Capital Asset Pricing Model Revisited:Empirical Studies on Beta Risks and Return

Simon G. M. Koo and Ashley OlsonDepartment of Mathematics and Computer Science

University of San DiegoSan Diego, CA 92110

{koo, ashleyo-07}@sandiego.edu

Keywords: Capital Asset Pricing Model, beta risk, expectedreturn, portfolio management.

AbstractThe Capital Asset Pricing Model (CAPM) has been the dom-inating capital market equilibrium model since its inceptionand continues to be widely used in practical portfolio man-agement and in academic research. Its central implication isthat the contribution of an asset to the variance of the marketis the correct measure of the asset’s risk and the only sys-tematic determinant of the asset’s return. However, studiesshowed that firm size appeared to be a significant determi-nant of stock returns that there is no cross-sectional relation-ship between beta risk and return once firm size and book-to-market ratios are included as explanatory variables.

In this study, we revisited the CAPM with empirical datafrom large firms. We gathered stock information for morethan 288 publicly traded companies with market cap largerthan 500 million dollars, price-earning-ratio less than ten, anda greater than zero profitability for over a one year period. Wealso categorize risk factors of the stocks into three categories:low (beta around point five), market (beta about one), andhigh (beta about two). Covariance and correlation relationsbetween the stocks as well as their risk factors were used tocreate optimal portfolios in hindsight. The goal is to test thehypothesis that the systematic risk of a portfolio as measuredby its market model beta is indeed a relevant measure of risk,and we would like to examine if beta is reliably related to thereturn of the portfolio conditional on the sign of the marketrisk premium, so we can justify the use of market model betasestimated from historical price.

INTRODUCTIONSince its inception in the 1960’s, the Capital Asset Pricing

Model (CAPM) (Sharpe 1964; Lintner 1965; Mossin 1966)has been a dominating capital market equilibrium model, andit is still widely used in academic research and practical port-folio management nowadays, even substantial criticism hasbeen raised (Roll 1977). Recently, researchers were able toidentify anomalies from empirical results. In particular, thesize of a firm appears to be a significant factor to determine

the return on its stock (Banz 1981; Daniel and Titman 1997),and it was demonstrated that there is no cross-sectional rela-tionship between systematic risk and return once firm sizeand book-to-market ratio are included as explanatory vari-ables (Fama and French 1992; Schlag and Wohlschieβ 1997).

In this study, we revisited the CAPM with empirical datafrom large firms by gathering stock information for more than288 publicly traded companies with market cap larger than500 million dollars, price-earning-ratio less than ten, and agreater than zero profitability for over a one year period fromNovember 2005 to November 2006. The goal is to test thehypothesis that the systematic risk of a portfolio as measuredby its market model beta is indeed a relevant measure of risk,and we would like to examine if beta is reliably related to thereturn of the portfolio conditional on the sign of the marketrisk premium, so we can justify the use of market model betasestimated from historical price.

The rest of the paper is organized as follows: We will firstreview the basis of CAPM in the next section, followed bythe methodology of our empirical study. We will then discussthe results of study and their implications, and conclude thepaper.

BASIC MODELThe central implication of CAPM is that the contribution

of an asset to the systematic risk (also known as beta risk) isthe correct measure of the asset’s risk and the only systematicdeterminant of the asset’s return. There are two main compo-nents of CAPM: the market portfolio M, and beta risk β of aportfolio, which correlates the portfolio to the rise and fall ofthe market. The basic relation for the CAPM is the following:

ri− r f = βi(rM− r f ) (1)

where ri is the rate of return on asset i, r f is the risk-free rate,and rM is the return of the market portfolio. The systematicrisk βi is the coefficient that describes how portfolio i willfollow the market, which is defined as:

βi =cov(ri,rM)

var(rM)(2)

Equation 1 can be interpret as follows: The risk-free rater f of the left-hand-side is the compensation given to the in-vestors for their time spent in the market, and the right-hand-side of the equation represents the risk of the portfolio andthe compensation the investor received for the risk they take.Under CAPM, it is also assumed that the expected residualreturn on portfolio i is zero, so we have

E[ri] = βiE[rM] (3)

which means that the expected excess return of a portfoliowith respect to the market is proportional to the portfolio’sbeta.

