The CAPM Lives for Expected Returns

Wei Liu∗, James W. Kolari†, and Seppo Pynnonen‡

January 16, 2017

Abstract

A major problem in testing the Capital Asset Pricing Model (CAPM) is that the theoreti-

cal equilibrium relation links expected returns to market risk as measured by beta but empir-

ical tests utilize realized returns. This paper proposes a simple approximation to the monthly

expected return based on the average of daily returns in any given month and comparatively

tests this estimated expected return against the monthly realized return. Simulation anal-

yses show that cross-sectional tests of market beta (β) are more likely to be significant for

estimated expected returns than realized returns, particularly when returns are averaged

over a long sample period. Empirical tests using U.S. CRSP stock return data corroborate

the simulated findings. Strikingly, using a full sample approach to cross-sectionally test the

market factor with estimated expected returns, the t-statistic on the market price of beta

risk exceeds 12, which is well beyond any statistical result obtained in prior literature. Based

on these and further robustness tests, we conclude that the CAPM is strongly supported for

estimated expected returns but not realized returns.

JEL: G12

∗Research Scientist, Texas A&M University, Department of Finance, College Station, TX 77843-4218Office phone: 979-845-3514Email address: yuna0831@gmail.com†The corresponding author is JP Morgan Chase Professor of Finance, Texas A&M University, Department of

Finance, College Station, TX 77843-4218Office phone: 979-845-4803Email address: j-kolari@tamu.edu‡Department of Mathematics and Statistics, University of Vaasa, P.O.Box 700, FI-65101, Vaasa, Finland

Office phone: +358-29-449-8311Email: sjp@uwasa.fi

The authors gratefully acknowledge financial support from the Center for International Studies, MaysBusiness School, Texas A&M University. Helpful comments have been received from Ali Anari, Donald Fraser,Julian Gaspar, Adam Kolasinski, Johan Knif, Austin Murphy, Ed Swanson, and Christopher Yost-Bremm.

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The CAPM Lives for Expected Returns

1 Introduction

The capital asset pricing model (CAPM) of Treynor (1961, 1962) Sharpe (1964), Lintner (1965),

and Mossin (1966) posits a linear relation between expected asset return and market beta (β).

A Nobel Prize was awarded to Professor Sharpe in 1990 in recognition of this innovative asset

pricing model. Unfortunately, after more than 50 years of extensive empirical tests, the renowned

CAPM has fallen into disrepute. In a well-known paper on the real world viability of the CAPM,

Fama and French (1996) concluded that, "In our view, the evidence that β does not suffice to

explain expected return is compelling. The average return anomalies of the CAPM are serious

enough to infer that the model is not a useful approximation."(1996, p.1957) Even early evidence

raised suspicions about the CAPM. Seminal studies by Black, Jensen, and Scholes (1972) and

Fama and MacBeth (1973) found a flatter relation between average returns and β than predicted

by theory. In the wake of the CAPM’s failure, a plethora of multi-factors have emerged to help

explain cross-sectional expected returns. Cochrane (2011) has facetiously characterized the

hundreds of new factors (e.g., size, value, momentum, capital investment, profitability, net stock

issues, idiosyncratic risk, etc.) proposed by researchers as a "factor zoo." More recently, Harvey,

Liu, and Zhu (2015) surveyed the vast CAPM and multi-factor literature and concluded that,

due to potential data mining issues, the hurdle rate for a new factor in asset pricing should be a

t-statistic of 3.0 or more for cross-sectional tests relating average returns to β or other factors’

loadings. According to their standards, the CAPM is dead.1

In the present paper, based on U.S. common stock returns, we report cross-sectional empirical

tests of β in the period 1965 yo 2015 that yield encouraging t-values in the 2.0 to 3.0 range and

1The significance of the market factor in cross-sectional tests typically falls below the 3.0 t-statistic standardin extant literature. Some studies have reported statistically significant tests of the market factor but fall belowthis threshold. For example, Kim (1997) implemented an adjustment to market beta to take into account thewell-known error-in-variables problem and obtained a t-value of 2.4 and 2.16 in cross-sectional tests of marketbeta using monthly and quareterly returns of individual stocks, respectively. Similar t-values were estimatedupon adding size, value, and earnings/price accounting variables to the model. Fama and French (2004) providean excellent review of CAPM theory and evidence. More recently, Bali, Engle, and Tang (2016) have reportedevidence that surpasses the 3.0 t-value threshold. They found that, while traditional empirical models basedon unconditional beta are dead, dynamic conditional beta is alive and well in the cross-section of stock returns.The authors focused their cross-sectional tests on the relation between one-day-ahead excess returns of individualstocks and time-varying conditional beta estimates. They also tested two-to-five-day-ahead cumulative excessreturns. Based on U.S. stock returns in the period July 1963 to December 2013, they obtained t-values for themarket price of beta risk in the range of 1.3 to 8.5 in different model specifications and out-of-sample tests. Seealso related studies by Murphy (1990), Kim (1997), Jostova and Philipov (2005), and Levy and Roll (2012).

2

beyond. Indeed, in one cross-sectional test we obtain a t-value equal to 12.7, which far exceeds

any statistical threshold for significance. These and other findings strongly support the CAPM.

To achieve these results, we depart from extant literature by utilizing estimated expected returns

equal to average daily returns in a one month period (rescaled to a monthly return). This simple

approximation to expected monthly returns differs from commonly-used realized monthly returns

that represent compounded daily returns in any given month. Additionally, we find that cross-

sectional tests that use a full sample approach with monthly estimated expected returns averaged

over a long period of time (e.g., 50 or 60 years) yield better estimates of expected returns than

a monthly rolling approach so often used in empirical tests. In full sample approach tests,

the t-values for β loadings dramatically increase compared to those for the monthly rolling

approach. The cross-sectional test results lead us to conclude that the CAPM lives for expected

returns rather than realized returns. A key insight is that realized returns are single estimates

of expected returns, whereas estimated expected returns are a sample-based value using (for

example) 21 daily realized returns to provide an estimate of the expected return. To the extent

that realized returns are poor proxies for expected returns, it is not possible to make inferences

about the empirical validity of the CAPM based on cross-sectional tests using realized returns.

Related work by Brav, Lehavy, and Michaely (2005) proxied expected returns using analysts’

forecasted target stock prices gathered by Value Line and First Call.2 The authors argued that

realized returns imperfectly measure ex ante expected returns due to noise, information sur-

prises, and complex learning effects. For about 1,500 firms per year in the period 1975 to 2001,

cross-sectional tests produced t-values for market beta in range of 4 to 6.5 in different model

specifications. Our study extends their analyses of ex ante returns forecasted by analysts to ex

post returns available in actual stock data. Despite using very different proxies for expected re-

turns, our empirical tests also find a significant positive relation between approximated expected

returns and market beta. Unlike their study, in cross-sectional tests adjusted for market beta

risk, we find that Fama and French’s size and book-to-market (value) multi-factors are both

positive and significantly priced and tend to drive alpha (i.e., mispricing error) to zero. When

the momentum factor is added to the three-factor model, not only is it positive and significantly

priced but it supplants the size factor, which becomes insignificant. We interpret these results

to suggest that zero-investment portfolios are needed to augment the CRSP market index in

2The authors cited a number of previous studies that utilized analyst forecasts to proxy expected returns forfirms, including Ang and Peterson (1985), Shefrin and Statman (2003), Botosan and Plumlee (2005), and others.

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order to better approximate the cross-section of expected returns.

The next section discusses the problem of using realized returns to proxy expected returns

in testing the CAPM. We propose average daily returns within one month as a potential proxy

for estimated expected returns. Section 3 conducts simulation experiments with synthetically-

created stock return data series and compares the empirical results for realized versus estimated

expected returns. Section 4 extends the analyses to U.S. CRSP stock return data. Our empirical

results tend to corroborate simulation findings. Also, we provide empirical tests of the popular

size, value, and momentum factors. Section 5 concludes.

2 The Problem of Realized Returns in Testing the CAPM

The CAPM posits the following equilibrium relationship for the expected return on the ith asset:

E(Ri)−Rf = βi[E(Rm)−Rf ], (1)

where Rf is the riskless rate of return, Rm is the market index rate of return, and βi is the

systematic market risk of the ith asset.

Testing the above equation with real world date turns out to be nontrivial due to fact that

investors’ expected returns cannot be directly observed. Instead, as noted by Brav, Lehavy, and

Michaely (2005), researchers typically utilize realized returns to proxy expected returns in tests

of the CAPM, which introduces a gap between empirical tests and CAPM theory. In an efficient

market, all assets should move with the general market simultaneously, such that the length of

the period to estimate expected returns is very small. Of course, for the smallest time scale,

realized returns converge to expected returns. Unfortunately, the CAPM does not specify the

length of the time period to compute expected returns.

In practice, the following empirical form of the CAPM known as the market model by

Markowitz (1959) and Sharpe (1963) is employed to estimate beta for common stocks:

Rit −Rft = αi + βi(Rmt −Rft) + εit, t = 1, · · ·T, (2)

where α is a mispricing term equal to zero in theory, εit is a random error term that represents

idiosyncratic behavior of the ith stock’s rate of return with εit ∼ N(0, σ2i,Id), and t is a time

4

subscript. In cross-sectional asset pricing tests, it is common to use realized monthly returns for

60 months to estimate the market model. Subsequently, stocks are formed into (for example)

25 test asset portfolios based on their ranked βi values and portfolio returns are computed for

the next 61st month (or next 12 months). In step one of cross-sectional Fama and MacBeth

(1973) tests, these monthly portfolio returns are used in a 60-month time-series regression to

estimate βp values for the 25 test assets. In step two, the 25 estimated portfolio betas are used

as independent variables in the following cross-sectional regression:

Rpt+1 −Rft+1 = α+ λβp + εpt+1, t = 1, · · ·T, (3)

where Rpt+1 − Rft+1 is the realized excess return of the pth test asset, and λ is the estimated

market price of risk associated with market beta. If λ is significantly different from zero, we

infer that beta loadings associated with the market factor are priced in the cross-section of

stocks. Also, the estimated intercept term α should be zero. A significant positive α would

imply mispricing in the sense that other factors are needed to fully price stocks.

As mentioned earlier, a potential problem in implementing the market model to test the

CAPM is the common usage of realized rates of return rather than expected rates of return.

