A combined sochasic programming and opimal conrol approach o personal finance and pensions Agnieszka Karolina Konicz agko@du.dk Kourosh Marjani Rasmussen kmra@du.dk David Pisinger dapi@du.dk Mogens Seffensen mogens@mah.ku.dk April 30, 2014 Absrac We combine a dynamic programming approach (sochasic opimal conrol) wih a muli-sage sochasic programming approach (MSP) in order o solve various problems in personal finance and pensions. Sochasic opimal conrol produces an opimal policy ha is easy o undersand and implemen. However, explici soluion may no exis, especially when we wan o deal wih consrains, such as limis on porfolio composiion, limis on he sum insured, an inclusion of ransacion coss or axes on capial gains, which are imporan issues regularly menioned in he lieraure. Boh opimizaion mehods are inegraed ino one MSP formulaion, and in a shor compuaional ime produce a soluion, which akes ino accoun he enire lifeime of an individual wih a focus on he pracical consrains during he firs years of a conrac. Two applicaions are considered: (A) opimal invesmen, consumpion and sum insured for an individual maximizing he expeced uiliy of consumpion and beques, and (B) opimal invesmen for a pension saver who wishes o maximize he expeced uiliy of reiremen benefis. Numerical resuls show ha among he considered pracical consrains, he presence of axes affecs he opimal conrols he mos. Furhermore, he individual s preferences, such as impaience level and risk aversion, have even a higher impac on he conrolled processes han he axes on capial gains. 1 Inroducion The purpose of his paper is o formulae and solve wo opimizaion problems relevan for personal finance and pensions in a muli-period sochasic framework. Problem (A) invesigaes he opimal invesmen, consumpion and sum insured for an individual maximizing he uiliy of consumpion and a erminal uiliy of leaving a posiive amoun of money upon deah over an uncerain lifeime. Problem (B) is relevan for a pension produc design in a defined conribuion plan. A pension saver wishes o maximize he uiliy of he fuure reiremen benefis by opimally conrolling he invesmen boh before and afer reiremen as well as he level of he benefis ha will be received afer Corresponding auhor a: DTU Managemen Engineering, Managemen Science, Technical Universiy of Denmark, Produkionsorve 426, 2800 Kgs. Lyngby, Denmark. Tel.: +45 4525 3109; Fax: +45 4525 3435; email: agko@du.dk 1 Elecronic copy available a: hp://ssrn.com/absrac=2432869
reiremen. The soluion deermines a pension produc ha reflecs he individual s risk preferences and impaience level. A classical approach for consumpion-invesmen opimizaion problems is sochasic opimal conrol, also referred o coninuous ime and sae dependen dynamic programming. Sochasic opimal conrol is common in financial and acuarial lieraure and focuses on deriving he explici (analyical) soluions o a given model. Probably he mos influenial papers in his field have been wrien by Meron (1969, 1971) who defined he original problem of opimizing uiliy of consumpion and erminal wealh over a fixed ime horizon for an invesor. This work has inspired many researchers who eiher expanded he original model or invesigaed differen objecive funcions, by inroducing for insance he lifeime uncerainy, sum insured and he labor income, Richard (1975), sochasic ineres rae, Munk and Sørensen (2004), salary uncerainy, Cairns e al. (2006), muliperson household, Bruhn and Seffensen (2011), borrowing consrains, Byung and Yong (2011), or consan linear axaion, Bruhn (2013). Applicaions wihin defined conribuion pension scheme wih a focus on he invesmen sraegy eiher during he accumulaion or pos reiremen phase (decumulaion) ogeher wih he opimal ime of annuiizaion, have been considered by Milevsky and Young (2007) and Gerrard e al. (2004, 2012). The main advanage of a sochasic opimal conrol approach is he analyical form of he opimal soluion, which is easy o undersand and implemen. However, he main drawback of his approach is ha he explici soluion in many cases does no exis, especially, when dealing wih more realisic consrains, such as limis on porfolio composiion, limis on he sum insured, an inclusion of ransacion coss or axes on capial gains, which are imporan issues regularly menioned in he lieraure. Therefore, o overcome hese limiaions, researchers mus apply numerical mehods. To focus on he pracicaliies of he problem, we choose in his paper a muli-sage sochasic programming (MSP) approach, which is broadly applied in operaions research. MSP is a general purpose framework for modelling opimizaion problems where an objecive funcion can ake a variey of forms. I is based on he scenario rees ha represen he range of possible oucomes for he uncerainies. Raher han finding a generic opimal policy, he opimal soluion is compued numerically a each node in he scenario ree, for he specified decision variables. MSP can easily address realisic consideraions and consrains, as long as hey have an algebraic form. A sochasic programming approach is more common wihin porfolio managemen and asseliabiliy managemen han wihin he areas concerning he individual invesors such as opimal invesmen-consumpion decisions. See e.g. Mulvey e al. (2006, 2007) who argue ha muli-period invesmen models combined wih Mone Carlo simulaion can address imporan consideraions for long-erm invesors, and Fersl and Weissenseiner (2010) who presen a sochasic linear programming model for opimal asse allocaion in a siuaion where a financial company wishes o minimize he condiional value a risk. Neverheless, applying sochasic programming or oher numerical approaches o find he opimal decisions from he individual invesors poin of view can also be found in he lieraure. See e.g. Horneff e al. (2008) considering an individual characerized by Epsein-Zin 2 Elecronic copy available a: hp://ssrn.com/absrac=2432869
