OPTIMAL CONSUMPTION AND INVESTMENT DECISIONS UNDER TIME-VARYING RISK ATTITUDES

Transcription

1 OPTIMAL CONSUMPTION AND INVESTMENT DECISIONS UNDER TIME-VARYING RISK ATTITUDES FELIX HENTSCHEL Absrac. As a change in he individual s risk aversion over he life-cycle is observed wihin several lieraure, differen approaches are made o exend he opimal consumpion and invesmen problem by a ime-varying risk aversion. This paper combines he wo approaches of including a habi level (see e.g. Consaninides (99)) and a coefficien of ime-varying risk aversion (see e.g. Seffensen (2)). The opimal consumpion and invesmen rules are derived in a complee marke seing. They are examined in a numerical analysis for differen magniudes of habi formaion and differen shapes of ime-varying risk aversion. Our findings show ha wih a coefficien of ime-varying risk aversion, he shape of he decision rules, raher han jus heir magniude, depend on he iniial wealh of he individual. In he numerical analysis, we find ha wih an increasing risk aversion and sufficienly large habi formaion a hump-shaped consumpion paern is achieved, as i is observed in he lieraure. Furhermore, in his case he invesmen ino he risky asse is decreasing over he life-cycle, as suggesed by financial advisers. Keywords: Opimal consumpion and invesmen, habi formaion, ime-varying risk aversion JEL: D9, G Dae: March 3, 25. Insiue of Insurance Science, Universiy of Ulm, Helmholzsrasse 2, 898 Ulm, Germany. E Mail: felix.henschel@uni-ulm.de.

2 . Inroducion Several lieraure observes ha he risk aversion of an individual is changing over he life-cycle. For example, Riley and Chow (992) find ha he risk aversion decreases wih age for individuals younger han 65 and increases for individuals older han 65. Ohers who find ha risk aversion increases wih age are e.g. Morin and Suarez (983), Bakshi and Chen (994), Bellane and Green (24), Al-Ajmi (28), Ho (29) and Yao e al. (2). A decreasing risk aversion, on he oher hand, is observed by e.g. Bellane and Saba (986) and Wang and Hanna (997). The lieraure hus agrees ha he risk aversion is changing over ime, however here is no consensus wheher i is increasing or decreasing. Differen approaches are used in he lieraure o accoun for a changing risk aversion wihin he opimal consumpion and asse allocaion problem, formulaed in Meron (969). One of hese approaches is o include a habi level for consumpion ino he model, accouning for he fac ha individuals ge used o heir sandard of living. The inclusion of a habi level for consumpion is inroduced in Sundaresan (989) and Consaninides (99) and furher analyzed by e.g. Ingersoll Jr (992), Deemple and Zapaero (99), Schroder and Skiadas (22) and Munk (28). Including a habi level for consumpion in he uiliy funcion leads o a relaive risk aversion ha is decreasing in he raio of acual consumpion o he habi level. Since his raio changes over ime, also he risk aversion changes over ime. Anoher echnique o accoun for changing behavior over ime, is o direcly modify he power uiliy, such ha he coefficien of relaive risk aversion is varying over ime. This problem is formulaed and solved wih differen approaches in Seffensen (2) and Aase (29). In his paper, we combine he wo aforemenioned approaches. We consider an addiive habi level wihin a power uiliy funcion wih a ime-varying coefficien of risk aversion. For his combinaion, he relaive risk aversion of he uiliy funcion consiss of wo imevarying pars: he firs par is solely he exogenously given coefficien of ime-varying risk aversion. Whereas he second par is driven by he consumpion o habi raio and is hus affeced by he curren and pas marke paricipaion. Wihin a complee marke seing, we are able o solve he problem in closed form, using he dualiy resul from Schroder and Skiadas (22). In a subsequen numerical analysis, we examine he effec of a habi level for consumpion and a ime-varying risk aversion 2

3 on he opimal decision rules. We compare differen magniudes of habi formaion and several shapes for he ime-varying risk aversion. Our main findings are: firs, by including a coefficien of ime-varying risk aversion, he shape of he consumpion and invesmen rules, respecively heir expecaions, depend on he iniial wealh of he individual. In conras o a ime-consan risk aversion, where he invesmen decision is unaffeced by he iniial wealh and he consumpion decision changes only in absolue values, bu no in shape, for differen iniial wealh. Second, he resuling opimal consumpion and invesmen decisions are more in line wih empirical observaions. Namely for an increasing risk aversion and a sufficienly large habi formaion, he consumpion exhibis a hump-shaped paern. This hump-shaped consumpion is observed by e.g. Carroll and Summers (99), Aanasio e al. (999), Gourinchas and Parker (22), Fernandez-Villaverde and Krueger (27) and Bullard and Feigenbaum (27). The opimal invesmen in he risky asse is decreasing over ime for an increasing risk aversion. This is consisen wih suggesions from financial advisers and lieraure on invesmen guides, e.g. Malkiel (999) and Morris e al. (998), who sugges ha individuals inves a larger proporion ino risky asses early in he life-cycle and reduce heir invesmen in risky asses as hey age. For a decreasing risk aversion, on he oher hand, he asse allocaion exhibis a hump-shaped paern, which is observed by e.g. Berau and Sarr (2), Ameriks and Zeldes (24) and Agnew e al. (23). The remainder of he paper is organized as follows. Secion 2 inroduces he model and he opimizaion problems: firs an opimal consumpion and invesmen problem for a ime-varying risk aversion wihou habi formaion, second an opimizaion problem including a habi level. Opimal soluions in a complee marke seing are presened for boh problems. Secion 3 conains numerical calculaions for differen shapes of he imevarying risk aversion and a deailed discussion of he resuls and he effecs on he opimal soluions. Secion 4 concludes. 2. Model and opimizaion problem In his secion, we inroduce he marke model and formulae he individual s opimizaion problems. We sar wih he problem wih a ime-varying risk aversion and wihou habi formaion and review he resuls of Seffensen (2) and Aase (29). Subsequenly, we consider he problem including habi formaion and, wih he previous resuls, we solve 3

