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A Test of the Efficiency of a Given Portfolio

Author(s): Michael R. Gibbons, Stephen A. Ross, Jay Shanken

Source:

Econometrica,

Vol. 57, No. 5 (Sep., 1989), pp. 1121-1152

Published by: The Econometric Society

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Econometrica, Vol. 57, No. 5 (September, 1989), 1121-1152

A TEST OF THE EFFICIENCY OF A GIVEN PORTFOLIO

MICHAEL

GIBBONS, STEPHEN

A. Ross,

AND JAY SHANKEN1

A test for the ex ante efficiency of a given portfolio of assets is analyzed. The relevant

statistic has a tractable small sample distribution. Its power function is derived and used to

study the sensitivity of the test to the portfolio choice and to the number of assets used to

determine the ex post mean-variance efficient frontier.

Several intuitive interpretations of the test are provided, including a simple mean-stan-

dard deviation geometric explanation. A univariate

test, equivalent to our multivariate-based

method, is derived, and it suggests some useful diagnostic tools which may explain why the

null hypothesis is rejected.

Empirical examples suggest that the multivariate approach can lead to more appropriate

conclusions than those based on traditional inference which relies on a set of dependent

univariate statistics.

KEYWORDS:

Asset pricing, CAPM, multivariate test, portfolio efficiency.

1. INTRODUCTION

The modern theory of finance has always been rooted in empirical analysis.

The mean-variance capital asset pricing model (CAPM) developed by Sharpe

(1964) and Lintner (1965) has been studied and tested in more papers than can

possibly be attributed here. This is only natural; the quality and quantity of

financial data, especially stock market price series, are the envy of other fields in

economics.

The theory is generally expressed in terms of its first-order conditions on the

risk premium. Expected returns on assets are linearly related to the regression

coefficients, or betas, of the asset returns on some index of market returns. In

other words, risk premiums in equilibrium

depend on betas. The standard tests of

the CAPM are based on regression

techniques with various adaptations. For

some notable examples, see Black,

Jensen, and Scholes (1972) and Fama and

MacBeth (1973). Usually, cross-sectional regressions are run of asset returns on

estimated beta

coefficients, and estimates of the slope are reported. Often the

data are grouped to reduce measurement

errors, and sometimes the estimation is

done at a

sequence

of time

points to create a time series of estimates from which

the precision of the overall

average can be determined.

Roll (1977, 1978), among others, has raised serious doubts whether these

procedures are, in fact, tests of the CAPM. Insofar as proxies are used for the

market

portfolio, the Sharpe-Lintner

theory is not being tested. Furthermore, as

Roll

emphasizes, the regression tests are probably of quite low power, and

We are

grateful

to Ted

Anderson,

Fischer

Black, Douglas Breeden,

Michael

Brennan, Gary

Chamberlain,

Dave

Jobson,

Allan

Kleidon,

Bruce

Lehmann,

Paul

Pfleiderer,

Richard

Roll,

and two

anonymous

referees as well as the seminar

participants

at Duke

University,

Harvard

University,

Indiana University, Stanford University, University of California at San Diego, University

of Illinois

at Urbana, and Yale

University

for

helpful

comments. We

appreciate

the research assistance of

Ajay

Dravid, Jung-Jin Lee,

and

Tong-sheng Sun.

Financial

support

was

provided

part by

the National

Science Foundation and the Stanford

Program

in Finance. This

paper supersedes

an earlier

paper

with the same title by Stephen Ross.

1121

1122

MICHAEL R. GIBBONS, STEPHEN A. ROSS,

AND JAY SHANKEN

These objections

leave the empirical

grouping may lower the power

further.

testing of the CAPM

in an odd state of limbo. If the proxy is not a

valid

are

investigations

then as tests of the CAPM the existing

empirical

surrogate,

beside the point.2

On the other hand, if the proxy

is valid, then

the

somewhat

of the tests are

unknown.

small sample

distribution and power

is one

opportunity. The CAPM

This is unfortunate and

indicative of a missed

couched

which suggest

quite specific

hypotheses

of many financial

theories

these hypotheses are an

terms of observables. The rich

data available for testing

at them.

In this paper

are

explicitly

directed

tests which

incentive

to develop

a test using multivariate statistical methods.

example of such

develop

a canonical

in tests of the CAPM.

one addressed

is the central

The problem

we consider

Since the theory is equivalent

to the assertion that the market portfolio

portfolio

mean-variance

efficient,

we wish to test whether any particular

efficient.

ex ante mean-variance

into seven sections, it also can be

viewed as

While the paper is organized

consisting of three parts.

The first part (Sections 2 through 4) considers

the prop-

efficiency and examines

for testing mean-variance

multivariate statistic

erties of such a test. The second part (Sections

5 and 6) studies the relation

based on a set of

approaches

between this multivariate test and alternative

the paper by

7 and 8) concludes

statistics. The third

part (Sections

univariate

for

and providing suggestions

to related

hypotheses

extending the framework

summary of each

section

follows.

detailed

future research. A more

portfolio.

for the efficiency of some

In Section 2 we recall a necessary condition

as a null hypothesis that can be tested using

a statistic

We use this implication

both the null and alternate

distribution under

sample

which

has a tractable finite

we relate

this statistic

to three alternative approaches

hypotheses.

In addition,

In the third section the multi-

which are based on asymptotic approximations.

deviation

interpretation in the mean-standard

variate test is given a geometric

to a data

are then applied

The method and geometry

space of portfolio theory.

we reaffirm and

in modern finance;

empirical papers

set from one of the classic

The fourth

section

and Scholes (1972).

Jensen,

the findings of Black,

complement

the sensitivity

of the test. Here

we consider

to the power

turns to issues relating

which is examined for efficiency and the

of the test to the choice

of the portfolio

frontier. A

of assets used to determine the ex post

efficient

effect of the number

that one's

new data base is analyzed

in this section, and we demonstrate

index can be altered by the type

the efficiency of a given

conclusions

regarding

assets used to construct the

post

frontier.

when the multi-

results

actual

to contrast

empirical

The fifth section attempts

based on a set of dependent

inference

informal

variate method

is used versus

the multivariate test

rejects

where

provide

Here

examples

univariate statistics.

We also have

seem

to be

significant.

none of the univariate statistics

even though

tests of the

do consider

and Stambaugh (1987)

and Shanken (1987b)

2Recent

work by Kandel

and the true

market

the proxy

on an assumption about

the correlation between

CAPM

conditional

portfolio.

PORTFOLIO

EFFICIENCY

1123

the reverse situation where there are a seemingly large number of "significant"

univariate statistics; yet, the multivariate test fails to reject at the traditional

levels of significance. In this section we also introduce another data set which

allows us to re-examine the size-effect anomaly. Section 6 develops an alternative

interpretation of the multivariate test. The statistic is equivalent to the usual

calculation for a t statistic on an intercept term in a univariate

simple regression

model, with the ex post efficient portfolio used as the dependent variable and the

portfolio whose ex ante efficiency is under examination as the explanatory

variable. This section also develops some useful diagnostics for explaining why

the null hypothesis may not be consistent with the data. Most of the

empirical

work in this section focuses on the size effect only in the month of January.

