Generalized two parameter asset pricing models

Journal

of Financial

GENERALIZED

Economics

6 (1978)

11-32.

TWO PARAMETER

0

North-Holland

ASSET

Publishing

PRICING

Company

MODELS

Some Empirical Evidence Robert

R. GRAUER*

Simon Fraser University, Burnaby, British Columbia VSA I.%, Canada Received

September

1976, revised

version

received

November

1977

A series of empirically refutable generalized two parameter asset pricing models that linearly relate risk and return are identified for the power (and quadratic) utility members of the linear risk tolerance capital asset pricing models. Five possible power utility models and the lhean variance model are tested to determine whether one model might provide a more accurate description of security pricing. The major empirical result is that the data do not allow us to distinguish between the models.

1. Introduction Empirical testing of positive theories of asset pricing has overwhelmingly rested on the mean variance capital asset pricing model (MV CAPM). However, the model itself has received less than full-fledged support in recent empirical tests [Friend and Blume (1970), Black, Jensen, and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973)]. Moreover, from a theoretical view it suffers from a number of deficiencies. These two factors have led several authors [Roll (1973), Hakansson (1974a), Rubinstein (1975), Kraus and Litzenberger (1972, 1975), Grauer (1976a)] either implicitly or explicitly to suggest that one of the power linear risk tolerance (LRT) utility functions may form the basis of a better positive theory of capital asset pricing. While most of the empirical work on CAPMs has been conducted from an MV perspective, it has not been exclusively concerned with tests of that model. Some, in fact, has compared an equilibrium formulation of the ‘growth optimal’ model [which is consistent with logarithmic utility (only) and which, in turn, can be viewed as a special case of the power LRT models]. Employing weekly stock market data from the 1960s Roll (1973) found the two models to be empirically indistinguishable. Essentially the same conclusion was reached by *The research was supported by grants from the Canada Council, Simon Flaser University’s President’s Research Fund and the University of Toronto Commerce and Finance Research Fund. I thank M. Brennan. P. Halpern, J. Herzog. R. Koopman, A. Kraus, J. MacBeth, J. Watts. and the referee, E. Fama, for helpful comments, and D. McKinnon and R. Whaley for computational assistance. Naturally 1 fall heir to any errors or omissions.

R. R. Grauec, Tests oJ IWO parameter asset pricing models

12

Fama and MacBeth (1974) and Kraus and Litzenberger (1975) employing monthly data. In a somewhat different context Friend and Blume (1975) working with a continuous-time form of the MV CAPM and employing cross-sectional data on household asset holdings reported that the data are consistent with an aggregate utility function on the order of ---w-i. Grauer (1976b), however, conducted tests of the various power LRT models based on their ability to predict the relative market values of portf6lios of common stocks and bonds and found that none of the models fared very well. In this paper conventional econometric techniques are employed to determine whether one of the power LRT CAPMs or generalized two parameter models may provide a better positive theory of security pricing than does the MV CAPM. Alternatively the research can,be viewed as an attempt to measure an ‘aggregate’ or ‘composite’ individual’s utility function based on the observed market behavior of investors. Section 2 reviews the LRT CAPMs; shows how the power LRT CAPMs can be formulated as empirically refutable generalized two parameter models, where risk (appropriately measured) and expected return are linearly related; and presents a hypothetical example to develop more fully the theory and to illustrate how the empirical tests will be conducted. Section 3 presents the testable implications of the generalized two parameter models. Section 4 contains an explanation of the empirical methodology. Section 5 reports and discusses the results of the empirical tests. Section 6 contains a summary and conclusions. 2. Theoretical background 2.1. Empirically

testable generalized

two parameter

models

derioed from

the

LRT utility functions

By invoking the standard assumptions that borrowing and lending can take place at the riskfree rate, all investments are infinitely divisible and can be sold short, taxes are non-existent, and all investors share homogeneous beliefs and identical decision horizons, we can derive the LRT CAPMs from the LRT utility functions for which the separation property holds, namely:

(1)

Ui(W) = l/y(!-V+L7i)‘,

Y <

ui(w) = -(a,-

y > 1 and ai large,

(2)

ai < 0,

(3)

w)y,

ui(w) = -exp(aiwi),

1,

where the ai are allowed to vary among investors to reflect differing risk aversion, but a common y value must be shared by all investors.’ ‘Quadratic y = 0.

utility

is a special

case of (2) where

y = 2 and (1) reduces

degrees of

to log (~+a,)

when

R.R. Grauer, Tests of two parameter

asset pricing models

13

A CAPM based on any of the polynomial families of utility functions in (2) (where a family corresponds to a given y value) will imply that risky assets are inferior goods, while a model based on any of the power families in (1) will imply that risky assets are normal goods. It is this feature of the power families which makes them desirable candidates for study.’ Let: zj be the dollar amount invested in asset j (asset 1 being riskless); wO and w1 = Cj=, ~,(r~--r)+~~(l +r) be initial and end-of-period wealth; rjr r, and rM be the rate of return on assetj, the riskless asset and the market portfolio respectively; II(.) be the utility function of a ‘composite’ individual defined on end-of-period wealth; u’( ‘) be the first derivative of the utility function; E(.) be the expectation operator; cov( .) be covariance; and S,,, be the aggregate present value of asset j. In LRT economies, where the aggregation property holds,3 Rubinstein (1974) has derived a generalized valuation equation equivalent to

Eq. (4) provides a general statement of how securities are priced in an LRT economy but it lacks empirical content as it contains unobservable utility information. However, it can be shown to yield empirically refutable results. Note that, because the ‘composite’ of all investors holds the unlevered market portfolio, ~1, = w,(l +yw). Assume that tastes are quadratic, then it is well known that (4) reduces to the familiar and empirically testable MV CAPM valuation equation E(rj)

=

r

+

[E(rM)

(5)

- r]Pj.

Now if we restrict attention to any one of the power utility functions first derivative of the utility function is given by

u’(w,) = [w$(l

in (I), the

+td+rM)C],

where c = y- 1 and cl = a/~~.

Substituting

this result into (4) yields

‘The pouer families allow the theorist to bypass the internal logical inconsistencies of a MV model based on normality. Specifically, normality conflicts with the empirical reality of limited liability, or more generally with the idea of finite returns in a finite time period, and with the opportunity for riskless borrowing and lending, On the other hand, to be able to make use of the expected utility theorem with power utility functions it is assumed that the von Neumann-Morgenstern postulates have been modified in such a way as to permit unbounded utility functions. ‘Rubinstein’s aggregation theorem provides alternative sets of sufficient conditions under w,hich equilibrium security rates of return are determined as if there existed only identical individuals whose resources, beliefs, and tastes are composites of the (at least partially) heterogeneous resources, beliefs, and tastes of actual individuals in the economy.

14

R.R. Grauer, Tests of fwo parameter

E(rj) = r+ [E(rM)-r]

assef pricing models

Riskj,

where Riskj = cov(rj, (1 +d+r,)‘)/cov(r,, (1 +d+r,)‘). Clearly, different values of c and d yield a whole series of simple empirically testable two parameter valuation equations that linearly relate ‘reward’ (expected return) to ‘risk’, but only for the special case of quadratic utility (c = 1, Ii = - 1) will the ‘risk’ measure be the familiar beta coefficient.4 While the MV CAPM describes how securities are priced in terms of two parameters, expected return and beta, the generalized two parameter models describe how, in given LRT economies including the MV or quadratic utility economy as a special case, securities are priced in terms of two parameters, expected return and Risk. A particularly interesting aspect of the models is that the specific Risk terms implicitly capture the effect of all the relevant moments (and comoments) that determine security pricing whether they are just the mean and variance in a quadratic utility model or all the moments in the power utility models.

