Optimal Life Insurance

Transcription

1 Opimal Life Insurance by Karl Borch Barch * 1. Inroducion 1.1. A person who wans o arrange his life insurance in he bes possible manner will probably be bewildered by he many differen offers available from insurance companies. he lieraure which should guide him hrough his jungle of offers is usually of lile help. Much of i is plain sales alk, alhough i may beof of high echnical qualiy, paricularly when i comes o explaining he ax advanages offered by life insurance in differen counries. here is cerainly lieraure which seeks o offer unbiased advice, bu his does no help much, since he auhors and would-be expers seem o disagree among hemselves Much of he heoreical lieraure on life insurance is based on Huebner's concep of "human" life value". his is essenially he presen value of he fuure income which will no be realized if he person dies. his is an asse - presumably an inangible one - which,according o o Huebner, a a raional person should cover by insurance. his is however conroversial. According o o Denenberg [2] "Huebner's "human life value" ideas have been endorsed and enshrined by he life insurance exs and prominenly feaured in he raining maerials of he American College of Life Under- Underwriers". ". On he oher hand, Josephson [5] saes flaly: "I believe i can be said unequivocally ha he Huebner concep did no influence he markeing philosophy of a single life insurance company". We shall no ry o sele his dispue, bu we should noe ha he idea of Huebner has is origin in propery insurance. Boh auhors seem o overlook ha he real problem in his field is no o evaluae he propery, bu o decide if he owner should carry some of he risk himself. Clearly his decision will depend on he cos of insurance cover, and calls for an economic analysis Life insurance has obvious relaions o economics, since i is essenially a form of saving. One gives up consumpion a a he presen ime, in order o provide for one's own old age, or for dependens one may leave a deah. If a person wans o know by how much he shoud cu down curren consumpion in order o provide beer for he fuure, he will ge lile help from economic lieraure. Economic heory is based on "consumer's sovereigny", ", and leaves i i o he individual o decide for himself how he will evaluae curren needs in relaion o he need for providing for fuure coningencies. In In he heory one does however sudy how his evaluaion is * Professor of insurance economics a a he Norwegian School of Economics and Business Adminisraion, Bergen. he auhor is graeful o Denis Moffe for many heaed discussions of his subjec. 3

2 made, since such informaion is is essenial in in he consrucion of general models for predicing saving and invesmen in in he economy as a whole. An economis is more likely o observe, han o preach hrif and frugaliy wih he enhusiasm of a good insurance salesman As life insurance is is aa form of saving, i i will have o compee wih oher forms of saving. he growing ineres in in porfolio heory over he las wo decades has brough much aenion o insurance. Life insurance policies obviously should have a place in an opimal porfolio. How prominen his place should be will clearly depend on he naure of he alernaive invesmens, and his leads o a number of ineresing problems, which recenly have been sudied by several auhors, i.a, i.a. by Fischer [3] and Richard [7]. We shall no ake up hese problems in heir full generaliy, and we shall find i convenien o begin our discussion by considering some older models. 2. he Simples saving saving - consumpion models 2.1. he models we shall sudy go back o he neo-classical schools, and were sudied in deail by Marshall [6] and by Fisher [4], who dedicaed his book o Böhm-Bawerk Bohm-Bawerk [1], "who laid he foundaion upon which I have endeavoured o build". he purpose of hese sudies was o deermine he heoreical relaionships beween savings and ineres raes in he economy. he noaion used by hese wriers appears cumbersome o a modern reader, and in he following we shall use a noaion which is due o Yaari [8] and [9] he given elemen in he models we shall sudy is he consumer's income sream y(), a funcion of ime. he problem is o deermine he opimal sream c() for his expendiure on consumpion. I is convenien o refer o he funcion c() as a consumpion plan. Any plan which can be carried ou for a given income sream will be called a feasible plan. If he consumer can neiher borrow money, nor save money for fuureconsump- consumpion, he feasible consumpion plans will be defined by he inequaliy c() < y() his simply says ha he consumer canno a any ime spend more han his income. Among he feasible plans, he plan c() = y() will appear as he opimal one, under he usual assumpion of non-sauraion, i.e. if we assume ha he consumer prefers o spend his income on consumpion, raher han leing i be wased In order o arrive a less rivial models, we mus assume ha he consumer has some possibiliies of of ransfering money from one poin of ime o anoher, i.e. i.e, ha he can reallocae his income over ime. he simples assumpion of of his kind is ha he consumer can borrow or save any amoun a a he same rae of ineres. For an arbirary consumpion plan, his accumulaed savings a a ime will hen be where 0 is he force of ineres. 4 S() = eö ~ Je-~s {y(s).c(s)} {y(s)-c(s)} ds 0o

