Cash management using multi-stage stochastic programming

Abstract

We consider a cash management problem where a company with a given financial endowment and given future cash flows minimizes the Conditional Value at Risk of final wealth using a lower bound for the expected terminal wealth. We formulate the optimization problem as a multi-stage stochastic linear program (SLP). The company can choose between a riskless asset (cash), several default- and option-free bonds, and an equity investment, and rebalances the portfolio at every stage. The uncertainty faced by the company is reflected in the development of interest rates and equity returns. Our model has two new features compared to the existing literature, which uses no-arbitrage interest rate models for the scenario generation. First, we explicitly estimate a function for the market price of risk and change the underlying probability measure. Second, we simulate scenarios for equity returns with moment-matching by an extension of the interest rate scenario tree.

Keywords:

• Dynamic stochastic programming
• Stochastic linear programming
• Cash management
• Market price of risk
• Change of measure
• Scenario generation

Acknowledgements

The authors gratefully acknowledge helpful comments by Alois Geyer, Michael Hanke Richard Stanton and two anonymous referees. We thank the Austrian Nationalbank for financial support (Jubiläumsfondsprojekt No. 13054).

Notes

† The data are referenced to a six-month tenor, except for the one-year maturity, which is based on a tree-month tenor (according to the International Swap and Derivatives Association, Inc. (ISDA) and Tullet Prebon). In all cases where the tenor was shorter than the available data, we interpolated with cubic splines.

† For easy comparability, our notation follows that in Stanton (1997). There, the dependence of the bond's return on the short rate is also omitted for the sake of brevity.