The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case

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Abstract

In this paper, we extend the analysis of the behaviour of pension contracts with guaranteed annuity conversion options (as presented in Ballotta and Haberman [Insurance: Math. Econ. 33 (2003) 87]) to the case in which mortality risk is incorporated via a stochastic model for the evolution over time of the underlying hazard rates. The pricing framework makes also use of a Black–Scholes/Heath–Jarrow–Morton economy in order to obtain an analytical solution to the fair valuation problem of the liabilities implied by these particular pension policies. The solution is not in closed form, and therefore, we resort to Monte Carlo simulation. Numerical results are investigated and the sensitivity of the price of the option to changes in the key parameters from the financial and mortality models is also analyzed.

Introduction

A large number of products offered by life insurance companies involve a range of complex contingent claims involving equity risk, interest rate risk and mortality risk. The presence of these products in the life insurance and pension liabilities of insurance companies has given rise to an increasing focus on the issues of capital adequacy and solvency requirements. As regulation in this area makes use of accounting information as a starting point, the fair valuation methodologies promoted by the International Accounting Standards Board (IASB) in the newly issued International Financial Reporting Standards (IFRS), are going to be adopted for the construction of consolidated financial statements, in an attempt to make the assessment process of companies’ financial performance much more realistic and reliable.

An example of such a complex contract, which also created instability in the life insurance market, is the so-called guaranteed annuity option (GAO). A GAO is a design feature attached to individual pension policies which provides the policyholder with the right to receive at retirement either a cash payment or an annuity which would be payable throughout the policyholder’s remaining lifetime and which is calculated at a guaranteed rate, depending on which has the greater value. This guaranteed conversion rate between cash and pension was a common feature of individual pension policies sold in the UK during the (late) 1970s and 1980s, with more than 40 companies involved in this market.

Until the early 1990s, the UK experience has been that the cash benefit was more valuable than the guaranteed annuity payment since a higher pension could be obtained by using the cash to buy the best annuity rates available in the market (the so-called “open market option”). Since the late 1990s, reductions in market interest rates and unanticipated falls in mortality rates at the oldest ages have meant that the position has changed and the guaranteed annuity has tended to be worth more than the cash benefit. As a result of these two combined effects, many UK insurance companies (which have sold policies with guaranteed annuity options) have experienced solvency problems, requiring the setting up of extra reserves, and leading one large mutual life insurer (Equitable Life, the world’s oldest life insurance company) to be closed to new business in 2000. Although pension policies with these guarantees are no longer being sold in the UK, these are a common feature of corresponding policies in other countries, for example, the US. Thus, currently in the US variable annuity market, there are guaranteed annuity rate (GAR) contracts and guaranteed minimum income benefit (GMIB) contracts. A GAR contract is identical to a GAO. A GMIB contract includes the additional feature that the cash benefit available at retirement is guaranteed to be at least a pre-specified amount.1

Although the new accounting directives promoted by IASB mentioned above focus essentially on the financial risk affecting life insurance contracts, the UK historical experience shows that very long-term products-like GAOs are significantly exposed to unanticipated changes over time in the mortality rates of the reference population (mortality risk). This means that the fair valuation techniques proposed in the IFRS need to be integrated with an accurate assessment of future mortality rates. Hence, in this paper, we propose a possible integrated framework for the market consistent valuation of GAOs, which incorporates mortality risk as well by means of a stochastic model for the evolution of mortality rates over time.

In this paper, we focus on unit-linked deferred annuity contracts purchased originally by a single premium. For simplicity, we ignore insurance company expenses, taxes, profit and pre-retirement death benefits in order to concentrate on the GAO. The analytical approach that we adopt follows the financial economics literature and exploits the well-known option valuation theory in order to obtain results for the pricing, reserving and hedging of the GAO. In particular, we follow Ballotta and Haberman (2003) and we use a single-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 parameters leading to arbitrage opportunities (Heath et al., 1992).

