Stochastic Dynamic Optimization Models for Pension Planning
PhD thesis advisor:
PhD thesis - Full text (PDF 760 K)
Synopsis (PDF 236 K)
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
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.
- models based on the expected utility maximization (models DAM and
- models minimizing the riskiness of the investment (models TRMM and
 S. Kilianová, G. Pflug (2008):
Optimal pension fund management under multi-period risk minimization. Annals of Operations Research, DOI:
 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)
 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)