8Managing a Book through Drawdown

Two portfolio managers are both down eight per cent from their peak in March. One runs a strategy that has had a bad few months; the other’s edge died a year ago and nobody has noticed. On the day each first reaches the eight per cent, the best statistical reading of their P&L gives the first a 24% chance of having broken and the second 39%: the numbers barely differ. To tell a strategy whose Sharpe ratio has fallen from 1 to 0 from one that is merely unlucky, at a false-alarm rate of one in ten years, takes on average more than fifteen years of data. The head of desk must decide in March.

8.1 What a drawdown says about a Sharpe ratio

A drawdown (One Quant Book 2, chapter 29) is the fall of cumulative P&L from its running peak; its maximum over a period and its duration are Book 7’s measures (chapter 22). What matters for management is how large a drawdown a healthy strategy produces. For daily P&L with a Sharpe ratio of 1 and a volatility of 10% a year on capital, the maximum drawdown over ten years has a median of 16.8% of capital and a 90th percentile of 25.4% (2 000 simulated paths): a healthy strategy of this kind will spend some years more than one year’s volatility below its peak. Magdon-Ismail, Atiya, Pratap and Abu-Mostafa give the distribution of the maximum drawdown of a Brownian motion with drift; the simulation reproduces its lesson, that drawdowns of one to two years’ volatility are the normal life of a Sharpe-1 strategy.

8.2 Stop rules and de-risking ladders

A stop rule makes two errors, and the drawdown limit of One Quant Book 8, chapter 28 already showed their shape: it stops live strategies in bad luck, and it lets dead ones run. The ladder is a two-step version of Grossman and Zhou’s rule, in which the position falls smoothly as the drawdown approaches a floor; the time stop watches duration instead of depth, because a dead strategy with a Sharpe ratio of zero drifts sideways rather than down.

def posterior(ret: np.ndarray, sr: float, sr_dead: float, vol: float, hazard: float) -> np.ndarray:
    """Forward filter of a two-state chain alive -> broken (absorbing, daily hazard `hazard`), Gaussian emissions."""
    d = vol / math.sqrt(DAYS)
    m1, m0 = sr * vol / DAYS, sr_dead * vol / DAYS
    n, T = ret.shape
    p = np.zeros(n)                     # P(broken | data so far)
    out = np.zeros((n, T))
    for t in range(T):
        prior = p + (1 - p) * hazard
        l1 = np.exp(-0.5 * ((ret[:, t] - m1) / d) ** 2)
        l0 = np.exp(-0.5 * ((ret[:, t] - m0) / d) ** 2)
        p = prior * l0 / (prior * l0 + (1 - prior) * l1)
        out[:, t] = p
    return out

8.3 Bad luck or broken: sequential evidence

A drawdown rule looks at one statistic of the P&L path. The whole path carries more: every day’s return is evidence about whether the strategy is still the one that was backtested.

The bound is Lorden’s (1971) result, cited in omsources; the CUSUM test itself is One Quant Book 7’s (chapter 13). Its arithmetic is the chapter’s central fact. A Sharpe ratio that falls from 1 to 0, detected at one false alarm per ten years ($A=2\,520$ days), takes $2\ln2\,520=15.7$ years to detect on average; at one per five years, 14.3. A fall to $-1$ takes 3.9 years and a fall to $-2$, 1.7 (Figure 8.1). An edge that disappears quietly cannot be distinguished from bad luck on its P&L in any useful time; only an edge that turns into a loss can.

The posterior of Proposition 8.3, run on two example paths, shows what the head of desk sees (Figure 8.2). The live strategy’s posterior wanders, reaching 0.80 in its fourth year; the broken strategy’s climbs slowly after its break at the start of year four and passes 0.9 only in its fifth year.

8.4 Comparing the rules

The chapter’s evaluation runs 2 000 live strategies and 2 000 that break at the start of year 4 to a Sharpe ratio of 0, for ten years, under each rule, with every stop permanent. It reports false stops per hundred live strategy-years, the share of broken strategies stopped after their break and the median delay, and the value kept per strategy over ten years: P&L less a capital charge of 2% of capital for each year run at full size (a 15% hurdle on capital of 13% of the strategy’s size), in per cent of capital.

The 10% stop-loss is the worst rule by far. It stops 98% of healthy strategies at some point in ten years, and two thirds of the broken ones before they break; it keeps 18.3% of capital per strategy against 45.1% for no rule at all. The posterior rule stops 28% of healthy strategies over ten years, catches 95% of the broken ones about two and a half years after the break, and keeps 44.8%, as much as doing nothing. Stopping a broken strategy whose Sharpe ratio is zero saves no P&L in expectation; it saves the capital it ties up and the risk it adds, which is why the capital charge is in the value. With a charge of zero no rule beats doing nothing; with a strategy that turns into a loser, every rule catches it sooner and the posterior rule is clearly best (exercise 7).

8.5 Restart, re-underwrite, retire

A stop is a decision under uncertainty; a restart rule is what makes it reversible. Because a healthy strategy is stopped by any rule that catches broken ones in useful time, the cost of false stops is paid in the months the strategy is flat. A restart rule that paper-trades a stopped strategy and brings it back at reduced size when its paper P&L recovers recovers much of that cost; a restart that happens automatically at the new year, as in the model of One Quant Book 8, chapter 28, recovers it only for the good strategies and restarts the dead ones too. Re-underwriting a strategy is reviewing it as if it were new: its thesis, its evidence, its capacity and its cost of capital. A strategy that fails that review is retired even if its P&L has not yet said so, the only way to beat the delays of Proposition 8.4.

8.6 Tutorial: two thousand strategies and four rules

What to change next. Break the strategies to a Sharpe ratio of $-1$ (exercise 7); set the hazard ten times too high and watch the posterior rule stop healthy strategies.

8.7 Build: drawdown rules

8.8 Exercises

8.9 Problem: Down Eight in March

8.10 Interview questions