Research proposals submitted to Inquire Europe

							                           Research proposal submitted to Inquire Europe


              Dynamic Portfolio Optimization under Tracking Error Constraints

Isabelle Bajeux-Besnainou
The George Washington University - Finance Department
2023 G St, N.W., Washington DC 20052
bajeux@gwu.edu
1-202-994-2559


Roland Portait
CNAM and ESSEC
2, rue Conté, CNAM - Chaire de Finance, Paris 75003
rolcot@wanadoo.fr
01-40-27-25-05


Abstract


The purpose of this research project is to derive dynamic portfolio allocation rules when deviations from a
benchmark are accounted for.
The first problem addressed will be to find closed-form solutions for optimal strategies maximizing an expected
value of terminal wealth subject to a constraint on the variance of this deviation. (Max E(WT) subject to Var(WT-

IT)= constant and WT, self-financing); where WT is the terminal wealth and IT is the chosen benchmark.


We already know from previous research (see Bajeux/Portait (1998) and Nguyen/Portait (2001)) that the optimal
terminal wealth is unbounded from below. A second optimization problem is thus considered where an

exogenous minimum wealth constraint (WTW0) in the same spirit as portfolio insurance- is included.


Finally, in a third model, we will investigate the case where a minimum performance level is required. In all
these different cases, we will try to derive closed form solutions, we will carry out numerical simulations and
compare the performance of optimal dynamic portfolios to those obtained from static strategies.


Research objectives, literature review and context.


The literature in portfolio management has been mainly focusing in recent years on asset allocation using a static
framework a la Markowitz. Although Merton’s (1971, 1973) fundamental papers deriving optimal dynamic
portfolio allocation rules in continuous time have been published in the 70’s, relatively little has been done since
then and till very recent years to find some practical applications for these rules. This is surprising, since
dynamic asset allocation advice is currently provided by most of the brokerage firms and financial advisors and
dynamic asset allocation strategies are carried out by active portfolio managers. Very recently, some researchers
and practitioners have started to question what can be drawn from the theory to help optimal dynamic portfolio
decision making. Wai Lee (2000) in his book on “Advanced Theory and Methodology of Tactical Asset
Allocation” describes the different techniques that may apply to Tactical Asset Allocation. Again, although he
addresses the problem of dynamic asset allocation, most of the developments and applications displayed in this
book are conducted using a static one-period model. This is not only a restrictive framework but it also yields, in
most cases, to a sub-optimal solution to the asset allocation problem (for example, it is shown in Bajeux and
Portait (1998) that the Dynamic Efficient Frontier derived from dynamically managed optimal portfolio
strategies outperform the standard Markowitz frontier derived from efficient static portfolio strategies).
The performance of mutual fund or pension fund managers are often evaluated by comparing the returns of the
managed portfolio and the returns of a benchmark portfolio, as an index. Roll (1992) solved the optimal asset
allocation problem when the objective is to minimize the variance of the tracking error in a static (buy and hold)
framework. He proved that, under most circumstances, the corresponding optimal portfolio is not mean-variance
efficient. Clarke, Krase and Statman (1994) argue that that the tracking error model should not be understood in
the standard Markowitz model but in a model involving aversion to regret. Regret would come when deviations
from a benchmark are going in the wrong direction. This would involve a “mental accounting framework” rather
than the standard mean-variance portfolio framework. Implementing this mental accounting framework happens
to be very similar to using a quadratic utility function (mean-variance optimization) calculated on the tracking
error instead than the return of the portfolio itself. This methodology has been applied to the case of multiple-
benchmarks by Wang (1999). In the single-benchmark case, Rudolf, Wolter and Zimmermann (1999) investigate
four different linear models (as opposed to quadratic models as in the three articles mentioned above) for
minimizing tracking errors. They show that these models are consistent with expected utility maximization and
thus provide a new explanation for the Roll’s paradox.




Description of the research method, expected form of the results and discussion of the practical value of
the project.


We anticipate to write one or two papers as an output of this research project depending on the results. We will
at least write an academic paper and depending on the practical importance of these results, we may write a
second paper for a professional journal.
We will associate to this research a professional person involved in quantitative asset management to monitor the
practical value of our project
The research methodologies will involve dynamic stochastic optimization using the Cox/Huang approach. This
methodology yields closed-form solutions for the optimal values of the portfolios and the optimal strategies
(weights) when particular stochastic processes are assumed for the stock prices. Simulations that may involve
Monte-Carlo techniques will also be conducted.
Proposed timetable.


September 2001- June 2002
Calculations of closed-form solutions of the proposed optimization programs, simulations and writing of the first
draft of the academic paper.


June 2002 - December 2002
Additional simulations and writing of the professional paper.


May 2003
Finalization of the project and potential presentation at an INQUIRE conference.


Enclosed is a copy of each researcher's resume.


Related papers


Related papers that have been sponsored by INQUIRE (1999-2000)
              An Asset Allocation Puzzle: Comment, American Economic Review, September 2001, by I.
    Bajeux, J. Jordan and R. Portait.
              Dynamic Allocations of Stocks, Bonds and Cash, forthcoming in Journal of Business, by I.
    Bajeux, J. Jordan and R. Portait
              Allocation stratégique d'actifs: l'apport de nouveaux modèles d'optimisation de portefeuille,
    Banque et Marchés, Juin 1999, by I. Bajeux and R. Portait
Other related papers
              Dynamic Asset Allocation in a Mean-Variance framework, Management Science, November
    1998 by I. Bajeux and R. Portait
              Dynamic Mean-Variance Efficiency and Asset Allocation with a Solvency Constraint,
    forthcoming, Journal of Economics, Dynamics and Control, P. Nguyen and R. Portait.


CONTACT:
Research proposals and further questions may be directed to:
Name: Theo Nijman
E-mail: Theo.Nijman@kub.nl
Postal: Dept. of Finance and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, the Netherlands
Phone: 31-13-4668367 (office)
Fax: 31-13-4662875