298 IEEE JOURNAL OF SELECTED TOPICS IN …ipollak/CAPM.pdf298 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012 Empirical Evidence Against CAPM: Relating

298 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012

Empirical Evidence Against CAPM: RelatingAlphas and Returns to BetasMayur Agrawal, Debabrata Mohapatra, and Ilya Pollak

Abstract—One of the consequences of the capital asset pricingmodel (CAPM) is that the expected excess return of a financial in-strument is proportional to the expected excess market return. Theproportionality constant, called the instrument’s beta, is the coeffi-cient in the linear least-squares fit of the excess return of the instru-ment with the excess return of the market. CAPM therefore im-plies that stocks with larger empirical estimates of beta will tend toproduce larger returns. We analyze this hypothesis using the stockreturn data for the S&P 500 constituents from 1966 to 2010. Weobtain several statistically significant results inconsistent with thehypothesis. These inconsistencies aremuch less pronounced duringthe last two decades of our dataset than before 1990.

Index Terms—Alpha, beta, capital asset pricing model (CAPM),finance, market, regression, statistical significance, stock.

I. BACKGROUND AND OUR CONTRIBUTIONS

W E empirically analyze the capital asset pricing model(CAPM) [31], [22], [29] in the context of the U.S.

stock market, by constructing and tracking over time severalportfolios consisting of shares of large publicly traded U.S.companies.CAPM has been highly influential both in academic fi-

nance (see, for example, [5] and [30] and references therein)and among industry practitioners [17], [28], [11], [15], [20].Among many reasons behind the large amount of literatureon CAPM are the debates regarding the validity of some ofthe model’s assumptions, as well as inconsistencies betweenCAPM’s predictions and empirical observations. In particular,there exists a significant body of empirical research contra-dicting CAPM [4], [26], [6], [12], [7], [17], [11]. The presentpaper contributes to this literature in the following ways.• In Section V, we propose a novel statistic to test CAPM.We use this statistic to show that one of the conclusions ofCAPM can be rejected with a high level of confidence.

Manuscript received July 15, 2011; revised January 21, 2012, March 22,2012; acceptedMay 16, 2012. Date of publication June 04, 2012; date of currentversion July 13, 2012. This work grew out of a semester project done by the firsttwo authors as part of a financial engineering course taught by the third authorat Purdue’s School of Electrical and Computer Engineering in Spring 2010. Apreliminary version of this paper appeared in the proceedings of the IEEE In-ternational Conference on Acoustics, Speech, and Signal Processing, Prague,Czech Republic, May 2011. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Ali Akansu.M. Agrawal and I. Pollak are with the School of Electrical and Computer

Engineering, Purdue University, West Lafayette, IN 47907 USA ([email protected]; [email protected]).D. Mohapatra is with the Microarchitecture Research Lab, Intel Corporation,

Santa Clara, CA 95054 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JSTSP.2012.2202635

• In Section VIII, we construct and analyze an imple-mentable investment strategy based on our test statistic.This strategy outperforms the market over the testingperiod (1971–2010) and has low correlation with themarket. We show that this strategy is robust to the sizeand price-to-book ratio of the underlying assets, and tothe transaction costs. However, the strategy’s performancesignificantly degrades during the last 20 years, as com-pared to the 1970s and 1980s.

• In Section VII, we carefully consider the implications ofthe heavy tails of returns and their temporal dependencefor our statistical analysis. We show that, despite these twoproperties of returns, the relevant -statistics that we con-struct in our analysis can be well modeled as normal.

• In Section VIII-E, we perform a testing procedure similarto those of [4] and [7] on the data from 1971–2010 (mostof this data appeared after the publication of [4]), and againobtain results inconsistent with CAPM.

Based on these results, we conclude that there is economicallyand statistically significant evidence contradicting CAPM in theU.S. stock market over the past 40 years.We start in Section II by defining the notation and termi-

nology used in the sequel. CAPM is introduced in Section III.The empirical procedure we use for testing CAPM is describedin Section IV. A novel statistic that we use in this testing isintroduced in Section V. Section VI provides an illustrationof the procedure. Section VII treats statistical modeling of re-turns and profit-and-loss sequences. It is empirically shown thatsuch sequences are often heavy-tailed and temporally depen-dent although uncorrelated. Two statistical models are intro-duced that account for these properties. Both models yield -sta-tistics which are approximately normal for a large number ofobservations. The results of the statistical tests are presented inSection VIII.We conclude with a short discussion in Section IX.

II. BASIC TERMINOLOGY: RETURNS,SHARPE RATIOS, AND PnL

The daily return of a portfolio on trading day is defined asthe difference between the portfolio’s closing prices on tradingdays and , divided by the closing price on day :

returnprice price

price(1)

Given a risk-free interest rate , the excess return of a port-folio is defined as the difference between the portfolio’s returnand the risk-free rate. We adopt a widely used assumption thatthe only risk-free asset is cash, and therefore the risk-free rateis zero. In this case, the excess return is the same as the return.(Another widely used proxy for the risk-free rate is the interestrate on the 1-month U.S. Treasury Bills [13]).

