Optimal investment with insurable background risk and nonlinear portfolio allocation frictions

Paper: arXiv 2303.04236

Authors: Hugo E. Ramirez, Rafael Serrano

Abstract

We study investment and insurance demand decisions for an agent in a theoretical continuous-time expected utility maximization model that combines risky assets with an (exogenous) insurable background risk. This risk takes the form of a jump-diffusion process with negative jumps in the return rate of the (self-financed) wealth. The main distinctive feature of our model is that the agent’s decision on portfolio choice and insurance demand causes nonlinear friction in the dynamics of the wealth process. We use the dynamic programming approach to find optimality conditions under which the agent assumes the insurable risk entirely, or partially, or purchases total insurance against it. In particular, we consider differential and piece-wise linear portfolio allocation frictions, with differential borrowing and lending rates as our most emblematic example. Finally, we present a mutual-fund separation result and illustrate our results with several numerical examples when the adverse jump risk has Beta distribution.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using stochastic optimal control, Hamilton-Jacobi-Bellman equations, and jump-diffusion processes to tackle portfolio selection with frictions. However, as a purely theoretical work, it lacks real-world empirical backtesting, relying instead on stylized numerical simulations using a Beta distribution.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal investment & insurance <br>under nonlinear frictions & jump risk"] --> B["Inputs: Wealth dynamics with negative jumps,<br>borrowing/lending rates & Beta-distributed risk"]
    B --> C["Apply Dynamic Programming & Hamilton-Jacobi-Bellman (HJB) Equations"]
    C --> D["Analyze Friction Models:<br>Differential & Piece-wise Linear Rates"]
    D --> E["Derive Optimality Conditions for Insurance:<br>None, Partial, or Full Coverage"]
    E --> F["Key Outcomes: Mutual-fund separation result<br>& numerical insights on optimal allocation"]