Valuation of guaranteed annuity conversion options

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Abstract

In this note we introduce a theoretical model for the pricing and valuation of guaranteed annuity conversion options associated with certain deferred annuity pension-type contracts in the UK. The valuation approach is based on the similarity between the payoff structure of the contract and a call option written on a coupon-bearing bond. The model makes use of a one-factor Heath–Jarrow–Morton framework for the term structure of interest rates. Numerical results are investigated and the sensitivity of the price of the option to changes in the key parameters is also analyzed.

Introduction

A guaranteed annuity option provides the holder of the contract the right to either receive at retirement a cash payment or receive an annuity which would be payable throughout his/her remaining lifetime and which is calculated at a guaranteed rate, depending on which has the greater value. This guarantee of the conversion rate between cash and pension income was a common feature of pension policies sold in the UK during the 1970s and 1980s. Thus, in a survey conducted by Bolton et al. (1997), annuity conversion guarantees were found to apply to just over 10% of the long term liabilities of the responding insurance companies.

Until recently, the cash benefit was more valuable than the guaranteed annuity payment since a higher pension could be obtained by applying the cash on the best annuity rates available in the market (the so-called “open market option”). After the reduction in the level of market interest rates over recent years, and particularly since 1998, the position has become reversed and the guaranteed annuity is now usually worth more than the cash benefit; improvements in mortality rates since these policies were issued have also made them more valuable to policyholders. As a result of these two combined effects, many insurance companies have experienced solvency problems requiring the setting up of extra reserves (using ad hoc methods) and leading one large life insurer (Equitable Life, the world’s oldest life insurance company) to be closed to new business.

In this paper, we concentrate on unit-linked deferred annuity contracts purchased originally by a single premium. The pricing of options embedded in insurance contracts with guarantees has been addressed in the literature over the past 25 years. Thus, Brennan and Schwartz (1976) and Boyle and Schwartz (1977) analyzed unit-linked life insurance contracts with maturity guarantees using an approach centered on financial economics theory while the MGWP (1980) used a simulation-based methodology. More recently, Grosen and Jørgensen (2000) have analyzed with profit policies, allowing for the bonus guarantee and surrender option.

The approach advocated in this paper follows the above-cited literature and exploits the traditional option valuation procedure in order to provide indications in terms of pricing, reserving and hedging of the guaranteed annuity option contract. In this regard, our methodology differs from that proposed by Yang et al. (2001), who use a simulation-based asset model which is not arbitrage free. The approach also differs from that of Bezooyen et al., 1998, Pelsser, 2002 and Yang et al. (2002) who each model the dynamics of the annuity price rather than modeling the term structure of interest rates. We believe that a methodology that begins with the term structure of interest rates is more sound in that it allows the effect of changes in the term structure on the value of the guaranteed annuity option contract to be explored. At the time of writing, we have become aware of the work of Boyle and Hardy (2002) which follows a similar methodology to the one described below. Other work in this area has compared the guaranteed annuity option to swaption contracts (Lee, 2001).

The option pricing approach we propose to valuation of these guarantees is based on the similarity between the payoff structure of the contract under consideration and a call option written on a coupon-bearing bond. The model makes use of a one-factor Heath–Jarrow–Morton framework for the term structure of interest rates. This choice is justified by the need to avoid dependence of the model on the market price of interest rate risk, which usually implies an arbitrary specification of the model’s parameters leading to arbitrage opportunities (Heath et al., 1992). Also, single-factor models allow a mathematically tractable solution to the coupon bond option pricing problem (Jamshidian, 1989). We present two alternative formulations of the HJM framework based on different specifications for the forward rate volatility. The first relies on the assumption of constant volatility, while the second uses an exponentially decaying volatility structure, typical of the Vasicek (1977) class of models. Under the additional assumption of an unsystematic mortality risk, independent of the financial risk, a general pricing framework is proposed and closed analytical formulae for the value of the guaranteed annuity option are obtained. In both models, the pricing formulae derived implicitly contain the dynamic investment strategy that replicates the contract. Numerical results for both models are investigated and the sensitivity of the price of the option to changes in the key parameters is also analyzed.

The paper is organized as follows. Section 2 describes the financial model. Section 3 presents the guaranteed annuity option as a contingent claim. Section 4 considers in details two models for the dynamic of the term structure (in the HJM framework) and obtains closed form solutions for the value of the guaranteed annuity option at inception of the contract. In Section 5, we present some numerical examples and sensitivity analysis results.

Section snippets

The financial model

Assume a frictionless market with continuous trading, no taxes, no transaction costs, no restrictions on borrowing or short sales and perfectly divisible securities. The insurance company invests the single premium paid by each policyholder in an equity fund, S, whose dynamic under the risk-neutral equivalent martingale measure P̂ is described by the following equation: dSt=rtStdt+σSStdẐt,where (Ẑt:t≥0) is a standard one-dimensional P̂-Brownian motion and σSR+. Assume also that the evolution

The guaranteed annuity option

We consider now a guaranteed annuity option, which is a contract giving the holder the right to receive at retirement the greater of (a) a cash payment equal to the current value of the investment in the equity fund, S, and (b) the expected present value of the life annuity obtained by converting this investment at the guaranteed rate. In other words, if at inception the policyholder is aged x, and if N is the normal retirement age, then the guaranteed annuity option payoff at maturity is CT=gST

Term structure movements and option pricing

This section presents two examples to illustrate in details the valuation procedure introduced in Section 3.

In the first example, we assume that the volatility of the forward rate process (1) is a positive constant, σf(t,T)=σf R+. This is a continuous time limit of Ho and Lee’s (1986) model which may prove useful in practical applications due to its computational simplicity. However, according to this model, all rates fluctuate in the same way. Another related disadvantage is that this model

Numerical results and sensitivity analysis

The results obtained in the previous section have been used to study the behavior of the guaranteed annuity option under different scenarios. Throughout the following analysis, unless otherwise stated, the basic set of parameters is S0=100,σS=0.2,ρ=1,g=0.111,x=50,T+x=N=65.In particular, the choice of the parameter g follows the indication of Bolton et al. (1997) as the most common parameter value in the UK. As far as the volatility function of the forward rate is concerned, we fix σf=0.001 for

Conclusions

In this paper we have introduced a theoretical model, based on the one-factor Heath–Jarrow–Morton term structure framework, for the valuation of guaranteed annuity conversion options attached to single premium deferred annuity contracts. The approach depends on the correspondence between the contingent claim under consideration and an option contract written on a coupon paying bond. Two set of results are derived for the cases of (a) constant volatility and (b) of exponentially decaying

Acknowledgements

The financial support from the Society of Actuaries Committee on Knowledge Extension Research and the Actuarial Education on Research Fund is gratefully acknowledged. The authors would like to thank Prof. Gerald Rickayzen for his assistance with various C++ implementations. Earlier versions of this work have been presented at the CeRP Conference “Developing an Annuity Market in Europe” (Turin), 8th International Vilnius Conference on Probability Theory and Mathematical Statistics, 6th

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