Optimal Consumption and Insurance: A ContinuousTime Markov Chain Approach


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1 Opimal Consumpion and Insurance: A ConinuousTime Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems abou consumpion and insurance are modelled in a coninuousime mulisae Markovian framework. The opimal soluion is derived and sudied. The model, he problem, and is soluion are exemplified by wo special cases: In one model he individual akes opimal posiions agains he risk of dying; in anoher model he individual akes opimal posiions agains he risk of losing income as a consequence of disabiliy or unemploymen. Key words: Personal finance, mulisae model, sochasic conrol, financial decision making, moraliydisabiliyunemploymen risk JELClassificaion: G11, G22, J65 Fachbereich Mahemaik, Universiä Kaiserslauern, ErwinSchrödingerSrasse, D Kaiserslauern, Germany Corresponding and presening auhor, Deparmen of Mahemaical Sciences, Universiesparken 5, DK2100 Copenhagen O, Denmark 1
2 1 Inroducion Opimal personal financial decision making plays an imporan role in modern financial mahemaics and economics. I covers broad disciplines like microeconomic equilibrium heory and porfolio opimizaion. Meron 1969,1971) iniiaed he area of coninuousime consumpioninvesmen problems which has become a paradigm for a large se of developmens and generalizaions. These are ofen driven by modificaions of financial marke models or individual preferences. In his aricle we disregard he invesmen decision of he individual and concenrae on he consumpion decision along wih he inroducion of insurance decisions of various general ypes. Meron 1969,1971) formulaed he coninuousime consumpioninvesmen problem. Imporan general resuls concern he opimal asse allocaion including he socalled muual fund heorem. Explici resuls on asse allocaion and consumpion are obained for lognormal prices and socalled HARA uiliy. Richard 1975) generalized boh he general and explici resuls o he case where he individual has an uncerain lifeime, income while being alive, and, apar from asse allocaion and consumpion, decides coninuously on a life insurance coverage. The uncerain lifeime is modelled, in consisency wih coninuousime life insurance mahemaics, by an agedependen moraliy inensiy. Acually, he idea of involving he life insurance decision in he personal decision making of an individual wih an uncerain lifeime daes furher back o Yaari 1965) who sudied he issue in a discree ime seing. The same year in which Meron 1969) firs published his ideas, Hoem 1969) demonsraed ha he coninuousime finiesae Markov chain is an ineviable ool in he consrucion of general life insurance producs and he modelling of general life insurance risk ha year queuing up for aking small seps ha proved o be gian leaps.) The finiesae Markov chain has been sudied in he conex of life insurance and vice versa since hen by Hoem 1988), Norberg 1991) and many ohers. I provides a model for various kinds of risk conneced o an individual s life. One imporan example o which we reurn in he core ex, is he risk of loss of income in connecion wih disabiliy or unemploymen. Oher examples are he risk of increasing moraliy inensiy in connecion wih sochasic deerioraion of healh and he risk conneced wih ones dependans following from e.g. marriage and parenhood. Financial decision making in connecion wih life and pension insurance has in he lieraure primarily been a maer for he life insurance company. Boh asse allocaion and adjusmen of nondefined paymens have been sudied as decision processes subjec o opimizaion. These decision problems have ofen been sudied in he conex of a quadraic loss funcion where deviaions of wealh and paymens from cerain arges are punished, see Cairns 2000) for a sae of he ar exposiion of he resuls. In ha area, he life insurance risk was unil recenly approximaed by normal disribuions wih reference o he fac ha he decisions concerned large porfolios of insurance conracs. However, he life insurance marke shows a rend owards a larger exen of individualizaion of hese decisions, e.g. in unilinked insurance conracs where decisions on he asse allocaion and composiion of paymens are parly or fully individualized. Seffensen 2003,2004) models he life insurance risk by a coninuousime Markov chain and solves he decision problem of he insurance company in he cases of quadraic loss and power uiliy preferences, respecively. Meron 1969,1971) solved he consumpioninvesmen problem by sochasic conrol heory. Also he decision problems of he insurance company presened by Cairns 2000) are approached by sochasic conrol heory. However, in all hese problems all risks are modelled as normal. Also in many exbooks on sochasic conrol heory, applicaions o primarily normal risk are sudied, see e.g. Fleming and Rishel 1975). This is in conras o Richard 1975) who works in 2
3 a survival model hrough inroducion of a moraliy inensiy. I is also in conras o Seffensen 2003,2004) who works in a finiesae Markov chain model. Davis 1993) exbook demonsraes he applicaion of sochasic conrol heory o nonnormal risks. As menioned a he end of he very firs paragraph we disregard he invesmen decision in his aricle. The reason is ha i does no add much insigh o his work given he resuls of Meron 1969,1971) and Richard 1975). Our focus is on he consumpion and insurance decisions. Richard sudies he consumpion and life insurance decisions in a survival model where he saving akes place on a privae accoun. We generalize his in wo direcions. Firsly we model he life insurance risk in a mulisae framework such ha e.g. insurance decisions wih respec o disabiliy and unemploymen can be sudied and an opimal posiion can be aken. This reflecs he variey and complexiy of real life financial decisions and insurance markes. This is he primary conribuion of his aricle. Secondly we allow for saving in he insurance company. Richard 1975) concludes his aricle by noing ha rich, old people opimally should be sellers of insurance while consuming heir wealh. In his aricle his life insurance is sold alhough he policy holder has no saved anyhing in he insurance company. In pracice his is no possible since he maximum life insurance sum he policy holder can sell is exacly he savings in he company. Taking his sum o be equal o he savings in he company is exacly wha happens when he policy holder holds a life annuiy. Therefore, in order o find he opimal posiion one has o inroduce he possibiliy of saving in a life insurance company. We model his by leing wealh consis of boh he balance of a privae accoun and he balance of an accoun in he insurance company. This is he secondary conribuion of his aricle. In Secion 2 we presen he finiesae Markov model, he insurance accoun and is relaion o classical life insurance mahemaics. In Secion 3 we presen he res of he model, he income process, and he consumpion process and how all paymen processes affec he privae accoun. In Secion 4 we formalize an opimizaion problem and presen is soluion. The derivaion appears in he Appendix. We sudy and inerpre aspecs of he opimal decisions in Secion 5. Secions 6 and 7 conain special sudies of wo imporan cases. The survival model is sudied in Secion 6, obaining as a special case he same resuls as Richard 1975) obained. The disabiliy/unemploymen model is sudied in Secion 7 providing furher resuls and insigh. 