Reserve-dependent benefits and costs in life and health insurance contracts

Premiums and benefits associated with traditional life insurance contracts are usually specified as fixed amounts in policy conditions. However, reserve-dependent surrender values and reserve-dependent expenses are common in insurance practice. The famous Cantelli theorem in life insurance ensures that under appropriate assumptions surrendering can be ignored in reserve calculations provided that the surrender payment equals the accumulated reserve. In this paper, more complex reserve-dependent payment patterns are considered, in line with insurance practice. Explicit formulas are derived for the corresponding reserve.     

Nonlinear reserving and multiple contract modifications in life insurance

Life insurance cash flows become reserve dependent when contract conditions are modified during the contract term on condition that actuarial equivalence is maintained. As a result, insurance cash flows and prospective reserves depend on each other in a circular way, and it is a non-trivial problem to solve that circularity and make cash flows and prospective reserves well-defined. In Markovian models, the (stochastic) Thiele equation and the Cantelli Theorem are the standard tools for solving the circularity issue and for maintaining actuarial equivalence. This paper expands the stochastic Thiele equation and the Cantelli Theorem to non-Markovian frameworks and presents a recursive scheme for the calculation of multiple contract modifications.     

Dependent interest and transition rates in life insurance

For market consistent life insurance liabilities modelled with a multi-state Markov chain, it is of importance to consider the interest and transition rates as stochastic processes, for example in order to consider hedging possibilities of the risks, and for risk measurement. In the literature, this is usually done with an assumption of independence between the interest and transition rates. In this paper, it is shown how to valuate life insurance liabilities using affine processes for modelling dependent interest and transition rates. This approach leads to the introduction of so-called dependent forward rates. We propose a specific model for surrender modelling, and within this model the dependent forward rates are calculated, and the market value and the Solvency II capital requirement are examined for a simple savings contract.     

Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs

Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.     

Stochastic utilities with subsistence and satiation: Optimal life insurance purchase,consumption and investment

We introduce stochastic utilities such that utility of any fixed amount of interest is a stochastic process or random variable. Also, there exist stochastic (or random) subsistence and satiation levels associated with stochastic utilities. Then, we consider optimal consumption, life insurance purchase and investment strategies to maximize the expected utility of consumption, bequest and pension with respect to stochastic utilities. We use the martingale approach to solve the optimization problem in two steps. First, we solve the optimization problem with an equality constraint which requires that the present value of consumption, bequest and pension is equal to the present value of initial wealth and income stream. Second, if the optimization problem is feasible, we obtain the explicit representations of the replicating life insurance purchase and portfolio strategies. As an application of our general results, we consider a family of stochastic utilities which have hyperbolic absolute risk aversion (HARA).     

Intensity-based framework for surrender modeling in life insurance

In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox–Ingersoll–Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed-form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.