Faculty Publications

Optimal Constrained Investment In The Cramer-Lundberg Model

Document Type

Article

Keywords

constrained investment, Cramer-Lundberg model, HJB equation, ruin probability, stochastic control

Journal/Book/Conference Title

Scandinavian Actuarial Journal

Issue

5

First Page

383

Last Page

404

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 short-selling, 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. For example, 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, or with the case of no short-selling and no borrowing studied in Azcue and Muler. © 2014 Copyright Taylor & Francis Group, LLC.

Department

Department of Mathematics

Original Publication Date

1-1-2014

DOI of published version

10.1080/03461238.2012.699001

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