Constrained portfolio choices in the decumulation phase of a pension plan M. Di Giacinto 1 S. Federico 2 F. Gozzi 3 E. Vigna 4 1 Universit` a degli Studi di Cassino - Italy 2 LUISS University, Roma - Italy 3 LUISS University, Roma - Italy 4 Universit` a degli studi di Torino - Italy Workshop on Stochastic Control in Finance Roscoff, March 18–23, 2010 M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 1 / 32
Plan of the talk Plan of the talk Motivations. State equation and optimization problems. (P1) Constraints on the strategies. ◮ Explicit solution and optimal feedback by verification. (P2) Constraints on the strategies and on the wealth. ◮ Viscosity approach. ◮ Regularity of the value function. ◮ Explicit solution and optimal feedback by verification in a special case. Future targets. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 2 / 32
Motivations Motivations Depending on the laws, in many countries the retiree is allowed for a certain period after retirement: 1 to withdraw a periodic income from the fund; 2 to invest the rest of the fund in the period between retirement and annuitization. Thus, in this period the pensioner can: 1 decide how much of the fund to withdraw at any time; 2 decide the strategy to adopt to invest the fund at her/his disposal. → Investment/consumption Merton problem, which can be solved using, e.g. stochastic optimal control techniques. We focus on the last problem: Fixed withdrawal/consumption rate. How to invest optimally? − → Portfolio allocation problem with special features. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 3 / 32
Motivations Motivations Depending on the laws, in many countries the retiree is allowed for a certain period after retirement: 1 to withdraw a periodic income from the fund; 2 to invest the rest of the fund in the period between retirement and annuitization. Thus, in this period the pensioner can: 1 decide how much of the fund to withdraw at any time; 2 decide the strategy to adopt to invest the fund at her/his disposal. → Investment/consumption Merton problem, which can be solved using, e.g. stochastic optimal control techniques. We focus on the last problem: Fixed withdrawal/consumption rate. How to invest optimally? − → Portfolio allocation problem with special features. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 3 / 32
Motivations Motivations Depending on the laws, in many countries the retiree is allowed for a certain period after retirement: 1 to withdraw a periodic income from the fund; 2 to invest the rest of the fund in the period between retirement and annuitization. Thus, in this period the pensioner can: 1 decide how much of the fund to withdraw at any time; 2 decide the strategy to adopt to invest the fund at her/his disposal. → Investment/consumption Merton problem, which can be solved using, e.g. stochastic optimal control techniques. We focus on the last problem: Fixed withdrawal/consumption rate. How to invest optimally? − → Portfolio allocation problem with special features. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 3 / 32
Motivations Motivations Depending on the laws, in many countries the retiree is allowed for a certain period after retirement: 1 to withdraw a periodic income from the fund; 2 to invest the rest of the fund in the period between retirement and annuitization. Thus, in this period the pensioner can: 1 decide how much of the fund to withdraw at any time; 2 decide the strategy to adopt to invest the fund at her/his disposal. → Investment/consumption Merton problem, which can be solved using, e.g. stochastic optimal control techniques. We focus on the last problem: Fixed withdrawal/consumption rate. How to invest optimally? − → Portfolio allocation problem with special features. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 3 / 32
The state equation and the optimization problems The state equation and the optimization problems t = 0 retirement time; T > 0 annuitization time (horizon of the problem); x 0 fund wealth at t = 0; X ( · ) process representing the fund wealth (state variable) ; π ( · ) process representing the amount of money invested in the risky asset (control variable) ; b 0 consumption rate of the pensioner; r , λ, σ usual market parameters in the Black-Scholes model. � dX ( s ) = [ rX ( s ) + σλπ ( s ) − b 0 ] ds + σπ ( s ) dB ( s ) , s ∈ [0 , T ] X (0) = x 0 . M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 4 / 32
The state equation and the optimization problems F ( · ) is a target we aim to reach. � � F ( s ) = b 0 F − b 0 e − r ( T − s ) , r + r where F ∈ (0 , b 0 / r ) is such that x 0 < F (0). NOTE: If we reach the target, investing the whole wealth in the riskless asset keeps the wealth on the target. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 5 / 32
The state equation and the optimization problems F ( · ) is a target we aim to reach. � � F ( s ) = b 0 F − b 0 e − r ( T − s ) , r + r where F ∈ (0 , b 0 / r ) is such that x 0 < F (0). NOTE: If we reach the target, investing the whole wealth in the riskless asset keeps the wealth on the target. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 5 / 32
The state equation and the optimization problems Cost functional: �� T � κ e − ρ s ( F ( s ) − X ( s )) 2 ds + e − ρ T ( F ( T ) − X ( T )) 2 J = E ≥ 0 , 0 where κ ≥ 0. NOTE: If we reach the target at time t , investing the whole wealth in the riskless asset from t on yields 0 in the remaining part of the functional above. We can say that F ( · ) is an optimal absorbing boundary for the problem. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 6 / 32
The state equation and the optimization problems Cost functional: �� T � κ e − ρ s ( F ( s ) − X ( s )) 2 ds + e − ρ T ( F ( T ) − X ( T )) 2 J = E ≥ 0 , 0 where κ ≥ 0. NOTE: If we reach the target at time t , investing the whole wealth in the riskless asset from t on yields 0 in the remaining part of the functional above. We can say that F ( · ) is an optimal absorbing boundary for the problem. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 6 / 32
The state equation and the optimization problems [Gerrard, Haberman & Vigna, 2004] minimize J without constraints on the strategies and on the wealth. We study the minimization of the functional in the cases (P1) constraint on the strategies (no short selling): � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted Admissible Strategies = ; (P2) constraint on the strategies (no short selling) and on the wealth (no ruin): Admissible Strategies = � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted | X ( t ; π ( · )) ≥ 0 , t ∈ [0 , T ] . M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 7 / 32
The state equation and the optimization problems [Gerrard, Haberman & Vigna, 2004] minimize J without constraints on the strategies and on the wealth. We study the minimization of the functional in the cases (P1) constraint on the strategies (no short selling): � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted Admissible Strategies = ; (P2) constraint on the strategies (no short selling) and on the wealth (no ruin): Admissible Strategies = � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted | X ( t ; π ( · )) ≥ 0 , t ∈ [0 , T ] . M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 7 / 32
The state equation and the optimization problems [Gerrard, Haberman & Vigna, 2004] minimize J without constraints on the strategies and on the wealth. We study the minimization of the functional in the cases (P1) constraint on the strategies (no short selling): � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted Admissible Strategies = ; (P2) constraint on the strategies (no short selling) and on the wealth (no ruin): Admissible Strategies = � � π ( · ) ∈ L 2 (Ω × [0 , T ]; R + ) adapted | X ( t ; π ( · )) ≥ 0 , t ∈ [0 , T ] . M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 7 / 32
(P1) Constraint on the strategies (P1) Constraint on the strategies We follow a classic dynamic programming approach to solve the problem, proceeding along the following steps: We define the value function V ( t , x ) as the optimum for generic initial data t ∈ [0 , T ] , x ≤ F ( t ). We associate to the value function the HJB equation. We find an explicit solution to the HJB equation. We prove a verification theorem which, as a byproduct: ◮ says that this solution is indeed the value function; ◮ gives a way to define an optimal strategy by this function. M. Di Giacinto, S. Federico, F. Gozzi, E. Vigna ( ) Constrained portfolio choices . . . 8 / 32
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