The capital-asset-pricing-model-capm75

Prepared byPrepared byKen HartviksenKen Hartviksen

INTRODUCTION TOINTRODUCTION TO CORPORATE FINANCECORPORATE FINANCELaurence Booth Laurence Booth •• W. Sean Cleary W. Sean Cleary

Chapter 9 – The Capital Asset Pricing Chapter 9 – The Capital Asset Pricing ModelModel

CHAPTER 9CHAPTER 9 The Capital Asset Pricing The Capital Asset Pricing

Model (CAPM)Model (CAPM)

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

Lecture AgendaLecture Agenda

• Learning ObjectivesLearning Objectives• Important TermsImportant Terms• The New Efficient FrontierThe New Efficient Frontier• The Capital Asset Pricing ModelThe Capital Asset Pricing Model• The CAPM and Market RiskThe CAPM and Market Risk• Alternative Asset Pricing ModelsAlternative Asset Pricing Models• Summary and ConclusionsSummary and Conclusions

– Concept Review QuestionsConcept Review Questions– Appendix 1 – Calculating the Ex Ante BetaAppendix 1 – Calculating the Ex Ante Beta– Appendix 2 – Calculating the Ex Post BetaAppendix 2 – Calculating the Ex Post Beta


Learning ObjectivesLearning Objectives

1.1. What happens if all investors are rational and risk What happens if all investors are rational and risk averse.averse.

2.2. How modern portfolio theory is extended to develop the How modern portfolio theory is extended to develop the capital market line, which determines how expected capital market line, which determines how expected returns on portfolios are determined.returns on portfolios are determined.

3.3. How to assess the performance of mutual fund How to assess the performance of mutual fund managersmanagers

4.4. How the Capital Asset Pricing Model’s (CAPM) security How the Capital Asset Pricing Model’s (CAPM) security market line is developed from the capital market line.market line is developed from the capital market line.

5.5. How the CAPM has been extended to include other risk-How the CAPM has been extended to include other risk-based pricing models.based pricing models.


Important Chapter TermsImportant Chapter Terms

• Arbitrage pricing theory Arbitrage pricing theory (APT)(APT)

• Capital Asset Pricing Capital Asset Pricing Model (CAPM)Model (CAPM)

• Capital market line Capital market line (CML)(CML)

• Characteristic lineCharacteristic line• Fama-French (FF) modelFama-French (FF) model• Insurance premiumInsurance premium• Market portfolioMarket portfolio• Market price of riskMarket price of risk

• Market risk premiumMarket risk premium• New (or super) efficient New (or super) efficient

frontierfrontier• No-arbitrage principleNo-arbitrage principle• Required rate of returnRequired rate of return• Risk premiumRisk premium• Security market line Security market line

(SML)(SML)• Separation theorumSeparation theorum• Sharpe ratioSharpe ratio• Short positionShort position• Tangent portfolioTangent portfolio

Achievable Portfolio CombinationsAchievable Portfolio Combinations

The Capital Asset Pricing Model The Capital Asset Pricing Model (CAPM)(CAPM)


Achievable Portfolio CombinationsAchievable Portfolio CombinationsThe Two-Asset CaseThe Two-Asset Case

• It is possible to construct a series of portfolios with It is possible to construct a series of portfolios with different risk/return characteristics just by varying the different risk/return characteristics just by varying the weights of the two assets in the portfolio.weights of the two assets in the portfolio.

• Assets A and B are assumed to have a correlation Assets A and B are assumed to have a correlation coefficient of -0.379 and the following individual coefficient of -0.379 and the following individual return/risk characteristicsreturn/risk characteristics

Expected ReturnExpected Return Standard Standard DeviationDeviation

Asset AAsset A 8%8% 8.72%8.72%Asset BAsset B 10%10% 22.69%22.69%

The following table shows the portfolio characteristics for 100 The following table shows the portfolio characteristics for 100 different weighting schemes for just these two securities:different weighting schemes for just these two securities:


Example of Portfolio Combinations and Example of Portfolio Combinations and CorrelationCorrelation

AssetExpected

ReturnStandard Deviation

Correlation Coefficient

A 8.0% 8.7% -0.379B 10.0% 22.7%

Weight of A Weight of BExpected

ReturnStandard 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 CharacteristicsThe 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 Example of Portfolio Combinations and CorrelationCorrelation

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

Exp

ecte

d R

etu

rn o

f th

e P

ort

foli

o


Two Asset Efficient FrontierTwo Asset Efficient Frontier

• Figure 8 – 10 describes five different Figure 8 – 10 describes five different portfolios (A,B,C,D and E in reference to the portfolios (A,B,C,D and E in reference to the attainable set of portfolio combinations of this attainable set of portfolio combinations of this two asset portfolio.two asset portfolio.

(See Figure 8 -10 on the following slide)(See Figure 8 -10 on the following slide)


Efficient FrontierEfficient FrontierThe Two-Asset Portfolio CombinationsThe 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

Ex

pe

cte

d R

etu

rn %

Standard Deviation (%)

A

E

B

C

D


Achievable Set of Portfolio CombinationsAchievable Set of Portfolio CombinationsGetting to the ‘n’ Asset CaseGetting to the ‘n’ Asset Case

• In a real world investment universe with all of the In a real world investment universe with all of the investment alternatives (stocks, bonds, money investment alternatives (stocks, bonds, money market securities, hybrid instruments, gold real market securities, hybrid instruments, gold real estate, etc.) it is possible to construct many estate, etc.) it is possible to construct many different alternative portfolios out of risky different alternative portfolios out of risky securities.securities.

• Each portfolio will have its own unique expected Each portfolio will have its own unique expected return and risk.return and risk.

• Whenever you construct a portfolio, you can Whenever you construct a portfolio, you can measure two fundamental characteristics of the measure two fundamental characteristics of the portfolio:portfolio:– Portfolio expected return (Portfolio expected return (ERERpp))– Portfolio risk (Portfolio risk (σσpp))


The Achievable Set of Portfolio The Achievable Set of Portfolio CombinationsCombinations

• You could start by randomly assembling ten You could start by randomly assembling ten risky portfolios.risky portfolios.

• The results (in terms of ER The results (in terms of ER pp and and σσpp ) )might might look like the graph on the following page:look like the graph on the following page:


Achievable Portfolio CombinationsAchievable Portfolio CombinationsThe First Ten Combinations CreatedThe First Ten Combinations Created

Portfolio Risk (σp)

10 Achievable Risky Portfolio Combinations

ERp


The Achievable Set of Portfolio The Achievable Set of Portfolio CombinationsCombinations

• You could continue randomly assembling You could continue randomly assembling more portfolios.more portfolios.

