Capital Asset Pricing and Arbitrary Pricing Theory

CHAPTER 7

Risk and diversification

Market risk is the only risk left after diversification Return that investors get in the market is rewarded for market

risk only, not total risk Hence market risk is relevant risk, and specific risk is

irrelevant risk In the market, higher beta gets higher return, not higher std

gets higher return.

Example:

std beta amount invested

IBM 40% 0.95 2000

AT&T 20% 1.10 4000

1. What is the beta of market portfolio

2. Does IBM have more or less risk than the market

3. Which stock has more total risk Which stock has more systematic risk

Which stock is expected to have higher return in the market

4. In the boom market, which stock do you choose

5. In the recession market, which stock do you choose

6. What is the beta of your portfolio

Capital Asset Pricing Model (CAPM)

How do investors know whether the return they get in the market is high enough to reward for the level of risk taken.

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Capital Asset Pricing Model (CAPM)

CAPM gives the relationship between risk and return. It gives the minimum return required by investors in order for

them to buy stock Example:

market risk premium = 0.08, Rf = 0.03 βx=1.25, βy=1.25

What is E(Rx), E(Ry), what is meaning of them?

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Graph of Sample Calculations

SML Relationships

= [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium

SML = rf + [E(rm) - rf]

Capital Asset Pricing Model (CAPM)

Remember earlier, we have

In CAPM, we have

What is the difference in meaning between the two expected return?

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Capital Asset Pricing Model (CAPM)

When forecasted E(R) > required E(R), stock is undervalued or the price is too low

When forecasted E(R) < required E(R), stock is undervalued or the price is too high

In equilibrium, forecasted E(R) = required E(R)

Capital Asset Pricing Model (CAPM) Example: E(Rm) = 14%, Rf = 6%

Stocks Beta E(R) (forecasted)IBM 1.2 17%ATT 1.5 14%

1. What is the market risk premium2. what is the risk premium on IBM and ATT3. According to CAPM, what is the required E(R) for IBM and

ATT4. Which stock is undervalued, which stock is overvalued

Determining the Expected Rate of Return for a Risky Asset

Let alpha (α) be the difference between the actual (forecasted) E(R) and the required E(R)

In equilibrium, all assets and all portfolios of assets should plot on the SML ( i.e., α = 0)

Any security with an estimated return that plots above the SML is underpriced (α > 0 )

Any security with an estimated return that plots below the SML is overpriced ( α < 0 )

Previous example: αIBM= 17 – 15.6 = 1.4 > 0. Actual E(R) is above the SML

αATT= 14-18 = -4 > 0. Actual E(R) is below the SML

Figure 7-2 The Security Market Line and Positive Alpha Stock

Chap. 7, Problem 19, p.236

Two investment advisers are comparing performance. One average a 19% return and the other a 16%. However, the beta of the first adviser was 1.5, while that of the second was 1.0.

a. Can you tell which adviser was better?

b. If the T-bill rate were 6%, and market return during the period were 14%, which would be better?

c. What if T-bill rate were 3% and market return 15%?

Estimating alpha and beta in practice (using index model)

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• alpha is the abnormal return = actual return – return predicted by CAPM

•According to CAPM, alpha should be = 0

•Beta is the systematic risk

Figure 7-4 Characteristic Line for GM

Characteristic Line for GM

All the points are actual values Line is the predicted relationship If there are a lot of specific risk, there will be a wide scatter of

points around the line. Hence, using market risk only in this case does not produce a precise estimate of expected return

If the points are close to the line, there is only small specific risk. Using market risk can explain most of the company return.

Table 7-2 Security Characteristic Line for GM: Summary Output

Security Characteristic Line for GM: Summary Output

R-square: 0.2866 ANOVA table

Total risk = systematic risk + unsystematic risk

7449.17 = 2224.696 + 5224.45

(100%) = 29.87% + 70.13%

Alpha = 0.8890 > 0 (positive alpha, undervalued or overvalued?) During the period Jan 99-Dec03: the risk-adjusted or abnormal return of

GM = 0.8990% or actual return is higher than CAPM predicted Is this value statistically different from 0? Is this still consistent with

CAPM 95% confidence interval (-1.5690 to 3.3470) Beta = 1.2384 Is beta statistically different from 0?

Implication of CAPM

CAPM is a benchmark about the fair (required) expected return on a risk asset. Investors calculate the return they actually earn based on their input and compare with the return they get from the CAPM

Compare the performance of the mutual fund: we use alpha or risk-adjusted return rather than regular return

Compute the cost of equity for capital budgeting

Does CAPM work in reality?

CAPM is only a theory Assumptions

Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations

Empirical test of CAPM

CAPM was introduced by Sharpe (1964) and later earned him a Nobel prize in 1990

It changes the world how we should perceive risk and the relationship between risk and return

However, it is only a theory, we need to test whether it works in practice

The test of CAPM falls into 2 categories Stability of the beta Slope of the SML

Empirical test of CAPM

Stability of beta betas of individual stocks are unstable

betas of a portfolio (> 10 stocks randomly selected) are reasonable stable

CAPM is a better concept for portfolio than for individual securities

Slope of SML positive relationship between beta and return (consistent with CAPM) Empirical slope is smaller than predicted (=market risk premium) CAPM says that beta is the only source of risk, no specific risk, however, the

empirical data show that there exists both risk (market and specific),

Current status of CAPM

CAPM is powerful at the conceptual level. It is a useful way to think about risk and return

Empirical data does not support CAPM fully but it is simple, logical, easy to use, so use CAPM with caution

Arbitrary pricing theory (APT)

CAPM is a single factor model. The market risk premium is the only factor

In CAPM, all the news, uncertainties affect the market, then the market affect the stock individually

In APT, there are n factors that can influence stock return so there will be n-sources of risk or n-channels of uncertainties

Empirical evidence support APT (more than 1 factor affect stock return), but unable to identify these factors.

So if the purpose is to get cost of capital only, then APT is appropriate

If we want to know sources of risk then APT is not useful

Fama-French three-factor model

Fama and French propose three factors: The excess market return, rM-rRF.

the return on, S, a portfolio of small firms (where size is based on the market value of equity) minus the return on B, a portfolio of big firms. This return is called rSMB, for S minus B.

the return on, H, a portfolio of firms with high book-to-market ratios (using market equity and book equity) minus the return on L, a portfolio of firms with low book-to-market ratios. This return is called rHML, for H minus L.

ri = rRF + (rM - rRF)bi + (rSMB)ci + (rHMB)di

bi = sensitivity of Stock i to the market return.cj = sensitivity of Stock i to the size factor.dj = sensitivity of Stock i to the book-to-market

factor.

The model is widely used in research and practice

Required Return for Stock i under the Fama-French 3-Factor Model