The capital-asset-pricing-model-capm75

Prepared by
Ken Hartviksen
INTRODUCTION TO
CORPORATE FINANCE
Laurence Booth • W. Sean Cleary
Chapter 9 – The Capital Asset Pricing
Model

CHAPTER 9
The Capital Asset Pricing
Model (CAPM)

CHAPTER 9 – The Capital Asset Pricing Model (CAPM) 9 - 3
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

Learning Objectives
1. What happens if all investors are rational and risk averse.
2. How modern portfolio theory is extended to develop the capital
market line, which determines how expected returns on
portfolios are determined.
3. How to assess the performance of mutual fund managers
4. 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 risk-based
pricing models.

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

Achievable Portfolio Combinations
The Capital Asset Pricing Model (CAPM)

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
Expected Return Standard Deviation
Asset A 8% 8.72%
Asset B 10% 22.69%
The following table shows the portfolio characteristics for 100
different weighting schemes for just these two securities:

Example of Portfolio Combinations and
Correlation
Asset
Expected
Return
Standard
Deviation
Correlation
Coefficient
A 8.0% 8.7% -0.379
B 10.0% 22.7%
Weight of A Weight of B
Expected
Return
Standard
Deviation
100% 0% 8.00% 8.7%
99% 1% 8.02% 8.5%
98% 2% 8.04% 8.4%
97% 3% 8.06% 8.2%
96% 4% 8.08% 8.1%
95% 5% 8.10% 7.9%
94% 6% 8.12% 7.8%
93% 7% 8.14% 7.7%
92% 8% 8.16% 7.5%
91% 9% 8.18% 7.4%
90% 10% 8.20% 7.3%
89% 11% 8.22% 7.2%
Portfolio Components Portfolio Characteristics
The first
combination
simply
assumes
you invest
solely in
Asset A
The second
portfolio
assumes 99%
in A and 1% in
B. Notice the
increase in
return and the
decrease in
portfolio risk!
You repeat this
procedure
down until you
have determine
the portfolio
characteristics
for all 100
portfolios.
Next plot the
returns on a
graph (see the
next slide)

Example of Portfolio Combinations and
Correlation
Attainable Portfolio Combinations for a
Two Asset Portfolio
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Standard Deviation of Returns
ExpectedReturnofthe
Portfolio

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)

Efficient Frontier
The Two-Asset Portfolio Combinations
A is not attainable
B,E lie on the
efficient frontier and
are attainable
E is the minimum
variance portfolio
(lowest risk
combination)
C, D are
attainable but are
dominated by
superior portfolios
that line on the line
above E
8 - 10 FIGURE
ExpectedReturn%
Standard Deviation (%)
A
E
B
C
D

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)

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:

The First Ten Combinations Created
Portfolio Risk (σp)
10 Achievable
Risky Portfolio
Combinations
ERp

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:

Thirty Combinations Naively Created
30 Risky Portfolio
Combinations
ERp

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:

ERp
More Possible Combinations Created
E
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.

The Efficient Frontier

Achievable Set of
Risky Portfolio
Combinations
ERp
Efficient Frontier (Set)
E
Efficient
frontier is the
set of
achievable
portfolio
combinations
that offer the
highest rate
of return for a
given level of
risk.

The New Efficient Frontier
Efficient Portfolios
Figure 9 – 1
illustrates
three
achievable
portfolio
combinations
that are
‘efficient’ (no
other
achievable
portfolio that
offers the
same risk,
offers a higher
return.)
Risk
9 - 1 FIGURE
Efficient Frontier
ER
MVP
A
B

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).

The New Efficient Frontier and
Separation Theorem

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)

Risk-Free Investing
– Expected return on a two asset portfolio made up of
risky asset A and RF:
The possible combinations of A and RF are found graphed on the following slide.
RF)-(ERRFER Ap w[9-1]

Attainable Portfolios Using RF and A
9 - 2 FIGURE
Risk
ER
RF
A
Ap  w[9-2]
Equation 9 – 2
illustrates
what you can
see…portfolio
risk increases
in direct
proportion to
the amount
invested in the
risky asset.
RF-)E(R
RFER
A
A
PP 
 





[9-3]
Rearranging 9
-2 where w=σ
p / σA and
substituting in
Equation 1 we
get an
equation for a
straight line
with a
constant
slope.
This means
you can
achieve any
portfolio
combination
along the blue
coloured line
simply by
changing the
relative weight
of RF and A in
the two asset
portfolio.