• Thirty risky portfolios Thirty risky portfolios might look like the might look like the graph on the following slide:graph on the following slide:


Achievable Portfolio CombinationsAchievable Portfolio CombinationsThirty Combinations Naively CreatedThirty Combinations Naively Created


30 Risky Portfolio Combinations

ERp


Achievable Set of Portfolio CombinationsAchievable Set of Portfolio CombinationsAll Securities – Many Hundreds of Different CombinationsAll Securities – Many Hundreds of Different Combinations

• When you construct many hundreds of When you construct many hundreds of different portfolios naively varying the weight different portfolios naively varying the weight of the individual assets and the number of of the individual assets and the number of types of assets themselves, you get a set of types of assets themselves, you get a set of achievable portfolio combinations as achievable portfolio combinations as indicated on the following slide: indicated on the following slide:



ERp

Achievable Portfolio CombinationsAchievable Portfolio CombinationsMore Possible Combinations CreatedMore 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 FrontierThe Efficient Frontier




Achievable Set of Risky Portfolio Combinations

ERp

Achievable Portfolio CombinationsAchievable Portfolio CombinationsEfficient Frontier (Set)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 FrontierThe New Efficient FrontierEfficient PortfoliosEfficient 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 FrontierER

MVP

A

B


Underlying AssumptionUnderlying AssumptionInvestors are Rational and Risk-AverseInvestors are Rational and Risk-Averse

• We assume investors are risk-averse wealth We assume investors are risk-averse wealth maximizers.maximizers.

• This means they will not willingly undertake fair This means they will not willingly undertake fair gamble.gamble.– A risk-averse investor prefers the risk-free situation.A risk-averse investor prefers the risk-free situation.– The corollary of this is that the investor needs a risk premium to The corollary of this is that the investor needs a risk premium to

be induced into a risky situation.be induced into a risky situation.– Evidence of this is the willingness of investors to pay insurance Evidence of this is the willingness of investors to pay insurance

premiums to get out of risky situations.premiums to get out of risky situations.

• The implication of this, is that investors will only The implication of this, is that investors will only choose portfolios that are members of the efficient set choose portfolios that are members of the efficient set (frontier).(frontier).

The New Efficient Frontier and The New Efficient Frontier and Separation TheoremSeparation Theorem



Risk-free InvestingRisk-free Investing

• When we introduce the presence of a risk-free When we introduce the presence of a risk-free investment, a whole new set of portfolio investment, a whole new set of portfolio combinations becomes possible.combinations becomes possible.

• We can estimate the return on a portfolio We can estimate the return on a portfolio made up of made up of RF RF asset and a risky asset asset and a risky asset A A letting the weight letting the weight ww invested in the risky invested in the risky asset and the weight invested in RF as asset and the weight invested in RF as (1 – w)(1 – w)


The New Efficient FrontierThe New Efficient FrontierRisk-Free InvestingRisk-Free Investing

– Expected return on a two asset portfolio made up of Expected return on a two asset portfolio made up of risky asset risky asset AA and and RFRF::

The possible combinations of A and RF are found graphed on the following slide.The possible combinations of A and RF are found graphed on the following slide.

RF) - (ER RF ER Ap w[9-1]


The New Efficient FrontierThe New Efficient FrontierAttainable Portfolios Using Attainable Portfolios Using RFRF and and AA

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

RF ER A

APP

[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.


The New Efficient FrontierThe New Efficient FrontierAttainable Portfolios using the Attainable Portfolios using the RF RF and and A, A, and and RFRF and and TT

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


The New Efficient FrontierThe New Efficient FrontierEfficient Portfolios using the Tangent Portfolio Efficient Portfolios using the Tangent Portfolio TT

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!


The New Efficient FrontierThe New Efficient FrontierLending PortfoliosLending 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


The New Efficient FrontierThe New Efficient FrontierBorrowing PortfoliosBorrowing 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 Efficient FrontierThe New Efficient FrontierThe New (Super) Efficient FrontierThe 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 New Efficient FrontierThe New Efficient FrontierThe Implications – Separation Theorem – Market PortfolioThe Implications – Separation Theorem – Market Portfolio

• All investors will only hold individually-determined All investors will only hold individually-determined combinations of:combinations of:– The risk free asset (RF) andThe risk free asset (RF) and– The model portfolio (market portfolio)The model portfolio (market portfolio)

• The separation theoremThe separation theorem– The investment decision (how to construct the portfolio of risky The investment decision (how to construct the portfolio of risky

assets) is separate from the financing decision (how much assets) is separate from the financing decision (how much should be invested or borrowed in the risk-free asset)should be invested or borrowed in the risk-free asset)

– The tangent portfolio T is optimal for every investor regardless of The tangent portfolio T is optimal for every investor regardless of his/her degree of risk aversion.his/her degree of risk aversion.

• The Equilibrium ConditionThe Equilibrium Condition– The market portfolio must be the tangent portfolio T if everyone The market portfolio must be the tangent portfolio T if everyone

holds the same portfolio holds the same portfolio – Therefore the market portfolio (M) is the tangent portfolio (T)Therefore the market portfolio (M) is the tangent portfolio (T)


σρ

ER

RF

M

CML

The New Efficient FrontierThe New Efficient FrontierThe Capital Market LineThe 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 ModelThe Capital Asset Pricing Model

The Hypothesized Relationship The Hypothesized Relationship between Risk and Returnbetween Risk and Return


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelWhat is it?What is it?

– An hypothesis by Professor William SharpeAn hypothesis by Professor William Sharpe• Hypothesizes that investors require higher rates of return for greater levels of Hypothesizes that investors require higher rates of return for greater levels of

relevant risk.relevant risk.• There are no prices on the model, instead it hypothesizes the relationship There are no prices on the model, instead it hypothesizes the relationship

between risk and return for individual securities.between risk and return for individual securities.• It is often used, however, the price securities and investments.It is often used, however, the price securities and investments.


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelHow is it Used?How is it Used?

– Uses include:Uses include:• Determining the cost of equity capital.Determining the cost of equity capital.

• The relevant risk in the dividend discount model to estimate a stock’s intrinsic The relevant risk in the dividend discount model to estimate a stock’s intrinsic (inherent economic worth) value.(inherent economic worth) value. (As illustrated below) (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

2iM

i,M

σ

COV )( iMi RFERRFk

gk

DP

c 1

0Is the stock fairly priced?


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelAssumptionsAssumptions

– CAPM is based on the following assumptions:CAPM is based on the following assumptions:1.1. All investors have identical expectations about expected All investors have identical expectations about expected

returns, standard deviations, and correlation coefficients for all returns, standard deviations, and correlation coefficients for all securities.securities.

2.2. All investors have the same one-period investment time All investors have the same one-period investment time horizon.horizon.

3.3. All investors can borrow or lend money at the risk-free rate of All investors can borrow or lend money at the risk-free rate of return (RF).return (RF).

4.4. There are no transaction costs.There are no transaction costs.5.5. There are no personal income taxes so that investors are There are no personal income taxes so that investors are

indifferent between capital gains an dividends.indifferent between capital gains an dividends.6.6. There are many investors, and no single investor can affect There are many investors, and no single investor can affect

the price of a stock through his or her buying and selling the price of a stock through his or her buying and selling decisions. Therefore, investors are price-takers.decisions. Therefore, investors are price-takers.

7.7. Capital markets are in equilibrium.Capital markets are in equilibrium.


Market Portfolio and Capital Market LineMarket Portfolio and Capital Market Line

• The assumptions have the following The assumptions have the following implications:implications:1.1. The “optimal” risky portfolio is the one that is The “optimal” risky portfolio is the one that is

tangent to the efficient frontier on a line that is drawn tangent to the efficient frontier on a line that is drawn from RF. This portfolio will be the same for all from RF. This portfolio will be the same for all investors.investors.

