Stochastic Models in Life Insurance

Frontmatter

Chapter 1. A General Life Insurance Model

Abstract

The life insurance market offers a wide range of different policies. It is, without expert knowledge, hardly possible to differentiate between all these policies. This is in particular due to the fact that the content of a life insurance is an abstract good.

Michael Koller

Chapter 2. Stochastic Processes

Abstract

In this section we will recall basic definitions from probability theory. These will be used throughout the book.

Michael Koller

Chapter 3. Interest Rate

Abstract

An important part of every insurance contract is the underlying interest rate. The so called technical interest rate describes the interest which the insurer guarantees to the insured. It is a significant factor for the size of the premiums. If the technical interest rate is too low it yields inflated premiums, if it is too high it might yield to insolvency of the insurance company.

Michael Koller

Chapter 4. Cash Flows and the Mathematical Reserve

Abstract

In the previous two chapters we introduced several types of insurances and their setup. Based on this we will now answer several fundamental questions.

Michael Koller

Chapter 5. Difference Equations and Differential Equations

Abstract

In this chapter we focus on the Markov model in continuous time. The differential equations are the continuous counter part to the difference equations of the discrete model.

Michael Koller

Chapter 6. Examples and Problems From Applications

Abstract

In this chapter we take a closer look at problems which appear in applications. Moreover, the examples will show the scope of the Markov model and illustrate some special modelling tricks. The examples are based on the discrete model, since this is most popular in applications.

Michael Koller

Chapter 7. Hattendorff’s Theorem

Abstract

Hattendorff’s Theorem (1868) states that the losses which incur for an insurance policy in different years are uncorrelated and have mean zero. When this theorem was first discovered it caused many discussions. Nowadays it is part of every introduction to stochastic modelling in insurance. In the following we present the theorem in its general form and its version for the Markov model.

Michael Koller

Chapter 8. Unit-Linked Policies

Abstract

Up to now we have mostly considered models with deterministic interest rate or with an interest rate given by a Markov chain on a finite state space. This helped us to keep the calculations simple. In this chapter we have a look at some more general models. On the one hand we consider models for policies whose actual value depends on the performance of an underlying unit (usually a fond), on the other hand we will discuss further models with stochastic interest rate.

Michael Koller

Chapter 9. Policies with Stochastic Interest Rate

Abstract

In the previous chapter we looked at unit-linked policies. Based on these we will now consider policies with a technical interest rate modelled by a stochastic process, which is for example given by a stochastic differential equation.

Michael Koller

Chapter 10. Technical Analysis

Abstract

In this chapter we discuss the technical analysis of life insurance policies. This is the actual technical analysis which has to be done at the end of each fiscal year. It is concerned with the question, if the underlying loaded assumptions resemble the reality or if these have to be adapted. This analysis is used for yearly inspections of the current portfolio.

Michael Koller

Chapter 11. Abstract Valuation

Abstract

A Stochastic Cash Flow is a sequence \(x=(x_k)_{k\in \mathbb{ N} } \in L^2{(\Omega ,\mathcal{ A} ,P)}^{\mathbb{ N} }\), which is \(\mathbb{ F} = (\mathcal{ F} _t)_{t\ge 0}\) adapted.

Michael Koller

Chapter 12. Policyholder Bonus Mechanism

Abstract

The aim of this chapter is to introduce the concept of policyholder bonus and the corresponding effects.

Michael Koller

Backmatter