Chapter 7
Capital Asset Pricing Model
(CAPM)
7.1 The Capital Asset Pricing Model
7-2
Capital Asset Pricing Model (CAPM)
•
Markowitz, Sharpe, Lintner and Mossin are researchers
credited with its development.
•
Predicts the relationship between the risk and
equilibrium expected returns on risky assets.
•
Approach the CAPM in a simplified setting and then add
complexity to the environment.
7-3
Simplifying Assumptions
Individuals are alike, with the notable exceptions of initial wealth
and risk aversion.
•
Individual investors are price takers.
: There are many investors, each with an endowment of
wealth that is small compared with the total endowment
of all investors.
•
Single-period investment horizon
•
Investments are limited to traded financial assets.
•
No taxes and no transaction costs
7-4

Simplifying Assumptions (cont.)
•
Information is costless and available to all investors.
•
Investors are rational mean-variance optimizers.
: All investors attempt to construct efficient frontier
portfolios.
•
Homogeneous expectations.
: All investors analyze securities in the same way and share
the same economic view of the world.
7-5
Hypothetical Equilibrium
•
All investors will hold the same portfolio for risky assets; the “market
portfolio”. Market portfolio contains all securities and the proportion of
each security is its market value as a percentage of total market value.
•
The market portfolio will be on the efficient frontier. It will be the
optimal risky portfolio, the tangency point of the capital allocation line
(CAL) to the efficient frontier.
•
The capital market line (CML), line from the risk-free rate through the
market portfolio, is the best attainable capital allocation line.
7-6
•
The risk premium on the market portfolio will be proportional to the
variance of the market portfolio and investors’ typical degree of risk
aversion.
E(rM) – rf = A* σM2

•
The risk premium on individual assets will be proportional to the risk
premium on the market portfolio (M) and to the beta coefficient of
the security on the market portfolio.
7-7
Hypothetical Equilibrium (cont.)
Why All investors Would Hold the Market Portfolio
•
With all assumptions, all investors should hold the same optimal risky portfolio.
•
They all derive identical efficient frontiers and find the same tangency portfolio
for the capital allocation line from the risk-free asset to that frontier.
•
With everyone choosing to hold the same risky portfolio, stocks will be
represented in the aggregate risky portfolio in the same proportion as they are
in each investor’s risky portfolio.
•
The market portfolio is the aggregate of all individual portfolios.
•
Each investor uses the market portfolio for the optimal risky portfolio, the CAL
in this case is called CML.
7-8
E(r)
rf
E(rM)
M
CML
σm
Capital Market Line
σ

M = The value weighted “Market”
Portfolio of all risky assets
Efficient
Frontier
7-9
The Passive Strategy is Efficient
•
The CAPM implies that a passive strategy, using the CML as the
optimal CAL, is a powerful alternative to an active strategy.
•
The market portfolio proportions are a result of a profit-oriented
“buy” and “ sell” orders that cease only when there is no more
profit to be made.
•
It implies that only one mutual fund of risky assets, the market
portfolio, is sufficient to satisfy the investment demands of all
investors.  called a mutual fund theorem
•
If a passive strategy is costless and efficient, why would anyone
follow an active strategy? But if no one does not any security
analysis, what brings about the efficiency of the market portfolio?
7-10
The Risk Premium of the Market Portfolio
•
Think about the decision of how much to invest in the market portfolio
M and how much in the risk-free asset.
•
In equilibrium, the risk premium on the market portfolio must be just
high enough to induce investors to hold the available supply of stocks.
•

Investors purchase stocks  their demand drives up prices  lower
expected rates of return and risk premium  Given lower risk premium,
relatively more risk-averse investors will move their funds to risk-free
asset from the risky market portfolio.

•
The risk premium on the market portfolio will be proportional to both the
risk of the market and risk aversion of the investor.
E(rM) – rf = A* σM2
7-11
Expected Returns on Individual Securities
•
The appropriate risk premium will be determined by its contribution
on the risk of investors’ overall portfolios.
•
The risk premium on individual securities is a function of the
individual security’s contribution to the risk of the market
portfolio.
•
What type of individual security risk will matter, systematic or
unsystematic risk?
: Nonsystematic risk can be diversified away through
diversification.
 Need to be compensated only for bearing systematic risk.
7-12
Expected Returns on Individual Securities
•
An individual security’s total risk (σ2i) can be partitioned into
systematic and unsystematic risk:
σ2i = βi2 σM2 + σ2(ei)

M = market portfolio of all risky securities
•
Individual security’s contribution to the risk of the market portfolio is
a function of the covariance of the stock’s returns with the market
portfolio’s returns and is measured by BETA.

With respect to an individual security, systematic risk can be
measured by βi = [COV(ri,rM)] / σ2M
7-13
Expected Returns on Individual Securities
•
The risk premium of an asset is proportional to its beta.

