INTRODUCTION TO
CORPORATE FINANCE
Laurence Booth • W. Sean Cleary
Prepared by
Ken Hartviksen
CHAPTER 9
The Capital Asset Pricing
Model (CAPM)
Lecture Agenda
•
•
•
•
•
•
•
Learning Objectives
Important Terms
The New Efficient Frontier
The Capital Asset Pricing Model
The CAPM and Market Risk
Alternative Asset Pricing Models
Summary and Conclusions
– Concept Review Questions
– Appendix 1 – Calculating the Ex Ante Beta
– Appendix 2 – Calculating the Ex Post Beta
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-3
Learning Objectives
1.
2.
What happens if all investors are rational and risk averse.
How modern portfolio theory is extended to develop the
capital market line, which determines how expected
returns on portfolios are determined.
3.
4.
How to assess the performance of mutual fund managers
How the Capital Asset Pricing Model’s (CAPM) security
market line is developed from the capital market line.
5.
How the CAPM has been extended to include other riskbased pricing models.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-4
Important Chapter Terms
• Arbitrage pricing theory
(APT)
• Capital Asset Pricing
Model (CAPM)
• Capital market line (CML)
• Characteristic line
• Fama-French (FF) model
• Insurance premium
• Market portfolio
• Market price of risk
• Market risk premium
• New (or super) efficient
frontier
• No-arbitrage principle
• Required rate of return
• Risk premium
• Security market line
(SML)
• Separation theorum
• Sharpe ratio
• Short position
• Tangent portfolio
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-5
Achievable Portfolio Combinations
The Two-Asset Case
• It is possible to construct a series of portfolios with
different risk/return characteristics just by varying the
weights of the two assets in the portfolio.
• Assets A and B are assumed to have a correlation
coefficient of -0.379 and the following individual
return/risk characteristics
Asset A
Asset B
Expected Return
8%
10%
Standard Deviation
8.72%
22.69%
The following table shows the portfolio characteristics for 100
different weighting schemes for just these two securities:
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-6
Example of Portfolio Combinations and
Correlation
You repeat this
procedure
down until you
have determine
the portfolio
characteristics
The first
for all
100
The
second
combination
portfolios.
portfolio
simply99%
assumes
Next
plot1%
the
in
A and
in
assumes
returns
onthe
a
B.
Notice
you
invest
graph
(see in
the
increase
solely
inthe
next
slide)
return
and
decrease
Asset Ain
portfolio risk!
Asset
A
B
Expected
Return
8.0%
10.0%
Portfolio Components
Weight of A Weight of B
100%
0%
99%
1%
98%
2%
97%
3%
96%
4%
95%
5%
94%
6%
93%
7%
92%
8%
91%
9%
90%
10%
89%
11%
Standard
Deviation
8.7%
22.7%
Correlation
Coefficient
-0.379
Portfolio Characteristics
Expected
Standard
Return
Deviation
8.00%
8.7%
8.02%
8.5%
8.04%
8.4%
8.06%
8.2%
8.08%
8.1%
8.10%
7.9%
8.12%
7.8%
8.14%
7.7%
8.16%
7.5%
8.18%
7.4%
8.20%
7.3%
8.22%
7.2%
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-7
Attainable Portfolio Combinations for a
Two Asset Portfolio
12.00%
Expected Return of the
Portfolio
Example of
Portfolio
Combinations
and
Correlation
10.00%
8.00%
6.00%
4.00%
2.00%
0.00%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
Standard Deviation of Returns
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-8
Two Asset Efficient Frontier
• Figure 8 – 10 describes five different portfolios
(A,B,C,D and E in reference to the attainable set
of portfolio combinations of this two asset
portfolio.
(See Figure 8 -10 on the following slide)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9-9
Efficient Frontier
The Two-Asset Portfolio Combinations
8 - 10 FIGURE
A is not attainable
B,E lie on the
Expected Return %
efficient frontier and
are attainable
A
B
E is the minimum
C
variance portfolio
(lowest risk
combination)
C, D are
E
D
Standard Deviation (%)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
attainable but are
dominated by
superior portfolios
that line on the line
above E
9 - 10
Achievable Set of Portfolio Combinations
Getting to the ‘n’ Asset Case
• In a real world investment universe with all of the
investment alternatives (stocks, bonds, money
market securities, hybrid instruments, gold real
estate, etc.) it is possible to construct many
different alternative portfolios out of risky
securities.
• Each portfolio will have its own unique expected
return and risk.
• Whenever you construct a portfolio, you can
measure two fundamental characteristics of the
portfolio:
– Portfolio expected return (ERp)
– Portfolio risk (σp)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 11
The Achievable Set of Portfolio
Combinations
• You could start by randomly assembling ten
risky portfolios.
• The results (in terms of ER p and σp )might look
like the graph on the following page:
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 12
Achievable Portfolio Combinations
The First Ten Combinations Created
ERp
10 Achievable
Risky Portfolio
Combinations
Portfolio Risk (σp)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 13
The Achievable Set of Portfolio
Combinations
• You could continue randomly assembling more
portfolios.
• Thirty risky portfolios might look like the graph
on the following slide:
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 14
Achievable Portfolio Combinations
Thirty Combinations Naively Created
ERp
30 Risky Portfolio
Combinations
Portfolio Risk (σp)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 15
Achievable Set of Portfolio Combinations
All Securities – Many Hundreds of Different Combinations
• When you construct many hundreds of different
portfolios naively varying the weight of the
individual assets and the number of types of
assets themselves, you get a set of achievable
portfolio combinations as indicated on the
following slide:
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 16
Achievable Portfolio Combinations
More Possible Combinations Created
ERp
E is the
minimum
variance
portfolio
Achievable Set of
Risky Portfolio
Combinations
The highlighted
portfolios are
‘efficient’ in that
they offer the
highest rate of
return for a given
level of risk.
Rationale investors
will choose only
from this efficient
set.
E
Portfolio Risk (σp)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 17
Achievable Portfolio Combinations
Efficient Frontier (Set)
ERp
Achievable Set of
Risky Portfolio
Combinations
E
Efficient
frontier is the
set of
achievable
portfolio
combinations
that offer the
highest rate
of return for a
given level of
risk.
Portfolio Risk (σp)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 18
The New Efficient Frontier
Efficient Portfolios
9 - 1 FIGURE
Efficient Frontier
ER
B
A
MVP
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Figure 9 – 1
illustrates
three
achievable
portfolio
combinations
that are
‘efficient’ (no
other
achievable
portfolio that
offers the
same risk,
offers a higher
return.)
9 - 19
Underlying Assumption
Investors are Rational and Risk-Averse
• We assume investors are risk-averse wealth maximizers.
