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Controlling Portfolio Skewness and Kurtosis Without Directly Optimizing Third and Fourth Moments Economics Letters 122 (2014) 154–158 Contents lists available at ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/ecolet Controlling portfolio skewness and kurtosis without directly optimizing third and fourth moments Woo Chang Kim a,∗, Frank J. Fabozzi b, Patrick Cheridito c, Charles Fox d a Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), 91 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea b EDHEC Business School, 393, Promenade des Anglais BP3116, 06202 Nice Cedex 3, Nice, France c Department of Operations Research and Financial Engineering, Princeton University, Sherrerd Hall, Princeton University, Princeton, NJ 08542, USA d Pimco, 840 Newport Circle Dr. Newport Beach, CA 92660, USA h i g h l i g h t s • We propose a new mean–variance approach that can control higher moments. • Our model does not directly impose higher moment terms in the formulation. • Our model employs robust formulation with a specific choice of uncertainty set. • We provide analytical proofs showing our model can control higher moments. • We present empirical results supporting the validity of our model. article info a b s t r a c t Article history: In spite of their importance, third or higher moments of portfolio returns are often neglected in portfo- Received 23 August 2013 lio construction problems due to the computational difficulties associated with them. In this paper, we Received in revised form propose a new robust mean–variance approach that can control portfolio skewness and kurtosis without 19 November 2013 imposing higher moment terms. The key idea is that, if the uncertainty sets are properly constructed, ro- Accepted 20 November 2013 bust portfolios based on the worst-case approach within the mean–variance setting favor skewness and Available online 1 December 2013 penalize kurtosis. ' 2013 Elsevier B.V. All rights reserved. JEL classification: G11 C61 C63 C65 Keywords: Portfolio selection Robust portfolio Higher moments Mean–variance framework 1. Introduction skewed, it is more likely to have an extreme left-tail event than one in the right-tail. Thus the typical investor prefers more posi- The traditional Markowitz model takes the expectation and tively skewed return distributions. For instance, a more positively variance of a portfolio return as the performance and risk measures skewed portfolio has better Sortino ratio (Sortino and van der and determines the optimal portfolio by solving a quadratic pro- Meer, 1991) and lower semi-deviation. Similarly, a portfolio with gram. From the practitioner's perspective, the skewness and kur- smaller kurtosis tends to have less extreme events, thus is pre- tosis are also important. When the portfolio return is negatively ferred by most investors. Obviously, as the traditional mean–variance approach exclu- sively deals with the first two moments, a revised approach is re- ∗ quired to incorporate and control the third and fourth moments Corresponding author. Tel.: +82 42 350 3129. when constructing a portfolio. The most straightforward approach E-mail addresses: [email protected], [email protected] (W.C. Kim), [email protected] (F.J. Fabozzi), [email protected] (P. Cheridito), would be to introduce new terms for the portfolio skewness and [email protected] (C. Fox). kurtosis in the objective function. To see this, let us first consider 0165-1765/$ – see front matter ' 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econlet.2013.11.024 W.C. Kim et al. / Economics Letters 122 (2014) 154–158 155 2 2 3 the traditional mean–variance formulation as follows. In addition, let µ D EXi;j; σ D E Xi;j − µ ; µ3 D E Xi;j − µ 4 0 0 D − D max µ w − βw Σw (1) and µ4 E Xi;j µ . Also, for i 1;:::; I, we define w2C VD O 0 − 0 O D N − 2 fi.w/ µiw βw Σiw Xi βSi ; where C is a convex subset of the hyperplane fw 2 nj P w D R i i 0 0 O g 2 n 2 n×n f .w/ VD min µO w − βw Σw D min fi.w/: 1 ; µ R ; Σ R are the expected return and covariance ma- O i .