ABSTRACT KEYWORDS 1. INTRODUCTION


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1 ON MERON S PROBLEM FOR LIFE INSURERS BY MOGENS SEFFENSEN ABSRAC his paper deals wih opimal invesmen and redisribuion of he free reserves conneced o life and pension insurance conracs in form of dividends and bonus. Formulaed appropriaely his problem can be viewed as a modificaion of Meron s problem of opimal consumpion and invesmen wih a very paricular form of consumpion and uiliy hereof. Boh are lined o a finie sae Marov chain. We disinuish beween uiliy opimizaion of dividends, where a semiexplici resul is obained, and uiliy opimizaion of bonus paymens. he laer connecs o he financial noion of durable oods and allows for an explici soluion only in very special cases. KEYWORDS Marov chain; Dividends and bonus; Power uiliy; Bellman equaion; Durable oods.. INRODUCION Life insurance companies ofen hold socalled free reserves. hese are he par of he oal reserves which are no se aside for uaraneed paymens. As reserves for uaraneed paymens we have he socalled mare reserve in our mind, see Seffensen (2000b). Whereas he free reserves belon o he policy holders as does he mare reserve for uaraneed paymens, he insurance company decides how o inves and when o pay ou hese free reserves wihin some leislaive consrains. In his paper we approach his decision problem wih ools from sochasic conrol heory and ideas from classical opimal invesmenconsumpion problems in finance. he life insurance policies ha we primarily have in mind, are socalled paricipain policies. Here, a se of uaraneed paymens are areed upon a he issuance of he policy. he uaraneed paymens are se under pruden assumpions on capial ains, moraliy ec. and herefore ive rise o a surplus which is acivaed a he ime of issuance if he uaraneed paymens are reserved for under a mare basis. Hereafer i is up o he insurance company o inves and redisribue his surplus in form of dividends and bonus paymens o he policy holders. See Norber (999) and Seffensen (2000a) for a deailed ASIN BULLEIN, ol. 34, No., 2004, pp. 525
2 6 MOGENS SEFFENSEN sudy of he noions of surplus, dividends and bonus. In his paper, he free reserve is he surplus acivaed a he issuance of he policy wih addiion of any capial ains and subracion of any paymens paid ou durin he erm of he policy. Mos of he ideas in he paper apply o pension fundin as well as paricipain insurance. In fac, pension fundin can be considered as paricipain insurance wih paricipaion in he downside as well as he upside. he precise conen of his saemen is iven in Seffensen (200). Sochasic conrol heory has played a role in pension fundin in many years, see Cairns (2000) and Seffensen (200) and references herein. he basic idea is o use an opimizaion crierion ha rewards sabiliy of he surplus and he paymens. he crierion is based on a quadraic disuiliy funcion ha punishes he surplus for deviaions from a surplus are and he paymen rae for deviaions from a paymen rae are. Worin wih quadraic disuiliy one can benefi from sudies on he linear reulaor in he lieraure on conrol heory, see e.. Flemin and Rishel (975). However, his approach has some disadvanaes concernin e.. counerinuiive invesmen sraeies, see Cairns (2000). Furhermore, he explici resuls obainable for pension fundin where dividends and bonus paymens are ypically unconsrained, do no carry over o he problems of paricipain insurance, where dividends and bonus paymens are consrained o be o he benefi of he policy holder. See Seffensen (200) for a deailed sudy. In he financial lieraure, he mos widely acceped approach o opimal invesmen seems o be he one aen by Meron (969,97). he problem formulaion has laer been referred o as Meron s problem. his is based on opimal uiliy of fuure wealh, or, in case of inroducion of consumpion, uiliy of fuure consumpion raes. Meron s approach has been eneralized and reformulaed in various direcions in order o mae he resuls more applicable o real life invesmen and consumpion problems. A number of hese eneralizaions are relevan for applicaions o he invesmen and redisribuion problems of he insurance company. Amon ohers, we menion Korn and Kreel (2002), where a predefined consumpion sream relaes o he uaraneed paymens, Korn and Kraf (200) and Mun and Sørensen (2002), where ineres raes are allowed o be sochasic. he focus in his paper is an adapaion of Meron s problem o he special paern of paymen sreams presen in life insurance. Basically, Meron (969, 97) iniialized hese sudies by considerin opimal lifeime consumpion. he uiliy approach o opimizaion in life insurance daes bac o Richard (975). He considered he decision problem of an insured choosin beween invesmen in financial asses lie bonds and socs and invesmen in a life insurance conrac. From he view poin of he life insurance company he uiliy approach has been sudied by Jensen and Sørensen (200) and Hansen (200). he approach in Hansen (200) is closely relaed o ours and he obains for a special class of insurance producs resuls similar o ours. he ouline of he paper is as follows. In Secion 2, we describe he uaraneed paymens of a life insurance conrac and he financial mare on which he insurance company invess he free reserves. In Secion 3, we consider opimal dividends by sain he conrol problem and he correspondin Bellman equaion.