An alternative approach based on modelling the dynamics of the annuity price, rather than the underlying term structure of interest rates, has been used by Bezooyen et al., 1998, Pelsser, 2003, and Wilkie et al. (2003). However, we argue that a methodology based on the term structure of interest rates is more sound in that, on one hand, it relies on quantities, like zero coupon bonds, that are fully traded in the financial market, and on the other hand, it facilitates the analysis of the effect on the GAO of changes in market interest rates and their term structure. In this respect, we believe that the model proposed in this paper is more consistent with the recommendation from IASB that the valuation techniques used to estimate fair values should maximize the use of market inputs (see also Jørgensen (2004), for a more detailed discussion an accounting standards for life insurance liabilities).

Under the additional assumption of a mortality risk that is independent of the financial risk, a general pricing model is proposed and a numerical procedure, using Monte Carlo techniques, for the estimation of the value of the guaranteed annuity option is implemented. Numerical results 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 develops the framework for the valuation of guaranteed annuity conversion options. In Section 3, we introduce a stochastic model for the mortality risk; Section 4 provides a model for the financial market and a pricing formula for the guaranteed annuity option. In Section 5, we discuss the numerical evidence produced and concluding remarks are offered in Section 6.

Section snippets

A valuation approach for guaranteed annuity options

A guaranteed annuity option (GAO) provides the holder of the contract the right to receive at retirement, at time T, either a cash benefit (equal to the current value of the reference portfolio, ST), or an annuity which would be payable throughout his/her remaining lifetime and which is calculated at a guaranteed rate, g, depending on which has the greater value.

Hence, if the policyholder is aged x0 at time 0, when the contract is initiated, and N is the normal retirement age, the GAO pays out

A stochastic approach to mortality risk: the basic model and its extensions

In the previous section, we defined τx to be a random variable which represents the remaining lifetime of the policyholder and which depends on the age, x, of the policyholder at time t. The survival function of the random variable τx is given bypxs=P(τx>s|Ft),where P is the objective probability measure. If we explicitly allow for the time dependence of the hazard rate, and we define μ(x+z,t+z) to be the hazard rate for an individual at time t+z then aged x+z, it follows thatpxs=E[e0sμ(x+z,t+

A model for the financial risk and the GAO valuation formula

In the frictionless market introduced in Section 2, assume that the insurer invests the single premium paid by each policyholder at the start of the contract into an equity fund, whose risk-neutral dynamic is described by the following stochastic differential equation.dSt=rtStdt+σSStdZˆt,S0R+,where σSR+ and (Zˆt:t0) is a standard one-dimensional Pˆ-Brownian motion. Thus, S0 is the single premium. As mentioned above, we assume that the evolution of the term structure of interest rates is

Numerical calculations and sensitivity analysis

As explained in the previous sections, we implement a numerical procedure to estimate the value of the guaranteed annuity option contract based on the valuation formula contained in Eq. (17) and here repeated for convenience:Vx0(x(t),t,Tt)=gStE˜e0Ttμ(x(t)+z,t+z)dzj=0w(T+x0)px(T)TjTPT(Tj)K+Ft,where E˜ is the expectation taken under the stock-risk-adjusted probability measure P˜. In particular, we recall that the pre-retirement mortality factore0Ttμ(x(t)+z,t+z)dzand the post-retirement

Conclusions

The aim of this work has been to extend the valuation model for guaranteed annuity options proposed by Ballotta and Haberman (2003) in order to allow for stochastic uncertainty in mortality trends. The behaviour of this contract with respect to changes in market conditions and mortality risk has been analyzed with numerical examples and the sensitivity analysis presented.

We have seen that the inclusion of stochastic mortality, through fluctuations around a trend (the parameter σh), or longevity

Acknowledgments

The financial support from the Society of Actuaries’ Committee on Knowledge Extension Research and the Actuarial Education and Research Fund is gratefully acknowledged. The authors express special thanks to Enrico Biffis, Luigi Colombo and Russell Gerrard for many useful discussions, and to Moshe Milevsky for useful advice.

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