4 i by using he dual approach presened in Schroder and Skiadas (22). For he following, we assume a filered probabiliy space (Ω, F, P, (F ) [,T ] ), wih F = {, Ω}, F T = F and finie ime horizon T <. 2.. The model. The individual has access o a complee financial marke consising of a risk-free asse B and a risky asse S, evolving according o he following dynamics: db = rb d, ds = µs d + σs dw. The ineres rae r >, he drif of he risky asse µ > and he volailiy of he risky asse σ > are assumed o be consan and W is a Brownian moion under P on he given probabiliy space. The marke price of risk is denoed by θ = µ r. Since he given σ marke is complee, here exiss a unique sae-price densiy ξ ha suffices he following equaion: dξ = ξ (rd + θdw ), i.e. ξ = exp {( r 2 θ2 ) θw }, wih ξ =. The individual has an iniial wealh x > and a ime [, T ] invess a proporion of her wealh π ino he risky asse and coninuously consumes c. The conrols (π, c) are called admissible, if he consumpion process (c ) [,T ] and he porfolio process (π ) [,T ] are measurable and [ adaped o he filraion (F ) [,T ]. Furhermore, if c P a.s. ( ) ] 2 T [, T ] and E c d < P a.s. By assuming ha he remaining fracion of wealh ( π ) is invesed ino he risk-free asse, we guaranee ha he porfolio process is self-financing. The se of admissible conrols wih iniial wealh x is denoed by A(x ). The dynamics of he wealh of he invesor, X, are given by: dx = (r + π (µ r)) X d c d + π σx dw, wih X = x. The individual bases her decisions on expeced uiliy and aims o maximize he expeced uiliy of consumpion over he life-cycle. The uiliy funcion is denoed by U(, c) and will be specified laer. Her ime preferences are represened by her individual discoun rae ρ >. The objecive funcion is hence o maximize he following expeced uiliy over he se of all admissible conrols: [ E ] e ρ U(, c )d. 4

5 2.2. Time-varying risk aversion wihou habi formaion. We sar wih an individual wih a power uiliy funcion ha exhibis a ime-varying risk aversion (RA). The uiliy funcion is given by U(, c) = γ c γ, wih γ he ime-varying relaive risk aversion. The RA γ : [, T ] R + is a deerminisic, coninuous funcion in ime. For noaional purpose, we denoe is reciprocal by Φ = γ. The opimal invesmen and consumpion problem wih ime-varying relaive risk aversion is solved in Seffensen (2), by applying he approach of dynamic conrols and solving he problem via he Hamilon-Jacobi-Bellman equaion. We review his resul by alernaively using he maringale approach presened in Cox and Huang (989) and Karazas e al. (987). This is done in a similar way by Aase (29) and can, in conras o he dynamic programming approach, also be applied o non-markov asse dynamics. For he maringale approach, he problem is rewrien in he following form: [ sup E e ρ (π,c) A(x ) [ ] subjec o: E ξ c d γ c γ ] d, x. (2.) The soluion of he opimizaion problem (2.) is given in Aase (29), bu since we need is soluions o solve he laer problem including he habi, we formulae he soluion in he following proposiion and review is proof in he Appendix. Proposiion 2.. The soluion o he opimizaion problem (2.) is given by c =e ρφ (yξ ) Φ, π = µ r σ 2 X = X Φ s e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) (yξ ) Φs ds, e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) (yξ ) Φs ds and y is he opimal Lagrangian-muliplier, which is implicily given by Proof: Given in he Appendix. e ρφ y Φ e ( Φ)( r 2 θ2 Φ ) d = x. For he case of γ =, we define he uiliy as he limiing case U(, c) = log(c). This does no affec he laer resuls, since he soluion only requires he derivaive of he uiliy funcion. 5

6 For he properies of he opimal consumpion, Seffensen (2) shows ha, for a fixed ime poin, he consumpion is increasing in wealh and. Furhermore, in he case of an increasing γ, i is a convex funcion in wealh. In he following, we examine furher properies of he opimal soluion. Firs, we calculae he derivaive of he consumpion wih respec o wealh, for a fixed ime poin, using he represenaion of he soluion from in Proposiion 2.. For he calculaion, we ake he sae-price-densiy a ime as a funcion of he wealh x which is implicily defined by x = e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) y Φs ξ Φs ds. Applying he implici funcion heorem, we ge he following derivaive of he consumpion wih respec o wealh for a fixed ime poin : x c = ( ( Φ s e ρsφs e ( Φs)(r+ 2 θ2 Φ )(s ) (yξ ) ds) Φs Φ e ρφ (yξ ) Φ). We recognize ha his derivaive corresponds o he one from Seffensen (2). Bu wih his represenaion in he complee marke seing, we see ha for a consan RA γ γ, he Lagrangian muliplier y in he derivaive cancels ou. This does no happen for a ime-varying RA which is no consan. Therefore he effec of wealh on he consumpion does depend on he Lagrangian muliplier y, and hus on he iniial wealh x, when a ime-varying RA is aken ino accoun. Nex, we examine he average developmen of he consumpion over ime. Therefore, we consider is expeced value: E[c ] = e (r ρ)φ y Φ e (+Φ) 2 θ2 Φ = e B(), wih B() := (r ρ)φ Φ log(y) + ( + Φ ) 2 θ2 Φ. The derivaive of he expeced value wih respec o ime is given by [ Φ log(y) + E[c ] = B() eb() = (r ρ + 2 θ2 ) (Φ + Φ ) + 2 θ2 ( 2Φ Φ + Φ 2 ) ] E[c ]. Since e B() is posiive, he sign of he derivaive only depends on B(). For he special case of a consan RA γ γ, we have Φ =. Given ρ r, his gives ha E[c ] >. So if he individual discoun is less han or equal o he risk-free rae on he marke, he consumpion is increasing in expecaion. Especially, for a consan γ, he derivaive does no depend on he Lagrangian muliplier y and hence no on he iniial wealh. By including a ime-varying RA, we have ha he shape of he developmen of he expeced consumpion over ime does depend on he iniial wealh x. For he case of a ime-varying RA, we can formulae a condiion for he Lagrangian muliplier y and hus implicily on he 6

