## Valuation of the interest rate guarantee embedded in defined contribution pension plans.(English)Zbl 1141.91554

Summary: We derive the valuation formulae for a defined contribution pension plan associated with the minimum rate of return guarantees. Different from the previous studies, we work on the rate of return guarantee which is linked to the $$\delta$$-year spot rate. The payoffs of interest rate guarantees can be viewed as a function of the exchange option. By employing W. Margrabe’s option pricing approach [The value of an option to exchange one asset for another. J. Finance 33, 177–186 (1978)], we derive general pricing formulae under the assumptions that the interest rate dynamics follow a single-factor HJM [D. Heath, R. Jarrow and A. Morton, Econometrica 60, No. 1, 77–105 (1992; Zbl 0751.90009)] interest rate model and the asset prices follow a geometric Brownian motion. The volatility of the forward rates is assumed to be exponentially decaying. The formula is explicit for valuing maturity guarantee (type-I guarantee). For multi-period guarantee (type-II guarantee), the analytical formula only exists when the guaranteed rate is the one-year spot rate. The accuracy of the valuation formulae is illustrated with numerical analysis. We also investigate the effect of mortality and the sensitivity of key parameters on the value of the guarantee. We find that type-II guarantee is much more costly than the type-I guarantee, especially with a long duration policy. The closed form solution provides the advantage in valuing pension guarantees.

### MSC:

 91B30 Risk theory, insurance (MSC2010)

Zbl 0751.90009
Full Text:

### References:

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