Idiosyncratic Risk and Security Returns∗
Burton G. Malkiel
Department of Economics
Princeton University
Yexiao Xu
School of Management
The University of Texas at Dallas
Email: yexiaoxu@utdallas.edu
This revision: July 13, 2000
Abstract
The traditional CAPM approach argues that only market risk should be incorporated into asset prices and command a risk premium. This result may not
hold, however, if some investors can not hold a market portfolio. For example, if
one group of investors fails to hold the market portfolio for exogenous reasons, the
remaining investors will also be unable to hold the market portfolio. Therefore,
idiosyncratic risk could also be priced to compensate rational investors for an
inability to hold the market portfolio. A variation of the CAPM model is derived
to capture this observation as well as to draw testable implications. Using both
constructed portfolios and equity mutual funds, we find that (1) idiosyncratic
volatility is useful in explaining cross-sectional expected returns; (2) returns from
constructed portfolios and equity mutual funds directly co-vary with idiosyncratic
risk; (3) the intercept and the beta estimated from the CAPM model are highly
negatively correlated, which reduces the overall explanatory power of beta in
cross-sectional study.
∗
We are grateful to Ted Day, Campbell R. Harvey, Larry Merville, Grant McQueen, Michael Pinegar, and Steven Thorley, seminar participants at Brighman Young University, and 1999 Econometrics
Society Conference for helpful comments. We acknowledge with thanks the support of Princeton’s
Bendheim Center and the Center for Policy Studies. The corresponding author’s address is: School
of Management; The University of Texas at Dallas; Richardson, TX 75083. Telephone (972)883-6703;
FAX (972)883-6522.
i
Idiosyncratic Risk and Security Returns
Abstract
The traditional CAPM approach argues that only market risk should be incorporated into asset prices and command a risk premium. This result may not
hold, however, if some investors can not hold a market portfolio. For example, if
one group of investors fails to hold the market portfolio for exogenous reasons, the
remaining investors will also be unable to hold the market portfolio. Therefore,
idiosyncratic risk could also be priced to compensate rational investors for an
inability to hold the market portfolio. A variation of the CAPM model is derived
to capture this observation as well as to draw testable implications. Using both
constructed portfolios and equity mutual funds, we find that (1) idiosyncratic
volatility is useful in explaining cross-sectional expected returns; (2) returns from
constructed portfolios and equity mutual funds directly co-vary with idiosyncratic
risk; (3) the intercept and the beta estimated from the CAPM model are highly
negatively correlated, which reduces the overall explanatory power of beta in
cross-sectional study.
Introduction
The usefulness of the well-celebrated CAPM theory of Sharpe, Lintner, and Black
to predict cross-sectional security and portfolio returns has been challenged by researchers such as Fama and French (1992, 1993)). It is still debatable, however, whether
Fama and French’s empirical approach has invalidated the CAPM (see, for example,
Loughran(1996) and Kothari, Shanken, and Sloan (1995)). Moreover, as Roll (1977)
has pointed out, it is difficult, if not impossible, to devise an adequate test of the theory.
Nevertheless, financial economists have worked in several directions to improve the theory of asset pricing. The first route has involved relaxing the underlying assumptions
of the model, including the introduction of a tax effect on dividends (e.g., Brennan
(1970)), non-marketable assets (e.g., Mayers (1972)), as well as accounting for inflation
and international assets (e.g., Stulz (1981)). A second route has been to extend the
one period CAPM to an intertemporal setting (e.g., Merton (1973)). Ross (1976) has
taken a different route by assuming that the stochastic properties of asset returns are
consistent with a factor structure, and our approach is in that tradition. We accept the
CAPM as a reasonable first order approximation, but find that the model has different
implications when we assume that not all investors are able to hold the market portfolio.
Starting from mean variance analysis,1 the traditional CAPM theory predicts that
only market risk should be priced in equilibrium; any role for idiosyncratic risk is
completely excluded. CAPM must surely hold if investors are alike and can hold a
combination of the market portfolio and a risk-free asset as the theory prescribes. In
practice, however, this assumption is often violated to some extent. When one group
of investors – called “constrained” investors – is unable to hold the market portfolio for
various reasons, such as transactions costs, liquidity constraints, or any other exogenous
factors, the remaining investors – labeled as “free” or “unconstrained” investors – will
also be unable to hold the market portfolio. This is so because the constrained investors’
holdings and the free investors’ holdings together make up the whole market. An
inability to hold the market portfolio will force investors to care about total risk not
simply market risk. Since the relative per capita supply will be high for those stocks that
the constrained investors do not hold or only hold in very limited amounts, the prices of
these stocks must be relatively low. In other words, an idiosyncratic risk premium can
1
This is consistent with expected utility maximization for any concave utility function (see Ross
(1978)).
1
be rationalized to compensate investors for the “over supply” or “unbalanced supply”
of some assets.
Still another intuition can be gained in terms of diversification. Suppose the actual market portfolio consists of only tradable securities. In other words, the market
portfolio is observable and measurable. Under our assumption that some investors are
constrained from holding all securities, the “available market portfolio” that unconstrained investors can hold will be less diversified than the actual market portfolio.
Therefore, its corresponding risk will be higher, and a larger risk premium will be required. When individual investors use the available market portfolio to price individual
securities, the corresponding risk premia tend to be higher than those under the CAPM
where all investors are able to hold the actual market portfolio. This is because some
of the systematic risk would be considered as idiosyncratic risk relative to the actual
market portfolio. Hence, we would find that idiosyncratic risk is priced in the market.
We will fully develop this idea below and investigate the empirical implications.
There are many reasons why individual investors might not be able to hold the
market portfolio. First, transactions costs are likely to prevent individual investors
from holding large numbers of individual stocks in their portfolios. More than half of
U.S. households have accounts with brokerage firms. Because of limited resources or
their desires to exploit the unique characteristics of individual stocks, these investors
normally only hold a handful of stocks.2 Also, in order to provide financial incentives
for their employees that are not charged against corporate income, more and more
companies now grant stock options to their employees. In general, holders of these
options are not allowed to liquidate their positions or to hedge the stocks of their
own firms, hence they tend to hold very unbalanced portfolios.3 Moreover, some stock
traders and arbitrageurs in markets take large bets on individual stocks. Finally, there
are several thousand actively managed mutual funds in existence. While these funds
are able to hold a market portfolio, they typically do not do so. Moreover, Day, Wang,
and Xu (2000) have demonstrated that the portfolios of equity mutual funds are not
even mean-variance efficient with respect to their holdings. These “active” portfolio
2
One may argue that the idiosyncratic volatility of a portfolio is close to zero when there are more
than 20 stocks. However, this conclusion is based on a random sampling. In reality, investors do not
randomly select their stocks. In addition, Campbell, Lettau, Malkiel, and Xu (2000) have shown that
a well-diversified portfolio must have at least 40 stocks in recent decades due to an increasing trend in
idiosyncratic volatility.
3
This practice is particularly prevalent in high technology industries.
2
managers are able to obtain large management fees because they claim to be able to
find “undervalued” securities and hence offer investors the possibility of risk-adjusted
returns superior to the market averages. While there is no evidence that they can
achieve this goal even before expenses (see Jensen (1968) and Malkiel (1995)), they do
affect the relative supply of stocks available for other investors. Equity mutual funds
hold portfolios comprising almost one-third of the total capitalization of the U.S. stock
market, and thus they have the potential to alter the supply of securities available to
other investors in an important way. The fact that investors are willing to pay the
high costs to invest in non-indexed mutual funds indicates that they do not choose to
allocate their portfolios between a market portfolio and a risk-free asset as the CAPM
theory assumes.
There is no doubt, therefore, that not every investor is willing or able to hold
the market portfolio. To what extent this distortion will affect the CAPM is purely
an empirical question. Our approach is not to conduct a direct investigation of the
portfolio holdings of investors. Instead, we will take as given that investors are unable
to hold the market portfolio. Starting from there, we investigate the consequences.
The role of idiosyncratic risk in asset pricing has been studied in the literature to
some extent (see for example, Fama and McBeth (1972)). Most discussions have been
focused on the effect of idiosyncratic (or uninsurable) income risk on asset pricing (see
for example, Heaton and Lucas (1996), Thaler (1994), Aiyagari (1994), Lucas (1994),
Telmer (1993), Franke, Stapleton, and Subrahmanyam (1992), and Kahn (1990)). In
addition, some indirect evidence exists on the role of idiosyncratic risk. Falkenstein
(1996) found some evidence that the equity holdings of mutual fund managers appeared
to be related to idiosyncratic volatility. In studying the volatility linkage between
national stock markets, King, Sentana, and Wadhwani (1994) provided evidence that
idiosyncratic economic shocks are priced and that the ‘the price of risk’ is different
across stock markets. Lehmann (1990) studied the significance of residual risk in the
context of statistical testing methodology. Perhaps the most relevant study to this
paper is Levy (1978). Based on assumptions similar to ours, he derived a modified
CAPM that revealed possible bias in the beta estimator as well as a possible role for
idiosyncratic risk. In contrast, our model demonstrates the explicit role of idiosyncratic
risk in asset pricing. Furthermore, we show that the beta estimator will be unbiased if
idiosyncratic risk is appropriately account for.
3
In this paper, we provide a theory of idiosyncratic risk and test some of the implications of our model with both constructed portfolio returns and equity mutual
fund returns. The empirical results not only support our model, but also show that
(1) idiosyncratic volatility is important in explaining cross-sectional expected returns;
(2) returns from more than 80% of the constructed portfolios from the universe of
NYSE/AMEX/NASDAQ stocks and two thirds of equity mutual funds co-vary with
aggregate idiosyncratic volatility; (3) the intercept and the beta estimated from the
CAPM model are highly negatively correlated, which reduces the overall explanatory
power of beta in cross-sectional studies.
The paper is organized as follows: A simple CAPM type of model with some constrained investors is constructed in the first section. After studying the implications of
the model, we discuss issues related to empirical testing and data construction in section
2. Section 3 presents cross-sectional evidence in the spirit of Fama and French. Times
series evidence concerning the role of idiosyncratic volatility is presented in Section 4.
Section 5 presents concluding comments.
4
1
The basic model and its implications
The Capital Asset Pricing Model is an equilibrium model in which the demand for
equity securities is determined under a mean-variance optimization framework. The
market clearing condition then equates demand and the exogenous supply to achieve
equilibrium. Since it is assumed that investors are homogenous and are able to hold
every asset in the market portfolio, their holdings will be similar in equilibrium. As a
result, investors’ holdings of risky stocks will be comprised of shares held in proportion
to the market portfolio, which is a value-weighted portfolio of all the stocks available
for investment. In other words, the market portfolio is always feasible and will be the
only portfolio held in equilibrium. Such an available market portfolio will be altered,
however, when a group of investors does not or cannot hold every stock for any of the
reasons described in the previous section.