2 Life Insurance Mahemaics In his secion we presen he finiesae Markov model and he insurance paymen process. We also inroduce an insurance accoun which urns ou o coincide wih a radiionally defined reserve in a special case of deerminisic paymen process coefficiens. We ake as given a probabiliy space Ω, F, P). On he probabiliy space is defined a process Z = Z )) 0 n aking values in a finie se J = {0,...,J} of possible saes and saring, by convenion, in sae 0 a ime 0. We define he J + 1dimensional couning process N = N k) k J by N k ) = # {s 0,,Z s ) k, Z s) = k}, couning he number of jumps ino sae k unil ime. Assume ha here exis deerminisic funcions µ jk ), j, k J, such ha N k admis he sochasic inensiy process µ Z )k ) ) 0 n for k J, i.e. M k ) = N k ) 0 µ Zs)k s)ds consiues a maringale for k J. Then Z is a Markov process. For each sae we inroduce he 3
4 indicaor process indicaing sojourn, I j ) = 1 [Z ) = j, and he funcions µ jk ), j, k J and he inensiy process µ Z )k ) are conneced by he relaion µ Z)k ) = j:j k Ij )µ jk ). The reader should hink of Z as he sae of life of an individual in a cerain sense of personal financial decision making which will be described in his secion. In order o fix ideas we already now offer he reader some examples of Z o have in mind. The simples model is he socalled survival model wih only wo saes, alive and dead. There, he individual jumps from he sae of being alive o he absorbing sae of being dead wih an agedependen inensiy. This model is illusraed in Figure 1. To solve problems in a survival model he seup of he finie sae Markov chain is like cracking a nu wih a sledge hammer, hough. Indeed, we have a much wider se of applicaions in mind. 0 alive 1 dead Figure 1: Survival model Consider he hree sae model illusraed in Figure 2. The absorbing sae 2 is he sae of being dead. The individual can jump beween wo saes of being alive, 0 and 1, wih cerain agedependen inensiies, possibly 0. From each of hese saes he individual can jump ino he sae of being dead wih an age and saedependen inensiy. Two examples of saes 0 and 1 which are boh very ap o hink of hroughou his paper are he following: If 0 is he sae of aciviy and 1 is he sae of disabiliy, we speak of a disabiliy model; if 0 is he sae of employmen and 1 is he sae of unemploymen, we speak of an unemploymen model. 0 acive/employed 1 disabled/unemployed ց 2 dead ւ Figure 2: Disabiliy/Unemploymen model We inroduce now an insurance paymen process B = B )) 0 n represening he accumulaed insurance ne paymens from he insurance company o he policy holder. The insurance paymen process is assumed o follow he dynamics db ) = j Ij )db j ) + k:k Z ) bz )k )dn k ), where B j ) is a sufficienly regular adaped process specifying accumulaed paymens during sojourns in sae j and b jk ) is a sufficienly regular predicable process specifying paymens due upon ransiions from sae j o sae k. We assume ha each B j decomposes ino an absoluely coninuous par and a discree par, i.e. db j ) = b j )d + B j ), where B j ) = B j ) B j ), when differen from 0, is a jump represening a lump sum payable a ime if he policy holder is hen in sae j. Posiive elemens of B are called benefis whereas negaive elemens are called premiums. In he survival model one can hink of a life insurance sum paid ou upon deah before erminaion. Alernaively, one can hink of a socalled deferred emporary life annuiy benefi saring upon reiremen a ime m and running for n m ime unis unil erminaion ime n or deah 4
5 whaever occurs firs. Such benefis or sreams of benefis can e.g. be paid for by a premium rae paid coninuously unil deah or reiremen whaever occurs firs. In he disabiliy/unemploymen model here are also several possible consrucions of insurance paymen processes: Insurance sums may be paid ou upon occurrence of disabiliy/unemploymen or raes of benefis may be paid ou as long as he individual is disabled/unemployed, i.e. socalled disabiliy/unemploymen annuiies. These insurances agains disabiliy/unemploymen are ypically paid for by a premium rae paid coninuously as long as he individual is acive/employed. All hese insurances can now be combined wih he differen insurances agains deah and/or survival menioned in he previous paragraph. Assume ha here exiss a consan ineres rae r. In general he reserve is defined as he condiional expeced presen value of fuure paymens, Y ) = E e rs ) db s) F ). 1) Here and in he res of he aricle we define n =,n. The coefficiens in he paymen process are seled in accordance wih he socalled equivalence principle saing ha Y 0 ) = 0 or, equivalenly, Y 0) = B 0 0), see e.g. Norberg 1991). The aserisk decoraion of E means ha he expecaion is aken wih respec o a valuaion measure which we denoe by P and which may be differen from he objecive measure. We assume ha Z is Markov also under his measure such ha we can paramerize his measure by he ransiion inensiies µ jk ), j k, j, k J. We inroduce a process Ỹ which is described by he following forward sochasic differenial equaion, dỹ ) = rỹ )d + k:k Z ) yz )k )dn k ), 2) j Ij ) db j ) + k:k j µjk ) b jk ) + y jk ) ) ) d Ỹ 0) = B 0 0). 3) The process y jk is here a sufficienly regular predicable process. We will now show prove a relaion saing ha if he erminal lump sum benefi is simply he value of Ỹ prior o erminaion, independenly of he sae, hen Y and Ỹ are equal. Proposiion 1 If B j n) = Ỹ n ) hen Y = Ỹ. We hen have ha y Z )k ) = E e rs ) db s) F ) {Z ) = k} Y ). 4) Proof. Firs realize ha, according o 2), we have ha e rn ) Ỹ n ) = Ỹ ) + n = Ỹ ) + n ) d e rs ) Ỹ s) re rs ) Ỹ s)ds + e rs ) dỹ s) ). 5
