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Capital Asset Pricing Model (CAPM)
Appendix A Capital Asset Pricing Model (CAPM)
In 1990, William Sharpe was awarded the Nobel Prize for Economics for his work on the capital asset pricing model (CAPM) published in 1964. Other authors who developed the CAPM simultaneously and independently were John Lintner (1965) and Jan Mossin (1966). Subsequently, the Economics Nobel Prize winner in 1997, Merton (1973), developed the model for continuous time. In this appendix, we show as simply as possible the model's development, its implications, and the assumptions on which it is based.1 The CAPM came about when answering the following question: What equity and bond portfolio should an investor who has risk aversion form? By risk aversion we mean, given equal expected return, an investor will always prefer a lower risk portfolio.
A.1. AN INVESTOR FORMS AN OPTIMAL PORTFOLIO An investor wishes to form an optimal portfolio.2 By optimal portfolio we mean that which has the lowest risk for a given expected return (the measure of the risk is the variance3 of the portfolio return). The investor forms a portfolio with N securities. The expected return of each security in the following period is Ri and the weight of each security in the portfolio is Wi . The sum of each security's weights in the portfolio is unity: N X
Wi 1
(A:1)
i1
1 Readers interested in an alternative derivation of the CAPM can see, for example, page 93 of Ingersoll (1987). Ingersoll derives the CAPM using vectors. 2 Also called the eYcient portfolio. 3 The volatility is the square root of the variance.
587
588
An Investor Forms an Optimal Portfolio
The portfolio's expected return, Rc , is Rc
N X
Wi R i
(A:2)
i1
The expected variance of the portfolio return is Var(Rc ) s2c
N X N X
Cov(Ri , Rj )Wi Wj
(A:3)
i1 j1
sc is the portfolio's expected volatility. Cov(Ri , Rj ) is the covariance of the expected return of company i with the expected return of company j. We want to Wnd the weight of each share (Wi ) which minimizes the expected variance of the portfolio return, for a given expected return R. Consequently, we have to solve: Min s2c with the conditions Rc R; and
N X
Wi 1
(A:4)
i1
For each expected return, there will be a diVerent portfolio with a minimum variance. This portfolio is usually called the eYcient portfolio. These eYcient portfolios, taken together, form the eYcient frontier (EF).4 This problem is solved by minimizing the following Lagrange equation: ! N X 2 Lagrange sc l(Rc R) 1 Wi 1 (A:5) i1
To minimize, the Lagrange equation is derived with respect to W1 , W2 , . . . WN and is made equal to zero for each of the N derivatives qs2c qRc l f 0: i 1, . . . , N qWi qWi We can simplify these expressions because: qRc Ri qWi N X qs2c Wj Cov(Ri , Rj ) W1 Cov(Ri , R1 ) W2 Cov(Ri , R2 ) . . . qWi j1
Cov(Ri ,
N X
Wj Rj ) Cov(Ri , Rc )
j1 4
DiVerent eYcient portfolios diVer in their composition (Wi ).
(A:6)
(A:7)
Appendix A
Capital Asset Pricing Model (CAPM)
589
Consequently, the derivatives become: Cov(Ri , Rc ) lRi 1 0;
I 1, 2, . . . , N
(A:8)
If one of the securities is un risk-free bond, with a yield Ri RF , its covariance with the portfolio is zero: Cov(RF , Rc ) 0. The equation (A.8) for the risk-free bond becomes: lRF 1 0
(A:9)
The partial derivative also must be applicable to the portfolio c as a whole. In this case, Ri Rc ; Cov(Rc , Rc ) Var(Rc ). Consequently, Var(Rc ) lRc 1 0;
as 1
lRF : Var(RC )
l(RC
RF );
The parameters l and 1 are l
Var(RC )=(RC
RF );
1 RF Var(RC )=(RC
RF )
(A:10)
Substituting the values of l and 1 gives: Cov(Ri , Rc )
Var(Rc ) Var(Rc ) Ri RF 0: (Rc RF ) (Rc RF )
i 1, 2, . . . , N
Isolating the expected return for the share i gives: Cov(Ri , Rc ) Ri RF (Rc RF ): i 1, 2, . . . , N Var(Rc ) If we call bi
Cov(Ri , Rc ) : i 1, 2, . . . , N Var(Rc )
This gives: Ri RF bi (Rc
RF ):
i 1, 2, . . . , N
(A:11)
It is important to stress that Ri , Cov(Ri , Rj ) and Var(Ri ) are our investor's expectations for the next period (which may be one year, one month, etc).