1.1
Asset returns in a traditional CAPM world
For illustrative purposes, we assume that there are three risky stocks4 denoted a, b,
and c that generate a return vector R = [Ra , Rb , Rc ] and one riskless bond that pays
interest r. The risk structure for the three stocks is represented by their varianceσa2 σab σbc


covariance matrix of returns, V =  σab σb2 σca . Each investor has the following
σca σbc σc2
utility function,
u(W ) = E(W ) −
1
V ar(W ),
2τ
(1)
where W represents future wealth and τ is the coefficient of risk tolerance. This particular utility function is consistent with the family of exponential utility functions when
future wealth has a normal distribution. If we denote Xj = [xa,j , xb,j , xc,j ] as investor
j’s dollar amount invested in the three stocks, the corresponding budget constraint can
be written as,
Wj = W0,j (1 + r) + Xj (R − r1),
4
(2)
None of the stocks are necessarily on the traditional mean-variance efficient frontier. We chose
three stocks for ease of exposition. Since we use vector notations, the final result will not depend on
the number of stocks assumed.
5
where W0,j is the initial endowment. Utility maximization of equation (1) subject to
the constraint represented by equation (2) leads to the following demand function,
Xj = τ V−1 (µ − r1),
(3)
where µ = E(R) is the vector of expected returns of individual stocks. Note that demand is independent of investors’ initial wealth because we do not constrain borrowing
at the rate r on riskless bond. Although the variances and covariances among stock
returns are fixed exogenously in this framework, the expected returns should be determined in equilibrium by total supply. In other words, the equity market clears with the
condition of
j
Xj = S, where S = [Sa , Sb , Sc ] is the total supply of individual stocks.
The equilibrium expected returns in this unrestricted world can thus be written as,
µ − r1 =
1
VS,
nτ
(4)
where n is the total number of investors. Following the convention, we define the market
portfolio as α =
1
S,
M
where M = S1 = Sa + Sb + Sc . Under this notation, the expected
2
market return and market volatility can be expressed as µm = α µ and σm
= α Vα.
Equation (4) can thus be converted into the traditional CAPM,
µ − r1 = β(µm − r),
where β = [βa , βb , βc ] =
1
2 Vα
σm
(5)
is the conventional measure of systematic risk. Sub-
stituting equation (5) back into equation (3), it is clear that, in an economy where
investors are homogenous and unconstrained, only a linear combination of the risk-free
bond and the (efficient) market portfolio m generated from all the individual stocks will
be held. What makes this single factor model a truly equilibrium model is the existence
of an equilibrium market portfolio, which is determined by the aggregate supply. Since
the variance-covariance structure and the supply of stocks are common knowledge, this
market portfolio can be constructed by an econometrician even when there are limited
investment opportunities.5
Equation (5) says that only systematic risk, represented by the scaled covariance
between individual stock returns and the market return, matters for valuation. Idiosyncratic risk can be diversified away in this framework, and will not command a risk
5
For illustrative purposes, if we assume that only the NYSE/AMEX/NASDAQ listed stocks form
the entire universe of investment assets, the conventional market index portfolio, such as the value
weighted NYSE/AMEX/NASDAQ index, is indeed the market portfolio and is observable to everyone.
Since we will study the effect of limited investment opportunities under such a scenario where we know
the market portfolio, Roll’s (1977) critique is inapplicable in this context.
6
premium. Based on equation (5), we can express individual stock returns as,
Ri,t − rf,t = βi (Rm,t − rf,t ) + i,t ,
(6)
where i,t is the idiosyncratic return. In the discussion that follows, we will use idiosyncratic volatility to measure idiosyncratic risk. In light of equation (6), idiosyncratic
volatility V can simply be calculated as,
2 β.
V = V − βσm
1.2
(7)
Asset returns under an imperfect market portfolio
The CAPM must prevail when all individuals are unconstrained in holding underlying
assets, but this assumption is frequently violated in practice. When some investors
can not or do not wish to hold every security for the reasons discussed in the previous
section, the CAPM will fail to hold. For ease of exposition, let us assume that there
are three groups of investors. While the “free” investors in the second group have full
investment opportunities and can hold all securities, the first and the third groups of
investors are assumed to be constrained from holding the first and the third stocks,
respectively. Following the same steps, we can derive demand equations similar to
equation (3) for representative investors in each group as,
X(1) = τ
0 0
0 Σ−1
bc
(µ − r1),
X(2) = τ V−1 (µ − r1),
X(3) = τ
where
Σab =
σa2 σab
σab σb2
Σ−1
0
ab
0 0
(µ − r1),
, and Σbc =
σb2 σbc
σbc σc2
.
Supposing there are n1 , n2 , and n3 number of investors in the first, second, and third
groups, respectively, the market clearing condition leads to the following,
S = n1 X(1) + n2 X(2) + n3 X(3) = nτ [η1.3 (V∗ )−1 + η2 V−1 ](µc − r1),
(8)
of expected
equilibrium
where µc is the vector
stock returns in this constrained world,
−1
0 0
Σab 0 −1
1
3
and V∗ = ( n1n+n
+ n1n+n
) is the aggregate variance-covariance
3
3
0 Σ−1
0 0
bc
7
matrix perceived by constrained investors. η1.3 = (n1 + n3 )/n is the proportion of
constrained investors and η2 = n2 /n = 1 − η1.3 is the proportion of “free” investors.
The expected equilibrium return vector µc is then determined by the following equation,
µc − r1 =
1
[η1.3 (V∗ )−1 + η2 V−1]−1 S.
nτ
(9)
Equation (9) says that in a constrained market, investors apply an altered variancecovariance matrix [η1.3 (V∗ )−1 + η2 V−1 ]−1 to price stocks instead of using the true
variance-covariance matrix V of stock returns. In other words, a CAPM would prevail
if the altered variance-covariance matrix represented the true risk structure. However,
since the risk structure is given, we should rewrite equation (9) alternatively as,
µc − r1 =
1
1
V[η1.3 (V∗ )−1 V + η2 I]−1 S =
VS∗ .
nτ
nτ
(10)
where S∗ is the effective supply. Therefore, equation (10) can be interpreted as if
investors are subject to an altered market portfolio6 α∗ (=
S∗
).
S∗ 1
In other words, a
CAPM type of relationship continues to hold with regard to the altered market portfolio
in equilibrium. But the CAPM relationship will not hold with respect to the actual
market portfolio.
Equation (10) can not be tested directly, however, since we – the econometricians
– do not know the distribution of investors among different groups. In other words,
it is impossible for us to construct such an altered market portfolio. When investors
determine the returns with respect to the altered market portfolio available to them,
the econometricians tend to observe an imperfect CAPM. The econometricians can only
use the market return Rm † derived from the true observable market portfolio weights
α constructed from all of the outstanding shares of stocks, that is Rm † = α R. The
net effect is that we will perceive that idiosyncratic risk is a priced factor with respect
to the actual observed market portfolio.
In order to illustrate the point, we rewrite equation (9) in the following way;
1
[V−1 − η1.3 (V−1 − V∗−1)]−1 S
nτ
η1.3
1
VS +
V[(I − V∗−1V)−1 − η1.3 I]−1 S
=
nτ
nτ
η1.3
1
VS +
Vω,
=
nτ
nτ
µc − r1 =
6
(11)
In the case of two risky stocks, it is the actual market portfolio less the holdings of the constrained
investors.
8
where ω is the supply adjustment. Equation (11) reveals that the equilibrium expected
returns will adjust both to the actual total supply, which is prescribed by the traditional
CAPM, and to the supply adjustment from constrained investors. When the number of
constrained investors is relatively small, i.e., η1.3 ≈ 0, the supply adjustment is trivial.
Expected returns will not deviate much from those predicted by a traditional CAPM
(equation (4)). However, if the aggregate demand from the constrained investors is
large, as we suggest it is, substantial adjustments will be required. Next, we multiply
both sides of equation (11) by the true market portfolio weights α, that is,
M M
M 2
M
2 β ω∗ ,
α Vα +
η1.3 α Vω∗ =
σm +
η1.3 σm
nτ
nτ
nτ
nτ
µm † − r = α µc − r =
1
ω
M
where ω ∗ =
(12)
is the relative supply adjustment. Substituting equation (12) back
into equation (11) and applying equation (7), we have the following result,
2
(µm † − r)/σm
µm † − r
+
η1.3 Vω∗
1 + η1.3 ω ∗ β
1 + η1.3 ω ∗ β
2
(µm † − r)/σm
2 = β(µm † − r) +
β ω∗]
η1.3 [Vω ∗ − βσm
1 + η1.3 ω ∗ β
= β(µm † − r) + κδSR V ω ∗ ,
µc − r1 = β
where κ =
η1.3
1+ω ∗ β
and δSR =
µm † −r
2
σm
(13)
is the market Sharpe Ratio. If we define the
undiversified market wide idiosyncratic return with respect to equation (7) as Im = ω ∗ ,
equation (13) can be rewritten as,
µc,i − r = βi (µm † − r) + βI,i µ ,
where βI,i =
Cov(Ri ,m )
V ar(Im )
(14)
is the sensitivity coefficient of the market wide undiversified
idiosyncratic risk factor in the spirit of an APT model, and µ = κV ar(Im )δSR is
the market wide undiversified idiosyncratic risk premium that arises because of the
constrained investors. Equation (14) says that, if investors can not hold a market
portfolio, the expected return for each individual stock will be related not only to the
observed market expected return through the conventional beta measure, but will also
depend on an extra risk premium which is due to the assumption of some undiversified
idiosyncratic risk forced by the constraints imposed. Of course, the portfolio ω ∗ is
unobservable to an econometrician, but we are able to construct an idiosyncratic risk
hedging portfolio in the spirit of Fama and French (1993) to approximate portfolio ω ∗
in our empirical work.
9
Our model also offers cross-sectional implications that can be directly tested. If
the idiosyncratic returns for individual stocks have very low pairwise correlations, i.e.