7 iniial wealh x, for which he expecaion of he consumpion is eiher sricly increasing or sricly decreasing over ime. Therefore, define for Φ : { z := exp [(r ρ + 2 ) Φ θ2 (Φ + Φ ) + ( ) ]} 2 θ2 2Φ Φ + Φ 2, [, T ]. Then we ge he following condiions for he sign of he derivaive of he expecaion: For Φ <, i.e. γ > : E[c ] > max z < y (2.2) E[c ] < y < min z For Φ >, i.e. γ < : E[c ] > y < min z E[c ] < max z < y (2.3) From condiion (2.2), we ge ha for an increasing risk aversion, he consumpion is in expecaion sricly decreasing for sufficienly small Lagrangian muliplier y, i.e. for sufficienly large iniial endowmen x. On he oher hand, if he iniial endowmen is small enough, i.e. y is big enough, he consumpion is sricly increasing in expecaion. For a decreasing risk aversion, i.e. condiion (2.3), he relaion is he oher way round. Since y is decreasing in x, given an iniial wealh x for which he Lagrangian muliplier y fulfills y < min z, he condiion is also fulfilled for larger values of x, i.e. he shape remains he same for larger x. 2 For he invesmen, we can no calculae he derivaives in a similar closed form. Is behavior is considered in he laer numerical analysis Time-varying risk aversion wih habi formaion. Afer describing he opimizaion problem for a ime-varying RA, we now exend he problem by including a habi level for he consumpion in he uiliy funcion. The habi level accouns for he fac ha 2 To show ha y is decreasing in x, define G(x, y) := e ρφ y Φ e ( Φ)( r 2 θ2 Φ ) d x. Then he Lagrangian muliplier is implicily defined by G(x, y) =. Since G is differeniable, we can apply he implici funcion heorem and ge for he derivaive of y wih respec o x : ( ) ( y = x y G(x T, y) G(x, y) = Φ e ρφ y Φ e ( Φ)( r 2 θ2 Φ d) ) < x Hence we have ha he Lagrangian muliplier is sricly decreasing in x. 7

8 individuals are assumed o ge used o heir sandard of living. Following Consaninides (99) we define he habi level h in he following way: h = h e β + α e β( s) c s ds, i.e. dh = (βh αc )d. The parameers α, β and h are required o be non-negaive. Furhermore, α < β is assumed o ensure ha he habi level is decreasing when he consumpion is equal o he habi level. Wih his definiion, he habi level is an exponenially smoohed average of pas consumpion. The exponenial weighs imply ha recen consumpion ges a higher weigh han earlier consumpion in he life-cycle. We assume ha consumpion and habi are addiive and ha he individual has a power uiliy wih a ime-varying coefficien of RA γ, as inroduced before. The uiliy of consumpion is hen given by U(, c, h) = γ (c h) γ. The Arrow-Pra measure of relaive risk aversion (RRA) for his uiliy funcion is given by: RRA = c 2U(,c,h) c 2 c U(,c,h ) = γ = γ + γ c h c c h. (2.4) Hence, he change of he RRA over ime is driven by wo pars: firs he ime-varying γ and second he relaion beween curren consumpion c and he habi h, which conains he pas consumpion. The firs summand, γ, does solely depend on exogenous changes in he aiude owards risk over he life-cycle. I is independen of he consumpion and he habi and hus of marke developmens. The second summand of he RRA, γ c, depends on h he ime-varying RA and also and he consumpion o habi raio. Hence, i is driven by he curren and pas consumpion of he individual and hus implicily by her curren and pas marke paricipaion. The magniude of he impac of he raio is eiher amplified or weakened, depending if γ is increasing or decreasing, respecively. Independen of he ime-varying RA, his par goes o infiniy, if he consumpion approaches he habi level. In his case he RRA goes o infiniy, wha ensures ha he consumpion remains above he habi level. If he consumpion o habi raio is large enough, e.g. he consumpion is wice he habi level, he effec of he second summand diminishes and he RRA is mainly driven by he ime-varying RA. As before, he individual s aim is o maximize he expeced uiliy of he consumpion over he life-cycle. Using he saic formulaion from Cox and Huang (989) and Karazas 8

9 e al. (987) he opimizaion problem is given by [ ] sup E e ρ (c h ) γ d (π,c) A(x ) γ [ ] subjec o: E ξ c d x. (2.5) To solve he problem, we use he resuls from Schroder and Skiadas (22) and rewrie he problem (2.5), called he primal problem, ino a dual problem, by defining a dual consumpion ĉ := c h. 3 We hen solve he dual problem and ransform is soluion o ge he soluion o he primal problem. The dual consumpion ĉ consiss of curren consumpion c reduced by he habi level h, which is some ficiious consumpion derived from pas consumpion. In his dual marke, an increase in curren consumpion, leads o an increase in he curren insan uiliy of he individual. Bu i also decreases fuure uiliy hrough a larger habi level, see e.g. Deemple and Zapaero (99). This has o be aken ino accoun, when pricing he consumpion in his dual marke and is refleced in he dual sae price densiy given by ˆξ = ξ (+αf ). The funcion F resuls from calculaing he coss of keeping he consumpion a he habi level. This can be seen when assuming ha from ime on, we consume exacly a he habi level, i.e. c s = h s, s >. Then he consumpion is deermined by dc s = (β α)c s ds, wih c = h, i.e. [ ] c s = e (β α)(s ) T h. The coss for his consumpion are given by E e (β α)(s ) ξ h s ξ ds, where E denoes he expecaion condiional on F. Wih ] F := ds = e (β α+r)(s ) ds = e (β α)(s ) E [ ξs ξ ( e (β α+r)(t ) ), α β r he coss o keep he consumpion a he habi level can be wrien as h F. This funcion F can also be inerpreed as he price of a coupon bond, for which he coupon is exponenially decreasing over ime, see e.g. Munk (28). Wih h F he cos o keep he consumpion a he habi level h, we ge a condiion for he iniial wealh a =. Namely, he iniial wealh needs o be sufficien o cover he iniial habi h, i.e. o finance a consumpion a leas a he iniial habi. Thus we require x > h F. 3 These primal and dual problems are no o be confused wih he dual problem formulaed for he maringale approach in Cox and Huang (989) and Karazas e al. (987), bu raher denoe a formulaion wih and wihou habi. 9

