Optimal Consumption and Investment with Habit Formation and Hyperbolic discounting Mihail Zervos Department of Mathematics London School of Economics Joint work with Alonso P´ erez-Kakabadse and Dimitris Melas 1
• The Standard Optimisation Problem (SOP) • A Model with Hyperbolic Discounting • Investment Decisions with Habit Formation and Hyperbolic Dis- counting • Portfolio Optimisation with Habit Formation as a Special Case of the SOP (Many references!...) 2
The Standard Optimisation Problem (SOP) • We consider the standard frictionless, arbitrage-free, complete mar- ket driven by a standard one-dimensional Brownian motion W . • The value process X of an admissible portfolio - consumption pair (Π , C ) satisfies the SDE dX t = rX t dt − C t dt + σ Π t θ dt + σ Π t dW t , (1) where r ≥ 0 is the short-term interest rate, σ is the volatility of the risky asset, and θ is the market relative risk. • Admissible choices of a strategy (Π , C ) are such that C > 0 and the process X given by (1) is well-defined and strictly positive. • The objective is to maximise �� T � U 1 ( s, C s ) ds + U 2 ( X T ) | X t = x (2) E t over all admissible pairs (Π , C ), where U 1 ( s, · ) and U 2 are given utility functions. 3
A Model with Hyperbolic Discounting • We want to solve �� T � v ( t, T, x ) = sup q ( s ) U 1 ( C s ) ds + U 2 ( X T ) | X t = x , (3) E (Π ,C ) t subject to dX t = rX t dt − C t dt + σ Π t θ dt + σ Π t dW t , (4) where U 1 ( c ) = c p U 2 ( x ) = ζx p , and (5) p for some p ∈ ]0 , 1[ and ζ > 0, and q ( t ) = (1 + βt ) − γ/β , (6) for some constants β, γ > 0. • This problem is a special case of the SOP. The interest in this problem arises from the hyperbolic discounting function q given by (24). Experimental evidence suggests that economic agents may have rel- atively high discounting rates over short horizons and relatively low discounting rates over long horizons: − ˙ q ( t ) γ q ( t ) = 1 + βt. (7) 4
• The HJB equation of this problem is given by w t ( t, T, x ) � 1 w x ( t, T, x ) + q ( t ) � 2 σ 2 π 2 w xx ( t, T, x ) + p c p � � + sup rx − c + σθπ ( π,c ) = 0 , (8) with boundary condition w ( T, T, x ) = ζx p . (9) In light of the standard theory regarding the SOP, the value function v of our optimal control problem satisfies the PDE 2 θ 2 v 2 v t ( t, T, x ) − 1 x ( t, T, x ) v xx ( t, T, x ) + rxv x ( t, T, x ) + 1 − p q 1 / (1 − p ) ( t ) v − p/ (1 − p ) ( t, T, x ) = 0 , (10) x p with boundary condition v ( T, T, x ) = ζx p . (11) • We look for a solution of the form v ( t, T, x ) = f ( t, T ) x p . (12) 5
• Substituting this expression for v in (10)–(11), we obtain f t ( t, T ) = − ξf ( t, T ) − h ( t ) f − p/ (1 − p ) ( t, T ) , (13) f ( T, T ) = ζ. (14) where θ 2 � � ξ = p 2(1 − p ) + r , (15) and h ( t ) = (1 − p ) p − p/ (1 − p ) q 1 / (1 − p ) ( t ) . (16) The solution of this ODE is given by � T � 1 − p � γ 1 − p + p − p e − ξ 1 1 − p s (1 + βs ) − β (1 − p ) ds f ( t, T ) = e − ξt ( ζe ξT ) . 1 − p t (17) 6
• Using the general theory, we can conclude that the value function of our problem is given by v ( t, T, x ) = f ( t, T ) x p , (18) the optimal investment strategy is given by θ Π ∗ σ (1 − p ) X ∗ t = t , (19) and the optimal consumption rate is given by � 1 / (1 − p ) � q ( t ) C ∗ X ∗ t = t . (20) pf ( t, T ) • The optimal strategy as well as the value function are non-stationary: they depend on both t and T , not just on the time to “maturity” T − t . Non-stationary models are indeed used in finance. For instance, the Hull and White interest rate model results in non-stationary discount bond prices. The non-stationarity of this investment model, which arises by the introduction of hyperbolic discounting, may be appropriate for indi- vidual investors, but may not be so for institutional investors such as pension funds. 7
