Monotonicity of Savings Function in Endogenous Gridpoint Method with Stochastic Portfolio Returns

This paper provides a comprehensive proof of monotonicity of the savings function in the application of the Method of Endogenous Gridpoints (EGM) to problems with stochastic portfolio returns. The proof contributes to the completeness of solutions by providing the sufficient condition for the application of EGM to problems with stochastic portfolio returns as seen in the literature.


Introduction
The method of endogenous gridpoints (EGM), introduced by Carroll (2006), provides significant computational efficiency in solving dynamic optimization problems compared to the traditional value function iteration methods (Iskhakov, 2015).These computational benefits have seen widespread use of EGM in the literature, including problems that involve stochastic asset returns.For instance, Love (2013) consider a dynamic model with stochastic asset returns and income shocks to solve for optimal consumption and portfolio allocation policy functions.Other works include Koijen et al. (2010) and Wu et al. (2023).
The monotonicity of pre-decision and post-decision state variables is a pivotal sufficient condition for the application of EGM (White, 2015).However, the application of EGM in problems with stochastic portfolio returns in the literature fails to provide a comprehensive monotonicity check.This omission represents a critical gap in the existing literature and may lead to potential inaccuracies in derived solutions (Iskhakov et al., 2017).Establishing a rigorous monotonicity proof not only strengthens the robustness of the method but also extends its applicability, offering insights that may facilitate further advancements in computational efficiency and solution accuracy (Fella, 2014).Therefore, this paper aims to provide the proofs for the monotonicity relationship between pre-decision and post-decision state variables when portfolio returns are stochastic.
The remainder of this paper is structured as follows.Section 2 introduces the specific problem setting and presents the corresponding monotonicity proofs.Section 3 discusses the key conclusions drawn from the proofs as well as suggestions for further research.Iskhakov et al. (2017) provide the proof of monotonicity of the savings function for dynamic optimization problems with deterministic portfolio returns.However, the literature has not extended this type of proof to include stochastic investment returns, a gap that we aim to fill.In Section 2.1 we formulate a representative problem with stochastic investment returns, derive the optimal policy functions and highlight the need for monotonicity in the EGM application.Section 2.2 then provides the proof of monotonicity.

An illustrative problem and the need for monotonicity
Consider a discrete-time, finite-horizon dynamic optimization problem with one state variable of pre-decision wealth   , for  = 0, 1, … ,  , and two continuous decision variables: consumption choice   and proportion of portfolio allocated to the risky asset   , for  = 0, 1, … ,  − 1.Let the constant risk-free return and the stochastic risky asset return be designated as  and R+1 , respectively, for  = 0, 1, … ,  −1.The risky asset's return generates a natural filtration, denoted as , where  0 is trivial.In particular,   acts as a dynamic decision variable constrained within F-adapted bounds π and π (i.e.,   ∈ [ π , π ]).For example, π and π can be set deterministically as 0 and 1, respectively.The portfolio's investment return,  +1 , for  = 0, 1, … ,  − 1, is then calculated as   × R+1 + (1 −   ) × .Alongside the consumption choice   , constrained within F-adapted bounds C and C (i.e.,   ∈ [ C , C ]; for example, C = 0 and C =   ), the wealth dynamic is given by ) is non-decreasing and 1-Lipschitz in   .We denote   =   −   as the post-decision state variable.
To ensure non-negative individual wealth, we require the portfolio's investment return   × R+1 + (1 −   ) ×  to be no less than −1 for all possible R+1 states.We define   = ess inf R+1 and   = ess sup R+1 .Sufficient conditions to ensure this requirement include   <  <   and − 1+ Consequently, by using the instantaneous non-decreasing concave differentiable utility function (⋅) and the subjective discount factor  ∈ (0, 1], the Bellman equation is given by: with the terminal condition as ) .Applying the Karush-Kuhn-Tucker (KKT) conditions to Eq. ( 1), we derive the following optimality theorem.
as the Lagrangian multipliers for the lower and upper constraints for consumption, respectively; also, denote  3  and  4  for the lower and upper constraints for the risky asset allocation proportion, respectively.For any )) )) (3) The optimal consumption and risky allocation strategies, and the Lagrange multipliers, of the constrained optimization problem are given by: where Proof.See Appendix.□ At each time  =  − 1,  − 2, … , 1, 0, both equations in (5) are fully non-linear with respect to the two decision variables,   and   .The essential idea of the EGM is that, rather than solving two variables   and   simultaneously in (5) as in standard Value Function Iteration methods, EGM defines an endogenous post-decision state variable   which acts as a separator between the decision variables and significantly improves computational efficiency (Carroll, 2012).
Given the savings function of   =   −  *  (  ), we can perform the 'relabeling' of the optimal policies as long as there is a one-toone correspondence between   and   .White (2015) calls this the 'weak monotonicity' condition of the savings functions and notes that this ensures that the decision and value functions can be approximated using interpolation on the endogenous grid points.