Cov(i , j ) ≈ 0, we can further simplify equation (13) as,
2
µci − r ≈ βi (µm † − r) + κδSR w∗,i σI,i
,
(15)
2
is the conventional measure of ith stock’s idiosyncratic volatility. The appearwhere σI,i
ance of the Sharpe Ratio, δSR , in addition to the idiosyncratic volatility makes perfect
sense in this context. Essentially it translates the idiosyncratic risk into a comparable
risk premium. Equation (15) is useful in understanding the cross-sectional implications
of the pricing of idiosyncratic risk. It suggests that the differences among individual
stocks’ expected returns will be related not only to their firms’ systematic volatilities
(β), but also to the firm’s idiosyncratic volatility. In other words, firms that are subject to large idiosyncratic shocks will tend to have high expected returns. This is the
theoretical foundation for the cross-sectional empirical study of section 3.
10
2
The data and idiosyncratic risk proxies
Two data sets are employed in this study. The first data set is from the 1998 version of
the CRSP (Center for Research in Security Prices) tape, which includes NYSE, AMEX,
and NASDAQ stock returns. Our study covers the post war period from 1952 to 1998
and thus extends the time period of the Fama and French (1992) study. We base our
cross-sectional tests on both individual stocks as in Fama and French (1992) and time
series tests on portfolios.7 Since there are so many AMEX and NASDAQ stocks that
have very low market capitalizations, portfolio breakdowns are determined by using
NYSE stocks only to avoid the small size portfolios from being too small. In June of
each year, all NYSE stocks on CRSP are sorted according to their size. The ten NYSE
size deciles are then used to split the whole sample. At the same time, the beta of
each stock is estimated using the previous 24 to 60 months of sample returns. Within
each size group for NYSE stocks only, stocks are sorted again by their betas into ten
equal number groups. Similarly, the break points thus obtained are used to sort all the
stocks in our sample. All portfolios are rebalanced at the middle of each year. The
100 portfolios thus constructed are very close to those used in Fama and French (1992).
Financial stocks are also excluded in our sample so that our results are comparable with
those of Fama and French (1992). In general, however, our results are stronger when
financial stocks are included. Finally, we use NYSE/AMEX/NASDAQ index returns
as the market returns and treasury-bill rates from Ibbotson Associates (1999) as the
risk-free rates.
Insert Table 1
Although our sample period and the stocks in our sample are different from those of
Fama and French (1992), the essential characteristics are close, as shown in Table 1. In
Panel A of Table 1, we report the average monthly returns for each of the 100 portfolios.
When compared with those of Fama and French (1992), they are close, except that the
return differences between the smallest size decile and the second smallest size decile
seems to have widened. This could be attributed to the fact that there was a dramatic
increase in the number of NASDAQ stocks in the 90’s. At the same time the variations
in the post-ranking betas among different portfolios stay almost the same as shown in
7
In both cases, we use post-ranking portfolio betas described in Fama and French (1992) in order
to reduce “errors in variables” problems.
11
Panel C. They range from 0.59 to 1.73. In general, they vary more within each size
decile than across size deciles. As expected, the average log market capitalization of the
100 portfolio has gone up from 4.11 through 1990 (see Fama and French (1992)) to 5.54
through 1998 (not shown in the table). As revealed in Panel B, the average log market
capitalization for our whole sample period is much larger (about 5.15). At the same
time, the increases are uniform across portfolios. Moreover, portfolio size is much more
uniformly distributed for beta portfolios within each size decile than in the original
Fama and French study. Therefore, our portfolios are representative and capture the
same degree of variations as did Fama and French.
The construction of the 100 portfolios may appear arbitrary as has been argued
by Ferson, Sarkissian, and Simin (1998). For a check of robustness, we have also
used a second data set from actual equity mutual fund returns. In particular, we
have used equity mutual fund returns provided by Lipper Analytical Services for the
sample period from 1971 to 1991. Since these funds are well diversified at different
risk levels, their portfolios contain a larger pool of stocks than is the case for our
constructed portfolios. Furthermore, since the mutual funds in our sample were actively
managed, their composition does not suffer from any potential selection problems that
the constructed 100 portfolios may have. We do realize, however, that different mutual
funds are likely to have some common holdings during certain, perhaps most, periods
of time. Therefore, we view that the empirical results based on both data sets are
complementary.
Since volatilities, especially idiosyncratic volatilities, are unobservable, we need estimates in order to perform empirical tests of equation (15). Presumably these estimates
can be obtained from the residuals of an asset pricing model. Empirically, however, it
is very difficult to interpret the residuals from the CAPM or even a multi-factor model
as solely reflecting idiosyncratic risk. One can always argue that these residuals simply
represent omitted factors. Therefore, we can only assert that the residuals from a market model as measuring idiosyncratic returns in the context of the model considered. In
other words, it would be legitimate to measure idiosyncratic volatility using the mean
square of residuals as suggested in the model discussion of the previous section. In fact,
this is the approach used in Fama and MacBeth (1973). We also realize that the current
literature has leaned toward a three-factor model of Fama and French (1993) such as
12
shown in equation (16) below,
Ri,t − Rf,t = βm,i (Rm,t − Rf,t ) + βsmb,i Rsmb,t + βhml,i Rhml,t + i,t ,
(16)
where Rm,t − Rf,t is the excess return on the market return factor, with Rsmb and Rhml
respectively representing the returns on portfolios formed to capture the size effect and
the book-to-market equity effect.8 Therefore, we actually use the residuals from this
three-factor model to estimate idiosyncratic volatility. In particular, at any month, we
fit the three-factor model to individual stocks using the current and previous five years
of monthly returns to compute idiosyncratic volatility.
Panel D of Table 1 summarizes the average monthly idiosyncratic volatility. Average
idiosyncratic volatilities range from 4.8% to 17.0% with an overall average of 8.29%.
Compared with the average size in Panel B, we see that idiosyncratic volatility and
size are highly correlated, as was shown in Malkiel and Xu (1997). In particular, both
statistics vary more across small size portfolios than large size portfolios. However,
different from the size measure, which is very stable within each size decile, idiosyncratic
volatility increases with beta for portfolios within each decile. In other words, stocks
with high betas tend to have high idiosyncratic volatility.
In order to perform time series tests on equation (14), we need a proxy for the market
wide undiversified idiosyncratic risk factor. We have constructed two proxies for robustness. The first proxy RI,t , is the idiosyncratic risk hedging portfolio, which follows the
logic of Fama and French (1993) by constructing six size-idiosyncratic volatility sorted
portfolios. We split the sample into two size groups. Within each size group, stocks
are sorted again by their idiosyncratic volatilities into three groups of equal numbers of
securities.9 The idiosyncratic volatility measure for each stock is estimated using the
mean squared residuals from the CAPM model over the previous 24 to 60 month period
in order to be consistent with our model. We denote B as the big size, S as the small
size, H as the high idiosyncratic volatility, M as the median idiosyncratic volatility, and
L as the low idiosyncratic volatility. The six portfolios can be characterized as the B/L
portfolio, the B/M portfolio, the B/H portfolio, the S/L portfolio, the S/M portfolio,
and the S/H portfolio. The return proxy for the market wide idiosyncratic risk factor
is then calculated as the difference between average returns for portfolios B/L and S/L
and average returns for portfolios B/H and S/H. The second proxy for idiosyncratic risk
8
9
We are grateful to Eugene Fama for making this data available to us.
The actual breakdowns are based on NYSE stocks only for the same reason discussed previously.
13
factor RI,SR,t is constructed directly according to equation (14). At any month, we first
compute the Sharpe Ratio for each of the 100 portfolios by dividing the excess return
by the portfolio total volatility. We then value weight these Sharpe Ratios to arrive at
the aggregate Sharpe Ratio. At the same time, we compute the aggregate idiosyncratic
volatility by value weighting each portfolio’s total idiosyncratic volatility, which is obtained by equal weighting each individual stocks’ idiosyncratic volatility obtained from
the three factor model. Finally, RI,SR,t is taken as the product of aggregate Sharpe
Ratio and the aggregate idiosyncratic volatility.
14
3
The cross-sectional evidence
Ever since doubts were raised about the CAPM model by Fama and French (1992),
considerable attention has been devoted to risk measurement. For example, Jagannathan and Wang (1996) have argued that a conditional CAPM behaves well. Some
have argued that the variables used by Fama and French are not robust (see for example
Tim (1996), and Kothari, Shanken, and Sloan (1995)). Others have suggested that a
mutifactor model in the spirit of the Ross’s (1976) APT provides a better explanation of
returns than a single factor CAPM model.10 As noted by Fama and French (1992) and
others, the most significant factors in “explaining” cross-sectional returns appear to be
the market capitalization (size) and book to market ratios. These are largely empirical
findings rather than equilibrium implications Therefore, it is difficult to understand why
these factors should matter in determining expected returns unless they are proxies for
other (systematic) risk factors. Moreover, combining time series evidence of return predictability and cross-sectional testing in a conditional framework, Ferson and Harvey
(1999) have rejected the three-factor model advocated by Fama and French (1993) as
a conditional asset pricing model. Meanwhile, Malkiel and Xu (1997) have found that
size and idiosyncratic volatility are highly correlated. Therefore, the so-called size effect
may just as well be attributed to idiosyncratic risk. Guided by our theoretical model,
we will study the empirical significance of idiosyncratic risk in addition to other factors
from both a cross-sectional and time series perspectives.
As suggested by our model (15), idiosyncratic volatilities for individual securities
and their expected returns will be related. In other words, we need cross-sectional evidence to conclude that return differences among securities can be partially explained
by differences in their idiosyncratic volatilities. In order to provide an overview of the
relationship, we first plot the average monthly returns versus the average idiosyncratic
volatility calculated from the residuals to the three-factor model for the ten decile portfolios in Figure 1. Clearly there is a positive association between idiosyncratic volatility
and average returns. The significance of such a relationship is further demonstrated in
the following cross- sectional tests.
Insert Figure 1
10
This does not mean that the market factor is unimportant, only that other factors are important
as well.
