Supplement to Models for Quantifying Risk, 5 th Edition Cunningham, Herzog, and London

Transcription

1 Supplemen o Models for Quanifying Risk, 5 h Ediion Cunningham, Herzog, and London We have received inpu ha our ex is no always clear abou he disincion beween a full gross premium and an expense augmened premium, which does no include a provision for profi. The 14 pages ha follow provide an expanded presenaion on his issue and can be used o replace he corresponding pages in he 2012 prining of Models for Quanifying Risk, 5 h Ediion.

2 208 CHAPTER NINE If he funding scheme is limied o years, where n, hen he APV of he m hly annuiy in ( m) he denominaor of he premium formula becomes a and he premium symbol is adjused o include he pre-subscrip as in he Equaion (9.8) se. For example, for -pay n-year erm insurance wih immediae paymen of claims we would have ( m) P A1 xn : x: A a. (9.29) The numerical calculaion of m hly funding paymens follows direcly from he approximae calculaion of he associaed m hly annuiy-due in he denominaor of he premium formula. The deails of such calculaions are presened in Secion and is associaed exercises. 1 x: n ( m) x: 9.6 FUNDIN PLANS INCORPORATIN EXPENSES Recall he observaion made earlier in his chaper ha he annual funding paymens deermined by he equivalence principle, which we called ne annual premiums in he life insurance conex, provide only for he coningen benefi paymen. In pracice, of course, he price of an insurance (or oher coningen paymen) produc mus be se higher han he ne premium in order o generae revenue o pay expenses of operaion and he coningen benefi paymens, as well as providing a profi margin o he insurer. The oal annual premium charged for an insurance produc is called he gross annual premium or he conrac premium. A premium deermined o cover benefis and expenses, bu no profi, is called he expense-augmened premium. In his ex we will use o denoe an expense-augmened premium and * o denoe a gross (or conrac) premium. I is a simple maer o exend he equivalence principle o incorporae expenses and herefore o calculae expense-augmened premiums. For ease of illusraion we assume ha he expenses allocaed o a paricular coningen conrac are fixed coss known in advance. Then he expense-augmened equivalence principle saes ha he APV of he expenseaugmened funding scheme equals he APV of he benefi paymen plus he APV of he expense charges allocaed o he conrac. For illusraion we assume ha expense charges allocaed o a conrac are of he following four ypes: (1) A percenage of he gross premium iself. (2) A fixed amoun per uni of benefi paymen. (3) A fixed (or percenage of benefi) amoun incurred when he benefi paymen is made. (4) A fixed amoun for he conrac iself, regardless of benefi amoun. The analysis of corporae operaional expenses leading o he deerminaion of expense charges o be included in he price of each produc is a complex issue ha will vary according o he ype of business under discussion. In any case he mechanics of his expense analysis are beyond he scope of his ex.

3 262 CHAPTER ELEVEN 11.4 INCORPORATION OF EXPENSES Recall ha all reserve expressions developed in Chaper 10 and hus far in his chaper are for benefi reserves only, by which we mean ha hey are based on benefi premiums (also called ne premiums). In Secion 9.6 we saw how o incorporae expense facors ino an expense-augmened equivalence principle o deermine expense-augmened premiums. I is now a simple maer o include he expense facors, along wih he expense-augmened premium, o deermine expense-augmened reserves. (When profi margins are also included in he premium, producing a gross (or conrac) premium, he resuling reserves are referred o as gross premium reserves.) The general prospecive formula for he h benefi reserve, given by Equaion (10.2), is now modified o read V ( APV of fuure benefis and expenses) ( APV of fuure expense- augmened premiums), (11.20) where he symbol V denoes he h expense-augmened reserve. This is illusraed in he following example. EXAMPLE 11.5 ive an expression for he h prospecive expense-augmened reserve for he whole life conrac described in Example 9.8. SOLUTION A duraion, given ha he coningen conrac is sill in effec, he APV of he fuure expense-augmened premium income is a x, where is defined in Example 9.8. The APV of he fuure percen of premium expense charges is.10 a x, and he APV of he fuure fixed per $1000 of benefi expense charges is 2 a x. The APV of he benefi paymen iself plus he selemen expense ogeher is 1020 Ax. Thus we have V 1020 A (.102) a a 1020 A (.902) a. x x x x x Reurning o Secion 9.6, we can separae he amoun of he level benefi premium P from he expense-augmened premium. The remainder of represens he amoun of annual premium needed o fund he expenses of adminisering he conrac. In oher words, we define o be he annual expense premium for he conrac. EP P (11.21) The noion of separaing he expense-augmened premium ino benefi premium and expense premium componens naurally exends o he reserve. As already covered exensively in

