Supplement Note for Candidates Using. Models for Quantifying Risk, Fourth Edition

Transcription

1 Supplement Note for Candidates Using Models for Quantifying Risk, Fourth Edition Robin J. Cunningham, Ph.D. Thomas N. Herzog, Ph.D., ASA Richard L. London, FSA Copyright 2012 by ACTEX Publications, nc. Posted with permission of ACTEX Publication, nc.

2 CHAPTER NNE FUNDN PLANS FOR CONTNENT CONTRACTS (ANNUAL PREMUMS) 9.6 FUNDN PLANS NCORPORATN EXPENSES Recall the observation made earlier in this chapter that the annual funding payments determined by the equivalence principle, which we called net annual premiums in the life insurance context, provide only for the contingent benefit payment. n practice, of course, the price of an insurance (or other contingent payment) product must be set higher than the net premium in order to generate revenue to pay expenses of operation and the contingent benefit payments, as well as providing a profit margin to the insurer. The total annual premium charged for an insurance product is called the gross annual premium or the contract premium. A premium determined to cover benefits and expenses, but not profit, is called the expense-augmented premium. n this text we will use to denote an expense-augmented premium and * to denote a gross (or contract) premium. t is a simple matter to extend the equivalence principle to incorporate expenses and therefore to calculate expense-augmented premiums. For ease of illustration we assume that the expenses allocated to a particular contingent contract are fixed costs known in advance. Then the expense-augmented equivalence principle states that the APV of the expenseaugmented funding scheme equals the APV of the benefit payment plus the APV of the expense charges allocated to the contract. For illustration we assume that expense charges allocated to a contract are of the following four types: (1) A percentage of the gross premium itself. (2) A fixed amount per unit of benefit payment. (3) A fixed (or percentage of benefit) amount incurred when the benefit payment is made. (4) A fixed amount for the contract itself, regardless of benefit amount. The analysis of corporate operational expenses leading to the determination of expense charges to be included in the price of each product is a complex issue that will vary according to the type of business under discussion. n any case the mechanics of this expense analysis are beyond the scope of this text. We will, however, illustrate the four types of expense charges listed above under a typical life insurance policy, since the pricing of life insurance is a common example of this type of actuarial analysis. 1

3 2 Percent of premium expenses arise as commissions paid to sales agents and state premium taxes. t is customary for sales commissions to be larger for the first several years of a contract and then smaller in later years. The premium tax is likely to be level over all years. The amount per unit of insurance might be expected to cover such things as underwriting (i.e., risk classification) expense, policy issue expenses, and subsequent policy maintenance expenses. Again it would be customary for these per unit expenses to be higher in the first year than in subsequent years of the policy. n our illustration we will consider the unit of insurance to be $1000. The expense associated with making the benefit payment, called the settlement expense, is a one-time charge incurred at the same time as the benefit is paid. Therefore it can be introduced into the gross premium calculation by simply adding it to the benefit payment amount. Since premiums need to be calculated per unit of benefit, such as per $1000, it is convenient to convert the fixed amount per policy expenses to the fixed amount per unit of insurance type by assuming an average policy size. 1 We illustrate the above discussion with the following example. EXAMPLE 9.8 Assume that all per policy expenses have been converted to the per $1000 of benefit type. Assume a whole life insurance contract issued to ( x ), with the following expenses: 75% of the first premium and 10% of all premiums thereafter; $10 at the beginning of the first year and $2 at the beginning of each year thereafter; $20 settlement expense. Find an expression for the expense-augmented premium of a $1000 benefit contract. SOLUTON f we let denote the expense-augmented premium, then the APV of the premium income is a x. The APV of the percent of premium expense charges is a x, since the 10% charge is incurred at the beginning of each year after the first only if the contract is still in force. The APV of the fixed per $1000 of benefit expense charges is 10 2 a x, by the same reasoning. The APV of the settlement expense is 20 A x, since it is incurred at the same time as the benefit payment. Of course the APV of the benefit payment itself is 1000 A x. Then the equivalence principle states that so a.75.10a 10 2a 1020A x x x x 1020Ax 10 2a a.75.10a x x x. 1 An alternative approach is to determine an annual policy fee that is independent of policy size. Actuaries who specialize in life insurance pricing will study this issue far more deeply than the introductory level presented here. n particular, they will explore this issue of policy fee determination and the important topic of expense analysis mentioned earlier.