A.2. OPTIMAL PORTFOLIO IF ALL INVESTORS HAVE HOMOGENEOUS EXPECTATIONS If all investors have the same time horizon and also identical return and risk expectations (volatility of each share and correlation with the other shares) for all shares, then the investors will have the same portfolio and this is the market portfolio M (composed of all the shares on the market). If E(RM ) is the market return expected by all investors (because they all have the same expectations):
590
Optimal Portfolio if all Investors have Homogeneous Expectations
E(Ri ) RF bi (E(RM )
RF ) i 1, 2, . . .
(A:12)
This is the expression of the capital asset pricing model (CAPM). In equilibrium, the investors will have shares in all companies and the portfolio c will be the stock market. All investors will have a portfolio composed of risk-free assets and the diversiWed portfolio, which is the market. Figure A.1 shows the line called capital market line (CML), whose equation is: E(Ri ) RF [(E(RM )
RF )=sM ]si
The expression [(E(RM ) RF )=sM ] is called the required return to risk. It can be shown that (E(RM ) RF ) depends on the investors' degree of risk aversion. If we call this parameter A, E(RM ) RF As2M . For example, if sM 20% and the degree of risk aversion is 2, E(RM ) RF 8%. Thus, according to the CAPM, the required return to an asset will be equal to its expected return and will be equal to the risk-free rate plus the asset's beta multiplied by the required market return above the risk-free rate. Figure A.2 is the representation of the CAPM. E(Ri) M E(RM)
EFEfficient frontier
RF
σ(Ri) 0
σ(RM)
Figure A.1 Capital asset pricing model. In equilibrium, if all investors have identical expectations, all of them will have the market portfolio M, which is on the eYcient frontier (EF). The straight line RF M is called capital market line (CML): E(Ri ) RF [E(RM ) RF ] [si =sM ].
Appendix A
Capital Asset Pricing Model (CAPM)
591
E(Ri) E(RM)
RF
βi
0
1 = βM
Figure A.2 Capital asset pricing model. In equilibrium, if all investors have identical expectations, each asset's expected return is a linear function of its beta. The straight line is called the security market line (SML).
Kei E(Ri ) RF bi [E(RM )
RF ]
(A:13)
A.3. BASIC ASSUMPTIONS OF THE CAPM The basic assumptions on which the model is based are 1. All investors have homogeneous expectations. This means that all investors have the same expectations about all assets' future return, about the correlations between all asset returns, and about all the assets' volatility. 2. Investors can invest and borrow at the risk-free rate RF . 3. There are no transaction costs. 4. Investors have risk aversion. 5. All investors have the same time horizon.
A.4. BASIC CONSEQUENCES OF THE CAPM 1. Any combination of risk-free bonds and the market portfolio prevails over any other combination of shares and bonds. 2. All investors will have a portfolio composed in part of risk-free bonds and in part of the market portfolio. The proportions will vary depending on their utility function. 3. The market portfolio is composed of all the assets that exist and each one's quantity is proportional to their market value.
