The sensitivity in tests of the efficiency of a portfolio and portfolio performance measurement

The QatarMy Review of I+konosnica ad Em, Vol. 95, No. 2, Sununer, 1995, pages 167406 Copyright0 1995 Trustees of the Unive~3tyof Illinois An lights of repaY&lctionin anyfom lvserwd ISSN 00955797

The Sensitivityin Tests of the Efficiency of a Portfolio and Portfolio Performance Measurexnent YOON K. CHOI University of Texas-Dallas

A theoretical rationale and empirical evidence fm the sensitivity of the test of tk eflzciaq of a given portfolio (ur the testof the CXPM zfapjmpiately designed) are provided. Stock and bond data are emplyed as the ‘left hand side’ assets to show that a misspecification in the ‘lzft-hand-side’ (or LHS) assets may cause tke sensitivity in testing tke ejjiciency of a given ~f~io and the blurt in po~fo~o penance. Also, the results support the use of art ‘asset ckzssfactor model in ~~~po~fo~o p~~an&e.

Gibbons, Ross, and Shanken (1989, GRS hereafter) examine the efficiency of a given portfolio, recognizing its relevance to the CAPM test. Specifically, they consider the sensitivity of the test with respect to the portfolio choice and the number of assets used to determine the ex-post efficient frontier. They are concerned with the statistical power of the test statistics in choosing an optimal number of assets, N and number of observations, T. This paper is concerned with a related but quite distinct issue regarding the sensitivity in tests of the efficiency of a portfolio. Basically, the paper addresses the issue of appropriate characteristics of the assets used to determine the ex-post efficient frontier. Whether a portfolio is mean-variance efficient or not depends on the kinds of assets against which the portfolio is evaluated. For example, a stock portfolio (e.g., Equal-Weighted CRSP Index) may be efficient with respect to a set of stocks, but the same portfolio may not be efficient with respect to a combined set of stocks and bonds. Stambaugh (1982) illustrates this point convincingly in the context of testing the CAPM.’ Therefore, in order to address the sensitivity issue unambiguously, a particular class of assets are examined first with respect to the efficiency of a portfolio. The particular class of assets we have in mind is the set of assets which are elements of the portfolio whose efficiency is being examined.* Thus, a ‘misspeci~cation’ of the LHS asset is said to occur when a researcher chooses some of the LHS assets 5~~s~~ethe restricted class of assets3 When additional assets are added in the test, the sample 187

188

QUARTERLY REVIEW OF ECONOMICS AND FINANCE

mean-variance

efficient frontier moves in general to the left in the mean-variance

space. As a result, the portfolio

being evaluated becomes

less efficient

additional assets. The magnitude of the impact of the misspecification the relative return/risk

characteristics

with the

depends on

of the additional assets. The primary contri-

bution of this paper is to address this sensitivity in the test of the efficiency due to the misspecification

of the LHS assets which has been overlooked in the literature

by

identifying the source and the direction of the sensitivity. Recently, a surge of multivariate statistical approaches used to test the efficiency of a portfolio (e.g., a market portfolio for the GAF’M test) since Gibbons (1982) and Stambaugh (1982) whose test statistics are based on asymptotic distribution have appeared.4 They employed the test statistics based on the asymptotic distributions whose approximate small sample results are reported to be very sensitive to the sample size and a particular test statistic used. Alternatively, GRS provide a multivariate approach whose F-test statistics are based on small sample distribution, MacKinlay (1985)

shows by simulation that the multivariate Ftest employed here is robust to

deviations from the normality assumption of stock returns. This is one of the advantages of Ftest statistics over the standard asymptotic test statistics when the sample size is finite. Furthermore, empiricists usually examine subperiods and aggregate the results for efftciency tests assuming constant parameters over these subperiods. Gibbons and Shanken (1987) support this practice by showing that aggregate power is substantially higher than that for a single subperiod. Likewise, mean-variance efficiency tests based on conditional moments drawn much attention

have

(e.g., Ferson, Kandel and Stambaugh 1987).5 Our approach

based on unconditional moments is thus limited in the sense that the estimation of potentially time-varying parameters is not accounted for. However, the objective here is to explain a source of the sensitivity in mean-variance efftciency tests, which is based on economics of the mean-variance efficient frontier, not on a specific test statistic. Thus, the argument for the sensitivity of test inferences advanced here applies to the conditional moments approach as well as the unconditional moments approach that is the focus of this paper. Furthermore, the conditional moments approach relies on maximum likelihood estimation with normality assumption and a large sample size. Thus,

the conditional

moments

methods

are subject

to the finite sample bias

mentioned earlier. Finally, our effort to match the class of the LHS assets with the portfolio in the mean-variance efficiency test has an important implication for measuring portfolio performance. Recently, Elton, Gruber, Das and Hlavka (1993) reported a bias in measuring selectivity performance of a managed portfolio. The bias comes from using an inappropriate performance benchmark. Their interesting empirical findings are reevaluated in the context of the LHS asset misspecification considered here.6 The ‘asset class factor model’ advocated by Sharpe (1988,1992) is also supported as a variant of multifactor models which reduces the LHS asset misspecification problem.

EFFICIENCY OF PORTFOLIO

PERFORMANCE

MEASUREMENT

189

In the next section, GRS test statistics are reviewed along with its geometric interpretation. Second, we study the characteristics of the effect of the omitted LHS assets on the effkiency test results. Third, the results of sensitivity analyses for the misspecification are reported. Fourth, we discuss the implication of the LHS asset misspecification

for the portfolio performance

GRS’s MULTIV..TE

measurement.

TEST OF EFFICIENCY

Consider the following multivariate linear regression:

sip + piprp,+ tzit i = l,...., N,

rit=

(1)

where ri, = the excess return on asset i in period t; % = the excess return on portfolio p whose efficiency is being examined; Eit = the disturbance term for asset i in period t. Assume that the disturbances are jointly normally distributed each period with mean zero and nonsingular covariance matrix C, conditional on the excess returns for portfolio p. The disturbances are also assumed to be independent over time and the covariance matrix C is assumed to be nonsingular. In order for a particular portfolio to be mean-variance efficient, first-order condition must be satisfied for each of the given N assets:

the following

Combining conditions in Equations 1 and 2 leads to the following null hypothesis: H;

c+=O,

i=l,.......,

N.