Before moving to the empirical tests we consider a hypothetical equilibrium example invohing three risky securities as a means of demonstrating the efficacy of the research technique and of introducing some further theoretical and empirical considerations that will reduce the area of search for relevant c, d combinations. Envision the following scenario. All the assumptions of the asset pricing models hold exactly. Investors perceive the ex ante return distribution to be as given in table 1 and can borrow and lend at the same riskless rate of interest. For purposes of illustration assume that everyone has a logarithmic utility function as in (1). In equilibrium each investor will hold the market portfolio of

*For quadratic utility saying that d G N wO = -I i\ a notational con\cniencc to emphasix the simllarlty bet\\een the models. As ib hho\rn belo\+, for the pcxrcr utlllty model\. the con\tant dis a v,ell defined economK quantity that guarantee\ borrowing and Icndlng canxl 111aggregate. In the ca\e of quadratic utility (/is not defined but ah It i, a con\tant it doe\ not atfect the cwariaIKe.

5The example mas constructed: (i) to illustrate the elIicacy of the empirical methodology, (in) to \ho\\ that beta is not the mcasurc of Rlhk in a poher LRT economy, and (iii) to >hok+ that the concept of a zero-beta portfolio is not meaningful if the market portfoho ib not MV etticicnt. In addition, the present draft shwrs that if the marhet portfolIo i\ not hZV efficient not onl! does 150~;) not equal I’ but the expected returns on zero-beta portfolios are not equal. It has recentl) come to my attention that in the context of the MV model, Mama (1976), Roll (1977) and Rosh (1977) habe ernpha\tred the importance of the market portfolio bclng MV etticient and the idea that II does not equal r if the marhet proxy is not the true mnrhet portfolio but is ~1111MV efficient. While the pre\ent xctlon war completely independently de\cloped, I habe no doubt that their lrork ha, influenced my interpretation of the empirical result5 111section 5. 1 thank a referee for polntmg out Fama’s work.

R.R. Grauer, Tests of two paramefer

risky assets in conjunction with some combination the example, the equilibrium weights in the market to be (0.2613, 0.5774, 0.1613).6 Table The ex ante return

I5

asset pricing models

of the riskless asset. In portfolio were calculated

1

distribution

for individual

securities

State Security

Return

1

2

3

4

5

6

1 2 3 4

l+r l+r,, l+r,, lfr,,

1.05 0. 1.0 0.6

1.05 0. 1.0 1.8

1.05 1.5 1.0 0.6

1.05 1.5 1.0 1.8

1.05 1.5 2.5 0.6

1.05 1.5 2.5 1.8

0.05

0.05

0.4

0.4

0.05

0.05

Probability

n,,

Summary statisfics E(~z) = 0.35 E(r3) = 0.15 E(r4) = 0.20

Skewness Skewness Skewness

Variance-covariance

2 3 4

(r2) = -2.67 (r3) = +2.67 0. (r.,) =

matrix

2

3

4

0.2025 0.0225 0.

0.0225 0.2025 0.

0. 0. 0.3600

Now suppose that a social scientist did not know that the true process of equilibrium price formation was based on logarithmic tastes but he was supplied with the true ex ante rates of return and asked to determine the pricing process. His first guess might be that the true pricing process is consistent with the MV CAPM. Thus he would run an empirical analogue of (5), (7)

6For the power

utility

functions

the individual’s

expected

utility

problem

was reformulated

as:

MaxE (U,)

l/v

i j=2

vj(rj-r)+l+r

,

Y <

1,

I’

where z‘, = z,‘/(w”~ + a,/(1 + r)), and the i’s indicate individuals. The expected utility problem was then solved by numerical analysis employing an alogrithm by Best (1975). The calculated solutions were (0.564, 1.247, 0.348). Because of the separation property exhibited by the LRT utility functions everyone holds the same mix of risky assets, which in equilibrium must be the market portfolio. Forming the set of ratios, [I,~/C,~,I,,], yielded the equilibrium market value weights.

R.R. Grauer, Tes/s of two parameter

16

asset pricing models

where fj denotes average return on assetj [but is actually identical to E(rj) in the example]. If the MV CAPM correctly describes the process a, will equal 0 and a, = I.’ Clearly something is amiss as the slope of the regression shown in the first panel of fig. 1 is negative. (Note this result occurs with ex ante data and involves none of the other econometric difficulties we face in the empirical section of the paper.) -

~000

-

74 L

-010

ij-

-020

iii I

r

-77-7-1

ii $

021

-

2

a=

k

_L

(I) 014 B i!

: -

007

-030

2

-040

5

,(_::I c

2

0 07

000 SIMPLE INTER-

7

026

=27 z

014

021

RISK X EXCESS MV MODEL 3304

0 26

RETURN I

SLOPE=-1

ON

0800

0 35

-0 50

-0 40 RISK

MARKET

-0 30

X EXCESS

RETURN

ZERO-BETA MODEL INTER=2012 SLOPE

RSO-3616

026

c

-0 20

-010 ON

r.4663

-0 00

MARKET

RSQ=.3618

-

021

-

0.21 -

nw 2 014

c

0.14 -

-

0.07 -

;

s ::

007

0 00

I 000

0 07

014

I

I

0 21

0 26

RISK X EXCESS RETURN LOG MODEL D=O INTER-. 2104 SLOPE=2 4568

PORTFOLIOS

ON

v

0.00 0 00

0.35

I

I

0 21

0 28

I

014

RISK X EXCESS RETURN LOG MODEL D--.5637 INTER - 00001 SLOPE: 99997

MARKET

RSQr.4544 cov

X

’ 0 07

(rj,

ON

0 35

MARKET

RSO-1,

(l+d+r,)o)

RlSKj E MARKET

q

Fig. 1. Estimated

cov (rM, (1 +d+rM)C) risk return

tradeoffs

for the three risky security

example

%r the example the risk measure is multiplied by the excess return so that the true slope and intercept will lie on the 45” line in fig. 1. In the empiricial work that follows we revert to the more standard form of regressing the excess return on the risk measure so that the results may be more easily compared to previous work on the MV CAPM.

R.R. Grauer, Tests of two parameter

asset pricing models

17

Given the result of his first test our social scientist may find the zero-beta mode1 an appealing alternative to the simple MV model because of its apparent empirical success in explaining security pricing in terms of a two factor model, rj = aj+bjr.,r+(l

-bj)rz+ej

(as in Black, Jensen, and Scholes), and,‘or because he may feel that the equilibrium pricing process he has been asked to study really involves rigidities in borrowing and lending. Thus, defining the return on the minimum variance zero-beta portfolio, Z, as rz, he would run the cross sectional regression fj-‘= an empirical

=

U,+a,Bj(r*,-~~)+~j,

analogue &rj)

=

of Black’s (1972) zero-beta

pricing

equation,

E(rZ)+(E(r.~)-E(r,))Pj.

In one sense this formulation might be considered to provide a better explanation of security pricing than does the simple MV model (because at least the mode1 is consistent with a positive ‘reward’ to ‘risk’ tradeoff) until a closer inspection of the second pane1 of fig. 1 indicates that all the securities, including the market portfolio are plotting in the third quadrant! This plot can be explained by recalling some of the properties of the zero-beta model. Black (1972), Merton (1972), and Sharpe (1970) have shown that the MV feasible frontier can be generated by two basic (but non-unique) MV frontier portfolios. In an equilibrium context we can choose the two portfolios to be the MV efficient market portfolio and the zero-beta portfolio, or alternatively stated the zero-beta MV feasible frontier portfolio (which incidentally is inefficient). Black noted two other properties of zero-beta portfolios: (i) r 6 E(r,) < E(rM), and (ii) every portfolio with beta equal to zero must have the same expected return. Because the zero-beta model has been derived in an MV framework it may be that all of the properties of the mode1 do not hold in a non MV model. What perhaps is surprising is that in the example: (i) The market portfolio is not MV efficient. [Given the same opportunity set, including unrestricted borrowing and lending at the same riskless rate, and an economy populated by MV decision-makers the market value weights would have been (0.6582,0.1519, 0.1899)]. (ii) For the minimum variance zero-beta portfolio that we calculated, E(r,) > E(r,).* (iii) For the three zero-beta portfolios calculated in this example, ‘The return on the zero-beta portfolio v,as calculated from the formula rZr = .~,r,~ +.Yxrx,, i, k = 1, 2, 3, where .Y, = -/7~~(D,-~J and SC = (I -.I,). The minimum variance zero beta portfolio was taken to be the one that minimized oZ2 when the o,*‘s from the j. k pairs (1, 2). (1, 3), (2, 3) were compared. In an MV model this selection process may not have identified the MV zero-beta feasible frontier portfolio (discussed below) but it does not affect the observations made in the text.