4 3 Since a a 1, this can also be written as x x 1020Ax 8 2ax..90 a.65 x Recall the present value of payment random variable Z x, defined in Section 7.1.2, and the present value of an immediate contingent unit payment stream random variable Y x, as defined in Section We can use Z x and x H The definition of x Y to now define a present value of expenses random variable, which we denote by x. H will depend on the specific pattern of expenses in each case. For the contract described in Example 9.8, for example, it would be defined as H (.7510) (.102) Y 20 Z, (9.30) x x x which the reader is asked to verify in Exercise Recall the whole life present value of loss random variable L x, defined by Equation (9.13). We can now define an expense-augmented present value of loss random variable, which we denote by L x, as the excess of the present value of benefits and expenses over the present value of expense-augmented premiums. For the whole life contract described in Example 9.8, we have where L 1000 Z H Y, (9.31) x x x x H x is defined by Equation (9.30), which the reader is asked to verify in Exercise The expense-augmented equivalence principle applied to determine the expense-augmented premium would then solve for such that EL [ x ] 0. For the contract of Example 9.8 we have Then L x 1000Z x H x Y x 1020 Z (.7510) (.102) Y Y. x x x x x x x EL [ ] 1020 A (.7510) (.102) a a 0 implies a.75.10a 10 2a 1020 A, x x x x as already established in Example 9.8.

5 4 Exercises 9.6 Funding Plans ncorporating Expenses 9-25 A special endowment contract issued to (25) pays a pure endowment of 150,000 for survival to age 65, and a refund of all gross annual premiums paid, with interest, at the end of the year of failure, for failure before age 65. The gross annual premium is 1.20 times the net level annual premium. Calculate the net level annual premium, given the values P.008, 40 p25.80, and i : Consider a 20-pay unit discrete whole life insurance, with expense factors of a flat amount.02 each year, plus an additional.05 in the first year only, plus 3% of each premium paid. Find the expense-augmented annual premium for this contract, given the values a 20, a 10, and d.04. x x: Verify Equation (9.30) Verify Equation (9.31).

6 CHAPTER TEN CONTNENT CONTRACT RESERVES (NET LEVEL PREMUM BENEFT RESERVES) 10.6 AN AND LOSS ANALYSS An important actuarial activity is the analysis of contingent contracts to determine financial gain or loss under such contracts, and, in certain cases, to identify the source of the gain or loss. We explore this analysis separately for contingent insurance and contingent annuity contracts CONTNENT NSURANCE CONTRACTS The recursive relationships of Section 10.2 are particularly useful for this analysis. n Equations (10.30a) through (10.30e), the symbols i, qx t, and px t refer to the interest and mortality assumed in the calculation of the benefit premium and the benefit reserves. Equation (10.30b) shows that if the interest and mortality actually experienced in the ( t 1) st contract year are the same as that assumed in premium and reserve calculation, then there is neither gain nor loss to the insurer in that year under that contract. This is illustrated as well if we rewrite Equation (10.30b) as ( tvp)(1 i) ( qxtpxt t1v) 0. (10.51) Now suppose the interest and mortality actually experienced in the ( t 1) st contract year are denoted by i, q xt,and p xt. n particular, suppose ii and qx t qxt. Then the amount on hand at time t 1 would be T ( VP)(1 i) ( q p V), (10.52) t xt xt t1 which would be a positive quantity and would represent a gain in the ( t contract year T for the insurer. We denote this gain by. (f i i and/or q q, then we would find T 0 and interpret the negative gain as a loss to the insurer.) xt xt 1) st We can define M ( VP)(1 i) ( q p V) (10.53) t xt xt t1 1

