Pension funding problem with regime‐switching geometric Brownian motion assets and liabilities

Abstract

This paper extends the pension funding model in (N. Am. Actuarial J. 2003; 7:37-51) to a regimeswitching case. The market mode is modeled by a continuous-time stationary Markov chain. The asset value process and liability value process are modeled by Markov-modulated geometric Brownian motions. We consider a pension funding plan in which the asset value is to be within a band that is proportional to the liability value. The pension plan sponsor is asked to provide sufficient funds to guarantee the asset value stays above the lower barrier of the band. The amount by which the asset value exceeds the upper barrier will be paid back to the sponsor. By applying differential equation approach, this paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In addition, we study the effects of different barriers and regime switching on the results using some numerical examples. The optimal dividend problem is studied in our examples as an application of our theory. can be significantly affected by the changes in political policies, the impact of economic news, etc., we may need to model the asset and liability value processes in a way with more flexibility.In the recent years, regime switching models have become popular in finance and related fields. This type of model is motivated by the intension to reflect the state of the financial market. For example, the state of the market can be roughly divided into 'bullish' and 'bearish' two regimes, in which the price movements of the stocks are quite different. Generally, in a regimeswitching model, the value of market modes is divided into a finite number of regimes. The key parameters, such as the bank interest rate, stocks appreciation rates, and volatility rates, will change according to the value of different market modes. Since the market state may change from one regime to another, both the nature of the regime and the change point should be estimated. If the market state process is modeled by a continuous time Markov chain with finite states, regime switching models are also referred to as Markov switching or Markov-modulated models in some literatures.With time-varying parameters, regime switching models are obviously more realistic than constant parameter model to reflect the random market environment. As discussed in Neftci [2], an appealing ability of these models is to account for the accumulating evidence that business cycles are asymmetric. Most of the studies indicate that regime-switching models perform well in some sense, for example, Hardy [3] used monthly data from the Standard and Poor's 500 and the Toronto Stock Exchange 300 indices to fit a regime switching lognormal model. In their paper, the fit of the regime switching model to the data was compared with other econometric models and they found that regime-switching models provided a significant improvement over all the other models in the sense of maximizing the likelihood function. In a special case, if the data are in lognor...

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