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245 PTP Option Valuation dividends This makes call options cheaper, because the replicating portfolio can use the incoming dividends to increase the stock part of the replicating portfolio This is described in Chapter 7, with the formula given in equation 7.36 for an option on the stock with price St at t, where dividends are paid continuously on the stock at a rate of d per year Using the standard Black-Scholes call option formula, with allowance for dividend income, the cost of the option at the inception of the contract, using PTP indexation, assuming index linking to a regular index without reinvested dividends, and where dividends are received continuously at a rate of d per year, is H0 ‫ס‬ ␣P S0 eϪd n ⌽(d1 ) Ϫ Kptp eϪrn ⌽(d2 ) S0 Ά ΂ ‫ ס‬P eϪd n ␣ ⌽(d1 ) Ϫ ΂ P Ϫ (1 Ϫ ␣ )΃ e G · Ϫrn (13.5) ⌽(d2 ) ΃ ‫ ␣ ס‬P eϪd n ⌽(d1 ) Ϫ ΘG Ϫ P(1 Ϫ ␣ )Ι eϪrn ⌽(d2 ) (13.6) (13.7) where d1 ‫ס‬ ‫ס‬ log(S0 ΋ Kptp ) ‫( ם‬r Ϫ d ‫ ␴ ם‬΋ 2)n ␴Ίn log(␣ P΋ (G Ϫ P (1 Ϫ ␣ ))) ‫( ם‬r Ϫ d ‫ ␴ ם‬΋ 2)n ␴Ίn (13.8) (13.9) and d2 ‫ ס‬d1 Ϫ ␴Ίn We can use the formula to value the option cost of equity indexation for a standard PTP EIA contract The details are as follows: Seven-year contract with PTP indexation Sixty percent participation rate A single premium of $100 Three percent per year minimum return guarantee, applied to 95 percent of the premium In addition, we need parameters for the Black-Scholes call option value; say, a risk-free force of interest of r ‫ ס‬percent per year, a dividend rate of d ‫ ס‬percent, and a volatility of ␴ ‫ 02 ס‬percent The cost of the call option for the contract is $11.567 Using a percent interest spread, the 246 EQUITY-INDEXED ANNUITIES TABLE 13.1 Break-even participation rates for seven-year PTP EIA Volatility, ␴ ‫ء‬ Interest Spread Available 0.20 0.25 0.30 1% 2% 3% 59.5% 81.3% 101.3% 51.1% 70.5% 88.4% 44.6% 62.1% 78.2% ‫ء‬ Net of non-option expenses funds available to fund the option are $17.41, which is substantially on the profitable side of the break-even point Most authors treat the participation rate as the variable controlling the cost of the embedded option The guaranteed minimum benefit is generally not used to adjust costs If the insurer approaches an external option vendor to provide a static hedge for the contract, the option price quoted may be based on a high volatility value, because option vendors use volatility margin for profit and contingencies In Table 13.1 we show the break-even participation rates for the seven-year PTP contract described in this section, assuming a $100 premium, of which the interest spread available to fund the option is between percent and percent, where an available interest spread of ␦ implies funds available of P Ϫ (0.95)PeϪ␦ n There is no allowance in these figures for lapses or deaths; incorporating these assumptions would reduce the cost of the option and increase the break-even participation rates As a relatively new contract, there is little lapse experience available Because the market offers a range of terms, and the standard contract is relatively short compared with most variable annuity contracts, it is expected that lower lapse rates will apply The participation rates in the middle row of Table 13.1 correspond approximately to those offered in the market; particularly for values of ␴ of around 25 percent Tiong (2001) Tiong (2001) is a well-known paper in U.S actuarial circles giving valuation formulae for some options, including PTP and CAR The break-even participation rates in Tiong’s work for the PTP option are somewhat different from those in Table 13.1, as is her valuation formula, and it is worth exploring briefly why Tiong values embedded options in EIAs by a more circuitous route than we have used, using Esscher transforms This is a device to find the market price For the PTP and CAR contracts there does not appear to be any advantage over the usual method of expectation under Q-measure 247 Compound Annual Ratchet Valuation for these contracts, although there may be for others Within the regular Black-Scholes-Merton framework, where stock prices follow a geometric Brownian motion process, the Esscher transform method must give the same results as the standard methodology of taking the Q-measure expectation of the discounted payoff So this is not a source of difference in the results The first difference between the results in this chapter and Tiong’s results is that Tiong assumes that the guaranteed minimum interest rate applies to 90 percent of the premium, where we have assumed 95 percent, so that the guarantee appears more expensive here Second, Tiong assumes that the entire difference between the risk-free rate and the guaranteed rate is available to fund the option, so that her figures correspond to the bottom row of the table The third difference—and this is the reason for the difference in the final valuation formulae—is that Tiong is valuing a different option in her section on PTP contracts The participation rate is applied to the loge of the stock index appreciation; that is, she values the payoff: ␣ ‫ם‬ ΂΂ ΃ ΃ Sn P S0 ϪG The first term can be expanded using the binomial theorem, showing that the standard payoff under the equity indexation, given in equation 13.1, corresponds to the first two terms in the power series The contract valued by Tiong is generally smaller than the true EIA payoff for ␣ Ͻ 1, because the third term in the binomial expansion of (Sn ΋ S0 )␣ is ␣ (␣ Ϫ 1) Sn Ϫ1 S0 2! ΂ ΃ which is negative for ␣ Ͻ All of these differences work the same way, so that the break-even participation rates in Tiong’s work are rather higher than those found in this section COMPOUND ANNUAL RATCHET VALUATION Viewed as a derivative security, the annual ratchet benefit is an option on an option That is, the payout