On This Page
- What is the Ulcer Index?
- How is the Ulcer Index calculated?
- Ulcer Index example: same drawdown, different paths
- Reading the expanding Ulcer Index chart
- Ulcer Index vs standard deviation and the Martin ratio
- Comparing positions and portfolios with the Ulcer Index
- Limitations of the Ulcer Index
- Bottom line
- FAQ
Key Takeaways
- The Ulcer Index is the root mean square of percentage drawdowns across all observations: one number for how deep and how long a position stays underwater.
- Squaring gives deeper declines more weight, and every period below the peak adds to the index.
- Peter Martin devised the measure in 1987 and published it with Byron McCann in 1989, naming it for the stress that sustained declines create in a long investor.
- Standard deviation measures return dispersion. The Ulcer Index measures distance below the running peak, including periods that gain while remaining below it.
- The Martin ratio, or Ulcer Performance Index, divides excess return by the Ulcer Index: return earned per unit of time spent underwater.
- Two paths with the same maximum drawdown can carry very different Ulcer Index values, because a V-shaped recovery spends far fewer periods underwater than a grinding one.
What is the Ulcer Index?
The Ulcer Index is a downside risk measure equal to the root mean square of percentage drawdowns from a running high. Because it squares every drawdown and counts every observation, over a fixed number of observations it rises with both the depth of declines and the time spent below the peak. Peter Martin devised the measure in 1987.
The index is zero at a sequence of running-high observations, including ties. Positive returns that remain below the peak still contribute nonzero drawdowns.
How is the Ulcer Index calculated?
At each observation, include the current value when updating the running high, calculate the percentage drawdown, square it and add it to the sample. The index is the square root of the mean squared drawdown across all observations, in percentage points.
Formula
Ulcer Index = √( sum of Di² / N ), where Di = 100 × (Pi − Hi) / Hi and Hi is the highest close through period i; Ulcer Performance Index (Martin ratio) = (total return − risk-free return) / Ulcer Index- Pi is the closing price in period i and Hi is the highest close observed from the start of the window through period i, so Hi never decreases.
- Di is zero at the running high, including an unchanged value equal to that high, and negative below it. Squaring removes the sign.
- N counts all observations, so every period below the peak adds to the index.
- The result is expressed in percentage points, so an Ulcer Index of 6.18 means the root mean square distance below the running high was 6.18 percent.
- This example divides total portfolio return minus the matching risk-free holding-period return by the Ulcer Index; Sharpe uses the arithmetic mean return instead.
- Compare two readings on the same data interval, window length, and market period.
Ulcer Index example: same drawdown, different paths
Two paths have 14 weekly closing observations spanning 13 weekly return intervals. Both start at $100, end at $102 and reach a 12% maximum drawdown. A is 100, 98, 95, 90, 88, 90, 94, 96, 92, 94, 97, 99, 100, 102. B is 100, 100, 100, 100, 88, 100, 100, 100, 100, 100, 100, 100, 100, 102.
- Path A percentage drawdowns from the running high of $100: 0, −2, −5, −10, −12, −10, −6, −4, −8, −6, −3, −1, 0, 0
- Path A squared drawdowns sum to 0 + 4 + 25 + 100 + 144 + 100 + 36 + 16 + 64 + 36 + 9 + 1 + 0 + 0 = 535, so the Ulcer Index is √(535 / 14) = √(38.21) = 6.18
- Path B has a single nonzero drawdown of −12, so its squared sum is 144 and its Ulcer Index is √(144 / 14) = √(10.29) = 3.21
- Sample standard deviation of the weekly returns (n minus 1 divisor): Path A 3.12 percent, Path B 5.27 percent, which annualize to 22.5 percent and 38.0 percent at the square root of 52
- With a 1% risk-free holding-period return across the same 13 intervals, the total-return Martin ratio is (2 − 1) ÷ 6.18 = 0.16 for A and (2 − 1) ÷ 3.21 = 0.31 for B.
Both paths gain 2% and have a 12% maximum drawdown. A has 11 of 14 observations below the running high and an Ulcer Index of 6.18; B has one and an index of 3.21. The 13 weekly returns have sample standard deviations of 3.12% and 5.27%, respectively.
Both paths start at $100, reach $88 and end at $102. They have the same 2% total return and 12% maximum drawdown.
Path A’s squared drawdowns sum to 535. Across 14 observations, its index is √(535 ÷ 14) = 6.18. Path B has one −12% observation, giving √(144 ÷ 14) = 3.21.
A has 11 observations below its peak, while B has one. Their different recovery paths produce different index values despite the same maximum drawdown.
Reading the expanding Ulcer Index chart
The plotted index uses every observation from the start through the current point. It peaks at 7.88 on observation six, then falls on observation seven even though the price is still below its peak.
That happens because the next squared drawdown is smaller than the current mean squared drawdown. The final new high contributes zero while the earlier drawdowns remain in the calculation.
Ulcer Index vs standard deviation and the Martin ratio
Fourteen weekly closes create 13 weekly returns. Their sample standard deviations are 3.12% for A and 5.27% for B. The conventional √52 scaling gives 22.5% and 38.0%, and the order flips: standard deviation calls B the riskier path, while the Ulcer Index shows B is the easier one to hold.
For the total-return Martin ratio shown here, use a 1% risk-free return over those same 13 intervals. The 1-percentage-point excess return divided by each Ulcer Index gives 0.16 for A and 0.31 for B, so B earned nearly twice the return per unit of drawdown.
Comparing positions and portfolios with the Ulcer Index
Use the same calendar interval and value-series definition across candidates. Pair the Ulcer Index with return, maximum drawdown and elapsed recovery time.
Limitations of the Ulcer Index
- Compare readings on one basis. Use the same data interval, window length and market period; a new start date or a rolling window changes the number.
- Pair it with return. The index measures distance below the running high and ignores gains above the peak, so read it with total return or fold both into the Martin ratio.
- Use a rolling window to track recent risk. An expanding window keeps earlier drawdowns after a full recovery: Path A ends at a new high with an index of 6.18.
- Use the index to see the grind that maximum drawdown hides. Both paths fall 12%, yet A’s 11 periods underwater score 6.18 against B’s 3.21.
- Plan the next drawdown in the trade itself. The index scores the path so far; the stop and the position size set how deep the next decline can cut.
Bottom line
The Ulcer Index condenses how deep and how long a value path stays below its high into one percentage figure. Compare it only across series with the same interval and window, and read it with total return, maximum drawdown and recovery time, or combine it with return in the Martin ratio.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Peter Martin: The Ulcer Index
- Financial Analysts Journal: Maximum Drawdown as a Predictor of Mutual Fund Performance and Flows
- Chekhlov, Uryasev, and Zabarankin: Drawdown Measure in Portfolio Optimization
How entries are sourced, calculated and updated: editorial methodology.