The purpose of this section is to illustrate via an example how the embedded optionality associated with consumer behavior can be quantified. One living benefit rider that was briefly mentioned in Chapter 8 was the GMAB rider. As the reader will recall, this rider can be added on by the policyholder to the basic GMDB product should the policyholder be interested in protecting the investment on the maturity date of the GMAB rider.

Although Figure 8.15 shows a GMAB rider with a roll-up and ratchet guarantee, for the ease of explanation, I will henceforth keep my discussion focused on the return-of-principal guarantee at maturity (i.e., no roll-ups or ratchets) GMAB rider. In this instance, as can be seen from Figure 8.15, the insurance company is exposed to the risk that the value of the investment drops below the initial deposit on GMAB maturity and the annuitant is alive to receive this benefit. To value this benefit, as in Chapter 8,1 will assume that In ST is normally distributed with a mean of (T – t) and variance of .

As outlined in Chapter 8, an investment into a unit of fund value with a GMAB rider without any roll-up or ratchet features that is attached to a standard GMDB policy can be reinterpreted as the sum of

■ Investment into a fund unit.

■ At-the-money put option maturing at the time of the GMAB rider, provided the annuitant is still alive.

■ At-the-money put option expiring at the time of death (where the rider falls away at the time of death provided this death happens before GMAB rider maturity).

Since I have already discussed the valuation of the death benefit option embedded in a standard GMDB product in Chapter 8,1 will focus my attention on the valuation of a GMAB rider and the impact of policyholder optionality on this valuation.

Using the notations of Chapter 8, assuming that the annuitant is x years old and the GMAB has a y-year maturity, the value of the embedded maturity option is given by the expression

(9.1)

Figure 9.1 shows how the present value of the 7-year option embedded in the GMAB rider changes across varying annuitant ages when the fund is assumed to grow at varying growth rates and no rider fee is charged. To produce Figure 9.1, for the sake of consistency, I used the implied information in Figures 8.5a and 8.5b for the term structure of volatilities and zero rates respectively.

Although Figure 9.1 is the consequence of implementing equation (9.1), the reader should be mindful that this is only an approximation. The reason for this stems from the fact that in putting this illustration together, I assumed no GMAB rider fees and 2 percent continuously compounded Μ & E charges. In practice, as mentioned in Chapter 8, the rider fees are taken off annually while the Μ & E charges are taken off daily from the fund returns before the fund values are reported. Rerunning the analysis that was used to produce Figure 9.1 for the seven-year GMAB rider when the rider fees are 50 basis points and the Μ & E charges are 1.5 percent,

FIGURE 9.1 Present Value of an Option Embedded in a GMAB Rider

FIGURE 9.2 Relative Errors from Approximating Annual Rider Fees

one obtains the results in Figure 9.2,^{[1]} which demonstrates the relative error arising from approximating the annual GMAB rider fee with 0.5 percent of M & E charges so as to result in a total of 2 percent of Μ & E charges.

As can be seen from Figure 9.2, the relative error associated with the approximation increases as the annuitant's age increases – reiterating the need to accurately capture the nuances associated with the cash flows especially when the annuitant gets older.

TABLE 9.1A Inputs Used to Value GMAB Rider Opdonality

Current Fund Value (St)

10

Continuously Compounded Growth Rate (g)

5.00%

Management Fees (q)

1.50%

Accumulation Benefit Fees

0.50%

Current Age

80

Term

7

# of units

1

Tables 9.1a and 9.2a show the inputs and simulations that have been put together to obtain the results in Figure 9.2.

As can be seen from Table 9.1b, a positive value in cell H13 implies that the premiums charged for the maturity benefit rider are insufficient to cover the benefit (when only mortalities and account-value returns are in play) for that particular path. To find the fair value of the rider, one has to do these runs over multiple simulated paths to ensure that on average the value of cell H13 turns out to be 0.