Section 7 extends the analysis to a case where one wishes to

investigate the

potential efficiency of some linear combination of a set of portfolios, where the

weights in the combination are not specified. This turns out to be a minor

adaptation of the work

in Section 2.

2. TEST STATISTIC FOR JUDGING

THE EFFICIENCY

OF A GIVEN PORTFOLIO

We assume throughout that there is a given riskless rate of interest,

Rft,

for

each time period. Excess returns are computed

by subtracting

Rft

from the total

rates of return. Consider the following multivariate linear regression:

(1)

flipppt

iit

'it=aip

1,. .., N,

where ri

the excess return on asset i in period t;

Fpp-

the excess return on the

portfolio whose efficiency is being tested; and

9it-

the disturbance term for asset

i in period t. The disturbances are assumed to be jointly

normally distributed

each period with mean zero and

nonsingular covariance matrix 2, conditional on

the excess returns for portfolio

p. We also assume independence of the distur-

bances over time. In order that

be nonsingular, Fp and the N left-hand side

assets must be linearly

independent.

If a particular

portfolio is mean-variance efficient (i.e., it minimizes variance

for a given level of expected

return), then the following first-order

condition must

be satisfied for the given N

assets:

(2)

(Fit)

fiPeQPt).

Thus, combining the first-order

condition in (2) with the distributional assump-

tion

given by (1) yields

the

following parameter restriction,

which is stated in the

form of a null

hypothesis:

(3)

Ho:

aip

=19,...,9N.

Testing the above null hypothesis is essentially the same proposal as in the

work by Black, Jensen, and Scholes

(1972), except that they replace

ip,

by a

portfolio

which

they call the market portfolio and refer to their test as a test of

the CAPM. In

addition, they do not report the joint significance of the estimated

1124

MICHAEL R. GIBBONS, STEPHEN

A. ROSS, AND JAY SHANKEN

values for

a?1p

across all N equations;

instead, they report N univariate

based on each

equation.

statistics

Given the

normality assumption,

the null

hypothesis

(3)

can be tested

using

"Hotelling's

T2 test,"

a multivariate

generalization of the univariate t-test

(e.g.,

see Malinvaud

(1980, page 230)). A brief derivation of the

equivalent

F test is

included for completeness and as a

means

introducing some

notation

that will

be needed

later. If we estimate

the multivariate

system of (1) using

ordinary least

for each individual

squares

equation, the estimated intercepts

have

multivariate

normal

distribution, conditional on

rp,

(Vt= 1,..., T), with

(4)

/T(+k)ap

NtT(

sT;

p)Op;

where T

number of time series

observations on returns;

P);

mean of r

and

sample

variance of

rp,

without

-plsp;

rp-

sample

adjustment for degrees of freedom.

Furthermore,

independent with

and T are

2)2 having a Wishart

distribution with parameters (T

2) and 2. These

facts imply (see Morrison

(1976, page 131)) that (T(T

1)/N(T

2))WJK has

a noncentral F

distribution with degrees

and (T

1), where

of freedom

(5)

apE lp/(l

and 2-

unbiased

residual covariance matrix.3

(The

corresponding

statistic

based

on the maximum

likelihood estimate of

will be denoted

as W.)

The

noncentral-

ity parameter,

is given

[T/(1+2

(6)

)]a

lap.

Under the null

hypothesis that

equals

zero,

0, and we have

a central

distribution. More

generally,

the distribution under the alternative

provides

way to study

the power

of the

test;

more will be said

about this

in a

later

section.

It is also interesting to note

that

under

the null hypothesis

the

statistic has a

central F distribution

for

the

parameters

unconditionally,

of this central F do not

depend

on rin

any

way.

However,

we do not know

the unconditional distribu-

tionofn

under the alternate, for the conditional

distribution depends on

the sample

values

of -p,

through 0.

has been viewed as

providing

Generally,

the

normality

assumption a

"good

working

approximation" to the distribution of

monthly

stock

returns

(see

Fama

(1976, Chapter

for a

summary of the relevant

There

is some

empinrcal

work).

that the true distributions are

slightly

evidence,

however,

relative

leptokurtic

the normal

distribution. While from

normality

of the disturbances in

departures

(1) will affect the

small-sample

distribution of the test

statistic,

simulation

evidence

by MacKinlay

that the F test is

fairly

robust

to such

(1985) suggests

of standard

misspecifications.4 This is important, since the

application

asymp-

totic tests to the

efficiency problem

can result in

faulty inferences,

given

the

sample

sizes often used in financial

empirical work.

Tests for normality of the

residuals of the size and

industry portfolios, which

are used

below,

reveal

excess

kurtosis and some These

skewness as well.

results are available on

request to the

authors.

that N is less than

or equal to T

2 so that I is

nonsingular.

4We

assume

PORTFOLIO EFFICIENCY

TABLE 1

1125

COMPARISON OF FOUR ASYMPTOTICALLY EQUIVALENT TESTS OF EX ANTE EFFICIENCY OF A

GIVEN PORTFOLIO.

THE

W STATISTIC IS DISTRIBUTED AS A TRANSFORM OF A CENTRAL

DISTRIBUTION

IN FINITE SAMPLES. THE WALD

TEST, THE LIKELIHOOD RATIO TEST

(LRT),

AND

THE LAGRANGE MULTIPLIER TEST

(LMT)

ARE MONOTONE TRANSFORMS OF

AND

EACH IS DISTRIBUTED

AS CHI-SQUARE WITH

DEGREES OF FREEDOM AS

APPROACHES

INFINITY.

P-Value Using

Exact Distribution

of W

P-Values Using

Asymptotic Approximations

Wald LRT

LMT

118

238

118

238

120

240

120

240

.05

.10

.008

.000

.023

.005

.000

.035

.109

.003

.000

.025

.000

.056

.017

.000

.076

.048

.009

.001

.000

.027

.007

.000

.038

.023

.003

.000

.044

.035

.017

.006

.000

.061

.019

.000

.081

.053

.010

.000

.090

.075

.041

.018

.000

.071

.094

.173

.403

.060

.070

.094

.122

.431

.055

.059

.069

.079

.123

.451

.122

.146

.216

.404

.111

.122

.147

.175

.432

.106

.111

.122

.133

.178

.452

Note: N is the number of assets used together with portfolio p to construct the ex

post frontier, and T is the

number of time series observations.

Table I illustrates

this problem

for the Wald, likelihood

ratio, and Lagrange

is asymptotically distributed as

chi-square with N

multiplier

tests, each

of which

degrees of freedom as T

oo.S Since the small-sample distribution of W is

known (assuming

realization of W can be inferred from

normality), the implied

the information in the first three columns of Table I

(i.e., N, T, and the

hypothetical p-value).

