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Robust Optimal Portfolio in a Mixture Setting with Partial Ambiguity

N. D. Shyamalkumar, Tianrun Wang

TL;DR

This paper addresses two robust portfolio optimization problems with partial ambiguity, where the loss function involves either variance or conditional value-at-risk (CVaR) and uses a projected subgradient descent algorithm to solve the optimization problems.

Abstract

Managing insurance and financial risk when data is limited is a key task in the insurance industry. In this paper, we focus on cases where the risk distribution is modeled as a mixture with some components estimable to high precision or known, and others, along with their weights, are not. Our paper addresses two robust portfolio optimization problems with partial ambiguity, where the loss function involves either variance or conditional value-at-risk (CVaR). We use a projected subgradient descent algorithm to solve the optimization problems. The problem reduces to a convex-nonconcave minimax problem. We show that, while the general problem converges at an $O(1/\sqrt{k})$ rate, where $k$ denotes the number of iterations, exponential convergence is possible in some cases. Lastly, we provide numerical examples to show the effectiveness of our approach and the attainment of a geometric convergence rate. This work aims to provide more effective solutions for actuarial decision-making under model uncertainty.

Robust Optimal Portfolio in a Mixture Setting with Partial Ambiguity

TL;DR

This paper addresses two robust portfolio optimization problems with partial ambiguity, where the loss function involves either variance or conditional value-at-risk (CVaR) and uses a projected subgradient descent algorithm to solve the optimization problems.

Abstract

Managing insurance and financial risk when data is limited is a key task in the insurance industry. In this paper, we focus on cases where the risk distribution is modeled as a mixture with some components estimable to high precision or known, and others, along with their weights, are not. Our paper addresses two robust portfolio optimization problems with partial ambiguity, where the loss function involves either variance or conditional value-at-risk (CVaR). We use a projected subgradient descent algorithm to solve the optimization problems. The problem reduces to a convex-nonconcave minimax problem. We show that, while the general problem converges at an rate, where denotes the number of iterations, exponential convergence is possible in some cases. Lastly, we provide numerical examples to show the effectiveness of our approach and the attainment of a geometric convergence rate. This work aims to provide more effective solutions for actuarial decision-making under model uncertainty.
Paper Structure (10 sections, 14 theorems, 86 equations, 2 algorithms)

This paper contains 10 sections, 14 theorems, 86 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mixture for Mean-Var Disutility function
  3. Proofs of Results
  4. Proof of Theorem \ref{['minimax']}
  5. Proof of Proposition \ref{['subgradientJ']}
  6. Proof of Lemma \ref{['propertieshJ']}
  7. Proof of Theorem \ref{['fastConvergeVar']}
  8. Proof of Theorem \ref{['oneOverD']}
  9. Mean-CVaR Disutility function
  10. Proofs of Results

Key Result

Theorem 1

The optimization problem in eq:main_problem is equivalent to the following min-max problem: where $I$ is a compact interval not dependent on $\mathbf{x}$. ∎

Theorems & Definitions (24)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 14 more