The Ulcer Index is a specialized volatility metric developed by Peter Martin and Byron McCann in 1987. Unlike traditional volatility measures like standard deviation that treat upside and downside movements equally, the Ulcer Index focuses exclusively on downside risk by measuring the depth and duration of price declines (drawdowns) from previous highs. The name "Ulcer Index" aptly reflects the anxiety investors experience during these periods of portfolio decline.
This metric is particularly valuable for risk-averse investors who are more concerned with avoiding significant drawdowns than with overall volatility. The Ulcer Index provides a more nuanced view of downside risk by penalizing deeper and longer-lasting drawdowns more heavily, aligning with the psychological reality that investors tend to feel progressively more pain the longer their portfolios remain below previous peak values.
Think of the Ulcer Index as a "pain meter" for investments. While standard deviation measures all price movements (both up and down) equally, the Ulcer Index only cares about how far an investment has fallen from its peak and how long it stays down—the two factors that cause investors the most distress.
Consider two portfolios with the same average returns and standard deviation over a year. Portfolio A experiences a single sharp 15% drop but quickly recovers, while Portfolio B suffers a 15% decline that persists for several months before recovering. Traditional volatility measures would rate these portfolios similarly, but the Ulcer Index would assign a higher "pain score" to Portfolio B because of the extended duration of its drawdown.
Health analogy: Standard deviation is like measuring someone's average body temperature fluctuations throughout the day. The Ulcer Index is more like tracking how far below normal a patient's temperature drops during an illness and how many days they remain sick. It's not just that the temperature is abnormal (volatility), but how severe the illness is and how long it lasts (drawdown depth and duration) that really matters for patient comfort.
The Ulcer Index is calculated by taking the square root of the mean of the squared percentage drawdowns from historical peak values:
where represents the percentage drawdown at time period and is the number of time periods.
For each time period , the percentage drawdown is calculated as:
where:
is the price (or portfolio value) at time period
is the maximum price (or highest portfolio value) up to time period
The key properties of this calculation are:
when price is at a new peak (no drawdown)
is always negative during drawdowns (but the squared value makes it positive in the calculation)
Squaring the drawdowns gives greater weight to larger percentage drops
A related measure is the Ulcer Performance Index (UPI), also known as the Martin Ratio, which is calculated by dividing excess returns by the Ulcer Index:
where is the portfolio return and is the risk-free rate. The UPI is analogous to the Sharpe ratio but uses the Ulcer Index as the risk measure instead of standard deviation.
Our portfolio analyzer calculates the Ulcer Index through the following process:
Price Series Analysis: We track the value of the portfolio over time, identifying the maximum value reached up to each point.
Drawdown Calculation: For each time period, we calculate the percentage decline from the highest peak to the current value.
Squaring Process: We square each percentage drawdown to penalize larger drops more heavily.
Averaging and Square Root: We take the mean of these squared drawdowns and then calculate the square root to produce the final Ulcer Index.
We also provide the Ulcer Performance Index (UPI) to give users a risk-adjusted performance measure that's particularly sensitive to drawdowns. For backtesting purposes, we allow users to view how the Ulcer Index evolves over time, helping identify periods of increased drawdown risk.
Let's calculate the Ulcer Index for a portfolio with the following month-end values over a 10-month period:
$100, $102, $104, $99, $101, $97, $95, $98, $102, $105
Track the highest value achieved up to each time period:
$100, $102, $104, $104, $104, $104, $104, $104, $104, $105
For each month, calculate how far the portfolio is below its previous peak:
Month 1: ($100 - $100)/$100 × 100% = 0% (no drawdown)
Month 2: ($102 - $102)/$102 × 100% = 0% (no drawdown)
Month 3: ($104 - $104)/$104 × 100% = 0% (no drawdown)
Month 4: ($99 - $104)/$104 × 100% = -4.81%
Month 5: ($101 - $104)/$104 × 100% = -2.88%
Month 6: ($97 - $104)/$104 × 100% = -6.73%
Month 7: ($95 - $104)/$104 × 100% = -8.65%
Month 8: ($98 - $104)/$104 × 100% = -5.77%
Month 9: ($102 - $104)/$104 × 100% = -1.92%
Month 10: ($105 - $105)/$105 × 100% = 0% (new peak)
0², 0², 0², 4.81², 2.88², 6.73², 8.65², 5.77², 1.92², 0² = 0, 0, 0, 23.14, 8.29, 45.29, 74.82, 33.29, 3.69, 0
This Ulcer Index of 4.34% represents the severity of drawdowns experienced by the portfolio over the 10-month period. The higher the index, the more severe the combination of drawdown depth and duration.