Attainable Portfolios using the RF and A, and RF and T
Which risky
portfolio
would a
rational risk-
averse
investor
choose in the
presence of a
RF
investment?
Portfolio A?
Tangent
Portfolio T?
9 - 3 FIGURE
Risk
ER
RF
A
T

Efficient Portfolios using the Tangent Portfolio T
9 - 3 FIGURE
Risk
ER
RF
A
T
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!

Lending Portfolios
9 - 3 FIGURE
Risk
ER
RF
A
T
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).
Lending Portfolios

Borrowing Portfolios
9 - 3 FIGURE
Risk
ER
RF
A
T
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.
Lending Portfolios Borrowing Portfolios

9 - 4 FIGURE
σρ
ER
RF
A2
T
A
B
B2
Capital Market Line
The New (Super) Efficient Frontier
The optimal
risky portfolio
(the market
portfolio ‘M’)
Clearly RF with
T (the market
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!
This is now
called the new
(or super)
efficient frontier
of risky
portfolios.
Investors can
achieve any
one of these
portfolio
combinations
by borrowing or
investing in RF
in combination
with the market
portfolio.

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)

σρ
ER
RF
M
CML
The Capital Market Line
The optimal
risky portfolio
(the market
portfolio ‘M’)
The CML is that
set of superior
portfolio
combinations
that are
achievable in
the presence of
the equilibrium
condition.

The Capital Asset Pricing Model
The Hypothesized Relationship between
Risk and Return

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.

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)
Determine Investment’s
Required Return
Estimate the
Investment’s Intrinsic
Value
Compare to the actual
stock price in the market
2i
M
i,M
σ
COV
 )( iMi RFERRFk 
gk
D
P
c 
 1
0
Is the stock
fairly priced?

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.

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’)

9 - 5 FIGURE
σρ
ER
RF
MERM
σM
P
M
M
P
RFER
RFk 
 




 

CML
The CML is that
set of achievable
portfolio
combinations
that are possible
when investing
in only two
assets (the
market portfolio
and the risk-free
asset (RF).
The market
portfolio is the
optimal risky
portfolio, it
contains all risky
securities and
lies tangent (T)
on the efficient
frontier.
The CML has
standard
deviation of
portfolio returns
as the
independent
variable.

The Market Portfolio and the Capital Market Line (CML)
– The slope of the CML is the incremental expected
return divided by the incremental risk.
– This is called the market price for risk. Or
– The equilibrium price of risk in the capital market.
RF-ER
CMLtheofSlope
M
M

[9-4]

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 :
– 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
)(
σ
- RFER
RFRE P
M
M
P 





[9-5]

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.

Expected and Required Rates of Return
A is an
undervalued
portfolio. Expected
return is greater
than the required
return.
Demand for
Portfolio A will
increase driving up
the price, and
therefore the
expected return will
fall until expected
equals required
(market equilibrium
condition is
achieved.)
Required
return on A
Expected
return on A
B is a portfolio that
offers and expected
return equal to the
required return.
9 - 6 FIGURE
σρ
ER
RF
B
C
A
CML
C is an overvalued
portfolio. Expected
return is less than
the required return.
Selling pressure will
cause the price to
fall and the yield to
rise until expected
equals the required
return.
Required
Return on C
Expected
Return on C

Risk-Adjusted Performance and the Sharpe Ratios
– William Sharpe identified a ratio that can be used to assess the risk-
adjusted performance of managed funds (such as mutual funds and
pension plans).
– It is called the Sharpe ratio:
– 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.
RF-ER
ratioSharpe
P
P

[9-6]

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.

Income Trust Estimated Values
Return σP Sharpe β
Median income trusts 25.83% 18.66% 1.37 0.22
Equally weighted trust portfolio 29.97% 8.02% 3.44 0.28
S&P/TSX Composite Index 8.97% 13.31% 0.49 1.00
Scotia Capital government bond index 9.55% 6.57% 1.08 20.02
Table 9-1 Income Trusts Estimated Values
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.

CAPM and Market Risk

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.

The CAPM and Market Risk
Portfolio Risk and Diversification
9 - 7 FIGURE
Number of Securities
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.

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.

Measuring Systematic Risk
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.

