Pricing of Renewable Term Life Products An Analysis from Microeconomic and Financial Engineering Perspectives

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(1)Pricing of Renewable Te rm Life Products —An Analysis from Microeconomic and Financial Engineering Perspectives. Taizo Ueda. Con te nts 1. Introduction 2. A Renewable Term Product Model Based on Microeconomic Analysis 3. Description of Renewable Term Products 4. Basic Data 5. Stochastic Process of Death Rates Based on Actuarial Finance 6. Estimation of Withdrawal Rates Based on Actuarial Finance 7. Estimation of Premiums 8. Conclusion 9. Bibliography. 1.

(2) ABS TRAC T As the variety of life insurance products expands, life insurers face a growing risk stemming from the behavior of policyholders. For example, healthy policyholders are prone to withdraw and enter less expensive new contracts, while unhealthy policyholders tend to stay with their present contracts. This adverse selection risk emerges clearly in renewable term life insurance, where premiums increase in stages. Thus in setting premiums for renewable term life insurance, I need to estimate the average death rate of remaining policyholders in the future while considering policyholders’ price preferences. To estimate the fair premium rate, I use tools from the world of practical finance—financial engineering theory, and a prepayment risk model for housing loans and deposit savings. My research results are applicable to new product development and stress tests.. Taizo Ueda Actuarial Department Nippon Life Insurance Co. [email protected]. 2.

(3) 1. Introduction Long-term insurance carries two types of risk—risk of change in short-term event risk, and an adverse selection risk in which persons with deteriorating health tend to renew contracts more than others. Insurers try to adjust premiums based on both types of risk, while consumers select and buy products that they find more advantageous. First, I present a simple microeconomic model for renewable term insurance. The model contains two periods—the initial contract period, and the subsequent period in which the policyholder’s health classification changes. It then shows the conditions under which consumers will buy the required amount of term insurance based on income and. health condition,. while insurance. companies. maintain financial. equilibrium(cf. Hendel and Lizzeri(2000)). Next, I describe the model from the perspective of insurance companies who set premiums. Given the proliferation of term products in the U.S., consumers are likely to focus on price in their purchasing decision. For example, healthy persons are more likely to withdraw from their existing policy and switch to a less expensive one, while persons with deteriorating health are likely to renew. This selection risk of policyholders is clearly revealed in renewable term insurance, where premiums increase in stages. Premium determination for renewable term insurance is complicated by the fact that insurers must estimate the future average death rate of continuing policyholders while considering their price preferences and other factors. While Shinohara (2001) presents a binomial model with option pricing theory, I use methods from the world of practical finance—financial engineering theory and a prepayment risk model for housing loans and deposits—and estimate the fair price of premiums using the following procedure. 1. I assume that the death rate conforms to an average regressive stochastic process, and depict the future distribution using a Monte Carlo simulation. 2. I assume a market in which policyholders will continue to be able to buy new contracts appropriate to their health status (death rate). Moreover, I assume that the withdrawal rate is determined based on a price comparison of existing and new contracts (existing contracts are more likely to be cancelled if new contracts are less expensive), and estimate the future withdrawal rate. 3. I simulate the death rate distribution of remaining policyholders in the future, 3.

