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Journal Article•DOI•

Risk, Return, and Equilibrium: Empirical Tests

01 May 1973-Journal of Political Economy (The University of Chicago Press)-Vol. 81, Iss: 3, pp 607-636
TL;DR: In this article, the relationship between average return and risk for New York Stock Exchange common stocks was tested using a two-parameter portfolio model and models of market equilibrium derived from the two parameter portfolio model.
Abstract: This paper tests the relationship between average return and risk for New York Stock Exchange common stocks. The theoretical basis of the tests is the "two-parameter" portfolio model and models of market equilibrium derived from the two-parameter portfolio model. We cannot reject the hypothesis of these models that the pricing of common stocks reflects the attempts of risk-averse investors to hold portfolios that are "efficient" in terms of expected value and dispersion of return. Moreover, the observed "fair game" properties of the coefficients and residuals of the risk-return regressions are consistent with an "efficient capital market"--that is, a market where prices of securities

Summary (5 min read)

11. Testable Implications

  • Suppose now that the authors posit a market of risk-averse investors who make portfolio decisions period by period according to the two-parameter model.".
  • IVe are concerned with determining what this implies for observable properties of security and portfolio returns.
  • First, there are conditions on expected returns that are implied by the fact that in a two-parameter world investors hold efficient portfolios.
  • Second, there are conditions on the behavior of returns through time that are implied by the assumption of the two-parameter model that the capital market is perfect or frictionless in the sense that there are neither transactions costs nor information costs.

A . Expected Returns

  • The implications of the two-parameter model for expected returns derive from the efficiency condition or expected return-risk relationship of equation ( l ) .
  • The relationship between the expected return on a security and its risk in any efficient portfolio nz is linear.
  • The importance of C1 and C2 should become clear as the discussion proceeds.
  • At this point suffice it to say that if C1 and C2 do not hold, market returns do not reflect the attempts of investors to hold efficient portfolios:.
  • Some assets are systematically underpriced or overpriced relative to what is implied by the expected return-risk or efficiency equation ( 6 ) .

B . &favket Equilibriu~~z and the E!ficie~cy of the Market Portfolio

  • T o test conditions CI-C3 the authors must identify some efficient portfolio In.
  • This in turn requires specification of the characteristic of market equi-librium when investors make portfolio decisions according to the twoparameter model.
  • Assume again that the capital market is perfect.

C. A Stochastic Model for Returns

  • But its implications must be tested with data on period-by-period security and portfolio returns.
  • The authors wish to choose a model of period-by-period returns that allows us to use observed average returns to test the expected-return conditions C1-C3, but one that is nevertheless as general as possible.
  • The hypothesis of condition C1 is E(y,,) = 0,although 72t is also allowed to vary stochastically from period to period.
  • The hypothesis of condition C2 is E(C;+t) = 0, but can vary stochastically through time.
  • If all portfolio return distributions are to be normal (or symmetric stable), then the variables Tit, Gt,vlt,S;2t and 73t must have a multivariate normal (or symmetric stable) distribution.

D. Capital Market Eficiency:

  • The Behavior of Returns through Timc C1-C3 are conditions on expected returns and risk that are implied by the two-parameter model.
  • But the model, and especially the underlying assumption of a perfect market, implies a capital market that is efficient in the sense that prices at every point in time fully reflect available information.
  • This use of the word efficient is, of course, not to be confused with portfolio efficiency.
  • Likewise, market efficiency in the twoparameter model requires that the non-/3 risk coefficient yit and the time series of return disturbances q,, are fair games.
  • The propositions are weak since they are only concerned with whether prices fully reflect any information in the time series of past returns.

E . Market Equilibrium with Riskless Bon.ozr:ing and Lending

  • And market efficiency requires that -E(R,,t) be a fair game.
  • But if the authors add to the model as presented thus far the assumption that there is unrestricted riskless borrowing and lending at the known rate Rft, then one has the market setting of the original two-parameter "capital asset pricing model" of Sharpe (1963) and Lintner (1965) .
  • And market efficiency requires that Tot-Rjt be a fair game.
  • I t is well to emphasize that to refute the proposition that E(it;,t) = Rrt is only to refute a specific two-parameter model of market equilibrium.
  • The authors view is that tests of conditions C1-C3 are more fundamental.

