تجزیه و تحلیل مزایای بازنشستگی با گزینه ها

This study applies the contingent claim approach to evaluate retirement benefits with the options of choosing the maximum defined benefit and defined contribution pension plans. A least-squares Monte Carlo simulation values complex retirement benefits that feature the properties of multiple variables, early exercise, stochastic interest rates, and several embedded options. Furthermore, this study examines the impacts of different forms of early decrements of the value of retirement benefits with options.

The world is facing increasingly salient issues of aging as the population of people aged older than 65 years continues to grow.1 In this context, retirement programs are becoming more important, because people must rely on the pension benefits to support themselves for the rest of their lives, after they retire from work. Retirement benefits are usually paid according to two approaches: defined benefits (DBs) or defined contributions (DCs). In Australia, Italy, and some states in the United States, retirement benefit programs are designed to offer the maximum of two alternative benefit values (DB and DC), equivalent to a better-of option (Smithson, 1998). Employees have the option to choose from the maximum of the two underlying assets when they retire (Johnson, 1987 and Stulz, 1982). When an employee is allowed to retire early, American options also are embedded in the retirement benefit program. Option pricing methodologies first were used in the valuation of investment guarantees by Boyle and Schwartz (1977). Wilkie (1989) discusses option pricing techniques for pension benefit payments in the United Kingdom. Shimko, 1989 and Shimko, 1992 develops a contingent claim approach to value insurance claims. Sherris (1995) extends the contingent claim approach to the case of retirement benefits. Ever since Black and Scholes (1973) conducted a renowned European option closed-form solution, the development and application of option evaluations have attracted concern from academia and practice. There also has been plenty of research and actual costs devoted to attempting to find a reasonable price of a derivative by the most efficient and precise method, to benefit both the exchange proceeding that of financial merchandises and analyzing strategies to manage risk. Except for the simplest European option, there are many other exotic options traded in the market. The Black and Scholes (1973) formula can only evaluate a simple European option, whereas several numerical methods exist to evaluate complicated exotic options. The use of numerical techniques for valuing options is discussed in Hull (2006). Three types of numerical techniques have been frequently applied for option valuation: (1) approximate the underlying stochastic process directly by Monte Carlo simulation as first introduced by Boyle (1977); (2) use various lattice (tree) approaches, such as Cox et al.'s (1979) binomial tree method; or (3) discretize a partial differential equation by finite difference methods, such as in Brennan and Schwartz (1978). Both the lattice model and finite difference method will be inefficient when the state variable number is large enough, because the memory space and computation time will grow exponentially as the state variable increases. Generally, lattice and finite difference methods become incapable of valuing options when the number of state variables increases beyond three. On the contrary, the simulation approach seems to be the best choice for pricing complex options. The most important concern for numerical methods is their efficiency and convergence. Because the DC depends on the history of the salary growth rate, it is not a simple function of the final value of the state variables. Considering the path dependency and the multi-variable problem in the pension pricing, the lattice and finite difference techniques are impractical. It is more efficient to deal with the valuation of retirement benefits by simulation. Boyle (1977) first proposed a simulation approach for pricing options. The simulation approach is very flexible and can be used to price complex European-style options. Before 1993, there were few published works on the use of simulation to value American options. Tilley (1993) was the first to develop such a technique, which included a procedure for incorporating an early exercise principle in the simulation. Tilley in his study mentions but does not demonstrate that it is possible to handle the pricing of American options with two or more state variables. Barraquand and Martineau (1995) subsequently proposed an approach that could track the conditional probabilities of path-specific outcomes in simulation. They used these values to make early-exercise decisions and value American put options. Broadie and Detemple (1996) also developed a simulation algorithm that produces two estimators for the true option value, one biased high and the other biased low, and both asymptotically unbiased as the number of simulations tends to infinity. By the convergence principle, these two estimates provide a conservative confidence interval for the option values. Grant et al. (1996) demonstrated how to determine the critical prices before maturity at the first step by backward induction in simulation. Knowing the critical prices at each possible early exercise time before maturity, these authors illustrated in a second step the pricing of put options by forward induction using simulations. Their model is able to incorporate the early-exercise feature into simulations, and they showed that their approach could accurately value plain vanilla American options, but they did not reveal whether their approach could be applied to pricing American options depending on multiple assets. Raymar and Zwecher (1997) extended Barraquand and Martineau's (1995) approach by increasing the factors that separate the simulation paths from one to two. Their approach can effectively eliminate the pricing error that Barraquand and Martineau (1995) committed and thus increase the accuracy of simulation for pricing American options. Longstaff and Schwartz (2001) developed a simple least-squares Monte Carlo simulation (hereafter LSM) method to price American options. LSM can easily handle complex option pricing problems with several variables and path-dependent exotic features. We apply LSM to solve the valuation problem of pension benefits in this article. Some additional issues have to be considered when applying option pricing methods in pension valuation, such as the consideration of decrements of resignation, death, and disability into the calculation, as well as the path-dependent form of the benefit payment. This article extends the study of Sherris (1995) by relaxing some impractical assumptions that can fit reality well. It is unfeasible and will cause serious biases by assuming a fixed interest rate environment for the valuation of pension benefit with maturity of more than ten years. One of the key extensions is that we assume that the risk-free interest rate follows the Ito process, which Sherris (1995) assumes is deterministic. The DC depends on the history of the salary growth rate, which means that DC is not a simple function of the final value of the state variables. Moreover, since an employee is allowed to retire earlier, the embedded option is a path-dependent American-style option. To build a practical model, we propose a LSM simulation approach, which can efficiently analyze retirement benefits with the properties of multi-variables, early exercise, and several embedded options. Series of sensitivity analyses of parameters to the value of retirement benefits and early retirement probabilities are presented. People can apply our model to evaluate the values of retirement benefits with options under different scenarios to make optimal retirement decisions. The remainder of this article is organized as follows. Section 2 constructs the model. The state variable processes, pension systems and least squares Monte Carlo simulation applied to retirement are introduced in this section. Section 3 presents the numerical analyses for various scenarios. The impacts of stochastic interest rate and state variable correlations to pension benefits will be discussed. We finally draw conclusions and discuss implications of our findings in Section 4.

We address the valuation of retirement benefits with better-of options by applying the contingent claim approach implemented with a least square Monte Carlo simulation. The numerical results suggest that our model can efficiently handle the valuation of retirement benefits with options from choosing the maximum of defined benefit and defined contribution pension plans. Also, our model can deal with different forms of early decrements to the value of retirement benefits. We demonstrate that the retirement benefits under the constant interest rate of Sherris (1995) are under-estimated. Moreover, Sherris's (1995) model is only a special case of ours, since he assumed that the interest rate is constant through time and employees cannot retire early. From the sensitivity analyses, we know that the stochastic interest rate, correlation ρfr, and entry age have significant impacts on the value of retirement benefits. Our model provides great contributions to both the employer and the employee, who can make better retirement decisions by following our model. Furthermore, our model could be applied to other types of retirement benefits with other complex exotic options.