The implied

asymptotic p-values given in the last three

the

Jobson

three tests

using

a simulation. They approximate

and Korkie also discuss these

(1982)

In -their

on Rao's

(1951) work.

distribution of the likelihood ratio

test with an F distribution based

the null hypothesis.

is available under

distribution

sample

they recognize that

a small

1985 paper

1126

MICHAEL R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

columns are then obtained using the fact that each test statistic is a monotonic

function of W.6

Consistent with the results of Berndt and Savin (1977), the p-values are always

lowest for the Wald test and highest for the Lagrange multiplier test with the

likelihood ratio test in between. Clearly, the asymptotic approximation becomes

worse as the number of assets, N, approaches the number of time series

observations, T. Shanken (1985) reaches similar conclusions based on an approxi-

mation when the riskless asset is not observable.

3. A GEOMETRIC INTERPRETATION OF THE TEST STATISTIC, W

So far, the primary motivation for the W statistic has been its well-known

distributional properties. For rigorous statistical inference such results are an

absolute necessity. Just as important, though, is the development of a measure

which allows one to examine the economic significance of departures from the

null hypothesis. Fortunately, our test has a nice geometric interpretation.

It is shown in the Appendix that:

?*2

(7)

-12

=2_ 1

where

is the ex

post price

of risk

(i.e.,

the maximum excess

sample

mean

return per unit of sample standard deviation) and

is the ratio of ex post

p~~~~~

average excess return on portfolio p to its standard deviation (i.e., p

-p/sp).

Note that

cannot be less than one since 0* is the slope of the ex post frontier

based on all assets used in the test (including portfolio p).

The curve in Figure la represents the (ex post) minimum-variance frontier of

the risky assets. When a riskless investment is available, the frontier is a straight

line emanating from the origin and tangent to the curve at m. 0* is the slope of

the tangent line whereas

is the slope of the line through p.

An examination of (7) suggests that

should be close to one under the null

hypothesis. When 0* is sufficiently greater than 0

the return per unit of risk for

portfolio p is much

lower than the ex

post

frontier

tradeoff,

and we will

reject

the

hypothesis that portfolio p is ex ante

mean-variance efficient. In

Figure

just the distance along the ex post frontier up

any given

risk

level, a,

divided

by the

similar distance

along

the line from the

origin through p.

The reader may wonder why the test is based on the square

of the

slopes

opposed to

the actual

slopes.

The reason is

straightforward.

Our null

hypothesis

only represents

necessary

condition for ex ante

efficiency.

This condition is

satisfied even if portfolio p is on the negative sloping portion

of the minimum-

variance frontier for all assets (including the risk-free security). Thus, only the

The relations are LRT= T ln(1 + W) and LMT= TW/(1 + W). Shanken (1985) has

discussed

this result for the case where the riskless asset does not exist. A proof of

the result in the case where

the riskless asset does exist is available upon request to

the authors. Bemdt and Savin (1977) discuss

similar relationships among alternative asymptotic tests

in a more general setting.

PORTFOLIO

EFFICIENCY

1127

p~~~~~~~~~~~

Standard

Deviation

of Excess

Return

la.)

Geometric

intuition for W. Note the drstance

2.4

2.2

+ 02,

and the distance Od is

1.2]

0.8-

0.6-]

0.4

0.2-

Standard

Deviation

of Excess

Return

lb.)

Ex post efficient frontier based on 10

beta-sorted

portfolios

and tihe CRSP

Equal-Weighted

Index

using monthly

data,

1931-1965.

Point p repre-

sents the CRSP

Equal-Weighted

Index.

FIGURE

1.-Various plots of ex post mean variance efficient frontiers.

1128

MICHAEL R.

GIBBONS,

STEPHEN A.

ROSS,

AND JAY SHANKEN

1.7

1.6

1.5

1.4

1.3

1.2

0.9

0.7-

XOB

0.8

0.5

0.4-

0.3-

Standard

Deviation of Excess

Return

ic.) Ex post efficient

frontier based on 12

industry

portfolios and the CRSP

Value-Weighted

1n(lex

using monthly

data,

1926-1982. Point

p repre-

sents the CRSP

Value-Weighted

Index.

1.2-

1.1

0.9

0.8

0.?

0.6

0.5

0.4-

0.3-

0.2-

0 2

Standard

Deviation of Excess

Return

Id.) Ex post efficient frontier based on 10 size-sorted

portfolios and the CRSP

Value-Weighted

Index

using monthly

data,

1926-1982. Point

p repre-

sents the CRSP

Value-Weighted

Index.

FIGURE

1.-Continued.

PORTFOLIO EFFICIENCY

TABLE II

1129

SUMMARY

STATISTICS ON BETA-SORTED PORTFOLIOS BASED ON MONTHLY DATA,

1931-65

(T=420).

ALL SIMPLE EXCESS RETURNS ARE NOMINAL AND IN PERCENTAGE FORM, AND THE

INDEX IS PORTFOLIO

CRSP

EQUAL-WEIGHTED

PARAMETER ESTIMATES

THE FOLLOWING

AND

Vt =1.

ARE FOR THE REGRESSION MODEL:

Pit

alp

Pip Fpt

9,t

420,

WHERE R2 IS THE COEFFICIENT OF DETERMINATION FOR EQUATION i.

Portfolio

Number

&,P

S(&,P)

/3P

s(Ap)

0.94

0.97

0.98

0.97

0.96

0.92

0.85

-0.19

-0.06

-0.09

-0.06

0.05

0.03

0.12

0.14

0.22

0.17

0.10

0.09

0.07

0.08

0.09

1.54

1.37

1.24

1.17

1.06

0.92

0.86

0.74

0.63

0.51

0.02

0.01

NOTE: For this sample period

and

are 0.166 and 0.227, respectively. These imply a value for Wu equal to

0.023, which has a

p-value

of 0.476. Under the

hypothesis that the CRSP Equal-Weighted Index is efficient,

'( WTu)

is 0.024 and SD( W,,) is 0 01 1.

is relevant for

our absolute value of the slope null

hypothesis, and our

test is then

based on the squared values.

Figure

lb is based

on a data set that is very similar to the one used by Black,

Jensen, and Scholes (1972) (hereafter,

BJS).7 Using monthly returns on 10

beta-sorted

portfolios

from

January, 1931 through 1965, 0*

0.227

December,

0.166. To

while the CRSP Equal-Weighted

which is portfolio

Index, p, has

judge whether these two slopes are statistically

different,

we can calculate

is 0.02333. Based on the results

in Section

(#2

1),

which

we can use a central

F distribution

with degrees of freedom 10 and 409 to judge the statistical

significance of this

difference in slopes.

The

resulting F statistic is 0.96, which

has

a p-value of 0.48. Our

multivariate test confirms the conclusion

reached by BJS

in that the ex ante

efficiency of the CRSP Equal-

for their overall time

period

Index

cannot Weighted be rejected; equivalently, if this Index

is taken

as the true

market portfolio, then the Sharpe-Lintner version of the CAPM cannot be

rejected. Table

provides some summary statistics

on the beta-sorted portfolios

that were used for

Figure

lb. Table II, when compared

with Table

in BJS,

that our data

base is very

similar

verifies

to the one used by BJS.