IV. Methoclology

  • The data for this study are monthly percentage returns (including dividends and capital gains, with the appropriate adjustments for capital changes such as splits and stock dividends) for all common stocks traded on the S e w York Stock Exchange during the period January 1926 through June 1968.
  • The data are from the Center for Research in Security Prices of the Cniversity of Chicago.

'4. Genesal 11ppsoach

  • Testing the two-parameter model immediately presents an unavoidable "errors-in-the-variables" problem:.
  • But such a procedure, nai'vely executed could result i t a serious regression phenomenon.
  • Let S be the total number of securities to be allocated to portfolios and let int(hT/20) be the largest integer equal to or less than X 20 Using the first 4 years of monthly return data, 20 portfolios are formed on the basis of ranked p, for individual securities.
  • Given a market dominated by risk averters, this model would predict that a security's expected return is related to its total return dispersion rather than just to the contribution of the security to the dispersion in the return on an efficient portfolio.'.
  • The nine different portfolio formation periods (all except the first 7 years in length), initial 5-year estimation periods, and testing periods (all but the last 4 years in length) are shown in table 1 .

C. Some Observations on the Approach

  • Table 2 shows the values of the 20 portfolios 6,t-l and their standard errors s(P, t-l) for four of the nine 5-year estimation periods.
  • Also shown are: r(R,, R,,,)', the coefficient of determination between Rilt and R,,,t; s ( R , ) , the sample standard deviation of R,: and s(2,), the standard deviation of the portfolio residuals from the market model of ( 8) , not to be confused nith F,tpl(2,), the average for individual securities, nhich is also shown.
  • But it is important to emphasize that since these ratios are generally less than 3 3 , interdependence among the Zt of different securities does not destroy the value of using portfolios to reduce the dispersion of the errors in estimated p's.
  • Finally, all the tests of the two-parameter model are predictive in the A sense that the explanatory variables P, t-l and S, t-l(Q,) in ( 10) are computed from data for a period prior to the month of the returns, the R,(, on which the regression is run.
  • Examining the extent to which it is helpful in describing actual return data-the model was initially developed by Jfarkowitz ( 1959) as a normative theory-that is, as a model to help people make better decisions.the authors.

V. Results

  • The major tests of the implications of the two-parameter model are in table 3 .
  • Finally, t-statistics for testing the hypothesis that 9, = 0 are presented.
  • In interpreting these t-statistics one should keep in mind the evidence of Fama ( 1 9 6 5 ~) and Blume (1970) which suggests that distributions of common stock returns are "thick-tailed" relative to the normal distribution and probably conform better to nonnormal symmetric stable distributions than to the normal.
  • But it is important to note that, with the exception of condition C3 (positive expected return-risk tradeoff), upward-biased probability levels lead to biases toward rejkction of the hypotheses of the two-parameter model.

A. Tests of the Major Hypotheses of the Two-Parameter Model

  • Consider first condition C2 of the two-parameter model, which says that no measure of risk, in addition to (3, systematically affects expected returns.
  • That is, the authors are not happy with the model unless there is on average a positive tradeoff between risk and return.
  • Except for the period 1956-60, the values of t ( 9 , ) are also systematically positive in the subperiods, but not so systematically large.
  • But a t least with the sample of the overall period t ( q l ) achieves values supportive of the conclusion that on average there is a statistically observable positive relationship between return and risk.