1932-4553/$31.00 © 2012 IEEE

AGRAWAL et al.: EMPIRICAL EVIDENCE AGAINST CAPM: RELATING ALPHAS AND RETURNS TO BETAS 299

We view the returns as random variables, denoting the meanand the variance of by and , respectively. A commonlyused performance metric for portfolio is its Sharpe ratio [32],[33], defined as the ratio of the expected value of the excessreturn to the standard deviation of the excess return. This per-formance metric is similar to the signal-to-noise ratio widelyused in signal processing applications. Typically, Sharpe ratiois reported as an annual quantity, i.e., the ratio of the mean an-nual return to the standard deviation of the annual return. Toannualize the Sharpe ratio computed from daily returns, it isusually assumed that daily returns are approximately uncorre-lated and additive [30]. Assuming that the (average) number oftrading days in a year is , we then have that the mean an-nual return is approximately equal to the mean daily return times. The standard deviation of the annual return is then equal to

the standard deviation of the daily return multiplied by .Hence, the daily Sharpe ratio multiplied by is the annual-ized Sharpe ratio.In Section VIII, we discuss investment strategies that have

both long and short positions in various stocks. A long positionin a stock is obtained by simply buying the shares of the stock.A short position in a stock is obtained by borrowing its sharesand selling them. Entering long positions in a set of stocks andshort positions in another set is equivalent to betting that thevalue of the former will grow relative to the latter. For example,a profit would result even if both groups of stocks lose value,provided that the second one loses more than the first. Suchstrategies oftentimes start with zero initial value (when the valueof all long positions is equal to the absolute value of all shortpositions), which makes their return undefined because of thezero denominator in (1). For such strategies, we define the dailyprofit on day (or PnL, for profit and loss) as

value value (2)

where value and value are the net values of all thepositions held by the strategy on days and , respectively.This quantity, by itself, is not very informative, as it scales withposition sizes. We treat the PnL as a random variable and com-pute its Sharpe ratio as the expected PnL divided by the standarddeviation of the PnL, annualized. Note that the Sharpe ratio doesnot depend on position sizes. In other words, multiplying all thepositions by a number does not change the Sharpe ratio. TheSharpe ratio computed this way is a very common way of char-acterizing the performance of investment strategies, especiallyin cases when the return in (1) is not well-defined.

III. CAPM: NOTATION AND TERMINOLOGY

We define the market return, , to be the return of a port-folio broadly representative of the overall market. We let itsmean and variance be and , respectively.Let be the return of portfolio . The linear least-squares

estimator [3] of based on is

where is the covariance between and the market return. This estimator can be rewritten as follows:

with

(3)

(4)

These two parameters are very widely used both in the financialindustry and literature, and are referred to as the beta and alphaof the portfolio . Portfolio could consist of a single stock. Inthis case, the parameters defined in (3) and (4) are called thebeta and alpha of that stock.CAPM consists of a number of assumptions on the structure

of themarket and the preferences of themarket participants [23],[30]. One consequence of the model is that, for any portfolio,

(5)

(6)

Both the assumptions of the model and the derivation of (5),(6) are laid out in a number of textbooks, e.g., [23], [30]. InSection VIII, we describe several experiments which show thatthe empirical behavior of the U.S. stock market over the pastfour decades disagrees with (5), (6) in a statistically significantmanner.

IV. CONSTRUCTING EMPIRICAL TESTS

There are several difficulties with empirically evaluating (5),(6), as none of the quantities involved are directly observable.Moreover, there is no single, universally accepted definition forone crucial ingredient of (6): the market return. Since we restrictour studies to the U.S. stock market, we choose to follow a largebody of literature (see, e.g., [13], [30], and references therein) inusing the return of a broad stock index as a proxy for the marketreturn. Specifically, we use the S&P 500 returns [34] as marketreturns.We use the following standard procedure to empirically esti-

mate the alpha and beta of any portfolio during a time intervalof trading days. We first use the observed daily returns toestimate the expectation as the empirical mean of the returns

:

We form the empirical covariance of the portfolio returnsand the market returns :


where is the empirical mean return for the market portfolio.The empirical market variance is

These are then used to construct estimates of and :

(7)

(8)

Equations (5) and (6) suggest the following testing strategy, de-veloped in [4].1) On day , compute the estimate of the beta for eachmember of the trading universe—in our case, the set ofthe constituents of the S&P 500 index—using the returnsfrom the preceding days.

2) Sort all the members of S&P 500 on day by their esti-mated betas calculated in Step 1, from the smallest to thelargest, and group the stocks into quantiles.

3) Form quantile portfolios and record their returns for days.

4) Increment by and go to Step 1. Iterate until the end ofthe available data is reached.

5) Analyze the returns on the quantile portfolios obtainedduring the entire testing period. The specifics of this stepdepend on the statistical tests one wishes to run. For ex-ample, [4] estimates the alphas of the quantile portfolios.If the stockmarket is consistent with (5), then the estimatedalphas would not be different from zero by a statisticallysignificant amount.

In our simulations, we use years, year,1 and.

V. STATISTIC FOR TESTING CAPM

In addition to testing whether or not alphas are equal to zero,we propose another test statistic. Suppose that and aretwo portfolios, with respective betas and , and respectiveexpected returns and . Then (6) implies

After multiplying the first of these equations by and thesecond by , their right-hand sides become the same, and hencethe left-hand sides are equal:

(9)

Hence, if the stock market is consistent with (6), then (9) wouldhold. In this case, an estimate of the quantity ob-tained through the five-step procedure outlined above, will notdeviate from zero in a statistically significant way.

1Note that the values of and measured in days are time-varying duringour simulations, since the number of trading days in a year varies.

In fact, as shown in the remainder of the paper, we observe theopposite: both (9) and (5) can be rejected with high confidence.