preferences, Cai and Ge (2012) invesigaing he asse allocaion for an invesor wih a loss aversion objecive, a predeermined objecive and a greedy objecive, Consigli e al. (2012) comparing invesmen opporuniies offered by radiional pension producs and uni-linked conracs wih variable life annuiies, and Blake e al. (2013) deriving he opimal invesmen for a loss averse pension saver wih an inerim and final arge. However, he main drawback of a muli-sage sochasic programming approach is ha he problem size grows quickly as a funcion of he number of periods and scenarios. In paricular, aking ino consideraion he enire lifeime of an individual can be challenging in erms of compuaional racabiliy. Several sudies invesigaed he mehods ha could overcome his issue. For example, Grinold (1977, 1983) presens and compares differen mehods of approximaing he general mulisage opimizaion problem upon horizon, and concludes ha a dual equilibrium echnique will give improving approximaions of he opimal soluion as he horizon increases, and perfec approximaions in he limi. Barro and Canesrelli (2005, 2006) show ha a dynamic porfolio problem wrien as a mulisage sochasic program can be rewrien as a discree ime opimal conrol problem, whereas Barro and Canesrelli (2011) furher exends his work o a broader class of mulisage sochasic programming problems, and reformulaes he problems as discree ime sochasic opimal conrol problems. Due o his reformulaion one can solve large opimizaion problems in a low compuaional ime. Finally, Geyer e al. (2009a) argue ha sochasic opimal conrol and sochasic programming can be combined and inegraed ino one muli-sage programming formulaion. The auhors model he opimal invesmen-consumpion problem using sochasic linear programming and combine i wih he closed-form soluion obained by Richard (1975). The mixed approach can accuraely replicae he firs-sage invesmen and consumpion decisions derived by Richard (1975). They argue ha sochasic programming and sochasic opimal conrol complemen each oher, especially in he areas where one or he oher does no perform well on is own. Thus, he main advanage of he mixed approach is he opimal soluion ha is obained under he realisic assumpions and wihin a shor compuaional ime. Inspired by he advanages of he mixed approach, we apply his mehod for wo opimizaion problems relevan for personal finance and pensions. Problem (A), similarly o Geyer e al. (2009a), is based on he explici soluion derived by Richard (1975). Our paper, however, differs from he model presened in Geyer e al. (2009a) in several aspecs. The mos imporan improvemen is adding he sum insured o he model. The explici soluions for opimal consumpion, invesmen and sum insured derived by Richard (1975) assume he life insurance policy in he model. Therefore, he sum insured mus be included in he MSP formulaion. Furhermore, we focus on he opimal conrols a all sages, and no only he firs-sage decision as done in Geyer e al. (2009a). Finally, insead of approximaing he objecive funcion by a piecewise-linear inerpolan, we solve he problem direcly using a nonlinear solver, hus removing he approximaion error from he resuls. Our work differs from Richard (1975) by including realisic consrains such as limis on porfolio 3
composiion, limis on he sum insured, an inclusion of ransacion coss and axes on capial gains. We show ha he opimal invesmen and sum insured derived by Richard (1975) is for a large variey of parameers problemaic from a pracical poin of view. Firs, he opimal invesmen implies shoring he risk-free asse and gearing he muual fund, which is ofen a limiaion for a privae invesor. Second, he sum insured is for mos of he individual s lifeime negaive, which canno happen in pracice. Third, while ransacion coss have a minor impac on he opimal conrols, he presence of axes on capial gains affecs he opimal invesmen sraegy significanly. Anoher conribuion of his paper is a formulaion and he soluion for problem (B). This problem is relevan for pension produc design in defined conribuion pension plans, and is uniqueness from he classical sochasic opimal conrol perspecive lies in defining he opimal invesmen for he period before and afer reiremen, whereas he opimal consumpion only during he period afer reiremen. All he aforemenioned sudies invesigaing he opimal invesmen-consumpion problems consider he opimal conrols over he same period; none of hem defines he opimal conrols over differen periods. Therefore, we derive he explici formula for he opimal value funcion and he opimal conrols using Hamilon-Jacobi-Bellman echniques for such a model. We furher solve problem (B) by using he mixed opimizaion approach. The paper is organized as follows. Secion 2 describes problems (A) and (B) in more deail ogeher wih heir opimal soluions obained via dynamic programming. Secion 3 explains how o build a bridge beween dynamic programming and muli-sage sochasic programming by incorporaing boh opimizaion mehods in one MSP formulaion. Secion 4 focuses on defining he MSP model, i.e. he objecive funcion and he consrains, and scenario ree generaion. Secion 5 presens he numerical resuls of he opimal conrols obained via wo differen opimizaion mehods, and he impac on he conrols of he modificaions considered during he firs years of he conrac. Finally, Secion 6 summarizes he paper and suggess he fuure work. The paper includes wo appendices: Appendix A explains he applicaion of he muual fund heorem, and Appendix B presens a derivaion of he opimal value funcion and conrols for problem (B). 2 Model descripion The secion presens he general model assumpions relevan for problems (A) and (B). One of he main decisions for he individual/pension fund is how o allocae he savings beween differen financial asses. We assume ha he savings can be invesed in N asses: one risk-free (bonds) and N 1 risky asses (socks). Given ha he economy is represened by a sandard Brownian moion W defined on he measurable space (Ω, F), where F is he naural filraion of W, 4