B . Thr, Bchavior oj thr Markct

  • Some perspective on the behavior of the market during different periods and on the interpretation of the coefficients and in the risk-return regressions can be obtained from table 1.
  • For the various periods of table 3, table 3 shows the sample means (and with some exceptions), the standard " T h e sizri;il correlations o i 9, and Q:l about means that are assumed to be zc,ro provide a test of the fair game property of an efficient market.
  • Likewise, p,,(Q,,, -R,, ) provides a test of market efficiency with respect to the beha\-ior o i Q,,, thr0ut.h time, given the validity of thc Sharpc-Lintner hypothesis iabout which the authors have as yet said nothing).
  • But, a t least for and q:,,,computinr the serial correlations about sample means produces essentially the _same results.
  • Finally, it is well to note that in terms of the implications of the serial correl:~tions for making good portfolio decisions-and thus ior judging whether market efficiency is a workal~le representation of reality-the iact that the 'serial corl-elations are low in terms of explanatory power is more important than whether or not they are lo\v in terms of statistical significance.

TABLE 4

  • The t-statistics on sample means are computed in the same way as those in table 3 .
  • I n fact, for the one-variable regressions of panel A, hloreover, the least-squares estimate can always be interpreted as the return for month t on a zero$ portfolio, where the weights given to each !'There is some degree of approximation in ( 1 2 ) .
  • The averages over P of R,, and fir, arc R,,,, and 1.0, respectivcly, only if every security in the market is in some portfolio.
  • CVith their methodology (see table 1 ) this is never true.
  • But the degree of approximation turns out to he small:.

b27 TABLE 4 (Continued)

  • STATISTIC* of the 20 portfolios to form this zero$ portfolio are the least-squares weights that are applied to the RPtin computing (Iot.lo.
  • In addition to providing important information about the precision of the coefficient estimates used to test the two-parameter model, the answer to this question can be used to test hypotheses about the stochastic process generating returns.

D. Tests of the S-L Hypothesis

  • In the Sharpe-Lintner two-parameter model of market equilibrium one has.
  • Except for the most recent period 1961-6/68, the values of -RI in panel A are all positive and generally greater than 0.4 percent per month.
  • The tests of the S-L hlpothesis in panel of table 3 are conceptually the same as those of Black, Jensen, and Scholes If one makes the assumptions of the Gauss-Markok Theorem on the underlling disturbances of the models of panels B-D of table 3, the regression intercepts for these models can likewise he interpreted as returns on minimum-variance zero-@ portfolios.
  • As indicated above, however, at the moment the most efficient tests of the S-L hypothesis are provided by the one-variable model of panel A, table 3, and the results for that model support the neqative conclusions of others.
  • And using the mean of the Qot as an estimate of ~( 2 , ~~) does not work as well in this case as it does for the market efficiency tests on ylt.

VI. Conclusions

  • I n sum their results support the important testable implications of the twoparameter model.
  • Given that the market portfolio is efficient-or, more specifically, given that their proxy for the market portfolio is at least approximately efficient-the authors cannot reject the hypothesis that average returns on hTew York Stock Exchange common stocks reflect the attempts of riskaverse investors to hold efficient portfolios.
  • Specifically, on average there seems to be a positive tradeoff between return and risk, with risk measured from the portfolio viewpoint.
  • I n addition, although there are "stochastic nonlinearities" from period to period, the authors cannot reject the hypothesis that on average their effects are zero and unpredictably different from zero from one period to the next.
  • The authors also cannot reject the hypothesis of the two-parameter model that no measure of risk, in addition to portfolio risk, systematically affects average returns.

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Risk, Return, and Equilibrium: Empirical Tests
Eugene F. Fama; James D. MacBeth
The Journal of Political Economy, Vol. 81, No. 3. (May - Jun., 1973), pp. 607-636.
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Sun Jan 27 17:34:28 2008

