# The conditional beta and the cross-section of expected returns.

We examine the cross-sectional relation between conditional*Subject to change; dependent upon or granted based on the occurrence of a future, uncertain event.*

A

*conditional payment*is the payment of a debt or obligation contingent upon the performance of a certain specified act. betas and expected stock returns for a sample period of July July: see month. 1963 to December December: see month. 2004. Our portfolio-level analyses and the firm-level cross-sectional regressions indicate a positive, significant relation between conditional betas and the cross-section of expected returns. The average return difference between high- and low-beta portfolios ranges between 0.89% and 1.01% per month, depending on the time-varying specification of conditional beta. After controlling for size, book-to-market, liquidity, and momentum, the positive relation between market beta and expected returns remains economically ec·o·nom·i·cal

*adj.*

**1.**Prudent and thrifty in management; not wasteful or extravagant. See Synonyms at sparing.

**2.**Intended to save money, as by efficient operation or elimination of unnecessary features; economic: and statistically significant.

**********

The Sharpe (1964), Lintner (1965), and Black (1972) capital asset pricing model Capital asset pricing model (CAPM)

An economic theory that describes the relationship between risk and expected return, and serves as a model for the pricing of risky securities. (CAPM CAPM

See: Capital asset pricing model

CAPM

See capital-asset pricing model (CAPM). ) implies (logic)

**implies**- (=> or a thin right arrow) A binary Boolean function and logical connective. A => B is true unless A is true and B is false. The truth table is

A B | A => B ----+------- F F | T F T | T T F | F T T | T

It is surprising at first that A => the mean-variance efficiency of the market portfolio in the sense of Markowitz

**Markowitz**- The author of the original Simscript language. (1959). The primary implication

**implication**

In logic, a relation that holds between two propositions when they are linked as antecedent and consequent of a true conditional proposition. Logicians distinguish two main types of implication, material and strict. of the CAPM is that there is a positive linear relation between expected returns on securities and their market betas, and that variables other than beta should not capture the cross-sectional variation in expected returns. However, over the last three decades, many studies have tested the empirical em·pir·i·cal

*adj.*

**1.**Relying on or derived from observation or experiment.

**2.**Verifiable or provable by means of observation or experiment.

**3.**performance of the static (or unconditional HEIR, UNCONDITIONAL. A term used in the civil law, adopted by the Civil Code of Louisiana. Unconditional heirs are those who inherit without any reservation, or without making an inventory, whether their acceptance be express or tacit. Civ. Code of Lo. art. 878.

UNCONDITIONAL. ) CAPM in explaining the cross-section of realized average returns. The findings of these earlier studies indicate that firm size, book-to-market ratio Book-To-Market Ratio

A ratio used to find the value of a company by comparing the book value of a firm to its market value. Book value is calculated by looking at the firm's historical cost, or accounting value. , earnings-to-price ratio, liquidity, and momentum have significant explanatory ex·plan·a·to·ry

*adj.*

Serving or intended to explain: an explanatory paragraph.

**ex·plan**power for average stock returns, but that market beta has little or no power.

Early tests of the CAPM are based on the cross-sectional regressions of average stock returns on estimates of individual stock betas. Two obvious problems with these tests are errors-in-variables and residual Residual

See:Residual value correlations. First, beta estimates for individual stocks are imprecise im·pre·cise

*adj.*

Not precise.

**impre·cisely**

*adv.*and generate a measurement error problem when they are used to explain average returns. To improve the accuracy of estimated betas, Blume Blume ,

**Judy**Born 1938.

American novelist best known for depicting the everyday problems of adolescence. Her works include

*Are You There God? It's Me, Margaret*(1970). (1970), Friend and Blume (1970), and Black, Jensen

**Noun**

**1.**

**Jensen**- modernistic Danish writer (1873-1950)

Johannes Vilhelm Jensen , and Seholes (1972) use portfolios instead of individual stocks in their cross-sectional tests. Since estimates of betas for diversified diversified (di·verˑ·s portfolios are more precise than estimates for individual stocks, using portfolios in the cross-section regressions of average returns on betas diminishes the errors-in-variables problem.

Second, the regression regression, in psychology: see defense mechanism.

**regression**

In statistics, a process for determining a line or curve that best represents the general trend of a data set. residuals Residuals

(1) Part of stock returns not explained by the explanatory variable (the market index return). Residuals measure the impact of firm-specific events during a particular period. have common sources of variation. Positive correlation

**Noun**

**1.**

**positive correlation**- a correlation in which large values of one variable are associated with large values of the other and small with small; the correlation coefficient is between 0 and +1

direct correlation in the residuals yields downward bias in the usual ordinary least squares (OLS OLS Ordinary Least Squares

OLS Online Library System

OLS Ottawa Linux Symposium

OLS Operation Lifeline Sudan

OLS Operational Linescan System

OLS Online Service

OLS Organizational Leadership and Supervision

OLS On Line Support

OLS Online System ) estimates of the standard errors of the cross-sectional regression slopes. Fama and MacBeth (1973) introduce a method

for addressing the inference (logic)

**inference**- The logical process by which new facts are derived from known facts by the application of inference rules.

See also symbolic inference, type inference. problem caused by correlation

**correlation**

In statistics, the degree of association between two random variables. The correlation between the graphs of two data sets is the degree to which they resemble each other. of the residuals in cross-sectional regressions. Rather than estimating a single cross-section regression of average monthly returns on betas, they estimate month-by-month cross-section of regressions of monthly returns on betas. The time-series averages of the monthly slopes and intercepts and their standard errors are used to test whether the average market risk premium is positive and the average intercept is equal to the risk-free rate Risk-free rate

The rate earned on a riskless asset. .

In cross-sectional tests, Douglas (1969), Black, Jensen, and Scholes

**Scholes**(/skowlz/ or /šowlz/) could refer to the following places:

United Kingdom:

- Scholes, Greater Manchester
- Scholes, South Yorkshire
- Scholes, Cleckheaton, Kirklees, West Yorkshire
- Scholes, Holmfirth, Kirklees, West Yorkshire

**1.**an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities.

**2.**on beta is less than the average excess market return and the intercept is greater than the average risk-free interest rate Risk-Free Interest Rate

Describes return available to an investor in a security somehow guaranteed to produce that return. The risk-free interest rate compensataes the investor for the temporary sacrifice of consumption. . In their widely cited study, Fama and French (1992) examine the static version of the CAPM and find both at the firm and portfolio level that the cross-sectional relation between market beta and average return is flat. (1) They interpret To run a program one line at a time. Each line of source language is translated into machine language and then executed. this fiat

*[*Latin, Let it be done.

*] In old English practice, a short order or warrant of a judge or magistrate directing some act to be done; an authority issuing from some competent source for the doing of some legal act.*relation as strong empirical evidence against the CAPM.

As indicated by Jagannathan and Wang (Wang Laboratories, Inc., Lowell, MA) A computer services and network integration company. Wang was one of the major early contributors to the computing industry from its founder's invention that made core memory possible, to leadership in desktop calculators and word processors. (1996), although a flat relation between the unconditional expected return Expected Return

The average of a probability distribution of possible returns, calculated by using the following formula: and the unconditional market beta may be evidence against the static CAPM, it is not necessarily evidence against the conditional CAPM. The CAPM was originally developed within the framework of a hypothetical

**is an adjective, meaning of or pertaining to a hypothesis. See:**

*Hypothetical*- Hypothesis
- Hypothetical
- Hypothetical (album)

There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time.3

In this paper, we investigate whether time-varying conditional betas can explain the cross-section of expected returns at the firm and portfolio level. There is substantial empirical evidence that conditional betas and expected returns depend on the nature of the information available at any given point in time and vary over time. Earlier studies use either a single or rolling long sample of monthly data in estimating beta. Instead, we use daily returns within a month to compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer. realized beta for each stock trading at the New York Stock Exchange

**New York Stock Exchange (NYSE)**

World's largest marketplace for securities. The exchange began as an informal meeting of 24 men in 1792 on what is now Wall Street in New York City. (NYSE NYSE

See: New York Stock Exchange ), American Stock Exchange (Amex), and Nasdaq for our sample period of July 1963 to December 2004. We propose three alternative specifications of expected future beta based on the past information on realized beta using autoregressive Autoregressive

Using past data to predict future data.

Notes:

Essentially it's forecasting, similar to the weather... Sometimes even the weatherman can be caught in an unexpected downpour. , moving average, and generalized gen·er·al·ized

*adj.*

**1.**Involving an entire organ, as when an epileptic seizure involves all parts of the brain.

**2.**Not specifically adapted to a particular environment or function; not specialized.

**3.**autoregressive conditional heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH)

A nonlinear stochastic process, where the variance is time-varying, and a function of the past variance. ARCH processes have frequency distributions which have high peaks at the mean and fat-tails, much like fractal distributions. (GARCH GARCH Generalized Autoregressive Conditional Heteroskedasticity )-in-mean models to obtain time-varying conditional betas for each stock.

We estimate conditional betas by using the entire history of returns on a stock. Hence, the high- and low-conditional beta portfolios we form cannot be exactly replicated by an investor at any given point in time. Our focus is more in the nature of a hypothesis

*An assumption or theory.*

During a criminal trial, a hypothesis is a theory set forth by either the prosecution or the defense for the purpose of explaining the facts in evidence. test that has asymptotic validity. Thus, our approach is somewhat different from standard practice, which identifies ex ante measures of risk based on information available at a given point in time that a particular portfolio will earn a higher return on average than another portfolio.

For each specification of conditional beta, we find that stocks with high (low) market betas have, on average, high (low) average returns. Our portfolio-level analyses and the firm-level cross-sectional regressions indicate that the positive relation between the conditional betas and the cross-section of average returns is economically and statistically significant. Average portfolio returns increase almost monotonically when moving from low- to high-beta portfolios. The [R.sup.2] values from the regression of average portfolio returns on average portfolio betas are in the range of 82% to 98% for 10, 20, 50, and 100 beta portfolios. When we form the equal-weighted decile decile

one of the groups when a series of ranked data is divided into ten equal parts, or dividing points between such groups. See also quartile. portfolios by sorting the NYSE/Amex/Nasdaq stocks based on conditional beta, we find that the average return difference between decile 10 (high beta) and decile 1 (low beta) portfolios ranges between 0.89% and 1.01% per month, depending on the time-varying specification of conditional beta. For 20, 50, and 100 beta portfolios, the average return difference ranges from 1.01% to 1.23% per month.

To check whether our findings are driven by small, illiquid Illiquid

An asset or security that cannot be converted into cash very quickly (or near prevailing market prices).

Notes:

A house is a good example of an illiquid asset.

See also: Cash, Liquidity

Illiquid

In the context of finance. , and low-price stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the well-known cross-sectional effects, including size and book-to-market (Fama and French, 1993, 1995, 1996), liquidity (Amihud, 2002; Pastor and Stambaugh

**Stambaugh**is a city and a township in Iron County, Michigan

- Stambaugh
- Stambaugh Township

*n.*

*New England & Upstate New York*

**1.**A runt, especially one of a litter of pigs.

**2.**A small person. See Regional Note at tit

^{1}. , 1993). After controlling for these effects, we estimate the cross-sectional beta premium as being in the range of 0.86% to 1.46% per month.

The paper is organized as follows. Section I contains the data and variable definitions. In Section II, we discuss the average raw returns and the average risk-adjusted returns on beta portfolios, and in Section III we present the firm-level cross-sectional regression results. Section IV provides a battery of robustness checks, including portfolio-level cross-sectional regressions, testing the long-term predictive power The

**predictive power**of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory. of conditional betas, and some additional tests after controlling for liquidity and momentum, after excluding the Amex and Nasdaq sample, and after controlling for microstructure mi·cro·struc·ture

*n.*

The structure of an organism or object as revealed through microscopic examination.

microstructure

*Noun*

a structure on a microscopic scale, such as that of a metal or a cell effects. In Section V, we investigate whether our main findings are robust for size/BM/beta portfolios. In Section VI, we discuss the cross-sectional implications of the conditional CAPM approach. Section VII concludes the paper.

I. Data and Variable Definitions

Our first data set comprises all NYSE, Amex, and Nasdaq financial and nonfinancial

**Adj.**

**1.**

**nonfinancial**- not involving financial matters

financial, fiscal - involving financial matters; "fiscal responsibility" firms. We obtain this information from the Center for Research in Security Prices This article or section needs

**sources or references**that appear in reliable, third-party publications. Alone, primary sources and sources affiliated with the subject of this article are not sufficient for an accurate encyclopedia article. (CRSP CRSP Collaborative Research Support Program (USA)

CRSP Collaborative Research Support Program

CRSP Center for Research in Security Prices

CRSP Center for Research in Security Prices ) for the period from July 1963 through December 2004. We use the daily stock returns to generate the conditional beta measures. Our second data set is Compustat '''Standard & Poor's Compustat

^{®}is a database of financial, statistical and market information on active and inactive companies throughout the world. Compustat

^{®}data has a reputation for extensive coverage, standardization, expertise and timeliness. , which we use primarily to obtain the book values for individual stocks.

For each month from July 1963 to December 2004, we compute the following variables for each firm in the sample.

A. Size

Following other studies, we measure firm size (ME) by the natural logarithm Natural logarithm

Logarithm to the base

*e*(approximately 2.7183). of the market value of equity (a stock's price times shares outstanding in millions of dollars) for each stock.

B. Book-to-Market

Following Fama and French (1992), we compute a firm's book-to-market ratio (BE/ME) by using its market equity at the end of June of year t - 1 and the book value of common equity plus balance-sheet-deferred taxes for the firm's latest fiscal year ending in calendar year t - 1. To avoid giving extreme observations heavy weight in our analysis, like Fama and French (1992), we set the smallest and largest 0.5% of the observations on book-to-market ratio equal to the next largest and smallest values of the ratio (the 0.005 and 0.995 fractiles).

C. Realized Beta

To estimate the monthly beta for an individual stock, we assume a single-factor return-generating process in the form of a market model:

[R.sub.i,d,t] = [[alpha].sub.i,t] + [[beta].sub.i,t] [R.sub.m,d,t] + [[epsilon].sub.i,d.t], (1)

where [R.sub.i,d,t] is the daily return on stock i on day d of month t, [R.sub.m,d,t] is the daily market return on day d of month t, [[epsilon].sub.i,d.t] is the residual term, (4) [[alpha].sub.i,t] is the intercept, and [[beta].sub.i,t] is the realized beta of stock i in month t. We define the realized beta as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE re·pro·duce

*v.*

**re·pro·duced**,

**re·pro·duc·ing**,

**re·pro·duc·es**

*v.*

*tr.*

**1.**To produce a counterpart, image, or copy of.

**2.**

*Biology*To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], (2)

where [[bar.R].sub.i,t] is the average daily return on stock i in month t, [[bar.R].sub.m,t] is the average daily market return in month t, and n is the number of daily return observations in month t. In our empirical analysis, we measure [R.sub.m,d,t] by the CRSP daily value-weighted index, that is, the daily value-weighted average returns of all stocks trading at the NYSE, Amex, and Nasdaq.

Although earlier studies generally use monthly returns to estimate beta and test the CAPM and other factor models, we use daily returns because in principle, we believe we can estimate betas more precisely with higher-frequency data, just as Merton Merton, outer borough (1991 pop. 161,800) of Greater London, SE England. The area is largely residential with some industry, including tanning and the manufacture of silk and calico prints, varnish and paint, and toys. (1980) observed for variances. In practice, using daily returns creates microstructure issues caused by nonsynchronous

**Adj.**

**1.**

**nonsynchronous**- not occurring together

unsynchronised, unsynchronized, unsynchronous

asynchronous - not synchronous; not occurring or existing at the same time or having the same period or phase trading. Nonsynchronous prices can have a big impact on short-horizon betas. Lo and MacKinlay (1990) show that small stocks react with a significant (i.e., a week or more) delay to common news, so a daily beta will miss much of the small-stock covariance Covariance

A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. with market returns. To mitigate mit·i·gate

*v.*

To moderate in force or intensity.

