The Impact of Negative Interest Rates on Optimal Capital Injections

TL;DR

This paper explores how negative interest rates influence the optimal timing and amount of capital injections for an insurance company with a surplus modeled by Brownian motion, revealing different strategies under positive and negative rates.

Contribution

It introduces a novel model incorporating Markov-switching interest rates and derives the optimal capital injection strategy, proving it to be of barrier type and providing an algorithm for its computation.

Findings

Optimal to inject capital only at insolvency when rates are positive.

Possibility of maintaining a positive reserve during negative interest periods.

The value function satisfies the Hamilton--Jacobi--Bellman equation.

Abstract

In the present paper, we investigate the optimal capital injection behaviour of an insurance company if the interest rate is allowed to become negative. The surplus process of the considered insurance entity is assumed to follow a Brownian motion with drift. The changes in the interest rate are described via a Markov-switching process. It turns out that in times with a positive rate, it is optimal to inject capital only if the company becomes insolvent. However, if the rate is negative it might be optimal to hold a strictly positive reserve. We establish an algorithm for finding the value function and the optimal strategy, which is proved to be of barrier type. Using the iteration argument, we show that the value function solves the Hamilton--Jacobi--Bellman equation, corresponding to the problem.