(3)

GRS employ “Hotelling’s T*” statistic to test the null hypothesis in Equation 3, given the normality assumption. They also derive an equivalent F test based on the noncentral Fdistribution, (TN-l) and

F= (T(T-N-l)/N(T2))W,

with degrees of freedom N and

(4) A

where C = unbiased residual covariance matrix; and dP is the ratio of ex-post average excess return on portfolio p to its standard deviation (i.e., bP = rP/sP). Finally, GRS show that the test statistic has a nice geometric follows:

dizF2-l=@U=+, w=

[

4-q

I

$*2

_

interpretation

as

g2

(5) P

190

QUARTERLY REVIEW OF ECONOMICS AND FINANCE

where & is the ex post price of risk (i.e., the maximum excess sample mean return per unit of sample standard deviation). Fr/\om the,mFimization

problem in deriving

the minimum variance set, they show that Cl**= 3 V’r, where iz = fit.,,7; ). rp is defined earlier; F,, is a column vector of mean excess returns on the original N assets included in the portfolio p. V is the variance-covariance portfolio.

Relationship

matrix of N + 1 assets including the

given in Equation 5 impliy

that I$* should be close to one

under the null hypothesis. GRS have shown that 0” turns out to be the maximum slope of the tangent line fro?

the origin in mean-standard

deviation space. The null

hypothesis is rejected when 8* is sufftciently greater than 6, because the return/risk ratio for portfolio p is much lower than the ex-post frontier return/risk ratio.

AN EFFICIENCYTESTAND

OMITTED ASSETS IN A PORTFOLIO

The CAPM can be derived either by solving an expected utility maximization or by solving the variance minimization assumption.

In the maximization

problem under a normality of asset return

(or minimization)

problem

for efficiency,

implicitly assumed that a security i (a LHS asset) is one component portfolio. Therefore,

the risk/expected

problem it is

of the efficient

return linear relationship only applies to the

security within the efficient portfolio. This is why the valid test of the CAPM is whether the true market portfolio is ex-ante mean variance efficient or not (See Roll 1977 for the mathematics

of the efficient frontier).

At a theoretical

level, the point mentioned

above does not carry any importance

at all simply because in the CAPM, the market portfolio, by definition,

includes all

marketable assets in the economy. However, this point plays an important role when we consider using a market proxy for testing the model. To be more general, we consider the efficiency test of a portfolio following the GRS approach. Of course, to the extent that the portfolio is a good proxy for the market, this test becomes a test of the CAPM. Again, to avoid potential confusion, we consider as a benchmark the situation where the LHS assets used in the efficiency test are the elements portfolio being examined for its mean-variance

of the

efficiency.

Suppose that there is a portfolio, p, which consists of N assets, the benchmark case. Furthermore, 7: klTO; 8f = ?i k’,,

suppose that there are K assets available in addition. Let 8: = where 7: = (Tt,,7; , -’rk ) and 7,’ = (Tt,,;I,’ ) ; Tp is defined earlier;

7” is a column vector of mean excess return on the original N assets included in the portfolio p; and Tk is a column vector of mean excess return on the K assets omitted

from the portfolio P, V,,is the variance-covariance

matrix of N + K+ 1 assets including

the omitted assets and the portfolio and Vais the variance-covariance matrix of N + 1 assets without the K assets. Then, 6: and 6: can be very different from each other,

depending upon the nature of the omitted assets, K, in the efftciency test, as shown by Proposition

1.

Proposition

1:

Given the variables defined above, consider two efficient frontiers: one

with all original assets (i.e., N assets) which are elements of a portfolio whose mean-variance efficiency is being tested and the other with additional assets omitted in theportfolio as well as with the original assets (i.e., N+K assets). The condition that these two different frontiers become iokntical in terms of the maximum tangent slope is:

wherevaak 2sa covariance matrix of the omitted and included assets. PROOF:

See Appendix.

Proposition 1 suggests that statistical test inference depends on the relative characteristics of the omitted assets in comparison with that of the original assets in the test. The condition derived in Proposition and risk characteristics the test. Therefore,

1 will be closely met when the return

of the omitted assets are very similar to the included assets in

a potentially serious problem in testing arises when a researcher

adds the assets that have very different risk/return characteristics in comparison with the original assets. A plausible example would be a situation where the portfolio p is a market index including stocks only, but a bond portfolio as well as stocks is employed in testing the efficiency of the portfolio p. A more interesting question is how significantly the misspecification

affects the test inference,

as examined in the next

section. It is important mean-variance

to recognize that the result of Proposition

spanning results in Huberman

whether K derived assets/portfolios

1 is not the same as

and Kandel (1987).

They examine

can span the N+K original assets. Therefore,

portfolios they consider are a set of a linear combination

K

of the N+K original assets.

However, in our paper, K assets are ‘the omitted assets’ which are neither part of nor derived from the original opportunity generating

N assets; that is, we consider

set. We derived a condition

the minimum variance frontier is eliminated in Proposition

The result in Proposition Arbitrage

a different

Pricing Theory

investment

when the effect of the omitted asset in 1.

1 is also conceptually related to the bias in testing the

mentioned

in Dybvig and Ross (1985).

For example,

assume that there is a factor, called land value, which occurs in only one asset, say, real estate. Suppose we use stocks and a real estate index in order to test the APT When stocks do not include the land value factor, the real estate index will appear not to conform to the APT because the land value factor variance in the real estate index will be mistakenly identified as idiosyncratic variance. In the factor model context, the condition in Proposition 1 is a way to check the consistency of the factor structure between the stocks and real estate index. Let us say rk is the return of the real estate index and ra is the vector of stock returns. Since the real estate index has its own factor independent

of stocks factor, the condition is not met and thus the bias

192

QUARTERLY REVIEW OF ECONOMICS AND FINANCE

will lead to the rejection

of the APT if the real estate is used with the stocks in a

pooling sample.7

SENSITIVITY Stambaugh

ANALYSIS (1982)

studies how tests of the CAPM are sensitive to different sets of

asset returns. He finds that the result of the test is more sensitive to the selection of the LHS assets than to the composition

of the market index and maintains that this

sensitivity with respect to the choice of the LHS assets is quite frequent in other asset pricing relations too. A clue to the sensitivity of Stambaugh’s

results can be traced to Proposition

Proposition 1 shows that the return/risk characteristics

of the omitted assets in testing

the efficiency of a given portfolio may affect the test inference. recognize

that the test statistic changes

because

1.