**miti·gation**

*n.*the problem, we exclude the Amex and Nasdaq stocks and control for the size effect using twodimensional size/beta portfolios based on the NYSE sample. Also, following Dimson (1979), we use both current and lagged market returns in the regressions. In Equation (3) we estimate the realized beta as the sum of the slopes ([[??].sup.1.sub.i,t] and [[??].sup.2.sub.i,t]):

[R.sub.i,d,t] = [[alpha].sub.i,t] + [[beta].sup.1.sub.i,t] [R.sub.m,d,t] + [[beta].sup.2.sub.i,t] [R.sub.m,d-1,t] + [[epsilon].sub.i,d,t], (3)

where the sum of the slopes, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], adjusts for nonsynchronous trading (see Scholes and Williams, 1977; Dimson, 1979).

We estimate the time-varying conditional betas based on the following autoregressive of order one AR(1), moving average of order one MA(1), and GARCH(1,1)-in-mean models:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

We drop the i subscript

**(1)**In word processing and scientific notation, a digit or symbol that appears below the line; for example, H

_{2}O, the symbol for water. Contrast with superscript.

**(2)**In programming, a method for referencing data in a table. in Equations (4) to (6) to save space. Here, E([[beta].sub.t] | [[OMEGA

**1.**(programming)

**Omega**- A prototype-based object-oriented language from Austria.

["Type-Safe Object-Oriented Programming with Prototypes - The Concept of Omega", G. Blaschek, Structured Programming 12:217-225, 1991].

**2.**].sub.t-1]) denotes the current conditional mean of realized beta estimated with the information set at time t - 1, [[OMEGA].sub.t-1]. In AR(1) and MA(1) models, we assume that the conditional variance In statistics,

**conditional variance**is a special form of the variance. If we have a conditional distribution

*Y*|

*X*the conditional variance is defined as

where of realized beta, denoted by E([[epsilon].sup.2.sub.t] | [[OMEGA].sub.t-1]), is constant.

We use the GARCH-in-mean model, which was originally introduced by Engle En´gle

**n.**

**1.**A favorite; a paramour; an ingle.

**v. t.**

**1.**To cajole or coax, as favorite.

I 'll presently go and engle some broker.

- B. Jonson. , Lilien, and Robins (1987), to model the conditional mean of asset returns as a function of the conditional volatility Volatility

1. A statistical measure of the tendency of a market or security to rise or fall sharply within a period of time.

2. A variable in option pricing formulas that denotes the extent to which the return of the underlying asset will fluctuate between now and the . In the GARCH-in-mean model, we assume that the conditional variance of realized beta follows the GARCH(1,1) model of Bollerslev (1986). We compare the conditional betas ([[beta].sup.AR.sub.t|t-1]], [[beta].sup.MA.sub.t|t-1], [[beta].sup.GARCH.sub.t|t-1] with the lagged realized beta [[beta].sup.realized.sub.t-1] in terms of their power to predict the cross-section of one-month-ahead average stock returns.

Earlier studies find significant persistence

**(1)**In a CRT, the time a phosphor dot remains illuminated after being energized. Long-persistence phosphors reduce flicker, but generate ghost-like images that linger on screen for a fraction of a second. in the conditional beta estimates for industry, size, or book-to-market portfolios (e.g., Braun Braun ,

**Eva**1912-1945.

German lover and later wife of Adolf Hitler. They began living together in 1936, but the liaison was kept secret, and she was never seen in public with him. They were married hours before their double suicide on April 30, 1945. , Nelson, and Sunier, 1995; Ang and Chen, 2007). However, these studies do not estimate conditional betas at the firm level. Ang, Chen, and Xing Xing Crossing (2006) compute realized beta at the firm level using daily returns over the past 12 months and propose alternative measures of downside risk Downside Risk

An estimation of a security's potential to suffer a decline in price if the market conditions turn bad.

Notes:

You can think of this as an estimate of the amount that you could lose on a stock or other investment. based on the unconditional realized betas. According to according to

*prep.*

**1.**As stated or indicated by; on the authority of: according to historians.

**2.**In keeping with: according to instructions.

**3.**their descriptive statistics descriptive statistics

see statistics. , the average AR(1) coefficient of realized betas is in the range of 0.077 to 0.675 depending on their specification of downside risk. We generate conditional beta estimates for each stock using AR(1) and MA(1) models given in Equations (4) and (5).

Jagannathan and Wang (1996) examine the relation between unconditional betas and the cross-section of unconditional expected returns by assuming that the conditional CAPM holds period by period. As described in Jagannathan and Wang (1996), when the conditional CAPM is assumed to hold for each period, cross-sectionally, the unconditional expected return on any asset is a linear function of its expected beta and its beta-premium sensitivity. In other words

**Adv.**

**1.**

**in other words**- otherwise stated; "in other words, we are broke"

put differently , the standard static (or unconditional) CAPM leads to a two-factor unconditional asset pricing model Asset pricing model

A model for determining the required or expected rate of return on an asset. Related: Capital asset pricing model and arbitrage pricing theory. , where the first factor is the unconditional market beta that measures average market risk and the second factor is the unconditional premium beta that measures beta-instability risk. According to this model, stocks with higher expected betas should have higher unconditional expected returns. Similarly, stocks with betas that are correlated cor·re·late

*v.*

**cor·re·lat·ed**,

**cor·re·lat·ing**,

**cor·re·lates**

*v.*

*tr.*

**1.**To put or bring into causal, complementary, parallel, or reciprocal relation.

**2.**with the market risk premium and hence are less stable over the business cycle should also have higher unconditional expected returns. Jagannathan and Wang (1996) indicate that the beta-premium sensitivity of an asset measures the instability instability /in·sta·bil·i·ty/ (-stah-bil´i-te) lack of steadiness or stability.

**detrusor instability**of the asset's beta over the business cycle.

We model the conditional mean of market beta as a function of its conditional variance as in the GARCH-in-mean specification. Equation (6) models the current conditional mean and conditional variance of realized betas as a function of the information set at time t - 1.

To provide an alternative justification

**justification**

In Christian theology, the passage of an individual from sin to a state of grace. Some theologians use the term to refer to the act of God in extending grace to the sinner, while others use it to define the change in the condition of a sinner who has received for our use of the GARCH-in-mean model, we compute the correlations between the realized beta ([[beta].sub.t]) and the conditional standard deviation of realized beta ([[sigma].sub.t]), the realized beta ([[beta].sub.t]) and the conditional variance of realized beta ([[alpha].sub.t]), the conditional mean of realized beta (E([[beta].sub.t] | [[OMEGA].sub.t-1])) and the conditional standard deviation of realized beta ([[sigma].sub.t]), and the conditional mean of realized beta (E([[beta].sub.t] | [[OMEGA].sub.t-1])) and the conditional variance of realized beta ([[sigma].sub.t]).

Table I presents the percentiles of the correlation measures for all stocks trading at the NYSE, Amex, and Nasdaq. The correlation statistics indicate a strong relation between the monthly realized betas and their conditional volatility, and a strong relation between the conditional mean of monthly realized betas and their conditional volatility. We also find that the estimated slope coefficients ([[??].sub.1]) in [[beta].sub.t] = [c.sub.0] + [c.sub.1][[sigma].sub.2t|t-1]] + [[epsilon].sub.t] are statistically significant. This result provides further justification of our use of the GARCH-in-mean model.

Table II presents percentiles of the time-series mean and standard deviation of realized and expected conditional betas. The statistics presented in Panel A of Table II are based on realized betas that we compute by using daily returns over the previous month without lagged market return. The statistics shown in Panel B of Table II are based on realized betas that we compute by using daily returns over the previous month with the lagged market return. In both panels, we only report the sample mean of the realized betas because theoretically, the mean of conditional betas is the same as the mean of realized beta. Theoretically, the means should be the same, but the discrepancies are caused by the filtration filtration: see sewerage; water supply.

Filtration

The separation of solid particles from a fluidsolids suspension of which they are a part by passage of most of the fluid through a septum or membrane that retains most of the solids of conditional beta. The standard deviation of realized betas is greater than the standard deviation of expected betas. In both panels, the standard deviation of conditional betas obtained from the GARCH-in-mean model is somewhat greater than the standard deviation of conditional betas that we obtain from the AR(1) and MA(1) models.

We compare the conditional betas ([[beta].sub.ARs.sub.t|t-1]], [[beta].sub.MA.sub.t|t-1]], [[beta].sup.GARCH.sub.t|t- 1]) with the lagged realized beta [[beta].sub.realized.sub.t-1] in terms of their power to predict the one-month-ahead realized beta, [[beta].sup.realized.sub.t]. Table III shows the percentiles of [R.sup.2] values from the regression of one-month-ahead realized betas on the lagged realized beta and conditional betas. The performance of conditional betas in predicting the one-month-ahead realized beta is much higher than the lagged realized beta. The 1 percentile of [R.sup.2] is 0.01% for [[beta].sup.realized.sub.t-1] and the 99 percentile of [R.sup.2] is 26.82%. The corresponding figures are 1.14% and 33.87% for [[beta].sup.GARCH.sub.t|t-1].

These results provide some explanation for why the earlier studies that use lagged realized beta or unconditional beta could not identify a positive and significant relation between market beta and expected stock returns. We think that to generate more accurate measures of expected futures betas and to explain the cross-sectional variation in stock returns, one needs to use conditional betas.

II. Average Returns and FF-3 Alphas on Beta Portfolios

This section presents univariate univariate

*adjective*Determined, produced, or caused by only one variable and bivariate bi·var·i·ate

*adj.*

*Mathematics*Having two variables: bivariate binomial distribution.

**Adj.**

**1.**portfolio-level analysis after controlling for size and book-to-market.

A. Univariate Portfolio-Level Analysis

Table IV presents the equal-weighted average returns of decile portfolios that are formed by sorting the NYSE/Amex/Nasdaq stocks based on the lagged realized beta, and the conditional AR(1), MA(1), and GARCH-in-mean betas. We base the results in Panel A on the realized betas that we compute using daily returns over the previous month without lagged market return.

When we sort portfolios based on the lagged realized beta, [[beta].sup.realized.sub.t-1], the average return difference between decile 10 (high beta) and decile 1 (low beta) is about -0.49% per month with the Newey-West (1987) t-statistic of -2.53. Although this result does not support the empirical validity of CAPM, it is not conclusive

*Determinative; beyond dispute or question. That which is conclusive is manifest, clear, or obvious. It is a legal inference made so peremptorily that it cannot be overthrown or contradicted.*, because as discussed earlier, [[beta].sup.realized.sub.t-1] is not a precise estimator of [[beta].sup.realized.sub.t]. The static CAPM predicts a contemporaneous con·tem·po·ra·ne·ous

*adj.*

Originating, existing, or happening during the same period of time: the contemporaneous reigns of two monarchs. See Synonyms at contemporary. positive relation between expected stock returns and market betas. However, we cannot use the current realized beta in empirical tests because of the statistical problems indicated by Miller and Scholes (1972) and Fama and MacBeth (1973).

When we examine the cross-sectional predictive power of the conditional beta measures, we see that they are more accurate estimators of [[beta].sup.realized.sub.t].When we sort decile portfolios based on [[beta].sup.AR.sub.t|t-1], [[beta].sup.MA.sub.t|t-1] and [[beta].sup.GARCH.sub.t|t-1], the average return difference between decile 10 (high beta) and decile 1 (low beta) is in the range of 0.74% to 0.92% per month and highly significant.

In addition to the average raw returns, Panel A of Table IV also shows the magnitude magnitude, in astronomy, measure of the brightness of a star or other celestial object. The stars cataloged by Ptolemy (2d cent. A.D.), all visible with the unaided eye, were ranked on a brightness scale such that the brightest stars were of 1st magnitude and the and statistical significance of the intercepts (FF-3 alphas) from the regression of the equal-weighted portfolio returns on a constant, excess market return, small minus big (SMB

**(1)**(

**S**mall to

**M**edium-sized

**B**usiness) Also called "SME" (small to medium-sized enterprise), it refers to companies that are larger than the small office/home office (SOHO), but not huge. ), and high minus low (HML HML Hämeenlinna (Finland)

HML Hawaii Medical Library

HML High Minus Low (Book to Market Value ratio)

HML Hard Money Lender (real estate)

HML Human Media Lab ) factors. If the conditional CAPM is right and FF-3 alphas do not adequately capture time variations in betas, then conditional-beta-sorted portfolios will have alphas different from zero. Panel A shows that the 10-1 difference in the FF-3 alphas is negative for [[beta].sup.realized.sub.t-1], but it is positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

Our results in Panel B of Table IV are based on the realized betas that we compute using daily returns over the previous month with the lagged market return. When we sort portfolios based on [[beta].sup.realized.sub.t-1] the average return difference between high- and low-beta portfolios is negative but marginally mar·gin·al

*adj.*

**1.**Of, relating to, located at, or constituting a margin, a border, or an edge: the marginal strip of beach; a marginal issue that had no bearing on the election results.

**2.**significant. When decile portfolios are sorted based [[beta].sup.AR.sub.t|t-1], [[beta].sup.MA.sub.t|t-1], and [[beta].sup.GARCH.sub.t|t-1], the average return difference between high- and low-beta portfolios is positive, in the range of 0.89% to 1.01% per month, and highly significant. Panel B also shows that the 10-1 difference in the FF-3 alphas is negative for [[beta].sup.realized.sub.t-1] but positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas. Overall, the results in Table IV indicate that the strong positive relation between the conditional betas and expected returns is robust to the measurement of realized betas. (5)

Due to space considerations, in the following sections we report results only from the realized beta measures estimated with the lagged market return.

B. Controlling for Size and Book-to-Market

We test whether there is a positive relation between conditional beta and expected returns after we control for size and book-to-market. We control for size by first forming decile portfolios ranked based on market capitalization Market Capitalization

A measure of a public company's size. Market capitalization is the total dollar value of all outstanding shares. It's calculated by multiplying the number of shares times the current market price. This term is often referred to as market cap. . Then, within each size decile, we sort stocks into decile portfolios, which we rank based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel A of Table V shows that in each size decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled "Average Returns" averages across the 10 size deciles to produce decile portfolios with dispersion

**dispersion, in chemistry**

dispersion, in chemistry, mixture in which fine particles of one substance are scattered throughout another substance. A dispersion is classed as a suspension, colloid, or solution. in market beta but containing all sizes of firms. This procedure creates a set of decile beta portfolios with near-identical levels of firm size, and thus these decile beta portfolios control for differences in size. After controlling for size, the average return difference between high- and low-beta portfolios is 1.41% per month with the Newey-West (1987) t-statistic of 3.48. Thus, market capitalization does not explain the high (low) returns to high (low) beta stocks.

We also control for book-to-market (BM) by first forming decile portfolios ranked based on the ratio of book value of equity to market value of equity. Then, within each BM decile, we sort stocks into decile portfolios, which we rank based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. Panel B of Table V shows that in each BM decile, the highest (lowest) beta decile has a higher (lower) average returns. The last two columns report the average returns and Newey-West (1987) t-statistics of 10 beta portfolios after controlling for BM. The average return difference between high- and low-beta portfolios is 1.16% per month with the Newey-West t-statistic of 3.69. Thus, book-to-market ratio does not explain the high (low) returns to high- (low-) beta stocks. (6)

Table VI shows the average return differences and FF-3 alphas on high-beta minus low-beta portfolios within each size and book-to-market decile. As shown in Panel A of Table VI, for all specifications of conditional beta, the average return differences and FF-3 alphas are positive and economically significant within each size decile. Except for the two biggest size portfolios (sizes 9 and 10), the average return differences and FF-3 alphas are also statistically significant at the 5% level or better. For example, for the smallest size decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 2.49% per month for [[beta].sup.AR.sub.t|t-1], 2.57% per month for [[beta].sup.MA.sub.t|t-1] and 2.50% per month for [[beta].sup.GARCH.sub.t|t-1]. The corresponding FF-3 alphas are 2.02% per month for [[beta].sup.AR.sub.t|t-1], 2.06% per month for [[beta].sup.MA.sub.t|t-1], and 2.02% per month for [[beta].sup.GARCH.sub.t|t-1]. This strong positive relation between market beta and expected return is present for size 1 to size 9 portfolios, and the relation becomes somewhat weaker for the largest size portfolio (size 10).