(4) and estimate the fair premium.. 2.. Term Insurance Model Based on Microeconomic Analysis. This. model. contains. three. agents:. insurance. companies,. insured. persons. (policyholders), and dependents of the insured. The insured buy term life insurance for the benefit of their families in case of death. While dependents play no active role in the model, they derive utility from the death benefit. I assume that perfect competition exists. The model has two periods. In period 1, the initial contract period, the death rates of all insured persons are identical. Period 1 is further subdivided into three stages. In stage 1, insurance companies offer contracts to the insured. In stage 2, the insured choose a contract. In stage 3, the insured pay premiums, and some of them incur an insured event with probability q (death). If insured persons die, insurance companies pay death benefits to dependents; otherwise, the insured continue to pay premiums. Period 2 is subdivided into four stages. In stage 1, there are i health statuses. Each health status occurs with a probability of ri. For each health status i, the death rate is. qi (q1 q2 . . . qN ) in ascending order. I also assume that each person’s death rate increases with age, such that qi (death rate in period 2) q (death rate in period 1). Insurance companies offer premiums based on each individual’s health status in stage 2, and the insured select their insurance contract in stage 3. In stage 4, the insured start paying premiums, and insured events start to occur. Insured persons seek to maximize their intertemporal utility function. If they survive, they obtain utility u(c) with respect to the net income for the relevant period. All insured persons expect that in the event of death, dependents will obtain utility v(d) from the death benefit. Utility functions u and v are strictly concave and twice differentiable. If the insured survive, they obtain income y in the first period and income y+g (g 0) in the second period. There is no credit market spanning the two periods. While all insured persons have the same preferences and same income in the initial period, income growth rate g varies. This parameter g allows future income to affect consumption decisions according to differing consumption needs. In period 1, I assume a premium P1 and face amount of insurance S1 . In period 2, 4.

(5) premiums are ( P2 , P2 ,L , P2 ) depending on health status, and insurance amounts 1. 2. N. are ( S 2 , S 2 ,L , S 2 ) . Moreover, premiums are paid in as long as contract conditions are 1. 2. N. not changed.. (1). Ch oo sing a Lo ng- Te r m or S hor t- Te r m In sur ance Po licy. If insurers and the insured commit to long-term insurance contracts, risk-neutral insurers will cover 100% of the insurance risk of risk-averse consumers. Moreover, the assumption of perfect competition drives profits to zero, and the allocation is Pareto efficient. On the other hand, with short-term contracts, the insurance is actuarially fair in each period. But while the short-term insurance risk is fully covered, the adverse selection risk is left completely uncovered. Actual consumer behavior lies somewhere between the long-term and short-term situations described above, and insurers must set premiums accordingly. I discuss this problem further after section 4.. (2). Mo de l fo r Ren e w abl e Te rm In sur ance. I formulate a model of how the insurance company and the insured act with regard to renewable term insurance. If the health classification in period 2 is determined, the insured must choose between renewing the insurance policy from period 1, or undergoing another medical exam and choosing a new term insurance policy. No penalty is incurred if the insured does not renew the policy and simply stops paying premiums. The model is described in Figure 1.. 5.

(6) Figure 1. Renewable Term Insurance Model. Period 1. Period 2. Insurance companies offer policies.. Health status of insured changes.. Consumers choose an offer.. Insurers offer policies for health status i.. Premiums are paid, insured events occur.. Insured choose new policy or renew current one. Premiums are paid, insured events occur.. Under the assumption that insurance companies break even, it is clear that competitive equilibrium is achieved by maximizing consumer utility as shown below.. . ; qv( S1 ) + (1 q )u ( y P1 ). P1 , S1 ,( P21 ,L, P2N ),( S21 ,L, S2N ). . N. + (1 q ) ri qi v( S 2i ) + (1 qi )u ( y + g P2i ) i =1. subject to the following constraints:. N P qS (1 q ) ri P2i qi S 2i = 0 + 1 1 i =1. P i q S i 0 (i = 1, 2,L , N ) i 2. 2. L L. The solution ( S1 , S 2 ) is obtained by differentiating with respect to S1 and S 2 . i. i. v '( S1 ) = u '( y P1 ). . v '( S 2i ) = u '( y + g P2i ). . Constraint indicates that all policies chosen by consumers in period 1 must achieve financial equilibrium for insurers throughout periods 1 and 2. Under constraint , insurers suffer losses for contracts made only in period 2. This is because in period 2, health classifications 1N have become apparent, and other insurers can offer better rates for persons whose classification has not worsened. If consumers act rationally, under the assumption of perfect competition, insurers cannot earn profits in period 2; thus any losses incurred in period 2 must be compensated for in period 1. Thus for renewals, the risk of adverse selection is unavoidable, and if consumers are 6.