18

R.R. Grauer, Tests of two parameter

asset prichg

morlels

the E(r,) values were (0.582, -0.915, 0.245). [For the corresponding MV economy the three E(rJ values were equal to five percent as they should have been, given the equal riskless borrowing and lending opportunity.] (iv) The minimum variance zero-beta portfolio that we calculated did not exhibit limited liability in two of the six states of nature. The moral to be derived from this example is straightforward: a test ofwhether E(r,)-r = 0 is a joint test of the assumption that there in unrestricted riskless borrowing and lending at the known rate r and that the market portfolio is MV efficient. In the empirical work that follows we generalize the concept of the zero-beta portfolio to zero-risk portfolios, with returns ro, and it is well to remember that tests of whether E(r,)-r = 0 are joint tests of whether there is unrestricted borrowing and lending and that the composition of the market portfolio is consistent with the specific tastes under consideration. Our social scientist, who is now batting 0 for 2, decides to fit the empirical analogue of (6) for various values of c and d, i.e., fj-r

= a,+a,

Riskj (?,-r)+ej.

(8)

Setting c = - 1 and d = 0 is equivalent to testing a ‘growth optimal’ model or to assuming that ‘composite’ investors have utility functions of the form In (wr). Even though the example was constructed where everyone had logarithmic utility, the regression results are less than satisfactory, as is shown in the third panel of fig. 1. However, with c = - 1 and d = -0.5637, the fourth panel indicates that the fit is essentially perfect. Immediately the question arises: Why is d = -0.5637? The answer is found in the market clearing condition for the riskless asset. In these models, with no exogenous government or foreign sector, borrowing must equal lending or the net supply of the riskless asset must be zero for (6) to hold; a condition satisfied if d = -0.5637.9 This example clearly illustrates that it is empirically possible to infer investor tastes from observed market return data. But it also indicates that one may have gFormally consider the market clearing conditions in an economy v.here there are I identical individuals. For thej- 1 risky assets, the conditions for demand to equal supply are

Zcjw,(l +d,‘(l +r))

j = 2, . . . . J,

= S,,,,

while for the riskless asset, demand equals supply (equals zero) :

I\+*, 1-(l+d/(l+r))

i j=2

In the example

1

Uj

= S,e = 0.

f. oj = 2.159. j=2

Substituting into the above equation d = -0.5637.

and simplifying,

confirms

that the market clears if

R.R. Grauer, Tests of two parameter

asset pricing models

19

to be close to clairvoyant to find the correct c, d combination. As computational costs prevent testing an unlimited number of possibilities we apply an eclectic mix of theoretical argument and empirical evidence to narrow the area of search. Although all the empirical work has tested the MV CAPM in a multiperiod setting or, more accurately, as if it held period by period in a multiperiod setting, it is really a single period equilibrium model.” While we cannot completely circumvent the single period equilibrium context, if we narrow the search to the power utility models where ai (and hence d) 2 0, not only will risky assets be normal goods but the models will be consistent with individual multiperiod discrete-time decision making. The next step is to provide some idea as to the range of d values. The example showed that d may be quite large and negative. This could be interpreted as implying that the ‘composite’ individuals had a positive subsistence level of wealth. However, there is a lower bound on how negative d may be. If u(w,) = I/Y(M’~+a)‘, then (n,,+nj(l +r)) must be positive and thus d s a/w0 > -( 1 t-r). On the other hand, Hakansson (1974b) has shown, in a multiperiod discrete-time wealth accumulation model, that if the utility function of terminal wealth is continuous monotone-increasing and bounded both above and below by an appropriately shifted power function, then the induced utility of wealth functions will converge to a power function of the form I/@‘, which implies that rl is close to zero.” A corroborating empirical finding by Friend and Blume (1975) was that the assumption of constant proportional risk aversion (d = 0) is as a first approximation a fairly accurate description of the market place Given these arguments, the strong theoretic appeal of a generalized logai-ithmic utility function, and the desirability of investigating a wide range of tastes, the following six c, d combinations were chosen for further study: c 1 -0.5 -1 -1

d -1

Utility function

Functional

U(W,) = -(aw1)2 u(w,) = -2H1, I’* u(w~) = Inwl L1(b+,l) = In(rr,,(0.5 + r,,,)) 11(11.,) = -WI-l

rc(w,) = - 1, WI- 5

-2

0

quadratic square root logarithmic generalized logarithmic negative one power

-6

0

negative fi\e power

0 0 -0.5

form

‘OIt is emphasircd that \\e are irorking ,,\ith a discrete-time model. Merton (1973) has derived a continuous-time b1V CAPM independent of the \pcci.!ic form of investor tastes. In discretetime the picture is not so clear. The l.eader is referred to Hakamson (1974a, 1977) for a discussion ot’the multiperiod nature of the MV CAPM in discrete time. “In a multiperiod setting the assumption that C/ is a constant. while it may provide a good approximation, is an approximation nonetheless for two reason\. First. C/ way vary with a change in tastes. For cuamplc. \\ith the v,ar babes becoming of in\ehtment age ae may find that aggregate in\c>tment beha\,iour exhibits a loher degree of risk aversion. Second, \bith no change in tabrcs, the value of ~1in the induced utility function, of \\ealth ikill change o\er time as investors mo\e closer to thcil decision horizons. [%x for cuumplc Hakansson (1974a).]

20

R.R. Grauer,-Tests

of two parameter

asset pricing models

3. Testable implications As statedearlier, the research is designedto determine whether one ofthepower LRT CAPMs may provide a better positive theory of security pricing than does the MV CAPM. Eq. (6) provides a framework for generating testable hypotheses about specific models and estimating the ‘true’ c, d parameters. The estimation and testing aspects are intimately entwined in the sense that, if we cannot reject hypotheses concerning a specific model, then that specific model provides a possible ‘true’ set of c, d values. We might then argue that from the set of nonrejected models the one that yields the closest parameter estimates is the ‘true’ model. Specifically, we begin with eq. (6) and generalize it to encompass a zero-risk form. Define r,, as the return on a zero-risk portfolio, then (6) becomes

E(rj) = E(ro) + (E(rM) -E(ro)) Riskj .

(9)

Generalizing the results of Black, Jensen and Scholes and Fama and MacBeth (1973) we propose the following stochastic generalization of (9),

rjr = a,,,

+a,,Riskj,

+a,,Risk:,

+ejr,

(10)

where sot and a,, are permitted to vary from period to period and are meant to be the counterparts of E(r,) and E(rM)-E(ro) respectively, while the Risk; term is added to test for linearity. In the context of the MV CAPM, Fama and MacBeth (1974) have shown that the least squares estimates a,, and a,, from the cross-sectional regressions (10) (with Riskf suppressed and Riskj = Beraj) are proxies for rz, and r,+,,-rrt respectively. It is easy to verify that this result generalizes and that in general the correspond to the return on a estimates a,, and a,, (with Risk; suppressed) zero-risk portifolio, ro,, and the excess return rMI - rO, respectively. In sum, with the stochastic generalization of (6) and (9) that is given by (lo), the testable implications of the generalized two parameter models are: (i) linearity, (ii) a positive expected return-risk tradeoff, and (iii) the joint hypothesis that there are no restrictions on riskless borrowing or lending and the composition of the market portfolio is consistent with the specific tastes under consideration. Finally, the model which most closely supports linearity, a,, = 0, a positive risk-return tradeoff, a,, > 0, and the joint hypothesis that there are no restrictions on riskless borrowing or lending and the composition of the market portfolio is consistent with the specific tastes under consideration, a o, = rt and a,, = rMt-r,, will be taken to be the ‘true’ model.

