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Actuarial Control Problems under a Stochastic Interest Rate

Actuarial Control Problems under a Stochastic Interest Rate

Julia Eisenberg (ORCID: 0000-0002-6995-6743)
  • Grant DOI 10.55776/V603
  • Funding program Elise Richter
  • Status ended
  • Start July 1, 2018
  • End December 31, 2022
  • Funding amount € 315,378
  • E-mail

Disciplines

Mathematics (100%)

Keywords

    Actuarial Mathematics, Optimal Stochastic Control, Hamilton-Jacobi-Bellman equation, Dividends, Capital Injections, Short Rate Models

Abstract Final report

Following the global financial crisis, short interest rates fell sharply in many countries, resulting even in negative rates in some European countries. While working with stochastic interest rates is the usual practice in financial mathematics, in non-life insurance mathematics it is a fairly new field. Indeed, the optimal control problems in insurance mainly assume a positive constant interest rate. One may believe that the changes in the interest rates mainly affect the life insurance branch, but economic and mathematical reports show that also non-life insurances suffer from the low interest rate environment. For instance, in Swiss Res sigma 4/2012, Facing the interest rate challenge, the authors investigate the question how interest rates affect insurers and explain why a rapid rise in or sustained low interest rates can be a challenge. In fact, not all lines of business are affected with the same severity. Under ultra-low nominal interest rates, life insurance companies suffer the most, having difficulties to meet their long-term liabilities, such as pensions or life insurance policies, offered at fixed nominal rates. However, in order to manage sudden rises or drops in interest rates, also the non-life insurers might have to increase their premia or to shorten the dividend payments. In the present project, we concentrate on the actuarial optimisation problems. We model the surplus process of an insurance entity via a Brownian motion with drift describing the random dynamics of the company`s earnings and losses. Additionally, the discounting factor will be given by a stochastic process, i.e. it will depend on the global macroeconomic situation, which cannot be assumed to be deterministic. At first, we consider two different risk measures: expected discounted dividends, and the expected discounted difference of dividends and capital injections (payments which are necessary to keep the wealth of the considered company non-negative). That is, our risk measure is the present expected value of the future cash flow, accounting for the impact of random changes in the interest rate. The interest rate follows an Ornstein-Uhlenbeck process, i.e. we allow for negative rates. This assumption appears more than realistic, considering the fact that the European Central Bank (ECB) cut the fixed rate to zero on the 16th of March 2016. In the first part of the problem, we approximate the underlying Ornstein-Uhlenbeck process by a sequence of random walks, providing more techniques for solving the optimisation problem explicitly. Then, we assume that the company can adjust the dividend/capital injection strategy discretely in time: at random times described by Poisson arrivals. Discretising the strategies will help in the construction of the solution. In the second topic, we pick up the problem of finding the optimal strategy in a time inhomogeneous setting. Usually, in such settings one fails to characterise the value function as a smooth solution to the corresponding HJB equation. A possible way out is to find a distance between a return function of an arbitrary strategy and the value function. This will, of course, help to investigate the goodness of a non-optimal control strategy in any control problem based on an Ito-process. Finally, we model the discounting factor as an exponential of an affine process. We start with a Cox-Ingersoll-Ross (CIR) process and choose the parameters in such a way that the CIR process can attain arbitrary positive values and converges to infinity in infinite time. Here, we target to maximise the value of expected discounted dividends.

