Given the comment in the earlier section, one can assume without a loss of generality that the policyholder purchases one unit of a fund instead of the N units (and hence deposits an amount of $St). From this, it readily follows that the value of the return of premium option (expiring at time T) can be obtained using the expression

(8.2)

whereandare as defined earlier.

As the reader will realize, equation (8.2) holds true only if the annuitant dies at time T. Equation (8.2) can alternatively be viewed as a put option that was described in equation (3.7b) with the dividend rate, qt,T in that equation, set to

Since the probability of the annuitant dying precisely at one moment (i.e., exactly at time T) is 0, it is more realistic to consider the probability of the annuitant dying in a time interval. More precisely, by assuming that the annuitant can die within the ith year of policy purchase (where i = 1,2,3,...), I will, for the purposes of option valuation, assume that the market risks associated with the death of the annuitant during the ith policy year can be hedged by an option that has a life of i years (where i = 1,2,3,...). It is intuitively reasonable to expect the probability of the annuitant dying during the ith policy year to change as the value i increases and for these probabilities to be a function of the annuitant's age.^{[1]}

TABLE 8.1 Sample Annual Mortality Rates for Annuitants Aged from 50 to 60

Given the above backdrop, if the age of the policyholder is x when the current time is t (i.e., at the policy inception), it readily follows from equation (8.2) that the value of the 116 – x options is given by the expression

(8.3a)

where denotes the probability of an x-year-old annuitant dying during the age x + у – 1 and x + y for у = 1, 2, 3, ....

Table 8.1 gives a snapshot of a typical mortality table capturing the probability of an individual dying within a year for a given age.

The 0.4518 percent entry in cell B1 of Table 8.1 refers to the probability that a 50-year-old would die in the coming year (i.e., before reaching age 51). Similarly, the 0.4938 percent entry in cell B2 refers to the probability that a 51-year-old will die before reaching 52, and so on. Since Table 8.1 is an example of what a typical mortality table looks like, one would need to arithmetically manipulate these rates to obtain the values of qx,x+i that are needed to solve equation (8.3a). Table 8.2 shows how all these components are put together to obtain the value of the options when the annuitant is 100 years old.^{[2]}

There are a few observations the reader should make (some obvious and some not so obvious) from Table 8.2.

Since the current age of the annuitant is 100, the mortality table goes until age 115 and assumes that any death happening in any one year can be hedged by a put option expiring at the end of the death year. It is easy to see that only 116 – 100 = 16 put options contributed to the total cost of the put options embedded in this GMDB. The costs associated with each of these options are given from cells G9:G24, and the total of these costs is given in cell G25. In the event that the age of the annuitant is now 50 (instead of 100), this would imply one would have a total of 116 – 50 = 66 put options (ranging in maturities from 1 to 66 years) that would be contributing toward the cost of these options.

The mortality rates (extracted directly from the mortality table) are given in cells B9:B24, where cell B9 refers to the probability of a 100- year-old dying before age 101 (i.e., q100,101)" cell B10 refers to the probability of a 101-year-old dying before age 102 (i.e., q101,102), and so on. To compute the qx>x+i factor that is present in equation (8.3a), namely q100,101, q100,102, q100,103, ..., q100,151 one has to derive them with the aid of the mortality rates given in cells B9:B24 using the following approach:

q100,101, the value in cell C9, refers to the probability that a 100-year-old would die before 101 and this is trivially the mortality probability given in cell B9.

q100,102, the value in cell C10, refers to the probability that a 100-year- old would die after 101 but before 102. This is trivially the probability of surviving until age 101 and then dying in the year of age 101. As a consequence,

q100,103, the value in cell C11, can similarly be obtained using the expression

TABLE 8.2 Valuation of Embedded Put Options When Annuitant Is 100 Years Old

FIGURE 8.2 Impact of Issue Age on Guaranteed Value

In general, q100,i (for i= 102, ..., 116) can be obtained using the expression

In practice, although most of the annuitants purchasing such investment products tend to be in their fifties, the exact age breakdown is not known to the insurance company a priori to launching the product. As a consequence, when developing these products, it is difficult for the insurance company to accurately forecast the actual breakdown of the age of potential purchasers of the product. Given this backdrop, an insurance company typically runs the type of analysis outlined in Table 8.2 across differing ages and then probability weighs^{[3]} the results across the ages to estimate how much these guarantees are in fact worth. Doing this across differing ages when the investment policy is issued, one can arrive at Figure 8.2.

FIGURE 8.3 Impact of Maturity on Option Premium

The younger the annuitant's age on the date of issue, the greater the number of options needed to quantify the risks accurately – which intuitively translates to the younger the annuitant, the larger the cumulative value of the options used to hedge the risks. Figure 8.2 seems to contradict this intuition and the reason for this lies in the way put-option values behave as the maturity of the option increases (a consequence of the Black-Scholes equation). To understand this better, I will digress for a bit to discuss the valuation of long-dated vanilla put options.

