Oimal Consumion and Invesmen wih Habi Formaion and Hyerbolic discouning Mihail Zervos Dearmen of Mahemaics London School of Economics Join work wih Alonso Pérez-Kakabadse and Dimiris Melas 1
The Sandard Oimisaion Problem (SOP) A Model wih Hyerbolic Discouning Invesmen Decisions wih Habi Formaion and Hyerbolic Discouning Porfolio Oimisaion wih Habi Formaion as a Secial Case of he SOP (Many references!...) 2
The Sandard Oimisaion Problem (SOP) We consider he sandard fricionless, arbirage-free, comlee marke driven by a sandard one-dimensional Brownian moion W. The value rocess X of an admissible orfolio - consumion air (Π, C) saisfies he SDE dx = rx d C d + σπ θ d + σπ dw, (1) where r 0 is he shor-erm ineres rae, σ is he volailiy of he risky asse, and θ is he marke relaive risk. Admissible choices of a sraegy (Π, C) are such ha C > 0 and he rocess X given by (1) is well-defined and sricly osiive. The objecive is o maximise [ T ] E U 1 (s, C s ) ds + U 2 (X T ) X = x over all admissible airs (Π, C), where U 1 (s, ) and U 2 are given uiliy funcions. (2) 3
A Model wih Hyerbolic Discouning We wan o solve v(, T, x) = su E (Π,C) subjec o [ T ] q(s)u 1 (C s ) ds + U 2 (X T ) X = x, (3) where dx = rx d C d + σπ θ d + σπ dw, (4) U 1 (c) = c for some ]0, 1[ and ζ > 0, and for some consans β, γ > 0. and U 2 (x) = ζx, (5) q() = (1 + β) γ/β, (6) This roblem is a secial case of he SOP. The ineres in his roblem arises from he hyerbolic discouning funcion q given by (24). Exerimenal evidence suggess ha economic agens may have relaively high discouning raes over shor horizons and relaively low discouning raes over long horizons: q() q() = γ 1 + β. (7) 4
The HJB equaion of his roblem is given by w (, T, x) { 1 + su (π,c) 2 σ2 π 2 w xx (, T, x) + [ rx c + σθπ ] w x (, T, x) + q() } c = 0, wih boundary condiion (8) w(t, T, x) = ζx. (9) In ligh of he sandard heory regarding he SOP, he value funcion v of our oimal conrol roblem saisfies he PDE v (, T, x) 1 2 θ2 v2 x(, T, x) v xx (, T, x) + rxv x(, T, x) wih boundary condiion We look for a soluion of he form + 1 q1/(1 ) ()vx /(1 ) (, T, x) = 0, (10) v(t, T, x) = ζx. (11) v(, T, x) = f(, T )x. (12) 5
Subsiuing his exression for v in (10) (11), we obain f (, T ) = ξf(, T ) h()f /(1 ) (, T ), (13) f(t, T ) = ζ. (14) where [ θ 2 ] ξ = 2(1 ) + r, (15) and h() = (1 ) /(1 ) q 1/(1 ) (). (16) The soluion of his ODE is given by [ f(, T ) = e ξ (ζe ξt ) 1 1 + 1 T e ξ 1 1 s (1 + βs) β(1 ) ds]. γ (17) 6
Using he general heory, we can conclude ha he value funcion of our roblem is given by v(, T, x) = f(, T )x, (18) he oimal invesmen sraegy is given by Π = and he oimal consumion rae is given by θ σ(1 ) X, (19) C = ( ) 1/(1 ) q() X. (20) f(, T ) The oimal sraegy as well as he value funcion are non-saionary: hey deend on boh and T, no jus on he ime o mauriy T. Non-saionary models are indeed used in finance. For insance, he Hull and Whie ineres rae model resuls in non-saionary discoun bond rices. The non-saionariy of his invesmen model, which arises by he inroducion of hyerbolic discouning, may be aroriae for individual invesors, bu may no be so for insiuional invesors such as ension funds. 7