3 ON MERON S PROBLEM FOR LIFE INSURERS 7 In Secion 4, we ive a semiexplici soluion o he dividend opimizaion problem. In Secion 5, we consider opimal bonus paymens by sain he conrol problem and he correspondin Bellman equaion. In Secion 6, we ive an explici soluion o he bonus opimizaion problem in a raher special infinie ime horizon case. 2. PAYMENS, RESERES, AND HE MARKE We ae as iven a probabiliy space (W, F, P). On he probabiliy space is defined a process Z =(Z()) 0 ain values in a finie se J = {0,, J} of possible saes and sarin in sae 0 a ime 0. We define he Jdimensional counin process N =(N ) J by! N () = #" s s ^ Z( s )!, Z() s =, counin he number of umps ino sae unil ime. Assume ha here exis deerminisic funcions m (),, J, such ha N admis he sochasic inensiy process (m Z( ) ()) 0 for J, i.e. N () # m Z() s () s ds 0 consiues a marinale for J. hen Z is a Marov process. he reader should hin of Z as a policy sae of a life insurance conrac, see Hoem (969) for a moivaion for he seup. One par of he paymen process of an insurance conrac is he uaraneed paymen process B. Denoin by B() he accumulaed uaraneed paymens o he policy holder over [0, ], he uaraneed paymens are described by!! db() Z () = b () d+ Z ( ) b () dn () + Z() u DB () de (),! J u!! 0, + B^0 h = 0. where e u ()=I( u) and riers a lump sum paymen a he deerminisic poin in ime u. Posiive elemens of B are called benefis whereas neaive elemens are called premiums or conribuions. he rae b Z() () is he rae of paymens a ime iven he policy sae Z(), b Z( ) () is he lump sum paymen a ime iven ha Z umps from Z( ) o he sae a ime, and DB 0 (0) and DB Z() are lump sum paymens a he issuance and he erminaion of he conrac, respecively, iven he saes of Z a hese poins in ime. For noaional convenience we resric lump sum paymens a deerminisic ime poins o ae place a ime 0 and ime, exclusively. One consrucion of he process Z and he paymen process B ha he reader could have in mind is illusraed in Fiure. he model is a disabiliy model where here are hree saes, 0 = acive,= disabled,2= dead.for an endowmen insurance wih disabiliy annuiy he consan premium rae
4 8 MOGENS SEFFENSEN durin periods of aciviy is iven by b 0 < 0, a consan disabiliy annuiy rae durin periods of disabiliy is iven by b > 0, a deah lump sum paid upon deah is iven by b 02 = b 2 > 0, finally, in case of survival unil ime a pension lump sum DB 0 ()=DB () > 0 is paid ou. 0 acive b 0 <0 DB 0 ()>0 m 0 m 0 disabled b >0 DB ()>0 m 02 m 2 Æ Æ b 02 >0 b 2 >0 2 dead Fiure : Endowmen insurance wih disabiliy annuiy. he insurance company lays down he paymen process on a socalled firs order basis. he firs order basis conains a consan firs order ineres rae r and a se of firs order ransiion inensiies mˆ,. he paymen process B will now be arraned in accordance wih he socalled equivalence principle, i.e. such ha he oal expeced discouned uaraneed paymens includin he iniial lump sum paymen DB 0 (0) equals zero under he firs order basis. In mahemaical erms, we define he saewise firs order reserves by # r ( s)! () = E; e db() s Z() = E, J, where E denoes expecaion wih respec o he probabiliy measure under which N admis he inensiy process (mˆ Z( ) ()) 0. he equivalence principle reads or, equivalenly, 0 ( 0 ) = 0, 0 0 () 0 =DB (). 0
5 ON MERON S PROBLEM FOR LIFE INSURERS 9 We shall also need a mare value of he paymen process B. A he end of his secion we shall inroduce a consan mare ineres rae r. Resricin ourselves o sae dependen mare values we e from Seffensen (2000b) ha he mare value can be wrien in he form # () = EQ er( s) ; db() s Z() = E,! J, where E Q denoes expecaion wih respec o he probabiliy measure under which N admis he inensiy process (m Z( );Q ()) 0. his probabiliy measure reflecs he mare aiude owards ris in Z. If he mare is neural wih respec o ris in Z, he measure equals he obecive measure P and m Z( );Q ()=m Z( ) (). See also Seffensen (2000b) for such a noion of mare value. he mare value of he uaraneed paymen process a he ime of issuance will in eneral be differen from zero. he insurance company ses aside a oal reserve on he policy. We spea of he difference beween he oal reserve and he mare value of uaraneed paymens as he free reserves. he oal reserve of he conrac mus equal 0 a ime 0. hus, he free reserve a ime 0 equals minus he mare value of he uaraneed paymens. I is required from he firs order basis ha his mare value of uaraneed paymens a ime 0 is neaive. his ives us a posiive iniial free reserve a ime 0 which we denoe by x 0. Noe ha since x 0 is he free reserve a ime 0, his does no include a possible iniial paymen aen from he free reserves a ime 0. he free reserves belon o he policy holder. he followin secions deal wih he problem of opimal invesmen and redisribuion of he free reserves. We could, insead, have wored wih he oal reserves and aen ino accoun he uaraneed paymens as a par of he problem. hen he uaraneed paymens could be considered as a predefined paymen sream and he soluion o an