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The impact of negative interest rates on optimal capital injections

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  • Eisenberg, Julia
  • Krühner, Paul

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.

Suggested Citation

  • Eisenberg, Julia & Krühner, Paul, 2018. "The impact of negative interest rates on optimal capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 1-10.
    • Handle: RePEc:eee:insuma:v:82:y:2018:i:c:p:1-10
    DOI: 10.1016/j.insmatheco.2018.06.004
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    Cited by:

    1. Brinker, Leonie Violetta & Eisenberg, Julia, 2021. "Dividend optimisation: A behaviouristic approach," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 202-224.
    2. Eisenberg, Julia & Fabrykowski, Lukas & Schmeck, Maren Diane, 2021. "Optimal Surplus-dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Center for Mathematical Economics Working Papers 648, Center for Mathematical Economics, Bielefeld University.
    3. Eisenberg, Julia & Krühner, Paul, 2022. "On Itô’s formula for semimartingales with jumps and non-C2 functions," Statistics & Probability Letters, Elsevier, vol. 184(C).
    4. Julia Eisenberg & Lukas Fabrykowski & Maren Diane Schmeck, 2021. "Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Risks, MDPI, vol. 9(4), pages 1-25, April.

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    More about this item

    Keywords

    Negative interest rate; Capital injections; Markov-switching; Optimal stochastic control; Hamilton–Jacobi–Bellman equation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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