R.R. Grauer, Tests of two parameter asset pricing models

21

4. Methodology 4.1. The general approach The difficulties in testing the simple MV two parameter model (5) have been well documented.” Clearly we face the same difficulty with the generalized risk measures employed in this paper. The first and most obvious problem is that the models are stated in ex ante terms while empirical work must be concerned with ex post data. Thus, it is assumed that investors correctly assess return distributions so that the data observed ex post are drawings from the ex ante distributions assessed by investors. The second problem is closely. related to the first but more subtle. Basically it is an ‘errors in variables’ problem, caused from having to estimate the true risk measures, that may cause the slope of the estimated risk-reward tradeoff to be downward biased. Almost all the empirical studies have employed a grouping technique to circumvent this problem. In this paper the actual method employed is similar to that employed by Fama and MacBeth (1973j. Three periods are used to construct a time series of risk return data needed to test the LRT CAPMs: a formation period (to group securities into portfolios based on individual security risk measures), an estimation period (to re-estimate the individual security risk measures used in calculating the portfolio risk measures), and a test period (over which the portfolio returns and risk measures are calculated). The tests themselves are conducted by running a series of cross-sectional regressions on the time series of portfolio returns and risk measures to determine if one of the ‘risk-return’ tradeoffs in (6) holds. 4.2. The details The monthly data used in the tests were taken from a merged University of Chicago Center for Research in Security Prices (CRSP) and Compustat data base. The monthly returns on the market portfolio were taken to be Fisher’s Arithmetic Performance Index. The proxy for the riskfree rate was the one month rate on U.S. Treasury Bills subsequent to 1941 and the monthly rate on Bankers’ Acceptances prior to 1941. For each of the 6 models rates of return and risk measures were generated for that model’s set of 20 (risk measure) ranked portfolios. The following steps were undertaken : (1) In each year for which there were at least 36 observations from the previous 60 months (the formation period), at least 36 observations from the ‘%ee Jensen (1972b) for a summary of the research. Blume (1970). Friend and Blume (1970), Miller and Scholes (1972), Black, Jensen and Scholes (1972), Blume and Friend (1973). Roll (1973) and Fama and MacBeth’(1973) have originated and developed the econometric procedures we employ here.

22

R.R. Grauer, Tesfs of two paramefer

asset pricing models

subsequent 60 months (the estimation period), and at least 1 observation from the 12 months subsequent to the estimation period (the test period), the 6 risk measures for each stock were calculated over both the formation and estimation periods. (The first formation and estimation periods were only 48 months long and the first test period 24 months long so that two more years of risk return data could be generated.) (2) The stocks were then ranked from largest to smallest for each of the 6 risk measures calculated in the formation period. (3) Twenty portfolios were selected for each of the 6 risk measures by assigning the highest 5 percent of the ranked stocks to portfolio 1, the next 5 percent to portfolio 2, and so on. [More accurately, if 12stocks were available for selection in year t, n/20 stocks were assigned to each of the 20 portfolios. The remainder fromj = mod (n, 20) was distributed such that one extra stock was included in each of the first j portfolios.] (4) Over the 12 month test period, for each of the 6 sets of 20 portfolios, monthly returns and the corresponding monthly portfolio risk measures were calculated as the averages of the returns (in the test period) and the risk measures (calculated over the estimation period) of the individual securities contained in the portfolios in each month. Thus in the event of the delisting of a stock in some month during the test period the portfolio return and risk measures were calculated as averages of the stocks remaining in the portfolio. (5) Steps 1 to 4 were repeated for each year until January 1966 so that 6 sets of monthly returns on 20 portfolios were created for the 456 month period from January 1934 to December 1971. For each of the 6 sets of20 portfolios in each month of the period from January 1934 to December 1971, a cross-sectional regression of the form 2

rpf = ao,+a,,R~sk,, +a2,R~skpr + ep,,

p=l

, . ..) 21,

(11)

was run in two forms with Risk: included and excluded. (The market portfolio provided the twenty-first observation.) The 456 month time-series of data on the ui,‘s provided input for the tests. The tests on the individual models were conducted using simple t tests while the Hotelling T2 test was employed in order to determine wherher the models differed.

5. Results Table 2 shows a profile of the risk-return characteristics of selected portfolios. To save space only the statistics for the 1934-71 period are shown for all 6 models. For the MV model the statistics for tLvo half periods and the 1935-6168 period (studied by Fama and MacBeth) are presented for comparative purposes. The table shows that for any particular model the average returns do not differ appreciably over the range of risk measures.

rask rate of return

risk rate of return

risk rate ofreturn

risk rate of return

risk rate of return

Mean Mean

Mean Mean

Mean Me.m

Mean Mean

Mean Mean

were

formed

Ncgolirr fir<, ur,lirv 1.424 I280 1.265 0.016 0.014 0.016

0.990 0.014

0.957 0.014

I.026 0.014

mud?/ I.188 0.015

12171 1.094 0.014

basis

of risk

1134 to 12171 1.13X I.101 0.015 0.014

model 1134 to 1.196 1.140 0.016 0.016

groupings.

1.063 0.015

1.053 0.014

_

The

I.008 0.013

0991 0.013

2

_

0.948 0.013

0.950 0.013

0.948 0.013

0.951 0.014

0.933 0.014

0.947 0016

1.020 0.013

0.897 0017

0.958 0.015

IO

for

twenty-first

0.987 0.013

0.990 0.014

0.985 0.014

0.998 0.013

1.000 0.014

1037 0.011

0.904 0.013

urilrry modrl I13 I to 1217 I I.200 I.141 1.099 1.04Y 0 497 0.015 0.015 0.015 0.014 0.013

1.071 0.012

0.976 0.015

0.965 0013

9

0.99X 0.014

to b/68 1.142 0.017

to I2171 I.186 I.114 0.013 0.013

1.005 0.017

1.007 0.013

8

Table profile

0.986 0.013

on the

c on@ urrlit.v I.?Yl 1265 0.014 0.015

Ntwlri 1.434 0.016

1.054 0.014

Logar~rh,nr<. urilirv wodel I!34 to 1217 1 1.433 1.293 1.268 1.205 I I25 1091 0.016 0015 0014 0.016 0.016 0.014

Ioporirhmic I.298 I 251 0014 0.016

1.054 0.014

roe, utilrry modci l/34 to I?/71 1.300 1.253 1.207 I.125 1.089 0.015 0.015 0.017 0.015 0.013

Square 1.436 0.016

Genrmlr:cd 1.432 0.016

IO33 0.016

1.088 0.015

I ariorrw nrodrl 1’35 1.302 1.256 I.190 0.016 0.016 0.018

Mzan 1.441 0.018

I!53 I.206 0.013

r~oriarwc fwdri I.315 1.249 0.012 0.013

10 I2152 LOX9 1.058 0.017 0.015

1.038 0.015

Mean 1.435 0.014

7

variance modeI l/34 I.%? 1.258 1.170 0.017 0.016 0.020

6

Mean 1.445 0.019

20 portfolios Index.

risk rate of return

Mean Medn

‘The first Performance

r,ak ra:e of return

Mean Mea

5

mriance vrodrl 1134 to 12/71 1.304 1.254 I.188 1.137 1.086 0.015 0.014 0.016 0.015 0.014

