Time Consistency in Portfolio Management

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1 1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland

2 Time Consisency in Porfolio Managemen 1 Moivaion There are a leas wo examples in porfolio managemen ha are ime inconsisen: Maximizing uiliy of ineremporal consumpion and final wealh assuming a non-exponenial discoun rae. Mean-variance uiliy. I means ha he agens may have an incenive o deviae from heir decisions ha were opimal in he pas. I is no ofen he case ha managemen decisions are irreversible; here will usually be many opporuniies o reverse a decision which, as imes goes by, seems ill-advised.

3 Time Consisency in Porfolio Managemen 2 Maximizing uiliy of ineremporal consumpion and final wealh We work in a Black and Scholes world, wih riskless rae r : ds() = S() [α d + σ dw ()], 0, Consider a self-financing porfolio. The oal value is X, he amoun invesed in he sock is ζ and he consumpion rae is c, hen wih µ = α r : dx ζ,c () = X ζ,c ()[(r + µζ() c()) d + σζ() dw ()]. The invesor a ime [0, T ] uses he crierion: [ T ] J(, x, ζ, c) E h(s )U(c(s)X ζ,c (s)) ds + a(t )U(X ζ,c (T )) Xζ,c () = x.

4 Time Consisency in Porfolio Managemen 3 The invesor s characerisics J(, x, ζ, c) E [ T ] h(s )U(c(s)X ζ,c (s)) ds + a(t )U(X ζ,c (T )) Xζ,c () = x. T is a sopping ime (deah of he invesor). In he sequel, we will ake i as deerminisic. U( ) is he uiliy funcion. In he sequel we will use U p (x) = xp p wih p < 1. h( ) is he psychological discoun rae. We assume ha 0 h() h(0) = 1, and h( ) = 0. a( ) is he beques coefficien.

5 Time Consisency in Porfolio Managemen 4 Examples J(, x, ζ, c) E [ T ] h(s )U(c(s)X ζ,c (s)) ds + a(t )U(X ζ,c (T )) Xζ,c () = x. h() = a() = exp ( ρ). This is he classical (Meron) problem. h() = exp ( ρ) and a() h(). h() = (1 + a) b a (hyperbolic discouning), a() = nh(). The las wo cases have srong empirical suppor, bu hey do no fall wihin he classical framework. Indeed, hey give rise o ime inconsisency on he par of he invesor, so ha here is no implemenable opimal porfolio.

6 Time Consisency in Porfolio Managemen 5 Time-inconsiency for dummies Consider an individual who wans o sop smoking: If he sops oday, he will suffer -1 oday (wihdrawal), bu gain +2 omorrow (healh). He has a non-consan discoun rae: a sream u is valued oday ( = 0) a u ρ u for some ρ ( 1 2 2, 1) =1 Sopping oday yields a uiliy of 1 + ρ < 0. Sopping omorrow yields a uiliy of ( 1+2ρ) 2 > 0. So he decides oday o sop omorrow. Unforunaely, when omorrow comes, i becomes oday, and he decides again o sop he nex day.

7 Time Consisency in Porfolio Managemen 6 Time-inconsisency for mahemaicians In he exponenial case, uiliies discouned a ime 0 and > 0 are proporional [ ] T E e ρ(s ) U(c(s)X ζ,c (s)) ds + e ρ(t ) U(X ζ,c (T )) = e ρ E [ T e ρs U(c(s)X ζ,c (s)) ds + e ρt U(X ζ,c (T )) In he non-exponenial case, his is no longer he case. The HJB equaion wrien for he invesor a ime is ]

8 Time Consisency in Porfolio Managemen 7 V s (, s, x)+sup ζ,c [ (r + µζ c)x V x (, s, x) σ2 ζ 2 x 2 2 V x ] (, s, x) 2 + h (s ) V (, s, x) + U(xc) = 0, h(s ) V (, T, x) = a(t )U(x) which obviously depends on (so every day he invesor changes his crierion of opimaliy).

9 Time Consisency in Porfolio Managemen 8 Markov Sraegies A Markov sraegy is a pair (F (, x), G(, x)) of smooh funcions. Invesmen and consumpion raes are given by: ζ() = F (, X()), c() = X() G(, X()), X() and he wealh hen is a soluion of he sochasic differenial equaion (SDE): dx(s) = [rx(s)+µf (s, X(s)) G(s, X(s))]ds+σF (s, X(s))dW (s). Subsiuing ino he crierion, we ge: J(, x, F, G) E [ T ] h(s )U(c(s)X ζ,c (s)) ds + a(t )U(X ζ,c (T )) Xζ,c () = x,

10 Time Consisency in Porfolio Managemen 9 Equilibrium Sraegies We say ha (F, G) is an equilibrium sraegy if a any ime, he invesor finds ha he has no incenive o change i during he infiniesimal period [, + ɛ]. Definiion: (F, G) is an equilibrium sraegy if a any ime, for every ζ and c : lim ɛ 0 J(, x, F, G) J(, x, ζ ɛ, c ɛ ) ɛ 0, where he process {ζ ɛ (s), c ɛ (s)} s [0,T ] is defined by:

11 Time Consisency in Porfolio Managemen 10 [ζ ɛ (s), c ɛ (s)] = [F (s, X(s)), G(s, X(s))] 0 s [ζ(s), c(s)] s + ɛ [F (s, X(s)), G(s, X(s))] + ɛ s T and he equilibrium wealh process is given by he SDE: dx(s) = [rx(s)+µf (s, X(s)) G(s, X(s))]ds+σF (s, X(s))dW (s).