592
When the Assumptions of the CAPM are not Met
A.5. WHEN THE ASSUMPTIONS OF THE CAPM ARE NOT MET A.5.1. INVESTORS HAVE DIFFERENT EXPECTATIONS One of the model's crucial assumptions is that all investors have homogeneous expectations. When this assumption is not met, the market will no longer be the eYcient portfolio for all investors. Investors with diVerent expectations will have diVerent portfolios (each one having the portfolio he considers most eYcient), instead of the market portfolio. There will also be investors (those who expect that the price of all shares will fall or have a return below the market return) who have no shares but only Wxed-income securities in their portfolio.5
A.5.2. CAPM IN CONTINUOUS TIME Merton (1973), the Nobel Economics Prize winner in 1997, derived the CAPM in continuous time, which has the following expression: Kei E(ri ) rF bi [E(rM )
rF ]
where rF is the instant risk-free rate, E(rM ) is the instant expected market return, and E(ri ) is the instant required return to assets. In Merton's model, the returns follow a lognormal distribution.
A.5.3. IF THE RISK-FREE RATE IS RANDOM Merton (1990) shows the expression of the CAPM when the risk-free rate is random: E(Ri ) RF b1i [E(RM )
RF ] b2i [E(RN )
RF ]
E(RN ) is the expected return of a portfolio N, whose return has a correlation of 1 with the risk-free asset.
A.5.4. THERE IS NO RISK-FREE RATE Black (1972) provided that in this case, the CAPM is as follows: E(Ri ) E(RZ ) bi [E(RM ) 5
Or short-sell stocks, or sell futures on a stock index.
E(RZ )]
Appendix A
Capital Asset Pricing Model (CAPM)
593
where E(RZ ) is the expected return of a portfolio Z which has zero beta, which means that its covariance with the market portfolio is zero. Black, Jensen, and Scholes (1972) showed that the portfolios with zero covariance had a return markedly above the risk-free rate.
A.6. EMPIRICAL TESTS OF THE CAPM There are many papers that seek to assess whether the predictions of the CAPM are met in reality. One of the most famous was written by Roll6 in 1977 and it says that that it is not possible to perform tests of the CAPM because two things are being analyzed at the same time: (1) that the market is an eYcient portfolio a priori,7 and (2) the expression of the CAPM. The expression normally used to perform tests of the CAPM is the following: ERit a0 a1 bi eit where ERit Rit RFt is the share or portfolio excess return over the riskfree rate. What should be obtained when the regressions are performed (if the CAPM were to be met exactly) is the following: 1. a0 0 2. a1 RMt
RFt the market return above the risk-free rate.
The most commonly mentioned tests of the CAPM are Friend and Blume (1970), Black, Jensen, and Scholes (1972), Miller and Scholes (1972), Fama and Macbeth (1973), Litzemberger and Ramaswamy (1979), Gibbons (1982), and Shanken (1985). In general, these studies and many others agree that a0 6 0 and that a1 < RMt Rft . This means that, on average, companies with a small beta have earned more than what the model predicted, and the companies with a large beta have earned less than what the model predicted. In addition, other factors appear that account for the shares' return: the company's size (small companies, on average, are more proWtable), the PER (companies with a small PER were more proWtable than what the model predicted), dividend yield, market-to-book value, etc.
6 Roll, R. (1977). ``A Critique of the Asset Pricing Theory's Tests: Part I: On Past and Potential Testability of Theory.'' Journal of Financial Economics, 4: 129±176. 7 A posteriori, the market portfolio is almost never eYcient: there has almost always been another portfolio with higher return and lower volatility.
594
Relationship between Beta and Volatility
A.7. FORMULAE FOR CALCULATING THE BETA A share's historical beta can be calculated by means of any of the following formulae: b Covariance (Ri , RM )=Variance (RM )
(A:14)
b Correlation coefficient (Ri , RM ) Volatility (Ri )=Volatility (RM ) (A:15) where: Ri security return and RM market return. To calculate a share's beta, a regression is normally performed between the share's return (Ri ) and the market return (RM ). The share's beta (ûi ) is the slope of the regression: Ri a ûi RM e e is the error of the regression. When estimation of beta is based on the CAPM the standard recommendation is to use 5 years of monthly data and a value-weighted index. But Bartholdy and Peare (2001) found that 5 years of monthly data and an equalweighted index, as opposed to the commonly recommended value-weighted index, provides a more eYcient estimate.