In this project, we consider three relevant actuarial topics: Optimisation problems in non -life insurance with non-deterministic interest rates; Optimisation problems in life insurance in the low-interest phase and their after-effects; The impact of COVID-19 on the insurance industry. For the ranking of insurance companies, the choice of a suitable risk measure is crucial. And it seems only natural to include the company`s cash flow into the valuation. However, the risk measures based on the cash flow have to take into account the payment times. After all, money today rarely has the same value as the same amount of money in the future. The changes in the time value are described by an interest rate. It is nave to assume that the interest rate will remain unchanged over the years. Therefore, it makes sense to model the interest rate by a stochastic process. Optimization problems with different risk measures, infinite or finite time horizon and a stochastic interest rate are the subject of the first part of the present project. In such models one considers objective functions, quantifying the risks of an insurance portfolio with a possibility of dividend payments or reinsurance. The difference in the technical handling of these problems compared to the models with a constant interest rate is that the objective function becomes multidimensional, which complicates the solution via the Hamilton -Jacobi- Bellman approach. Nevertheless, the problems considered in this project have been solved, either explicitly or by recursion methods.\smallskip In the second part of the project, the interest rates play only an implicit role. The problems in the life insurance industry, severely affected by longevity and falling birth rates, are further amplified in periods of low interest rates and their aftermath. Can the expected, almost certain, losses be mitigated by timely investments? How to replace guarantees in annuity products? What are the reput ational risks of a pension insurer who doesn`t offer guarantees and wants to lower pensions due to a bad economic situation? All these questions are modelled mathematically, and answered within the created framework. The last part of the project emerged during the first phase of COVID-19, in March 2020. Back then, it quickly became clear that the insurance industry could not cope with the tsunami of claims caused by the lockdown situation. Now, insurers are convinced: the next pandemic will come for sure. Is pandemic insurance possible? What are the legal and actuarial consequences of COVID-19, how to model the pandemic costs? These and other questions have been addressed in an edited volume as well as in a separate article.

Research institution(s)
  • Technische Universität Wien - 100%
International project participants
  • Kais Hamza, Monash University - Australia
  • Hanspeter Schmidli, University of Copenhagen - Denmark
  • Yuliya Mishura, Taras Shevchenko National University of Kyiv - Ukraine