opimizaion problem of invesmen of he oal reserves would also conain an approach o he problem of hedin opimally he uaraneed paymens. However, in his paper we shall no ae his sarin poin bu concenrae on he free reserves only. By considerin only he free reserve, we indirecly assume ha whaever ains and losses are conneced o he uaraneed paymens, hese ains and losses affec he equiy of he company and no he free reserve. Gains and losses occur if he insurance company does no hede for whaever reason he uaraneed paymens. Whereas he uaraneed paymens are specified for he individual policy, he oal free reserves on a porfolio of policies belon o he porfolio as a whole. Here, he erm porfolio could cover all policies in an insurance company bu could also be a se of policies wih common characerisics in some sense. I could even be an individual policy. In fac, when speain abou uaraneed paymens below, he reader should hin of he oal uaraneed paymens for he porfolio of policies o which he iven free reserves can be said o belon. Sraeies which evenually empy he free reserves o he porfolio of policies can be said o be fair iven a concern abou fairness beween he porfolio
6 0 MOGENS SEFFENSEN holders as a roup and he owners of he company. A compleely differen quesion is wheher a iven redisribuion sraey is fair iven a concern abou fairness muually beween policy holders in he porfolio. I is by no means clear which fairness crierion o pu up here. In fac, in he followin we shall no pay any aenion o he fairness muually beween policy holders. For decision on an invesmen sraey for he free reserves, we specify a financial mare as follows: On he probabiliy space (W, F, P) is also defined a Brownian moion W. We consider a mare wih wo asses which are coninuously raded. he mare is described by he sochasic differenial equaion, ds 0 () = rs 0 ()d S 0 (0) =, ds () = (r +ls)s ()d + ss ()dw(), S (0) = s 0, where, r, l, s > 0 are consans. his is he classical BlacScholes mare. I is possible o eneralize he resuls below o more eneral financial mares, e.. mulidimensional diffusion mares. 3. UILIY OPIMIZAION OF DIIDENDS One way of repayin he free reserves o he policy holders is simply o pay ou dividends in cash. When dividends are paid ou cash one speas of cash bonus. In his secion we consider an insurance company maximizin expeced uiliy of dividends. In case of cash bonus his approach maes sense since he policy holder acually receives hese paymens. In anoher repaymen scheme, he dividends are ep wihin he company and raded ino fuure bonus paymens. hen uiliy opimizaion of dividends may no be he appropriae approach o ae. In Secion 5 we consider uiliy opimizaion of bonus. he uaraneed paymen process B consiues ypically only one par of he oal paymen process. In case of cash bonus he insurance company adds o he uaraneed paymens an addiional dividend paymen process dependin on various condiions in he insurance mare, hereunder he financial mare on which paymens are invesed. he insurance company decides on he invesmen profile and on his addiional paymen process wihin any leislaive consrains here may be. We formalize he dividend paymen process B by!! u db() = b() d+ b () dn () + DB() de (),! J u!! 0, + () B^0 h = 0, where he processes b, b, J, and DB are decided by he insurance company. Here b is a dividend paymen rae, b is a lump sum dividend paymen riered by ransiion ino sae, and DB is a lump sum dividend paid ou a a deerminisic poin in ime. I should be emphasized ha i is he processes
7 ON MERON S PROBLEM FOR LIFE INSURERS b and b and he quaniies DB ha are decided by he insurance company and no he process B iself. he free reserve is he source of dividend paymens and can be considered as a wealh process from which he dividends are paid ou as consumpion. he invesmen behavior of he insurance company is modelled by a porfolio process p denoin he proporion of he free reserve invesed in he asse S. Resricin ourselves o selffinancin porfoliodividend processes, he free reserve process follows he sochasic differenial equaion dx() = ^r + plsh X() d + psx() dw() db(). (2) he sochasic differenial equaion for he free reserves can be considered as a conrolled sochasic differenial equaion wih he conrol bein he porfoliodividend process (p,b) in he sense ha i is he b, b, J, and DB which are conrollable and no he paymen process B iself. he insurance company is allowed o choose a porfoliodividend process such ha here exiss a nonneaive soluion o he sochasic differenial equaion (2), i.e. X() 0, 0. Such porfoliodividend processes are said o belon o a se A. he consrain X() 0, 0, can be inerpreed as a solvency consrain on he life insurance company. We consrain he dividend process o be nondecreasin conformin wih he usual requiremen ha dividends and bonus should be o he benefi of he policy holders. We impose no consrains on he porfolio process p. When no imposin any borrowin consrains on p, we have he realisic siuaion in mind ha he insurance company can, when invesin he free reserves, borrow from is own posiion in risfree asses held o cover he uaraneed paymens. hen, iven a porfoliodividend process (p, B) A, he conrolled sochasic differenial equaion describin he wealh is iven by p, B p, B p, B dx () = ( r + pls) X () d + psx () dw() db(), Xp, B( 0 ) = x. 0 We assume ha he insurance company chooses a porfoliodividend process o maximize imeaddiive power uiliy of he policy holder in he sense of he followin opimizaion problem: R sup E S v c, Z(), b() d v u(, Z(), DB()) de u # _ i +! ()! S 0 u!! + + v, Z( ), b() dn _ i (), G (, p B) A 0, #! 0 (3) where
