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TABLE 6.3 105 y Fund cash flows under example scenario assuming contract is in force. 0–1 100.00 0.0417 0.427% 99.32 80 0 1–2 99.32 0.0414 4.70% 103.73 80 0 2–3 103.73 0.0432 0.770% 102.67 80 0 3–4 102.67 0.0428 1.685% 100.69 80 0 4–5 100.69 0.0420 1.428% 99.00 80 0 5–6 99.00 0.0413 1.530% 100.27 80 0 6–7 100.27 0.0418 8.098% 108.12 80 0 7–8 108.12 0.0450 6.316% 101.03 80 0 8–9 101.03 0.0421 0.879% 99.89 80 0 9–10 99.89 0.0416 10.708% 110.31 80 0 10–11 110.31 0.0460 6.302% 103.40 80 0 . . . 23–24 148.47 0.0619 7.356% 158.99 80 0 24–25 158.99 0.0662 1.917% 161.63 158.99 0 25–26 161.63 0.0673 7.004% 149.94 158.99 9.05 26–27 149.94 0.0625 4.738% 156.65 158.99 2.34 27–28 156.65 0.0653 0.546% 157.11 158.99 1.88 . . . 141–142 107.01 0.0446 12.339% 119.91 158.99 39.08 142–143 119.91 0.0500 1.251% 121.11 158.99 37.88 143–144 121.11 0.0505 1.206% 122.26 158.99 36.73 144–145 158.99 0.0662 1.649% 155.98 158.99 3.01 145–146 155.98 0.0650 4.362% 162.38 158.99 0 . . . 263–264 471.99 0.1967 6.755% 512.61 158.99 0 GMAB Example tt t t t = = = Ϫ Ϫ ttF M I F GGF(1) ( ) F MtI F Ft M F 11 In Table 6.3, we show the fund at the start of the month, before management charges are deducted, ; the income from the risk premium, ; the interest rate earned on the fund in the th month, ; and the end- year fund, , after deducting management charges and adding the year’s interest. All these figures are calculated assuming that the contract is still in force. In this table starts at $100 at time 0. The total management charge deducted at the start of the year is 0.25, of which 0.0417 ( ) is received as risk-premium income to offset the guarantee cost. The net fund 0 427 percent, leading to an end-year fund of $99.32. This is still greater than the current guarantee of $80, so there is no guarantee liability for death benefits in the first month. ϪϪ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ t t t t ם Ϫ Ϫ – ϪϪ 1 1 0 0 1 1 after expenses is $99.75, which earns a return of I. = 106 = = = = Ϫ F. p q p. p. t All through the first two years, the fund exceeds the guarantee at the end of each month. At the end of the 24th month the first renewal date applies. In this scenario 158 99, compared with the guarantee of $80. There is, therefore, no survival benefit due, and the guarantee value is increased for the renewed 10-year contract to the month-end fund value, $158.99. In the 10 years following the first renewal under this single stock return scenario, the index rises very slowly. After the guarantee has been reset to the fund value, the fund value drifts below the new guarantee level, leaving a potential death benefit liability. In fact, over the entire 10-year period the accumulation is only 3.8 percent. Since expenses of 0.25 percent per month are deducted from the fund, by the end of 144 months the fund has fallen $36.73 below the guarantee that was set at the end of 24 months. At the second renewal, then, the insurer must pay the difference to make the fund up to the guarantee, provided the policy is still in force. Therefore, at the start of the 145th month the fund has been increased to the guarantee value of $158.99. Since the fund was less than the guarantee at the renewal date, the guarantee remains at $158.99 for the final 10 years of the contract. After the 145th month the fund is never again lower than the guarantee value, and there is no further liability. However, the risk-premium portion of the management charge continues to be collected at the start of each month. In Table 6.4, we show the liability cash flows under this particular scenario. Each month a negative cash flow comes from the income from the risk-premium management charge. The amount from the third column of Table 6.3 is multiplied by the survival probability for the expected cash flow. )isgreater than zero at the month end. For example, if the policyholder dies in the ) $9.05. Since we allow for mortality deterministically, we value this death benefit at the month end by multiplying by the probability of death in the 26th month, , which is an expected payment of $0.00273. The probability of the policyholder’s surviving, in force, to the second renewal date is 0 35212, and the payment due under the survival benefit is $36.73, leading