Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution

https://doi.org/10.1016/j.ejor.2013.07.011Get rights and content

Highlights

  • This paper extends Stein’s lemma for the multivariate extended skew-t distribution.

  • Efficient portfolios are located on a mean–variance–skewness efficient surface.

  • This surface is a direct extension of Markowitz’ efficient frontier.

  • The multivariate models introduced by Simaan admit the same properties.

  • There are also mean–variance–skewness efficient hyper-surfaces.

Abstract

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.

Introduction

If X is a random vector which has a full rank multivariate normal distribution N(μ, Σ) and h(·) is a scalar valued function of X which satisfies certain regularity conditions, then cov{X, h(X)} = ΣE{h(X)}, where ∇h(X) is the vector first derivatives of h(·) with respect to the elements of X. This result is the multivariate generalisation of Stein’s lemma, Stein’s, 1973, Stein’s, 1981 and is reported in Liu (1994). Liu also points out that similar results hold for other distributions and gives some examples. In particular, he provides an expression for cov{X, h(X)} for the case where the scalar random variable X has Student’s t distribution. Landsman’s, 2006, Landsman and Nešlehová’s, 2008 extend these results, with the latter paper providing an extension of Stein’s lemma for multivariate elliptically symmetric distributions. (For general background see Fang, Kotz, & Ng, 1990.) This is an important class of distributions for portfolio theory. If the multivariate distribution of asset returns is member of the elliptically symmetric class, then the distributions of returns on a portfolio of the assets, which is an affine transformation of the vector of returns, is a member of the same class. Of arguably greater importance is the implication of Landsman and Nešlehova’s extension of Stein’s lemma. For portfolio selection h(X) = U(wTX), where U(·) is a utility function and w is the vector of portfolio weights. Under elliptical symmetry, and subject only to regularity conditions, the portfolios of expected utility maximisers will be located on Markowitz’ mean–variance efficient frontier. Since Markowitz’ original paper, mean–variance portfolio selection has been the subject of great interest and many articles, books and monographs. The method has been the subject of both praise and criticism. Nonetheless, the fact that the efficient frontier arises under conditions which are far more general than quadratic utility or normally distributed returns is one of many the reasons for its longevity and a tribute to the robustness of the original theory.

As well as non-normality, almost always in the form of fat tails, it has long been accepted that returns on some financial assets are not symmetrically distributed. Indeed, there is a long literature about skewness in asset returns. This dates back at least to the foundation papers of Samuelson’s, 1970, Arditti and Levy’s, 1975, Kraus and Litzenberger’s, 1976. In addition to these and other theoretical papers, there are numerous articles which report empirical studies of skewness. There are also numerous papers which are concerned with portfolio selection in the presence of skewness. Well known works include, but are not limited to, papers by Chunhachinda et al.’s, 1997, Sun and Yan’s, 2003, de Athayde and Flôres’s, 2004, Briec et al.’s, 2007, Li et al.’s, 2010, Goh et al.’s, 2012, Matmoura and Penev’s, 2013. Many of these have been summarised recently in the review paper by Adcock, Eling, and Loperfido (2012). Some papers (Jondeau & Rockinger, 2006, for example) employ Taylor series methods to justify the inclusion of a cubic term in an approximation to a utility function. It is natural therefore to enquire to what extent there is a mean–variance–skewness extension of Markowitz’ efficient frontier; that is, a single mean–variance–skewness surface on which the portfolios of all expected utility maximisers will be located.

The view taken in this paper is that the starting point for the development of portfolio selection theory is a coherent multivariate probability distribution for asset returns. In principle, this allows the computation of expected utility and thus makes explicit the nature of the relationship between a set of efficient portfolio weights, the parameters of the distribution and the utility function itself. For symmetrically distributed returns, members of the elliptically symmetric class are attractive because they are tractable with respect to the requirements of efficient portfolio selection. The same requirements imply that skewed multivariate distributions should also be selected for portfolio selection, at least in part, because of their tractability. A well-known skewed multivariate distribution which meets the requirement of tractability is the multivariate skew-normal distribution (MSN henceforth). This was introduced in its original form by Azzalini and Dalla-Valle (1996). The derivation considers a random vector of length (n + 1), (UT, V), U an n-vector and V a scalar, which has a multivariate normal distribution with mean vector (μT,0) and full rank covariance matrix which is arbitrary except for the diagonal entry corresponding to V, which is unity. The multivariate skew-normal arises as the conditional distribution of U given that V > 0. The MSN distribution was also reported as a conditional distribution in Azzalini and Capitanio (1999).

