ABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION

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1 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed by a diffusion process. Here we obain semiexplici soluions for quadraic opimizaion in he case where he paymen process is driven by a finie sae Markov chain model commonly used in life insurance mahemaics. The opimal paymens are affine in he surplus wih sae dependen coefficiens. Also consrains on paymens and surplus are sudied. KEYWORDS Markov chain, Regulaion of paymens, Linear regulaor, Bellman equaions, Consrains. 1. INTRODUCTION Sochasic conrol in life and pension insurance concenraed unil recenly on conrol of pension funds. Defined conribuion schemes and defined benefi schemes leave he benefis and he conribuions, respecively, as variables parly decided by he fund manager. In addiion, decisions on allocaion of asses may be inegraed in he problem. The insiuional condiions for pension funds may be raher involved. I is by no means clear how he obecives of he fund manager, he employer who pays (pars of) he premium, and he employed who receives he benefis, should be refleced in he obecive of he conrol problem. The usual framework of conrol of pension funds is he one given in probably he mos sudied conrol problem, he linear quadraic opimal conrol problem or he linear regulaor problem. The obec in his class of conrol problems is o conrol, a he same ime, he posiion of a cerain process and he force wih which his process is regulaed. The obec funcion punishes quadraic deviaions from some arges of he conrolled process and he conrolled rae of regulaion, respecively. This obec funcion is widely used parly because Asin Bullein 36 (1), doi: /AST by Asin Bullein. All righs reserved.

2 246 M. STEFFENSEN of is mahemaical racabiliy and parly because i makes sense in cerain engineering applicaions. In he conex of pension funding he regulaed process represens some noion of surplus whereas he regulaion iself represens paymens. These are premiums or benefis depending on he ype of scheme. Obviously, only paymens which are allowed o depend on he performance of he pension fund, are open for regulaion. Defined paymens like e.g. guaraneed benefis do no coun as decision variables. A sae of he ar exposiion of sochasic conrol of pension funds is given in Cairns (2000) which is parly a survey aricle gahering resuls of several auhors. The lieraure conains soluions o several varians of he problem. From he reference lis in Cairns (2000) we draw he reader s aenion o he conribuions by O Brien (1986), Dufresne (1989), and Haberman e al. (1994). The linear regulaor approach has been sandard in engineering and has found is applicaion in insurance hrough pension funding. However, i was no widely used as an approach o dynamic financial decision problems like e.g. consumpion-invesmen problems. There, he mos popular approach is he one aken by Meron (1969, 1971). This is based on opimal uiliy of fuure wealh or surplus, or, in case of inroducion of consumpion, uiliy of fuure consumpion raes. In Seffensen (2004), his uiliy opimizaion approach o financial decision making was applied o he problem of he life insurance company regulaing surplus by adusing regulaive paymens. There he se-up differs from he classical one in finance by formalizing he process of accumulaed consumpion as an insurance paymen sream. This sream includes paymen raes and lump sum paymens linked o he sae of an insurance policy (porfolio). Modelling he policy by a general finie sae Markov chain allows for various applicaions in various ypes of insurance and on various levels of individualizaion of policies in he porfolio. In he linear regulaor approach o sochasic conrol of pension funds, he paymens are usually modelled on an aggregae porfolio level by modelling he risk in paymens by a diffusion erm. Seffensen (2004) shows ha for power uiliy opimizaion, he srucure of he obec funcion is refleced in a saedependen value funcion and sae-dependen opimal paymens. A naural quesion is now: Taking he linear regulaor approach, will he srucure of he obec funcion again be refleced in a sae-dependen value funcion and sae-dependen opimal paymens? This aricle answers yes o his quesion. This answer is a par of he moivaion for his aricle. A srong conclusion is ha he insurance company can apply he quadraic opimizaion crieria for regulaion of paymens a any sub-porfolio level, even a he level of he individual, and mainain a simple regulaion rule. This is useful if he insurance company wishes o, or is forced o, accoun for or manage each sub-porfolio separaely. A weaker conclusion is ha if he insurance company applies he linear regulaion of paymens based on diffusion modelling, hen his regulaion can parly be argued for even a a sub-porfolio level. Apar from hese