BJS provide various scatter

plots of average returns versus

estimated betas to

data to the linear relation if the

CRSP

judge

the fit of the

expected

Equal-Weighted

Chicago,

it is not possible to replicate their The CRSP

the University data. tapes are continually

revised to reflect data errors, and one would need the same version of the CRSP file to perfectly

duplicate a data base. For

example,

we were able to find more firms per year than reported in Table 1

of BJS because of corrections to the data base. Also we relied on Ibbotson and Sinquefield (1979) for

the return of US Treasury Bills as the riskless rate. This latter data base was not used by BJS.

in forming the 10 portfolios that

were

However, we followed the grouping procedure outlined in BJS

1 and

Table

II.

used in constructing Figure

While

on the

data BJS relied from the Center for Research in Security Prices (hereafter, CRSP) at

1130

MICHAEL R. GIBBONS, STEPHEN

A. ROSS, AND JAY SHANKEN

Index is efficient. We view figures like our Figure lb as complementary to these

scatter plots, for they summarize the multivariate test in a manner familiar to

financial economists. The advantage of the scatter plots in BJS is that they may

provide some information as to which asset or which set of assets is least

consistent with the hypothesis that the index is efficient; figures like

Figure

really do not provide such information. On the other hand, the scatter plots in

BJS can be difficult to interpret due to heteroscedasticity across the different

portfolios as well as contemporaneous cross-sectional dependence. Section 6 will

suggest some other types of diagnostic information based on the multivariate

framework.

To understand further the behavior of our measure of efficiency,

4?,

its small

sample distribution given in Section

2 is

helpful. Since a linear transform of

has a central F distribution with degrees of freedom N and (T

1), we can

use the first two moments of the central F to calculate:

(8)

and

-1) = [

T-N-3

(9)

SD(P2_1)

=[T-N-3

T-N-5

The first moment for

only exists if T

N + 3 while the second moment for

only exists

if T> N + 5. These last two

equations

for the moments can be

applied to the BJS data set for 1931-1965 where

10 and T=

420, so

1) and the standard deviation of

are 0.024 and

0.011, respectively.

6,(+2-

the realized value of

is less than its

expectation, it is not surprising that

the ex ante efficiency of the

Equal-Weighted

Index cannot be

rejected

for this

time period.

This measure, 4, is a new variant of the geometry developed to examine

portfolio performance.

past procedures the efficient frontier has been taken as

given, and a distance such as mb

Figure la has been used as a measure of p's

performance. Note that mb is simply the return differential of the ex post

optimal portfolio over p, computed

at the

sample

standard deviation of the ex

post optimal portfolio.

Another

suggestion

has been to use the difference in their

slopes

as a measure of

p's

relative

performance.

How the true ex ante

frontier is to be known is unclear, and if the ex post frontier is

used,

then we face

the statistical

problem

of this

paper.

FOR EFFICIENCY

OF THE MULTIVARIATE TEST

4. THE POWER

The empirical illustration in the previous section fails to

reject

the ex ante

efficiency

of the

Equal-Weighted

Index when

using

10 beta-sorted

portfolios

as in

BJS.8 Such a result may occur because the null hypothesis is

in fact

true,

or it

the results of the multivariate test across these four

subperiods,

we can

reject

ex ante

efficiency

usual levels of

significance.

This confirms the conclusions reached

BJS.

We have also examined our data base

using

the same

subperiods

as in BJS. When we

aggregate

PORTFOLIO

EFFICIENCY

1131

may be due to the use

of a test which is not powerful enough to detect

of the Index. Questions

from efficiency

deviations

economically

important

power for various types of test statistics

have been a long standing

concern

will

others). This

section

among

financial economists (e.g.,

see Roll

(1977),

among

of the multivariate test.

focus on the power

both the null and alternate hypotheses

From Section

2 we know that under

of freedom

42,

has an F distribution with degrees

simple transform

and T

1. The F distribution is noncentral with

the noncentrality parameter

the noncentrality parameter is

the null hypothesis

(6); under

given by equation

under the alternative is

zero. It deserves emphasis

that the F distribution

parameter

conditional

on the returns of portfolio p since the noncentrality

on a

Thus, we

will be

studying

the

power

function conditional

depends on

function.

value for

not the unconditional power

The probability of rejecting a false

null hypothesis increases as the noncentral-

holding constant the numerator and denominator de-

increases,

ity parameter

that

page 193)).

Studying the factors

(Johnson and Kotz

(1970,

grees

of freedom

will give

some

the power

the noncentrality parameter, X, guidance about

affect

sum of

the multivariate test. From equation (6) we can see that

A is a weighted

matrix is the inverse

deviations about the point

=0. The weighting

squared

estimators for

ap.

Thus,

of the covariance matrix of the ordinary least squares

to the variability of

according

estimated

departures from the null are weighted

and the cross-sectional dependence among the estimators.

the estimator

economic interpre-

The noncentrality parameter can also be given

an intuitive

of equation

(23)

in the

Appendix would hold for the

tation. The derivation

of the

sample

that a'

-ap=

estimates, so it is also true

counterparts

population

9*2

It follows

directly

that

[

T/(I

"p2

)] (

@*2

0p2)

of the test will increase as the ex ante inefficiency of

Not surprisingly, the power

of the

slope

of the relevant

as measured in terms

opportu-

portfolio p

increases

of the estimator for

declines,

so the

the precision

nity sets. If

increases,

power

of the test decreases.

When

of the test is affected by

and

Op.

2 summarizes how the power

Figure

equal

one,

the null

the proportion of potential efficiency

(i.e.,

Op/O*)

approaches zero, the given portfolio

hypothesis is true.

As this proportion

2 is based

on values

for the

significance level,

less efficient.

Figure

becoming

work on asset

pricing

for existing

models;

and T that are common

empirical

significance level.

have used N

10 or 20 and T= 60 or 120 and a five percent

estimates of

between 0 and 0.4.

work on the CRSP Indexes

reports

Empirical

and 0*. In

We have used this range to guide our selection of a grid for

2 is based to eliminate one of the

on the assumption that

Figure

addition,

that

our

calculations of

power

that affect

this

assumption suggests

parameters

of the

underlying population.

representative

the

sample

are for situations where

Even within the range of parameters that

consider,

the probability

ranges

from five

percent

to nearly 100 percent

rejecting the null hypothesis

measures of

slope.

For

depending

on the difference between the

two relevant

1132

MICHAEL R.