(7) perfectly rational, all healthy persons would withdraw from their present contracts in favor of new contracts from other insurers offering lower rates, while only persons with a worsening health classification would choose to renew. This situation would destroy the viability of insurance products. In reality, however, since a new contract entails medical exams and other procedures, many healthy consumers can be expected to renew. Based on the above assumptions, I analyze after section 4 how insurers should set premiums from the perspective of financial engineering.. 3.. Renewable Term Insurance. With renewable term insurance, when contracts are renewed, premiums after renewal are offered on paper at the time of renewal. In Japan, premiums after renewal are the same as rates of new life policies at the time of renewal. Thus adverse selection will eliminate profit opportunities for insurance companies after renewal. In the U.S., on the other hand, there are products that stipulate at the time of contract future renewal premiums (Figure 2), or that guarantee a maximum premium (Figure 3). Under the latter, policyholders take medical exams annually or every few years, and allows them to renew at a low premium if their health status is good, or at a maximum premium otherwise. In Figure 3, for example, for a 35-year-old policyholder can renew in the next year at a $680 premium if his health status is good; if the medical exam reveals a worse status or he does not take a medical exam, he can still renew at a $790 premium. Amid advances in risk classification and price competition, if consumers choose policies they find most advantageous, it is possible that the only policyholders who will renew are those that are disadvantageous to insurance companies. Thus assuming that perfect competition exists in the insurance market and that policyholders are rational, the process of adverse selection will eliminate profit opportunities for insurance companies, and the market as a whole will fail to operate. In this case, insurance companies cannot price premiums fairly in the renewal period and still break even, and so must offset future losses by pricing premiums above their fair value in the first half. However, it is unlikely that all policyholders will automatically withdraw from present contracts in favor of more advantageous new contracts. In reality, policyholders are more inclined to withdraw from present contracts if new contracts offer lower 7.

(8) premiums, and less inclined to do so if premiums are higher. Assuming this irrational aspect of policyholders, even if premiums are not set high enough in the first period, fair premiums can be set in each period. To set premiums, I need only to estimate the average death rate of living policyholders. And for this, I need the future distribution of health status (death rate) of policyholders, and the corresponding withdrawal rate. Here I must note that the future average death rate of living policyholders also changes depending on premiums set in the initial contract. Using financial engineering theory as practiced in the world of finance, I attempt to estimate fair premiums through simulations.. Figure 2. Renewable Term Life Insurance in the U.S. (Initial Medical Exam Only) - Joins at age 40; annual renewable term insurance with $500,000 coverage - Future premiums stipulated in initial contract (initial medical exam only) - Policyholder can withdraw without penalty - For nonsmoker in excellent health. $ "$. . .

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(14). . . Renewable Term Life Insurance in the U.S. (Recurring Medical Exam) Product characteristics: - Annual renewable term insurance with $500,000 coverage - Future premiums stipulated in initial contract (maximum premiums) - Policyholder can receive annual medical exam; good health status reduces premium - Policyholder can withdraw without penalty - For nonsmoker in excellent health.

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(28) . . . . . Note: Contracts offered in July 1997 to non-smoking male by Prudential Insurance for $500,000 coverage. Source: I. Hendel and A. Lizzeri (2000), “The Role of Commitment in Dynamic Contracts: Evidence from Life Insurance,” NBER Working Paper 7470.. 8.