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)

The Characteristic Line for Security A
9 - 8 FIGURE
6
4
2
0
-2
-4
-6
Security A Returns (%)
-6 -4 -2 0 2 4 6 8
MarketReturns(%)
The slope of
the regression
line is beta.
The line of
best fit is
known in
finance as the
characteristic
line.
The plotted
points are the
coincident
rates of return
earned on the
investment
and the market
portfolio over
past periods.

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:
,
2i
M
iMi
M
i,M
σ
COV


 [9-7]

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

Canadian BETAS
Selected
Company Industry Classification Beta
Abitibi Consolidated Inc. Materials - Paper & Forest 1.37
Algoma Steel Inc. Materials - Steel 1.92
Bank of Montreal Financials - Banks 0.50
Bank of Nova Scotia Financials - Banks 0.54
Barrick Gold Corp. Materials - Precious Metals & Minerals 0.74
BCE Inc. Communications - Telecommunications 0.39
Bema Gold Corp. Materials - Precious Metals & Minerals 0.26
CIBC Financials - Banks 0.66
Cogeco Cable Inc. Consumer Discretionary - Cable 0.67
Gammon Lake Resources Inc. Materials - Precious Metals & Minerals 2.52
Imperial Oil Ltd. Energy - Oil & Gas: Integrated Oils 0.80
Table 9-2 Canadian BETAS
Source: Research Insight, Compustat North American database, June 2006.

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.
Weights of individual assets are found by dividing the value of
the investment by the value of the total portfolio.
... nnBBAAP www  [9-8]

The Security Market Line

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:
– 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)
)( iMi RFERRFk [9-9]

The Security Market Line (SML)
9 - 9 FIGURE
βM = 1
ER
RF
β
M
ERM
iMi RFERRFk )( 
The SML is
used to
predict
required
returns for
individual
securities
The SML
uses the
beta
coefficient as
the measure
of relevant
risk.

9 - 10 FIGURE
βA
ER
RF
β
B
A
βB
SML
The SML and Security Valuation
iMi RFERRFk )( 
Required returns
are forecast using
this equation.
You can see that
the required
return on any
security is a
function of its
systematic risk (β)
and market
factors (RF and
market premium
for risk)
A is an
undervalued
security because
its expected
return is greater
than the required
return.
Investors will
‘flock’ to A and bid
up the price
causing expected
return to fall till it
equals the
required return.
Required
Return A
Expected
Return A
Similarly, B is an
overvalued
security.
Investor’s will sell
to lock in gains,
but the selling
pressure will
cause the market
price to fall,
causing the
expected return to
rise until it equals
the required
return.

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.

Alternative Pricing Models

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)

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.

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.
– 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.
...11110 niniii FbFbFbaER [9-10]

The Arbitrage Pricing Theory – the Model
– Underlying factors represent broad economic forces which are
inherently unpredictable.
– 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.
...11110 niniii FbFbFbaER [9-10]

The Arbitrage Pricing Theory – Challenges
– Underlying factors represent broad economic forces
which are inherently unpredictable.
– Ross and Roll identify five systematic factors:
1. Changes in expected inflation
2. Unanticipated changes in inflation
3. Unanticipated changes in industrial production
4. Unanticipated changes in the default-risk premium
5. 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.

Summary and Conclusions
In this chapter you have learned:
– How the efficient frontier can be expanded by introducing risk-
free 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 Fama-
French Model and the Arbitrage Pricing Theory.

Concept Review Questions

Concept Review Question 1
Risk Aversion
What is risk aversion and how do we know investors
are risk averse?

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.

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

• Under the theory of the Capital Asset Pricing Model total risk is
partitioned into two parts:
– Systematic risk
– Unsystematic risk – diversifiable risk
• Systematic risk is non-diversifiable risk.
• Systematic risk is the only relevant risk to the diversified
investor
• The beta coefficient measures systematic risk
Systematic Risk Unsystematic Risk
Total Risk of the Investment

The Formula
ReturnsMarkettheofVariance
markettheandreturnsi''stockbetweenReturnsofCovariance
Beta 
,
2i
M
iMi
M
i,M
σ
COV


 [9-7]

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

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.