is the greater of the ratcheted premium, in the compound case: n Ά ΂ ΂S P Β ‫ ם‬max ␣ t‫1ס‬ St t Ϫ1 ΃ ΃· ‫ ס‬RP Ϫ ,0 (say) 248 EQUITY-INDEXED ANNUITIES and the fixed interest guarantee G ‫ 59.0 ס‬P(1.03)n So we can write the option benefit, which is the payout required in addition to the fixed interest guarantee, as H ‫ ס‬max(RP Ϫ G, 0) which is clearly a call option on the benefit RP, which is a function of the stock index St However, the payout RP itself involves an option, referred to as a ratchet or ladder option in the exotic derivatives literature In the compound case, we can calculate the value of the ratcheted premium as an option, even with the cap applied The simple ratcheted premium option cannot be valued analytically The difference is that multiplying lognormal random variables in the compound case gives another lognormal random variable, but adding them in the simple case requires the distribution of the sum of dependent lognormal random variables, which has no manageable analytic form However, even in the compound form, the additional guarantee of G, called the life-of-contract guarantee by Boyle and Tan (2002), means that no analytic form for the replicating portfolio is available In this section, we will derive the formula for the compound ratchet and explore whether this may be used as an approximation for the CAR with life-of-contract guarantee, which is the most common form of annual ratchet EIA CAR without Life-of-Contract Guarantee Without the life-of-contract guarantee, the benefit under the CAR contract is the ratcheted premium RP, which can be written: n Ά ΂ ΂S RP ‫ ס‬P Β ‫ ם‬max ␣ t‫1ס‬ St t Ϫ1 ΃ ΃· Ϫ ,0 (13.10) To find the value of the ratcheted premium using Black-Scholes principles we take expectation under the risk-neutral distribution of the discounted payout The only result we need is the standard Black-Scholes call option formula Under the normal Black-Scholes assumptions, St ΋ StϪ1 are independent and identically distributed for t ‫ ,2 ,1 ס‬under the unique Q-measure This means that we can replace each term in the product in equation 13.10 with its expectation, using independence, and that all the expectations are the same, because the annual accumulations St ΋ StϪ1 are identically 249 Compound Annual Ratchet Valuation distributed So the value of the RP option is H ‫ ס‬EQ ͫ eϪrn (RP)ͬ ΄Β Ά n ‫ ס‬P EQ ΂ ΂S eϪr ‫ ם‬max ␣ t‫1ס‬ n (13.11) Ά ͫ St t Ϫ1 ΂ ΂S St ‫ ס‬P Β eϪr ‫ ם‬EQ eϪr max ␣ t‫1ס‬ Now ͫ EQ eϪr max ΂΂S St t Ϫ1 ΃ ΃· ΅ Ϫ ,0 t Ϫ1 ΃ ΃ͬ· Ϫ ,0 (13.12) (13.13) ΃ ΃ͬ Ϫ ,0 is the value of a one-year call option on the stock St , with initial stock value and strike price both equal to 1.0 This comes from the fact that St ΋ StϪ1 has the same distribution as S1 ΋ S0 , and if we assume (without losing generality because it is an index) that S0 ‫ ,1 ס‬the expectation becomes EQ ͫ eϪr max ΘS1 Ϫ 1, 0Ιͬ (13.14) which is clearly the one-year call option value, with unit strike and unit current stock price Using the Black-Scholes call-option formula, allowing for dividends of d per year, we have Ά ␣ EQ ͫ eϪr max ΘS1 Ϫ 1, 0Ιͬ ‫ ␣ ס‬eϪd⌽(d1 ) Ϫ eϪr⌽(d2 ) · (13.15) where d1 ‫ס‬ r Ϫ d ‫2 ␴ ם‬΋ and d2 ‫ ס‬d1 Ϫ ␴ ␴ So the value of the ratcheted premium option is Ά · H ‫ ס‬P eϪr ‫΂ ␣ ם‬eϪd⌽(d1 ) Ϫ eϪr⌽(d2 )΃ n (13.16) In Table 13.2, we show the results for an initial premium of P ‫,001 ס‬ using the following parameters: r ‫ ;60.0 ס‬d ‫ ;20.0 ס‬the term n is years; the volatility is ␴ ‫ ,52.0 ,2.0 ס‬and 0.3; and we show a range of participation rates The value given is the market value of the entire ratcheted premium payout What these figures show is how much it would cost to provide the 250 EQUITY-INDEXED ANNUITIES TABLE 13.2 Ratchet premium option values, $100 initial premium RP Value Participation Rate ␴ ‫02.0 ס‬ ␴ ‫52.0 ס‬ ␴ ‫03.0 ס‬ 0.4 0.5 0.6 0.7 87.24 93.48 100.10 107.11 92.01 99.84 108.24 117.24 97.02 106.60 116.97 128.20 ratcheted premium payout, under the standard Black-Scholes assumptions So, if an insurer is purchasing option coverage for the benefit from an external vendor that uses a 25 percent volatility assumption in pricing the contract, it would cost them $99.84 for 50 percent participation; that is, leaving $0.16 of the premium for the insurer The insurer has no remaining liability unless the option vendor defaults Clearly, the insurer cannot afford a participation rate higher than around 60 percent, because this would cost more than the premium received for ␴ Ն 20 percent It is really quite straightforward to adapt the RP formula to allow for slightly more complicated products For example, under the scheme above, the ratcheted premium is guaranteed to increase each year by the lesser of 1.0 and 1‫( ␣ם‬St ΋ StϪ1 Ϫ 1) Suppose that instead of a minimum accumulation factor of 1.0 we applied a minimum accumulation factor of, say, eg for some g Then in place of ͫ Ά EQ eϪr ‫ ␣ ם‬max ΂S St t Ϫ1 Ϫ 1, ΃· ͬ we have ͫ ΂ ΂S Ά EQ eϪr ‫ ם‬max ␣ St t Ϫ1 ΃ ΃· ͬ Ϫ , eg Ϫ (13.17) ‫ ס‬EQ ͫ eϪr Ά ‫ ם‬max Θ␣ (S1 Ϫ 1), e g Ϫ 1Ι· ͬ ͫ (13.18) ΃ ΃· ͬ (13.19) e g Ϫ (1 Ϫ ␣ ) ,0 ␣ ΂ ΂ Ά ‫ ס‬EQ eϪr ‫( ם‬e g Ϫ 1) ‫ ␣ ם‬max S1 Ϫ ΂ ‫ ס‬e gϪr ‫ ␣ ם‬BSC K ‫ס‬ e g Ϫ (1 Ϫ ␣ ) ,n ‫1 ס‬ ␣ ΃ (13.20) where BSC(K, n) is the Black-Scholes call-option price with strike K, starting stock price 1.0, and term n years Substituting the appropriate Black-Scholes 251 Compound Annual Ratchet Valuation option formula gives equation 13.22, below: Ά ΂ H ‫ ס‬P e gϪr ‫ ␣ ם‬eϪd⌽(d1 ) Ϫ eϪr e g Ϫ (1 Ϫ ␣ ) ⌽(d2 ) ␣ ΂ ΃ Ά ‫ ס‬P ␣eϪd⌽(d1 ) ‫ ם‬e gϪr⌽(Ϫd2 ) ‫ ם‬eϪr (1 Ϫ ␣ )⌽(d2 ) ΃· · n (13.21) n (13.22) where K1 ‫ס‬ e g Ϫ (1 Ϫ ␣ ) ␣ (13.23) and log(1΋ K1 ) ‫ ם‬r Ϫ d ‫ ␴ ם‬΋ and d2 ‫ ס‬d1 Ϫ ␴ ␴ d1 ‫ס‬ (13.24) Substituting some numbers gives a table of results comparable with Table 13.2, but with a percent annual ratchet guarantee, that is e g ‫.30.1 ס‬ The results are