4 RESERVES AS FINANCIAL LIABILITIES 263 Chaper 10, he prospecive benefi reserve is he APV of fuure benefis minus he APV of fuure benefi premiums. Similarly we define he h prospecive expense reserve o be E V ( APV of fuure expenses) ( APV of fuure expense premiums). (11.22) I should be clear ha where B V V V (11.23) B E, h V is he benefi reserve previously denoed by simply. EXAMPLE 11.6 ive an expression for he h prospecive expense reserve for he whole life conrac described in Example 9.8. V SOLUTION Firs we find he level expense premium as EPa.75.10a 10 2a 20 A, x x x x where is defined in Example 9.8. Then he expense reserve is E V 20 A (.102) a EPa. x x x Recall how he concep of he presen value of loss (a issue) random variable, L x, inroduced in Secion 9.2, was easily exended o he presen value of loss (a duraion ) random variable, L x, defined in Secion In he same way, he expense-augmened presen value of loss (a issue) random variable, L x, defined in Secion 9.6, is easily exended o he expense-augmened presen value of loss (a duraion ) random variable, which we denoe by L. This is pursued in Exercise x Recall ha we expanded Equaion (10.30b), which had presumed level benefi premiums and a level failure benefi, ino Equaion (11.16b), which generalized Equaion (10.30b) for nonlevel benefis and benefi premium. Now we generalize furher o include expenses. Le denoe he expense-augmened premium for he h conrac year, r denoe he percen-of-premium expense facor for ha year, e denoe he fixed expense for ha year, and s denoe he selemen expense associaed wih a benefi paid a he end of he h conrac year. Then Equaion (11.16b) is generalized o [ V (1 r ) e ](1 i ) ( b s ) q p V. (11.24) x x 1 Noe ha Equaion (11.24) allows for he reserve ineres rae o also vary by conrac year, for maximum generaliy. In many applicaions, i 1 will be se as a consan.

5 268 CHAPTER ELEVEN 11.7 AIN AND LOSS ANALYSIS We now coninue our analysis of financial gain or loss under a coningen conrac which we began in Secion 10.6, his ime in he more realisic environmen of gross premiums and eiher gross premium reserves or expense-augmened reserves. We begin wih Equaion (11.24), found in Secion 11.4, and modify i o read * x x 1 [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.30a) which uses he gross premium bu expense-augmened (raher han gross premium) reserves. Recall ha his expression is wrien wih maximum generaliy o allow he benefi, gross premium, ineres rae, and all expense facors o vary by conrac year. In pracice, many of hese parameers would be consan over conrac years. An imporan difference beween Expression (11.30a) and is ne counerpar given by he lef side of Equaion (10.51) is ha he ne case expression always equals zero bu his gross case profi expression does no necessarily equal zero, since he gross premium has been se o include a profi margin, bu he expense-augmened reserves do no consider profi. When he gross case Expression (11.30a) is evaluaed using he parameers anicipaed o apply in he ( 1) s conrac year, we refer o he resuling value as he anicipaed profi for he ( 1) s year, which we denoe by P (0). Tha is, * x x 1 P(0) [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.30b) where we use unprimed symbols for anicipaed experience. Nex we consider he noion of gain or loss by source, inroduced in Secion 10.6 in he ne premium case. Here we have hree poenial sources of gain or loss (ineres, moraliy, and expenses), and we adop a differen approach o calculaing hem han was used in Secion In his case we mus specify he order in which he gain-by-source calculaions are o be made. We illusrae ha noion here by choosing he order ineres, hen moraliy, and hen expenses. Now we evaluae Expression (11.30a) using he acual ineres rae earned in he ( year, bu sill using anicipaed experience for moraliy and expenses, producing * x x 1 P(1) [ V (1 r ) e ](1 i ) [( b s ) q p V ], (11.31a) where we use i 1 in place of i 1 o denoe he acual ineres rae earned in he ( year. Nex we evaluae Expression (11.30a) using acual ineres and moraliy, bu sill using anicipaed expenses, producing * x x 1 P(2) [ V (1 r ) e ](1 i ) [( b s ) q p V ]. (11.31b) 1) s 1) s