2.2. This optimal risky portfolio will be the This optimal risky portfolio will be the market market portfolioportfolio (M) which contains all risky securities. (M) which contains all risky securities.

(Figure 9 – 4 illustrates the Market Portfolio ‘M’)(Figure 9 – 4 illustrates the Market Portfolio ‘M’)


The Capital Market LineThe Capital Market Line

9 - 5 FIGURE

σρ

ER

RF

MERM

σM

PM

MP

RFERRFk

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 Capital Asset Pricing ModelThe Capital Asset Pricing ModelThe Market Portfolio and the Capital Market Line (CML)The Market Portfolio and the Capital Market Line (CML)

– The slope of the CML is the incremental expected The slope of the CML is the incremental expected return divided by the incremental risk.return divided by the incremental risk.

– This is called This is called the market price for risk. Orthe market price for risk. Or– The equilibrium price of risk in the capital market.The equilibrium price of risk in the capital market.

RF - ER

CML theof Slope M

M

[9-4]


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelThe Market Portfolio and the Capital Market Line (CML)The Market Portfolio and the Capital Market Line (CML)

– Solving for the expected return on a portfolio in the presence of Solving for the expected return on a portfolio in the presence of a RF asset and given the a RF asset and given the market price for risk :market price for risk :

– Where:Where:• ERERMM = expected return on the market portfolio M = expected return on the market portfolio M

• σσMM = the standard deviation of returns on the market portfolio = the standard deviation of returns on the market portfolio

• σσPP = the standard deviation of returns on the efficient portfolio being = the standard deviation of returns on the efficient portfolio being

consideredconsidered

)( σ

- RFER RFRE P

M

MP

[9-5]


The Capital Market LineThe Capital Market LineUsing the CML – Expected versus Required ReturnsUsing the CML – Expected versus Required Returns

– In an efficient capital market investors will require a In an efficient capital market investors will require a return on a portfolio that compensates them for the return on a portfolio that compensates them for the risk-free return as well as the market price for risk.risk-free return as well as the market price for risk.

– This means that portfolios should offer returns along This means that portfolios should offer returns along the CML.the CML.


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelExpected and Required Rates of Return 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


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelRisk-Adjusted Performance and the Sharpe RatiosRisk-Adjusted Performance and the Sharpe Ratios

– William Sharpe identified a ratio that can be used to assess the risk-William Sharpe identified a ratio that can be used to assess the risk-adjusted performance of managed funds (such as mutual funds and adjusted performance of managed funds (such as mutual funds and pension plans).pension plans).

– It is called the Sharpe ratio:It is called the Sharpe ratio:

– Sharpe ratio is a measure of portfolio performance that describes how Sharpe ratio is a measure of portfolio performance that describes how well an asset’s returns compensate investors for the risk taken.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’s value is the premium earned over the RF divided by portfolio risk…so it is measuring valued added per unit of risk.it is measuring valued added per unit of risk.

– Sharpe ratios are calculated ex post (after-the-fact) and are used to rank 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 portfolios or assess the effectiveness of the portfolio manager in adding value to the portfolio over and above a benchmark.value to the portfolio over and above a benchmark.

RF - ER

ratio Sharpe P

P

[9-6]


The Capital Asset Pricing ModelThe Capital Asset Pricing ModelSharpe Ratios and Income TrustsSharpe Ratios and Income Trusts

– Table 9 – 1 (on the following slide) illustrates return, Table 9 – 1 (on the following slide) illustrates return, standard deviation, Sharpe and beta coefficient for standard deviation, Sharpe and beta coefficient for four very different portfolios from 2002 to 2004.four very different portfolios from 2002 to 2004.

– Income Trusts did exceedingly well during this time, Income Trusts did exceedingly well during this time, however, the recent announcement of Finance however, the recent announcement of Finance Minister Flaherty and the subsequent drop in Income Minister Flaherty and the subsequent drop in Income Trust values has done much to eliminate this Trust values has done much to eliminate this historical performance.historical performance.


Income Trust Estimated ValuesIncome Trust Estimated Values

Return σP Sharpe β

Median income trusts 25.83% 18.66% 1.37 0.22Equally weighted trust portfolio 29.97% 8.02% 3.44 0.28S&P/TSX Composite Index 8.97% 13.31% 0.49 1.00Scotia 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 RiskCAPM and Market Risk



Diversifiable and Non-Diversifiable RiskDiversifiable and Non-Diversifiable Risk

• CML applies to efficient portfoliosCML applies to efficient portfolios• Volatility (risk) of Volatility (risk) of individual security returnsindividual security returns are are

caused by two different factors:caused by two different factors:– Non-diversifiable risk (system wide changes in the economy and Non-diversifiable risk (system wide changes in the economy and

markets that affect all securities in varying degrees)markets that affect all securities in varying degrees)– Diversifiable risk (company-specific factors that affect the returns Diversifiable risk (company-specific factors that affect the returns

of only one security)of only one security)

• Figure 9 – 7 illustrates what happens to portfolio risk Figure 9 – 7 illustrates what happens to portfolio risk as the portfolio is first invested in only one as the portfolio is first invested in only one investment, and then slowly invested, naively, in more investment, and then slowly invested, naively, in more and more securities. and more securities.


The CAPM and Market RiskThe CAPM and Market RiskPortfolio Risk and DiversificationPortfolio 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 RiskRelevant RiskDrawing a Conclusion from Figure 9 - 7Drawing a Conclusion from Figure 9 - 7

• Figure 9 – 7 demonstrates that an individual securities’ Figure 9 – 7 demonstrates that an individual securities’ volatility of return comes from two factors: volatility of return comes from two factors:– Systematic factorsSystematic factors– Company-specific factorsCompany-specific factors

• When combined into portfolios, company-specific risk When combined into portfolios, company-specific risk is diversified away.is diversified away.

• Since all investors are ‘diversified’ then in an efficient Since all investors are ‘diversified’ then in an efficient market, no-one would be willing to pay a ‘premium’ market, no-one would be willing to pay a ‘premium’ for company-specific risk.for company-specific risk.

• Relevant risk to diversified investors then is systematic Relevant risk to diversified investors then is systematic risk.risk.

• Systematic risk is measured using the Beta Coefficient.Systematic risk is measured using the Beta Coefficient.

Measuring Systematic Risk Measuring Systematic Risk The Beta CoefficientThe Beta Coefficient



The Beta CoefficientThe Beta CoefficientWhat is the Beta Coefficient?What is the Beta Coefficient?

• A measure of systematic (non-diversifiable) A measure of systematic (non-diversifiable) riskrisk

• As a ‘coefficient’ the beta is a pure number As a ‘coefficient’ the beta is a pure number and has no units of measure.and has no units of measure.


The Beta CoefficientThe Beta CoefficientHow Can We Estimate the Value of the Beta Coefficient?How Can We Estimate the Value of the Beta Coefficient?