Expected returns are well defined in mathematical terms but not in empirical terms. Assume

that a stock’s price Pt follows geometric Brownian motion across time t, or

dPt = θPtdt+ σPtdWt, (4)

where parameters θ and σ > 0, and Wt denotes the Wiener process. Applying Ito’s lemma, we

have

d[log(Pt)] =dPtPt− 1

2

σ2P 2t dt

P 2t

=

(θ − σ2

2

)dt+ σdWt, (5)

such that the log-price process becomes an arithmetic Brownian motion with drift parameter

µ = θ − σ2/2, (6)

5

which gives

Pt = P0eµt+σWt . (7)

Consequently, the expected price at time t becomes

E(Pt) = P0e(µ+ 1

2σ2)t ≡ P0e

θt. (8)

Empirically speaking, for any given observed time series price data (P0, P1, · · ·PT ), parameters

µ and σ are estimated consistently by (e.g., see Campbell, Lo, and MacKinlay (1997, Sections

9.3.1 and 9.3.2)

µ =1

T

T∑t=1

Xt (9)

σ2 =1

T − 1

T∑t=1

(Xt − µ)2, (10)

where

Xt = log

(PtPt−1

). (11)

Equation (8) shows that the expected return eθt is greater than the realized return eµt due to the

variance term. The usage of realized returns to proxy expected returns may not conform to how

investors estimate expected returns in the real world. If not, then empirical evidence against

the CAPM based on realized returns would be suspect. In this regard, there are a number

of potential alternative methods by which investors could compute expected returns. Previous

research does not explore other expected return possibilities in asset pricing tests of the CAPM.

Suppose that a random variable follows a standard normal distribution. How would a rea-

sonable person go about computing the mean of observed variable data? The simple mean is a

logical choice for many individuals. Assuming a stock’s rate of return follows Brownian motion

and is normally distributed with mean rate R and standard deviation σAV G, the realized rate

of return in period T (T >> 1) is R1→T =∏Tt=1(1 + Rt) − 1 ≈ exp

(RT − σ2AV GT/2

)− 1.3 In

3This result can be readily derived by means of a Taylor series expansion of the natural log function of eachperiod’s gross return (1 + Rt), which can be approxiated by dropping higher order terms.

6

simple mean terms, we have:

µ ' R− σ2AV G/2, (12)

where

R =1

T

T∑t=1

Rt (13)

σ2AV G =1

T − 1

T∑t=1

(Rt − R)2. (14)

Comparing equations (6) and (12), we see that θ is proxied by R. From equation (8), an

approximation for expected returns is possible by substituting R for θ.

It is clear from equation (12) that the simple average value of the observed return R is a

possible proxy for θ. The simple mean R is easy for investors to use also. Over time only

one path of realized returns is observed by the investor. This path represents one draw from

a distribution of many potential paths for realized returns. It is impossible for investors to

know the true values of expected returns and volatilities. As discussed shortly, investors (for

example) may well perceive that R computed as the average of daily returns in a month provides

a reasonable estimate of daily expected returns.

Another issue concerns compounding effects in realized returns. If investors prefer the simple

mean return as the expected return, then using the realized return as a proxy for expected return

in empirical tests could muddle the order of betas among different assets. Consider the following

scenario. Assume that there are two stocks 1 and 2 with β1 = 2β2. According to CAPM theory,

the expected return of stock 1 is larger than stock 2 assuming that the expected return of the

market index portfolio is positive and larger than the riskless rate for some period of time T . If

we utilize realized returns to proxy expected returns, the market model relation above may not

hold due to the fact that realized returns contain components beyond expected returns, including

idiosyncratic errors as well as other risk factors that may exist. The latter extra components

can change the order of realized returns for different test assets. In our simple example, based

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on the market model, realized returns can be defined as:

Rrealized1 = exp

(β1RT −

1

2β21σ

2m,AV GT −

1

2σ21,IdT

)− 1

Rrealized2 = exp

(β2RT −

1

2β22σ

2m,AV GT −

1

2σ22,IdT

)− 1 (15)

Here we see that realized returns have one expected return component and two unexpected return

components. Particularly relevant to cross-sectional asset pricing tests, the latter unexpected

noise components 12 β

2i σ

2m,AV GT and 1

2σ2i,IdT can cause the order of expected returns across beta-

sorted stocks (e.g., stocks i = 1 and 2) to differ from the order of their realized returns. Notice

that the second term 12 β

2i σ

2m,AV GT implies that higher estimated betas can make the ordering

of realized returns less stable. Also, the third idiosyncratic term 12σ

2i,IdT can be relatively

large due to substantial firm-specific risks. Forming portfolios as test assets can mitigate the

influence of unexpected return components but does not necessarily rule out problems due to

the limited number of stocks in the marketplace. Also, when daily returns are compounded

in the computation of monthly realized returns, the unexpected noise component is multiplied

over time and, therefore, can become quite large in a one-month return.4 These component

return concepts suggest that cross-sectional tests relating beta-sorted test asset portfolios to

their realized returns may well fail due to ranked orders of realized returns that differ from their

respective ranked orders of expected returns.

Furthermore, compounding can cause the estimated beta for a stock to differ from its true

beta. Given the ith stock with true market beta βi, if market returns are normally distributed

N(µm, σ2m) and idiosyncratic volatility is set to zero (i.e., ρim = 1), then stock i has distribution

N(βiµm, β2i σ

2m). Empirically, when beta is estimated using expected compounded returns in

period [0, T ] for both the stock and market portfolio, we have

E(RciT ) = βiE(RcmT )

eβiµmT − 1 = βi(eµmT − 1), (16)

wherein the estimated βi varies with the time frame T used (e.g., monthly or quarterly data give

4Noise in realized returns is well recognized. In this respect, in their paper testing analysts’ expected stockreturns, Brav Lehavy, and Michaely (2005) cited numerous studies that discuss potentially large noise in realizedreturns, including Blume and Friend (1973), Sharpe (1978), Elton (1999), and Lewellen and Shanken (2002)).

8

different estimated betas). Compounding worsens the problem so that βi increasingly diverges

from true βi. For this reason we specify a time invariant estimation of expected returns as

follows:

E(RiT ) = βiE(RmT )

βiµmT = βi(µmT ), (17)

where βiµmT empirically is the stock’s expected return during period [0, T ], and µmT is the

market portfolio’s expected return during the period.

In an attempt to fill the gap between CAPM theory based on expected returns and empirical

tests using realized returns as proxies for expected returns, we propose the following empirical

model:

Rit −Rft = αi + βi(Rmt −Rft) + εit, t = 1, · · ·T, (18)

where Rit and Rmt are estimated expected returns for the ith asset and market index m, respec-

tively, in period t (e.g., monthly returns in most asset pricing tests), and relation (17) holds

such that βi is time invariant. We estimate expected returns in month t for the ith individual

common stock by computing its average daily return in the month as follows:

RiT =1

M

M∑T=1

RdailyiT , (19)

where M is the number of trading days in the month (e.g., M = 21 for many months). For the

purpose of forthcoming cross-sectional tests, this average daily return is scaled up to monthly

returns denoted Rit in equation (18).5

The estimated expected return Rit differs from realized monthly returns Rit computed by

cumulatively compounding the daily returns for M days, i.e.,∏

(1 + RdailyiT=1)(1 + RdailyiT=2)...(1 +

RdailyiT=M ). Instead, it is based on mean daily returns computed from a random sample of M

5Factually, the one-month return Rit equals the simple sum of daily returns ignoring the effects of compounding,otherwise known as the simple monthly rate, in contrast to the effective monthly rate with daily compounding.Scaling up average daily returns to one month returns does not change the significance of forthcoming cross-sectional tests with either simulated or actual stock returns. This scaling only changes the magnitude of themarket price of risk from daily to monthly risk premiums. Typically, asset pricing studies utilize monthly returnsand report monthly risk premiums in cross-sectional tests.

9

daily returns assumed to be normally distributed. Given that the CAPM is based on expected

returns, the average daily return within one month is a potential proxy for expected returns that

has not previously been considered in asset pricing tests. Also, because the CAPM does not

explicitly take into account compounding period-by-period returns in its derivation, average daily

rates may well be more consistent with CAPM theory than commonly-used effective monthly

returns. While the realized return captures the actual investor return in a one-month period

of time, estimated monthly return Rit filters out daily return compounding effects to focus on

mean daily returns. Another way to interpret Rit is as a rescaled mean daily return that is

computed each month. Every 21 or trading days or so, we estimate the mean daily return and

repeatedly estimate this mean return for all months in the sample period. As such, we essentially

utilize mean daily returns within one month for cross-sectional asset pricing tests as opposed

to monthly realized returns. Whether investors price systemic market risk using mean daily

returns or compounded daily returns is an empirical question that the remainder of this paper

addresses. As we will see in forthcoming analyses, mean daily returns but not realized returns

are closely related to beta risk in cross-sectional tests.

Lastly, other possible methods of computing expected returns could be explored, such as a

target expected return based a mean reverting return process over time (e.g., an autoregression,

or AR, model). We prominently cited in the introduction section the study by Brav, Lehavy,

and Michaely (2005), who utilized analysts’ forecasts of expected returns. Based on ex ante

return data gathered by Value Line and First Call from analysts’ target prices for the period

1975 to 2001, these authors tested their cross-sectional relationship to Value Line market betas

and found a t-value as high as 5.1 in different model specifications. Using betas estimated from

realized stock returns, the t-value was as high as 5.6. They did not report test results using

betas estimated with their ex ante analyst returns. Various robustness checks supported their

main finding that beta was positively priced for ex ante returns forecasted by analysts. This

paper is important to the present study to the extent that: (1) realized returns are shown to

be potentially poor proxies for expected returns, and (2) approximated expected returns are

positively related to market beta in the cross-section of common stocks but not realized returns.