Panel B of Table VI shows that for all specifications of conditional beta, the average return differences and FF-3 alphas are positive and economically significant within each BM decile. The average return differences and FF-3 alphas are also statistically significant at the 5% level or better. For example, for the lowest BM decile, the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.50% per month for [[beta].sup.AR.sub.t|t-1], 1.49% per month for [[beta].sup.MA.sub.t|t-1], and 1.29% per month for [[beta].sup.GARCH.sub.t|t-1]. The corresponding FF-3 alphas are 1.60% per month for [[beta].sup.AR.sub.t|t-1], 1.54% per month for [[beta].sup.MA.sub.t|t-1], and 1.36% per month for [[beta].sup.GARCH.sub.t|t-1]. Although there is no obvious pattern, the strong positive relation between market beta and expected return is more pronounced for BM I to BM 9 portfolios, and the relation becomes weaker for the highest BM portfolio.

III. Firm-Level Cross-Sectional Regressions

Here, we present the time-series averages of the slope coefficients from the cross-section of average stock returns on the lagged realized beta, conditional beta, size, and BM. The average slopes provide standard Fama-MacBeth (1973) tests for determining, on average, which explanatory variables have nonzero non·ze·ro

*adj.*

Not equal to zero.

nonzero

Not equal to zero. expected premiums. We run monthly cross-sectional regressions for the following econometric e·con·o·met·rics

*n.*

*(used with a sing. verb)*

Application of mathematical and statistical techniques to economics in the study of problems, the analysis of data, and the development and testing of theories and models. specifications:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

In Equations (7) to (10) [R.sub.i,t] is the realized return Realized return

The return that is actually earned over a given time period. on stock i in month t, [logME.sub.i,t-1] is the natural logarithm of market equity for firm i in month t - 1, log([BE.sub.i,t-1]/[ME.sub.i,t-1]) is the natural logarithm of the ratio of book value of equity to market value of equity for firm i in month t - 1, [[beta].sup.realized.sub.i,t-1] is the lagged realized beta of stock i in month t - 1, and [[beta].sup.AR.sub.i,t|t-1], [[beta].sup.MA.sub.i,t|t-1], and [[beta].sup.GARCH.sub.i,t|t-1] are the conditional expected beta of stock i in month t estimated with the information set at month t - 1.

Table VII reports the time-series averages of the slope coefficients [[gamma].sub.i,t] (i = 1, 2, 3) over the 498 months from July 1963 to December 2004. The Newey-West (1987) adjusted t-statistics are given in parentheses. The results show a negative but insignificant relation between the lagged realized beta and the cross-section of average stock returns. The average slope, [[gamma].sub.1,t], from the monthly regressions of realized returns on [[beta].sup.realized.sub.i,t-1] alone is about -0.07% with a t-statistic of -1.19. The univariate regression results indicate a significant positive relation between average stock returns and conditional betas. The average slope, [[gamma].sub.1,t], from the monthly regressions of realized returns on [[beta].sup.AR.sub.i,t|t-1], [[beta].sup.MA.sub.i,t|t-1], or [[beta].sup.GARCH.sub.i,t|t-1] in the range of 0.44% to 0.50% and statistically significant at the 5% or better. These values imply a reasonable expected market risk premium of 5.33% to 5.87% per annum Per annum

Yearly. .

The univariate regression results also indicate a significant negative relation between average stock returns and firm size. The average slope, [[gamma].sub.2,t], from the monthly regressions of realized returns on [logME.sub.i,t-1] alone is about -0.24% with a t-statistic of-4.75. The parameter

**(1)**Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. estimates show a significant positive relation between average stock returns and BM ratio. The average slope, [[gamma].sub.3,t], from the monthly regressions of realized returns on log([BE.sub.i,t-1]/[ME.sub.i,t-1]) alone is about 0.42% with a t-statistic of 5.98. The findings of negative size and positive BM effect in Fama-MacBeth (1973) regressions are consistent with Fama and French (1992) and related studies.

We find that the strong positive relation between conditional beta and expected stock returns is robust across different econometric specifications. When we add size to the univariate regressions, the average slope coefficient on [[beta].sup.AR.sub.i,t|t-1], [[beta].sup.MA.sub.i,t|t-1], or [[beta].sup.GARCH.sub.i,t|t-1] is about 0.8% and statistically significant at the 1% level. When we add BM to the univariate regressions, the average slope coefficient on conditional betas is in the range of 0.65% to 0.68% and statistically significant at the 1% level. When we include both size and BM in the univariate regressions, the average slope coefficient on the conditional betas is about 0.9% and statistically significant at the 1% level.

The [R.sup.2] values from the univariate regressions of realized returns on conditional beta are in the range of 2.02% to 2.13%. When we add size and BM to these univariate regressions, the [R.sup.2] values increase to 4.7% to 4.87%. Although the [R.sup.2] values from univariate and multivariate cross-sectional regressions are small, they are consistent with the earlier studies that report [R.sup.2] for the firm-level cross-sectional regressions.

IV. Robustness Check

This section presents results from a battery of robustness checks.

A. Alternative Portfolio Partitions

We compute the equal-weighted average returns of 20, 50, and 100 portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCH-in-mean betas. Although not presented in the paper, the average return difference between high- and low-beta portfolios is in the range of 0.83% to 1% per month [[beta].sup.AR.sub.i,t|t-1], 0.89% to 1.11% per month for [[beta].sup.MA.sub.t|t-1], and 1.06% to 1.31% per month [[beta].sup.GARCH.sub.t|t-1] All these average return differences are statistically significant at the 5% level or better. In addition to the average raw returns, we also find the magnitude and statistical significance of the FF-3 alphas. The 10-1 difference in the FF-3 alphas is positive and highly significant for the AR(1), MA(1), and GARCH-in-mean betas.

In addition to the firm-level Fama-MacBeth (1973) regressions, we examine the cross-sectional relation between conditional beta and expected returns at the portfolio level. We present the time-series averages of the slope coefficients from the cross-section of average portfolio returns on the conditional portfolio beta:

[R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.AR.sub.p,t|t-1] + [[epsilon].sub.i,t], (11)

[R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.MA.sub.p,t|t-1] + [[epsilon].sub.i,t], (12)

[R.sub.p,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[beta].sup.GARCH.sub.p,t|t-1] + [[epsilon].sub.i,t]. (13)

In Equations (11) to (13), [R.sub.p,t] is the realized return on portfolio p in month t calculated as the equal-weighted average returns of all stocks in portfolio p, and [[beta].sup.AR.sub.p,t|t-1], [[beta].sup.MA.sub.p,t|t-1], and [[beta].sup.GARCH.sub.p,t|t-1] are the conditional expected betas of portfolio p that we obtain from the equal-weighted average conditional beta of all stocks in portfolio p in month t estimated with the information set at month t - 1.

First, we form 10, 20, 50, and 100 portfolios by sorting the NYSE/Amex/Nasdaq stocks based on the conditional AR(1), MA(1), and GARCH-in-mean betas. Then, for each month from July 1963 to December 2004, we compute each portfolio's return as the equal-weighted average return of all stocks in the portfolio, and we calculate the portfolio's conditional beta as the equal-weighted average conditional beta of all stocks in the portfolio. We run the univariate regressions of average portfolio returns on the average conditional portfolio beta for each month from July 1963 to December 2004.

We calculate the time-series averages of the slope coefficients and the Newey-West (1987) adjusted t-statistics. The univariate regression results indicate a significant positive relation between average portfolio returns and average portfolio betas. Although not presented here, for 10 beta portfolios, the average slopes from the monthly regressions of average portfolio returns on [[beta].sup.AR.sub.p,t|t-1], [[beta].sup.MA.sub.p,t|t-1] and [[beta].sup.GARCH.sub.p,t|t-1] are about 0.44%, 0.47%, and 0.53%, respectively. These average slope coefficients have Newey-West t-statistics of 2.25, 2.37, and 2.61, respectively. A notable point is that for 20, 50, and 100 beta portfolios, the average slope coefficients are similar. In other words, the results are robust across different portfolio formations. These slope coefficients, which imply an expected market risk premium of 5.28% to 6.36% per annum, are also similar to our earlier findings from the firm-level cross-sectional regressions. The [R.sup.2] values are much higher for the portfolio-level regressions: about 58% to 59% for 10 beta portfolios, 48% to 49% for 20 beta portfolios, 35% to 36% for 50 beta portfolios, and 25% to 26% for 100 beta portfolios.

In addition to the month-by-month Fama-MacBeth (1973) regressions, we take the time-series average of the monthly portfolio returns and the monthly portfolio betas and compute overall average portfolio return and overall average portfolio beta for 10, 20, 50, and 100 portfolios. We plot the average portfolio return against the average portfolio beta and find a strong positive relation between market beta and expected returns for all portfolio partitions. The [R.sup.2] value is 97.94% for 10 beta portfolios, 95.97% for 20 beta portfolios, 88.47% for 50 beta portfolios, and 81.5% for 100 beta portfolios. We also find the slope coefficients for each portfolio partition A reserved part of disk or memory that is set aside for some purpose. On a PC, new hard disks must be partitioned before they can be formatted for the operating system, and the Fdisk utility is used for this task. . The results are similar to our earlier findings from the month-by-month firm-level and portfolio-level Fama-MacBeth regressions. The slope on average portfolio beta is almost identical for different portfolio partitions: 0.53% for 10, 50, and 100 beta portfolios, and 0.52% for 20 beta portfolios.

B. Long-Term Predictive Power of Conditional Betas

Table VIII presents the equal-weighted returns of decile portfolios that we form by sorting the NYSE/Amex/Nasdaq stocks based on the conditional GARCH-in-mean betas. (7) The column labeled "[[beta].sup.GARCH.sub.t|t-1]" repeats our earlier result for one-month-ahead predictability: when we sort decile portfolios based on [[beta].sup.GARCH.sub.t|t-1] the average return difference between decile 10 (high beta) and decile 1 (low beta) is 1.01% per month with the Newey-West (1987) t-statistic of 2.83. To test three-month-ahead predictability, we form decile portfolios by sorting stocks based on their conditional betas at time t - 2 obtained from the information set at time t - 3, [[beta].sup.GARCH.sub.t-2|r-3] average return difference between high- and low-beta portfolios is 0.93% per month with the t-statistic of 2.6. As shown in Table VIII, the conditional beta can predict up to 12 months ahead because the average return difference between high- and low-beta portfolios is 0.75% per month with a t-statistic of 2.09.

Table VIII also presents the magnitude and statistical significance of the FF-3 alphas from the regression of the equal-weighted portfolio returns on a constant, excess market return, SMB and HML factors. The 10-1 difference in the FF-3 alphas is positive and significant at the 5% level or better up to nine-month-ahead predictability. However, the economic and statistical significance of FF-3 alpha gradually grad·u·al

*adj.*

Advancing or progressing by regular or continuous degrees: gradual erosion; a gradual slope.

*n.*

*Roman Catholic Church*

**1.**reduce to 0.43% per month with the t-statistic of 1.88 for 12-month-ahead returns.

C. Controlling for Liquidity and Momentum

Following Amihud (2002), we measure stock illiquidity as the ratio of absolute stock return to its dollar volume:

[ILLIQ.sub.i,t] = [absolute value of [R.sub.i,t]] / [VOLD.sub.i,t], (14)

where [R.sub.i,t] is the return on stock i in month t, and [VOLD.sub.i,t] is the respective monthly volume in dollars. This ratio gives the absolute percentage price change per dollar of monthly trading volume Trading volume

The number of shares transacted every day. As there is a seller for every buyer, one can think of the trading volume as half of the number of shares transacted. That is, if A sells 100 shares to B, the volume is 100 shares. . As in Amihud (2002), [ILLIQ.sub.i.t] follows the Kyle's (1985) concept of illiquidity, that is, the response of price to the associated order flow or trading volume. The measure of stock illiquidity given in Equation (14) represents the price response associated with one dollar of trading volume. Thus, it serves as a rough measure of price impact.

We control for liquidity by first forming decile portfolios ranked based on Amihud's (2002) measure of illiquidity. Then, within each illiquidity decile, we sort stocks into decile portfolios, which we rank based on the GARCH-in-mean betas so that decile 1 (10) contains stocks with the lowest (highest) market beta. In each illiquidity decile, the highest (lowest) beta decile has a higher (lower) average returns. The column labeled "Illiquidity" in Table IX presents the average returns across the 10 illiquidity deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-identical levels of illiquidity. Thus, these decile beta portfolios control for differences in illiquidity. After controlling for illiquidity, we find that the average return difference between high- and low-beta portfolios is 1.20% per month with the Newey-West (1987) t-statistic of 3.05. Thus, liquidity does not explain the high (low) returns to high- (low-) beta stocks.

When we measure liquidity of individual stocks using dollar trading volume, we obtain similar results. The column labeled "Volume" presents the average returns across the 10 volume deciles to produce decile portfolios with dispersion in Table IX market beta. After controlling for dollar trading volume, we find that the average return increase monotonically from 0.92% to 2.38% when moving from low- to high-beta portfolios. The average return difference between high- and low-beta portfolios is 1.46% per month with the Newey-West (1987) t-statistic of 3.68. Thus, trading volume does not explain the high (low) returns to high- (low-) beta stocks either.

We control momentum by first forming decile loser-winner portfolios ranked based on the past six-month average returns of individual stocks. Then, within each six-month momentum portfolio, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "MOM6" in Table IX presents the average returns across the 10 momentum deciles to produce decile portfolios with dispersion in market beta. This procedure creates a set of decile beta portfolios with near-identical levels of past average six-month returns. Thus, these decile beta portfolios control for differences in momentum. After controlling for momentum, the average return difference between high-and low-beta portfolios is 0.99% per month with the Newey-West (1987) t-statistic of 2.89. Thus, momentum does not explain the high (low) returns to high- (low-) beta stocks. We obtain similar results when we form loser-winner portfolios based on the past 12-month average returns (MOM12). The average return difference between high- and low-beta portfolios is 0.86% per month with a t-statistic of 2.74.

After controlling for liquidity, momentum, size, and BM, we investigate whether the positive relation between conditional beta and the cross-section of expected returns holds in the firm-level Fama-MacBeth (1973) regressions.

Table X presents the time-series averages of the slope coefficients and the Newey-West (1987) adjusted t-statistics in parentheses. The regression results indicate a significant positive relation between average stock returns and the conditional GARCH-in-mean betas after controlling for illiquidity, trading volume, past average 6- and 12-month returns with and without size, and BM.

The average slope coefficient on [[beta].sup.GARCH.sub.t|t-1] is about 0.51% with [ILLIQ.sub.t-1] and 0.52% with [VOL VOL Volume

VOL Volunteer

VOL Volcano

VOL Volvo (stock symbol)

VOL Verdingungsordnung für Leistungen (German)

VOL Volatile Organic Liquid

Vol Volscan (linguistics) .sub.t-1], and both coefficients are highly significant. The average slope coefficient on [[beta].sup.GARCH.sub.t|t-1] is about 0.41% and significant at the 5% level when we add [MOM6.sub.t-1] or [MOM12.sub.t- 1] along with [[beta].sup.GARCH.sub.t|t-1] in the cross-sectional regressions.

When we include alternative measures of liquidity and momentum along with [[beta].sup.GARCH.sub.t|t-1], size, and BM, the average slope coefficient on [[beta].sup.GARCH.sub.t|t-1] becomes stable in the range of 0.62% to 0.66% for different specifications. As shown in for all these multivariate regressions with liquidity, momentum, size, and BM, the average slope coefficient on [[beta].sup.GARCH.sub.t|t-1] is statistically significant at the 1% level.

At an earlier stage of the study, we replicated our results presented in using the conditional beta estimates obtained from the AR(1) and MA(1) specifications. The results turn out to be similar to those from [[beta].sup.GARCH.sub.t|t-1]. We do not present our findings here from [[beta].sup.AR.sub.i,t|t-1] and [[beta].sup.MA.sub.i,t|t-1]. They are available on request.

D. Results from the NYSE Sample

To check the robustness of our findings, we exclude the Amex and Nasdaq stocks from our sample and form the beta portfolios by sorting only the NYSE stocks based on the conditional GARCH-in-mean betas. Table XI shows that for the univariate sort of NYSE stocks, the average return difference between high- and low-beta portfolios is about 0.86% with the Newey-West (1987) t-statistic of 2.79. The 10-1 difference in the FF-3 alphas is 0.37% with a t-statistic of 2.44.

