[Paper Review] Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management

Zuzana Macova, Daniel Ševčovič|ArXiv.org|May 1, 2009

Stochastic processes and financial applications3 references17 citations

TL;DR

This paper develops a weakly nonlinear analysis of the Hamilton-Jacobi-Bellman (HJB) equation for optimal dynamic asset allocation in pension savings, using asymptotic expansions and bounds to approximate the optimal stock-to-bond proportion. The key contribution is a first-order analytical approximation of the optimal strategy that enables sensitivity analysis and validation against discrete-time models, with numerical results showing accumulated wealth of 5.2× and 8.1× the final salary for Slovakian and Bulgarian pension systems, respectively.

ABSTRACT

The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as well as quantitative point of view. We construct useful bounds of solution yielding estimates for the optimal value of the stock to bond proportion in the portfolio. Furthermore we construct asymptotic expansions of a solution in terms of a small model parameter. Finally, we perform sensitivity analysis of the optimal solution with respect to various model parameters and compare analytical results of this paper with the corresponding known results arising from time-discrete dynamic stochastic optimization model.

Motivation & Objective

• To formulate the optimal portfolio selection problem in pension savings as a fully nonlinear parabolic HJB equation.
• To derive analytical bounds and asymptotic expansions for the solution of the HJB equation under weakly nonlinear assumptions.
• To perform sensitivity analysis of the optimal stock-to-bond proportion with respect to model parameters.
• To validate the continuous-time analytical results against a known discrete-time dynamic stochastic optimization model.
• To provide a practical approximation method for the optimal investment strategy in pension fund management.

Proposed method

• Formulate the optimal asset allocation problem as a fully nonlinear parabolic HJB equation with terminal condition $ V(T,y) = U(y) $, where $ U $ is a concave, bounded, and increasing utility function.
• Apply weakly nonlinear analysis by expanding the solution in powers of a small parameter $ \varepsilon $, leading to a first-order asymptotic approximation of the value function and optimal control.
• Derive the optimal control $ \theta^* $ via the first-order condition on the Hamiltonian, resulting in an implicit equation involving the ratio of derivatives of $ A_\varepsilon $ and $ B^2 $ with respect to $ \theta $.
• Construct bounds on the solution using monotonicity and convexity properties of the drift and diffusion coefficients under given assumptions.
• Use Monte Carlo simulations of the wealth process $ y_t $ under the optimal strategy to validate the analytical results numerically.
• Compare the continuous-time analytical solution with a discrete-time dynamic programming model from prior work, assessing differences in the optimal stock-to-bond proportion $ \hat{\theta}(t,y) $.

Experimental results

Research questions

• RQ1 How can the optimal stock-to-bond proportion in a pension savings portfolio be approximated analytically using weakly nonlinear methods?
• RQ2 What are the bounds and asymptotic behavior of the solution to the fully nonlinear HJB equation arising in dynamic pension fund management?
• RQ3 How sensitive is the optimal strategy to changes in model parameters such as risk aversion and the small parameter $ \varepsilon $?
• RQ4 How do the analytical results from the continuous-time HJB model compare with those from a discrete-time dynamic stochastic optimization model?
• RQ5 What is the impact of different values of $ \varepsilon $ (e.g., 0.09 for Slovakia, 0.14 for Bulgaria) on the expected accumulated wealth at retirement?

Key findings

• The asymptotic expansion of the solution to the HJB equation yields a first-order approximation that enables qualitative analysis of the optimal strategy's dependence on model parameters.
• The optimal stock-to-bond proportion $ \hat{\theta}(t,y) $ is derived implicitly via the inverse of a function $ G $, which depends on the derivatives of the drift and diffusion coefficients with respect to the control $ \theta $.
• For the Slovakian pension system ($ \varepsilon = 0.09 $), the expected accumulated wealth at retirement is approximately 5.2 times the final yearly salary.
• For the Bulgarian pension system ($ \varepsilon = 0.14 $), the expected accumulated wealth is approximately 8.1 times the final yearly salary, indicating a significant impact of higher $ \varepsilon $ on savings performance.
• The maximum difference between the continuous-time and discrete-time models in the optimal proportion $ \hat{\theta}(t,y) $ is 0.33, occurring at $ t=1 $, $ y \approx 1.3 $, showing strong qualitative agreement.
• The numerical scheme used to solve the HJB equation exhibits second-order convergence and is applicable to a broader class of utility functions beyond the CRRA family.