EFN412 Advanced Managerial Finance
Marco Elia
2. The Pricing of Risky Assets and the Capital Asset Pricing Model
(CAPM)
Reading and Assumed Knowledge
Reading
Berk and DeMarzo 5th ed. - Chapter 10 (10.5 – 10.8)
Chapter 11 (11.5 – 11.8)
Assumed Knowledge
Topic 2 builds on the ideas about CAPM introduced in the
Managerial Finance course and on those in the Topic 1
lecture.
Goal
Measure the effect of risk on expected return.
How prices of individual securities are determined?
Learning Objectives - Critical Concepts
• Systematic and Unsystematic Risk
• Systematic Risk and Beta
• The Capital Market Line (CML)
• The Security Market Line (SML) and the
Capital Asset Pricing Model (CAPM)
• Fama-French Model
Assumptions of Capital Market Theory
1. All investors are Markowitz efficient investors.
2. Investors can borrow or lend any amount of money at the risk-free
rate.
3. All investors have homogeneous expectations, i.e. they estimate
identical probability distributions for future rates of return.
4. All investors have the same one-period time horizon, for eg. one
month, one-year or ten years.
5. All investments are infinitely divisible which means that it is possible
to buy or sell fractional shares of any asset in the portfolio.
6. There are no taxes or transaction costs involved in buying or selling
assets.
7. There is no inflation or any changes in interest rates, or inflation is
fully anticipated.
8. Capital markets are in equilibrium, ie. we begin with all investments
properly priced in line with their risk levels.
The Risk-Free Asset
 The
risk-free asset is critical to asset pricing.
 The
expected return on a risk-free asset is
entirely certain, thus, the standard deviation of
its expected return is zero.
 RF  0


Risk free assets only exist in theory.
Empirically, we use government bonds or
treasury bills as proxies for risk free asset.
Covariance with the Risk-Free Asset

Recall the covariance between two sets of returns
is:
n
Covij   [ Rit  E ( Rit )][ R jt  E ( R jt )] / (n  1)
t 1
n
Covij   [ Rit  E ( Rit )][ R jt  E ( R jt )] pij
t 1
Because the returns for the risk-free asset are certain,
the standard deviation is zero, R-E(R) will equal zero
and the above expression will equal zero.
 Thus, the covariance of the risk-free asset with any
risky asset or portfolio will always equal zero.
 Therefore, the correlation between any risky asset and
the risk-free asset would be zero.
C o v R F ,i
 R F ,i 
 0

6

R
F

i
Combining the Risk-Free Asset
with a Risky Portfolio – ‘Expected Return’

The expected return for a portfolio that includes a
risky asset and the risk-free asset is the weighted
average of the two returns:
E(R
port
)  w R F ( R F )  (1  w R F ) E ( Ri )
where
w R F  the proportion of the portfolio invested in the rf asset.
E(R i )  the expected rate of return on risky Portfolio i
R F  the rate of return of the risk-free asset.
7
Combining the Risk-Free Asset
with a Risky Portfolio – ‘Standard Deviation’
Recall, the variance for a two-asset portfolio is:

2
 port
 wA2 A2  wB2 B2  2 wA wBCov A,B

Substituting the risk-free asset for Security
A, and the risky asset i for Security B, the
above formula becomes:
2
 port
 wR2  R2  (1  wR )2  i2  2wR (1  wR )CovR
F
8
F
F
F
F
F
,i
Combining the Risk-Free Asset
with a Risky Portfolio – ‘Standard Deviation’

We know that (i) the variance of the risk-free asset is
zero and (ii) the correlation (and covariance) between
the risk-free asset and any risky asset is also zero. So:
2
 port
 wR2  R2  (1  wR ) 2  i2  2 wR (1  wR )CovR
F
F
F
F
F
F
,i
2
 port
 (1  wR ) 2  i2
F
 port  (1  wR ) 2  i2
F
 port  (1  wR ) i
F

9
The standard deviation of a portfolio that combines the
risk-free asset with a risky asset is the linear proportion
of the standard deviation of the risky asset.
Portfolio Possibilities Combining the RiskFree Asset with Risky Portfolios on the Eff.
Frontier
E(R)
M
D
B
Rf
10
A
port
Risk-Return Combinations
11

A number of portfolio possibilities exist
when a risk-free asset is combined with
alternative risky portfolios on the Markowitz
efficient frontier.