Invesmen Decisions wih Habi Formaion and Hyerbolic Discouning We wan o solve v(, T, x, a) [ T = su E (Π,C) subjec o ] q(s)u 1 (C s A s ) ds + U 2 (X T ) X = x, A = a, (21) dx = rx d C d + σπ θ d + σπ dw, da = δa d + C d, (22) where U 1 (c a) = (c a) for some ]0, 1[ and ζ > 0, and and U 2 (x) = ζx, (23) for some consans β, γ > 0. q() = (1 + β) γ/β, (24) This roblem is no obviously a secial case of he SOP. 8
The HJB equaion of his roblem is given by { 1 w (, T, x, a) + su (π,c) 2 σ2 π 2 w xx (, T, x, a) + [ rx c + σθπ ] w x (, T, x, a) + (c δa)w a (, T, x, a) + q() } (c a) = 0, (25) wih boundary condiion Incororaing he firs order condiions and w(t, T, x, a) = ζx. (26) σ 2 πw xx (, T, x, a) + σθw x (, T, x, a) = 0, (27) w a (, T, x, a) w x (, T, x, a) + q()(c a) 1 = 0, (28) ha arise from he choice of π and c, we obain w (, T, x, a) 1 2 θ2 w2 x(, T, x, a) w xx (, T, x, a) + (rx a)w x(, T, x, a) + (1 δ)aw a (, T, x, a) + q()(1 ) wih boundary condiion ( ) /( 1) (wx w a )(, T, x, a) = 0, (29) q() w(t, T, x, a) = ζx. (30) 9
We look for a soluion of he form w(, T, x, a) = f(, T ) [ x + k(, T )a ]. (31) Subsiuing his exression for v in (29) (30), we obain [ θ 2 f (, T ) 2( 1) + k ] (, T )a + rx a + (1 δ)k(, T )a f(, T ) x + k(, T )a ( ) /( 1) q()(1 ) [1 k(, T )]f(, T ) + = 0, (32) q() f(t, T ) = ζ and k(t, T ) = 0. (33) To eliminae a from (32), we se k (, T )a + rx a + (1 δ)k(, T )a = r[x + k(, T )a]. (34) This exression and (32) imly ha [ θ 2 ] f (, T ) 2( 1) + r f(, T ) ( ) /( 1) q()(1 ) [1 k(, T )]f(, T ) + = 0. (35) q() Also, (34) imlies ha k (, T ) = (1 δ r)k(, T ) + 1. (36) 10
We can calculae ha he soluion of he sysem of ODEs (35) and (36) wih boundary condiions (33) is given by k(, T ) = and 1 e(1 δ r)(t ) 1 δ r (37) f(, T ) = e ξ { (ζe ξ ) 1 1 + 1 1 T e ξ 1 s [ 1 1 e(1 δ r)(t ) 1 δ r ] 1 (1 + βs) γ β(1 ) ds} 1. (38) Building on hese calculaions, we can derive exlici exressions for he value funcion and he oimal sraegy. Again, hese are nonsaionary. 11
Porfolio Oimisaion wih Habi Formaion as a Secial Case of he SOP Consider he differenial equaion Given C > 0, he rocess C defined by da = δa d + C d. (39) C = e (δ 1) [A 0 + e (δ 1) C + 0 ] e (δ 1)s Cs ds (40) and he corresonding rocess A defined by (39) saisfy C A = C. Problem 1. The objecive of he SOP is o maximise he erformance crierion [ T ] J,T,x (Π, C) = E Ũ 1 (s, C s ) ds + U 2 ( X T ) X = x, (41) over all admissible airs (Π, C), subjec o d X = r X d C d + σπ θ d + σπ dw. (42) Problem 2. The objecive of he orfolio oimisaion roblem wih habi formaion is o maximise he erformance crierion [ T ] J,T,x,a (Π, C) = E U 1 (s, C s A s ) ds + U 2 (X T ) X = x, A = a, subjec o dx = rx d C d + σπ θ d + σπ dw, (43) da = δa d + C d. (44) 12
Using Iô s formula, we calculae d(x + k(, T )A ) = r(x + k(, T )A ) d If we le k saisfy and we define + [k (, T ) (δ + r)k(, T )]A d [1 k(, T )]C d + σπ θ d + σπ dw. (45) k (, T ) (δ + r)k(, T ) = 1 k(, T ), (46) X = X + k(, T )A and C = [1 k(, T )](C A ), (47) hen (...) Furhermore, if hen and rovided ha d X = r X d C d + σπ θ d + σπ dw. (48) k(t, T ) = 0, (49) X T = X T, (50) J,T,x,a (Π, C) = J,T,x+k(,T )a (Π, C), (51) Ũ 1 (s, z) = U 1 ( s, z/[1 k(s, T ) ). (52) I follows ha solving he orfolio oimisaion roblem wih habi formaion is a secial case of he SOP! 13