3 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 247 immediae applicaions o individualizaion of accouns, he aricle is a conribuion o he general discussion on opimal design of paymens. One general drawback of quadraic approaches o invesmen problems is he couner-inuiive conclusion ha if he surplus is above he surplus arge, hen one should ry o loose money on he financial marke. This drawback appears in quadraic hedging approaches in finance as well as in linear regulaion of pension funds. One could hen choose o ac only when he surplus is below is arge. In Seffensen (2001), his problem is resolved by punishing deviaions of he deflaed surplus insead of he nominal surplus. In his aricle, we resolve he problem simply by disregarding he asse allocaion as a decision variable. Anyway, our obec is o generalize he modelling of paymens. In general, linear regulaor problems lose heir mahemaical racabiliy when inroducing consrains on he conrols or he conrolled processes. Some consrains can be allowed for while some oher consrains, wih clear applicaions, make he problem much harder. An imporan example is o consrain he regulaion of paymens o be o he policy holder s benefi. This means ha he fund manager or insurance company is allowed o pay ou posiive surplus only (by increasing benefis or decreasing premiums) and is no allowed o collec deficis. Seffensen (2001) obains resuls in his direcion and also shows ha a erminal expecaion condiion is easily aken care of by a Lagrange muliplier. We approach some racable consrains on he surplus and he paymens a he end of he aricle. The ouline of he aricle is as follows. In Secion 2 he dynamics of he surplus are inroduced, and in Secion 3 hese dynamics are moivaed by considering some noions of surplus inroduced previously in he lieraure. In Secion 4 he preferences are formalized in he obec funcion. Secion 5 conains he main resuls of he aricle. In Secion 6 and Secion 7 we show how o handle cerain consrains on he paymens and he surplus, respecively. A he end of Secions 3, 5, 6, and 7 we presen a cross-secional coninued example which presens he machinery a work. This example also serves as moivaion. 2. THE DYNAMICS OF THE SURPLUS We ake as given a probabiliy space (W,F,P). On he probabiliy space is defined a process Z=(Z()) 0 T aking values in a finie se J = {0,,J} of possible saes and saring in sae 0 a ime 0. We define he J-dimensional couning process N=(N k ) k! J by N k () =#{s s! (0,], Z(s )! k,z(s) =k}, couning he number of umps ino sae k unil ime. Assume ha here exis deerminisic funcions m k (),, k! J, such ha N k admis he sochasic inensiy process (m Z()k ()) 0 T for k! J, i.e.

4 248 M. STEFFENSEN k k M ] g= N - # m Z() s k () s ds consiues a maringale for k! J. Then Z is a Markov process. The reader should hink of Z as a policy sae of a life insurance conrac, see Hoem (1969) for a moivaion for he se-up. Based on he probabiliy heoreical framework above we now go direcly o he dynamics of he surplus. This will allow he reader o accep he dynamics and comprehend he conrol problem wihou necessarily having i grounded in he noions of surplus sudied by Norberg (1999) and Seffensen (2000). In he following secion we link he surplus dynamics inroduced below wih he noions of surplus sudied here. However, already now we need some clarificaion of erminology: Throughou he aricle, he conribuions are added o he surplus. Working wih e.g. he noion of surplus inroduced in Secion 3, hese conribuions sem from he realized paymens compared o wha is aken ino accoun in he liabiliy valuaion. This is in conras o he usual erminology of pension funding where he conribuions are usually he premium paymens. The dividends, which may in general be posiive or negaive, are subraced from he surplus. The dividends regulae he paymens ha are aken ino accoun in he liabiliy valuaion, and adap hese paymens o he developmen of he policy. We inroduce he nominal surplus process X given by 0 dx () = r()x ()d + dc () dd (), (1) X (0 ) =0, where r is a deerminisic ineres rae process and he conribuions C and he dividends D follow he dynamics!! Z] g Z - k k Z n dc = c ] g d + c ] g ] g ] g dn + DC de, k! J n!! 0, T+!! k k n dd = d d + d dn + D D de, k! J n!! 0, T+ where e n () =I ( n) indicaes ha n. Here he coefficiens of he conribuions c (), c k (), and DC are deerminisic funcions. The coefficien c () represens he rae of conribuions during soourn in sae a ime. The coefficien c k () represens he lump sum conribuion when umping from sae o sae k a ime. Finally, he coefficien DC () represens a lump sum conribuion a he deerminisic ime poin during soourn in sae. We allow for lump sum conribuions a deerminisic ime poins only a ime 0 and T. In (1) he iniial condiion X (0 ) = 0 defines he surplus us prior o ime 0 such ha he surplus a ime 0 can be expressed hrough he dynamics of C and D, namely, X (0) = DC (0) DD (0). The source of surplus conribuions