A.8. RELATIONSHIP BETWEEN BETA AND VOLATILITY The relationship between the beta and the volatility (s) is given by: s2i û2i s2M s2e
(A:16)
where si is the volatility of the return Ri, which can be inferred from the graph below, which represents the relationship between the market risk or nondiversiWable risk (ûi , sM ), and the non-systematic or diversiWable risk (se ):
σε
σi
Volatility (total risk)
Diversifiable risk
β i σ M Non-diversifiable risk
Appendix A
Capital Asset Pricing Model (CAPM)
595
A.9. IMPORTANT RELATIONSHIPS DERIVED FROM THE CAPM ûi Covariance (Ri , RM )=s2M Correlation (Ri , RM )si =sM
(A:17)
R Correlation (Ri , RM ) Covariance (Ri , RM )=(sM =si )
(A:18)
R Correlation (Ri , RM ) ûi sM =si R2 1 s2e s2i
û2i s2M s2i
s2e =s2i R2 s2i s2i (1
(A:19) (A:20)
R2 )
(A:21)
Viceira (2001) provides support for the popular recommendation by investment advisors that employed investors should invest in stocks a larger proportion of their savings than retired investors. Econometric help to deal with beta calculations can be found in Arino and Franses (2000).
SUMMARY The CAPM is an asset valuation model that relates the risk and the expected return. The CAPM enables us to determine which is the equity and bond portfolio that gives the highest expected return, assuming a given risk, or has the lowest risk for a given expected return. In the CAPM, as in any other model, certain assumptions are used to simplify reality and study it better. Almost all the assumptions are reasonable (they are based above all on the fact that investors demand a higher return for accepting a higher risk) except for all investors having homogeneous expectations.
REFERENCES Arino, M. A., and P. H. Franses (2000), ``Forecasting the Levels of Vector Autoregressive LogTransformed Time Series,'' International Journal of Forecasting, 16, pages 111±116. Bartholdy, J., and P. Peare (2001), ``The Relative EYciency of Beta Estimates,'' Social Science Research Network, Working Paper No. 263745. Black, F. (1972), ``Capital Market Equilibrium with Restricted Borrowing,'' Journal of Business, July 1972, pp. 444±455. Black, F., M. Jensen, and M. Scholes (1972), ``The Capital Asset Pricing Model: Some Empirical Findings,'' in Studies in the Theory of Capital Markets, (M. Jensen, ed.), New York: Praeger.