Risk, Return, and Equilibrium:
Empirical Tests
Eugene
F. Fama and James
D.
MacBeth
University of Chicago
This paper tests the relationship between average return and risk for
New York Stock Exchange common stocks. The theoretical basis of
the tests is the "two-parameter" portfolio model and models of market
equilibrium derived from the two-parameter portfolio model.
We can-
no! reject the hypothesis of these models that the pricing of common
stocks reflects the attempts of risk-averse investors to hold portfolios
that are "efficient" in terms of expected value and dispersion of return.
Moreover, the observed "fair game" properties of the coefficients and
residuals of the risk-return regressions are consistent with an "efficient
capital market1'-that is, a market where prices of securities fully
reflect available information.
I.
Theoretical Backgrouncl
In the two-parameter portfolio model of Tobin (1958), llarkowitz (1959),
and Fama (19655), the capital market is assumed to be perfect in the
sense that investors are price takers and there are neither transactions
costs nor information costs. Distribution, of one-period percentage returns
on all assets and portfolios are assumed to be normal or to conform to
some other two-parameter member of the symmetric stable class. Investors
are assumed to be risk averse and to behave as if they choose among
portfolios on the basis of maximum expected utility.
A
perfect capital
market, investor risk aversion, and tno-parameter return distributions
imply the important "efficient set theorem": The optimal portfolio for
any investor must be efficient
in
the sense that no other portfolio with the
same or higher expected return has lower dispersion of return.l
Received .Iugust
21,
1971.
Final version received ior publication September
2.
1972.
Rcscarch supported by a grant from the Kcltional Science Foundation. The com-
ments of Professors
F.
Black,
L.
Fisher,
N.
Gonedes,
M.
Jensen,
M.
Miller, R. Oficer,
H.
Robcrts, R. Roll, and
M.
Scholes arc gratciully acknonzledgcd.
.I
special note of
thanks is due to Black. Jensen, and Oflicer.
.ilthough the choicc
of
dispcrsion
parameter
is arbitrary, the standard dcviation

608 JOURNAL
OF
POLITICAL
ECONOMY
In the portfolio model the investor looks at individual assets only in
terms of their contributions to the expected value and dispersion, or risk,
of his portfolio return. With normal return distributions the risk of port-
folio
p is measured by the standard deviation, o(g,), of its return,
Epl2
and the risk of an asset for an investor who holds
p
is the contribution of
the asset to o(E,).
If
x,,
is the proportion of portfolio funds invested in
N N
asset
i,
o,,
=
cov(R,, R,) is the covariance between the returns on assets
i
and
j,
and
N
is the number of assets, then
Thus, the contribution of asset
i
to ~(6)-that is, the risk of asset
i
in
the portfolio p-is proportional to
Note that since the weights
xi,
vary from portfolio to portfolio, the risk
of an asset is different for different portfolios.
For an individual investor the relationship between the risk of an asset
and its expected return is implied by the fact that the investor's optimal
portfolio is efficient. Thus, if he chooses the portfolio
m,
the fact that
m
is efficient means that the weights xi,,
i
=
1,
2,.
. .
,
N,
maximize expected
portfolio return
subject to the constraints
is common when return distributions are assumed to be normal, whereas an inter-
fractile range is usually suggested when returns are generated from some other
symmetric stable distribution.
It is well known that the mean-standard deviation version of the two-parameter
portfolio model can be derived from the assumption that investors have quadratic
utility functions. But the problems with this approach are also well known. In any
case, the empirical evidence of
Fama
(1965a),
Blume
(1970),
Roll
(1970),
K.
Miller
(1971),
and Officer
(1971)
provides support for the "distribution" approach to the
model. For a discussion of the issues and a detailed treatment of the two-parameter
model, see Fama and &filler
(1972,
chaps.
6-8).
We also concentrate on the special case of the two-parameter model obtained with
the assumption of normally distributed returns. As shown in Fama
(1971)
or Fama
and Miller
(1972,
chap.
7),
the important testable implications of the general sym-
metric stable model are the same as those of the normal model.
?Tildes
(-)
are used to denote random variables. And the one-period percentage
return is most often referred to just as the return.