7 2 to be the gain from mortality. Note that i is used in Equation (10.53) rather than i, even in cases where i i. Similarly, we can define to be the gain from interest, where qx t and ( VP)(1 i) ( q p V) (10.54) t xt xt t1 p are used rather than q x t and p x t. x t M T n this setting of net premium and net premium reserves, it follows that, T M when,, and are calculated by Equations (10.52), (10.53), and (10.54), respectively. (See Exercise ) EXAMPLE 10.9 Consider a unit discrete whole life insurance issued at age 40 with benefit premium and benefit reserves given by P , 10 V , and 11 V (These values were calculated from the life table in Appendix A at 6% interest.) Suppose, in the eleventh contract year, the actual earned interest rate is i.065 and the actual experienced mortality T M rate is q Calculate each of, and, and show that M T. SOLUTON We calculate we calculate and we calculate T from Equation (10.52) as T ( )(1.065) (.99700)( ) , M from Equation (10.53) as M ( )(1.06) (.99700)( ) , from Equation (10.54) as ( )(1.065) (.99644)( ) Then M , T which is the same as.

8 3 n practice, of course, contingent insurance contracts are not issued on a net premium and net premium reserve basis. We have chosen to introduce the topic of gain and loss analysis in this environment of only two factors (interest and mortality) for simplicity and ease of understanding. We will revisit this notion of gain or loss by source in Section 11.7, when expense factors and gross premiums are also present CONTNENT ANNUTY CONTRACTS n the case of an annual premium deferred annuity contract, we distinguish the two subcases of (a) the contract being within the deferred period, and (b) the contract being after the deferred period (i.e., during the payout period when there are no further premiums). Again we are considering here the simpler net premium case, with the more realistic gross premium case considered in Section The simplest case is the contingent annuity model presented in Section and revisited in Example 10.5, where the failure benefit during the deferred period is the year-end reserve. We have seen that mortality is not a factor under such a contract, so any gain or loss during the deferred period can be due to interest only. f the interest rate anticipated to be earned in the ( t 1) st contract year is i, and the rate actually earned is i, then the interest gain in that year is ( ii) V, (10.55) where t V is the contract reserve at the beginning of the ( t year. Next we consider an annual premium deferred annuity-due which is in its payout period. f the annual annuity benefit is B, then the recursive relationship for the ( t 1) st contract year is ( V B)(1 i) p V, t xt t1 since there are no premiums, the benefit is paid at the beginning of the year under an annuity-due, and the year-end reserve needs to be established only if the contract holder survives the year. Then, just as for an insurance contract in Section , if i and px t refer to the interest and mortality assumed in the reserve calculation, we have The total gain would be given by the gain from mortality would be given by t xt t1 t 1) st ( V B)(1 i) p V 0. (10.56) T ( VB)(1 i) p 1V, (10.57) M t x t t t x t t 1 ( VB)(1 i) p V, (10.58)

9 4 and the gain from interest would be given by Again we see that ( VB)(1 i) p V. (10.59) t xt t1 M T (see Exercise 10-32). Exercises 10.6 ain and Loss Analysis M T Show that, where the three gains are calculated by Equations (10.52), (10.53), and (10.54) Show that the gain from mortality, as given by Equation (10.53), can also be written as M ( q q )(1 V). xt xt t Similarly, show that the gain from interest, as given by Equation (10.54), can also be written as ( VP)( i i). t Show that the gains from mortality and interest, given by Equations (10.58) and (10.59), respectively, sum to the total gain, as given by Equation (10.57).

