M.9.5 Ruin & Drawdown

Basic Prerequisites: 04 · Kelly Criterion.

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1. Intuition & Practical Objective

Kelly gives the growth-optimal fraction. This page quantifies what goes wrong when you miss it - and what still goes wrong when you hit it exactly. Two distinct failure channels matter:

1. Ruin proper. Past the critical fraction $f_c$, the growth rate goes negative and the account tends to zero almost surely. Even at full Kelly, the drawdown distribution has a heavy left tail: full Kelly will, with probability $\frac{1}{2}$ (in the $r=0$ case), at some point be worth half what it started at. In the leveraged S&P calibration, full Kelly’s median maximum drawdown over 30 years is $\approx 71\%$, and its 95th-percentile drawdown is $\approx 89\%$.
2. Behavioural ruin. Investors do not sit through an $89\%$ drawdown; they liquidate at the bottom, which converts a temporary drawdown into a permanent loss. In practice “ruin” is a psychological threshold, not just $W=0$.

The practical objective is to internalise that drawdown is a first-class output of a strategy, not noise, and that the correct sizing is not “Kelly” but “fractional Kelly” - a deflated fraction chosen precisely to keep drawdowns survivable.

Takeaway. Never trade at full Kelly on estimated parameters. Full Kelly’s own drawdown law is brutal ($\mathbb{P}(\text{ever}\le x)=x$ in the zero-rate case); half Kelly cuts the chance of ever halving your capital from $\frac{1}{2}$ to $\frac{1}{8}$ while keeping $\frac{3}{4}$ of the growth. Fractional Kelly is not a compromise - it is the risk-managed optimum.

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2. Mathematical Ground Truth & Derivations

2.1 Gambler’s ruin (fixed stakes)

A gambler starts with $i$ units, wins $1$ with probability $p$ and loses $1$ with $q=1-p$ each trial, and stops at $0$ (ruin) or $N$ (goal). The classic result (Feller) is

$$\mathbb{P}(\text{ruin})=\{\begin{array}{ll}\frac{{\left(q/p\right)}^i-{\left(q/p\right)}^N}{1-{\left(q/p\right)}^N},& p\ne \frac{1}{2},\\ 1-\frac{i}{N},& p=\frac{1}{2}.\end{array}$$

For a fair game ($p=\frac{1}{2}$) the probability of losing $i$ before reaching $N$ is $1-i/N$: from $i=50$ toward $N=100$ this is exactly $\frac{1}{2}$. Fixed-stake ruin is a linear boundary problem.

2.2 Drawdown law for a growth process

Model $\ln W(t)$ as Brownian motion with drift $g_{\infty }$ (the growth rate) and per-unit-time variance $\mathrm{Var}(G_{\infty })$. The classic two-barrier result (Cox & Miller; Thorp eq. 7.12) says: if $x<1<y$, then for a fraction $f=c{f}^{\ast }$,

$$\mathbb{P}\left(W\text{ reaches }y{W}_0\text{ before }x{W}_0\right)=\frac{1-x^a}{1-(x/y{)}^a},a=\frac{2g_{\infty }}{\mathrm{Var}(G_{\infty })}.$$

Letting $y\to \infty$ gives the drawdown law

$$\boxed{\mathbb{P}\left(\text{ever}\le x\right)=x^a,a=\frac{2g_{\infty }}{\mathrm{Var}(G_{\infty })}}$$

For the continuous Kelly model, $\mathrm{Var}(G_{\infty }(f))=s^2{f}^2$ and $g_{\infty }=r+f(m-r)-\frac{1}{2}{s}^2{f}^2$. Two clean special cases (Thorp §7.4):

• Zero rate, full Kelly ($c=1$): $a=1\Rightarrow \mathbb{P}(\text{ever}\le x)=x$. So $\mathbb{P}(\text{ever halve})=50\%$, and more generally the probability of ever dropping to a fraction $x$ equals that fraction.
• Zero rate, half Kelly ($c=\frac{1}{2}$): $a=3\Rightarrow \mathbb{P}(\text{ever}\le x)=x^3$. So $\mathbb{P}(\text{ever halve})=\frac{1}{8}=12.5\%$ - a quarter of the risk, for $\frac{3}{4}$ of the growth.