3 I is naural o assume ha here are some resricion on he consumer's borrowing. One resricion of his kind would be a condiion: (1) S()>OS >: 0 W'hich which says ha he consumer mus be solven a ime. We can inerpre as his "planning.c horizon" -- or haof of his crediors. A A sricer condiion would be: : foro. S()>O If his condiion is imposed, he consumer is is never allowed o o be in deb, and i (2) S () >: 0 for 0 < <. is no necessary o assume ha ineres raes are he same for borrowing and saving. Eiher of he condiions (1) or (2) will give a seof of feasible consumpion plans, and i is eviden ha (2) will give he more resriced se. If no valueis is assigned o consumpion beyond he horizon, we we will have S S () () = = 00 for for he opimal plan -- under he non-sauraion assumpions I is no necessary o assume ha y () and c () are coninuous flows. We can define Y () and C () as accumulaed income and consumpion up o ime.. Accumulaefaed savings a ime will hen Accumu be S() Se) = e e~ { f Je-~ e dy(s)-f Je-~se dc(s)}. o 0 here is lile o be gained by his rival generalizaion, and i will no be used in iii he following discussion o selec he opimal plan, we need a preference ordering over he se of feasible consumpion plans. I I is convenien if if his ordering can be represened by a uiliy funcional, i.e, i.e. a mapping from he se of feasible plans o he real line, which can be expressed in a simple analyical manner. In economic heory one usually assumes ha he uiliy assigned o an arbirary consumpion plan is is given by an expression of he form: 13) U(c)= = J J e u[c()j u[c()] d. o0 Here i' y represens he consumer's "impaience", ", i.e. his preference for consump- consumpion early raher han lae in he planning period. For he insananeous uiliy funcion II u (c) one usually assumes: : u' (c) > 0 and U" u"(c)<o. < o. Expression (3) clearly conains aa number of of "heroic assumpions", ", and he main mori meri of models of his ype may be ha hey lead o problems which can be solved hyrelaively by simple mahemaical mehods. One should however noe ha i is difficul even o o describe oher, and possibly more realisic preference orderings over les ses of consumpion plans We have now arrived a he problem of maximizing (3), subjec o some condiion, mch such as (1) or (2), which will keep he maximand finie. 5

4 or From he definiion of accumulaed savings S (1) () we find S' (1) () = ~S os () () + y () -- c () c () = y () + as ~S (1) () - S' (). (1). Subsiuing his in in (3) we arrive a he problem of maximizing: r Je- S e" y U[y()+bS()-S'()] u[y()+5s()s'()] d. o0 his expression is is of he form r JF[,x(),x'()] ()J d o so he problem is reduced o a problem in he classical calculus of variaion. A soluion o he problem mus saisfy he Euler equaion f3f d (i3f ôxdk3x' 2.7. In our problem he Euler equaion akes he form which reduces o (4) or (4') ~ d[e_l u'[c()1 _y ()~ d[e- y u'[c()]] ue 5e u'[c()] c~+ + d =0, d Here K is an arbirary posiive consan, which occurs because he soluion of he problem remains he same if he uiliy funcion is muliplied by a posiive consan. From (4) i follows ha c (1) () will be consan if r = a, d, i.e. i.e, if he rae of impaience is equal o he rae of ineres. If If r y> > 6, ~, c (1) () will be monoonic decreasing, and if r ' < a, d, c (1) () will be monoonic increasing. If by pure chance he soluion of he problem should be c (1) () = y Y (), he consumer wifi will neiher borrow nor save. In general i will however be o his advanage o ransfer income from one period o anoher I may be useful o give a simple example o illusrae he resuls above. Le u (c) = log c. he differenial equaion (4) hen becomes c' () = (6 (~ - y) r) C c () wih he soluion c () = Ae@). Ae(&-r). he consan of inegraion A = c (0) mus be deermined so ha condiion (1) is saisfied, wih sric equaliy, i.e, i.e. we mus have 6 ~: = :(:~)., u'[c()] c() c'()- - (y u'[c()j (V c5) u"[c()] u"[c()j u' ii' [c ()] = Ke(1 Ke - al. 5e'y()d= Je- 1 = Je- Sec()d=AJed. 1 ='Af e-y d. 0O' 0 0 Le us furher assume y () = y = consan for 0. Le us furher assume y () = y = consan for 0 < <.