We further examine the cross-sectional relation by forming the beta portfolios within each size and BM decile. Table XI shows that the average return difference between high- and low-beta portfolios is 0.84% after we control for size and 0.78% after we control for BM. Both return differences are statistically significant at the 1% level. The 10-1 differences in the FF-3 alphas are also positive and highly significant. These results indicate that excluding the Amex and Nasdaq sample has almost no effect on our previous findings. These results remain the same for alternative specifications of conditional beta ([[beta].sup.AR.sub.t|t-1] and [[beta].sup.MA.sub.t|t-1]).

E. Controlling for Microstructure Effects and NYSE Breakpoint The location in a program used to temporarily halt the program for testing and debugging. Lines of code in a source program are marked for breakpoints. When those instructions are about to be executed, the program stops, allowing the programmer to examine the status of the program

Above, we excluded the Amex and Nasdaq stocks and presented the return/beta estimates from the portfolios of NYSE stocks formed based on the NYSE breakpoints. However, these results may be contaminated contaminated,

*v*

**1.**made radioactive by the addition of small quantities of radioactive material.

**2.**made contaminated by adding infective or radiographic materials.

**3.**an infective surface or object. by microstructure effects because there is only a one-month gap between the conditional beta estimates and portfolio returns. Here, we follow Fama and French (1992) by skipping skip

*v.*

**skipped**,

**skip·ping**,

**skips**

*v.*

*intr.*

**1.**

**a.**To move by hopping on one foot and then the other.

**b.**To leap lightly about.

**2.**the month following portfolio formation to avoid microstructure effects and use the NYSE breakpoints to generate beta portfolios of NYSE/Amex/Nasdaq stocks with a relatively more balanced average market share. Since there are so many small-cap Small-cap

A stock with a small capitalization, meaning a total equity value of less than $500 million.

small-cap

**1.**Of or relating to the common stock of a relatively small firm having little equity and few shares of common stock Nasdaq stocks, we determine portfolio breakdowns by using only NYSE stocks. Doing so enables us to avoid the beta portfolios that contain small stocks from being too small in terms of average market share.

Table XII presents the average returns on the beta portfolios of NYSE/Amex/Nasdaq stocks with NYSE breakpoints after we skip the month following portfolio formation. When we sort portfolios based on the lagged realized beta, the average return difference between high- and low-beta portfolios is economically and statistically nonsignificant non·sig·nif·i·cant

*adj.*

**1.**Not significant.

**2.**Having, producing, or being a value obtained from a statistical test that lies within the limits for being of random occurrence. . When we form portfolios based on the AR(1), MA(1), and GARCH-in-mean beta estimates, the average return difference between deciles 10 and 1 is about 0.7%, 0.72%, and 0.92% per month, respectively. Similar to our earlier findings, for all conditional beta estimates, the 10-1 differences in average returns and FF-3 alphas are positive and highly significant. Overall, the results in Table XII indicate that forming portfolios with CRSP or NYSE breakpoints and skipping the month following portfolio formation does not affect our main conclusions.

V. Results from Size/BM/Beta Portfolios

When we construct beta portfolios, we control for size or BM ratio, but not both. Here, we test whether the significantly positive relation between conditional beta and expected returns remains intact after we control simultaneously si·mul·ta·ne·ous

*adj.*

**1.**Happening, existing, or done at the same time. See Synonyms at contemporary.

**2.**

*Mathematics*for size and BM.

Table XIII presents the average returns and FF-3 alphas on the quintile quin·tile

*n.*

**1.**The astrological aspect of planets distant from each other by 72° or one fifth of the zodiac.

**2.**

*Statistics*The portion of a frequency distribution containing one fifth of the total sample. portfolios of realized and conditional betas after we control for size and BM. At the beginning of each month t from July 1963 to December 2004, we first sort all NYSE/Amex/Nasdaq stocks into five size (market equity) portfolios. Then within each size portfolio, stocks are sorted into five BM (book-to-market equity ratio) portfolios. Finally, within each portfolio formed based on the intersections of five size and five BM portfolios, we sort stocks into five beta portfolios based on their realized and conditional betas in month t - 1.

Table XIII shows that when we sort stocks in the 5 x 5 size/BM portfolios into five realized beta ([[beta].sup.realized.sub.t-1]) portfolios, the average return difference between high- and low-beta portfolios is about 0.5% per month with a t-statistic of 0.04. Similar to our earlier findings from the univariate and bivariate sorts, there is no significant relation between lagged realized beta and the cross-section of expected returns from trivariate sorts. When we sort the stocks in the 5 x 5 size/BM portfolios into five AR(1), MA(1), and GARCH-in-mean beta portfolios, the average return differences between high- and low-beta portfolios are about 0.97%, 1.01%, and 1.06% per month, respectively. These return differences are statistically significant at the 1% level. Moreover, for all conditional beta estimates, the 5-1 differences in FF-3 alphas are positive and highly significant. Overall, the results in Table XIII indicate that the significant positive relation between conditional beta and the cross-section of expected returns remains the same after we control simultaneously for both size and BM.

To provide further evidence for the significant positive link between conditional beta and expected returns on size/BM/beta portfolios, we run the Fama-MacBeth (1973) regressions using the 125 (5 x 5 x 5) portfolios of size, BM, and beta. First, we compute the monthly realized beta for each of the 125 portfolios, using daily returns within a month. Then, we generate the conditional beta estimates for each of the 125 portfolios using the AR(1), MA(1), and GARCH-in-mean specifications. We use the average firm size and average BM ratio of each portfolio as additional controls in Fama-MacBeth regressions. Table XIV shows that the average slope coefficients on conditional betas are positive and highly significant with and without controlling for the portfolios' size and BM. Confirming the earlier findings from firm-level regressions, the average slopes on size and BM turn out to be significantly negative and positive, respectively.

Overall, we can conclude that the Fama-MacBeth regressions at the firm level and at the portfolio level yield similar results on the relation between market beta and expected returns.

To check whether the cross-sectional relation still holds after we control for the time-series relation between conditional betas and expected returns, we run the pooled panel regressions using both the cross-section and time series of 125 portfolio returns and betas. Table XV presents the parameter estimates and the t-statistics that are corrected for heteroskedasticity, first-order

**first-order**- Not higher-order. autocorrelation Autocorrelation

The correlation of a variable with itself over successive time intervals. Sometimes called serial correlation. , and contemporaneous cross-correlations in the error terms. Similar to our earlier findings, the pooled panel regressions indicate a positive and highly significant relation between conditional beta and expected returns, but the relation between lagged realized beta and expected returns is not significant. These results hold after controlling for size and BM in cross-section and time-series setting.

VI. Cross-Sectional Implications of the Conditional CAPM

The static (or unconditional) CAPM of Sharpe (1964), Lintner (1965), and Black (1972) indicates that there is a positive linear relation between expected returns on securities and their market betas:

E([R.sub.i,t]) = [[beta].sub.i] E ([R.sub.m,t]), (15)

where E([R.sub.i,t]) is the unconditional expected excess return of asset i, E([R.sub.m,t]) is the unconditional expected excess return of the market portfolio, and [[beta].sub.i] = Cov([R.sub.i,t], [R.sub.m,t])/Var([R.sub.m,t]) is the unconditional beta of asset i.

Fama and French (1992) and related studies find that the unconditional market beta cannot explain the cross-sectional variation in expected stock returns. The unconditional CAPM was derived de·rive

*v.*

**de·rived**,

**de·riv·ing**,

**de·rives**

*v.*

*tr.*

**1.**To obtain or receive from a source.

**2.**by examining the behavior of investors in a hypothetical model in which they live for only one period, but in the real world, investors live for many periods. Hence, in an empirical examination of the CAPM that uses data from the real world, it is necessary to make certain assumptions. One of the most common assumptions in the static CAPM framework is that the betas of the assets remain constant over time. However, this assumption is not reasonable, because the relative risk of a firm's cash flow is likely to vary over the business cycle. As indicated by Harvey Harvey, city (1990 pop. 29,771), Cook co., NE Ill., a suburb S of Chicago; inc. 1895. Its manufactures include steel castings, metal products, chemicals, machinery, and electronic equipment. Harvey has an oil research center. The city was founded by Turlington W. (1989), Shanken (1990), Jagannathan and Wang (1996), Ferson and Harvey (1991, 1999), and Lettau and Ludvigson (2001), betas and expected returns generally depend on the nature of the information available at any given point in time, and thus will vary over time.

The conditional version of the CAPM imposes the restriction

**restriction**- A bug or design error that limits a program's capabilities, and which is sufficiently egregious that nobody can quite work up enough nerve to describe it as a feature. that conditionally con·di·tion·al

*adj.*

**1.**Imposing, depending on, or containing a condition. See Synonyms at dependent.

**2.**

*Grammar*Stating, containing, or implying a condition.

**3.**expected returns on assets are linearly related to the conditionally expected return on the market portfolio in excess of the risk-free rate. The coefficient in the linear relation is the asset's conditional beta or the ratio of the conditional covariance of the asset's return with the market to the conditional variance of the market:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

where E([R.sub.i,t+1] | [[OMEGA].sub.t]) is the conditional expected excess return of asset i, E([R.sub.m,t+1] | [[OMEGA].sub.t]) is the conditional expected excess return of the market portfolio, [[beta].sub.i,t+1] = Cov([R.sub.i,t+1], [R.sub.m,t+1] | [[OMEGA].sub.t]) / Var ([R.sub.m,t+1] | [[OMEGA].sub.t]) is the conditional market beta of asset i, and [OMEGA].sub.t] denotes the information set at time t.

We can rewrite re·write

*v.*

**re·wrote**,

**re·writ·ten**,

**re·writ·ing**,

**re·writes**

*v.*

*tr.*

**1.**To write again, especially in a different or improved form; revise.

**2.**Equation (16) to simplify the follow-up follow-up,

*n*the process of monitoring the progress of a patient after a period of active treatment.

follow-up

subsequent.

**follow-up plan**expressions:

E ([R.sub.i,t+1] | [[OMEGA].sub.t] = [A.sub.m,t+l] x [[beta].sub.i,t+1], (17)

where [A.sub.m,t+1] : E([R.sub.m,t+1] | [OMEGA].sub.t]) is the time t + 1 conditional expected market risk premium.

Taking the unconditional expectation of both sides of Equation (17), we obtain the unconditional implication of the conditional CAPM:

E[[R.sub.i,t+l]] = [[bar.A].sub.m] x [[bar.[beta]].sub.i] + Cov([A.sub.m,t+l], [[beta].sub.i,,t+l]), (18)

where Cov([A.sub.m,t+1], [[beta].sub.i,t+1]) denotes the unconditional covariance, and E[[A.sub.m,t+1] = [[bar.A].sub.m] and E[[beta].sub.i,t+1] = [[bar.[beta]].sub.i] are the unconditional means of the corresponding conditional estimates.

We note that the last term in Equation (18) depends only on the part of the conditional beta that is in the linear span of the market risk premium, which motivates Jagannathan and Wang (1996) to decompose de·com·pose

*v.*

**de·com·posed**,

**de·com·pos·ing**,

**de·com·pos·es**

*v.*

*tr.*

**1.**To separate into components or basic elements.

**2.**To cause to rot.

*v.*

*intr.*

**1.**the conditional beta of any asset i into two orthogonal components by regressing the conditional beta on the market risk premium. For each asset i, we run the following regression:

[[beta].sub.i,t+1] = [[bar.[beta]].sub.i] + [[lambda].sub.i] ([A.sub.m,t+1] - [[bar.A].sub.m]) + [u.sub.i,t+1], (19)

where [[lambda].sub.i] = Cov([A.sub.m,t+l], [[beta].sub.i,t+1])/Var([A.sub.m,t+1]) is the unconditional market beta-premium sensitivity that measures the sensitivity of conditional beta to the market risk premium.

Substituting (19) into (18) gives:

E[[R.sub.i,t+1]] = [[bar.A].sub.m] x [[bar.[beta]].sub.i] + [[lambda].sub.i] x Var([A.sub.m.t+l]). (20)

Hence, cross-sectionally, the unconditional expected excess return on any asset i is a linear function of the unconditional average of its conditional market beta ([[bar.[beta]].sub.i]) and its unconditional market beta-premium sensitivity ([[lambda].sub.i]). Equation (20) implies that stocks with higher expected betas have higher unconditional expected returns, as do stocks with betas that are prone to vary with the market risk premium and hence less stable over the business cycle. Hence, the one-factor conditional CAPM leads to a two-factor model Two-factor model

Usually, Fischer Black's zero-beta version of the capital asset pricing model. It may also refer to another type of model whereby expected returns are generated by any two factors. for unconditional expected returns.

A complete test of the conditional CAPM specification requires that we estimate the expected beta ([[bar.[beta]].sub.i]) and beta-premium sensitivity ([[lambda].sub.i]). Here, we use the average conditional beta estimates obtained from AR(1), MA(1), and GARCH-in-mean specifications as a proxy See proxy server.

(networking)

**proxy**- A process that accepts requests for some service and passes them on to the real server. A proxy may run on dedicated hardware or may be purely software. for [[bar.[beta]].sub.i]. We estimate beta-premium sensitivity [[lambda].sub.i] using the lagged market return as a proxy for the expected market risk premium, that is, [[lambda].sub.i] = Cov([R.sub.m,t], [[beta].sub.i,t+1])/Var([R.sub.m,t]), where we use the lagged market return, [R.sub.m,t], as a proxy for the time t + 1 conditional expected market risk premium, [A.sub.m,t+1] = E([R.sub.m,t+1] | [[OMEGA].sub.t]) = [R.sub.m,t].

For each month, we run the following cross-sectional Fama-MacBeth (1973) regressions:

[R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.AR.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.AR.sub.i] + [[epsilon.sub.i,t], (21)

[R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.MA.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.MA.sub.i] + [[epsilon.sub.i,t], (22)

[R.sub.i,t] = [[gamma].sub.0,t] + [[gamma].sub.1,t] [[bar.[beta]].sup.GARCH.sub.i] + [[gamma].sub.2,t] x [[lambda].sup.GARCH.sub.i] + [[epsilon.sub.i,t], (23)

In Equations (21) to (23), [[bar.[beta]].sup.AR.sub.i] , [[bar.[beta]].sup.MA.sub.i], and [[bar.[beta]].sup.GARCH.sub.i] are the time-series average [[bar.[beta]].sup.AR.sub.i,t|t-1,], [[bar.[beta]].sup.MA.sub.i,t|t-1], and [[beta]].sup.GARCH.sub.i,t|t-1] respectively. Here, [[lambda].sup.MA.sub.i], [[lambda].sup.GARCH.sub.i] are obtained from the regression of [[beta]].sup.AR.sub.i,t|t-1] [[beta]].sup.MA.sub.i,t|t-1] and [[beta]].sup.GARCH.sub.i,t|t-1] on [R.sub.m,t], respectively. The lagged return on the CRSP value-weighted index is our proxy for [R.sub.m,t].

We compute the time-series averages of the slope coefficients and their Newey-West (1987) t-statistics from the monthly cross-sectional Fama-MacBeth (1973) regressions of stock returns on their average conditional beta and beta-premium sensitivity. The average slopes on [[bar.[beta]].sup.AR.sub.i], [[bar.[beta]].sup.MA.sub.i] and [[bar.[beta]].sup.GARCH.sub.i]are about 0.622, 0.6189, and 0.5705 with the Newey-West t-statistic of 2.89, 2.85, and 2.67, respectively. However, the average slope coefficients on beta-premium sensitivity are economically and statistically nonsignificant for all specifications of the conditional beta measures. The results indicate a significant positive relation between average conditional beta and the cross-section of expected returns within the conditional CAPM framework.

To provide further evidence on the correlation between conditional beta and market risk premium, we investigate the correlations between the conditional betas and the Chicago

**Chicago, city, United States**

Chicago (shĭkä`gō, shĭkô`gō), city (1990 pop. 2,783,726), seat of Cook co., NE Ill., on Lake Michigan; inc. 1837. Fed National Activity Index (CFNAI CFNAI Chicago Fed National Activity Index ), which is a weighted average of 85 existing monthly indicators of national economic activity constructed to have an average value of zero and a standard deviation of one.