•
The ratio of risk premium to beta should be the same for any two
securities or portfolios.
•
If we were to compare the ratio of risk premium to systematic risk
for the market portfolio, which has a beta of 1, with the
corresponding ratio for any stock, its relationship is the following.
[E(rM) – rf] / 1 = [E(ri) – rf] / βi
•
CAPM’s expected return-beta relationship:
E(ri) = rf + βi[E(rM) – rf]
7-14
Expected Returns on Individual Securities
•
CAPM’s expected return-beta relationship:
E(ri) = rf + βi[E(rM) – rf]
 The rate of return on any asset exceeds the risk-free rate by

a risk premium equal to the asset’s systematic risk
measure (Beta) times the risk premium of the market portfolio.
 Only systematic risk matters to investors who can diversify
and that systematic risk is measured by the beta of the security.
•
In reality, even if one does not hold the precise market portfolio, a well-
diversified portfolio will be so highly correlated with the market that a
stock’s beta relative to the market still be a useful risk measure.
7-15
The Security Market Line
•
Consider the expected return-beta relationship as a reward-risk
equation.
•
The beta of a security is the appropriate measure of its risk in that
beta is proportional to the risk the security contributes to the
optimal risky portfolio.
•
The beta of a stock measures the stock’s contribution to the standard
deviation of the market portfolio.
•
Expect the required risk premium to be a function of beta.
•
The security’s risk premium is directly proportional to both the beta and
the risk premium of the market portfolio; βi[E(rM) – rf]
7-16
Individual Stocks: Security Market Line
Slope SML =
=
Equation of the SML (CAPM)

E(ri) = rf + βi[E(rM) - rf]
(E(rM) – rf )/ βM
price of risk for market
E(r)
E(rM)
rf
SML
M
ß
ß
= 1.0
7-17
The Security Market Line
•
CML
- Graphs the risk premiums of efficient portfolios as a function of portfolio
standard deviation.
- Standard deviation is a valid measure of risk for portfolios that are
candidates for an investor’s complete portfolio.
•
SML
- Graphs individual asset risk premiums as s function of asset risk.
- The relevant measure of risk for an individual asset is not the asset’s
standard deviation.
- The contribution of the asset to the portfolio standard deviation is
measured by the asset’s beta.
- The SML is valid both for portfolios and individual assets.
7-18
Capital Market Line (CML)
E(r)

rf
E(rM)
M
CML
σm
σ
Efficient
Frontier
7-19
E(r)
E(rM)
rf
SML
M
ß
ß
= 1.0
7-19
Security Market Line (SML)
Sample Calculations for SML
E(rm) - rf = 0.08 rf = 0.03
βx = 1.25
E(rx) =
βy = .6
E(ry) =
Equation of the SML
E(ri) = rf + βi[E(rM) - rf]
0.03 + 1.25(.08) = .13 or 13%
0.03 + 0.6(0.08) = 0.078 or 7.8%
If β = 1? Also, If β = 0?

7-20
E(r)
SML
ß
ßM
1.0
RM=11%
3%
Rx=13%
ßx
1.25
Ry=7.8%
ßy
.6
.08
Graph of Sample Calculations
If the CAPM is correct, only β
risk matters in determining the
risk premium for a given slope
of the SML.
7-21
Whenever the CAPM holds, all
securities must lie on the SML
in market equilibrium.
Suppose a security with a β of
____ is offering an expected
return of ____
According to the SML, the E(r)
should be _____
1.25

15%
13%.
E(r) = 0.03 + 1.25(.08) = 13%
Is the security under or overpriced?
13%
7-22
Expected rate of return > Required rate of return

People want to hold a stock.

A stock price will go up.

Its expected return will go down.
 Therefore, the stock is underpriced, and you should buy it now.
Suppose a security with a β of
____ is offering an expected
return of ____
According to the SML, the E(r)
should be _____
1.25
15%
13%.
Underpriced: It is offering too high of a rate of return for its level of risk
The difference between the return required for the risk level as measured
by the CAPM in this case and the actual return is called the stock’s _____
denoted by __
What is the __ in this case?
E(r) = 0.03 + 1.25(.08) = 13%
Is the security under or overpriced?
α = +2% Positive α is good, negative α is bad

+ α gives the buyer a + abnormal return
alpha
α
α
13%
7-23
More on alpha and beta
E(rM) = 14% βS = 1.5 rf = 5%
Required return = rf + βS[E(rM) – rf]
= 5 +1.5[14-5] = 18.5%
If you believe the stock will actually provide a return of 17%,
what is the implied alpha?
α = 17% - 18.5% = -1.5%
A stock with a negative alpha plots below the SML
& gives the buyer a negative abnormal return.
7-24
Portfolio Betas
βP =
If you put half your money in a stock with a beta of 1.5 and 30% of your money in a stock
with a beta of 0.9 and the rest in T-bills, what is the portfolio beta?
βP =
0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02
ΣWi βi
•
All portfolio beta-expected return combinations
should also fall on the SML.
•
All (E(ri) – rf) / βi should be the same for all stocks.
7-25