he asse prices S i can be modeled by a geomeric Brownian moion, ds i = α i S i d + σ i S i dw i, (1) S i 0 = s i 0 > 0, where α i and σ i are consans, and {i,j}=1,...,n 1 dw i and dw j are correlaed wih a coefficien corr ij. The risk-free asse is defined by α N = r and σ N = 0. Meron (1971) shows ha wihou loss of generaliy we can assume ha all he risky asses are included in one muual fund, he prices of which are modeled by a geomeric Brownian moion, ds = αs d + σs dw, where α is he expeced rae of reurn on he risky fund and σ is he volailiy of he risky fund. See App. A for deails. We also assume ha he individual is allowed o borrow money a he risk-free rae r in order o buy he risky asses, as well as o ake a shor posiion in hose asses. Furher, assume ha P and P are equivalen probabiliy measures on he measurable space (Ω, F). P denoes he objecive measure, whereas P is used by he pension fund for pricing boh marke and life insurance risk. Thus, consisenly wih Richard (1975) s assumpions, we consider life insurance policies as sandard radable financial conracs. 1 The individual has an uncerain lifeime modeled by a finie sae Markov chain Z, defined on a measurable space (Ω, F). The sae process Z indicaes wheher or no he person is alive; i akes values in {0, 1}, and sars in 0 a ime 0, i.e. he person is alive. The moraliy raes, µ and µ (respecively, under P and P ) are defined by he jump inensiies of he process Z. They are assumed o be coninuous and deerminisic (as defined laer in Sec. 5) and saisfy µ and µ, which implies ha lim P(Z = 1) = lim P (Z = 1) = 1. The cash-flows accompanying he savings accoun are formalized by he coninuous processes: a deerminisic conribuion process l ha, depending on he problem, is inerpreed eiher as he labor income ha is conribued o he accoun or he percenage of he labor income ha is paid o he reiremen savings (premiums), 2 and a consumpion process c deermining he consumpion of he savings for he privae purposes or he benefis ha he pension fund pays o he reiree. Finally, we assume ha he individual is risk averse and has a CRRA uiliy funcion u characerized by a consan relaive risk aversion 1 γ, consan elasiciy of ineremporal subsiuion 1/(1 γ) (EIS) and ime dependen weighs w reflecing he imporance of presen consumpion in 1 A similar approach o life insurance conracs can also be found in Kraf and Seffensen (2008). The assumpion abou radabiliy of life insurance is no subsanially differen from considering a case where policy holders are allowed o make aleraions o heir conracs. Apar from realisic issues wih healh and oher ypes of assymeric informaion (which do no appear in our model), his is cerainly wha appears in pracice. 2 The model could be exended by adding a sochasic conribuion process. However, he explici soluions o problems (A) and (B) can be derived only if he labor income is assumed o be spanned by he sock risk. Oherwise, he explici soluions o he conrol problems do no exis. 5
conras o fuure consumpion characerized by he impaience facor ρ: u(, c) = 1 γ w1 γ c γ, where w 1 γ = e ρ. Parameer γ is defined for (, 1)\{0}, whereas for γ = 0 we have he case of he logarihmic uiliy. Below, we define he model seup specific for he paricular problems. 2.1 Problem (A) - opimal invesmen, consumpion and sum insured We keep he original seings defined in Richard (1975) and recall only he mos imporan assumpions and resuls ha are crucial for his paper. Savings dynamics. The individual has a beques moive and upon deah her heirs receive an amoun on he savings accoun plus he sum insured, X + I. The premium for he coverage is µ I, where µ is he naural premium inensiy decided by he life insurance company, also called pricing moraliy. In paricular, we allow for negaive I, which means ha he person sells he amoun I o he pension fund and purchases life annuiies. The wealh dynamics while he person is alive develop as follows: dx = ( r + π (α r) ) X d + π σx dw + l d c d µ I d, (2A) X 0 = x 0, where r is he reurn on he risk-free asse, α is he expeced rae of reurn on he muual fund, σ is he volailiy of he muual fund, π is he proporion invesed in he muual fund, and 1 π in he risk-free asse. Opimizaion problem. he amoun ha she will leave upon deah o her heirs: Since he individual has a beques moive, she obains he uiliy from U(, x) = 1 γ v1 γ x γ, where v is he weigh for his uiliy. I sounds reasonable o define v in erms of he weighs for he uiliy of consumpion w, hence v denoes he weigh pu on her heir s consumpion relaive o her own, v 1 γ = λ γ w 1 γ, where λ is a consan. The individual can conrol consumpion, invesmen and sum insured, in order o maximize he uiliy of consumpion and beques. Given he wealh dynamics, X, he problem 6
is mahemaically formulaed as: V A (, x) = V A ( T, x) = 0, [ T sup E,x e ( ] ) s µτ dτ u(s, c s ) + µ s U(s, X s + I s ) ds, (3A) π,c,i Q[, T ) where E,x is he condiional expecaion under P, given ha he person is alive a ime and holds wealh X = x, and Q[, T ) is he se of conrol processes for he ime [, T ) which are admissible a ime. T is a fixed ime poin a which he invesor is dead wih cerainy. Boh uiliies are muliplied by facor e s µτ dτ denoing he probabiliy ha he individual survives unil ime s >, given she has survived unil ime. Furhermore, he uiliy of beques is muliplied by he moraliy inensiy rae µ s, which represens he probabiliy ha he person dies wihin a shor period afer ime s. The uiliy funcions u and U are assumed o be sricly concave in c and X + I, respecively. The opimal value funcion for his problem is given by V A (, x) = 1 γ f A() 1 γ( x + g A () ) γ, (4A) where f A () = g A () = T T e 1 1 γ s ) [ ( ) ] 1/(1 γ) (µ τ γ(µ τ +ϕ) dτ µs w s + (µ s) γ v s ds, e s (r+µ τ )dτ l s ds, ϕ = r + (α r)2 2σ 2 (1 γ), (5A) (6A) and he opimal invesmen, consumpion and sum insured are of he form: π = α r X +g A () σ 2 (1 γ) X, c = w f A ()( X + g A () ), I = ( µ µ ) 1/(1 γ) v f A () (X + g A ()) X. Equaions (4A)-(7A) correspond o equaions (25) and (40)-(44) in Richard (1975). (7A) 2.2 Problem (B) - opimal invesmen wih opimal life annuiies The applicaion of his problem is relevan for a pension produc design in a defined conribuion pension scheme, boh occupaional and privae. As seen in mos of he European pension markes, premiums are defined as a fixed percenage of he salary. In he occupaional pension schemes his percenage is ypically decided by he employer, whereas privae schemes allow he cusomer o choose boh he size and he frequency of he premiums. When enering he conrac, he individual is ofen given a choice of differen pension producs characerized by various invesmen sraegies and a ype of benefis. For example, one can choose beween a conservaive, moderae or aggressive 7