5 > r, we will have c () < y in he beginning of he period, and If in his case ô> ~ y, we will have c () <y in he beginning of he period, and c () > y owards he end. his plan will be feasible, also if he sricer condiion (2) is imposed. If on he oher hand 6 d < y. r, we have c () > y in he beginning of he period, and his plan is no feasible under condiion (2). he opimal feasible plan will hen be c () = y. In his case he inroducion of credi faciliies would relax he condiion (2), and allow he consumer o adop a beer consumpion plan. 3. Models wih uncerainy and Insurance 3.1. In a world of complee cerainy here are no compelling reasons for no giving credi o a consumer, provided ha his fuure income is sufficien o repay he deb wih ineres. If however here is a possibiliy ha he income sream may be cu off a any ime, poenial crediors may impose he sric condiion (2), and no allowing he consumer o be in deb. o formulae his idea in a simple manner, we shall assume ha he income sream y (r) () is given, bu ha i a any ime can drop o zero for he res of he planning period. Le r si () (r) be he probabiliy ha he income sream flows a ime. We have n n(0) = 1. As a concree inerpreaion we can hink. of siuaions in which he income ceases because he consumer becomes permanenly disabled, or unemployed. he radiional purpose of saving is jus o build up a reserve for such evens he problem of he consumer mus now be reformulaed, and i is naural o assume ha he will maximize expeced uiliy: (5) J i e" (x()u[c()+y()] {n()u[c() + +[l-1{)]u[c()]} [1 x()]u[c()]} d o subjec o some condiions such as (1) or (2). As in para 2.6. we can finda a differenial equaion which accumulaed savings mus saisfy, and solve a problem in he calculus of variaion. his soluion o he problem seems inefficien, since i i may lead o an accumulaion of savings which will no be "needed" if if he income sreamis is mainained unil ime, i.e. unil he end of he planning period. he inroducion of insurance will make i possible for he consumer o obain an income c () for consumpion if, and only if, his earnings y () sops. his possibiliy may hen help he consumer, jus as credi faciliies did in para o make hings simple, we shall assume ha he hands his whole income y () over o an insurance company. In reurn he receives a sream c (), which he can use for consumpion. he principle of equivalence implies ha we mus have r r (6) J1()y()e- f ir()y()e_ fl d- f JC()e-M c()e d = 0. O. o0 0 7

6 his condiion which defines he feasible consumpion plans, plays he same role as condiion (1) in he previous secion. he prospecive reserve of he insurance arrangemen a ime 1 is: : Ve) V() = edic(s)e-as"ds-e J ds e5 8f In(s)y(s)e- i(s)y(s)e cls ds. For any insurance conrac one usually requires ha he reserve shall be non- nonnegaive, i.e, i.e. ha (7) V() v (I) >O. >0. his is obviously equivalen o o condiion (2), and he condiions (2) and (7) are usually imposed for similar reasons Insurance has removed he uncerainy, so he problem of he consumer is now r (8) max J j e ufc()j u[c()]d subjec o condiion (6), or o he sricer condiion (7). From he definiion of he reserve we find o0 V'() (I) = dv ov() (1)+ + 11;,r()y() (I) Y (1) - c(f). C (1). We can use his o subsiue for c C (1), (), and obain a problem in he calculus of variaion. o compare he wo problems (5) and (8), we noe ha expeced income a ime 1 is (5') E{y()} = x()y(). n()y(). Hence he former problem can be wrien and he second maxee{u(ös_s+y)} d (8') max] f e- 1u[~V-Jl'+E{y()}]d. u[öv - V'+E{y()}] o Formally he wo problems are idenical, and hey will have idenical soluions, provided ha he sricer condiions (2) and (7) do no become effecive. As we have assumed ha u (c) is concave, i i follows from Jensen's inequaliy ha we have E{u(y)} <u(e{y}) so ha problem (8) gives a higher uiliy han problem (5) We have inroduced insurance in a raher arificial way. A more convenional way would be o assume ha he consumer agrees o pay a sream of premiums p (I) () as long as his income flows. In reurn he receives an income sream c (I) () if he sream y (I) () should sop. he problem is hen o deermine an insurance plan which will make i possible o carry ou he bes possible consumpion plan. 8 max] e-, E{u(~S-S'+y)} d o

