Minimax Portfolio Optimization Under Interval Uncertainty
Meng Yuan*, Xu Lin*, Junzo Watada*, and Vladik Kreinovich**
*Graduate School of Information, Production, and Systems, Waseda University
2-7 Hibikino, Wakamatsu, Kitakyushu 808-0135, Japan
**Department of Computer Science, University of Texas at El Paso
500 W. University, El Paso, TX 79968, USA
In the 1950s, Markowitz proposed to combine different investment instruments to design a portfolio that either maximizes the expected return under constraints on volatility (risk) or minimizes the risk under given expected return. Markowitz’s formulas are still widely used in financial practice. However, these formulas assume that we know the exact values of expected return and variance for each instrument, and that we know the exact covariance of every two instruments. In practice, we only know these values with some uncertainty. Often, we only know the lower and upper bounds on these values – i.e., in other words, we only know the intervals that contain these values. In this paper, we show how to select an optimal portfolio under such interval uncertainty.
-  D. J. Sheskin, “Handbook of Parametric and Nonparametric Statistical Procedures,” Chapman & Hall/CRC, Boca Raton, Florida, 2011.
-  H. Markowitz, “Portfolio selection,” J. of Finance, Vol.7, No.60, pp. 77-91, 1952.
-  D. P. Bertsekas, “Nonlinear Programming,” Athena Scientific, Cambridge, Massachisetts, 1999.
-  D. Berleant, L. Andrieu, J.-P. Argaud, F. Barjon, M.-P. Cheong, M. Dancre, G. Sheble, and C.-C. Teoh, “Portfolio management under epistemic uncertainty using stochastic dominance and Information-Gap Theory,” Int. J. of Approximate Reasoning, Vol.49, No.1, pp. 101-116, 2008.
-  M. Salahi, F. Mehrdoust, and F. Piri, “CVaR Robust Mean-CVaR Portfolio Optimization,” ISRN Applied Mathematics, Article ID 570950, 2013.
-  V. DeMiguel, L. Galappi, and R. Uppal, “1/N,” Proc. of the 33rd Annual Meeting of the European Finance Association EFA’2006, Zurich, Switzerland, August 23-26, 2006; available at
-  G. Gigerenzer, “Rationality for Mortals: How People Cope with Uncertainty,” Oxford University Press, New York, 2008.