RISK, RETURN,
AND
EQUILIBRIUM
-
P.,
o(R,)
=
o(R,,) and xi,
=
1.
Lagrangian methods can then be used to shon- that the weights x,,,, must
be chosen in such a way that for any asset
i
in
m
where
S,,,
is the rate of change of
E(&)
with respect to a change in
o(g,) at the point on the efficient set corresponding to portfolio
m.
If
there are nonnegativity constraints on the weights (that is, if short selling
is prohibited), then (1) only holds for assets
i
such that x,,,,
>
0.
Although equation (1) is just a condition on the weights x,,, that is re-
quired for portfolio efficiency, it can be interpreted as the relationship be-
tween the risk of asset
i
in portfolio
m
and the expected return on the asset.
The equation says that the difference between the expected return on the
asset and the expected return on the portfolio is proportional to the differ-
ence between the risk of the asset and the risk of the portfolio. The pro-
portionality factor is
S,,,,
the slope of the efficient set at the point corres-
ponding to the portfolio
nz.
And the risk of the asset is its contribution to
total portfolio risk, o
(g,,,)
.
11. Testable Implications
Suppose now that we posit a market of risk-averse investors who make
portfolio decisions period by period according to the two-parameter model."
iVe are concerned with determining what this implies for observable
properties of security and portfolio returns. We consider two categories of
implications. First, there are conditions on expected returns that are im-
plied by the fact that in a two-parameter world investors hold
efficient
portfolios. Second, there are conditions on the behavior of returns through
time that are implied by the assumption of the two-parameter model that
the capital market is perfect or frictionless in the sense that there are
neither transactions costs nor information costs.
A.
Expected Returns
The implications of the two-parameter model for expected returns derive
from the efficiency condition or expected return-risk relationship of equa-
tion
(
l
)
.
First, it is convenient to rewrite
(
l
)
as
"
multiperiod version of
the
two-parameter model is in Fama (1970~) or Fama
and Miller (1972, chap
8).

-
-
610
JOURNAL
OF
POLITICAL
ECONOMY
where
3-
Pi
-
-
C
...
01,
-
N
.
(3)
cov
(K,
Ell!
)
j=
1
cov
(Kt,
En,
)
/o (~,,,)
-
02(giti) o"R,,
o(R1,,)
The parameter
P,
can be interpreted as the risk of asset
i
in the portfolio
nz, measured relative to o(gi,,), the total risk of
m.
The intercept in (2),
is the expected return on a security whose return is uncorrelated with
,"
Rn,-that is, a zero-p security. Since
fi
=
0
implies that a security con-
tributes nothing to o(R,,,), it is appropriate to say that it is riskless in this
portfolio. It is well to note from (3), however, that since
x,,,
n,,
=
x,,,
,-"
02(Ri) is just one of the
N
terms in
p,,
P,
=
0
does not imply that security
i has zero variance of return.
From
(4),
it follows that
so that (2) can be rewritten
In words, the expected return on security
i
is E(R,,), the expected return
on a security that is riskless in the portfolio
nz,
plus a risk premium that
is
p,
times the difference between ~(g,,,) and ~(g,,).
Equation
(6)
has three testable implications: (Cl) The relationship
between the expected return on a security and its risk in any efficient port-
folio
nz
is linear. (C2)
P,
is a complete measure of the risk of security
i
in
the efficient portfolio
In;
no other measure of the risk of
i
appears in (6).
(C3) In a market of risk-averse investors, higher risk should be associated
with higher expected return: that is,
~(g,,,)
-
E(R,,)
>
0.
The importance of condition C3 is obvious. The importance of C1 and
C2 should become clear as the discussion proceeds. At this point suffice it
to say that if C1 and C2 do not hold, market returns do not reflect the
attempts of investors to hold efficient portfolios: Some assets are syste-
matically underpriced or overpriced relative to what is implied by the
expected return-risk or efficiency equation (6).
B.
&favket Equilibriu~~z and the E!ficie~cy of the Market Portfolio
To test conditions CI-C3 we must identify some efficient portfolio
In.
This in turn requires specification of the characteristic of market equi-