to an expected cash flow under the survival benefit of 36 73 $12.93. In the final column, the cash flows from the th month are discounted to the start of the projection at the assumed risk-free force of interest of 6 percent per year. The management charge income is discounted from the start of the month, and any death or survival benefit is discounted from the end of the month. x t t d x x x ͉ Ϫ ␶ ␶ ␶ MODELING THE GUARANTEE LIABILITY 24 () 1 26 () 25 () 144 () 144 Adeathbenefitliabilityarisesinmonthsforwhich( GF – 26thmonth,thedeathbenefitdueatthemonthendwouldbe( GF – TABLE 6.4 107 yyyyy yyyyy yyyyy Expected nonfund cash flows allowing for survivorship. 0–1 1.00000 0.000287 0 0.0417 0.0417 1–2 0.99307 0.000288 0 0.0411 0.0409 2–3 0.98619 0.000289 0 0.0426 0.0422 3–4 0.97934 0.000289 0 0.0419 0.0413 4–5 0.97255 0.000290 0 0.0408 0.0400 5–6 0.96580 0.000290 0 0.0398 0.0389 6–7 0.95909 0.000291 0 0.0401 0.0389 7–8 0.95243 0.000292 0 0.0429 0.0414 8–9 0.94581 0.00029 0 0.0398 0.0383 9–10 0.93923 0.000293 0 0.0391 0.0374 10–11 0.93270 0.000293 0 0.0429 0.0408 . . . 23–24 0.85157 0.000301 0 0.0527 0.0470 24–25 0.84561 0.000301 0 0 0.0560 0.0497 25–26 0.83970 0.000302 0.00273 0.0538 0.0475 26–27 0.83382 0.000303 0.00071 0.0514 0.0451 27–28 0.82797 0.000303 0.00057 0.0535 0.0467 . . . 141–142 0.36032 0.000359 0.01402 0.0021 0.0010 142–143 0.35757 0.000359 0.01360 0.0043 0.0021 143–144 0.35483 0.000359 0.01319 12.932 12.9276 6.2925 144–145 0.35212 0.000359 0.00183 0.0222 0.0108 145–146 0.34942 0.000360 0 0.0228 0.0110 . . . 263–264 0.12938 0.000351 0 0 0.0254 0.0068 ␶ GMAB Example dt tt tt xx ttp q CCv In-Force Mortality Expected Expected Probability Probability Death Survival ( 1) Benefit Benefit 11 For this example scenario, the net present value (NPV) of the guarantee liability is $2.845. The contribution of the death benefit guarantee is $1.338, and the survival benefit expected present value is $6.295. The management charge income offsets these expenses by $4.788. In fact, this example is unusual; in most scenarios there is no survival benefit at all, and the management charge income generally exceeds the expected outgo on the death benefit, leading to a negative NPV of the guarantee liability. Ϫ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ϪϪ ͉ ϪϪ The NPV of the Liability STOCHASTIC SIMULATION OF LIABILITY CASH FLOWS 108 ס ם ס םם ם Fx x x Fx , x,x, ,x x x Fx Fx f xx For a stochastic analysis of the guarantee liability, we repeat the calculations described in the previous section many times using different sequences of investment returns. If we consider a contract with monthly cash flows over, say, 22 years (such as the example above), applying 10,000 different simulations will give a lot of information and there are different ways of analyzing the output. In this section, we examine how to summarize that information and give an example of the simulated liability for the GMAB contract of the example in Tables 6.3 and 6.4. One method of summarizing the output is to look at the simulated NPVs for the liability under each simulation. As an example, we have repeated the GMAB example above for 10,000 simulations, all generated using the same stock return model. The range of net liability present values generated The principle of stochastic simulation is that the simulated empirical distribution function is taken as an estimate of the true underlying distribu- tion function. This means that, for example, since 8,620 projections out of 10,000 produced a negative NPV, the probability that the NPV is negative is estimated at 0.8620. We can, therefore, generate a distribution function ˜ for the NPVs. Let ( ) denote the empirical distribution function for the NPV at some value . Then Number of simulations giving NPV ˜ () 10 000 This gives the distribution function in Figure 6.1. It may be easier to visualize the distribution from the simulated density function. The density can be estimated from the distribution using the procedure: Partition the range of the NPV output into, say, 100 intervals, indicated by ( ). The intervals do not have to be equal; for best results use wider intervals in the tails and smaller intervals in the center of the distribution. The estimated density function at the partition midpoints is ˜˜ ()() ˜ 2 tt t t tt 1. 