It is important

the sample efficient

frontier

to is

affected by the LHS asset misspecification in a&it&z to the change in sample size. While GRS focus on the effect of the sample size, N and T, we will concentrate on the effect of the LHS asset misspecification, We follow the GRS procedure

given a sample size.’

to form 10 stock portfolios as the LHS portfolios

for the period 1931 to 1985. We select this particular period so as to compare our results with earlier studies by GRS and Elton et al. First of all, for a consistency check in our data, we use a data set similar to the one used by Black, Jensen and Scholes (19’72) and show in Table 1 summary statistics on the 10 beta-sorted portfolios from January 1931 through December In each portfolio,

1965, which are very consistent with the GRS result.

additional asset (the Ibbotson-Sinquefield

long-term corporate

bond index or the long-term government bond index) is added as a part of the LHS

Tabb 1. SUMMARY STATISTICS ON BETA-SORTED PORTFOLIOS ON MONTHLY DATA, 1931-65(T = 420) Portfolio number

s(o\p) -0.0020 1

BASED

@IpI

0.0018

1.51

0.020

0.93

0.0010

1.38

0.011

0.97

2

-0.002

3

-0.0015

0.0009

1.23

0.010

0.97

4

-0.0004

0.0007

1.19

0.008

0.98

5

-0.0010

0.97

0.0008

1.08

0.009

6

0.0004

0.0008

0.93

0.008

0.97

7

0.0010

0.0008

0.87

0.009

0.96

8

0.0004

0.0008

0.75

0.009

0.95

9

0.001 I

0.0010

0.65

0.011

0.89

0.0013

0.0009

0.53

0.010

0.87

10

EFFICIENCY OF fORTFOLIO P~O~CE

193

M~~~E~

7’ubk 2. EFFICIENT SLOPES FOR EFFICIENT FXONTIERS WlTH THE 10 BETA-SORTED STOCK PORTFOLIOS IN THE PERIOD FROM 1931 to 1985 Time Period(T)

0:

Gc

1931-1940(120)

0.137

1941-1950(120)

0.173

1951-1960(120) 1961-1970(120)

@:,

0;

w,

WC

w,

0.220

0.214

0.017

0.118

0.20*

0.194*

0.244

0.198

0.095

0.071

0.136

0.094

0.337

0.363

0.354

0.095

0.221”

0.245*

0.237*

0.168

0.190

0.197

0.018

0.147

0.169**

0.176**

1$71-1980( 120)

0.046

0.055

0.056

0.012

0.033

0.042

0.043

198 l-1985 (60)

0.424

0.430

0.441

0.027

0.387**

0.392**

0.403**

assets in turn in order to examine how these extra assets affect the maximum slope and thus the test of efficiency. This exercise will shed some light on the implications of Proposition 1. Table 2 illustrates the impact of the LHS asset misspeci~cation on the test of efficiency. The influence of the misspeci~cation is reIlected in the slope estimates obtained by including the omitted assets, corporate bonds and government bonds, in constructing the ex-post efficient frontiers. Adding these assets unambiguously increases the squared slope estimates throughout the sample period, 1931 to 1985. When a corporate bond portfolio is added in constructing an efficient frontier, the squared slope increases substantially especially in the periods 1931 to 1940 and 1941 to 1950, for example, from 0.137 to 0.22 and from 0.173 to 0.244, respectively. For the period 1931-1940, the efftciency of the CRSP Equal-Weighted Index is not rejected when the 10 beta-sorted stock portfolios (W = 0.118) are used, but the same Index is rejected for its efficiency with the W statistic of 0.20 which is significant at the 1% level when another

asset is added, for example,

a corporate

bond or

government bond. Although there is a sharp increase in the W values in the period 1941-50,

it is not sufficiently

large to reject the efficiency of the CRSPEqual-Weighted

Index. We have another reversal of inference for the period 1961-70; changes from 0.147 to 0.169, which is significant at the 5% level.g

the W value

Now, we examine the return/risk characteristic ofeach omitted asset, a long-term corporate and govermnent bond, in order to understand the magnitude of the sensitivity. Before 1960, the impact of the long-term corporate bond is greater than the government bond, while that relationship is reversed after 1960. Table 3 contains the information about the return/risk characteristics of the corporate and government bond relative to the CRSP index and the original assets. The risk/return differential (DIFF), Ti -Vi;;, @ar,, measures the degree of the impact. The larger the absolute value of the differential, the larger the impact will be. Of course, to be exact,

194

QUARTERLY REYIEW OF ECONOMICS AND FINANCE Table 3. THE BETUBN/BISK CHABACTEKISTIC OF THE LT CORPORATE AND GOVERNMENT BOND

Time Period(T)

DIFFLT corporate bond

DJPP LT government bond 0.448

1931-1940(120)

0.438

1941-1950(120)

0.160

0.132

1951-1960(120)

-0.209

-0.191

1961-1970(120)

-0.223

-0.339

1971-1980(120)

-0.209

-0.239

1981-1985 (60)

-0.118

-0.117

Nolts: DIFF measures the degree of similarity of the omiuwd assetsrelative to the original assets. The numbers are in percenrage. DIFF = rk - VA%%,. where rk is the mean excess return on the either LT corporate bond or LT government bond. r, is the mean excess return vector on the original assets (the 10 stock portfolios and the market proxy). V&and V, are the covatince and variance for the aset re~~nns denoted in the a~bsc~ipt.

the differential

is to be ‘divided’ (or weighted) by a variance factor of the omitted

asset (i.e., the F matrix). Indeed, Table 3 shows that the absolute values of the differential (smaller)

for the corporate

1931-1940

and 1981-1985.

the LT corporate 1931-1940,

bond index before For example,

bond and 0.448%

while those are 0.118%

government

(after)

are larger

1960 except for the periods

the absolute value of DIFF is 0.438% for

for the LT government and 0.117%

bond in the period

for the LT corporate

bond, respectively, in the period 1981-1985.

and the LT

This can be explained by

the fact that the exact impact of the omitted assets on the maximum slope (or the W statistics) is obtained by dividing DIFF by the weighted variance and covariance of the assets (i.e., Vkk -V&+$,& LT government LT corporate

I n f ac t , m * the 1931-1940

period, the variance of the

bond’s excess return was only slightly higher with 0.027% bond with 0.023%.