given in Table 13.3 We can see that if the option is priced at a volatility rate of 25 percent, then the participation rate must be less than 40 percent for the contract to break even A participation rate of 36.8 percent will exactly break even Now if we add an annual cap rate—that is, a maximum amount by which the premium is ratcheted up each year of ec Ϫ 1—the valuation formula becomes: Ά P ␣ eϪd Θ⌽(d1 ) Ϫ ⌽(d3 )Ι ‫ 1( ם‬Ϫ ␣ )eϪr Θ⌽(d2 ) Ϫ ⌽(d4 )Ι ‫ם‬egϪr⌽(Ϫd2 ) ‫ ם‬ecϪr⌽(d4 )· n (13.25) TABLE 13.3 Ratchet premium option values with percent annual minimum ratchet, $100 initial premium RP Value Participation Rate ␴ ‫02.0 ס‬ ␴ ‫52.0 ס‬ ␴ ‫03.0 ס‬ 0.3 0.4 0.5 0.6 91.03 97.05 103.60 110.62 94.05 103.60 110.86 119.86 98.83 108.86 118.53 129.73 252 EQUITY-INDEXED ANNUITIES where d1 and d2 are defined in equation and K2 ‫ס‬ ec Ϫ (1 Ϫ ␣ ) ␣ (13.26) d3 ‫ס‬ log(1΋ K2 ) ‫ ם‬r Ϫ d ‫ ␴ ם‬΋ and d4 ‫ ס‬d3 Ϫ ␴ ␴ (13.27) With a cap of c where ec ‫ ,1.1 ס‬all the values in Table 13.3 are reduced to between $90 and $96 The vulnerability to both the stock price volatility and the participation rate are very much reduced because the process is constrained at both ends, with a percent floor and a 10 percent ceiling This was demonstrated earlier in this chapter, where we showed that a seven-year CAR contract purchased on January 1, 1995, would have an RP benefit that is the same for any participation rate above 53 percent, because the returns in each year are either negative (so that the floor applies) or greater than 18.7 percent (so that the ceiling applies provided the participation rate is greater than 0.1/0.187 ‫ 35 ס‬percent) The break-even participation rate for the CAR using 25 percent volatility is 180 percent, a dramatic increase on the rate of less than 40 percent without a cap Increasing the cap quickly reduces the break-even participation rate; using 14 percent in place of 10 percent reduces the break-even participation rate from 180 percent to 52 percent Even this relatively high cap is a very effective way of reducing the guarantee costs, compared with offering unlimited upside annual ratchet Some readers will notice that the participation rates quoted here are lower than some of those quoted in the market For example, a selection of annual ratchet contracts from a few different companies featured on www.annuityratewatch.com currently (as at June 2002) shows: Contract Participation Rate Annual Cap A B C D 75% 70% 55% 100% 12% 11% none none All of these companies offer an annual floor rate of g ‫ ס‬percent as well as a life-of-contract minimum guarantee of percent per year Without the life-of-contract guarantee, and assuming 25 percent volatility and a compound ratchet benefit, the break-even participation rates for these contracts are greater than 100 percent for contracts A and B, and 50.1 percent for contracts C and D So it appears that contracts A and B are Compound Annual Ratchet Valuation 253 comfortably profitable, at least before the life-of-contract guarantee cost is added, whereas contracts C and D are not Clearly contract D stands out here—how can the insurer offer such generous terms? One answer is in the use of simple rather than compound annual ratcheting We saw earlier in this chapter that the simple annual ratchet is cheaper than the compound version Also, contract D uses averaging in determining the indexation This means that the index value for determining the annual reset is averaged, either on a monthly or a daily basis, over the year prior to maturity This decreases the volatility of returns greatly and makes the option cheaper, although it does not necessarily reduce payouts to policyholders, providing lower returns in rising markets and higher returns in falling markets CAR with Life-of-Contract Guarantee The simple annual ratchet contract and the addition of a life-of-contract guarantee are not amenable to the analytic approach A simple method of valuing the option in these cases is by stochastic simulation, also called the Monte Carlo method Recall that the Black-Scholes valuation of any derivative contract is the expected value of the discounted payoff under the risk-neutral distribution In the standard Black-Scholes context that we are using in this chapter, the risk-neutral distribution is lognormal, with independent and identically distributed increments, and with parameters for the annual log-return distribution of r Ϫ d Ϫ ␴ ΋ and ␴ , where the d is the continuously compounded dividend yield rate We will simulate the payoff under the option for, say, 100,000 projections of the stock price process, and discount using the risk-free rate of interest The mean value is the estimated Black-Scholes price of the option We will use the Monte Carlo method in this section for the compound ratchet option with life-of-contract guarantee, as well as in the next section for the simple annual ratchet with life-of-contract guarantee Following the earlier results of this chapter, we ignore mortality and lapses We have used a control variate to improve the accuracy of the Monte Carlo simulation This calibrates the simulation by using the same random variables for the option and for some related value, which can also be calculated exactly by analytic methods This value is the control variate The simulated value of the option is adjusted by the difference between the actual and estimated values of the control variate The method is described in detail with examples and in Chapter 11 It is an obvious method to use here because the value of the compound annual ratchet benefit option with life-of-contract guarantee will be very close to the value of the annual ratchet benefit without life-of-contract guarantee, since in the great majority of cases the option will mature in-the-money 254 EQUITY-INDEXED ANNUITIES For an example, we look at