Asimple formula for optimal management of individual pension accounts.
Aleš Černý and Igor Melicherčík
This document contains abrief rationale for the research, summary of the main results and description of the accompanying EXCEL spreadsheet.
The model is based on maximization of the expected utility of the terminal value of the pension savings:Max E(U(WT)). The most commonly used function is so-called CRRA (constant relative risk-aversion) utility
whereis client’s coefficient of relative risk aversion (commonly somewhere between 2 and 10).
In acontinuous-time setting with lognormally distributed returns this is adynamic programming problem that leads to theBellman’s equation of optimality and subsequently gives rise to astatic mean-variance optimization
,( 1 )
whereis the vector of expected risky returns (cells A10:A11), ris the risk-free rate (cell F10) andVis the covariance matrix[1] of the risky returns(cells C10:D11). The task is to find the optimal portfolio weights . This is aclassical result going back to Samuelson and Merton. In the case of no additional constraints on the portfolio weights it is simple to evaluate the optimal investment for an investor with unit risk aversion from the formula
,( 2 )
where V-1is the matrix inverse of V. For an investor with risk aversion the optimal investment is proportionally smaller, . This value can be found numerically in cells F17:F18.
Formula (2) is only appropriate if all the money to be invested is available up front, or if one can borrow against future contributions. Neither of these conditions is met for anindividual pension savings plan. The pension contributions must be invested gradually and the risky investment can never exceed the savings already accumulated in the account. The new situation leads to adifferent dynamic programming problem whose Hamilton-Jacobi-Bellman equation does not immediately reduce to asimple static mean-variance problem like (1). Instead, the fully optimal solution has to be computed numerically using mathematically sophisticated methods[2].
In theresearch highlighted here, Černý and Melicherčík (2013) have shown that one can nonetheless devise a nearly optimal strategy based on the static optimization (1)by including an additional constraint
,
where is the the value of current pension savings as afraction of the total of the current savings plusthe present value of future contributions. For illustration, an individual whohas been making pension constributionsfor 20 years out of an estimated 40-year working life will have of roughly ½, although the actual value of at that point will depend on the performance of the funds already invested as well as any potential non-linearity in the time profile of contributions.Inclusion of the constraint is crucial to achieve near optimality and it is the key insight of this reasearch. Alongside, one can consider other institutional constraints such as exclusion of short sales,.
The ensuing static optimization then takes the form
.( 3 )
This is aquadratic programming problem, implemented in the accompanying spreadsheet by using the Solver interface.The cell to be optimized is A17, its value corresponding to equation (3). The resulting optimal portfolio weights are found in cells H17:H18.
The weights represent proportions out of the total value of all contributions, including the stream of future contributions. If one wishes to obtain investment weights out of the current savings then the resulting value of must be divided by The rescaled result is shown in cells J17:J18.
The proposed methodology has been successfully implemented by Allianz DSS – a pension savings provider in Slovakia. The modeller available on Allianz DSSclient web pages does not use EXCEL solver as in the example discussed above, but instead makes use ofexplicit formulae given in Černý and Melicherčík (2013). For example, if no positivity constraint is binding the optimal portfolio weights in (3)are given by
,
with being the unconstrained Merton-Samuelson weights from equation (2), and 1 ad-dimensional vector of ones.
REFERENCES
[1]Cairns, A. J. G., D. Blake, and K. Dowd (2006). Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans. Journal of Economic Dynamics and Control 30(5), 843–877.
[2] Černý, A. and I. Melicherčík (2013). Optimal management of individual pension plans. Cass Business School Working Paper
[1] The accompanying EXCEL file contains auxiliary data in the form of the correlation matrix (cells A22:B23) and volatilities (cells D22:D23) of the risky assets, which are combined to form the covariance matrix V.
[2]There is arich academic literature that does this with all kinds of embellishments, in particular allowing the contributions themselves to be stochastic, see Cairns et al. (2006) and references therein.