15
3.1
A discussion of cross-sectional models
To put our results in perspective, we will use a framework similar to that of Fama
and French (1992) to show that idiosyncratic volatility has a useful role in explaining
differences in security returns. However, we interpret our results as indicating to what
degree the inability of holding the whole market portfolio will affect asset returns. In
particular, we estimate the following equations cross-sectionally for each month:
I:
II :
Ri,t = γ0,t + γβ,t βi + i,t ,
Ri,t = γ0,t + γM E,tlog(MEi,t ) + i,t ,
III :
Ri,t = γ0,t + γIV,t vi,I,t + i,t ,
IV :
Ri,t = γ0,t + γβ,t βi + γM E,t log(MEi,t ) + i,t ,
V :
VI :
V II :
Ri,t = γ0,t + γβ,t βi + γIV,t vi,I,t + i,t ,
˜ i,t ) + γIV,tvi,I,t + i,t
Ri,t = γ0,t + γme,t log(ME
˜ i,t ) + γIV,t vi,I,t + i,t ,
Ri,t = γ0,t + γβ,t βi + γme,t log(ME
˜ i,t ) is the log size variable, and vi,I is the idiosyncratic volatility for
where log(ME
stock i, calculated as the mean square error from the three-factor model for the reasons
discussed in the previous section. Note that βi is the portfolio beta assigned to each
individual stock within the size-beta portfolio. It is estimated in the same fashion as
in Fama and French (1992). The first three models are intended to investigate the
significance of individual factors including beta, size and the idiosyncratic volatility
measure. For comparison, Fama and French’s model (Model IV) is also included. As
predicted by equation (15), in Model V we include idiosyncratic volatility in addition
to the beta factor. Model VII examines the additional explanatory power provided by
idiosyncratic volatility relative to the Fama and French model. Since in most cases, the
beta factor is insignificant, Model VI studies the importance of size and idiosyncratic
volatility alone.
It is useful to assess the overall explanatory power of a variable utilizing the coefficient of determinant. However, this is very challenging in cross-sectional studies (see
Kandel and Stambaugh (1995)). As suggested by Jagannathan and Wang (1996), we
compute R2 s for cross-sectional regression of the average returns of each of the corresponding 100 portfolios on the time average of the portfolio variables, such as size and
idiosyncratic volatility.11 Although our cross-sectional study is performed on individual
11
If we denote R̄i and ēi as the time series average of the returns and residuals for portfolio i, we
16
stocks as is in Fama and French (1992), this R2 measure from portfolio returns serves
as a proxy.
3.2
The empirical results
For comparison, we report results for both the Fama and French (1992) period (1963.71990.12) in Panel A, and our whole sample period (1952.7-1998.12) in Panel B of Table
2. In particular, the time series averages of each model’s cross-sectional estimates are reported. For example, we calculate the time series average γ̄x =
1
T
T
γ̂x,t for the cross√
sectional estimates γ̂x,t . The corresponding t ratio is defined as tγ = T γ̄x /std(γ̂x,t).
t=1
Insert Table 2
The first equation in Panel A of Table 2 shows the statistically insignificant estimate
for the beta variable when used alone, which is comparable to Fama and French’s (1992)
results. Furthermore, we see that the Rp2 is rather small (.2% for the Fama and French
sample and .5% for the whole sample). This provides supplemental evidence that beta
does not appear to be helpful in explaining the cross-sectional variation of average stock
returns. However, as noted by Fama and French (1992), the size variable does appear
to explain the cross sectional variability of returns as shown in the second equation
with an estimate of −.156 and a t−value of −2.79. This is very close to Fama and
French’s (1992) estimates for the same sample period. Over our whole sample period,
the estimates are even stronger with a t−value of −3.70. Moreover, Rp2 s are about
44% for both sample periods. When both beta and size are included, the multiple
regression of equation IV in Table 2 shows only size is significant and no explanatory
power remains for beta. Note that the beta variable even has a negative sign. The
estimates are very similar to those of Fama and French (1992), and are robust with
respect to both sample periods. However, the Rp2 increases to 59% for the Fama and
French sample period and 52% for the whole sample period. This result suggests that
beta may have additional explanatory power when used with other variables.12 We will
discuss the issue as to why beta has a negative sign later.
When the idiosyncratic volatility variable vi,I computed from the residuals of a
three factor model is used alone, equation III shows a very significant and positive
can define Rp2 = (V ar[R̄i ] − V ar[ēi ])/V ar[R̄i ].
12
The small t value could be attributed to high correlation between size and beta variables.
17
relationship. Similar to size variable, the Rp2 measure goes up from 0.5% when beta
alone is used to 27% for the whole sample period (and from 0.2% to 15% for the Fama
and French sample period), which represents a significant improvement. This suggests
that idiosyncratic volatility helps to explain the cross-sectional variability in average
returns in an important way. As predicted by equation (15), high idiosyncratic risk,
measured by idiosyncratic volatility is associated with high returns on average. It is also
surprising to see that in a multiple regression of equation V, not only the idiosyncratic
risk variable but also the beta variable are statistically significant with an Rp2 measure
as high as 46% for the whole sample period (and 31% for the Fama and French sample
period). However, the sign on beta variable is “wrong” as was the case in the regressions
using beta and size.
As demonstrated by Malkiel and Xu (1997), portfolio size is strongly related to idiosyncratic volatility. Portfolios of smaller sized stocks tend to have larger idiosyncratic
risk than portfolios of larger stocks. Thus, the size effect in Fama and French and
our idiosyncratic volatility effect may be substitutes. The important question is which
variable has the strongest relationship to portfolio returns. In equations VII of Panel A
and Panel B of Table 2, we have included both the size and the idiosyncratic volatility
variables. For both sample periods, the size variable in the regressions is insignificant
when the idiosyncratic volatility variable is used. It seems that the idiosyncratic volatility variable has taken away the explanatory power of size variable. Furthermore, when
the beta variable is also included in regressions shown in the last two equations of Table
2, the size variable is still insignificant. However, the Rp2 is as high as 56% for the
whole sample period and 59% for the Fama and French’s sample period. Idiosyncratic
volatility appears to be a more useful predictor of returns than size.
Since the Fama and French sample period is included in our whole sample period,
a more appropriate way to investigate the robustness of our results is by dividing the
whole sample period into two equal length sub-sample periods. In particular, the first
sub-sample period runs from July 1952 to June 1975, and the second sub-sample period
covers July 1975 to December 1998. The results are reported in Panel C and Panel D
of Table 2. The overall results for the second sub-sample period (see Panel D of Table
2) are in line with that of the whole sample period. The general conclusion remains
the same with comparable estimates and larger t-value for the idiosyncratic volatility
variable but smaller t-value for the size variable. However, the results are somewhat
different for the first sub-sample period (see Panel C of Table 2), where only half of the
18
sub-sample period overlaps with that of Fama and French. The coefficient estimate on
the beta variable is larger than that of the Fama and French sample period but remains
insignificant. At the same time, idiosyncratic volatility continues to be statistically
significant. It is also important to note that size variable remains insignificant once
the idiosyncratic volatility variable is included. Therefore, it is reasonable to conclude
that idiosyncratic risk factor is more robust than the size variable in explaining the
cross-sectional difference of asset returns over the different sample periods considered
here. The size variable per se has a somewhat stronger explanatory power in the later
period than was the case in the earlier period.
A similar testing methodology can also applied to our quarterly equity mutual fund
sample. Due to the availability of data, we will only examine models I, III, and V.
We construct the same idiosyncratic volatility measure vi,I , i.e., the mean square error
from Fama and French three factor model. Note that the quarterly return proxies for
SMB and HML are simply the three month averages. We require a fund to have
at least 20 quarterly returns to be considered in our sample. There are also several
huge outliers in our equity mutual fund sample. By applying the criterion of requiring
positive cumulative excess returns, we have reduced our sample to 476 funds.13 Table
3 reports the empirical results. Once again beta does not seem to explain the crosssectional return differences of mutual funds. In contrast, the idiosyncratic volatility
measure alone is almost significant at a 5% level. When both beta and idiosyncratic
volatility measures are included, idiosyncratic volatility is significant at a 5% level while
beta remains insignificant. These results further support the implications of our model.
Insert Table 3
While there are some (usually short) periods in which beta measures do appear to
play some role in explaining the cross- sectional pattern of returns in accordance with
the CAPM, the general conclusion is that size and especially idiosyncratic volatility are
the only variables that show a consistently strong relationship with returns. Of course, it
is possible to interpret our measures of idiosyncratic volatility as simply approximating
for size as well as omitted systematic risk factor(s). However, at the very least, our
model seems to offer a consistent and empirically supported explanation for some of
13
We do not have information as to why these 18 funds with negative cumulative excess returns
have survived over the twenty-year period. However, with these outliers the average residual kurtosis
is twice as large as that of the normal residuals.
19
the deficiencies of popular asset pricing models. In any case, what is clear is that the
beta of the capital asset pricing model appears to be a very imperfect measure of the
risk that is related to securities returns.
3.3
A further look at the size and idiosyncratic Volatility Variables
As popularized by Fama and French (1992, 1993), size variables are considered to be
important in empirical studies of security returns. However, as shown above, the power
of size variables in explaining cross-sectional returns differences weakens once some
measure of idiosyncratic risk is also included. Despite the empirical evidence, we cannot
definitely say that size approximates idiosyncratic volatility, or vice versa, since they
are highly correlated with each other.
In order to reach a conclusion on this issue, we have designed the following test.
If indeed a size variable approximates idiosyncratic volatility, we should observe that,
within idiosyncratic volatility sextiles,14 a size variable is unimportant in explaining
the cross-sectional returns. At the same time we should also find that within size
sextiles idiosyncratic volatility still explains the cross-sectional returns. The actual
implementation of this idea involves forming six groups of portfolios. In “Scheme I,” we
sort all stocks on the monthly CRSP tape into six groups in June of each year according
to the size breakdowns that use NYSE stocks only as described above.15 Stocks within
each group are then sorted into 40 portfolios according to their betas. There are between
4 to 64 stocks in each of the 240 size-beta portfolios. “Scheme II” is similar to “Scheme
I” except we use the idiosyncratic volatility in the first sorting instead.16
In Table 4, we report the average cross-sectional results for Model II (size only),
Model III (idiosyncratic volatility only), and Model VI (both size and idiosyncratic
volatility) both across and within each of the six groups under different sorting schemes.
The results seem to confirm our belief that the size variable approximates the idiosyn14
There is no particular reason for choosing six groups except that Fama and French (1993) used a
similar grouping. Results are very similar if four groups are used.
15
Stocks in the first group have the smallest market capitalization, and stocks in the sixth group
have the largest market capitalization.
16
The idiosyncratic volatilities are computed from the CAPM residuals using the previous two to
five years of monthly returns. Stocks in the first group have the smallest idiosyncratic volatility, and
stocks in the sixth group have the largest idiosyncratic volatility.
20
cratic volatility variable. In particular, within each of the six idiosyncratic volatility
sextiles, i.e. “Sorting Scheme II,” the size variable is insignificant within the first two
groups of portfolios, but significant at a 5% level for the third group, and very significant for the last three groups. At the same time, the idiosyncratic volatility is still very
significant at an almost 0% level within all the six groups. The risk premium in terms
of per unit change in the log size increases from the group with the least idiosyncratic
volatility (e.g. .016%) to the group with the most idiosyncratic volatility (e.g. .345%).
