The Ulcer Index

Standard deviation treats a sharp rally and a sharp crash as equally "risky," since both are just deviations from the mean. That's not how an investor actually experiences a portfolio — nobody loses sleep over an unusually good month. What actually causes stress is being underwater: down from a previous peak, and staying down. The Ulcer Index, named for exactly that feeling, measures risk using only the depth and duration of drawdowns, ignoring upside volatility entirely.

Squaring every drawdown, not just the worst one

$$\text{UI} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} D_i^2}, \qquad D_i = \frac{P_i - \max_{j \le i} P_j}{\max_{j \le i} P_j} \times 100 .$$

In words: at every point in time $i$, $D_i$ is the percentage drawdown from the running peak so far — zero if the portfolio is at a new high, negative and growing the longer and deeper it stays underwater. Squaring each $D_i$ before averaging (then taking the square root, the same root-mean-square structure as a standard deviation) means a long, moderate drawdown accumulates just as much penalty as a short, sharp one of similar total depth — because it's summed over every day spent below the peak, not measured once at the bottom.

Worked example 1 — computing it by hand

A five-day equity curve: 100, 110, 105, 108, 95. Running peaks: 100, 110, 110, 110, 110. Drawdowns: day 1, $0\%$ (at its own peak); day 2, $0\%$ (new peak); day 3, $(105-110)/110 = -4.5\%$; day 4, $(108-110)/110 = -1.8\%$; day 5, $(95-110)/110 = -13.6\%$. Squaring: $0, 0, 20.5, 3.3, 185.6$. Mean of squares: $(0+0+20.5+3.3+185.6)/5 = 41.9$. Ulcer Index: $\sqrt{41.9} \approx 6.47$.

Worked example 2 — same max drawdown, different duration

Two portfolios both hit a maximum drawdown of exactly $-10\%$ at some point, so any risk measure based only on the worst single dip would call them identical. Portfolio A recovers within 2 days: drawdowns roughly $0, -5\%, -10\%, -3\%, 0$, giving squared values $0, 25, 100, 9, 0$, mean $26.8$, Ulcer Index $\approx 5.18$. Portfolio B takes 5 days to recover, hovering near its worst level the whole time: drawdowns $0, -6\%, -10\%, -9\%, -7\%, -2\%, 0$, giving squared values $0, 36, 100, 81, 49, 4, 0$, mean $\approx 38.6$, Ulcer Index $\approx 6.21$. The two share the same maximum drawdown but Portfolio B's Ulcer Index is noticeably higher, correctly reflecting the longer stretch spent underwater.

What this means in practice

Managers running strategies where recovery speed matters — retirement drawdown portfolios, tactical allocation funds sold on "smooth ride" positioning — quote the Ulcer Index alongside or instead of standard deviation, because it directly captures the thing clients actually complain about: being stuck below a prior high for a long stretch, not ordinary day-to-day wiggle. It's also commonly paired with return to form the "Ulcer Performance Index," a Sharpe-like ratio using the Ulcer Index instead of standard deviation in the denominator.