3.8 Drawdowns

Motivation

Investors and traders do not experience growth rates; they experience drawdowns. A fund with a 30% drawdown loses clients; a proprietary trader with a 10% drawdown loses their book. The distribution of drawdowns, not the growth rate, often decides survival.

Drawdowns of Brownian log-wealth#

Setup

For a process $X_t$ (log-wealth), the running maximum is $M_t = \sup_{u\le t}X_u$ , the drawdown is $D_t = M_t - X_t \ge 0$ , and the maximum drawdown over $[0, T]$ is $\text{MDD}_T = \sup_{t\le T}D_t$ . In wealth terms a log-drawdown $d$ is a fall of $1 - e^{-d}$ from the peak.

Proposition 3.15stationary drawdown of Brownian log-wealth

Let $X_t = mt + sB_t$ with $m > 0$ . As $t \to \infty$ , $D_t$ converges in distribution to an exponential law with rate $2m/s^2$ :

\lim_{t\to\infty}P(D_t > d) = e^{-2md/s^2}. \tag{3.20}

Proof

Fix $t$ . $D_t = \sup_{0\le u\le t}(X_u - X_t) = \sup_{0\le v\le t}\big(-(X_t - X_{t-v})\big)$ . The process $\tilde X_v = X_t - X_{t-v}$ , $0 \le v \le t$ , has independent stationary Gaussian increments with mean $mv$ and variance $s^2v$ and starts at 0, so it is a Brownian motion with drift $m$ on $[0, t]$ (time reversal). Hence $D_t \overset{d}{=} \sup_{v\le t}(-\tilde X_v)$ , which increases to $\sup_{v\ge0}(-\tilde X_v)$ as $t \to \infty$ . By Proposition 3.7, $P(\sup_{v\ge0}(-\tilde X_v) > d) = P(\inf_v\tilde X_v < -d) = e^{-2md/s^2}$ .∎

For a fractional-Kelly bettor, $2m/s^2 = 2/c - 1$ (Proposition 3.8), so in the long run the fraction of time spent more than a fraction $1 - x$ below the peak is $x^{2/c - 1}$ . A full-Kelly bettor is more than 50% below their previous peak half of the time. Half Kelly: one-eighth of the time.

Expected maximum drawdown#

Maximum drawdown over a horizon For Brownian motion with drift the distribution of $\text{MDD}_T$ has no elementary closed form. Magdon-Ismail, Atiya, Pratap and Abu-Mostafa (2004) give its expectation as a series and the large- $T$ asymptotic

E[\text{MDD}_T] \approx \frac{2s^2}{m}\Big(\tfrac14\ln\frac{m^2T}{2s^2} + 0.49088\Big), \qquad \frac{m^2T}{2s^2} \gg 1. \tag{3.21}

Expected maximum drawdown grows only logarithmically in $T$ , at a rate $s^2/(2m)$ set by the same ratio as in (3.20).

Worked example 3.8.1a Sharpe-1 strategy

A strategy with arithmetic excess return 15% and volatility 15% a year has log drift $m = 0.15 - 0.15^2/2 = 0.139$ and $s = 0.15$ .

Probability of ever being 20% below the starting capital: $e^{-2(0.139)(0.223)/0.0225} = 0.064$ .
Long-run share of time in a drawdown deeper than 20%: also 6.4%.
Expected maximum drawdown over 10 years from (3.21): $\frac{2(0.0225)}{0.139}(\frac14\ln 4.28 + 0.491) = 0.277$ in log terms, a 24% fall. A Monte Carlo of 20,000 ten-year paths with daily steps gives 0.278, and a 71% probability that the 10-year maximum drawdown exceeds 20%.

The last number is the one to tell investors: a strategy with Sharpe 1, a level few funds sustain, will probably have a 20% drawdown within a decade.

Worked example 3.8.2buy and hold Nifty

With $m = 8.75\%$ and $s = 20\%$ (the Nifty sample, M02 · 2.2), simulated 20-year paths have an expected maximum drawdown of 0.64 in log terms (a 47% fall), and a 33% chance of a maximum drawdown worse than 50%. The asymptotic formula gives 0.60. The Nifty sample's actual maximum drawdown was 59.9%, from 6,287.85 on 8 Jan 2008 to 2,524.20 on 27 Oct 2008; the index regained its January 2008 close only in November 2010.

Drawdowns of a random walk#

Proposition 3.16random-walk drawdown excursions

For the $\pm1$ walk with $p > q$ , standing at a running maximum, the probability that it falls $k$ units below that maximum before setting a new maximum is

\pi_k = \frac{\rho^k(1-\rho)}{1 - \rho^{k+1}}, \qquad \rho = q/p. \tag{3.22}

Proof

Shift coordinates so the current maximum is $k$ and the drawdown level is 0. A new maximum is reached at $k + 1$ . This is gambler's ruin from $a = k$ with $N = k + 1$ : $P_k = (\rho^k - \rho^{k+1})/(1 - \rho^{k+1})$ .∎

Successive excursions below the running maximum are independent, so the number of excursions before the first $k$ -unit drawdown is geometric with parameter $\pi_k$ . Over $n$ steps the walk makes about $n(p - q)$ new maxima, so $P(\text{MDD}_n < k) \approx (1 - \pi_k)^{n(p-q)}$ , and the typical maximum drawdown grows like $\ln n/\ln(1/\rho)$ , the discrete analogue of (3.21). For $p = 0.55$ : $\pi_5 = 0.095$ and $\pi_{10} = 0.027$ .

Practitioner notes

Maximum drawdown in a backtest is a single draw from a wide distribution. Report its distribution by simulation or by bootstrap of the trade list, not its single value (3.9).
Drawdowns are path-dependent, so volatility clustering (M02 · 2.7) and positive autocorrelation in stress lengthen them beyond the iid estimates. Treat (3.20)–(3.22) as lower bounds.

The lab builds the wealth of a Nifty 50 holding since 17 Sep 2007 at a constant leverage $L$ , rebalanced daily ( $W_t = W_{t-1}(1 + L R_t)$ , financing cost and dividends ignored, so wealth is measured against cash with $r = 0$ ). It draws the underwater curve $1 - W_t/\max_{s \le t} W_s$ , the share of days spent more than a depth $d$ below the running peak, and the Brownian prediction $x^{2m/s^2}$ with $x = 1 - d$ , $m = L\mu - \tfrac12 L^2\sigma^2$ , $s = L\sigma$ from the sample $\mu$ and $\sigma$ . The table gives $f^* = \mu/\sigma^2$ and the implied Kelly multiple $c = L/f^*$ , so that the exponent equals $2/c - 1$ (Propositions 3.8 and 3.15).

What to look at The 2008 drawdown and its recovery date (compare Worked example 3.8.2), how the curve deepens as $L$ rises, and whether the empirical share of time below each depth sits above or below the Brownian line.

Drawdowns of Brownian log-wealth#

Expected maximum drawdown#

Drawdowns of a random walk#

Nifty drawdowns at constant leverage

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