The Formula Approach to Measuring the Beta
)Var(k
)kCov(k
Beta
M
Mi

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

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
Possible
Returns on
the Stock
Possible
Returns on
the Market
Boom 25.0% 28.0% 20.0%
Normal 50.0% 17.0% 11.0%
Recession 25.0% -14.0% -4.0%
By observation
you can see the
range is much
greater for the
stock than the
market and they
move in the
same direction.
Since the beta
relates the stock
returns to the
market returns,
the greater range
of stock returns
changing in the
same direction as
the market
indicates the beta
will be greater
than 1 and will be
positive.
(Positively
correlated to the
market returns.)

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
Possible
Returns on
the Stock
Possible
Returns on
the Market
Boom 25.0% 28.0% 20.0%
Normal 50.0% 17.0% 11.0%
Recession 25.0% -14.0% -4.0%
100.0%

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) (3) (4)
Possible
Future State
of the
Economy Probability
Possible
Returns on
the Stock (4) = (2)*(3)
Boom 25.0% 28.0% 0.07
Normal 50.0% 17.0% 0.085
Recession 25.0% -14.0% -0.035
Expected return on the Stock = 12.0%

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) (3) (4)
Possible
Future State
of the
Economy Probability
Possible
Returns on
the Market (4) = (2)*(3)
Boom 25.0% 20.0% 0.05
Normal 50.0% 11.0% 0.055
Recession 25.0% -4.0% -0.01
Expected return on the Market = 9.5%

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) (3) (4) (5) (6) (7)
Possible
Future State
of the
Economy Probability
Possible
Returns on
the Stock (4) = (2)*(3) Deviations
Squared
Deviations
Weighted
and
Squared
Deviations
Boom 25.0% 0.28 0.07 0.16 0.0256 0.0064
Normal 50.0% 0.17 0.085 0.05 0.0025 0.00125
Recession 25.0% -0.14 -0.035 -0.26 0.0676 0.0169
Expected return (stock) = 12.0% Variance (stock)= 0.02455
Standard Deviation (stock) = 15.67%

Measuring Variances, Standard Deviations of
the Forecast Market Returns
Now do this for the possible returns on the market
(1) (2) (3) (4) (5) (6) (7)
Possible
Future State
of the
Economy Probability
Possible
Returns on
the Market (4) = (2)*(3) Deviations
Squared
Deviations
Weighted
and
Squared
Deviations
Boom 25.0% 0.2 0.05 0.105 0.011025 0.002756
Normal 50.0% 0.11 0.055 0.015 0.000225 0.000113
Recession 25.0% -0.04 -0.01 -0.135 0.018225 0.004556
Expected return (market) = 9.5% Variance (market) = 0.007425
Standard Deviation (market)= 8.62%

Covariance
From Chapter 8 you know the formula for the covariance between
the returns on the stock and the returns on the market is:
Covariance is an absolute measure of the degree of ‘co-movement’
of returns.
)-)((Prob
_
,
1
_
,i BiB
n
i
iiAAB kkkkCOV 
[8-12]

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:
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)
BA
AB
AB
COV

 [8-13]

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) (4) (5) (6) (7) (8) "(9)
Possible
Future
State of the
Economy Prob.
Possible
Returns
on the
Stock
(4) =
(2)*(3)
Possible
Returns on
the Market (6)=(2)*(5)
Deviations
from the
mean for
the stock
Deviations
from the
mean for
the market (8)=(2)(6)(7)
Boom 25.0% 28.0% 0.07 20.0% 0.05 16.0% 10.5% 0.0042
Normal 50.0% 17.0% 0.085 11.0% 0.055 5.0% 1.5% 0.000375
Recession 25.0% -14.0% -0.035 -4.0% -0.01 -26.0% -13.5% 0.008775
E(kstock) = 12.0% E(kmarket) = 9.5% Covariance = 0.01335

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:
8.1
007425.
01335.
Var
Cov
Beta
M
MS,

• 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%.

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) (4) (5) (6) (6) (7) (8)
Possible
Future
State of the
Economy Prob.
Possible
Returns
on the
Market
(4) =
(2)*(3)
Possible
Returns
on the
Market (6)=(2)*(5)
Deviations
from the
mean for
the stock
Deviations
from the
mean for
the market
(8)=(2)(6)(7
)
Boom 25.0% 20.0% 0.05 20.0% 0.05 10.5% 10.5% 0.002756
Normal 50.0% 11.0% 0.055 11.0% 0.055 1.5% 1.5% 0.000113
Recession 25.0% -4.0% -0.01 -4.0% -0.01 -13.5% -13.5% 0.004556
E(kM) = 9.5% E(kM) = 9.5% Covariance = 0.007425
Since the variance of the returns on the market is = .007425 …the beta for
the market is indeed equal to 1.0 !!!
0.1
007425.
007425.
Var
Cov
Beta
M
M`M,


Proving the Beta of Market = 1
If you now place the covariance of the market with itself value in
the beta formula you get:
0.1
007425.
007425.
)Var(R
Cov
Beta
M
MM

The beta coefficient of the market will always be
1.0 because you are measuring the market returns
against market returns.