GIBBONS,

STEPHEN

ROSS,

AND JAY

SHANKEN

~~~0

OTENTIAL

EFFICIENC'

2a.)

and rr

60.

ONI0

PRP

ENIA

ION OF p0TrNTIAL

2b.) N

20 and

'r-

60.

FIGURE

2.-Sensitivity

of the

power

of the test to the choice of the index. Each

figure

is based on a

different combination of the number of assets

(N)

and the number of time series observations

(T).

all cases a critical level of five

percent

is used.

high

relative

example,

to the

average

equals .2 (which

from

1926-1982)

and if N

20 and T

60,

then

the

probability

rejecting

a false null

hypothesis

from ten

percent

ranges

(for

9* =

.3)

to 98

percent

(for

9* =

1.0).

Given the data bases

that are

available,

empiricist

always

faced with the

question of the

appropriate

sizes for N and

T. For

example,

with the CRSP

monthly

file we have a data

base which

extends

back to 1926 for

every

firm

the New York Stock

Exchange.

This would

permit

the

empiricist

to use around

700 time series

observations

2000 firms

(i.e., T)

and

well over

(i.e., N). However,

the actual N used

may

be restricted

by T

keep

estimates of covariance

matrices

nonsingular,

and the actual T used is

constrained

concerns

over

PORTFOLIO

EFFICIENCY

1133

PROPORTION

ION

FOTENTIAL

EFFICIENC-y

2c.) N 1 arid

2d.)

20 and T

120.

FIGURE

2.-Continued.

stationarity. It is not uncommon

to see published work where

T is around 60

monthly observations and N is

between 10 and 20. While

these numbers for

and T are common, we

are not aware of any formal

attempts to study the

appropriate values to select. We

will now examine this issue in

the context of the

specific hypothesis of ex ante

efficiency. While the analysis is

focused on an

admittedly special case, our hope is

that it may shed some

light on other cases as

well.

To get more intuition

about the impact of N on equation

(6), consider a case

which

has a constant value down the

diagonal and a constant (but

different)

value for all off-diagonal

elements. Since

represents the

contemporaneous

covariances across assets after

the "market effect" has been

removed, sucha

1134

MICHAEL R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

structure includes the Sharpe (1963) diagonal model as a special case when the

off-diagonal terms are zero. The more general case where the off-diagonal terms

are constant but are nonzero is motivated by the work of Elton and Gruber

(1973) and Elton, Gruber, and Urich (1978).9 Under this structure we can

parameterize T as:

(10)

= (1

-P)

2IN +

'I,

where p

the correlation between

eit

and

ij,;

the variance of

eit; IA,

identity matrix of dimension N; and

I-

a 1 x N vector of l's. The inverse10 of

this patterned matrix is (Graybill (1983)):

(lP)2

[IN

(N-l)p

LNLNb

in equation (6) gives:

Substituting the above equation for .1

(11) =

T/(1+

(1p)W2

)

[N2

(1-p)+Np

where y-

ap)/N

and

(a'a

)/N. One could view

as a measure of the

"average" misspecification across assets while

indicates the noncentral disper-

sion of the departures from the null hypothesis across assets.

When N is relatively large and p is not equal to zero, equation (11) implies:

(12)

X/N_

A)y2

(1(

)VAR(ap),

where VAR

(ap)

denotes the cross-sectional variance of the elements of

ap.

Thus,

X is approximately proportional to N and T."1 Alternatively, if either p = 0 or

Unfortunately,

this is still

then

exactly proportional

to N and T.12

not adequate to determine the impact of changing N and T, for these two

parameters affect not only the noncentrality parameter but also the degrees of

freedom.

We have evaluated the power of the multivariate test for various combinations

of X, N, and T.13

These numerical results provide some guidance on the proper

9Strictly speaking, the Sharpe diagonal model allows for heteroscedasticity in the disturbances of

the market model equations; our formulation assumes homoscedasticity. Also, the constant correla-

tion model of Elton and Gruber is usually applied to the correlation matrix for total returns; we are

assuming constant correlation after the market effect has been removed.

Necessary and sufficient conditions for this inverse to exist are that p

1 and p

see Graybill (1983, page 190-191). In addition, the matrix should be positive definite; this would

require that p>

l/(N

1).

"1 In general, since p < 1, X/N is less than or equal to the right side of (12) when p ? 0.

If the Equal-Weighted Index is portfolio p, then we would expect

to approach zero as N

becomes large.

These numerical calculations require

evaluation of a noncentral F distribution and an inverse of

a central F distribution. The latter calculation relied

on the MDFI subroutine provided by IMSL. The

former calculations are based on a subroutine written by J. M. Bremner (1978), and a driver program

written by R. Bohrer and T. Yancey of the University of Illinois at Champaign-Urbana. Each

subroutine was checked by verifying its output with the published tables reported

in Tang (1938) and

Titu (1967).

PORTFOLIO

EFFICIENCY

1135

0 .8-

0.7

0.6

SA=O.1T

0-4

0.002NT

0.3

0.00002NT

0.2

0.4

0.6

0.8

3a.)

60.

Ratio of N Divided

by T

0.9

0.8

0.7-

0.6-

0.5-

0.4-

0.3-

0.2

]

0.002NT

0.1

]

A-

0.00002NT

0.2

0.4

0.6

O.B

Ratio

of N Divided

by T

3b.)

120.

of the number of assets

(N) gives

of the test to the choice

3.-Sensitivity of the power

FIGURE

fixed number of

time series observations (T).

that X is proportional to NT, and Figure

choice of

N and T. We assumed

We selected this

provides various

values for the

constant

of proportionality.14

based

equation

(11)

when

= 0. In this case, the

constant

of proportionality

the cross-sectional

)].

We then

2/[2(1

replaced

constant

and

with

data set. We also know

that

from an actual

of and

6i2,

respectively,

averages

Index (1931-1965) and 0.109 for

the

is 0.166 for the CRSP

Equal-Weighted

of the

(1987) studies the power of

the test using alternative parameterizations

MacKinlay

noncentrality parameter.

1136

MICHAEL R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

0.9

0.7

A=O.1T

0.002NT

0.5

0.4

0.3

0.2

0.002N

0.4

0.6 0.8

Ratio of N Divided by T

3c.) T

240.

FIGURE

3.-Continued.

guide to typical

This provides a rough

CRSP Index (1926-1982).

Value-Weighted

values for the constant of proportionality. The constant is

0.004 using the

beta-sorted

portfolios, and it is 0.002 using a set of industry

portfolios. For

size-sorted portfolios the constant is 0.004 using all months and 0.763 for

and

on how the

industry portfolios

January. (The

details

data only using

monthly

later

in this paper.)

were will be provided

created

size-sorted portfolios

is 0.00002

and 0.002, which

In Figure

3, we look at cases where

the constant

are small relative to the above calculations. For purposes of comparison, Figure

=.1T.

This

by N; instead

we set

also includes

a case where

is not affected

where an investigator has one asset that violates the null

a situation

represents

with the

hypothesis,