(29) 4. Basic Data In the U.S., a wide range of products already exists, and death rates are available for nonsmokers, smokers, and other selection factors. For my analysis, I use U.S. data for model policyholders (Figure 4). Figure 4. Fair Value Estimation of Insurance Product, Model Policyholder, and Basic Statistics. 0.$3#2 --3!+%-%5!"+%%0,!'% 7%!0/.+)#7-.0%#300)-',%$)#!+%6!, -130%$ ,!+%-.-1,.*%0(%!+2(7 .02!+)27 ./3+!2). !+%.-1,.*%0!"+% -130%$. !+%.-1,.*%0!"+%8 % + #2 ) .- !#2 .0 1 %+%#2).-&!#2.01. %'

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(45). . . . . . Stochastic Process of Death Rates Based on Financial Engineering G en er al S to ch astic Proc e ss. When modeling stock price movements, a common assumption is that the future state depends entirely on the present state, and not on past processes or processes leading up to the present. This is called a Markov property, and movements with this property are said to conform to the Markov process. Two commonly used Markov process models are the Wiener process and Ito process. The Ito process generalizes on the Wiener 9.

(46) process. Wiener process Ito process. dq = adx + bdz dq = m(q, x)dx + s (q, x)dz. (Eq. 1) (Eq. 2). Here, q is the death rate, x is age, a and m( q, x) are shifts in unit time (drift), b and s ( q, x). 2. 2. are variances (probability terms), and dz = dt and are random. numbers with a standard normal distribution.. (2). Me an Rev er si on. Due to the effect of medical exams, the death rate of the group of policyholders at the time of contract is below the national average. However, as the years pass and the effect of medical exams wanes, the death rate tends to converge to the national average. Dukes and MacDonald (1980) illustrate this characteristic of the death rate in Figure 5. Moreover, coefficients for selection factors used in the U.S. are designed to approach 1 as the years pass by, and assume the mean reversion of the death rate (Figure 4).. Figure 5. Mean Reversion of Death Rate. Source: J. Dukes and A. MacDonald, “Pricing a Select and Ultimate Annual Renewable Term Product,” Transactions of the Society of Actuaries, Vol. XXXII.. Meanwhile, in the world of finance, studies have also observed the same property of mean reversion in interest rates. Two well-known stochastic process models that. 10.

(47) address mean reversion are the Vasicek(1996) and Hull-White(1990) models. The Vasicek model is,. dr = a (b r )dt + dz. where r is the interest rate, a is the speed of adjustment, and b is the average interest rate. The equation thus describes the property in which interest rate r is pulled in the direction of equilibrium interest rate b at the speed a.. The Hull-White model is,. where. (t ) dr = a ( r )dt + dz a. (t ) is the equilibrium level, and is a function of time t. The Hull-White model a. generalizes on the Vasicek model. Referring to these two models, I made the following assumptions and constructed a stochastic process model of the death rate. Assumptions: 1. For empirical data, I use average death rate trends of new policyholders. 2. Individual death rates revert to the national average death rate. 3. The more actual values deviate from the mean, the stronger is the reversion effect. 4. The death rate level and rate of change are normally distributed.. 11.

(48) Figure 6. Mean Reversion and Stochastic Process of the Death Rate ". ". ! #!. ( x) Q a ve ( x ) " ". ". !

(49) ! #!. . . . In Figure 6, the average death rate nationwide for persons age x is ( x) , and the average death rate of the universe of persons who acquired insurance at age 40 and are age x is Qave ( x) .. Mo d el 1: G en er al me an r eve rsio n d e ath r ate mo d el Eq. . dq = m(q, x)dx + s (q, x)dz m(q, x) = a ( ( x) q ( x, i ))dx + n( x)dx s ( q, x ) = ( x ). Eq. Eq. . where x is age, q(x,i) is the death rate of policyholder i at age x, ( x) is the average death rate nationwide at age x, a is the adjustment speed toward equilibrium level. ( x) , n( x) is the independent residual drift, and ( x) 2 is the variance of the death rate at age x. Assuming a discrete time t of 0.01 (years), adjustment speed a of 0.05 (for full reversion in 20 years), ( x) of 0.5Qave ( x) , and furthermore, estimating n( x) such that E x [ q ( x, i )] = Qave ( x) , I performed 10,000 Monte Carlo scenario simulations (to simulate 10,000 policyholders). The future distribution of death rates is shown in Figure 7.. 12.