4

Xfean 1.440 0.017

3

2

return-risk

1

Expected

portfolio

0.920 0.013

0930 0.013

0.934 0.014

0.924 0.013

0 93’) 0.014

0.911 0.014

0.981 0.012

0.X54 0015

0.917 0.013

11

twenty-one

is this

0.917 0.013

0909 0.013

0.908 0.013

0.907 0.013

0.897 0.017

0.895 0.013

0.965 0.011

0.X45 0.015

0.905 0.013

12

14

proxy

0.X55 0.013

OR19 0.012

0.X41 0012

0 X25 0.012

0.826 0.012

0.809 0.013

0.908 0.013

0.718 0.013

0.813 0.013

_

study’s

0.401 0.015

0.883 0.013

0 886 0.014

0.X71 0.013

0.X71 0.013

0.X64 0.013

0 954 0.010

0794 0.014

0.874 0.012

13

portfolios.e

for

0.794 0012

0.812 0.011

0.7’)‘) 0.011

0.795 0.011

0.X00 0.011

0.791 0.012

O.SYO 0.012

0.716 0.011

0803 0.011

15

the

0.710 0.012

0.709 0.013

0.680 0.013

0659 0.013

0 670 0013

0.661 0.013

0.659 0.013

0.623 0.014

0 700 0.013

0 595 0.013

0.647 0.013

18

portfolIo

0.71 I 0.012

0.701 0013

0.700 0.013

0.690 0 012

0 790 O.OIO

0.624 0015

0.707 0.013

17

market

0.774 0.012

0726 0012

0.743 0.012

0.730 0012

0.724 0.012

O.hY2 0013

0.7X2 0.012

0.613 0.014

0697 0.013

16

1.000 0.013

1.000 0.013

1000 0.013

1.000 0.013

I.000 0.013

1000 0.014

IO00 0.011

I.000 0.016

IO00 0.013

21

Arlfhmetic

0.529 0.010

0.504 0.009

0.514 0.00’)

0.495 0.009

0.493 0009

0.457 0.010

0 %I 0.010

0.416 0.009

0.488 0.009

20

- Fisher’s

0.615 0.010

0.598 0.010

0 597 0.011

0.597 0.010

0.?9? 0010

0.568 0.010

0.646 0.009

0.527 0.010

0.587 0.010

19

24

R.R. Crauer, Tests of two pcramerer

asset pricing models

As expected, the six models differed somewhat in the assignment of securities to the 20 portfolios, but not dramatically so. Moreover, the average rates of return ranked by portfolio are similar, but slight differences exist there as well. Finally, the risk measures generated from the various models, while similar in magnitude, reveal that the more risk averse utility functions assign higher risk values to low (risk) ranked portfolios than do their less risk averse counterparts. Table 3 contains the data for testing the major implications of the asset pricing models. The table contains statistics for 7 time periods, the 6 risk measures, and the 2 versions of test equation (11). Panel A suppresses the Risk; term while panel B reports the results for (I 1j itself. For each period and each model the table shows: “j, the average of the month-by-month regression coefficient estimates, a,,; s(aj), the standard deviation of the monthly estimates; and R: and s(R:), the mean and standard deviation of the month-by-month coefficients of determination, R:, which are adjusted for degrees of freedom. Finally, r-statistics for testing the hypothesis that Cj = 0 (or some other value) are presented. The t-statistics are of the form:

where dj, is the deviation of aj, from zero (or r or r,-r as the case may be) and n is the number of months in the period, which is also the number of estimates dj, used to compute aj and s(dj). As is always the case in the asset pricing literature, we caution the reader that, because the underlying variables are most likely not normally distributed, care should be taken not to interpret the results of the r-tests too literally.

5.1.

Tests of the major hypotheses

of the generalized

two parameter

models

We consider the three testable implications of the models: (i) linearity, (ii) positive expected returnrisk tradeoffs, and (iii) the joint hypothesis that there are no restrictions on riskless borrowing or lending and that the composition of the market portfolio is consistent with the specific tastes under consideration, in order. Linearity.

For each model the test of linearity is fundamental. It is equivalent to testing whether the composition of the market portfolio is consistent with investment decisions generated by the specific tastes under consideration. If the linearity hypothesis is rejected for the MV model we can conclude that the market portfolio (or more accurately market portfolio proxy) is not MV efficient. For the power models rejection of linearity implies that the market portfolio is not priced as if all investors were driven by the specific type of utility function being assumed and that the market for borrowing and lending does not clear.

R.R. Grauer, Tests of twoparameter asset pricing models

25

The t(aJ values in table 3 panel B indicate that a2 is not significantly different from zero in any but the 6/1943-52 subperiod for the MV, square root, and logarithmic utility models. Thus, we cannot reject the hypothesis of linearity for any of the models. Positive risk-return

To provide a viable positive theory of security tradeoff. pricing the models should, at a minimum, exhibit a positive risk-return tradeoff. But the reader is cautioned that this hypothesis is intimately related to linearity. Roll (1977, p. 136) points out that in the MV model if the market portfolio is known to be MV efficient then linearity and a positive risk-return tradeoff are both implied. Almost the same result holds for the power models. If we know that the market portfolio is consistent with (say) logarithmic utility decisionmaking and also know the d value that guarantees market clearing in the bond market then linearity and a positive risk-return tradeoff obtains. However, if we are studying a world where there is some exogenous supply of the riskless asset so that borrowing and lending among investors does not cancel any generalized logarithmic model will predict the correct relative market value weights but it will not necessarily lead to a positive linear risk-return tradeoff. (Note the numerical example illustrates exactly this phenomenon.) The results in panel A of table 3 show that there is a positive tradeoff for all the models except in the 1953-6/62 subperiod for the MV, square root, and logarithmic models. But the slope is too ‘flat’ in all cases.

Joint hypothesis:

No restrictions

market portfolio

is consistent with the specific tastes under consideration.‘3

ofp borrowing

or lending and composition

of the

In the ex ante example it was shown that one could infer the true process of price determination by finding the exact risk-return tradeoff that matched (9). In a similar manner the empirical test of whether _&a,,) = r, is the crucial test in attempting to determine which of the 6 models best fits the historic record. We may concentrate on testing whether the average intercept differs from the average risk-free rate because it can be shown that (from the nature of the construction of the market index and the properties of ordinary least squares regression) a test of the hypothesis that &a,,) = r1 is simultaneously a test of the hypothesis that _!?(a,,) = E(rMf) -rt. [See Kraus and Litzenberger (1976, p. 1097).14] In particular, we employ the values of the intercept to help identify which of the 6 models may provide the more accurate description of security pricing. ‘%I the MV model Roll (1977) also recognizes the nature of the joint hypothesis. But he is critical ofjoint hypotheses in general. l“The equality of the tests holds strictly only if the market proxy return is constructed to be the average of the n stock returns in the sample. As this is not exactly the case in the current study we present the slope values and r-statistics in table 3 but concentrate on the intercept test in the text.

7/1943-5i 1953 -6172 711962-71 1934 -52 1953 -71 1935 -6168

model

0.00664 0.01265 o.oc479

0.00872 0.00457 0.00823

0.00677 0.00755 0.00652

0.00656 0.01306 0.00433 - ~o.ml9 0.00904 0.00870 0.00442 O.Oa808

0.00646 0.01209 0.00441 -0.00008 0.00941 000825 0.00467 0.00801

0.00716 0.00436

EZ?% 0.00667

1934 1953 1935

-52 -71 -6168

0.00725 0.00399 0.00961 0.01201 0.00339

0.00739 0.00502 0.00960 0.01189 0.00303 0.0073 I 0.00746 0.00678

1934 -71 1934 -6/43 7/1943%52 1953 -6172 711962-7 I

Square roof model

M V model 1934 -71 1934 -6143 711943 -52 1953 -6172 111967-7’1 1934 -5i 1953 -71 1935 -6/68

-

0.00557 0.00421 0.00846 n.01015 0.00055 0.00633 0.00480 0.00519

X:EE 0.01021 -0.ooO32 0.00636 0.00494 0.00534

0.00565

0.00579 0.00487 O.Oa887 0.01010 -0.00068 0.00687 0.00471 0.00545 0.034 0.043 0.027 0.025 O.C.37 0.036 0.032 0.032

0.034 0.043 0.028 0.025 0.037 0.036 0.032 0.032

-0.00510 0.034 - 0.0048 I 0.044 -0.00810 0.027 - 0.00870 0.025 O.OaI21 0 037 - 0.00615 0.036 0.032 -0.00375 0.032 - 0.00468

