Stochastic Dynamic Optimization Models for Pension Planning
Soňa Kilianová
PhD thesis advisor: Daniel Ševčovič
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Summary: The goal of this thesis was to develope a model for determining an optimal strategy for the fund selection in pension planning in three-pillar pension systems. We assumed that savers are given a possibility to choose one from a set of various pension funds, differing in their risk profiles. We proposed two types of models: Model DAM assumes a 0-1 type selection of funds, it means, the savers may choose only one of them at a given time. We showed that this problem leads to a Bellman equation. A solution to it can be found backwards. The PIAM model is a modification of the DAM model. It finds an optimal weight of stocks in the fund selection, which may attain values from the interval $[0,1]$. The saver can then choose a fund with a risk profile that is closest to the optimal weight of stocks. We derived a nonlinear partial differential equation for the value function $V$ from the Bellman equation, for the case when the time step epsilonin the PIAM model tends to $0$. The first of the risk minimizing models is the TRMM model (Terminal Risk Minimizing Model). It is based on minimizing the insecureness of the terminal value of savings under an assumption that a target terminal value of savings is given. We minimize the average value-at-risk deviation risk measure, which is also known under the name conditional value-at-risk deviation. The second risk minimizing model is the MRMM model (Multi-period Risk Minimizing Model). We assume that the saver aims to minimize the riskiness of their savings throughout the whole period of saving. This can be reasoned by the fact that, in the case of early death, the saved sum becomes a subject of heritage. We use the multi-period average value-at-risk deviation measure as the objective function. If the saver makes a decision about the fund weights every year, both models TRMM and MRMM lead to a problem of linear programming. However, this assumption causes memory requirement problems in implementation. Therefore, we modified the models and assumed that the saver makes decisions only once during a period of several years. This modification changes the problems to nonlinear ones. We proposed numerical schemes for both types of models and implemented them for the case of Slovak Republic. We investigated the sensitivity of the results with respect to varying parameters using the DAM model. We also investigated the influence of the alpha parameter from the average value-at-risk definition on the results of the TRMM model. Based on experiments, we summarize the qualitative properties of the optimal solutions.
Related papers

[1] S. Kilianová, G. Pflug (2008): Optimal pension fund management under multi-period risk minimization. Annals of Operations Research, DOI: 10.1007/s10479-008-0405-3

[2] S. Kilianová, I. Melicherčík, D. Ševčovič (2006): Dynamic and Static Strategies for the Funded Pillar of the Slovak Pension System. Finance a uver - Czech Journal of Economics and Finance, 56, No 11-12, 506-521.
Full text    (PDF 166 K)

[3] S. Kilianová and D. Ševčovič (2004): The second pillar of the Slovak pension system - interest rate targeting. Journal of Electrical Engineering Vol.~57, No. 12/s, 2006.
Full text    (PDF 161 K)