One critical element that has still not been factored in the pricing of the GMAB rider benefit is that associated with lapses or surrenders (something I had briefly discussed in Chapter 8). More precisely, to do the above calculations, I implicitly assumed that the policyholder will not surrender the policy or withdraw any amount from the policy before the GMAB rider maturity date. In practice, to keep the cost of the guarantees down, insurance companies make assumptions about how their policyholders behave to ensure that their pricing reflects the fact that a fraction of them would not be around by the time the GMAB maturity date comes around. Depending on the product design, there are products in the marketplace which allow a policyholder to either surrender the GMAB rider without having to surrender the GMDB contract or surrender the entire contract (i.e., the underlying GMDB and the GMAB rider). Given this backdrop, for the purposes of my discussion, I will henceforth assume that the policyholder is able to surrender only the GMAB rider without impacting the underlying GMDB contract. For the ease of explanation, I will also make the simplifying assumption that the policyholder is only allowed to make a full surrender.^{[2]}

TABLE 9.1B Simulation Layout to Value GMAB Rider Optionality

FIGURE 9.3A Impact of Surrenders on the Present Value of GMAB Riders for Varying Growth Rates

FIGURE 9.3B Impact of Surrenders on the Present Value of GMAB Riders for Varying Lapse Rates

Figure 9.3a shows the impact of the surrenders on the present value of the GMAB rider for varying fund growth rates when the lapse rate is 5 percent.

Consistent with intuition, Figure 9.3a shows that the higher the growth rate of the underlying fund, the lower the value of the embedded option in the GMAB (as the option becomes highly out-of-the-money). Figure 9.3b shows the impact of surrenders on the present value of the GMAB rider for varying

FIGURE 9.3C Impact of Surrenders on Net Present Value of GMAB Riders for Varying Growth Rates When Lapse Rate Is 5 Percent

lapse rates^{[3]} (i.e., 0 percent per year, 5 percent per year, and 10 percent per year) when the growth rate is 5 percent.

As can be seen from Figure 9.3b, the higher the surrender rate, the lower the present value of the GMAB rider – confirming the intuition that the higher the lapse rate, the lower the risks. Redoing the analysis that was done to produce Figures 9.3a and 9.3b so as to understand the impact on the net present value of the GMAB rider (where the net present value is defined to be the difference of the present value of embedded options obtained in Figures 9.3a and 9.3b and the present value of rider fees collected during the life of the GMAB rider), one can arrive at Figures 9.3c, 9.3d, and 9.3e.

As can be observed from Figures 9.3a to 9.3e the insurance company is exposed to risks when the annuitant is alive on GMAB maturity date and when either realized lapse rates turn out to be much lower than what was used to price the guarantee decrease or the underlying fund value turns out to be lower than that initial deposit at the end of seven years. Furthermore, the greater the expected growth rate, expected lapse rates, or rider premium charged, the lower the expected breakeven issue age. As a consequence, one can conclude that if the target market for this product is much younger (e.g., 50), based on the economics associated with the GMAB rider, the insurance company should be charging more than 50 basis points or use a fund with

FIGURE 9.3D Impact of Surrenders on Net Present Value of GMAB Riders for Varying Lapse Rates When Growth Rate Is 5 percent

a higher growth rate or encourage surrendering as much as possible so the breakeven age shifts more to the left (i.e., gets lower).

Although I used a constant lapse rate to illustrate the impact of deterministic lapse rates (as seen in Figures 9.3a to 9.3e), one can also use dynamic lapse rates where the surrenders are driven by factors linked to market behavior. In fact, dynamic lapse rates can be neatly broken down into two categories: economic rationality (i.e., the policyholder is better off surrendering if surrendering offers the policyholder a higher payoff than continuing – a decision that is strictly based on financial market dynamics) and economic irrationality (i.e., the policyholder surrenders based on some

FIGURE 9.3E Impact of Surrenders on Net Present Value of GMAB Riders for Varying Lapse Rates When Rider Fee Is Increased by 50 Bps and Growth Rate Is 5 percent

predefined criteria that are not economically rational – e.g., if the fund value exceeds 120 percent of the deposit). Given the above backdrop, for ease of understanding, I will decompose the entire spectrum of lapse-rates into the following three groups:

1. Economic rational behavior.

2. Noneconomic rational behavior (which also includes static lapse rates).

3. A hybrid of economic and noneconomic rationality.

[1] The relative errors were computed using the expression where exact refers to the number obtained using simulations to precisely capture the annual rider-fee deduction at the beginning of each year and approx refers to the number obtained using equation (9.1).

[2] The consequence of this assumption is that a policyholder is not allowed to partially withdraw from the contract – an assumption that is NOT made by an insurance company in practice. More precisely, in addition to making an assumption on the percentage of full surrenders in each policy year, assumptions on the frequency, amount and timing of partial withdrawals are also made.

[3] These types of lapse rates are called static lapse rates. In practice, practitioners also use dynamic lapse rates (where the word dynamic is used to refer to factors like remaining life, value of the underlying funds and the in-the-moneyness of the rider – among other things).

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