5 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 249 is a possible difference beween an anerior measure and a poserior measure of a se of assumed paymens. As fuure paymens urn ino pas paymens, he conribuions are realized. Depending on he measures hese conribuions may be sysemaic and/or purely erraic. In he nex secion we consider a surplus definiion wih a concree example of hese measures. The sochasic differenial equaion for he surplus (1) can be considered as a conrolled sochasic differenial equaion wih he conrol being he coefficiens in he dividend process D. The insurance company is allowed o choose hese coefficiens such ha here exiss a soluion o he sochasic differenial equaion (1). Then we say he dividend process D belongs o a se A. We have decoraed he nominal processes above wih a prime o ease he noaion for he corresponding discouned processes inroduced below. We namely, insead of working wih nominal conribuions, nominal dividends, and nominal surplus, work wih he discouned versions defined by # - r dc = e 0 dc ] g Z Z k k Z n = c ] g ] -g d + c dn + DC ] g de, # - r dd = e 0 dd ] g k k n = dd + d dn + DDde. #! k! J - r X ] g = e 0 X ] g.!! k! J n!! 0, T+! n!! 0, T+ Noe ha he deerminisic quaniies c 0 () = e c (), c k 0 () = e c k () ec. are hereby defined. Then, given a dividend process D! A, he conrolled sochasic differenial equaion describing he surplus is given by - # r dx D () =dc() dd(), (2) X D (0 ) =0. Noe ha, in conras o he usual siuaion in finance where he surplus (wealh) and he dividend paymens (consumpion) are consrained o be posiive, we impose no such consrains a his sage. This is one fundamenal difference beween he se-up in his aricle and he se-up in Seffensen (2004). There he surplus and he dividend paymens were consrained o be posiive such ha a cerain solvency consrain was fulfilled and such ha dividends were o he benefi of he policy holder. The absence of consrains on he surplus and he dividends (and he echnical valuaion basis) limis our resuls o pension funding. This is in conras o paricipaing life insurance where he insurance company would need a posiive surplus o fulfill cerain solvency requiremens and where he dividends are resriced o be o he benefi of he policy holder. In pracice here may - # r

6 250 M. STEFFENSEN also be consrains in pension funding. Though similar in srucure o paricipaing life insurance, hey will be less sric and we choose o disregard hese. See also Seffensen (2000) for a similar clear disincion beween paricipaing life insurance and pension funding. Noe ha depending on he final form of D, he dividends may boh change premiums and/or benefis. Thus, we do no specify wheher we have a so-called defined conribuions scheme, using his erm for he siuaion where he premiums are fixed and dividends affec he benefis only, or a so-called defined benefis scheme, using his erm for he siuaion where he benefis are fixed and dividends affec he premiums only. Below we see how hese differen cases are obained by an according specificaion of he preferences. 3. THE SURPLUS AND LIFE INSURANCE PAYMENT STREAMS In his secion we link he surplus dynamics inroduced in he previous secion wih some noions of surplus sudied in Norberg (1999) and Seffensen (2000). This is o be seen as examples of how he coefficiens of he conribuion process could be specified. One par of he paymen process of an insurance conrac is he guaraneed paymen process. Denoing by B() he accumulaed guaraneed paymens o he policy holder over [0, ], he guaraneed paymens are described by!! Z Z - k k Z n db] = b ] ] g g g d + b dn + DB ] g de. k! J n!! 0, T+ See Seffensen (2004) for an inerpreaion of he various elemens of B, noing ha here he process of guaraneed paymens B is denoed by Bˆ. The guaraneed paymen process B consiues ypically only one par of he oal paymen process. The insurance company adds o he guaraneed paymens an addiional dividend paymen process depending on he performance of he insurance policy or a se of policies. The insurance company decides on his addiional paymen process wihin any legislaive consrains here may be. The dividend process was inroduced in he previous secion. The guaraneed paymens B and he dividend paymens D consiue he oal paymen process experienced by he policy holder. Since differen noaion has been used for guaraneed paymens and dividends in he lieraure we presen here a small noaion ranslaor: Seffensen (2000, 2004) Norberg (1999), here Guaraneed paymens Bˆ B Dividends Bˆ D Toal paymens B=Bˆ +Bˆ B+D.

7 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 251 We inroduce a noion of surplus along he lines of Seffensen (2000). There, he surplus is defined for a given deerminisic valuaion basis, i.e. a se of discoun rae and inensiy processes (r *, m * ), by g =- # r s ^ g + gh - * Z] g, (3) 0 - X ] # e d B] s D] s V where he saewise reserves according o he valuaion basis (r *, m * ) are given by T * * # s # - r* V = E ; e db ] sg Z = E,! J. Here, E * denoes expecaion wih respec o he probabiliy measures under which N k admis he inensiy processes (m *Z( )k ()) 0 T. Definiion (3) corresponds o he surplus inroduced in Seffensen (2000) for he case where he paymens are invesed in a porfolio wih reurn rae on invesmen r. Definiion (3) follows he lines of he (individual) surplus inroduced in Norberg (1999) as well. However, in Norberg (1999) he dividends paid in he pas are no accouned for on he asse side and he valuaion basis (r *, m * ) is fixed o be he firs order valuaion basis (rˆ, mˆ) inroduced below. The insurance company lays down he guaraneed paymen process B on a so-called firs order valuaion basis (rˆ, mˆ). This means ha he guaraneed paymens are se o fulfill he so-called equivalence relaion V *0 (0 ) = 0 for (r *, m * ) = (rˆ, mˆ ). In paricipaing life insurance one would usually impose a consrain on he firs order basis such ha he firs order reserves V * () are on he safe side, i.e. larger han some corresponding marke values. However, in his aricle where we have pension funding in mind, such consrains are no needed. The surplus defined in (3) will follow he dynamics given by (1) wih he following specificaion of he conribuions and iniial surplus (see Seffensen (2000)), * * * k k * k c = ^ r - r hv + ` m - m R +! k k m R* ] kk ;! g, k c k =-R*, DC ] 0g 0 0 =-DB ] 0g -V * ] 0g, DC ] Tg = 0,! kk ;! (4) where R *k () =b k ()+V *k () V * ().