Proofs of monotonicity under the EGM application
Inspired by Iskhakov et al. (2017), we can study the monotonicity of the savings function by modifying the Bellman equation ( 1), given an allocation to risky asset   and for  = 0, 1, … ,  − 1: In the remaining of this paper, we make the following assumption.
Assumption 2.1.The problem is convex, i.e., the value function   (⋅) is globally concave.□ This assumption ensures that the KKT conditions are sufficient for generating a global optimum.Moreover, this assumption is in line with the literature, such as Love (2013), Love and Phelan (2015), and Wu et al. (2023), which study a similar optimality problem with risky asset allocation decisions; see also Hintermaier and Koeniger (2010), White (2015), and Ludwig and Schön (2018).We note that concavity typically exists in the value functions of models that have only continuous decision variables, as well as a concave and differentiable utility (Iskhakov et al., 2017).
To examine the monotonicity of the savings function, we present the following theorem.) is weakly monotonic with respect to   .□ Proof.As a function of   and   , the maximand in ( 8) is given by, where   is the decision variable, When applying the KKT conditions to this optimization problem, the optimal post-decision balance  * can either belong to the interior feasible set (i.e.,  * ∈ ( − C,  − C)) or sit on the boundary.If  * sits on the boundary, it is obvious to see that weak monotonicity exists as  * ()  ≥ 0 due to the assumption that C and C are 1-Lipschitz in .Otherwise, if  * is in the interior feasible set then, following Edlin and Shannon (1998) (Theorem 1), we need to show that: 1.  (, ) is differentiable and has continuous first-order partial derivatives (i.e.,  (, ) is  1 ); 2. the optimal result  * lies within the interior of the interval At time  =  − 1, partially differentiating Eq. ( 9) we get: and where . To prove that  (  −1 ,   −1 ) is  1 , we need to check whether both partial derivatives exist and following equalities hold: , (⋅) is differentiable, and   and   are finite, the partial derivatives and the equalities in Eq. ( 12) hold.Thus, condition 1 is satisfied.Further, differentiating Eq. ( 10) with respect to   −1 , shows that the result of  (,) Hence, condition 3 is satisfied.Finally, since is negative,  * is a local maximum and this ensures that condition 2 is satisfied.Thus, all three conditions are satisfied, and the monotonicity theorem in Edlin and Shannon (1998) applies at time  =  − 1.

Conclusions
In this paper, we have provided the monotonicity proof for a simple and fairly general life-cycle problem with stochastic portfolio returns.We show the need for the monotonicity proof in applying EGM in such problems, and hence provide an important sufficient condition for the application of EGM.
Our proof can easily be modified to dynamic optimization problems involving stochastic investment returns and deterministic portfolio choices.With a little effort, the proof can also be adopted to more sophisticated models as those presented in Love (2013) and Wu et al. (2023). 1 Through this proof, we expect further application of EGM in more sophisticated settings in the future.
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