7 Formally he problem is as before o maximize Je- y u[c()]d. 0o Under his plan accumulaed savings a a ime will be: Se) S() = e~j{y(s)-c(s)-p(s)} e'5j' {y(s)c(s)p(s)} e- 6s ds. o0 he rerospecive reserve under he insurance conrac a a ime will be: Ve)= V() = e3'f e 6 J1(s)p(s)e- z(s)p(s)e cjs ds-emj[l-n(s)]c(s)e-cjsis. - e"j [1 x(s)jc(s)e ds. Differeniaing hese wo equaions, and eliminaing p (), we obain an expression for c () which can be subsiued ino he maximand. he problem hen akes a familiar form. If we impose no oher condiions on S () and V () han simple boundary condiions, such as and S (0) == = S () = 0 V V(0)=V()=0 = V () = 0 o 0 he problem can be solved by classical calculus of variaion. I can be shown ha under he opimal plans c () + p () (1) = y () i.e, i.e. S () (1) = 0, so ha all saving akes place by building up reserves in he insurance company. his resul should no be surprising, since insurance does no lead o any savings which may no be needed he premium is deermined by he principle of equivalence, i.e. i.e, he equaion V V() () = 0 or J1()p()e-6 x()p()e d = J{l-1()}c()e- {1 - iv()}c()e cj d. 0o 0 Any funcion p () which saisfies his equaion represens a feasible premium plan, alhough one would usually also require ha V () > O. alhough one would usually also require ha V () > 0. In convenional insurance conracs he funcion p () is usually of a simple form, for insance p () = p = consan. In such cases he difference y (1) (I) -- p P (0 () - c (I) (f) will be saved, or if negaive mus be covered from accumulaed savings. Hence an inflexible plan for paying premiums may force he consumer o savein inhe heconven- convenional way. he poin may have some pracical ineres. Why should a consumer have o pu money ino he bank in order o pay fuure "level premiums" o his insurance company? 4. LIfe Life Insurance 4.1. Le us now assume ha he income sream ceases only wih he consumer's deah. If we assume ha his need for consumpion dies wih him, he models in he preceding secion do do no longer apply. he upper limi in.he inegral in he objecive funcion (3), was inerpreed as he consumer's planning horizon. His life ime is a 9

8 very naural planning horizon indeed, so so le le us us inerpre as he ime of he consumer's deah. Clearly is a sochasic variable, and is probabiliy densiy is f() 1() ='-- Jx+ Here i x is he moraliy able, and I-x is ne ile force of moraliy. I is really unnecessary o o inroduce he symbol x, x, represening, he consumer's age, bu i i is convenien o o do so, since his makes i possible o use he sandard acuarial noaion he expeced uiliy of he life ime consumpion is hen obained from (3) u =J{fevurc()] d} + Jx+ d = o0 0 Hence he problem is o maximize (9) fe fe_ u[c()] u{c()) d. u{c()] d subjec o some condiions. he principle of equivalence requires ha he expeced discouned value of income mus be equal o ha of he expendiure on consumpion, i.e. i.e, U, X+ (10) fe-~ lx+ {y()-c()} {y(r)c()} d = 0. O. As in pam para 3.3 we can assume ha he consumer agrees o pay his whole income y (1) () o an insurance company, and in reurn receives funds c (1) () for consumpion. he prospecive reserve for his insurance conrac is V() = e3fes e~fe-~s +s ±! {c(s)y(s)} - x {c(s)-yes)} ds r.: 'X+ and we find (11) V' () (1) = (ô (~ + x+) V () (1) + y () (1) - c () I is usual o require ha he reserve for an insurance conrac mus be non. nonnegaive, i.e, i.e. ha (12) V V()0. (1) ~ o We have now arrived a he problem of maximizing (9) subjec o he condiion (12). If we carry ou he maximaion under only he weaker condiion (10), we can deermine c C () (1) from (11), subsiue in in (9), proceed as as in Secion 2, and arrive a he same resul. We find ha he opimal consumpion plan is deermined by he differenial equaion (4) e'() c'() - = (y (y- 5) ö) u'{c()] u'[c()] u"[c()j u"[c()] ,,+ U =f{f e-yu[c()]d} ": J.lx+ d = o 00 i, x fe- 1 ;: o0