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3 Simulation Tests

In this section we utilize simulation analyses to better understand how realized versus estimated

expected returns affect commonly-used Fama-Macbeth cross-sectional asset pricing tests. To do

this we randomly generate sample data with controlled parameters. Because the properties of

the generated data are known ex ante, the analyses should confirm these properties if the testing

method is valid. The steps to create sample data are as follows:

1. Generate 60 years of daily data (21 × 12 × 252 = 15, 120 days) for one hidden variable

drawn from a normal distribution N(µh(t), σ2h(t)) with t running from 1 to 15,120 and

µh(t) = 0.00035 constant across time. We set σh(t) = 0.012 + 0.006sin(πt/2, 520) to

produce a five-year, time-varying volatility cycle. The overall average values of µh and σh

are very similar to actual values for the CRSP market index over the past 60 years.

2. Generate 1,000 stocks’ daily data for the 60-year sample period based on the simple model

Rit = αi(t) + βi(t)Rht + εit, (20)

where Rht is the time series data for the hidden variable generated in the first step,

i = 1, · · · , 1, 000 and t = 1, · · · , 15, 120. In the next section, we impose different assump-

tions about the time-varying characteristics of αi(t), βi(t) and σi,Id(t) (i.e., the standard

deviation of εit) to simulate their effects on cross-sectional tests. These parameters are

fixed in the computation of each stocks’ daily returns.

3. Calculate the realized daily returns for the (equal-weighted) market index as

Rdailymt =1

1000

1000∑i=1

Rit, where t = 1, · · · , 15, 120,

where monthly returns are computed by cumulating every 21 consecutive days of daily

returns from the beginning of the sample period.

4. Steps 1 to 3 produce one path to mimic the world. To produce more paths, we repeat

these steps. If the generated average monthly return of the market index is less than 0.3%

or larger than 1%, we discard the path and generate a new one until the average monthly

11

return is larger than 0.3% and less than 1%. We produce a total of 1,000 valid paths using

the fixed parameters αi(t), βi(t) and σi,Id(t) for each stock (see step two above).

3.1 Cross-Sectional Tests Based on Simulated Realized Returns

Here we report the results of cross-sectional tests using simulated realized returns. Using daily

returns from equation (20), for each path we compute monthly realized returns for all individuals

stocks by compounding 21 days of daily returns. Individual stock betas are estimated using 60

months of monthly realized returns. Subsequently, 25 beta-sorted test assets are formed. Equal-

weighted one-month-ahead monthly realized returns are computed (out-of-sample) for all 25

portfolios for month 61. By rolling forward one month at a time and repeating this procedure

to the end of the 60-year sample period, realized returns for 25 test asset portfolios are created.

This process is repeated 1,000 times to obtain 25 beta-sorted test assets’ returns for each of the

1,000 paths. We seek to explore the sensitivity of cross-sectional test results to different tests

assets created by means of varying market model parameter settings. The following variations

in parameters are investigated:

Assets 1. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and

constant in the sample period; σi,Id(t) = 0

Assets 2. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 4.1] linearly distributed in this range of values and

constant in the sample period; σi,Id(t) = 0

Assets 3. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values

and constant in the sample period; σi,Id(t) = σi,Id(t) ∈ [0, 5]× σm linearly distributed

where σm is the overall standard deviation of market index returns

Assets 4. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and

constant in the sample period; σi,Id(t) = σmexp(x) with x ∼ N(0, 1) (lognormal distributed)

Assets 5. αi(t) = 0; βi(t) ∈ [0.1, 2.1]× x(t) with x(t) varying over time following a mean

reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)

Assets 6. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 2.1]× x(t) with x(t) varying

over time following a mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)

Assets 7. αi(t) follows a mean reversion process; βi(t) = [1+x(t)]×y(t) with x(t) ∼ N(0, 0.5)

and y(t) varying over time following a mean reversion process; σi,Id(t) = σmexp(x) with

x ∼ N(0, 1)

12

Assets 8. αi(t) follows a mean reversion process; βi(t) = [1+x(t)]×y(t) with x(t) ∼ N(0, 0.5)

and y(t) varying over time following the mean reversion process; σi,Id(t) = σmexp(x) with

x ∼ N(0, 1); add another factor Rkt ∼ N(0.0003, 0.015) for daily return generation

and βk(t) ∼ N(0, 0.25)

Assets 9. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 4.1]× x(t) with x(t) varying

over time following a mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1);

add another factor Rkt ∼ N(0.0003, 0.015) for daily return generation and βk(t) ∼ N(0, 0.25)

Assets 10. Same as Assets 9 except βk(t) ∼ N(0, 1).

In test assets 5–8, holding all else the same, we introduce more general simulated data scenarios,

wherein we let βi(t) and αi(t) follow mean reversion processes of the general Ornstein-Uhlenbeck

form:

dxt = κx[µx − xt] + σxdWxt, (21)

where κx > 0, µx > 0, and σx > 0 are parameters, and Wxt denotes the Weiner process

(i.e., random walk in continuous time). These test assets enable insights into the sensitivity of

cross-sectional tests to time-varying beta and time-varying alpha. In the above equation, the

parameters to generate β are κβ = 0.6, µβ = βi(0), and σβ = 0.25, and the parameters to

generate alpha are κα = 0.5, µα = 0, and σα = 0.25× Rm, where Rm is the average daily return

of the market index in the entire sample period.

Cross-sectional tests are conducted using monthly rolling and full sample approaches. For

monthly rolling tests, we regress the 25 tests assets’ monthly returns in the first month of the

sample period on their betas (estimated for the entire 60-year sample period) to obtain λ(1)

in the first month, then roll forward one month at a time to obtain the series λ(T ) for T=1,

..., 720, and compute the average market price of risk denoted as ˆλ1. For full sample tests, we

regress the average monthly returns over the 60-year sample period of the 25 test assets on their

betas (estimated in the 60-year period as before).6 The resultant average market price of risk is

denoted as λ2.

6Relevant to cross-sectional tests using realized returns, to be consistent with the calculation of monthly realizedreturns, it would seem reasonable to compute compounded monthly returns in the 60-year sample period, whichis the realized return for the entire sample period, and then take the geometric mean of this total realized returnsto get the mean monthly return. However, previous studies use the simple arithmetic mean of monthly returnsrather than geometric mean returns for the full sample period. We follow this standard practice in cross-sectionaltests.

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The results for the 10 test asset cases defined above are shown in Table 1. Superiorly,

using 1,000 different paths to repeatedly perform cross-sectional tests, the full sample tests

indicate that virtually 100 percent of the time λ2 is significantly different from zero (with t-

values exceeding 16 on average). For all 10 test assets, the estimated average market prices of

risk are close to the actual market return (i.e., the maximum difference is only 0.05 percent

per month). By contrast, while the monthly rolling tests normally find that ˆλ1 is significantly

different from zero with t-values exceeding 2.0 from 80 percent to 90 percent of the time, and

the average market prices of risk is the same as those for the full sample approach, the results

are not as compelling as the full sample tests. Notice that the monthly rolling tests for Assets

2 with a wider range of beta values are frequently unable to price market risk, with ˆλ1 having

significant t-values exceeding 2.0 only about 17 percent of the time. As discussed in the previous

section, realized returns contain the unexpected component 12 β

2i σ

2mT , which becomes larger as

beta increases, all else the same. By increasing unexpected returns via this component, it is

more difficult to price market risk. This result is interesting due to the fact that most researchers

broaden the range of betas among test assets in an effort to significantly price market risk. In

this regard, the full sample tests are not sensitive to the range of test assets’ betas, which lends

further support for this approach to cross-sectional tests.

3.2 Cross Sectional Tests Based on Simulated Estimated Expected Returns

The results in the previous section document that full sample λ2s are significant more frequently

in 1,000 simulated paths than monthly rolling λ1s. A plausible reason for this difference is

that average monthly returns for the 60-year sample period in the full sample approach are

good approximations for expected returns, which boosts cross-sectional test results. To improve

the monthly rolling test results, we propose to replace realized monthly returns with expected

monthly returns approximated as defined earlier in equation (19). Using a 60-month estimation

period, beta is estimated for each stock by regressing its estimated expected monthly returns on

the estimated expected monthly return for the market index. Next, stocks are sorted on betas

to form 25 portfolios, after which expected returns for the 25 test asset portfolios are computed

in month t = 61 (i.e., out-of-sample). This process is rolled forward one month as a time to

generate 25 test assets’ estimated expected monthly returns from month 61 to the last month

of the 60-year sample period.

14

Using these approximated expected returns, the monthly rolling and full sample cross-

sectional test results for Assets 1–10 are shown in Table 2. The monthly rolling results for

ˆλ1 are somewhat stronger than Table 1 using realized returns. Both the average t-values and

percentage of 1,000 paths with t-values exceeding 2.0 are higher for all test assets compared to

results in Table 1. Turning to full sample results, we see that the average λ2s across test assets

are similar in magnitude to Table 1 and again 100 percent have t-values exceeding 2.0. In sum,

these simulation results indicate that: (1) the full sample approach provides cross-sectional test

results that more reliably favor CAPM theory than the monthly rolling approach, especially

for wider ranges of betas among test assets, and (2) estimated expected returns yield more

significant cross-sectional test statistics than realized returns.

4 Cross-Sectional Tests Using Actual U.S. Stock Returns

In this section we conduct cross-sectional Fama-MacBeth tests of actual U.S. stock returns.

First, we construct different beta-sorted test assets and perform monthly rolling and full sample

approach tests. Second, we repeat these tests using size and book-to-market (BM) sorted as well

as industry portfolios downloaded from Kenneth French’s website. Third, and last, we provide

tests of popular multi-factors, including size, value, and momentum.

4.1 Beta-Sorted Test Assets

Our data consist of all U.S. common stocks (share codes 10 and 11) in both CRSP and COMPU-

STAT databases with at least 60 consecutive months of returns from January 1960 to December

2015. In any given month, we drop stocks in the bottom 10 percent market capitalization of

this population to avoid liquidity issues associated with very small stocks. From these data we

construct four sets of test assets comprised of 25 portfolio returns from January January 1965

to December 2015:

1. Test assets I are value-weighted realized returns for 25 test asset portfolios based on esti-

mated betas using market model equation (2) for individual stocks and the CRSP market

index. For all stocks with 60 monthly returns available from January 1960 to December

1964, 25 beta-sorted portfolios are formed and value-weighted one-month realized returns

for these portfolios are computed in the subsequent month of January 1965. This pro-

15

cedure is repeated by rolling forward one month at a time until the end of the sample

period.

2. Test assets II are equal-weighted realized returns for 25 test asset portfolios based on

estimated betas using market model equation (2) for individual stocks and the CRSP

market index.