If the portfolio had an annualized return of 7% during this period and the risk-free rate was 2%, the Ulcer Performance Index (UPI) would be:
A UPI of 1.15 indicates the portfolio generated 1.15 units of excess return per unit of drawdown risk.
The Ulcer Index is particularly valuable in several investment contexts:
Retirement Portfolio Management: Retirees making regular withdrawals are especially vulnerable to sequence-of-returns risk. The Ulcer Index helps identify strategies that minimize drawdown depth and duration, critical for preserving longevity in retirement accounts.
Risk-Averse Client Portfolios: For investors with lower risk tolerance, focusing on minimizing the Ulcer Index rather than standard deviation can better align with their psychological comfort and prevent panic selling during market declines.
Tactical Asset Allocation: The Ulcer Index can be used to evaluate the effectiveness of tactical strategies designed to reduce drawdown risk, providing a more appropriate metric than standard deviation for these approaches.
Manager Evaluation: When comparing investment managers, particularly those with capital preservation mandates, the Ulcer Index and UPI provide more relevant performance metrics than conventional Sharpe ratios.
Portfolio Construction: Including the Ulcer Index as a minimization objective in portfolio optimization can help construct portfolios that specifically target drawdown reduction rather than just overall volatility reduction.
Downside focus: Exclusively measures downside risk, aligning with investors' primary concern of avoiding losses rather than reducing overall volatility.
Duration sensitivity: Captures the length of time investments remain underwater, acknowledging the psychological impact of extended drawdown periods.
Squared penalties: By squaring drawdowns, the index disproportionately penalizes larger drops, reflecting the non-linear nature of risk perception.
Path dependency: Accounts for the specific sequence of returns rather than just their overall distribution, acknowledging the importance of when losses occur.
Intuitive interpretation: Directly measures what causes investors the most pain—deep, prolonged declines from previous peaks.
Lookback limitations: The index depends heavily on the chosen time period; too short a period may miss significant historical drawdowns.
Peak-anchoring bias: Always references the previous peak, potentially overstating risk if that peak was an anomalous high point.
Frequency sensitivity: Results can vary significantly based on the frequency of data sampling (daily vs. weekly vs. monthly).
Lesser adoption: Not as widely recognized or used as standard deviation or VaR, making comparisons across different information sources challenging.
Forward-looking limitations: Like all historical risk measures, the Ulcer Index makes no predictions about future drawdowns and may underestimate risk during regime changes.
While standard deviation measures dispersion of returns around the mean (both positive and negative), the Ulcer Index only considers negative deviations from peak values. Standard deviation treats a 5% gain and a 5% loss as equivalent in risk terms, whereas the Ulcer Index completely ignores positive movements and focuses on the depth and duration of drawdowns.
Maximum drawdown only captures the single worst peak-to-trough decline, while the Ulcer Index incorporates all drawdowns, their depths, and durations. Two portfolios might have identical maximum drawdowns of 20%, but if one recovered quickly while the other languished near the bottom for months, the Ulcer Index would be much higher for the second portfolio.
Martin, P., & McCann, B. (1987). "The Investor's Guide to Fidelity Funds: Winning Strategies for Mutual Fund Investors." John Wiley & Sons.
Martin, P. (2008). "The Ulcer Index: Use it to avoid indigestion when investing." Technical Analysis of Stocks & Commodities Magazine, 26(7), 58.
Magdon-Ismail, M., & Atiya, A. (2004). "Maximum drawdown." Risk Magazine, 17(10), 99-102.
Bacon, C. R. (2013). Practical Risk-Adjusted Performance Measurement. Wiley.
Rollinger, T., & Hoffman, S. (2013). "Sortino ratio: A better measure of risk." Futures Magazine, 1(2), 40-42.