• There are two basic approaches to There are two basic approaches to estimating the beta coefficient:estimating the beta coefficient:

1.1. Using a formula (and subjective forecasts)Using a formula (and subjective forecasts)

2.2. Use of regression (using past holding period returns)Use of regression (using past holding period returns)

(Figure 9 – 8 on the following slide illustrates the characteristic line used to estimate (Figure 9 – 8 on the following slide illustrates the characteristic line used to estimate the beta coefficient) the beta coefficient)


The CAPM and Market RiskThe CAPM and Market RiskThe Characteristic Line for Security AThe 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

Ma

rke

t R

etu

rns

(%

) 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 CoefficientThe Formula for the Beta Coefficient

Beta is equal to the covariance of the returns of the Beta is equal to the covariance of the returns of the stock with the returns of the market, divided by the stock with the returns of the market, divided by the variance of the returns of the market:variance of the returns of the market:

,

2iM

iMi

M

i,M

σ

COV

[9-7]


The Beta CoefficientThe Beta CoefficientHow is the Beta Coefficient Interpreted?How is the Beta Coefficient Interpreted?

• The beta of the market portfolio is ALWAYS = 1.0The beta of the market portfolio is ALWAYS = 1.0

• The beta of a security compares the volatility of its returns to the volatility The beta of a security compares the volatility of its returns to the volatility of the market returns:of the market returns:

ββss = 1.0 = 1.0 -- the security has the same volatility as the market the security has the same volatility as the market as a wholeas a whole

ββss > 1.0 > 1.0 -- aggressive investment with volatility of returns aggressive investment with volatility of returns greater than the marketgreater than the market

ββss < 1.0 < 1.0 -- defensive investment with volatility of returns less defensive investment with volatility of returns less than the marketthan the market

ββss < 0.0 < 0.0 -- an investment with returns that are negatively an investment with returns that are negatively correlated with the returns of the marketcorrelated with the returns of the market

Table 9 – 2 illustrates beta coefficients for a variety of Canadian InvestmentsTable 9 – 2 illustrates beta coefficients for a variety of Canadian Investments


Canadian BETASCanadian BETASSelectedSelected

Company Industry Classification Beta

Abitibi Consolidated Inc. Materials - Paper & Forest 1.37Algoma Steel Inc. Materials - Steel 1.92Bank of Montreal Financials - Banks 0.50Bank of Nova Scotia Financials - Banks 0.54Barrick Gold Corp. Materials - Precious Metals & Minerals 0.74BCE Inc. Communications - Telecommunications 0.39Bema Gold Corp. Materials - Precious Metals & Minerals 0.26CIBC Financials - Banks 0.66Cogeco Cable Inc. Consumer Discretionary - Cable 0.67Gammon Lake Resources Inc. Materials - Precious Metals & Minerals 2.52Imperial 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 PortfolioThe Beta of a Portfolio

The beta of a portfolio is simply the weighted average of the The beta of a portfolio is simply the weighted average of the betas of the individual asset betas that make up the portfolio.betas of the individual asset betas that make up the portfolio.

Weights of individual assets are found by dividing the value of Weights of individual assets are found by dividing the value of the investment by the value of the total portfolio.the investment by the value of the total portfolio.

... nnBBAAP www [9-8]

The Security Market LineThe Security Market Line



The CAPM and Market RiskThe CAPM and Market RiskThe Security Market Line (SML)The Security Market Line (SML)

– The SML is the hypothesized relationship between return (the The SML is the hypothesized relationship between return (the dependent variable) and systematic risk (the beta coefficient).dependent variable) and systematic risk (the beta coefficient).

– It is a straight line relationship defined by the following formula:It is a straight line relationship defined by the following formula:

– Where:Where:kkii = the required return on security ‘i’ = the required return on security ‘i’

ERERMM – RF = market premium for risk – RF = market premium for risk

ΒΒi i = the beta coefficient for security ‘i’ = the beta coefficient for security ‘i’ (See Figure 9 - 9 on the following slide for the graphical representation)(See Figure 9 - 9 on the following slide for the graphical representation)

)( iMi RFERRFk [9-9]


The CAPM and Market RiskThe CAPM and Market RiskThe Security Market Line (SML)The Security Market Line (SML)

9 - 9 FIGURE

βM = 1

ER

RF

β

MERM

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 CAPM and Market RiskThe CAPM and Market RiskThe SML and Security ValuationThe 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 SummaryThe CAPM in SummaryThe SML and CMLThe SML and CML

– The CAPM is well entrenched and widely used by The CAPM is well entrenched and widely used by investors, managers and financial institutions.investors, managers and financial institutions.

– It is a single factor model because it based on the It is a single factor model because it based on the hypothesis that required rate of return can be hypothesis that required rate of return can be predicted using one factor – systematic riskpredicted using one factor – systematic risk

– The SML is used to price individual investments and The SML is used to price individual investments and uses the beta coefficient as the measure of risk.uses the beta coefficient as the measure of risk.

– The CML is used with diversified portfolios and uses The CML is used with diversified portfolios and uses the standard deviation as the measure of risk.the standard deviation as the measure of risk.

Alternative Pricing ModelsAlternative Pricing Models



Challenges to CAPMChallenges to CAPM

• Empirical tests suggest:Empirical tests suggest:– CAPM does not hold well in practice:CAPM does not hold well in practice:

• Ex post SML is an upward sloping lineEx post SML is an upward sloping line• Ex ante Ex ante y (vertical)y (vertical) – intercept is higher that RF – intercept is higher that RF• Slope is less than what is predicted by theorySlope is less than what is predicted by theory

– Beta possesses no explanatory power for predicting stock returns Beta possesses no explanatory power for predicting stock returns (Fama and French, 1992)(Fama and French, 1992)

• CAPM remains in widespread use despite the foregoing.CAPM remains in widespread use despite the foregoing.– Advantages include – relative simplicity and intuitive logic.Advantages include – relative simplicity and intuitive logic.

• Because of the problems with CAPM, other models have Because of the problems with CAPM, other models have been developed including:been developed including:– Fama-French (FF) ModelFama-French (FF) Model– Abitrage Pricing Theory (APT)Abitrage Pricing Theory (APT)


Alternative Asset Pricing ModelsAlternative Asset Pricing ModelsThe Fama – French ModelThe Fama – French Model

– A pricing model that uses three factors to relate A pricing model that uses three factors to relate expected returns to risk including:expected returns to risk including:

1.1. A market factor related to firm size.A market factor related to firm size.

2.2. The market value of a firm’s common equity (MVE)The market value of a firm’s common equity (MVE)

3.3. Ratio of a firm’s book equity value to its market value of equity. Ratio of a firm’s book equity value to its market value of equity. (BE/MVE)(BE/MVE)

– This model has become popular, and many think it This model has become popular, and many think it does a better job than the CAPM in explaining ex does a better job than the CAPM in explaining ex ante stock returns.ante stock returns.


Alternative Asset Pricing ModelsAlternative Asset Pricing ModelsThe Arbitrage Pricing TheoryThe Arbitrage Pricing Theory

– A pricing model that uses multiple factors to relate expected A pricing model that uses multiple factors to relate expected returns to risk by assuming that asset returns are linearly related returns to risk by assuming that asset returns are linearly related to a set of indexes, which proxy risk factors that influence to a set of indexes, which proxy risk factors that influence security returns.security returns.