• This means they will not willingly undertake fair gamble.
– A risk-averse investor prefers the risk-free situation.
– The corollary of this is that the investor needs a risk premium to
be induced into a risky situation.
– Evidence of this is the willingness of investors to pay insurance
premiums to get out of risky situations.
• The implication of this, is that investors will only choose
portfolios that are members of the efficient set (frontier).
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 20
Risk-free Investing
• When we introduce the presence of a risk-free
investment, a whole new set of portfolio
combinations becomes possible.
• We can estimate the return on a portfolio made
up of RF asset and a risky asset A letting the
weight w invested in the risky asset and the
weight invested in RF as (1 – w)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 21
The New Efficient Frontier
Risk-Free Investing
– Expected return on a two asset portfolio made up of
risky asset A and RF:
[9-1]
ER p  RF  w (ER A - RF)
The possible combinations of A and RF are found graphed on the following slide.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 22
The New Efficient Frontier
Attainable Portfolios Using RF and A
9 - 2 FIGURE
ER
A
[9-3]
 pA 
w A
) - RF
[9-2]  E(R
ER P  RF  
 P

A


RF
This means
you
can 9 – 2
Equation
Rearranging
9
achieve
any
illustrates
-2 where w=σ
portfolio
what
you can
p / σA and
combination
see…portfolio
substituting in
along
the blue
risk
increases
Equation
1 we
coloured
line
in direct
get
an
simply
by to
proportion
equation for
a
changing
the
the amount
straight
line
relative
weight
invested
with a in the
of
RFasset.
and A in
risky
constant
the two asset
slope.
portfolio.
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 23
The New Efficient Frontier
Attainable Portfolios using the RF and A, and RF and T
9 - 3 FIGURE
Which risky
portfolio
would a
rational riskaverse
investor
choose in the
presence of a
RF
investment?
ER
T
A
RF
Portfolio A?
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Tangent
Portfolio T?
9 - 24
The New Efficient Frontier
Efficient Portfolios using the Tangent Portfolio T
9 - 3 FIGURE
ER
T
A
RF
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Clearly RF with
T (the tangent
portfolio) offers
a series of
portfolio
combinations
that dominate
those produced
by RF and A.
Further, they
dominate all but
one portfolio on
the efficient
frontier!
9 - 25
The New Efficient Frontier
Lending Portfolios
9 - 3 FIGURE
ER
Lending Portfolios
T
A
RF
Portfolios
between RF
and T are
‘lending’
portfolios,
because they
are achieved by
investing in the
Tangent
Portfolio and
lending funds to
the government
(purchasing a
T-bill, the RF).
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 26
The New Efficient Frontier
Borrowing Portfolios
9 - 3 FIGURE
ER
Lending Portfolios Borrowing Portfolios
T
A
RF
The line can be
extended to risk
levels beyond
‘T’ by
borrowing at RF
and investing it
in T. This is a
levered
investment that
increases both
risk and
expected return
of the portfolio.
Risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 27
The New Efficient Frontier
The New (Super) Efficient Frontier
9 - 4 FIGURE
Capital Market Line
ER
B2
T
B
A2
RF
A
σρ
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
This is now
called
Clearlythe
RFnew
with
(or
super)
T (the
market
The optimal
efficient
frontier
portfolio)
offers
risky portfolio
of
risky
a series
of
(the market
portfolios.
portfolio
portfolio ‘M’)
combinations
Investors can
that dominate
achieve any
those produced
one of these
by RF and A.
portfolio
combinations
Further, they
by
borrowing
or
dominate
all but
investing
in RF
one portfolio
on
in
thecombination
efficient
with
the market
frontier!
portfolio.
9 - 28
The New Efficient Frontier
The Implications – Separation Theorem – Market Portfolio
• All investors will only hold individually-determined
combinations of:
– The risk free asset (RF) and
– The model portfolio (market portfolio)
• The separation theorem
– The investment decision (how to construct the portfolio of risky
assets) is separate from the financing decision (how much
should be invested or borrowed in the risk-free asset)
– The tangent portfolio T is optimal for every investor regardless of
his/her degree of risk aversion.
• The Equilibrium Condition
– The market portfolio must be the tangent portfolio T if everyone
holds the same portfolio
– Therefore the market portfolio (M) is the tangent portfolio (T)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 29
The New Efficient Frontier
The Capital Market Line
The CML is that
set of superior
The optimal
portfolio
risky portfolio
combinations
(the market
that
are ‘M’)
portfolio
achievable in
the presence of
the equilibrium
condition.
CML
ER
M
RF
σρ
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 30
The Capital Asset Pricing Model
What is it?
– An hypothesis by Professor William Sharpe
• Hypothesizes that investors require higher rates of
return for greater levels of relevant risk.
• There are no prices on the model, instead it
hypothesizes the relationship between risk and
return for individual securities.
• It is often used, however, the price securities and
investments.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 31
The Capital Asset Pricing Model
How is it Used?
– Uses include:
• Determining the cost of equity capital.
• The relevant risk in the dividend discount model to estimate a stock’s intrinsic
(inherent economic worth) value. (As illustrated below)
Estimate Investment’s
Risk (Beta Coefficient)
i 
COVi,M
σ M2
Determine Investment’s
Required Return
ki  RF  ( ERM  RF )  i
Estimate the
Investment’s Intrinsic
Value
D1
P0 
kc  g
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Compare to the actual
stock price in the market
Is the stock
fairly priced?
9 - 32
The Capital Asset Pricing Model
Assumptions
– CAPM is based on the following assumptions:
1. All investors have identical expectations about expected
returns, standard deviations, and correlation coefficients for all
securities.
2. All investors have the same one-period investment time
horizon.
3. All investors can borrow or lend money at the risk-free rate of
return (RF).
4. There are no transaction costs.
5. There are no personal income taxes so that investors are
indifferent between capital gains an dividends.
6. There are many investors, and no single investor can affect
the price of a stock through his or her buying and selling
decisions. Therefore, investors are price-takers.
7. Capital markets are in equilibrium.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 33
Market Portfolio and Capital Market Line
•
The assumptions have the following
implications:
1. The “optimal” risky portfolio is the one that is
tangent to the efficient frontier on a line that is drawn
from RF. This portfolio will be the same for all
investors.
2. This optimal risky portfolio will be the market
portfolio (M) which contains all risky securities.
(Figure 9 – 4 illustrates the Market Portfolio ‘M’)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 34
The Capital Market Line
9 - 5 FIGURE
ER
CML
ERM
M
 ERM  RF 
k P  RF  
 P
 M

RF
σρ
The CML is that
setThe
of achievable
market
portfolio
The
portfolio
CMLishas
the
combinations
optimal
standard
risky
that
deviation
portfolio,
are possible
of
it
contains
portfolio
when investing
all
returns
risky
securities
in as
only
the
two
and
lies
independent
assets
tangent
(the(T)
market
on variable.
the efficient
portfolio
and frontier.
the risk-free
asset (RF).
