# Modeling Future Fund Value Movements

As in the earlier chapters, I assume that index-value movements best mimic fund movements. As a consequence, I will use a diffusion equation that is a slight modification of equation (3.5) that is given by the expression

(8.1)

where

*S* is the value of the fund.

*g* is the annualized continuously compounded growth rate.

*q* is the annualized continuously compounded fees.^{[1]}

σ is the annualized volatility of the fund return.

*dz* is the random variable drawn from a standard normal probability density function.

*dS* is the small change in the fund value over a small time interval *dt.*

As the reader will recall, the characterization in equation (8.1) is equivalent to assuming that In *Sj* is normally distributed with a mean of In , and, variance of, whererepresents the continuously compounded growth rate that is applied from time *t* to T, and the rest of the notations are consistent with those used before.

# Payoff Associated with the Guarantees

First observe that the beneficiary stands to receive from the annuitant's death a benefit of *N* * max(St,ST), where *N* represents the number of fund units, *S*t represents the fund unit value at time *t* (inception of the contract), and *S*T represents the fund unit value net of fees at time T (where T represents the time of the annuitant's death).

*N* * *max*(*S*t*,S*T) can be rewritten *as N ** [Sт + maх(St -ST, 0)], from which it can be easily seen that an investment in a GMDB can be alternatively viewed as the sum of investment value^{[2]} at the time of death, and *N* units of put option (which has a strike of *S*t and a life of (*T* – *t*)).

Thus, purchasing a basic GMDB is equivalent to purchasing *N* fund units and *N* put options (where each at-the-money spot option has a life of *T – t* years) – assuming that the annuitant dies at time T and the fund units are redeemed only at time *T.*

- [1] This is used to represent the sum of the fund management fees as well as the fees associated with the basic death benefit. The fact that this is continuously compounded is only a simplifying assumption. See the section on Other Details Associated with GMDB Products later in this chapter.
- [2] This is actually
*N*times the fund unit value at the time of death (i.e., T).