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The Ulcer Index — Quantifying Stress

Can you answer 'Will this investment let me sleep at night?' with a number? The Ulcer Index is a risk metric that converts the pain of your account being underwater into a number.

Why is it named after an 'ulcer'?

'Ulcer' refers to a stomach ulcer. The name is blunt, isn't it? The heartburn we feel when our account slides from a peak and can't recover for a long time—it puts that stress right into the metric's name.

The Ulcer Index was devised by Peter Martin in 1987 and first introduced to the world in the 1989 book 'The Investor's Guide to Fidelity Funds,' co-written with Byron McCann.

Since it was originally made to evaluate mutual funds, it has a clear viewpoint from the start: 'loss = risk.' Rising is a good thing, so it isn't risk; what truly hurts is falling and being unable to climb back for a long time.

How is it different from standard deviation?

The widely used risk metric standard deviation (volatility) treats upward and downward movements equally as risk. But think about it—isn't that odd? My assets suddenly rising 20% isn't stress but joy, yet standard deviation calculates that as 'high risk' too.

The Ulcer Index views this point differently. It doesn't count the rises at all, and measures only the drawdown from the peak. That's why it's called a 'downside risk' metric.

There's one more key difference. The Ulcer Index reflects not only the 'depth' of the drawdown but also 'how long you were underwater.' Between a -30% that recovered in a month and a -20% that stayed underwater for three years, the latter can be more painful. Standard deviation struggles to capture this pain of time, but the Ulcer Index captures it.

The biggest difference is that standard deviation treats rises and falls symmetrically, while the Ulcer Index measures only declines (drawdowns) asymmetrically.

The formula is simpler than it sounds

The name is scary, but the calculation idea takes just three steps.

① At each point, find 'what % it fell from the highest peak so far.' (0% if it's at a peak.) ② Square each of those drawdown percentages. ③ Average the squared values, then take the square root.

Written as a formula: UI = √( (drawdown₁² + drawdown₂² + … + drawdownₙ²) / N )

The point is the squaring. Squaring doesn't enlarge small drawdowns much, but it enlarges big drawdowns sharply. So it assigns a far bigger penalty to 'deep, long-lasting large losses.' That's why an asset that occasionally collapses hard has a much higher Ulcer Index than one that wobbles gently. The closer the value is to 0, the more at ease you were; the larger it is, the more it churned your stomach.

The UPI, which even measures 'return per unit of pain'

Peter Martin went one step further. Having measured risk with the Ulcer Index, he used it to also calculate 'return relative to risk.' That's the Ulcer Performance Index (UPI), also known as the Martin Ratio.

The formula is: UPI = (return − risk-free rate) / Ulcer Index

If the Sharpe ratio uses standard deviation in the denominator, the UPI puts the Ulcer Index in that spot. So it shows 'how much excess return you earned per unit of the heartburn you endured.' The risk-free rate is usually substituted with something like the 3-month Treasury rate. The higher the value, the more you earned for the same pain, so it's viewed favorably.

That said, be sure to remember that both the Ulcer Index and the UPI are numbers that summarize the 'past.' No one knows how much you'll hurt or how much you'll earn in the future. This isn't a prophecy but a report card of the road already traveled.

よくある質問

Q. If the Ulcer Index value is low, is it automatically a good investment?

A low value means it was less mentally taxing in the past, not that returns were high. An asset that neither rose nor fell at all would have a very low Ulcer Index. So don't look only at risk (the ulcer)—look at returns together. What shows both at once is the UPI (Martin Ratio).

Q. Is it the same as maximum drawdown (MDD)?

Similar, but different. Maximum drawdown tells you only the depth of the single deepest valley. The Ulcer Index gathers all the drawdowns and reflects, on top of how deep they were, 'how long you were underwater.' So even with the same MDD, the side with a slower recovery has a higher Ulcer Index.

Q. Can I pick safe stocks by looking only at this metric?

No. The Ulcer Index is just a number summarizing past data—it's not a tool that predicts the future or recommends particular stocks. Here too, we don't tell you to buy or sell. But as a reference for comparing 'how much this asset historically churned your stomach,' it's very useful.