"Valuing Variable Annuities" (in Japanese)

In this paper we propose a framework to evaluate variable annuities. We show that the invested capital to a variable annuity can be decomposed into: (i) the reserve money in the account, (ii) options, (iii) fees paid to the mutual fund companies, and (iv) margin accruing to the insurance company. The first two components comprise value to the insured, and the last two accrue to the supply side companies. This view provides a convenient method to double-check the computation of various components of value. We also show that death benefit option attached to the most popular variable annuities is a portfolio of European put options of differing maturities. Assuming that investment value follows a geometric Brownian motion, this component can be valued applying the Black-Scholes formula and using the "death rates statistics" published by the Ministry of Heath, Labour and Welfare. Options on the income benefit are also valued using the Black-Scholes formula. @Stepped-up death benefit is a form of look-back options, which we value using a trinomial lattice. We value some typical products assuming that a person purchases them at age 40 and at age 50. We then examine how various value components would change in response to the volatilities of the investment products, the length of the contract and so on.