A Simple, Adjustably Robust, Dynamic Portfolio Policy under Expected Return Ambiguity Mustafa Ç. Pınar Department of Industrial Engineering Bilkent University 06800 Bilkent, Ankara, Turkey March 16, 2012 Abstract In an economy with a CARA utility investor facing a set of risky assets with normally distributed returns over multiple periods, we consider the problem of making an ambiguityrobust dynamic portfolio choice when the expected return information is uncertain. We pose the problem in the Adjustable Robust Optimization framework under ellipsoidal representation of the expected return uncertainty, and provide a closed-form solution in the form of a simple, dynamic, partially myopic portfolio policy that is affine in the estimate of the expected return vector. The result provides a guideline in the form of an upper bound for the choice of the parameter controlling the uncertainty. Keywords: Dynamic portfolio selection, ambiguity, robustness, adjustable robust optimization. 1 Introduction Portfolio selection has been a core subject of mathematical finance and operations research for six decades starting with the ground-breaking work of Markowitz [20] who initiated the socalled Mean-Variance (MV) portfolio theory. The Mean-Variance portfolio selection based on maximizing an expected utility function that depends on the mean and variance/covariance of asset returns offers a sufficiently simple and yet powerful model to analyze risk/return trade-offs in financial portfolios. An alternative approach, but not as popular as the MV theory, takes the view that the investor is maximizing an expected utility function of his final wealth. In the case of mustafap@bilkent.edu.tr. 1
single period portfolio choice, the MV approach and the expected utility approach with negative exponential utility function under normally distributed asset returns yield identical portfolio choices. Extensions of the MV portfolio theory to multi-period portfolio selection problems have also been studied, perhaps with less success as the problems tend to get much harder in comparison to the single period case. An exception is Li and Ng [19] where the authors solve in closed form the multi-period mean-variance portfolio selection problem. The paper is quite technical and relies on a dynamic programming based method; the solution has a complicated form, too. For the expected utility maximization of final wealth, it appears that some simple dynamic portfolio choice rules have been derived already in the sixties [21]; these contributions are discussed in section 3.3 of [6]. Against this background (to which we cannot hope to do justice in a short review, hence we refer the reader to the excellent texts on the subject e.g., [11, 18, 20, 25], and to the more recent review [27]) the contribution of the present brief paper is to study the multi-period portfolio selection problem for an investor with a negative exponential utility function operating in an environment where the asset returns are normally distributed but the investor is suspicious of the accuracy of the expected return data. We term this situation expected return ambiguity. It is well-known and well-documented that the portfolio weights exhibit higher sensitivity to shifts in expected return data, see e.g. [9, 10, 12]. Thus, the investor who is well justified in worrying about inaccuracies in expected return information resorts to a max-min approach confining the ambiguity in expected return to an ellipsoidal uncertainty set around an available forecast of the expected return vector in the spirit of Robust Optimization of Ben-Tal and Nemirovski [1, 2]. First, inspired by [16] where the authors work in a single-period MV setting under expected return ambiguity for a set of risky assets, we consider the single period problem which yields a very simple solution. Then we pass to the multi-period case where we formulate the problem within the framework of the Adjustable Robust Optimization (ARO) of [5, 28]. Using the single-period solution we obtain a simple, dynamic portfolio policy very similar to the single-period ambiguity-robust portfolio selection rule. The fact that the single-period portfolio rule is independent of the initial wealth makes the multi-period result possible. It is somewhat surprising to obtain a simple-closed form portfolio policy in a ARO setting since it is known in general that ARO is a hard problem and such results are rather rare. Exceptions are for instance [13, 23]. In the latter, the authors study the existence of robust profit opportunities in single period, and then in multiple period ARO framework. They show that the problems remain tractable. In [13] the ARO problem is solved in closed-form for multi-period portfolio selection problem based on minimizing a risk function of the final wealth under robustness considerations quite different from those of the present paper. The authors in [13] take as given 2