8 252 M. STEFFENSEN Noe ha wih his specificaion of surplus conribuions we can wrie he dynamics of he surplus conribuions for >0 as ] g ^ ] g ] gh ] g! ` ] g ] g ] g * Z Z k Z k Z k dc = r - r V * ^ h d + m* ^ h ^ h - m R* ^ h d kk ;! Z^h Z - k k -! R* ^ h dm, k! J (5) hereby decomposing he conribuion ino a sysemaic incremen and a maringale incremen. For comparison wih Norberg (1999), noe ha he uses he leer C o denoe he process which conains only he sysemaic par of C. We emphasize ha he coefficiens of he conribuion process specified in (4) is us an example coming ou of defining he surplus as in (3). One can easily imagine oher specificaions of he surplus conribuions in he previous secion. One obvious choice is inspired direcly from he previous paragraph: Disregard he maringale erm of (5) and define insead c ^ r r hv ` R, * k k k = - * + m* - m * kk ;! k c ] g = 0. The corresponding surplus could naurally be called he sysemaic surplus. Example 1. We consider he survival model wih wo saes corresponding o a policy holder being alive (sae 0) or dead (sae 1). For he sake of simpliciy, we consider a T-year endowmen insurance. In his case we can simplify he noaion: N / N 1, m / m 01, and for all oher quaniies and funcions he specificaion of sae 0 is skipped, i.e. b / b 0 (he negaive premium rae), b 1 / b 01 (he life insurance sum), DB / DB 0 (he endowmen sum), c () / c 0 (), c 1 () =c 01 () ec. Assuming ha we wish o conrol he sysemaic surplus, we have ha where! c () = ((r r * )V * ()+(m * () m()) (b 1 V * ())), c 1 () =0, T s r* * T - # + m m* 1 - # r * + m # ` * D. V* = e s b + b ds + e ] g ] g B If, in paricular he echnical basis and he marke basis coincide, we ge ha c () = 0, and hus C() = 0.

9 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS PREFERENCES AND UTILITY PROCESSES Our scope is o search for opimal dividend paymen processes. We are going o formulae our preferences over paymen processes in erms of a so-called uiliy process U. Denoing by U() he accumulaed uiliies over [0, ], he uiliy process is described by! -! k! J n!! 0, T+ Z Z k k Z n du] = u ] ] g g g d + u dn + DU ] g de. In he uiliy process, u () specifies he rae of uiliy in sae, each u k () specifies a lump sum uiliy upon ransiions from sae o sae k, and DU () specifies a lump sum uiliy during soourns in sae. Below we specify how he paymen process D affecs U. Our erminal goal is o find he paymen process D among a se of allowable paymen processes ha maximizes he expeced oal uiliy, E; # T du]g s E. (6) 0- Noe ha by he inroducion of he uiliy process U, we can wrie he expeced oal uiliy in a similar way as we usually wrie he expeced fuure paymens when defining he saewise reserves. In ha respec we can speak of (6) as he uiliy reserve a ime 0. A uiliy process of he presen form was inroduced in Seffensen (2004). There, u (), u k (), and DU () were defined as cerain sae-dependen power funcions of d (), d k (), and DD (). This se of preferences was inspired by he classical Meron problem of opimal consumpion and invesmen. In his aricle he preferences are inspired by he classical pension fund opimizaion problem. Thus, insead we work wih a quadraic dis-uiliy funcion ha punishes quadraic deviaions of paymens from he paymens in an arificial paymen arge process combined wih quadraic deviaions of surplus from zero. For specificaion of he dis-uiliy semming from he paymen process D and he surplus X, we inroduce hree furher processes A, P, and Q. These processes are called he paymen arge process, he paymen weigh process, and he surplus weigh process, respecively. The processes A, P, and Q are given by!! Z Z - k k Z n da] = a ] ] g g g d + a dn + DA ] g de, k! J n!! 0, T+!! Z Z - k k Z n dp] = p ] ] g g g d + p dn + DP ] g de, k! J n!! 0, T+!! Z Z - k k Z n dq] = q ] ] g g g d + q dn + DQ ] g de, k! J n!! 0, T+ and we assume ha P and Q are increasing, i.e. all coefficiens are posiive.