the simulated value of a compound annual ratchet option with life-of-contract guarantee as follows: One-hundred dollar single-premium contract Seven-year term Sixty percent participation rate Zero percent annual floor Ten percent annual cap Life-of-contract guarantee of percent per year on 95 percent of the premium We use the following assumptions: Risk-free rate of return of percent per year continuously compounded Volatility ␴ ‫.52.0 ס‬ Dividend yield of percent per year continuously compounded Lapses and mortality ignored Then, using 100,000 simulations, the estimated value of the option before allowing for the control variate is $86.630; the estimated value of the ratcheted premium using the same simulations is $85.912 The true value of the ratcheted premium is $85.937, using equation 13.25 So the stochastic simulation appears to be valuing the option a little low, and we adjust by adding the difference (85.937 Ϫ 85.912) back to the original estimated option value, to give a value of $86.655 for the option including the life-of-contract part The value of the complete benefit is estimated at, say, $86.66 The value of the ratchet-only part, without the life-of-contract benefit, is $85.94, so the additional cost of the life-of-contract benefit is around $0.72, relatively small as we would expect It is worth noting that the ratchet-only part with a percent annual floor costs $95.48 for a $100 premium, considerably more than the percent annual floor and percent per year life-of-contract minimum benefit; therefore it is not possible to use the ratchet floor in place of the life-of-contract guarantee The reason for this conclusion is quite clear from an example; suppose that returns in three successive years are 25 percent, Ϫ5 percent, and 15 percent Consider a three-year contract with $100 premium, 10 percent cap, percent floor, 60 percent participation rate, and a percent life-of-contract benefit with no initial expense deduction (just to make things simpler) The ratchet hits the ceiling in the first year, hits the floor in the second, and falls in between in the third, giving the ratcheted premium value of 100(1.1)(1.0)(1 ‫991.1 ס ))51.0(6.0 ם‬ 272 APPENDIX B Clearly EQ [Ht1 eϪrt1 ] ‫ ס‬PS0 (t1 ) Also EQ [Ht2 eϪrt2 ] ‫ ס‬EQ [EQ [Ht2 eϪrt2 ͉Ft‫]] ם‬ ‫ ס‬EQ [Ft‫ ם‬eϪrt1 P(t2 Ϫ t1 )] Ϫ ‫ ס‬EQ [(Ft1 ‫ ם‬Ht1 )eϪrt1 ] P(t2 Ϫ t1 ) Ϫmt1 ‫( ס‬S0 e ‫ ם‬PS0 (t1 )) P(t2 Ϫ t1 ) And, similarly, EQ [Ht3 eϪrt3 ] ‫ ס‬EQ [EQ [Ht3 eϪrt3 ͉Ft‫]] ם‬ ‫ ס‬EQ [Ft‫ ם‬eϪrt2 ] P(t3 Ϫ t2 ) Ϫ ‫ ס‬EQ [(Ft2 ‫ ם‬Ht2 )eϪrt2 ] P(t3 Ϫ t2 ) ‫͕ ס‬EQ [Ft‫ ם‬eϪm(t2 Ϫt1 )Ϫrt1 ] ‫ ם‬EQ [Ht2 eϪrt2 ]͖P(t3 Ϫ t2 ) Ϫ ‫͕ ס‬EQ [(Ft1 ‫ ם‬Ht1 )eϪrt1 ]eϪm(t2 Ϫt1 ) ‫ ם‬EQ [Ht2 eϪrt2 ]͖P(t3 Ϫ t2 ) ‫͕ ס‬S0 eϪmt2 ‫ ם‬EQ [Ht1 eϪrt1 ]eϪm(t2 Ϫt1 ) ‫ ם‬EQ [Ht2 eϪrt2 ]͖ P(t3 Ϫ t2 ) ‫͕ ס‬S0 eϪmt2 ‫ ם‬eϪm(t2 Ϫt1 ) PS0 (t1 ) ‫( ם‬S0 eϪmt1 ‫ ם‬PS0 (t1 )) P(t2 Ϫ t1 ))͖ P(t3 Ϫ t2 ) This gives a total option price of PS0 (t1 ) ‫( ם‬S0 eϪmt1 ‫ ם‬PS0 (t1 ))(1 ‫ ם‬P(t3 Ϫ t2 ))P(t2 Ϫ t1 ) ‫ם‬P(t3 Ϫ t2 ) (S0 eϪmt2 ‫ ם‬eϪm(t2 Ϫt1 ) PS0 (t1 )) APPENDIX C Actuarial Notation e have generally used standard actuarial notation in this book, with the exception that we are generally measuring term and duration in months Standard actuarial notation uses the following conventions: W t px is the probability that a life currently aged x survives to age x ‫ ם‬t t qx is the probability that a life currently aged x dies before age x ‫ ם‬t ␮x,t is the force of mortality at age x ‫ ם‬t for a life currently age x The force of mortality is also known as the mortality transition intensity or hazard rate It is defined as Ϫ d t px t px dt ¨ ax:n is the expected present value of an annuity of per time unit, paid at the start of each time unit until the life age x dies, or until n time units expire, whichever is sooner For an interest rate of r, continuously compounded, the equation for the annuity is n1 ă ax:n t px er t t‫0ס‬ The force of interest is the continuously compounded interest rate v is the annual discount factor; for a force of interest r, v ‫ ס‬eϪ r Tx is the random future lifetime of a life currently aged x years 273 274 APPENDIX C In this book we have used these symbols adapted to allow for the two decrements, death and withdrawal The superscript ␶ indicates that both decrements are allowed for; d indicates decrement by death and w indicates decrements by withdrawal The specific notation used is it is assumed to take the value t ‫.0.1 ס‬ ␶ t px,u is the probability that a policyholder currently aged x years and u months survives and does not withdraw for a further t months ␶ ␶ t qx,u is Ϫ t px,u w t qx is the probability that a policyholder currently aged x years withdraws before t months expire d t qx is the probability that a policyholder currently aged x years dies in force before t months expire d u ͉t qx is the probability that a policyholder aged x years is still in force after u months, but dies in force before the expiry of a further t months If the t is omitted, it is assumed to take the value t ‫.0.1 ס‬ ␶ t qx is the probability that a policyholder aged x years dies or lapses the policy before t months expire (d ␮x,t) is the force of mortality experienced by a life aged x years and t months ăx a␶ :n iЈ is the value of an annuity of per month paid monthly in advance for n months, contingent on the survival, in force (the ␶ indicates the double decrement function), of a life age x The rate of interest is iЈ per month, which means that the discount factor for the payment due at t is (1 ‫ ם‬iЈ)Ϫt References Akaike, H (1974) A new look at statistical model identification IEEE Trans Aut Control, 19, 716–723 Annuity Guarantee Working Party (AGWP) (1997) Reserving for Annuity Guarantees Published by the Faculty of Actuaries and Institute of Actuaries Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D (1997, November) Thinking