and all the remaining assets that are added are consistent

3 is based for

on specific values

Figure

portfolio p. While

efficiency of some

given

the general pattern that is observed is consistent

the constant of proportionality,

here.15

of choices but did not report

that we tried

with a wide range

of time series observations, Figure

3 demonstrates that

For a fixed number

Even

though

decision to be made

the

empiricist. there may be an important

it is not

necessarily appro-

increases as N

increases, the noncentrality parameter

possible.

Given our

particular parameterization

priate

to choose the maximum

a third to one half of

roughly

of the problem, it appears that

N should

data are used, 20 to 30 assets may

appropriate.

when five years of monthly

of N

values

is so small

for all

possible

When the constant is very

low,

the

power

F for

very high

values

so we have little

the noncentral

We were not able to evaluate

If the

high.

the constant of proportionality

of the

power

function when

knowledge about

the shape

= -

T 2 be

solution of setting N

it is conceivable that a corner

may

constant is large enough,

appropriate.

PORTFOLIO

EFFICIENCY

1137

that it is not an important decision.

Alternatively, if the noncentrality parameter

is proportional to T and not affected by N,

clearly

setting

1 is the preferred

In this case

adding

strategy. securities does not

provide

more information about

departures from the null hypothesis;

additional securities increase

however,

the

number

of unknown

parameters to be estimated. It deserves

emphasis that these

conclusions about

the

proper choice

of N

may

not be appropriate for all

possible

and models.

situations

The choices of N and T are not the

only decisions

facing the empiricist

the econometric

Since N must

always

designing

analysis. be less than T

(unless

highly structured covariance matrices

are entertained), the empiricist must also

decide

how to select the assets

to maximize the power

of the test.

Given N and T

we wish to maximize

the quadratic form

a-T'ap,

or equivalently 9*; however,

these parameters are unobservable. A common

approach

is to use

beta-sorted

in betas is useful in

decreasing

portfolios. While dispersion

the asymptotic

in estimates

standard error

of the expected return

on the zero-beta

asset (Gib-

bons (1980) and Shanken

(1982)), such sorting need not maximize

departures

from the null hypothesis as measured by X.16

Empirical examples presented below

illustrate the effect

that

different asset

sets

can have on the outcome of the test.

First,

we consider

a set of 12

industry

seems reasonable on economic

portfolios.17

industry

grouping

and

grounds

also captures some of the important correlations

To measure the

among

stocks.

return from a "buy-and-hold"

investment strategy, the relative market

values of

the securities are used

to weight the returns. Almost

return on the

every

monthly

CRSP

tape from

1926 is

included,

1982 which should

minimize

through

problems

with survivorship

bias.18

Table III provides some summary

statistics on the

industry

portfolios.

The multivariate F statistic rejects

the hypothesis of ex ante

efficiency at about

the one

percent

level.

The relevant

F statistic

is 2.13 with

degrees

significance

freedom 12 and 671; its p-value is 0.013.19

To complement

these numerical

results,

Figure Ic, which

is similar

to Figures

la and lb, provides

a geometrical

summary.

assets into N portfolios; we could

form N portfolios so that

they

have very little

dispersion in their

beta values with no impact

on the

power. This follows from

the

well-known result in the

multivariate

statistics

literature that our test is invariant to linear

transformations of the data (Anderson (1984,

pages 321-323)). Of course,

the selection of the original subset

of assets to be analyzed is important

even though the way they are into

portfolios is not (given

aggregated

that

the number of portfolios is

the same

as the number of original assets).

For the details of the data see Breeden, Gibbons, and

Litzenberger

base,

who

developed

(1987),

these data for tests of the consumption-based

asset pricing

model. The industry

grouping closely

follows a classification used

by Sharpe (1982).

with a SIC

number of 39 (i.e., miscellaneous manufacturing

However, all firms

industries) are

excluded to avoid any possible problems with a

singular

covariance matrix when the CRSP

Value-Weighted Index

is used as portfolio p.

While not

reported

we also

analyzed

this

data set across

various

Based on five

here,

subperiods.

for the F statistic is less than in 7 out

of 11

cases,

year five

subperiods, the p-value

is less than

percent

in 9 out of 11 cases,

10 percent

and rejects when across

aggregated the

subperiods.

the

rejection Thus,

of the overall

is confirmed by the subperiods as well.

period

In fact, for a given

set of N securities, the multivariate

test is invariant to how we group

these

1138

MICHAEL

R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

To understand this low p-value, consider

the fact that for this time period

0.109 while the slope of the opportunity

set using the ex post optimal

than double with a value

of 0.224. With these numbers we

portfolio, 9*, is more

For N= 12 and T=

684,

e(42)

1.018 with

SD(

can calculate

as 1.038.

three standard deviations from

of 0.007. Thus, the realized value

nearly

Index is truly ex ante efficient.

value

if the CRSP

Value-Weighted

its expected

of greater interest

the null

Perhaps is the fact that the multivariate test rejects

all 12 univariate t statistics fail to

hypothesis at the one percent level even though

builds on such contrasting

reject at even the five percent level.

The next section

to summarize across

results by analyzing why univariate test may be difficult

different assets.

5. THE PROBLEM WITH UNIVARIATE

TESTS

Table

suggests that high beta portfolios earn too little and low beta

Index is presumed

to be efficient;

portfolios too much if the Equal-Weighted

for the zero-beta version of

similar evidence was used by BJS to garner

support

is difficult to

interpret.

The

upper

the CAPM.

triangular portion

Yet, this pattern

of the market model

correlation matrix residuals

of Table

IV provides

the sample

in Table II. A

very distinctive

based on the regressions that are summarized

are

the residuals of portfolios with

similar betas

positively

pattern emerges in that

correlated

while those of portfolios with very different betas are negatively

for

in equation

Based on the variance-covariance matrix

(4), it is

correlated.

the same pattern of correlation. Thus, it

clear that the estimators for

aip

will

have

in estimated

pattern values of a 's is

is difficult to infer whether the observed

TABLE III

ON INDUSTRY-SORTED PORTFOLIOS

BASED ON MONTHLY DATA,

SUMMARY STATISTICS

FORM,

1926-82 (T

684). ALL SIMPLE EXCESS RETURNS ARE NOMINAL AND IN PERCENTAGE

AND THE CRSP

VALUE-WEIGHTED INDEX IS PORTFOLIO

PARAMETER