(50) Figure 7. Stochastic Process of Basic Values and Death Rates (Mean Reversion, Change has Normal Distribution; Age 40, 15 Year Period). (&($%)!( $#-+ #')&*+. .

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(62) . Figure 8. Death Rate Distribution at Age 50 (Based on Figure 7). . (3). . . . . . .

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(67). . . Re g ard ing th e No nn eg ativ e Ch ar ac ter isti c of De ath R ate s. Figure 7 shows some negative death rate results, which are not commonly observed in reality. This occurs because Equation 3 assumes a normal distribution for the death rate level, such that for average values near zero, the standard deviation is relatively large. Reducing the standard deviation would reduce the possibility of generating negative values, but also limit the death rate’s upward fluctuation and hamper appropriate calculations. Since stock prices and interest rates are also subject to nonnegative constraints, in financial engineering several nonnegative models have been proposed. For example, with stock prices, a common assumption is that the rate of change is normally distributed (stock prices have a lognormal distribution).. 13.

(68) d ln St + dt d ln St = ( μ . 2 )dt + dz 2. On the other hand, for interest rates, the stochastic term is assumed to have a standard deviation proportional to the square root of the interest rate, and the CIR model described below is typical.1. dr = a (b r )dt + rdz Referring to the above, I describe a nonnegative stochastic process with mean reversion as follows. Mo d el 2: R ate o f ch ang e in d e ath rate con f orm s to n orm al d i str i bu ti on. 1 d ln qx + dx d ln qx = ( μ (q, x) r ( x) 2 )dx + r ( x)dz 2. μ ( q, x ) =. a ( ( x) q ( x, i )) + μind q ( x, i ). Eq. 6 Eq. 7. where μ ( q, x) is the rate of change in unit time, the first term on Equation 7’s right hand side is the rate of change due to regression, μind is the rate of change independent of regression, and r ( x). 2. is the variance of the rate of change.. Assuming that the period of convergence to the overall mean is 20 years, I estimated the. parameters. so. that. the. mean. of. the. distribution. of. death. rates. is. E x [q ( x, i )] = Qave ( x) , and conducted simulations. The future distribution satisfies the condition of nonnegativity (Figure 9).. The possibility remains that negative values will be generated if Monte Carlo simulations are done. Nonetheless, since the values are quite close to zero, depending on the constraint equations, if negative, I can write a program that replaces negative values with very small positive ones such as 0.0001%, so that the effect on results will be minimal. 1. 14.

(69) Figure 9. Stochastic Process of Basic Values and Death Rate (Death Rate: Mean Reversion, Rate of Change is Normally Distributed). '%'#$( '#",* "&(%)*. .

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(85). Figure 10. Death Rate Distribution— Age 50 (Based on Figure 9). . . . . . . . . . . . . . . . . . . Mo d el 3: Stoch astic te rm o f d e ath r ate i s adju ste d Eq. 8. dq = m(q, x)dx + s (q, x)dz m(q, x) = a ( ( x) q ( x, i ))dx + n( x)dx. Eq. 9 Eq. 10. s ( q , x ) = ( x ) q ( x, i ). Equation 8 is identical to Equation 3. However, the stochastic term in Equation 10 has been altered from Equation 5 by adding. q ( x, i ) . Since variance decreases as the. death rate approaches zero, nonnegativity can be expressed. Similar to previous models, simulations of the death rate distribution were conducted, and results are shown in Figure 11.. 15.

(86) Figure 11. Stochastic Process of Basic Values and Death Rate (Death Rate: Mean Reversion, Nonnegativity Constraint on Stochastic Term). '%' #$( '#",* "&(%)*. .

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(106) . Figure 12. Death Rate Distribution—Age 50 (Based on Figure 11).