invesmen sraegy, bu i is he pension fund ha decides how o inves he savings such ha i reflecs he cusomer s preferences. In a differen kind of produc, such as a uni-linked, i is he pension saver who can decide on he porfolio allocaion and adjus i during he conrac as she wishes. Boh ypes of producs ypically offer an opion o add a guaranee o he conrac, bu in his work we assume ha he benefis are direcly linked o he marke and no guaranees are provided. In erms of he benefis, one can ypically choose beween a lump sum paymen ( T = T ), life annuiies ( T ) or emporary life annuiies (10-25 years, T < ). The size of he benefis is calculaed by he pension fund according o he acuarially fair principles. The considered opimizaion problem allows for conrolling he invesmen sraegy boh before and afer reiremen, as well as he size and disribuion of he benefis afer reiremen. The opimal soluion deermines a produc ha is cusomized o he individual s risk and impaience preferences. In case of deah, he pension fund inheris all he individual s savings. Savings dynamics. We spli he problem ino a before reiremen period (accumulaion phase, < T ) and an afer reiremen period (decumulaion phase, T ). In pracice he pension saver is no allowed o wihdraw he pension savings for he consumpion reasons or receive he benefis before reiremen, and i is also reasonable o assume ha he premiums paid o he pension fund are se o 0 afer reiremen. Thus, c = 0 for [ 0, T ) and l = 0 for [T, T ). Since he person has no beques moive, she has an addiional income of µ X, which is he amoun ha he pension fund pays o be her only inherior. Wih hese assumpions he definiion of savings dynamics are of he form { (r + π (α r) + µ ) X d + π σx dw + l d, [ 0, T ), dx = (r + π (α r) + µ ) X d + π σx dw c d, [T, T ), X = x 0. Opimizaion problem. (2B) The pension saver wishes o maximize he uiliy of pension benefis, or in oher words, he uiliy of consumpion afer reiremen, while being able o conrol he invesmen sraegy and he disribuion of he benefis. She obains no uiliy from consumpion before reiremen, i.e. u(, c) = 0 for [ 0, T ), hence he problem is formulaed as follows: V B (, x) = V B ( T, x) = 0. [ ] T sup E,x e s µτ dτ u(s, c s )ds, (3B) π Q[, T ), c Q[max(,T ), T ) max(,t ) where, as before, E,x is he condiional expecaion under P, given ha he person is alive a ime and holds savings X = x, and Q[, T ) is he se of conrol processes for ime [, T ) which are admissible a ime. The conrol processes in his problem are he proporion in he risky fund 8
π and he benefis c. As in problem (A), we ake ino consideraion he uncerain lifeime of he individual by muliplying he uiliy funcion by he facor e s µτ dτ. A ime T he invesor is assumed o be dead wih cerainy. The uiliy funcion u is assumed o be sricly concave in c. Even hough he individual obains no uiliy from consumpion before reiremen, he invesmen process π is conrolled boh before and afer reiremen. Thus, he novely of his problem from he sochasic opimal conrol poin of view lies in defining he opimal consumpion and invesmen over differen (bu parially overlapping) periods. The research o dae has ended o focus on deriving he explici soluion for he conrols over he same period; eiher unil reiremen, see e.g. Cairns e al. (2006) and Bruhn and Seffensen (2011), afer reiremen, see e.g. Gerrard e al. (2004, 2006) and He and Liang (2013), or generally over he life cycle, see e.g. Meron (1969, 1971), Richard (1975), Milevsky and Young (2007), and Kraf and Seffensen (2008). Because, o he bes of our knowledge, neiher Problem (B) or he case of deriving opimal conrols over differen periods has been considered by oher researchers, we derive he opimal value funcion and he opimal conrols, see Appendix B for deails. The opimal value funcion is of he form: V B (, x) = 1 γ f B() 1 γ( x + g B () ) γ, where f B () = e 1 1 γ f B () = T e 1 1 γ T (µτ γ(µ τ +ϕ))dτ T s ( µ τ γ(µ τ +ϕ) 1 s T e 1 γ T ( ) µ τ γ(µ τ +ϕ) dτ ws ds, [ 0, T ), ) dτ ws ds, [T, T ), (4B) (5B) { g B () = T e s (r+µ τ )dτ l s ds, [ 0, T ), g B () = 0, [T, T ), (6B) and he opimal conrols are given by π = α r X +g B () σ 2 (1 γ) X, [, T ), c = w f B () X, [T, T ). (7B) Since he uiliy funcion u is concave in c, he reiremen benefis mus be non-negaive. Furhermore, we mus have ha X T = 0, which ensures ha all he savings ha belong o he pension saver have been paid ou during he disribuion phase. In boh problems he funcions g A () and g B () are of he same form. The firs funcion has been defined by Richard (1975) as human capial and represens he expeced presen value of fuure earnings, or, in oher words, wha an individual s labor force is worh in he financial marke. Funcion g B () represens he presen value of he fuure premiums, i.e. he fracion of he human capial ha 9
is conribued o he pension savings accoun. 3 Combined MSP and opimal conrol In his paper we furher invesigae a combined sochasic programming and sochasic opimal conrol approach ha was originally presened in Geyer e al. (2009a). We canno apply purely sochasic opimal conrol because he explici soluion can only be derived for simple models, which ofen canno be applied in pracice, and as soon as we add some realisic consrains, explici soluions do no exis. In paricular, we show in Sec. 5.1 ha he opimal invesmen and sum insured derived by Richard (1975) is for a large variey of parameers problemaic from a pracical poin of view. Firs, he opimal invesmen implies shoring he risk-free asse and gearing he muual fund. Second, he sum insured is negaive for mos of he individual s lifeime. Third, while ransacion coss have a minor impac on he opimal conrols, he presence of axes on capial gains affecs he opimal invesmen sraegy significanly. To include he aforemenioned consrains, one can choose he MSP approach, however, he main drawback of his approach is he limied abiliy o handle many ime periods under sufficien uncerainy. The scenario ree grows exponenially wih he number of ime periods, and solving he problem soon becomes compuaionally inracable. For example, o deal wih a long horizon one has o choose long and increasing inervals beween he decisions, see e.g. Carino e al. (1998), Dempser e al. (2003) and Consigli e al. (2012), add a seady sae erminal value erm o he objecive funcion and approximae he infinie horizon problem using he dual equilibrium echnique, see e.g. Grinold (1977, 1983) and Carino e al. (1998), or apply differen scenario reducion algorihms, see e.g. Heisch and Römisch (2009a,b). Each of he menioned mehods can handle he long ime horizon problems, however, each mehod has also some drawbacks. For example, choosing long inervals beween he sages implies ha he individual does no have an opporuniy o change her decisions for a considerable amoun of ime, and he opimal decisions may differ from hose made more frequenly. Furhermore, he impac of choosing increasing rebalancing inervals relaive o consan inervals remains unknown. The dual equilibrium echnique implies he approximaion of he infinie horizon according o some assumed discoun facor, and does no include he uncerainies during he seady sae. Finally, despie he increasing research on scenario reducion algorihms, modelling he enire lifeime of an individual wih yearly inervals sill remains compuaionally inracable, unless one simplifies he ree during he laer years o he branching facor of one. However, if he branching facor a any node is lower han he number of non-redundan asses, he scenarios may allow for arbirage opporuniies, which would affec he opimal decisions, see e.g. Kouwenberg (2001), Klaasen (2002) and Geyer e al. (2014). In he ligh of hese limiaions, he mixed approach sounds appealing. I allows us o model he enire lifeime of he individual under realisic assumpions, wih shor inervals beween he 10
