Investment Analysis and
Portfolio Management
Lecture 7
Gareth Myles
The Capital Asset Pricing Model
(CAPM)
The CAPM is a model of equilibrium in
the market for securities.
 Previous lectures have addressed the
question of how investors should choose
assets given the observed structure of
returns.
 Now the question is changed to:


If investors follow these strategies, how will
returns be determined in equilibrium?
The Capital Asset Pricing Model
(CAPM)

The simplest and most fundamental model of
equilibrium in the security market


Builds on the Markowitz model of portfolio choice
Aggregates the choices of individual investors

Trading ensures an equilibrium where returns
adjust so that the demand and supply of
assets are equal
 Many modifications/extensions can be made

But basic insights always extend
Assumptions

The CAPM is built on a set of assumptions
 Individual investors




Investors evaluate portfolios by the mean and
variance of returns over a one period horizon
Preferences satisfy non-satiation
Investors are risk averse
Trading conditions



Assets are infinitely divisible
Borrowing and lending can be undertaken at the
risk-free rate of return
There are no taxes or transactions costs
Assumptions
The risk-free rate is the same for all
 Information flows perfectly


The set of investors
All investors have the same time horizon
 Investors have identical expectations

Assumptions

The first six assumptions are the Markowitz
model
 The seventh and eighth assumptions add a
perfect capital market and perfect information
 The final two assumptions make all investors
identical except for their degree of risk
aversion
Direct Implications

All investors face the
same efficient set of
portfolios
rp
rf
r MVP
 MVP
p
Direct Implications

All investors choose
a location on the
efficient frontier
 The location
depends on the
degree of risk
aversion
 The chosen portfolio
mixes the risk-free
asset and portfolio M
of risky assets
rp
More risk
averse
Less risk
averse
M
rf
r MVP
 MVP
p
Separation Theorem

The optimal combination of risky assets is
determined without knowledge of preferences



All choose portfolio M
This is the Separation Theorem
M must be the market portfolio of risky assets



All investors hold it to a greater or lesser extent
No other portfolio of risky assets is held
There is a question about the interpretation of this
portfolio
Equilibrium

The only assets that need to be marketed are:




The risk-free asset
A mutual fund representing the market portfolio
No other assets are required
In equilibrium there can be no short sales of
the risky assets



All investors buy the same risky assets
No-one can be short since all would be short
If all are short the market is not in equilibrium
Equilibrium

Equilibrium occurs when the demand for
assets matches the supply



This also applies to the risk-free
Borrowing must equal lending
This is achieved by the adjustment of asset
prices
 As prices change so do the returns on the
assets
 This process generates an equilibrium
structure of returns
The Capital Market Line

All efficient portfolios
must lie on this line
rM  r f

Slope =

Equation of the line
rM
M
 rM  rf
rp  rf  
 M
rp

 p

rf
M
p
Interpretation

rf is the reward for "time"
Patience is rewarded
 Investment delays consumption

 rM  rf
M
is the reward for accepting "risk"
The market price of risk
 Judged to be equilibrium reward
 Obtained by matching demand to supply

Security Market Line
Now consider the implications for
individual assets
 Graph covariance against return

The risk on the market portfolio is  M
 The covariance of the risk-free asset is zero
 The covariance of the market with the
market is  M2

Security Market Line

Can mix M and the riskfree asset along the line



If there was a portfolio
above the line all
investors would buy it
No investor would hold
one below
The equation of the line
is
 rM  r f 
ri  r f  
 iM
2
  M 
rp
rM
M
rf
 M2
 iM
Security Market Line
 iM
 2
M

Define

The equation of the line becomes
 iM


ri  rf  rM  rf  iM

This is the security market line (SML)
Security Market Line

There is a linear
trade-off between
risk measured by  iM
and return ri
 In equilibrium all
assets and portfolios
must have risk-return
combinations that lie
on this line
rp
rf
 iM
Market Model and CAPM
Market model uses  iI
 CAPM uses  iM
  iI is derived from an assumption about
the determination of returns

it is derived from a statistical model
 the index is chosen not specified by any
underlying analysis

 iM
is derived from an equilibrium theory
Market Model and CAPM

In addition:
I is usually assumed to be the market index,
but in principal could be any index
 M is always the market portfolio

There is a difference between these
 But they are often used interchangeably
 The market index is taken as an
approximation of the market portfolio

Estimation of CAPM

Use the regression equation
ri  r f   iM  iM rM  r f   i

Take the expected value
E ri  r f  iM  iM E rM  r f

The security market line implies




 iM  0

It also shows

 iI  1  iM r f

CAPM and Pricing

CAPM also implies the equilibrium asset prices
 The security market line is


ri  rf  rM  rf iM

But
pi 1  pi 0
ri 
pi 0
where pi(0) is the value of the asset at time 0
and pi(1) is the value at time 1
CAPM and Pricing

So the security market line gives
p i 1  p i 0
 r f   iM rM  r f
p i 0


This can be rearranged to find
p i 1
p i 0  
1  r f   iM rM  r f




The price today is related to the
expected value at the end of the holding
period
CAPM and Project Appraisal
Consider an investment project
 It requires an investment of p(0) today
 It provides a payment of p(1) in a year
 Should the project be undertaken?
 The answer is yes if the present
discounted value (PDV) of the project is
positive

CAPM and Project Appraisal
If both p(0) and p(1) are certain then the
risk-free interest rate is used to discount
 The PDV is
p 1
PDV   p 0  
1 rf


The decision is to accept project if
p 1
p 0  
1 rf
CAPM and Project Appraisal
Now assume p(1) is uncertain
 Cannot simply discount at risk-free rate if
investors are risk averse
p 1
 For example using
PDV   p 0  
1 rf
will over-value the project
 With risk aversion the project is worth less
than its expected return
U ( p(1))  EU ( p(1))

CAPM and Project Appraisal
One method to obtain the correct value
is to adjust the rate of discount to reflect
risk
 But by how much?
 The CAPM pricing rule gives the answer
 The correct PDV of the project is

p (1)
PDV   p (0) 
1  r f   p [rM  r f ]