– It is based on the no-arbitrage principle which is the rule that two It is based on the no-arbitrage principle which is the rule that two otherwise identical assets cannot sell at different prices.otherwise identical assets cannot sell at different prices.

– Underlying factors represent broad economic forces which are Underlying factors represent broad economic forces which are inherently unpredictable.inherently unpredictable.

... 11110 niniii FbFbFbaER [9-10]


Alternative Asset Pricing ModelsAlternative Asset Pricing ModelsThe Arbitrage Pricing Theory – the ModelThe Arbitrage Pricing Theory – the Model

– Underlying factors represent broad economic forces which are Underlying factors represent broad economic forces which are inherently unpredictable.inherently unpredictable.

– Where:Where:• ERERii = the expected return on security i = the expected return on security i• aa00 = the expected return on a security with zero systematic risk = the expected return on a security with zero systematic risk• bbii = the sensitivity of security i to a given risk factor = the sensitivity of security i to a given risk factor• FFii = the risk premium for a given risk factor = the risk premium for a given risk factor

– The model demonstrates that a security’s risk is based on its sensitivity The model demonstrates that a security’s risk is based on its sensitivity to broad economic forces.to broad economic forces.

... 11110 niniii FbFbFbaER [9-10]


Alternative Asset Pricing ModelsAlternative Asset Pricing ModelsThe Arbitrage Pricing Theory – ChallengesThe Arbitrage Pricing Theory – Challenges

– Underlying factors represent broad economic forces Underlying factors represent broad economic forces which are inherently unpredictable.which are inherently unpredictable.

– Ross and Roll identify five systematic factors:Ross and Roll identify five systematic factors:1.1. Changes in expected inflationChanges in expected inflation2.2. Unanticipated changes in inflationUnanticipated changes in inflation3.3. Unanticipated changes in industrial productionUnanticipated changes in industrial production4.4. Unanticipated changes in the default-risk premiumUnanticipated changes in the default-risk premium5.5. Unanticipated changes in the term structure of interest ratesUnanticipated changes in the term structure of interest rates

• Clearly, something that isn’t forecast, can’t be used Clearly, something that isn’t forecast, can’t be used to price securities today…they can only be used to to price securities today…they can only be used to explain prices after the fact.explain prices after the fact.


Summary and ConclusionsSummary and Conclusions

In this chapter you have learned:In this chapter you have learned:

– How the efficient frontier can be expanded by introducing risk-How the efficient frontier can be expanded by introducing risk-free borrowing and lending leading to a super efficient frontier free borrowing and lending leading to a super efficient frontier called the Capital Market Line (CML)called the Capital Market Line (CML)

– The Security Market Line can be derived from the CML and The Security Market Line can be derived from the CML and provides a way to estimate a market-based, required return for provides a way to estimate a market-based, required return for any security or portfolio based on market risk as measured by any security or portfolio based on market risk as measured by the beta.the beta.

– That alternative asset pricing models exist including the Fama-That alternative asset pricing models exist including the Fama-French Model and the Arbitrage Pricing Theory.French Model and the Arbitrage Pricing Theory.

Concept Review QuestionsConcept Review Questions



Concept Review Question 1Concept Review Question 1Risk AversionRisk Aversion

What is risk aversion and how do we know What is risk aversion and how do we know investors are risk averse?investors are risk averse?

Estimating the Ex Ante (Forecast) BetaEstimating the Ex Ante (Forecast) Beta

APPENDIX 1APPENDIX 1


Calculating a Beta Coefficient Using Ex Ante Calculating a Beta Coefficient Using Ex Ante ReturnsReturns

• Ex Ante means forecast…Ex Ante means forecast…• You would use ex ante return data if historical rates of You would use ex ante return data if historical rates of

return are somehow not indicative of the kinds of return are somehow not indicative of the kinds of returns the company will produce in the future.returns the company will produce in the future.

• A good example of this is Air Canada or American A good example of this is Air Canada or American Airlines, before and after September 11, 2001. After Airlines, before and after September 11, 2001. After the World Trade Centre terrorist attacks, a the World Trade Centre terrorist attacks, a fundamental shift in demand for air travel occurred. fundamental shift in demand for air travel occurred. The historical returns on airlines are not useful in The historical returns on airlines are not useful in estimating future returns.estimating future returns.


Appendix 1 AgendaAppendix 1 Agenda

• The beta coefficientThe beta coefficient• The formula approach to beta measurement The formula approach to beta measurement

using ex ante returnsusing ex ante returns– Ex ante returnsEx ante returns– Finding the expected returnFinding the expected return– Determining variance and standard deviationDetermining variance and standard deviation– Finding covarianceFinding covariance– Calculating and interpreting the beta coefficientCalculating and interpreting the beta coefficient


The Beta CoefficientThe Beta Coefficient

• Under the theory of the Capital Asset Pricing Model Under the theory of the Capital Asset Pricing Model total risk is partitioned into two parts:total risk is partitioned into two parts:– Systematic riskSystematic risk– Unsystematic risk – diversifiable riskUnsystematic risk – diversifiable risk

• Systematic risk is non-diversifiable risk.Systematic risk is non-diversifiable risk.• Systematic risk is the only relevant risk to the Systematic risk is the only relevant risk to the

diversified investordiversified investor• The beta coefficient measures systematic riskThe beta coefficient measures systematic risk

Systematic Risk Unsystematic Risk

Total Risk of the Investment


The Beta CoefficientThe Beta CoefficientThe FormulaThe Formula

ReturnsMarket theof Variance

market theand returns i''stock between Returns of CovarianceBeta

,

2iM

iMi

M

i,M

σ

COV

[9-7]


The Term – “Relevant Risk”The Term – “Relevant Risk”

• What does the term “relevant risk” mean in the context of the CAPM?What does the term “relevant risk” mean in the context of the CAPM?– It is generally assumed that all investors are wealth maximizing risk It is generally assumed that all investors are wealth maximizing risk

averse peopleaverse people– It is also assumed that the markets where these people trade are highly It is also assumed that the markets where these people trade are highly

efficientefficient– In a highly efficient market, the prices of all the securities adjust instantly 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 to cause the expected return of the investment to equal the required returnreturn

– When E(r) = R(r) then the market price of the stock equals its inherent When E(r) = R(r) then the market price of the stock equals its inherent worth (intrinsic value)worth (intrinsic value)

– In this perfect world, the R(r) then will justly and appropriately In this perfect world, the R(r) then will justly and appropriately compensate the investor only for the risk that they perceive as compensate the investor only for the risk that they perceive as relevant…relevant…

– Hence investors are only rewarded for systematic risk.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 SystematicThe Proportion of Total Risk that is Systematic

• Every investment in the financial markets vary with Every investment in the financial markets vary with respect to the percentage of total risk that is respect to the percentage of total risk that is systematic.systematic.

• Some stocks have virtually Some stocks have virtually no systematicno systematic risk. risk.– Such stocks are not influenced by the health of the economy in Such stocks are not influenced by the health of the economy in

general…their financial results are predominantly influenced by general…their financial results are predominantly influenced by company-specific factors.company-specific factors.