2. ΂΃ Ͻ Ϫ Ϫ MODELING THE GUARANTEE LIABILITY 0 1 100 11 1 is – $24.6to$37.0.ThenumberofNPVsabovezero(implyingarawloss onthecontract)is1,380.ThemeanNPVis – $4.0. •••• ••• • ••••• •••••• ••••• ••••• ••••• ••••• ••••• •••• ••••• ••••• ••• •••••• •• ••••• ••••• •••• ••••• ••••• •••• ••• ••• ••••• ••••• •••• •• ••• •••• ••• ••• ••• •• ••• ••• ••• •••• ••• •• •• •••• •••• •• •• ••• •• ••• ••••• • ••• •••• ••• ••• •• •••• •• ••• •••• ••• •• ••••• •••• •• ••• • ••• •• ••• ••• •• •• •• •• •• •• •• •• •• •• •• • ••• •• • •• • ••• ••• •• •• •• •• •• •• •• •• •• •• •• •• • • ••• •• • •• •• •• • •• •• ••• •• • •• •• •• •• •• ••• • •• • • •• •• •• • •• • •• •• •• • •• •• •• • •• •• •• • •• •• •• •• •• •• • •• •• ••• •• •• • •• • •• •• •• •• •• • ••• • •• •• •• • •• • • •• • • • •• • •• ••• •• •• •• • • • • • •• •• • ••• •• • •• • •• • •• •• • •• • •• •• • • • •• • •• •• • •• • • •• •• • • •• •• • •• • • •• • • • •• •• • •• • •• • •• • • • • •• • •• • • • • • •• • •• • •• • • • •• • •• • •• • •• • •• • •• • • •• • •• • •• • •• • •• •• • •• • •• •• • • • •• • •• • •• • •• • • • • •• • •• • •• • •• • •• • • • •• •• • •• • •• • • •• • •• • •• • •• • • •• • •• • •• • • • • •• • • •• • •• • •• • • •• • • •• • • •• • • • •• • • • • • •• • • • • •• • • •• • •• • • •• • • •• • • • • • •• • •• • • •• • •• • • •• • • •• • • •• • • •• • • •• • •• • • •• • • •• • • • •• • • • • • • • • • •• • • • • • • • •• • • •• • • •• • •• • • • • •• • • • •• • • • •• • • • •• • •• • • •• • • •• • • • • •• • • • •• • • •• • • •• • • • •• • • • • •• • • •• • • • • •• • • •• • • • • •• • • • • • • •• • • • •• • • • •• • • • • •• • • • • •• • • • •• • • • • • •• • • •• • • • • •• • • • •• • • • •• • • • •• • • • • • • • • •• • • • •• • • • •• • • •• • • • •• • • • •• • • •• • • • •• • • • • •• • • • •• • • • •• • • • •• • • •• • • • •• • • • • • •• • • • • • •• • • • • •• • • • •• • • • • • • • • •• • • • • • • • •• • • • •• • • •• • • • • • • • •• • • • •• • • • • •• • • •• • • • • • • • • •• • • • • • •• • • • • • • • • • •• • • • • • •• • • • •• • • • •• • • • • • •• • • • • • • • • • • •• • • •• • • • • •• • • • • • • •• • • • •• • • • • •• • • • • •• • • • • •• • • • • • •• • • • •• • • • • • •• • • • •• • • • •• • • • • • •• • • • • •• • • • • • •• • • • •• • • • • • • • • • • • • •• • • • • •• • • • • •• • • • • • • • •• • • • • • •• • • • •• • • • •• • • • • • • • • • •• • • • •• • • •• • • • • •• • • • • •• • • • • •• • • • • •• • • • • • • • • • • •• • • • • • • • •• • • • •• • • • • • • •• • • • • • •• • • • • • •• • • • •• • • • • •• • • • • •• • • • • • • • • • • • •• • • • • • • • • • • • • • • • • •• • • • •• • • • • • •• • • • •• • • • • •• • • • • • • •• • • • •• • • • • • • • • • • • • • •• • • • • • • •• • • • • •• • • • • • •• • • • • •• • • • • •• • • • • • •• • • • • • • • • • • • • •• • • • • • • •• • • • • • • • • • • • • • • • • • • • •• • • • • •• • • • • • •• • • • • • •• • • • • • •• • • • • • •• • • • •• • • • •• • • • • •• • • • • • • •• • • • • • •• • • • • •• • • • • • • •• • • • •• • • • • • • •• • • •• • • • • • •• • • • • •• • • • • • • • • •• • • • • • • •• • • • • • • • • • • • • •• • • • • •• • • • • • • • • • •• • • • • •• • • •• • • • • • • • • • • •• • • • • • • •• • • • • •• • • • • •• • • • •• • • • • •• • • • • • •• • • • • • •• • • • • • •• • • • • •• • • • • • • • • • • • •• • • • • • • •• • • • • •• • • • • •• • • • • • •• • • • • •• • • • • • • • • • • • •• • • • • • •• • • • • • •• • • • • • •• • • • • •• • • • • • • •• • • • •• • • • • •• • • • •• • • • • • • • • • • •• • • • • • •• • • • • • •• • • • • •• • • • •• • • • •• • • • •• • • • • • • •• • • • • •• • • • • •• • • • •• • • • • •• • • • • •• • • • • • • • • • • • • • • • • • •• • • • • • • • •• • • • •• • • • •• • • • • • •• • • • • •• • • • •• • • • • •• • • • • • • •• • • • • • • • • •• • • • • • • •• • • • • •• • • • • • •• • • • •• • • • • • • • • • • •• • • • • • • • •• • • • • •• • • • • •• • • • • • • • •• • • • • • • • • • •• • • • •• • • • • •• • • • •• • • • •• • • •• • • •• • • • •• • • •• • •• • •• • • •• • •• • • • •• • •• • •• • • • • •• • •• • • •• • • • • •• • •• • • • • • • •• • • •• • • •• • • •• • • • •• • • •• • •• • • •• • • • • • •• • • •• • •• • • •• •• • • • •• • •• • •• • • •• •• • • •• • •• • •• • •• • •• • • •• • •• •• • •• • •• •• • •• • •• • • •• • •• •• •• •• •• • •• • •• • •• •• • •• •• •• •• • •• •• ••• •• •• •• •• •• •• • •• ••• •• •• ••• •• • •• ••• • •• ••• •• •••• •• •• ••• •• •• •• •• ••• •• •••• • •• •••• ••• ••• ••• •• •• ••• •• •••• •• ••• •• •• •• • •• ••• •• ••• •••• ••• •• •••• •••• •• •••• ••• ••• ••• ••• •• •••• •••• •••• ••••• ••••• •• •••• •••• ••••• •••• ••••• •• •••• •••• ••• ••••• ••••• ••••• ••••• •••••• •••• ••••• •••••• ••••• ••••• ••• • •• • NPV