On Consistent Decision Making. and the Theory of Continuous-Time. Recursive Utility

Transcription

1 On Consistent Decision Making and the Theory of Continuous-Time Recursive Utility Mogens Ste ensen Köln, April 23, 202 /5

2 Outline: Quadratic and collective objectives Further comments on consistency Recursive utility Putting the pieces together 2/5

3 Quadratic and collective objectives Portfolio selection db (t) = rb (t) dt ds (t) = S (t) dt + S (t) dw (t) dx (t) = (r + (t) ( r)) X (t) dt + (t) X (t) dw (t) Hamilton-Jacobi-Bellman equation for the continuum of problems V t = inf V (T; x) = u (x) V (t; x) = sup E t;x [u (X (T ))] (r + ( r)) xv x x 2 V xx 3/5

4 Mean-variance optimization mean-variance utility quadratic optimization inf V ar [X (T )] sup E [X (T )] V ar [X (T )] :E[X(T )]k 2 How should they be embedded in a continuum of problems? a) Classical but problematic: V (t; x) = sup E t;x X (T ) :E[X(T )]=k 2 (X (T ) k)2 b) 200 and onwards: (Basak and Chabakauri, Björk and Murgoci, Kryger and Steffensen) V (t; x) = sup sup E t;x [X (T )] 2 V ar t;x [X (T )] h ae t;x [X (T )] + be t;x X 2 (T ) i + c E t;x [X (T )] 2 4/5

5 Kryger and Ste ensen: Simple market, general objective including: mean-variance, mean-std, endogenous habit formation, collective utility V (t; x) = sup Pseudo-Bellman equation (V t f t ) = inf 0 0 x; E t;x [g (X (T ))]; E t;x [h (X (T ))] {z } G (r + ( r)) x (V x f x ) {z } H U = f GG G 2 x + 2f GH G x H x + f HH H 2 x CC AA x 2 (V xx U) 5/5

6 Quadratic objectives sup sup sup E t;x [X (T )] E t;x [X (T )] sup 2 V ar t;x [X (T )] 2x V ar t;x [X (T )] 2 Std t;x [X (T )] : = (t) x : = (t) E t;x [X (T )] : = 0 X E t;x (T ) xe (T t) 2 : = (t) 6/5

7 Collective objectives n sup u Et;x [u (X (T ))] + u 2 Et;x [u 2 (X (T ))] o : = (t) For power utility = (t) 2 r with (t) being a weighted average of and 2 Nice mathematics - interesting economics! 7/5

8 Further comments on consistency Consider a two period model: linear objective - sup over all strategies t = 0 t = The t = 0 strategy 50% 60% The t = strategy 60% non-linear objective - sup over all strategies t = 0 t = The t = 0 strategy 40% 50% The t = strategy 45% non-linear objective - sup over consistent strategies t = 0 t = The t = 0 strategy 42% 45% The t = strategy 45% 8/5

9 a) linear objective : V (t; x) = sup E t;x [u (X (T ))] b) non-linear objective : V (t; x) = sup f E t;x [u (X (T ))] The solution to the linear problem a) is time-consistent The solution to the non-linear problem b) is time-inconsistent if we choose among all strategies But we can restrict the set of strategies to include only time-consistent strategies Then we nd a solution to b) which is, per construction, time-consistent Of course, the value function is smaller than if there were no restriction! 9/5

10 Recursive utility Wealth dynamics dx (t) = (r + (t) (a r)) X (t) dt + (t) X (t) dw (t) c (t) dt Classical value function based on time-additive utility V (t; x) = sup c; E t;x " Z n t e (s t) c (s) ds # But is more than risk aversion: The deterministic problem ( = 0) V (t; x) = sup c " Z n t e (s t) c (s) ds is well-posed and has a nice explicit solution. So, is also related to Elasticity of Intertemporal Subtitution # 0/5

11 Recursive Utility Classic (power utility): V (t; x) = {z} W aggregator W = c (t) ; ( ) E t;x [V (t + ; X (t + ))] 4 c (t) + e {z } certainty equivalent of value function 0 C A ( ) Et;x [V (t + ; X (t + ))] {z } certainty equivalent of value function 3 7 C A 7 5 Complicating problems with di erentiability of the certainty equivalent and the aggregator! Du e and Epstein (992), Kraft and Seifried (200), Kraft, Seifried and Ste ensen (200) /5

12 Putting the pieces together V (t; x) = sup ;c 2 Z n 6 4 t e (s t) 0 ( ) E t;x " c (s) #! {z } certainty equivalent of consumption C A ds Since the value function is messed up by a series of non-linearities, we have to think carefully about consistency! We look for a solution in the set of consistent strategies! This explains why Du e and Epstein could prove consistency AND why the global formulation has not been proposed before! (?) 2/5

13 One more time: V (t; x) = sup ;c 2 Z n 6 4 t e (s t) 0 ( ) E t;x " c (s) #! {z } certainty equivalent of consumption C A ds Recognize time-additive utility as special case ( = ) " Z " n V (t; x) sup e (s t) E t;x ;c t c (s) " Z " n = sup E t;x e (s t) ;c t c (s) # # ds ds # # 3/5

14 The good news: For a simple (Black-Sholes) nancial market the solution to the local and the global formulations coincide: The value function is essentially equivalent with the solution to V t = inf f (c; V ) ((r + ( r)) x c) V c; x x 2 V xx + U f (c; V ) = V = 00 c (( ) V ) C A C A This classical continuous-time aggregator f (c; V ) corresponds to U = 0 4/5

15 The bad (?) news: This classical continuous-time aggregator f (c; V ) does not correspond to U = 0. So, we are doing something di erent than Du e and Epstien, (992), Chacko and Viceira (2005), Kraft, Seifried and Ste ensen (20) and others... But how are the two problems and theis solutions related? 5/5