Determine an optimal downside-risk weight

Determine the optimal value of the downside-risk weighting parameter $\kappa$ according to a well-defined decision criterion that balances cumulative portfolio wealth against worst-case portfolio CVaR across the mean-worst-case expected downside risk, mean-worst-case CVaR, and worst-case CVaR of balanced portfolio loss models.

Background

The parameter κ\kappa weights the portfolio’s downside risk term (ω⊤X−d)+(\boldsymbol{\omega}^{\top}\boldsymbol{X}-d)_+. The paper’s numerical experiments show that changing κ\kappa produces different and sometimes opposing effects on cumulative portfolio wealth and worst-case portfolio CVaR, depending on the portfolio model.

The authors therefore do not identify a universally optimal value of κ\kappa. They note that the appropriate choice depends on the investor’s risk preference and requires an explicit decision criterion capable of resolving the trade-off between return and risk.

References

The problem of identifying the optimal $\kappa$ according to a well-defined decision criterion remains an important question for future research.

— Conditional value-at-risk under reward-penalty mechanism with applications to robust portfolio management  (2610.09246 - Cai et al., 7 Oct 2026) in Section 6, “Numerical illustrations of the portfolio selection models using real data,” discussion following Figure 2