8 2 MOGENS SEFFENSEN vc, Z(), b() az() _ i = _ () i _ b() i, v, Z( ), b() az( ) = () b _ i _ i _ () i,! J, vu, Z(), DB() DAZ() _ i = _ () i _ DB() i, u! " 0,,. (4) he opimizaion problem above disinuishes iself from he classical Meron s problem in wo imporan direcions. Firsly, we have o ae ino accoun he paricular paern of dividend paymens iven in (). We have o decide he measure of uiliy of a combinaion of dividend raes and lump sum dividends. In he formulaion in (3) we simply ae uiliy of raes and lump sums and add up o measure he oal uiliy. Since benefi raes are raes and no paymens, his mih seem o be a criicizable approach. However, i conforms wih Meron s approach o he lifeime consumpion problem. Acually, he opimizaion problem (3) eneralizes he formulaion in Meron (969). Furhermore, one can arue ha uiliy of paymen raes and uiliy of a lump sum is usually simply added up whenever he opimal uiliy of consumpion problem is combined wih uiliy of erminal wealh, he socalled beques funcion. In our formulaion he beques funcion corresponds o he uiliy of he erminal dividend paymen D B(). Secondly, (3) conains socalled sochasic uiliy since we allow he uiliy of dividends o depend on he sae of he process Z. he idea becomes clear in he specificaion of he uiliy funcions (4). We hin of a siuaion where he policy holder saes his preferences over ime and evens in his life hisory by specificaion of a se of nonneaive weih funcions a (), a (), DA (),. One can hin of he weih funcions as bein componens of an arificial nondecreasin paymen sream iven by!! Z () Z ( ) Z() da() = a () d + a () dn () + Da () de (),! J u!! 0, + A^0 h = 0, However, such a paymen sream plays no oher role han specifyin a se of weih funcions. he paymen process A is experienced by neiher he insurance company nor he policy holder. Since he policy holder does no sae direcly a se of weih funcions, he insurance company needs o decide on a se of weih funcions. An obvious idea here would be o use he se of uaraneed paymens, since hese are o some exen decided by he policy holder, and his paymen sream indirecly saes his preferences. We sues wo differen ses of weih funcions based on he uaraneed paymens. Firsly, assume ha he policy holder demands a cerain profile of benefis for a iven premium paymen process. his consrucion relaes o he noion of defined conribuions. hen he benefi profile specifies a se of weih funcions by definin da() db () / _ db() i. = + + u
9 Secondly, assume ha he policy holder demands a cerain premium profile for a iven benefi paymen process. his consrucion relaes o he noion of defined benefis. hen he premium profile specifies a se of weih funcions by definin da() =db () / _ db() i. We shall laer see ha he propery of he paymen process A bein nondecreasin will lead o a nondecreasin opimal dividend process such ha he consrain on he dividend process is auomaically fulfilled by he opimal one. he weih funcions in (4) are aen o he power of wihou loss of eneraliy and us for noaional convenience in he followin. Obviously, one could add o he weih funcions suesed above a ime dependence lie e.. a r () = e b + ` () ec. ain he facor e r o he power of, r would be he usual parameer specifyin ime preference beyond wha has already been specified indirecly in he process B. We define he opimal value funcion by (, x) sup E azs () = x,,; # _ () s i b () s ds + + ON MERON S PROBLEM FOR LIFE INSURERS 3 (, p B)! A # #! u! azs ( ) () s b () s dn _ i _ i () s DAZs () () s DB() s de u! () s W _ i _ i,! 0, + W X where E,x, denoes condiional expecaion iven ha X()=x and Z()=. We can spea of (,x) as he saewise opimal value funcion. A fundamenal differenial sysem of equaliies or inequaliies in conrol heory is he Bellman sysem for he opimal value funcion. he Bellman sysem is here iven as he infimum over admissible conrols of parial differenial equaions for he opimal value funcion. We shall no derive he Bellman equaion here bu refer o Seffensen (2000b) for a derivaion of parial differenial equaions for relevan condiional expeced values. I can be realized ha (, x) inf x (, x) r pls x b 2 = ; _^ + h i xx(, x) p s x p, bb,,!, DB a () b m () R! (, x), W! W X (, x) = sup DA() DB() + ;_ i _ i _, xdb() ie, DB (5) where