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(111). . . . While I have proposed three models for the stochastic process of the death rate, none of them is accurate all of the time. Thus the model that better approximates empirical analysis and data should be used. The following analysis uses results from Figure 11, which are derived from Model 3.. 6.. Estimation of Withdrawal Rate Based on Financial Engineering. I assume that strong price competition exists, and that products in the market match up with the health status (death rate) of insureds (Figure 13).. 16.

(112) Figure 13 " $ " % " & ". Health Status of Insureds and Premium Levels. . #

(113) # # . ". ! ". . . . . . . I assume that present contract is an annual renewable term insurance product with face amount ¥10 million, additional premiums are not considered, and premiums are paid annually.. In Figure 13, one year after making a contract, person A can find a less expensive new product, while person C finds it more expensive to do so. If I assume as in economic and options theory that insureds are perfectly rational, A will definitely withdraw from the present contract, while C will maintain the present contract. However, no matter how insurance agents and premium comparison sites on the Internet proliferate, this assumption is extreme. In reality, the characteristic is that people who can enter a new product less expensive will be more inclined to withdraw, while those that find it more expensive are less likely to do so. In the world of finance as well, there are products whose withdrawal is affected by future environmental changes. For example, with housing loans, when market interest rates decrease, more people tend to take out new loans at a lower interest rate, and pay off their existing loans. On the other hand, with deposits, when market rates rise, people tend to withdraw from deposit contracts with low interest rates, and transfer funds to deposits with higher rates. Particularly in the U.S., where mortgage issuance and transactions are active, many empirical studies have been done on the prepayment risk of home loans. The withdrawal rate is often modeled as in Equation 11.. w = a (arctan(br c) + ) 2. Eq. 11. 17.

(114) Figure 14. Estimation Model for Home Loan Withdrawal Rate (Equation 11). #!$!#$. . . . . . . . . . . . . . . . !#!" .

(115) . . #!"#!#!'! ) $ &(% " # &*. In Equation 11, w is the withdrawal rate, r is the interest rate spread between the present loan and a new loan (new rate minus existing rate). Figure 14 shows that the more advantageous it is to withdraw from the present policy and obtain a new one, the more people tend to withdraw. Parameters a, b, and c are usually estimated from actual data. Note that r changes significantly near. c , b. while b is proportional to the slope. As for a, the larger it becomes, the larger is the range of the withdrawal rate. I adapted Equation 11 to estimate the withdrawal rate of term insurance. Assuming that withdrawal rate depends on a comparison of the price of the existing contract with that of a newly purchased different product (rate of price increase or decrease), I estimated it using Equation 12. As for the estimation of parameters, due to difficulty obtaining withdrawal rate data corresponding to price comparisons, I assumed that the withdrawal rate for when there is no price difference with old products is equivalent to 2001 U.S. LIMRA Term Laps. I also assumed that of the persons who can obtain an equivalent product to their existing contract at half the price, half of them will withdraw. The results are shown in Figure 15.2. 2 Since this type of cancellation rate model assumes that investors and policyholders are irrational, it is difficult to theoretically estimate parameters. Thus the estimated values are most appropriate to empirical distribution data. Objective data on how cancellation rates are affected by death rate changes (health status) is more difficult to obtain than for home loan and deposit cancellations due to market interest rate changes. However, because of the advanced state of classification in U.S. term insurance markets, collection of such data is underway at companies.. 18.

(116) Wi th dr aw al r ate e sti m atio n m o del. w( x, i ) = a ( x)(arctan(b( x). Figure 15. Pnew ( x, i ) Pold ( x) c( x)) + ) Pold ( x) 2. Eq. 12. Estimated Parameters and Price Comparisons. 2 * &! , !* + "(* /$ ,# * / % *, ! !+, $ & , $ (' *&!,!* 0 . . . 0 . 0 . . . .