σM
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 35
The Capital Asset Pricing Model
The Market Portfolio and the Capital Market Line (CML)
– The slope of the CML is the incremental expected
return divided by the incremental risk.
[9-4]
Slope of the CML 
ER M - RF
M
– This is called the market price for risk. Or
– The equilibrium price of risk in the capital market.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 36
The Capital Asset Pricing Model
The Market Portfolio and the Capital Market Line (CML)
– Solving for the expected return on a portfolio in the presence of a
RF asset and given the market price for risk :
[9-5]
 ERM - RF 
E ( RP )  RF  
 P
 σM

– Where:
• ERM = expected return on the market portfolio M
• σM = the standard deviation of returns on the market portfolio
• σP = the standard deviation of returns on the efficient portfolio being
considered
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 37
The Capital Market Line
Using the CML – Expected versus Required Returns
– In an efficient capital market investors will require a
return on a portfolio that compensates them for the
risk-free return as well as the market price for risk.
– This means that portfolios should offer returns along
the CML.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 38
The Capital Asset Pricing Model
Expected and Required Rates of Return
9 - 6 FIGURE
Required
Return on C
ER
Expected
return on A
CML
A
C
Required
return on A
RF
B
Expected
Return on C
σρ
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
C is an
A
B
a portfolio
overvalued
that
undervalued
offers
portfolio.
andExpected
expected
portfolio.
return
equal
is
less
Expected
tothan
the
return
required
the
required
is greater
return.
return.
than the required
Selling pressure will
return.
cause the price to
Demand
fall
and the
foryield to
Portfolio
rise
until A
expected
will
increase
equals
the
driving
required
up
the price, and
return.
therefore the
expected return will
fall until expected
equals required
(market equilibrium
condition is
achieved.)
9 - 39
The Capital Asset Pricing Model
Risk-Adjusted Performance and the Sharpe Ratios
– William Sharpe identified a ratio that can be used to assess the riskadjusted performance of managed funds (such as mutual funds and
pension plans).
– It is called the Sharpe ratio:
[9-6]
Sharpe ratio 
ER P - RF
P
– Sharpe ratio is a measure of portfolio performance that describes how
well an asset’s returns compensate investors for the risk taken.
– It’s value is the premium earned over the RF divided by portfolio
risk…so it is measuring valued added per unit of risk.
– Sharpe ratios are calculated ex post (after-the-fact) and are used to
rank portfolios or assess the effectiveness of the portfolio manager in
adding value to the portfolio over and above a benchmark.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 40
The Capital Asset Pricing Model
Sharpe Ratios and Income Trusts
– Table 9 – 1 (on the following slide) illustrates return,
standard deviation, Sharpe and beta coefficient for
four very different portfolios from 2002 to 2004.
– Income Trusts did exceedingly well during this time,
however, the recent announcement of Finance
Minister Flaherty and the subsequent drop in Income
Trust values has done much to eliminate this
historical performance.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 41
Income Trust Estimated Values
Table 9-1 Income Trusts Estimated Values
Median income trusts
Equally weighted trust portfolio
S&P/TSX Composite Index
Scotia Capital government bond index
Return
σ
Sharpe
β
25.83%
29.97%
8.97%
9.55%
18.66%
8.02%
13.31%
6.57%
1.37
3.44
0.49
1.08
0.22
0.28
1.00
20.02
P
Source: Adapted from L. Kryzanowski, S. Lazrak, and I. Ratika, " The True
Cost of Income Trusts," Canadian Investment Review 19, no. 5 (Spring
2006), Table 3, p. 15.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 42
Diversifiable and Non-Diversifiable Risk
• CML applies to efficient portfolios
• Volatility (risk) of individual security returns are caused
by two different factors:
– Non-diversifiable risk (system wide changes in the economy and
markets that affect all securities in varying degrees)
– Diversifiable risk (company-specific factors that affect the returns
of only one security)
• Figure 9 – 7 illustrates what happens to portfolio risk as
the portfolio is first invested in only one investment, and
then slowly invested, naively, in more and more
securities.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 43
The CAPM and Market Risk
Portfolio Risk and Diversification
9 - 7 FIGURE
Total Risk (σ)
Unique (Non-systematic) Risk
Market (Systematic) Risk
Market or
systematic
risk is risk
that cannot
be eliminated
from the
portfolio by
investing the
portfolio into
more and
different
securities.
Number of Securities
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 44
Relevant Risk
Drawing a Conclusion from Figure 9 - 7
• Figure 9 – 7 demonstrates that an individual securities’
volatility of return comes from two factors:
– Systematic factors
– Company-specific factors
• When combined into portfolios, company-specific risk is
diversified away.
• Since all investors are ‘diversified’ then in an efficient
market, no-one would be willing to pay a ‘premium’ for
company-specific risk.
• Relevant risk to diversified investors then is systematic
risk.
• Systematic risk is measured using the Beta Coefficient.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 45
The Beta Coefficient
What is the Beta Coefficient?
• A measure of systematic (non-diversifiable) risk
• As a ‘coefficient’ the beta is a pure number and
has no units of measure.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 46
The Beta Coefficient
How Can We Estimate the Value of the Beta Coefficient?
•
There are two basic approaches to estimating
the beta coefficient:
1. Using a formula (and subjective forecasts)
2. Use of regression (using past holding period returns)
(Figure 9 – 8 on the following slide illustrates the characteristic line used to estimate
the beta coefficient)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 47
The CAPM and Market Risk
The Characteristic Line for Security A
9 - 8 FIGURE
Security A Returns (%)
6
2
-6
-4
-2
0
0
2
4
6
-2
-4
8
Market Returns (%)
4
The
The slope
plotted
of
the
points
regression
are the
line
coincident
is beta.
rates of return
earned
The line
on of
the
investment
best fit is
andknown
the market
in
finance
portfolioasover
the
characteristic
past periods.
line.
-6
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 48
The Formula for the Beta Coefficient
Beta is equal to the covariance of the returns of the
stock with the returns of the market, divided by the
variance of the returns of the market:
[9-7]
COVi,M i , M  i
i 

2
σM
M
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 49
The Beta Coefficient
How is the Beta Coefficient Interpreted?