3. Test assets III are value-weighted estimated expected returns for portfolios based on esti-

mated betas using the proposed market model equation (18) for individual stocks and the

CRSP market index.

4. Test assets IV are equal-weighted estimated expected returns for portfolios based on esti-

mated betas using the proposed market model equation (18) for individual stocks and the

CRSP market index.

Statistical properties of these test assets in the sample period are shown in Table 3. For value-

weighted returns in test assets I and II, the mean CRSP index realized return and estimated

expected return are the same at 0.89 percent per month. However, for equal-weighted returns in

test assets II and IV, the mean CRSP index realized return is lower at 1.15 percent per month

compared to 1.60 percent per month for CRSP index estimated expected returns. This difference

is consistent with the definition of expected return in equation (8) wherein the expected return

should be greater than the realized return. It is noteworthy that the pattern of estimated

expected returns in test assets IV exhibits a linearly increasing trend from low to high beta

portfolios (i.e, portfolios 1 to 25). Other test assets do not exhibit this increasing relation between

portfolios betas and returns. Not only are estimated expected returns consistently higher than

realized returns (especially for equal-weighted returns), but the Sharpe ratios for test assets

using equal-weighted returns are larger than those for value-weighted returns. Consistent with

this finding, the CRSP index has a Sharpe ratio of 0.21 for equal-weighted estimated expected

returns compared to only 0.11 for value-weighted realized and estimated expected returns. We

interpret these results to mean that the equal-weighted market portfolio dominates the value-

weighted portfolio and, therefore, is closer to the efficient frontier.7 According to CAPM theory,

7Value-weighted portfolio returns introduce a size effect in the computation of returns that weights large stocksmore heavily than small stocks. Since the equal-weighted CRSP index places more emphasis on smaller stocksand has a much larger Sharpe ratio than the value-weighted CRSP index, it appears that small stocks are neededto better approximate efficient portfolios.

16

the most efficient market portfolio should be used in empirical tests. Iin forthcoming cross-

sectional tests of the CAPM, it is worthwhile to recall that the equal-weighted CRSP index is

more efficient than the value-weighted CRSP index.

Cross-sectional Fama-MacBeth procedure results for these test assets are provided in Table

4. Results for both monthly rolling and full sample approaches are reported. Panel A contains

traditional test results with value-weighted realized returns for both test asset portfolios and the

CRSP market index. Using the monthly rolling and full sample approaches, we obtain ˆλ1 = 0.05

with t = 0.18 and λ2 = 0.05 with t = 1.06, respectively, both far from significance. Also, using

these two approaches, we have ˆα1 = 0.45 with t = 2.43 and α2 = 0.45 with t = 9.35, respectively,

which indicate highly significant mispricing errors, and the goodness-of-fit is very low with R-

squared at only 0.04.8 These results are comparable to those of many previous studies that fail

to accept the CAPM based on empirical tests. Similar findings are obtained with equal-weighted

realized returns in Panel B. Using value-weighted estimated expected returns for both test asset

portfolios and the CRSP market index in Panel C improves matter somewhat with a significant

(at the one percent level) market price of risk using the full sample approach, viz., λ2 = 0.15

with t = 2.98, and higher R-squared at 0.27. Combining test assets III and IV by using equal-

weighted (value-weighted) estimated expected returns for test asset portfolios (CRSP index)

in Panel D greatly improves the results with a highly significant market price of risk for the

full sample approach of λ2 = 0.20 with t = 7.18 and relatively high R-squared value at 0.68;

however, ˆλ1 is not significant for the monthly rolling approach.

Strikingly, using equal-weighted estimated expected returns for both test asset portfolios and

the CRSP index in Panel E of Table 4, for the full sample approach the market price of risk is

extremely significant at λ2 = 0.44 with t = 12.71. This cross-sectional t-value far exceeds any

published evidence on the CAPM. By contrast, for the monthly rolling approach, market beta

is marginally significant (at the 10 percent level) at λ2 = 0.44 with t = 1.66. Moreover, the

R-squared value indicates strong goodness-of-fit at 0.87.9 In both Panels D and E, significant

mispricing exists (e.g., in the full sample approach in Panel E, we obtain α2 = 0.54 with t = 3.78),

which suggests that other factors are needed to more fully price test assets. In the next section

8This R-squared value is based on the full sample approach. When researchers report R-squared values for themonthly rolling approach, they typically report the value using the full sample approach. We follow this standardmethodology.

9By comparison, using ex ante returns forecasted by analysts, Brav, Lehavy, and Michaely (2005) estimatedR-squared values in the range of 0.06 to 0.35.

17

we conduct further tests with size, value, and momentum multi-factors in an attempt to mitigate

mispricing errors. In sum, the results in Table 4 corroborate our simulation analyses results in

Tables 1 and 2; that is, the full sample approach with estimated expected returns dominates the

monthly rolling approach and realized returns in terms of detecting a significant market price

of risk in support of the CAPM.

Figure 1 plots selected full sample approach results in Table 4. Using value-weighted portfolio

returns as in Panel A of Table 4, the graph on the left plots the average beta values for the 25

beta-sorted portfolios in the sample period with respect to their average excess realized returns.

By casual inspection, there is little or no discernable relationship evident. In stark contrast, using

equal-weighted portfolio returns as in Panel E of Table 4, the graph on the right of the figure

illustrates the relation between average beta values and average estimated expected returns.

Now a linear relation is clearly apparent between market beta and average excess estimated

expected returns. The latter graph of the empirical CAPM relation is almost a perfect depiction

of CAPM theory. No previous studies to the authors’ knowledge have produced any statistical

or graphical evidence in favor of the CAPM as compelling as those in Table 4 and Figure 1.

4.2 Fama-French Test Assets and Industry Portfolios

Table 5 repeats the above cross-sectional tests of realized and estimated expected returns for

Fama-French tests assets and industry portfolios. From Kenneth French’s website, we down-

loaded value- and equal-weighted returns for 25 test asset portfolios sorted on quintiles of size

and book-to-market(BM) as well as 30 industry portfolios. Estimated expected returns are

computed from daily realized returns. Traditional test results in Panel A with value-weighted

realized returns reveal the following monthly rolling (full sample) approach results: ˆλ1 = −0.33

with t = −0.77 (λ2 = −0.33 with t = −1.20). As in Table 4, the market factor is not signifi-

cantly priced in the cross-section of average stock returns. Again, mispricing errors are highly

significant at the one percent level, and R-squared is relatively low at 0.06. The results for equal-

weighted realized returns in Panel B are similar to Panel A, except that the signs of the market

price of risk switch from negative to positive. Using value-weighted estimated expected returns

in Panel C, now the market factor is significantly priced in the full sample test with 25 size-BM

sorted test assets but the sign is negative instead of positive. In Panel D the market price of

risk λ2 remains negative for the 25 size-BM sorted test assets but the estimated coefficient is

18

insignificant. For the 30 industry portfolios, however, the market price of risk is significant at

λ2 = 0.47 with t = 2.42.

Using equal-weighted estimated expected returns (rather than realized returns) for industry

portfolios, the market factor is even more significant in Panel E of Table 5 at λ2 = 0.53 with

t = 3.03. In this last panel, we see that all test statistics for the market price of risk for both

monthly rolling and full sample approaches are significant for different test assets, including

combined Fama-French and industry test assets. Also, R-squared values are higher in the range

of 0.16 to 0.24 across test assets. These results are encouraging in terms of supporting the

CAPM. Some researchers (e.g., Lewellen, Nagel, and Shanken (2010) and Daniel and Titman

(2012)) have recommended using industry portfolios as test assets (as well as combining them

with Fama-French portfolios) due to their exogeneity with respect to model factors, but empirical

tests using industry portfolios have typically not supported the CAPM. Contrary to Fama and

French’s (1992, 1993, 1995, 1996) studies of market beta using realized returns, based on equal-

weighted estimated expected returns in Panel E, the market factor is significantly priced with

the correct positive sign for 25 size-BM sorted test assets. In sum, we interpret this evidence as

corroborating the use of estimated expected returns to test the CAPM, especially equal-weighted

returns.

4.3 Cross-Sectional Tests of Multi-Factors

We next augment the market model with Fama and French’s size and value factors plus the

momentum factor. The results for the popular Fama-French three-factor model with market,

size, and value factors are reported in Table 6. Using the monthly rolling approach with value-

weighted realized returns, Panel A contains the following results: the market factor is either not

priced or negatively priced (at the 10 percent level) for the 25 size-BM portfolios, size is not

priced for any test assets, value is positively priced (at the 1 percent level) for the 25 size-BM

sorted portfolios, and R-squared values are fairly high at 0.68 for the 25 size-BM portfolios

but low for industry portfolios at 0.07. The full sample approach results for realized returns in

Panel B are similar, with the added findings for the combined size/BM and industry portfolios

that the market factor is negatively priced (at the 1 percent level) and size and value factors

are positively priced (at the 1 percent level). No factors are significantly priced for industry

portfolios. Using the full sample approach with equal-weighted estimated expected returns,

19

Panel C gives results similar to Panels A and B, except that size is priced (at the 5 percent

level) for industry portfolios. The full sample results for equal-weighted estimated expected

returns in Panel D are unchanged for the most part. Contrary to Panel E in Table 5 in which

the market factor is priced with a positive sign in the market model, the market factor is not

priced or is negatively priced in Table 6 when the size and value factors are used to augment

the market model. These results suggest that multicollinearity between the market factor and

size and value multi-factors could be causing the market beta coefficient to change erratically.

Concerning potential multicollinearity, in our sample period the estimated correlations for

value-weighted realized returns between the market factor and both size and value factors are

0.31 and -0.31, respectively. The size and value factors have an estimated correlation of -0.23.

Correlations using estimated expected returns are similar or higher (e.g., the market and size

factors have a correlation coefficient of 0.56). Due to this collinearity issue, we re-tested the size

and value factors as follows:

• Using the entire sample period, the market model is estimated by regressing test asset

portfolios’ monthly returns on CRSP market index returns (i.e., value- or equal-weighted

realized and estimated expected returns).