Research Output

  • 59 Citations
  • 29 Publications
Publications
  • 2022
    Title On Itô’s formula for semimartingales with jumps and non- C 2 functions
    DOI 10.1016/j.spl.2022.109369
    Type Journal Article
    Author Eisenberg J
    Journal Statistics & Probability Letters
    Pages 109369
    Link Publication
  • 2024
    Title Special Issue "Interplay Between Financial and Actuarial Mathematics II"
    DOI 10.3390/risks12110177
    Type Journal Article
    Author Constantinescu C
    Journal Risks
  • 2024
    Title Financial impact of pandemics on pension sustainability: an application for Spain
    DOI 10.1007/s10203-024-00482-w
    Type Journal Article
    Author Boado-Penas M
    Journal Decisions in Economics and Finance
    Pages 1-26
    Link Publication
  • 2023
    Title Managing reputational risk in the decumulation phase of a pension fund
    DOI 10.1016/j.insmatheco.2022.12.005
    Type Journal Article
    Author Boado-Penas M
    Journal Insurance: Mathematics and Economics
    Pages 52-68
    Link Publication
  • 2022
    Title Pandemics: Insurance and Social Protection
    DOI 10.1007/978-3-030-78334-1
    Type Book
    editors Boado-Penas M, Eisenberg J, Şahin‬‬‬ Ş
    Publisher Springer Nature
    Link Publication
  • 2022
    Title Some Optimisation Problems in Insurance with a Terminal Distribution Constraint
    DOI 10.48550/arxiv.2206.04680
    Type Preprint
    Author Colaneri K
  • 2023
    Title Measuring the Suboptimality of Dividend Controls in a Brownian Risk Model
    Type Journal Article
    Author Eisenberg
    Journal Advances of Applied Probability
  • 2022
    Title Some optimisation problems in insurance with a terminal distribution constraint
    DOI 10.1080/03461238.2022.2142156
    Type Journal Article
    Author Colaneri K
    Journal Scandinavian Actuarial Journal
    Pages 655-678
    Link Publication
  • 2021
    Title Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model
    DOI 10.3390/risks9040073
    Type Journal Article
    Author Eisenberg J
    Journal Risks
    Pages 73
    Link Publication
  • 2020
    Title Nachhaltige Altersvorsorge in Zeiten niedriger Zinsen - ein Ansatz für ein neues Produktmodell
    Type Journal Article
    Author Boado-Penas
    Journal Der Aktuar
    Pages 4-11
    Link Publication
  • 2020
    Title Optimal Dividends Paid in a Foreign Currency for a Lévy Insurance Risk Model
    DOI 10.1080/10920277.2020.1805633
    Type Journal Article
    Author Eisenberg J
    Journal North American Actuarial Journal
    Pages 417-437
    Link Publication
  • 2020
    Title First Quarter Chronicle of COVID-19: An Attempt to Measure Governments’ Responses
    DOI 10.3390/risks8040115
    Type Journal Article
    Author Sahin S
    Journal Risks
    Pages 115
    Link Publication
  • 2020
    Title First quarter chronicle of COVID-19: an attempt to measure governments’ response
    DOI 10.1101/2020.09.20.20198242
    Type Preprint
    Author Sahin S
    Pages 2020.09.20.20198242
    Link Publication
  • 2021
    Title Maximizing with-profit pensions without guarantees
    DOI 10.1002/asmb.2661
    Type Journal Article
    Author Boado-Penas M
    Journal Applied Stochastic Models in Business and Industry
    Pages 308-322
  • 2021
    Title COVID-19: A Trigger for Innovations in Insurance?
    DOI 10.1007/978-3-030-78334-1_1
    Type Book Chapter
    Author Boado-Penas M
    Publisher Springer Nature
    Pages 1-12
  • 2021
    Title All-Hands-On-Deck!—How International Organisations Respond to the COVID-19 Pandemic
    DOI 10.1007/978-3-030-78334-1_7
    Type Book Chapter
    Author Boado-Penas M
    Publisher Springer Nature
    Pages 127-142
  • 2020
    Title Authors’ Reply on the Discussion of Krafft and Pankratz
    DOI 10.1007/s13385-020-00231-4
    Type Journal Article
    Author Boado-Penas C
    Journal European Actuarial Journal
    Pages 25-27
  • 2020
    Title Optimal Dividends Paid in a Foreign Currency for a Lévy Insurance Risk Model
    DOI 10.48550/arxiv.2001.03733
    Type Preprint
    Author Eisenberg J
  • 2020
    Title Optimising dividends and consumption under an exponential CIR as a discount factor
    DOI 10.1007/s00186-020-00714-w
    Type Journal Article
    Author Eisenberg J
    Journal Mathematical Methods of Operations Research
    Pages 285-309
    Link Publication
  • 2021
    Title Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate
    DOI 10.3390/math9182257
    Type Journal Article
    Author Eisenberg J
    Journal Mathematics
    Pages 2257
    Link Publication
  • 2021
    Title Special Issue “Interplay between Financial and Actuarial Mathematics”
    DOI 10.3390/risks9080139
    Type Journal Article
    Author Constantinescu C
    Journal Risks
    Pages 139
    Link Publication
  • 2021
    Title Dividend optimisation: A behaviouristic approach
    DOI 10.1016/j.insmatheco.2021.08.008
    Type Journal Article
    Author Brinker L
    Journal Insurance: Mathematics and Economics
    Pages 202-224
  • 2021
    Title Two Approaches for a Dividend Maximization Problem under an Ornstein-Uhlenbeck Interest Rate
    DOI 10.48550/arxiv.2108.00234
    Type Preprint
    Author Eisenberg J
  • 2021
    Title Transforming public pensions: A mixed scheme with a credit granted by the state
    DOI 10.1016/j.insmatheco.2020.11.005
    Type Journal Article
    Author Boado-Penas M
    Journal Insurance: Mathematics and Economics
    Pages 140-152
    Link Publication
  • 2018
    Title Suboptimal Control of Dividends under Exponential Utility
    DOI 10.48550/arxiv.1809.01983
    Type Preprint
    Author Eisenberg J
  • 2018
    Title An Exponential Cox-Ingersoll-Ross Process as Discounting Factor
    DOI 10.48550/arxiv.1808.10355
    Type Preprint
    Author Eisenberg J
  • 2019
    Title Maximising with-profit pensions without guarantees
    DOI 10.48550/arxiv.1912.11858
    Type Preprint
    Author Boado-Penas M
  • 2019
    Title A new approach for satisfactory pensions with no guarantees
    DOI 10.1007/s13385-019-00220-2
    Type Journal Article
    Author Boado-Penas M
    Journal European Actuarial Journal
    Pages 3-21
    Link Publication
  • 2019
    Title Transforming public pensions: A mixed scheme with a credit granted by the state
    DOI 10.48550/arxiv.1912.12329
    Type Preprint
    Author Boado-Penas M

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