•
The beta of the market portfolio is ALWAYS = 1.0
•
The beta of a security compares the volatility of its returns to the volatility of
the market returns:
βs = 1.0
-
the security has the same volatility as the market as a whole
βs > 1.0
-
aggressive investment with volatility of returns greater than
the market
βs < 1.0
-
defensive investment with volatility of returns less than the
market
βs < 0.0
-
an investment with returns that are negatively correlated with
the returns of the market
Table 9 – 2 illustrates beta coefficients for a variety of Canadian Investments
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 50
Canadian BETAS
Selected
Table 9-2 Canadian BETAS
Company
Abitibi Consolidated Inc.
Algoma Steel Inc.
Bank of Montreal
Bank of Nova Scotia
Barrick Gold Corp.
BCE Inc.
Bema Gold Corp.
CIBC
Cogeco Cable Inc.
Gammon Lake Resources Inc.
Imperial Oil Ltd.
Industry Classification
Beta
Materials - Paper & Forest
Materials - Steel
Financials - Banks
Financials - Banks
Materials - Precious Metals & Minerals
Communications - Telecommunications
Materials - Precious Metals & Minerals
Financials - Banks
Consumer Discretionary - Cable
Materials - Precious Metals & Minerals
Energy - Oil & Gas: Integrated Oils
1.37
1.92
0.50
0.54
0.74
0.39
0.26
0.66
0.67
2.52
0.80
Source: Research Insight, Compustat North American database, June 2006.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 51
The Beta of a Portfolio
The beta of a portfolio is simply the weighted average of the
betas of the individual asset betas that make up the portfolio.
[9-8]
 P  wA  A  wB  B  ...  wn  n
Weights of individual assets are found by dividing the value of
the investment by the value of the total portfolio.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 52
The CAPM and Market Risk
The Security Market Line (SML)
– The SML is the hypothesized relationship between return (the
dependent variable) and systematic risk (the beta coefficient).
– It is a straight line relationship defined by the following formula:
[9-9]
ki  RF  ( ERM  RF )  i
– Where:
ki = the required return on security ‘i’
ERM – RF = market premium for risk
Βi = the beta coefficient for security ‘i’
(See Figure 9 - 9 on the following slide for the graphical representation)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 53
The CAPM and Market Risk
The Security Market Line (SML)
9 - 9 FIGURE
ER
ki  RF  ( ERM  RF ) i
TheSML
SMLis
The
uses
usedthe
to
beta
predict
coefficient
requiredas
the
measure
returns
for
of
relevant
individual
risk.
securities
M
ERM
RF
βM = 1
β
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 54
The CAPM and Market Risk
The SML and Security Valuation
9 - 10 FIGURE
ki  RF  ( ERM  RF ) i
ER
SML
Expected
Return A
A
Required
Return A
B
RF
βA
βB
β
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Similarly,
Required
A
is an B
returns
is an
are forecast using
undervalued
overvalued
this equation.
security
security.
because
its expected
You can see
Investor’s
willthat
sell
return is greater
thelock
to
required
in gains,
than the required
return
but
theon
selling
any
return.
security iswill
pressure
a
function
Investors
cause
the
ofwill
market
its
systematic
‘flock’
price
to
tofall,
A and
risk bid
(β)
andthe
up
causing
market
price
the
factors (RF
causing
expected
expected
return
and to
market
return
rise
until
topremium
fall
it equals
till it
for risk)
equals
the
required
the
required return.
return.
9 - 55
The CAPM in Summary
The SML and CML
– The CAPM is well entrenched and widely used by
investors, managers and financial institutions.
– It is a single factor model because it based on the
hypothesis that required rate of return can be
predicted using one factor – systematic risk
– The SML is used to price individual investments and
uses the beta coefficient as the measure of risk.
– The CML is used with diversified portfolios and uses
the standard deviation as the measure of risk.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 56
Challenges to CAPM
• Empirical tests suggest:
– CAPM does not hold well in practice:
• Ex post SML is an upward sloping line
• Ex ante y (vertical) – intercept is higher that RF
• Slope is less than what is predicted by theory
– Beta possesses no explanatory power for predicting stock
returns (Fama and French, 1992)
• CAPM remains in widespread use despite the foregoing.
– Advantages include – relative simplicity and intuitive logic.
• Because of the problems with CAPM, other models have
been developed including:
– Fama-French (FF) Model
– Abitrage Pricing Theory (APT)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 57
Alternative Asset Pricing Models
The Fama – French Model
– A pricing model that uses three factors to relate
expected returns to risk including:
1. A market factor related to firm size.
2. The market value of a firm’s common equity (MVE)
3. Ratio of a firm’s book equity value to its market value of equity.
(BE/MVE)
– This model has become popular, and many think it
does a better job than the CAPM in explaining ex
ante stock returns.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 58
Alternative Asset Pricing Models
The Arbitrage Pricing Theory
– A pricing model that uses multiple factors to relate expected
returns to risk by assuming that asset returns are linearly related
to a set of indexes, which proxy risk factors that influence
security returns.
[9-10]
ERi  a0  bi1 F1  bi1 F1  ...  bin Fn
– It is based on the no-arbitrage principle which is the rule that two
otherwise identical assets cannot sell at different prices.
– Underlying factors represent broad economic forces which are
inherently unpredictable.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 59
Alternative Asset Pricing Models
The Arbitrage Pricing Theory – the Model
– Underlying factors represent broad economic forces which are
inherently unpredictable.
[9-10]
ERi  a0  bi1 F1  bi1 F1  ...  bin Fn
– Where:
•
•
•
•
ERi = the expected return on security i
a0 = the expected return on a security with zero systematic risk
bi = the sensitivity of security i to a given risk factor
Fi = the risk premium for a given risk factor
– The model demonstrates that a security’s risk is based on its sensitivity
to broad economic forces.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 60
Alternative Asset Pricing Models
The Arbitrage Pricing Theory – Challenges
– Underlying factors represent broad economic forces
which are inherently unpredictable.
– Ross and Roll identify five systematic factors:
1.
2.
3.
4.
5.
•
Changes in expected inflation
Unanticipated changes in inflation
Unanticipated changes in industrial production
Unanticipated changes in the default-risk premium
Unanticipated changes in the term structure of interest rates
Clearly, something that isn’t forecast, can’t be used
to price securities today…they can only be used to
explain prices after the fact.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 61
Summary and Conclusions
In this chapter you have learned:
– How the efficient frontier can be expanded by introducing riskfree borrowing and lending leading to a super efficient frontier
called the Capital Market Line (CML)
– The Security Market Line can be derived from the CML and
provides a way to estimate a market-based, required return for
any security or portfolio based on market risk as measured by
the beta.
