Folio Lab/Documentation

Minimum Worst Realization

Minimise the worst single-day loss of the portfolio over the run window. This is Young's minimax rule: choose the weights whose worst day in the sample was the least bad.

Overview

Young (1998) proposed the minimax rule as a linear-programming alternative to mean-variance. It is the limit of CVaR as the confidence level approaches 1: the average of the worst fraction of days becomes the single worst day.

The result depends on one observation per candidate portfolio, so it is the most sensitive of the tail measures to the exact sample. A crash day in the window usually decides the allocation.

FolioLab implements this method with skfolio's mean-risk estimator, usingRiskMeasure.WORST_REALIZATION with the objective set to minimise risk. The problem is a linear programme, solved by CVXPY with CLARABEL first, then HiGHS, then SCS. 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

Worst realization

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

Linear programme

One variable carries the worst loss, with one inequality per day.

Worst realization vs CVaR

CVaR at 95% averages the worst 5% of days. Worst realization keeps only the single worst day, so it ignores every other loss in the window.

Two portfolios with the same worst day are equal under this measure, even when one of them loses more on every other bad day.

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

  • Simple: The objective is one number a reader can check in the return series.
  • Coherent: A coherent risk measure, the limit of CVaR.
  • Linear: Solved as a linear programme.

Limitations

  • One observation: The allocation depends on the single worst day in the sample.
  • Many solutions: Several weight vectors can share the same worst day; the solver returns one of them.
  • History-bound: A worse day outside the window is not considered.

References

  • Young, M. R. (1998). "A Minimax Portfolio Selection Rule with Linear Programming Solution." Management Science, 44(5), 673-683.
  • Cajas, D. (2021). Riskfolio-Lib documentation - risk measures.
  • skfolio documentation - skfolio.RiskMeasure.WORST_REALIZATION.

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