10 254 M. STEFFENSEN The preferences over he se of paymens are now given by he following disuiliy funcions u () = p ()(d() a ()) 2 + q ()X() 2, u k () = p k ()(d k () a k ()) 2 + q k ()X() 2, DU () =DP ()(DD() DA ()) 2 + DQ ()X() 2. The coefficien u () represens he rae of dis-uiliies during soourn in sae a ime. This rae of dis-uiliies sems from a deviaion of d() from a () weighed wih p () and from a deviaion of X() from 0 weighed wih q (). The coefficien u k () represens he lump sum dis-uiliy when umping from sae o sae k a ime. This lump sum dis-uiliy sems from a deviaion of d k () from a k () weighed wih p k () and from a deviaion of X() from 0 weighed wih q k (). Finally, he coefficien DU () represens a lump sum dis-uiliy a a deerminisic poin in ime during soourn in sae. This lump sum dis-uiliy sems from a deviaion of DD() from DA () weighed wih DP () and from a deviaion of X() from 0 weighed wih DQ (). We allow for lump sum dis-uiliy a he deerminisic ime poins 0 and T only. One may find i odd o add lump sum dis-uiliy corresponding o q k and DQ. Acually, hese are also a burden from a mahemaical poin of view as we see below. However, for he sake of symmery, we keep hem as far as we can. Noe ha he paymen processes A, P, and Q are no in general real paymen processes experienced by he policy holder or he insurance company. Their only role is o specify he preferences over paymen sreams D. Thus, we could simply have inroduced all he coefficiens of A, P, and Q direcly as sae-dependen funcions. However, for he comprehension of he srucure of hese coefficiens i is beneficial o have hese arificial paymen processes in mind. Furhermore, whereas P and Q really have no much o do wih paymens, he process A may be equal or relaed o a real paymen process. One may sugges he inroducion of a surplus arge process Y, say, wih dynamics given by!! Z Z - k k Z n dy] = y ] ] g g g d + y dn + DY ] g de, k! J n!! 0, T+ and replace X() 2 by (X() Y()) 2 in he coefficiens of he uiliy process. This formulaion, however, is covered by he consrucion above by simply redefining he surplus and he conribuion processes by Xˆ = X Y, Ĉ = C Y. We end his secion wih a commen on he idea of penalizing deviaions of X from 0. A he end of he previous secion we emphasized ha one could choose

11 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 255 o work wih several noions of surplus. Obviously, conrolling differen noions of surplus would have o be moivaed in differen ways. Consider he individual surplus inroduced in he previous secion, conaining boh a sysemaic and an erraic erm. A policy holder conribues o his surplus in wo differen ways. Firsly, he conribues o he surplus sysemaically according o he difference beween he echnical assumpions and realized moraliy and reurn. Secondly, he conribues erraically according o his own course of life. If we conrol he individual surplus, we hink ha his dividends should be affeced, no only by he sysemaic conribuions, bu also by his course of life. Conrolling he individual surplus reduces he risk of he oher paries involved i.e. he insurance company and/or he employers, compared o he alernaive below. Alernaively, consider he sysemaic surplus explained a he end of he previous secion. This surplus simply disregards he erraic erm of he individual surplus. Now, he policy holder does no conribue o he surplus erraically by his course of life. Only sysemaic conribuions are accouned for and, hus, disribued in erms of dividends. This consrucion leaves all he unsysemaic risk o he oher paries involved. One canno say ha one consrucion is righ and he oher is wrong. They are us differen consrucions based on differen ideas wih differen levels of insurance in he sense of averaging away he risk of he policy holder. As he preferences over he surplus should be inerpreed differenly for differen noions of surplus, so should also possible consrains on X. We reurn o consrains on X in Secion MARKOV CHAIN PENSION FUND OPTIMIZATION We define he opimal value funcion V by T x,, # D! A V ], xg = inf E ; du ] sge, (7) where E,x, denoes condiional expecaion given ha X() =xand Z() =. We can speak of V (,x) as he saewise opimal value funcion. A fundamenal sysem of differenial equaions in conrol heory is he Bellman sysem for he opimal value funcion. The Bellman sysem is here given 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 (2000) for a derivaion of parial differenial equaions for relevan condiional expeced values. I can be realized ha for all! J, V ], xg = sup :-V ], xg` c -d - p ` d-a - q x k dd,, k! - x! m k; k! k V k R ], xg W, W X 2 2 (8)

12 256 M. STEFFENSEN and for!{0,t}, 0 = sup 8- DR ], xgb, (9) DD where subscrip denoes he parial derivaive and where R k (,x) = p k ()(d k a k ()) 2 + q k ()(x + c k () d k ) 2 + V k (,x + c k () d k ) V (,x), DR (,x) = DP ()(DD DA ()) 2 + DQ ()(x + DC () DD) 2 + V (, x + DC () DD) V (, x). The differenial equaliy in (8) maximizes he parial derivaive in a any poin in he sae space and he equaliy in (9) maximizes he ump in he value funcion a ime 0 and a ime T. Togeher he equaliies minimize he condiional expeced value in (7) and hence characerize he value funcion. I should be emphasized ha he Bellman sysem is acually a sysem of J differenial equaions wih J condiions a ime 0 and a ime T. The Bellman sysem conains he erms presen in he Bellman equaion for he classical pension fund opimizaion problem and an addiional erm semming from he uncerainy in he process Z. The sysem of J differenial equaions is comparable wih he classical socalled Thiele s differenial equaion for he sae wise reserves, see e.g. Seffensen (2000). This moivaes parly he noaion V and R k : The saewise reserve is usually denoed by V and he risk sum in Thiele s differenial equaion is usually denoed by R k. Here, he conens of hese erms is differen bu he srucure is parly he same. The erm DR (, x) has similariies wih a risk sum and is used o specify he developmen of he uiliy reserve a deerminisic poins in ime wih a lump sum dis-uiliy. For a given lump sum DD, he relaion DR (, x) = 0 updaes he uiliy reserve a such a poin in ime. E.g. a ime T, since V (T, x) = 0 for all x, he relaion gives he erminal condiion V (T, x) = DP (T)(DD DA (T)) 2 + DQ (T)(x + DC (T) DD) 2. The Bellman equaion 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 works wih he verificaion resul saing ha a sufficienly nice funcion solving 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. One us needs a so-called viscosiy soluion wih relaxed requiremens on differeniabiliy which will hen coincide wih he opimal value funcion.