µ,O Σ/2U O trix, respectively, and β > 0 is the parameter to reflect investor's .µ,O Σ/ risk aversion. We first give a limit result for f .w/ when J ! 1. 0 i Then, given a coskewness matrix M3 D ET.r−Er/.r−Er/ ⊗.r− 0 n×n2 n 2 Er/ U 2 R for the random return vector r 2 R and Kronecker Theorem 2.1. For every w C, one has ⊗ 0 product , the third central moment of portfolio return w r is p d 2 2 4 2 0 0 3 0 J [fi.w/ − Efi.w/] −! N.0; σ − β σ − 2βµ3 C β µ4/ S .w; M3/ VD E w r − Ew r D w M3 .w ⊗ w/. Similarly, for a 0 0 for J ! 1: cokurtosis matrix M4 D E .r − Er/.r − Er/ ⊗ .r − Er/ ⊗ .r− 0 n×n3 0 Er/ 2 R , its fourth central moment is K .w; M4/ VD E w r− Proof. Consider 0 4 0 Ew r D w M4 .w ⊗ w ⊗ w/. p J [fi.w/ − Efi.w/] Consequently, the mean–variance formulation could be modi- ! fied by adding third and fourth central moments and solving the p N β X N 2 p 2 D J Xi − Xi;j − Xi − J µ − βσ optimization problem J − 1 j 0 0 max µ w − βw Σw C γ S .w; M3/ − δK .w; M4/ " ( )# w2C p N β X N 2 2 D J Xi − Xi;j − µ − Xi − µ C S J i for parameters β; γ ; δ ≥ 0: (2) j − p − 2 While problem (2) is intuitive, there are two critical issues. First, J µ βσ the third central moment term S .w; M3/ is a cubic function, thus it p N β X 2 N 2 D J Xi − Xi;j − µ − 2J Xi − µ makes problem (2) non-convex, which causes a significant increase J j in computational cost. Second, as the coskewness matrix M3 is 3-dimensional, the number of parameters is of the order of n3, C N − 2 C 2 − p − 2 which makes it practically impossible to obtain reliable estimators. J Xi µ Si J µ βσ The same issues apply to the kurtosis term, only with more diffi- " culties. The main objective of this study is to develop an approach p 1 X β X 2 D J Xi;j − Xi;j − µ that can control higher moments of portfolio returns within the J J j j mean–variance framework without directly imposing third and # fourth moment terms in the formulation. N 2 β 2 p 2 C β Xi − µ − S − J µ − βσ J i 2. Models and theories " # p 1 X 2 2 n D J fXi;j − β.Xi;j − µ) g − (µ − βσ / Let r 2 R be the random vector representing returns of n risky J assets. We assume it has finite moments up to order four. Also, let j n | {z } ri;j 2 R for i D 1;:::; I, and j D 1;:::; J be independent and .a/ identically distributed (i.i.d.) samples of r. Let µO D 1 P r and i J j i;j p N 2 β 2 0 C Jβ.Xi − µ) − p S : ΣO D 1 P r − O r − O be the sample mean and sample J i i J−1 j i;j µi i;j µi | {z } covariance matrix for the i-th sample set, respectively. We also de- .b/ | {z } .c/ O O D O O fine a joint uncertainty set of µ, Σ ; U µ,O ΣO µ1; Σ1 ;:::; h n oi . / 1 P 2 2 Since E Xi;j − β Xi;j − µ D µ − βσ , by the Cen- O O O O J j µi; Σi ;:::; µI ; ΣI Now, let us consider the robust version h −!d − − 2 of problem (1). tral Limit Theorem (CLT), .a/ N 0; Var Xi;j β Xi;j µ . N − −!P O 0 − 0 O Next, Since Xi µ 0 by the Weak Law of Large Numbers, and max min µ w βw Σw: (3) p d d w2C µ,O ΣO 2U N − −! 2 −! . / .µ,O ΣO / J Xi µ N 0; σ by CLT, by Slutsky's Theorem, .b/ 0. P P Finally, since p1 ! 0 and S2 −! σ 2;.c/ −! 0. Thus, again by For a feasible portfolio w of problem (3), let J i Slutsky's Theorem, 0 Xi;j D w ri;j; p d h 2 J [fi.w/ − fi.w/] −! N 0; ar Xi;j − β Xi;j − µ : J J E V N 1 X 0 1 X Xi D w ri;j D Xi;j; h 2i h 2i J J − D − D − jD1 jD1 Since Cov Xi;j; Xi;j µ µ3 and Var Xi;j µ µ4 4 J ! σ , 2 1 0 X 0 D − O − O h 2i Si w ri;j µi ri;j µi w J − 1 Var Xi;j − β Xi;j − µ jD1 h 2i 2 h 2i J D Var Xi;j − 2β Cov Xi;j; Xi;j − µ C β Var Xi;j − µ 1 X N 2 D Xi;j − Xi : J − 1 D 2 − C 2 − 2 4 jD1 σ 2βµ3 β µ4 β σ : 156 W.C.
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