Figure 1: The firs years decisions are solved by he MSP model, whereas he decision over he remaining lifeime of he individual are solved using HJB echniques. decisions, and in a shor compuaional ime. The decisions are made under enough uncerainy boh during he period modeled by MSP, and in he laer years modeled by sochasic opimal conrol. Even hough his approach assumes ha he more realisic consrains are imposed only during he firs years, and no he enire lifeime, he empirical resuls (Sec. 5) show ha he impac of he consrains is visible even if hey are imposed only for a shor period of ime, such as 6 years. We argue ha he iniial decisions are he mos imporan because he cusomer needs an advice abou her personal finance and pension a he presen momen. A financial advisor or a pension fund ypically would hold regular meeings wih he cusomer and rerun he model for a differen se of parameers han hose iniially chosen. This is necessary no only because he expecaions abou he economy change, bu also because he cusomer migh change her risk and impaience preferences. To wha exend i is possible o neglec cerain consrains and having a simpler model in he long run, is no rivial and lef o fuure research. To define he objecive funcion for he mixed approach we spli he lifeime of he individual ino wo periods: [ 0, T MSP ) and [T MSP, T ). We apply he MSP approach during he firs inerval and sochasic opimal conrol during he remaining lifeime of he individual, see Fig. 1. The mahemaical formulaion for problem (A), defined in eq. (3A), can be rewrien using is recursive properies in he following way: V A (, x) = sup π,c,i Q[,T MSP ) E,x [ TMSP e ( ) s µτ dτ u(s, c s ) + µ s U(s, X s + I s ) ds + e ] T MSP µ τ dτ V A (T MSP, X TMSP ). The above definiion is convenien because i separaes he problem ino wo inervals, which allows for applying differen opimizaion mehodologies in each inerval. The crucial par of he model is o inser he opimal value funcion derived explicily ino he objecive funcion of he MSP model, hus we can ensure ha he MSP formulaion incorporaes he decisions for he enire lifeime and no only for he firs years. We sar wih he laes period, [T MSP, T ), and calculae he opimal value funcion a ime T MSP, i.e. V A (T MSP, X TMSP ) according o (4A). Then, we inser his funcion ino eq. (8A) and solve he opimizaion problem using a muli-period sochasic model. In his way we consruc one MSP formulaion ha covers he enire lifeime of he individual. We proceed (8A) 11
analogously wih problem (B). The mahemaical formulaion for he opimizaion problem defined in eq. (3B) can be rewrien as V B (, x) = sup π Q[,T MSP ), c Q[max(,T ),T MSP ) E,x [ TMSP where V B (T MSP, X TMSP ) is given in eq. (4B). max(,t ) e s µτ dτ u(s, c s )ds + e ] T MSP µ τ dτ V B (T MSP, X TMSP ), (8B) 4 MSP formulaion The main elemens of muli-sage sochasic programming are a scenario ree and an opimizaion model. A scenario ree consiss of nodes n N represening he range of possible oucomes for he uncerainies. All he nodes are uniquely assigned o sages = 0,..., T MSP such ha a he firs sage here is a roo node n 0, a he subsequen sages, n N, > 0, each node has a unique ancesor n, and a all he sages excep for he final sage, n N, < T MSP, each node has children nodes n +. The nodes a he final sage T MSP are called he leaves. A scenario S n consiss of a leaf n and all is predecessors n, n,..., n 0, or equivalenly, a single branch from he roo o he leaf. The number of scenarios in he ree equals he number of leaves. Each node has a probabiliy pr,n, so ha n N pr,n = 1, and he probabiliy of each scenario S n is he produc of he probabiliies of all he nodes in he scenario, pr S n = pr TMSP,n pr TMSP 1,n pr T MSP 2,n... pr 0,n 0. The opimal decisions along he ree are compued numerically a each node of he ree, given he informaion available a ha poin. Decisions do no depend on he fuure observaions bu anicipae possible fuure realizaions of he random vecor. Afer he oucomes have been observed, he decisions for he nex period are made and depend boh on he realizaions of he random vecor and he decisions made in he previous sage. This combinaion of anicipaive and adapive models in one mahemaical framework makes his approach paricularly appealing in financial applicaions; he invesor can specify he composiion of a porfolio by aking ino accoun possible fuure movemens of asse reurns (anicipaion), and rebalance he porfolio (ake recourse decisions) as prices change, see Zenios (2008). The applicaions of MSP specifically in individual asse liabiliy managemen can be found e.g. in Ziemba and Mulvey (1998), Kim e al. (2012) and Konicz and Mulvey (2013), whereas for a general inroducion o sochasic programming we refer he reader o Birge and Louveaux (1997) and Ruszczyński and Shapiro (2003). 4.1 Objecive and consrains The firs years decisions for he opimizaion problems considered in his paper are modeled by a T MSP -period model. The decision variables are defined wih respec o he nodes n in he scenario 12