• Excess returns for (zero-cost investment) portfolios are computed for the entire sample

period as

Rexpt = Rpt − βpRMt. (22)

• Risk-adjusted excess returns Rexpt are regressed on the SMB and HML factors to obtain

βSMB and βHML as

Rexpt = αp + βpSMBSMBt + βpHMLHMLt + εpt. (23)

• Based on the second step in the Fama-MacBeth testing method, the market prices of risk

λSMB and λHML are estimated.

Table 7 reports the results of this risk-adjusted excess return approach for testing the size

and value factors. In Panels A and B using realized returns for the 25 size/BM portfolios, the

20

market prices of the size and value factors are positive and more significant than in Table 6’s

three-factor model (e.g., in Panel B’s full sample approach results, the t-values for size and value

prices are highly significant at 3.94 and 6.92, respectively, compared to 1.40 and 3.03 in Table

6). Importantly, mispricing error is insignificant now.10 And, the estimated R-squared is fairly

good at 0.68, which implies that the size and value factors help to explain risk-adjusted returns

after taking into account market beta. We infer from this evidence that the size and value

factors are better supported via our risk-adjusted excess return procedure than the traditional

three-factor model. An advantage of this risk-adjusted procedure is that the incremental pricing

effects of multi-factors can be tested to determine if they contribute beyond the market factor

in explaining the cross-section of average stock returns. Turning to the estimated expected

return results in Panels C and D, the results using the 25 size and BM portfolios and combined

portfolios with industries are somewhat stronger than Panels A and B in favor of the size and

value factors. Also, for the industry test assets, the size factor is negatively priced in Panels A

and B (at the 10 percent and 1 percent levels, respectively), but the value factor is not priced.

Weaker results for industry portfolios are consistent with other studies.

We repeated the above analyses for the four-factor Carhart (1997) model with market, size,

value, and momentum factors in Table 8. Most of the results for the size and value factors are

similar to Table 7’s results, with the exception that size is no longer priced in Panels B, C,

and D with estimated expected returns. It appears that the momentum factor has supplanted

size, as momentum is always priced in Panels A to D for both size and BM portfolios as well as

combined portfolios with industries.11 Due to low estimated correlations in the sample period

between size and momentum (i.e., -0.00 and -0.06 for realized and estimated expected returns,

respectively), multicollinearity does not appear to be the reason for size dropping out.12 Like

the three-factor model, no mispricing error exists for tests using estimated expected returns, but

mispricing occurs in some test assets using realized returns. Also, the momentum factor is not

priced in industry portfolios. In this regard, based on our earlier results in Tables 5 and 6, the

only factor that is priced for industry portfolios is the market factor, which in particular was

10Brav, Lehavy, and Michaely (2005) found significant alphas in all of their empirical tests using ex ante analysts’returns. They interpreted this finding to mean that other factors are missing from their models.

11Recent work by Asness, Frazzini, Israel, Moskowitz, and Pederssen (2015) has shown that the size effect ismasked to some extent by the quality of firms. Upon controlling for quality, they found that the returns to size aremore stable and significant than otherwise. Also, momentum did not explain the returns to size-quality portfolios.Interactions between multi-factors are recommended for further study by the authors.

12The estimated correlations between momentum and the excess market returns are -0.13 and -0.27 for realizedand estimated expected returns, respectively.

21

found for equal-weighted estimated expected returns and the full sample approach.

Overall, these results suggest that zero-investment portfolios are needed to augment the

CRSP market index in order to better approximate the cross-section of expected stock returns.

4.4 Interpretation of Empirical Results

Our t-value result of 12.7 for the cross-sectional relation between one-month-ahead, equal-

weighted estimated expected returns and market beta using the full sample approach is re-

markable. No previous studies in the literature report results anywhere near this significance

in empirical tests of the CAPM’s market factor. Indeed, most empirical studies infer that the

CAPM is dead. Our results beg the question: Why is the market factor priced in the monthly

cross-section of average daily stock returns rather than daily compounded realized rates of re-

turn? A number of possible explanations are plausible.

First, in the cross-section, investors price different stocks’ return and risk profiles in the

marketplace for a fixed time period. The CAPM does not specify the time period used by

investors. If it is one day, then using compounded daily returns will not capture investors’

decision-making process.

Second, compound returns incorporating earnings on previous earnings are a mathematical

result of the force of interest. This mathematical outcome may not be priced by investors,

only the period-by-period returns. By analogy, assume a newly-issued bond has a par value of

$1,000. Investors price bonds to pay coupons bearing a simple interest rate that reflects the

bond’s riskiness. The yield-to-maturity will equal the coupon rate at issuance. However, after

one period, interest payments can be reinvested to earn compound interest in the next period.

Because interest rates are constantly changing over time, coupon interest will likely be reinvested

at a different rate of return than the original coupon rate. In turn, the yield-to-maturity on

the bond will differ from the coupon rate, e.g., higher (lower) than the coupon rate if interest

rates have increased (decreased) in the next period. It is clear that realized returns such as

the yield-to-maturity come about in multi-period settings. Nonetheless, the coupon rate that

represents the underlying risk of the bond is fixed over time. This simple interest rate was priced

at the time the bond was issued based on expected future cash flows over the life of the bond.

Likewise, rating agencies such as Standard & Poor’s and Moody’s assign credit rating grades of

bonds in view of their current and expected future probabilities of payment. As a rule, after

22

issuance the agencies only change a bond rating at some future date in the case of a material

change in the issuer’s financial condition that would alter the credit quality of its bonds.

Third, cross-sectional tests are period-by-period tests, not time-series tests. In this context,

the average daily rate (per month) is more appropriate than a compound rate. Popular monthly

rolling approach tests divide up a long sample period of many years into one-month increments

or snapshots. Also, full sample approach tests utilize the simple arithmetic average of monthly

returns for portfolios and relate these returns to their estimated betas in the sample period.

Fourth, the classical CAPM is a single-period model13 rather than a multi-period model. In

a single-period, cross-sectional world, our empirical results strongly suggest that "... the capital

market operates ‘as if’ the assumptions of the CAPM were satisfied." (Merton(1973, p. 867))

Fifth, investors frequently trade in stocks. If a stock is not held for long durations and

dividend payments are not reinvested, compound returns on an individual stock may not be

relevant to many investors. For frequent traders, pricing occurs in short periods of time, such

that the focus is on simple returns rather than effective compounded returns. Investors that

buy-and-hold for long durations of time play a minor role in the day-to-day trading process of

pricing stocks.

Sixth, and last, compounding realized rates of return multiplies the unexpected (noise) com-

ponent actual returns. Referring back to equation (15), large increases or decreases in daily

returns will further magnify noise and cause large differences between simple returns and effec-

tive returns. In so doing, as discussed in Section 2, the order of realized returns for portfolios

across their respective betas can become muddled. We earlier saw in Table 3 that average

equal-weighted estimated expected returns for 25 beta-sorted portfolios (i.e., test assets IV)

monotonically increase from low to high beta portfolios. This linear relation is not observed for

realized returns (i.e., test assets I and II). With no observable pattern or trend observable in

the test assets, it is not surprising that market beta is not priced with realized returns.

These potential explanations for the strength of our cross-sectional test results using average

daily returns (scaled up to one month) are not exhaustive. For example, behavioral attributes

of stock market investors may be relevant and therefore worth consideration. Certainly robust

support for the CAPM using monthly average daily returns challenges our understanding of

13Merton (1973) developed the Intertemporal Capital Asset Pricing Model (ICAPM), which is a continuoustime alternative to the CAPM. In his paper, Merton offers the ICAPM to help overcome criticisms of the single-period CAPM and provide a model that is more consistent with empirical evidence (e.g., he cites weak evidencefor the CAPM by Black, Jensen, and Scholes (1972)).

23

general market equilbrium and modern portfolio theory. Regardless of personal views about

monthly average daily versus realized returns, the fact is that the CAPM lives for estimated

expected returns rather than realized returns.

5 Conclusion

This paper proposed that empirical tests of the CAPM are flawed due the gap between theory

based on expected returns and real world tests using realized returns. A number of issues were

discussed in using realized returns to proxy expected returns, including their mathematical in-

equality, the relation between expected returns and simple mean returns, the decomposition of

realized returns into expected and unexpected (noise) components, compounding effects within

realized returns on unexpected return components and beta estimates, and investor perceptions

of expected returns. Importantly, realized returns over time represent only one possible path

drawn from a distribution of return paths. In the real world, how investors arrive at expected

returns for assets is an open question. A number of alternative mean return measures are rea-

sonable suspects. To estimate expected returns we proposed a simple solution – namely, the

average daily return within the month represents an approximation. This estimated expected

return is easy for investors to compute and does not compound unexpected return components.

Alternatively, previous work by Brav, Lehavy, and Michaely (2005) approximated ex ante ex-

pected U.S. stock returns using analysts’ forecasts gathered by Value Line and First Call. in

the period 1975 to 2001, the authors found that market beta was significantly priced in the

cross-section of ex ante returns.

We began our investigation of realized and estimated expected returns by conducting con-

trolled simulation experiments based on randomly-generated sample data. The results for both

realized and estimated expected returns showed that two-step, cross-sectional Fama-MacBeth

tests using a full sample approach are more likely to detect a significant market price of risk

than the commonly-used monthly rolling approach. For the latter approach, introducing wider

ranges of beta values among the test assets tended to weaken the ability to detect market risk,

which we attribute to its impact on increasing the unexpected component of returns. By con-

trast, the range of betas did not affect full sample approach tests. Particularly relevant to the

present study, estimated expected returns yielded more frequently significant cross-sectional test

24

statistics than realized returns.

Extending the simulation analyses, we repeated cross-sectional tests for actual U.S. common

stock returns from the CRSP database for the sample period 1965 to 2015. Upon construct-

ing beta-sorted portfolios of stocks as test assets, different monthly portfolio return series were

created for this sample period based on value- and equal-weighted realized returns as well as

estimated expected returns. In general, the results corroborated our simulation findings. Like

so many previous studies, in the case of realized returns, the market price of beta risk was

insignificant, positive pricing error was highly significant, and R-squared values were relatively

low. These lackluster findings have motivated researchers to conclude that the CAPM is dead.