Simulated Distribution Function –20 –10 0 10 0.0 0.2 0.4 0.6 0.8 1.0 FIGURE 6.1 109 Simulated distribution function for GMAB NPV example. Stochastic Simulation of Liability Cash Flows Altering the partition will give more or less smoothness in the function. The simulated density function for the 10,000 simulations of the GMAB NPV of the liability is presented in the first diagram of Figure 6.2; in the right-hand diagram we show a smoothed version. The density function demonstrates that although most of the distribu- tion lies in the area with a negative liability value, there is a substantial right tail to the distribution indicating a small possibility of quite a large liability, relative to the starting fund value of $100. We can compare the distribution of liabilities under this contract with other similar contracts—for example, with a two-year contract with no renewals, otherwise identical to that projected in Figures 6.1 and 6.2. A set of 10,000 simulations of the two-year contract produced a range when renewals are taken into consideration. Thus, at first inspection it looks advantageous to incorporate the renewal option—after all, if the contract continues for 20 years, that’s a lot more premium collected with only a relatively small risk of a guarantee payout. But, when we take risk into consideration, the situation does not so clearly favor the with-renewal ofoutcomesfortheNPVoftheliabilityof–$1.6to$37.1,compared with–$24.6to$37.0forthecontractincludingrenewals.Themeanof theNPVsunderthetwo-yearcontractis–$0.30comparedwith–$4.00 NPV Simulated Density Function –20 –10 0 10 20 30 0.0 0.05 0.10 0.15 NPV Smoothed Simulated Density Function –20 –10 0 10 20 30 0.0 0.05 0.10 0.15 FIGURE 6.2 Liability Cash-Flow Analysis 110 Simulated probability density function for GMAB NPV example; original and smoothed. contract. The simulated probability of a positive liability NPV under the two-year contract is 7.5 percent, compared with 13.8 percent for the contract with renewals. So, if we ignore the renewal option, we ignore both upside (an extra 20 years of premiums) and downside (two further potential liabilities under the maturity guarantee). In addition to the NPV, which is a summary of the nonfund cash flows for the contract, we can use simulation to build a picture of the pattern of cash flows that might be expected under a contract. In the GMAB example, the nonfund cash flows are the management charge income, the death benefit outgo, and the maturity benefit outgo. Any picture of all three sources is dominated by the rare but relatively very large payments at the renewal dates. In Figure 6.3, we show 40 example projections of the cash flows for the GMAB contract. The income and the death benefit outgo are on the same scale, but the maturity benefit outgo is on a very different scale. For this contract, the death benefit rarely exceeds the management charge. An interesting feature of the death benefit outgo is the fact that the larger payments increase after each renewal. As the guarantee moves to the fund level, both the frequency and severity of the death benefit liability increase. In most projections there is no maturity benefit outgo, but when there is a liability, it may be very much larger than the management charge income. The cash flows plotted allow for survival and are not discounted. This type of cash-flow analysis can help with planning of appropriate asset strategy, as well as product design and marketing. We can also examine the projections to explore the nature of the vulnerability under the contract. For a simple GMMB with no resets or renewals, the risk is clearly that returns over the entire contract duration are very low. For the GMAB, there is an additional risk that returns start high but become weaker after the fund and guarantee have been equalized at a renewal date. By isolating the