5.7.4 Fractional Kelly, Drawdowns & the Risk of Ruin From Overbetting

Basic Prerequisites: 03 · Growth & Optimality and 05 · Ruin & Drawdown.

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1. Intuition & Practical Objective

Page 03 proved full Kelly is growth-optimal. This page confronts the cost: full Kelly is aggressive, and anything past it is ruinous. The objective is to make the risk profile concrete and turn it into a sizing rule change.

Three facts drive practice:

1. The $c(2-c)$ law. Half Kelly ($c=\frac{1}{2}$) keeps $\frac{3}{4}$ of the growth rate at half the volatility. The first 25% of growth you give up buys a 50% cut in risk - an extraordinary asymmetry in the risk dimension even though growth is symmetric around $f^{\ast }$.
2. Overbetting risk is not symmetric. Because $g(f)$ is concave and flat near the top, betting $f^{\ast }+\delta$ costs more growth than betting $f^{\ast }-\delta$, and the drawdown cost of overbetting compounds with quadratic volatility. Past the critical fraction $f_c$ the growth rate goes negative: ruin is near-certain with a positive edge.
3. Ruin and drawdown have quantitative laws. Any fixed-fraction bettor runs a constant-proportional-betting random walk, so drawdowns follow an explicit distribution (Thorp §6; the ergodicity folder’s ruin page); full Kelly has a large, fixed drawdown law that the median investor finds unacceptable.

Takeaway. Fractional Kelly is not a hedge against being wrong about $f^{\ast }$ - it is the rational response to the asymmetry that overbetting is punished far harder than underbetting, in a world where $p,m,s$ are estimated. The standard practitioner rule: bet at half Kelly ($c=0.5$) to start, de-rate further when parameter uncertainty is high.

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2. Mathematical Ground Truth & Derivations

2.1 The fractional-Kelly growth and risk

With $f=c{f}^{\ast }$:

$$\frac{g_{\infty }(c{f}^{\ast })-r}{g_{\infty }(f^{\ast })-r}=c(2-c)(\text{exact for the excess growth; the raw ratio equals }c(2-c)\text{ only when }r=0),\frac{\mathrm{SD}(G_{\infty }(c{f}^{\ast }))}{\mathrm{SD}(G_{\infty }(f^{\ast }))}=c.$$

Growth is a quadratic that is flat at the top; risk is linear in $c$. Hence half Kelly: $g/g^{\ast }=0.75$, risk $=0.5$. Double Kelly ($c=2$): $g/g^{\ast }=0$, risk $=2$. The ratio of return to risk is monotonically worse the further you scale above $c=1$.

2.2 The critical fraction and certain ruin

For the even-money game ($p=0.55$), $f_c=0.1987$ solves $g(f_c)=0$, i.e. $\mathbb{E}[\ln (1+R(f))]=0$. For $f>f_c$ the log-growth is negative, and by the law of large numbers the time-average wealth drifts to zero almost surely. Because $f_c\approx 2\times f^{\ast }$, a $2\times$ sizing error turns a profitable edge into a guaranteed-loss strategy - not a riskier one, a losing one.

2.3 Drawdown law (continuous)

For continuous Kelly, the drawdown from a peak follows an exponential-law tail controlled by the growth-rate-to-variance ratio. The key qualitative result: even at full Kelly the expected maximum drawdown is large, so a full-Kelly investor must tolerate 50%+ drawdowns as routine - a “growth-optimal but stomach-churning” profile.

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3. Computational Implementation - fractional Kelly, ruin, and drawdowns in simulation

Stdlib only: simulate fixed-fraction betting at half / full / double Kelly and measure three outcomes - median terminal wealth, probability of ruin (account < 10% of start), and average maximum drawdown from the running peak.

The pattern is unmistakable. Double Kelly ($c=2$, right at the critical $f_c=0.199$): half the terminal wealth of half-Kelly earnings, 22% of accounts ruined, and terminal drawdown 100%. Full Kelly: 0.9% ruin but 99% max-drawdown. Half Kelly: zero ruin in 20,000 paths and still compounds positively. The overbet is strictly worse on every risk metric - more ruin and lower median wealth - which is exactly the failure mode of page 05. Fractional Kelly is the only one of the three that is actually deployable.

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4. Failure Modes & First-Principles Breakdowns

1. “Kelly says bet 22%.” On an estimated edge, $f^{\ast }$ is a fiction; betting the estimated full Kelly on a slightly-wrong $p$ lands you at $2\times$ the true optimum - straddling $f_c$ with double-Kelly ruin probabilities (see the page 05 overbetting demo).
2. Treating drawdown as a nuisance. Full Kelly’s ~99% max drawdown is not a bug - it is the growth-optimal risk profile. Funds and individuals almost universally de-rate because the median path of full Kelly is brutally volatile.
3. Believing $c=2$ “geometrically” doubles returns. The growth curve is concave: beyond $f^{\ast }$ you gain nothing and lose risk. There is no compensation for overbetting.
4. Ignoring the continuous/leverage case. In securities, $f^{\ast }=(m-r)/s^2$ can exceed 1 (leverage); fractional Kelly then governs how much leverage, and over-leverage near $f_c$ in the continuous case means forced liquidation at drawdowns, compounding the ruin.

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5. References

• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• MacLean, Ziemba & Blazenko: Growth versus Security in Dynamic Investment Analysis, Management Science 38(11) (1992)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion (2011) - “bad properties of Kelly” (aggressiveness, drawdown).
• Browne, S.: The Return on Investment from Proportional Portfolio Strategies and the Optimal Investment Horizon (1999)

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6. Connected Graph Bridges

• Back: 03 · Growth & Optimality · Index Hub
• Apply: 05 · Failure Modes & Practice
• Ruin foundation: 05 · Ruin & Drawdown (foundations) · 04 · Kelly Criterion
• Risk metrics: VaR & Expected Shortfall