In contrast, the risk premium on idiosyncratic volatility seems to be more evenly distributed across each group except for the group with largest idiosyncratic volatility.
When we include both size and idiosyncratic volatility variables, only the size variable
within the last group is significant, while the idiosyncratic volatility variable continues
to be strongly significant.
Insert Table 4
We are only able to conclude at this point that the size factor is generally insignificant
after controlling for the idiosyncratic volatility factor. Therefore, we perform similar
tests the other way around. In other words, we can first sort stocks according to
size, i.e. under “Sorting Scheme I.” From Table 4, we see that even though the size
variable across groups is significant, as it should be from our previous results, it does
not explain return difference within each size group except for the smallest size group.
This is consistent with Loughran’s (1996) finding. However, the idiosyncratic volatility
variable behaves very similarly. Except for the first group, we do not find idiosyncratic
volatility variable to be significant within the size groups. In general, we can say that
the idiosyncratic volatility variable does not appear to approximate the size variable.
Otherwise we would expect that the size variable would be more significant than the
idiosyncratic volatility variable within each idiosyncratic volatility sextile.
3.4
A possible explanation for the poor performance of the
beta variable
It seems odd that a market model fits portfolio returns very well over time with R2 s
ranging from 60% to 80%. Yet, beta alone does not have any cross-sectional explanatory
power. At the same time, empirical results from previous sections show that it can
21
increase overall explanatory power when used with other variables, such as size or
idiosyncratic volatility. In this section, we suggest a simple explanation.
Suppose a multifactor model that includes the market factor holds from an empirical
perspective. Then expected return for any stock i can be written as,
E(Ri − Rf ) = αi + βi E(Rm − Rf ),
(17)
where E(Rm −Rf ) is the market risk premium, and αi is the expected return from other
factors. Therefore, if the time series counterpart of equation (17) fits well, βi should
also be significant in the cross sectional regression unless αi and βi are highly negatively
correlated. To see why, suppose,
αi = a − bβi + ui,
(18)
where βi and ui are orthogonal. Equation (17) can then be rewritten as,
E(Ri − Rf ) = a + βi [E(Rm − Rf ) − b] + ui.
(19)
Obviously, when E(Rm − Rf ) ≈ b, the relationship between E(Ri ) and βi tends to be
flat. This might explain the empirical findings that β is unable to explain the crosssectional return differences. An equally important question is why variables like size
and idiosyncratic volatility, which are also highly correlated with beta variable, are
significant in cross sectional regression. Moreover, why does the explanatory power
of these variables increase when used in conjunction with the beta variable in crosssectional regressions? We suggest that if variables like idiosyncratic volatility are highly
correlated with ui in equation (19), they will be important in cross-sectional regression.
At the same time, since [E(Rm − Rf ) − b] is not exactly equal to zero, adding βi in
cross- sectional regression reduces noise and improves the performance.
Insert Table 5
In order to test our conjecture, we study returns for the 100 size-beta sorted portfolios constructed in Section 2. In other words, cross-sectional regressions are performed
on portfolios instead of individual stocks. In Panel A of Table 5, we report the crosssectional regressions of α on β, M̄ E, and v̄I separately and jointly. α and β are estimated
using the CAPM for each portfolio,17 while M̄ E and v̄I are the time series average of
17
In order to be comparable with Fama and French (1992), β is the sum of the coefficients on the
current and previous period market excess returns.
22
the aggregate portfolio size and idiosyncratic volatility estimated from a three factor
model, respectively. The first regression equation shows that α and β are highly correlated with an R2 of 45% for the whole sample period (22% for the first sub-sample
period and 54% for the second sub-sample period). This evidence supports our conjecture stated in equation (18). More importantly, the estimated coefficient b (= −.590%)
is very close to the average excess market return (R̄m − R̄f ) of .635%. According to the
above analysis, this explains why the beta variable is insignificant in the cross-sectional
regression of average return on beta as shown in the first equation of Panel B of the
same table. Therefore, our explanation for a flat relationship between expected return
and beta appears to be plausible.
Meanwhile, although α is virtually uncorrelated with M̄ E and v̄I , as shown in the
second and the third equations of Panel A, no matter which sample period we examine,
they are very useful in explaining the cross-sectional return differences with an R2 of
44% for the size variable and 26% for the idiosyncratic volatility variable (see Panel B)
alone. This is because each of these variables is highly correlated with the beta variable
and is capable of isolating the alphas in expected returns. When β and M̄ E, or β and
v̄I , are used together in explaining cross-sectional differences in α, the R2 jumps to over
70% from the fourth and the fifth equations in Panel A. This can be understood by
examining equation (18). For variable v̄I , the positive correlation with u cancels out the
negative correlation with −bβ, which leaves an overall zero correlation with α. However,
by including an extra variable β in the regression is equivalent to controlling for the
negative part of the correlation. Therefore, the results are much stronger. The same
intuition works for the cross-sectional regressions for returns. In fact, as we expected,
the estimates on M̄E and v̄I , from the fourth to the sixth equations in Panel A, are
exactly the same as those from the corresponding equations in Panel B.
23
4
Time Series Empirical Evidence on Idiosyncratic
Risk
Generally speaking, two approaches have been applied in testing the CAPM. Recent
empirical studies have shifted our attention to testing the cross-sectional return and risk
implications of the CAPM. This is what we have accomplished in the previous section
by showing that idiosyncratic volatility is an independent pricing factor from crosssectional regressions. As our model predicts, however, it is also important to examine
an alternative approach that utilizes the constraint on the intercept of a market model.
In other words, we will study the explanatory power of idiosyncratic volatility from an
ex post perspective.
4.1
The time series empirical models
We expand the CAPM model to include the idiosyncratic risk factor as described by
equation (14). In particular, we will investigate the following empirical models:
I:
II :
III :
IV :
V :
VI :
V II :
V III :
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
R̃i,t − Rf,t
=
=
=
=
=
=
=
=
αi + βi (R̃M,t − Rf,t ) + i,t ,
αi + βi (R̃M,t − Rf,t ) + βsmb,i Rsmb,t + βhml,i Rhml,t + i,t ,
αi + βi (R̃M,t − Rf,t ) + βI,i RI,t + i,t ,
αi + βi (R̃M,t − Rf,t ) + βISR,i RI,SR,t + i,t ,
(20)
αi + βi (R̃M,t − Rf,t ) + βhml,i Rhml,t + βI,iRI,t + i,t ,
αi + βi (R̃M,t − Rf,t ) + βhml,i Rhml,t + βISR,i RI,SR,t + i,t ,
αi + βi (R̃M,t − Rf,t ) + βsmb,i Rsmb,t + βhml,i Rhml,t + βI,iRI,t + i,t ,
αi + βi (R̃M,t − Rf,t ) + βsmb,i Rsmb,t + βhml,i Rhml,t + βISR,i RI,SR,t + i,t ,
where R̃i , Rf,t , and R̃M are the returns for security i, the risk free rate, and the market
portfolio respectively; where RI,t is the return from the idiosyncratic risk hedging portfolio discussed in section 1; where RI,SR,t is the Sharpe Ratio adjusted return proxy for
idiosyncratic volatility in equation (14); where Rsmb,t is the return proxy for the size
factor; and where Rhml,t is the return proxy for the book-to-market factor. Model I
is the benchmark CAPM model. For comparison, we also include Fama and French’s
(1993) three factor model as Model II. Model III and Model IV are directly adopted
from equation (14) with different idiosyncratic risk measures. It is also important to
investigate the explanatory power of idiosyncratic risk by controlling for other known
factors, such as size and book-to-market variables. In Model V and Model V I, we control for the book-to-market factor only. We also control for both factors by including the
24
return proxy Rsmb and Rhml in model V II and model V III with different idiosyncratic
risk proxies. We note that both idiosyncratic risk factor proxies are highly correlated
with the market index returns. In order to investigate the additional contribution of
these proxies, we only use the orthogonal part of the proxies in the actual estimation.
We have included an intercept term αi in each of the above models. It is interpreted
as indicating mispricing in the literature. For example, if the CAPM holds, αi should
be zero in Model I. Similarly, if the other multi-factor models discussed above are true,
the corresponding αs should also be zero or statistically insignificant from zero.
Model I through model V III are estimated for each portfolio. In order to characterize the overall performance, we have used three different types of aggregation methods
to compute the summary statistics.18 In the first aggregation method, we pool the
data across portfolios and estimate each model. Such regressions are adjusted for cross
portfolio heteroscedasticity using an iterative procedure. The second approach simply
averages estimates from regressions performed on each individual portfolio. In the time
series regression for each portfolio, the significance of the estimates are based on NewyWest variance-covariance matrices to account for autocorrelation and heteroscedasticity
over time. In the last approach, we adopt a non-parametric method by counting the
number of portfolios with statistically significant estimates at a 5% significance level
across portfolios. Presumably, if there is no relationship at all between returns and
different independent variables, at a 5% significance level, one will only find about 5%
of the portfolios with significant estimates.
4.2
Results from individual constructed portfolios
Our empirical results in this section are based on the portfolio returns of the 100 sizebeta sorted portfolios obtained by equally weighting each security’s return in the portfolio. For comparison, we report estimates for both the Fama and French sample period
(1963.7-1990.12) and our whole sample period (1952.7-1998.12) in Table 6. Although
the average R2 s across all the portfolios are relatively high with 71.5% and 67.5% for
each of the sample period respectively, the CAPM has failed to hold because of the
significant intercept α (.056% per month for the whole sample and .168% per month
18
In addition to the methods used here, one could apply Gibbons, Ross, and Shanken’s (1989) Fstatistic. However, that statistic only applies to test the joint significance of the intercepts. In addition,
we would face a situation where the number of portfolios is greater than the number of time periods.
25
for the whole sample). However, with the inclusion of the idiosyncratic risk proxies in
equation III (or equation IV ), the aggregate α not only decreases, but also becomes
insignificant.19 More importantly, the idiosyncratic risk factor is very significant in both
sample periods using different proxies no matter what aggregation method we choose.
In particular, the signs of the estimates are positive, indicating that each percentage
increase in the idiosyncratic hedging proxy RI,t (or Sharpe Ratio adjusted proxy RI,SR,t )
is associated with .4% (or .9%) increase in return, on average, over the whole sample
period. For individual portfolios, the idiosyncratic hedging proxy is significant at a 5%
level in 90% of these portfolios, while the Sharpe Ratio adjusted proxy is significant in
97% of the individual portfolios. At the same time, the average adjusted R2 across all
the portfolios jumps to 76.8% using RI,t and 72.8% with RI,SR,t . Similar results carry
over to the Fama and French sample period. Therefore, the model in equation (14) is
supported empirically.