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(139) 1!* 1!* $,# */%,!. . . . . ' $.$ -%$"!!*+$+,!'1,- 1 )) . In Equation 12, w( x, i ) is the withdrawal rate of policyholder i at age x, Pnew ( x, i ) is the premium for the same policyholder if he obtains a new product, and Pold ( x) is the premium for the existing contract at age x. Thus. Pnew ( x, i ) Pold ( x) is the rate of Pold ( x). premium increase or decrease when obtaining a new product. For example, based on the above assumptions and supposed death rate, the withdrawal rate for policyholder i is shown in Figure 16. In the premium estimation of the next section, I assume that policyholder i is 45 years old and has a 30.4% probability of continuing his contract.. Figure 16. Death Rate and Withdrawal Rate. "##!##(!# (!. . . . . . . . . . . . .

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(144) 7.. Premium Estimation. I modeled the stochastic process of the death rate for policyholders in Section 5, and the estimated withdrawal rate in Section 6. By applying future premiums stipulated at the time of contract, I can estimate the average death rate of remaining policyholders who have not withdrawn or died. If the stipulated premiums are below appropriate levels, the death rate of remaining policyholders has risen above initial projections, and become a factor of mortality loss. On the other hand, if premiums are higher than the appropriate level, insurance companies can expect to benefit from mortality profit, but the higher premiums could hurt product competitiveness in the market. Appropriate premiums are at the level at which average death rates assumed at the calculation of net premiums and the average death rate of remaining policyholders are equal (or close). However, since this is analytically difficult to calculate, I simulated an approximation as in Figure 17. The results are shown in Figure 18. In the U.S., to increase the rate of continuation, many products guarantee premiums for a period ranging from five to 20 years. For example, 15-year death insurance that guarantees a fixed premium for ten years followed by a 5-year period of step-up increases, for which it is possible to estimate appropriate premiums using the same procedure as above. Results are shown for a scheduled interest rate of 4% in Figure 19.. 20.

(145) Figure 17. Calculation of Fair Premiums. Simulate death rate trend of policyholders 10,000 policies generated under Model. Step 1. 3 . Step. Estimate fair premium P41 for age 41. 2For average death rate of 41-year-old policyholder, initial values are set: Value less than fair value (e.g., 0.0001) qsmall(41) Value larger than fair value (e.g., 0.1) qlarge(41). Appropriate death rate is assumed to be qave(41): qave(41). = {qsmall(41) + qlarge(41)} / 2. Using above death rate qave(41) as basic rate, calculate premium P41 for age 41:. If q’ave(41) – qave(41) > 0.00001, replace qave(41). P41 = Insured amount qave(41). qsmall(41). with. If q’ave(41) – qave(41) < 0.00001, For premium P41 estimate withdrawal rate and average death rate for remaining policyholders q’ave(41).. replace qlarge(41) with qave(41). If NO. Confirm whether following condition is satisfied: | q’ave(41) – qave(41) | < 0.00001. If YES Calculated premium P41 is fair. (Step. In same way, estimate fair premiums for age 42-54 in. 3). sequence.. 21.

(146) Figure 18 "$&&) "$"!%$!(&$( (&(&$(. !%'$! "'!& $ '. Estimation of Fair Premium . . . . .

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(157) . Conclusion. I have proposed a simulation-based pricing method for renewable term life insurance. This method can be used not only for annual renewable insurance, but also for products renewable in ten years as described in Section 7, and for calculating maximum premiums for products with recurring medical exams. Products whose premiums rise significantly at renewal are likely to become unacceptable to customers. However, even in such cases, my method can be used to develop new products more acceptable to both customers and insurance companies (such as products that charge higher premiums in the first half of the contract period, and products whose insurance amount gradually decreases). Application of my method in Japan is hampered by the lack of empirical data to estimate parameters for the stochastic process of death rates and withdrawal rate models. In such cases, while designing insurance products based on conventional methods, my method can be used to simulate future profits (stress tests, etc.) for several scenarios and parameters. In addition, application of my method entails the following practical issues, all of which warrant further research. 1. Contract classification My assumption that the insurance amount does not affect the withdrawal rate (adverse 22.

