Folio Lab/Documentation

Minimum First Lower Partial Moment

Minimise the first lower partial moment (FLPM): the average amount by which daily portfolio returns fall short of the daily risk-free rate. Days above the threshold do not count, so only the downside is penalised.

Overview

Lower partial moments come from Bawa (1975) and Fishburn (1977). The first moment is the expected shortfall below a minimum acceptable return. It is the denominator of the Omega ratio, so a minimum-FLPM portfolio is the one with the smallest Omega denominator.

FolioLab sets the threshold to the run's risk-free rate converted to a daily rate, the same threshold the Omega ratio in the results uses. skfolio's own default threshold is the sample mean, which measures something different, so FolioLab passes the risk-free rate explicitly.

FolioLab implements this method with skfolio's mean-risk estimator, usingRiskMeasure.FIRST_LOWER_PARTIAL_MOMENT 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

First lower partial moment

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

Linear programme

r_f is the daily risk-free rate of the run, from the same conversion the metrics use.

FLPM vs semivariance

Both penalise only the downside. FLPM counts a shortfall by its size and semivariance by its square, so semivariance reacts more strongly to the single worst days.

Because the threshold is the risk-free rate, a portfolio that only beats cash by a small margin each day still has a low FLPM.

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

  • Downside only: Gains above the risk-free rate are not penalised.
  • Consistent with Omega: Uses the same threshold as the Omega ratio in the results.
  • Linear: Solved as a linear programme.

Limitations

  • Threshold choice: A different minimum acceptable return gives a different portfolio.
  • Mean of shortfalls: Does not distinguish many small shortfalls from one large one.
  • History-bound: The measure is the in-sample FLPM of the run window.

References

  • Bawa, V. S. (1975). "Optimal Rules for Ordering Uncertain Prospects." Journal of Financial Economics, 2(1), 95-121.
  • Fishburn, P. C. (1977). "Mean-Risk Analysis with Risk Associated with Below-Target Returns." American Economic Review, 67(2), 116-126.
  • Keating, C., & Shadwick, W. F. (2002). "A Universal Performance Measure." Journal of Performance Measurement, 6(3), 59-84.
  • skfolio documentation - skfolio.RiskMeasure.FIRST_LOWER_PARTIAL_MOMENT.

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