Ignoring the mortality component in Table 8.2, Figure 8.3 shows the impact of the option life on the option value for varying growth rates when all other inputs (fund value, risk-free rate, volatility, management fees) are kept the same.

As can be seen from Figure 8.3, regardless of the fund growth rate, the option value increases as option maturity increases until a certain point in time and then starts decreasing to 0.^{[4]} In reality, even if one could possibly buy a 100-year-option, the value of the 100-year option will definitely not be anywhere close to zero – underlining the importance of using the Black- Scholes model prudently. As in Ravindran and Edelist (1977), I will assume that the option premiums are bounded from below (i.e., floored). Figure 8.4 illustrates the consequence of flooring these premiums.

FIGURE 8.4 Impact of Maturity on a Floored Option Premium

Before applying the concept of flooring to redo the analysis required to reproduce Figure 8.2, one first needs to apply some of the ideas discussed in Chapter 6. More precisely, one has to imply inputs from the floored option premiums used to produce Figure 8.4. To do this, I will first imply the volatility of the options from the floored premiums and then use this implied volatility to price the embedded at-the-money options. Figure 8.5a shows the implied volatilities extracted from the at-the-money option premiums shown in Figure 8.4 over varying maturities.

FIGURE 8.5A Term Structure of Implied Volatilities for At-The-Money Options

FIGURE 8.5B Term Structure of Implied Risk-Free Rates for At-The-Money Options

As can be seen from Figure 8.5a, the implied volatility keeps increasing as the option maturity increases (an intuitively reasonable result) and then flattens out after a period of 30 years or more. The reason for this is simply due to the fact that since the start time of the flattening is also the start time for which no implied volatility exists (i.e., no volatility can be high enough to match the option premium), I had for convenience kept the implied volatility flat for this time period. As a consequence, I needed to calibrate another input of the option pricing model (e.g., the continuously compounded risk-free rate) so as to ensure that the option premiums in Figure 8.4 can be reproduced. To do this, I started with the term structure of zero rates and then calibrated it in conjunction with the volatility term structure given in Figure 8.5a, so that the option premiums in Figure 8.4 can be reproduced. Doing this yields the implied term structure of zero rates as given in Figure 8.5b.

Using the implied parameters in Figures 8.5a and 8.5b, one can now revalue the embedded options in the GMDB. Figure 8.6 illustrates the difference in premiums obtained by using the implied information (i.e., volatilities and risk-free rates) and the non-implied information (i.e., same as what was used to generate the premiums in Figure 8.2).

As can be seen from Figure 8.6, consistent with intuition, the difference in the total value of the embedded option premiums decrease as the annuitant's age gets higher – illustrating the fact that implied information affects longer dated options. Furthermore, in approximating a death incurring in a year by an option expiring at the end of the year, I have made a

FIGURE 8.6 Impact of Using Implied Information to Value GMDB Guarantees Across Varying Growth Rates

departure from the way these policies are settled in practice (as seen in Figure 8.1). More precisely, the insurance company is contractually on the hook for any downturn in investment performance at market closing on the day of death.^{[5]} Ravindran and Edelist (1997) discuss the use of continuous pdfs and their impact on the total-option premium when compared to the annual approximation. In practice, instead of using an annual time-step approximation, insurance companies use a monthly time-step approximation that I discuss next.

[1] In practice, life insurance companies do this using mortality tables. Mortality tables are created by the life insurance sector, using historical death patterns, to quantify the probability of dying in any given year for any given age (where this age can go as long as 115). These country-specific industry-based tables are typically revised when the industry, as a whole, feels that that has been a marked deviation from what has been produced. Although the life insurance industry as a whole produces different mortality tables for male smokers, male nonsmokers, female smokers, female nonsmokers, and so on, insurance companies in practice revise these industry-based mortality tables to reflect their company specific experience.

[2] I used 100 here for the purposes of illustration so that the exhibit associated with the illustration of the number of put options valued is manageable.

[3] Since the development of any product is done with the input of the sales force (or insurance agents), the estimation of the age distribution of potential clients is done using inputs from its agents, distribution of competitors' sales (if available), positioning of the product (e.g., if the product can be purchased in a retirement account using pretax funds or using after-tax dollars), and its own desire to target a certain age segment of the market – just to name a few.

[4] This is contrary to what is usually presented in many finance textbooks. More precisely, one of the properties of the Black-Scholes model, as described in this literature, is that option premiums are nondecreasing functions of option lives – which although true in practice is not the consequence of the Black-Scholes model.

[5] It is not uncommon for there to be a time lag between the actual time of the annuitant's death and the time when the insurance company is notified of the death. While in some insurance contracts this can be as long as a year, there are also instances when the insurance companies are never notified. As a practical matter, it is important for the reader to note that even if the beneficiary informs the insurance company of the passing of the annuitant months after the death, it is often the case that the insurance company is contractually required to pay the beneficiary using the closing market values at the time of the annuitant's death and not at the time the death notification is received by the insurance company.

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