Since economic activity tends toward trend growth rate over time, a positive index reading corresponds to growth above trend, and a negative index reading corresponds to growth below trend.

We expect to find a positive relation between expected stock returns and innovations in output (or growth above trend). Actual increases in real economic activity, if greater than expected (or greater than the trend), may increase agents' expectations of future growth. Forecasts of higher economic growth should make stocks more attractive and thus cause an immediate jump in share prices. That is, the positive relation between expected returns and the CFNAI makes economic sense. Since there is a positive relation between conditional betas and expected returns, we also expect to find a positive link between the conditional beta and the CFNAI. Figure 1 shows that the sample correlations for almost all of the 125 portfolios are positive for the AR(1), MA(1), and GARCH-in-mean beta estimates. These results provide further evidence on the capability of conditional betas to predict the time-series and cross-sectional variation in stock returns.

[FIGURE 1 OMITTED]

VII. Conclusion

In this paper, we investigate the cross-sectional relation between conditional betas and expected stock returns for the sample period of July 1963 to December 2004. First, we use daily returns within a month to compute realized beta for each stock trading at the NYSE, Amex, and Nasdaq and then use autoregressive, moving average, and GARCH-in-mean models to obtain time-varying conditional betas for each stock.

For each specification of conditional beta, we find that the average portfolio returns increase almost monotonically when moving from low-beta to high-beta portfolios. The portfolio-level analyses and the firm-level cross-sectional regressions indicate that the positive relation between the conditional betas and the cross-section of average returns is economically and statistically significant. For the NYSE/Amex/Nasdaq sample, the average return difference between high- and low-beta portfolios is in the range of 0.89% to 1.01% per month, depending on the time-varying specification of conditional beta.

To check whether our findings are driven by small, illiquid, and low-price stocks, we exclude the Amex and Nasdaq stocks and form the beta portfolios by sorting only the NYSE stocks based on the conditional betas. The results indicate that excluding the Amex and Nasdaq sample has almost no effect on our original findings. We also control for the cross-sectional effects of size, BM, liquidity, and momentum. After controlling for these effects, we estimate the cross-sectional beta premium to be in the range of 0.86% to 1.46% per month. These results are robust across different measures of conditional beta.

We thank Bill Christie (the Editor) and two other anonymous Nameless. See anonymous post and anonymous Web surfing. referees for their extremely helpful comments and suggestions. We also benefited from discussions with Hadiye Asian, Ozgur Demirtas, Armen

**Armen**may refer to:

- Armens, ancient Armenian tribes

**Surname**

- Robert Armen
- Rosy Armen, an Armenian-French singer
- Garo Armen

**First name**

- Armen Akopyan
- Armen Alchian
- Armen Ambartsumyan
- Armen Babalaryan

**Henry Martyn**1837-1923.

American army engineer and parliamentary authority. He designed the defenses for Washington, D.C., during the Civil War and later wrote

*Robert's Rules of Order*(1876).

**Noun**

**1.**Whitelaw, and seminar participants at Barueh College, Graduate School, and University Center of the University of New York There is no institution of higher education in the State of New York or the United States of America that bears the name

**University of New York**. However, in confusion, it is possible that such a reference may regard the following:

**Kenneth Ronald "Ken" French**(born March 10, 1954) is the Carl E. and Catherine M. Heidt Professor of Finance at the Tuck School of Business, Dartmouth College. He has previously been a faculty member at MIT, the Yale School of Management, and the University of Chicago Graduate for making a large amount of historical data publicly available in his online data library.

References

Amihud, Y., 2002, "Illiquidity and Stock Returns: Cross-Section and Time-Series Effects," Journal of Financial Markets 5, 31-56.

Ang, A. and J. Chen, 2007, "CAPM over the Long-Run: 1926-2001," Journal of Empirical Finance 14, 1-40.

Ang, A.J., J. Chen, and Y. Xing, 2006, "Downside Risk," Review of Financial Studies 19, 1191-1239.

Bali Bali (bä`lē), island and (with two offshore islets) province (1990 pop. 2,777,356), c.2,200 sq mi (5,700 sq km), E Indonesia, westernmost of the Lesser Sundas, just E of Java across the narrow Bali Strait. The capital is Denpasar. , T.G., 2008, "The Intertemporal Relation between Expected Returns and Risk," Journal of Financial Economics 87, 101-131.

Bali, T.G., N. Cakici, X. Yan, and Z. Zhang, 2005, "Does Idiosyncratic Risk Idiosyncratic Risk

Risk that affects a very small number of assets, and can be almost eliminated with diversification. Similar to unsystematic risk.

Notes:

This is news that is specific to a small number of stocks. One example is a sudden strike by employees. Really Matter?" Journal of Finance 60, 905-929.

Black, F., 1972, "Capital Market Equilibrium equilibrium, state of balance. When a body or a system is in equilibrium, there is no net tendency to change. In mechanics, equilibrium has to do with the forces acting on a body. with Restricted Borrowing," Journal of Business 45, 444-455.

Black, F., M. Jensen, and M. Scholes, 1972, "The Capital Asset Pricing Model: Some Empirical Tests," in M. Jensen, Ed., Studies in the Theory of Capital Markets, New York

**New York, state, United States**

New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , Praeger.

Bloomfield Bloomfield.

**1**Town (1990 pop. 19,483), Hartford co., N Conn., a suburb of Hartford, in a tobacco and dairy region; settled c.1642, inc. 1835. Helicopter parts are made there.

**2**City (1990 pop. 45,061), Essex co., NE N.J. , R. and R. Michaely, 2004, "Risk or Mispricing? From the Mouths of Professionals," Financial Management 33, 61-81.

Blume, M., 1970, "Portfolio Theory: A Step Towards Its Practical Application," Journal of Business 43, 152-174.

Blume, M. and I. Friend, 1973, "A New Look at the Capital Asset Pricing Model," Journal of Finance 28, 19-33.

Bollerslev, T., 1986, "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics econometrics, technique of economic analysis that expresses economic theory in terms of mathematical relationships and then tests it empirically through statistical research. 31, 307-327.

Bollerslev, T., R.F. Engle, and J.M. Wooldridge, 1988, "A Capital Asset Pricing Model with Time Varying Covariances," Journal of Political Economy 96, 116-131.

Braun, P.A., D.B. Nelson, and A.M. Sunier, 1995, "Good News, Bad News, Volatility, and Betas," Journal of Finance 50, 1575-1603.

Brav, A., R. Lehavy, and R. Michaely, 2005, "Using Expectations to Test Asset Pricing Models," Financial Management 34, 31-64.

Campbell

**Campbell, city, United States**

Campbell, city (1990 pop. 36,048), Santa Clara co., W Calif., in the fertile Santa Clara valley; founded 1885, inc. 1952. , J.Y., M. Lettau, B.G. Malkiel, and Y. Xu, 2001, "Have Individual Stocks Become More Volatile With regard to computer memory, it means "temporary" and not "highly changeable," which is the usual meaning of the word. See volatile memory.

**1.**(programming)

**volatile**- volatile variable.

**2.**(storage)

**volatile**- See non-volatile storage. ? An Empirical Exploration of Idiosyncratic Risk," Journal of Finance 56, 1-43.

Campbell, J.Y. and T. Vuolteenaho, 2004, "Bad beta, Good Beta," American Economic Review 94, 1249-1275.

Chan, C.K. and N. Chen, 1988, "An Unconditional Test of Asset Pricing and the Role of Firm Size as an Instrumental Variable for Risk," Journal of Finance 63, 309-325.

Dimson, E., 1979, "Risk Measurement When Shares Are Subject to Infrequent in·fre·quent

*adj.*

**1.**Not occurring regularly; occasional or rare: an infrequent guest.

**2.**Trading," Journal of Financial Economics 7, 197-226.

Douglas, G.W., 1969, "Risk in the Equity Markets: An Empirical Appraisal

*A valuation or an approximation of value by impartial, properly qualified persons; the process of determining the value of an asset or liability, which entails expert opinion rather than express commercial transactions.*of Market Efficiency," Yale Economic Essays 9, 3-45.

Dybvig, P.H. and S.A. Ross Ross , Sir

**Ronald**1857-1932.

British physician. He won a 1902 Nobel Prize for proving that malaria is transmitted to humans by the bite of the mosquito. , 1985, "Differential Information and Performance Measurement Using a Security Market Line," Journal of Finance 40, 383-400.

Engle, R.E, D.M. Lilien, and R.P. Robins, 1987, "Estimation

**estimation**

In mathematics, use of a function or formula to derive a solution or make a prediction. Unlike approximation, it has precise connotations. In statistics, for example, it connotes the careful selection and testing of a function called an estimator. of Time Varying Risk Premia

**Premia**is a

*comune*(municipality) in the Province of Verbano-Cusio-Ossola in the Italian region Piedmont, located about 140 km northeast of Turin and about 40 km northwest of Verbania, on the border with Switzerland. in the Term Structure: The ARCH-M Model," Econometrica

**is an academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society via Blackwell Publishing. 55, 391-407.**

*Econometrica*Fama, E.F. and K. French, 1992, "Cross-Section of Expected Stock Returns," Journal of Finance 47, 427-465.

Fama, E.F. and K. French, 1993, "Common Risk Factors in the Returns on Stocks and Bonds," Journal of Financial Economics 33, 3-56.

Fama, E.F. and K. French, 1995, "Size and Book-to-Market Factors in Earnings and Returns," Journal of Finance 50, 131-155.

Fama, E.F. and K. French, 1996, "Multifactor Explanations for Asset Pricing Anomalies," Journal of Finance 51, 55-84.

Fama, E.F. and K. French, 1997, "Industry Costs of Equity," Journal of Financial Economics 43, 153-193.

Fama, E.F. and J.D. MacBeth, 1973, "Risk and Return: Some Empirical Tests," Journal of Political Economy 81, 607-636.

Ferson, W.E. and C.R. Harvey, 1991, "The Variation of Economic Risk Premiums," Journal of Political Economy 99, 385-415.

Ferson, W.E. and C.R. Harvey, 1999, "Conditioning Variables and the Cross-Section of Stock Returns," Journal of Finance 54, 1325-1360.

French, K.R., G.W. Schwert, and R.F. Stambaugh, 1987, "Expected Stock Returns and Volatility," Journal of Financial Economics 19, 3-29.

Friend, I. and M. Blume, 1970, "Measurement of Portfolio Performance under Uncertainty," American Economic Review 60, 607-636.

Goyal John Fowles refers to a goyal as a tiny, chimneyed Greek cottage with a roof that is barrel vaulted.

**References**

John Fowles, The Magus, 1965, Published by Little Brown & Company Greek architecture , A. and P. Santa-Clara, 2003, "Idiosyncratic Risk Matters!" Journal of Finance 58, 975-1008.

Hansen Han·sen ,

**Gerhard Henrik Armauer**1746-1845.

Norwegian physician and bacteriologist who discovered (1869) the leprosy bacillus. , L.P. and S.F. Richard, 1987, "The Role of Conditioning Information in Deducing Testable Restrictions Implied

*Inferred from circumstances; known indirectly.*

In its legal application, the term

*implied*is used in contrast with express, where the intention regarding the subject matter is explicitly and directly indicated. by Dynamic Asset Pricing Models," Econometrica 50, 587-613.

Harris Harris, Scotland: see Lewis and Harris. , R.S., F.C. Marston Mar·ston ,

**John**1575?-1634.

English playwright whose works include

*The Malcontent*and

*The Dutch Courtezan*(both 1604). , D.R. Mishra

**Mishra**or

**Misra**is an Indian surname, normally associated with the Brahmin mostly it is similar to Mitra or Maitreya or Maitra or Maitri friend Mishra and Mitra written in devnagri script look almost identical and both have same meaning. , and T.J. O'Brien O'Bri·en ,

**Edna**Born 1932.

Irish writer whose works, including

*The Lonely Girl*(1962) and

*Johnny I Hardly Knew You*(1977), explore the lives of women in modern-day Ireland.

**Noun**

**1.**, 2004, "Ex Ante Cost of Equity Estimates of S&P 500 Firms: The Choice between Global and Domestic CAPM," Financial Management 32, 51-66.

Harvey, C.R., 1989, "Time-Varying Conditional Covariances in Tests of Asset Pricing Models," Journal of Financial Economics 24, 289-317.

Harvey, C.R., 2001, "The Specification of Conditional Expectations," Journal of Empirical Finance 8, 573-637.

Jagannathan, R. and Z. Wang, 1996, "The Conditional CAPM and the Cross-Section of Expected Returns," Journal of Finance 51, 3-53.

Jegadeesh, N., 1992, "Does Market Risk Really Explain the Size Effect," Journal of Financial and Quantitative Analysis Quantitative Analysis

A security analysis that uses financial information derived from company annual reports and income statements to evaluate an investment decision.

Notes: 27, 337-351.

Jegadeesh, N. and S. Titman, 1993, "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency," Journal of Finance 48, 65-91.

Jostova, G. and A. Philipov, 2005, "Bayesian Analysis Bayesian analysis A decision-making analysis that '…

*permits the calculation of the probability that one treatment is superior based on the observed data and prior beliefs…subjectivity of beliefs is not a liability, but rather explicitly allows*of Stochastic By guesswork; by chance; using or containing random values.

**stochastic**- probabilistic Betas," Journal of Financial and Quantitative Analysis 40, 747-778.

Kyle <noinclude></noinclude>

''This article or section is being rewritten at

One derivation of the surname is from the Scottish Highland word

*caol*, 'channel', or 'strait'. There are other possible derivations (see below). , A., 1985, "Continuous Auctions and Insider Trading," Econometrica 53, 1315-1335.

Lettau, M. and S. Ludvigson, 2001, "Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia Are Time Varying," Journal of Political Economy 109, 1238-1287.

Lewellen, J. and S. Nagel

**Nagel**can refer to:

**People**

- Ernest Nagel (1901-1985) Philosopher of science
- Patrick Nagel (1945-1984) American artist
- Steven R. Nagel (born 1946) American astronaut
- Thomas Nagel (born 1937) Professor of Philosophy and Law at New York University

Lintner, J., 1965, "The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics 47, 13-37.

Lo, A. and A.C. MacKinlay, 1990, "When Are Contrarian Contrarian

An investment style that goes against prevailing market trends by buys assets that are performing poorly and selling when they perform well.

Notes:

A contrarian investor believes that the people who say the market is going up do so only when they are fully Profits Due to Stock Market Overreaction o·ver·re·act

*intr.v.*

**o·ver·re·act·ed**,

**o·ver·re·act·ing**,

**o·ver·re·acts**

To react with unnecessary or inappropriate force, emotional display, or violence. ?" Review of Financial Studies 3, 175-205.

Markowitz, H., 1959, Portfolio Selection: Efficient Diversification Efficient diversification

The organizing principle of portfolio theory, which maintains that any risk-averse investor will search for the highest expected return for any particular level of portfolio risk. of Investments, New York, Wiley

**Wiley**may refer to:

- Wiley, Colorado, a U.S. town
- Wiley-Kaserne, a district of the city of Neu-Ulm, Germany
- USS Wiley (DD-597), a U.S. destroyer from the nineteenth century named after William Wiley
- Wiley College, a college in Texas founded by Isaac Wiley

Merton, R., 1980, "On Estimating the Expected Return on the Market: An Exploratory Investigation," Journal of Financial Economics 8, 323-361.

Miller, M.H. and M. Scholes, 1972, "Rates of Return in Relation to Risk: A Re-Examination of Some Recent Findings," in Michael

**Michael, archangel**

Michael (mī`kəl) [Heb.,=who is like God?], archangel prominent in Christian, Jewish, and Muslim traditions. In the Bible and early Jewish literature, Michael is one of the angels of God's presence. C. Jensen, Ed., Studies in the Theory of Capital Markets, New York, Praeger.

Newey, WK. and K.D. West, 1987, "A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix In statistics and probability theory, the

**covariance matrix**is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable. ," Econometrica 55, 703-708.

Pfistor, L. and R.F. Stambaugh, 2003, "Liquidity Risk and Expected Stock Returns," Journal of Political Economy 111, 642-685.

Petkova, R. and L. Zhang, 2005, "Is Value Riskier than Growth?" Journal of Financial Economics 78, 187-202.