Insert Table 6
Fama and French (1993) have argued that a size proxy and a book-to-market proxy
are also very important in explaining the return fluctuation over time. Their threefactor model is reconfirmed in equation II of Table 6. As expected, both the size
factor and the book-to-market factor are very significant in both sample periods and
the alpha estimates are much smaller. Moreover, the average adjusted R2 has been improved dramatically to 78.9% for the whole sample period. At the same time, given the
cross-sectional evidence, the size factor may very well be measuring the idiosyncratic
risk. In this case, the idiosyncratic risk factor will be more appropriate, which is represented by equations V and V I with different proxies. For the whole sample period,
we see that both the idiosyncratic hedging factor and the book-to-market factor are
very significant with similar R2 s. Although the number of individual portfolios showing
statistical significant estimates on the two factors at a 5% level resembles that of the
three factor model, the number of portfolios with significant alphas actually dropped
significantly from 31 to 26. This is also true for the Fama and French sample period.
Therefore, idiosyncratic volatility also appears to capture the size effect in the time
series framework.
19
Since we have used the orthogonal part of the idiosyncratic risk proxy, the reported α in the table
is almost the same as that from the CAPM. Without orthogoalization, they are insignificant.
26
One might argue that our idiosyncratic risk proxies are simply capturing the size
and book-to-market factors. Thus, it is important to put our model in perspective relative to these two additional factors. In equations V II and V III, we control for both
the size and book-to-market effect with different measure of idiosyncratic risk factor. It
is evident that, for the whole sample period, not only is the average alpha insignificant
and small, but also the number of portfolios with statistical significant estimates on
the idiosyncratic risk factor is comparable to that of the book-to-market factor. For
example, in equation V II, there are 86% and 75% of the portfolios with statistically significant estimates on the size and the book-to- market factors, respectively, while there
are 74% of the portfolios showing statistically significant estimates on RI,t proxy and
60% on RI,SR,t proxy. In other words, there is a strong connection between idiosyncratic
risk and individual portfolio returns, even after controlling for size and book-to-market
effect. For the Fama and French sample period, results are very similar, except that
the estimated alpha is significant although very small.
We can further check the robustness of our results by splitting the whole sample
period into two sub-sample periods as we did in the cross-sectional study. The results
(not shown in the table) show that the aggregate estimates on the different factors
are significant at a 99% significance level for each of the models using both the first
and the second aggregation methods. In general, alphas are much smaller for the
second sub- sample period than for the first sub-sample period. However, the number
of portfolios showing statistical significant estimates on each of the factors is very similar
to that reported in Table 6. The evidence from individual portfolios thus supports our
theoretical prediction that idiosyncratic volatility appears to be important in explaining
asset return fluctuations over time. Note also that the significant beta estimates from
the time series regressions do not contradict Fama and French’s (1992) findings that
were discussed in our cross-sectional study.
4.3
Evidence from quarterly mutual fund returns
The empirical evidence from 100 constructed portfolios supports our theoretical model
well. However, they may still be subject to some selection problems. We may have
overemphasized the size effect when grouping portfolios according to size, or selection
problems could have resulted from purely arbitrary factors as suggested by Ferson,
Sarkissian, and Simin (1998). One natural remedy for such potential problems is to
27
perform similar tests on equity mutual fund returns. We do note, however, that this
approach may also have problems. Even though mutual fund portfolios are well diversified at different risk levels, there are overlapping holdings of many stocks among different
portfolios. Therefore, we should view the mutual fund approach as complementary in
studying the role of idiosyncratic risk.
We require each equity mutual fund to be in existence during the whole sample
period of 1971 to 1991. There are 206 such equity mutual funds with quarterly return
data. For consistency, we used the same types of models. Depending on the nature of
the proxies, the return proxies for the market index, the risk free rate, and the Sharpe
adjusted proxy were compounded. At the same time, proxies for the size and book-tomarket factors and the idiosyncratic hedging factor were averaged over each quarter from
monthly returns. The empirical results are listed in Table 7 for different models. Similar
to Table 6, both return proxies for idiosyncratic volatility have significant explanatory
power. Not only do we find significant estimates in both model II and model III, based
on either aggregation method, but also the average adjusted R2 increases by one and a
half percentage points. Although the average improvement in the R2 is not as impressive
as before, it still may be economically important as discussed in Xu (1998). In fact,
since the R2 for the base model (model I) is as high as 83%, such an improvement is not
trivial. Moreover, when we examine the individual regression estimates for each mutual
fund, more than 45% of them have had significant beta estimates for the idiosyncratic
risk proxy at a 5% significance level.
Insert Table 7
When we control for the book-to-market factor in equation V , about half of the
individual funds show statistically significant beta estimates for the idiosyncratic risk
proxy. However, the book-to-market factor has negative average beta estimates. Similar
results hold when using the Sharpe Ratio adjusted idiosyncratic proxy. When including
both the size and book-to-market factors in addition to the idiosyncratic risk factor
in equation V II and equation V III, the beta estimates for idiosyncratic risk proxies
remain statistically significant. Although the sign turns negative for the idiosyncratic
hedging proxy, it is correct when using Sharpe Ratio adjusted proxy. The unexpected
negative sign could be attributed to the high correlation between the size factor and
idiosyncratic risk factor. Overall, however, the evidence reconfirms the conclusions from
our analysis of constructed portfolios.
28
The mispricing or excess return term from the CAPM is labeled as Jensen’s alpha
in the mutual fund literature. It measures the degree of superior performance of a
fund. Although our sample includes only funds that survived, which are undoubtedly
the better performing funds, the aggregate Jensen’s alpha is statistically insignificant
with cross-sectional heteroscedasticity adjustment but is significant when using a simple average. The mutual fund sample, biased by survivorship, tended to outperform by
about 0.1% per quarter after accounting for market risk. But Jensen’s alpha becomes
much smaller and statistically insignificant after including either proxy measuring idiosyncratic risk.20 However, when both size and book-to-market factors are used, Jensen’s
alpha becomes even larger and more significant. This is why Jensen’s alpha remains
significant when an idiosyncratic risk proxy is also included in equation V II or in equation V III. One can view this result as indirect evidence on the importance of including
idiosyncratic risk in the evaluation of mutual funds.
One may argue that the role we ascribe to idiosyncratic volatility may result from
our inability to define the market portfolio. Perhaps idiosyncratic risk is just an artifact
of approximation error. We view our argument here as deepening the critique of Roll.
In this paper, we have argued that even if investors are shortsighted and only pay
attention to tradable financial assets, in which case Roll’s critique is irrelevant, we may
still find an imperfect CAPM from an econometrician’s point of view. The essence of our
argument is that such an imperfection can be ameliorated by including an idiosyncratic
risk measure.
20
This conclusion is based on regressions without orthogonalization on the idiosyncratic risk proxies.
29
5
Concluding comments
In this paper, we have made some progress in understanding the role of idiosyncratic
risk in asset pricing both theoretically and empirically. Other things being equal, idiosyncratic risk will affect asset returns when not every investor is able to hold the market
portfolio. Even after controlling for factors such as size and book-to-market, evidence
from both individual stocks and a sample of equity mutual funds supports the predictions of our model. Most importantly, the cross-sectional results demonstrate that
idiosyncratic volatility variable is more powerful than either beta or size measures in
explaining the cross section of returns. We also suggested that beta performs poorly
in cross- sectional regression because beta and alpha estimates are highly negatively
correlated. At the very least, our results provide a unique perspective in understanding
the possible role of idiosyncratic risk in asset pricing.
30
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34
Table 1: Portfolio Averages (July 1952 to Decomber 1998)
This table reports the average monthly returns, average log size, post-ranking beta, average idiosyncratic volatility in percentage for each of the 100 size-beta sorted portfolios. In June of each year,
NYSE/AMEX/NASDAQ stocks are sorted into 10 groups according to the size break point obtained
by sorting NYSE stocks only. Stocks within each group are further sorted according to beta break
point obtained by sorting NYSE stocks only. Financial stocks with SIC code between 6000 and 6999
are excluded. Pre-ranking betas are estimated from the CAPM using the current and previous two to
five years of monthly returns. Idiosyncratic volatilities are taken as mean square errors from Fama and
French’s (1993) three factor model.