and trustworthy both the mean and variance/covariance of the asset returns, and study an ambiguity-robust choice where ambiguity is defined in the sense of an imprecise distribution. Therefore, their min-max problem minimizes the maximum risk of falling short of a target return over all distributions having the afore-mentioned mean and variance information. They also obtain a closed-form dynamic policy based on their single-period result. The afore-mentioned type of ambiguity in distribution is studied by several other authors; see e.g. [14, 22, 24]. Delage and Ye [14] report that the distribution free robust portfolios are also very sensitive to inaccuracies in moments and propose a framework addressing this drawback. Unlike our approach, the distribution free robustness as advocated in [13, 24] does not usually allow control of conservatism. Furthermore, the worst-case distribution attaining the bounds can be a twopoint distribution (i.e. the probability mass is concentrated on two realizations, e.g. [13]), a case which may be deemed somewhat unrealistic for portfolio optimization. In our case, the control mechanism allows one to fall back to a MV portfolio at one extreme whereas at the other extreme one obtains a completely risk-less portfolio, thereby allowing a whole spectrum of controlled robust portfolios. While our result can be criticized for assuming normally distributed returns (an assumption which may fail to hold often in practice) or for leaving out other important practical aspects of portfolio choice such as short sale restrictions, and/or transaction costs (see e.g., [3] for a multi-period robust portfolio selection model using conic optimization), it offers a very simple closed-form solution to the time honoured problem of multi-period portfolio selection in the context of the adjustable robustness paradigm. Our results are valid under the condition that the parameter controlling the uncertainty set be bounded above by the slope of the Capital Market Line in classical Markovitz MV theory, which provides a natural interval for the choice of the parameter. Interestingly, the closed-form solution is lost in the absence of a risk-less asset, or when one adopts the Mean-Variance approach within the ARO framework. The robust portfolio result of the present paper has repercussions in other areas akin to portfolio choice. We shall report the impact of single-period robust portfolio rule on delegated portfolio management in [15]. References [1] A. Ben-Tal and A. Nemirovski, 1999, Robust Solutions to Uncertain Linear Programming Problems, Operations Research Letters, 25(1), 1 13. [2] A. Ben-Tal and A. Nemirovski, 1998, Robust Convex Optimization, Mathematics of Operations Research, 23(4), 769 805. 3
[3] A. Ben-Tal, T. Margalit, and A. N. Nemirovski, 2002, Robust Modeling of Multi-stage Portfolio Problems. In H. Frenk, K. Roos, T. Terlaky, and S. Zhang, editors, High Performance Optimization, 303 328, Kluwer Academic Publishers, New York. [4] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, SIAM-MPS Series on Optimization, 2001. [5] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, 2004, Adjustable Robust Solutions to Uncertain Linear Programs, Mathematical Programming 99(2), 351 376. [6] D.P. Bertsekas, 1976, Dynamic Programming and Stochastic Control, Academic Press, New York. [7] D. Bertsimas, D. A. Iancu, P. A. Parrilo, 2010, Optimality of Affine Policies in Multistage Robust Optimization. Mathematics of Operations Research, 35(2), 363-394. [8] D. Bertsimas, D.B. Brown and C. Caramanis, 2011, Theory and Applications of Robust Optimization, SIAM Review, 53(3), 464-501. [9] M. Best and R. Grauer, 1991, Sensitivity Analysis for Mean-Variance Portfolio Problems, Management Science, 37 (8), 980-989. [10] M.J. Best and R.R. Grauer, 1991, On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results, The Review of Financial Studies, 4(2), 315-342. [11] M. Best, 2010, Portfolio Optimization, Chapman & Hall/CRC, Financial Mathematics Series, Boca Raton. [12] F. Black and R. Litterman, 1992, Global Portfolio Optimization, Financial Analysts Journal 48(5), 2843. [13] L. Chen, S. He, and Sh. Zhang, 2011, Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection, Operations Research, 59(4), 847 865. [14] E. Delage, and Y. Ye, 2010, Distributionally Robust Optimization under Moment Uncertainty with Application to Data-Driven Problems, Operations Research, 58(3), 596 612. [15] A. Fabretti, S. Herzel and M. Ç. Pınar, Robust Delegated Portfolio Management, in preparation, preliminary version presented at the XIII Workshop on Quantitative Finance, L Aquila, Italy, Jan. 2012. 4
[16] L. Garlappi, R. Uppal and T. Wang, 2007, Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach, Review of Financial Studies, vol. 20, no. 1, 41 81. [17] D. Goldfarb and G. Iyengar, 2003, Robust Portfolio Selection Problems, Mathematics of Operations Research 28(1), pp. 138. [18] J. Ingersoll, 1987, Theory of Financial Decision Making, Rowman-Littlefield, Maryland. [19] D. Li and W.-L. Ng, 2000, Optimal Dynamic Portfolio Selection: Multiperiod Mean- Variance Formulation, Mathematical Finance, 10(3), 387 406. [20] H. Markovitz, 1959, Portfolio Selection: Efficient Diversification of Investment, Wiley, New York. [21] J. Mossin, 1968, Optimal Multi-period Portfolio Policies, Journal of Business, 41, 215 229. [22] K.D. Natarajan, M. Sim and J. Uichanco, 2010, Tractable Robust Expected Utility and Risk Models for Portfolio Optimization, Mathematical Finance, 20(4), 695 731. [23] M.Ç. Pınar and R. Tütüncü, 2005, Robust Profit Opportunities in Risky Financial Portfolios, Operations Research Letters, 33, 331 340. [24] I. Popescu, 2005, Robust Mean-Covariance Solutions for Stochastic Optimization, Operations Research, 55(1), 98 112. [25] S. Ross, 1999, An Introduction to Mathematical Finance: Options and Other Topics, Cambridge University Press, Cambridge. [26] R.F. Stambaugh, 1997, Analyzing Investments whose Histories Differ in Length, Journal of Financial Economics, 45, 285 331. [27] M. Steinbach, 2001, Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis, SIAM Review, 43(1), 31-85. [28] A. Takeda, S. Taguchi and R. Tütüncü, 2008, Adjustable Robust Optimization Models for a Non-linear Two-Period System, Journal of Optimization Theory and Applications, 136 (2), pp.275-295. 5