Time Consistent Multi-period Worst-Case Risk Measure in Robust Portfolio Selection
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Time Consistent Multi-period Worst-Case Risk Measure in Robust Portfolio Selection Jia Liu1 · Zhi-Ping Chen1 · Yong-Chang Hui1
Received: 9 March 2017 / Revised: 18 July 2017 / Accepted: 16 December 2017 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this paper, we first construct a time consistent multi-period worst-case risk measure, which measures the dynamic investment risk period-wise from a distributionally robust perspective. Under the usually adopted uncertainty set, we derive the explicit optimal investment strategy for the multi-period robust portfolio selection problem under the multi-period worst-case risk measure. Empirical results demonstrate that the portfolio selection model under the proposed risk measure is a good complement to existing multi-period robust portfolio selection models using the adjustable robust approach. Keywords Distributionally robust optimization · Multi-period risk measure · Dynamic portfolio selection · Conditional value-at-risk Mathematical Subject Classification 91G10 · 93E20
This paper is dedicated to Professor Duan Li in celebration of his 65th birthday. This research was supported by the National Natural Science Foundation of China (Nos. 71371152 and 11571270).
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Zhi-Ping Chen [email protected] Jia Liu [email protected] Yong-Chang Hui [email protected]
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Department of Computing Science, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
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J. Liu et al.
1 Introduction The standard assumption in traditional risk measure and portfolio selection is that the probability distribution of random return is known beforehand and only the realizations are unknown at the time of decision making. However, the underlying distribution of the random investment return or loss cannot be exactly found in practice. The distributionally robust optimization technique was proposed to measure the ambiguity of the distribution and has been successfully applied in many kinds of management problems under uncertainty. This technique gives a robust estimation with respect to the worst-case distribution from an uncertainty set. Early research about distributionally robust optimization can be found in Scarf [1] and Žácková [2]. More recently, the distributionally robust optimization technique has been widely used in finance area, especially, it is very useful for constructing worst-case risk measures and formulating robust portfolio selection models. Lobo and Boyd [3] used the worst-case method to estimate the uncertain variance and demonstrated that the problem of minimizing the worst-case variance can be transformed into a semi-definite programming problem. After that, the worst-case technique has been frequently used to measure the investment risk when the ambiguity of the return (loss) distribution is considered. El Ghaoui et al. [4] defined the worst-case value-at-risk (VaR), and showed that the worst-case VaR constraint
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