MODELING THE GUARANTEE LIABILITY 0 50 100 150 200 250 0.0 0.10 0 50 100 150 200 250 0.0 0.10 0.20 Management Charge Income Projection Month % of Initial Fund 0.20 Projection Month % of Initial Fund Death Benefit Outgo 0 50 100 150 200 250 0 5 15 25 Projection Month % of Initial Fund Maturity Benefit Outgo FIGURE 6.3 111 Simulated projections of nonfund cash flows for GMAB contract. Stochastic Simulation of Liability Cash Flows stock return projections for those cases where a maturity benefit was paid, we may be able to identify more accurately what the risks are in terms of the stock returns. In Figure 6.4, we show the log stock index for the simulations leading to a maturity benefit at the first, second, and third renewal date. In the final diagram we show 100 paths where there was no maturity benefit liability. The risk for the two-year maturity benefit is, essentially, a catastrophic stock return in the early part of the projection. This is simply a two-year put option, well out-of-the-money because at the start of the projection the guarantee is assumed to be only 80 percent of the fund value. For the second and third maturity benefits, the stock index paths are flat or declining, on average, from the previous renewal date to the payment date. For this contract the 10-year accumulation factor has a substantial influence on the overall liability. In addition, the two-year accumulation factor plays the major role in the liability at the first renewal date. The calibration procedure discussed in Chapter 4 considers accumulation factors between 1 and 10 years to try to capture this risk. However, the right-tail risk is not tested in that procedure. 0 50 100 150 200 250 0 2 4 6 Projection Month Log Simulated Stock Index Stock Index for Early Maturity Benefit 0 50 100 150 200 250 0 2 4 6 Stock Index for Middle Maturity Benefit Projection Month Log Simulated Stock Index 0 50 100 150 200 250 0 2 4 6 Stock Index for Final Maturity Benefit Projection Month Log Simulated Stock Index 0 50 100 150 200 250 0 2 4 6 Stock Index, no Maturity Benefit Projection Month Log Simulated Stock Index FIGURE 6.4 THE VOLUNTARY RESET 112 Simulated projections of log-stock index separated by maturity benefit liability. A common feature of the more generous segregated fund contracts in Canada is a voluntary reset of the guarantee. The policyholder may opt at certain times to reset the guarantee to the current fund value, or some percentage of it; the term would normally be extended. The simple way to explain the voluntary reset is as a lapse and reentry option. Suppose that a policyholder is six years into a GMAB contract, with, say, two rollover dates before final maturity. The next rollover date is in four years. Stocks have performed well, and the separate fund is now worth, say, 180 percent of the guarantee. If the same contract is still offered, the policyholder could lapse the contract, receive the fund value, and immediately reinvest in a new contract with the same fund value but with guarantee equal to the current fund value. The term to the next rollover under a new contract would generally be 10 years, so the policyholder replaces the rollover in 4 years with another in 10 years with a higher guarantee. MODELING THE GUARANTEE LIABILITY TABLE 6.5 113 Quantiles for the NPV of the guarantee liability for a GMAB contract with resets; percentage of starting-fund value. No resets 10.7 7.0 5.2 3.3 5.1 2 resets per year 115% 9.9 6.2 4.2 1.1 7.8 No limit 105% 9.5 5.8 3.9 1.1 8.2 No limit 115% 9.7 6.2 4.2 1.3 8.0 No limit 130% 10.1 6.5 4.4 1.6 7.6 The Voluntary Reset Reset Assumption Threshold 5% 25% 50% 75% 95% Perhaps in order to avoid the lapse and reentry issue, many insurers wrote the option into the contract. A typical reset feature would allow the policyholder to reset the guarantee to the current fund value; the next rollover date is, then, extended to 10 years from the reset date. The number of resets per year may be restricted, or the option may be available only on certain dates. The reset feature can be incorporated in the liability modeling without too much extra effort, although we need to make some somewhat speculative assumptions about how policyholders will choose to exercise the option. The assumptions used to produce the figures in this section