6 RESERVES AS FINANCIAL LIABILITIES 269 Nex we evaluae Expression (11.30a) using acual ineres, moraliy, and expenses, producing * x x 1 P(3) [ V (1 r ) e ](1 i ) [( b s ) q p V ]. (11.31c) (Noe ha he benefi, gross premium, and expense-augmened reserve parameers are consan hroughou hese calculaions.) Finally, we define he gain from ineres o be he gain from moraliy o be and he gain from expenses o be Noe ha I P(1) P(0), (11.32a) M P(2) P(1), (11.32b) E P(3) P(2). (11.32c) T I M E P(3) P(0), (11.33) which shows ha he oal gain can be calculaed by subracing he anicipaed profi from he profi calculaed using acual experience hroughou. The heory developed here is illusraed in he following example. EXAMPLE 11.9 Consider a block of fully discree whole life policies issued a age 40 wih face amoun 50,000. On he assumed (or anicipaed) moraliy, ineres, and expense bases, he gross annual premium per policy is , he enh-year expense-augmened reserve is , and he elevenh-year expense-augmened reserve is The assumed ineres rae is 6%, he assumed moraliy rae for he elevenh year is q , and he assumed expenses are 5% of he gross premium and 300 o process a deah claim. In he elevenh year, here are 1000 policies in force a he beginning of ha year and five deahs occur in he year. Acual expenses in he elevenh year are 6% of he gross premium and 100 o process each deah claim, and he acual earned ineres rae is 6.5%. Calculae, in order, he gain from moraliy, he gain from expenses, and he gain from ineres on a single policy. SOLUTION Noe firs ha he beginning-of-year expense is percen of premium only, so he erm 1 e can be ignored. Using assumed experience hroughou, we calculae

7 RESERVES AS FINANCIAL LIABILITIES 271 Noe also ha we have aken a numerical approach here o he calculaions of each gain by source. We could also calculae each gain by a formula ha would bypass he need o do he various profi calculaions and hen find he various gains by subracing appropriae profi amouns. This approach is pursued in Exercise The same approach presened in his secion can also be applied o oher ypes of coningen conracs. This is explored in Exercises and There is an alernaive model ha we need o examine here. Suppose he gross premium, which has been defined o include a profi margin, is calculaed wihou using explici profi facors, bu raher has an implici profi margin buil ino i by using more conservaive pricing assumpions. In his case here is no disincion beween he gross premium and he expense-augmened premium (i.e., *). When he reserves are hen calculaed from his premium using he same assumpions, we likewise find no disincion beween gross premium reserves and expense-augmened reserves. The imporan oucome in his case is ha P (0) will again be zero. Then when Expression (11.30a) is evaluaed using one or more acual facors, he profi (or loss) is revealed. In his case we ac as if we do no anicipae any profi, alhough we cerainly do because of he conservaive pricing assumpions EXERCISES 11.1 Modified Benefi Reserves 11-1 If benefi premiums are modified for he enire premium paying period of n years, show each of he following: (a) NLP M V ( ) : V P a xn xn : xn : (b) V 1 ( d) a M xn : xn : 11-2 Consider a fully coninuous uni whole life insurance issued a age x, under which modified coninuous reserves accumulae using modified benefi premium rae () r a ime r, for 0r 5, and modified benefi premium rae a ime r 5. The premium rae () r is defined as (0).25, increasing linearly o (5). Show ha Ax. a.75 a.15( Ia) x x:5 x: For an h-pay, n-year uni endowmen insurance issued a age x, wih reserves calculaed by he FPT mehod, show ha where h n. h FPT h 1 NLP V : 1, xn V x1: n1

