The Capital Asset Pricing
Model
Review
Review of portfolio diversification
Capital Asset Pricing Model
Capital Market Line (CML)
Security Market Line (SML)
Capital Asset Pricing Model (CAPM)
It is the equilibrium model that underlies all
modern financial theory.
Derived using principles of diversification with
simplified assumptions.
Markowitz, Sharpe, Lintner and Mossin are
researchers credited with its development.
Assumptions
Single-period investment horizon.
Investors forecasts agree with respect to
expectations, standard deviations, and
correlations of the returns of risky securities
- Therefore all investors hold risky assets in the same
relative proportions
Investors behave optimally
- In equilibrium, prices adjust so that aggregate
demand for each security is equal to its supply
Market Portfolio
Since every investor’s relative holdings of
the risky security is the same, the only way
the asset market can clear is if those optimal
relative proportions are the proportions in
which they are valued in the market place
All investors will hold the same portfolio
for risky assets – market portfolio.
Capital Market Line (CML) and the CAPM
CAPM says that in equilibrium, any
investor’s relative holding of risky assets
will be the same as in the market portfolio
Depending on their risk aversions, different
investors hold portfolios with different
mixes of riskless asset and the market
portfolio
Example (online two stocks)
Capital Market Line
Err 
Erm  rf
slope 
m
 r  rf
Erm  rf
m
Capital Market Line
E(r)
E(rM)
M
CML
rf
m

What determines the market risk premium?
 Risk premium on the the market depends on the
average risk aversion (A) of all market participants.
 Example:
E (rm)  rf  A 
2
M
Erm  0.14,  m  0.20, rf  0.06,
Erm  rf  A
2
m
 A
0.14  0.06
A
 2.0
2
0.20
Erm  rf
 m2
Beta and Security Market Line
 If risk is defined as the measure such that as it
increases, a risk-averse investor would have to be
compensated by a larger expected return in order
for her to hold it in her optimal portfolio, then the
measure of a security’s risk is its beta, b, not its
standard deviation!
b tells you how much the security’s rate of return
changes when the return on the market portfolio
changes
CAPM Risk Premium on any Asset
 According to the CAPM, in equilibrium, the risk
premium on any asset is equal to the product of
- b (or ‘Beta’), and
- the risk premium on the market portfolio
 im
bi  2
 m
Eri  rf  Erm  rf b i  Eri  rf  Erm  rf b i
The Security Market Line
- The plot of a security’s risk premium Eri-rf (or
sometimes security returns) against security
beta b is the security market line
Note that the slope of the security market line is the
market premium
By CAPM theory, all securities must fall precisely
on the SML (hence its name)
Security Market Line (SML)
E(r)
SML
E(rM)
rf
bM= 1.0
b
Security Market Line (SML) Relationships
SML:
ri = rf + bi[E(rm) - rf]
bi = [COV(ri,rm)] / m2
Slope SML =
E(rm) - rf
= market risk premium
The Beta of a Portfolio in CAPM
When determining the risk of a portfolio
- using standard deviation results in a formula
that’s quite complex
 w r  w r ...  w r
11
2 2
n n


2
   wi i   2 wi w j i j i , j 
i j
 i 1,n

- using beta, the formula is linear
b w1r1 w2r2 ...  wn rn  w1b1  w2 b2  ...  wn bn   wi bi
i
1
2
Examples for SML
E(rm) - rf = .08 rf = .03
bx = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .6
E(ry) = .03 + .6(.08) = .078 or 7.8%
Graph of Sample Calculations
E(r)
SML
Rx=13%
.08
Rm=11%
Ry=7.8%
3%
b
.6
by
»By
1.0
1.25
bx
Disequilibrium Example
E(r)
SML
15%
Rm=11%
rf=3%
b
1.0
1.25
Disequilibrium Example
Suppose a security with a b of 1.25 is
offering expected return of 15%.
According to SML, it should be 13%.
Under-priced: offering too high of a rate of
return for its level of risk.
Alpha
Underpriced stocks plot above the SML
Overpriced stocks plot below the SML
The difference between the fair and actually
expected rates of return on a stock is called
the stock’s alpha.
- Example: an alpha fund with alpha=1%, beta=0.5,
standard deviation=0.15, market return=0.14, risk
free rate is 0.06, market standard deviation=0.2