3.7 Ruin under fixed-fractional betting

Motivation

Fixed-fractional betting (risk a fixed fraction $f$ of current capital) cannot reach zero in finitely many bets. The meaningful question is the probability of falling to a fraction $x$ of starting capital, a drawdown from which the trader might stop or be stopped (by a margin call, a fund mandate, or nerve).

The Lundberg bound#

Setup

Bets $X_i$ are iid with $P(X_i < 0) > 0$ , $1 + fX_i > 0$ a.s., and $E\ln(1 + fX) > 0$ . Let $L_n = \ln(W_n/W_0) = \sum_{i \le n} Y_i$ with $Y_i = \ln(1 + fX_i)$ , and let $\tau_x = \inf\{n : W_n \le xW_0\}$ for $x \in (0, 1)$ .

Proposition 3.13Lundberg bound

There is a unique $\theta > 0$ with

E\big[(1 + fX)^{-\theta}\big] = 1, \tag{3.18}

and

P(\tau_x < \infty) \le x^{\theta}. \tag{3.19}

Proof

Let $\varphi(\theta) = E e^{-\theta Y} = E(1 + fX)^{-\theta}$ . $\varphi$ is convex, $\varphi(0) = 1$ , $\varphi'(0) = -E[Y] < 0$ , and $\varphi(\theta) \to \infty$ as $\theta \to \infty$ because $P(Y < 0) > 0$ . So $\varphi$ dips below 1 and returns above it, crossing 1 at exactly one $\theta > 0$ . With this $\theta$ , $M_n = e^{-\theta L_n} = (W_n/W_0)^{-\theta}$ is a non-negative martingale with $M_0 = 1$ . For each $n$ , optional stopping at the bounded time $\tau_x \wedge n$ gives

1 = E[M_{\tau_x\wedge n}] \ge E[M_{\tau_x};\ \tau_x \le n] \ge x^{-\theta}P(\tau_x \le n),

since $W_{\tau_x} \le xW_0$ implies $M_{\tau_x} \ge x^{-\theta}$ . Let $n \to \infty$ .∎

The bound is tight up to the overshoot (the amount by which wealth jumps below $xW_0$ ). For small bets the overshoot is small and the bound is nearly exact; in Brownian motion there is no overshoot and (3.19) is the equality (3.12).

Ruin at the Kelly fraction#

Proposition 3.14 $\theta = 1$ at Kelly

If $f = f^*$ maximises $E\ln(1 + fX)$ in the interior, then $\theta = 1$ . Hence, for any bet distribution, a Kelly bettor's probability of ever falling to a fraction $x$ of initial wealth is at most $x$ . For small bets at $f = cf^*$ , $\theta \approx 2/c - 1$ .

Proof

The first-order condition is $E[X/(1 + f^*X)] = 0$ . Then $E[(1 + f^*X)^{-1}] = E\big[\frac{1 + f^*X - f^*X}{1 + f^*X}\big] = 1 - f^*E\big[\frac{X}{1 + f^*X}\big] = 1$ , so $\theta = 1$ solves (3.18), and it is the unique positive root. For the small-bet approximation, expand $E(1 + fX)^{-\theta} = 1 - \theta fm + \frac12\theta(\theta+1)f^2v + O(f^3)$ with $m = E X$ , $v = E X^2$ . Setting this to 1 and dividing by $\theta f$ : $\theta + 1 \approx 2m/(fv)$ . With $f = cf^* \approx cm/v$ , $\theta \approx 2/c - 1$ .∎

The identity $E[1/(1 + f^*X)] = 1$ says that $1/W_n$ is a martingale for the Kelly bettor: Kelly wealth is a numéraire under which all other fixed-fraction strategies' wealth processes are supermartingales. This property underlies the growth-optimal portfolio literature (Long 1990) and connects Kelly to the risk-neutral pricing of M08.

Worked example 3.7.1the swing system

For $p = 0.4$ , $b = 2.5$ (R-multiples $+2.5$ , $-1$ ), solving (3.18) numerically:

Risk per trade $f$	Growth per trade	$\theta$	$P$ (ever $-20\%$ ) $\le$	$P$ (ever halve) $\le$
0.5%	0.0020	59.8	$10^{-6}$	$10^{-18}$
1%	0.0038	29.5	0.0014	$10^{-9}$
2%	0.0074	14.3	0.041	$5\times10^{-5}$
5%	0.0163	5.19	0.31	0.027
8%	0.0229	2.91	0.52	0.13
16% (Kelly)	0.0300	1.00	0.80	0.50
24%	0.0233	0.36	0.92	0.78

These bounds concern falls from the starting capital. Falls from a later peak (drawdowns) are more frequent; 3.8 and the bootstrap in 3.9 treat them. All of these numbers assume the R distribution is exactly $\{+2.5, -1\}$ with $p = 0.4$ . A single gap of $-3R$ in the true distribution changes them materially (Problem 9).