10 In he dual marke, we ge for he dual iniial wealh ˆx = x h F +αf. This definiion accouns for he fac ha h F is required o finance he minimal consumpion a he habi level and he adjused pricing of he consumpion. 4 Wih he above consideraions, we have compleely defined he dual marke and can formulae he following dual problem: 5 [ ] sup E e ρ ĉ γ d (ˆπ,ĉ) A(ˆx ) γ [ ] subjec o: E ˆξ ĉ d ˆx. (2.6) This dual problem corresponds o he problem (2.) and is soluion is presened in Proposiion 2.. Hence, we ge he soluion o he dual problem from he previously presened resuls and can ransform hem o he soluion for he primal problem (2.5). The following proposiion gives he opimal soluion o he primal problem. Proposiion 2.2. Assuming x > h F, he soluion o he opimizaion problem (2.5) wih ime-varying γ and habi level h is given by c = e ρφ y Φ ( + αf ) Φ ξ Φ + h, h = e (α β) h + α π = θ σ X X = h F + e (α β)( s) e ρsφs (yξ s ) Φs ( + αf s ) Φs ds, Φ s e ρsφs ( + αf s ) Φs e ( Φs)( r 2 θ2 Φ s)(s ) (yξ ) Φs ds, e ρsφs ( + αf s ) Φs e ( Φs)( r 2 θ2 Φ s)(s ) (yξ ) Φs ds, where he Lagrangian muliplier y is implicily defined by e ρφ y Φ ( + αf ) Φ e ( Φ)( r 2 θ2 Φ ) d = x h F. In he following, we use he dualiy resul from Schroder and Skiadas (22) o prove he Proposiion. 4 The special case of a consan habi equal o he iniial habi h, is achieved by seing α =. The only difference beween he primal and dual marke is hen he adjusmen of he iniial wealh by h F, which is required o finance he consan habi. 5 In Schroder and Skiadas (22) a dual marke price of risk ˆθ is used. Here we have ˆθ = θ, since our marke parameers are consan.

11 Proof: Using Proposiion 2., ˆξ = ξ (+αf ) and ˆx = x h F +αf, he opimal consumpion ĉ for he dual problem (2.6) is given by ĉ = e ρφ Φ Φ y ˆξ = e ρφ y Φ ξ Φ ( + αf ) Φ. Wih ĉ = c h, he opimal consumpion for he primal problem is given by c = e ρφ y Φ ( + αf ) Φ ξ Φ + h, where h is he habi level of he opimal consumpion. We know ha h fulfills he differenial equaion dh = (βh αc )d. Following he idea from Proposiion in Schroder and Skiadas (22) and plugging c = ĉ + h ino he differenial equaion, we ge dh = ((α β)h + αĉ )d. This equaion is solved by ĥ := e (α β) h + α e (α β)( s) ĉ s ds. By he uniqueness of he soluion, we ge ha h (c ) = ĥ(ĉ ) and can express h in he following form: h = e (α β) h + α e (α β)( s) e ρsφs (yξ s ) Φs ( + αf s ) Φs ds. For he Lagrangian muliplier y, we ransform he consrain from he dual problem and ge ha i is implicily given by e ρφ y Φ ( + αf ) Φ e ( Φ)( r 2 θ2 Φ ) d = x h F. The condiion α < β ensures ha F is non-negaive in [, T ]. Hence, he lef-hand-side of he above equaion is coninuous and sricly decreasing in y. Furhermore, i goes o infiniy for y approaching zero and goes o zero for y approaching infiniy. Wih he condiion ha x > h F, he soluion o y exiss. The opimal wealh of he dual problem is given by ˆX = e ρsφs y = Φs Φs ˆξ E ( ˆξs ˆξ ) Φs ( + αf e ρsφs y Φs ξ Φs s ) Φs ds + αf e ( Φs)( r 2 θ2 Φ s)(s ) ds.