coherently RISK, 10, 68–71 Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D (1999) Coherent measures of risk Mathematical Finance, 9(3), 203–228 Bacinello, G., & Ortu, F (1993) Pricing equity-linked life insurance with endogenous minimum guarantees Insurance: Mathematics and Economics, 12, 245–257 Bakshi G., Cao, C., & Chen, Z (1999) Pricing and hedging long-term options Journal of Econometrics, 94, 277–183 Black, F., & Scholes, M., (1973) The pricing of options and corporate liabilities Journal of Political Economy, 81, 637–654 Bollen, N P B (1998) Valuing options in regime switching models Journal of Derivatives, 6, 38–49 Bollerslev, T (1986) Generalized autoregressive conditional heteroskedasticity Journal of Econometrics, 31, 307–327 Boyle, P P (1977) Options: A Monte Carlo approach Journal of Financial Economics, 4(4), 323–338 Boyle, P P., & Boyle, F P (2001) Derivatives: The tools that changed finance United Kingdom: Risk Books Boyle, P P., Broadie, M., & Glasserman, P (1997) Monte Carlo methods for security pricing Journal of Economic Dynamics and Control, 21, 1267–1321 Boyle, P P., Cox, S., Dufresne, D., Gerber, H., Mueller, H., Pedersen, H., Pliska, S., Sherris, M., Shiu, E., Tan, K S (1998) Financial economics Chicago: The Actuarial Foundation Boyle, P P & Emmanuel, D (1980) Discretely adjusted option hedges Journal of Financial Economics, 8, 259–282 275 276 REFERENCES Boyle, P P., & Hardy, M R (1996) Reserving for maturity guarantees (96-18) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Boyle, P P., & Hardy, M R (1998) Reserving for maturity guarantees: Two approaches Insurance: Mathematics and Economics, 21, 113–127 Boyle, P P., & Schwartz, E S (1977) Equilibrium prices of guarantees under equity-linked contracts Journal of Risk and Insurance, 44(4), 639–660 Boyle, P P., Siu, T K., & Yang, H (2002) A two level binomial tree for risk measurement Research Report 325 University of Hong Kong, Dept of Statistics and Actuarial Science Boyle, P P., & Tan, K S (2002) Valuation of ratchet options (02) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Boyle P P., & Tan, K S (2003) Quasi Monte Carlo methods with applications to actuarial science Monograph sponsored by Actuarial Education and Research Fund; Forthcoming Boyle, P P., & Vorst, T (1992) Option replication in discrete time with transaction costs Journal of Finance, 47(1), 271–294 Brennan, M J., & Schwartz, E S., (1976) The pricing of equity-linked life insurance policies with an asset value guarantee Journal of Financial Economics, 3, 195–213 Cairns, A J G., (2000) A discussion of parameter and model uncertainty in insurance Insurance: Mathematics and Economics, 27, 313–330 Campbell, J Y., Lo, A W., & MacKinlay, A C (1996) The econometrics of financial markets Princeton, NJ: Princeton University Press Canadian Institute of Actuaries (1999) Call for papers Symposium on stochastic modelling for variable annuity/segregated fund investment guarantees Canadian Institute of Actuaries Chambers, J M., Mallows, C L., & Stuck, B W (1976) A method for simulating stable random variables Journal of the American Statistical Association, 71, 340–344 Chan, T (1998) Some applications of Levy processes to stochastic investment models for actuarial use ASTIN Bulletin, 28, 77–93 Cox, J., Ingersoll, J., and Ross, S (1985) A theory of the term structure of interest rates Econometrica, 53, 385–487 David, H A (1981) Order statistics (2nd ed.) New York: Wiley Engle, R F (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation Econometrica, 50, 987–1006 Exley, J., & Mehta, S (2000, March) Asset models and the Ballard-Mehta stochastic investment model Presented to the Conference in Honour of David Wilkie, Heriot-Watt University References 277 Faculty of Actuaries Solvency Working Party (1986) The solvency of life insurance companies Transactions of the Faculty of Actuaries, 39, 251 Finkelstein, G (1995) Maturity guarantees revisited British Actuarial Journal, 3(2), 411–482 French, K R., Schwert, G W., & Stambaugh, R F (1987) Journal of Financial Economics, 19, 3–29 Gerber, H U (1979) An introduction to mathematical risk theory Huebner Foundation Monograph No Philadelphia: University of Pennsylvania, Wharton School Gilks, W R., Richardson, S., & Spiegelhalter, D J (Eds.) (1996) Markov chain Monte Carlo in practice London: Chapman and Hall/CRC Hamilton, J D (1989) A new approach to the economic analysis of non-stationary time series Econometrica, 57, 357–384 Hamilton, J D., & Susmel, R (1994) Autoregressive conditional heteroskedasticity and changes in regime Journal of Econometrics, 64, 307–333 Hancock, G H (2001) Policy liabilities using stochastic valuation methods Canadian Institute of Actuaries Segregated Fund Symposium Hancock, G H (2002) Private communication Hardy, M R (1998) Maturity guarantees for segregated fund contracts; hedging and reserving (98-07) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Hardy, M R (1999) Stock return models for segregated fund investment guarantees (99-12) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Hardy, M R (2001) A regime switching model of long term stock returns North American Actuarial Journal, 5(2), 41–53 Hardy, M R (2002) Bayesian risk management for equity-linked insurance Scandinavian Actuarial Journal, 3, 185–211 Hardy, M R., & Hardy, P G (2002) Regime switching lognormal model (rsln.xls) Excel workbook available from www.soa.org Harris, G R (1999) Markov chain Monte Carlo estimation of regime switching vector autoregressions ASTIN Bulletin, 29, 47–80 Huber, P P (1997) A review of Wilkie’s stochastic asset model British Actuarial Journal, 3, 181–210 Hull, J C (1989) Options futures and other derivative securities New Jersey: Prentice Hall Klugman, S A., Panjer, H H., & Willmot, G E (1998) Loss models; From data