However, the covariances

than the

(with the 10 stock

portfolios and the equal weighted index) of the LT corporate bond were two or three times larger than those of the LT government 1981-1985

can be also explained

bond. The exception

in the period

similarly. DIFFs are almost the same for both

omitted assets in this period. However, the correlations

between the LT government

bond and the original assets are slightly higher in most cases than those of the LT corporate bond for this period, while the variances are almost identical. Therefore, we conclude that the return/risk characteristics of the omitted LHS assets explain the sensitivity of the test inference

in addition tojust the number of the LHS assets.

And since the return/risk characteristics

change over time, the sensitivity in the test

of efficiency also changes even if the omitted assets are the same. Finally, we construct a new composite index by equally weighing the CUSP index and the long-term corporate bond (and long-term government bond) and show the

Table4. EFFICIENCY SLOPES FOR EFFICIENT FRONTIERS WITH THE 10 BETASORTED STOCK PORTFOLIOS IN THE PERIOD FROM 1931to 1985

1931-1940(120)

0.017

0.025

0.023

0.190*

0.187*

1941-1950(120)

0.095

0.113

0.108

0.118

0.081

1951-1960(120)

0.095

0.074

0.079

0.269*

0.255*

1961-1970(120)

0.018

0.008

0.006

0.181*

0.190*

1971-1980(120)

0.012

0.005

0.004

0.050

0.052

1981-1985(60)

0.027

0.036

0.031

0.380**

0.398**

No&:

, T. A new composite portfolio p The resulu are based on the regression model: q, = c$, + f$rl,, + Q V i = 1, , N and Vt = 1, slope. ep) and gwemment is obtained w combining the CRSP Equal-Weighted Indrx and the corporate (iu squared return/risk long-term bond index (its squared return/risk slope, e’$. c3; is from Table 2. *Significant

at the 1% level.

**Significant

at the 5% level.

W-statistic in efficiency

4. This

confirms Stambaugh’s

insensitive to

statistics WC

choices of

Wp are

the same,

that the

market proxy.

of

each sub-period,

in the

test inference.

IMPLICATION FOR PERFORMANCE MEASURE OF MANAGED PORTFOLIOS Since the Jensen’s

alpha measuring portfolio performance

is closely related to the

GRS statistic reviewed earlier, there may be important implications of the LHS asset misspecification Equation

for performance

measurement.

In fact, the null hypothesis given in

3 can be restated in a univariate test and interpreted

performance.

Then Proposition

ance measurement.

as no abnormal

1 directly applies to the case of portfolio perform-

The multivariate test discussed in second section of this paper is

particularly appropriate in measuring the sizc~basedportfolios because the residuals are highly correlated

in size-sorted portfolios and this correlation

in estimating alpha.” funds performance

may inject a bias

This observation is particularly useful in evaluating mutual

because size is a major classification of many mutual funds (e.g.,

small aggressive stock funds) .ll Elton et al. (1993; EGDH hereafter) due to an inappropriate

benchmark.

benchmark portfolio is correctwhen

point out a bias in measuring performance They argue that the S&P 500 index as a

measuring the performance

of a portfolio which

consists of the S&P 500 stocks (e.g., long-term growth stock funds). However, the S&P 500 Index is not an appropriate benchmark for small-stock funds, bond funds or international

funds, for example, because many of these funds are not part of the

S&P 500 stocks. The above argument

is exactly the same point advanced in the

196

QUARTERLY REVIEW OF ECONOMICS AND FINANCE

previous

sections.

risk/return

Again,

the magnitude

characteristics

and direction

of the omitted

of the bias depends

assets relative to the benchmark

on the used.

The Omitted Assets Bias EGDH show that the size of alpha systematically non-S&P paper).

stocks which are small capitalization That

with negative same

is, the size of alpha is the largest signs in the Jensen

pattern

1965-84.

except

period

those two periods. periods

and

Proposition

In this section,

explain

their

1. Due to the significant

we employ

the GRS multivariate

the following

discussion

The following performance

respectively. Then, 0;’ - 0:’ = a

portfolios studied

p,orplio/\ag:iyt obtained

monthly returns identical

portfolios

in the two context

of

in the residuals,

on the Jensen

alpha in

of various portfolios.12 the significance

of the

LHS assets. the ‘omitted’ assets,

is the intercept estimate in the multiple

employed previously, return returns

t$at there,,are deviation

two components

(i.e., 4, = ‘p/Q.

As noted before, each

by regressing

to determine DIFF, is equivalent to be consistent

data into the excess

is being

portfolio

benchmark

with those in EGDH’s

The portfolio

the efficient

it)is the

Algebraically,

term in the simple

which

portfolio,

each individual

is pa;t

p, and a,

of the is the

omitted asset on the set of

set. It turns

out that the variable

to &,. with EGDH’s annual

returns

study, we transform simply by adding

each year.13 We run the simple regression

and find that the magnitude

(e.g., the S&P500

that potentially

measured.

a,, is the intercept

individual

p, on the particular

obtained

results to the bias resulting portfolio

performance

which performanze

by regressing

portfolio, estimate

Now, in order monthly

among

in the

8G2and O,, where 8, is the ratio of ex-post average excess

p to its standard

Cl:* - 6; = C$ C-‘C+, + CC;F’a,.

intercept

on the alphas

correlations

F ‘cx~, wherea

2, we o$serve between

on portfolio

benchmark

period

See Appendix.

affect the difference

regression,

the small stocks behavior

impacts

VkL - Vu;h’bclk.

From Proposition

kenchmart

shows exactly the

how the small stocks affect the alphas in

wit+ all /\as>etsRnd with assets excluding

regression and &=

return

decile

8: and 8: be the maximum slope of the tangent line from the origin for

the efficient fronti

PROOF.

of the

in his sample

provides a statistic to measure

bias due to the omitted

Proposition 2.

