An Exact Test of the Improvement of the Minimum Variance Portfolio

• Date: 01 March 2019
• Published date: 01 March 2019
• DOI: http://doi.org/10.1111/irfi.12173
• Author: Paskalis Glabadanidis

An Exact Test of the Improvement

of the Minimum Variance

Portfolio*

PASKALIS GLABADANIDIS

Business School, University of Adelaide, Adelaide, South Australia, Australia

ABSTRACT

I propose an exact finite sample test of the risk reduction of the global mini-

mum variance (GMV) portfolio. The GMV test statistic is proportional to the

reduction in the variance of the GMV portfolio and has a straightforward

geometric and portfolio interpretation and complements the celebrated GRS

test in Gibbons et al. (1989). In practical applications, the GMV test leads to

a rejection of the null hypothesis of no improvement in the GMV portfolio

more often than the GRS test rejects the null hypothesis of no improvement

in the risk-return profile of the tangent portfolio. The power of the GMV test

increases with the variance reduction of the GMV portfolio. Using test asset

returns scaled by predetermined predictive variables is equivalent to increas-

ing the overall number of test assets and leads to substantial power gains.

JEL Codes: G11; G12

Accepted: 15 December 2017

I. INTRODUCTION

Following the pioneering work on the mean–variance trade-off in Markowitz

(1952) the question of mean–variance efficiency of an asset or a set of proposed

asset-pricing factors has been of great importance to investment practitioners as

well as finance researchers. Early theoretical work on the first two moments of

risky asset returns focused on preference-free methods of ranking like first- and

second-order stochastic dominance as well as necessary and sufficient paramet-

ric restrictions on investors’preferences or the data-generating process of risky

asset returns that would lead to a meaningful equilibrium trade-off between the

first two return moments. Early empirical tests relied mostly on asymptotic

econometric tests on relatively short historical records of a small set of

* I would like to thank Ding Ding, Jozef Drienko, Hardy Hulley, Phong Ngo, Talis Putnins, Jianxin

Wang and Takeshi Yamada as well as seminar participants at Australian National University Research

School of Finance, Actuarial Studies and Statistics and UTS Business School as well as participants in

the 2017 Auckland Finance Meeting for their valuable suggestions and comments. I would especially

like to thank the editor, Huining Cao, and one anonymous referee for their very detailed and

thoughtful comments. Any remaining errors are my own.

© 2018 International Review of Finance Ltd. 2018

International Review of Finance, 19:1, 2019: pp. 45–82

DOI: 10.1111/irfi.12173

portfolios and individual stock returns (see Gibbons 1982, Jobson and Korkie

1982, 1985, 1989, and MacKinlay (1987), for example). It was not until the

ground-breaking work in Gibbons et al. (1989) (GRS) that presented a finite

sample test of mean–variance efficiency of one or a set of assets with respect to

another set of basis securities, facilitating reliable inferences in empirical work

using only a limited historical time series of returns. Even more importantly,

GRS presented the finite sample distribution of their test under both the null

hypothesis of mean–variance efficiency and the alternative hypothesis of cer-

tain level of inefficiency. Their test is based on the proportional improvement

in one plus the squared Sharpe ratio of the tangent portfolio following the addi-

tion of the test asset or assets to the mean–variance frontier consisting of the

base securities. The power of the GRS test increases in the relative improvement

in the risk-return trade-off offered by the inclusion of the test assets. Recent

work extending the classic GRS test statistic includes Chou and Zhou (2006),

Barillas and Shanken (2015), and Hwang and Satchell (2015) among others.

In this paper, I propose a new exact test of the reduction of the variance of

the global minimum variance (GMV) portfolio based on the proportional

improvement in the variance of the GMV portfolio following the addition of

the test assets to the mean–variance frontier constructed with the base assets.

The GMV test has an identical finite sample distribution with the GRS test

under both the null and the alternative hypothesis. Furthermore, the GMV test

often produces higher values for the test statistic than the GRS test. The reason

for this is the fact that the proportional change in the squared value of a given

variable is always higher than the proportional change in one plus the same

squared variable. I present a graphical illustration to illustrate the intuition

behind this straightforward mathematical fact. Furthermore, I present a sub-

stantial amount of empirical evidence documenting that superior value of the

GMV test over the GRS test using various sets of US and international stock

portfolios in both an unconditional and several conditional forms.

The contributions of this paper are as follows. First, I derive a finite-sample

GMV improvement test based on the risk reduction in the GMV portfolio follow-

ing the addition of the test assets to the set of base assets. Second, I offer a geo-

metric and a portfolio intuition behind the GMV test and present an intuitive

explanation as to the reason it will typically lead to larger values than the GRS

test. Thirdly, I present the power function of the GMV test and offer thoughts on

experimental design cases where the test will be more powerful. Finally, I apply

both the GRS and the GMV tests to a large set of US and international portfolios

sorted by various characteristics and past returns as well as by industry. I perform

both unconditional and conditional tests using several predetermined predictive

variables. The empirical evidence overwhelmingly support the theoretical intui-

tion behind the derivation of the GMV test. In practice, this raises suspicion over

the validity of many unconditional versions of the market model of Sharpe

(1964), the three-factor model of Fama and French (1992) and the four-factor

model of Carhart (1997) while many conditional models are either marginally

rejected or fail to be rejected when using the GMV test.

© 2018 International Review of Finance Ltd. 201846

International Review of Finance

The paper proceeds as follows. Section II presents the derivation of the GMV

test. Section III discusses the geometric interpretation while Section IV demon-

strates the logic of the GMV statistic in a mean–variance optimal portfolio

framework. Section V briefly discusses issues relating to the power of the GMV

test. Section VI presents the empirical findings of applying both the GMV and

the GRS tests with several robustness checks discussed in Section VII. Finally,

Section VIII offers a few concluding thoughts.

II. TEST STATISTIC DERIVATION

Let R

1

be the T×Kmatrix of realized excess returns on the Ktest assets over

Tperiods. Similarly, let R

2

be the T×Nmatrix of realized excess returns on the

Nbasis assets. Consider the following linear regression of the basis assets

returns on the test assets returns

R0

2t=α+βR0

1t+ϵt,ð1Þ

where βis a N×Kmatrix of loadings of the test assets on the basis assets and

Σ=var.(ϵ

t

). Expressing the above regression in matrix form we have

R2=XB +E,ð2Þ

where X is a T×(K+ 1) matrix with a typical row of 1, R=

1t

hi

,B=[α,β]

0

, and Eis

aT×Nmatrix with ϵ’

tas a typical row. The maximum likelihood estimate of

Band Σare

^

B=X0XðÞ

−1X0R2

ðÞ,ð3Þ

^

Σ=1

TR2−X^

B



0R2−X^

B



:ð4Þ

In what follows it is helpful to introduce the following linear transform of β:

δ=1

N−β1K,ð5Þ

where 1

M

is an M-element column-vector of ones. The relative improvement of

the tangent and GMV portfolios on the frontier generated by R

1

(the bench-

mark frontier) and the frontier generated by R

2

(the expanded frontier) depend

crucially on the values of αand δ, respectively. The mean–variance efficiency

test in Gibbons et al. (1989) is focused on testing the null hypothesis HGRS

0:

α=0

Nversus the alternative hypothesis HGRS

A:α6¼ 0Nwhere 0

M

is an M-element

© 2018 International Review of Finance Ltd. 2018 47

GMV Test