The Capital Asset Pricing Model (CAPM) Imp

8/9/2019 The Capital Asset Pricing Model (CAPM) Imp

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Prepared byPrepared byKen HartviksenKen Hartviksen

INTRODUCTION TO

CORPORATE FINANCE

Laurence Booth W. Sean Cleary

Chapter 9 The Capital Asset PricingModel


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CHAPTER 9 The Capital Asset Pricing

Model (CAPM)


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

Lecture AgendaLecture Agenda

Learning ObjectivesLearning Objectives Important TermsImportant Terms The New Efficient Frontier The New Efficient Frontier

The Capital Asset Pricing Model The Capital Asset Pricing Model The CAPM and Market Risk The 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


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Learning ObjectivesLearning Objectives

1.1. What happens if all investors are rational and risk averse.What happens if all investors are rational and risk averse.2.2. How modern portfolio theory is extended to develop theHow modern portfolio theory is extended to develop the

capital market line, which determines how expectedcapital market line, which determines how expectedreturns on portfolios are determined.returns on portfolios are determined.

3.3. How to assess the performance of mutual fund managersHow to assess the performance of mutual fund managers4.4. How the Capital Asset Pricing Models (CAPM) securityHow the Capital Asset Pricing Models (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.


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Important Chapter TermsImportant Chapter Terms

Arbitrage pricing theoryArbitrage pricing theory(APT)(APT)

Capital Asset PricingCapital Asset PricingModel (CAPM)Model (CAPM)

Capital market lineCapital 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) efficientNew (or super) efficient

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

(SML)(SML) Separation theorumSeparation theorum

Sharpe ratioSharpe ratio

Short positionShort position Tangent portfolio Tangent portfolio


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Achievable Portfolio CombinationsAchievable Portfolio Combinations

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


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Example of Portfolio Combinations andExample of Portfolio Combinations andCorrelationCorrelation

AssetExpected

ReturnStandardDeviation

CorrelationCoefficient

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

Weight of A Weight of B

Expected

Return

Standard

Deviation100% 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

combinationsimplyassumes youinvest solelyin Asset A

The second

portfolioassumes 99%in A and 1% inB. Notice the

increase inreturn and the

decrease inportfolio risk!

You repeat thisprocedure

down until youhave determine

the portfoliocharacteristics

for all 100

portfolios.

Next plot thereturns on a

graph (see thenext slide)


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Example of Portfolio Combinations andExample of Portfolio Combinations andCorrelationCorrelation

Attainable Portfolio Combinations for aTwo 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

E x p e c t e

d R e

t u r n o

f t h e

P o r t

f o l i o


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Two Asset Efficient Frontier Two Asset Efficient Frontier

Figure 8 10 describes five differentFigure 8 10 describes five differentportfolios (A,B,C,D and E in reference to theportfolios (A,B,C,D and E in reference to the

attainable set of portfolio combinations of thisattainable set of portfolio combinations of thistwo asset portfolio.two asset portfolio.

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


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Efficient Frontier Efficient Frontier The Two-Asset Portfolio CombinationsThe Two-Asset Portfolio Combinations

A is not attainable

B,E lie on theefficient frontier andare attainable

E is the minimumvariance portfolio(lowest riskcombination)

C, D areattainable but aredominated bysuperior portfoliosthat line on the lineabove E

8 - 10 FIGURE

E

xpected Return %

Standard Deviation (%)

A

E

B

C

D


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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 theIn a real world investment universe with all of theinvestment alternatives (stocks, bonds, money marketinvestment alternatives (stocks, bonds, money marketsecurities, hybrid instruments, gold real estate, etc.) itsecurities, hybrid instruments, gold real estate, etc.) itis possible to construct many different alternativeis possible to construct many different alternativeportfolios out of risky securities.portfolios out of risky securities.

Each portfolio will have its own unique expected returnEach portfolio will have its own unique expected returnand risk.and risk.