(1989)

statistics, instead of focusing

of abnormal

proposition

Ippolito

is the opposite

we examine

differential

on the influence

for the stocks in the smallest

1945-64.

that the sign of alpha

However, they do not fully explain

depends

stocks (see Table 1 and Table 2 in their

using the excess annual

and the sign of the Jensen

results (see Panel A ofTable

from omitting

alphas

are almost

5). EGDH attribute

small stocks in constructing

index used here).

the excess up 12 excess

these

the benchmark

Based on the result of Proposition

2,

EFFICIENCY OF PORTFOLIO PERFORMANCE MEA!WREMENT Table 5.

197

THE ALPHA ESTIMATES BY ALTERNATIVE GROUP

Panel A: The alpha estimates by size deck

1965-1984 Ippolito Period

1945-1964 Jensen Period

Decile by size

-1.02

0.26

Largest

1.35

1.12

2 3

-1.74

3.65*

4

-1.71

4.60*

5

-3.00

5.64*

6

-3.32

7.19*

7

-4.58

9.13*

8

-3.08

9.29*

9

-I.04

10.80’

10

-4.88

12.84*

h’olp.

(I)

NI

alphas

at-e olxainrrl

hg regressing

the annual

PXCCSS wfurn

on rach

decilr

agamst

S&P500

index.

I‘hey are expressed

in

percenGtgl?. (2) *~l.i~lue

1s gleam

than 2.0.

Panel B: CX~by size decile

1965-1984 Ippolito Period

1945-1964 Jensen Period

Decile by size 6

-1.91

1.51

7

-0.37

1.93

8

-1.10

1.54*

9

-0.48

2.84* 2.31

1.95

10

we estimate

the variable,

DIFF, which is the alpha estimate

each of the omitted

LHS portfolios

As shown in Panel

B of Table V, aq’s are negative

period,

while they are positive

confirms

EGDH’s

estimating Jensen

alpha

especially

of the stock portfolio

period,

the small and large stock portfolio was 14.4%,

returns:

while the mean excess return

However, the small stock’s performance sive in the period

of Ippolito

was only 1.9%, while the return This stark performance values in the Ippolito

(1965-84):

in the Ippolito

for the Ippolito’s

period.

index.

in the Jensen

assets, small stocks here,

period.

inject The

This

a bias in

risk/return

for the two periods.

in the mean annual excess returns

In the between

the mean annual excess return on S&P500 on the smallest compared

stock portfolio

to the S&P500

the S&P5OO’s mean

on the smallest stock portfolio

in the small stock portfolio period.

by regressing

in the S&P500

and insignificant

is quite contrasting

there was little difference

obtained

portfolios

and weakly significant

claim that the omitted

the Jensen

characteristic

on the included

explains

was 17.8%.

index was impres-

annual

excess

return

was as much as 16%. the significant

alpha

198

QUARTERLY REVIEW OF ECONOMICS AND FINANCE

Tabb 6. MAXIMUM PORTFOLIOS

SQUARED SLOPE BY DIFFERENT NUMBER OF S&P index

Portfolios

1945-64

1965-84

1945-64

1965-84

0.558 0.561 0.675 0.749 0.784 0.785 0.790 0.795 0.637

0.297 0.304 0.304 0.368 0.467 0.484 0.499 0.527 0.013

0.623 0.720 0.732 0.745 0.752 0.752 0.780 0.780 0.601

0.296 0.307 0.317 0.420 0.544 0.552 0.592 0.602 0.019

&3) &4) G(5) &6) &7, &S) &9) &lO) 6: Nofr:

VI%’is the CR9 thr table.

value-weighted

05 (i) is the maximum

VW index

squared

index.

t$ is the squared

slope of rbe elfcienr

slope (return/risk)

frontier

of the market index.

with die i largest siresorted

Excess annual

returns

are used in

portfolios.

Table 6 showsjhat the effect of the small stock portfolio on the maximum squared slope parameter, 6z2. If only the large stock portfolios are used to test the efficiency of the SsCP500, the efI$iency cannot be rejected. For example, when only the 5 largest portfolios are used, 0E2 is equal to 0.675 which F very close to 4; which is 0.637 in 1945-1964,

where p is the S&P 500 Index, while t3E2is equal to 0.795 when all the 10

portfolios are used in the test. Therefore,

the efficiency of the S&P 500 Index which

consists of only large-size stocks can be rejected when all ten portfolios which include the small-size portfolios are used. That is, using the S&P500 as the benchmark, overall abnormal performance

the

of the first five largest stock portfolios is indeed zero.

In other words, the S&P500 is an unbiased benchmark for measuring the large-stock mutual funds. The Jensen Measure and the Maximum Slope Measure

In a multivariate framework, there is no abnormal portfolio performance is close enough

to 6: when the benchmark

is indeed efficient.

Therefore,

if 6z2 it is

important to examine the efficiency of the benchmark before portfolio performance is measured against it. Also, note that the efficiency of the benchmark time as shown in Table 2. It indicates that CRSP Equally-Weighted efficient in the periods 1951-1960

and 1981-1985,

changes over Index is not

while it is efficient for the rest of

the time. In order to compare the Jensen measure for individual portfolios with the slope measure of efficiency, we calculate the Jensen’s alphas 0: the 10 beta-sorted stock portfolios for the periods with the smallest and the largest t3E2in Table 7. In 1981-85,

Table 7. ALPHA ESTIMATES BY BETA DECILE FOR THE FOUR lo-YEAR SUBPERIODS IN 1941-1985 Decile

by beta

194160

1951-60

1971-80

1981-85

-0.0039

a.o074*

0.0009

a.o063*

2

-0.0031

-0.0043*

0.0003

-0.0071*

3

a.0015

-0.0017*

0.0005

a.0015

largest

4

0.0009

-0.0008

0.0008

-0.0022

5

0.0000

-0.0010

0.0005

a.0005

6

-0.0002

7

0.0017

0.0004 0.0016*

0.0007

0.0017

-0.0006

0.0005

8

0.0024*

0.0035*

-0.0002

0.0027

9

0.0026*

0.0039*

-0.0003

0.0054*

0.0022

0.0036*

0.0003

0.0079*

0.173

0.337*

0.046

0.424*

10

No/r:

0: is the maximum

squared

slope of the efficient

frontier

with the 10 beta-sorted

portfolios.