ree, where N denoes he se of nodes corresponding o sage, and J is he se of available financial asses. For each period = 0,..., T MSP, node n N, and asse class i J we define he following decision variables and parameers: Parameers: x 0 iniial value of he savings, l labor income/premiums paid o he savings accoun a ime, µ moraliy rae for a y + -year old individual, q y+ qy+ probabiliy ha a y + -year old individual dies during he following period, probabiliy ha a y + -year old individual dies during he following period used for pricing life insurance/life annuiies, pr,n probabiliy of being a node n, n N, obained via scenario generaion, see Sec. 4.3, r i,,n reurn on asse i a node n corresponding o sage, obained via scenario generaion. Decision variables: P i,,n 0 amoun of asse i purchased in period and node n, S i,,n 0 amoun of asse i sold in period and node n, X i,,n X,n holdings of asse i a he beginning of period, a node n, afer rebalancing, holdings in all asses a he beginning of period, a node n, before rebalancing, X 0,n 0 = x 0, X,n = i J (1 + r i,,n) X i, 1,n, = 1,..., T MSP, n N, C,n consumpion/benefi in period and node n, Ĩ,n sum insured in period and node n (problem (A) only). We use capial leers wih he ilde o denoe he variables of he MSP formulaion as opposed o he lowercase leers denoing he parameers for he model. Expression 1 {( )=} denoes an indicaor funcion equal o 1 if ( ) = and 0 oherwise. The enire MSP formulaion ha replicaes he assumpions made in dynamic programming model seup, consiss of he objecive funcion and hree consrains: he budge consrain, he asse invenory consrain and he consrain on posiiviy of cerain variables. Wih only hese hree consrains we can replicae he coninuous ime models presened in Sec. 2. Aferwards, in Sec. 4.2, we modify hese equaions in order o invesigae he impac of various facors, such as limis on porfolio composiion, limis on he sum insured, ransacion coss, and axes on capial gains. 4.1.1 Problem (A) The problem of opimizing he expeced uiliy funcion of consumpion and a erminal uiliy of leaving a posiive amoun of money upon deah, where he invesmen, consumpion and sum insured are he conrolled processes, can be modeled wih he se of he following equaions. 13
The objecive funcion is obained by subsiuing he inegrals wih he sums and he expecaion operaor E wih is discree definiion in he eq. (8A). The budge consrain (10A), or alernaively, he cash flow balance consrain specifies ha he amoun invesed in he purchase of new securiies plus consumpion mus be equal o he amoun gained from he sale of he securiies plus labor income ha is paid o he savings accoun plus any iniial savings x 0 ha he person has a he beginning of he conrac. Moreover, we add he erm q y+ĩ,n o he lef hand side of he equaion, which is he price for he life insurance ha he invesor pays a each period. The inclusion of he sum insured variable Ĩ,n in he budge consrain and in he objecive funcion for problem (A) is an imporan par of he model ha disinguishes our work from Geyer e al. (2009a). Consrain (11A) calculaes he value of he savings. I is equal o he accumulaed capial gains/losses on he asses held in porfolio from he previous period plus any amoun purchased in he curren period minus sold securiies. The coninuous ime versions of he problems assume he uiliy funcion u o be sricly concave in c, and U o be sricly concave in X + I. Furhermore, for γ < 0, u(0) = U(0) =, herefore we define he posiiviy consrains (12A). For γ (0, 1) we have u(0) = 0 and U(0) = 0, so he posiiviy consrains are subsiued wih non-negaiviy consrains. This leads o he model: T MSP 1 max s= 0 + n N s pr s,n e n N TMSP pr TMSP,n e [ s µ τ dτ 0 u(s, C ( s,n ) + q y+s U s, X )] s,n + Ĩs,n ( TMSP µ τ dτ 0 V A T MSP, X ) T MSP,n, (9A) P i,,n + C,n + qy+ĩ,n = x 0 1 {=0 } + i J i J S i,,n + l, = 0,..., T MSP 1, n N, (10A) X i,,n = ( 1 + r i,,n ) Xi, 1,n 1 {>0 } + P i,,n S i,,n, = 0,..., T MSP 1, n N, i J, (11A) C,n > 0, X,n + Ĩ,n > 0, = 0,..., T MSP 1, n N. (12A) 4.1.2 Problem (B) The problem of maximizing he expeced uiliy of reiremen benefis, where he invesmen process is conrolled boh before and afer reiremen, whereas he benefis are conrolled only afer reiremen, can be modeled using MSP formulaion as follows. 14
The objecive funcion (9B) is a discree version of eq. wih he sums and he expecaion operaor E wih is discree definiion. (8B) where he inegrals are subsiued The budge consrain (10B) specifies ha he amoun invesed in he purchase of new securiies plus consumpion mus be equal o he amoun gained from he sale of he securiies plus premium ha is paid o he savings accoun plus any savings x 0 ha he person has a he beginning of he conrac. We also add he erm qy+ X,n o he righ hand side of he balance equaion, denoing he price ha he pension fund pays he pension saver o be her only heir. Furhermore, we add he pension benefi o he side of he ougoing paymens for T. Consrain (11B) calculaes he value of he savings on he pension accoun. This consrain is idenical o he asse invenory balance for problem (A). The uiliy funcion u is assumed o be sricly concave in c, and for γ < 0, we have u(0) = U(0) =. Therefore, we assume ha he benefis are posiive, (12B), or, for γ (0, 1), non-negaive. max This leads o he following MSP formulaion: + T MSP 1 s=max( 0,T ) n N s pr s,n e n N TMSP pr TMSP,n e s 0 µ τ dτ u(s, Cs,n ) ( TMSP µ τ dτ 0 V B T MSP, X ) T MSP,n, (9B) P i,,n + C,n 1 { T } = x 0 1 {=0 } + i J i J S i,,n + l + q y+ X,n, = 0,..., T MSP 1, n N, (10B) X i,,n = ( 1 + r i,,n ) Xi, 1,n 1 {>0 } + P i,,n S i,,n, = 0,..., T MSP 1, n N, i J, (11B) C,n > 0, = max( 0, T ),..., T MSP 1, n N. (12B) 4.2 Addiional consrains The reason for modelling he firs years decisions wih MSP is ha his opimizaion mehod can easily capure he pracical consrains such as limis on porfolio composiion, limis on he sum insured, ransacion coss, and axes on capial gains. These consrains are commonly used in financial applicaions solved wih MSP approach, see e.g. Geyer e al. (2009a) and Fersl and Weissenseiner (2011), bu rare in invesmen-consumpion problems solved wih sochasic opimal conrol. Below, we presen how o modify he original consrains or add oher consrains o he MSP formulaion presened in Sec. 4.1. 15