VI. TESTING CAPM: AN ILLUSTRATION

In this section, we provide some intuition for (5), (6) and forthe five-step testing procedure outlined in Section IV. We alsopreview our empirical results of Section VIII. We use the returndata for the S&P 500 constituent stocks and for the S&P 500index itself, from January 1, 1966 until December 31, 2010. Thedata is obtained from CRSP [1] and Compustat through WRDS[36].On the first trading day of each year, we take all the S&P 500

constituent stocks2 and use the five preceding years to estimatethe beta for each stock using (7). We then form ten decile portfo-lios based on the estimated betas. For example, the lowest-decileportfolio contains those 10% of the S&P 500 constituents thathave the lowest estimated betas. We allocate the same dollaramount to each stock within a portfolio. We then calculate thedaily returns of each portfolio over the following year. For ex-ample, during the year 1971 we calculate the daily returns of tenportfolios formed on the basis of betas estimated over the years1966–1970.We perform this procedure for all years from 1971 to 2010.3

Note that this can be regarded as a simulation of ten investmentstrategies: we construct the ten decile portfolios at the beginningof a year, invest in each of them, and track the returns for oneyear. After one year, we reestimate all the betas, rebalance theten portfolios accordingly, hold the new portfolios for one moreyear, etc.The average annual returns of the ten decile portfolios over

the entire 40-year testing period are shown in Fig. 1(a). The av-erage annual S&P 500 return over the same period is 10.2%.To explain how these numbers were computed, we take an ex-ample: the lowest-beta decile portfolio whose average annualreturn is 13.2%. This means that someone who invested $1 intothe lowest-beta portfolio on January 1, 1971 and kept fully rein-vesting into the lowest-beta portfolio every year, would end upwith on December 31, 2010.The weight of each company in the S&P 500 index is based

on the company’s market capitalization (also commonly re-ferred to as the size), defined as the share price multiplied bythe number of shares outstanding [34]. Since our portfolios areequal-weighted, it is reasonable to also compare our portfolios’performance with that of an equal-weighted S&P 500 portfolio.The equal-weighted S&P 500 return for day is the arithmeticaverage of the daily returns on day of all the constituents ofS&P 500. The average annual equal-weighted S&P 500 returnover our 40-year testing period is 13.5%.The plot in Fig. 1(a) shows a trend from large returns for low-

beta portfolios to small returns for high-beta portfolios. Thistrend has been pointed out in the literature—for example, in[4], [5, p. 297], [11, pp. 103–105], [17], and [28]—and goes

2On each rebalancing day, we use the S&P 500 constituent list as of that day.For example, when we simulate portfolio rebalancing for January 2, 1980, weuse the S&P 500member list from January 2, 1980.We thus avoid “survivorshipbias” as we never use future index compositions in our simulations.3Due to our five-year training window, this requires data over 1966–2010.


Fig. 1. (a) Average annual returns for the ten beta decile portfolios, 1971–2010. The average annual S&P 500 and equal-weighted S&P 500 returns over the sameperiod are 10.2% and 13.5%, respectively. (b) Annualized Sharpe ratios for the ten beta decile portfolios, 1971–2010. The annualized Sharpe ratios for the S&P500 and equal-weighted S&P 500 portfolios over the same period are 0.65 and 0.81, respectively. (c) Time series plots of the values for the highest-beta portfolioand the lowest-beta portfolio during 1971–2010.

squarely against (6) which says that the mean portfolio returnmust be proportional to the beta of the portfolio. In order toanalyze the statistical significance of this finding, test statisticsmust be constructed and assumptions regarding their statisticalbehavior must be made. This is done in Section VIII.The annualized Sharpe ratios for the ten decile portfolios over

the period 1971–2010 are shown in Fig. 1(b). The annualizedSharpe ratios for the S&P 500 and equal-weighted S&P 500portfolios over the same period are 0.65 and 0.81, respectively.Fig. 1(c) shows the time series plots during 1971–2010 of the

total values for the lowest-beta decile portfolio and the highest-beta decile portfolio. They briefly converge during the end ofthe dot-com bubble in 1999–2000; however, apart from that anda few other brief periods, the lowest-beta portfolio consistentlyoutperforms the highest-beta portfolio.

VII. INFERRING STATISTICAL SIGNIFICANCE FROMSEQUENCES OF PnLs AND RETURNS

One of our mechanisms of testing CAPM will be to constructan investment strategy which has zero theoretical beta. If CAPMholds, then (6) would mean that such a strategy must have zeroexpected PnL. On the other hand, if a zero-beta strategy hasnonzero expected PnL, then (6) is violated, which is inconsistentwith CAPM. Therefore, in order to find out whether historicalobservations of our strategy’s PnL are consistent with CAPM,we must have a procedure for determining whether the meanPnL is different from zero by a statistically significant margin.