– An example is cigarette companies…people consume cigarettes An example is cigarette companies…people consume cigarettes because they are addicted…so it doesn’t matter whether the because they are addicted…so it doesn’t matter whether the economy is healthy or not…they just continue to smoke.economy is healthy or not…they just continue to smoke.

• Some stocks have a high proportion of their total risk Some stocks have a high proportion of their total risk that is systematicthat is systematic– Returns on these stocks are strongly influenced by the health of Returns on these stocks are strongly influenced by the health of

the economy.the economy.– Durable goods manufacturers tend to have a high degree of Durable goods manufacturers tend to have a high degree of

systematic risk.systematic risk.


The Formula Approach to Measuring the BetaThe Formula Approach to Measuring the Beta

)Var(k

)kCov(kBeta

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 Ex ante Return Data A Sample A Sample

A set of estimates of possible returns and their respective A set of estimates of possible returns and their respective probabilities looks as follows: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% The Total of the Probabilities must Equal 100%

This means that we have considered all of the possible outcomes This means that we have considered all of the possible outcomes in this discrete probability distributionin this discrete probability distribution

Possible Future State

of the Economy Probability



Boom 25.0% 28.0% 20.0%

Normal 50.0% 17.0% 11.0%

Recession 25.0% -14.0% -4.0%

100.0%


Measuring Measuring Expected Return on the StockExpected Return on the Stock From Ex Ante Return DataFrom Ex Ante Return Data

The expected return is weighted average returns from The expected return is weighted average returns from the given ex ante datathe 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 Measuring Expected Return on the MarketExpected Return on the Market From Ex Ante Return DataFrom Ex Ante Return Data

The expected return is weighted average returns from The expected return is weighted average returns from the given ex ante datathe 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 Measuring Variances, Standard Deviations of the Forecast Stock Returnsthe Forecast Stock Returns

Using the expected return, calculate the deviations away from the mean, square Using the expected return, calculate the deviations away from the mean, square those deviations and then weight the squared deviations by the probability of those deviations and then weight the squared deviations by the probability of their occurrence. Add up the weighted and squared deviations from the mean their occurrence. Add up the weighted and squared deviations from the mean

and you have found the variance!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 0.0064Normal 50.0% 0.17 0.085 0.05 0.0025 0.00125Recession 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 Measuring Variances, Standard Deviations of the Forecast Market Returnsthe Forecast Market Returns

Now do this for the possible returns on the marketNow 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 0.002756Normal 50.0% 0.11 0.055 0.015 0.000225 0.000113Recession 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%


CovarianceCovariance

From Chapter 8 you know the formula for the covariance From Chapter 8 you know the formula for the covariance between the returns on the stock and the returns on the between the returns on the stock and the returns on the market is:market is:

Covariance is an absolute measure of the degree of ‘co-Covariance is an absolute measure of the degree of ‘co-movement’ of returns. movement’ of returns.

)-)((Prob _

,1

_

,i BiB

n

iiiAAB kkkkCOV

[8-12]


Correlation CoefficientCorrelation Coefficient

Correlation is covariance normalized by the product of the standard Correlation is covariance normalized by the product of the standard deviations of both securities. It is a ‘relative measure’ of co-movement of deviations of both securities. It is a ‘relative measure’ of co-movement of returns on a scale from -1 to +1.returns on a scale from -1 to +1.

The formula for the correlation coefficient between the returns on the stock The formula for the correlation coefficient between the returns on the stock and the returns on the market is:and the returns on the market is:

The correlation coefficient will always have a value in the range of +1 to -The correlation coefficient will always have a value in the range of +1 to -1.1.

+1 – is perfect positive correlation (there is no diversification potential when combining these two +1 – is perfect positive correlation (there is no diversification potential when combining these two securities together in a two-asset portfolio.)securities together in a two-asset portfolio.)

- 1 - is perfect negative correlation (there should be a relative weighting mix of these two - 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) securities in a two-asset portfolio that will eliminate all portfolio risk)

BA

ABAB

COV

[8-13]


Measuring CovarianceMeasuring Covariancefrom Ex Ante Return Datafrom Ex Ante Return Data

Using the expected return (mean return) and given data measure the Using the expected return (mean return) and given data measure the deviations for both the market and the stock and multiply them deviations for both the market and the stock and multiply them

together with the probability of occurrence…then add the together with the probability of occurrence…then add the products up.products up.

(1) (2) (3) (4) (5) (6) (7) (8) "(9)

Possible Future

State of the Economy Prob.


(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% 0.0042Normal 50.0% 17.0% 0.085 11.0% 0.055 5.0% 1.5% 0.000375Recession 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 MeasuredThe Beta MeasuredUsing Ex Ante Covariance (stock, market) and Market VarianceUsing Ex Ante Covariance (stock, market) and Market Variance

Now you can substitute the values for covariance and the Now you can substitute the values for covariance and the variance of the returns on the market to find the beta of variance of the returns on the market to find the beta of

the stock:the stock:

8.1007425.

01335.

Var

CovBeta

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 1.0Lets Prove the Beta of the Market is 1.0

Let us assume we are comparing the possible market Let us assume we are comparing the possible market returns against itself…what will the beta be?returns against itself…what will the beta be?

(1) (2) (3) (4) (5) (6) (6) (7) (8)

Possible Future

State of the Economy Prob.


(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% 0.002756Normal 50.0% 11.0% 0.055 11.0% 0.055 1.5% 1.5% 0.000113Recession 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 !!!

Since the variance of the returns on the market is = .007425 …the beta for the market is indeed equal to 1.0 !!!

0.1007425.

007425.

Var

CovBeta

M

M`M,


Proving the Beta of Market = 1Proving the Beta of Market = 1

If you now place the covariance of the market with itself If you now place the covariance of the market with itself value in the beta formula you get:value in the beta formula you get:

0.1007425.

007425.

)Var(R

CovBeta

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 LineUsing the Security Market Line

Expected versus Required ReturnExpected versus Required Return


How Do We use Expected and Required How Do We use Expected and Required Rates of Return?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 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.can plot them on the SML and see if the stock is under or overpriced.


How Do We use Expected and Required How Do We use Expected and Required Rates of Return?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.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 This is what we would expect to see ‘normally’ or most of the time in an efficient

market where securities are properly priced.market where securities are properly priced.


Use of the Forecast BetaUse of the Forecast Beta

• We can use the forecast beta, together with an estimate of the We can use the forecast beta, together with an estimate of the risk-free rate and the market premium for risk to calculate the risk-free rate and the market premium for risk to calculate the investor’s required return on the stock using the CAPM:investor’s required return on the stock using the CAPM:

• This is a ‘market-determined’ return based on the current risk-This is a ‘market-determined’ return based on the current risk-free rate (RF) as measured by the 91-day, government of Canada 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 T-bill yield, and a current estimate of the market premium for risk (k(kMM – RF) – RF)

RF]k[EβRF Mi )( Return Required


ConclusionsConclusions

• Analysts can make estimates or forecasts for the Analysts can make estimates or forecasts for the returns on stock and returns on the market portfolio.returns on stock and returns on the market portfolio.