10 4 MOGENS SEFFENSEN R(, x) = a () b, x b b _ i + _ i (, x) l, and where subscrip denoes he parial derivaive. In principle, we would have o wrie (p, b, b,, DB) R ({0} R + ) +2 under he infimum in (5) bu, hrouhou, we shall sip he specificaion of he decision variable se. I should be emphasized ha he Bellman sysem is acually a sysem of J differenial equaions wih J erminal condiions. he Bellman sysem conains he erms presen in he Bellman equaion for Meron s problem and an addiional erm semmin from he process Z. he erm involvin R (, x) corresponds o he classical ris erm in he socalled hiele s differenial equaion for he saewise reserves, see Seffensen (2000b) where R is spoen of as he ris sum correspondin o a ump from o. he Bellman sysem plays wo differen roles in conrol heory. One role is ha if he opimal value funcion is sufficienly smooh, hen his funcion saisfies he Bellman sysem. However, usually i is very difficul o prove a priori he smoohness condiions. Insead one ofen wors wih he verificaion resul sain ha a sufficienly nice funcion solvin he Bellman sysem acually coincides wih he opimal value funcion. In fac, i is no even necessary o come up wih a classical soluion o he Bellman sysem. I is sufficien o come up wih a socalled viscosiy soluion wih relaxed requiremens on differeniabiliy which will hen coincide wih he opimal value funcion. 4. EXPLICI RESULS ON DIIDEND OPIMIZAION We shall now uess a soluion o he Bellman equaion based on a separaion of x in he same way as in he classical case. We ry a soluion in he form (, x) = f () x _ i, where f is a deerminisic funcion searched for below. his form leads o he followin lis of parial derivaives, (, x) x = d n f (), f () (, x) = ` f () x, x xx (, x) = ^ h ` f () x. 2 A candidae for he opimal (p, B) is found by solvin (5) for he supremums wih respec o he decision variables (p, b, b,, DB), i.e = f () x lsx( ) f () x ps x, 0 = f () x a () b,
11 ON MERON S PROBLEM FOR LIFE INSURERS = a () ( b ) f () ( xb ),!, = ` DA () _ DB() i ` f () _ xdb() i. his leads o he candidaes p (, x) =, s l a () b (, x) = x, f () a () b (, x) = x, a () + f () DB (, x) = DA () x. f () + DA () where he noaion is eviden and exposes p, b, b, DB as funcions of, X(), and Z(). Here we see ha he opimal proporion invesed in he risy asse is independen of he sae of Z and equals he classical proporion in Meron s problem. As for he opimal dividends we see ha boh b (, x), b (, x), and DB (, x) are linear funcions of wealh as is consumpion in Meron s problem wih consumpion. However, he proporionaliy facors involve he weih funcions in he arificial paymen process A and he funcion f. We shall now derive a differenial equaion and a sochasic represenaion for f (). Inserin he opimal candidae in he Bellman sysem ives he parial differenial equaion for f (), f ;! f () = r* f () a () m () R (), f ( ) = A ( ), f ( 0 ) = f ( 0) + A ( 0),! (6) where r* r, 2 l 2 = + b l f R ; a () f () f () f (). = + ` ` his sysem of differenial equaions has similariies wih hiele s differenial equaion, see Seffensen (2000b). However, he quaniy R f; is no a ris sum in he same sense as in hiele s differenial equaion. Neverheless, i is possible
12 6 MOGENS SEFFENSEN o derive a sochasic represenaion formula for he soluion o he differenial equaion. We shall realize ha f can be wrien as a condiional expecaion of he discouned arificial paymen sream A under a very paricular measure P * o be specified below, i.e. *, # r*( s) f () = E ; e da(), s E (7) where E * denoes expecaion wih respec o he measure P *. Define he lielihood processes L and he correspondin ump ernel by Z ( ) Z () dl () = L ( )! () `dn () m () d, () =! J `a () + f () `f () f (),!. a () + f () f () hen we can chane measure from P o P * by he definiion L = dp dp *, and i follows from Girsanov s heorems (see e.. Bör (994) ha N under P * admis he inensiy process We can finally wrie Z ( ) Z ( ) Z ( ) m * () = ` + () m ().! f ;! f ; f () = r* f () a () m* () R * (), R * () = a () + f () f (). his is precisely a version of hiele s differenial equaion for a reserve defined by (7). he calculaions above mae sense only if here exiss a soluion o he differenial equaion (6). Such an exisence relaes o he fac ha he lielihood process L acually defines a new probabiliy measure and ha he condiional expeced value in (7) is finie and sufficienly differeniable. hese requiremens pu consrains on he coefficiens in he weih process, which we shall no pursue any furher here. he represenaion (7) allows us o inerpre f as some ind of uiliyadused value of he arificial paymen sream A. he uiliyadused value is aen o be a condiional expeced value, under some ind of uiliyadused measure, of discouned paymens, under some ind of uiliyadused discoun facor. his leads o an inerpreaion of he opimal conrol. he opimal rae of dividends equals he rae of paymens in A per uiliyadused