The ten portfolios

are equal-weighted.

GE2(=0.424) is significant compared to 6: (=0.027) and the Jensen alphas have &values greater than 2.0 in absolute value in 4 cases out of the 10 portfolios. For the period 1951-60, the maximum squared slope is 0.337 which is also statistically significant and 7 portfolios have significant Jensen measures. From this result, it can be concluded that the inefficiency of the benchmark generates spurious performance in the Jensen measure. Let us make the same comparison for the period when the maximum squared slope measure is insignificant compared to &, implying the efftciency of the benchmark. For 1941-50, the maximum squared slope is 0.173 and only two portfolios have significant alphas. Interestmgly, none has a significant tvalue for the Jensen’s measure for period 1971-80 (t3E2 = 0.046). We extend this exercise for the Jensen (1945-64)

and the Ippolito period (1965-84)

for comparison purpose.

As shown in Table 8, the S&P 500 Index and the value-weighted NYSE Index are efficient for the period 1945-1964, while they are significantly inefficient in 1965-84. Therefore, we need to find or construct an efficient benchmark with respect to the LHS assets.14 Basically, we are concerned with spurious ‘abnormal performance’ in naive stock portfolios (e.g., the significant Jensen alpha particularly for the Ippolito period). The abnormal alpha may be due to the inefficiency of the benchmark in that period. Therefore, we examine the four 5-year subperiods of the period 1965-84 to identify a particular period in which the benchmark portfolio is efficient. Unfortunately, we find that the benchmarks (e.g., the S&P 500 and the Value-Weighted Index) are inefficient in all the subperiods. Interestingly, however, the intercepts in two subperiods (1970-74 and 1980-84) are not significantly different from zero. This implies that even inefficient benchmarks can produce a zero Jensen measure and thus, a careful interpretation of the Jensen measure is required. The problem in the single factor model leads to a discussion of multifactor models in the next section.

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AND FINANCE

Tubb 8. ALPHA ESTIMATES BY SIZE DECILE FOR THEJENSEN (1945-64) AND THE IPPOLITO (1965-84) PERIODS FOR AITERNATIYE BENCHMARK INDICES S&P index

Decile by size Largest 2 3

1945-64

1965-84

1945-64

1965-84

VW index 1945-64

-0.0004

-0.0008

0.0017

-0.0032*

0.0002

0.0007

0.0013

X1.0026*

0.0019

0.0003

-0.0018*

-0.0006

-0.0005

4

-0.0006

0.0034*

5

-0.0008

0.0026

6

-0.0004

7

-0.0013

8

-0.0016

9

EW index

0.0005

0.0000

-0.0001 0.0001

1965-84 -0.0012 0.0003 0.0015

-0.0005

-0.0008

0.0029*

-0.0002

-0.0015*

-0.0009

0.0022

0.0052*

0.0000

0.0010

-0.0006

0.0048*

0.0050*

-0.0011*

0.0002

-0.0015

0.0045*

0.0061*

-0.0016*

0.0013

-0.0018

0.0056*

0.0076*

0.0004

0.0025*

0.0003

0.0071*

10

-0.0006 0.0857

0.0011

0.0653

0.0114

0.0819

0.0017

,“g

0.1182

0.1082

0.1154

0.1100

0.1203

0.1091

0.0299

0.1070*

0.0470

0.0974”

0.0355

0.1072*

0.0095*

-0.0012

0.0040*

-0.0008

0.0089*

Sensitivity of the CAPM and APT Benchmark15

Lehman and Modest (1987) show that the evahtation of mutual funds is sensitive to alternative benchmarks including the standard CAPM benchmark and a variety of the APT benchmarks. Theoretically, Green (1986) shows that the relative rankings of portfolio performance can be reversed for any inefficient CAPM benchmark. Further, Grinblatt and Titman (1987) provide an equivalency relationship

between

mean-variance efficiency and the APT. Thus, in this section, we discuss the sensitivity of the alternative benchmarks in view of the efficiency of a benchmark. It is well established that the market portfolio in the CAPM or a single index benchmark can be decomposed into multi-beta portfolios. Therefore, when the single market proxy used does not represent the whole market but only part of it, the APT multiple benchmarks may explain the sensitivity of the single index model. Indeed, EGDH employ a multifactor approach to control for the omitted asset and show that the Jensen’s alpha becomes insignificant after controlling for the small-firm portfolios and the bond portfolio.‘6 We take a similar multifactor approach to estimate theJensen’s alpha for the Ippolito period (1965-1984) in which the simple regression intercepts are signifi-

EFFICIENCY OF PORTFOLIO PERFORMANCE MEUUREMENT

201

Table 9. ALPHA ESTIMATES AND THE ADJUSTED COEFFICIENT OF DETERMINATION (R2) BY SIZE DECILE IN THE SINGLE AND MULTI-FACTOR MODEL FOR THE PERIOD OF 1965-1984 u

Decile

t-value

Ril

@ 0.96

-0.0003

-0.59

0.96

2

-0.0004

-0.49

0.93

0.91

3

-0.0000

-0.05

0.92

0.85

Largest

4

0.0009

5

-0.0009

6

-0.0015

-0.96

0.93

0.84

J.I.99

0.93

0.78

1.79

0.95

0.80

7

0.0002

0.24

0.95

0.75

8

0.0009

0.95

0.95

0.70

9

0.0014

0.96

0.96

0.67

10

0.0019

2.04

0.96

0.59

Mean

0.0002

0.20

0.95

0.79

h’ott: e and ti are the ndjwl~l index model. respectively.

candy large.17 In controlling

coeflicients

of determination

for the multifactor

model

and the single

for the omitted assets we use the Ibbotson-Sinquefield

small-stock index to proxy the small-firm benchmark

and a bond index (50% of the

long-term corporate bond index return and 50% of the long-term government bond index return) in addition to the S&P 500 index return. Table 9 shows that all the intercepts

are not significantly different from zeros except for the smallest decile

portfolio, which is quite a different result from the single index model. This suggests that the APT benchmarks

are generally efficient and thus are more appropriate

benchmarks for portfolio performance

measurement.

