainty: Improving the Kelly Criterion." Decision Analysis 10(3), 189–199. doi:10.1287/deca.2013.0271
5. Doctor, J. N., Wakker, P. P. & Wang, T. V. (2020). "Economists' views on the ergodicity problem." Nature Physics 16, 1168. doi:10.1038/s41567-020-01106-x
6. Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., ch. XIV. Wiley. ISBN 978-0-471-25708-0