13 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 257 We now guess a soluion o he Bellman sysem based on a separaion of x in he same way as in he classical case. We ry he soluion V (, x) =f ()(x g ()) 2 + h (). This form leads o he following lis of parial derivaives, V (, x) = f ()(x g ()) 2 2f ()g ()(x g ()) + h (), V x (, x)= 2f ()(x g ()). A candidae for he opimal D is found by solving (8) for he suprema wih respec o he decision variables in D, for, k! J, k!, 0=2f ()(x g ()) 2p ()(d a ()), 0=2p k ()(d k a k ()) 2q k ()(x + c k () d k ) 2f k ()(x + c k () d k g k ()), 0=2DP ()(DD DA ()) 2DQ ()(x + DC () DD) 2f ()(x + DC () DD g ()). This leads o he candidaes, abbreviaing S k () =p k () +q k () +f k () and DS () = DP () + DQ () +f (), d (,x) =a f ()+ p (x g ()), ] g d k k p (,x) = k a k k q ()+ k (x + c k k f ()) + S S S k (x + c k () g k ()), (10) ] g DD DP (,x) = DA DQ ()+ (x + DC f ()) + (x + DC () g ()), DS DS DS where he noaion is eviden and exposes d, d k, DD as funcions of (,,x). The opimal conrol variables in (10) can be inerpreed as follows: d () is equal o is arge a () adused wih a correcion erm which akes ino accoun he fuure. X() is correced owards g f () (), and he raio deermines p () he weigh of his correcion. If p () is large (relaive o f ()), here is a high consideraion for he presen preference o have d close o a, and vice versa. d k () is a weighed average of hree consideraions. Firsly, d k () is preferred o be close o a k () and his is weighed wih p k (). Secondly, afer a possible ump from o k, X( ) +c k () (he posiion afer he ump bu before conrolling) is preferred o be close o 0 and his is weighed wih q k (). Thirdly, also he fuure afer he ump mus be aken ino consideraion and for his

14 258 M. STEFFENSEN X( ) +c k () should be correced owards g k (), and f k () deermines he weigh of his correcion. DD () is a weighed average somewha similar o d k (). Firsly, DD () is preferred o be close o DA () and his is weighed wih DP (). Secondly, a ime,x( ) + DC () (he posiion a ime bu before conrolling) is preferred o be close o 0 and his is weighed wih DQ (). Thirdly, also he fuure mus be aken ino consideraion and for his X( ) + DC () should be correced owards g (), and f () deermines he weigh of his correcion. We see ha boh d, d k, and DD are linear funcions of he surplus as he conrollable parameer is i in he classical case. However, he coefficiens involve he paymen processes A and C and he funcions f and g. Insering he opimal candidae in he Bellman sysem gives, afer several rearrangemens, he following parial differenial equaions for f () and g (), 2 f k f; k f = - q -! m R, p k; k! f; 0 = DR,!! 0, T+, ; g * r g () c a m k g k ] g = ] g + ] g - ] g - R, k; k! (11) g; 0 = DR,!! 0, T+, (12)! where r * ] g = q, f and he risk sums in he differenial equaions for f and g are given by R DR f; k f; k p k k = k ` q + f - f, S DP = `DQ + f - f ( -), DS (13) R g ; k = = k p J k k ] g q + f k k k a c g S K ` ] g - ] g - ] g + ] g f L f; k R + f k k `a - c - g f f; k R + f + k k q + f k f k g, f f f N ] g k g O P k