To develop such a procedure, suppose that the observed his-torical PnLs form a sequence of pairwise uncorrelated randomvariables with common mean and standarddeviation . The sample mean then has mean and standarddeviation . The standard deviation is commonly esti-mated as the sample standard deviation .Conditioned on the mean PnL being zero (i.e., on ), the

expectation of is zero, and the -statistic

(10)

tells us by how many standard deviations an observed value ofis different from zero. Suppose that, for some value , the

conditional probability of the event given , isextremely small. This would make the hypothesis veryunlikely. Thus, if we observed , we would say thatby a statistically significant margin. In order to make such anargument, however, we must be able to compute conditionalprobabilities of events of the form , given that. This means that we must know the conditional probabilitydistribution for given that .In order to model this distribution, typically some as-

sumptions are made about the underlying random variables. The most commonly made assumption is that

they are uncorrelated, identically distributed normal randomvariables. Then the -statistic has a student distribution with


Fig. 2. (a) Probability density function for a -distribution with 3.04 degrees of freedom (red), superimposed onto the histogram of the PnLs of the investmentstrategy described in Section VIII-B. (b) Q-Q plot of the PnLs against the quantiles of a -distribution with 3.04 degrees of freedom (blue). Straight red line signifiesa perfect fit. (c) Q-Q plot of the PnLs against normal quantiles (blue). Straight red line signifies a perfect fit.

degrees of freedom [18], which is well approximated bya normal distribution for large .Unfortunately, it is easily shown that the returns of financial

assets and PnLs of investment strategies are typically far fromnormally distributed, and are temporally dependent althoughuncorrelated. The remainder of this section describes both thesephenomena and explains how we account for them in our sta-tistical significance calculations. We construct two probabilisticmodels for the PnLs: one accounts for the non-normal tails ofthe marginal distributions but not for the temporal dependence;the other accounts for both but produces an inferior approxima-tion of the tails of the marginal distribution. In both cases, theconditional distribution of the resulting -statistic givenis shown to be well approximated by a normal distribution forlarge .

A. Non-Normality

As one of many empirical pieces of evidence for non-nor-mality, consider S&P 500’s one-day return of % on“Black Monday” of October 19, 1987. As reported in [24],assuming that the daily S&P 500 returns are independent,identically distributed (i.i.d.) normal random variables, andestimating their mean and variance using the historical data

from 1981–1991 implies that 8 10 is the probabilityof the event that a daily return’s absolute value is 22.8% orabove. Using our own data period 1971–2010, the daily meanreturn is 0.044% and the standard deviation of the daily returnsis 1.078%. We thus obtain a similar probability of reachingor exceeding the absolute value of Black Monday’s return:6.5 10 . Thus, the normal i.i.d. assumption predicts thatthe Black Monday essentially cannot happen. The fact thatit did happen suggests that the S&P 500 returns are not i.i.d.normal.Similarly, the PnL for our strategy described below in

Section VIII-B is 0.05727 on Black Monday, which is about11 standard deviations away from the mean. Under a normalassumption, an event which is at least 11 standard deviationsaway from the mean has probability 3.8 10 . This impliesthat the probability of occurrence of such an event at least onceduring 1000 years (about 252 000 trading days) is about 10 ,assuming that the PnLs are temporally i.i.d. Dropping theindependence assumption and instead assuming that the PnLsare temporally dependent would make the occurrence of suchan event during a fixed time period even less likely. Yet, duringour 40-year testing period, this event occurs twice: the PnLon January 3, 2001 is 0.06216, which is about 12 standard


Fig. 3. (a) Histogram of 100 000 -statistics, each simulated from a sequence of 2500 i.i.d. zero-mean random variables with 3.04 degrees of freedom (blue),with a normal density fit (red). (b) Q-Q plot of the -statistics from plot (a) against normal quantiles (blue). Straight red line signifies a perfect fit.

deviations away from the mean. This shows that the normalmodel for the PnL marginals is inadequate for our strategy.Some measure of success has been achieved through mod-

eling returns and PnLs using heavy-tailed marginals, such as-distributions [30], [24]. Fig. 2(a) shows the probability densityfunction for a -distribution with 3.04 degrees of freedom (inred), superimposed onto the histogram of our strategy’s PnLs.As evident from the figure, the fit is very good. This is confirmedby the Q-Q plot of Fig. 2(b). Note that the Q-Q plot againstnormal quantiles in Fig. 2(c) shows a very poor fit. Note alsothat the -distribution fit actually makes conservative estimatesof the tails: as evident from Fig. 2(b), the -distribution fit some-what overestimates tail probabilities.Moreover, the probability of an event whose magnitude at-

tains or exceeds Black Monday is now 0.0015. The absence ofany such event over the 40 years spanned by our historical data(10 095 trading days) is extremely unlikely: assuming temporalindependence of PnLs, the probability to always stay within 11standard deviations of the mean for 10 095 trading days is nowonly . In fact, even if temporal dependence of PnLsexists and lasts for 100 trading days (i.e., about five months), wewould still have about independent randomvariables in our data set. The probability to exceed 11 stan-dard deviations at least once during 40 years would then bebounded from below by . Based on thisevidence, we conclude that it is reasonable to model the PnLs ofour strategy as random variables with 3.04 degrees of freedom.

B. PnL Model 1: Temporally Independent Heavy-TailedRandom Variables

Amodel for the marginals of is not sufficientto construct a probability distribution for the -statistic : wealso need tomodel the dependency structure of .The simplest possible model is that these random variables

are independent. While we are aware of no closed-form ex-pression for the distribution of the -statistic for the samplemean of independent random variables, this distribution hasbeen shown to be asymptotically normal for two or more de-grees of freedom [16], [9]. Moreover, Fig. 3 shows that it isclose to a normal distribution for a large . Since some of our

calculations below involve 10 years of daily PnL data, corre-sponding to approximately trading days, we haveused in these plots. In Fig. 3(a) we show the his-togram of simulated -statistics obtained from 100 000 inde-pendent realizations of independent zero-meanrandom variables with 3.04 degrees of freedom. In other words,we simulated 2500 independent zero-mean random variableswith 3.04 degrees of freedom, computed the -statistic of theirsample mean, and repeated this experiment 100 000 times. Thehistogram of the resulting 100 000 -statistics is shown in bluein Fig. 3(a), and a normal density fit is shown in red. Fig. 3(b)shows the Q-Q plot of these -statistics against normal quantiles.If the PnLs were temporally independent, thecloseness of this fit would justify treating the distribution of the-statistic as a normal distribution, even though the random vari-ables themselves are not normal. The normaldistribution fit for (corresponding to the entire40-year period of our data) is also very close and therefore weomit it.