10 CHAPTER ELEVEN CONTNENT CONTRACT RESERVES (RESERVES AS FNANCAL LABLTES) 11.4 NCORPORATON OF EXPENSES Recall that all reserve expressions developed in Chapter 10 and thus far in this chapter are for benefit reserves only, by which we mean that they are based on benefit premiums (also called net premiums). n Section 9.6 we saw how to incorporate expense factors into an expense-augmented equivalence principle to determine expense-augmented premiums. t is now a simple matter to include the expense factors, along with the expense-augmented premium, to determine expense-augmented reserves. (When profit margins are also included in the premium, producing a gross (or contract) premium, the resulting reserves are referred to as gross premium reserves.) The general prospective formula for the t th benefit reserve, given by Equation (10.2), is now modified to read t V ( APV of future benefits and expenses) ( APV of future expense- augmented premiums), (11.20) where the symbol tv denotes the t th expense-augmented reserve. This is illustrated in the following example. EXAMPLE 11.5 ive an expression for the t th prospective expense-augmented reserve for the whole life contract described in Example 9.8. SOLUTON At duration t, given that the contingent contract is still in effect, the APV of the future expense-augmented premium income is a x t, where is defined in Example 9.8. The APV of the future percent of premium expense charges is.10 a x t, and the APV of the future fixed per $1000 of benefit expense charges is 2 a x t. The APV of the benefit payment itself plus the settlement expense together is 1020 Ax t. Thus we have V 1020 A (.102) a a 1020 A (.902) a. t xt xt xt xt xt 251

11 252 CHAPTER ELEVEN Returning to Section 9.6, we can separate the amount of the level benefit premium P from the expense-augmented premium. The remainder of represents the amount of annual premium needed to fund the expenses of administering the contract. n other words, we define to be the annual expense premium for the contract. EP P (11.21) The notion of separating the expense-augmented premium into benefit premium and expense premium components naturally extends to the reserve. As already covered extensively in Chapter 10, the prospective benefit reserve is the APV of future benefits minus the APV of future benefit premiums. Similarly we define the t th prospective expense reserve to be E V ( APV of future expenses) ( APV of future expense premiums). (11.22) t t should be clear that where t B B tv tv tv E, (11.23) th V is the t benefit reserve previously denoted by simply. EXAMPLE 11.6 ive an expression for the t th prospective expense reserve for the whole life contract described in Example 9.8. SOLUTON First we find the level expense premium as EPa.75.10a 10 2a 20 A, x x x x where is defined in Example 9.8. Then the expense reserve is E V 20 A (.102) a EPa. t xt xt xt Recall how the concept of the present value of loss (at issue) random variable, L x, introduced in Section 9.2, was easily extended to the present value of loss (at duration t) random variable, tl x, defined in Section n the same way, the expense-augmented present value of loss (at issue) random variable, L x, defined in Section 9.6, is easily extended to the expense-augmented present value of loss (at duration t) random variable, which we denote by L. This is pursued in Exercise t x Recall that we expanded Equation (10.30b), which had presumed level benefit premiums and a level failure benefit, into Equation (11.16b), which generalized Equation (10.30b) for nonlevel benefits and benefit premium. Now we generalize further to include expenses. t V

12 RESERVES AS FNANCAL LABLTES 253 Let t denote the expense-augmented premium for the t th contract year, r t denote the percent-of-premium expense factor for that year, e t denote the fixed expense for that year, and s t denote the settlement expense associated with a benefit paid at the end of the t th contract year. Then Equation (11.16b) is generalized to [ V (1 r ) e ](1 i ) ( b s ) q p V. (11.24) t t1 t1 t1 t1 t1 t1 xt xt t1 Note that Equation (11.24) allows for the reserve interest rate to also vary by contract year, for maximum generality. n many applications, it 1 will be set as a constant.