to decisions New York: Wiley Kolkiewicz, W A., & Tan, K S (1999) Unit linked life insurance contracts with lapse rates depending on economic factors (99-09) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research 278 REFERENCES Lee, H (2002) Pricing equity-indexed annuities embedded with exotic options Contingencies, Jan/Feb 2002, 34–38 Leland, H (1995) Option pricing and replication with transactions costs Journal of Finance, 40, 1283–1301 Lin, X S., & Tan, K S (2002) Valuation of equity-indexed annuities under stochastic interest rates (02) Ontario, Canada: University of Waterloo, Institute for Insurance and Pensions Research Manistre, B J., & Hancock, G H (2002) Variance of the CTE estimator Working paper, MMC Enterprise Risk Consulting, Toronto Maturity Guarantees Working Party (MGWP) (1980) Journal of the Institute of Actuaries, 107, 103–209 McCulloch, J H (1996) Financial applications of stable distributions Handbook of Statistics, 14, 393–425 Merton, R C (1973) Theory of rational option pricing Bell Journal of Economics and Management Science, 4, 141–183 Morgan, M S (1990) The history of econometric ideas Cambridge: Cambridge University Press Nolan, J P (1998) Parameterization and modes of stable distributions Statistics and Probability Letters, 38, 187–195 Nolan, J P (2000) Information on stable distributions [On-line] Available: http://www.cas.american.edu/jpnolan/stable.html Nonnemacher, D J F., & Russ, J (1997) Equity linked life insurance in Germany: Quantifying the risk of additional policy reserves Proceedings of the 7th AFIR conference, Cairns, 719–738 Pagan, A R., & Schwert, G W (1990) Alternative models for conditional stock volatility Journal of Econometrics, 45, 267–290 Panjer, H H., & Sharp, K P (1998) Report on Canadian Economic Statistics Canadian Institute of Actuaries Panjer, H H., & Tan, K S (1995) Graduation of Canadian individual insurance mortality experience: 1986–1992 Canadian Institute of Actuaries Panneton, C.-M.(1999) The impact of the distribution of stock market returns on the cost of the segregated fund long term guarantees Segregated Funds Symposium Proceedings Canadian Institute of Actuaries Pelsser, A (2002) Pricing and hedging guaranteed annuity options via static option replication Working Paper, Erasmus University at Rotterdam, Netherlands [On-line] Available: http://www.few.eur.nl/few/people/ pelsser Persson, S.-A., & Aase, K (1994) Valuation of the minimum guaranteed return embedded in life insurance products Journal of Risk and Insurance, 64(4), 599–617 Press, W H., Teukolsky, S A., Vetterling, W T., & Flannery, B P (1992) Numerical recipes in C Cambridge: Cambridge University Press References 279 Roberts, G O (1996) Markov chain concepts related to sampling algorithms In Gilks, W R., Richardson, S., & Spiegelhalter, D J (Eds.) Markov chain Monte carlo in practice (pp 45–57) London: Chapman and Hall/CRC Ross, S M (1996) Simulation San Francisco: Morgan Kaufmann Publishers Schwartz, G (1978) Estimating the dimension of a model Annals of Statistics, 6, 461–464 Segregated Funds Task Force (SFTF) (2002) Report of the Task Force on Segregated Fund Investment Guarantees Canadian Institute of Actuaries [On-line] Available: http://www.actuaries.ca/publications/2002/ 202012e.pdf Streiff, T F., & DiBiase, C A (1999) Equity indexed annuities Dearborn Financial Publishing, USA Tiong, S (2001) Valuing equity-indexed annuities North American Actuarial Journal, 4(4), 149–170 Wang, S X (1995) Insurance pricing and increased limits ratemaking by proportional hazard transforms Insurance: Mathematics and Economics, 17, 43–54 Webber, N., & James, J (2000) Interest rate modelling: Financial engineering London: Wiley Wilkie, A D (1986) A stochastic investment model for actuarial use Transactions of the Faculty of Actuaries, 39, 341–381 Wilkie, A D (1995) More on a stochastic asset model for actuarial use British Actuarial Journal, (V), 777–964 Windcliffe, H., Le Roux, M., Forsythe, P., Vetzal, K (2002) Understanding the behavior and hedging of segregated funds offering the reset feature North American Actuarial Journal, 6(2), 107–124 Windcliffe, H., Forsythe, P., Vetzal, K (2001) Valuation of segregated funds shout options with maturity extensions Insurance: Mathematics and Economics, 29, 1–21 Wirch, J L., & Hardy, M R (1999) A synthesis of risk measures for capital adequacy Insurance: Mathematics and Economics, 25, 337–347 Wright, I D (1997) A stochastic approach to pension scheme funding and asset allocation Ph.D thesis, Heriot-Watt University, Edinburgh, Scotland Yang, S (2001) Reserving, pricing and hedging for guaranteed annuity options Ph.D thesis, Heriot-Watt University, Edinburgh, Scotland Zhang, P G (1998) Exotic options: A guide to second generation options (2nd Ed.) River Edge, NJ: World Scientific Publishing Company Index a(55) mortality table, 225 Acceptance probability (for MCMC), 83, 85, 86 Acceptance-rejection method (for MCMC), 82 Accumulation factors, 68–76, 94, 125 Actuarial approach (to risk management), 3, 12 capital requirements for GMAB with actuarial risk management, 180 emerging cost analysis using actuarial risk management, 178 for guaranteed annuity options, 228 risk measure for GMAB using actuarial risk management, 170 risk measure for VA GMDB using actuarial risk management, 173 Actuarial notation, 273–274 Administration fees See Management expense ratio Akaike information criterion (AIC), 61 American options, Annual ratchet (equity-indexed annuities), 240 Antithetic variates, 204 Arbitrage See No-arbitrage assumption Asian options, 7, 131 Asymptotic MLE results, 50 At-the-money, Autocorrelation, 16, 26, 27 Autoregression, 17 Autoregressive (AR) model, 27, 55, 61, 73, 82 Autoregressive conditionally heteroscedastic (ARCH) models ARCH(1), 28, 29, 31, 47, 56, 61, 82 GARCH(1,1), 29, 