9 As u' (c) > 0 and u" (c) < 0 i follows ha c' () (I) < ( 0 if y " > ô. ~. hisis is Fisher's case of high impaience. he consumer will be eager for early consumpion, and his opimal plan will be represened by a decreasing funcion c (). (I). his is no a reasonable resul. I may of course happen ha a person has a high consumpion early in life, and ha his consumpion decreases seadily. I is however difficul o o accep ha a consumpion paern of his form should be he oucome of deliberae lifelong planning. Mos people seem o o plan for progress, and o improve heir lo. he case ~ > y 'Y does no appear much more reasonable, alhough a person may conceivably plan so ha he can enjoy an increasing consumpion all hrough his life o hrow more ligh on he wo quesions raised above, we shall sudy a simple special case. Le as in para 2.8. u ii (c) = log c. he differenial equaion which deermines c (1) () hen has he soluion c (I) () = c C (0) e(a-r). e(). Le us furher assume ha y () = y for 0 < is a case of a ypical old age pension. he iniial value c (0) is deermined by (10), i.e. yj'e2!± e; d = c(0)fe7, he premium o be paid for his pension plan is p()=yc() for0<n. n, and y () = 0 for n <. his Le us furher assume ha y (1) = Y for 0 < 1 < n, and y (I) = 0 for n <. his I ex> yfe- 8 lx+ d = c(o){e-y lx+ d. o 0 P (I) = Y - C (1) for 0< 1 < n: Under condiion (12), he plan will no be feasible if if C c (0) > y. In his casei may be opimal o consume he whole income early in life, wih good inenions of beginning o pay ino a pension plan a some laer dae. I is difficul o accep ha he consumpion plan we have found is superior o he more convenional plan wih c (I) () = c = consan. he feasible consan consumpion level will hen be deermined by and i will be paid for by he level premium p p = = yy -- c up o ime n. he paradoxes we have found are due o our raher arbirary assumpions abou he uiliy funcion. We shall however no sudy uiliy funcions which lead o more reasonable resuls. Insead we shall discuss a more real paradox Le us assume ha he consumer has no income in he firs par of he planning period. He may hen borrow money for consumpion, and secure he loan by life insurance. Le c () and p () (1) be he flow funcions represening respecively consumpion and premium paymen. A ime 1 he accumulaed deb deb of of he he consumer will be will be (13) S() Se) = eö(c(s)+p(s)}e emj {c(s)+p(s)}e- 8s ds. o C 11

10 he premium mus be sufficien o o pay for a life insurance wih an amoun S () payable if he consumer dies a ime. he principle of equivalence hen gives he relaion X+söa S fp( S) lx+s-~s ds = JS(s) fs(s) lx+si+5 -~s.,, Jlx+ s e d ds.. fp(s) x x We see ha his equaion is saisfied for p () = S () lx +r his + corresponds o a pure risk insurance, or a "renewable erm" conrac. he reserve for his insurance is zero, so ha henon-negaiviy condiion (12) is saisfied. Subsiuing he expression for p () in (13), we obain S() = e J {c(s) + p+s(s)}e' ds. Differeniaing we obain he differenial equaion which has he soluion S'() = (o (0 + ++)S() f.x+) S () +c() + c Se)= S() = Jc(s)e s 0o or in he sandard acuarial noaion o 0.f SCi} = e~ J{c(s)+Jlx+~(s)}e-~S ds. o * ls s...j(~+ P:e +,.)4r ±! ds Se) S() = f D J x + s c(s) ds. Dx+ 0o 4.6. In he problem discussed above here are no obvious limis neiher o c () nor o. Hence a person wih no prospecive income should be able o mainain an arbirary high level of consumpion for any finie period of of ime, if if he is allowed o play he "insurance game" we have oulined. Apparenly nobody loses in he game. he insurance company will receive he premiums, and pay he corresponding amoun on he deah of he consumer. he lender wil will receive he loan back, wih compound ineres when he consumer dies. here are no obvious insiuional difficulies. here should for insance be no legal objecions if if he lender himself akes ou life insurance on he consumer, pays he premiums and adds hem, wih ineres o he loan. Yaari [9] recognizes he paradox, bu dismisses i i wih an assumpion ha he "company will refuse o issue life insurance afer he consumer reaches a cerain age". his is correc, bu he age limi is in mos counries beween 70 and 90, so his is no a very saisfacory explanaion. o a man in his early wenies, half a cenury of no work and unresrained consumpion, even if if a day of reckoning is bound o come, may seem an aracive prospec Fisher [4] does no discuss he paradox expliciely, bu he recognizes he legiimacy of "consumpion " loans o anicipae improvemen in financial condiion ". I is however clear from his book ha Fisher would have ried o explain he paradox bymacro- macroeconomic argumens. Nobody can consume more han his income, unless somebody else is willing o o save a par of his income and lend i o he impaien consumer. In 12