Using the Security Market Line
Expected versus Required Return

How Do We use Expected and Required Rates
of Return?
% Return
Risk-free Rate = 3%
BM= 1.0
E(kM)= 4.2%
Bs = 1.464
R(ks) = 4.76%
E(Rs) = 5.0%
SML
Since E(r)>R(r) the stock is underpriced.
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.

How Do We use Expected and Required Rates
of Return?
% Return
Risk-free Rate = 3%
BM= 1.0
E(RM)= 4.2%
BS = 1.464
E(Rs) = R(Rs) 4.76%
SML
• 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.

Use of the Forecast Beta
• We can use the forecast beta, together with an estimate of the risk-free rate
and the market premium for risk to calculate the investor’s required return
on the stock using the CAPM:
• 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)
RF]k[EβRF Mi  )(ReturnRequired

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.

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.

Historical Beta Estimation
The Approach Used to Create the Characteristic Line
Period HPR(Stock) HPR(TSE 300)
2006.4 -4.0% 1.2%
2006.3 -16.0% -7.0%
2006.2 32.0% 12.0%
2006.1 16.0% 8.0%
2005.4 -22.0% -11.0%
2005.3 15.0% 16.0%
2005.2 28.0% 13.0%
2005.1 19.0% 7.0%
2004.4 -16.0% -4.0%
2004.3 8.0% 16.0%
2004.2 -3.0% -11.0%
2004.1 34.0% 25.0%
Cha r a c te r istic Line (Re gr e ssion)
-15.0%
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
-40.0% -20.0% 0.0% 20.0% 40.0%
Returns on TSE300
ReturnsonStock
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.
The regression line is a line of ‘best
fit’ that describes the inherent
relationship between the returns on
the stock and the returns on the
market. The slope is the beta
coefficient.
The ‘cloud’ of plotted points
represents ‘diversifiable or company
specific’ risk in the securities returns
that can be eliminated from a portfolio
through diversification. Since
company-specific risk can be
eliminated, investors don’t require
compensation for it according to
Markowitz Portfolio Theory.

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.

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)

Characteristic Line for Imperial Tobacco
An Example of Volatility that is Primarily Company-Specific
Returns on
the Market %
(S&P TSX)
Returns on
Imperial
Tobacco %
Characteristic
Line for Imperial
Tobacco
• High alpha
• R-square is very
low ≈ 0.02
• Beta is largely
irrelevant

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.

Characteristic Line General Motors
A Positive Beta with Predictive Power
Returns on
the Market %
(S&P TSX)
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

An Unusual Characteristic Line
A Negative Beta with Predictive Power
Returns on
the Market %
(S&P TSX)
Returns on a
Stock %
Characteristic Line for a stock
that will provide excellent
portfolio diversification
(high R2)
• Positive alpha
• R-square is
very high
• Beta is negative
<0.0 and > -1.0

Diversifiable Risk
(Non-systematic Risk)
• Volatility in a security’s returns caused by company-specific
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.

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)

Demonstration Through Example
The following slides will be based on
Alcan Aluminum (AL.TO)

Five Year Stock Price Chart for AL.TO

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.

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

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
Volume of
trading done
in the stock
on the TSE in
the month in
numbers of
board lots
The day,
month and
year
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
of the month

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 Close
01-May-02 59.22
01-Apr-02 57.9
01-Mar-02 63.03
01-Feb-02 64.86
02-Jan-02 61.85

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.

Raw Company Data
Alcan Example
Date
Issued
Capital
Closing Price
for Alcan
AL.TO
Cash
Dividends
per Share
01-May-02 321,400,589 $59.22 $0.00
01-Apr-02 321,400,589 $57.90 $0.15
01-Mar-02 321,400,589 $63.03 $0.00
01-Feb-02 321,400,589 $64.86 $0.00
02-Jan-02 160,700,295 $123.70 $0.30
01-Dec-01 160,700,295 $119.30 $0.00
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.