are described below, but it should be emphasized that modeling policyholder behavior is an enormous open problem. So, we adapt the GMAB contract described in the previous section to incorporate resets. We assume the same true term for the contract, and that the policyholder does not reset in the final 10 years. We assume also that the policyholder will reset when the ratio of the fund to the guarantee hits a certain threshold—we explore the effect of varying this threshold later in this section. We also assume the effect of restricting the maximum number of resets each year. The figures given are for a GMAB with a 10-year nominal term (between rollover terms, if the policyholder does not reset) and a 30-year effective term. The starting fund to guarantee ratio is 1.0. In Table 6.5, some quantiles of the NPV distributions are given for the various reset assumptions. These result from identical sets of 10,000 scenarios. Figures are per $100 starting fund. This table shows that the effect of the reset option is not very large, although the right-tail difference is sufficiently significant that it should be taken into consideration. This will be quantified in Chapter 9. The effect of different threshold choices is relatively small, as is the choice in the policy design of restricting the number of resets permitted per year, although that will clearly affect the expenses associated with maintaining the policy. Having a restricted number of possible resets does not matter much because infrequent use of the reset appears to be the best strategy. Ϫ ϪϪϪ ϪϪϪϪ ϪϪϪϪ ϪϪϪϪ Ϫ ϪϪϪ 0 50 100 150 200 250 0 10 20 30 Cash Flows, No Resets Projection Month % o f Initia l Fun d 0 50 100 150 200 250 0 10 20 30 Projection Month % o f Initia l Fun d Cash Flows, with Resets FIGURE 6.5 114 Simulated cash flows, with and without resets. Resetting every time the fund exceeds 105 percent of the guarantee may lead to lost rollover opportunities, so that the contract may pay out less than the contract without resets. From these figures it does not appear that the reset feature is all that valuable, on average, but the tail risk is significantly increased (as repre- sented by the 95th percentile). In addition, the reset will constrain the risk management of the contract, for two major reasons. The first is a liquidity issue—without the reset option, the maturity benefit is due at dates set at issue. Allowing resets means that the maturity benefit dates could arise at any time after the first 10 years of the contract have expired. This will make planning more difficult. For example, in Figure 6.5 we show 50 simulated cash flows from a contract without resets; then, with everything else equal, the same contract cash flows are plotted if resets are permitted, and a threshold of 105 percent is used as a reset threshold. The other problem with voluntary resets is that the option has the effect of concentrating risk across cohorts. Consider a GMAB policy written in 2000 and another written in 2003. Without resets, there is a certain amount of time diversification here, because the first rollover dates for these contracts are 2010 and 2013, respectively, and it is unlikely that very poor stock returns will affect both contracts. Now assume that both policies carry the reset option and that stocks have a particularly good year in 2004. Both policyholders reset at the end of 2004, which means that both now have identical rollover dates at the end of 2014, and the time diversification is lost. In the light of these problems, the voluntary reset feature is becoming less common in new policy design. For a more technical discussion of the financial engineering approach to risk management for the reset option see Windcliff et al. (2001) and (2002). MODELING THE GUARANTEE LIABILITY [...]... the moment, we will derive the hedge portfolio at time t = 0 for a GMAB with renewals at t1 > 0 and at t2 > t1 , maturing at t3 > t2 The starting guarantee is G, and the starting separate fund is F0 At t1 , if the fund value is more than G, then the guarantee is reset to the fund level On the other hand, if the guarantee is greater than the fund at t1 , then the insurer pays the difference into the. .. 