Scholes, M. and J.T. Williams, 1977, "Estimating Betas from Nonsynchronous Data," Journal of Financial Economics 5, 309-327.

Shanken, J., 1990, "Intertemporal Asset Pricing: An Empirical Investigation," Journal of Econometrics 45, 99-120.

Shanken, J., 1992, "On the Estimation of Beta-Pricing Models," Review of Financial Studies 5, 1-33.

Sharpe, W.F., 1964, "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk," Journal of Finance 19, 425-442.

(1) Jegadeesh (1992) obtains results similar to Fama and French (1992).

(2) This is because an asset that is on the conditional mean-variance frontier frontier, in U.S. history, the border area of settlement of Europeans and their descendants; it was vital in the conquest of the land between the Atlantic and the Pacific. need not be on the unconditional frontier, as Dybvig and Ross (1985) and Hansen and Richard (1987) point out. Also see Chan and Chen (1988) who indicate that even when betas vary over time, unconditional CAPM can hold.

(3) An incomplete list includes Bollerslev, Engle, and Wooldridge (1988), Harvey (1989, 2001 ), Shanken (1990, 1992), Ferson and Harvey (1991, 1999), Fama and French (1997), Lettau and Ludvigson (2001), Campbell and Vuolteenaho (2004), Jostova and Philipov (2005), Petkova and Zhang (2005), Ang and Chen (2007), Lewellen and Nagel (2006), and Bali (2008).

(4) French, Schwert, and Stambaugh (1987), Campbell, Lettau, Malkiel, and Xu (2001), Goyal and Santa-Clara (2003), and Bali, Cakici. Yan, and Zhang (2005) use within-month daily returns to estimate the monthly market variance

*The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial.*

*In*Zoning

*law, an official permit to use property in a manner that departs from the way in which other property in the same locality*or the monthly idiosyncratic id·i·o·syn·cra·sy

*n.*

*pl.*

**id·i·o·syn·cra·sies**

**1.**A structural or behavioral characteristic peculiar to an individual or group.

**2.**A physiological or temperamental peculiarity.

**3.**or total volatility of each stock trading at the NYSE, Amex, and Nasdaq.

(5) Brav, Lehavy, and Michaely (2005) use analysts' expected rates of return instead of realized rates of return as a proxy for expected return and identify a positive, robust relation between expected return and market beta. Based on their experimental study, Bloomfield and Michaely (2004) find that market professionals expect firms with higher betas to be riskier investments and to generate higher returns. Harris, Marston, Mishra, and O'Brien (2003) estimate the relation between market beta and expected returns for S&P 500 stocks, and their findings indicate that for estimating the cost of equity, the choice between the domestic and global CAPM may not be a material issue for many large US firms.

(6) In Table V, we form decile portfolios based on the GARCH-in-mean beta estimates. The results from the AR(I) and MA(1) models are similar to those in Table V and are available from the authors on request.

(7) We do not present the results from [[beta].sup.AR.sub.i,t|t-1] and [[beta].sup.MA.sub.i,t|t-1] which are similar to those in the table. They are available on request.

Turan Turan (trän`), desert lowland, shared by Kazakhstan, Uzbekistan, and Turkmenistan, S and E of the Aral Sea. G. Bali, Nusret This article is about the Turkish minelayer. For the Pakistani musician, see Nusrat Fateh Ali Khan

**also known as**

*Nusret***is a Turkish minelayer that strongly influenced the course of the 1915 Battle of Gallipoli. Cakici, and Yi Tang**

*Nusrat***tang, in zoology**

tang: see butterfly fish. *

* Turan G. Bali is the David Krell

**David Krell**is a founder and President & CEO of ISE, LLC and ISE Holdings. From 1997 to 1998, he was Chairman and co-founder of K-Squared Research, LLC, a financial services consulting firm. Chair Professor of Finance at the University of New York in New York NE and Visiting Professor of Finance at Koc University, Turkey. Nusret Cala'ci is a Professor of Finance at Fordham University Fordham University (fôr`dəm), in New York City; Jesuit; coeducational; founded as St. John's College 1841, chartered as a university 1846; renamed 1907. Fordham College for men and Thomas More College for women merged in 1974. in New York, NY. Yi Tang is an Assistant Professor of Finance at Fordham University in New York, NE