Investment Decisions with Habit Formation and Hyperbolic Discounting • We want to solve v ( t, T, x, a ) �� T � = sup q ( s ) U 1 ( C s − A s ) ds + U 2 ( X T ) | X t = x, A t = a , E (Π ,C ) t (21) subject to dX t = rX t dt − C t dt + σ Π t θ dt + σ Π t dW t , dA t = − δA t dt + C t dt, (22) where U 1 ( c − a ) = ( c − a ) p U 2 ( x ) = ζx p , and (23) p for some p ∈ ]0 , 1[ and ζ > 0, and q ( t ) = (1 + βt ) − γ/β , (24) for some constants β, γ > 0. • This problem is not obviously a special case of the SOP. 8
• The HJB equation of this problem is given by � 1 2 σ 2 π 2 w xx ( t, T, x, a ) + � � w t ( t, T, x, a ) + sup rx − c + σθπ w x ( t, T, x, a ) ( π,c ) � + ( c − δa ) w a ( t, T, x, a ) + q ( t ) p ( c − a ) p = 0 , (25) with boundary condition w ( T, T, x, a ) = ζx p . (26) Incorporating the first order conditions σ 2 πw xx ( t, T, x, a ) + σθw x ( t, T, x, a ) = 0 , (27) and w a ( t, T, x, a ) − w x ( t, T, x, a ) + q ( t )( c − a ) p − 1 = 0 , (28) that arise from the choice of π and c , we obtain 2 θ 2 w 2 w t ( t, T, x, a ) − 1 x ( t, T, x, a ) w xx ( t, T, x, a ) + ( rx − a ) w x ( t, T, x, a ) + (1 − δ ) aw a ( t, T, x, a ) � p/ ( p − 1) + q ( t )(1 − p ) � ( w x − w a )( t, T, x, a ) = 0 , (29) p q ( t ) with boundary condition w ( T, T, x, a ) = ζx p . (30) 9
• We look for a solution of the form � p . � w ( t, T, x, a ) = f ( t, T ) x + k ( t, T ) a (31) • Substituting this expression for v in (29)–(30), we obtain θ 2 � 2( p − 1) + k t ( t, T ) a + rx − a + (1 − δ ) k ( t, T ) a � f t ( t, T ) − p f ( t, T ) x + k ( t, T ) a � p/ ( p − 1) + q ( t )(1 − p ) � p [1 − k ( t, T )] f ( t, T ) = 0 , (32) p q ( t ) f ( T, T ) = ζ and k ( T, T ) = 0 . (33) To eliminate a from (32), we set k t ( t, T ) a + rx − a + (1 − δ ) k ( t, T ) a = r [ x + k ( t, T ) a ] . (34) This expression and (32) imply that θ 2 � � f t ( t, T ) − p 2( p − 1) + r f ( t, T ) � p/ ( p − 1) + q ( t )(1 − p ) � p [1 − k ( t, T )] f ( t, T ) = 0 . (35) p q ( t ) Also, (34) implies that k t ( t, T ) = − (1 − δ − r ) k ( t, T ) + 1 . (36) 10
• We can calculate that the solution of the system of ODEs (35) and (36) with boundary conditions (33) is given by k ( t, T ) = 1 − e (1 − δ − r )( T − t ) (37) 1 − δ − r and �� ζe ξt � 1 f ( t, T ) = e − ξt 1 − p � p � T � 1 − p 1 − 1 − e (1 − δ − r )( T − t ) p − 1 � − γ e − ξ + p − 1 1 − p s β (1 − p ) ds (1 + βs ) . 1 − p 1 − δ − r t (38) • Building on these calculations, we can derive explicit expressions for the value function and the optimal strategy. Again, these are non- stationary. 11
Portfolio Optimisation with Habit Formation as a Special Case of the SOP • Consider the differential equation dA t = − δA t dt + C t dt. (39) Given ˜ C > 0, the process C defined by � t � � A 0 + e ( δ − 1) t ˜ e − ( δ − 1) s ˜ C t = e − ( δ − 1) t C t + C s ds (40) 0 and the corresponding process A defined by (39) satisfy C − A = ˜ C . • Problem 1 . The objective of the SOP is to maximise the perfor- mance criterion �� T � J t,T,x (Π , ˜ ˜ U 1 ( s, ˜ ˜ C s ) ds + U 2 ( ˜ X T ) | ˜ C ) = E X t = x , (41) t over all admissible pairs (Π , ˜ C ), subject to d ˜ X t = r ˜ X t dt − ˜ C t dt + σ Π t θ dt + σ Π t dW t . (42) • Problem 2 . The objective of the portfolio optimisation problem with habit formation is to maximise the performance criterion �� T � J t,T,x,a (Π , C ) = E U 1 ( s, C s − A s ) ds + U 2 ( X T ) | X t = x, A t = a , t (43) subject to dX t = rX t dt − C t dt + σ Π t θ dt + σ Π t dW t , dA t = − δA t dt + C t dt. (44) 12
• Using Itˆ o’s formula, we calculate d ( X t + k ( t, T ) A t ) = r ( X t + k ( t, T ) A t ) dt + [ k t ( t, T ) − ( δ + r ) k ( t, T )] A t dt − [1 − k ( t, T )] C t dt + σ Π t θ dt + σ Π t dW t . (45) If we let k satisfy k t ( t, T ) − ( δ + r ) k ( t, T ) = 1 − k ( t, T ) , (46) and we define ˜ ˜ X t = X t + k ( t, T ) A t and C t = [1 − k ( t, T )]( C t − A t ) , (47) then (...) d ˜ X t = r ˜ X t dt − ˜ C t dt + σ Π t θ dt + σ Π t dW t . (48) Furthermore, if k ( T, T ) = 0 , (49) then ˜ X T = X T , (50) and J t,T,x,a (Π , C ) = ˜ J t,T,x + k ( t,T ) a (Π , ˜ C ) , (51) provided that ˜ � � U 1 ( s, z ) = U 1 s, z/ [1 − k ( s, T ) . (52) • It follows that solving the portfolio optimisation problem with habit formation is a special case of the SOP! 13
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