Uniform Pessimistic Risk and its Optimal Portfolio

TL;DR

The paper introduces Uniform Pessimistic Risk (UPR), defining $\varrho(Y; \varphi_u)$ as the integral of $\alpha$-risk over $\alpha\in(0,1)$ to capture a continuum of pessimistic outcomes in portfolio optimization. It develops three complementary viewpoints—UPR as a limit of composite quantile risk, as a proper scoring-rule formulation, and as a law-invariant coherent risk with a DRO interpretation—and provides a practical algorithm that minimizes UPR via a spline-parameterized quantile function. The approach is analyzed theoretically for finiteness, consistency, and extensions to Beta distortions, and is validated with real-market data (S&P500, CSI500, KOSPI200), where UPR-based portfolios exhibit superior out-of-sample performance and tail-risk control compared with several benchmarks and DRO variants. The work offers a robust, pessimism-aware alternative to CVaR-based methods, with potential extensions to nonparametric models and broader distortion families for increased flexibility and applicability.

Abstract

The optimal allocation of assets has been widely discussed with the theoretical analysis of risk measures, and pessimism is one of the most attractive approaches beyond the conventional optimal portfolio model. The $α$-risk plays a crucial role in deriving a broad class of pessimistic optimal portfolios. However, estimating an optimal portfolio assessed by a pessimistic risk is still challenging due to the absence of a computationally tractable model. In this study, we propose an integral of $α$-risk called the \textit{uniform pessimistic risk} and the computational algorithm to obtain an optimal portfolio based on the risk. Further, we investigate the theoretical properties of the proposed risk in view of three different approaches: multiple quantile regression, the proper scoring rule, and distributionally robust optimization. Real data analysis of three stock datasets (S\&P500, CSI500, KOSPI200) demonstrates the usefulness of the proposed risk and portfolio model.

Figures (4)

• Figure 1: Derivative functions of distortion functions. $\nu_{0.1}(t)$ and $\phi(t)$ denote the distortion functions of $\alpha$-risk at the level $10\%$ and UPR, respectively.
• Figure 2: Cumulative wealth of the portfolio models.
• Figure 3: (a) Joint PDF of the Clayton copula with $\tau=2/3$. (b) - (d) Empirical quantile functions of portfolio returns with quantile level within the range $(0, 0.1]$.
• Figure 4: Out-of-sample evaluation via quantile function matching in KOSPI200 dataset.

Theorems & Definitions (14)

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