– That alternative asset pricing models exist including the FamaFrench Model and the Arbitrage Pricing Theory.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 62
Estimating the Ex Ante (Forecast) Beta
APPENDIX 1
Calculating a Beta Coefficient Using Ex Ante
Returns
• Ex Ante means forecast…
• You would use ex ante return data if historical rates of
return are somehow not indicative of the kinds of returns
the company will produce in the future.
• A good example of this is Air Canada or American
Airlines, before and after September 11, 2001. After the
World Trade Centre terrorist attacks, a fundamental shift
in demand for air travel occurred. The historical returns
on airlines are not useful in estimating future returns.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 64
Appendix 1 Agenda
• The beta coefficient
• The formula approach to beta measurement
using ex ante returns
–
–
–
–
–
Ex ante returns
Finding the expected return
Determining variance and standard deviation
Finding covariance
Calculating and interpreting the beta coefficient
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 65
The Beta Coefficient
• Under the theory of the Capital Asset Pricing Model total
risk is partitioned into two parts:
– Systematic risk
– Unsystematic risk – diversifiable risk
Total Risk of the Investment
Systematic Risk
Unsystematic Risk
• Systematic risk is non-diversifiable risk.
• Systematic risk is the only relevant risk to the diversified
investor
• The beta coefficient measures systematic risk
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 66
The Beta Coefficient
The Formula
Beta 
Covariance of Returns between stock ' i' returns and the market
Variance of the Market Returns
[9-7]
COVi,M i , M  i
i 

2
σM
M
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 67
The Term – “Relevant Risk”
•
What does the term “relevant risk” mean in the context of the CAPM?
– It is generally assumed that all investors are wealth maximizing risk
averse people
– It is also assumed that the markets where these people trade are highly
efficient
– In a highly efficient market, the prices of all the securities adjust instantly
to cause the expected return of the investment to equal the required
return
– When E(r) = R(r) then the market price of the stock equals its inherent
worth (intrinsic value)
– In this perfect world, the R(r) then will justly and appropriately
compensate the investor only for the risk that they perceive as
relevant…
– Hence investors are only rewarded for systematic risk.
NOTE: The amount of systematic risk varies by investment. High systematic risk
occurs when R-square is high, and the beta coefficient is greater than 1.0
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 68
The Proportion of Total Risk that is Systematic
• Every investment in the financial markets vary with
respect to the percentage of total risk that is systematic.
• Some stocks have virtually no systematic risk.
– Such stocks are not influenced by the health of the economy in
general…their financial results are predominantly influenced by
company-specific factors.
– An example is cigarette companies…people consume cigarettes
because they are addicted…so it doesn’t matter whether the
economy is healthy or not…they just continue to smoke.
• Some stocks have a high proportion of their total risk that
is systematic
– Returns on these stocks are strongly influenced by the health of
the economy.
– Durable goods manufacturers tend to have a high degree of
systematic risk.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 69
The Formula Approach to Measuring the Beta
Cov(k i k M )
Beta 
Var(k M )
You need to calculate the covariance of the returns between the
stock and the market…as well as the variance of the market
returns. To do this you must follow these steps:
• Calculate the expected returns for the stock and the market
• Using the expected returns for each, measure the variance
and standard deviation of both return distributions
• Now calculate the covariance
• Use the results to calculate the beta
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 70
Ex ante Return Data
A Sample
A set of estimates of possible returns and their respective
probabilities looks as follows:
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
25.0%
50.0%
25.0%
Possible
Possible
Returns on Returns on
the Stock
the Market
28.0%
17.0%
-14.0%
20.0%
11.0%
-4.0%
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
Since the beta
relates the stock
By
observation
returns
to the
market
returns,
you can
see the
the
greater
range
range
is much
of stock returns
greater
for the
changing in the
stock
than theas
same direction
market
and they
the market
indicates
the beta
move in the
will
be direction.
greater
same
than 1 and will be
positive.
(Positively
correlated to the
market returns.)
9 - 71
The Total of the Probabilities must Equal 100%
This means that we have considered all of the possible outcomes in
this discrete probability distribution
Possible
Future State
of the
Economy
Probability
Boom
Normal
Recession
25.0%
50.0%
25.0%
Possible
Possible
Returns on Returns on
the Stock
the Market
28.0%
17.0%
-14.0%
20.0%
11.0%
-4.0%
100.0%
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 72
Measuring Expected Return on the Stock
From Ex Ante Return Data
The expected return is weighted average returns from the
given ex ante data
(1)
(2)
Possible
Future State
of the
Probability
Economy
(3)
(4)
Possible
Returns on
the Stock (4) = (2)*(3)
Boom
25.0%
28.0%
Normal
50.0%
17.0%
Recession
25.0%
-14.0%
Expected return on the Stock =
0.07
0.085
-0.035
12.0%
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 73
Measuring Expected Return on the Market
From Ex Ante Return Data
The expected return is weighted average returns from the
given ex ante data
(1)
(2)
Possible
Future State
of the
Probability
Economy
(3)
(4)
Possible
Returns on
the Market (4) = (2)*(3)
Boom
25.0%
20.0%
Normal
50.0%
11.0%
Recession
25.0%
-4.0%
Expected return on the Market =
0.05
0.055
-0.01
9.5%
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 74
Measuring Variances, Standard Deviations of
the Forecast Stock Returns
Using the expected return, calculate the deviations away from the mean, square those
deviations and then weight the squared deviations by the probability of their
occurrence. Add up the weighted and squared deviations from the mean and you
have found the variance!
(1)
(2)
Possible
Future State
of the
Probability
Economy
Boom
Normal
Recession
25.0%
50.0%
25.0%
(3)
(4)
Possible
Returns on
the Stock (4) = (2)*(3)
0.28
0.17
-0.14
Expected return (stock) =
(5)
Deviations
(6)
(7)
Squared
Deviations
Weighted
and
Squared
Deviations
0.16
0.0256
0.05
0.0025
-0.26
0.0676
12.0% Variance (stock)=
Standard Deviation (stock) =
0.07
0.085
-0.035
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
0.0064
0.00125
0.0169
0.02455
15.67%
9 - 75
Measuring Variances, Standard Deviations of
the Forecast Market Returns
Now do this for the possible returns on the market
(1)
(2)
Possible
Future State
of the
Probability
Economy
Boom
Normal
Recession
25.0%
50.0%
25.0%
(3)
(4)
Possible
Returns on
the Market (4) = (2)*(3)
0.2
0.11
-0.04
Expected return (market) =
(5)
Deviations
(6)
(7)
Squared
Deviations
Weighted
and
Squared
Deviations
0.105 0.011025
0.015 0.000225
-0.135 0.018225
9.5% Variance (market) =
Standard Deviation (market)=
0.05
0.055
-0.01
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
0.002756
0.000113
0.004556
0.007425
8.62%
9 - 76
Covariance
From Chapter 8 you know the formula for the covariance
between the returns on the stock and the returns on the
market is:
n
[8-12]
_
_
COV AB   Prob i (k A,i  ki )( k B ,i - k B )
i 1
Covariance is an absolute measure of the degree of ‘comovement’ of returns.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 77
Correlation Coefficient
Correlation is covariance normalized by the product of the standard
deviations of both securities. It is a ‘relative measure’ of co-movement of
returns on a scale from -1 to +1.