The idea behind CAPM is that investors are compensatedfor taking necessary risks, but not for taking unnecessaryrisks. In particular, market risk is necessary and inevitable.On the other hand, the residual risk of a portfolio i with re-spect to the market ωi, which is defined as:

ωi =√

σ2i −β2

i ×σ2M (4)

where σ2i is the variance of portfolio i, σ2

m is the variance ofthe market, and βi is the beta risk of the portfolio. The residualrisk is self-imposed and could be avoided.

The objective of this study is to validate the accuracy ofusing beta in predicting portfolio performance. We will usethe return of the S&P 500 index as the benchmark for marketportfolio when evaluating a portfolio’s excess return.

METHODOLOGY AND RESULTSTo validate the CAPM’s accuracy, we performed an empir-

ical experiment to test how well the beta measures portfoliorisks with respect to the market portfolio (S&P 500 index inour study.) We collected data from 288 publicly traded com-panies for one year from November 1, 2005 to November 1,2006. In order to make sure that the companies we chose werestable and having a high profit-to-earing ratio, each companywe picked had a price over earnings ratio less than 10, prof-itability greater than zero, and a market cap larger than 500million dollars. We then built different portfolios in hindsight.There were three different sets of portfolios built for the stud-ies, ranging from low to high risks. We have portfolios withsmall beta (around 0.5), market beta (around one), and largebeta (around two). There were sixteen portfolios in each betagroup, and each portfolio consisted of six stocks.

To justify the accuracy of CAPM, we have the followinghypotheses:

(Group A, β = 0.5) H0 : µA = 0.5×µM

(Group B, β = 1.0)H1 : µB = µM

(Group C, β = 2.0) H2 : µC = 2×µM

Table 1. Results for hypothesis testings

Group A Group B Group Cµi 0.167 0.087 0.078si 0.372 0.119 0.115ti 0.266 0.423 1.712

p-value 0.206 0.322 0.893

where µM ( = 0.137 in this study) is the average market return,and µA, µB and µC represent the average return of portfoliosfrom Group A, B, and C, respectively.

The hypotheses were tested by Student’s t-test, and the re-sults are show in Table 1. Large p-values obtained from theexperiments suggest that the there are significant informationto reject all three hypotheses, indicating that CAPM are notapplicable in all the three cases.

DISCUSSIONS AND IMPLICATIONSFrom our experiments, it is evident that the CAPM is not

the perfect model for portfolio management. At best, thep-value is about twenty percent, and at worst the p-valueis around ninety percent. This experiment shows how theCAPM is not a good model for portfolio management. Ourstudy also suggests that the CAPM is (relatively) more ac-curate with smaller betas, and progressively gets worse withlarger betas. Therefore, for a risk-aggressive portfolio (largebeta), the CAPM is not a good model to explain its perfor-mance.

There are some procedures which could improve upon thisstudy and bring further depth to this experiment. While com-piling any portfolio, diversification is always a necessary pre-caution. Complete diversification among stocks in any givenportfolio is difficult to obtain, and in this study it is possiblethat our portfolios were not as diversified as they could havebeen. Therefore, to expand upon this study, it would be ben-eficial to ensure that very diversified stocks were collected.Also, this study only used publicly traded stocks to be thecomponent of a portfolio instead of using bonds, real estate,foreign exchange, or a hybrid of the above. With a hybridof investments, it is possible to expose different results andprovide more insights for the validity of using CAPM in ap-proximating expected return with beta risk.

CONCLUSIONIn this study, we collected stock information over a period

of one year and evaluated the accuracy of CAPM based of thedata we collected. Our results suggest that the systematic riskof a portfolio, as measured by its market model beta is nota relevant measure of risk, and beta is statistically unreliablyrelated to the return of the portfolio. If we were to take an

aggressive portfolio with a high beta, there is a 89.3% chancethat this beta will not be the appropriate risk we wanted totake, and our portfolio will not perform as preficted. More-over, we have shown that the CAPM is not a good model forexpected return for risk-aggresive portfolios.

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