-0.00518 -0.OiM39 - O.OO856 -“00X76 O.OOQ98 -- 0.00647 - 0.00389 -0.00482

-0.00528 -0.00537 -0.00848 - O.OOR65 0.00136 -0.00692 -0.00364 -0.00490

0065 0.100 0.053 0.034 0.053 0.080 0.044 0.062

0.065 0.102 0.053 0.034 0.052 0.081 0.044 0.062

0.064 0.100 0.052 0.034 0.053 0.079 0.044 0.061

-

-

-

0.034 0044 0.027 0.025 0.037 0.036 0.032 0.032

0.034 0.043 0.027 0.025 “037 _._~ 0.036 0.032 0.032

0.034 0.043 0.027 0.025 0.037 0.036 0.032 0.032

0.035 0.044 0.028 0.025 0.037 0.037 0.032 0.033

0.034 0.043 0.028 0.025 0.037 0.036 0.032 0.033

0.034 0.042 0.029 0.025 0.037 0.036 0.032 0.033

4.49 I .06 3.62 5.08 0.92 2.81 3.60 4.05

4 57 0:99 3.74 5.13 0.99 2.84 3.69 4.19

4.66 1.26 3.73 5.03 0.88 3.08 3.54 4.23

2.19 1.35 0.97 -0.04 1.88 1.65 1.55 2.68

2.15 1.37 0.88 -0.06 1.84 1.62 1.51 2.61

-0.02 1.91 1.57 1.59 2.61

-

-

-

-0.16 2.63 2.27 3.22

3.48 1.03 4.30 3.33

3.55 0.95 3.46 4.35 0.09 2.66 2.36 3.35

3.64 1.22 3.45 4.26 - 0.20 2.90 2.22 3.40

0.35 -2.63 - 1.78 -2.83

-3.15 -1.16 -3.05 -3.66

-3.24 - 1.09 -3.21 -3.71 0.29 -2.68 - 1.85 -2.98

-3.30 - 1.35 -3.14 -3.63 0.39 -2.88 ~ 1.72 -2.98

0.24 0.29 0.23 0.26

0.26 0.30 0.27 0.21

0.26 0.30 0.27 0.21 0.24 0.28 0.23 0.26

0.26 0.29 0.27 0.22 0.24 0.28 0.23 0.26

0.23 0.27 0.24 0.26

0.26 0.28 ;:I;

0.25 0.28 0.25 0.25 0.24 0.27 0.24 0.25

0.25 0.28 0.25 0.24 0.24 0.27 0.24 0.26

utility model o.oO701 0.00435 0.00874 0.01175 0.00321 0.00654 0.00748 0.00626

uriliw modrl 0.00639 0.00333 0.00745 0.01177 0.00301 0.00539 0.00739 000553

power

Minusfiw power 1934 -71 1934 -6143 7/1943-52 1953 -6172 711962-71 1934 -52 1953 -71 1935 -6/68

f;;

1934 -6143 711943-52 1953 -6172 7/l 962-7 I 1934 -52 1953 -71 1935 -6/68

r3y

Generalized logarithmic model 1934 -71 0.00697 1934 -6143 0 00502 7/1943-5i 0.0081 I 1953 -6/72 0.01155 711962-71 0.00322 1934 -52 0.00657 1953 -71 0.00738 1935 -6168 0.00621

0.00733 0.01342 0.00644 0.00007 0.00941 0.00993 0.00474 0.00911

0.00679 0.01261 0.00525 O.OooO8 0.00922 0.00893 0.00465 0.00848

0.00680 0.01182 0.0&7 O.OiJO28 0.00921 0.00885 0.00475 0.00850

-

-

-.

-

0.00479 0.00317 O.MKi72 0.00997 0.00070 0.00494 0.004b4 0.00421

0.00541 0.00420 0.00800 0.00995 -0.00051 0.00610 0.00472 0.00493

0.00537 0.00486 0.00738 0.00975 ~0.00050 0.00612 0.00463 0.00488

0.00441 - 0.00404 0.00645 000850 0.00136 0.00524 - 0.00357 000380

-0.00495 0.00484 PO.00763 0.00X49 0.00116 PO.00624 0.00367 ~0.00442

-0.00494 -0.00563 -~0.00701 -0.00829 0.001 IS 0.00632 ~0.00357 0.00441

0.036 0.046 0030 0 025 0.037 0039 0.032 0.034

0.034 0.043 0.028 0.025 0 036 0.036 0032 0.032

0.034 0.043 0.029 0.026 0.037 0037 0.032 0.032

0.067 0.104 0.057 0.034 0 054 0.084 0.045 0.065

0.064 0.099 0.054 0.034 0.053 0.080 0.044 0.062

0.065 0.100 0.055 0.034 0.053 0.081 0.045 0.062

-

-

-I

-

-

0.035 0.044 00x1 0.026 0.037 0.037 0.032 0.033

0.035 0044 0.03 0 026 0.037 0.038 0.032 0.034

0.037 0.032 0.033

0.036 0.032 0.032 0.036 0.046 0.030 0.02h 0.037 0.039 0.032 0.034

0.034 0.043 0.029 0025

0.034 0.043 0.028 0.025

0036 0036

0.034 0.043 0.028 0.026 0.037 0.037 0.032 0.032

I 4.Y4 0 87 2.10 3.48 3.29

3.84 0.77

2.66

4.42 1.08 3.35 4.96 0.94 2.73 3.58 3.88

4.34 I .23 3.04 4.83 0.93 2.70 3.49 3.85

2.33 I.37 I.21 0.02 1.X8 1.79 1.60 2.82

2.25 1.36 1.05 0.02 1.x7 I.69 1.58 2.73

1

2.23 1.26 I4 0.09 I.86 I.66 I.61 2.73

-

-

-

-

-

-

-

2.88 0.74 2.40 4.17 ~- 0.20 1.93 2.1X ?.?O

3.40 I.04 3.08 4.19 --“.I5 ii5 2.25 3.05

3.33 1.20 2.77 4.06 -0.14 2.52 2.18 3.02

-2.67 -0.98 -2.20 - 3.53 0.39 - 2.08 - 1.67 -2.24

-3.08 - 1.20 -2.81 -3.56 0.34 -ii6 1.75 -2.68

-3.03 -1.38 --2 51 -3.43 0.33 -2.56 -1.68 -2.67

0.25 0.29 0.27 0.20 0.24 0.28 0.22 0.25

0.25 0.30 0.27 0.20 0.24 0.29 0.22 0.26

0.26 0.31 0.27 0.20 0.24 0.29 0.22 0.26

0.25 0.28 0.26 0.23 0.23 0.27 0.23 0.25

0.26 0.28 0.26 0.24 0.23 0.27 0.23 0.26

0.25 0.28 0.26 0.23 0.23 0.27 0.23 0.25

0.00210 0.00499 -0.00491 0.00963 -0.00130 0.00004 0.00416 0.00211

model 0.00204 0.00453 -0.00432 0.00972 -0.00178 o.Oa310 0.00397 o.OQ200

Lc&ri!$j(;ir 1934 -6143 7/1943-52 1953 -6162 7/1962X71 1934 -52 1953 -71 1935 -6/68

model

0.00331 0.00950 -0.00481 0.00914 -0.ooO61 0.00234 0.00427 0.00276

Square root 1934 -7, 1934 -6!43 711943-52 1953 -6162 711962-71 1934 -52 1953 -71 1935 -6168

M V model 1934 -71 lY34 -6143 7 1943-52 1953 -6162 711962L71 1934 -s2 1953 -71 1935 -b/68