However, when we repeated the cross-sectional tests using estimated expected returns, the mar-

ket price of risk became significant, with t-values in the 2.0 to 3.0 range. In line with simulation

outcomes, the highest estimated cross-sectional t-value for market beta was 12.71 using the full

sample approach with equal-weighted estimated expected returns, a remarkable finding that

implies β does a very good job of explaining expected returns. Additionally, R-squared values

were relatively high, but significant mispriciing error was found in all tests.

Further analyses using Fama-French size and book-to-market (BM) portfolios as well as

industry portfolios tended to confirm our beta-sorted portfolio results. While most researchers

cannot price the market factor using industry portfolios, we found that (using estimated expected

returns and the full sample approach) β was significantly priced with t-value slightly more than

3.0. Market beta was priced for size-BM-sorted portfolios also. Interestingly, for these portfolios,

mispricing error was insignificant, which suggested that the size and value factors may be priced

in the cross-section of average stock returns.

Subsequently, we tested multi-factor models by augmenting the market model with size,

value, and momentum factors in addition to utilizing size-BM and industry portfolios as test

assets. Based on Fama-French three-factor model tests using realized returns, the value factor

is positively priced for size-BM portfolios but not industry portfolios. When these tests were

performed for estimated expected returns, both size and value factors were positively priced.

In both of these tests, the market factor was negatively priced and mispricing errors were sig-

nificantly positive. To take into account multicollinearity arising from the correlation of the

market factor with size and value factors, we repeated the three-factor analyses by regressing

risk-adjusted returns (after taking into account the market factor) on the size and value fac-

25

tors. Cross-sectional results for the size and value factors were stronger than in the three-factor

model, and mispricing errors became insignificant. Lastly, adding the momentum factor to the

latter risk-adjusted analyses revealed that size was supplanted by momentum as a priced factor

for the most part. We infer that multi-factors are important in helping to better explain the

cross-section of estimated expected returns.

Consistent with empirical tests of market beta by Brav, Lehavy, and Michaely (2005) using ex

ante analysts’ returns, we conclude from both simulated and real world stock returns that market

beta is priced in the cross-section of estimated expected returns but not realized returns. In cross-

sectional tests using the full sample approach, good estimates of expected monthly returns can

be obtained by using average daily returns for any given month and then averaging estimated

expected monthly returns over a long sample period. A key insight is that cross-sectional

tests based on realized returns employ a single estimate of expected returns; by contrast, the

proposed estimated expected returns are sample-based estimates using (for example) 21 daily

realized returns to obtain an estimate of the expected return. An important implication of

our study is that investors in the marketplace appear to unconditionally price systemic market

risk through average daily returns (per month) rather than daily compounded realized returns.

Further research is recommended on this empirical feature of investor behavior, including other

methods of possibly computing expected returns, surveys of investor attitudes, and controlled

laboratory experiments. Also, the fact that the market model fails for realized returns based on

compounded returns implies that a different asset pricing model than the static (unconditional)

expected return CAPM may be needed to price realized returns.

26

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29

Table 1: Cross-Sectional Fama-MacBeth Tests for the Market Factor Using Realized Returnsfor Different Test Assets Generated for 1,000 Simulated Paths

Ten different tests assets are generated from 1,000 stocks’ returns by varying the market modelparameter settings as shown below. The test assets are realized returns for 25-beta sortedportfolios created by simulation experiments.

Assets 1. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = 0Assets 2. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 4.1] linearly distributed in this range of values and constant thesample period; σi,Id(t) = 0Assets 3. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = σi,Id(t) ∈ [0, 5] × σm linearly distributed where σm is the overall STD ofthe market indexAssets 4. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = σmexp(x) with x ∼ N(0, 1) (lognormal distributed)Assets 5. αi(t) = 0; βi(t) ∈ [0.1, 2.1] × x(t) with x(t) varying over time following a mean reversionprocess; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 6. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 2.1] × x(t) with x(t) varying over timefollowing the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 7. αi(t) follows a mean reversion process; βi(t) = [1 + x(t)]× y(t) with x(t) ∼ N(0, 0.5) and y(t)varying over time following the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 8. αi(t) follows a mean reversion process; βi(t) = [1 + x(t)] × y(t) with x(t) ∼ N(0, 0.5) andy(t) varying over time following the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1); addanother factor RK(t) ∼ N(0.0003, 0.015) for daily returns generation and βK(t) ∼ N(0, 0.25)Assets 9. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 4.1] × x(t) with x(t) varying overtime following a mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1); add another factorRK(t) ∼ N(0.0003, 0.015) for daily returns generation and βK(t) ∼ N(0, 0.25)Assets 10. Same as Assets 9 except βi(t) ∈ [0.1, 2.1]× x(t) with x(t) varying over time following a meanreversion process and βK(t) ∼ N(0, 1)

Using the monthly rolling Fama-MacBeth approach, the estimated market price of risk is based onregressing the first month’s returns for the 25 test assets on estimated betas obtained using the entire60-year sample period, and then rolling forward one month at a time to generate a time series of λ1(t)

estimates which are then averaged to get ˆλ1. Note that the beta for each test asset is the same in eachmonthly cross-sectional regression. Using the full sample approach, the estimated market price of riskdenoted as λ2 is obtained by regressing average monthly returns (over the entire 60-year sample period)for the 25 assets on the estimated full sample betas from time series regression. The term Rvw

m − Rf isthe equal-weighted average excess realized return per month in the sample period. The results shown areaverages from 1,000 valid simulated paths.

Market beta price (λ1s and λ2s) (T = 660 months, 1000 paths)

Portfolios ˆλ1 t-value t > 2(%) λ2 t-value t > 2(%) Rvwm −Rf

Assets 1 0.73 2.96 89.3 0.73 >10 100 0.73Assets 2 0.78 1.66 16.2 0.78 >10 100 0.79Assets 3 0.70 2.76 84.3 0.70 >10 100 0.73Assets 4 0.72 2.80 85.0 0.72 >10 98.9 0.74Assets 5 0.70 2.77 85.7 0.70 >10 98.5 0.73Assets 6 0.72 2.81 84.7 0.72 >10 98.6 0.74Assets 7 0.67 2.84 83.5 0.67 >10 98.5 0.70Assets 8 0.67 2.82 82.8 0.67 >10 97.7 0.70Assets 9 0.74 1.57 14.6 0.74 >10 97.0 0.79Assets 10 0.70 2.65 78.8 0.70 >10 97.9 0.73

30

Table 2: Cross-Sectional Fama-MacBeth Tests for the Market Factor Using Estimated ExpectedReturns for Different Test Assets Generated for 1,000 Simulated Paths

Ten different tests assets are generated from 1,000 stocks’ returns by varying the market modelparameter settings as shown below. The test assets are estimated expected returns for 25-betasorted portfolios created by simulation experiments.

Assets 1. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = 0Assets 2. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 4.1] linearly distributed in this range of values and constant thesample period; σi,Id(t) = 0Assets 3. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = σi,Id(t) ∈ [0, 5] × σm linearly distributed where σm is the overall STD ofthe market indexAssets 4. αi(t) = 0; βi(t) = βi(0) ∈ [0.1, 2.1] linearly distributed in this range of values and constant inthe sample period; σi,Id(t) = σmexp(x) with x ∼ N(0, 1) (lognormal distributed)Assets 5. αi(t) = 0; βi(t) ∈ [0.1, 2.1] × x(t) with x(t) varying over time following a mean reversionprocess; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 6. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 2.1] × x(t) with x(t) varying over timefollowing the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 7. αi(t) follows a mean reversion process; βi(t) = [1 + x(t)]× y(t) with x(t) ∼ N(0, 0.5) and y(t)varying over time following the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1)Assets 8. αi(t) follows a mean reversion process; βi(t) = [1 + x(t)] × y(t) with x(t) ∼ N(0, 0.5) andy(t) varying over time following the mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1); addanother factor RK(t) ∼ N(0.0003, 0.015) for daily returns generation and βK(t) ∼ N(0, 0.25)Assets 9. αi(t) follows a mean reversion process; βi(t) ∈ [0.1, 4.1] × x(t) with x(t) varying overtime following a mean reversion process; σi,Id(t) = σmexp(x) with x ∼ N(0, 1); add another factorRK(t) ∼ N(0.0003, 0.015) for daily returns generation and βK(t) ∼ N(0, 0.25)Assets 10. Same as Assets 9 except βi(t) ∈ [0.1, 2.1]× x(t) with x(t) varying over time following a meanreversion process and βK(t) ∼ N(0, 1)

Using the monthly rolling Fama-MacBeth approach, the estimated market price of risk is based onregressing the first month’s returns for the 25 test assets on estimated betas obtained using the entire60-year sample period, and then rolling forward one month at a time to generate a time series of λ1(t)

estimates which are then averaged to get ˆλ1. Note that the beta for each test asset is the same in eachmonthly cross-sectional regression. Using the full sample approach, the estimated market price of riskdenoted as λ2 is obtained by regressing average monthly returns (over the full 60-year sample period) forthe 25 assets on the estimated entire sample betas from time series regression. The term Rew

m −Rf is theequal-weighted average estimated excess expected return per month in the sample period. The resultsshown are averages from 1,000 valid simulated paths.

Market beta price (λ1 and λ2) (T = 660 months, 1000 paths)

Portfolios ˆλ1 t-value t > 2(%) λ2 t-value t > 2(%) Rewm −Rf

Assets 1 0.73 3.01 90.5 0.73 >10 100 0.73Assets 2 0.80 1.75 29.7 0.80 >10 100 0.80Assets 3 0.70 2.80 84.0 0.70 >10 100 0.73Assets 4 0.72 2.91 88.5 0.72 >10 100 0.74Assets 5 0.72 2.90 88.4 0.72 >10 100 0.73Assets 6 0.72 2.91 87.3 0.72 >10 100 0.74Assets 7 0.68 2.96 87.7 0.68 >10 100 0.70Assets 8 0.68 2.94 87.9 0.68 >10 100 0.70Assets 9 0.80 1.73 27.3 0.80 >10 100 0.79Assets 10 0.70 2.44 68.5 0.70 >10 98.2 0.73

31

Table 3: Statistical Properties of Test Asset Portfolios Sorted on Market Beta

Based on CRSP data, test assets are created by estimating individual stock betas using themarket model with U.S. common stocks’ monthly returns for the 60-year sample period January1965 to December 2015 and then forming the stocks into 25 beta-sorted portfolios, i.e., portfolio1 (25) has the lowest (highest) beta. Both value- and equal-weighted CRSP market index returnsare employed, where returns are either realized or estimated expected returns.