The set of portfolio possibilities along the
RF-M line dominates ALL portfolios below
Point M.
The Capital Market Line (CML)
E(R)
The new efficient frontier when you
include a risk-free asset.
CML
M
Rf
port
The Market Portfolio (Point M)





13
Portfolio M lies at the point of tangency therefore everybody will
want to invest in Portfolio M and borrow or lend to be
somewhere on the CML.
Theoretically, this Market Portfolio (Point M) must include ALL
risky assets in the entire world.
If a risky asset is not in this portfolio in which everyone wants to
invest, there would be no demand for it and therefore it would
have no value.
Because the market is in equilibrium, it is also necessary that all
assets are included in this portfolio in proportion to their market
value.
Because the Market Portfolio contains all risky assets, it is
‘completely diversified’, which means that all the risk unique to
individual assets in the portfolio is diversified away. Specifically,
the unique risk of any single asset is offset by the unique
variability of all other assets in the portfolio.
Features of the CML

CML shows only efficient portfolios (max
expected return for every level of risk)

M is the market portfolio
Slope of the CML line is [E(Rm) - Rf]/ M

E( RP ) = R f
14
  E( R M ) - R f
+ 
M

 
 P

Risk-Return Possibilities with Leverage



15
An investor may want to have a higher
return than available at Point M
You do this by adding leverage to the
portfolio by borrowing money at the risk-free
rate and investing the proceeds in the risky
asset portfolio at Point M
Beware: higher return as well as higher risk
The Capital Market Line (CML)
Borrowing (dotted line)
E(R)
15%
CML
Lending (solid line)
12%
M
Rf
16
6%
port
Systematic and
Unsystematic
Risk
17
Unsystematic Risk
A risk that specifically affects a single security or a
small group of securities. Can be diversified away in
a large portfolio.
It is also referred to as idiosyncratic risk.
Examples:
• Volkswagen’s emission scandal
• Alitalia’s pilots go on strike
• A competitor comes up with a lower cost product
Systematic Risk
Variability in all risky assets caused by macroeconomic variables. It cannot be diversified away
through the formation of a portfolio. It remains in
the market portfolio M.
Examples:
•
•
•
•
Pandemic
Energy prices go up
Hurricane
War
Decomposition of Risk
Total Firm risk = Systematic Risk + Unsystematic Risk
Systematic risk is driven by market-wide factors
Unsystematic risk is firm-specific
Investors can diversify (20-40 securities) and eliminate (or
reduce substantially) unsystematic risk
Investors require higher returns as compensation for higher
systematic risk
Diversification
 The standard deviation of returns is a measure of total
firm risk
 Adding more stocks reduces volatility
 This reduction in risk arises because worse than
expected returns from one asset are offset by better than
expected returns from another
 However, there is a minimum level of risk that cannot be
diversified away and that is the systematic portion
Diversification
Standard
Deviation
of
Return
diversifiable risk
(unsystematic, unique or idiosyncratic risk)
Total
Risk
undiversifiable risk
(non-diversifiable, systematic, beta or market risk)
Number of Assets
in the Portfolio
Diversification
Unsystematic risk is eliminated by diversification
There is not a reward for bearing risk unnecessarily.
Diversification cannot eliminate systematic risk. Investors
must be rewarded for bearing this risk.
The expected return on an asset depends only on
that asset’s systematic risk
ONLY SYSTEMATIC RISK MATTERS FOR THE
COST OF CAPITAL
From the CML to the CAPM

How should we measure the risk of an individual asset
(stock)?

The relevant risk measure for risky assets is their
covariance with the M portfolio which is known as their
‘Systematic Risk’ or Beta.

We can proceed to use an asset’s covariance to determine
an appropriate expected rate of return on a risky asset.
24
Beta and the CAPM

For an efficient asset, there is no diversifiable risk,
therefore total risk will be equal to systematic risk.

For an inefficient asset, total risk will be greater than
systematic risk.

25
Given that the CML only plots efficient assets (whose
total risk = sys risk), we must modify it to get a model
which allows us to price efficient and/or inefficient
assets.
The Role of Beta

Diversifiable risk can be eliminated
- Only systematic risk is rewarded

For an efficient asset, there is no diversifiable
risk
- The CML only plots efficient assets

To model all assets we need a measure of
systematic risk
Let’s look at the role of beta
26
The CAPM’s Security Market Line (SML)

The return for the market portfolio M should be
consistent with its own risk, which is the
covariance of the market with itself.