C. PnL Model 2: Temporal Dependence and Heavy Tails

There exists a large amount of very convincing evidence thatreturns and PnLs are not temporally independent [30], [35],[24]. Fig. 4 presents some of this evidence. Figs. 4(a) and (b)show the empirical autocorrelation functions of, respectively,squared daily returns and absolute daily returns of the S&P 500index during 1971–2010. The calculations are normalized sothat each point in each plot is an empirical correlation coeffi-cient. The unit correlation coefficients for the zero delays areomitted from the plots. Both these figures strongly suggest thepresence of temporal correlations, implying the presence of tem-poral dependence among the returns. Fig. 4(c) suggests that thereturns themselves are temporally uncorrelated. Figs. 4(d)–(e)show similar plots for the PnLs of the investment strategy de-scribed in Section VIII-B. These empirical observations suggestmodeling returns and PnLs as sequences of temporally uncorre-lated but dependent random variables.Modeling temporal dependency structure of returns and PnLs

is an open research problem. In particular, we are not aware ofany model that can be used to accurately fit both the marginals


Fig. 4. Sample autocorrelation functions (ACFs) computed from daily data for 1971–2010. Each ACF is normalized so that each point in each plot is a correlationcoefficient. In each plot, the value of 1 at zero is omitted. (a) ACF of squared returns of S&P 500. (b) ACF of absolute returns of S&P 500. (c) ACF of returns ofS&P 500. (d) ACF of squared PnLs of the investment strategy described in Section VIII-B. (e) ACF of absolute PnLs of the investment strategy. (f) ACF of PnLsof the investment strategy.

Fig. 5. (a) Estimated marginal distribution for a GARCH(2,1) model obtained through a Monte Carlo simulation (red), superimposed onto the histogram of thePnLs of the investment strategy described in Section VIII-B. (b) Q-Q plot of the PnLs against the quantiles of the simulated GARCH(2,1) marginal. Straight redline signifies a perfect fit.

of the PnLs and the correlation structures of the PnLs, squaredPnLs, and absolute PnLs all at the same time. While no con-sensus methodology has emerged so far, GARCH and relatedmodels [24] are extensively used for the purpose of modelingthe temporal dependence of returns and PnLs.We have used GARCH(2,1) model to fit the daily PnLs of our

investment strategy described in Section VIII-B, over the entire40-year simulation period from 1971 until 2010. This modelassumes the following form for the PnLs :

for (11)

where• are zero-mean i.i.d. -distributed randomvariables with degrees of freedom;

• and are independent random variables for each;

• the nonrandom parameter is the mean of ;• the random variable is given by the followingrecursion:


Fig. 6. (a) Empirical autocorrelation functions of the squared PnLs of the investment strategy described in Section VIII-B (blue) and of the simulated squared PnLsfor the GARCH(2,1) model with maximum likelihood parameters (red). (b) Empirical autocorrelation functions of the absolute PnLs of the investment strategydescribed in Section VIII-B (blue) and of the simulated absolute PnLs for the GARCH(2,1) model with maximum likelihood parameters (red).

Fig. 7. (a) Histogram of 100 000 -statistics, each simulated from a 2500-sample realization of GARCH(2,1) (blue), with a normal density fit (red). (b) Q-Q plotof the -statistics against normal quantiles (blue). Straight red line signifies a perfect fit.

There is no closed-form formula for the marginal distributionof GARCH processes. We therefore use Monte Carlo simula-tions to evaluate the marginal distribution fit provided by thismodel. We first compute the maximum-likelihood parameter es-timates using the daily PnLs of our strategy during 1971–2010.These estimates are:

, and .We then perform a Monte Carlo simulation of the estimatedGARCH(2,1) model and calculate the histogram of the esti-mated samples. The resulting fit and the corresponding Q-Q plotare shown in Fig. 5. While generally good, this fit is poorer thanthe one obtained with the i.i.d. random variables and illus-trated in Figs. 2(a) and (b). The estimated probability of a BlackMonday event (i.e., 11 or more standard deviations away fromthe mean) is , about 6.5 times less likely than theprobability of obtained under the i.i.d. -distributionmodel. The empirical probability to be at least 11 standard de-viations away from the mean at least once in 40 years is about0.21.