A modified version of this distribution, which has an additional parameter, was reported by Arnold and Beaver (2000). For this version of the distribution, before truncation the mean of the scalar variable V is arbitrary. This distribution was also reported independently in Adcock and Shutes (2001), who were the first to employ it in finance. This modification, which is generally known as the multivariate extended skew-normal (MESN) distribution, is attractive for applications in finance. The additional parameter offers more flexibility in modelling higher moments than the MSN. Explicit formulae for the moments are in Adcock and Shutes (2001).

From the perspective of portfolio selection the MESN distribution also admits an extension to Stein’s lemma. Adcock (2007) shows that under the MESN distribution there is a single mean–variance–skewness efficient surface. As is well known, the derivation of both the MSN and MESN distributions employs a single unobserved or hidden variable which is non-negative. From a finance perspective, this implies that there is a single source of asymmetry in returns; a one factor model for skewness. A more complex skew-normal model in which there is more than one hidden non-negative variable is described in Sahu et al.’s, 2003, Gonzàlez-Farías et al.’s, 2004, Arellano-Valle and Azzalini’s, 2006. This is generally known as the closed skew-normal (CSN) distribution. Adcock (2007) describes an extended version of this distribution, denoted CESN, and shows that there is an extension of Stein’s lemma for it. Under this distribution, skewness is generated by multiple factors and there is a single mean–variance–skewness hyper-surface for expected utility maximisers, with a dimension for each skewness factor. The CSN distribution is employed in portfolio selection by Harvey, Liechty, Liechty, and Müller (2010), who extend Sahu et al.’s (2003) model and thus deal with co-skewness between assets.

Both the MESN and CESN distributions are tractable from the perspective of portfolio theory. However, even though the additional parameter(s) give flexibility in modelling moments, it is debatable whether these distributions will always adequately deal with the fat tails effects observed in empirical data. The first objective of this paper, therefore, is to present results for portfolio selection based on multivariate skew-Student-t distributions. These are increasingly well-known extensions of the multivariate skew-normal. Initial papers by Branco and Dey’s, 2001, Azzalini and Capitanio’s, 2003 have been followed by articles by several authors including Azzalini and Genton’s, 2008, Arellano-Valle et al.’s, 2006. A multivariate extended skew-Student-t, MEST, distribution and its properties are described in Adcock’s, 2002, Adcock’s, 2010, Arellano-Valle and Genton’s, 2010. Like its skew-normal counterpart, the MEST distribution employs one truncated variable.

The notable paper by Sahu et al. (2003), already referred to above, presents a general multivariate skew-elliptical distribution. This assumes that a 2n-vector (UT, VT) has a multivariate elliptically symmetric distribution and then conditions on all elements of U being positive. The expected values of the elements of U are assumed to be equal to zero. Their general results are exemplified using the skew-normal and skew-Student cases. Arellano-Valle and Genton (2010) also describe a version of the skew-Student t distribution, CEST henceforth, which has more than one truncated variable. In this case, the expected values of the elements of U are not restricted. In their paper, the CEST distribution extends related earlier work by Arellano-Valle and Azzalini (2006).

The second objective of this paper is to present corresponding results for multivariate models of asset returns which are members of the class defined by Simaan’s, 1987, Simaan’s, 1993. He proposes that the n-vector of returns on financial assets should be represented as X = U + λV. The n-vector U has a multivariate elliptically symmetric distribution and is independent of the non-negative univariate random variable V, which has an unspecified skewed distribution. The n-vector λ, whose elements may take any real value, induces skewness in the return of individual assets. Adcock and Shutes (2012) describe multivariate versions of the normal-exponential and normal-gamma distributions. These are specific cases of the model proposed by Simaan’s, 1987, Simaan’s, 1993, which have not appeared explicitly in the literature before.