8 272 CHAPTER ELEVEN 11-4 As an exension of Example 11.2, show ha, under he wo-year FPT reserving F F mehod, 1 V 2 V 0 and 2. P x 11.2 Benefi Reserves a Fracional Duraions 11-5 Show ha he expression for r given in Equaion (11.9) reduces o r 1 s under he UDD assumpion Show ha he h year mean reserve for a uni insurance can be wrien as V 1 (1 v p ) V v q. 2 1/2 x1 x eneralizaion o Non-Level Benefis and Premiums 11-7 Wrie he general rerospecive formula which is he counerpar o he prospecive formula given by Equaion (11.15) A 3-year erm insurance issued o (x) has a decreasing failure benefi, paid a he end of he year of failure. The ineres rae is i.06. Calculae he iniial reserve for he second year, given he following values: b 200 b 150 b q.03 q.06 q.09 x x1 x A 2-year endowmen conrac issued o (x) has a failure benefi of 1000 plus he reserve a he end of he year of failure and a pure endowmen benefi of iven ha i.10, q.10, and q 1.11, calculae he ne level benefi premium. x x A whole life conrac issued o (40) pays a benefi, a he end of he year of failure, of b k for failure in he k h year. The ne premium P is equal o P 20, and he benefi reserves saisfy V V20, for 0,1,,19. Furhermore, q 40 q 20.01, for k 0,1,,19. iven ha 11 V and q , calculae b 11. k k A coninuously decreasing 25-year erm insurance issued o (40) has benefi rae b 1000 for failure a ime. The coninuous ne premium rae is P 200. a 25 iven also ha i.05 and A.60, find he benefi reserve a ime :15

9 RESERVES AS FINANCIAL LIABILITIES Solve Thiele s differenial equaion, given by Equaion (11.19), o reach he rerospecive reserve expression given by Equaion (11.18) Incorporaion of Expenses For he 20-pay whole life insurance described in Exercise 9-26, find he expenseaugmened reserve a (a) duraion 10 and (b) duraion 20, given he addiional values a x , a x , and 7. a x 10:10 E Show ha V V V, where V is given in Example 11.5 and E V is given in Example We now define L x as he expense-augmened presen value of prospecive loss * measured a ime, given ha K x (i.e., he conrac has no ye failed a ime ). For he whole life conrac described in Example 9.8, show ha E * Lx Kx V, he expense-augmened reserve, as deermined in Example Consider he expense-augmened premium recursion relaionship given by Equaion (11.24). Suppose he premium is paid coninuously a annual rae a ime, and fixed expenses are paid coninuously a annual rae e a ime. Sae Thiele s differenial equaion in his general case including expenses Inroducion o Universal Life Insurance The accoun value roll forward process under a universal life conrac is ofen done on a monhly basis. Suppose he conrac in Example 11.7 receives annual conribuions of 5000, earns ineres a i (12).03, assesses expense charges a 50% of conribuion plus 10 per monh, and esimaes monhly moraliy raes a 1/12 he corresponding annual rae. Calculae he accoun values a he ends of each of he firs hree monhs In pracice, he ne amoun a risk under a universal life conrac paying a failure benefi fixed a amoun B is ofen defined as he excess of B over he prior period ending accoun value plus he curren period ne conribuion before deducing fixed expenses. The cos of insurance is hen defined as he moraliy rae imes he ne amoun a risk, wihou he discoun facor. Rework Example 11.8 under hese definiions of NAR and COI.