12 Following Schroder and Skiadas (22), we obain he following opimal wealh of he primal problem: X = h F + ( + αf ) ˆX = h F + e ρsφs y Φs ξ Φs ( + αf s ) Φs e ( Φs)( r 2 θ2 Φ s)(s ) ds. From Proposiion 2., we ge he opimal dual invesmen: ˆπ = ˆθ ) Φs T Φ s e ρsφs Φs Φs y ˆξ E ( ˆξs ds σ ˆξ = θ σ ˆX ˆX ( + αf Φ s e ρsφs y Φs ξ Φs s ) Φs e ( Φs)( r + αf 2 θ2 Φ s)(s ) ds. Following Schroder and Skiadas (22) and he represenaion in Munk (28), he opimal ( ) invesmen for he primal problem is given by π = h F ˆπ X. Wih ˆX = X h F +αf we ge he opimal porfolio process for he primal problem: ( ) X π = h F θ T ( + αf Φ s e ρsφs y Φs ξ Φs s ) Φs e ( Φs)( r σ + αf ge = θ σ X X X h F +αf Φ s e ρsφs y Φs ξ Φs ( + αf s ) Φs e ( Φs)( r 2 θ2 Φ s)(s ) ds. 2 θ2 Φ s)(s ) ds To compare our resuls o he previous lieraure, we se γ = γ (i.e. Φ = ) and hence γ X h F = e ρs γ (yξ ) γ ( + αfs ) γ e ( γ )( r 2 θ2 γ )(s ) ds (yξ ) X γ = h F. e ρs γ ( + αfs ) γ e ( γ )( r 2 θ2 γ )(s ) ds This gives he following resuls for he opimal conrols π = θ σ γ X h F, X c = e ρ γ (yξ ) γ ( + αf ) γ + h = ( + αf ) X γ h F e ρ(s ) γ ( + αfs ) γ e ( γ )( r 2 θ2 γ )(s ) ds + h, which correspond o he soluions presened in Munk (28) for consan θ and consan r. 2

13 Similar as before, we calculae he expeced value of he opimal consumpion, o examine he change over ime of he average consumpion: E[c ] =e ρφ y Φ ( + αf ) Φ E [ ] ξ Φ + e (α β) h + α =e (r ρ)φ y Φ ( + αf ) Φ e (+Φ) 2 θ2 Φ + e (α β) h + α =e B() + e (α β) h + α e (α β)( s) e ρsφs y Φs ( + αf s ) Φs E [ ] ξs Φs ds e (α β)( s) e (r ρ)sφs y Φs ( + αf s ) Φs e (+Φs) 2 θ2 Φ ss ds e (α β)( s) e B(s) ds wih B() := (r ρ)φ + ( + Φ ) 2 θ2 Φ Φ log(y) Φ log( + αf ) The firs derivaive wih respec o ime is given by d d E[c ] = (α + dd ) ( B() e B() + (α β) e (α β) h + α ) e (α β)( s) e B(s) ds, wih db() d = (r ρ + 2 ) θ2 (Φ + Φ ) + ) 2 θ2 (2Φ Φ + Φ 2 ) Φ log(y) (Φ αf log( + αf ) + Φ. + αf Here we have, as in he case wihou habi, ha he shape of he average consumpion over ime depends on he iniial wealh, implicily hrough he Lagrangian muliplier y. For a consan RA γ, we have ha Φ = and hus ge db() d >, since F <. Hence, he derivaive would be posiive, as long as (α + dd ) ( B() e B() > (α β) e (α β) h + α ) e (α β)( s) e B(s) ds, where he righ hand side is posiive, since α β <. This means ha he expeced value would be increasing, as long as he curren expeced consumpion is large enough compared o he expeced habi level. Especially he sign of he derivaive would no depend on he iniial wealh. By including he ime-varying RA, he average consumpion may also decrease depending on he RA and he Lagrangian muliplier, i.e. he iniial wealh x. 3

14 As before, we can no calculae a closed form for he expecaion of he opimal invesmen. In he following numerical secion we calculae i numerically and examine he effecs on he decision rules for specific choices of ime-varying RA. We see ha he effecs may change for differen iniial wealh. 3. Numerical resuls In his secion, we analyze he effec of specific choices for he ime-varying RA and he combinaion wih habi formaion on he opimal soluions. Firs, we examine he changes over ime by numerically calculaing some descripive saisics. Second, we analyze he effec of wealh, by numerically calculaing he decision rules as funcions of wealh. For he financial marke, we fix he parameers similar o he ones used in Munk (28): r = 2%, µ = 8%, σ = 2%, θ = 3% and ake an individual discoun of ρ = r, a ime horizon of T = 3 years and an iniial wealh of x =. Comparing he resuls for differen iniial endowmens x, we find ha he resuls in case of x = are similar for even bigger endowmens. Therefore, we decide for his value and aferwards discuss he effec of smaller iniial endowmens. For he habi formaion, we use parameers similar o he ones in Munk (28) and choose he iniial habi h proporional o our larger iniial wealh. In oal, we consider four differen parameerizaions of he habi level, shown in Table, represening differen magniudes of habi formaion. No habi Low habi Medium habi High habi α = α =. α =.3 α =.4 β = β =.2 β =.4 β =.5 h = h = 2 h = 3 h = 4 Table. Habi formaion parameers. For he changes in risk aversion over he life-cycle, as noed in he inroducion, he lieraure does no agree on he direcion he risk aversion is changing. Therefore, we consider boh increasing and decreasing risk aversions. As base case, we choose a consan risk aversion of γ = 3, o compare our observaions o previous resuls in he lieraure. The oher parameerizaions are chosen such ha he average over he life-cycle of he risk 4

15 aversion, i.e. γ d, is he same for all funcions γ. 6 Their parameers are given in he following Table 2 and heir shapes are shown in Figure. Oher parameerizaions and possible differen shapes are discussed in he following. Consan Linear increasing Convex increasing Linear decreasing γ = 3 γ = γ = γ = Table 2. Parameerizaions for he ime-varying risk aversion. 6 Consan 6 Linear increasing 6 Convex increasing 6 Linear decreasing Figure. Differen shapes for RA γ over he life-cycle of he individual. For he cases where we can no find closed forms of he saisics, we perform Mone- Carlo simulaions wih N = pahs and pariion each year ino 5 subinervals. 3.. Effec of ime on he opimal decisions. In he following, we examine he changes of he opimal consumpion and invesmen over ime. Therefore, we calculae heir expeced value and variance The base case: consan risk aversion. We sar wih he base case of a consan RA of γ 3, o compare our resuls o previous lieraure. Figure 2 shows he corresponding expeced value and he variance for he opimal consumpion c and he expeced value for he corresponding opimal habi level h. The four images show he four differen parameerizaions for he habi formaion wih no habi on he very lef and a high habi 6 As shown in equaion (2.4), he relaive risk aversion (RRA) is no he same in he case wihou and wih habi. Comparing differen RRA values for he case wihou habi, we find ha he shape of he resuls remains similar, only heir absolue values change. Therefore we decide o ake he same γ for he case wihou and wih habi. 5