The paper presents extensions to Stein’s lemma for both the MEST and CEST distributions. This work extends Adcock (2007), which presents versions of Stein’s lemma for the MESN and CESN distributions. Similar to the results for these skew-normal distributions, a consequence of the single truncated variable employed in the MEST distribution is that the lemma offers results which are not complicated to compute for a given function h(·). As is shown below, the extension to Stein’s lemma for the MEST distribution leads to a single mean–variance–skewness surface for expected utility maximisers. The lemma also leads to insights into the nature of skewness preference under this distribution. For the CEST distribution, the extension to the lemma leads to a single mean–variance–skewness hyper-surface. For general applications, this particular extension to the lemma is mainly of theoretical interest. This is because of the necessity to compute integrals of multivariate Student-t distributions and because parameter estimation will not be a trivial task. However once parameter estimates are available, it is shown in Section 7 that portfolio selection may be performed using quadratic programming. Notwithstanding its complexities, and as discussed in Section 5, a case may be made for a useful role for the CEST distribution in finance. Accordingly use of this distribution is a topic for future research. The paper then presents some analogous results for Simaan-type models. Given the more general nature of the models that Simaan proposes, these results are less elegant mathematically than those for the MEST and CEST distributions. Nonetheless, they show that, even under more general conditions, there is a single mean–variance–skewness surface for one factor models. When the models are extended to include more than one non-negative variable, the single mean–variance–skewness hyper-surface arises. Thus, this papers shows that there is a comprehensive set of multivariate probability distributions which incorporate asymmetry and which lead to straightforward extensions to Markowitz’ efficient frontier. It is shown in Section 7 that portfolios on these surfaces may also be determined by quadratic programming.

The structure of the paper is as follows. Section 2 summarizes the MEST and CEST distributions. A number of basic properties of the CEST distribution and specific results that are required in the rest of the paper are summarised in Appendix A Properties of the MEST and CEST distributions, Appendix B An integral of the multivariate student distribution. Many of these results may also be found in a different notation in Arellano-Valle and Genton (2010). Section 3 contains two lemmas which show that the MEST and CEST distributions may be expressed as scale mixtures of the MESN and CESN distributions, respectively. These results, which are believed to be new, are used in the subsequent discussion of skewness preference referred to above. Section 4 presents the main result of the paper, Stein’s lemma for the CEST distribution. This section also contains several corollaries to the result, including Stein’s lemma for the MEST distribution. Section 5 contains a discussion of portfolio selection and skewness preference under the MEST and CEST distributions. Section 6 presents the analogous results for the models of Simaan’s, 1987, Simaan’s, 1993. Section 7 describes how portfolio selection may be performed using quadratic programming and presents an example. Section 8 concludes. Technical results and proofs are in appendices at the end of the paper. The notations 0m and 0mn denote respectively an m-vector and m × n matrix of zeros. For m-vectors v and a the notation v  a means that vi  ai for i = 1,  , m. The standard normal distribution function evaluated at x is Φ(x). Other notation is either that in standard use or is defined in the paper.

Section snippets

Multivariate skew-Student distributions

The multivariate skew-Student-t distribution was introduced by Azzalini and Capitanio (2003) and is an extension to the MSN distribution first described in Azzalini and Dalla-Valle (1996). The standard form may be obtained by considering random vector of length (n + 1), (UT, V), U an n-vector and V a scalar, which has a multivariate Student distribution with υ degrees of freedom, location parameter vector (μT, 0) and full rank scale matrix which is arbitrary except for the diagonal entry

Representation of the MEST and DEST distributions as scale mixtures

For symmetrically distributed returns, the multivariate Student-t distribution may be represented as a scale mixture of the multivariate normal. For a vector time series of asset returns, this representation may be interpreted as a manifestation of changing volatility and contributes to an explanation of why Student’s t distribution is often a better model for returns than the normal. This section of the paper presents two results which show that the MEST and CEST distributions may also be

Extensions of Stein’s lemma

This section presents the main results of the paper, the extensions to Stein’s lemma. Lemma 5, Lemma 6 are provided for background and for completeness. The main result is in Lemma 7 and its corollaries. Lemma 5 is a special case of the results in Landsman and Nešlehová (2008) and an extension to that in Liu (1994). As reported in Liu, the lemma may be established using integration by parts. Lemma 6 is believed to be a new result. The proofs of Lemma 6, Lemma 7 are in Appendix D.

