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


 Judith Golden
 1 years ago
 Views:
Transcription
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. consumpioninvesmen 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 seup 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 saedependen opimal paymens. A naural quesion is now: Taking he linear regulaor approach, will he srucure of he obec funcion again be refleced in a saedependen value funcion and saedependen 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 subporfolio 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 subporfolio 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 subporfolio 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 counerinuiive 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 crosssecional 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 Jdimensional 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 seup. 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 seup in his aricle and he seup 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 socalled defined conribuions scheme, using his erm for he siuaion where he premiums are fixed and dividends affec he benefis only, or a socalled 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 socalled firs order valuaion basis (rˆ, mˆ). This means ha he guaraneed paymens are se o fulfill he socalled 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 Tyear 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 socalled 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 saedependen 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 disuiliy 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 disuiliy 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 disuiliies during soourn in sae a ime. This rae of disuiliies 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 disuiliy when umping from sae o sae k a ime. This lump sum disuiliy 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 disuiliy a a deerminisic poin in ime during soourn in sae. This lump sum disuiliy 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 disuiliy a he deerminisic ime poins 0 and T only. One may find i odd o add lump sum disuiliy 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 saedependen 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 ` da  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 disuiliy. 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 socalled 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 Jdimensional 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 AC under he paricular saedependen 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.
Optimal Consumption and Insurance: A ContinuousTime Markov Chain Approach
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
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationABSTRACT KEYWORDS 1. INTRODUCTION
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
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationThis page intentionally left blank
This page inenionally lef blank MarkeValuaion Mehods in Life and Pension Insurance In classical life insurance mahemaics, he obligaions of he insurance company owards he policy holders were calculaed
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationMarkov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension
Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationOptimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach
28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 1113, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A
More informationOn Valuation and Control in Life and Pension Insurance. Mogens Steffensen
On Valuaion and Conrol in Life and Pension Insurance Mogens Seffensen Supervisor: Ragnar Norberg Cosupervisor: Chrisian Hipp Thesis submied for he Ph.D. degree Laboraory of Acuarial Mahemaics Insiue for
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationOptimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime
Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics,
More informationA martingale approach applied to the management of life insurances.
A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 LouvainLaNeuve, Belgium.
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationA MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.
A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B1348, LouvainLaNeuve
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationLIFE INSURANCE MATHEMATICS 2002
LIFE INSURANCE MATHEMATICS 22 Ragnar Norberg London School of Economics Absrac Since he pioneering days of Black, Meron and Scholes financial mahemaics has developed rapidly ino a flourishing area of science.
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationWHAT ARE OPTION CONTRACTS?
WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationCLASSIFICATION OF REINSURANCE IN LIFE INSURANCE
CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics FriedrichAlexanderUniversiy
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationMULTIPERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN
Journal of he Operaions Research Sociey of Japan 27, Vol. 5, No. 4, 463487 MULTIPERIOD OPTIMIZATION MODEL FOR A HOUSEHOLD, AND OPTIMAL INSURANCE DESIGN Norio Hibiki Keio Universiy (Received Ocober 17,
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationPREMIUM INDEXING IN LIFELONG HEALTH INSURANCE
Far Eas Journal of Mahemaical Sciences (FJMS 203 Pushpa Publishing House, Allahabad, India Published Online: Sepember 203 Available online a hp://pphm.com/ournals/fms.hm Special Volume 203, Par IV, Pages
More informationOn the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes
On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationThe Uncertain Mortality Intensity Framework: Pricing and Hedging UnitLinked Life Insurance Contracts
The Uncerain Moraliy Inensiy Framework: Pricing and Hedging UniLinked Life Insurance Conracs Jing Li Alexander Szimayer Bonn Graduae School of Economics School of Economics Universiy of Bonn Universiy
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationFair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration
Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichlexanderUniversiy
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationResearch. Michigan. Center. Retirement. Behavioral Effects of Social Security Policies on Benefit Claiming, Retirement and Saving.
Michigan Universiy of Reiremen Research Cener Working Paper WP 2012263 Behavioral Effecs of Social Securiy Policies on Benefi Claiming, Reiremen and Saving Alan L. Gusman and Thomas L. Seinmeier M R R
More information