The formula for the correlation coefficient between the returns on the stock
and the returns on the market is:
[8-13]
 AB 
COV AB
 A B
The correlation coefficient will always have a value in the range of +1 to -1.
+1 – is perfect positive correlation (there is no diversification potential when combining these two
securities together in a two-asset portfolio.)
- 1 - is perfect negative correlation (there should be a relative weighting mix of these two
securities in a two-asset portfolio that will eliminate all portfolio risk)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 78
Measuring Covariance
from Ex Ante Return Data
Using the expected return (mean return) and given data measure the
deviations for both the market and the stock and multiply them
together with the probability of occurrence…then add the products
up.
(1)
(2)
(3)
Possible
Future
State of the
Economy
Prob.
Possible
Returns
on the
Stock
Boom
25.0%
Normal
50.0%
Recession 25.0%
28.0%
17.0%
-14.0%
E(kstock) =
(4)
(4) =
(2)*(3)
0.07
0.085
-0.035
12.0%
(5)
(6)
(7)
Deviations
Possible
from the
Returns on
mean for
the Market (6)=(2)*(5) the stock
20.0%
11.0%
-4.0%
E(kmarket ) =
0.05
0.055
-0.01
9.5%
(8)
"(9)
Deviations
from the
mean for
the market
(8)=(2)(6)(7)
16.0%
10.5%
5.0%
1.5%
-26.0%
-13.5%
Covariance =
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
0.0042
0.000375
0.008775
0.01335
9 - 79
The Beta Measured
Using Ex Ante Covariance (stock, market) and Market Variance
Now you can substitute the values for covariance and the
variance of the returns on the market to find the beta of the
stock:
Beta 
CovS, M
VarM

.01335
 1.8
.007425
• A beta that is greater than 1 means that the investment is aggressive…its
returns are more volatile than the market as a whole.
• If the market returns were expected to go up by 10%, then the stock
returns are expected to rise by 18%. If the market returns are expected
to fall by 10%, then the stock returns are expected to fall by 18%.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 80
Lets Prove the Beta of the Market is 1.0
Let us assume we are comparing the possible market returns
against itself…what will the beta be?
(1)
(2)
(3)
Possible
Future
State of the
Economy
Prob.
Possible
Returns
on the
Market
Boom
25.0%
Normal
50.0%
Recession 25.0%
20.0%
11.0%
-4.0%
E(kM) =
(4)
(5)
(6)
Possible
Returns
`M, M
on the
Market
(6)=(2)*(5)
M
Cov
Beta (4) =
Var
(2)*(3)
0.05
0.055
-0.01
(6)
(7)
(8)
Deviations
from the
mean for
the stock
Deviations
from the
mean for
the market
(8)=(2)(6)(7
)
.007425

 1.0
.007425
20.0%
11.0%
-4.0%
9.5% E(kM) =
0.05
0.055
-0.01
9.5%
10.5%
10.5%
1.5%
1.5%
-13.5%
-13.5%
Covariance =
0.002756
0.000113
0.004556
0.007425
Since the variance of the returns on the market is = .007425 …the beta for
the market is indeed equal to 1.0 !!!
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 81
Proving the Beta of Market = 1
If you now place the covariance of the market with itself
value in the beta formula you get:
Cov MM
.007425
Beta 

 1 .0
Var(R M ) .007425
The beta coefficient of the market will always be
1.0 because you are measuring the market returns
against market returns.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 82
How Do We use Expected and Required Rates
of Return?
Once you have estimated the expected and required rates of return, you can
plot them on the SML and see if the stock is under or overpriced.
% Return
E(Rs) = 5.0%
R(ks) = 4.76%
SML
E(kM)= 4.2%
Risk-free Rate = 3%
BM= 1.0
Bs = 1.464
Since E(r)>R(r) the stock is underpriced.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 83
How Do We use Expected and Required Rates
of Return?
•
•
The stock is fairly priced if the expected return = the required return.
This is what we would expect to see ‘normally’ or most of the time in an efficient market
where securities are properly priced.
% Return
E(Rs) = R(Rs) 4.76%
SML
E(RM)= 4.2%
Risk-free Rate = 3%
BM= 1.0
BS = 1.464
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 84
Use of the Forecast Beta
•
We can use the forecast beta, together with an estimate of the riskfree rate and the market premium for risk to calculate the investor’s
required return on the stock using the CAPM:
Required Return  RF  βi [E(k M )  RF]
•
This is a ‘market-determined’ return based on the current risk-free
rate (RF) as measured by the 91-day, government of Canada T-bill
yield, and a current estimate of the market premium for risk (kM – RF)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 85
Conclusions
• Analysts can make estimates or forecasts for the returns
on stock and returns on the market portfolio.
• Those forecasts can be analyzed to estimate the beta
coefficient for the stock.
• The required return on a stock can then be calculated
using the CAPM – but you will need the stock’s beta
coefficient, the expected return on the market portfolio
and the risk-free rate.
• The required return is then using in Dividend Discount
Models to estimate the ‘intrinsic value’ (inherent worth)
of the stock.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 86
Calculating the Beta using Trailing
Holding Period Returns
APPENDIX 2
The Regression Approach to Measuring the
Beta
•
You need to gather historical data about the stock and the market
•
You can use annual data, monthly data, weekly data or daily data.
However, monthly holding period returns are most commonly used.
• Daily data is too ‘noisy’ (short-term random volatility)
• Annual data will extend too far back in to time
•
You need at least thirty (30) observations of historical data.
•
Hopefully, the period over which you study the historical returns of the
stock is representative of the normal condition of the firm and its
relationship to the market.
•
If the firm has changed fundamentally since these data were produced
(for example, the firm may have merged with another firm or have
divested itself of a major subsidiary) there is good reason to believe
that future returns will not reflect the past…and this approach to beta
estimation SHOULD NOT be used….rather, use the ex ante approach.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 88
Historical Beta Estimation
The Approach Used to Create the Characteristic Line
In this example, we have regressed the quarterly returns on the stock against the
quarterly returns of a surrogate for the market (TSE 300 total return composite
index) and then using Excel…used the charting feature to plot the historical
points and add a regression trend line.