6 Plugging his relaion ino 1), using ha B j n) = Ỹ n ), and applying 2) now gives he resul ha he reserve equals Ỹ ), Y ) = E e rs ) db s) + e rn ) Ỹ n ) F ) = Ỹ ) + E e rs ) k:k Zs ) b Zs )k s) + y Zs )k s) ) dm k s) F ). Here, M k is a maringale under P such ha he las erm vanishes. We know ha Ỹ upon ransiion of Z o k a ime equals Ỹ ) + yz )k ). We also know from he definiion of Y ha Y upon ransiion of Z o k a ime can be wrien as E e rs ) db s) F ) {Z ) = k}. Bu if Y = Ỹ hese observaions ogeher give 4). This proposiion has he consequence ha we can skip he decoraion of Ỹ such ha Y follows he SDE given in 2). Thus, alhough Ỹ is a rerospecively calculaed accoun, i coincides wih he reserve 1). We emphasize ha his relaion is relies heavily on he fac ha Ỹ n ) is paid ou upon erminaion such ha Ỹ n) = 0. Henceforh, we use only he leer Y wihou he decoraion. Since his represens he savings in he insurance company we call Y for insiuional wealh. We emphasize ha he resuls above hold even for nondeerminisic db j ), b jk ) and y jk ). I is in his assumpion ha hey differ from similar classical resuls in life insurance mahemaics where he processes B j and b jk are assumed o be deerminisic, see e.g. Norberg 1991) for a derivaion of he following classical resuls. If B j and b jk are deerminisic, hen, by he Markov propery, he reserve is fully specified by he socalled saewise reserves, Y j ) = E e rs ) db s) Z ) = j, 5) since Y ) = j I j )Y j ). In his case he reserve jump y jk is, in accordance wih 4), y jk ) = Y k ) Y j ). The dynamics of Y are, in accordance wih 2), given by dy ) = I j )dy j ) + ) Y k ) Y Z ) ) dn k ) j j dy j ) = ry j ) d db j ) k j µjk ) b jk ) + Y k ) Y j ) ) d, Y j n) = 0. In his paper we consider a decision problem where, a ime, he policy holder decides on db Z) ), b Z)k ) and y Z)k ) for all k Z ). This is really an unconvenional consrucion and o a reader wih a life insurance background, his may look like a very awkward decision problem. Deciding on db Z) ) and b Z)k ) may seem reasonable bu wha does i mean ha he policy holder decides on he reserve jump y Z)k )? In pracice he policy holder decides on a se of fuure paymens, i.e. db j s), b jk s), s >, j k, and on he basis of hese, he insurance company calculaes he reserve jumps y Z)k ), 6
7 k Z ). These reserve jumps are he only consequence of he fuure paymens ha he policy holder realizes on his accoun a ime. Bu his means ha he policy holder in pracice indirecly decides on he reserve jumps hrough specificaion of he fuure paymens. In he decision problem sudied here he indirec decision on he reserve jump is made direc by reversing he procedure: We assume ha he policy holder decides direcly a ime on he reserve jumps y Z)k ), k Z ). In principle he policy holder could hen jus inform he insurance company abou he reserve jump he wans insured. Bu his is hen wha he does indirecly by demanding a se of fuure paymens, such ha he reserve jump is obained upon a possible ransiion. I.e. for a given decision y Z)k ), k Z ), he or someone else wih he required experise like e.g. he insurance company) calculaes a se of fuure paymens db j s), b jk s), s >, j k such ha 4) is fulfilled. There is ypically a coninuum of fuure paymen processes leading o he same reserve jump so he fuure paymens are no uniquely deermined by he above procedure. However, he can ake any one of hese as long as i leads o he righ reserve jumps. When he fuure urns ino he presen, as ime goes by, he decides hen coninuously on he acual paymens o be realized, db Z) ) and b Z)k ), k Z ). In he example secion we demonsrae how his machinery works. 3 The Model and he Decision Processes In his paragraph, we inroduce an income process A = A)) 0 n represening he accumulaed income of he individual. The income process is assumed o follow he dynamics da) = a Z) )d + k:k Z ) az )k )dn k ), where a j ) and a jk ) are assumed o be deerminisic funcions. Here, a j ) is he rae of income given ha he individual is in sae j a ime and a jk ) is he lump sum income a ime given ha he individual jumps from sae j o sae k a ime. By income we hink primarily of labor income which makes sense o he saedependen income rae a j in he disabiliy and unemploymen models. The lump sum income is aken ino accoun for he sake of generaliy, and more creaive models could acually defend he possibiliy of having a lump sum income upon ransiion: Consider he hree sae model where sae 0 is he sae where he rich uncle o whom he individual is he only inherior, is sill alive and sae 1 is he sae where he rich uncle has passed away. Noe ha in his case µ 10 should be se o zero jus as µ 20 and µ 21 are defaul se o zero. In his paragraph, we inroduce a consumpion process C = C )) 0 n represening he accumulaed consumpion of he individual. The consumpion process is assumed o follow he dynamics dc ) = c Z) )d + k:k Z ) cz )k )dn k ). Here, c j ) is he rae of consumpion given ha he individual is in sae j a ime and c jk ) is he lump sum consumpion a ime given ha he individual jumps from sae j o sae k a ime. The processes c j ) and c jk ) are decision processes chosen a he discreion of he individual. As for he income process he rae of consumpion has an obvious inerpreaion while a lump sum consumpion mus be moivaed by a more creaive paern of hinking. The personal wealh is accouned for on a bank accoun of he individual. This bank accoun is assumed o earn ineres a rae r, he same rae ha is earned on he insurance accoun, and, 7
8 hereafer, i accouns for he hree paymen processes A, B, and C. Thus he bank accoun has he following dynamics, dx ) = rx )d + da) + db ) dc ), 6) X 0) = x 0. Thus, apar from earning ineress, his bank accoun simply accouns for labor income and insurance benefis as income and consumpion as ougo. Finally, we can add up he insiuional and personal wealhs o derive he dynamics of he oal wealh, d X ) + Y )) = r X ) + Y )) d + da) dc ) + ) b Z )k ) + y Z )k ) dm Z)k ). k:k Z ) These dynamics have he following inerpreaion. Firsly, he oal wealh earns ineres a rae r. Secondly, he income process and he consumpion process affec he oal wealh direcly. Thirdly, upon a ransiion from j o k he oal wealh increases by b jk ) + y jk ). From his amoun, b jk ) is paid from he insurance insiuion o he individual and added o he bank accoun. The amoun, y jk ) is also paid from he insurance company o he individual bu kep by he insurance company by adding i o he insurance accoun. For his oal wealh incremen of b jk )+y jk ), he individual pays a naural premium a rae µ jk b jk ) + y jk ) ). From he dynamics of X + Y here are hree imporan poins o make. Assume ha X and Y appear in he objecive funcion of he decision problem hrough heir sum only. Then i seems possible o replace