Limis on porfolio composiion. A feaure ha is ineresing from he poin of view of a privae invesor, a pension saver and a pension fund, is he limi on he porfolio composiion. For insance, no pension fund allows for borrowing or shor selling of he asses. Alernaively, if he invesor has cerain preferences abou he minimum and maximum percenage of her wealh invesed in a cerain asse, we would like o include i in he opimizaion model. The limis on he porfolio composiion can be incorporaed in he MSP formulaion by adding he following consrains: X i,,n d i j J X j,,n, Xi,,n u i j J X j,,n, = 0,..., T MSP 1, n N, i J, (16) where d i and u i are he lower and upper limis for he holdings of asse i. In paricular, d i and u i can be exended o be ime dependen, which would be suiable for a person wih specific preferences only for a cerain year. Limis on he sum insured. Richard (1975) s model does no assume any consrain on he size or he sign of he sum insured I. Including such a consrain in he dynamic programming approach is definiely no rivial o solve and o our knowledge he explici soluion for his case has no been derived. The MSP model allows for adding he limis on he sum insured in a sraighforward way: Ĩ,n d ins, Ĩ,n u ins, = 0,..., T MSP 1, n N. (17) As shown laer in Sec. 5.1, being able o conrol he sign of he sum insured is especially imporan from he pracical poin of view. A negaive sum insured means ha i is he individual who sells he life insurance o he pension fund for he price of q y+ĩ,n. Specifically, one should give up he par Ĩ,n of he savings upon deah in reurn for he exra premium q y+ĩ,n. Theoreically, we could inerpre his siuaion as invesing a par of he savings in a life annuiy, neverheless, from a pracical poin of view having a negaive sum insured sounds srange. Transacion coss. more realisic. imporance. Similarly, he presence of ransacion coss c i makes he considered problems Invesigaing how he ransacion coss affec he opimal conrols is of pracical We add he coss as a percenage of he raded amoun by modifying he budge equaions (10A) and (10B). We subrac he coss from he amoun of he asses sold in a given period, and add he coss o he amoun purchased. The modified budge equaions are defined as follows: P i,,n (1 + c i ) + C,n + qy+ĩ,n = x 0 1 {=0 } + S i,,n (1 c i ) + l, i J i J P i,,n (1 + c i ) + C,n 1 { T } = x 0 1 {=0 } + S i,,n (1 c i ) + l + q X y+,n, i J i J (10A ) (10B ) 16
for = 0,..., T MSP 1 and n N. Taxes on capial gains. Worh consideraion are also axes on capial gains τ i. Generaing a scenario ree for he MSP model allows us o subrac axes only from he posiive capial gains, i.e. ner i,,n = { r i,,n (1 τ i ), r i,,n > 0, (18) r i,,n, r i,,n 0. This requires changing he asse invenory balance consrains (11A) and (11B) as follows: X i,,n = ( ) 1 + ne r i,,n Xi, 1,n 1 {>0 } + P i,,n S i,,n, = 0,..., T MSP 1, n N, i J. (11A,11B ) This is also convenien in conras o he sochasic opimal conrol approach, where he asse prices are modeled by he Black Scholes model, and one can only adjus he expeced reurns and volailiy of he asse reurns by inroducing α i,ne = α i (1 τ i ) and σ i,ne = σ i (1 τ i ), see e.g. Bruhn (2013). The axes are hen subraced from boh posiive (gains) and negaive (losses) reurns. The laer can be inerpreed as a possibiliy o deduc he axes from he negaive capial income. Oher modificaions. There are pleny of economic facors whose impac on he opimal conrol would be ineresing o invesigae bu are beyond he scope of his paper. Probably he mos relevan from a pracical perspecive is modelling he asse reurns using a differen model han Black Scholes. Geyer e al. (2009b) exended heir previous work by applying he firs-order unresriced vecor auoregressive process (VAR) o model asse reurns. They find ha here is a subsanial difference in asse allocaion which reflecs he impac of ime-varying invesmen opporuniies, which, no surprisingly, shows he sensiiviy of he model o he assumpions in he scenario generaion. 4.3 Scenario generaion The uncerainy associaed wih he marke reurns is modeled by an N 1-dimensional random process. The mulivariae reurn process evolves in discree ime, and he underlying probabiliy disribuions are approximaed by discree disribuions in erms of a scenario ree. We have esed a number of scenario generaion mehods for sochasic programming, including sampling, simulaion, scenario reducion echniques and he mehods based on maching he saisical properies of he underlying process, see e.g.kau and Wallace (2005) and Heisch and Römisch (2009a). Since we combine he sochasic programming wih he opimal conrol approach in one mahemaical framework, our goal is o generae a scenario ree wih he prices for he securiies following he Black Scholes model defined in eq. (1). Specifically, we aim for consrucing a muliperiod scenario ree wih a discree represenaion of a normal disribuion N ( α i σ 2 ) i /2, σ i, where 17
he reurns in wo adjacen periods are independen and idenically disribued. Therefore, he mos suiable approach in our sudy is he momen maching mehod. The momen maching approach has been firs described in Høyland and Wallace (2001) and is based on solving a nonlinear opimizaion problem where boh he asse reurns and he probabiliies of each node are he decision variables defining he scenario ree. This algorihm has laer been improved in Høyland e al. (2003) where, insead of solving a nonlinear opimizaion problem, he auhors sugges o perform a number of ransformaions ha ensure ha he required momens are mached. Since hen, furher improvemens o his mehod have been made. Gülpınar e al. (2004) sugges a combined simulaion and opimizaion approach. The asse reurns are firs simulaed and hen fixed in he opimizaion model, so ha he only decision variables in he model are he probabiliies associaed wih each node. This mehod can moreover be applied o generae he enire ree a once, and no node by node, as in he aforemenioned approaches. Ji e al. (2005) is he firs o propose a linear programming (LP) momen maching approach, hough only for a single period ree. Similarly as in he previous paper, he oucomes of asse reurns mus be predeermined. Inspired by ha work, Xu e al. (2012) design a new approach ha combines simulaion, he K- means clusering approach, and linear momen maching o generae he muli-sage scenario ree. This mehod ensures ha he saisical properies are mached well, he generaed scenario ree has a moderae size, he soluion ime is reduced, and a leas wo branches are derived from each nonleaf node. Finally, Chen and Xu (2013) improve his work by removing he simulaion componen and applying he K-means clusering mehod direcly ono he hisorical daase combined wih LP momen maching. This approach significanly reduces he compuaional ime while preserving he required saisical properies. Despie he advanages of he newes scenario generaion mehods described above, for he purpose of our