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Published in: Financial Markets and Portfolio Management 1/2018

05-02-2018

What really happens if the positive definiteness requirement on the covariance matrix of returns is relaxed in efficient portfolio selection?

Author: Clarence C. Y. Kwan

Published in: Financial Markets and Portfolio Management | Issue 1/2018

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Abstract

The Markowitz critical line method for mean–variance portfolio construction has remained highly influential today, since its introduction to the finance world six decades ago. The Markowitz algorithm is so versatile and computationally efficient that it can accommodate any number of linear constraints in addition to full allocations of investment funds and disallowance of short sales. For the Markowitz algorithm to work, the covariance matrix of returns, which is positive semi-definite, need not be positive definite. As a positive semi-definite matrix may not be invertible, it is intriguing that the Markowitz algorithm always works, although matrix inversion is required in each step of the iterative procedure involved. By examining some relevant algebraic features in the Markowitz algorithm, this paper is able to identify and explain intuitively the consequences of relaxing the positive definiteness requirement, as well as drawing some implications from the perspective of portfolio diversification. For the examination, the sample covariance matrix is based on insufficient return observations and is thus positive semi-definite but not positive definite. The results of the examination can facilitate a better understanding of the inner workings of the highly sophisticated Markowitz approach by the many investors who use it as a tool to assist portfolio decisions and by the many students who are introduced pedagogically to its special cases.

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Appendix
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Footnotes
1
A symmetric \(n\times n\) real matrix \(\varvec{V}\) is said to be positive semi-definite if the scalar \(\varvec{x}^{\prime }\varvec{Vx},\) where the prime denotes matrix transposition, is never negative for any column vector \(\varvec{x}\) of n real numbers. Here, \(\varvec{x}\) can have all zero elements, for which \(\varvec{x}^{\prime }\varvec{Vx}\) is zero. For a set of n random variables represented by an n-element column vector \(\varvec{R},\) the corresponding \(n\times n\) covariance matrix is, by definition, the expected value of \((\varvec{R}-\varvec{ \mu })(\varvec{R}-\varvec{\mu })^{\prime },\) where \(\varvec{\mu } \) is the expected value of \(\varvec{R}.\) If \(\varvec{V}\) is a covariance matrix, as \(\varvec{x}^{\prime }\varvec{Vx}\) is the expected value of \([\varvec{x}^{\prime }(\varvec{R}-\varvec{\mu } )]^{2},\) which is never negative, it must be positive semi-definite. A symmetric \(n\times n\) real matrix \(\varvec{V}\) is said to be positive definite if the scalar \(\varvec{x}^{\prime }\varvec{Vx}\) is always positive for any nonzero column vector \(\varvec{x}\) of n real numbers. Thus, a positive definite matrix \(\varvec{V}\) is also positive semi-definite; however, the converse of the statement is not true. In a portfolio context, suppose that \(\varvec{V}\) is the covariance matrix of security returns and \(\varvec{x}\) is a portfolio weight vector. Individual elements of \(\varvec{x},\) which sum to one, represent the proportions of investment funds as allocated to the corresponding securities. In this context, the scalar \(\varvec{x} ^{\prime }\varvec{Vx}\) represents the corresponding variance of portfolio returns.
 
2
In the constant correlation model based on n risky securities, the covariance of returns between securities i and j is characterized as \( \sigma _{ij}=\rho \sigma _{i}\sigma _{j},\) for \(i,j=1,2,\ldots ,n\) and \( i\ne j,\) where \(\rho \) is a constant—known as the constant correlation—and \(\sigma _{i}\) and \(\sigma _{j}\) are the standard deviations of returns of securities i and j,  respectively. To implement the constant correlation model for portfolio construction, the constant \(\rho \) is estimated by the average of the \(n(n{-}1)/2\) pairwise correlations of returns. In the single index model based also on n risky securities, the random return \(R_{i}\) of each security i is characterized as \(R_{i}=\alpha _{i}+\beta _{i}I+\epsilon _{i},\) for \(i=1,2,\ldots ,n,\) where I is the random return of an index, \(\alpha _{i}\) and \(\beta _{i}\) are constants, and \(\epsilon _{i}\) is random noise. Under the assumption of \(Cov(\epsilon _{i},\epsilon _{j})=Cov(\epsilon _{i},I)=0,\) the covariance of returns between securities i and j is \(\sigma _{ij}=\beta _{i}\beta _{j}Var(I),\) for \(i,j=1,2,\ldots ,n\) and \(i\ne j.\) Here, \(Var(\cdot )\) and \(Cov(\cdot ,\cdot )\) are the variance and the covariance of the random variables involved, respectively. In each of the two models, the covariance matrix of returns can be written as \(\varvec{V}=\varvec{BB}^{\prime }+ \varvec{D},\) where \(\varvec{B}\) is an n-element column vector, \( \varvec{D}\) is an \(n\times n\) diagonal matrix with all positive diagonal elements, and the prime denotes matrix transposition. As \(\varvec{x} ^{\prime }\varvec{Vx}\) is always positive for any nonzero column vector \(\varvec{x}\) of n real numbers, the positive definiteness of \( \varvec{V}\) is assured.
 
3
For a convex objective function \(f(\varvec{x})\) where the column vector \( \varvec{x}\) represents a set of n decision variables \( x_{1},x_{2},\ldots ,x_{n},\) the approximated objective function is of the quadratic form \((\varvec{x}-\varvec{x}_{0})^{\prime }\varvec{H} _{0}(\varvec{x}-\varvec{x}_{0}).\) Here, \(\varvec{x}_{0}\) is a column vector representing the values of \(x_{1},x_{2},\ldots ,x_{n}\) at which each element of the Hessian matrix \(\varvec{H}_{0}\) is evaluated, and the prime denotes matrix transposition. To solve the corresponding constrained optimization problem by following the Markowitz approach, it is more convenient to use \(\varvec{y}= \varvec{x}-\varvec{x}_{0}\) instead as the n decision variables. With such variable changes, if each portfolio weight in the Markowitz model is required to be in the range between zero and an upper investment limit, then each decision variable here will be in the range between its minimum and maximum values instead. However, the consequences of relaxing the positive definiteness requirement for the covariance matrix in the Markowitz model will still carry over to the Hessian matrix here. For example, the Swiss Solvency Test and European Union’s Solvency II stipulate the minimal levels of financial resources that individual insurers must maintain, as safeguards against adverse events. Each insurer’s convex objective function for satisfying the Solvency Capital Requirement, if approximated in the above manner, will lead to a positive semi-definite Hessian matrix as evaluated locally for some preset values of the decision variables. If this Hessian matrix fails to satisfy the positive definiteness requirement, an examination of the corresponding optimization results can reveal whether the failure is consequential locally.
 
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Metadata
Title
What really happens if the positive definiteness requirement on the covariance matrix of returns is relaxed in efficient portfolio selection?
Author
Clarence C. Y. Kwan
Publication date
05-02-2018
Publisher
Springer US
Published in
Financial Markets and Portfolio Management / Issue 1/2018
Print ISSN: 1934-4554
Electronic ISSN: 2373-8529
DOI
https://doi.org/10.1007/s11408-018-0306-7

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