Fig. 6(a) compares the empirical autocorrelation of thesquared PnLs [depicted in Fig. 4(d)] with the empirical autocor-relation of the squared PnLs from the estimated GARCH(2,1)model obtained through a Monte Carlo simulation. The com-parison of the empirical autocorrelations for the absolute PnLsis shown in Fig. 6(b). The fits are far from perfect; however,the GARCH model successfully captures the long-range corre-lations of both the squared PnLs and the absolute PnLs.4

Just like the i.i.d. model of Section VII-B, the GARCHmodelyields a -statistic whose conditional distribution given iswell approximated by a normal distribution. This is illustratedin Fig. 7. Specifically, Fig. 7(a) shows a histogram of 100 000-statistics, each calculated from a simulated realization of ourGARCH(2,1) model. The size of each simulated realization is

samples, corresponding to approximately 10 years’worth of daily data. The corresponding Q-Q plot against normal

4We have experimented with GARCH models of different orders. Increasingthe order beyond GARCH(2,1) does not appreciably improve the autocorrela-tion fit and reduces the quality of the marginal distribution fit.


quantiles is shown in Fig. 7(b). From these plots, we concludethat the statistic can be accurately modeled as normal if theunderlying PnLs are distributed according to GARCH(2,1).

VIII. STATISTICAL SIGNIFICANCE OF THE RESULTS

A. Portfolios With the Lowest Beta and the Highest Beta

We construct our first statistical test using (9). Referring tothe five-step testing procedure described in Section IV and il-lustrated in the previous section, we define portfolio to be thelowest-beta decile portfolio—i.e., the portfolio that, on any dayduring a calendar year, contains the stocks whose empirical betaestimated from the preceding five calendar years, is in the lowestdecile. We similarly define , the highest-beta decile portfolio.As described in the previous section, the composition of bothportfolios remains unchanged during each calendar year.We let and be the daily returns on trading dayof portfolios and , respectively. We let and be

their respective betas, and we let and be their respectiveexpected returns. For each trading day during our 40-yearinvestment period, we form the following quantity:

(12)

We assume that all ’s have mean and standard deviation. Note that their mean is equal to , which isthe left-hand side of (9). Therefore, verifying (9) is equivalentto testing the hypothesis .We use (7) over a calendar year to obtain the estimates of the

betas of portfolios and held during that calendar year, anduse these estimates in (12) to compute ’s for that year. Wethen form the sample mean and sample standard deviationof ’s over the entire 40-year investment period. From ourassumption that the mean of each is it follows that themean of is also . We assume that ’s are uncorrelated.Empirical evidence presented in Section VII supports this as-sumption. From this assumption, it follows that the standard de-viation of the samplemean is , where is thetotal number of trading days during the 40 years. We estimatethe standard deviation of as . The specific realizationsfor our data are:

. Conditioned on , the expectation of iszero, and therefore the -statistic is

which is a very compelling piece of evidence for rejecting thehypothesis . Empirical evidence presented in Section VIIsupports the assumption that the -statistic is approximatelynormal, even though the underlying ’s may be heavy-tailedand temporally dependent. Under this assumption, the null hy-pothesis is rejected at significance level 1.89 10 .

B. Mean PnL of a Beta-Hedged Portfolio

We can use the preceding discussion to construct an imple-mentable investment strategy which has the property of beingbeta-hedged. Specifically, note that if today we are able to enter

a long position in portfolio worth dollars and a short po-sition in portfolio worth dollars, then (12) would be to-morrow’s one-day PnL of our portfolio. The beta of our port-folio, computed through (3), would then be zero. However, inorder to construct such a portfolio on some day , we wouldneed estimates of and that only use observations on orbefore day . Hence, we cannot use the estimates utilized in thestatistical analysis of the previous subsection, as they were ob-tained using the data from days .In order to construct a valid, implementable investment

strategy, we instead compute the estimates and on dayusing five previous years of data. The quantity

is then the PnL on day from entering into the followingpositions on day : buying worth ofportfolio and selling short worth of portfolio. The time series of , over the entire investment period,

is the PnL stream of a strategy that makes an investment in thismanner every trading day and liquidates after each trading day.(A similar strategy has been considered in [11].) Note that theinitial gross investment (i.e., the dollar amount of long positionsplus the dollar amount of short positions) for this strategy foreach trading day is always . The sample mean and samplestandard deviation of are, respectively, and

resulting in a -statistic of 5.94, annualized Sharperatio of 0.94, and correlation coefficient with S&P 500 returnsof 0.22 and with equal-weighted S&P 500 returns of 0.26.Following the arguments in Section VII, we assume that the-statistic is approximately normal. As shown in Section VII,this assumption is valid both if ’s are modeled as i.i.d.random variables and if they are modeled as a temporally

dependent GARCH(2,1) process. Under this assumption, thenull hypothesis that the expected PnL of our strategy is zero, isrejected at significance level . We therefore haveconstructed a strategy which has a very low correlation withthe market yet produces returns which are above zero by astatistically significant margin, and which in fact has a betterSharpe ratio than the market.In Table I, we summarize the performance of this strategy

over several sub-periods of the testing period. The returns ofthe strategy are statistically significant during 1971–1990. Theyare barely distinguishable from zero over 1991–2000 with a-statistic of only 0.86. The returns rebound in 2001–2010; how-ever, a -statistic of 1.84 hardly constitutes incontrovertible ev-idence of statistical significance, as it leads us to reject the nullhypothesis at significance level 0.0329, again assuming that the-statistic is approximately normal.

C. Trading Costs

Any trading activity influences the price of the asset that isbeing traded. In addition, fees must be paid for every trade.Therefore, it is unrealistic to assume in the simulations of anyinvestment strategy that the trades required to implement thestrategywould have been executed at the actual historical prices.Modeling and estimating the price impact of trading is notori-ously difficult and is the subject of much research [10], [2], [19],


TABLE IPERFORMANCE OF THE BETA-NEUTRAL STRATEGY ON S&P 500

Fig. 8. Sharpe ratio of the investment strategy of Section VIII-B as a function of the trading cost parameter : (a) over the entire testing period 1971–2010;(b) over 1971–1990 and 1991–2010; (c) over 1971–1980, 1981–1990, 1991–2000, and 2001–2010.