13 CHAPTER ELEVEN CONTNENT CONTRACT RESERVES (RESERVES AS FNANCAL LABLTES) 11.7 AN AND LOSS ANALYSS We now continue our analysis of financial gain or loss under a contingent contract which we began in Section 10.6, this time in the more realistic environment of gross premiums and either gross premium reserves or expense-augmented reserves. We begin with Equation (11.24), found in Section 11.4, and modify it to read * t t1 t1 t1 t1 t1 t1 xt xt t1 [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.30a) which uses the gross premium but expense-augmented (rather than gross premium) reserves. Recall that this expression is written with maximum generality to allow the benefit, gross premium, interest rate, and all expense factors to vary by contract year. n practice, many of these parameters would be constant over contract years. An important difference between Expression (11.30a) and its net counterpart given by the left side of Equation (10.51) is that the net case expression always equals zero but this gross case profit expression does not necessarily equal zero, since the gross premium has been set to include a profit margin, but the expense-augmented reserves do not consider profit. When the gross case Expression (11.30a) is evaluated using the parameters anticipated to apply in the ( t 1) st contract year, we refer to the resulting value as the anticipated profit for the ( t 1) st year, which we denote by P (0). That is, * t t1 t1 t1 t1 t1 t1 xt xt t1 P(0) [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.30b) where we use unprimed symbols for anticipated experience. Next we consider the notion of gain or loss by source, introduced in Section 10.6 in the net premium case. Here we have three potential sources of gain or loss (interest, mortality, and expenses), and we adopt a different approach to calculating them than was used in Section n this case we must specify the order in which the gain-by-source calculations are to be made. We illustrate that notion here by choosing the order interest, then mortality, and then expenses. 1

14 2 Now we evaluate Expression (11.30a) using the actual interest rate earned in the ( t year, but still using anticipated experience for mortality and expenses, producing * P(1) [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.31a) t t1 t1 t1 t1 t1 t1 xt xt t1 where we use it 1 in place of it 1 to denote the actual interest rate earned in the ( t year. Next we evaluate Expression (11.30a) using actual interest and mortality, but still using anticipated expenses, producing * t t1 t1 t1 t1 t1 t1 xt xt t1 P(2) [ V (1 r ) e ](1 i ) [( b s ) q p V ]. (11.31b) Next we evaluate Expression (11.30a) using actual interest, mortality, and expenses, producing * t t1 t1 t1 t1 t1 t1 xt xt t1 P(3) [ V (1 r ) e ](1 i ) [( b s ) q p V ]. (11.31c) (Note that the benefit, gross premium, and expense-augmented reserve parameters are constant throughout these calculations.) Finally, we define the gain from interest to be 1) st 1) st the gain from mortality to be and the gain from expenses to be P(1) P(0), (11.32a) M P(2) P(1), (11.32b) Note that E P(3) P(2). (11.32c) T M E P(3) P(0), (11.33) which shows that the total gain can be calculated by subtracting the anticipated profit from the profit calculated using actual experience throughout. The theory developed here is illustrated in the following example. EXAMPLE 11.9 Consider a block of fully discrete whole life policies issued at age 40 with face amount 50,000. On the assumed (or anticipated) mortality, interest, and expense bases, the gross annual premium per policy is , the tenth-year expense-augmented reserve is , and the eleventh-year expense-augmented reserve is The assumed interest rate is 6%, the assumed mortality rate for the eleventh year is q , and the assumed expenses are 5% of the gross premium and 300 to process a death claim. n the eleventh year, there are 1000 policies in force at the beginning of that year and five deaths occur in the year. Actual expenses in the eleventh year are 6% of the gross premium and 100 to process each death claim, and the actual earned interest rate is 6.5%. Calculate, in order, the gain from mortality, the gain from expenses, and the gain from interest on a single policy.