596
References
Fama, E. F., and J. D. MacBeth (1973), ``Risk, Return and Equilibrium: Empirical Tests,'' Journal of Political Economy, Vol. 81, pp. 607±636. Friend, I., and M. Blume (1970), ``Measurement of Portfolio Performance under Uncertainty,'' American Economic Review, September, pp. 561±575. Gibbons, M. R. (1982), ``Multivariate Tests of Financial Models: A New Approach,'' Journal of Financial Economics, (March), pp. 3±28. Ingersoll, J. (1987), Theory of Financial Decision Making, Totowa, NJ: Rowman & LittleWeld. Lintner, J. (1965), ``The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,'' Review of Economics and Statistics, Vol. 47, pp. 13±37. Litzenberger, R., and K. Ramaswamy (1979), ``The EVects of Personal Taxes and Dividends on Capital Asset Prices: Theory and Empirical Evidence,'' Journal of Financial Economics, Vol. 7 (June), pp. 163±195. Merton, R. C. (1973), ``An Intertemporal Capital Asset Pricing Model,'' Econometrica, Vol. 41, No. 5, pp. 867±887. Merton, R. C. (1990), Continuous-Time Finance, Cambridge, MA: Blackwell. Miller, M., and M. Scholes (1972), ``Rates of Return in Relation to Risk: A Re-examination of Some Recent Findings,'' in Studies in the Theory of Capital Markets, (M. Jensen, ed.), New York: Praeger. Mossin, J. (1966), ``Equilibrium in a Capital Asset Market,'' Econometrica, Vol. 34, pp. 768±783. Roll, R. (1977), ``A Critique of the Asset Pricing Theory's Tests: Part I: On Past and Potential Testability of Theory,'' Journal of Financial Economics, 4:129±176. Shanken, J. (1985), ``Multivariate Tests of the Zero-Beta CAPM,'' Journal of Financial Economics, Vol. 14 , pp. 327±348. Sharpe, W. (1964), ``Capital Asset Prices: A Theory of Capital Market Equilibrium under Conditions of Risk,'' Journal of Finance, Vol. 19, pp. 425±442. Viceira, L. (2001), ``Optimal Portfolio Choice for Long-Horizon Investors with Nontradable Labor Income,'' Journal of Finance, 56, No. 2, pp. 433±470.
In 1990, William Sharpe was awarded the Nobel Prize for Economics for his work on the capital asset pricing model (CAPM) published in 1964. Other authors who developed the CAPM simultaneously and independently were John Lintner (1965) and Jan Mossin (1966). Subsequently, the Economics Nobel Prize winner in 1997, Merton (1973), developed the model for continuous time. In this appendix, we show as simply as possible the model's development, its implications, and the assumptions on which it is based.1 The CAPM came about when answering the following question: What equity and bond portfolio should an investor who has risk aversion form? By risk aversion we mean, given equal expected return, an investor will always prefer a lower risk portfolio.
A.1. AN INVESTOR FORMS AN OPTIMAL PORTFOLIO An investor wishes to form an optimal portfolio.2 By optimal portfolio we mean that which has the lowest risk for a given expected return (the measure of the risk is the variance3 of the portfolio return). The investor forms a portfolio with N securities. The expected return of each security in the following period is Ri and the weight of each security in the portfolio is Wi . The sum of each security's weights in the portfolio is unity: N X
Wi 1
(A:1)
i1
1 Readers interested in an alternative derivation of the CAPM can see, for example, page 93 of Ingersoll (1987). Ingersoll derives the CAPM using vectors. 2 Also called the eYcient portfolio. 3 The volatility is the square root of the variance.
587
588
An Investor Forms an Optimal Portfolio
The portfolio's expected return, Rc , is Rc
N X
Wi R i
(A:2)
i1
The expected variance of the portfolio return is Var(Rc ) s2c
N X N X
Cov(Ri , Rj )Wi Wj
(A:3)
i1 j1
sc is the portfolio's expected volatility. Cov(Ri , Rj ) is the covariance of the expected return of company i with the expected return of company j. We want to Wnd the weight of each share (Wi ) which minimizes the expected variance of the portfolio return, for a given expected return R. Consequently, we have to solve: Min s2c with the conditions Rc R; and
N X
Wi 1
(A:4)
i1
For each expected return, there will be a diVerent portfolio with a minimum variance. This portfolio is usually called the eYcient portfolio. These eYcient portfolios, taken together, form the eYcient frontier (EF).4 This problem is solved by minimizing the following Lagrange equation: ! N X 2 Lagrange sc l(Rc R) 1 Wi 1 (A:5) i1
To minimize, the Lagrange equation is derived with respect to W1 , W2 , . . . WN and is made equal to zero for each of the N derivatives qs2c qRc l f 0: i 1, . . . , N qWi qWi We can simplify these expressions because: qRc Ri qWi N X qs2c Wj Cov(Ri , Rj ) W1 Cov(Ri , R1 ) W2 Cov(Ri , R2 ) . . . qWi j1
Cov(Ri ,
N X
Wj Rj ) Cov(Ri , Rc )
j1 4
DiVerent eYcient portfolios diVer in their composition (Wi ).