10 274 CHAPTER ELEVEN 11.6 Inroducion o Deferred Variable Annuiies A five-year deferred variable annuiy is issued o (60) who makes annual conribuions of 5,000 each. The percen-of-conribuion expense rae is 60% in he firs year and 10% in subsequen years. There is an annual expense charge of 2% of he prior year s accoun value, assessed a he beginning of each year. For convenience, assume he ineres rae credied o he accoun is consan a 8%, and he failure benefi is he accoun value. A age 65 he accoun value is used o purchase an annual annuiy-due based on 6% ineres and he survival model of Appendix A. Calculae he annual annuiy paymen ain and Loss Analysis Referring o Example 11.9, wrie each of P(0), P(1), P(2), and P (3) symbolically and do he appropriae subracions o reach each of he following resuls: M (a) ( qxqx)( bs 1V) E (b) *( rr)(1 i) ( ss) q I (c) [ V *(1 r)]( i i) x A block of 1000 fully discree 20-year erm insurance policies of face amoun 10,000 were issued o independen lives all age 40, of which 990 remain in force afer hree policy years. The gross premium and expense-augmened reserves are * 90, 3 V 100, and 4 V 125. For he fourh policy year, he anicipaed ineres rae is i.05, he anicipaed moraliy rae is q , and he anicipaed percen-of-premium expense rae is r.03. In he fourh policy year, he acual ineres rae, moraliy rae, and percen-of-premium expense rae were.04,.002, and.025, respecively. Calculae, in order, each of he following gains by source for he 990 policies ogeher: (a) ain from ineres (b) ain from moraliy (c) ain from expenses An annual premium deferred annuiy is now in is payou phase, paying 10,000 a he end of each year. The conrac holder is currenly age 70. The only expense is 5% of he benefi paymen, payable a he end of he year for surviving conrac holders only. The conrac reserves are calculaed from he life able in Appendix A a i.06. For he year of age from 70 o 71, he anicipaed ineres and moraliy raes are.06 and.02, respecively, and he acual ineres and moraliy raes are.055 and.025, respecively. Calculae, in order, (a) he gain from moraliy and (b) he gain from ineres.

11 334 CHAPTER FOURTEEN under a single coningen conrac (such as an insurance policy) or for a block of such conracs. The resuls are known as projeced asse shares. Suppose we have a coningen paymen conrac funded by a level annual conrac premium *. The conrac pays in he even of he failure (such as deah) of a specified eniy of ineres or in he even of wihdrawal from he coningen conrac. The paymen due in he (1) even of failure in Year k is denoed b k and he paymen due in he even of wihdrawal is denoed b (2) k ; in eiher case he benefi is paid a he end of he year. As in Example 14.3, expenses are paid a he beginning of each year and are of boh he percen-of-premium and conrac consan ypes. The projeced asse share a duraion k, which is he acuarial accumulaed value of premiums minus expeced benefis and expenses, is denoed by k AS. All noaion used in his secion is summarized in he following able. Symbol Concep * Annual conrac premium Benefi paid a end of Year k for failure during Year k (1) bk (2) bk r k e i k (1) qx k 1 (2) qx k 1 ( ) p xk 1 k AS Benefi paid a end of Year k for wihdrawal during Year k Percen-of-premium expense facor paid a beginning of Year k Fixed conrac expense paid a beginning of Year k Effecive annual rae of ineres (presumed consan) Condiional probabiliy of failure during Year k, given ha he conrac is sill in force a ime k 1 Condiional probabiliy of wihdrawal during Year k, given ha he conrac is sill in force a ime k 1 Condiional probabiliy of he conrac saying in force hrough Year k, given ha i is sill in force a ime k 1 The projeced asse share associaed wih he conrac a he end of Year k We denoe he iniial asse share a ime 0 by 0 AS, and noe ha 0 AS may or may no equal zero. Successive values of k AS are hen found recursively by expanding he discussion in Secion 11.4 o include muliple decremens. For k 1 we have so For k in general we have so (1) (2) AS *(1 r) e (1 i) b q b q AS p, (14.5a) (1) (2) ( ) x 1 x 1 x (1) (2) *(1 ) (1 ) AS r e i b q b q. (14.5b) (1) (2) x 1 x 1 AS ( ) px (1) (1) (2) (2) ( ) AS *(1 r ) e (1 i) b q b q AS p (14.6a) k1 k k k xk1 k xk1 k xk1