[8] which is beyond the scope of our paper. We adopt a simpleapproach of imposing a linear trading cost on our strategy, toencompass both the fixed fees and the loss incurred due to theprice impact of trading. Specifically, we assume that every timewe trade (buy or sell any shares), we lose some fraction of thehistorical price. For example, suppose yesterday’s and today’sprices of one share of stock XYZ are and , respectively.Suppose yesterday we bought two shares of this stock, and todaywe are selling both shares. The simulations above assume thatwe make a profit of . Now we assume thatthe profit is only , where

is our trading cost associated with buying two shares ofXYZ using the historical price of per share and selling themat per share.The Sharpe ratio of the investment strategy of Section VIII-B

over 1971–2010 is shown in Fig. 8(a) as a function of . For ex-ample, with zero trading costs, the mean PnL is 3.0 10 , thestandard deviation is , the -statistic is 5.94, andthe annualized Sharpe ratio is 0.94. For %, the meanPnL goes down to , the standard deviation remainsat , the statistic goes down to 5.62, and the annual-ized Sharpe ratio goes down to 0.89. As shown in the figure, theperformance degradation is gradual: the Sharpe ratio goes downfrom 0.94 with no trading costs to 0.43 for 1% trading costs.We therefore conclude that this result is robust to trading costsand thus is economically significant. Fig. 8(b) shows the depen-dence of the Sharpe ratio on over 1971–1990 and 1991–2010.Fig. 8(c) shows the same dependence broken down by decade.These results suggest that both economic and statistical signifi-cance of our strategy’s profits hold over the entire testing periodand over its first two decades, but not over the last two decades.

We disregard the trading costs in the remainder of the paper, i.e.,present all the remaining simulation results with .

D. Influence of Company’s Size and Price-to-Book Ratio

It has been shown [14] that company’s stock returns are in-fluenced by the company’s market capitalization, as well as bythe price-to-book ratio (P/B), defined as the ratio of the marketcapitalization to the company’s value. (Book value is the ac-counting value of the company.)To provide evidence that the effect described in

Section VIII-B is distinct from both the size and P/B effects,we repeat the experiment described in Section VIII-B usinguniverses of stocks with similar sizes and similar P/B ratios,namely, the following four universes:• 200 members of S&P 500 with the smallest marketcapitalization;

• 200 members of S&P 500 with the largest marketcapitalization;

• 200 members of S&P 500 with the smallest P/B ratio;• 200 members of S&P 500 with the largest P/B ratio.

For each 200-member universe, portfolios L and H are definedsimilarly to Section VIII-B: specifically, they contain 20 stockswith, respectively, the lowest and highest estimated betas overthe previous five years. The summary statistics over the entiretesting period 1971–2010 are given in Table II, and indicate thatthe statistically significant deviation of fromzero largely holds even after controlling for companies’ sizesand P/B ratios. For convenience, we reproduce in this table theresults from the first line of Table I as well as the summarystatistics for the S&P 500 index over the same time period.As further evidence of the fact that the effect described in

Section VIII-B is distinct from the size and P/B effects, we


TABLE IIPERFORMANCE OF THE BETA-NEUTRAL STRATEGY ON VARIOUS SUBSETS OF S&P 500, 1971–2010

TABLE IIICORRELATION OF THE BETA-NEUTRAL STRATEGY ON

S&P 500 WITH SMB AND HML

compute the correlation coefficients between the PnLs of theinvestment strategy of Section VIII-B and the PnLs of theFama–French size and P/B portfolios [14] constructed out ofthe S&P 500 constituents. Specifically, the Fama–French SMBportfolio is long small companies and short large companies.The Fama–French HML portfolio is long companies withsmall P/B ratio and short companies with large P/B ratio.5 Thecorrelation coefficients between the PnLs of our strategy andSMB and HML portfolios, over several different time periods,are shown in Table III. Most of these correlation coefficientshave small absolute values.

E. Alphas

In order to evaluate the statistical significance of our alpha es-timates, we make the following modeling assumptions, whichare significantly more restrictive than those made in the pre-ceding sections. Suppose that we are evaluating the statisticalsignificance over a time window of duration trading days,say, days . We assume that the market returns for allthese days are deterministic and that we have the following re-lationship between the returns of the th quantile portfolio andthe market returns:

for (13)

where are i.i.d. zero-mean normal randomvariables with variance . It can then be shown [18] that (7),(8) are unbiased estimates of and , respectively, and thatthe covariance matrix of the vector is

(14)

5Note that the Fama-French size and P/B portfolio data available from Prof.Kenneth French’s website (mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html), is obtained from a universe which is much larger than ouruniverse of S&P 500 constituents. We therefore construct our own size and P/Bportfolios using the S&P 500 universe on each rebalancing date of our beta-neu-tral portfolio. To construct these portfolios, we use the Fama–French method-ology described in [14].

where is a matrix whose first column consists of all1’s and whose second column is the -dimensional vector ofmarket returns, . An unbiased estimate ofis obtained as follows [18]:

