# American Institute of Mathematical Sciences

January  2008, 4(1): 81-94. doi: 10.3934/jimo.2008.4.81

## Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China 2 Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China, China

Received  August 2006 Revised  November 2007 Published  January 2008

The optimal portfolio problem under a VaR (value at risk) constraint is reviewed. Two different formulations, namely with and without consumption, are illustrated. This problem can be formulated as a constrained stochastic optimal control problem. The optimality conditions can be derived using the dynamic programming technique and the method of Lagrange multiplier can be applied to handle the VaR constraint. The method is extended for inventory management. Different from traditional inventory models of minimizing overall cost, the cashflow dynamic of a manufacturer is derived by considering a portfolio of inventory of raw materials together with income and consumption. The VaR of the portfolio of assets is derived and imposed as a constraint. Furthermore, shortage cost and holding cost can also be formulated as probabilistic constraints. Under this formulation, we find that holdings in high risk inventory are optimally reduced by the imposed value-at-risk constraint.
Citation: K. F. Cedric Yiu, S. Y. Wang, K. L. Mak. Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains. Journal of Industrial & Management Optimization, 2008, 4 (1) : 81-94. doi: 10.3934/jimo.2008.4.81
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