12 MULTIPLE-DECREMENT MODELS (APPLICATIONS) 335 k AS (1) (1) (2) (2) AS *(1 r ) e (1 i) b q b q k1 k k k xk1 k xk1 ( ) p xk1. (14.6b) EXAMPLE 14.4 Consider he five-year endowmen insurance described in Example If he conrac 2 premium is * and he iniial asse share is 0 AS 50, find k AS for k 1,2,3,4,5. SOLUTION Using he recursive relaionship given by Equaion (14.6b) he following values are obained. (The deails of he calculaions are lef o he reader as an exercise. 3 ) k k AS The excess (if any) of he asses associaed wih a coningen conrac over he conrac liabiliy may be inerpreed as an amoun of surplus generaed by ha conrac. The liabiliy is given by he expense-augmened reserve, and he associaed asse value is given by he projeced asse share. Then he projeced surplus a he end of Year k is given by U AS V k k k. (14.7) EXAMPLE 14.5 Find he surplus U k, for k 1,2,3,4,5, for he five-year endowmen conrac described in Examples 14.3 and SOLUTION Noe ha alhough he projeced asse shares are deermined using he acual conrac premium of , he reserves are deermined using he expense-augmened premium of , which is he premium necessary o cover he benefis and expenses ha consiue he conrac liabiliy. The following resuls are obained direcly from he resuls of Example 14.3(b) and Example Noe ha U0 50 since 0 AS 50 and 0 V 0. 2 The premium migh exceed he value calculaed in Example 14.3 o reflec consideraions of compeiion and profi. 3 The complee soluion can be found on he ACTEX Publicaions websie.

13 364 CHAPTER FOURTEEN 14.6 AIN AND LOSS ANALYSIS We consider, for he hird and final ime, he noion of gain or loss by source, his ime in a muliple-decremen and gross premium environmen. For he ( 1) s conrac year, he general profi expression is an expansion of Expression (11.30a) o include, say, wo decremens, producing (1) (1) (1) (2) (2) (2) ( ) x 1 1 x x 1 [ V * (1 r ) e ](1 i ) [( b s ) q ( b s ) q p V ], (14.39a) (1) where we assume an expense of s 1 o sele a benefi claim due o Cause 1 and an expense (2) of s o sele a benefi claim due o Cause 2 in he ( 1) s year. 1 The reader should by now undersand he ensuing calculaions. We firs calculae (1) (1) (1) (2) (2) (2) ( ) x 1 1 x x 1 P(0) [ V *(1 r ) e ](1 i ) [( b s ) q ( b s ) q p V ] (14.39b) using he fixed gross premium, expense-augmened reserves, and Cause 1 and Cause 2 benefi amouns, and anicipaed experience for all four facors of ineres, expenses, Cause 1 failure rae, and Cause 2 failure rae. Then he order of calculaing each gain by source is esablished, and we calculae P (1) by subsiuing acual for anicipaed experience for he facor whose gain is o be calculaed firs. Then we calculae P (2) using acual experience for he wo facors whose gains are o be calculaed firs and second, bu anicipaed experience for he oher wo facors. Then we calculae P (3) using acual experience for he hree facors whose gains are o be calculaed firs, second, and hird, bu anicipaed experience for he fourh facor. Finally we calculae P (4) using acual experience hroughou. Then he gain from he facor whose gain is calculaed firs is he gain from he facor whose gain is calculaed second is he gain from he facor whose gain is calculaed hird is and he gain from he facor whose gain is calculaed fourh is As before, he oal 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)