16 on he righ side. The lower row shows he expeced value and variance for he opimal invesmen π. No habi Low habi Medium habi High habi 8 E[c * ] Var(c * ) 8 E[c * ] Var(c * ) E[h * ] 8 E[c * ] Var(c * ) E[h * ] 8 E[c * ] Var(c * ) E[h * ] E[π * ] Var(π * ).6.5 E[π * ] Var(π * ).6.5 E[π * ] Var(π * ) E[π * ] Var(π * ) Figure 2. Saisics for he opimal consumpion, habi and invesmen for a consan risk aversion over ime. The opimal consumpion increases in expecaion over ime for all presened parameerizaions. The inclusion of a habi level leads o a smooher consumpion, as seen by he smaller variance of c. For larger habi parameers, he average consumpion sicks closer o he habi level. Therefore, he pah of he expeced consumpion and habi level are more similar as for he lower habi parameers. The opimal invesmen is consan and equal o µ r σ 2 γ = 5% for he case of no habi. Including he habi level, leads o a decrease of he expeced invesmen over ime. I decreases sronger owards he end of he life-cycle, o ensure sufficien wealh for a consumpion above he habi. We hus observe ha by he inclusion of a habi level, we achieve a decreasing shape of he average opimal invesmen. The opimal consumpion is smooher when a habi level is included, bu is increasing in expecaion. 6

17 3..2. Linear increasing risk aversion. Nex, we examine he case of an increasing RA and presen he resuls for he linear increasing γ in Figure 3. 5 No habi E[c * ] Var(c * ) 5 Low habi E[c * ] Var(c * ) E[h * ] 5 Medium habi E[c * ] Var(c * ) E[h * ] 5 High habi E[c * ] Var(c * ) E[h * ] E[π * ] Var(π * ).7.6 E[π * ] Var(π * ).7.6 E[π * ] Var(π * ).7.6 E[π * ] Var(π * ) Figure 3. Saisics for he opimal consumpion, habi and invesmen for a linear increasing risk aversion over ime. For he case of no habi, he opimal consumpion is increasing over ime in expecaion. Tha follows direcly from condiion (2.2) for hese parameer combinaions. When a habi level is included, he consumpion sars a smaller values and is changes over ime are much smooher. The average habi level exhibis a hump-shaped form in all hree cases. Bu for he low habi, he expeced consumpion is far above he habi level and does no adap is shape. Unlike for he wo cases of larger habi formaion, where he consumpion follows he shape of he habi level and hus exhibis a hump-shaped paern. The opimal invesmen, in he case wihou any habi, is decreasing in expecaion, since he RA is increasing. For he cases wih habi, he increasing RA leads o a srong decrease of he invesmen in he firs years. This is differen o he case of a ime-consan RA, where he invesmen decreases very slow in he firs years. As before, we can see ha by including he habi level, we sar a a smaller iniial invesmen. This follows from he 7

18 higher relaive risk aversion, resuling from he desire o ensure a consumpion above he habi. For differen parameers of he linear increasing γ he overall shape remains similar, only he magniude of he resuls changes. Differen shapes for γ would be a convex or a concave increasing one. For he firs one, we ge a more disinc hump in he shape of he expeced consumpion. The average invesmen in his case has a convex shape and is decreasing even sronger. The second one, a concave increasing shape, leads o consumpion and invesmen decisions which sar in expecaion a very large values, heavily drop down in he firs years and hen slowly decrease in he laer years. Since he resuls for he firs case are similar o he linear shape and for he concave case seem no in line wih empirical observaions, we do no presen hese wo cases in more deail. The combinaion of a habi level wih an increasing RA hus leads o a furher smoohed consumpion. The average consumpion evenually exhibis a hump-shaped paern, depending on he magniude of he habi formaion. The invesmen is sricly decreasing in expecaion, which is necessary o ensure he smooh consumpion above he habi level Linear decreasing risk aversion. Las, we consider he case of a decreasing RA over ime. The resuls are presened in Figure 4. In conras o he previous resuls, we observe a very srong increase over ime in he expecaion of he consumpion and habi level. This increase is reduced for higher habi formaions. Considering he difference beween he consumpion and he habi level, hey are closes a he beginning and hen drif apar a he end. This follows from he smaller RA a he end of he life-cycle, compared o he beginning and is he oher way round for he increasing RA. In addiion o he expeced value, also he variance is increasing, resuling in a highly volaile consumpion a he end of he life-cycle. The opimal invesmen is increasing in expecaion. Bu in he case of a habi level, begins o decrease again owards he end and hus exhibis a hump-shaped paern. This resuls from he need o ensure ha he large and owards he end highly volaile consumpion remains above he habi level. Regarding differen parameerizaions for γ, we ge similar shapes for he expeced consumpion and invesmen, only he absolue values are differen. Also for differen shapes of γ, e.g. convex or concave decreasing, we ge similar resuls. For he firs one he resuls are even more exreme, whereas for he laer one, hey are more moderae. 8