Realized Returns Estimated Expected ReturnsI. Value-Weighted II. Equal-Weighted III. Value-Weighted IV. Equal-Weighted

Portf µ σµ−Rf

σ µ σµ−Rf

σ µ σµ−Rf

σ µ σµ−Rf

σ

1 0.69 4.15 0.07 0.81 3.10 0.13 0.79 4.09 0.09 0.99 3.13 0.192 0.83 3.77 0.11 0.90 2.98 0.16 0.92 3.56 0.15 0.96 2.95 0.193 0.88 3.69 0.13 1.00 3.11 0.19 0.85 3.57 0.12 1.06 3.09 0.214 0.95 3.70 0.15 1.04 3.39 0.19 0.99 3.71 0.16 1.21 3.37 0.245 0.80 3.90 0.10 1.06 3.68 0.18 0.91 3.91 0.13 1.21 3.70 0.226 0.96 3.92 0.14 1.13 3.85 0.19 0.94 3.90 0.14 1.25 3.88 0.227 0.91 4.06 0.13 1.16 4.08 0.18 1.03 4.27 0.15 1.29 4.06 0.228 0.97 4.39 0.13 1.11 4.28 0.16 1.05 4.34 0.15 1.26 4.23 0.209 0.95 4.39 0.13 1.14 4.48 0.16 0.93 4.35 0.12 1.31 4.51 0.2010 0.90 4.47 0.11 1.18 4.69 0.17 1.09 4.38 0.16 1.30 4.64 0.1911 1.01 4.58 0.13 1.22 4.86 0.17 0.94 4.64 0.12 1.32 4.80 0.1912 0.88 4.74 0.10 1.11 5.01 0.14 1.01 4.72 0.13 1.36 4.96 0.1913 0.87 4.87 0.10 1.21 5.14 0.16 0.85 5.01 0.09 1.30 5.13 0.1714 0.80 5.04 0.08 1.28 5.37 0.16 0.96 4.96 0.11 1.42 5.38 0.1915 1.02 5.12 0.12 1.26 5.57 0.15 0.98 5.29 0.11 1.43 5.49 0.1916 0.94 5.38 0.10 1.13 5.75 0.13 1.03 5.33 0.12 1.38 5.64 0.1717 0.94 5.70 0.09 1.23 6.03 0.14 0.87 5.68 0.08 1.34 5.94 0.1618 0.83 5.93 0.07 1.21 6.36 0.13 1.12 5.78 0.12 1.39 6.20 0.1619 0.95 5.86 0.09 1.11 6.55 0.11 1.04 6.18 0.10 1.43 6.41 0.1620 0.94 6.38 0.08 1.21 6.95 0.12 1.03 6.36 0.10 1.42 6.67 0.1521 0.79 6.68 0.06 1.16 7.22 0.10 0.88 6.59 0.07 1.50 7.18 0.1522 0.93 7.23 0.07 1.14 7.81 0.09 0.89 7.10 0.07 1.50 7.63 0.1423 0.81 7.69 0.05 1.13 8.36 0.09 1.16 7.92 0.10 1.45 8.15 0.1324 0.99 8.36 0.07 0.95 9.23 0.06 0.99 8.41 0.07 1.56 8.92 0.1325 0.94 9.42 0.06 1.00 10.71 0.06 1.21 9.54 0.08 1.72 10.46 0.13CRSP 0.89 4.49 0.11 1.15 5.68 0.13 0.89 4.45 0.11 1.60 5.60 0.21

32

Table 4: Cross-Sectional Fama-MacBeth Tests for the Market Factor Using Realized and Esti-mated Expected U.S. Stock Market Returns for Beta-Sorted Test Asset Portfolios

Test assets are created by estimating individual stock betas using the market model with monthlyreturns for the 60-year sample period January 1965 to December 2015 and then forming thestocks into 25 beta-sorted portfolios. Test assets are formed using both value- and equal-weightedCRSP market index returns. Implementing the Fama-MacBeth test procedure, in the first time-series regression step, the market model with monthly returns is used to estimate the marketbetas for 25 test asset portfolios for the 60-year sample period. Here both equal-weighted andvalue-weighted returns are used for test asset portfolios as well as the CRSP market index. Also,both realized returns and estimated expected returns are used. In the second cross-sectional step,the market price of risk denoted as λ1 is estimated by regressing the returns for test assets inthe first month of the sample period on their estimated full-sample betas. Subsequently, rollingforward one month at a time, this second step is repeated until the last month of the sample

period to generate a time series of λ1(t) estimates, which are then averaged to obtain ˆλ1 thatis tested for significant difference from zero. Using this monthly rolling approach, the portfoliobeta for each test asset is the same in each monthly cross-sectional regression. Using the fullsample approach, step two is changed by regressing the average monthly returns over the fullsample period for each test asset on estimated full-sample betas to obtain the estimated marketprice of risk denoted as λ2.

Panel A. Value-weighted realized returns

ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

0.45 2.43 0.05 0.18 0.45 9.35 0.05 1.06 0.04

Panel B. Equal-weighted realized returns

ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

0.65 4.45 0.07 0.25 0.65 10.74 0.07 1.09 0.05

Panel C. Value-weighted estimated expected returns

ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

0.43 2.31 0.15 0.57 0.43 7.99 0.15 2.98 0.27

Panel D. Equal-(value-)weighted expected returns for 25 portfolios (CRSP index)

ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

0.72 4.86 0.20 0.85 0.72 22.83 0.20 7.18 0.68

Panel E. Equal-weighted estimated expected returns

ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

0.54 3.78 0.44 1.66 0.54 16.27 0.44 12.71 0.87

33

Table 5: Cross-Sectional Fama-MacBeth Tests for the Market Factor Using Realized and Esti-mated Expected Returns for Fama-French Test Assets and Industry Portfolios

From Kenneth French’s website, we downloaded daily and monthly returns for 25 size and book-to-market(BM) portfolios as well as 30 industry portfolios for the 60-year sample period January1965 to December 2015. Value- and equal-weighted portfolio and CRSP market index returns aredownloaded. Implementing the Fama-MacBeth test procedure, in the first time-series regressionstep, the market model with monthly returns is used to estimate the market betas for 25 testasset portfolios for the 60-year sample period. Here both equal-weighted and value-weightedreturns are used for test asset portfolios as well as the CRSP market index. Also, both realizedreturns and estimated expected returns are used. In the second cross-sectional step, the marketprice of risk denoted as λ1 is estimated by regressing the returns for test assets in the firstmonth of the sample period on their estimated full-sample betas. Subsequently, rolling forwardone month at a time, this second step is repeated until the last month of the sample period to

generate a time series of λ1(t) estimates, which are then averaged to obtain ˆλ1 that is testedfor significant difference from zero. Using this monthly rolling approach, the portfolio beta foreach test asset is the same in each monthly cross-sectional regression. Using the full sampleapproach, step two is changed by regressing the average monthly returns over the full sampleperiod for each test asset on estimated full-sample betas to obtain the estimated market priceof risk denoted as λ2.

Panel A. Value-weighted realized returns

Portfolios ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

FF25 1.06 2.76 -0.33 -0.77 1.06 3.59 -0.33 -1.20 0.06Ind30 0.66 2.84 -0.07 -0.23 0.66 4.51 -0.07 -0.49 0.01FF25+Ind30 0.75 3.08 -0.10 -0.34 0.75 5.05 -0.10 -0.75 0.01

Panel B. Equal-weighted realized returns

Portfolios ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

FF25 0.71 2.49 0.06 0.17 0.71 3.29 0.06 0.25 0.00Ind30 0.72 3.31 0.05 0.17 0.72 4.37 0.05 0.32 0.00FF25+Ind30 0.72 3.10 0.05 0.18 0.72 5.50 0.05 0.41 0.00

Panel C. Value-weighted estimated expected returns

Portfolios ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

FF25 1.23 2.44 -0.53 -1.32 1.23 4.70 -0.53 -2.19 0.17Ind30 0.69 3.05 -0.12 -0.41 0.69 4.90 -0.12 -0.88 0.03FF25+Ind30 0.84 3.53 -0.21 -0.71 0.84 6.03 -0.21 -1.64 0.05

Panel D. Value-(equal-)weighted expected returns for CRSP index (25 portfolios)

Portfolios ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

FF25 1.4 3.94 -0.41 -1.09 1.4 2.19 -0.41 -0.71 0.02Ind30 0.70 3.00 0.47 1.54 0.70 3.23 0.47 2.42 0.17FF25+Ind30 0.94 3.72 0.15 0.47 0.94 2.98 0.15 0.52 0.01

Panel E. Equal-weighted estimated expected returns

Portfolios ˆα1 t-value ˆλ1 t-value α2 t-value λ2 t-value R-squared

FF25 0.16 0.57 0.87 2.56 0.16 0.40 0.87 2.10 0.16Ind30 0.69 3.33 0.53 1.79 0.69 3.88 0.53 3.03 0.24FF25+Ind30 0.34 1.54 0.79 2.61 0.34 1.61 0.79 3.68 0.21

34

Table 6: Cross-Sectional Fama-MacBeth Tests for the Three-Factor Model Using Realized andEstimated Expected Returns for Fama-French Test Assets and Industry Portfolios

From Kenneth French’s website, we downloaded daily and monthly returns for 25 size andbook-to-market(BM) portfolios as well as 30 industry portfolios for the 60-year sample periodJanuary 1965 to December 2015. Value- and equal-weighted portfolio and CRSP market indexreturns are downloaded in addition to the Fama-French size ((SMB) and value (HML) factors.Implementing the Fama-MacBeth test procedure, in the first time-series regression step, thethree-factor model with monthly returns is used to estimate market, size, and value betas for 25test asset portfolios for the 60-year sample period. Here both equal- and value-weighted returnsare used for test asset portfolios as well as the CRSP market index. Also, both realized returnsand estimated expected returns are used. In the second cross-sectional step, the market pricesof risk denoted as λ1 for the factors are estimated by regressing the returns for test assets inthe first month of the sample period on their estimated full sample betas. Subsequently, rollingforward one month at a time, this second step is repeated until the last month of the sampleperiod to generate a time series of λ1(t) estimates for each factor, which are then averaged to

obtain ˆλ1 that is tested for significant difference from zero. Using this monthly rolling approach,the estimated portfolio CRSP index, SML, and HML betas for each test asset are the same ineach monthly cross-sectional regression. Using the full sample approach, step two is changedby regressing the average monthly returns over the full sample period for each test asset onestimated full-sample betas to obtain the estimated market price of risk denoted as λ2 for thethree factors.