11 Fisher's world consumers who wan o o play he "insurance game" may be unable o find lenders. his is however no a complee explanaion. he ineres rae should bring abou equilibrium beween demand from borrowers and supply from lenders, bu i is clear ha higher ineres would no deer a consumer who waned o play he game. An increase in he ineres rae would jus drive ou of he marke hose who wan o borrow for invesmen purposes. Hence life insurance backing for consumer loans mus creae some inflaionary pressure, simply because i makes possible consumpion which oherwise would have o be posponed. he real paradox behind hese observaions may be ha life insurance companies, whose very exisence seems o o be hreaened by inflaion, conribue o he inflaion by selling erm insurance o cover loans which may accelerae consumpion. We shall no discuss his quesion furher, since i i seems o meri a separae paper. I may however be of ineres o menion ha Fisher observes ha "such loans are made perhaps mos ofen in Grea briain" ([4], page 358), a counry which has he world's mos developed insurance insiuions, and also an unenviable rae of inflaion. 5. Insurance for he benefi of survivors 5.1. We have so far considered life insurance only as a mean o smoohen a flucuaing income sream over an uncerain life ime. he soluion o he problems considered made cerain ha he consumer lef no unspen savings a his deah. Much life insurance is however wrien for he explici purpose of leaving liquid asses as "bequess" o heirs. We mus herefore conclude ha some people assign uiliy o leaving such bequess. o bring his elemen ino he model we can assume ha he person'sconsumpion- consumpioninsurance plan consiss of wo elemens: (i) a consumpion plan for his life ime c ();; (ii) an amoun B () payable as beques o he consumer's heirs if he should die a ime. Wih given resources, i.e. when he income sream y () is given, a se of pairs {c (), B ()} will appear as feasible. he firs problem is hen o esablish a preference ordering over a se of such pairs. Wih he assumpions we have made earlier, i is naural o assume ha he uiliy assigned o an arbirary pair is ex> 00 (14) fe i + u[c()]d +fr() i+ w[b()ld, (14) fe-y ;: u[c()]d+f P() ;: Jl.x+ w[b()]d. o 0 Here we have wrien {3 fi () raher han e-~, e, becausei i seemsa a lile arificial o assume ha bequess should be discouned a a consan rae. he funcion w (B) represens he uiliy of a beques B. Expression (14) is Yaari's crierion funcion in sandard acuarial noaion. 13

