On the surplus management of funds with assets and liabilities in presence of solvency requirements

TL;DR

This paper analyzes optimal dividend strategies for a company with assets and liabilities modeled as correlated geometric Brownian motions, considering solvency constraints and capital injections, deriving explicit solutions and optimal barrier strategies.

Contribution

It introduces a novel analysis of dividend strategies under solvency and capital injection scenarios, with explicit barrier strategies based on asset-liability ratios.

Findings

Optimal strategies are of barrier type.

Explicit closed-form value functions are derived.

Optimal barriers depend on asset-liability ratios.

Abstract

In this paper we consider a company whose assets and liabilities evolve according to a correlated bivariate geometric Brownian motion, such as in Gerber and Shiu (2003). We determine what dividend strategy maximises the expected present value of dividends until ruin in two cases: (i) when shareholders won't cover surplus shortfalls and a solvency constraint (as in Paulsen, 2003) is consequently imposed, and (ii) when shareholders are always to fund any capital deficiency with capital (asset) injections. In the latter case, ruin will never occur and the objective is to maximise the difference between dividends and capital injections. Developing and using appropriate verification lemmas, we show that the optimal dividend strategy is, in both cases, of barrier type. Both value functions are derived in closed form. Furthermore, the barrier is defined on the ratio of assets to liabilities, which mimics some of the dividend strategies that can be observed in practice by insurance companies. Existence and uniqueness of the optimal strategies are shown. Results are illustrated.