# The Capital Asset Pricing Model (CAPM)

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1 The Capital Asset Pricing Model (CAPM) Tee Kilenthong UTCC c Kilenthong 2016 Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 1 / 36

2 Main Issues What is an equilibrium implication if all investors construct portfolios as we studied? How should we measure risk of any asset? Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 2 / 36

3 CAPM: Simple Derivation The efficient frontier without riskless asset. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 3 / 36

4 CAPM: Simple Derivation The efficient frontier with riskless asset. The straight line is called the capital market line. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 4 / 36

5 CAPM: Simple Derivation The equation of the capital market line is R e = R F + R m R F σ m σ e (1) where σ e is an efficient portfolio on the capital market line. R m R F σ m represents the market price of risk. σ e represents the amount of risk. That is, Expected return = (Market price of risk) (Amount of risk) (2) Problem: This equation does not describe equilibrium return on nonefficient portfolios or individual securities. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 5 / 36

6 Only R i and β i Matter From Single-Index Model, the investor should hold a very well-diversified portfolio. Therefore, only relevant risk is systematic risk measured by β. The only dimensions of a security that need be of concern are expected return R i and β i. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 6 / 36

7 Nonefficient Portfolios and Arbitrage Opportunity From the above figure, one can imagine that they can make money from the following arbitrage strategy: Cash Invested Expected Return β Portfolio C Portfolio D Arbitrage Portfolio Note: we use β P = i X iβ i. Main point: there is a portfolio involving zero risk and zero net investment that has a positive expected return. There is an arbitrage opportunity. This should not be the case in an equilibrium. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 7 / 36

8 Nonefficient Portfolios and Arbitrage Opportunity Therefore, all investors must hold efficient portfolios. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 8 / 36

9 CAPM Equation The security market line can be represented by R i = R F + ( R m R F ) βi (3) Key Insight: systematic risk (measured by β) is the only important ingredient in determining expected returns and that nonsystematic risk plays no role. In other words, investors get rewarded for bearing systematic risk. Important: this implication is empirically testable. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 9 / 36

10 CAPM Equation: Alternative We can represent risk by covariance of asset return and market return: where we use β i = σ im. σm 2 R i = R F + ( Rm R F σ m ) σim σ m (4) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 10 / 36

11 CAPM: A More Rigorous Derivation Recall: an optimal condition for an efficient portfolio is where σ ki = Cov (R k, R i ). R k R F = λ (X 1 σ 1k X N σ Nk ) (5) Homogeneous expectation implies that the solution to this problem or the optimal portfolio here must be the market portfolio. As a result, R m = N R i X i (6) i=1 Then, we can show that X 1 σ 1k X N σ Nk = Cov (R k, R m ). (7) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 11 / 36

12 CAPM: A More Rigorous Derivation So, we can write R k R F = λcov (R k, R m ) (8) Since the market portfolio is one of an efficient portfolio, we can write R m R F = λcov (R m, R m ) = λσ 2 m = λ = R m R F σ 2 m (9) Finally we can get the CAPM: R k R F = R m R F σm 2 σ km = ( ) σ km R m R F σm 2 R i R F = ( ) Rm R F βi (10) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 12 / 36

13 Keys Assumptions for the Derivation of CAPM 1 No transaction costs, no personal income taxes. 2 Perfect competition. 3 Assets are infinitely divisible. 4 Investors have mean-variance preferences. 5 There exist a riskless asset. 6 Homogeneous expectation. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 13 / 36

14 CAPM without Riskless: Simple Derivation Portfolios in expected return β space. This is the case without riskless asset. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 14 / 36

15 CAPM without Riskless: Simple Derivation First, combinations of two risky portfolios lie on a straight line connecting them in expected return β space. Second, consider C and D. Using an arbitrage argument as before, we can show that all portfolios and securities must plot along the straight line, as in the figure. This line can be represented by Our job is to find what are a and b? R i = a + bβ i (11) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 15 / 36

16 CAPM without Riskless: Simple Derivation Let R Z be the expected return on a zero beta portfolio. This portfolio must also be on the straight line: R Z = a + b 0 = a = R Z (12) Again, the market portfolio must be on the line as well: R m = R Z + b = b = R m R Z (13) Hence, we have an alternative CAPM (without riskless asset): R i = R Z + ( R m R Z ) βi (14) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 16 / 36

17 CAPM without Riskless: A Rigorous Derivation Recall: an optimal condition for an efficient portfolio is R k R F = λ (X 1σ 1k X N σ Nk ) (15) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 17 / 36

18 CAPM without Riskless: A Rigorous Derivation Using the result we derived earlier, we can write We then can show that X 1 σ 1k X N σ Nk = Cov (R k, R m ), (16) R k R F = λcov (R k, R m ) (17) R i = R F + ( R m R F ) βi (18) Issue: This equation holds for any zero beta expected return R F. But we should use the least risky zero beta portfolio. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 18 / 36

19 CAPM without Riskless: A Rigorous Derivation The least risky zero beta portfolio is the zero beta portfolio that has the least total risk, the minimum variance zero beta portfolio Z. The CAPM now is R i = R Z + ( Rm R Z ) βi (19) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 19 / 36

20 Testing CAPM Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 20 / 36

21 Main Issues How can we test CAPM model? Are those tests reliable? Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 21 / 36

22 CAPM is in an Ex-ante Form The basic CAPM model can be written as E (R i ) = R F + β i [E(R m ) R F ] (20) If lending and borrowing at the risk free rate is not possible or there is no risk free rate, then the CAPM becomes E (R i ) = E(R Z ) + β i [E(R m ) E(R Z )] (21) Notice: these models are in an expectation form. This is suppose to be about future values. An expectation means that we are thinking about the situation before the uncertainty is realized. We call this ex-ante. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 22 / 36

