Worst-case values of target semi-variances with applications to robust portfolio selection

TL;DR

This paper derives closed-form solutions for worst-case target semi-variances under uncertainty, extending previous work, and applies these results to develop robust portfolio selection methods tested on real financial data.

Contribution

It provides new closed-form expressions for worst-case target semi-variances under mean-variance uncertainty and introduces robust portfolio optimization techniques based on these measures.

Findings

Closed-form solutions for worst-case target semi-variances derived.

Robust portfolio selection methods outperform existing models in numerical experiments.

New insights into downside risk management under distributional uncertainty.

Abstract

The expected regret and target semi-variance are two of the most important risk measures for downside risk. When the distribution of a loss is uncertain, and only partial information of the loss is known, their worst-case values play important roles in robust risk management for finance, insurance, and many other fields. Jagannathan (1977) derived the worst-case expected regrets when only the mean and variance of a loss are known and the loss is arbitrary, symmetric, or non-negative. While Chen et al. (2011) obtained the worst-case target semi-variances under similar conditions but focusing on arbitrary losses. In this paper, we first complement the study of Chen et al. (2011) on the worst-case target semi-variances and derive the closed-form expressions for the worst-case target semi-variance when only the mean and variance of a loss are known and the loss is symmetric or non-negative. Then, we investigate worst-case target semi-variances over uncertainty sets that represent undesirable scenarios faced by an investors. Our methods for deriving these worst-case values are different from those used in Jagannathan (1977) and Chen et al. (2011). As applications of the results derived in this paper, we propose robust portfolio selection methods that minimize the worst-case target semi-variance of a portfolio loss over different uncertainty sets. To explore the insights of our robust portfolio selection methods, we conduct numerical experiments with real financial data and compare our portfolio selection methods with several existing portfolio selection models related to the models proposed in this paper.