All
Low-β
ALL
ME-S
ME-2
ME-3
ME-4
ME-5
ME-6
ME-7
ME-8
ME-9
ME-H
1.18
1.58
1.20
1.20
1.18
1.19
1.18
1.07
1.14
1.08
1.01
1.18
1.59
1.09
1.12
1.27
1.21
1.22
1.11
1.16
1.05
1.01
β2
Panel
1.20
1.53
1.17
1.11
1.21
1.49
1.23
1.17
1.03
1.02
1.08
ALL
ME-S
ME-2
ME-3
ME-4
ME-5
ME-6
ME-7
ME-8
ME-9
ME-H
5.17
2.25
3.57
4.09
4.53
4.96
5.37
5.79
6.29
6.88
8.00
5.15
2.12
3.56
4.08
4.55
4.97
5.37
5.80
6.32
6.85
7.93
5.18
2.21
3.57
4.08
4.54
4.99
5.37
5.80
6.30
6.90
8.05
ALL
ME-S
ME-2
ME-3
ME-4
ME-5
ME-6
ME-7
ME-8
ME-9
ME-H
1.16
1.38
1.31
1.27
1.24
1.18
1.13
1.12
1.04
0.99
0.94
0.72
1.04
0.89
0.83
0.74
0.67
0.61
0.66
0.59
0.59
0.59
0.85
1.17
1.12
1.00
0.88
0.80
0.76
0.78
0.67
0.63
0.71
β3
β4
β5
β6
β7
β8
β9
A: Average Monthly Returns (in Percent)
1.25 1.28 1.24 1.24 1.13 1.16 1.16
1.54 1.74 1.59 1.59 1.47 1.59 1.74
1.34 1.36 1.25 1.29 1.17 1.13 1.22
1.39 1.18 1.33 1.39 1.15 1.14 1.27
1.24 1.45 1.15 1.33 1.06 1.09 1.13
1.31 1.22 1.18 1.17 1.20 1.26 0.94
1.30 1.28 1.31 1.13 1.09 1.04 1.15
1.01 1.08 1.20 1.01 1.13 1.22 0.86
1.14 1.19 1.28 1.13 0.92 1.19 1.25
1.20 1.17 1.15 1.26 1.03 1.04 1.03
1.08 1.12 1.00 1.07 1.09 0.87 0.98
Panel B: Average size (ln(ME))
5.19 5.19 5.19 5.19 5.18 5.17 5.16
2.25 2.24 2.28 2.28 2.29 2.26 2.28
3.58 3.58 3.58 3.58 3.57 3.56 3.57
4.10 4.10 4.10 4.10 4.09 4.10 4.09
4.54 4.56 4.53 4.54 4.52 4.52 4.53
4.96 4.97 4.96 4.96 4.96 4.96 4.93
5.37 5.38 5.37 5.36 5.37 5.35 5.35
5.80 5.79 5.79 5.79 5.80 5.81 5.77
6.31 6.30 6.30 6.28 6.28 6.29 6.30
6.90 6.89 6.90 6.90 6.85 6.86 6.87
8.05 8.07 8.09 8.09 8.12 7.99 7.91
Panel C: Post-Ranking βs
1.00 1.10 1.17 1.20 1.26 1.31 1.41
1.24 1.30 1.37 1.40 1.44 1.53 1.64
1.13 1.19 1.29 1.34 1.35 1.46 1.61
1.11 1.20 1.28 1.25 1.39 1.36 1.53
1.12 1.13 1.18 1.32 1.42 1.44 1.48
1.05 1.18 1.19 1.26 1.28 1.36 1.44
0.91 1.07 1.19 1.16 1.21 1.31 1.45
0.98 1.07 1.17 1.20 1.21 1.28 1.34
0.88 0.96 1.04 1.08 1.19 1.23 1.28
0.83 0.97 1.03 0.97 1.13 1.16 1.22
0.79 0.93 0.93 1.00 1.01 1.01 1.08
35
High- β
1.00
1.43
0.98
0.94
0.91
0.97
1.04
0.93
1.09
0.83
0.84
5.13
2.28
3.56
4.06
4.52
4.94
5.36
5.78
6.27
6.83
7.76
1.59
1.72
1.73
1.73
1.70
1.60
1.61
1.57
1.48
1.38
1.36
Table 1–Continued
All
ALL
ME-S
ME-2
ME-3
ME-4
ME-5
ME-6
ME-7
ME-8
ME-9
ME-H
8.29
13.6
10.4
9.24
8.72
8.05
7.49
7.18
6.64
6.19
5.45
ALL
ME-S
ME-2
ME-3
ME-4
ME-5
ME-6
ME-7
ME-8
ME-9
ME-H
24
95
28
20
17
15
14
13
12
12
12
Low-β
β2
β3
β4
β5
β6
β7
β8
β9
High- β
Panel D: Average Monthly Idiosyncratic Volatility (in Percent)
6.83
7.02 7.44 7.72 7.90 8.24 8.48 8.94 9.41
10.9
12.4
12.4 12.2 12.6 12.7 13.4 13.4 14.5 15.2
17.0
8.70
8.70 9.50 9.80 9.90 10.2 10.7 11.1 11.5
13.4
7.30
7.70 8.20 8.50 8.90 9.30 9.70 9.90 10.7
12.2
7.00
7.30 7.80 7.90 8.20 8.80 9.10 9.60 10.1
11.4
6.20
6.60 7.10 7.60 7.70 8.00 8.30 8.90 9.30
10.8
5.80
6.10 6.60 6.90 7.10 7.50 7.80 8.10 8.70
10.3
5.80
6.00 6.40 6.60 6.90 7.00 7.30 7.80 8.30
9.70
5.20
5.30 6.00 6.40 6.40 6.50 6.80 7.10 7.60
9.10
5.10
5.20 5.60 5.80 6.00 6.30 6.20 6.70 6.90
8.10
4.80
4.90 5.00 5.10 5.20 5.40 5.50 5.70 5.80
7.10
Panel E: Average Number of Stocks in Each Portfolio
33
23
21
21
20
20
21
22
23
36
168
93
83
76
75
75
77
82
84
141
34
26
24
23
24
23
23
26
28
48
22
18
17
18
17
18
17
18
20
35
18
16
16
15
15
16
15
16
17
27
15
13
14
14
14
14
14
14
15
22
15
12
13
13
12
13
13
13
13
20
14
12
12
12
12
12
13
13
13
17
13
11
11
12
12
12
12
12
12
17
14
11
11
11
11
11
11
11
11
16
14
11
11
11
11
11
11
11
11
15
36
Table 2: Cross-Sectional Regressions for Individual Stocks
This table shows the cross-sectional relationship between returns and relevant variables for individual
stocks. βi is the post ranking portfolio beta assigned to individual stocks within each size-beta portfolio.
vi,I is the mean square error from a three-factor model with market return, size proxy, and book-tomarket proxy for stock i using the current and previous five year returns. M Ei denotes the log of the
June market capitalization each year for stock i. Rp2 is calculated using average portfolio returns on
time averages of independent variables.
Model
I
II
III
IV
V
VI
VII
I
II
III
IV
V
VI
VII
Coefficient
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Const.
βi
M Ei vi,I
Rp2
Panel A: (1963.7-1990.12)
1.13
.086
0.2
4.94
.259
.000
.796
1.79
-.156
44.6
3.74
-2.79
.000
.006
.753
38.8 15.0
2.85
3.55
.005
.000
2.58
-.457 -.190
59.1
7.43
-1.51 -3.72
.000
.132
.000
1.53
-.604
43.0 30.5
7.46
-2.25
4.95
.000
.025
.000
.910
-.038 35.0 51.4
2.45
-.801 3.98
.015
.424 .000
2.13
-.783 -.084 39.0 59.1
6.11
-2.84 -1.71 4.98
.000
.005
.087 .000
Panel C: (1952.7-1975.6)
.799
.217
3.34
.624
.001
.533
1.51
-.111
3.01
-1.84
.003
.067
.808
43.5
3.38
2.42
.001
.016
1.87
-.186 -.133
5.45
-.616 -2.49
.000
.539
.013
1.02
-.211
51.2
4.91
-.768
3.53
.000
.443
.001
1.04
-.049 40.8
2.78
-.985 2.56
.006
.325 .011
1.66
-.412 -.080 47.9
4.94
-1.51 -1.62 3.47
.000
.132
.106 .001
37
Const.
βi
M Ei vi,I
Rp2
Panel B: (1952.7-1998.12)
1.14
.121
0.5
7.08
.507
.000
.612
1.84
-.143
43.8
5.63
-3.70
.000
.000
.820
39.5 26.4
4.50
4.25
.000
.000
2.38
-.324 -.166
51.9
9.83
-1.46 -4.61
.000
.144
.000
1.55
-.608
44.6 45.7
10.7
-3.08
5.78
.000
.002
.000
.864
-.016 37.9 44.3
3.41
-.484 4.56
.001
.629 .000
1.91
-.719 -.053 42.2 55.9
7.96
-3.57 -1.56 5.79
.000
.000
.118 .000
Panel D: (1975.7-1998.12)
1.53
-.016
7.18
-.047
.000
.962
2.17
-.173
5.14
-3.60
.000
.000
.832
35.6
3.04
6.55
.003
.000
2.82
-.439 -.195
8.64
-1.37 -4.10
.000
.171
.000
2.00
-.961
38.0
10.2
-3.41
7.44
.000
.001
.000
.691
-.016 35.0
2.02
-.362 6.54
.044
.717 .000
2.05
-.976 -.020 36.7
6.15
-3.33 -.444 7.22
.000
.001
.658 .000
Table 3: Fama-MacBeth Type Cross-Sectional Regressions for Equity Mutual
Funds (1971-1991)
This table shows the cross-sectional relationship between returns and relevant variables for mutual
fund returns. βi is the coefficient estimate on the market factor from the Fama and French’s (1993)
three factor model, and vi,I is the mean square error from the same model for fund i.
Model
I
III
V
Coefficient
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Constant
1.927
4.737
0.000
2.977
3.438
0.001
1.740
3.969
0.000
38
β
1.633
1.263
0.210
1.422
1.159
0.250
vi,I
286.2
1.922
0.058
270.8
1.926
0.057
Table 4: Cross-Sectional Regressions for Portfolios Sorted According to Size
as well as Idiosyncratic Volatility
This table shows the cross-sectional relationship between returns and relevant variables for various
portfolios. In “Sorting Scheme I,” stocks are first sorted into six groups according to their market
capitalization. Within each group, stocks are sorted again into 40 portfolios according to their betas.
In “Sorting Scheme II,” stocks are also sorted into six groups, first according to their idiosyncratic
volatilities. Within each group, stocks are then sorted into 40 portfolios according to their betas. vi,I
is the time varying idiosyncratic volatility measure. M Ei denotes the log of the market capitalization
of portfolio i.
Group
Number
1
2
3
4
5
6
All
1
2
3
4
5
6
All
1
2
3
4
5
6
All
Sorting Scheme I
Estimates
t-Value
M Ei
vi,I
M Ei
vi,I
-.462
-6.31
-.068
-0.53
.014
0.10
-.158
-1.30
-.044
-0.48
.026
0.68
-.143
-3.69
48.8
5.04
7.35
0.62
3.19
0.22
-.195
-0.01
8.82
0.44
-22.6
-0.98
39.5
4.25
-.185 47.9 -2.66 4.88
-.085 7.19 -0.67 0.60
.053
2.86
0.42
0.20
-.178 -.413 -1.52 -0.02
-.048 8.11 -0.56 0.40
.006 -23.7 0.15 -0.99
-.016 37.9 -0.48 4.55
39
Sorting Scheme II
Estimates
t-Value
M Ei vi,I M Ei vi,I
-.016
-0.70
-.050
-1.69
-.066
-2.07
-.097
-2.78
-.139
-3.39
-.345
-6.11
-.142
-3.69
407.
5.26
585.
8.61
644.
10.6
545.
11.6
471.
12.9
50.6
6.35
39.5
4.24
-.001 4.15 -0.02 5.34
-.022 5.79 -0.73 8.63
-.031 6.35 -0.98 10.5
-.053 5.41 -1.54 11.7
-.069 4.73 -1.73 13.1
-.192 .470 -3.48 6.05
-.016 .379 -0.48 4.55
Table 5: Alpha in Fama-MacBeth Type Cross-Sectional Regressions for 100
Portfolios
This table shows the cross-sectional relationship between alphas (or average returns) and relevant
variables for 100 portfolio constructed using Fama and French (1992) methodology. βi is the post
ranking beta for portfolio i. Similarly alpha is the post ranking portfolio alpha from the CAPM. v̄i,I
is the time average for equally weighted average mean square errors from a three-factor model using
market return, a size proxy, and a book-to-market proxy for each stock in portfolio i using the current
and previous five year returns. M¯E i denotes the time average of log market capitalization for portfolio
i.