13 ON MERON S PROBLEM FOR LIFE INSURERS 7 value of fuure paymens in A imes he free reserves. he opimal lump sum paymens upon ransiion equals he ransiion paymen in A per uiliyadused value of fuure paymens in A (includin he ransiion paymen iself) imes he free reserves. he idea of his sraey is very easy o undersand and implemen in pracice, e.. for he examples of A described above in erms of he uaraneed paymens. Since he funcion f appears in he ump ernel iself, he represenaion (7) can no be direcly used as a consrucional ool for deerminaion of f. One would have o approach he differenial equaion (6) by numerical mehods. We shall no pursue his furher here. However, in one special case we can direcly e a sep furher, and we shall end his secion by briefly menionin ha case. he case of loarihmic uiliy can be obained by lein = 0 above such ha P * equals P. If e.. da()=e r db + (), (7) reduces o f () = E ; e db (), s E, # rs ( ) + which can be inerpreed as he mare value of fuure uaraneed benefis. his expeced value has an explici soluion in erms of he ransiion probabiliies of Z. 5. UILIY OPIMIZAION OF BONUS Dividends are no always direcly paid ou o he policy holders in form of cash bonus. Ofen hey are ep wihin he insurance company and raded ino fuure bonus paymens. In his secion we consider an insurance company maximizin expeced uiliy of bonus paymens. Insead of measurin uiliy of dividends we measure uiliy of he bonus paymens ino which he dividends are raded. However, i is sill he dividends ha are o be decided by he insurance company. See Norber (999) and Seffensen (2000a) for a deailed sudy of dividends and bonus. We shall now inroduce a nondecreasin paymen process A which plays a somewha differen role in his secion and in Secion 6 han in Secions 3 and 4. he paymen process is described by!! Z () Z ( ) Z() u da () = a () d+ a () dn () + DA () de (),! J u!! 0, + A^0 h= 0. In Secions 3 and 4, dealin wih uiliy opimizaion of dividends, he coefficiens of A only occur in he uiliy funcion as a specificaion of he preferences of he policy holder over ime and evens in he hisory of he policy. In his secion, dealin wih uiliy opimizaion of dividends, A specifies he profile of he bonus paymens in he followin sense: When dividends are ep wihin he company, dividends are used as sinle premiums o buy amouns of he addiional paymen process A. We denoe by
14 8 MOGENS SEFFENSEN K() he number of paymen processes bouh unil ime and le K(0 ) = 0. Over he shor ime inerval (, + d], he dividend paymen is iven by db() and he number of processes bouh equals dk(). hen, by definin # A E r ( s) () = ; e da () s Z () = E,! J, he equivalence principle for he insurance conrac bouh over (, + d] ives he followin relaion beween B and K, Z () A db() = dk() (). (8) Noe ha in conras o he siuaion in Secion 3 where B follows he sochasic differenial equaion (), we impose no a priori srucure of he dividend process B in he presen secion. he dividend paymen db(), which plays he role as a premium payin for he fuure bonus paymen process (dk()a()) <, is aen from he free reserves. However, he rade also riers an immediae conribuion o he free reserve. By definin rs ( ) A # () = E ; e da () s Z () = E,! J, as he mare value of he paymen process A, he conversion of dividend paymens ino bonus paymens under he firs order basis conribues o he free reserves wih Z () Z () A A dk() a () () such ha he ne effec on he free reserves is Z () A Z () A Z () () d () dk () A () Z () B + a () () A = d B, () which has an inerpreaion as he mare value of he dividend paymen bouh over (, + d]. In his secion, we impose he same consrains on he free reserves and on he dividend process as in Secion 3. Consrainin dividends o be nonneaive and havin assumed A o be nondecreasin will lead o a nondecreasin process K, followin o (8). One may acually wish o relax he consrain on dividends such ha K, in eneral, is nonneaive and no necessarily nondecreasin. his would allow he insurance company o cancel previously added bonus by payin ou a correspondin amoun of neaive dividends. However, here we shall ae he view poin ha previously added bonus has he saus as uaraneed paymens.
15 ON MERON S PROBLEM FOR LIFE INSURERS 9 hen, iven a porfoliodividend process (p, B), he dynamics of he free reserve is iven by he followin sochasic differenial equaion, Z () A Z () A p, B p, B p, B () dx () = ( r + pls) X () d + psx () dw() db(), () p, B X ( 0 ) = x. 0 In addiion o he uaraneed paymen process, he policy holder will receive he bonus paymen process. he bonus paymens over he ime inerval (, + d] add up o bonus paymens iven by K( ) da(), (9) Obvious examples of he paymen process A are he same as in Secion 3. Firsly, consider he consrucion da()=(db()) +. hen only he benefis are increased, and one could spea of defined conribuions wih proporionally increasin benefis where defined conribuions refer o he fac ha premiums are no chaned durin he erm of he policy. In Fiure, his scheme would lead o a proporional increase of disabiliy annuiy rae, deah lump sum and pension lump sum. Secondly, consider he consrucion da() = (db()). hen only premiums are chaned, and one could spea of defined benefis wih proporionally decreasin conribuions where defined benefis refer o he fac ha benefis are no chaned durin he erm of he policy. In Fiure, his scheme would lead o a decreasin premium rae. We assume ha he insurance company chooses a porfoliodividend process o maximize imeaddiive power uiliy of he policy holder in he sense of he followin opimizaion problem: R J N sup E S vc, Z(), K() d vu(, Z(), K()) de u # + () B! 0K ^ h! S O u! L! + P + v, Z( ), K( ) dn ^ h () G (, p ) A 0, #! 0 (0) where vc Z, (), K () az() ^ h = _ () K () i, v Z, ( ), K ( ) az( ) = ^ h _ ( K ) ( ) i,! J, vu Z, (), K () D AZ() ^ h = _ () K () i, u! " 0,,. ()