2005.1
19.0%
7.0%
2004.4
-16.0%
-4.0%
The regression line is a line of ‘best
2004.3
8.0%
16.0%
fit’ that describes
the inherent
2004.2
relationship-3.0%
between the-11.0%
returns on
2004.1
34.0%
25.0%
the stock and the returns on the
market. The slope is the beta
coefficient.
Ch a r a c te r istic L in e (Re gr e ssio n )
30.0%
25.0%
Returns on Stock
The ‘cloud’ of plotted points
Period
HPR(Stock)
HPR(TSE
300)
represents
‘diversifiable
or company
2006.4
1.2%
specific’ risk-4.0%
in the securities
returns
2006.3
-16.0%
that can be eliminated
from-7.0%
a portfolio
through diversification.
Since
2006.2
32.0%
12.0%
risk can
2006.1company-specific
16.0%
8.0%be
eliminated,-22.0%
investors don’t
require
2005.4
-11.0%
compensation
2005.3
15.0%for it according
16.0% to
Portfolio Theory.
2005.2Markowitz
28.0%
13.0%
20.0%
15.0%
10.0%
5.0%
-40.0%
0.0%
-20.0% -5.0%0.0%
20.0%
40.0%
-10.0%
-15.0%
Returns on TSE 300
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 89
Characteristic Line
•
The characteristic line is a regression line that represents the
relationship between the returns on the stock and the returns on the
market over a past period of time. (It will be used to forecast the
future, assuming the future will be similar to the past.)
•
The slope of the Characteristic Line is the Beta Coefficient.
•
The degree to which the characteristic line explains the variability in
the dependent variable (returns on the stock) is measured by the
coefficient of determination. (also known as the R2 (r-squared or
coefficient of determination)).
•
If the coefficient of determination equals 1.00, this would mean that
all of the points of observation would lie on the line. This would mean
that the characteristic line would explain 100% of the variability of
the dependent variable.
•
The alpha is the vertical intercept of the regression (characteristic
line). Many stock analysts search out stocks with high alphas.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 90
Low R2
• An R2 that approaches 0.00 (or 0%) indicates that the
characteristic (regression) line explains virtually none of the
variability in the dependent variable.
• This means that virtually of the risk of the security is
‘company-specific’.
• This also means that the regression model has virtually no
predictive ability.
• In this case, you should use other approaches to value the
stock…do not use the estimated beta coefficient.
(See the following slide for an illustration of a low r-square)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 91
Characteristic Line for Imperial Tobacco
An Example of Volatility that is Primarily Company-Specific
Returns on
Imperial
Tobacco %
Characteristic
Line for Imperial
Tobacco
• High alpha
• R-square is very
low ≈ 0.02
• Beta is largely
irrelevant
Returns on
the Market %
(S&P TSX)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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High R2
• An R2 that approaches 1.00 (or 100%) indicates that the
characteristic (regression) line explains virtually all of the
variability in the dependent variable.
• This means that virtually of the risk of the security is
‘systematic’.
• This also means that the regression model has a strong
predictive ability. … if you can predict what the market will
do…then you can predict the returns on the stock itself with a
great deal of accuracy.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Characteristic Line General Motors
A Positive Beta with Predictive Power
Returns on
General
Motors %
Characteristic
Line for GM
(high R2)
• Positive alpha
• R-square is
very high ≈ 0.9
• Beta is positive
and close to 1.0
Returns on
the Market %
(S&P TSX)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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An Unusual Characteristic Line
A Negative Beta with Predictive Power
Returns on a
Stock %
Characteristic Line for a stock
that will provide excellent
portfolio diversification
• Positive alpha
(high R2)
• R-square is
very high
• Beta is negative
<0.0 and > -1.0
Returns on
the Market %
(S&P TSX)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 95
Diversifiable Risk
(Non-systematic Risk)
• Volatility in a security’s returns caused by companyspecific factors (both positive and negative) such as:
– a single company strike
– a spectacular innovation discovered through the company’s R&D
program
– equipment failure for that one company
– management competence or management incompetence for that
particular firm
– a jet carrying the senior management team of the firm crashes (this
could be either a positive or negative event, depending on the
competence of the management team)
– the patented formula for a new drug discovered by the firm.
• Obviously, diversifiable risk is that unique factor that
influences only the one firm.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 96
OK – lets go back and look at raw data
gathering and data normalization
• A common source for stock of information is Yahoo.com
• You will also need to go to the library a use the TSX Review (a
monthly periodical) – to obtain:
– Number of shares outstanding for the firm each month
– Ending values for the total return composite index (surrogate for the
market)
• You want data for at least 30 months.
• For each month you will need:
–
–
–
–
Ending stock price
Number of shares outstanding for the stock
Dividend per share paid during the month for the stock
Ending value of the market indicator series you plan to use (ie. TSE
300 total return composite index)
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Demonstration Through Example
The following slides will be based on
Alcan Aluminum (AL.TO)
Five Year Stock Price Chart for AL.TO
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 99
Spreadsheet Data From Yahoo
Process:
– Go to http://ca.finance.yahoo.com
– Use the symbol lookup function to search for the
company you are interested in studying.
– Use the historical quotes button…and get 30 months
of historical data.
– Use the download in spreadsheet format feature to
save the data to your hard drive.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 100
Spreadsheet Data From Yahoo
Alcan Example
The raw downloaded data should look like this:
Date
Open
High
Low
Close
Volume
01-May-02
57.46
62.39
56.61
59.22
753874
01-Apr-02
62.9
63.61
56.25
57.9
879210
01-Mar-02
64.9
66.81
61.68
63.03
974368
01-Feb-02
61.65
65.67
58.75
64.86
836373
02-Jan-02
57.15
62.37
54.93
61.85
989030
03-Dec-01
56.6
60.49
55.2
57.15
833280
01-Nov-01
49
58.02
47.08
56.69
779509
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Spreadsheet Data From Yahoo
Alcan Example
The raw downloaded data should look like this:
Date
01-May-02
01-Apr-02
The day,
month and
year
Open
57.46
62.9
High
62.39
63.61
Low
56.61
56.25
Close
Volume
59.22
753874
57.9
879210
Opening price per share, the
highest price per share during the
month, the lowest price per share
achieved during the month and the
closing price per share at the end
CHAPTER
9 – The
Capital Asset Pricing Model (CAPM)
of the
month
Volume of
trading done
in the stock
on the TSE in
the month in
numbers of
- 102
board 9lots
Spreadsheet Data From Yahoo
Alcan Example
From Yahoo, the only information you can use is the closing
price per share and the date. Just delete the other
columns.
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
Close
59.22
57.9
63.03
64.86
61.85
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Acquiring the Additional Information You Need
Alcan Example
In addition to the closing price of the stock on a per share basis,
you will need to find out how many shares were outstanding at
the end of the month and whether any dividends were paid
during the month.
