A MARKOVIAN APPROXIMATED SOLUTION TO A PORTFOLIO MANAGEMENT PROBLEM



There were, however, differences in the investment
schedules: as expected19 , the volatility troughs
triggered higher investment levels. A pension fund
management example illustrated the use of the
method for a non HARA objective function; it
also highlighted the insufficiency of a mean value
as an optimisation criterion. A cautious strategy
was computed with very low VaR and CVaR.

All solutions were practical in that they could be
applied to real life situations describable by the
portfolio model. The method of obtaining them
is ready to be applied to other scenarios of the
model parameters (including a variable discount
rate,
etc.).

Some optimisation runs (on a Pentium II PC)
lasted up to 30 hours (for
δ = .02, h = 100).
However, an “intelligent” state space search (à
la
[12] or [16] ) could be implemented to accelerate
the algorithm convergence.

7. REFERENCES

[1] Andersson, F., S. Uryasev, 1999, “Credit Risk Op-
timization with Conditional Value-at-Risk Criterion”,
working paper available from authors.

[2] Fleming, W. H., R. W. Rishel, 1975, Deterministic
& Stochastic Optimal Control,
Springer-Verlag, New
York etc.

[3] Haurie, A., J. B. Krawczyk, M. Roche, 1994,
“Monitoring Cooperative Equilibria in a Stochastic
Differential Game”,
JOTA, Vol. 81, No 1 (April), pp.
73-95.

[4] Karatzas, I., S. E. Shreve, 1992, Methods of Math-
ematical Finance,
Springer-Verlag, New York etc.

[5] Kloeden, P. E., E. Platen, 1992, Numerical Solution
of Stochastic Differential Equations,
Springer-Verlag,
Berlin etc.

[6] Krawczyk, J. B., 1999, “Approximated Numerical
Solutions to a Portfolio Management Problem”. Pa-
per prepared for, and presented at,
Stanford Institute
for Theoretical Economics 1999
Summer Workshop on
Computational Economics в Economic Theory.

[7] Krawczyk, J. B., 1998, “Finite-Horizon Continuous-
Time Stochastic Optimisation via a Markovian Chain
Approximation”, Preprints of the Society for Com-
putational Economics
1998 Conference on Computa-
tion in Economics, Finance and Engineering: Economic
Systems, (CEFES’98), Cambridge, England, paper No
33, 7 pages.

[8] Krawczyk, J. B. & A. Windsor, 1997, An Approxi-
mated Solution to Continuous-Time Stochastic Optimal
Control Problem Through Markov Decision Chains, Eco-
nomic Working Papers Archive, comp/971001

[9] Kushner, H.J., 1990, “Numerical Methods for
Stochastic Control Problems in Continuous Time”,
SIAM J. Contr. & Optim., Vol. 28, No. 5, pp. 999-
1048.

[10] Merton, R. C., 1971, “Optimum Consumption and
Portfolio Rules in a Continuous-Time Model”,
J. of
Economic Theory,
3, pp. 373-413.

[11] MθRCK, R., E. Schwarz & D. Stangeland, 1989,
“The Valuation of Forestry Resources under Stochas-
tic Prices and Inventories”,
J. of Financial and Quan-
titative Analysis,
Vol. 24, No. 4, pp. 473-487.

[12] Pereira, M.V.F. & L.M.V.G Pinto , 1991, “Multi-
stage Stochastic Optimization Applied to Energy
Planning”,
Mathematical Programming, Series B, 52,
pp. 359-375.

[13] Rust, J, 1997, “Using Randomization to Break the
Curse of Dimensionality”,
Econometrica, Vol. 65, No.
3 (May), pp. 487-516.

[14] Rust, J, 1997, A Comparison of Policy Iteration
Methods for Solving Continuous-State, Infinite-Horizon
Markovian Decision Problems Using Random, Quasi-
random, and Deterministic Discretizations,
Economic
Working Papers Archive, comp/9704001.

[15] Tapiero, C., 1998, Applied Stochastic Models and
Control for Finance and Insurance,
Kluwer, Boston, etc.

[16] Tolwinski, B. & R. Underwood, 1991, An Algorithm
to Estimate the Optimal Evolution of an Open Pit Mine,
Work. Paper MCS-91-05, Colorado School of Mines,
Golden, Colorado.

[17] Sarkar S., 2000, “On the investment-uncertainty
relationship in a real options model”,
J. of Economic
Dynamics and Control,
24, pp. 219-225.

[18] Windsor, A. & J. B. Krawczyk, 1997, SOCSol-I:
a Matlab Package for Approximating the Solution to a
Continuous-Time Stochastic Optimal Control Problem ,
Econonomic Working Papers Archive, comp/9701002.

19There are situations where the conclusion could have
been opposite, see [17].

16



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