sudy we choose one of he older algorihms, namely Høyland and Wallace (2001), which we furher improve by adding a priori he non-arbirage consrains, see Klaasen (2002). This mehod maches he saisical properies of a geomeric Brownian moion beer han oher algorihms in a siuaion when having as low a branching facor as possible is a prioriy. The algorihms described in Xu e al. (2012) and Chen and Xu (2013) are definiely more efficien and can capure more complex models for asse reurns, such as he vecor auoregressive and mulivariae generalized auoregressive condiional heeroscedasiciy models, bu hey require a larger branching facor. For our choice of asses and he disribuion parameers, he saisfacory saisical mach is obained for a branching facor of a leas 8, whereas he algorihm presened in Høyland and Wallace (2001) allows o generae he required scenario ree wih only 4 branches. Our prioriy is o invesigae he impac of some realisic consrains ha are hard o implemen in he sochasic opimal conrol approach, herefore o sudy more ime periods, we choose a scenario ree wih fewer branches. 18
5 Numerical resuls We firs presen realisic examples illusraing problems (A) and (B) in he original model seup (coninuous ime) and he soluions obained by he MSP model presened in Sec. 4.1. Aferwards, we add or modify he paricular consrains as described in Sec. 4.2 and invesigae heir impac on he opimal conrols during he firs years of he conrac. Parameers. If no specified in he capions of he ables and figures, he following parameers have been chosen for esing he models: Marke: he number of asses, N = 3; wo risky and one risk-free asse, he expeced raes of reurns on he asses are, respecively, α 1 = 0.05, α 2 = 0.07, r = α 3 = 0.02, he volailiy of he asses, σ 1 = 0.2, σ 2 = 0.25, σ 3 = 0, and he correlaion beween he risky asses is corr 12 = corr 21 = 0.5; all parameers are adjused for inflaion. Uiliy funcion: risk aversion, 1 γ = 4, corresponding o he opimal proporion in risky asse afer reiremen π equal o 25%, he impaience facor for he uiliy weighs, ρ = 0.04, inuiively chosen such ha ρ r, and he weigh on he uiliy of beques relaively o he uiliy of consumpion, λ = 5, implying ha he opimal amoun which he inheriors receive upon deah is approximaely equal o hree and a half years of he person s consumpion while she is alive, X + I 3.5c (λ is only relevan for problem (A)). Conrac: age a he beginning of he conrac, age 0 = 45, reiremen age, age T = 65, he final age a which he person is assumed o be dead wih cerainy, T = 110, ype of reiremen benefis (problem (B)): life annuiy. Lifeime uncerainy: he moraliy inensiy model is of he form µ = µ = θ + 10 β+δ(y+) 10, where y is he age of he person a ime 0, and θ, β and δ are consans. 3 The model has been calibraed o he moraliy raes among Danish women in 2010 over age 40, Finansilsyne (2010), where θ = 0.0, β = 4.59364, δ = 0.05032. We approximae he probabiliy ha a y + -year old individual dies during he nex year by he moraliy rae, i.e. q y+ µ and qy+ µ. Scenario ree: number of MSP sages, T MSP = 6, branching facor in a single ree (number of nodes), bf = 4, number of rees no ree = 50, which implies in oal 4 5 50 = 51, 200 scenarios. Cash-flows: 3 In principle he moraliy inensiy model does no assume ha an individual is dead a ime T wih probabiliy 1. However, for T = 110, his error is negligible. 19
Problem (A). Labor income l = 27, 000 EUR, corresponding o he average Danish disposable income (afer axes) for a 45-year old individual, Danmarks Saisik (2010), average savings of a 45-year old individual, x 0 = 60, 000 EUR, Problem (B). Before reiremen: premiums, l = 4, 000 EUR, corresponding o 15% of he salary. Assuming ha he person has been conribuing o he pension accoun for around 15 years, accumulaed wih some capial gains, gives he value of he iniial savings of x 0 = 75, 000 EUR; Afer reiremen: premiums, l = 0, he value of he savings is esimaed from he expeced savings for a 70 year old person (see before reiremen case), i.e. x 0 = E[X25 B ] = 225, 000 EUR. Similarly as Geyer e al. (2009a), we have chosen o consider he small-scale opimizaion problems raher han he large-scale problems. On he firs place, we are ineresed in replicaing he opimal conrols obained from he coninuous ime models. The small-scale models are sufficien o evaluae wheher a similar soluion can be achieved by running he MSP model. Wih only hree asses: one risk-free and wo risky asses, he firs four momens of a normal disribuion can be approximaed wih 4 nodes. 6 periods wih 4 nodes each give in oal 4 5 = 1, 024 scenarios. Addiionally, we rerun each program no ree = 50 imes for differen scenario rees, and presen he resuls in erms of means and sandard errors from sampling. We use he noaion X, π, c and I for he expeced opimal value of savings, invesmen, consumpion, and sum insured (average across all he scenarios) boh from he Richard (1975) s model and from he combined model. Each problem has been implemened using Malab 8.2.0.701 (R2013b) for calculaion of he explici soluions, GAMS 24.1.3 wih CONOPT 3 (3.15L) solver for scenario generaion, and GAMS wih MOSEK 7.0.0.75 solver for he MSP formulaion and soluion of he sochasic programming models. The running ime of each model for one scenario ree akes only a few seconds on a Dell compuer wih an Inel Core i5-2520m 2.50 GHz processor and 4 GB RAM. Due o he lineariy of he consrains, one can also approximae he objecive funcion by a piecewise-linear inerpolan, see e.g. Konogiorgis (2000) and Rasmussen (2011), and solve he problem using a linear solver. We have esed a number of boh linear and nonlinear solvers, and we find ha for he considered problems, in erms of speed, robusness and accuracy, MOSEK, which uses an inerior poin algorihm, is he bes suied solver. 5.1 Numerical resuls for problem (A) We sar wih analyzing he opimal conrols obained explicily by solving Richard (1975) s model and he MSP soluions replicaing his model (no addiional consrains). Fig. 2 (op lef and righ) shows he developmen in he expeced values of he savings accoun, X, opimal consumpion, c, and sum insured, I, in 1000 EUR (lef figure), as well as he opimal proporion in boh risky asses, π, and opimal proporion in he firs risky asse π 1 (righ figure) during he enire lifeime of he individual. The markers, which represen he opimal decisions obained from he MSP 20