he wo sae variables X and Y by heir sum S = X + Y. Oherwise we sill need he wo sae variables. One siuaion where X and Y will no appear hrough heir sum only, is if here are specific consrains on e.g. Y. To keep insurance business separaed from banking loaning) business, one could have he consrain ha Y ) 0 for all while here could be no such consrain X. Tha is, loaning akes place in he bank, no in he insurance company. We solve he unconsrained problem below bu we are sill able o carry ou special sudies wih cerain consrains. If b Z )k ) and y Z )k ) appear in he objecive funcion hrough heir sum only, hey will no be deermined uniquely. Below, b Z )k and y Z )k will no appear in he objecive funcion a all, so herefore we ge a nonunique soluion. One siuaion where b Z )k and y Z )k will no appear hrough heir sum only, is if here are specific consrains on e.g. b Z )k. To preven individuals from selling life insurance on heir own lives, one could have he consrain ha b jk ) 0. We solve he unconsrained problem below bu we are sill able o carry ou special sudies wih cerain consrains. If he coninuous insurance paymen rae b Z) ) does no appear in he objecive funcion, hen ha paymen rae canno be deermined by solving he opimizaion problem since he paymen rae has vanished from he dynamics of X + Y. The paymen rae can appear in differen forms in he objecive funcion, e.g. hrough consrains. We solve he unconsrained problem below, hough, and b Z) ) will herefore no be deermined. Above we have menioned possible consrains like Y ) 0 and b jk ) 0 which boh can be moivaed by no mixing insurance and he crediworhiness of he individual. If such consrains are fulfilled, he insurance company does no need o worry abou wheher he individual can afford 8
9 he insurance conrac he eners ino. Crediworhiness is compleely lef up o he bank o decide. This moivaion is closely linked o he way banking and life insurance regulaion is carried ou in pracice. We emphasize ha below we solve he unconsrained problem in general. Bu due o he nonuniqueness of b j, b jk, and y jk, we can sill solve cerain relevan consrained problems as i is also seen in he examples. 4 The Conrol Problem and Is Soluion In his secion we presen he conrol problem and is soluion. Inroduce a uiliy process wih dynamics given by ) du ) = u Z), c Z) ) d + ) k:k Z ) uz )k, c Z )k ) dn k ) + U Z ), X ),Y ))dε, n). Here, u j, c) is a deerminisic uiliy funcion which measures uiliy of he consumpion rae c given ha he individual is in sae j a ime and u jk, c) is a deerminisic uiliy funcion which measures uiliy of he lump sum consumpion c given ha he individual jumps from sae j o sae k a ime. Finally, U j n, x, y) is a deerminisic funcion which measures uiliy of he erminal lump sum payou from he wo accouns x and y given ha he individual is in sae j a ime n. We assume ha he individual chooses a consumpioninsurance process o maximize uiliy in he sense of supe du ). 0 where he supremum is aken over b j, b jk, y jk, c j, c jk, j k. We specify furher he uiliy funcions appearing in he uiliy process. We are ineresed in solving he problem for an individual wih preferences represened by he power uiliy funcion in he sense of u j, c) = 1 γ wj ) 1 γ c γ, u jk, c) = 1 γ wjk ) 1 γ c γ, U j, x, y) = 1 γ W j )x + y) γ. Here, w j ) is a weigh process which gives weigh o power uiliy of he consumpion rae c given ha he individual is in sae j a ime, w jk ) is a weigh process which gives weigh o power uiliy of he lump sum consumpion c given ha he individual jumps from sae j o sae k a ime. Finally, W j ) is a weigh funcion which gives weigh o power uiliy of lump sum consumpion given ha he individual is in sae j a ime. The weigh funcions appear simply as facors in he uiliy funcions. However, i is convenien o hink of hese weigh funcions as semming from a weigh process wih dynamics given by dw ) = w Z) )d + k:k Z ) wz )k )dn k ) + W Z) )dε, n). This arificial process  arificial in he sense ha i is no a paymen process anyone pays  exposes he symmery in srucure across all appearing processes. For presenaion of he resuls we inroduce he abbreviaing funcion 9
10 µ h jk jk ) 1/1 γ) ) ) = µ jk. 7) ) The quoien of inensiies in 7), µ jk ) /µ jk ), is acually he reciprocal of one plus he socalled Girsanov kernel ha characerizes he measure ransformaion from P o P in probabilisic erms. In financial applicaions, minus he Girsanov kernel is called he marke price of risk. Calculaions in Appendix A show ha he opimal consumpion and insurance sraegies are given by he following feedback funcions for c j ), c jk ), b jk ), and y jk ), c j, x, y) = c jk, x, y) = b jk, x, y) + y jk, x, y) = wj ) x + y + g j f j ) ), 8a) ) wjk ) f j ) hjk ) x + y + g j ) ), fk ) + w jk ) f j h jk ) x + y + g j ) ) ) a jk ) + x + y + g k ) ) 8b) 8c) wih f and g given below. In he nex secion we sudy hese opimal conrol funcions in deails. Here we jus specify he funcions g and f such ha he opimal conrols are fully specified by 8). They saisfy he sysems of ordinary differenial equaions given by g j ) = rgj ) a j ) k:k j µjk ) a jk ) + g k ) g j ) ), 9) g j n) = 0, f j ) = r j ) f j ) w j ) k:k j µjk ) w jk ) + f k ) f j ) ) 10) f j n) = W j n), wih µ jk ) = µ jk )h jk ) γ = µ jk ) h jk ), γ δ = 1 γ, r j ) = δr δ µ j ) µ j ) ) + µ j ) µ j ). The soluion o he sysem of ordinary differenial equaions for g has he FeynmanKac represenaion g j ) = e rs ) das) = E,j n e rs ) k p jk, s) a k s) + ) l:l k µkl s)a kl s) ds. 11) Thus, g j ) is he condiional expeced presen value of he fuure income process where he expecaion is aken under P. This is, in oher words, he financial value of he fuure income. The soluion o he sysem of ordinary differenial equaions for f has he FeynmanKac represenaion f j ) = Ẽ,j e R s r Zτ) τ)dτ dw s). 12) 10