Panel A. Monthly rolling approach results using value-weighted realized returns

Portfolios ˆα1 t-val ˆλ1MKT t-val ˆλ1SMB t-val ˆλ1HML t-val R-squared

FF25 1.15 4.00 -0.63 -1.83 0.18 1.40 0.36 3.03 0.68Ind30 0.74 2.63 -0.12 -0.36 0.01 0.07 -0.12 -0.78 0.07FF25+Ind30 1.00 4.24 -0.44 -1.48 0.17 1.28 0.20 1.58 0.29

Panel B. Full sample approach results using value-weighted realized returns

Portfolios α2 t-val λ2MKT t-val λ2SMB t-val λ2HML t-val R-squared

FF25 1.15 4.00 -0.63 -1.83 0.18 1.40 0.36 3.03 0.68Ind30 0.74 2.63 -0.12 -0.36 0.01 0.07 -0.12 -0.78 0.07FF25+Ind30 1.00 5.69 -0.44 -2.55 0.17 3.21 0.20 3.11 0.29

Panel C. Monthly rolling approach results using equal-weighted estimated expected returns

Portfolios ˆα1 t-val ˆλ1MKT t-val ˆλ1SMB t-val R-squared

FF25 2.31 8.71 -0.99 -2.86 0.76 5.79 0.49 4.08 0.89Ind30 1.40 5.26 -0.09 -0.27 0.53 2.19 -0.21 -1.16 0.55FF25+Ind30 1.65 8.07 -0.36 -1.20 0.76 5.73 0.29 2.28 0.75

Panel D. Full sample approach results using equal-weighted estimated expected returns

Portfolios α2 t-val λ2MKT t-val λ2SMB t-val R-squared

FF25 2.31 5.93 -0.99 -2.73 0.76 12.24 0.49 7.04 0.89Ind30 1.40 6.59 -0.09 -0.48 0.53 3.71 -0.21 -1.72 0.55FF25+Ind30 1.65 7.49 -0.36 -1.78 0.76 12.40 0.29 4.07 0.75

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Table 7: Cross-Sectional Fama-MacBeth Tests for the Size and Value Factors Using Realizedand Estimated Market Beta Risk-Adjusted Excess Returns for Fama-French Test Assets andIndustry Portfolios

From Kenneth French’s website, we downloaded daily and monthly returns for 25 size and book-to-market(BM) portfolios as well as 30 industry portfolios for the 60-year sample period January1965 to December 2015. Value- and equal-weighted portfolio and CRSP market index returns aredownloaded in addition to the Fama-French size ((SMB) and value (HML) factors. Implementingthe Fama-MacBeth test procedure, in the first time-series regression step, the market model withmonthly returns is used to estimate market betas βp for 25 test asset portfolios for the 60-yearsample period. Here both equal- and value-weighted returns are used for test asset portfolios aswell as the CRSP market index. Also, both realized returns and estimated expected returns areused. Excess monthly returns for each portfolio are computed as follows: Rexpt = Rpt − βpRmt,where Rmt is the monthly CRSP index return. Subsequently, these excess returns are regressedon the SMB and HML factors to estimate these multi-factors’ loadings for the full sample period.In the second cross-sectional step, the market price of risk denoted as λ1 for each multi-factor isestimated by regressing the excess returns for test assets in the first month of the sample periodRexpt=1 on their estimated full sample multi-factor betas. Rolling forward one month at a time,this second step is repeated until the last month of the sample period to generate a time series

of λ1(t) estimates for each multi-factor, which are then averaged to obtain ˆλ1 that is tested forsignificant difference from zero. Using this monthly rolling approach, the estimated portfolioSML and HML betas for each test asset are the same in each monthly cross-sectional regression.Using the full sample approach, step two is changed by regressing the average monthly returnsover the full sample period for each test asset on estimated full-sample betas to obtain theestimated market price of risk denoted as λ2 for the SML and HML factors.

Panel A. Monthly rolling approach results using value-weighted realized returns

Portfolios α1 t-value λ1SMB t-value λ1HML t-value R-squared

FF25 -0.00 -0.00 0.06 1.42 0.51 4.05 0.67Ind30 0.15 4.07 -0.39 -1.88 -0.14 -0.21 0.25FF25+Ind30 0.07 2.26 -0.02 -0.17 0.28 2.16 0.17

Panel B. Full sample approach results using value-weighted realized returns

Portfolios α2 t-value λ2SMB t-value λ2HML t-value R-squared

FF25 -0.00 -0.00 0.06 3.94 0.51 6.92 0.67Ind30 0.15 3.82 -0.39 -2.87 -0.04 -0.29 0.25FF25+Ind30 0.07 1.86 -0.02 -0.34 0.28 3.37 0.17

Panel C. Monthly rolling approach results using equal-weighted estimated expected returns

Portfolios α1 t-value λ1SMB t-value λ1HML t-value R-squared

FF25 -0.03 -1.00 0.35 2.18 0.71 5.20 0.67Ind30 0.03 0.99 -0.23 -0.78 -0.12 -0.62 0.03FF25+Ind30 -0.02 -0.97 0.35 2.09 0.49 3.41 0.38

Panel D. Full sample approach results using equal-weighted estimated expected returns

Portfolios α2 t-value λ2SMB t-value λ2HML t-value R-squared

FF25 -0.03 -0.46 0.35 2.47 0.71 5.42 0.67Ind30 0.03 0.55 -0.23 -0.82 -0.12 -0.61 0.03FF25+Ind30 -0.02 -0.58 0.35 2.72 0.49 4.09 0.38

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Table 8: Cross-Sectional Fama-MacBeth Tests for the Size, Value, and Momentum Factors UsingRealized and Estimated Market Beta Risk-Adjusted Excess Returns for Fama-French Test Assetsand Industry Portfolios

From Kenneth French’s website, we downloaded daily and monthly returns for 25 size andbook-to-market(BM) portfolios as well as 30 industry portfolios for the 60-year sample periodJanuary 1965 to December 2015. Value- and equal-weighted portfolio and CRSP market indexreturns are downloaded in addition to the Fama-French size ((SMB) and value (HML) factorsand momentum factor. Implementing the Fama-MacBeth test procedure, in the first time-seriesregression step, the market model with monthly returns is used to estimate market betas βpfor 25 test asset portfolios for the 60-year sample period. Here both equal- and value-weightedreturns are used for test asset portfolios as well as the CRSP market index. Also, both realizedreturns and estimated expected returns are used. Excess monthly market returns for eachportfolio are computed as follows: Rexpt = Rpt − βpRmt, where Rmt is the monthly CRSP indexreturn. Subsequently, these excess returns are regressed on the SMB, HML, and MOM factorsto estimate these multi-factors’ loadings for the full sample period. In the second cross-sectionalstep, the market price of risk denoted as λ1 for each factor is estimated by regressing the excessreturns for test assets in the first month of the sample period Rexpt=1 on their estimated full samplefactor betas. Rolling forward one month at a time, this second step is repeated until the lastmonth of the sample period to generate a time series of λ1(t) estimates for each factor, which are

then averaged to obtain ˆλ1 that is tested for significant difference from zero. Using this monthlyrolling approach, the estimated portfolio SML, HML, and MOM betas for each test asset arethe same in each monthly cross-sectional regression. Using the full sample approach, step twois changed by regressing the average monthly returns over the full sample period for each testasset on estimated full-sample betas to obtain the estimated market price of risk denoted as λ2for the three factors.

Panel A. Monthly rolling approach results using value-weighted realized returns

Portfolios α1 t-val λ1SMB t-val λ1HML t-val λ1MOM t-val R-squared

FF25 0.06 2.37 -0.08 -0.54 0.64 4.97 3.25 4.85 0.77Ind30 0.15 3.92 -0.36 -1.61 -0.01 -0.07 0.27 0.44 0.27FF25+Ind30 0.10 2.84 -0.05 -0.34 0.36 2.65 0.97 2.08 0.31

Panel B. Full sample approach results using value-weighted realized returns

Portfolios α2 t-val λ2SMB t-val λ2HML t-val λ2MOM t-val R-squared

FF25 0.06 1.44 -0.08 -1.14 0.64 8.66 3.25 3.06 0.77Ind30 0.15 3.90 -0.36 -2.47 -0.01 -0.10 0.27 0.77 0.27FF25+Ind30 0.10 2.67 -0.05 -0.75 0.36 4.42 0.97 2.97 0.31

Panel C. Monthly rolling approach results using equal-weighted estimated expected returns

Portfolios α1 t-val λ1SMB t-val λ1HML t-val λ1MOM t-val R-squared

FF25 0.02 0.74 -0.11 -0.53 1.03 6.52 3.08 3.94 0.71Ind30 0.04 1.32 -0.39 -1.17 0.07 0.32 1.53 2.11 0.27FF25+Ind30 -0.01 -0.25 0.08 0.39 0.67 3.99 1.74 2.52 0.47

Panel D. Full sample approach results using equal-weighted estimated expected returns

Portfolios α2 t-val λ2SMB t-val λ2HML t-val λ2MOM t-val R-squared

FF25 0.02 0.35 -0.11 -0.35 1.03 4.44 3.08 1.65 0.71Ind30 0.04 0.85 -0.39 -1.57 0.07 0.38 1.53 3.11 0.27FF25+Ind30 -0.01 -0.16 0.08 0.55 0.67 5.41 1.74 3.09 0.47

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Figure 1: The Relation Between Average Excess Returns and Market BetaThe graph on the left shows the results for 25 beta-sorted test asset portfolios with averagemarket betas on the X-axis and average excess (value-weighted) realized returns on the Y-axis. The graph on the right shows the results using average excess (equal-weighted) estimatedexpected returns on the Y-axis. The graphs are based on results in Panels A and E of Table4 using the full sample approach for cross-sectional tests in the sample period January 1965 toDecember 2015.

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