Lemma 5

Stein’s lemma for the multivariate Student-t distribution

Let X be an

Portfolio selection and skewness preference

In this section, Lemma 7 and Corollary 2 are used to derive the first order conditions for portfolio selection. Results are presented first for the MEST distribution and then for the more complicated CEST model. Henceforth, the general notations δ and Θ are used to denote the vector of expected returns and covariance matrix respectively.

Under the MEST distribution, the first order conditions for portfolio selection depend on the expressionθ1δ+θ2{Σ+λλT}w+θ3λ.The scalars θ1,2,3 are defined asθ1=E

Portfolio selection using Simaan type models

As noted in the introduction, Simaan’s, 1987, Simaan’s, 1993 proposes that the n-vector of returns on financial assets should be represented as X = U + λV, where U has a multivariate elliptically symmetric distribution and is independent of the non-negative univariate random variable V, which has an unspecified skewed distribution. If the scale matrix for the distribution of U is now denoted by Σ, the independence of U and V means that the first order conditions for portfolio selection depend onθ1δ+

First order conditions and practical portfolio selection

When asset returns follow a multivariate normal distribution and noting that they are scale invariant, the first order conditions for portfolio selection, ignoring terms required to deal with constraints, areθδ-Θw,with the scalar θ defined asθ1=-E{U(Xp)}/E{U(Xp)},where denotes expectation over the multivariate normal distribution N(δ, Θ). In general θ is a non-linear function of w the vector of portfolio weights. However, the consequence of Stein’s lemma is that all solutions lie on the

Concluding remarks

It is tempting to write a very short conclusion to this paper: Markowitz rules! The theory developed by him and reported initially in Markowitz (1952) is robust to numerous departures from multivariate normality. Elliptically symmetric distributions lead to the efficient frontier. As reported in this paper, several multivariate skew-elliptical distributions lead to an extension to the frontier, which includes an extra dimension or dimensions for skewness. Both the general nature of Simaan’s

Acknowledgements

Thanks are due to Marc Genton for reading a previous version of this paper. They are also due to Zinoviy Landsman for stimulating correspondence about developments of Stein’s lemma and to Frank Schuhmacher for discussions of asymmetric location scale distributions. Catherine Jackson kindly provided the data used to generate the example shown in Fig. 2. Very grateful thanks to the two referees who reviewed the previous version of this paper with great care. They pointed out errors and made

References (45)

  • Y. Matmoura et al.

    Multistage optimization of option portfolio using higher order coherent risk measures

    European Journal of Operational Research

    (2013)
  • J. Mencía et al.

    Multivariate location–scale mixtures of normals and mean–variance–skewness portfolio selection

    Journal of Econometrics

    (2009)
  • Q. Sun et al.

    Skewness persistence with optimal portfolio selection

    Journal of Banking and Finance

    (2003)
  • Adcock, C. J. (2002). Asset pricing and portfolio selection based on the multivariate skew-student distribution. In...
  • C.J. Adcock

    Extensions of Stein’s lemma for the skew-normal distribution

    Communications in Statistics – Theory and Methods

    (2007)
  • Adcock, C. J. (2009). Moments and distribution of quadratic forms under the multivariate extended skew-normal and...
  • C.J. Adcock

    Asset pricing and portfolio selection based on the multivariate extended skew-student-t distribution

    Annals of Operations Research

    (2010)
  • Adcock, C. J., Eling, M., & Loperfido, N. (2012). Skewed distributions in finance and actuarial science: A review. The...
  • C.J. Adcock et al.

    Portfolio selection based on the multivariate skew normal distribution

  • C.J. Adcock et al.

    On the multivariate extended skew-normal, normal-exponential and normal-gamma distributions

    The Journal of Statistical Theory and Practice

    (2012)
  • F.D. Arditti et al.

    Portfolio efficiency in three moments: The multiperiod case

    Journal of Finance

    (1975)
  • R.B. Arellano-Valle et al.

    On the unification of families of skew-normal distributions

    Scandinavian Journal of Statistics

    (2006)
  • Cited by (51)

    • Portfolio selection: A target-distribution approach

      2023, European Journal of Operational Research
    • Reconciling mean-variance portfolio theory with non-Gaussian returns

      2022, European Journal of Operational Research
    View all citing articles on Scopus
    View full text