11 Thus, f j ) is he condiional expeced value of he fuure weigh process where expecaion is aken under an arificial measure P under which N k admis he inensiy process µ Z)k ). This is, in oher words, an arificial financial value of he fuure weighs in he sense ha an arificial sochasic ineres rae process and an arificial valuaion measure are applied. 5 Sudies of he Opimal Conrols In his secion we sudy in deail he opimal conrols derived in he previous secion. We give inerpreaions of he opimal conrols in he general forms in 8). In he succeeding wo secions we sudy wo imporan special consrucions of he underlying process Z. There we pay furher aenion o he opimal conrols. We noe wo imporan poins on uniqueness. These relae o he remarks a he end of Secion 3. Firsly, here is no condiion on he coninuous paymen rae b j ). This was foreseen in Secion 3. Secondly, he decision processes b jk ) and y jk ) are no uniquely deermined since here is only one equaion for heir sum. Also his was foreseen in Secion 3. Special cases are, of course, he cases where he one or he oher is defaul se o zero. If we pu b jk ) or y jk ) equal o zero in 8c), respecively, we ge unique opimal conrols for y jk ) and b jk ), respecively. This illusraes how he lack of uniqueness, makes i possible o sudy cerain consrained conrol problems afer all. We now ake a closer look a he opimal conrols. Firs we give inerpreaions of hem as hey appear in 8). In all hree formulas appear he sum x + y + g j ). This can be inerpreed as he oal wealh of he individual given ha he is in sae j a ime. This oal wealh consiss of personal wealh x, insiuional wealh y, and human wealh g j. Recall ha g j is he financial value of fuure income given ha he individual is in sae j. Furhermore, in 8c) appears he sum a jk ) + x + y + g k ). This can be inerpreed as he oal wealh of he individual upon ransiion from sae j o sae k a ime before he effec of insurance. This wealh consiss of he lump sum income upon ransiion a jk ) and hen again of personal wealh x, insiuional wealh y, and human wealh g k ). Here he human wealh is measured given ha he individual is in sae k a ime. We emphasize ha his is he wealh before a possible insurance sum is paid ou or a reserve jump has been added o he insiuional wealh. Wih hese inerpreaions of oal wealh in mind we can now inerpre he hree conrol funcions: The opimal coninuous consumpion rae in 8a) is a fracion of oal wealh. The fracion w j )/f j ) measures he uiliy of presen consumpion agains uiliy of consumpion in he fuure. Recall ha f j ) is an arificial value of he fuure weighs. The opimal consumpion rae is in relaed problems ypically formed by a similar fracion of oal wealh. The opimal lump sum consumpion upon ransiion in 8b) is also a fracion of wealh. The fracion h jk )w jk ) /f j ) consiss of wo elemens. The fracion w jk )/f j ) measures he uiliy of consumpion upon ransiion agains uiliy of fuure consumpion. However, fuure consumpion is calculaed given ha he individual is in sae j a ime and no given ha he individual is in sae k a ime. This is explained by he fac ha he risk conneced o he jump is parly insured away. The price of his insurance is, ogeher wih he individual aiude owards risk, hidden in h jk. This explains he firs elemen of he facor h jk )w jk )/f j ). The opimal insurance sum plus reserve jump upon ransiion in 8c) can be inerpreed as a proecion of wealh. In he opimal decision one should no disinguish beween an 11
12 insurance sum ha is a sum added o personal wealh, and a reserve jump ha is a sum added o insiuional wealh. The allocaion of he oal jump should be deermined by oher consideraions. However, how does his opimal oal insurance sum proec wealh? The opimal sum measures he difference beween a fracion h jk ) f k ) + w jk ) ) /f j ) of presen wealh x+y+g j ) and wealh upon ransiion a jk )+x+y+g k ). If he fracion h jk ) f k ) + w jk ) ) /f j ) is 1 hen his difference reduces o a jk ) + g k ) g j ) ) which is minus he human wealh sum a risk. Thus, his is really he wealh ha is poenially los upon ransiion and which should be proeced by an opposie insurance posiion. However, in he calculaion of he opimal proecion wo furher consideraions should be aken ino accoun: 1) The uiliy of fuure wealh in case of no ransiion is measured agains he uiliy of fuure wealh in case of ransiion in he raio f k ) + w jk ) ) /f j ). If uiliy of fuure wealh given a ransiion is lower han wihou ransiion, i.e. f k ) + w jk ) ) /f j ) < 1, hen one should underinsure ones wealh under risk, a jk ) + g k ) g j ) ), and vice versa. 2) If he proecion is expensive, i.e. h jk < 1, hen one should also underinsure ones wealh under risk in order o pick up some of his marke price of risk. These underinsurances are implemened by weighing he presen human wealh wih he raio f k ) + w jk ) ) /f j ) and h jk ). Now, ake a closer look a he conrols c j and c jk. For fixed Z ) = j, we can sudy he opimally conrolled processes X j and Y j ha solve he following ordinary differenial equaions d d Xj ) = rx j ) + a j ) + b j ) c j ), d d Y j ) = ry j ) b j ) k:k j µjk ) b jk ) + y jk ) ). Since X j and Y j evolve deerminisically, we can sudy he saewise conrols c j, X j ), Y j ) ) and c jk, X j ),Y j ) ) as funcions of ime. Wih a sligh abuse of noaion we denoe hese deerminisic funcions by c j ) and c jk ). Furhermore, we consider he opimal wealh upon ransiion before consumpion which is given by q jk, x, y) = b jk, x, y) + y jk, x, y) + a jk ) + x + y + g k ) = fk ) + w jk ) f j h jk ) x + y + g j ) ). ) Also q jk, X ),Y )) can be sudied as a funcion of ime, accordingly denoed by q jk ). From 8) we can, by using c j ) = cj, X j ),Y j ) ) + x cj, X j ),Y j ) ) dx j ) d + y cj, X j ),Y j ) ) dy j ) d and similar formulas for c jk and q jk, derive he following simple exponenial differenial equaions 12
13 for c j ), c jk ), and q jk ), c j ) = c j 1 ) r + µ j ) µ j ) ) ) + wj ) 1 γ w j, ) c jk ) = c jk 1 ) r + µ j ) µ j ) ) ) + wjk ) 1 γ w jk ) + hjk ) h jk, ) q jk ) = q jk 1 ) r + µ j ) µ j ) ) ) + fk ) + wjk ) 1 γ f k ) + w jk ) + hjk ) h jk. ) By he definiion of h in 7) and inroducing µ jk ) = 1 + Γ jk ) ) µ jk ), we can calculae ha h jk )/h jk ) = Γ jk )/ 1 γ) 1 + Γ jk ) )). If we define he weighs according o he usual impaience facor, i.e. w j ) 1 γ = exp ι) we can furhermore calculae ha w j )/wj ) = ι/ 1 γ). Plugging in hese relaions and inroducing Γµ) j ) = k:k j Γjk )µ jk ), we ge he following simple differenial equaions for he opimal conrols c j ) and c jk ), c j ) = c j 1 ) ) r ι + Γµ) j ), 1 γ ) c jk ) = c jk 1 ) r ι + Γµ) j ) Γjk ) 1 γ 1 + Γ jk. ) 6 The Survival Model In his secion we specialize he resuls in Secion 4 ino he case of a survival model. The idea is o sudy opimal consumpion and insurance decisions of an individual who has uiliy of consumpion while being alive including uiliy of lump sum consumpion upon erminaion. Furhermore he or raher his inheriors) has uiliy of consumpion upon deah before erminaion. In Figure 3, we have specified a se of income process coefficiens and a se of uiliy weigh coefficiens. 0 alive w 0 0, W 0 0 a 0 0, A 0 = 0 µ w 01 0 a 01 = 0 1 dead w 1 = W 1 = 0 a 1 = A 1 = 0 Figure 3: Survival model wih income and uiliy weighs All saewise coefficiens are zero in he sae dead. This means ha here is no income and no uiliy of consumpion in ha sae. Weighs on uiliy of consumpion in he sae alive are specified by he coefficiens w 0 and W 0, and weigh on uiliy of a lump sum paymen upon deah is specified by w 01. Income is specified by he rae a 0 and oher income coefficiens are se o zero such ha here is no lump sum income upon deah or upon survival unil erminaion. We sar ou by specifying he funcions f and g for his special case. According o 10) f 1 = 0 and f 0 is characerized by f 0 ) = w0 ) + f 0 )r ) µ ) w 01 ) f 0 ) ), f 0 n) = W 0 n), r = δr δ µ ) µ)) + µ ) µ ) 13