5 10 20 60 80 100 120 0.0 062 0.0393 0.1395 0.3329 0.0307 0.1194 0.3154 0 .64 26 0.0957 0.2758 0 .60 58 1.1045 138 DYNAMIC HEDGING FOR SEPARATE ACCOUNT GUARANTEES so the first part is the risk-free asset portion of the hedge portfolio, whereas the second part is the risky asset portion The GMDB costs are rather less than the GMMB for this sample contract, even for an in -the- money option, because the mortality... and the option have the same payout at time t = 1, then they must, by the no-arbitrage principle, also have the same price at time t = 0 Hence the price of the option at t = 0 must be the same as the price of the matching portfolio at t = 0; the option price is 2.78114 119 Replication and No-Arbitrage Pricing A very interesting feature of the result is that we never needed to know or specify the probability... to reflect the fact that the underlying asset is not the stock price itself, but the segregated fund value, and that this differs from the stock price through the deduction of the management charge In addition, for the GMDB, the option matures at death, giving a random term to maturity rather than the fixed term of a European option For the GMAB, payable on death or maturity, the payoff of the option... the valuation date The table shows that, even for a fund that is significantly less than the guarantee (i.e., the option is in -the- money) at the valuation date, if the term is long enough, the hedge cost is small This happens because the cost of a put option decreases over the long term (though it increases in the short term) and because of the survival effect On the other hand, the shorter-dated options... substantial cost, even when the guarantee is only 80 percent of the fund value at the valuation date Black-Scholes Formula for the GMDB Under the GMDB the liability is identical to the GMMB, except that the maturity date is contingent on the policyholder’s death rather than his or her survival The term of the option is, therefore, itself a random variable Let BSP0 (T ) denote the cost at time 0 of a put... goes down, because in that case the purchaser receives 100 under the contract compared with 85 in the market If the stock price goes up, the purchaser can sell the asset in the market for 110 and, therefore, has no incentive to exercise the option and sell for only 100 The option seller then has a liability of K – Sd = 15 if the market goes down (since they have to buy the stock at K but end up with... ‫ ס‬Ϫ0 .6 (7.4) (7.5) This solution means that if the option seller buys the portfolio at time 0 that consists of a short holding of –0 .6 units of stock (with price – $60 , since S0 = 100) and a long holding of aeϪr = 62 .78114 in the risk-free asset, then whether the stock goes up or down, the portfolio will exactly meet the option liability The option is perfectly hedged by this portfolio Since the portfolio... to fund Pu if the market rises and Pd if the market falls Then at time 1 we rearrange the portfolio, investing au eϪr in the risk-free asset and bu Su in the risky asset if the risky-asset value rises, or ad eϪr in the risk-free asset and bd Sd in the risky-asset if the risky asset price falls Either way, no extra money is required at time 1 The rearranged portfolio will exactly meet the option liability... than the strike price K, the option holder buys the share for K, and may immediately sell for ST , giving a payoff at maturity of (ST – K) Obviously, if K > ST , then the option is not exercised and the contract expires with zero payoff The price of the call option at time t < T is found in the same way as for the put option, shown earlier, by taking the expectation under the risk-neutral measure of the . 121.11 0.0505 1.2 06% 122. 26 158.99 36. 73 144–145 158.99 0. 066 2 1 .64 9% 155.98 158.99 3.01 145–1 46 155.98 0. 065 0 4. 362 % 162 .38 158.99 0 . . . 263 – 264 471.99 0.1 967 6. 755% 512 .61 158.99 0 GMAB Example tt. 0 9–10 99.89 0.04 16 10.708% 110.31 80 0 10–11 110.31 0.0 460 6. 302% 103.40 80 0 . . . 23–24 148.47 0. 061 9 7.3 56% 158.99 80 0 24–25 158.99 0. 066 2 1.917% 161 .63 158.99 0 25– 26 161 .63 0. 067 3 7.004% 149.94. consideration, the situation does not so clearly favor the with-renewal ofoutcomesfortheNPVoftheliabilityof–$1.6to$37.1,compared with–$24.6to$37.0forthecontractincludingrenewals.Themeanof theNPVsunderthetwo-yearcontractis–$0.30comparedwith–$4.00 NPV Simulated Density