14 MULTIPLE-DECREMENT MODELS (APPLICATIONS) 365 Under a life insurance policy, i is ofen he case ha Cause 1 is deah and Cause 2 is surrender of (or wihdrawal from) he conrac. The profi expression given by Expression (14.39a) presumes ha boh deah and wihdrawal can occur a any ime hroughou he conrac year. Alernaively, we migh assume ha deah can occur hroughou he year, bu wihdrawal can occur only a year end. In his case he anicipaed profi expression is (1) (1) (1) (2) (2) (1) (2) ( ) 1 1 x 1 1 x x x 1 P(0) [ V * (1 r ) e ](1 i ) [( b s ) q ( b s )(1 q ) q p V ], (14.42) since, wih wihdrawal no possible wihin he conrac year, moraliy is operaing in a single-decremen environmen and he policyholder mus survive deah hroughou he year in order for he Cause 2 (i.e., wihdrawal) benefi o be paid.) The conceps presened in his secion are reviewed in Exercises and EXERCISES 14.1 Acuarial Presen Value 14-1 A company hires all new employees a age 25. An employee can leave he company via deah while employed (Decremen 1), resignaion prior o age 65 (Decremen 2), or reiremen a age 65. The company provides he following benefis for is employees: (a) Employees who reire a age 65 receive coninuous reiremen income a an annual rae of 500 for each year of employmen wih he company. (b) Employees who die while employed receive a one-ime deah benefi of 200,000 a he precise ime of deah. (c) Employees who resign prior o age 65, bu survive on o age 65, receive coninuous reiremen income a an annual rae of 400 for each year of employmen wih he company (including parial years). Wrie expressions involving coninuous annuiies and/or inegrals for he APV a ime of hire for each of he hree benefis Asse Shares 14-2 Show ha (1) (1) (2) (2) AS AS (1 r ) e (1 i) q ( b AS) q ( b AS). k k1 k k xk1 k k xk1 k k (This relaionship shows ha he difference beween he wihdrawal value and he asse share is imporan o he progression of he asse share values. If he asse share were paid as a wihdrawal value, hen he asse share values would progress independenly of he wihdrawal risk.)

15 370 CHAPTER FOURTEEN benefi payable immediaely, wihou reducion, if he employee has a leas five years of service. The deah benefi requires en years of service, and is se a 50% of he hen accrued benefi, reduced as for early reiremen. Assume he surviving beneficiary is hree years younger han he employee. Wrie he APV formulas for each of (a) normal reiremen, (b) early reiremen, (c) wihdrawal, (d) disabiliy, and (e) deah. Assume early reiremen, wihdrawal, disabiliy, or deah occur half way hrough he year of age, on average Assume he employee of Exercise is now exac age 56, wih a salary of 150,000 in he year from age 55 o age 56 and a salary of 156,000 in he year from age 56 o age 57. Deermine each of he following: (a) The benefi accrual for he year from age 56 o age 57. (b) The uni credi normal cos. (c) The accrued liabiliy under he uni credi cos mehod ain and Loss Analysis Consider he general double-decremen profi expression given by Expression (14.39a). Le he acual earned ineres rae in he ( 1) s year be denoed by i * 1, and he acual Cause 2 decremen probabiliy be denoed by q *(2) x. If he gain from ineres is calculaed firs and he gain from he Cause 2 decremen is calculaed second, show ha he gain from he Cause 2 decremen is (2) (2) (2) (2) *(2) x x. b s V q q A block of 1000 fully discree insurances, issued a age 70, are in force a age 79. The gross premium is * 16, he ninh gross premium reserve is , he enh gross premium reserve is , he enh year deah benefi is 1000, he enh year wihdrawal benefi is 110, and he assumed ineres rae is.06. Expenses are 3 per policy, incurred a he beginning of he year, and here are no claim selemen expenses. Wihdrawals can occur only a he end of he conrac year. The assumed ( d) ( w) decremen raes are q and q During he enh conrac year here are 15 deahs and 100 wihdrawals. Calculae, in order, (a) he gain from moraliy and (b) he gain from wihdrawal on his block of policies. [Noe ha, excep for rounding, in his problem P(0) 0.]