The covariance of any asset with itself is its
variance.

The equation for the risk-return line (SML) is…
27
Security Market Line (SML)
In equilibrium the expected return on a risky asset (or
inefficeint portfolio) is:
E ( Ri )  RF 
E(R
RM m)- RRFF

 RF 
Defining
Covi , M
Covi,M

2
M
2
M

2
M
(Covi , M )
((E(R
RM m)-RRFF))
as beta, (i ), this equation can be stated:
CAPM
E(R i )  RF   i ((E(R
RM m)-RR
))=
 CAPM
FF
Linear relationship between beta and expected returns
28
CAPM
E(Ri)
SML
Rm
Rf
Slope = E(Rm) - Rf
Intercept = Rf
E(Ri ) R f  i [E(Rm )  R f ]
29
ßi
The slope of the line

Remember the slope of a line is given by the
difference in the “y” coordinates divided by the
difference in the “x” coordinates
• (y2 - y1) / (x2 - x1)
Using the points Rf and Rm
• [E(Rm) - Rf] / ( 1 - 0) = [E(Rm) - Rf]
30
CAPM







31
CAPM: assets are priced according to systematic
risk
Risk of asset relative to market risk
In equilibrium: each risky asset priced so that it
plots on the SML
CAPM and SML are ‘ex-ante’ models
Based on expectations (forward looking)
The return to be earned on an investment will
depend on its beta
CAPM applies to individual assets and to portfolios
(betas linearly additive)
Measuring Systematic Risk
What does beta tell us?
i 
Cov  Ri , RM 
 M2
 A beta of 1 : firm has the same systematic risk as the
overall market
 A beta < 1 : firm has less systematic risk than the overall
market
 A beta > 1 : firm has more systematic risk than the
overall market
NOTE - Additivity

Variance is not linearly additive
BUT
Return
 Covariance
 Beta
are linearly additive.

33
CML and SML





34
SML plots all assets, both efficient and inefficient,
but CML only plots efficient portfolios which are
combinations of the market portfolio and the risk
free asset.
For efficient assets, the SML reduces to the CML.
Systematic risk = Cov(i,m)/ M2.
The correlation between an efficient asset and
the market must be +1.
For efficient assets, systematic risk = total risk.
CML and SML - Equations
For any portfolio P:
 P,M 
Cov( P, M )
 2M
 p ,m p m  p ,m p


2
 M
M
Thus, the SML can be written as:
 E[ RM ]  RF 
E[ R p ]  RF  
 (  p ,m p )
M


For efficient assets  p , m  1 so SML reduces to CML:
E( R P ) = R f
35
  E( R M ) - R f
+ 
M

 
 P

CML and SML - Equations

An asset lies on the SML and CML if
 p ,m  1

An asset p only lies on the SML and is not a combination
of the risk free asset and the market portfolio if
 p ,m  1
36
ESTIMATION OF CAPM
USING THE MARKET MODEL
(MM)
37
CAPM Estimation with ‘Market Model’

38
To test CAPM theory and apply it to real world
situations, an ‘ex-post’ or historical model is used.
 The most common approach is to use the
market model proposed by Markowitz (1959).
 Model identifies different components of return.
 Return due to market wide (systematic)
factors.
 Return due to asset unique (non-systematic)
factors.
The Market Model - MM
Variable Return not
explained by Rm
(error term)
Total
Return
Ri   i   i Rm   i
The Average
Return Unique
to Asset “i”
39
Asset Return due to
Market wide Factors
The Market Model in Words



Ri is the return on any asset “i” and is made
up of returns influenced by market wide
factors plus returns due to asset unique
factors
Rm is the return on the market portfolio
i and i are parameters unique to “i”
 i is the unpredictable part of Ri
40
Estimating the Parameters (i and i )


To calculate the parameters for an asset, say a
share in Apple Inc., we perform a regression of
the returns on Apple Inc. with the returns on
the market.
Example (Apple Inc.)
 Historical
data for Apple and the SP500 index
(Value weighted index) are collected and a time
series of both returns are calculated.
 An OLS regression is then performed.
 The regression provides values for the parameters
and a plot of the observations may be made.
41
The Market Model Graphically
Ri
Rm