15 3 SOLUTON Note first that the beginning-of-year expense is percent of premium only, so the term et 1 can be ignored. Using assumed experience throughout, we calculate P(0) [ (1.05)](1.06) [(50,300)(.00592) (.99408)( )] ( ) The actual mortality rate is q Using actual mortality, but assumed expenses and interest, we calculate P(1) [ (1.05)](1.06) [(50,300)(.00500) (.99500)( )] ( ) Next we use actual mortality and expenses, but assumed interest, to calculate P(2) [ (1.06)](1.06) [(50,100)(.00500) (.99500)( )] ( ) Finally we use actual experience throughout to calculate P(3) [ (1.06)](1.065) [(50,100)(.00500) (.99500)( )] ( ) Then we calculate the gain from mortality as the gain from expenses as and the gain from interest as M P(1) P(0) 42.04, E P(2) P(1) 6.26, (The negative value of which can also be found as P(3) P(2) E shows a loss from expenses.) The total gain is T M E , T P(3) P(0)

16 4 Note carefully that our notation of P (1), P (2), and P (3) refer to the profit calculations with the first, second, and third factors changed from assumed to actual, respectively. n Equations (11.31a), (11.31b), and (11.31c), the order was interest, mortality, expenses. n Example 11.9, the order was mortality, expenses, interest. n all cases we use P (0) to denote the anticipated profit, which uses assumed experience throughout. Note also that we have taken a numerical approach here to the calculations of each gain by source. We could also calculate each gain by a formula that would bypass the need to do the various profit calculations and then find the various gains by subtracting appropriate profit amounts. This approach is pursued in Exercise The same approach presented in this section can also be applied to other types of contingent contracts. This is explored in Exercises and There is an alternative model that we need to examine here. Suppose the gross premium, which has been defined to include a profit margin, is calculated without using explicit profit factors, but rather has an implicit profit margin built into it by using more conservative pricing assumptions. n this case there is no distinction between the gross premium and the expense-augmented premium (i.e., *). When the reserves are then calculated from this premium using the same assumptions, we likewise find no distinction between gross premium reserves and expense-augmented reserves. The important outcome in this case is that P (0) will again be zero. Then when Expression (11.30a) is evaluated using one or more actual factors, the profit (or loss) is revealed. n this case we act as if we do not anticipate any profit, although we certainly do because of the conservative pricing assumptions. Exercises 11.7 ain and Loss Analysis Referring to Example 11.9, write each of P(0), P(1), P(2), and P (3) symbolically and do the appropriate subtractions to reach each of the following results: M (a) ( qxtqxt)( bs t1v) E (b) *( rr)(1 i) ( ss) q (c) [ V *(1 r)]( i i) t x t A block of 1000 fully discrete 20-year term insurance policies of face amount 10,000 were issued to independent lives all age 40, of which 990 remain in force after three policy years. The gross premium and expense-augmented reserves are * 90, 3 V 100, and 4 V 125. For the fourth policy year, the anticipated interest rate is i.05, the anticipated mortality rate is q , and the anticipated percent-of-premium expense rate is r.03. n the fourth policy year, the actual interest

17 5 rate, mortality rate, and percent-of-premium expense rate were.04,.002, and.025, respectively. Calculate, in order, each of the following gains by source for the 990 policies together: (a) ain from interest (b) ain from mortality (c) ain from expenses An annual premium deferred annuity is now in its payout phase, paying 10,000 at the end of each year. The contract holder is currently age 70. The only expense is 5% of the benefit payment, payable at the end of the year for surviving contract holders only. The contract reserves are calculated from the life table in Appendix A at i.06. For the year of age from 70 to 71, the anticipated interest and mortality rates are.06 and.02, respectively, and the actual interest and mortality rates are.055 and.025, respectively. Calculate, in order, (a) the gain from mortality and (b) the gain from interest.