19 No habi Low habi Medium habi High habi 6 5 E[c * ] Var(c * ) 6 5 E[c * ] Var(c * ) E[h * ] 6 5 E[c * ] Var(c * ) E[h * ] 6 5 E[c * ] Var(c * ) E[h * ] E[π * ] Var(π * ).8 E[π * ] Var(π * ).8 E[π * ] Var(π * ) E[π * ] Var(π * ) Figure 4. Saisics for he opimal consumpion, habi and invesmen for a linear decreasing risk aversion over ime. The average opimal consumpion for a decreasing RA is hus differen o he observaions in he lieraure. On he oher hand, he average invesmen exhibis a hump-shaped form as observed in some lieraure presened in he inroducion The effec of he iniial wealh. As previously discussed, he shape of he expeced consumpion depends on he iniial wealh. For he case of no habi level, we can even express he dependence in closed form. For example for a linear increasing risk aversion, no habi level and he above menioned parameers i holds ha he expeced value of he opimal consumpion is increasing in x and decreasing in x For values of x in beween, he expeced value is increasing in he firs years and decreasing in he laer years. We have herefore compared he resuls for differen values of he iniial wealh x. When he RA is consan, i.e. γ γ, he shape of he decision rules are similar, only he absolue values change proporionally o he change in he iniial endowmen. Bu when a ime-varying RA is considered, in paricular for an iniial wealh of x =, he 9

20 resuls for a linear increasing and linear decreasing γ are almos reversed. 7 Namely, when a habi level is included, he iniial wealh is x = and he RA is linear increasing, he resuls are similar as for he case of x = and linear decreasing RA. Excep ha he average consumpion increases more moderae and he invesmen is decreasing sronger owards he end, when x =. On he oher hand, when x = and he RA is linear decreasing, he resuls are similar as for he case of x = and linear increasing RA. Only for he case of x = he hump in he expeced consumpion is much more disinc and he average invesmen is smaller and hence decreases less srong. In he siuaion wihou a habi level, x = and RA linear increasing, he opimal invesmen is sill decreasing in expecaion. On he oher hand, he opimal consumpion is increasing, similar as for a decreasing RA and x =, only more moderae. For x = and RA linear decreasing, he average invesmen is increasing as one would expec. Bu he average consumpion is decreasing as for he case of x = and increasing RA. Thus he resuls when a habi level is included and linear increasing and decreasing RA is considered are almos opposie for iniial endowmen of one and. For he case wihou a habi level, only he consumpion is opposie, he invesmen sill has a similar shape Effec of wealh on he opimal decisions. Nex we examine he effec of wealh on he opimal consumpion and invesmen decision. Since we can no express eiher c nor π in dependence of he opimal wealh, we numerically calculae hem as funcions of he wealh. As shown before, he shapes depend heavily on he iniial wealh x. Therefore, we consider boh an iniial wealh of x = and x = in he following. Since he resuls are similar for he hree habi levels, we limi he following o he case of a medium habi parameerizaion. For he RA, we compare he case of a consan RA, linear increasing RA and linear decreasing RA, as before. As fixed ime poins we choose = 5 and = 2, o compare he resuls early and lae in he life-cycle. For ime poins closer o mauriy, e.g. = 25, he effecs are similar as for = 2. 7 When he iniial wealh is reduced, also he iniial habi has o be reduced. For x =, we choose iniial habi of h =.2, h =.3, h =.4 for he differen habi formaions respecively. 2

21 3.2.. Opimal consumpion. The opimal consumpion c as funcion of wealh is shown in Figure 5. The lef image shows = 5 and he righ one = 2. The op row shows he case of an iniial endowmen of x = and he boom row of x =. I sands ou ha he consumpion in he case of a habi formaion can no go below a cerain wealh level. This level corresponds o he minimum wealh h F, which is required o finance he fuure habi level = 5 consan γ linear increasing linear decreasing including habi level = 2 c *(X).3 c *(X) X X c *(X) 4 c *(X) X X Figure 5. Opimal consumpion c as funcion of wealh for fixed ime poins = 5 (lef) and = 2 (righ) for iniial endowmen x = (op row) and x = (boom row) for hree differen ime-varying RA. Lines wihou a round marker represen consumpion wihou a habi level and lines wih a round marker he consumpion including a habi level. We observe ha for all presened cases he consumpion is increasing in wealh and i increases sronger laer in he life-cycle. Whereas he increases is less srong when a habi level is included, compared o no habi formaion. The laer resuls, since an increase in consumpion leads o an insan addiional uiliy, bu a decreased fuure uiliy hrough a higher habi level. For he differen RA we see ha for a consan risk aversion of γ 3 he consumpion is linear in wealh, for an increasing RA convex in wealh and for a 2

22 decreasing RA concave in wealh. This was shown for he case wihou habi in Seffensen (2) and is also observable for he consumpion including a habi level. So far he observaions are similar for he wo iniial endowmens. When considering he differences beween he resuls for differen values of x, in he case of a consan RA, he consumpion changes proporional o he change of he iniial wealh. Bu when considering he order of he lines represening he consumpion for differen ime-varying RA, we see ha he order is reversed for he wo iniial endowmens. For x = he consumpion is highes for a decreasing RA, for x = i is highes for an increasing RA. This is similar o he previous secion, where we see ha he effec of increasing and decreasing RA is reversed for he wo differen iniial endowmens Opimal invesmen. Nex, we show he effec of wealh on he opimal invesmen π. Figure 6 presens he resuls in he same way as previously for he consumpion..7 = 5.7 = π *(X) consan γ linear increasing. linear decreasing including habi level X π *(X) X π *(X).4.3 π *(X) X X Figure 6. Opimal invesmen π as funcion of wealh for fixed ime poins = 5 (lef) and = 2 (righ) for iniial endowmen x = (op row) and x = (boom row) for hree differen ime-varying RA. Lines wihou a round marker represen consumpion wihou a habi level and lines wih a round marker he consumpion including a habi level. 22

