Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 5 (2010), 135 -- 149

DYNAMIC SHORTFALL CONSTRAINTS FOR OPTIMAL PORTFOLIOS

Daniel Akume, Bernd Luderer and Ralf Wunderlich

Abstract. We consider a portfolio problem when a Tail Conditional Expectation constraint is imposed. The financial market is composed of n risky assets driven by geometric Brownian motion and one risk-free asset. The Tail Conditional Expectation is calculated for short intervals of time and imposed as risk constraint dynamically. The method of Lagrange multipliers is combined with the Hamilton-Jacobi-Bellman equation to insert the constraint into the resolution framework. A numerical method is applied to obtain an approximate solution to the problem. We find that the imposition of the Tail Conditional Expectation constraint when risky assets evolve following a log-normal distribution, curbs investment in the risky assets and diverts the wealth to consumption.

2010 Mathematics Subject Classification: 91G10, 93E20, 91B30, 37N40.
Keywords: Portfolio optimization; Risk management; Dynamic risk constraints; Tail Conditional Expectation.

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Daniel Akume Bernd Luderer
Mathematics Department, Faculty of Mathematics,
University of Buea, Chemnitz University of Technology,
P.O Box 63 Buea, Cameroon. 09107, Chemnitz, Germany.
e-mail: d-akume@yahoo.ca e-mail: b.luderer@mathematik.tu-chemnitz.de

Ralf Wunderlich
Mathematics Department,
Zwickau University of Applied Sciences,
08012, Zwickau, Germany.
e-mail: ralf.wunderlich@fh-zwickau.de

http://www.utgjiu.ro/math/sma