THE FOLLOWING

ESTIMATES ARE

FOR THE REGRESSION MODEL:

,,=

a?ip

+/ApFpt

AND

=1,.12

Vt=

1.684,

WHEREI

IS THE COEFFICIENT OF DETERMINATION FOR EQUATION i.

Industry Portfolio

s(fi

0.17

-0.05

0.03

0.00

0.12

-0.17

0.10

-0.17

0.05

0.00

0.43

-0.03

s(&,p)

Petroleum

Financial

Consumer Durables

Basic Industries

Food and Tobacco

Construction

Capital Goods

Transportation

Utilities

Trade and Textiles

Services

Recreation

0.14

0.09

0.00

0.07

0.12

0.08

0.14

0.09

0.00

0.37

0.13

0.93

1.19

1.29

1.09

0.76

1.20

1.08

1.20

0.74

0.94

0.80

1.22

0.02

0.01

0.02

0.01

0.02

0.06

0.02

0.69

0.89

0.90

0.94

0.83

0.85

0.91

0.78

0.76

0.77

0.19

0.78

NOTE:

For this sample period

and 6* are 0.109 and 0.224, respectively. These imply a value for Wu equal to

0.038, which has a p-value of 0.013. Under the hypothesis

that the CRSP

Value-Weighted Index is efficient, 9'( W")

is 0.018 and

SD(W")

is 0.007.

PORTFOLIO EFFICIENCY

TABLE IV

1139

SAMPLE

CORRELATION

MATRIX OF RESIDUALS FROM MARKET MODEL REGRESSIONS USING

EXCESS RETURNS.

THE UPPER TRIANGULAR

PORTION OF THE TABLE IS BASED ON 10 BETA-SORTED PORTFOLIOS

FOR THE DEPENDENT

VARIABLES AND THE

CRSP

EQUAL-WEIGHTED

INDEX FOR PORTFOLIO

ALL MONTHLY

DATA FROM

1931-65

420)

ARE USED. TABLE II

SUMMARIZES THE

OTHER

PARAMETER ESTIMATES FOR THIS REGRESSION

MODEL.

THE

LOWER

TRIANGULAR

PORTION OF THE TABLE IS BASED ON 10

SIZE-SORTED PORTFOLIOS FOR THE DEPENDENT

VARIABLES

AND THE

CRSP

VALUE-WEIGHTED

INDEX FOR PORTFOLIO

ALL MONTHLY

DATA FROM

1926-82

684)

ARE USED. TABLE V SUMMARIZES THE OTHER PARAMETER

ESTIMATES FOR THIS REGRESSION

MODEL.

1 2

4 5

Portfolio Number:

6 7

9 10

.52

.62

.72 .68

.66

.70

.66

.63

.61

.63

.41

.52

.39 .39

.28 .35

-.54 -.59

.39

.38

.75

.70

.68

.57

.51

.21

-.68

.03

.01

.08

.72

.65

.55

.46

.27

-.66

-.32

-.16

-.06

.72

.62

.50

.26

-.68

-.51

-.35

-.25

-.07

.21

.67

.59

.36

-.68

-.64

-.50

-.37

-.11

.25

.34

.52

.27

-.61

-.60

-.52

-.43

-.13

.12

.36

.43

.30

-.66

-.64

-.53

-.46

-.32

.03

.26

.46

.49

-.38

-.50

-.55

-.49

-.22

- .16

.06

.23

.27

.56

NOTE:

For the upper triangular portion of the table, portfolio 1 consists of firms with the highest values for

historical estimates of beta while portfolio 10 contains the firms with the lowest values. For the lower triangular

portion of the table, portfolio 1 is a value-weighted portfolio of firms whose market capitalization is in the lowest

decile of the NYSE while portfolio 10 contains firms in the highest decile.

in the true

in the estimation or to the actual

pattern due to correlation error

parameters.

in financial economics could

also be cited

from

Other empirical work

examples

of Banz

difficult to interpret. Since the work and

(1981)

where univariate tests

are

a great

deal of attention. (For

has received

Reinganum (1981), the "size effect"

see Schwert

more information about this research

(1983),

who summarizes the

a useful bibliography.) While most of the

existing evidence and also provides

in this area

now focuses on returns in January, we begin

by looking at

research

which did

not distinguish between January and

non-January

evidence

the original

returns.

We have created

a data base of monthly

stock returns

using the CRSP file.

based on the relative

market value of their

Firms were sorted

into 10 portfolios

In other we ranked

firms

total equity

outstanding.

market values

by their

words,

in December, 1925 (say), and we then formed 10 portfolios where the first

in the lowest decile of firm

size and the tenth

portfolio contains all those

firms

in the

highest

of firm size on the New York

companies

decile

consists

portfolio

and the firms are

value-weighted,

Each of the ten

portfolios

Stock Exchange.

for five

years.

market values

not resorted Thus,

the returns on these 10

by their

1926

through December,

1930

represent

the returns

portfolios

from

January,

for five this

portfo- from a buy-and-hold strategy without years;

any rebalancing

to represent a low transaction cost investment strat-

lio formation was adopted

1140

MICHAEL R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

TABLE V

SUMMARY STATISTICS ON

SIZE-SORTED PORTFOLIOS BASED ON MONTHLY

DATA,

1926-82

684).

ALL SIMPLE EXCESS RETURNS ARE NOMINAL AND IN PERCENTAGE FORM, AND THE

CRSP

VALUE-WEIGHTED INDEX IS PORTFOLIO

THE FOLLOWING PARAMETER ESTIMATES

,,fprpt

Eit

Vi =

10 AND Vt =1.

MODEL:

ARE FOR THE REGRESSION

684,

Fit

WHERE R2

IS THE COEFFICIENT OF DETERMINATION FOR EQUATION i.

Portfolio

Number

/,p

s(6,p)

f3,lip

s(ip)

0.28

0.34

0.25

0.18

0.19

0.18

0.08

0.00

-0.01

0.24

0.18

0.14

0.13

0.10

0.09

0.07

0.06

0.00

0.05

1.59

1.45

1.40

1.36

1.27

1.25

1.18

1.17

1.16

0.94

0.04

0.03

0.02

0.01

0.00

0.68

0.78

0.84

0.85

0.89

0.91

0.93

0.95

0.96

0.98

NOTE:

Portfolio 1 is a value-weighted portfolio of firms whose market capitalization is in the lowest decile of the

NYSE while portfolio 10 contains firms in the highest decile. For this sample period

and #* are 0.109 and 0.172,

respectively. These imply a value for Wu equal to 0.017, which has a p-value of 0.301. Jnder the hypothesis that the

is 0.007.

CRSP Value-Weighted

Index is efficient, 6(W") is 0.015 and SD(PWV)

egy. The resorting and rebalancing

occurred in December of

1925, 1930,...,1980.

for the entire

Table

V summarizes the

behavior of the returns on these portfolios

time

period.

Given the existing evidence on the size effect, some readers

may find it

somewhat surprising

that,

in the overall

period

from 1926

through 1982, the

of the CRSP

Value-Weighted

Index at

multivariate test fails to reject

efficiency

reports

The first row of Table the statistic and

the usual

levels of

significance.

a geometrical interpretation for

its corresponding p-value;

Figure

Id provides

this overall

period.20

The correlation

matrix of the market model residuals of the size portfolios

exhibits a distinctive of Table IV provides

pattern. The lower triangular portion

this information based on the overall

period. However, the pattern is identical

in Table

ten

year

across

every subperiod reported VI, and a similar pattern is also

described

by Brown, Kleidon,

and Marsh

(1983, page 47) and Huberman and

The correlation is positive

Kandel

(1985b).

and high

among

the low decile

firms.

The correlation declines as one compares portfolios from very different deciles.

Even more striking

is the fact that the highest decile portfolio has negative

with all other decile portfolios.

sample correlation (In some of the subperiods,

for the ninth decile as

well.) Thus,

if we

occurred

this negative correlation

observe that the lowest decile performs well (i.e., estimated alphas that are

that the highest decile would do poorly

we would then

expect positive), (and

vice

in the

period 1946-1955, where

This is the case, for example,

five out of

versa).

The subperiod results in Table VI are consistent with the conclusions of Brown, Kleidon, and