It is also noteworthy to observe

that the adjusted R2s for the multifactor model are consistently very high with the mean of 0.95, while the mean adjusted R2 for the single index is 0.79 and the value is decreasing

as the size of the LHS portfolio decreases. For example, the adjusted

R2 is 0.96 for the largest portfolio and is only 0.59 for the smallest portfolio. This adds further evidence that the multifactor

index model explains the LHS assets much

better than the single index model.”

CONCLUSIONS This paper emphasizes the importance of selecting the LHS assets from the ‘restricted class’ of assets defined in the paper when a portfolio’s mean-variance efficiency is examined. We have derived two propositions which explain the sensitivity in the test statistics to the choice of the LHS asset in testing portfolio efficiency and

202

QUARTERLY REVIEW OF ECONOMICS

measuring

portfolio

performance.

The magnitude

depend on the risk/return characteristics original assets. We employ beta-sorted corporate

bond and a government

Equal-Weighted

AND FINANCE

of the sensitivity is shown to

of the omitted LHS assets relative to the

stock portfolios

(the original assets) and a

bond index (the omitted assets) and the CRSP

Index as portfolio p and provide empirical evidence to support our

results regarding the sensitivity in the test of its efficiency, It is shown in two lo-year subperiods

that the efficiency

beta-sorted

stock portfolios

efficiency

of the CRSP EW Index is not rejected are used, but the same index

when either the corporate

or government

when the 10

is rejected

for its

bond index is added to the

test. This sensitivity problem can be serious, depending assets. The exact nature of the sensitivity is explained

on the type of the omitted by the condition

in Propo-

sition 1. Our results have important

implications

for portfolio

performance

measure-

ment. We employ the GRS multivariate statistic which is a nonlinear function of the Jensen’s

alpha. This statistic has advantages in measuring

overall performance,

especially for size-sorted portfolios since the residuals are highly correlated. correlation

may inject a bias in estimating the alphas for individual portfolios. Using

the maximum apparent injected

slope parameter,

abnormal index).

the results of EGDH are reinterpreted

performance

by omitting

S&P500

This

in size-sorted

the small stocks from

For example,

only when the large portfolios

the efficiency

can explain the seemingly

by Ippolito

the market

in that the

are due to the bias benchmark

of the S&P500

cannot

(i.e.,

the

be rejected

(without the small stocks) are used in the test. It

is shown that a very distinct behavior (1965-84)

portfolios

of the small stocks in the Ippolito huge abnormal

performance

period

documented

(1989).

Also, the empirical analysis reveals that the efficiency of an index changes over time and inefficiency Jensen

of the benchmark

generates

spurious performance

measure. This result warrants a caution in choosing a benchmark

different

time periods in measuring

abnormal

return in portfolio

in the index in

performance.

Further, the results support the use of the ‘asset class factor model’ in portfolio management

measurement.

Finally, an extension

proach would be an interesting

tional moments approach to mean-variance Acknowledgment:

to the conditional

We thank Theodore

efficiency. Day, Kevin Dougherty, David Emanuel,

Larry Merville, Richard Green and the participants Association meeting for many valuable comments. provided excellent

moments ap-

project although the focus here is on the uncondi-

at the 1994 Midwest Finance Hyunrin Shin and Rob Maurer

research assistance. A previous version of the paper, “Tests of the

Efficiency of a Portfolio and Optimal Choices OfAssets,” was selected as the Outstanding Paper in Investments at the 1994 Midwest Finance meeting.

EFFICIENCY OF PORTFOLIO PE~O~CE

GAEL

203

APPENDIX PROOF

of Proposition

Applying

1

an inversion

of a partitioned

matrix,

NOTES *Direct all correspondence Management,

to: Yoon K. Choi, University of Texas at Dallas, School of

2601 N. Floyd Road, Richardson, TX 750834638.

1. An example is when four preferred stocks and bond portfolios are used as part of the assets in testing the efficiency of a broad market index which does not contain these assets as in Stambaugh (1982). The broad market index consists of NYSE Index, corporate bonds, government bonds, and real estate properties. Actnally, Stambaugh does not test the efftciency of the true market portfolio. Instead, he tests the CAF’M with some market proxies. These two tests become identical only when the market proxy used is a ‘correct’ proxy for the true market

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204

portfolio. Therefore,

REVIEW OF ECONOMICS

AND FINANCE

for Stambaugh’s purpose, his test is not misspecified. We argue the test

is misspecified when the purpose is to test the efficiency of the broad market index. 2.

The emphasis on this particular class of assets is consistent with the mathematics of

the efficient

set shown in Roll (1977).

relationship

only applies to the individual assets used to construct

That is, it is implicit that the linear beta/return the efficient

set (see

Corollary 6 in the Appendix in his paper). Previous research has focused on misspecilications due to a market proxy-a

problem of

the omitted assets from the market portfolio construction. For example, Mayers (1973) points out human capital as an important

component,

often omitted, for the aggregate wealth

portfolio in the CAPM test. Fama and Schwert (1977) and Jagannathan

and Wang (1993)

employ labor incomes as a proxy for human capital in their tests of the CAPM. 3.

For example, when we want to test the efficiency of the S&P500 index, a well-specified

question is to ask whether the S&P500 index would lie on the efficient frontier constructed with the 500 stocks or the portfolios derived from them. 4.

Also, see Kandel (1984)) Roll (1985)) Shanken (1985), Ma&inlay

and Stambaugh 5.

Also, see Bodurhta and Mark (1984), Roll (1985)) Shanken (1985)) Ma&inlay

and Kandel and Stambaugh 6.

(1987), and Kandel

(1987) for more multivariate tests examples. (1987),

(1987) for more multivariate tests examples.

Green (1986) argues that inefficiency ofa benchmark can generate spurious abnormal

performance.

We emphasize a proper selection of the LHS assets in testing the efficiency of

a benchmark. 7.

This point is more relevant with the equilibrium APT Grinblatt and Titman (1987)

show the equivalency relationship between mean-variance efficiency and the APT 8.