18 CHAPTER FOURTEEN MULTPLE-DECREMENT MODELS (APPLCATONS) 14.6 AN AND LOSS ANALYSS We consider, for the third and final time, the notion of gain or loss by source, this time in a multiple-decrement and gross premium environment. For the ( t 1) st contract year, the general profit expression is an expansion of Expression (11.30a) to include, say, two decrements, producing tv *(1 rt1) et1(1 it1) (1) (1) (1) (2) (2) (2) ( ) t1 t1 xt t1 t1 xt xt t1 ( b s ) q ( b s ) q p V, (14.39a) (1) where we assume an expense of st 1 to settle a benefit claim due to Cause 1 and an expense (2) of s to settle a benefit claim due to Cause 2 in the ( t 1) st year. t 1 The reader should by now understand the ensuing calculations. We first calculate P(0) tv *(1 rt1) e t1(1 it1) (1) (1) (1) (2) (2) (2) ( ) ( bt1st1) qxt( bt1st1) qxt pxt t1v (14.39b) using the fixed gross premium, expense-augmented reserves, and Cause 1 and Cause 2 benefit amounts, and anticipated experience for all four factors of interest, expenses, Cause 1 failure rate, and Cause 2 failure rate. Then the order of calculating each gain by source is established, and we calculate P (1) by substituting actual for anticipated experience for the factor whose gain is to be calculated first. Then we calculate P (2) using actual experience for the two factors whose gains are to be calculated first and second, but anticipated experience for the other two factors. Then we calculate P (3) using actual experience for the three factors whose gains are to be calculated first, second, and third, but anticipated 1

19 2 experience for the fourth factor. Finally we calculate P (4) using actual experience throughout. Then the gain from the factor whose gain is calculated first is the gain from the factor whose gain is calculated second is the gain from the factor whose gain is calculated third is and the gain from the factor whose gain is calculated fourth is As before, the total gain is T F 1 P(1) P(0), (14.40a) F 2 P(2) P(1), (14.40b) F 3 P(3) P(2), (14.40c) F 4 P(4) P(3). (14.40d) F F F F P P (4) (0). (14.41) Under a life insurance policy, it is often the case that Cause 1 is death and Cause 2 is surrender of (or withdrawal from) the contract. The profit expression given by Expression (14.39a) presumes that both death and withdrawal can occur at any time throughout the contract year. Alternatively, we might assume that death can occur throughout the year, but withdrawal can occur only at year end. n this case the anticipated profit expression is P(0) tv * (1 rt1) e t1(1 it1) (1) (1) (1) (2) (2) (1) (2) ( ) ( bt1st1) qxt( bt1st1)(1 qxt) qxt pxt t1v, (14.42) since, with withdrawal not possible within the contract year, mortality is operating in a single-decrement environment and the policyholder must survive death throughout the year in order for the Cause 2 (i.e., withdrawal) benefit to be paid.) The concepts presented in this section are reviewed in Exercises and Exercises 14.6 ain and Loss Analysis Consider the general double-decrement profit expression given by Expression (14.39a). Let the actual earned interest rate in the ( t 1) st year be denoted by i * t 1, and the actual Cause 2 decrement probability be denoted by q *(2) x t. f the gain from interest is calculated first and the gain from the Cause 2 decrement is calculated second, show that the gain from the Cause 2 decrement is (2) (2) (2) (2) *(2) t1 t1 t1 xt xt. b s V q q

20 A block of 1000 fully discrete insurances, issued at age 70, are in force at age 79. The gross premium is * 16, the ninth gross premium reserve is , the tenth gross premium reserve is , the tenth year death benefit is 1000, the tenth year withdrawal benefit is 110, and the assumed interest rate is.06. Expenses are 3 per policy, incurred at the beginning of the year, and there are no claim settlement expenses. Withdrawals can occur only at the end of the contract year. The assumed ( d) ( w) decrement rates are q and q During the tenth contract year there are 15 deaths and 100 withdrawals. Calculate, in order, (a) the gain from mortality and (b) the gain from withdrawal on this block of policies. [Note that, except for rounding, in this problem P(0) 0.]