Conditional Drawdown at Risk

Max Drawdown tells you the single worst peak-to-trough loss in a track record, and that is its whole problem: it is one number, drawn from one unlucky (or lucky) path, and it is entirely blind to how often nasty drawdowns happen or how deep the second- and third-worst ones were. A fund with one brutal 40% drawdown and otherwise placid history, and a fund that regularly grinds through 25% drawdowns every year, can report a similar max drawdown while being completely different animals to hold. Conditional Drawdown at Risk (CDaR) fixes this the same way expected shortfall fixed VaR: instead of reporting the worst point, it averages the worst stretch of points.

The analogy

Think of a hiker rating trails by "worst single step down a cliff." That number misses whether the trail has one bad ledge or a mile of them. A better rating averages the depth of every genuinely steep descent along the whole hike, not just the single steepest one. CDaR is that better rating applied to an equity curve: instead of "what was my worst drawdown," it asks "on average, how deep are my worst drawdowns, the ones in the tail of severity?"

Building the formula

First recall a drawdown at time $t$ is how far the current equity value $E_t$ sits below its running peak so far:

$D_t = \frac{\max_{s \le t} E_s - E_t}{\max_{s \le t} E_s}.$

In words: look at every high-water mark reached to date, take the most recent one, and measure the percentage drop from it to today. This produces a whole time series $D_t$, mostly zero (at new highs) with dips whenever the fund is below a prior peak.

CDaR at confidence level $\alpha$ (commonly 95%) is the average of the worst $(1-\alpha)$ fraction of that drawdown series, exactly mirroring how expected shortfall averages the worst $(1-\alpha)$ fraction of losses:

$\text{CDaR}_\alpha = \mathbb{E}\left[D_t \mid D_t \ge \text{DaR}_\alpha\right],$

where $\text{DaR}_\alpha$ ("Drawdown at Risk") is the $\alpha$-quantile of the drawdown series, the threshold that the worst $(1-\alpha)$ of drawdown readings exceed. In words: find the cutoff depth that only the worst 5% of your daily drawdown readings breach, then average everything past that cutoff. Unlike max drawdown, this uses every day the fund spent underwater in the tail, not just the single deepest instant.

Generate a few paths here and picture drawing the running peak as a dashed ceiling above each path, then shading the gap between the ceiling and the path whenever the path is below it. CDaR is the average height of the deepest shaded regions across the path, not just the single deepest point of shading.

Worked example 1: a short daily drawdown series

Suppose ten days of drawdown readings (percent below peak): $0, 2, 5, 8, 12, 9, 3, 0, 6, 15$.

Max drawdown here is simply $15\%$, the single worst reading. To get $\text{CDaR}$ at $\alpha = 70\%$ (worst 30% of readings, i.e. worst 3 of the 10 days): sort the readings descending: $15, 12, 9, 8, 6, 5, 3, 2, 0, 0$. The worst 3 are $15, 12, 9$. Average them: $(15+12+9)/3 = 12\%$.

So while the single worst moment was 15%, the typical bad moment (averaging the worst 30% of the sample) was 12%. That 12% is a far more honest number to size a stop-loss or an allocation around, because it reflects that deep drawdowns were not a one-off freak event in this series, they clustered near the top of the range.

Worked example 2: comparing two funds with the same max drawdown

Fund A's drawdown readings over 10 days: $0,0,0,0,0,0,0,0,0,40$ — perfectly calm except one 40% crash. Fund B's: $18,22,25,30,28,24,20,19,17,40$ — a rough, choppy fund that also happens to hit 40% once.

Both report an identical max drawdown of 40%. At $\alpha=70\%$ (worst 3 readings): Fund A's worst three are $40, 0, 0$, averaging $13.3\%$. Fund B's worst three are $40, 30, 28$, averaging $32.7\%$. CDaR correctly reveals Fund B as the far riskier fund to hold day-to-day, information that "max drawdown" alone completely hid since both funds tied on that single statistic.

What this means in practice

CDaR shows up as an objective function in portfolio optimization precisely because, unlike max drawdown, it is convex, mathematically tractable, and can be plugged into a linear program alongside return targets, the same reason expected shortfall replaced VaR as the preferred optimization target. Allocators use it to compare strategies whose max drawdown numbers tie, and to build "worst-case-aware" portfolios that avoid concentrating in assets that share deep, prolonged drawdown periods together.