16 20 MOGENS SEFFENSEN he opimizaion problem above disinuishes iself from he classical formulaion in hree imporan direcions. Firsly, as in he case of uiliy opimizaion of dividends, we wan o ae ino accoun he special form of he bonus paymen iven in (9). For his we add up uiliy of paymen raes and uiliy of lump sum paymens. his leads o he sochasic ineral in (0). Secondly, as in he case of uiliy opimizaion of dividends, we ae he process Z ino accoun in he uiliy. We can hen direcly ae power uiliy of he acual bonus paymen raes and he lump sum bonus paymens by he forms (). hirdly, i should be emphasized ha whereas uiliy is aen of he acual bonus paymens, i is sill he dividend paymens ha are o be decided. he dividend paymens and he bonus paymens are conneced by he relaions (8) and (9). his siuaion relaes o he financial noion of durable oods. Durable oods mean ha a consumpion oday leads o uiliy in he fuure. One hen needs o specify how he uiliy of oday s consumpion is disribued over ime. his is precisely he siuaion in case of uiliy of bonus. he dividend paymen oday leads o uiliy of bonus paymens in he fuure. he way hese bonus paymens are disribued over ime is specified by he paymen funcion A. Uiliy opimizaion of durable oods has been sudied by Hindi and Huan (993). he resuls in Hindi and Huan (993) are no direcly applicable because of he presence of Z in our siuaion. Neverheless, we shall no o ino echnical deails here bu refer o Hindi and Huan (993) for he ideas i aes o wor ou hese deails. In he nex secion, we shall refer o Hindi and Huan (993) for explici soluions in some special cases where he resuls of Hindi and Huan (993) apply almos direcly. We define he opimal value funcion by (, x, ) sup E Zs () = x,,, ; # `a () s K() s ds (, p B)! A Zs ( ) + #! `a () s K( s ) dn () s D Z ( ) + ` A ( ) K( ) E he Bellman sysem for he opimal value funcion is now iven by a variaional inequaliy where one inequaliy conains an infimum over admissible invesmen conrols. I can be realized ha for all J, (, x, ) inf x (, x, ) ( r pls) x xx # 8 ^ + h (, x, ) p s x, p `a ()! m () R (, x, ) G, (, x, ) x (, x, ) $, ()!
17 ON MERON S PROBLEM FOR LIFE INSURERS 2 0 inf x (, x, ) ( r pls) x xx = 8 ^ + h (, x, ) p s x, p! `a () m () R (, x, ) (, x, ) G #! R S (, x, ) W x (, x, ), S () W X, x, D ^ h = ` A ( ), (2) where R (, x, ) = c `a () + (, x, ) (, x, ) m. he produc in (2) maes sure ha, a any poin in he sae space, a leas one of he wo inequaliies above is an equaliy. Alhouh we sip he deailed derivaion of he Bellman sysem by reference o Hindi and Huan (993), we come up wih a heurisic arumen for is consrucion. If B is no required o be absoluely coninuous beween he umps of Z, hen he Bellman sysem for he opimal value funcion is iven as he infimum over admissible conrols of parial differenial equaions for he opimal value funcion. I can hen be realized ha for all J, J () N (, x, ) inf x (, x, ) ( r pls) x b xx(,, ), () = 8 + x p s x p, b K O L P (, x, ) b a () () R (, x, ), A () `! m W! W X, x, D ^ h = ` A ( ). he infimum is searched for by differeniain wih respec o b, and he problem here, in opposiion o he siuaion in Secion 3, is ha he sysem is linear in b. Assume ha > 0 and > 0. hen if (, x, ) x (, x, ) = () $ 0, he coefficien in fron of b is posiive and infimum is obained by puin b =0.If x (, x, ) (, x, ) () # 0,
18 22 MOGENS SEFFENSEN he coefficien in fron of b is neaive and infimum is obained by puin b = unil, loosely speain, he inequaliy is an equaliy. his heurisic arumen also indicaes he opimal dividend policy. he insurance company should eep an eye on a cerain surplus boundary which in eneral depends on (, Z(), K()). If he surplus exceeds he boundary he company should immediaely brin he surplus bac o he boundary by payin ou dividends. An inriuin fac abou he opimal sraey is ha his dividend payou should happen so fas ha he surplus never becomes sricly larer han he boundary. his leads o a combinaion of a socalled local ime ype of dividend paymens beween he umps of Z and ump dividend paymens upon a ump of Z. Whenever he free reserve his he surplus boundary which depends on he sae of Z in beween umps of Z, i aes a local ime dividend paymen o eep he surplus below he boundary. When Z umps, he surplus boundary conneced o he presen sae of Z may chane such ha he surplus, if no conrolled, lies above he boundary. hen i aes a ump paymen o brin he surplus immediaely below he surplus boundary. Apar from he ump imes