Table I. Correlation between Realized Beta and Conditional Volatility This table presents the percentiles of the correlation measures for all stocks trading at the NYSE, Amex, and Nasdaq. The correlation statistics indicate a strong relation between the monthly realized betas and their conditional volatility, and a strong relation between the conditional mean of monthly realized betas and their conditional volatility. [[beta].sub.t], [[beta].sub.t], Correlation [[sigma].sub.t] [[sigma].sup.2.sub.t] 1% -0.625 -0.650 5% -0.390 -0.422 10% -0.272 -0.301 20% -0.150 -0.168 30% -0.069 -0.080 40% 0.002 -0.004 50% 0.067 0.067 60% 0.130 0.135 70% 0.196 0.204 80% 0.273 0.287 90% 0.390 0.412 95% 0.493 0.520 99% 0.696 0.711 E([[beta].sub.t] E([[beta].sub.t] [[OMEGA].sub.t-1]), [[OMEGA].sub.t-1]), Correlation [[sigma].sub.t] [[sigma].sup.2.sub.t] 1% -0.988 -0.989 5% -0.971 -0.973 10% -0.944 -0.946 20% -0.838 -0.826 30% -0.505 -0.474 40% 0.073 0.066 50% 0.583 0.549 60% 0.791 0.773 70% 0.882 0.877 80% 0.929 0.931 90% 0.963 0.967 95% 0.979 0.981 99% 0.992 0.993 Table II. Time-Series Mean and Standard Deviation of Realized and Conditional Betas This table presents the percentiles of the time-series mean and standard deviation of realized and conditional betas for the sample period of July 1963 to December 2004. In Panel A, we compute the realized beta for each stock trading at NYSE\Amex\Nasdaq by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP value-weighted index as our proxy for the market portfolio. We estimate conditional betas based on the AR(1), MA(1), and GARCH-in-mean models: AR(1): [[beta].sub.t] = [a.sub.0] + [a.sub.1][[beta].sub.t-1] + [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.AR.sub.t\t-1] = [??.sub.0] + [??.sub.1] [[beta].sub.t-1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], MA(1): [[beta].sub.t] = [b.sub.0] + [b.sub.1][[epsilon].sub.t-1] + [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.MA.sub.t\t-1] = [??.sub.0] + [??.sub.1] [[epsilon].sub.t-1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], GARCH-in-mean: [[beta].sub.t] = [c.sub.0] + [c.sub.1] [[sigma].sub.t\t-1] + [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.GARCH.sub.t\t-1] = [??.sub.0] + [??.sub.1][[sigma].sup.2.sub.t\t-1], E([[epsilon].sup.2.sub.t]| [[OMEGA].sub.t-1]) = [[sigma].sup.2.sub.t\t-1] = [[gamma].sub.0] + [[gamma].sub.1][[epsilon].sup.2.sub.t-1] + [[gamma].sub.2] [[sigma].sup.2.sub.t-1]. Panel A. Realized Beta Is Estimated without the Lagged Market Return Mean 1% 5% 10% 20% [[beta].sup.realized.sub.t] -0.1426 0.0347 0.1112 0.2325 SD 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.3929 0.5348 0.6360 0.7841 [[beta].sup.AR.sub.t\t-1] 0.0020 0.0109 0.0220 0.0442 [[beta].sup.GARCH.sub.t\t-1] 0.0014 0.0101 0.0164 0.0454 30% 40% 50% [[beta].sup.realized.sub.t] 0.3503 0.4610 0.5642 SD 30% 40% 50% [[beta].sup.realized.sub.t] 0.9236 1.0641 1.2175 [[beta].sup.AR.sub.t\t-1] 0.0657 0.0891 0.1154 [[beta].sup.GARCH.sub.t\t-1] 0.0747 0.1071 0.1431 60% 70% 80% [[beta].sup.realized.sub.t] 0.6862 0.8163 0.9777 SD 60% 70% 80% [[beta].sup.realized.sub.t] 1.3879 1.5970 1.8609 [[beta].sup.AR.sub.t\t-1] 0.1439 0.1830 0.2369 [[beta].sup.GARCH.sub.t\t-1] 0.1880 0.2435 0.3251 90% 95% 99% [[beta].sup.realized.sub.t] 1.2420 1.4954 2.0480 SD 90% 95% 99% [[beta].sup.realized.sub.t] 2.3190 2.8373 4.2165 [[beta].sup.AR.sub.t\t-1] 0.3378 0.4527 0.8169 [[beta].sup.GARCH.sub.t\t-1] 0.4693 0.6275 1.1644 Panel B. Realized Beta Is Estimated with the Lagged Market Return Mean 1% 5% 10% 20% [[beta].sup.realized.sub.t] -0.1450 0.0692 0.1744 0.3217 SD 1% 5% 10% 20% [[beta].sup.realized.sub.t] 0.5428 0.7126 0.8439 1.0454 [[beta].sup.AR.sub.t\t-1] 0.0026 0.0117 0.0234 0.0472 [[beta].sup.GARCH.sub.t\t-1] 0.0059 0.0101 0.0221 0.0461 Mean 30% 40% 50% [[beta].sup.realized.sub.t] 0.4587 0.5885 0.7111 SD 30% 40% 50% [[beta].sup.realized.sub.t] 1.2325 1.4229 1.6222 [[beta].sup.AR.sub.t\t-1] 0.0723 0.0976 0.1260 [[beta].sup.GARCH.sub.t\t-1] 0.0802 0.1189 0.1635 Mean 60% 70% 80% [[beta].sup.realized.sub.t] 0.8392 0.9726 1.1352 SD 60% 70% 80% [[beta].sup.realized.sub.t] 1.8551 2.1360 2.4868 [[beta].sup.AR.sub.t\t-1] 0.1609 0.2071 0.2737 [[beta].sup.GARCH.sub.t\t-1] 0.2201 0.2877 0.3853 Mean 90% 95% 99% [[beta].sup.realized.sub.t] 1.3826 1.6126 2.2350 SD 90% 95% 99% [[beta].sup.realized.sub.t] 3.0675 3.6365 5.4468 [[beta].sup.AR.sub.t\t-1] 0.3971 0.5455 1.0561 [[beta].sup.GARCH.sub.t\t-1] 0.5551 0.7714 1.5448 Table III. Performance of Lagged Realized and Conditional Betas in Predicting Future Realized Beta This table presents the percentiles of the [R.sup.2] values from the regression of one-month-ahead realized betas on the lagged realized and conditional betas for our sample period of July 1963 to December 2004. In Panel A, we compute the realized beta for each stock trading at NYSE/Amex/Nasdaq by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP value-weighted index as our proxy for the market portfolio. We estimate conditional betas based on the AR(1), MA(1), and GARCH-in-mean models. We run the following OLS tests to obtain the [R.sup.2] values: [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.realized.sub.t-1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.AR.sub.t\t-1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.MA.sub.t\t-1] + [[epsilon].sub.t], [[beta].sup.realized.sub.t] = [d.sub.0] + [d.sub.1] [[beta].sup.GARCH.sub.t\t-1] + [[epsilon].sub.t], [R.sup.2] 1% 5% 10% 20% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t-1] 0.01% 0.11% 0.22% 0.69% [[beta].sup.AR.sub.t\t-1] 0.74% 2.04% 2.87% 4.58% [[beta].sup.MA.sub.t\t-1] 0.63% 1.94% 2.67% 4.48% [[beta].sup.GARCH.sub.t\t-1] 1.14% 2.46% 3.33% 4.93% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t-1] 0.01% 0.03% 0.08% 0.22% [[beta].sup.AR.sub.t\t-1] 1.29% 2.28% 3.12% 4.22% [[beta].sup.MA.sub.t\t-1] 1.29% 2.35% 3.14% 4.15% [[beta].sup.GARCH.sub.t\t-1] 1.17% 2.12% 3.09% 4.23% [R.sup.2] 30% 40% 50% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t-1] 1.18% 1.94% 3.22% [[beta].sup.AR.sub.t\t-1] 6.16% 7.88% 10.31% [[beta].sup.MA.sub.t\t-1] 5.95% 7.65% 9.98% [[beta].sup.GARCH.sub.t\t-1] 6.75% 8.90% 11.01% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t-1] 0.46% 0.81% 1.21% [[beta].sup.AR.sub.t\t-1] 5.29% 6.10% 7.13% [[beta].sup.MA.sub.t\t-1] 5.19% 6.04% 7.13% [[beta].sup.GARCH.sub.t\t-1] 5.04% 6.02% 7.10% [R.sup.2] 60% 70% 80% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t-1] 4.66% 6.89% 9.94% [[beta].sup.AR.sub.t\t-1] 13.12% 15.77% 19.24% [[beta].sup.MA.sub.t\t-1] 12.54% 15.39% 18.72% [[beta].sup.GARCH.sub.t\t-1] 13.80% 16.70% 20.67% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t-1] 1.73% 2.65% 4.12% [[beta].sup.AR.sub.t\t-1] 8.38% 10.20% 12.66% [[beta].sup.MA.sub.t\t-1] 8.40% 10.19% 12.50% [[beta].sup.GARCH.sub.t\t-1] 8.31% 10.50% 12.58% [R.sup.2] 90% 95% 99% Panel A. Realized Beta Is Estimated without the Lagged Market Return [[beta].sup.realized.sub.t-1] 13.26% 17.67% 26.82% [[beta].sup.AR.sub.t\t-1] 23.90% 27.76% 33.82% [[beta].sup.MA.sub.t\t-1] 23.26% 26.70% 32.32% [[beta].sup.GARCH.sub.t\t-1] 24.88% 27.88% 33.87% Panel B. Realized Beta Is Estimated with the Lagged Market Return [[beta].sup.realized.sub.t-1] 6.63% 9.54% 16.76% [[beta].sup.AR.sub.t\t-1] 16.20% 19.60% 24.77% [[beta].sup.MA.sub.t\t-1] 15.92% 19.16% 24.00% [[beta].sup.GARCH.sub.t\t-1] 16.99% 19.73% 24.62% Table IV. Equal-Weighted Portfolios Sorted by Realized and Conditional Beta We form equal-weighted decile portfolios every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on realized and conditional beta. In Panel A, we compute the realized beta for each stock by using daily returns over the previous month without lagged market return. In Panel B, we compute the realized beta by using daily returns over the previous month with the lagged market return. We use the CRSP value-weighted index as our proxy for the market portfolio. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "High-Low" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the Fama-French (1993) model. Newey-West (1987) adjusted t-statistics appear in parentheses. AR(1): [[beta].sub.t] = [a.sub.0] + [a.sub.1][[beta].sub.t-1] + [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.AR.sub.t\t-1] = [??.sub.0] + [??.sub.1] [[beta].sub.t-1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], MA(1): [[beta].sub.t] = [b.sub.0] + [b.sub.1][[epsilon].sub.t-1] + [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.MA.sub.t\t-1] = [??.sub.0] + [??.sub.1] [[epsilon].sub.t-1], E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], GARCH-in-mean: [[beta].sub.t] = [c.sub.0] + [c.sub.1] [[sigma].sup.2.sub.t\t-1] | [[epsilon].sub.t], E([[beta].sub.t] | [[OMEGA].sub.t-1]) = [[beta].sup.GARCH.sub.t\t-1] = [??.sub.0] + [??.sub.1][[sigma].sup.2.sub.t\t-1], E([[epsilon].sup.2.sub.t] | [[OMEGA].sub.t-1]) = [[sigma].sup.2.sub.t\t-1] = [[gamma].sub.0] + [[gamma].sub.1][[epsilon].sup.2.sub.t-1] + [[gamma].sub.2] [[sigma].sup.2.sub.t-1]. Panel A. Realized Beta Estimated without the Lagged Market Return [[beta].sup.realized [[beta].sup.AR .sub.t-1] .sub.t\t-1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.56 -1.65 1.09 0.08 2 1.36 -1.37 1.21 0.24 3 1.38 -0.01 1.30 0.37 4 1.38 0.23 1.41 0.48 5 1.35 0.47 1.37 0.58 6 1.41 0.72 1.44 0.69 7 1.34 1.01 1.51 0.81 8 1.27 1.37 1.53 0.96 9 1.25 1.91 1.58 1.16 10 High [beta] 1.07 3.40 1.83 1.55 High to low -0.49 ** 0.74 ** (-2.53) (2.33) Alpha -0.48 *** 0.50 ** (-2.85) (2.17) [[beta].sup.MA [[beta].sup.GARCH .sub.t\t-1] .sub.t\t-1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.07 0.02 1.11 0.00 2 1.21 0.25 1.15 0.23 3 1.31 0.37 1.27 0.36 4 1.36 0.48 1.29 0.47 5 1.41 0.59 1.38 0.58 6 1.46 0.70 1.41 0.69 7 1.46 0.82 1.49 0.82 8 1.54 0.96 1.51 0.97 9 1.60 1.15 1.67 1.18 10 High [beta] 1.85 1.62 2.03 1.71 High to low 0.78 ** 0.92 *** (2.47) (2.65) Alpha 0.53 ** 0.60 ** (2.33) (2.60) Panel B. Realized Beta Estimated with the Lagged Market Return [[beta].sup.realized [[beta].sup.AR .sub.t-1] .sub.t\t-1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.44 -2.25 1.13 0.03 2 1.32 -0.55 1.15 0.33 3 1.33 -0.08 1.27 0.47 4 1.39 0.24 1.38 0.60 5 1.42 0.53 1.36 0.71 6 1.41 0.85 1.41 0.83 7 1.36 1.21 1.49 0.95 8 1.38 1.65 1.57 1.08 9 1.25 2.32 1.62 1.27 10 High [beta] 1.08 4.25 2.02 1.74 High to low -0.33 * 0.89 *** (-1.92) (2.66) Alpha -0.35 ** 0.63 *** (-2.36) (2.76) [[beta].sup.MA [[beta].sup.GARCH .sub.t\t-1] .sub.t\t-1] Decile Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.13 0.05 1.10 0.02 2 1.15 0.33 1.19 0.32 3 1.25 0.48 1.25 0.46 4 1.38 0.60 1.28 0.59 5 1.37 0.72 1.39 0.71 6 1.42 0.83 1.40 0.83 7 1.49 0.95 1.44 0.95 8 1.58 1.08 1.55 1.09 9 1.64 1.26 1.69 1.29 10 High [beta] 2.00 1.70 2.11 1.83 High to low 0.87 ** 1.01 *** (2.54) (2.83) Alpha 0.60 ** 0.71 *** (2.62) (3.06) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table V. Equal-Weighted Portfolios Sorted by GARCH-in-Mean Beta after Controlling for Size and BM In Panel A, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their market capitalizations. Then, within each size decile, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "Average Returns" averages across the 10 size deciles to produce decile portfolios with dispersion in market beta and with near-identical levels of firm size, and thus these decile beta portfolios control for differences in size. In Panel B, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their book-to-market ratios (BM). Then, within each BM decile, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The column labeled "Average Returns" averages across the 10 BM deciles to produce decile portfolios with dispersion in market beta and with near-identical levels of BM. Thus, these decile beta portfolios control for differences in BM. Panel A. Equal-Weighted Returns on Beta Portfolios after Controlling for Size Small Size Size 2 Size 3 Size 4 1 Low [beta] 2.85 0.61 0.48 0.63 2 2.12 0.98 1.03 0.83 3 2.63 0.98 1.06 0.86 4 2.40 1.31 1.13 1.05 5 2.93 1.29 1.20 1.19 6 3.18 1.51 1.17 1.18 7 3.39 1.79 1.55 1.29 8 3.49 1.83 1.62 1.65 9 4.46 2.18 1.88 1.63 10 High [beta] 5.36 3.04 2.31 2.23 Size 5 Size 6 Size 7 Size 8 1 Low [beta] 0.68 0.78 0.88 0.83 2 0.88 0.86 1.11 1.00 3 0.81 0.84 0.99 1.10 4 0.96 0.98 1.12 1.07 5 1.13 1.12 1.24 1.24 6 1.15 1.18 1.16 1.13 7 1.17 1.17 1.19 1.11 8 1.53 1.40 1.27 1.34 9 1.72 1.69 1.28 1.26 10 High [beta] 2.05 2.13 2.03 1.75 Big Average t- Size 9 Size Returns statistic 1 Low [beta] 0.84 0.80 0.94 *** 5.07 2 1.05 0.88 1.07 *** 5.26 3 0.96 0.93 1.12 *** 4.76 4 1.03 0.99 1.20 *** 4.67 5 1.01 1.02 1.34 *** 4.72 6 1.06 0.97 1.37 *** 4.58 7 1.28 0.99 1.49 *** 4.67 8 1.11 1.01 1.63 *** 4.54 9 1.28 1.00 1.84 *** 4.56 10 High [beta] 1.51 1.08 2.35 *** 4.55 High [beta] 1.41 *** 3.48 to low [beta] Panel B. Equal-Weighted Returns on Beta Portfolios after Controlling for BM Low BM BM 2 BM 3 BM 4 1 Low [beta] 0.35 0.40 0.77 0.86 2 0.19 0.65 0.72 0.90 3 0.51 0.79 0.93 1.04 4 0.59 0.70 1.11 1.10 5 0.69 1.01 1.19 1.11 6 0.45 0.92 0.90 1.20 7 0.85 1.09 1.39 1.31 8 1.00 1.19 1.34 1.40 9 1.03 1.39 1.34 1.58 10 High [beta] 1.64 1.62 1.68 1.90 BM 5 BM 6 BM 7 BM 8 1 Low [beta] 0.82 1.15 1.21 1.42 2 1.03 1.05 1.22 1.49 3 1.02 1.16 1.19 1.32 4 1.05 1.22 1.47 1.49 5 1.13 1.27 1.45 1.57 6 1.20 1.36 1.57 1.64 7 1.27 1.37 1.52 1.68 8 1.17 1.67 1.54 1.70 9 1.63 1.73 1.89 2.05 10 High [beta] 2.04 2.07 2.44 2.31 High Average t- BM 9 BM Returns statistic 1 Low [beta] 1.35 1.91 1.02 *** 4.55 2 1.31 1.95 1.05 *** 4.62 3 1.60 1.83 1.14 *** 4.79 4 1.90 2.28 1.29 *** 5.14 5 1.84 2.13 1.34 *** 5.04 6 1.71 1.85 1.28 *** 4.69 7 1.79 2.22 1.45 *** 4.87 8 1.91 2.26 1.52 *** 4.76 9 2.43 2.43 1.75 *** 4.93 10 High [beta] 2.94 3.16 2.18 *** 4.80 High [beta] 1.16 *** 3.69 to low [beta] *** Significant at the 0.01 level. Table VI. Average Return Differences and FF-3 Alphas within Each Size and BM Deciles In Panel A, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their market capitalizations. Then, within each size decile, we sort stocks into decile portfolios ranked based on conditional beta so that decile 1 (10) contains stocks with the (lowest) highest market beta. We report the average return differences and alphas along with their Newey-West (1987) adjusted t-statistics in parentheses for each size decile. In Panel B, we first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked based on their book-to-market ratios (BM). Then, within each BM decile, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. We report the average return differences and alphas along with their Newey-West (1987) adjusted t-statistics in parentheses for each BM decile. Panel A. Average Return Differences and FF-3 Alphas within Size Deciles [[beta].sup.AR.sub.t\t-1] Decile High [beta] to Low [beta] Alpha Small size 2.49 *** 2.02 *** (5.79) (5.54) Size 2 2.17 *** 1.65 *** (4.94) (4.55) Size 3 1.92 *** 1.40 *** (4.00) (3.55) Size 4 1.67 *** 1.25 *** (3.65) (3.48) Size 5 1.54 *** 1.14 *** (3.46) (3.56) Size 6 1.49 *** 1.17 *** (3.25) (3.59) Size 7 1.19 *** 1.01 *** (2.68) (3.26) Size 8 0.72 * 0.64 ** (1.67) (2.34) Size 9 0.60 0.63 ** (1.33) (2.09) Big size 0.24 0.42 (0.60) (1.58) [[beta].sup.MA.sub.t\t-1] Decile High [beta] to Low [beta] Alpha Small size 2.57 *** 2.06 *** (6.09) (6.05) Size 2 2.22 *** 1.69 *** (5.01) (4.67) Size 3 1.88 *** 1.33 *** (3.88) (3.36) Size 4 1.68 *** 1.23 *** (3.57) (3.44) Size 5 1.52 *** 1.13 *** (3.40) (3.57) Size 6 1.51 *** 1.16 *** (3.21) (3.57) Size 7 1.08 ** 0.88 *** (2.39) (2.87) Size 8 0.82 * 0.73 *** (1.87) (2.64) Size 9 0.65 0.67 ** (1.39) (2.13) Big size 0.24 0.43 (0.60) (1.64) [[beta].sup.GARCH.sub.t\t-1] Decile High [beta] to Low [beta] Alpha Small size 2.50 *** 2.02 *** (5.89) (5.35) Size 2 2.43 *** 1.83 *** (5.34) (5.14) Size 3 1.83 *** 1.27 *** (3.82) (3.40) Size 4 1.60 *** 1.05 *** (3.46) (3.44) Size 5 1.37 *** 0.94 *** (3.06) (3.05) Size 6 1.35 *** 0.93 *** (2.83) (2.69) Size 7 1.15 ** 0.88 *** (2.56) (2.91) Size 8 0.92 ** 0.87 *** (2.09) (3.05) Size 9 0.67 0.65 ** (1.45) (2.13) Big size 0.28 0.50 * (0.68) (1.91) Panel B. Average Return Differences and FF-3 Alphas within Book-to-Market Deciles [[beta].sup.AR.sub.t\t-1] Decile High [beta] to Low [beta] Alpha Low BM 1.50 *** 1.60 *** (3.75) (4.59) BM 2 1.27 *** 1.27 *** (3.44) (3.83) BM 3 0.89 ** 0.80 *** (2.58) (2.71) BM 4 0.98 *** 0.68 *** (2.90) (2.59) BM 5 1.18 *** 0.71 *** (3.48) (2.71) BM 6 0.90 *** 0.44 * (2.90) (1.73) BM 7 0.90 ** 0.47 (2.46) (1.53) BM 8 0.82 ** 0.41 (2.27) (1.39) BM 9 1.51 *** 1.01 *** (4.09) (3.25) High BM 0.60 (0.15) (1.62) (-0.46) [[beta].sup.MA.sub.t\t-1] High [beta] to Low [beta] Alpha Low BM 1.49 *** 1.54 *** (3.74) (4.50) BM 2 1.31 *** 1.33 *** (3.66) (4.12) BM 3 0.82 ** 0.70 ** (2.37) (2.37) BM 4 0.93 *** 0.65 ** (2.76) (2.41) BM 5 1.22 *** 0.75 *** (3.48) (2.73) BM 6 0.84 *** 0.39 (2.66) (1.56) BM 7 0.90 ** 0.46 (2.39) (1.45) BM 8 0.74 ** 0.30 (2.15) (1.07) BM 9 1.41 *** 0.94 *** (3.86) (2.98) High BM 0.49 (0.24) (1.32) (-0.73) [[beta].sup.GARCH.sub.t\t-1] High [beta] to Low [beta] Alpha Low BM 1.29 *** 1.36 *** (3.27) (4.21) BM 2 1.22 *** 1.19 *** (3.28) (3.57) BM 3 0.91 ** 0.73 *** (2.61) (2.67) BM 4 1.03 *** 0.67 ** (3.06) (2.52) BM 5 1.22 *** 0.79 *** (3.22) (2.65) BM 6 0.92 *** 0.37 (2.83) (1.53) BM 7 1.24 *** 0.81 ** (3.08) (2.38) BM 8 0.89 ** 0.37 (2.38) (1.25) BM 9 1.59 *** 1.14 *** (4.10) (3.52) High BM 1.25 *** 0.59 * (3.21) (1.75) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table VII. Firm-Level Cross-Sectional Regressions Panel A presents the firm-level cross-sectional regression results for the NYSE/Amex/Nasdaq stocks for our sample period of 23193 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCH-in-mean specification of the realized beta measures. We calculate the realized beta of each stock by using daily data over the previous month with the lagged market return. Here, log[ME.sub.t-1] is the last month's log market capitalization (size), and log([BE.sub.t-1]/[ME.sub.t-1]) is the last fiscal year's log book-to-market ratio. The time-series average slope coefficients are reported in each row. Newey-West (1987) adjusted t-statistics appear in parentheses. The last column presents the average [R.sup.2] values. [[beta].sub. [[beta].sub.AR [[beta].sub.MA [[beta].sub. realized.sub. .sub.t\t-1] .sub.t\t-1] GARCH.sub. t-1] t\t-1] -0.0678 (-1.19) 0.4443 ** (2.09) 0.4890 ** (2.26) 0.4565 ** (2.32) -0.0124 (-0.21) 0.7985 *** (3.07) 0.8063 *** (3.09) 0.8010 *** (3.05) -0.0357 (-0.68) 0.6694 *** (3.34) 0.6796 *** (3.36) 0.6470 *** (3.41) 0.0045 (0.08) 0.8967 *** (3.65) 0.9041 *** (3.66) 0.8929 *** (3.60) log[ME.sub.t-1] log [R.sup.2] ([BE.sub.t-1]/ [ME.sub.t-1]) 1.04% 2.06% 2.02% 2.13% -0.2373 *** 1.81% (-4.75) 0.4185 *** 1.00% (5.98) -0.2029 *** 0.2396 *** 2.69% (-3.83) (3.21) -0.2229 *** 2.79% (-4.18) -0.2907 *** 4.34% (-4.69) -0.2901 *** 4.31% (-4.68) -0.2869 *** 4.27% (-4.52) 0.4328 *** 1.85% (6.67) 0.4721 *** 2.83% -7.39 0.4718 *** 2.79% -7.37 0.4771 *** 2.89% (7.44) -0.1861 *** 0.2765 *** 3.50% (-3.31) (4.03) -0.2587 *** 0.2745 *** 4.70% (-4.01) (4.29) -0.2579 *** 0.2740 *** 4.70% (-4.00) (4.28) -0.2526 *** 0.2872 *** 4.87% (-3.86) (4.62) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table VIII. Long-Term Predictive Power of Conditional Beta This table presents the equal-weighted average returns, average return differences, and alphas from the 1- to 12-month-ahead predictability of stock returns. We form equal-weighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on the GARCH-in- mean beta estimates conditional on time t - 1 to t - 12. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or expected betas. The row High-Low refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the Fama-French (1993) model. Newey-West (1987) adjusted t-statistics appear in parentheses. [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t\t-1] t-1\t-2] t-2\t-3] t-3\t-4] 1 Low P 1.10 1.11 1.12 1.12 2 1.19 1.16 1.18 1.22 3 1.25 1.24 1.22 1.26 4 1.28 1.29 1.29 1.23 5 1.39 1.41 1.38 1.41 6 1.40 1.37 1.41 1.43 7 1.44 1.46 1.48 1.42 8 1.55 1.55 1.52 1.56 9 1.69 1.69 1.69 1.66 10 High [beta] 2.11 2.06 2.05 2.04 High to low 1.01 *** 0.95 *** 0.93 *** 0.92 ** (2.83) (2.69) (2.60) (2.55) Alpha 0.71 *** 0.64 *** 0.62 *** 0.60 *** (3.06) (2.82) (2.69) (2.58) [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t-4\t-5] t-5\t-6] t-6\t-7] t-7\t-8] 1 Low P 1.12 1.16 1.16 1.17 2 1.25 1.20 1.20 1.20 3 1.22 1.23 1.25 1.27 4 1.31 1.30 1.31 1.29 5 1.37 1.42 1.33 1.37 6 1.44 1.41 1.48 1.44 7 1.44 1.43 1.42 1.44 8 1.56 1.58 1.57 1.53 9 1.67 1.64 1.64 1.65 10 High [beta] 2.04 2.03 2.00 1.98 High to low 0.92 ** 0.87 ** 0.84 ** 0.81 ** (2.54) (2.42) (2.33) (2.23) Alpha 0.61 *** 0.56 ** 0.53 ** 0.49 ** (2.63) (2.4l) (2.28) (2.10) [[beta].sup. [[beta].sup. [[beta].sup. [[beta].sup. Decile GARCH.sub. GARCH.sub. GARCH.sub. GARCH.sub. t-8\t-9] t-9\t-10] t-10\t-11] t-11\t-12] 1 Low P 1.19 1.21 1.21 1.22 2 1.21 1.23 1.24 1.25 3 1.25 1.24 1.27 1.27 4 1.30 1.28 1.33 1.32 5 1.37 1.40 1.36 1.39 6 1.43 1.43 1.44 1.44 7 1.42 1.42 1.40 1.38 8 1.53 1.55 1.56 1.54 9 1.67 1.64 1.66 1.67 10 High [beta] 1.96 1.98 1.97 1.97 High to low 0.77 ** 0.77 ** 0.76 ** 0.75 ** (2.l3) (2.l6) (2.l2) (2.09) Alpha 0.46 ** 0.44 * 0.43 * 0.43 * (l.96) (l.93) (l.88) (l.88) *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table IX. Average Returns on Beta Portfolios after Controlling for Liquidity and Momentum This table presents the equal-weighted average returns and average return differences on beta portfolios after controlling for liquidity and momentum. We first form decile portfolios of NYSE/Amex/Nasdaq stocks ranked according to their illiquidity, dollar trading volume, past average 6-month (MOM6), and past average 12-month (MOM12) returns. Then, within each illiquidity, volume, MOM6, and MOM12 decile, we sort stocks into decile portfolios ranked based on GARCH-in-mean beta so that decile 1 (10) contains stocks with the lowest (highest) market beta. The average returns reported below are the averages across the 10 illiquidity, volume, MOM6, and MOM12 deciles to produce decile portfolios with dispersion in market beta and with near-identical levels of illiquidity, volume, MOM6, and MOM12. Newey-West (1987) adjusted t-statistics appear in parentheses. Decile Illiquidity Volume MOM6 MOM12 1 Low [beta] 0.99 *** 0.92 *** 0.86 *** 1.05 *** (5.23) (4.92) (3.89) (4.82) 2 1.13 *** 1.05 *** 0.94 *** 1.18 *** (5.30) (5.13) (3.99) (5.10) 3 1.18 *** 1.09 *** 1.02 *** 1.22 *** (5.06) (4.72) (4.19) (5.17) 4 1.19 *** 1.19 *** 1.16 *** 1.34 *** (4.63) (4.70) (4.53) (5.34) 5 1.29 *** 1.32 *** 1.30 *** 1.53 *** (4.74) (4.82) (5.02) (6.01) 6 1.38 *** 1.33 *** 1.36 *** 1.55 *** (4.61) (4.51) (5.00) (5.71) 7 1.37 *** 1.45 *** 1.39 *** 1.52 *** (4.44) (4.51) (4.82) (5.43) 8 1.57 *** 1.58 *** 1.45 *** 1.60 *** (4.44) (4.43) (4.61) (5.21) 9 1.75 *** 1.83 *** 1.57 *** 1.67 *** (4.36) (4.51) (4.47) (4.83) 10 High [beta] 2.20 *** 2.38 *** 1.85 *** 1.91 *** (4.33) (4.58) (4.11) (4.34) High to low 1.20 *** 1.46 *** 0.99 *** 0.86 *** (3.05) (3.68) (2.89) (2.74) *** Significant at the 0.01 level. Table X. Firm-Level Cross-Sectional Regressions with Size, BM, Liquidity, and Momentum This table presents the firm-level cross-sectional regression results for the NYSE/Amex/Nasdaq stocks for the sample period of July 1963 to December 2004. We estimate the monthly conditional beta based on the GARCH-in-mean specification. Here, [ILLIQ.sub.t-1] is the last month's illiquidity measure of each stock, [VOL.sub.t-1] is the last month's dollar trading volume, [MOM6.sub.t-1] is the past 6-month average return, [MOM12.sub.t-1] is the past 12-month average return, log[ME.sub.t-1] is the last month's log market capitalization (size), and log([BE.sub.t-1]/[ME.sub.t-1]) is the last fiscal year's log book-to-market ratio. The time-series average slope coefficients are reported in each row. Newey-West (1987) adjusted t-statistics appear in parentheses. The [R.sup.2] column presents the average [R.sup.2] values. [[beta].sup.GARCH.sub.t|t-1] [ILLIQ.sub.t-1] [VOL.sub.t-1] 0.5092 *** 6.4938 *** (2.45) (3.41) 0.5183 *** -- (2.46) 6.3239 *** (-2.57) 0.4123 *** (2.06) 0.4058 *** (2.12) 0.6453 *** 3.1506 ** (3.00) (2.20) 0.6609 *** 3.2395 ** (3.l9) (2.22) 0.6235 *** (0.00) (2.92) (-0.01) 0.6361 *** 0.11 (3.09) (0.08) [[beta].sup.GARCH.sub.t|t-1] [MOM6.sub.t-1] [MOM12.sub.t-1] 0.5092 *** (2.45) 0.5183 *** (2.46) 0.4123 *** -3.6474** (2.06) (-2.33) 0.4058 *** 1.31 (2.12) (0.61) 0.6453 *** -4.6308 *** (3.00) (-3.25) 0.6609 *** (0.09) (3.l9) (-0.05) 0.6235 *** -4.7820 *** (2.92) (-3.41) 0.6361 *** (0.31) (3.09) (-0.16) [[beta].sup.GARCH.sub.t|t-1] log[ME.sub.t-1] log([BE.sub.t-1]/ [ME.sub.t-1]) 0.5092 *** (2.45) 0.5183 *** (2.46) 0.4123 *** (2.06) 0.4058 *** (2.12) 0.6453 *** -0.1480 ** 0.3141 *** (3.00) (-2.51) (4.46) 0.6609 *** -0.1659 *** 0.3065 *** (3.l9) (-2.84) (4.55) 0.6235 *** -0.1893 *** 0.3215 *** (2.92) (-2.98) (4.58) 0.6361 *** -0.2087 *** 0.3136 *** (3.09) (-3.31) (4.67) [[beta].sup.GARCH.sub.t|t-1] [R.sup.2] 0.5092 *** 2.87% (2.45) 0.5183 *** 2.64% (2.46) 0.4123 *** 3.45% (2.06) 0.4058 *** 3.53% (2.12) 0.6453 *** 6.27% (3.00) 0.6609 *** 6.33% (3.l9) 0.6235 *** 6.14% (2.92) 0.6361 *** 6.20% (3.09) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XI. Equal-Weighted Portfolios of NYSE Stocks Sorted by Conditional Beta We form equal-weighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE stocks based on the conditional GARCH-in-mean beta. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "High-Low" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the Fama-French (1993) model. Newey-West (1987) adjusted t-statistics appear in parentheses.Decile Average Return Average Return Average Return (Univariate Sort) (After (After Controlling Controlling for Size) for BM) 1 Low [beta] 1.11 1.06 1.11 2 1.21 1.20 1.15 3 1.34 1.24 1.22 4 1.36 1.32 1.25 5 1.35 1.37 1.33 6 1.35 1.37 1.37 7 1.45 1.45 1.30 8 1.51 1.56 1.46 9 1.54 1.65 1.58 10 High [beta] 1.97 1.90 1.89 High to low 0.86 *** 0.84 *** 0.78 *** (2.79) (2.81) (2.94) Alpha 0.37 ** 0.43 *** 0.34 ** (2.44) (2.87) (2.23) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XII. Equal-Weighted Portfolios Using NYSE Breakpoints and Controlling for Microstructure Effects We form equal-weighted decile portfolios for every month from July 1963 to December 2004 by sorting the NYSE/Amex/Nasdaq stocks based on the realized and conditional beta. We compute the realized beta by using daily returns over the previous month with the lagged market return. We generate portfolios based on the NYSE breakpoints and skipping the month following portfolio formation. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) realized or conditional betas. The row "High-Low" refers to the difference in monthly returns between portfolios 10 and 1. The row "Alpha" reports Jensen's alpha with respect to the Fama-French (1993) model. Newey-West (1987) adjusted t-statistics appear in parentheses. Decile [[beta].sup.realized. [[beta].sup.AR. sub.t-2] sub.t-1|t-2] Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.31 -1.56 1.17 0.21 2 1.31 -0.17 1.30 0.52 3 1.36 0.18 1.40 0.65 4 1.36 0.45 1.39 0.76 5 1.38 0.70 1.44 0.86 6 1.40 0.96 1.45 0.95 7 1.38 1.24 1.58 1.04 8 1.35 1.59 1.57 1.14 9 1.31 2.09 1.65 1.29 10 High [beta] 1.23 3.82 1.87 1.73 High to low (0.08) 0.70 ** (-0.51) (2.24) Alpha -0.09 0.48 ** (-0.82) (2.32) Decile [[beta].sup.MA.sub. [[beta].sup.GARCH.sub. t-1|t-2] t-1|t-2] Average Average Average Average Return Beta Return Beta 1 Low [beta] 1.15 0.22 1.14 0.19 2 1.32 0.53 1.24 0.50 3 1.38 0.66 1.33 0.64 4 1.41 0.77 1.43 0.75 5 1.43 0.86 1.36 0.85 6 1.48 0.95 1.43 0.94 7 1.53 1.04 1.49 1.03 8 1.59 1.14 1.57 1.14 9 1.67 1.28 1.68 1.30 10 High [beta] 1.87 1.69 2.06 1.80 High to low 0.72 ** 0.92 *** (2.29) (2.76) Alpha 0.49 ** 0.63 *** (2.38) (3.07) *** Significant at the 0.01 level. ** Significant at the 0.05 level. Table XIII. Equal-Weighted Beta Portfolios after Controlling for Size and Book-to-Market Simultaneously This table presents average returns for each beta quintile, the average return differences between high- and low-beta portfolios, and the FF-3 alpha differences between high- and low-beta portfolios. We report the results for realized and conditional betas after controlling for size and book-to-market. At the beginning of month t, we first sort the NYSE/Amex/Nasdaq stocks into five size (market equity) portfolios. Then within each size portfolio, we sort the stocks into five BM (book-to-market equity ratio) portfolios. Finally, within each portfolio formed from the intersections of five size and five BM portfolios, we sort the stocks into five beta portfolios, based on their realized and conditional betas in month t-1. Newey-West (1987) adjusted t-statistics appear in parentheses. [[beta].sup. [[beta].sup.AR. Quintile real?zed.sub.t-1] sub.t|t-1] Low [beta] 1.3016 0.9565 (4.79) (4.70) 2 1.3720 *** 1.1579 *** (5.44) (4.76) 3.00 1.3938 *** 1.3315 *** (5.23) (4.87) 4.00 1.4153 *** 1.5181 *** (4.83) (4.91) High [beta] 1.3066 *** 1.9297 *** (3.69) (4.81) Return dif. 0.0050 *** 0.9731 *** High [beta] to low [beta] (0.04) (3.75) Alpha diff. -0.1138 *** 0.6903 *** High [beta] to low [beta] (-1.02) (3.92) [[beta].sup.MA. [[beta].sup.GARCH. Quintile sub.t|t-1] sub.t|t-1] Low [beta] 0.9380 0.9351 (4.59) (4.69) 2 1.1623 *** 1.1771 *** (4.79) (4.88) 3.00 1.3190 *** 1.2629 *** (4.80) (4.65) 4.00 1.5217 *** 1.5126 *** (4.95) (4.89) High [beta] 1.9492 *** 1.9944 *** (4.83) (4.84) Return dif. 1.0112 *** 1.0593 *** High [beta] to low [beta] (3.84) (3.83) Alpha diff. 0.7272 *** 0.7552 *** High [beta] to low [beta] (4.07) (4.08) *** Significant at the 0.01 level. Table XIV. Fama-MacBeth (1973) Cross-Sectional Regressions Using 125 Size/BM/Beta Portfolios This table presents the cross-sectional regression results from the 125 size/BM/beta portfolios for the sample period of July 1963 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCH-in-mean specification of the realized beta measures. We calculate the realized beta of each portfolio by using daily data over the previous month with the lagged market return. Here, [logME.sub.t-1] is the last month's log market capitalization (size) of each portfolio, and [log(BE.sub.t-1]/[ME.sub.t-1]) is the last fiscal year's log book-to-market ratio of each portfolio. We report the time-series average slope coefficients in each row. Newey-West (1987) adjusted t-statistics appear in parentheses. The [R.sup.2] column presents the average [R.sup.2] values. [[beta].sup. [[beta].sup. [[beta].sup. realized.sub. AR.sub.t|t-1] MA.sub.t|t-1] t-1] -0.0759 (-1.25) 0.3507 * (l.82) 0.3876 ** (l.96) -0.0183 (-0.28) 0.8091 *** (2.83) 0.8580 *** (2.94) -0.0389 (-0.69) 0.5709 *** (2.97) 0.610l *** (3.10) -0.0074 (-0.12) 0.8472 *** (3.04) 0.8942 *** (3.l5) [[beta].sup. (log(Be.sub.t-1]/ GARCH.sub.t|t-1] [logMe.sub.t-1] [ME.sub.t-1]) 0.4359 ** (2.l3) -0.1864 *** (-3.43) -0.2673 *** (-3.97) -0.2698 *** (-3.99) 0.8093 *** -0.2583 *** (2.78) (-3.88) 0.7236 *** (6.64) 0.7860 *** (7.24) 0.7891 *** (7.25) 0.6356 *** 0.7856 *** (3.12) (7.22) -0.1222 ** 0.5424 *** (-2.17) (4.9l) -0.2081 *** 0.4918 *** (-2.99) (4.82) -0.2106 *** 0.4914 *** (-3.01) (4.8l) 0.8451 *** -0.1973 *** 0.5045 *** (2.98) (-2.88) (5.l1) [R.sup.2] 7.98% 13.88% 13.72% 14.41% 23.05% 30.33% 30.26% 30.60% 15.51% 20.38% 20.28% 20.91% 28.98% 35.27% 35.23% 35.36% *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. Table XV. Pooled Panel Regressions Using 125 Size/BM/Beta Portfolios This table presents the pooled panel regression results from the 125 size/BM/beta portfolios for the sample period of July 1963 to December 2004. We estimate the monthly conditional betas based on the AR(1), MA(1), and GARCH-in-mean specification of the realized beta measures. We calculate the realized beta of each portfolio by using daily data over the previous month with the lagged market return. Here, [logME.sub.t-1] is the last month's log market capitalization (size) of each portfolio, and [log(BE.sub.t-1] /[ME.sub.t-1]) is the last fiscal year's log book-to-market ratio of each portfolio. We report the slope coefficients in each row. We adjust the t-statistics given in parentheses for heteroskedasticity, first-order autocorrelation, and contemporaneous cross-correlation in the error term. [[beta].sup. realized.sub. [[beta].sup. [[beta].sup. t-1] AR.sub.t|t-1] MA.sub.t|t-1] -0.0505 ** (-2.44) 0.3067 *** (3.8l) 0.3250 *** (4.03) (0.02) (-0.95) 0.3061 *** (6.65) 0.6251 *** (6.68) (0.03) (-1.30) 0.4372 *** (5.49) 0.4615 *** (5.79) (0.01) (-0.56) 0.6194 *** (6.83) 0.6435 *** (6.89) [[beta].sup. (log(Be.sub.t-1]/ GARCH.sub.t|t-1] [logMe.sub.t-1] [ME.sub.t-1]) 0.2906 *** (3.74) -0.2003 *** (-8.33) -0.1960 *** (-8.01) -0.1887 *** (-7.60) 0.5147 *** -0.1781 *** (5.92) (-7.13) 0.5279 *** (11.77) 0.5546 *** (12.16) 0.5310 *** (11.46) 0.4103 *** 0.5378 *** (5.32) (12.05) -0.1368 *** 0.4582 *** (-5.66) (10.15) -0.1283 *** 0.4704 *** (-5.24) (10.41) -0.1227 *** 0.4436 *** (-4.92) (9.60) 0.5360 *** -0.1098 *** 0.4578 *** (6.l8) (-4.36) (l0.31) *** Significant at the 0.01 level. ** Significant at the 0.05 level.

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Author: | Bali, Turan G.; Cakici, Nusret; Tang, Yi |
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Publication: | Financial Management |

Geographic Code: | 1USA |

Date: | Mar 22, 2009 |

Words: | 19202 |

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