-0.00586 0.00088 -0.01684 - 0.003 10 -0.00437 - 0.00798 -- 0.00374 -0.00534

-0.00593 -0.00049 -0.OlSb4 -0.00288 -0.00473 -0.00806 -0.00381 -0.00532

0.01796 0.01295 0.03468 0.00514 0.01909 0.02381 0.01211 0.01830

-0.00448 0.00592 - 0.01672 -0.00361 -0.00351 -0.00540 --0.00356 -0.00447

0.01786 0.01123 0.03652 0.00547 0.01822 0.02388 0.01 I85 0.01825

0.01521 0.00130 0.03640 0.00650 0.01665 0.0188s 0.01158 0.01669

0.00044 0.00437 - 0.00506 0.00793 - 0.00549 - 0.00034 0.00122 0.00067

o.ooo5o O.CO484 - 0.00564 0.00783 - 0.00501 -0.00040 0.00141 0.00079

0.0073s -0.00432 0.00190 0.00151 0.00143

0.00622 -o.M)450 0.02179 -0.00343 0.01103 0.00864 0.00380 0.00539

0.00612 -0.00622 0.02364 -0.00310 0.01017 0.00871 0.00353 0.00534

0.00347 -0.01615 0.02352 -0.00207 0.00860 0.00368 0.00326 0.00379

0.067 nn71 0.060 0.044 0.086 0.066 0.068 0.062

0.067 0.069 0.062 0.044 0.087 0.065 0.069 0.063

0.068 0.067 0.062 0.045 0.091 0.065 0.071 0.063

0.159 0.194 0.159 0.104 0.167 0.177 0.139 0.154

0.158 0.185 0.161 0.104 0.170 0.174 0.141 0.153

0.157 0.174 0.162 0.109 0.175 0.169 0.145 0.151

0.074 OORT 0.077 0.054 0.078 0.081 0.067 0.070

0.075 0.083 0.079 0.054 0.079 0.081 0.068 0.071

0.076 0.084 0.080 0.056 0.082 0.083 0.070 0.071

0.067 0.071 0.060 0.044 0.086 0.066 0.068 0.062

0.067 0.069 0.062 0.044 0.087 0.065 0.069 0.063

0.068 0.067 0.063 0.044 0.091 0.065 0.071 0.063

0.141 0.155 0.140 0.098 0.163 0.148 0.134 0.133

0.142 0.152 0.144 0.098 0.166 0.148 0.136 0.134

0.145 0.151 0.145 0.101 0.172 0.149 0.141 0. I35

0.65 0 68 -0.77 2.36 -0.22 0.02 0.88 0.64

0.67 0.77 -0.85 2.34 -0.16 0.01 0.91 0.68

1.03 1.51 -082 2.19 -0.07 0.54 0.90 0.88

2.33 0.53 1.22 2.03 1.31 2.38

i:::

2.41 0.65 2.42 0.56 1.15 2.07 1.27 2.39

2.06 0.08 2.39 0.64 1.02 1 .b9 1.20 2.22

I::;: -1.51 -0.86 - 1.52

-%

-1.71

-1.51

-0:Sj

-z

0.11 - 2.29 -0.61

_ 1.68

- 0.99 -0.77 - 1.26

I::$

-2.24

0.14 0.65 - 0.90 I .93 -0.68 - 0.08 0.27 0.22

0.16 0.75 -0.98 1.90 -0.61 -0.09 0.31 0.25

0.53 1.49 - 0.95 1.77 -0.51 0.44 0.32 0A6

0.94 -0.31 1.66 -0.37 0.72 0.88 0.43 0.81

0.39 0.80

-% I-l x9 _._.

0.92 -0.44 1.76

0.51 -1.14 1.73 -0.22 0.53 0.37 0.35 0.56

0.26 0.28 0.26 0.24

0.28 0.32 0.31 0.25 0.26 0.31 0.25 0.29

ii::: 0.24 0.26

0.26 0.28 0.25 0.24 0.24 0.27 0.24 0.25

0.25 0.28 0.26 0.23 0.24 0.27 0.24 0.26 0.28 0.32 0.30 0.25 0.26 0.31 0.25 0.28

0.29 0.32 0.31 0.26 0.27 0.31 0.26 0.29

0.01604 OooY71 003197 000810 0.01437 0 02084 0.01 I24 0.02043

0.01573 0.00815 0.01062 O.OOh56 0.0175Y 0 01939 001208 0.01642

G<~nr~ralr:cd /<‘gu, rrh,,,rr mudpI I934 -71 0 002X8 iY34 -6,43 0.00652 7,lY33&52 0.0037x 1953 6 62 0.00814 7.lY62L71 0.00065 I’),4 -52 0.00137 IYS3 -71 0.00440 lY35 -6’68 O.OOOY2

i,t,l,r,, ,,,odel 0.00?93 0 00660 0 no279 0 00890 0.000’)x 0 00191 0.001Yh 0 00264

ulilrrv ,rwdcl 0 00400 0012YO 0.00766 0 00476 0.00140 0 02593 0 OOK05 0.00882 oon171 0.0121 I 0 003 I3 0.01534 0.00488 0 01046 0.00135 0.01858

.Ili~ii,\
bl~nrr~ /II L’ pvwcr I’)34 -71 1934 -6’43 7:1943-52 IV53 -6 62 7:1962-7, lY34 -57 1953 -71 1935 -6/68

0.00305 0.00408 ~0.01017 - 0.00472 0.0013Y ~ 0.00304 - 0.00306 0.00507

0.00470 0.00190 0.01322 - 0.00342 0.00105 - 0.00566 .0.00374 ~0.00419

- 0.00490 0.00065 -mo.o1360 0.00414 ~O.oo252 0.00647 -- 0.00333 0.00634

0.001 I6 -0.01269 0.01304 0.00025 0.00405 0.00018 000215 0.00568

--0.00201 0.00954 0.00422 0.00377 0.00351

0.0071 I - 0.00470 0.00146 0.00121 0.00131 0.00741 0.00751 mo.oo213 0.00625 0.00200 0.00269 000213 0.00002

0.00399 -0.00910 0.01774

000430 -0.00775 0.01908 ~0.00047 0.00632 0.00567 0.00292 0.0075?

0.00133 0.00645 0.00353

0.00128 0.00636 - 0.0045 I 0.00635 - 0.00306 0.00093 0.00164 ~- 0.00040

0.175 0.223 0.182 0.106 0.171 0.204 0.142 0.172

0.108 0.165 0.185 0.139 0.157

0.046 0.084 0.071 0.068 0.064 0.076 0.093 0.069 0.046 0.087 0.082 0.069 0.072

0.164 0.169 0.200

0.171 0.222 0.170 0.106 0.165 0.198 0.138 0.169

0.06’) 0.065 0.076

0.072 0.087 0.065 0.046 0.087 0.077 0.067 0.069

0.083 0.101 O.OY2 0.055 0.078 0.097 0.068 0.081

0.056 0.078 0.085 0.06X 0.073

0.077 0.082 0.088

0.080 0.097 0.083 0.056 0.078 0.090 0.067 0.078

0.076 0.093 0.069 0.046 0.087 0.082 0.069 0.072

0.046 0.084 0.071 0.068 0.064

0.06Y 0.065 0.076

0.072 0.087 0.065 0.046 O.OR2 0.077 0.067 0.06’)

0.160 0.197 0.164 0.101 0.165 0.181 0.137 0.153

0.103 0.161 0.157 0.135 0.138

0 146 0.163 0.150

0.153 0.187 0.151 0.102 0.160 0.170 0.134 0.148

I.13 0.88 0.22 I .88 0.21 0.58 I .06 0.38

2.07 -0.13 0.41 0.88 0.82

0.9 I -0.46 0.93

0.85 0.80 -0.62 1.89 0.08 0.27 1.00 0.27

-0.78 0.43 I.18 -0.91 ~0.19 -0.47 0.68 -1.26

-0.65 -0.56 - I.01 0.83 I.15

0.65 I.14 1.58 I.31 2.09 I.57 0.23 I..52 0.89 0.76 I.14 I.11 2.16

1.31 0.23 1.73

1.32 0.07 PI.75 -0.79 -0.35 -~ 1.08 -0.75 1.63

2.05 0.44 I .Y4

2.01 0.47 2.00 0.81 0.93 1.59 1.23 2.42

0.68 0.86 0.33 I .46 0.2s 0.49 0.46 0.00

I .65 -0.60 0.31 0.27 0.41

0.41 0.91 0.58

0.38 0.78 -0.74 I.48 -0.40 0.18 0.37 m-O.12

0.15 -0.69 0.85 0.03 0.26 0.01 0.24 0.74

0.21 0.63 0.41 0.42 0.51

0.58 -0.61 1.26

0.60 -0.44 1.35 -0.05 0.42 0.50 0.33 1.02

0.27 0.31 0.30 0.24 0.25 0.31 0.24 0.28

0.24 0.25 0.32 0.25 0.28

0.28 0.33 0.31

0.28 0.33 0.31 0.24 0.26 0.32 0.25 0.29

0.25 0.28 0.26 0.12 0.23 0.27 0.22 0.25

0.23 0.24 0.27 0.24 0.26

0.26 8:;;

0.25 0.28 0.26 0.22 0.23 0.27 0.23 0.25

30

R.R.