of Z a ump paymen may also ae place a ime 0. If he iniial free reserve x 0 lies above he boundary correspondin o he iniial sae 0 of Z, he opimally conrolled process should be brouh o his boundary by a lump sum paymen a ime EXPLICI RESULS ON BONUS OPIMIZAION In eneral, one mus approach he Bellman sysem by numerical mehods. However, for he life annuiy and for he erm insurance which are insurances on an infinie ime horizon, explici soluions can be obained in case of consan moraliy. In a survival model as illusraed in Fiure 2 a life annuiy is a rae of benefis b 0 > 0 unil deah whereas a erm insurance is a lump sum benefi b 0 > 0 upon deah. One nice hin abou his siuaion is ha he sysem reduces o one variaional equaliy since he opimal value funcion afer ransiion o he sae dead becomes zero. Furhermore, due o he infinie ime horizon and he consan moraliy we e rid of he ime dependence. 0 alive b 0 >0 m 0 b 0 >0 dead Fiure 2: Life annuiy and erm insurance
19 For a life annuiy he Bellman sysem reduces o 0 # inf6 (, x )( ^r+ pls) xh (, x ) x (, x ) $. p x xx ps (, x ) x ( a) + m(, x ), E In order o mae he sysem loo lie he sysem in Hindi and Huan (993), we chane he variable u = a, such ha x ON MERON S PROBLEM FOR LIFE INSURERS 23 0 # inf6 (,)( x u ^r+ pls) xh p u x xx ups (,) x x u + m (,), x u E (,) x u $ b (,), x u wih b = a. he boundary condiion for x =0 is ^ 0, uh= # 3 e s ( a) ds = m u 0 m his sysem is equal o he sysem obained in Hindi and Huan (993). We can herefore refer direcly hereo for he derivaion of he opimal sraey and us quoe heir resul wih our parameers. For he derivaion, Hindi and Huan (993) mae he followin assumpions m > r 2, l 2 b + l ( b ) > mr. Given hese assumpions he value funcion aes he form x (, ) = * 2 c c + c x, # d, 3 a c `a + x, $ d, where c, c 2, c 3 are consans which are deermined by he model parameers, and where c x a x a
20 24 MOGENS SEFFENSEN c d = l + r+ b + u 2 ( r + b) 2 2 ` = c c. b l + r+ b + u 4^r+ bh m, 2 ( r + b) he opimal sraey is iven by a consan invesmen in he risy asse similar o he classical soluion bu wih replaced by he consan c iven above by he model parameers, i.e. p =. c s l As for opimal dividend paymens, hese should eep he surplus below he boundary ad = c c. For a erm insurance, he Bellman sysem reduces o 0 # inf6 (, x )( ^ r+ p^ rh) xh p x xx ps mb (, x ) x ( a) (, x ) le, (, x ) x (, x ) $. In order o mae he sysem loo lie he sysem in Hindi and Huan (993), we chane he variable u= m, such ha he sysem can be wrien as above wih b = a Given his b, he opimal value funcion, opimal invesmen and opimal dividend sraey are as in he life annuiy case. m. ACKNOWLEDGMENS he auhor would lie o han he referees and Peer Holm Nielsen for heir commens on he firs version of his paper.
21 ON MERON S PROBLEM FOR LIFE INSURERS 25 REFERENCES BJÖRK,. (994) Soasis alyl och apialmarnadseori. Lecure noes, Royal Insiue of echnoloy, Socholm. CAIRNS, A.J.G. (2000) Some noes on he dynamics and opimal conrol of sochasic pension fund models in coninuous ime, ASIN Bullein 30(), 955. FLEMING, W.H. and RISHEL, R.W. (975) Deerminisic and Sochasic Opimal Conrol, Spriner erla. HANSEN, M.S. (200) Opimal porfolio policies and he bonus opion in a pension fund. echnical repor, Deparmen of Finance, Universiy of Odense. HINDI, A. and HUANG, C. (993) Opimal consumpion and porfolio rules wih durabiliy and local subsiuion. Economerica 6(), 852. HOEM, J.M. (969) Marov chain models in life insurance. Bläer der Deuschen Gesellschaf für ersicherunsmahemai 9, 907. JENSEN, B.A. and SØRENSEN, C. (200) Payin for minimum ineres rae uaranees: Who should compensae who? European Financial Manaemen 7, 832. KORN, R. and KRAF, H. (200) A sochasic conrol approach o porfolio problems wih sochasic ineres raes. SIAM Journal of Conrol and Opimizaion 40(4), KORN, R. and KREKEL M. (2002) Opimal porfolios wih fixed consumpion or income sreams. echnical repor, Fraunhofer IWM, Germany. MERON, R.C. (969) Lifeime porfolio selecion under uncerainy: he coninuous ime case. Review of Economics and Saisics 5, MERON, R.C. (97) Opimum consumpion and porfolio rules in a coninuous ime model. Journal of Economic heory 3, 37343; Erraum 6 (973); NORBERG, R. (999) A heory of bonus in life insurance. Finance and Sochasics 3(4), RICHARD, S.F. (975) Opimal consumpion, porfolio and life insurance rules for an uncerain lived individual in a coninuous ime model. Journal of Financial Economics 2, SEFFENSEN, M. (2000a) Coninen claims analysis in life and pension insurance. Proceedins AFIR 2000, SEFFENSEN, M. (2000b) A no arbirae approach o hiele s differenial equaion. Insurance: Mahemaics and Economics 27, SEFFENSEN, M. (200) On valuaion and conrol in life and pension insurance. Ph.D.hesis, Laboraory of Acuarial Mahemaics, Universiy of Copenhaen. MOGENS SEFFENSEN Laboraory of Acuarial Mahemaics Insiue of Mahemaical Sciences Universiy of Copenhaen Universiesparen 5 DK200 Copenhaen Ø, Denmar
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