Normalizing the Raw Company Data
Alcan Example
Date
Issued
Capital
Closing
Price for
Alcan
AL.TO
Cash
Dividends
per Share
Adjustment
Factor
Normalized
Stock Price
Normalized
Dividend
01-May-02 321,400,589 $59.22 $0.00 1.00 $59.22 $0.00
01-Apr-02 321,400,589 $57.90 $0.15 1.00 $57.90 $0.15
01-Mar-02 321,400,589 $63.03 $0.00 1.00 $63.03 $0.00
01-Feb-02 321,400,589 $64.86 $0.00 1.00 $64.86 $0.00
02-Jan-02 160,700,295 $123.70 $0.30 0.50 $61.85 $0.15
01-Dec-01 145,000,500 $111.40 $0.00 0.45 $50.26 $0.00
The adjustment factor is just the value in the issued
capital cell divided by 321,400,589.

Calculating the HPR on the stock from the
Normalized Data
Date
Normalized
Stock Price
Normalized
Dividend HPR
01-May-02 $59.22 $0.00 2.28%
01-Apr-02 $57.90 $0.15 -7.90%
01-Mar-02 $63.03 $0.00 -2.82%
01-Feb-02 $64.86 $0.00 4.87%
02-Jan-02 $61.85 $0.15 23.36%
01-Dec-01 $50.26 $0.00
Use $59.22 as the ending price, $57.90 as the
beginning price and during the month of May, no
dividend was declared.
%28.2
$57.90
$0.00$57.90-$59.22
)(
0
101





P
DPP
HPR

Now Put the data from the S&P/TSX Total
Return Composite Index in
Date
Normalized
Stock Price
Normalized
Dividend HPR
Ending
TSX
Value
01-May-02 $59.22 $0.00 2.28% 16911.33
01-Apr-02 $57.90 $0.15 -7.90% 16903.36
01-Mar-02 $63.03 $0.00 -2.82% 17308.41
01-Feb-02 $64.86 $0.00 4.87% 16801.82
02-Jan-02 $61.85 $0.15 23.36% 16908.11
01-Dec-01 $50.26 $0.00 16881.75
You will find the Total Return S&P/TSX Composite
Index values in TSX Review found in the library.

Now Calculate the HPR on the Market Index
Date
Normalized
Stock Price
Normalized
Dividend HPR
Ending
TSX
Value
HPR on
the TSX
01-May-02 $59.22 $0.00 2.28% 16911.33 0.05%
01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34%
01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02%
01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63%
02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16%
01-Dec-01 $50.26 $0.00 16881.75
Again, you simply use the HPR formula using the
ending values for the total return composite index.
%05.0
16,903.36
16,903.36-16,911.33
)(
0
01




P
PP
HPR

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.

Regression
Defining the Data Ranges
Date
Normalized
Stock Price
Normalized
Dividend HPR
Ending
TSX
Value
HPR on
the TSX
01-May-02 $59.22 $0.00 2.28% 16911.33 0.05%
01-Apr-02 $57.90 $0.15 -7.90% 16903.36 -2.34%
01-Mar-02 $63.03 $0.00 -2.82% 17308.41 3.02%
01-Feb-02 $64.86 $0.00 4.87% 16801.82 -0.63%
02-Jan-02 $61.85 $0.15 23.36% 16908.11 0.16%
01-Dec-01 $50.26 $0.00 16881.75
The independent variable is the returns on the Market.The dependent variable is the returns on the Stock.

Now Use the Regression Function in Excel to
regress the returns of the stock against the
returns of the market
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.05300947
R Square 0.00281
Adjusted R Square -0.2464875
Standard Error 5.79609628
Observations 6
ANOVA
df SS MS F Significance F
Regression 1 0.3786694 0.37866937 0.011271689 0.920560274
Residual 4 134.37893 33.5947321
Total 5 134.7576
CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%
Intercept 59.3420816 2.8980481 20.4765686 3.3593E-05 51.29579335 67.38836984 51.2957934 67.38837
X Variable 1 3.55278937 33.463777 0.10616821 0.920560274 -89.35774428 96.46332302 -89.3577443 96.46332
Beta
Coefficient is
the X-
Variable 1
The alpha is the
vertical intercept.
R-square is the
coefficient of
determination =
0.0028=.3%

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.

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.

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.

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