15 DR g; QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 259 DP J ] g DQ f = ] g + ] g ] g A C g ( ) DS K `D ] g - D ] g ] g f ( -) L f; DR + f ( -) = `DA - DC - g ( -) f ( -) f; DR + f ( -) f + g DQ + f f ( -) f N ] g g f ( -) O P f = DA - DC - g ( - ) + g. (14) DQ + f The las equaliy follows from (11). The sysem of differenial equaions for f is a J-dimensional Riccai equaion. For a given erminal condiion, his has a unique posiive soluion under cerain assumpions on he coefficiens. We see ha he erminal condiion for f mus be f (T) = 0, whereas we ake he erminal condiion for g o be g (T) = 0 by convenion. We emphasize ha we could inroduce any erminal condiion for g. Following (11), (12), (13), and (14), he erminal condiions f (T) = 0 and g (T) = 0 lead o DP ( T) DQ ( T) f ( T- ) =, DP ( T) + DQ ( T) g ( T- ) = DA ( T) - DC ( T). Given f, he sysem of differenial equaions for g has similariies wih Thiele s differenial equaion, see Seffensen (2000). However, he quaniy R g;k is no a risk sum in he same sense as in Thiele s differenial equaion and DR g; () does no lead o a usual adusmen of he condiional expeced value for a lump sum paymen a a deerminisic poin in ime. Neverheless, i is possible o derive a sochasic represenaion formula for he soluion o he differenial equaion in he case where q k () = DQ () = 0. This is done in he res of his secion. In he case q k () = DQ () = 0, R f;k (), DR f; (), R g;k () and DR g;k () above simplify o f; k k R = f - f, f; DR = f - f ( -), k k g; k p f k k k R = k k `a - c - g + g, p + f f g; DR = DA - DC - g ( - ) + g. One realizes hen ha g can be wrien as a condiional expecaion of he presen value of he paymen process A-C under he paricular sae-dependen discoun rae r * and under a paricular measure P * defined below, i.e.

16 260 M. STEFFENSEN T s * Z u * - # r u du ] =, # ] ] g g g ] - g] g, g E ; e d A C s E (15) where E * denoes expecaion wih respec o he measure P *. Define he likelihood process L and he corresponding ump kernel by! Z ( -) k k Zk () dl = L ( -) g `dn - m d, k! J k k k p f g = k k - 1,! k. p + f f Then we can change measure from P o P * by he definiion L T = dp * dp, and i follows from Girsanov s heorems (see e.g. Börk (2004)) ha N k under P * admis he inensiy process m * Z ( - ) k ] Z 1 g ( - ) k Z m ( - ) k g = ` + ] g. We can finally wrie g * r g c a * ; m k g * k ] g = ] g ] g + ] g - ] g - R, k; k! g ( T- ) = DA ] Tg - DC ] Tg, g; * k k k k R = a - c + g - g,! which is precisely a version of Thiele s differenial equaion for a reserve defined by (15). Example 2. We now coninue Example 1. We are now ineresed in paying ou dividends opimally and need for his purpose o specify he hree processes A, P, and Q. We ake he arge process A o be 0. This means ha he policy holder has a arge process for his oal paymens B + D equal o B. For he weigh process P, we ake he coefficiens corresponding o sae 1 (dead) o be posiive. We skip he specificaion of sae 0 for he oher elemens of P and ake p / p 0, p 1 / p 01, and DP / DP 0 o be consan. For he weigh process Q we ake he coefficiens corresponding o sae 1 o be zero. Furhermore, we ake q / q 0,q 1 / q 01, and DQ / DQ 0 o be consan. Plugging in all hese coefficiens in he differenial equaions for f and g, we see ha he differenial equaions corresponding o sae 1 are solved by f 1 =g 1 = 0. Hereafer, he differenial equaions and erminal condiions for f / f 0 and g / g 0 are reduced o

17 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 261 ] g J f p K p + q L DPDQ ft ] - g =, DP + DQ q g () = d + m* g + c, f ] gn ] g ] g ] g gt ] g = f pq ] g = - m K f] go - q, N O P The Riccai equaion for f mus be solved numerically, while he soluion for g is given by T s q # # - + m f g ]g=- e csds (). We can now wrie down he opimal dividend paymens in erms of f and g, d ] x, g = f ] p g ^x - g] gh, 1 d () x = 1 q 1 1x, p + q 1 DQ DD () x = x. DP + DQ The lump sum dividend paymens upon deah or erminaion whaever occurs firs, have he same srucure. I is easy o verify ha hese dividend paymens simply minimize he final lump sum penalies given by p 1 (d 1 ) 2 + q 1 (x d 1 ) 2 and DPDD 2 + DQ (x DD) 2, respecively. The raio q 1 /(p 1 + q 1 ) deermines he preferences beween wo exreme siuaions: Eiher one could ge d 1 =0by having no preferences for X i.e. q 1 =0, or one could obain X =0afer deah or erminaion whaever occurs firs, by having no preferences for d 1, i.e. p 1 =0. A similar inerpreaion goes for he raio DQ/(DP + DQ). 6. CONSTRAINED PAYMENTS In secion 5 we had no consrains on he dividend paymens. In his secion we show how i is possible o solve problems where cerain paymens are consrained o be equal o cerain values. One can hink of several examples where such consrains are relevan. Consider he opimizaion problem under he consrain ha for s u d() =â Z() ().

