Optimal Constrained Investment in the Cramer-Lundberg model

TL;DR

This paper analyzes an optimal investment strategy for an insurance company modeled by the Cramer-Lundberg process, considering constraints on risky asset investments, to minimize ruin probability, revealing complex policy behaviors influenced by model parameters.

Contribution

It introduces a constrained investment model within the Cramer-Lundberg framework and characterizes the optimal policy and ruin probability using HJB equations, highlighting novel counterintuitive strategies.

Findings

Optimal policies depend critically on model parameters.

Counterintuitive strategies like short-selling high-return stocks can be optimal.

Explicit characterization of ruin probabilities under constraints.

Abstract

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when shortselling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. E.g., for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum (2000), or with the case of no shortselling and no borrowing studied in Azcue and Muler (2009).