[Paper Review] A Composite Risk Measure Framework for Decision Making under Uncertainty

Pengyu Qian, Zizhuo Wang+1 more|arXiv (Cornell University)|Jan 6, 2015

Risk and Portfolio Optimization35 references3 citations

TL;DR

This paper proposes a composite risk measure (CRM) framework that unifies stochastic programming, robust optimization, and distributionally robust models by minimizing a two-layer risk measure: the inner measure captures decision risk under a known distribution, while the outer measure quantifies risk from distributional uncertainty using Bayesian posterior distributions. The framework yields less conservative solutions with probabilistic guarantees, validated through portfolio selection experiments showing improved risk-adjusted returns.

ABSTRACT

In this paper, we present a unified framework for decision making under uncertainty. Our framework is based on the composite of two risk measures, where the inner risk measure accounts for the risk of decision given the exact distribution of uncertain model parameters, and the outer risk measure quantifies the risk that occurs when estimating the parameters of distribution. We show that the model is tractable under mild conditions. The framework is a generalization of several existing models, including stochastic programming, robust optimization, distributionally robust optimization, etc. Using this framework, we study a few new models which imply probabilistic guarantees for solutions and yield less conservative results comparing to traditional models. Numerical experiments are performed on portfolio selection problems to demonstrate the strength of our models.

Motivation & Objective

• To address the lack of a unified framework that integrates stochastic programming, robust optimization, distributionally robust optimization, and worst-case risk models.
• To fill the gap in existing literature by applying risk measures not only to the objective function but also to the expectation functional under distributional uncertainty.
• To incorporate Bayesian methods for modeling distributional uncertainty in decision-making under uncertainty.
• To develop new, practically meaningful models within the CRM framework that offer probabilistic guarantees and reduced conservatism.

Proposed method

• The framework minimizes a composite of two risk measures: an inner risk measure (e.g., CVaR or VaR) applied to the objective function given a known distribution, and an outer risk measure applied to the distributional uncertainty.
• The outer risk measure uses Bayesian posterior distributions over model parameters to quantify the risk of estimation error, enabling a principled treatment of distributional uncertainty.
• The model is shown to be convex under mild conditions when both inner and outer risk measures are convex, ensuring tractable optimization.
• The framework generalizes existing models: stochastic programming (outer risk = expectation), robust optimization (outer risk = worst-case), and distributionally robust optimization (outer risk = worst-case expectation over ambiguity set).
• New models are constructed, including VaR-Expectation, CVaR-Expectation, and CVaR-CVaR, which balance risk and return with probabilistic guarantees.
• Numerical experiments use historical market data and dynamic portfolio rebalancing over 300 days, comparing CRM models against DRO, worst-case VaR, and single-stock benchmarks.

Experimental results

Research questions

• RQ1 Can a unified framework be developed that subsumes stochastic programming, robust optimization, and distributionally robust optimization under a single risk-based structure?
• RQ2 How can risk measures be applied not only to the objective function but also to the expectation functional under distributional uncertainty?
• RQ3 What is the impact of using Bayesian posterior distributions for modeling distributional uncertainty in decision-making frameworks?
• RQ4 Do CRM-based models yield less conservative solutions while maintaining the same level of probabilistic guarantees compared to traditional models?
• RQ5 How do CRM-based portfolio models perform in real-world dynamic trading scenarios compared to existing benchmarks?

Key findings

• The CRM framework is convex under mild conditions when both inner and outer risk measures are convex, ensuring tractable optimization.
• The VaR-Expectation and CVaR-Expectation models achieve higher average daily returns (0.096%) than DRO (0.088%) and worst-case VaR (0.087%) models.
• The CVaR-CVaR model achieves the lowest volatility (1.17×10⁻²) among all models, outperforming the single-stock benchmark (2.17×10⁻²).
• Computation times remain reasonable even for large portfolios: the CVaR-CVaR model for n=50 stocks averages 29.33 seconds per solve with N=5000 samples.
• The optimal solution converges as sample size increases, indicating stability and robustness of the CRM framework.
• Numerical results show that CRM models provide less conservative solutions than traditional models while maintaining equivalent probabilistic guarantees.