18 262 M. STEFFENSEN Under his consrain, we have ha he erm in du() involving d() for s u is given by p Z() ()(â Z() () a Z() ()) 2 d. For a given weigh p Z() () and a given arge a Z() (), his is a deerminisic funcion of (,Z()) and herefore plays no role for he decision of he opimal sraegy. We can herefore choose p Z() () and a Z() () freely, and in paricular search for coefficiens such ha he unconsrained problem has a soluion where he consrain is fulfilled for he opimal dividend process. If we find such, we have hen a soluion for our consrained problem. From he opimal dividends in he unconsrained problem (10) we see ha if we for s u choose p Z() () =n, a Z() () = â Z() (), (16) and le n " 3, hen we ge in he limi ha for s u (and all oher weighs and arges fixed), and, hus, f p Z () Z () = 0, d() =â Z() (), such ha he consrain is fulfilled. The arificial se of weigh and arge funcions given in (16) has he obvious inerpreaion ha, in he limi, deviaions from he arge given by he consrain are punished infiniesimally severely. Obviously, o avoid an infinie value funcion in he limi, he consrain is herefore fulfilled by he opimal conrol. For consrains on lump sum paymens upon ransiion and a deerminisic poins in ime he argumen goes in almos he same way. Here we go hrough he argumen for a consrained lump sum paymen upon ransiion. Consider he opimizaion problem under he consrain ha d k () =â Z( )k (). Under his consrain, we have ha he erm in du() involving d k () is given by! p Z( )k () (â Z( )k () a Z( )k ()) 2 dn k (). k! J For a given weigh p Z( )k () and a given arge a Z( )k (), his is a deerminisic funcion of (,Z( )) and herefore plays no role for he decision of he opimal

19 QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS 263 sraegy. We can herefore choose p Z( )k () and a Z( )k () freely. If we find coefficiens such ha he unconsrained problem has a soluion where he consrain is fulfilled for he opimal dividend proces, hen we have a soluion for our consrained problem. From he opimal dividends in he unconsrained problem (10) we see ha if we choose p Z( )k () =n, a Z( )k () =â Z( )k (), (17) and le n " 3, hen we ge in he limi ha (for all oher weighs and arges fixed) p S k k k k q f = 1, k = 0, k = 0, S S and, hus, d k () =â Z( )k (), such ha he consrain is fulfilled. Again, he inerpreaion of he weigh and arge funcions given in (17) is ha deviaions from he arge are punished infiniesimally severely in he limi. To avoid an infinie value funcion, he consrain is herefore fulfilled by he opimal conrol. Example 3. We now coninue Example 2. Consider he case where he dividend rae is consrained o be zero. This case could appropriaely be spoken of as Defined Conribuion since he premium rae is no regulaed hrough dividends bu benefis are. This is handled by considering he conrols for p " 3. The opimal dividends in he limi urn ino d ] x, g = 0, 1 q d ] x, g = 1 1 x, p + q 1 DQ DD ] T, xg = x. DP + DQ Alernaively, consider he case where lump sum dividends are consrain o be zero. This case could appropriaely be spoken of as Defined Benefis since he benefis are no regulaed hrough dividends bu he premium rae is. This is handled by considering he conrols for p 1 " 3 and DP " 3. Then he opimal dividends in he limi urn ino x, f ] p g d ] g = ^x - g] gh, 1 d ] x, g = 0, 1 DD ] T, xg = 0. 1

20 264 M. STEFFENSEN The differenial equaion for f becomes f] g f p ft ] - g = DQ. 2 1 ] g = - m`q - f] g - q, 7. CONSTRAINED SURPLUS In his secion we explain how i is possible o solve problems where he erminal surplus is consrained o equal zero. If X is he sysemaic surplus, his relaes o he individual fairness crierion as described by Norberg (1999) since his consrain sees o i ha he surplus is empied compleely a erminaion for a given insurance conrac or porfolio of conracs. Thus, we consider he opimizaion problem under he consrain ha X(T) =0. Under his consrain, we have ha he erm in du(t) involving X(T) equals 0. Therefore, for a given weigh DQ (T), his plays no role for he decision of he opimal sraegy. We can herefore choose DQ (T) freely. If we find a coefficiens such ha he unconsrained problem has a soluion where he consrain is fulfilled for he opimal dividend process, hen we have a soluion for our consrained problem. From he opimal dividends in he unconsrained problem (10) we see ha if we choose DQ (T) =n, (18) and le n " 3, hen we ge in he limi ha (for all oher weighs and arges fixed) and, hus, leading o DP ( T) DQ ( T) f ( T) = 0, = 1, = 0, DS ( T) DS ( T) DS ( T) DD (T) =X(T ) + DC (T), X(T) =X(T ) + DC Z(T) (T) DD Z(T) (T) =0. which exacly obeys our consrain. The arificial weigh given in (18) has he obvious inerpreaion ha in he limi, deviaion from zero is punished infiniesimally severely. Obviously, o avoid an infinie value funcion in he limi, he consrain is hen fulfilled.

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