Gm~rer,Tests

oftwo parameter

nssrt pricing

models

Turning to table 3, it is apparent from the f tests that the hypothesis ~?(a,,) = r, must be rejected for all the models except in the first and fourth quarters of the sample period. Therefore, we reject the joint hypotheses that there is riskless borrowing and lending and that the market proxy is consistent with investment decisions determined by any of the LRT CAPMs considered here. But whether one wishes to make a blanket indictment of the LRT CAPMs, based on these results, depends on how close he believes the market proxy is to the true market portfolio. My judgment (strongly bolstered by reading Roll) is that w)e should be highly skeptical of the results, and further research employing either different data containing a better proxy for the true market portfolio or a different research design is called for before the issue can be settled.

5.2. Do the models differ ? A key part of the research was not only to determine which model might best describe security pricing but also whether the 6 models differed. With even a casual glance at table 3 one is struck by the similarity between the models across all the statistics. But somewhat tenuously we might argue that the more risk-averse models offer the better description of security pricing because the intercepts are smaller, the coeflicients on the squared risk term are never significantly different from zero, and the slopes are never negative. (On the other hand, the adjusted coeficients of determination are slightly lower.) Of course the argument would carry more weight if we could establish a statistical difference between the models. With a large sample it is sometimes possible to detect significant differences between apparently small quantities. The highly significant t-values on the intercept coefficients is a case in point. To determine whether the models differed, we tested for differences in the 6 time series of intercepts generated from eq. (I 1) (with Risk’ excluded,). The tests must allow for the interdependence between the models arising from time dependent measurements. The tests were conducted employing the I-iotelling T2 statistic because it allows for the interdependence between the models induced by time and does not make the strong assumptions regarding the homogeneity of variances and covariances between models as uould say a t\vo-way analysis of variance design. [See Morrison (1967, ch. 4).] The results (not shown) revealed no differences between the models. The close similarity of the models may be best described by the fact that the intercepts in panel A of table 3 were correlated on the order of 0.95 to 0.99. Clearly the tests failed to differentiate between the models. Unfortunately, ambiguity again surrounds any conclusions we may wish to draw. Probably, the majority of researchers \\ould argue that the models are empirically identical, citing either the approximate normality of return distributions or a ‘compact return’ distribution argument to back up their claim that in the stock market

R.R. Grauer, Tests of two parameter

asset pricing models

31

all risk averters are approximately MV decision-makers. On the other hand, one might believe that there is an underlying difference between the models and it is only the particular research design (accentuated by a poor market proxy) that has caused the failure to detect a significant difference between the models. Again, this appears to be an area calling for further research.

6. Summary and conclusions A series of empirically refutable generalized two parameter asset pricing models that linearly -relate risk and return were identified for the power (and quadratic) utility members of the LRT CAPMs. Five possible power utility models and the MV model were tested to determine whether one model might provide a more accurate description of security pricing. Three interrelated hypotheses were tested: linearity, a positive risk-return tradeoff, and a joint hypothesis of no restrictions on riskless borrowing and lending and that the composition of the market portfolio is consistent with decision-making governed by the specific LRT tastes under consideration. All the models passed the first two tests and failed the third. There was a slight indication that the more risk averse models better described security pricing but no statistically significant differences between the 6 models were detected. Finally it was noted (echoing Roll) that any conclusions drawn from the results of the tests should be greeted with a healthy dose of skepticism because the market proxy did not correspond to the true market portfolio.

References Best, Michael J., 1975, A feasible conjugate direction method to solve linearly constrained optimization problems, Journal of Optimization Theory and Applications 16, July, 25-38. Black, Fischer, 1972, Capital market equilibrium with restricted borrowing, Journal of Business 45, July,444-455. Black, Fischer, Michael C. Jensen and Myron Scholcs, 1972, The capital asset pricing model: Some empirical tests, in: Michael C. Jensen, ed., Studies in the theory of capital markets. Blume, Marshall E., 1970, Portfolio theory: A step towards its practical application, Journal of Business43, April, 152-173. Blume, Marshall E. and Irwin Friend, 1973, A new look at the capital asset pricing model, Journal of Finance 28, March, 19-34. Fama, Eugene F., 1976, Foundations of finance (Basic Books, New York). Fama, Eugene F. and James D. MacBeth, 1973, Risk, return, and equilibrium: empirical tests, Journal of Political Economy 81, May, 607-636. Fama, Eugene F. and James D. MacBeth, 1974, Long-term growth in a short-term market, Journal of Finance 29, June, 857-885. Friend, Irwin and Marshall Blume, 1970, Measurement of portfolio performance under uncertainty, American Economic Review 60, September, 561-575. Friend, lrwin and Marshall Blume, 1975, The demand for risky acsets, American Economic Review 65, December, 900-922. Grauer, Robert R., 1976a, Beta as a measure of risk in linear risk tolerance economies, Finance Working Paper No. 49, Institute of Business and Economic Research, University of California, Berkeley.

Grauer, Robert R., 1976b, The inference of tastes and bclisfs from bond and stock market data, Simon Fraser Untvcrsity Department of Economics and Con~n~crcc Discussion Paper. Forthcoming in the Journal of Financial and Quantitative Analysis. Hakansson. Nils H., 1977. The capital asset pricing model: Some open and closed ends, Finance Working Paper No. 22, Institute of Business and I-conomic Research, University of California, Bcrhclcy, June 1974a. and in: Irwin Friend and James Bickslcr, cds., Risk and return in finance, vol. I (Bnllingct-. Cambridge. MA). Hakansson, Nils H., 1974, Convcrpcnce to isoclastic utility and policy in multiperiod portfolio choice, Journal of Financial Economics I. September. 201-224. Jensen, Michael C., ed., l972a. Studies in the theory of capital marlets (Praegcr, New York). Jensen, Michael C., 1972b. Capital marhcts: Theory and evidence, Bell Journal of Economics and Management Science 3, Autumn, 3S7-398. Kraus, Alan and Robert Litzcnbcrgcr, 1075, Market equilibrium in a multiperiod state preference market with logarithmic utility. Journal of Finance 30. Dcccmher, 1213-1227. Kraus, Alan and Robert Litzcnbergcr. 1976. Skewness preference and the valuation of risky assets, Research Paper No. 130, Graduate School of Business, Stanford University, December 1972, and Journal of Finance 31, September, 1085--l 100. Merton, Robert C., 1972. An analytic dcrixation of the cfhcicnt portfolio frontier, Journal of Financialand Quantitati\eAnnlysis7, September, 1851~1871. Merton. Robert C., 1973, An intertemporal capital asset pricing model, Econometrica 41, September, 867-887. Miller, Mcrton H. and Myron Scholes, 1972, Rates ofretmn in relation to risk: A reexamination of some recent findings, in: Michael C. Jensen, ed., Studies in the theory of capital markets. Morrison, Donald F., 1967, blultivariate statistical methods (McGraw-Hill, New York). Roll, Richard, 1973, Evidence on the growth optimum model, Journal of Finance 28, June, 551-566. Roll, Richard, 1977, A critique of the asset pricing theory’s tests: Part I - On past and potential testability of the theory, Journal ofFinancial Economics4, Mat-ch. 129-176. Ross, Stephen A., 1977, The capital asset pricing model (CAPM): Short-sale restrictions and related issues, Journal of Finance 32, March, 177-183. Rubinstein, Mark, 1974. An aggregation theorem for securities markets, Journal of Financial Economics 1,September, 225-244. Rubinstein, Mark, 1976, The strong case for the generalized logarithmic utility model as the premier model of financial markets, Finance Working Paper No. 34, Institute of Business and Economic Research, University of California, Berkeley, February 1975, and Journal of Finance 3 1, May. Sharpe, William F., 1970, Portfolio analysis and capital markets (McGrawHill, New York).