Marsh (1983) who find the size effect is not constant across all subperiods.

PORTFOLIO EFFICIENCY

TABLE VI

1141

TESTING THE EX ANTE

EFFICIENCY OF THE

CRSP VALUE-WEIGHTED INDEX (I.E., PORTFOLIO

PORTFOLIOS. ALL SIMPLE EXCESS

RETURNS ARE NOMINAL

RELATIVE TO 10 SIZE-SORTED

AND IN PERCENTAGE FORM. OVERALL

PERIOD IS BASED ON ALL MONTHLY

DATA FROM

1926-82.

THE FOLLOWING MODEL IS ESTIMATED

AND TESTED:

alp+

Aprpt

9,t

10 AND Vt= 1,...,T.

Ho:

a,p

Vi=

1,0.

Time

Period

(T)

(P-Value)

Number

?t(ap)

1.96

1926-1982

(684)

1926-1935

(120)

1936-1945

(120)

1946-1955

(120)

1956-1965

(120)

1966-1975

(120)

1976-1982

(84)

0.109

0.065

0.146

0.308

0.216

-0.019

0.093

0.172

0.354

0.286

0.469

0.604

0.408

0.538

0.018

(0.301)

0.119

(0.227)

0.059

(0.765)

0.113

(0.264)

0.302

(0.001)

0.164

(0.065)

0.275

(0.039)

NOTE: on the CRSP

Value-Weighted Index divided by its

excess return

is the ratio of the sample average

sample standard

deviation,

and #* is the maximum value

possible

of the ratio

of the

sample

average

excess return

divided

by the sample

standard deviation.

(#*2

)/(1

2),

and it is distributed as a transform of a central

with

degrees

of freedom 10 and T

11 under the null

hypothesis.

W should

converge

F distribution

to zero as T

the

p-values

for the

approaches

infinity

if the CRSP

Value-Weighted Index

is ex ante efficient.

By converting

for a standardized normal

statistics to an implied realization

random the results

across the 6

subperiods

variable,

and standardized normals

can be summarized by summing

and dividing root of

the 6

independent

by the square

in Shanken

6 as suggested This

quantity

is 2.87 which

implies

across

the

subperiods

rejection

at the usual

(1985).

levels of significance.

ten portfolios have

significant alphas

but the multivari-

(at

the five

percent level),

ate test cannot

reject

the

efficiency

of the

Value-Weighted

Index.

Even though summarizing the results of

univariate tests can be difficult,

work continues to

report

such statistics.

This is

only natural,

applied empirical

for univariate tests are more

intuitive

because

and

(perhaps

they

are used

more)

seem to give more

diagnostic information about

the nature of the

departure from

the null

hypothesis

when

it is

rejected.

Part of the

goal

of this

paper

is to

provide

some intuition

behind

multivariate tests. Section 3 has already done this to some

extent by

demonstrating that the multivariate test can be viewed

as a particular

in mean-standard deviation

space

portfolio theory.

The next

measurement

section

shows

that

the multivariate test is equivalent to a

test" on the

intercept

in a particular regression which

should be intuitive. A way to generate

diagnostic

about the nature

information

of the departures from the null

hypothesis

is also

provided.

6. ANOTHER

INTERPRETATION OF THE TEST

STATISTIC,

Vi is violated

if and

only

if some

linear The hypothesis combina-

that

aoip

of the N assets

has a

tion of the a's is

zero;

i.e.,

if and

only

if some

portfolio

1142

MICHAEL R. GIBBONS, STEPHEN A. ROSS, AND JAY SHANKEN

nonzero intercept when its excess

returns are regressed on those of portfolio

the portfolio

which,

in a

given

With this in mind, it is interesting to consider

of the usual t statistic

the square for the intercept. It is well

sample,

maximizes

on multivariate statistics

that this maximum value is

known in the literature

statistic,

our

TW,.

In this section

we focus on the

composition

Hotelling's

the maximizing portfolio,

a, and its economic

interpretation.

Thus, let

Frat

a'F2, where F2,

is an N x 1 vector with typical element

Vi-1,..., N. Let a& be the N x

vector of regression intercept estimates. Then

a'&t and

VAR(

)a'>a/T

by (4) above.21 Therefore,

[&A]2

(A'&)2AY

(13)

[S(^)]=

Since

we can multiply

a by any scalar

without of

t2,

we shall

changing the value

= c, where c is any constant

different

from

adopt the normalization that a'&t

zero. With this normalization,

T(a'&t)2

and (1 +

0p2)

(13) are fixed

given the

is equivalent to the following minimization prob-

sample. Hence, maximizing

lem:

min:

a'Ta

=c.

a'ap

to:

subject

Since

the above

problem is similar to the standard portfolio

problem, the form

the solution

is:

a p

for a into equation

Substituting this solution

(13),

becomes:

this equation with

(5)

establishes that

TWa.

Not surprisingly,

Combining

the

is not Student

t, for portfolio a was formed after

examining

distribution

the data.

The derivation of

suggests some additional information to summarize

work on ex ante

efficiency. Given the actual

value of a based on the empirical

sample, one

will know the

particular

linear combination which led to the

of the null

hypothesis. If the

null

hypothesis is rejected, then a may

give

rejection

us some constructive information about a better

how to create

model.

Portfolio a has an economic

basis as well. When this portfolio is combined

with portfolio

properly p, the combination turns

out to be ex post efficient. In

Since we are

working

with returns in excess of the riskless

rate,

a need not

equal

for the

short)

asset

will be held

(long

so that all wealth is invested.

riskless

PORTFOLIO EFFICIENCY

1143

of k,

other words,

for some

value

(14)

= (1

-k)Fpt

krat,

Ft*

For convenience, we set

is the return on this ex post efficient portfolio.

where

Ft*

means of

rpt,

are all equal.

The

equivalence

c so that the sample

rat,

and

Ft*

means

requires that:

these three

aCf

a2rp

For the remainder of the paper, we will refer to portfolio a as the "active"

of our methodology, portfolio p will be a

portfolio. In many applications

our methods

"passive" portfolio, i.e., a buy-and-hold investment strategy. While

in its

where portfolio p is not passive, certainly

are applicable to situations

In such a

setting

to tests of the CAPM, portfolio p

will be

passive.

application

for it represents a way

portfolio,

portfolio a is naturally interpreted as an active

of "active" and

to improve the efficiency

of portfolio p. The terminology

has been used

by Treynor and Black

among others.

(1973),

"passive"

and portfolios

between the ex post efficient portfolio

To establish this relation

for the weights of an efficient portfolio,

a and p, we first recall the equation w*.

and setting

m in that

equation

rp:

Using equation (22) in the Appendix

(15)

V can be parameterized as:

V- X

2tS

Using the formula for the

inverse of a partitioned matrix (see equation (24) in the

on the

last expression and substituting this

into equation (15) for

V-1,

Appendix)

but

straightforward,

after some

tedious,

algebra.

equation (14)

can be derived

The previous paragraphs have established

that the square of the usual t

statistic for the estimated intercept,

&a,

equals the

test statistic,

TW,.

with the

can be established as well

for the ex

post efficient portfolio

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