14 This differenial equaion has he soluion and FeynmanKac represenaion, respecively, f 0 ) = n = Ẽ,0 e R s r +eµ w 0 s) + µs)w 01 s) ) ds + e R n r +eµ W 0 n), 13) e R s r τ)dτ dw s). According o 9) g 1 = 0 and g 0 is characerized by g 0 ) = rg 0 ) a 0 ) + µ )g 0 ), 14) g 0 n) = 0. This differenial equaion has he soluion and FeynmanKac represenaion, respecively, n g 0 ) = e R s r+µ a 0 s)ds 15) = E,0 e rs ) das). We can now specify he opimal conrols in erms of f 0 and g 0. We ge he conrols c 0, x, y) = c 01, x, y) = b 01, x, y) + y 01, x, y) = w0 ) x + y + g 0 f 0 ) ), ) w01 ) f 0 ) h01 ) x + y + g 0 ) ), w01 ) f 0 ) h01 ) x + y + g 0 ) ) x y = c 01, x, y) x y. 16) Wha happens upon deah in ha case is ha he benefi b 01, x, y) is paid ou and c 01, x, y) is consumed. If x and y are he accouns jus prior o deah, hese accouns upon deah will hen be x + b 01, x, y) c 01, x, y) and y + y 01, x, y), respecively. Bu according o 16) hese accouns are he same wih opposie sign. These accouns are now rolled forward earning ineres bu experiencing no cash flow o ime n where Y n ) is paid ou posiive or negaive). Bu his benefi exacly covers he defici a he bank accoun so boh accouns close a zero. We can now specify a se of paymens afer ime which gives he righ reserve jump y 01, x, y) in accordance wih 4). There are several soluions from which we ake he paymen sream specifying ha he sum B 1 n) is paid ou a ime n if he policy holder is dead hen, i.e. db s) = B 1 s)i 1 s)dε n s), s >. For his paymen process for we ge he following calculaion for isolaion of B n) in 4), y 01, x, y) = E e rs ) db s) Z ) = 1 y = e rn ) B 1 n) y B 1 n) = e rn ) y 01, x, y) + y ). 17) We now have wo equaions 16) wih hree unknowns b 01, x, y),y 01, x, y), B 1 n) ) and all soluions are equally opimal. The naural one o ake is he one where B 1 n) = 0, y 01, x, y) = y, and b 01, x, y) = c 01, x, y) x, such ha all accouns are se o zero upon deah. 14
15 We also specify he simple exponenial differenial equaion characerizing he saewise consumpions, c 0 ) = c 0 1 ) r ι + Γ )µ)), 18) 1 γ c 01 ) = c 01 1 ) r ι + Γ )µ) Γ ) ). 19) 1 γ 1 + Γ ) We now consider wo examples, where he conrol in his special survival case can be specified furher. These examples correspond o he cases where here is no bank accoun and no insurance accoun, respecively. Example 2 No insurance accoun We can pu he insurance accoun equal o zero wihou losing expeced uiliy by specifying ha he naural premium for he opimal deah sum is he only paymen o he insurance accoun, i.e. b ) = µ )b 1 ). Realize from 2) and y 01, x, y) = y ha hen Y ) = 0 for all. Then we can pu y = 0 in all conrols and skip he dependence on y, i.e. c 0, x) = c 01, x) = b 01, x) = c 01 ) x. w0 ) x + g 0 f 0 ) ), ) w01 ) f 0 ) h01 ) x + g 0 ) ) The formulas are idenical o hose by Richard 1975,42,43)). In comparison we menion ha Richard 1975) uses he following noaion Richard noaion noaion here): a f 1 γ, b g, h w 1 γ, m w 1) 1 γ, λ µ, µ µh γ 1. In Richard 1975) he moraliy is modelled such ha he probabiliy of survival unil erminaion n, exp n µ), is zero for all. This is obained by µ for n. Furhermore, i is assumed ha µ )/µ) 1 for n. Bu hen he las erm of 13) is zero and W n) is superfluous: If we know ha we will no survive ime n, he uiliy of consumpion a ime n plays no role for our decision. Wih W n) = 0, 13) and 15) are idenical o hose by Richard 1975,41,25)). One problem, from a pracical poin of view, wih his consrucion is ha he opimal insurance sum may become negaive. If he individual and his inheriors) has relaively large uiliy from consuming while being alive compared o consuming upon deah, he should opimally risk losing pars of his wealh as he grows old. When here is no insiuional wealh he does so by selling life insurance. Bu he way individuals sell life insurance in pracice is insead by holding life annuiies based on insiuional wealh. Therefore, a much more realisic special case is given now in an example wih no bank accoun. Example 3 No bank accoun We can pu he bank accoun equal o zero wihou losing expeced uiliy by specifying ha income minus consumpion goes direcly ino he insurance accoun, i.e. B = C A. In his concree case his corresponds o leing b 0 ) = a 0 ) c 0 ), i.e. he excess of income over consumpion is paid as premium on he insurance conrac, and b 01 ) = c 01 ), i.e. upon 15
16 deah he insurance benefi is consumed by he inheriors). Realize from 6) and 16) ha hen X ) = 0 for all and we can pu x = 0 and skip he dependence on x in all conrols, i.e. c 0, y) = c 01, y) = w0 ) f 0 y + g )), 20) ) w01 ) f 0 h )y + g )) ) = b 01 ). Noe ha since now b 01 = c 01, he differenial equaion 19) holds also for he opimal deah sum. Remark 4 The opimal consumpion rae 20) solves he problem of opimal design of a life annuiy. If from ime here are no more incomes, i.e. g ) = 0, he opimal life annuiy rae is given by he fracion w 0 )/f 0 ) of he reserve. If w 0 ) is consan and here is no uiliy from benefis upon deah or erminaion, i.e. w 01 ) = W 0 n) = 0, we ge he opimal annuiy rae equals he reserve divided by n e R s r +eµ ds. This is jus he presen value of he life annuiy wih ineres r and moraliy rae µ. For he logarihmic invesor γ = 0) we ge he simpler n R e s µ ds. I is ineresing o see how he life annuiy rae evolves over ime, bu his quesion is answered by 18), c 0 ) = c 0 ) If he insurance is priced fair, his simplifies o 1 r ι + Γ )µ)). 1 γ c 0 ) = c 0 ) r ι 1 γ. Wheher his annuiy is decreasing or increasing depends on wheher he impaience facor ι is larger or smaller han he ineres rae r. Typically, one would ake he impaience facor o be larger han he ineres rae and hen he opimal annuiy rae decreases exponenially. 7 The Disabiliy/Unemploymen Model In his secion we specialize he resuls in Secion 4 ino he special case of a survival model. The idea is o sudy he opimal consumpion and insurance decisions of an individual who has uiliy of consumpion as long as he is alive. The uiliy may change, however, as he jumps ino a sae where he looses his income. This sae may be inerpreed as a disabiliy sae or unemploymen sae, depending on he sudy. In Figure 3, we have specified a se of income process coefficiens and a se of uiliy weigh coefficiens. All saewise coefficiens are zero in he sae dead. This means ha here is no income and no uiliy of consumpion in ha sae. Weigh on uiliy of consumpion in he sae acive/employed are specified by he coefficien w 0. Lump sum consumpion upon erminaion or ransiion beween saes is given no weigh, i.e. W 0 = W 1 = w 01 = 0. Weigh on uiliy of consumpion in he sae disabled/unemployed is specified by he coefficien w 1. The income in ha sae is se o zero, a 0 = 0. Leing ρ = 0 may be less realisic in an unemploymen inerpreaion han in a disabiliy inerpreaion bu is neverheless assumed o obain explici resuls. 16