• Those forecasts can be analyzed to estimate the beta Those forecasts can be analyzed to estimate the beta coefficient for the stock. coefficient for the stock.

• The required return on a stock can then be calculated The required return on a stock can then be calculated using the CAPM – but you will need the stock’s beta using the CAPM – but you will need the stock’s beta coefficient, the expected return on the market coefficient, the expected return on the market portfolio and the risk-free rate.portfolio and the risk-free rate.

• The required return is then using in Dividend Discount The required return is then using in Dividend Discount Models to estimate the ‘intrinsic value’ (inherent Models to estimate the ‘intrinsic value’ (inherent worth) of the stock.worth) of the stock.

Calculating the Beta using Trailing Calculating the Beta using Trailing Holding Period ReturnsHolding Period Returns

APPENDIX 2APPENDIX 2


The Regression Approach to Measuring the The Regression Approach to Measuring the BetaBeta

• 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 EstimationHistorical Beta EstimationThe Approach Used to Create the Characteristic LineThe 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%

Characteristic Line (Regression)

-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 TSE 300

Ret

urn

s o

n S

tock

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 LineCharacteristic Line

• The characteristic line is a regression line that represents the The characteristic line is a regression line that represents the relationship between the returns on the stock and the returns on 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 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 future, assuming the future will be similar to the past.)

• The The slope of the Characteristic Lineslope of the Characteristic Line is the Beta Coefficient. is the Beta Coefficient.

• The degree to which the characteristic line explains the variability The degree to which the characteristic line explains the variability in the dependent variable (returns on the stock) is measured by in the dependent variable (returns on the stock) is measured by the coefficient of determination. (also known as the the coefficient of determination. (also known as the RR22 (r-squared (r-squared or coefficient of determination)).or coefficient of determination)).

• If the coefficient of determination equals 1.00, this would mean If the coefficient of determination equals 1.00, this would mean that all of the points of observation would lie on the line. This that all of the points of observation would lie on the line. This would mean that the characteristic line would explain 100% of the would mean that the characteristic line would explain 100% of the variability of the dependent variable.variability of the dependent variable.

• The The alpha alpha is the vertical intercept of the regression is the vertical intercept of the regression (characteristic line). Many stock analysts search out stocks with (characteristic line). Many stock analysts search out stocks with high alphas.high alphas.


Low RLow R22

• An RAn R22 that approaches 0.00 (or 0%) indicates that the that approaches 0.00 (or 0%) indicates that the characteristic (regression) line explains virtually none of the characteristic (regression) line explains virtually none of the variability in the dependent variable.variability in the dependent variable.

• This means that virtually of the risk of the security is This means that virtually of the risk of the security is ‘company-specific’. ‘company-specific’.

• This also means that the regression model has virtually no This also means that the regression model has virtually no predictive ability.predictive ability.

• In this case, you should use other approaches to value the In this case, you should use other approaches to value the stock…do not use the estimated beta coefficient.stock…do not use the estimated beta coefficient.

(See the following slide for an illustration of a low r-square)(See the following slide for an illustration of a low r-square)


Characteristic Line for Imperial TobaccoCharacteristic Line for Imperial TobaccoAn Example of Volatility that is Primarily Company-SpecificAn 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 alpha

• R-square is very low ≈ 0.02

• Beta is largely irrelevant


High RHigh R22

• An RAn R22 that approaches 1.00 (or 100%) indicates that the that approaches 1.00 (or 100%) indicates that the characteristic (regression) line explains virtually all of the characteristic (regression) line explains virtually all of the variability in the dependent variable.variability in the dependent variable.

• This means that virtually of the risk of the security is This means that virtually of the risk of the security is ‘systematic’. ‘systematic’.

• This also means that the regression model has a strong This also means that the regression model has a strong predictive ability. … if you can predict what the market will predictive ability. … if you can predict what the market will do…then you can predict the returns on the stock itself with do…then you can predict the returns on the stock itself with a great deal of accuracy.a great deal of accuracy.


Characteristic Line General MotorsCharacteristic Line General MotorsA Positive Beta with Predictive PowerA 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

• Positive alpha

• R-square is very high ≈ 0.9

• Beta is positive and close to 1.0


An Unusual Characteristic LineAn Unusual Characteristic LineA Negative Beta with Predictive PowerA Negative Beta with Predictive Power


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

• Positive alpha

• R-square is very high

• Beta is negative <0.0 and > -1.0


Diversifiable RiskDiversifiable Risk(Non-systematic Risk)(Non-systematic Risk)

• Volatility in a security’s returns caused by company-Volatility in a security’s returns caused by company-specific factors (both positive and negative) such as:specific factors (both positive and negative) such as:– a single company strikea single company strike

– a spectacular innovation discovered through the company’s R&D programa spectacular innovation discovered through the company’s R&D program

– equipment failure for that one companyequipment failure for that one company

– management competence or management incompetence for that management competence or management incompetence for that particular firmparticular firm

– a jet carrying the senior management team of the firm crashes (this could 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 be either a positive or negative event, depending on the competence of the management team)the management team)

– the patented formula for a new drug discovered by the firm.the patented formula for a new drug discovered by the firm.

• Obviously, diversifiable risk is that unique factor that Obviously, diversifiable risk is that unique factor that influences only the one firm. influences only the one firm.


OK – lets go back and look at raw data OK – lets go back and look at raw data gathering and data normalizationgathering and data normalization

• A common source for stock of information is Yahoo.comA common source for stock of information is Yahoo.com• You will also need to go to the library a use the TSX Review You will also need to go to the library a use the TSX Review

(a monthly periodical) – to obtain:(a monthly periodical) – to obtain:– Number of shares outstanding for the firm each monthNumber of shares outstanding for the firm each month– Ending values for the total return composite index (surrogate for the Ending values for the total return composite index (surrogate for the

market)market)

• You want data for at least 30 months.You want data for at least 30 months.• For each month you will need:For each month you will need:

– Ending stock priceEnding stock price– Number of shares outstanding for the stockNumber of shares outstanding for the stock– Dividend per share paid during the month for the stockDividend per share paid during the month for the stock– Ending value of the market indicator series you plan to use (ie. TSE Ending value of the market indicator series you plan to use (ie. TSE

300 total return composite index)300 total return composite index)

Demonstration Through ExampleDemonstration Through Example

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


Five Year Stock Price Chart for AL.TOFive Year Stock Price Chart for AL.TO


Spreadsheet Data From YahooSpreadsheet Data From Yahoo

Process:Process:

– Go to Go to http://ca.finance.yahoo.comhttp://ca.finance.yahoo.com– Use the symbol lookup function to search for the Use the symbol lookup function to search for the

company you are interested in studying.company you are interested in studying.– Use the historical quotes button…and get 30 months Use the historical quotes button…and get 30 months

of historical data.of historical data.– Use the download in spreadsheet format feature to Use the download in spreadsheet format feature to

save the data to your hard drive.save the data to your hard drive.