42
Sometimes this is called the ‘Characteristic Line’.
The Market Model Graphically
Ri
Slope is
Beta
Rm
Intercept
is Alpha
Error Term
Epsilon
43
The Market Model Graphically
Ri
Upward sloping
beta is positive
Rm
Downward sloping
beta is negative
Monthly Apple and SP500 returns
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
25-May-79
14-Nov-84
7-May-90
28-Oct-95
Return_APPL
45
19-Apr-01
Return_SP500
10-Oct-06
1-Apr-12
Example: regress Return of APPL on
SP500 return (1981-2015)
Return_SP500 Line Fit Plot
0.6
Y=0.01233+1.3974*X
0.4
Return_APPL
0.2
-0.25
0
-0.2
Return_SP500
Standard
Error
-0.05
0
-0.4
t Stat
0.01233
0.006159 2.001735
1.397398
0.13995 9.985018
Adjusted R Square 0.194025
46
-0.1
-0.2
Coefficients
Intercept
-0.15
-0.6
Return_SP500
Return_APPL
-0.8
Predicted Return_APPL
0.05
0.1
0.15
Beta Estimates of Australian Stocks
47
Name of Firm
Industry Sector
Beta
ANZ Banking Group
Banks
1.16
BHP Billiton
Materials
1.32
Coca-Cola Amatil
Food, beverage and
tobacco
0.41
Amcor
Packaging
0.75
Fairfax Media
Media
1.16
Harvey Norman
Retailing
1.03
QBE Insurance
Insurance
0.95
Woolworths
Food and Staples
Retailing
0.46
CAPM & MM

Both CAPM and MM provide expressions of
returns on an asset.
CAPM & MM - Differences
 CAPM/SML plotted in E(Ri)/ i space and the
slope is [E(Rm) - Rf]
MM plotted in Ri/Rm space and the slope is i
 CAPM is in expectations.
MM is in realizations.
 Treatment of Non-Systematic Risk (NSR)
48
CAPM and NSR (Diversifiable Risk)
49

CAPM is formulated under the assumption that
individuals will diversify.

It follows that diversifiable risk will not attract a
premium (will not be rewarded).

Only systematic risk is recognized in the
formula, that is, β.
MM and NSR (Diversifiable Risk)
Ri   i   i Rm   i
Var ( R i )  Var { i   i R m   i }
Var ( R i )  Var (  i )  Var (  i R m )  Var (  i )
Var ( R i )  0   i2 Var ( R m )  Var (  i )
Total Risk = Systematic Risk plus Non-Systematic Risk
50
Applications of the CAPM

Since its origins in the 1960’s the CAPM has
been widely employed in the real-world and in
academic applications.
 Gives
51
cost of capital for project – calculate NPV
ALTERNATIVES
TO THE CAPM:
APT and FAMA-FRENCH
52
Timeline
53

Early Testing, up to 1980s

Asset Pricing Anomalies

Arbitrage Pricing Theory

Fama and French Multifactor Model
Asset Pricing Anomalies








54
In most cases the CAPM has been rejected.
Other things have been found to matter (appear).
Small firm effect.
Day of week effect.
Within-the-month effect.
January effect.
So what?
These anomalies appear to persist and such factors
are not correctly priced by the CAPM.
Arbitrage Pricing Theory (APT)






Alternative models have been suggested
APT is the most prominent
CAPM limits pricing to one factor, the market
APT allows for several factors (measures of
systematic risk) including Rm
CAPM identifies the measure of systematic risk
(the market) as a result of an equilibrium argument.
APT does not use an equilibrium argument, so it
does not identify factors
APT  ri   i  1Factor 1  2 Factor 2  ..  n Factor n
55

Researchers do not agree on the APT risk factors
Fama-French Model

Fama & French (1992, 1996) suggest the following
multifactor model:
E  Rit   R ft   i M  E  RMt   R ft   i S E  SMB  + i h E  HML 


56
The model Includes:
- Market factor
- SMB: a small minus big portfolio factor
- HML: a high minus low book-to-market portfolio
(value – growth stocks)
Carhart (1997) adds Momentum to the factors
Summary – Key Points
Goal
Measure the effect of risk on
expected return.
• Total firm risk can be divided in systematic (driven by
market-wide factors) and unsystematic risk (risk specific to the
particular firm)
• Only systematic risk matters for the cost of capital
• Use the CAPM to measure to effect of risk on expected
returns
E(RE) = Rf + E x [E(RM)- Rf]
• Fama-French model
• Stock prices and firm betas:http://www.google.com/finance