Actuarial Control Problems under a Stochastic Interest Rate

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 reputational 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.