23 CAPM is tested using Ex-post Data On the other hand, we usually perform tests of CAPM models using realized (historical) data. These values are said to be ex-post values. How can we justify using ex-post data to test ex-ante model? Defense: using the law of large number, we should be able to estimate consistently (unbiased) the ex-ante expectation using sample mean of ex-post data. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 23 / 36

24 Ex-post test of Ex-ante CAPM model That is, testing a CAPM model is a simultaneous test of all three following hypothesises: 1 The market model holds 2 The CAPM model holds 3 β i is stable over time NOTE: if there is no risk free rate, we will test the following model R it = R Zt + β i [ R mt R Zt ] + ẽ it (22) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 24 / 36

25 Simple Test of CAPM Sharpe and Cooper (1972) divide stocks into ten portfolios using beta as a criterion. β at each point in time uses past 60 months of data. They also calculate average returns of each portfolio. They then estimate a linear equation with R 2 > R i = β i (23) Key Points: the relationship between return and β is linear. Though the intercept (supposed to represent the risk free rate) is too high. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 25 / 36

26 Simple Test of CAPM Lintner performed a similar but more systematic test of CAPM. First, run a time-series linear regression R it = α i + b i R mt + e it (24) using data of all stocks from 1954 to Then, run a cross-sectional regression R i = a 1 + a 2 b i + a 3 S i + ϵ i (25) where S i = Var(e it ) is the residual variance, representing residual risk. The result shows that a 3 = and statistically significant. This implies that CAPM is violated. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 26 / 36

27 More Advanced Test of CAPM Miller and Scholes Test Black, Jensen, and Scholes Test Fama and MacBeth Test Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 27 / 36

28 Fama and MacBeth Test They use the same procedure as Black et al. to form 20 portfolios. The difference is in the cross-sectional regression: 1 They run R it = ˆγ 0t + ˆγ 1t β i ˆγ 2t β 2 i + ˆγ 3t S et + η it (26) 2 This regression is run each month, month by month. Therefore, they can study how the parameters change over time. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 28 / 36

29 Fama and MacBeth Test This new form allow them to test the following hypotheses: 1 E(ˆγ 3t ) = 0: residual risk does not affect the return 2 E(ˆγ 2t ) = 0: test the linearity structure of CAPM 3 E(ˆγ 1t ) = 0: test the positivity of the price of risk If the first two hold, then we can conclude that a CAPM (either the standard or zero beta version) holds. Table 15.3 confirms that the first two hypothesises hold. In addition, they run a regression without those two terms and get better estimates. We then can conclude that residual risk has no effect. This is the opposite of Litner. the main reason is the measurement error. That is, using portfolios instead of securities reduce the error significantly (use the argument of Miller and Scholes). Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 29 / 36

30 Fama and MacBeth Test Given that a CAPM holds, then we can further distinguish between the standard and zero-beta CAPM using E(ˆγ 0t ) and E(ˆγ 1t ). If the zero beta model is the true model, the deviation of ˆγ 0t from its mean E(R Z ) and the deviation of ˆγ 1t from its mean E(R m ) E(R Z ) must be random. Since we know that E(R Z ) > R F, if E(ˆγ 0t R F ) > 0 and E(ˆγ 1t E(R m ) + R F ) < 0, we will then conclude that the zero beta model is the true model. They find that 1 price of risk is positive, 2 ˆγ 0 is greater than R F and ˆγ 1 is less than R m R F These results support the zero beta CAPM. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 30 / 36

31 Fama and MacBeth Test We can also test if the market operates as a fair game (efficient market). If CAPM is a true model, the expected value of ˆγ 2t and ˆγ 3t at time t + 1 should be zero, regardless of past values. Fama and MacBeth test this implication by looking at the correlation of ˆγ 3t with its lags values. They found that the correlation is not statistically different from zero. They also found a similar result for ˆγ 2t. They then conclude that the market operates as a fair game. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 31 / 36

32 Fama and MacBeth Test of Thailand Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 32 / 36

33 Fama and MacBeth Test of Thailand Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 33 / 36

34 CAPM is Not Testable? If any ex-post mean variance efficient portfolio p selected as the market portfolio, and β are computed using this portfolio as the market proxy, then R i = R ZP + β ip ( Rp R ZP ) (27) must hold. That is, testing CAPM is not meaningful. Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 34 / 36

35 Roll s Proof Consider again the first order condition of an optimal portfolio problem: λ (X 1 σ 1k X N σ kn ) = R k R F (28) Let p be the optimal portfolio, hence we can write λσ kp = R k R F (29) which must be true for any asset or portfolio. That is, it must be true for the optimal portfolio p as well: Hence, we have R i = R F + σkp σ 2 p λσ 2 p = R p R F λ = R p R F σ 2 p ( R p R F ) = RF + β kp ( R p R F ) (30) (31) Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 35 / 36

36 Roll s Proof Using a similar argument as before, we can have a model without riskless: R i = R Zp + β kp ( R p R Zp ) (32) where R Zp is the mean return of the minimum varaince zero beta portfolio. This proves that we can write a zero beta CAPM model with any efficient portfolio as a market proxy. But the true CAPM is the one with the true market portfolio. If we cannot observed the true market portfolio, then we cannot test the CAPM! Tee Kilenthong UTCC The Capital Asset Pricing Model (CAPM) 36 / 36

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