Folio Lab/Documentation

Minimum Ulcer Index

Minimise the ulcer index: the root mean square of the portfolio's drawdowns over the run window. Squaring makes deep drawdowns cost more than shallow ones, and the mean makes long ones cost more than short ones.

Overview

Peter Martin introduced the ulcer index in 1987 (Martin and McCann, 1989) as a measure of the stress of holding an investment. It sits between average drawdown, which weighs every drawdown linearly, and maximum drawdown, which keeps only the deepest.

The square root makes the problem a second-order cone programme rather than a linear one. FolioLab solves it on uncompounded cumulative returns, the same convention as the other two drawdown methods.

FolioLab implements this method with skfolio's mean-risk estimator, usingRiskMeasure.ULCER_INDEX with the objective set to minimise risk. The problem is a second-order cone programme, solved by CVXPY with CLARABEL first, then SCS, then ECOS. Per-name weight bounds, asset exclusions, and group sum limits such as sector caps pass to the estimator and enter the same programme.

Mathematical Formulation

Ulcer index

is the vector of daily asset returns on day, and is the number of days in the run window.

Second-order cone programme

The norm of the drawdown vector is a second-order cone, solved by CLARABEL first, then SCS, then ECOS.

Ulcer index vs average drawdown

Both use the whole drawdown path. The ulcer index squares each drawdown first, so it penalises depth more than average drawdown does.

The ulcer index is the denominator of the Martin ratio, as the standard deviation is of the Sharpe ratio.

What the result is, and is not

The weights minimise the measure on the returns of the run window, and only on them. They are not a forecast that the measure will be lowest in the future. Like every method on FolioLab, this one is not validated out of sample, and no claim is made that it performs better than any other method.

Before release, FolioLab compared these weights with the weights riskfolio-lib 7.0.1 produces for the same measure on the same returns.

Advantages & Limitations

Advantages

  • Depth and duration: Penalises both how far and how long the portfolio is under water.
  • Whole path: Uses every day of the drawdown path.
  • Convex: Solved as a second-order cone programme.

Limitations

  • Not linear: A cone programme, slower than the linear drawdown methods.
  • Uncompounded: Differs slightly from a compounded ulcer index.
  • History-bound: The measure is the in-sample value of the run window.

References

  • Martin, P. G., & McCann, B. B. (1989). The Investor's Guide to Fidelity Funds. John Wiley & Sons.
  • Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown Measure in Portfolio Optimization." International Journal of Theoretical and Applied Finance, 8(1), 13-58.
  • skfolio documentation - skfolio.RiskMeasure.ULCER_INDEX.

Not investment advice. Past performance is not indicative of future results.