17 0 acive/employed w 0 0, W 0 = 0 a 0 0, A 0 = 0 µ ց w 02 = 0 a 02 = 0 σ ρ=0 w 01 = 0 a 01 = 0 2 dead w 2 = W 2 = 0 a 2 = A 2 = 0 ν ւ w 12 = 0 a 12 = 0 1 disablied/unemployed w 1 0, W 1 = 0 a 1 = A 1 = 0 Figure 4: Disabiliy/Unemploymen model According o 10) f 2 = 0 and f 1 and f 0 are characerized by f 1 ) = r 1 ) + ν ) ) f 1 ) w 1 ),f 1 n) = 0, r 1 ) = δr δ ν ν) + ν ν, f 0 ) = r 0 ) + µ ) + σ ) ) f 0 ) w 0 ) σ )f 1 ),f 0 n) = 0 r 0 ) = δr δ µ ) + σ ) µ ) σ )) + µ) + σ ) µ ) σ ). This differenial equaion has he soluion and FeynmanKac represenaion, respecively, f 1 ) = f 0 ) = n = Ẽ,1 n = Ẽ,0 e R s r 1 +eν w 1 s)ds e R s r 1 dw s), e R s r 0 +eµ+eσ w 0 s) + σ s)f 1 s) ) ds e R s r Zτ) τ)dτ dw s). We specify furher his soluion in he special case where σ = σ, ν = µ and ν = µ. This means ha disabiliy/unemploymen risk is priced by he objecive measure and ha he moraliy risk is no changed when jumping ino sae 1. In ha case we have ha r 1 = r 0 r and σ = σ. If we furhermore have ha he uiliy is he same for he saes 0 and 1, i.e. w 0 = w 1 w, we ge ha f 0 = f 1 is given by f 0 ) = n = Ẽ,0 e R s r +eµ w s)ds e R s r τ)dτ dw s). According o 9) g 2 = g 1 = 0 and g 0 is characerized in he same way as in 14) and 15) wih µ replaced by µ + σ. Assume now, as in he previous secion, ha he insurance accoun is se o zero upon deah, i.e. y 02, x, y) = y 12, x, y) = y. 21) 17
18 We can now specify he opimal conrols in erms of f and g. The consumpion upon ransiion is zero since we have no uiliy of consumpion upon ransiion. Furhermore, he opimal life insurance sum becomes minus he personal wealh, i.e. c 01 = c 02 = c 12 = 0, b 02 = b 12 = x. 22) The more ineresing conrols are he consumpion raes as acive and disabled and he opimal proecion agains disabiliy risk, c 0, x, y) = c 1, x, y) = b 01, x, y) + y 01, x, y) = w0 ) x + y + g 0 f 0 ) ), ) w1 ) f 1 x + y), ) f1 ) f 0 ) h01 ) x + y + g 0 ) ) x + y). 23) Concerning he ransiions ino sae 2 we have from 21) and 22) he relaions b 02, x, y) + y 02, x, y) = x + y), 24) b 12, x, y) + y 12, x, y) = x + y), which have he same inerpreaions as in Secion 6: Upon deah he bank accoun balance x + b 02, x, y) or x + b 12, x, y) if he policy holder dies as disabled) equals minus he insurance accoun balance y + y 02, x, y) or y + y 12, x, y) if he policy holder dies as disabled). This jus means ha hese accouns, one being negaive he oher, earn ineres unil ime n where he erminal benefi Ỹ n) is paid o he bank such ha boh accouns close a zero. As in Secion 6 we can now find paymens upon deah and afer deah such ha 24) holds. Bu hen we can jus as well ake he insurance accoun o zero already upon deah by requiring ha y 02, x, y) = y 12, x, y) = y. Before looking closer a he insurance decision we specify he simple exponenial differenial equaion characerizing he saewise consumpions, c 0 ) = c 1 ) = 1 c0 ) r ι + Γ 02 )µ) + Γ 01 σ ) ), 1 γ 1 c1 ) r ι + Γ 12 )ν ) ). 1 γ The opimal proecion agains loss of income is given in 23). There we see ha for he special case wih he same uiliy in saes 0 and 1, i.e. f f 0 = f 1, he opimal proecion reduces o h 01 ) x + y + g 0 ) ) x + y). If furhermore, he price of his proecion is calculaed by he Pinensiy, i.e. h 01 = 1, hen he proecion reduces o g 0 ). Thus, under hese circumsances he individual should fully proec he financial value of fuure income. If uiliy of consumpion as disabled is lower han uiliy of consumpion as acive and/or if he proecion is expensive in he sense of h 01 > 1, hen one should underinsure he poenial loss. We now consider wo examples, where he conrol in his special disabiliy/unemploymen case can be specified furher. These examples correspond o he cases where here is no bank accoun and no insurance accoun, respecively. Example 5 No insurance accoun. 18
19 We can pu he insurance accoun equal o zero wihou loss of value by specifying ha he naural premium for he opimal deah sum is he only paymen o he insurance accoun, i.e. b 0 ) = b 1 ) = y 12 ) = 0. σ )b 01 ) + µ )b 02 ), µ )b 12 ), Realize from 2) and y 02, x, y) = y 12, x, y) = y ha hen Y ) = 0 for all and we can pu y = 0 and skip he dependence on y in all conrols, i.e. c 0, x) = w0 ) x + g 0 f 0 ) ), ) c 1, x) = w1 ) f 1 ) x, b 01, x) = f1 ) f 0 ) h01 ) x + g 0 ) ) x. The remark a he and of Example 2 applies again here: The insurance sum may becomes negaive, and in pracice negaive insurance sums are no obained by and individual s selling of life insurance bu by puing he wealh saved in he insiuion a risk hrough some annuiy conrac. Therefore, a much more realisic special case is given now in an example wih no bank accoun. Example 6 No bank accoun. We can pu he bank accoun equal o zero wihou losing expeced uiliy by specifying ha income minus consumpion goes direcly ino he insurance accoun, i.e. B = C A. In his concree case his corresponds o leing b 0 ) = a 0 ) c 0 ), i.e. as acive he excess of income over consumpion is paid as premiums on he insurance conrac, b 1 ) = c 1 ), i.e. as disabled he annuiy benefi is fully consumed, and b 01 ) = c 01 ) a 01 ) = 0, i.e. here is no lump sum deah benefi paid ou. Realize from 6) ha hen X ) = 0 for all. We can hen pu x = 0 in all conrols and skip he dependence on x, i.e. c 0, y) = c 1, y) = y 01, y) = w0 ) y + g 0 f 0 ) ), ) w1 ) f 1 ) y, f1 ) f 0 ) h01 ) y + g 0 ) ) y. 25) The quesion is now, wha should he policy holder acually do in order o demand he opimal reserve jump y 01, y). Le us consider he case where he policy holder demands o opimal reserve jump by purchasing an opimal disabiliy annuiy. In general, we have ha he disabiliy annuiy rae solves he equivalence principle upon ransiion y + y 01, y) = which by he opimizaion relaion 25) leads o n f 1 ) f 0 ) h01 ) y + g 0 ) ) = 19 e R s r+ν b 1 s)ds, n e R s r+ν b 1 s)ds.
