Academic journal article Journal of Risk and Insurance

Guarantees in With-Profit and Unitized With-Profit Life Insurance Contracts: Fair Valuation Problem in Presence of the Default Option

Academic journal article Journal of Risk and Insurance

Guarantees in With-Profit and Unitized With-Profit Life Insurance Contracts: Fair Valuation Problem in Presence of the Default Option

Article excerpt

ABSTRACT

The purpose of the article is to apply contingent claim theory to the valuation of the type of participating life insurance policies commonly sold in the UK. The article extends the techniques developed by Haberman, Ballotta, and Wang (2003) to allow for the default option. The default option is a feature of the design of these policies, which recognizes that the insurance company's liability is limited by the market value of the reference portfolio of assets underlying the policies that have been sold. The valuation approach is based on the classical contingent claim pricing "machinery," underpinned by Monte Carlo techniques for the computation of fair values. The article addresses in particular the issue of a fair contract design for a complex type of participating policy and analyzes in detail the feasible set of policy design parameters that would lead to a fair contract and the trade-offs between these parameters.

INTRODUCTION

Participating contracts of various types make up a significant part of the life insurance market of many countries including the United States, Japan, Australia, Canada, and several members of the European Union. The modeling, valuation, and pricing of these contracts are important subjects for scientific analysis. This is because of the need by actuaries for appropriate and robust methods of internal financial risk management; the need by insurance companies to demonstrate solvency and the ability to pay claims (and hence benefits); the need to offer customers a "fair price" and be able to demonstrate this; and the need to measure profitability. The task, however, is made difficult by the nature of these life insurance contracts which incorporate a wide range of guarantees and option-like features.

In recent years, a series of studies have applied classical contingent claim theory, building on the pioneering work of Brennan and Schwartz (1976) on unit-linked policies, to different types of participating contracts: see, for example, Bacinello (2001, 2003), Grosen and Jorgensen (2000, 2002), Jensen, Jorgensen, and Grosen (2001), Miltersen and Persson (1999), and Persson and Aase (1997).

In this article, our approach is to consider the most common policy design used in the UK for unitized with-profit contracts and use classical contingent claim methodology to solve the valuation problem related to this contract. In particular, the policy design under examination incorporates both a reversionary-type bonus and a terminal bonus, and hence differs from that considered by other authors. The article is a companion to Haberman, Ballotta, and Wang (2003), but with the added feature of the default option which is discussed in more detail in the section "Equilibrium Condition for Fair Valuation."

The article is organized as follows. The following section describes the model and the next section describes the valuation framework and the equilibrium condition for fair valuation, considering the separate claims of the policyholders and the equityholders. The subsequent section provides the numerical results and the final section provides some concluding comments.

MODEL OF A LIFE INSURANCE COMPANY WITH PARTICIPATING CONTRACTS

The aim of this section is to set up a simple valuation model for the liabilities of a life insurance company implied by participating policies. Assume that at time t = 0, the insurance company acquires an asset portfolio A and finances this portfolio with the (single) premium, P0, received from a policyholder and with paid-in capital, E0, as illustrated in the company's balance sheet in Table 1. In return for the payment of the single premium, the policyholder is entitled to a fixed guaranteed benefit, together with the so-called reversionary bonus which is added periodically (i.e., a variable component reflecting the individual policyholder's smoothed share of the insurance company's profits), and a second variable component, which is based on the final surplus earned by the insurance company. …

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