Using this to replace in (14) gives the following standarderror for the estimate :

where is the first diagonal entry of the matrix .We test, for the portfolios and , whether their alpha

estimates are different from zero by a statistically significantmargin. Our null hypothesis, for each portfolio, is , andtherefore the -statistic is

The alternative hypothesis for portfolio is . The -testat significance level is to reject the null hypothesis if

, where is the -CDF with degrees offreedom. We perform this test for each calendar year from 1971until 2010. In seven out of the 40 years (1975, 1976, 1979, 1981,1982, 1985, 1993) the null hypothesis is rejected at significancelevel 1%. In 1976 and 1982, it can be rejected at significancelevel 0.01%.The alternative hypothesis for portfolio is . The-test at significance level is to reject the null hypothesis if

. In four out of the 40 years (1973, 1984, 1985,1986), the null hypothesis is rejected at significance level 1%.In 1985, it can be rejected at significance level 0.001%. Thesefindings, like the findings in the previous subsections, contradict(6) and therefore contradict CAPM. However, note that again,most statistically significant results inconsistent with CAPM areobtained from the 1970s and 1980s data.

IX. CONCLUSION AND DISCUSSION

Our conclusion that empirical data from theU.S. stockmarketcontradicts CAPM has been previously pointed out and dis-cussed in literature [4], [26], [6], [12], [7], [17], [11]. We haveproposed a new statistic for testing this conclusion and have an-alyzed an implementable investment strategy which is based onthis statistic. If CAPM applied to the U.S. stock market, thenthe strategy’s expected profit would be zero. In fact, the samplemean of the profit is above zero by a statistically significant


amount. This finding is not explained by the trading costs orby other factors—company size and price-to-book ratio—thathave been proposed as improvements to CAPM. We moreoverhave shown that, despite the fact that the strategy’s profits areheavy-tailed and temporally dependent, the -statistic of theirsample mean is well modeled as a normal distribution. This jus-tifies our calculations of significance levels.The strategy outperforms themarket and has a low correlation

with the market. The low correlation with the market is impor-tant, as it implies that the strategy can provide diversificationcompared to a pure market portfolio. Additionally, the strategyis less volatile than the market, in the sense that its largest dailyloss over the 40-year testing period is about 12 daily standarddeviations, whereas the market’s largest loss is about 21 stan-dard deviations. The tail behavior of investment strategies is avery important consideration and is a cornerstone of risk anal-ysis [25], [27], [21].The dramatic reduction of the profitability of this strategy

over the last 20 years suggests that the U.S. stock market haslately become much more consistent with CAPM than it usedto be. This point is confirmed by our analysis of yearly alphaswhich fails to reject the hypothesis that alpha is zero at signifi-cance level 1% in any years starting with 1994.

ACKNOWLEDGMENT

The authors would like to thank Purdue University Li-braries for subscribing to CRSP, Compustat, and other financialdatabases through WRDS and thus enabling us to access thedata used in this research. They would also like to thank theanonymous reviewers for many valuable comments.

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Mayur Agrawal received the B.Tech. degree inelectronics and electrical communication engi-neering from the Indian Institute of TechnologyKharagpur, in 2007. He is currently pursuing thePh.D. in electrical engineering at Purdue University,West Lafayette, IN.During the summers of 2008 and 2009, he waswith

the Qualcomm Research Center, San Diego, and theMotorola iDEN Group, Fort Lauderdale, FL, respec-tively. He spent the summer of 2011 as a QuantitativeResearcher with the Portfolio Construction Group at

Citadel Investment Group, Chicago, IL. His research interests include the de-sign of modern communication systems, stochastic processes, and time-seriesdata.Mr. Agrawal was a recipient of the Magoon Award for Teaching Excellence

in 2008 and 2009.


Debabrata Mohapatra received the B.Tech. degreein electrical engineering from the Indian Institute ofTechnology, Kharagpur, in 2005 and the Ph.D. degreefrom Purdue University, West Lafayette, IN, in April2011.Since the Ph.D. degree, he has been working

as a Research Scientist in the MicroarchitectureResearch Lab, Intel Corporation, Santa Clara, CA.His research interests include design of low-powerand process-variation aware hardware for errorresilient applications.

Ilya Pollak received the B.S. and M.Eng. degrees in1995 and Ph.D. degree in 1999, all in electrical en-gineering. from the Massachusetts Institute of Tech-nology, Cambridge.In 1999–2000, he was a Post-Doctoral Researcher

at the Division of Applied Mathematics, Brown Uni-versity, Providence, RI. Since 2000, he has been withPurdue University, West Lafayette, IN, where he iscurrently Associate Professor of Electrical and Com-puter Engineering. He has held visiting positions atINRIA (The French National Institute for Research

in Computer Science and Control) in Sophia Antipolis; at Tampere Universityof Technology, Finland; and at Jefferies, Inc., New York. His research interestsare in signal and image processing and financial engineering.Dr. Pollak received a CAREER award from the National Science Foundation

in 2001. He received the Eta Kappa Nu Outstanding Faculty Award in 2002and in 2007, and the Chicago-Area Alumni Young Faculty Award in 2003. Hewas Lead Guest Editor of the IEEE Signal Processing Magazine Special Issueon Signal Processing for Financial Applications which appeared in September2011. He is Co-Chair of the SPIE/IS&T Conference on Computational Imaging.

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298 IEEE JOURNAL OF SELECTED TOPICS IN …ipollak/CAPM.pdf298 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 4, AUGUST 2012 Empirical Evidence Against CAPM: Relating