Spreadsheet Data From YahooSpreadsheet Data From YahooAlcan ExampleAlcan Example

The raw downloaded data should look like this:The raw downloaded data should look like this:

Date Open High Low Close Volume01-May-02 57.46 62.39 56.61 59.22 75387401-Apr-02 62.9 63.61 56.25 57.9 87921001-Mar-02 64.9 66.81 61.68 63.03 97436801-Feb-02 61.65 65.67 58.75 64.86 83637302-Jan-02 57.15 62.37 54.93 61.85 98903003-Dec-01 56.6 60.49 55.2 57.15 83328001-Nov-01 49 58.02 47.08 56.69 779509


Spreadsheet Data From Yahoo Spreadsheet Data From Yahoo Alcan ExampleAlcan Example

The raw downloaded data should look like this:The raw downloaded data should look like this:

Date Open High Low Close Volume01-May-02 57.46 62.39 56.61 59.22 75387401-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

Volume of trading done

in the stock on the TSE in the

month in numbers of board lots

The day, month and year

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

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


Spreadsheet Data From Yahoo Spreadsheet Data From Yahoo Alcan ExampleAlcan Example

From Yahoo, the only information you can use is the From Yahoo, the only information you can use is the closing price per share and the date. Just delete the closing price per share and the date. Just delete the other columns.other columns.

Date Close01-May-02 59.2201-Apr-02 57.901-Mar-02 63.0301-Feb-02 64.8602-Jan-02 61.85


Acquiring the Additional Information You Need Acquiring the Additional Information You Need Alcan ExampleAlcan Example

In addition to the closing price of the stock on a per share In addition to the closing price of the stock on a per share basis, you will need to find out how many shares were basis, you will need to find out how many shares were outstanding at the end of the month and whether any outstanding at the end of the month and whether any dividends were paid during the month.dividends were paid during the month.

You will also want to find the end-of-the-month value of the 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 S&P/TSX Total Return Composite Index (look in the green pages of the TSX Review)pages of the TSX Review)

You can find all of this in You can find all of this in The TSX ReviewThe TSX Review periodical. periodical.


Raw Company Data Raw Company Data Alcan ExampleAlcan Example

DateIssued Capital

Closing Price for Alcan

AL.TO

Cash Dividends per Share

01-May-02 321,400,589 $59.22 $0.0001-Apr-02 321,400,589 $57.90 $0.1501-Mar-02 321,400,589 $63.03 $0.0001-Feb-02 321,400,589 $64.86 $0.0002-Jan-02 160,700,295 $123.70 $0.3001-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 Normalizing the Raw Company Data Alcan ExampleAlcan Example

DateIssued 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 $0.0001-Apr-02 321,400,589 $57.90 $0.15 1.00 $57.90 $0.1501-Mar-02 321,400,589 $63.03 $0.00 1.00 $63.03 $0.0001-Feb-02 321,400,589 $64.86 $0.00 1.00 $64.86 $0.0002-Jan-02 160,700,295 $123.70 $0.30 0.50 $61.85 $0.1501-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 Calculating the HPR on the stock from the Normalized DataNormalized Data

DateNormalized 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

DPPHPR


Now Put the data from the S&P/TSX Total Now Put the data from the S&P/TSX Total Return Composite Index inReturn Composite Index in



Ending TSX

Value01-May-02 $59.22 $0.00 2.28% 16911.3301-Apr-02 $57.90 $0.15 -7.90% 16903.3601-Mar-02 $63.03 $0.00 -2.82% 17308.4101-Feb-02 $64.86 $0.00 4.87% 16801.8202-Jan-02 $61.85 $0.15 23.36% 16908.1101-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.

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 IndexNow Calculate the HPR on the Market Index



Ending TSX

ValueHPR 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.

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

PPHPR


Regression In ExcelRegression In Excel

• If you haven’t already…go to the tools If you haven’t already…go to the tools menu…down to add-ins and check off the VBA menu…down to add-ins and check off the VBA Analysis PacAnalysis Pac

• When you go back to the tools menu, you When you go back to the tools menu, you should now find the Data Analysis bar, under should now find the Data Analysis bar, under that find regression, define your dependent that find regression, define your dependent and independent variable ranges, your output and independent variable ranges, your output range and run the regression.range and run the regression.


RegressionRegressionDefining the Data RangesDefining the Data Ranges



Ending TSX

ValueHPR 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 independent variable is the returns on the Market.The dependent variable is the returns on the Stock.The dependent variable is the returns on the Stock.


Now Use the Regression Function in Excel to Now Use the Regression Function in Excel to regress the returns of the stock against the regress the returns of the stock against the

returns of the marketreturns of the market

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.05300947R Square 0.00281Adjusted R Square -0.2464875Standard Error 5.79609628Observations 6

ANOVAdf SS MS F Significance F

Regression 1 0.3786694 0.37866937 0.011271689 0.920560274Residual 4 134.37893 33.5947321Total 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 67.38837X 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 Finalize Your Chart Alcan ExampleAlcan Example

• You can use the charting feature in Excel to create a You can use the charting feature in Excel to create a scatter plot of the points and to put a line of best fit scatter plot of the points and to put a line of best fit (the characteristic line) through the points.(the characteristic line) through the points.

• In Excel, you can edit the chart after it is created by In Excel, you can edit the chart after it is created by placing the cursor over the chart and ‘right-clicking’ placing the cursor over the chart and ‘right-clicking’ your mouse.your mouse.

• In this edit mode, you can ask it to add a trendline In this edit mode, you can ask it to add a trendline (regression line)(regression line)

• Finally, you will want to interpret the Beta (X-Finally, you will want to interpret the Beta (X-coefficient) the alpha (vertical intercept) and the coefficient) the alpha (vertical intercept) and the coefficient of determination.coefficient of determination.


The Beta The Beta Alcan ExampleAlcan Example

• Obviously the beta (X-coefficient) can simply Obviously the beta (X-coefficient) can simply be read from the regression output.be read from the regression output.– In this case it was 3.56 making Alcan’s returns more In this case it was 3.56 making Alcan’s returns more

than 3 times as volatile as the market as a whole.than 3 times as volatile as the market as a whole.– Of course, in this simple example with only 5 Of course, in this simple example with only 5

observations, you wouldn’t want to draw any serious observations, you wouldn’t want to draw any serious conclusions from this estimate.conclusions from this estimate.


CopyrightCopyright

Copyright © 2007 John Wiley & Copyright © 2007 John Wiley & Sons Canada, Ltd. All rights Sons Canada, Ltd. All rights reserved. Reproduction or reserved. Reproduction or translation of this work beyond that translation of this work beyond that permitted by Access Copyright (the permitted by Access Copyright (the Canadian copyright licensing Canadian copyright licensing agency) is unlawful. Requests for agency) is unlawful. Requests for further information should be further information should be addressed to the Permissions addressed to the Permissions Department, John Wiley & Sons Department, John Wiley & Sons Canada, Ltd.Canada, Ltd. The purchaser may The purchaser may make back-up copies for his or her make back-up copies for his or her own use only and not for distribution own use only and not for distribution or resale.or resale. The author and the The author and the publisher assume no responsibility publisher assume no responsibility for errors, omissions, or damages for errors, omissions, or damages caused by the use of these files or caused by the use of these files or programs or from the use of the programs or from the use of the information contained herein.information contained herein.

Documents

The capital-asset-pricing-model-capm75