(A:6)
(A:7)
Appendix A
Capital Asset Pricing Model (CAPM)
589
Consequently, the derivatives become: Cov(Ri , Rc ) lRi 1 0;
I 1, 2, . . . , N
(A:8)
If one of the securities is un risk-free bond, with a yield Ri RF , its covariance with the portfolio is zero: Cov(RF , Rc ) 0. The equation (A.8) for the risk-free bond becomes: lRF 1 0
(A:9)
The partial derivative also must be applicable to the portfolio c as a whole. In this case, Ri Rc ; Cov(Rc , Rc ) Var(Rc ). Consequently, Var(Rc ) lRc 1 0;
as 1
lRF : Var(RC )
l(RC
RF );
The parameters l and 1 are l
Var(RC )=(RC
RF );
1 RF Var(RC )=(RC
RF )
(A:10)
Substituting the values of l and 1 gives: Cov(Ri , Rc )
Var(Rc ) Var(Rc ) Ri RF 0: (Rc RF ) (Rc RF )
i 1, 2, . . . , N
Isolating the expected return for the share i gives: Cov(Ri , Rc ) Ri RF (Rc RF ): i 1, 2, . . . , N Var(Rc ) If we call bi
Cov(Ri , Rc ) : i 1, 2, . . . , N Var(Rc )
This gives: Ri RF bi (Rc
RF ):
i 1, 2, . . . , N
(A:11)
It is important to stress that Ri , Cov(Ri , Rj ) and Var(Ri ) are our investor's expectations for the next period (which may be one year, one month, etc).
A.2. OPTIMAL PORTFOLIO IF ALL INVESTORS HAVE HOMOGENEOUS EXPECTATIONS If all investors have the same time horizon and also identical return and risk expectations (volatility of each share and correlation with the other shares) for all shares, then the investors will have the same portfolio and this is the market portfolio M (composed of all the shares on the market). If E(RM ) is the market return expected by all investors (because they all have the same expectations):
590
Optimal Portfolio if all Investors have Homogeneous Expectations
E(Ri ) RF bi (E(RM )
RF ) i 1, 2, . . .
(A:12)
This is the expression of the capital asset pricing model (CAPM). In equilibrium, the investors will have shares in all companies and the portfolio c will be the stock market. All investors will have a portfolio composed of risk-free assets and the diversiWed portfolio, which is the market. Figure A.1 shows the line called capital market line (CML), whose equation is: E(Ri ) RF [(E(RM )
RF )=sM ]si
The expression [(E(RM ) RF )=sM ] is called the required return to risk. It can be shown that (E(RM ) RF ) depends on the investors' degree of risk aversion. If we call this parameter A, E(RM ) RF As2M . For example, if sM 20% and the degree of risk aversion is 2, E(RM ) RF 8%. Thus, according to the CAPM, the required return to an asset will be equal to its expected return and will be equal to the risk-free rate plus the asset's beta multiplied by the required market return above the risk-free rate. Figure A.2 is the representation of the CAPM. E(Ri) M E(RM)
EFEfficient frontier
RF
σ(Ri) 0
σ(RM)
Figure A.1 Capital asset pricing model. In equilibrium, if all investors have identical expectations, all of them will have the market portfolio M, which is on the eYcient frontier (EF). The straight line RF M is called capital market line (CML): E(Ri ) RF [E(RM ) RF ] [si =sM ].
Appendix A
Capital Asset Pricing Model (CAPM)
591
E(Ri) E(RM)
RF
βi
0
1 = βM
Figure A.2 Capital asset pricing model. In equilibrium, if all investors have identical expectations, each asset's expected return is a linear function of its beta. The straight line is called the security market line (SML).