20 If he disabiliy annuiy demanded is consan his leads o he opimal annuiy rae b 1 = f1 ) f 0 ) h01 ) y + g 0 ) r+ν ds. n e R s This rae becomes paricularly simple in he special case where preferences in he saes 0 and 1 are equal and where insurance is priced fair, i.e. h 01 )f 1 )/f 0 ) = 1. However in ha case one could also come up wih a very simple nonconsan soluion. The differenial equaion for Y 0 ) + g 0 ), d Y 0 ) + g 0 ) ) d = r + µ))y 0 ) c 0 ) + a 0 ) σ )y 01, Y 0 ) ) + r + µ ) + σ ))g 0 ) a 0 ) = r + µ)) Y 0 ) + g 0 ) ) c 0 ) should be equal o he differenial equaion for he value of he fuure annuiy benefis, d n ) e R n s r+µ b 1 s)ds = r + µ)) e R s r+µ b 1 s)ds b 1 ). d Bu hese are he same exacly if b 1 ) = c 0 ). Thus he policy holder obains he opimal reserve jump by demanding a disabiliy annuiy wih a ime dependen paymen rae corresponding o his opimal consumpion rae given ha he is sill in sae 0. I is very inuiive ha he hen ges full proecion if he disabiliy rae equals his opimal consumpion in sae 0 since his gives him he opporuniy, in case of disabiliy, o coninue consuming as if nohing had happened. If insead he policy holder is underinsured, i.e. h 01 )f 1 )/f 0 ) < 1, because he has lower uiliy from consumpion as disabled han as acive and/or because he proecion is expensive, hen he would have o demand a correspondingly lower disabiliy annuiy. References Cairns, A. J. G. 2000). Some noes on he dynamics and opimal conrol of sochasic pension fund models in coninuous ime. ASTIN Bullein, 301): Davis, M. H. A. 1993). Markov Models and Opimizaion. Chapman and Hall. Fleming, W. H. and Rishel, R. W. 1975). Deerminisic and Sochasic Opimal Conrol. Springer Verlag. Hoem, J. 1988). The versaliy of he Markov chain as a ool in he mahemaics of life insurance. Transacions of he 23rd Congress of Acuaries, pages Hoem, J. M. 1969). Markov chain models in life insurance. Bläer der Deuschen Gesellschaf für Versicherungsmahemaik, 9: Meron, R. C. 1969). Lifeime Porfolio Selecion under Uncerainy: The Coninuous Time Case. Review of Economics and Saisics, 51: Meron, R. C. 1971). Opimum Consumpion and Porfolio Rules in a Coninuous Time Model. Journal of Economic Theory, 3: ; Erraum );
21 Norberg, R. 1991). Reserves in life and pension insurance. Scandinavian Acuarial Journal, pages Richard, S. F. 1975). Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuous ime model. Journal of Financial Economics, 2: Seffensen, M. 2004). On Meron s problem for life insurers. ASTIN Bullein, 341):5 25. Seffensen, M. 2006). Quadraic opimizaion of life and pension insurance paymens. ASTIN Bullein, 361): Yaari, M. 1965). Uncerain lifeime, life insurance, and he heory of he consumer. Review of Economic Sudies, 32: Appendix A For soluion of he conrol problem we inroduce a value funcion V, x, y) = supe j,x,y du s), where E j,x,y denoes condiional expecaion given ha X ) = x, Y ) = y, and Z ) = j. The HJB equaion for his value funcion is as follows, V j, x, y) = wih [ inf 1 γ wj ) 1 γ c j ) γ Vx j, x, y) rx + a j ) + b j ) c j ) ) Vy j, x, y) ry b j ) k:k j µjk ) b jk ) + y jk ) )) 1 k:k j µjk ) γ wjk ) 1 γ c jk ) γ + V k, x jk ),y + y jk ) ) ) V j, x, y) x jk ) = x + a jk ) + b jk ) c jk ). We now guess ha he HJB equaion is solved by he following funcion wih according derivaives, V j, x, y) = 1 γ fj ) 1 γ x + y + g j ) ) γ, V j, x, y) = 1 γ γ fj ) γ f j ) x + y + g j ) ) γ +f j ) 1 γ x + y + g j ) ) γ 1 g j ), V j x, x, y) = f j ) 1 γ x + y + g j ) ) γ 1, V j y, x, y) = f j ) 1 γ x + y + g j ) ) γ 1. Firsly we consider he firs order condiions for he elemens of he consumpion process. The condiions for c and c k becomes c j ) = wj ) x + y + g j f j ) ), ) c jk ) = w jk ) f k ) + w jk ) a jk ) + b jk ) + x + y + y jk ) + g k ) ). 26) 21
22 Here, one should noe ha b jk and y jk appear in he firs place in he relaion for c jk. Secondly we consider he firs order condiions for he elemens of he insurance conrac. These condiions are given by he following relaion b jk + y jk = hjk )f k ) f j ) x + y + g j ) ) x + a jk ) c jk + y + g k ) ). 27) Here, here are several poins o make. Firsly, here is no condiion on b j. Secondly, b jk and y jk are no uniquely deermined since he firs order condiion only pus a condiion on heir he sum. Thirdly, c jk appears on he righ hand side. Thus, we have o calculae he soluion of wo equaions 26) and 27) wih wo unknowns c jk ) and b jk ) + y jk ). The soluion is c jk ) = wjk ) f j ) hjk ) x + y + g j ) ), b jk + y jk = fk ) + w jk ) f j h jk ) x + y + g j ) ) ) a jk ) + x + y + g k ) ). Plugging hese conrol candidaes and he value funcion and is derivaives ino he HJB equaion leads o he ordinary differenial equaions for f and g presened in 9) and 10). Tha hese ordinary differenial equaions acually have soluions, see 11) and 12), verifies ha he suggesed value funcion is he righ one and ha he conrol candidaes above are indeed he opimal ones. 22
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