Company Value with Ruin Constraint in Lundberg Models

Christian Hipp

doi:10.3390/risks6030073

Abstract

In this note we study the problem of company values with a ruin constraint in classical continuous time Lundberg models. For this, we adapt the methods and results for discrete de Finetti models to time and state continuous Lundberg models. The policy improvement method works also in continuous models, but it is slow and needs discretization. Better results can be obtained faster using the barrier method for discrete models which can be adjusted for Lundberg models. In this method, dividend strategies are considered which are based on barrier sequences. In our continuous state model, optimal barriers can be computed with the Lagrange method leading to a backward recursion scheme. The resulting dividend strategies will not always be optimal: in the case without ruin constraint, there are examples in which band strategies are superior. We also develop equations for optimal control of dynamic reinsurance to maximize the company value under a ruin constraint. These identify the optimal reinsurance strategy in no action regions and allow for an interactive computation of the value function. We apply the methods in a numerical example with exponential claims.

Highlights

We consider a classical Lundberg model for the surplus S(t) of an insurer at time t: S(t) = s + ct − X1 − ... − X N (t), t ≥ 0, (1)with initial surplus s ≥ 0, premium rate c and claim sizes X, X1, X2, ... which are iid
For the construction of optimal dividend strategies the barrier method has been used for time and state discrete models in Hipp (2018)
The function V ( x ) = V ( x, ψn ( x )) satisfies Equation (30) above. This implies that the optimal reinsurance strategy for maximizing the company value with ruin constraint equals a∗ ( x ) for 0 ≤ x < Bn, i.e., the same strategy as without ruin constraint, as long as we are in the no action region

Summary

Introduction

Dividend optimization with a ruin related constraint has been considered in several earlier papers: Albrecher and Thonhauser (2007), Hernandez et al (2017) as well as Junca et al (2018) deal with the time value of ruin In these problems both objective functions are discounted which allows for explicit solutions. The optimal dividend problem is often solved by a barrier strategy, i.e., there exists a constant B∗ ≥ 0 such that. In our numerical example below (exponential claims with mean 1, premium rate 2 and discount rate 0.03) the dividend value U (s, a, b) is considerably smaller than our numerical values for V (s, α) whenever ψ(s, a, b) ≤ α For this we consider the surplus s = 2 and the allowed ruin probability α = 0.2. We checked our source code with the numerical results in Albrecher et al (2005)

Policy Improvement without Bellman Equation

Barrier Method

Reinsurance Control for Company Value with Ruin Constraint

Numerical Example