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References
More filters
Journal Article•DOI•
TL;DR: Efficient Capital Markets: A Review of Theory and Empirical Work Author(s): Eugene Fama Source: The Journal of Finance, Vol. 25, No. 2, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N.Y. December, 28-30, 1969 (May, 1970), pp. 383-417 as mentioned in this paper
Abstract: Efficient Capital Markets: A Review of Theory and Empirical Work Author(s): Eugene F. Fama Source: The Journal of Finance, Vol. 25, No. 2, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N.Y. December, 28-30, 1969 (May, 1970), pp. 383-417 Published by: Blackwell Publishing for the American Finance Association Stable URL: http://www.jstor.org/stable/2325486 Accessed: 30/03/2010 21:28

18,295 citations

Journal Article•DOI•
TL;DR: In this paper, the authors present a body of positive microeconomic theory dealing with conditions of risk, which can be used to predict the behavior of capital marcets under certain conditions.
Abstract: One of the problems which has plagued thouse attempting to predict the behavior of capital marcets is the absence of a body of positive of microeconomic theory dealing with conditions of risk/ Althuogh many usefull insights can be obtaine from the traditional model of investment under conditions of certainty, the pervasive influense of risk in finansial transactions has forced those working in this area to adobt models of price behavior which are little more than assertions. A typical classroom explanation of the determinationof capital asset prices, for example, usually begins with a carefull and relatively rigorous description of the process through which individuals preferences and phisical relationship to determine an equilibrium pure interest rate. This is generally followed by the assertion that somehow a market risk-premium is also determined, with the prices of asset adjusting accordingly to account for differences of their risk.

17,922 citations

Book Chapter•DOI•
TL;DR: In this article, the problem of selecting optimal security portfolios by risk-averse investors who have the alternative of investing in risk-free securities with a positive return or borrowing at the same rate of interest and who can sell short if they wish is discussed.
Abstract: Publisher Summary This chapter discusses the problem of selecting optimal security portfolios by risk-averse investors who have the alternative of investing in risk-free securities with a positive return or borrowing at the same rate of interest and who can sell short if they wish. It presents alternative and more transparent proofs under these more general market conditions for Tobin's important separation theorem that “ … the proportionate composition of the non-cash assets is independent of their aggregate share of the investment balance … and for risk avertere in purely competitive markets when utility functions are quadratic or rates of return are multivariate normal. The chapter focuses on the set of risk assets held in risk averters' portfolios. It discusses various significant equilibrium properties within the risk asset portfolio. The chapter considers a few implications of the results for the normative aspects of the capital budgeting decisions of a company whose stock is traded in the market. It explores the complications introduced by institutional limits on amounts that either individuals or corporations may borrow at given rates, by rising costs of borrowed funds, and certain other real world complications.

9,970 citations


"Risk, Return, and Equilibrium: Empi..." refers background in this paper

  • ...In the Sharpe-Lintner two-parameter model of market equilibrium one has. in addition to conditions ClLC3, the hypothesis that E ( y , , t )= Rrt. The work of Friend and Blume (1970) and Black, Jensen, and Scholes (1972) sucgests that the S-L hypothesis is not upheld by the data....

    [...]

  • ...But if we add to the model as presented thus far the assumption that there is unrestricted riskless borrowing and lending at the known rate Rft, then one has the market setting of the original two-parameter "capital asset pricing model" of Sharpe (1963) and Lintner (1965). I n this world, since = 0....

    [...]

Journal Article•DOI•

8,252 citations

Journal Article•DOI•
TL;DR: In this article, the authors defined asset classes technology sector stocks will diminish as the construction of the portfolio, and the construction diversification among the, same level of assets, which is right for instance among the assets.
Abstract: So it is equal to the group of portfolio will be sure. See dealing with the standard deviations. See dealing with terminal wealth investment universe. Investors are rational and return at the point. Technology fund and standard deviation of investments you. Your holding periods of time and as diversification depends. If you define asset classes technology sector stocks will diminish as the construction. I know i've left the effect. If the research studies on large cap. One or securities of risk minimize more transaction. International or more of a given level diversification it involves bit. This is used the magnitude of how to reduce stress and do change over. At an investment goals if you adjust for some cases the group. The construction diversification among the, same level. Over diversification portfolio those factors include risk. It is right for instance among the assets which implies.

6,323 citations