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(159) selection) needs to be verified. Moreover, the withdrawal rate is likely to be affected by the insured’s income level and macroeconomic factors (market interest rates, inflation, etc.) at the time of renewal. Regarding withdrawal and non-renewal factors, further empirical research is needed to improve the model.. 2.. Stochastic processes. While empirical data exists for average death rates, death rates of individuals are extremely difficult to estimate. In practice, using medical exam results, it is possible to classify policyholders, and then calculate average death rates for each class. In this case, I are dealing not with the stochastic process for death rate fluctuations of individual policyholders, but for movements across health status classes.. 3.. Markov process assumption. I have assumed that death rates conform to a Markov process, but two individuals A and B can start out with the same death rate of 1% and have different future distributions. I make the Markov process assumption because empirical data is unavailable on this characteristic. However, as more data becomes available, series models such as the ARMA model or GARCH model will become applicable.. Figure 20

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(161) . Death Rate Trends. . . .

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(163) . . . . . 23. .

(164) 4.. Probability distribution of death rates. I have assumed that the death rate level and rate of change are normally distributed. However, in reality these can surge suddenly due to accident or illness. The use of stochastic processes without normal distributions and jump processes are conceivable.. 5.. Withdrawal rate estimation model. I estimated the withdrawal rate using a prepayment risk model for home loans. However, while interest rate spread data for home loans is readily available, health status is difficult to ascertain even for the individuals concerned. Thus a clearer and accessible analogy needs to be found. In addition, while I have ignored cost, recurring medical exams cost the insurance company money and are also time consuming for policyholders. This consideration needs to be addressed as a practical matter.. 9.. Bibiography. [1] Albert,F.,D.Bragg. and J. Bragg.(1997)“Mortality Rates as a Function of Lapse Rates”,Society of Actuaries. [2] Cawley ,J. and T. Philipson.(1996) “An Enprical Examination of Information Barriers to trade in Insurance,”NBER Working Paper 5669. [3] Dukes ,J. and A. MacDonald.(1980), “Pricing a Select and Ultimate Annual Renewable Term Product,” in Transactions of the Society of Actuaries, Vol. XXXII, pp. 547-567. [4] Hendel ,I. and A. Lizzeri.(2000) “The Role of Commitment in Dynamic Contracts: Evidence From Life Insurance,”NBER Working Paper 7470. [5] Hull,J., Options, Futures, & Other Derivatives , Prentice-Hall International, Inc,2000. [6] Hull,J. and A. White.(1990), “Pricing Interest Rate Derivative Securities,” Review of Financial Studies, vol. 3, no. 4, pp. 573-592. [7] Leimberg, S. and R.Doyle., The Tools and Techniques of Life Insurance Planning, National Underwriter Co,1999. [8] Lemaire, J. ,K. Subramanian. ,K. Armstrong, and D. Asch,(2001)”Pricing Term Insurance in the Presense of a Family History of Breast or Ovarian Canter”,North American Actuarial Journal,vol.4,num.2. [9] Shinohara, T.(2000) “Pricing of Renewable Insurance Schemes,” (in Japanese), 24.

(165) Kaiho Bessatsu, Institute of Actuaries of Japan , vol. 53, no. 2. [10] “The Valuation of Individual Renewable Term Insurance “,Canadian Institute of Actuaries,Valuation Technique Paper #2,1986. [11] Vasicek ,O.(1997) “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, Vol. 5 (1997), pp. 177-188 [12] Young, Virginia R. and T. Zariphopolou.(2001) “Pricing Dynamic Insurance Risks. Using the Principle of Equivalent Utility”, in Actuarial Research Conference at the Ohio State University.. 25.

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