You will also want to find the end-of-the-month value of the
S&P/TSX Total Return Composite Index (look in the green
pages of the TSX Review)
You can find all of this in The TSX Review periodical.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Raw Company Data
Alcan Example
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
Issued
Capital
321,400,589
321,400,589
321,400,589
321,400,589
160,700,295
160,700,295
Cash
Closing Price
Dividends
for Alcan
per Share
AL.TO
$0.00
$59.22
$0.15
$57.90
$0.00
$63.03
$0.00
$64.86
$0.30
$123.70
$0.00
$119.30
Number of shares doubled and share price fell by half between
January and February 2002 – this is indicative of a 2 for 1 stock split.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Normalizing the Raw Company Data
Alcan Example
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
Issued
Capital
321,400,589
321,400,589
321,400,589
321,400,589
160,700,295
145,000,500
Closing
Price for
Cash
Alcan
Dividends Adjustment
AL.TO
per Share
Factor
$59.22
$0.00
1.00
$57.90
$0.15
1.00
$63.03
$0.00
1.00
$64.86
$0.00
1.00
$123.70
$0.30
0.50
$111.40
$0.00
0.45
Normalized Normalized
Stock Price
Dividend
$59.22
$0.00
$57.90
$0.15
$63.03
$0.00
$64.86
$0.00
$61.85
$0.15
$50.26
$0.00
The adjustment factor is just the value in the issued
capital cell divided by 321,400,589.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Calculating the HPR on the stock from the
Normalized Data
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
Normalized
Stock Price
$59.22
$57.90
$63.03
$64.86
$61.85
$50.26
Normalized
Dividend
$0.00
$0.15
$0.00
$0.00
$0.15
$0.00
HPR 
HPR
2.28%
-7.90%
-2.82%
4.87%
23.36%
( P1  P0 )  D1
P0
$59.22 - $57.90  $0.00
$57.90
 2.28%

Use $59.22 as the ending price, $57.90 as the
beginning price and during the month of May, no
dividend was declared.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Now Put the data from the S&P/TSX Total
Return Composite Index in
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
Normalized Normalized
Stock Price
Dividend
$59.22
$0.00
$57.90
$0.15
$63.03
$0.00
$64.86
$0.00
$61.85
$0.15
$50.26
$0.00
HPR
2.28%
-7.90%
-2.82%
4.87%
23.36%
Ending
TSX
Value
16911.33
16903.36
17308.41
16801.82
16908.11
16881.75
You will find the Total Return S&P/TSX Composite
Index values in TSX Review found in the library.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Now Calculate the HPR on the Market Index
HPR 
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
( P1  P0 )
P0
16,911.33 - 16,903.36

16,903.36
 0.05%
Normalized Normalized
Stock Price
Dividend
$59.22
$0.00
$57.90
$0.15
$63.03
$0.00
$64.86
$0.00
$61.85
$0.15
$50.26
$0.00
HPR
2.28%
-7.90%
-2.82%
4.87%
23.36%
Ending
TSX
HPR on
Value the TSX
16911.33
0.05%
16903.36
-2.34%
17308.41
3.02%
16801.82
-0.63%
16908.11
0.16%
16881.75
Again, you simply use the HPR formula using the
ending values for the total return composite index.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Regression In Excel
• If you haven’t already…go to the tools
menu…down to add-ins and check off the VBA
Analysis Pac
• When you go back to the tools menu, you should
now find the Data Analysis bar, under that find
regression, define your dependent and
independent variable ranges, your output range
and run the regression.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 110
Regression
Defining the Data Ranges
Date
01-May-02
01-Apr-02
01-Mar-02
01-Feb-02
02-Jan-02
01-Dec-01
Normalized Normalized
Stock Price
Dividend
$59.22
$0.00
$57.90
$0.15
$63.03
$0.00
$64.86
$0.00
$61.85
$0.15
$50.26
$0.00
HPR
2.28%
-7.90%
-2.82%
4.87%
23.36%
Ending
TSX
HPR on
Value the TSX
16911.33
0.05%
16903.36
-2.34%
17308.41
3.02%
16801.82
-0.63%
16908.11
0.16%
16881.75
The dependent
independentvariable
variableis isthe
thereturns
returnsonon
the
the
Stock.
Market.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 111
Now Use the Regression Function in Excel to
regress the returns of the stock against the
returns of the market
SUMMARY OUTPUT
R-square is the
coefficient of
determination =
0.0028=.3%
Regression Statistics
Multiple R
0.05300947
R Square
0.00281
Adjusted R Square
-0.2464875
Standard Error
5.79609628
Observations
6
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
Beta
Coefficient is
the XVariable 1
SS
MS
1 0.3786694 0.37866937
4 134.37893 33.5947321
5
134.7576
F
Significance F
0.011271689 0.920560274
CoefficientsStandard Error t Stat
59.3420816 2.8980481 20.4765686
3.55278937 33.463777 0.10616821
P-value
Lower 95%
3.3593E-05 51.29579335
0.920560274 -89.35774428
Upper 95% Lower 95.0%Upper 95.0%
67.38836984 51.2957934 67.38837
96.46332302 -89.3577443 96.46332
The alpha is the
vertical intercept.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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Finalize Your Chart
Alcan Example
• You can use the charting feature in Excel to create a
scatter plot of the points and to put a line of best fit (the
characteristic line) through the points.
• In Excel, you can edit the chart after it is created by
placing the cursor over the chart and ‘right-clicking’
your mouse.
• In this edit mode, you can ask it to add a trendline
(regression line)
• Finally, you will want to interpret the Beta (X-coefficient)
the alpha (vertical intercept) and the coefficient of
determination.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
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The Beta
Alcan Example
• Obviously the beta (X-coefficient) can simply be
read from the regression output.
– In this case it was 3.56 making Alcan’s returns more
than 3 times as volatile as the market as a whole.
– Of course, in this simple example with only 5
observations, you wouldn’t want to draw any serious
conclusions from this estimate.
CHAPTER 9 – The Capital Asset Pricing Model (CAPM)
9 - 114
Copyright
Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights
reserved. Reproduction or translation of this work beyond that
permitted by Access Copyright (the Canadian copyright licensing
agency) is unlawful. Requests for further information should be
addressed to the Permissions Department, John Wiley & Sons
Canada, Ltd. The purchaser may make back-up copies for his or her
own use only and not for distribution or resale. The author and the
publisher assume no responsibility for errors, omissions, or
damages caused by the use of these files or programs or from the
use of the information contained herein.
CHAPTER 19 – Equity and Hybrid Instruments
19 - 115