Money, Banking & Finance
Lecture 5
The Capital Asset Pricing Model
CAPM
Aims
• Analyse the determinants of the equilibrium
expected return on an individual security.
• To show how the risk premium on an asset
is determined.
• Explain the capital asset pricing model.
• Show that the riskiness of an individual
asset is given by its ‘beta’.
The Capital Market Line
• Assume once again that the investor can borrow
and lend at the same rate.
• The transformation line that is tangential to the
efficient set is the capital market line CML
• This defines the single composition of risky assets
the investor wants to hold. This is called the
market portfolio M
• The CML is the transformation line that is
tangential to the efficient frontier.
CML
• The market portfolio represents the point on the
efficient frontier which maximises the slope of the
CML.
• The optimal proportions of risky assets at M
maximise expected return E(Rm)-Rf.
• If all investors are at M then they earn the same
excess return per unit of risk.
• The slope of the CML represents the market price
of risk.
Capital Market Line
• The CML provided a linear relation between
expected return and risk that describes the
proportion of a risk-free asset and an efficient
portfolio of assets (market portfolio) that an
investor can hold.
• The same basic function can be used to derive an
expression for the expected return on an
inefficient investment other than the market
portfolio.
• Or indeed for a single stock
Individual stock return
• At the Market portfolio ‘M’ all the risky assets are held in
the optimal proportions by all investors. This represents a
market equilibrium.
• Since all ‘n’ assets are held at M, there is a set of expected
returns E(Ri) corresponding to point M on the efficient
frontier.
• The equation representing the equilibrium returns for asset
‘i’ recognises that when held as part of a wider portfolio it
could reduce the risk of the total portfolio depending on
the covariance.
• The riskiness of asset ‘i’ when considered as part of a
diversified portfolio is not its own variance but the
covariance between Ri and the market return Rm.
CML and the market portfolio
CML
E(Rp)
M
E(Rm)
E(Rm)-Rf
Rf
α
σm
Slope of CML
tan  
E ( Rm )  R f
m
So to recap - The Capital
Market Line
• The transformation line that is tangential to
the efficient set and has intercept at the riskfree rate Rf is the capital market line CML
• This defines the single composition of risky
assets the investor wants to hold. This is
called the market portfolio M
• The CML is the transformation line that is
tangential to the efficient frontier.
CML and SML
• CML: E(Rp) = Rf + λσp, where λ is the slope.
• Interpretation of the CML is that it represents the
return available to an investor with no risk or the
additional return that can be expected as a reward
for holding the investment’s risk – this is λσp and
is known as the risk premium.
• The risk premium is the product of the market
price of risk λ and the amount of risk taken given
by σp.
• The amount of market risk at ‘M’ is σM
SML
• The SML is similar to the CML.
• The individual expected return of a share consists
of 2 elements. The risk-free return and a risk
premium.
• The risk premium for an individual share is not the
product of the market price of risk λ and the risk
of the share σi but the covariance relationship
between the share and the market portfolio which
is defined by - Beta
Beta
• Beta represents an asset’s systematic (market or
non-diversifiable) risk
• The CAPM at the point ‘M’ on the efficient
frontier gives the risk adjusted equilibrium return
on asset ‘i’
• E(Ri) – Rf = βi[E(Rm) – Rf]
• βi = Cov(Ri,Rm)/σ2m
• The risk premium is βi[E(Rm) – Rf] and represents
the reward for taking risk above that of the riskfree rate
Deriving the SML
• The covariance between the returns of an
individual share return and the market
return will tell us how much the inclusion of
that asset in the portfolio will reduce the
risk on the portfolio as a whole.
• To derive the SML let us look at a 2-asset
portfolio an asset A and the market M
• E(Rp) = ωE(RA) + (1 – ω)E(Rm)
Two-asset portfolio
E(Rp)
CML
E(Rm)
M
Rf
A
σm
σp
The expected return from a
marginal investment in the
inefficient portfolio is
dE ( R p )
d
 E ( RA )  E ( Rm )
The marginal risk produced by
a marginal investment in A is
p 
d p
   (1   )   2 (1   )Cov( R , R )
    (1   )   2 (1   )Cov( R , R )
2
1
2
2
A
2
2
2
A
2
m
2
A
2
m
m
 12
d
2 A2  2(1   ) m2  2Cov( RA , Rm )  4Cov( RA , Rm )

A
m

At the point M ω=0
d p
d
d p
d
  2Cov( R , R
 
1
2

1
2 2
m
A
Cov( RA , Rm )  
 
2
m
1
2
2
m
m

)  2
2
M

Cov( RA , Rm )  
m
2
m
Tangent at point M on the
frontier AM is:
dE ( R p )
d p
d
d

dE ( R p )
d p

E ( RA )  E ( Rm )  m

Cov( RA , Rm )  
2
m
E ( RA )  E ( Rm )

2
Cov( RA , Rm )   m  m
The CML is also tangent to
frontier at M. So:
E ( RA )  E ( Rm ) m
CovRA , Rm    m2
E ( RA )  E ( Rm


E ( Rm )  R f

E(R
) 
m

2


)

R
Cov
R
,
R


m
f
A
m
m
 m2
CovRA , Rm 
 m2
E ( RA )  E ( Rm )   E ( Rm )  R f  E ( Rm )  R f 

E ( RA )  R f   E ( Rm )  R f


Security Market Line
E(Rp)
SML
E(Rm)
Rf
1
β
Interpreting Beta

CovR A , Rm 
 A, m 

2
m
CovR A , Rm 
 A m
 A,m A

m
Beta
• The numerator represents the systematic risk of
asset A
• The denominator represents the total risk of the
market portfolio
• Beta is an index of the amount of share A’s
systematic risk relative to the market portfolio
• The beta value will tell us how much the expected
return on a share should rise or fall relative to the
market.
If the expected return on the
market rises by 10%
•
•
•
•
E(Ri) will rise > 10% if β > 1
E(Ri) will rise < 10% if β < 1
E(Ri) will rise = 10% if β = 1
Higher beta shares will outperform the
market in a bull run and lower beta shares
will under-perform
• Conversely high beta shares will fall faster
in a bear market.
Measurement of beta
• Plot the risk premium of the asset against the risk
premium of the market – slope is beta
• Regress risk premium of the asset against the risk
premium of the market – beta is the regression
coefficient
• (Ri – Rf) = αi + βi(Rm – Rf) + ui; αi = 0
• αi = 0 means that when the market risk premium is
zero so should the individual shares that make up
the portfolio have zero risk premium.
• So what if αi ≠ 0?
Alpha
• Over a long period alpha values should be zero.
• But if not it can used to provide investor advice.
• αi < 0 shares should be sold and αi > 0 should be
bought. Meaning share price of {i} is mispriced.
• Negative alpha means that returns are below the
equilibrium predicted by the CAPM, therefore
share prices will fall until yields rise to the
equilibrium. The act of selling will drive down the
price.
• Vice versa for positive alpha
Summary
• We have examined the CAPM framework
• The CAPM is used to measure the systematic risk
of an individual share.
• The beta value measures the degree of
responsiveness of the expected return on the share
relative to movements in the expected return of the
market.
• Alpha is interpreted as an indication of mispricing
and has been used as justification for investment
strategy