23 For mos cases he opimal invesmen is increasing in wealh. Only for an increasing RA wih a habi level, i is decreasing in wealh lae in he life-cycle, i.e. a = 2. One of he mos significan observaions is ha he invesmen decision for he case of habi formaion goes owards zero early in he life-cycle, when he wealh is close o he minimal wealh level required o finance he fuure habi. This effec vanishes laer in he life-cycle (i.e. a = 2), since he coss of he habi level are much smaller a ha ime poin. Regarding he case of no habi level and comparing he resuls for he differen iniial endowmens, we see ha he differences are bigger a = 5, han a = 2, showing ha he effec of he iniial endowmen is larger early in he life-cycle han laer, where he invesmen decision is more influenced by he marke han by he iniial endowmen. For he case including a habi level, he difference beween he iniial endowmens are large early and lae in he life-cycle, bu have differen effecs. Early in he life-cycle he order for a decreasing and increasing RA are reversed, as discussed before. Lae in he life-cycle, he invesmen decisions for he differen RA are similar when x =, bu are far apar, when x =. 4. Conclusion In his paper, we combine a ime-varying risk aversion wih a habi level in he opimal consumpion and invesmen problem. In a complee marke, we firs review he problem wihou a habi level. Using hese resuls and he dual approach from Schroder and Skiadas (22), we subsequen find he soluion for he problem wih a ime-varying risk aversion and a habi level. In a numerical analysis, we compare he effecs of differen shapes of ime-varying risk aversion and differen magniudes of habi formaion on he opimal soluion. We compare boh increasing and decreasing shapes for he risk aversion, as is observed in several lieraure. Our findings show ha he shape of he opimal decision rules depends on he iniial wealh, wha is no he case for a ime-consan risk aversion. Furhermore, wih an increasing risk aversion we achieve a hump-shaped consumpion paern and decreasing invesmen in he risky asse wih increasing ime. The resuls are hus more in line wih empirical observaions. 23

24 In our seing, we consider a simple marke model and neglec furher fuure endowmens by e.g. labor income. Furher research migh consider a more complex marke model or consider he inclusion of labor income ino he presened seing. The habi level is considered as addiive, considering a non-addiive habi level migh be of furher ineres. The presened uiliy funcion omis he ime separabiliy of uiliy, bu sill he conceps of risk aversion and elasiciy of ineremporal subsiuion are mixed. Furher research migh ry o disenangle hese effecs furher and for example ransfer he concep of ime-varying risk aiudes o recursive uiliy funcions. References Aase, K. K. (29): The invesmen horizon problem: A resoluion. Available a SSRN: hp://dx.doi.org/.239/ssrn Agnew, J., Balduzzi, P. and Sundén, A. (23): Porfolio choice and rading in a large 4(k) plan. The American Economic Review 93, Al-Ajmi, J. Y. (28): Risk olerance of individual invesors in an emerging marke. Inernaional Research Journal of Finance and Economics 7, Ameriks, J. and Zeldes, S. P. (24): How do household porfolio shares vary wih age. Working paper, Columbia Universiy Business School: hp:// Aanasio, O. P., Banks, J., Meghir, C. and Weber, G. (999): Humps and bumps in lifeime consumpion. Journal of Business & Economic Saisics 7, Bakshi, G. S. and Chen, Z. (994): Baby boom, populaion aging, and capial markes. Journal of Business 67, Bellane, D. and Green, C. A. (24): Relaive risk aversion among he elderly. Review of Financial Economics 3, Bellane, D. and Saba, R. P. (986): Human capial and life-cycle effecs on risk aversion. Journal of Financial Research 9, 4 5. Berau, C. C. and Sarr, M. (2): Household porfolios in he Unied Saes. Available a SSRN: hp://dx.doi.org/.239/ssrn Bullard, J. and Feigenbaum, J. (27): A leisurely reading of he life-cycle consumpion daa. Journal of Moneary Economics 54,

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26 Munk, C. (28): Porfolio and consumpion choice wih sochasic invesmen opporuniies and habi formaion in preferences. Journal of Economic Dynamics and Conrol 32, Riley, W. B. and Chow, K. V. (992): Asse allocaion and individual risk aversion. Financial Analyss Journal 48, Schroder, M. and Skiadas, C. (22): An isomorphism beween asse pricing models wih and wihou linear habi formaion. Review of Financial Sudies 5, Seffensen, M. (2): Opimal consumpion and invesmen under ime-varying relaive risk aversion. Journal of Economic Dynamics and Conrol 35, Sundaresan, S. M. (989): Ineremporally dependen preferences and he volailiy of consumpion and wealh. Review of Financial Sudies 2, Wang, H. and Hanna, S. (997): Does risk olerance decrease wih age? Financial Counseling and Planning 8, Yao, R., Sharpe, D. L. and Wang, F. (2): Decomposing he age effec on risk olerance. The Journal of Socio-Economics 4, Appendix Proof of Proposiion 2.. Following he mehodology described in Cox and Huang (989), he firs order condiion for he opimal soluion o c is given by e ρ c γ = yξ, which gives an opimal consumpion of c = e ρφ (yξ ) Φ. Plugging c Lagrangian muliplier y: ino he consrain, we ge he following equaion implicily defining he The opimal wealh a ime is given by X = [ ] E ξ s c ξ sds = e ρφ y Φ e ( Φ)( r 2 θ2 Φ ) d = x. [ e ρsφs y Φs E ( ξ ] s ) Φs ξ Φs ds ξ = e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) (yξ ) Φs ds, where we changed inegraion and condiional expecaion as done in e.g. Munk (28) and used E [ ( ξ s ξ ) Φs ] = e ( Φs)(r+ 2 θ2 Φ s)(s ). In order o ge he opimal asse allocaion, we follow Chen e al. (2) and calculae he opimal amoun invesed ino he sock α 26 = X S and hen he opimal proporion

27 π = α S. Therefore, we reformulae he sae price densiy as a funcion of he sock X price process S, i.e. ξ =: g(, S ). Then α is given as he dela-hedge: α = X S = = µ r σ 2 S For he opimal proporion π π = α S X = µ r σ 2 X e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) y Φs g(, S ) Φs ds S Φ s e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) (yξ ) Φs ds. follows: Φ s e ρsφs e ( Φs)(r+ 2 θ2 Φ s)(s ) (yξ ) Φs ds. 27