31, 47, 57, 61, 76, 82, 97, 126 using ARCH and GARCH models, 29 Averaging, 253 Bayesian parameter estimation, 77–94 See also Markov chain Monte Carlo Bayes’ theorem, 77 Bias of an estimator, 50 Binomial model example for option pricing, 117–124 Black-Scholes hedge, 125 Black-Scholes-Merton assumptions, 124, 133 Black-Scholes-Merton theory, 10, 115–132 Black-Scholes option pricing formula, 10, 124, 145 European put option (BSP), 126 European call option (BSC), 128, 250 European call option with dividends, 130, 245 for the GMAB, 139 for the GMDB, 136 for the GMMB, 134 Bootstrap method for quantiles, 69 Brownian bridge, 39 Burn-in (for MCMC), 80 281 282 Calibration See Left-tail calibration, or parameter estimation Call option, Canadian calibration table, 67 Canadian Institute of Actuaries Task Force on Segregated Funds (SFTF), 17, 65–69, 169 Candidate distribution (for MCMC), 82 Cap rate (equity-indexed annuities), 241, 251 Cash-flow analysis, 193 Cauchy distribution, 38 Certificate of deposit, 237, 243 Coherence criteria for risk measures, 168 Compound annual ratchet (CAR), 240, 247 Conditional tail expectation (CTE), 158, 163–176, 181, 200, 208, 230, 262 Conditionally heteroscedastic models See Autoregressive conditionally heteroscedastic (ARCH) models Confidence interval for simulated quantile risk measure, 160 Consols (U.K government bonds), 224 Control variate, 161, 207–211, 253 Counterparty risk, 11, 236 Cox-Ingersoll-Ross model, 223 Cramer-Rao lower bound for the variance of an estimator, 51 Data mining, 45 Delta method (of maximum likelihood estimation), 51 Deterministic methods deterministic techniques, 2, 3, 15 deterministic valuation, Discrete hedging error See hedging error Dividends, effect on Black-Scholes option price, 129, 135 INDEX Dynamic hedging, 3, 11, 13, 120 capital requirements for GMAB with dynamic hedging, 184 emerging costs analysis with dynamic hedging, 179 for equity-indexed annuities, 260 for guaranteed annuity options, 230 risk measures with dynamic hedging, 170 for separate account guarantees, 133–156 for VA death benefits, 174 Efficient market hypothesis, 17, 45 Emerging cost analysis, 177–194 Empirical model, 36 Equitable Life (U.K.), 13 Equity-indexed annuities (EIA), 1, 6, 10, 130, 237–263 Equity participation, Esscher transforms, 246 European option, European call option (BSC), 128, 250 European call option with dividends, 130, 245 European put option (BSP), 126 for segregated fund guarantees, 134 Exotic options, 130 Expected information, 50 Expected shortfall, 158 Family-of-funds benefit, Floor rate, 240, 250 FTSE All Share index, 225 Fund-by-fund benefit, Generalized-ARCH (GARCH) model See Autoregressive conditionally heteroscedastic (ARCH) models Geometric Brownian motion (GBM), 16, 24, 125 Gibbs sampler, 81 Guaranteed annuity option (GAO), 5, 13, 221–236 283 Index Guaranteed annuity rate (GAR), 222 Guaranteed minimum accumulation benefit (GMAB), 4, 5, 6, 16, 21 Black-Scholes formula, 139, 271 control variate method, 208 dynamic hedging for GMAB, 133 emerging costs for GMAB, 189 with hedging error and transactions costs, 151 modeling the guarantee liability, 102, 104–108, 110 model uncertainty, 220 option price, 271–272 parameter uncertainty, 217 risk measures, 169, 171 sampling error, 197 solvency capital for GMAB example, 180 with voluntary reset, 112, 171 Guaranteed minimum death benefit (GMDB), 4, 5, hedge formula, 136 with hedging error and transactions costs, 151 modeling the guarantee liability, 99, 101–102, 151 parameter uncertainty, 215 quantile risk measure, 163 risk measures, 158, 173 Guaranteed minimum income benefit (GMIB), 5, 6, 221 See also Guaranteed annuity option Guaranteed minimum maturity benefit (GMMB), 4, 5, 6, 9, 16 Black-Scholes formula, 134 CTE risk measure, 167 hedge costs, 136 hedge error, 145 historical evidence, 23 modeling the guarantee liability, 99–102 unhedged liability, 151 Guaranteed minimum surrender benefit (GMSB), Hedging error, 144, 146–149, 152 High water mark (HWM) (equityindexed annuity), 242, 258 Hurdle rate, 190 Importance sampling, 211 Indexation benefit, 237 Information matrix, 50, 54, 56 Insurance risk, Interest rate modeling, 39, 42, 224 Interest rate risk, 223 Interest spread, 243 In-the-money, Invariant (stationary) distribution for Markov regime-switching process, 34, 58 Joint probability density function, 47, 49 KPTP (equity-indexed annuity, point-to-point strike price), 244 Law of one price See No-arbitrage Left-tail calibration, 65–76, 220 Levy process, 37–38 Life annuity, 6, 222 Life-contingent risks, 1, Life-of-contract guarantee, 248 Likelihood-based model selection, 60 Likelihood function, 47–49, 78, 83 Likelihood ratio test, 60 Lognormal model, 16, 24, 53, 61, 66, 70 Log-return random variable, 27, 67 Lookback option, 131, 259 Low discrepancy sequences, 212 Management expense ratio (MER), 5, 99, 134 Margin offset, 99, 100, 133, 143, 158 Markov chain Monte Carlo parameter estimation (MCMC), 77–94 burn-in, 80 candidate distribution, 82 Gibbs sampler, 80 284 Markov chain Monte Carlo parameter estimation (continued) Metropolis-Hastings Algorithm (MHA), 80–85 parameter uncertainty, 213 for the RSLN model, 85–89 Maturity Guarantees Working Party (MGWP) U.K., 12, 17, 39 Maximum likelihood estimation (MLE), 47–63, 65, 66, 72, 73, 78 AR(1) model, 55 ARCH and GARCH models, 56 asymptotic minimum variance, 50 asymptotic normal distribution, 51 asymptotic unbiasedness, 50 conditions for asymptotic properties, 49, 52 delta method, 51 lognormal model, 53 RSLN model, 57 Metropolis-Hastings Algorithm (MHA), 80–85 Minimum variance estimator, 50 Model selection, 60 Model uncertainty and model error, 150, 195, 219–220 Moment matching for parameter estimation, 63 variance reduction technique, 203 Monte Carlo method for option pricing, 131, 253, 258 Mortality and