Strictly speaking, we are not holding the sample size, N, constant because of the

additional omitted assets in the analysis. However, since we are adding just one or two assets, we believe that the effect of these additional assets on the analysis is minimal. Again, we are interested in the risk/return characttistics notjust 9.

of the added assets and their effect on the sensitivity,

the effect of the numberof the LHS asset. We also tried the 20 and 40 beta-sorted portfolios and found that in every period, the

efficiency of the CRSP Index is not rejected. One reason may be that the weight assigned to the omitted asset sharply decreases as the number of the stock portfolios doubles. It also demonstrates that the power of the test falls when the number of the LHS assets increases. 10.

GRS find that the correlation

of the market model residuals of the size portfolios

changes systematically (Table lV in their paper). That is, the correlation is positive and high among the large-size decile portfolios.

The correlations

are low among different

decile

portfolios. Interestingly, the smallest decile portfolio has negative sample correlation with all other decile portfolios. 11.

For the mutual fund analysis, we use size-based portfolios similar to GRS. That is, we

form 10 portfolios based on the relative market value of their total equity outstanding. Each portfolio is value-weighted and resorted by their market values every five years. We resorted and rebalanced the 10 portfolios in December 1925,1930,..., 1980. 12. We examine the correlation of the residuals from the simple regression between each portfolio’s return and the benchmark returns, using the dame data as in EGDH.,For the period from 1945-64

(the Jensen Period), the correlations are positive and very high (.60 to .96 as

compared to .21 to .75 in the GRS’ period of 1926 to 1982) for portfolio 2 and portfolio 10

EFFICIENCY

OF PORTFOLIO

PERFORMANCE

MEASURFMENT

205

(the smallest decile portfolio). The largest decile portfolio has negative correlation with 5 other portfolios and positive correlation with 4 other portfolios. 13. We use the annual return data in this section only to make a proper comparison with EGDH’s results, which leaves us with only 20 observations in the regression. the monthly returns with more observations for more powerful tests. 14. In view of Proposition in measuring the performance

Otherwise,

we use

2, it becomes clear which benchmark portfolio is appropriate of a mutual fund. EGDH claim that the Small Cap Index is

more appropriate than the S&P 500 when we measure the performance of a small stocks mutual fund. Proposition 2 theoretically supports their argument. That is, by using the Small Cap Index, 15.

the omitted

assets bias can be avoided.

The relationship

between

the CAPM and the APT discussed

in this section

is not to

be exactly described. Therefore, readers should take the discussion in this section as a diagnostic nature. See Dybvig and Ross (1985) and Shanken (1985) for details for empirical tests of the APT 16. See Connor and Korajczyk (1986) and Grinblatt and Titman (1987) for a theoretical support for the APT in measuring portfolio measurement. Dybvig and Ross (1985) derive a relationship 17.

between

the CAPM market portfolio

EGDH use “orthogonalized”

portfolios

and the APT multi-factor in order

to separate

model.

a unique

power of each portfolio from each other. Since we are not interested in estimating sensitivity beta of each index, multicollinearity problems can be ignored.

explanatory the unique

18. This result provides additional support for the practice of using an asset class factor model in measuring portfolio management. For example, see Sharpe (1988,1992).

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[ 121 Grinblatt, M. and S. Titman. (1987). “The Relation Between Mean Variance Efficiency and Arbitrage Pricing.” Journal of Business, 60: 97-112. [ 131 Huberman, G. and S. Kandel. (1987). “Mean Variance Spanning.” Journal of Finance, 42: 383-388. [14] Ippolito, R. (1989). “Efficiency with Costly Information: A Study of Mutual Fund Performance.” Quarterly Journal of Economics, 104: l-23. [15] Jagannathan, R. and Z. Wang. (1993). The CAPM is Alive and Well. Working paper, University of Minnesota. [ 161 Kandel, S. (1984). “The Likelihood Ratio Test Statistic of Mean-Variance Efficiency without a Riskless Asset.” Journal ofFinancial Economics, IA 575-592. [ 171 Kandel, S. and R. Stambaugh. (1987). “On Correlations and Inferences about Mean-Variance Efficiency.” Journal of Financial Economics, 18: 61-90. [ 181 Lehman, B. and D. Modest. (1987). “Mutual Fund Performance Evaluation: A Comparison of Benchmarks and Benchmark Comparisons.” Journal of Finance, 42: 233-265. [ 191 MacKinley, A. (1985). An Analysis of Multivariate Financial Tests. Ph.D. dissertation, Graduate School of Business, University of Chicago. (1987). “On Multivariate Tests of the CAPM.” Journal of Financial Economics, 18: [201 -. 341-372. [21] Mayers, D. (1973). “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty.” Pp. 223-248 in Studies in the Theory of Capital Markets, edited by Michael C. Jensen. New York: Praeger. [22] Ng, L. (1991). “Tests of the CAPM with Time-Varying Covariances: A Multivariate GARCH Approach.” Journal ofFinance, 46: 1507-1521. [23] Ng, V., R. Engle and M. Rothschild. (1992). “A Multi-Dynamic-Factor Model for Stock Returns.” Journal ofEconomettics, 52: 245-266. [24] Roll, R. (1977). “A Critique of the Capital Asset Pricing Theory’s Tests; Part I: On Past and Potential Testability of the Theory.” Journal of Financial Economics, 4: 129-l 76. [25] -. (1985). “A Note on the Geometry of Shanken’s CSR T* Test for Mean/Variance Efficiency.” Journal of Financial Economics, 14: 349-358. [26] Shanken, J. (1985). “Multi-Beta CAPM or Equilibrium APT? A Reply.” Journal of Finance, 40.1189-1196. (1985). “Multivariate Tests of the Zero-Beta CAPM.“Journal ofFinancialEconomics, [271 -. 14: 327-348. [28] Sharpe, W. (1988). “Determining a Fund’s Effective Asset Mix.” Investment Management Review, (December): 59-69. (1992). “Asset Allgcation: Management Style and Performance Measurement.” WI -. Journal of Portfolio Management, (Winter): 7-19. [30] Stambaugh, R. (1982). “On the Exclusion of the Assets from Tests of the Two-Parameter Model: A Sensitivity Analysis.” Journal of Financial Economics, 10. 237-268. [31] Turtle, H., A. Buse and B. Korkie. (1994). “Tests of Conditional Asset Pricing with Time-Varying Moments and Risk Prices.” Journal of Financial and Quantitative Analysis, 29: 15-29.