Kei E(Ri ) RF bi [E(RM )
RF ]
(A:13)
A.3. BASIC ASSUMPTIONS OF THE CAPM The basic assumptions on which the model is based are 1. All investors have homogeneous expectations. This means that all investors have the same expectations about all assets' future return, about the correlations between all asset returns, and about all the assets' volatility. 2. Investors can invest and borrow at the risk-free rate RF . 3. There are no transaction costs. 4. Investors have risk aversion. 5. All investors have the same time horizon.
A.4. BASIC CONSEQUENCES OF THE CAPM 1. Any combination of risk-free bonds and the market portfolio prevails over any other combination of shares and bonds. 2. All investors will have a portfolio composed in part of risk-free bonds and in part of the market portfolio. The proportions will vary depending on their utility function. 3. The market portfolio is composed of all the assets that exist and each one's quantity is proportional to their market value.
592
When the Assumptions of the CAPM are not Met
A.5. WHEN THE ASSUMPTIONS OF THE CAPM ARE NOT MET A.5.1. INVESTORS HAVE DIFFERENT EXPECTATIONS One of the model's crucial assumptions is that all investors have homogeneous expectations. When this assumption is not met, the market will no longer be the eYcient portfolio for all investors. Investors with diVerent expectations will have diVerent portfolios (each one having the portfolio he considers most eYcient), instead of the market portfolio. There will also be investors (those who expect that the price of all shares will fall or have a return below the market return) who have no shares but only Wxed-income securities in their portfolio.5
A.5.2. CAPM IN CONTINUOUS TIME Merton (1973), the Nobel Economics Prize winner in 1997, derived the CAPM in continuous time, which has the following expression: Kei E(ri ) rF bi [E(rM )
rF ]
where rF is the instant risk-free rate, E(rM ) is the instant expected market return, and E(ri ) is the instant required return to assets. In Merton's model, the returns follow a lognormal distribution.
A.5.3. IF THE RISK-FREE RATE IS RANDOM Merton (1990) shows the expression of the CAPM when the risk-free rate is random: E(Ri ) RF b1i [E(RM )
RF ] b2i [E(RN )
RF ]
E(RN ) is the expected return of a portfolio N, whose return has a correlation of 1 with the risk-free asset.
A.5.4. THERE IS NO RISK-FREE RATE Black (1972) provided that in this case, the CAPM is as follows: E(Ri ) E(RZ ) bi [E(RM ) 5
Or short-sell stocks, or sell futures on a stock index.
E(RZ )]
Appendix A
Capital Asset Pricing Model (CAPM)
593
where E(RZ ) is the expected return of a portfolio Z which has zero beta, which means that its covariance with the market portfolio is zero. Black, Jensen, and Scholes (1972) showed that the portfolios with zero covariance had a return markedly above the risk-free rate.
A.6. EMPIRICAL TESTS OF THE CAPM There are many papers that seek to assess whether the predictions of the CAPM are met in reality. One of the most famous was written by Roll6 in 1977 and it says that that it is not possible to perform tests of the CAPM because two things are being analyzed at the same time: (1) that the market is an eYcient portfolio a priori,7 and (2) the expression of the CAPM. The expression normally used to perform tests of the CAPM is the following: ERit a0 a1 bi eit where ERit Rit RFt is the share or portfolio excess return over the riskfree rate. What should be obtained when the regressions are performed (if the CAPM were to be met exactly) is the following: 1. a0 0 2. a1 RMt
RFt the market return above the risk-free rate.
The most commonly mentioned tests of the CAPM are Friend and Blume (1970), Black, Jensen, and Scholes (1972), Miller and Scholes (1972), Fama and Macbeth (1973), Litzemberger and Ramaswamy (1979), Gibbons (1982), and Shanken (1985). In general, these studies and many others agree that a0 6 0 and that a1 < RMt Rft . This means that, on average, companies with a small beta have earned more than what the model predicted, and the companies with a large beta have earned less than what the model predicted. In addition, other factors appear that account for the shares' return: the company's size (small companies, on average, are more proWtable), the PER (companies with a small PER were more proWtable than what the model predicted), dividend yield, market-to-book value, etc.