No
I
II
III
IV
V
VI
I
II
III
IV
V
VI
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
Estimate
t − value
p − value
(1952.7-1998.12)
(1952.7-1975.6)
(1975.7-1998.12)
R̄m − R̄f = .635
R̄m − R̄f = .539
R̄m − R̄f = .730
M¯E i
M¯E i
M¯E i
βi
v̄i,I
R2
βi
v̄i,I
R2
βi
v̄i,I
Panel A: Alpha as the dependent variable
-.590
44.5 -.399
21.9 -.781
-8.87
-5.23
-10.7
.000
.000
.000
-.024
2.3
-.040
6.0
-.011
-1.50
-2.50
-.596
.136
.014
.552
-6.86 1.8
-14.9 2.3
-5.34
-1.35
-1.50
-1.30
.180
.136
.198
-.849 -.096
73.3 -.684 -.107
53.4 -1.00 -.086
-16.1 -10.2
-9.93 -8.10
-15.9 -7.98
.000
.000
.000
.000
.000
.000
-1.05
36.9 69.8 -1.14
99.3 47.2 -1.15
22.8
-14.8
9.00
-9.09
6.83
-15.9
8.10
.000
.000
.000
.000
.000
.000
-.976 -.066 16.6 75.5 -.866 -.082 33.6 54.5 -1.12 -.047 13.2
-14.6 -4.72 2.93
-6.38 -3.92 1.56
-15.7 -2.84 3.06
.000
.000
.004
.000
.000
.123
.000
.006
.003
Panel B: Average return as the dependent variable
.045
0.5
.139
3.3 -.052
.670
1.82
-.718
.504
.072
.474
-.078
43.8
-.093
39.5
-.065
-8.74
-8.01
-6.22
.000
.000
.000
19.4 26.4
39.1 19.2
12.4
5.92
4.82
4.93
.000
.000
.000
-.214 -.096
51.9 -.145 -.107
42.2 -.273 -.086
-4.05 -10.2
-2.11 -8.08
-4.33 -7.97
.000
.000
.037
.000
.000
.000
-.419
36.8 45.7 -.599
99.2 34.6 -.423
22.8
-5.87
8.98
-4.79
6.82
-5.82
8.09
.000
.000
.000
.000
.000
.000
-.340 -.066 16.6 55.9 -.328 -.082 33.5 43.6 -.388 -.047 13.2
-5.10 -4.71 2.92
-2.41 -3.92 1.55
-5.44 -2.83 3.05
.000
.000
.004
.018
.000
.123
.000
.006
.003
40
R2
54.1
0.4
1.7
72.3
72.6
74.8
0.5
28.3
19.9
39.9
40.6
45.2
Table 6: Times Series Evidence from Individual Portfolio Returns
This table reports estimates for Model I through Model V III. RV W,t and Ri,t denote returns from a
value-weighted index and the i−th portfolio respectively; RI,t and RI,SR,t denote the return proxies for
idiosyncratic volatility from a hedging portfolio and the Sharpe ratio adjusted proxy respectively using
the procedure described in Section 2; RSMB,t and RHML,t denote proxies for the size and book-tomarket factors respectively. “AM I” pools data for individual portfolios and adjusts for cross-sectional
heteroscedasticity. “AM II” simply averages estimates for individual portfolios. “Ind.” counts the
percentage of individual regressions for each portfolio with significant coefficients at 5% level. Numbers
in the parentheses denote standard deviations.
Model
Method
AM I
I
AM II
Ind.(%)
AM I
II
AM II
Ind.(%)
AM I
III
AM II
Ind.(%)
AM I
IV
AM II
Ind.(%)
AM I
V
AM II
Ind.(%)
AM I
VI
AM II
Ind.(%)
AM I
VII
AM II
Ind.(%)
AM I
VIII
AM II
Ind.(%)
α
.168
(.017)
.188
(.026)
32
.032
(.016)
.012
(.019)
15
.177
(.017)
.186
(.026)
45
.187
(.016)
.185
(.026)
39
.085
(.017)
.079
(.019)
20
.178
(.017)
.183
(.021)
29
.039
(.017)
.027
(.018)
11
.081
(.017)
.050
(.022)
29
RV W,t
1.08
(.004)
1.14
(.025)
100.
1.02
(.004)
1.04
(.017)
100.
1.12
(.004)
1.14
(.025)
100.
1.08
(.004)
1.14
(.025)
100.
1.15
(.004)
1.18
(.024)
100.
1.08
(.004)
1.14
(.023)
100.
1.04
(.005)
1.07
(.025)
100.
1.03
(.004)
1.04
(.017)
100.
(1963.7-1990.12)
RSM L,t
RHM L,t
RI,t
.536
(.006)
.598
(.053)
91
RI,SR,t
.184
(.007)
.184
(.008)
62
82.3
(.67)
.401
(.005)
.488
(.047)
86
75.3
(.83)
1.23
(.018)
1.45
(.087)
94
.465
(.012)
.460
(.039)
86
.432
(.008)
.522
(.060)
84
.192
(.007)
.205
(.021)
67
.018
(.007)
.005
(.017)
28
.186
(.007)
.187
(.020)
65
.115
(.007)
.127
(.021)
51
R2 (%)
71.5
(.97)
.407
(.005)
.485
(.047)
88
77.0
(.68)
81.6
(.67)
1.21
(.020)
1.45
(.085)
93
.074
(.011)
.137
(.041)
66
77.4
(.64)
83.3
(.61)
.510
(.025)
.411
(.055)
55
83.1
(.56)
α
.056
(.013)
.058
(.026)
34
-.021
(.012)
-.035
(.021)
31
.050
(.012)
.056
(.026)
45
.069
(.012)
.058
(.026)
39
-.023
(.013)
-.030
(.021)
26
.035
(.013)
.020
(.021)
24
-.021
(.012)
-.034
(.020)
32
-.002
(.012)
-.023
(.021)
34
RV W,t
1.03
(.003)
1.09
(.024)
100.
1.00
(.003)
1.02
(.019)
100.
1.07
(.003)
1.09
(.024)
100.
1.03
(.003)
1.09
(.024)
100.
1.09
(.003)
1.12
(.023)
100.
1.05
(.003)
1.10
(.023)
100.
1.01
(.003)
1.04
(.025)
100.
1.01
(.003)
1.02
(.019)
100.
(1952.7-1998.12)
RSM L,t
RHM L,t
RI,t
.545
(.005)
.622
(.051)
92
RI,SR,t
.191
(.005)
.201
(.018)
71
78.9
(.73)
.402
(.046)
.498
(.001)
90
76.8
(.85)
.956
(.012)
1.14
(.064)
97
.484
(.009)
.488
(.039)
86
.459
(.006)
.562
(.059)
92
.159
(.005)
.177
(.017)
70
.071
(.005)
.078
(.014)
43
.188
(.005)
.195
(.018)
75
.155
(.005)
.174
(.019)
64
R2 (%)
67.5
(1.1)
.406
(.004)
.496
(.046)
90
72.8
(.82)
78.0
(.76)
.923
(.012)
1.10
(.061)
96
.065
(.008)
.134
(.043)
74
73.2
(.77)
80.3
(.62)
.349
(.015)
.264
(.047)
60
79.8
(.58)
Table 7: Times Series Evidence from Equity Mutual Fund Returns (19711991) with Different Measures of Idiosyncratic Volatilities
This table reports estimates for Model I through Model V III applied to mutual fund returns. RV W,t
and Ri,t denote quarterly returns from a value-weighted index and the i − th mutual fund respectively; RI,t and RI,SR,t denote the return proxies for idiosyncratic volatility from a hedging portfolio
and Sharpe ratio adjusted proxy respectively using the procedure described in Section 2; RSMB,t and
RHML,t denote proxies for size and book-to-market factors respectively. “AM I” pools data for individual portfolios and adjusts for cross-sectional heteroscedasticity. “AM II” simply averages estimates
for individual portfolios. “Ind.” counts the percentage of individual regressions for each portfolio with
significant coefficients at 5% level. Numbers in the parentheses denote standard deviations.
Model
Method
Aggregation I
I
Aggregation II
Individual (%)
Aggregation I
II
Aggregation II
Individual (%)
Aggregation I
III
Aggregation II
Individual (%)
Aggregation I
IV
Aggregation II
Individual (%)
Aggregation I
V
Aggregation II
Individual (%)
Aggregation I
VI
Aggregation II
Individual (%)
Aggregation I
VII
Aggregation II
Individual (%)
Aggregation I
VIII
Aggregation II
Individual (%)
Jensen’s α
.026
(.026)
.111
(.043)
18.0
.113
(.028)
.277
(.051)
21.4
.030
(.026)
.111
(.043)
18.4
.038
(.026)
.111
(.043)
17.5
.100
(.028)
.254
(.050)
22.3
.148
(.027)
.329
(.052)
18.9
.117
(.028)
.279
(.050)
22.3
.147
(.028)
.310
(.051)
21.4
RV W,t
.927
(.003)
.985
(.013)
100.
.895
(.004)
.894
(.008)
100.
.929
(.003)
.985
(.013)
100.
.926
(.003)
.985
(.013)
100.
.919
(.003)
.959
(.011)
100.
.910
(.003)
.946
(.011)
100.
.876
(.005)
.886
(.011)
100.
.902
(.004)
.902
(.008)
100.
42
RSML,t
RHML,t
.222
(.015)
.577
(.056)
56.8
-.176
(.016)
-.336
(.051)
56.3
RI,t
RI,SR,t
87.47
(.73)
.108
(.013)
.387
(.043)
45.1
85.77
(.77)
.245
(.016)
.411
(.038)
38.8
.378
(.032)
.645
(.059)
37.4
.086
(.019)
.445
(.089)
45.6
-.150
(.016)
-.274
(.051)
56.3
-.231
(.017)
-.419
(.052)
51.9
-.188
(.016)
-.342
(.050)
56.8
-.231
(.017)
-.395
(.051)
53.4
R2 (%)
84.23
(.86)
.117
(.013)
.399
(.043)
49.1
85.18
(.82)
87.10
(.75)
.310
(.017)
.549
(.039)
44.2
-.148
(.026)
-.065
(.038)
21.8
86.50
(.79)
87.68
(.72)
.258
(.021)
.253
(.026)
23.3
87.67
(.73)