12 Time Consisency in Porfolio Managemen 11 The inegral equaion An equilibrium sraegy is given by F (, x) = µ v x (, x), G(, x) = I σ 2 2 v x (, x) 2 ( ) v (, x), I = (U ) 1. x where v saisfies he inegral equaion: [ T ] v(, x) = E h(s )U(G(s, X(s))) ds + a(t )U(X(T )) X() = x, where {X(s)} s [0,T ] is given by he SDE dx(s) = [rx(s)+µf (s, X(s)) G(s, X(s))]ds+σF (s, X(s))dW (s).

13 Time Consisency in Porfolio Managemen 12 In a differenial form he inegral equaion has a non-local erm. ( v! (, x)+ rx I E [ T ( )) v v µ2 (, x) (, x) x x h (s )U ( I 2σ 2 [ v ( )) v x (s, X,x (s)) x (, x)]2 2 v x (, x) +U 2 ( I ( )) v (, x) x ds + a (T )U(X,x (T )) For he special case of exponenial discouning i coincides wih he HJB equaion since ]. = E [ T h (s )U ( I ( )) v x (s, X,x (s)) ds + a (T )U(X,x (T )) ] = ρv(, x).

14 Time Consisency in Porfolio Managemen 13 Ansaz Assume U(x) = U p (x) = xp p. Le us look for he value funcion v of he form v(, x) = λ()x p, so ha: F (, x) = µx σ 2 (1 p), G(, x) = [λ()] 1 p 1 x. This linearizes he equilibrium wealh dynamics: ] dx() [r X() = (α r)2 1 + σ 2 1 p [λ()] 1 p 1 d+ (α r) σ 1 dw () 1 p The inegral equaion becomes

15 Time Consisency in Porfolio Managemen 14 λ() = T +a(t )e K(T ) e λ(t ) = 1, where h(s )e K(s ) [λ(s)] K = p R T p[λ(u)] 1 ( r + p p 1 e p 1 du «µ 2 ) 2(1 p)σ 2. R s p[λ(u)] 1 «p 1 du ds+ This equaion has a unique smooh soluion.

16 Time Consisency in Porfolio Managemen 15 Logarihmic uiliy Take U(x) = ln x (corresponding o p = 0) and a(t ) = nh(t ). We ge an explici formula: λ() = T h(s )e K(s ) ds + nh(t )e K(T ), and he equilibrium policies are F (, x) = µ σ 2 x, G(, x) = [λ()] 1 x.

17 Time Consisency in Porfolio Managemen 16 Numerical Resuls Le us consider he following discoun funcions: exponenial-h 0 (), pseudo-exponenial ype I-h 1 () and pseudo-exponenial ype II-h 2 () h 0 () = a 0 () = exp( ρ), h 1 () = a 1 () = λ exp( ρ 1 )+(1 λ) exp( ρ 2 ), h 2 () = a 2 () = (1 + λ) exp( ρ). For α = 0.12, σ = 0.2, r = 0.05, he discoun facors ρ 1 = 0.1, ρ 2 = 0.3 and he weighing parameer λ = 0.25, CRRA p = 1 we graph equilibrium consumpion raes

18 Time Consisency in Porfolio Managemen 17 1 (a) c (1) (2) (4) 0.4 (3) (1) ρ 2 exp; (2) λ, ρ 1, ρ 2 ype I ; (3) ρ 1 exp; (4) λ, ρ 1 ype II

19 Time Consisency in Porfolio Managemen 18 Take CRRA p = (b) c (1) (2) 0.4 (4) 0.3 (3) (1) ρ 2 exp; (2) λ, ρ 1, ρ 2 ype I ; (3) ρ 1 exp; (4) λ, ρ 1 ype II

20 Time Consisency in Porfolio Managemen 19 Take CRRA p = 0 1 (c) c 0.6 (1) (2) (4) 0.3 (3) (1) ρ 2 exp; (2) λ, ρ 1, ρ 2 ype I ; (3) ρ 1 exp; (4) λ, ρ 1 ype II

21 Time Consisency in Porfolio Managemen 20 Take CRRA p = (d) (1) 0.9 (2) c 0.6 (3) 0.5 (4) (1) ρ 2 exp; (2) λ, ρ 1, ρ 2 ype I ; (3) ρ 1 exp; (4) λ, ρ 1 ype II

22 Time Consisency in Porfolio Managemen 20 Consumpion Puzzle Sandard models predic ha consumpion will grow smoohly over ime (or i will decrease smoohly). Household daa indicae ha consumpion is hump-shaped. This inconsisency is known as he consumpion puzzle. Hyperbolic discouning can explain his puzzle. Le us consider: h() = (1 + a) b a, a() = n(1 + a) b a.

23 Time Consisency in Porfolio Managemen h(x) wih h(1)=0.3 a=5 a=10 a=

24 Time Consisency in Porfolio Managemen 22 Le us graph equilibrium consumpion raes 1.2 n=1,t= a=5 a=10 a=

25 Time Consisency in Porfolio Managemen n=10,t= a=5 a=10 a=

26 Time Consisency in Porfolio Managemen n=30,t= a=5 a=10 a=

27 Time Consisency in Porfolio Managemen 25 Mean Variance Uiliy The risk crierion is: J(, x, ζ) = E[X ζ (T ) X ζ () = x] γ 2 V ar(xζ (T ) X ζ () = x). Time inconsisency arrises from he risk crierion non-lineariy wr expeced value of he erminal wealh. Bjork and Murgoci found ha he equilibrium invesmen is F (, x) = 1 γ µ σ 2 e r(t ).

28 Time Consisency in Porfolio Managemen 26 The End! Thank You!

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