6 Roll, R. (1977). ``A Critique of the Asset Pricing Theory's Tests: Part I: On Past and Potential Testability of Theory.'' Journal of Financial Economics, 4: 129±176. 7 A posteriori, the market portfolio is almost never eYcient: there has almost always been another portfolio with higher return and lower volatility.
594
Relationship between Beta and Volatility
A.7. FORMULAE FOR CALCULATING THE BETA A share's historical beta can be calculated by means of any of the following formulae: b Covariance (Ri , RM )=Variance (RM )
(A:14)
b Correlation coefficient (Ri , RM ) Volatility (Ri )=Volatility (RM ) (A:15) where: Ri security return and RM market return. To calculate a share's beta, a regression is normally performed between the share's return (Ri ) and the market return (RM ). The share's beta (ûi ) is the slope of the regression: Ri a ûi RM e e is the error of the regression. When estimation of beta is based on the CAPM the standard recommendation is to use 5 years of monthly data and a value-weighted index. But Bartholdy and Peare (2001) found that 5 years of monthly data and an equalweighted index, as opposed to the commonly recommended value-weighted index, provides a more eYcient estimate.
A.8. RELATIONSHIP BETWEEN BETA AND VOLATILITY The relationship between the beta and the volatility (s) is given by: s2i û2i s2M s2e
(A:16)
where si is the volatility of the return Ri, which can be inferred from the graph below, which represents the relationship between the market risk or nondiversiWable risk (ûi , sM ), and the non-systematic or diversiWable risk (se ):
σε
σi
Volatility (total risk)
Diversifiable risk
β i σ M Non-diversifiable risk
Appendix A
Capital Asset Pricing Model (CAPM)
595
A.9. IMPORTANT RELATIONSHIPS DERIVED FROM THE CAPM ûi Covariance (Ri , RM )=s2M Correlation (Ri , RM )si =sM
(A:17)
R Correlation (Ri , RM ) Covariance (Ri , RM )=(sM =si )
(A:18)
R Correlation (Ri , RM ) ûi sM =si R2 1 s2e s2i
û2i s2M s2i
s2e =s2i R2 s2i s2i (1
(A:19) (A:20)
R2 )
(A:21)
Viceira (2001) provides support for the popular recommendation by investment advisors that employed investors should invest in stocks a larger proportion of their savings than retired investors. Econometric help to deal with beta calculations can be found in Arino and Franses (2000).
SUMMARY The CAPM is an asset valuation model that relates the risk and the expected return. The CAPM enables us to determine which is the equity and bond portfolio that gives the highest expected return, assuming a given risk, or has the lowest risk for a given expected return. In the CAPM, as in any other model, certain assumptions are used to simplify reality and study it better. Almost all the assumptions are reasonable (they are based above all on the fact that investors demand a higher return for accepting a higher risk) except for all investors having homogeneous expectations.
REFERENCES Arino, M. A., and P. H. Franses (2000), ``Forecasting the Levels of Vector Autoregressive LogTransformed Time Series,'' International Journal of Forecasting, 16, pages 111±116. Bartholdy, J., and P. Peare (2001), ``The Relative EYciency of Beta Estimates,'' Social Science Research Network, Working Paper No. 263745. Black, F. (1972), ``Capital Market Equilibrium with Restricted Borrowing,'' Journal of Business, July 1972, pp. 444±455. Black, F., M. Jensen, and M. Scholes (1972), ``The Capital Asset Pricing Model: Some Empirical Findings,'' in Studies in the Theory of Capital Markets, (M. Jensen, ed.), New York: Praeger.
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References
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