12 5.2. In order o deermine he feasible pairs, we can again assume ha he consumer pays his whole income y (1) () o an insurance company, which in in reurn gives him a pair {c (), (1), B (I)}. ()}. he principle of equivalence hen requires ha (15) fe y()d fe ± {c()+px+ B()}d. he prospecive reserve of his insurance conrac is From his we obain V() = eaje- eofe' {c(s)+z+3 B(s)y(s)} - as x + s {c(s)+jlx+s B(s)-y(s)} ds. lx+ x+ V'() = {ö {b+jlx+} +p+1} V()+y()-{c()+Jlx+B()}. We can use his o find an expression for c (I) () + u f..x++ B (I), (), and subsiue in (14). he problem of maximizing (14) is hen reduced o a problem in he classical calculus of variaion. he problem becomes more complicaed if if we impose he naural condiion: (15) fe-a lx+ y()d = fe-a lx+ {c()+jlx+ B()}d. e, ; o 0 00 v V (I) () ;;:: > 0. O I is no possible o discuss he shape of he soluion in any deail, wihou making some assumpions abou he funcions u (c), w (B) and fi fj (). (1). I is however no easy o decide which assumpions one should reasonably make, and he lieraure we have referred o has lile o say abou his quesion. If he purpose of he insurance is o provide income for a surviving widow, i may be naural o pu B () (r) = b~+, bi +, where z is he age of he wife when he insurance arrangemen is made. he funcion c (1) () and b mus saisfy (15), and be deermined so ha (14), or some oher crierion funcion is maximized. his arrangemen will give he widow a lump sum, sufficien o buy a life-long annuiy for an amoun b he arrangemen we have oulined may be inefficien for wo reasons: (i) (1) if he widow does no wan a consan consumpion plan, she will afer he deah of he husband have o solve he problem discussed in Secion 4, and deermine her own opimal consumpion-insurance plan; (ii) he wife may die before he husband, and in his case he beques will be be ""wased" in he same way as he convenional savings discussed in Secion 2. he more general approach would consis in specifying hree consumpion plans c1 c 1 (I), (), c2 c 2 (1) () and c3 c 3 (I) () for respecively he couple, he surviving widow, and he surviving widower. he principle of equivalence will hen give he feasible plans, which mus saisfy he condiion Jy() d = f {c1() c2() c3()} d +fc2() d +5 c3() d. 14

13 he opimal riple mus hen be deermined so ha some crierion funcion is maximized. I does no seem realisic o assume ha a family shall be able o specify is preferences for fuure consumpion in in he form of a crierion funcion, even more complicaed han (14). Such assumpions mus however be made by hose who design or sell pension plans for group of families. Governmenal plans are usually esablished hrough a democraic process, and privae plans are sold, so we mus assume ha he plans we find in real life, in some sense are close o opimal. hese plans do however differ considerably from one counry o o anoher, and from one group o anoher. If hey all are opimal, here musbe be wide differencesin in he underlying preferences. If we don' accep his conclusion, we should examine exising plans criically. I may be possible o improve hem, wihou violaing he principle of equivalence, which simply says ha you pay for wha you ge o generalize our model, we can consider he family as he uni, and inroduce he noaion: : 7I: s () = he probabiliy ha he family shall be in sae s a ime. Here "Lns()= i3()= 1. Y$(r) y8 () = he family's income a ime, if i is in sae s. c3 () (I) = he family's consumpion a ime 1,, if i is in sae s. s, C s.1=1 he principle of equivalence hen gives n 00 L Je-cln,(){Ys()-c,()}d= e S ir3()y() - c3()}d 0. O. s1 s=1 0 his equaion gives he se of feasible consumpion plans c9 C s () for he differen saes. he given elemens are he income sreams y Ys () and he sae probabiliies 71:$ (). (r), In order o deermine he opimal plan, we need informaion abou he family's preferences, and his may be difficul o obain in an ariculae form. If he purpose of life insurance is o provide income for he family in differen saes, here is no need for he convenional insurance conrac wih a lump sum payable a deah - or or more generally- - whenhe he family makesa a ransiion from one sae o anoher. his conrac is however flexible in he sense ha i makes i possible for he family o readjus is consumpion plan afer a change of sae. he flexibiliy may be worh some heoreical loss in efficiency, a a leas o he normal family unable o specify a complee preference ordering over fuure consumpion in all possible saes

14 REFERENCES 1. BOHM-BAWERK, E., B., Posiive heorie des Kapials, Vienna, DENENBERG, H. H. S., S., "Auhor's Reply", ", he Journal of Risk and Insurance, 1970, FISCHER, S., "A Life Cycle Model of of Life Insurance Purchasing ", Inernaional Economic Review, 1973, FISHER, I., he heory of Ineres, Macmillan, JOSEPHSON, H. H. D., D., "A New Concep of he Economics of Life Value and he Human Life Value: : Commen ", ", he Journal of Risk and Insurance, 1970, MARSHALL, A., Principles of Economics, Macmillan, RICHARD, S. F., F., "Opimal Consumpion, Porfolio and Life Insurance. Rules for an Uncerain Lived Individual in a Coninuous ime Model ", Journal of Financial Economics, 1975, YAARI, M.-E., "On he Exisence of an Opimal Plan in a Coninuous ime Allocaion Process ", Economerica, 1964, YAARI, M~ M. E., B., "Uncerain Lifeime, Life Insurance, and he heory of he Consumer ", Review of Economic Sudies, 1965,