Equivalently, for general $c$: $a=\frac{2}{c}-1$, hence $\mathbb{P}(\text{ever}\le x)=x^{2/c-1}$.

Doubling before halving ($x=\frac{1}{2},y=2$): full Kelly $\to \frac{1-1/2}{1-1/4}=\frac{2}{3}$; half Kelly $\to \frac{1-1/8}{1-1/64}=\frac{8}{9}$.

2.3 Why fractional Kelly wins the risk-adjusted argument

From page 04 and the box above: scaling by $c$ multiplies the growth rate by $c(2-c)$ (for $r=0$) and multiplies the drawdown exponent $a$ by $\frac{1}{c}$-ish, i.e. it raises the exponent $2/c-1$, crushing the tail probability of deep drawdowns. There is no fraction that both maximises growth and is survivable; the practitioner chooses $c\in [0.25,0.5]$.

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3. Computational Implementation - ruin, the drawdown law, and 30 years of drawdown

Stdlib only. Block 1 confirms the classic gambler’s ruin and the drawdown law $x^a$ (using the exact Brownian-bridge crossing probability for continuous monitoring). Block 2 simulates 30 years of leveraged Kelly equity and measures the realised max drawdown.

The simulation reproduces $x^a$ to the third decimal at every level and both fractions - the drawdown law is not just an asymptotic abstraction. Note the exponent difference: $\frac{1}{2}$-Kelly’s $a=7.32$ versus full Kelly’s $a=2.08$. That one number is why half Kelly’s chance of ever halving is $0.0065$ instead of $0.2355$ at these parameters.

Half Kelly gives up about a third of the terminal wealth ($21.1\times$ vs $32.0\times$) to cut the median drawdown from $71\%$ to $38\%$ and the 95th-percentile drawdown from $89\%$ to $55\%$. The $+52\%$ more wealth at full Kelly comes with a drawdown most real investors cannot hold through - that is the entire case for fractional Kelly, in two lines.

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4. Failure Modes & First-Principles Breakdowns

1. Treating $f_c$ as a soft threshold. Below $f_c$ you grow a.s.; above it you decay a.s. The penalty is not gradual - crossing $f_c$ flips the sign of the growth rate.
2. Ignoring the drawdown law when choosing $c$. Even full Kelly ($c=1$) has $\mathbb{P}(\text{ever halve})=\frac{1}{2}$ at zero rates. If a $50\%$ drawdown would force liquidation, full Kelly is not optimal - the objective must include survival.
3. Convexity of ruin to leverage. Required leverage scales like $1/(1-T)$ under taxes and like $(m-r)/s^2$ under return estimates; small errors in $m$ or $s$ move you across $f_c$ because the growth function is flat at the top and steep at the edges.
4. Fat tails invalidate the drawn-down law’s Gaussian core. Real returns jump (Tsay; Merton in BSM · Extensions); a single $-30\%$ gap can leap over a monitored barrier. The bridge simulation above assumes continuous paths - reality is worse, so treat the $x^a$ numbers as optimistic.
5. Behavioural ruin as true ruin. The mathematically “temporary” drawdowns of full Kelly are psychologically permanent. Sizing must respect the human, not just the diffusion.

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5. References

• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Feller, William: An Introduction to Probability Theory and Its Applications, Vol. I
• Cox, D. R. & Miller, H. D.: The Theory of Stochastic Processes
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion (2011)

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6. Connected Graph Bridges

• Back: 04 · Kelly Criterion
• Forward: 06 · Advanced Extensions · Index Hub
• Theory: Brownian Motion & First Passage (barrier hitting) · Probability & Measure Theory
• Applications: Quantitative Risk · Extreme Value Theory & Fat Tails · Portfolio Optimization