Cash flows and policyholder behaviour in the semi-Markov life insurance setup

Abstract

Within the setup of a semi-Markov process in a finite state space, we consider a life insurance contract. First, without the modelling of policyholder behaviour, we show how to calculate the expected cash flow associated with future payments, and to that end we present a version of Kolmogorov's forward integrodifferential equation. The semi-Markov model is then extended to include modelling of surrender and free policy behaviour, and the main result is a modification of Kolmogorov's forward integro-differential equation, such that the cash flow can be calculated without significantly more complexity than the cash flow without policyholder modelling. The result is also demonstrated for the traditional Markov case where there is no duration dependence, and numerical examples are studied.

Life Insurance Cash Flows with Policyholder Behavior

“…Second, we consider the disability model and present certain ordinary differential equations that solve the problem. This second method has recently been suggested in a more general semi-Markov setup in [1]. We discuss how the integral expressions originating from the survival model can be used to approximate the more correct modeling in the disability model, for a very simple, yet effective type of policyholder behavior modeling.…”

“…In practice, we would have to solve partial instead of ordinary differential equations. We present the main result from [1] that allows us to effectively dismiss the duration dependence and to calculate cash flows with policyholder behavior by simply calculating a slightly modified version of Kolmogorov's forward (ordinary) differential equation. The complexity of the calculations is therefore not increased significantly by inclusion of policyholder behavior.…”

“…This model is presented in [1] for the general semi-Markov setup, and here, we present the special case of a Markov process for the disability model with recovery. Within this setup, the transition probabilities are first calculated using Kolmogorov's forward differential equation, and then, the cash flow can be determined.…”

“…In this paper, we study policyholder behavior in life and pension insurance with a focus on two so-called policyholder options: first, the surrender option, where the policyholder may surrender the contract, canceling all future payments and instead receiving a single payment corresponding to the value of the contract on a technical basis; second, the free policy option 1 , where the policyholder may cancel the future premiums and have the benefits reduced according to the technical basis. Policyholder 1 The free policy option is sometimes referred to as a "paid-up policy" in the literature.…”

“…Policyholder 1 The free policy option is sometimes referred to as a "paid-up policy" in the literature.…”