survival probabilities, 265 Mortality risk, 135 Move-based strategy for rebalancing hedge, 144 Multivariate models Wilkie, 39– 45 vector autoregression, 45 Mutual fund, Net present value of future loss (NPVFL), 190 Net present value of liability (NPV), 107, 108, 113 INDEX No-arbitrage, 8, 9, 116 Nondiversifiable risk, Nonoverlapping data, 68 Nonstationary models, 52 October 1987 stock market crash, 16, 26 Office of the Superintendent of Financial Institutions (OSFI) in Canada, 15, 16, 169 Options, 7–11 American, 7, 10 Asian 7, 10 Black-Scholes-Merton pricing theory, 115–129 in equity-linked insurance, in-the-money, 8, 120 out-of-the-money, Parameter estimation, 47–63, 77–94 Parameter uncertainty, 77, 195, 213–219 Participation rate, 6, 239, 246, 250, 251 Path-dependent benefit, 16 Periodicity of random number generators, 97 Physical measure See P-measure P-measure, 11, 115, 120, 147, 159, 223 Point-to-point indexation (PTP), 239 Policyholder behavior, 96, 113 Posterior distribution, 78, 80, 86, 88, 90 p-quantile, 66 Predictive distribution, 79, 90, 94, 214 Premium principles, 158 Pricing and capital requirements, 14 Pricing using B-S-M valuation, 142 Prior distribution, 78, 81 Profit testing See Emerging cost analysis Put-call parity, 9, 10, 128 Put option, Index Q-measure, 11, 115, 119, 125–126, 147, 150, 159, 223 Quantile, 66–76 Quantile matching, 66 Quantile risk measure, 158, 159–163, 167–173, 198 Random number generators, 97, 104 Random walk (stock price process), 17 Random-walk Metropolis algorithm, 85 Ratcheted premium, 241 Real-world measure See P-measure Rebalancing the replicating (hedge) portfolio, 115 Regime-switching lognormal (RSLN) model, 30–36, 47, 57, 77 comparison with other models, 61 hedging and the RSLN model, 152 invariant (stationary) distribution for regime process, 34, 58 left-tail calibration, 74 Markov chain Monte Carlo parameter estimation, 85–89 maximum likelihood estimation, 57 parameters for examples, 104 probability function for RSLN model, 34 simulation, 98 sojourn distribution probability function, 33, 74 stress testing for parameter uncertainty, 218 transition matrix, 32 Regime-switching autoregressive (RSAR) model, 59, 225 Reinsurance, 11 Replicating portfolio, 10, 11, 115, 116 Reset option for segregated fund policies, 112–114, 171 Risk management actuarial approach, 3, 12, 13, 158 ad hoc approach, 13 dynamic hedging, 3, 11, 158 reinsurance, 11 285 Risk measures, 12, 157–176 Risk-neutral measure (Q-measure), 11, 22, 115, 119, 125, 150, 159 Sample paths (for MCMC), 84, 89, 91 Sampling error, 195, 196–201 Sampling variability, 75, 76, 160 S&P 500 total return index, 18–25 AR(1) model, 55 ARCH and GARCH models, 56 likelihood-based model selection, 61 lognormal model, 53 maximum likelihood parameter estimation, 53–64 MCMC parameter estimation, 86–90 RSLN model, 35, 57 S&P/TSX-Composite index, 18 See also TSE 300 index Schwartz-Bayes criterion (SBC), 60 Segregated fund contracts, 1, 2, 5, 9, 11, 21, 65, 67, 133 See also GMAB, GMDB, and GMMB Self-financing hedge, 123, 150 Separate account insurance, 2, 65, 133 Simple annual ratchet (SAR), 257 Sojourn time (R), 32 Solvency capital, 158 Stable model, 37, 61 Standard error CTE estimate, 165, 183 expected value, 197 quantile estimate, 160 Static hedge, 123 Static replication for guaranteed annuity options, 235 Stationary distributions, 49 Stochastic simulation for left-tail calibration, 75 Stochastic simulation of liabilities, 16, 108 actuarial approach, 95–114 cash-flow analysis, 110, 154 CTE risk measure, 165 distribution function, 108–109 286 Stochastic simulation of liabilities (continued) density function, 108–110 quantile risk measure, 159 stock return process, 97 Stochastic volatility models, 38 Stock price index, Stress testing (for parameter uncertainty), 217 Strike price, 7, 121 Systematic risk, Systemic risk, Tail risk, Tail-VaR, 158 Task Force on Segregated Funds See Canadian Institute of Actuaries Task Force on Segregated Funds (SFTF) Term structure of interest rates, 223 Time-based strategy for rebalancing hedge, 144 Tracking error See hedging error Transactions costs, 149 Transactions costs and hedging error reserve (t V T &H ), 180 Transition matrix (for RSLN model), 32 Trinomial lattice approximation, 256 TSE 300 total return index, 18–25, 72 AR(1) model, 55 ARCH and GARCH models, 56 calibration table, 67 empirical evidence for quantiles, 68 likelihood-based model selection, 61 lognormal model, 53 INDEX maximum likelihood parameter estimation, 53–64, 218 MCMC parameter estimation, 86–90 RSLN model, 35, 57 U.K FTSE All-Share total return index, 59 Unbiased estimator, 50 Unhedged liability, 133, 143 Unit-linked insurance, 1, 6, 133, 221 Universal life, Value-at-risk (VaR), 12, 158 Variable annuity (VA), 1, 2, 6, 10, 133, 138, 143, 158 Variable-annuity death benefits, 173– 176, 215 See also Guaranteed minimum death benefit (GMDB) Variable-annuity guaranteed living benefits (VAGLB), See also Guaranteed minimum maturity benefit (GMMB) Variance reduction, 131, 201–213 Vector autoregressive model, 45 Volatility, 18, 22, 28, 30, 38 general stochastic volatility models, 38 market (implied) volatility, 22 stochastic volatility, 26, 28, 30, 38, 150 volatility bunching, 26, 27, 37 White noise process, 27 Wilkie model, 17, 39– 45 Withdrawals, 96, 100 ... 0.98928 0.12223 0.1 2107 0.11992 0.11877 0.11764 0.11651 0.11539 0.11428 0.11317 0.11208 0. 1109 9 0 .109 91 0 .108 84 0 .107 77 0 .106 71 0 .105 66 0 .104 62 0 .103 59 0 .102 56 0 .101 54 0 .100 53 0.09953 0.09853... ␶ t px d t ͉1 qx t p␶ x,t ␶ t px d t ͉1 qx 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128... into the segregated fund required at renewal dates The annual charge is 100 m percent compounded continuously At the renewal and maturity dates, if the fund has fallen below the previous renewal