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:
IBM
AT&T
1.
2.
3.
std
40%
20%
beta
0.95
1.10
amount invested
2000
4000
What is the beta of market portfolio
Does IBM have more or less risk than the market
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.
E ri  : expected return of stock i
rf : risk - free rate
Erm  : expected return of the market portfolio
RPm : market risk premium  Erm   rf
it is the premium demanded by investors to hold the market portfolio rather tha n T - bill
E ri   rf  RPi  rf   i RPm  rf   i Erm   rf 
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?
Graph of Sample Calculations
E(r)
SML
.08
Rx=13%
Rm=11%
Ry=7.8%
3%
.6 1.0 1.25
ßy ßm ßx
ß
SML Relationships
 = [COV(ri,rm)] / sm2
Slope SML = E(rm) - rf
= market risk premium
SML = rf + [E(rm) - rf]
Capital Asset Pricing Model (CAPM)

Remember earlier, we have
n
E ri    pi ri
i 1

In CAPM, we have
E ri   rf  i Erm   rf 

What is the difference in meaning between the two expected
return?
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)

1.
2.
3.
4.
Example: E(Rm) = 14%, Rf = 6%
Stocks Beta
E(R) (forecasted)
IBM
1.2
17%
ATT
1.5
14%
What is the market risk premium
what is the risk premium on IBM and ATT
According to CAPM, what is the required E(R) for IBM and
ATT
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)


ri  rf   i   i rm  rf  ei
ri : return on asset i
ri  rf : exess return or risk premium of asset i
 i ,  i : are intercept and slope of the regression
rm : return on market
r
m

 rf : exess return or risk premium of the market
ei : residual which measures firm specfic effects.
• 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-tomarket ratios (using market equity and book equity) minus
the return on L, a portfolio of firms with low book-tomarket ratios. This return is called rHML, for H minus L.
Required Return for Stock i
under the Fama-French 3-Factor Model
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