Whenever you construct a portfolio, you can measureWhenever you construct a portfolio, you can measuretwo fundamental characteristics of the portfolio:two fundamental characteristics of the portfolio: Portfolio expected return (Portfolio expected return ( ER ER p p )) Portfolio risk (Portfolio risk ( p p ))


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The Achievable Set of PortfolioThe Achievable Set of PortfolioCombinationsCombinations

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

The results (in terms of ER The results (in terms of ER pp andand pp ))might lookmight looklike the graph on the following page:like the graph on the following page:


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

Portfolio Risk ( p)

10 AchievableRisky PortfolioCombinations

ER p


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The Achievable Set of PortfolioThe Achievable Set of PortfolioCombinationsCombinations

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

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


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Achievable Portfolio CombinationsAchievable Portfolio CombinationsThirty Combinations Naively CreatedThirty Combinations Naively Created

Portfolio Risk ( p)

30 Risky PortfolioCombinations

ER p


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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 weightdifferent portfolios naively varying the weightof 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 asachievable portfolio combinations asindicated on the following slide:indicated on the following slide:


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Portfolio Risk ( p)

ER p

Achievable Portfolio CombinationsAchievable Portfolio CombinationsMore Possible Combinations CreatedMore Possible Combinations Created

E

E is the

minimumvarianceportfolio Achievable Set of

Risky PortfolioCombinations

The highlightedportfolios areefficient in thatthey offer thehighest rate of return for a given

level of risk.Rationale investorswill choose onlyfrom this efficientset.


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The Efficient Frontier The Efficient Frontier



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Portfolio Risk ( p)

Achievable Set of Risky PortfolioCombinations

ER p

Achievable Portfolio CombinationsAchievable Portfolio CombinationsEfficient Frontier (Set)Efficient Frontier (Set)

E

Efficientfrontier is theset of achievable

portfoliocombinationsthat offer thehighest rateof return for agiven level of risk.


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The New Efficient Frontier The New Efficient Frontier Efficient PortfoliosEfficient Portfolios

Figure 9 1illustratesthreeachievable

portfoliocombinationsthat areefficient (noother achievable

portfolio thatoffers thesame risk,offers a higher return.)

Risk

9 - 1 FIGURE

Efficient Frontier ER

MVP

A

B


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

We assume investors are risk-averse wealth maximizers.We assume investors are risk-averse wealth maximizers. This means they will not willingly undertake fair gamble. This means they will not willingly undertake fair 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 beThe corollary of this is that the investor needs a risk premium to beinduced into a risky situation.induced into a risky situation.

Evidence of this is the willingness of investors to pay insuranceEvidence of this is the willingness of investors to pay insurancepremiums to get out of risky situations.premiums to get out of risky situations.

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


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The New Efficient Frontier andThe New Efficient Frontier and

Separation TheoremSeparation Theorem The Capital Asset Pricing Model The Capital Asset Pricing Model

(CAPM)(CAPM)


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Risk-free InvestingRisk-free Investing

When we introduce the presence of a risk-freeWhen we introduce the presence of a risk-freeinvestment, a whole new set of portfolioinvestment, a whole new set of portfoliocombinations becomes possible.combinations becomes possible.

We can estimate the return on a portfolioWe can estimate the return on a portfoliomade up of made up of RFRF asset and a risky assetasset and a risky asset AAletting the weightletting the weight ww invested in the riskyinvested in the risky

asset and the weight invested in RF asasset and the weight invested in RF as (1 w)(1 w)


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The New Efficient Frontier The New Efficient Frontier Risk-Free InvestingRisk-Free Investing

Expected return on a two asset portfolio made up of riskyExpected return on a two asset portfolio made up of riskyassetasset A A andand RF RF ::

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

RF)-(ER RFER A p w+=[9-1]


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The New Efficient Frontier The New Efficient Frontier Attainable Portfolios UsingAttainable Portfolios Using RF RF andand A A

9 - 2 FIGURE

Risk

ER

RF

A

A p w=[9-2]

Equation 9 2illustrateswhat you canseeportfoliorisk increasesin directproportion tothe amountinvested in therisky asset.

RF-)E(R

RFER A

A P P

+=[9-3]

Rearranging 9-2 where w=

p /

Aand

substituting inEquation 1 weget anequation for astraight line

with aconstantslope.

This meansyou canachieve anyportfolio

combinationalong the bluecoloured linesimply bychanging therelative weightof RF and A inthe two assetportfolio.


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The New Efficient Frontier The New Efficient Frontier Attainable Portfolios using theAttainable Portfolios using the RF RF andand A, A, andand RF RF andand T T

Which riskyportfoliowould arational risk-averseinvestor choose in thepresence of aRF investment?

Portfolio A?

TangentPortfolio T ?

9 - 3 FIGURE

Risk

ER

RF

A

T


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The New Efficient Frontier The New Efficient Frontier Efficient Portfolios using the Tangent PortfolioEfficient Portfolios using the Tangent Portfolio T T

9 - 3 FIGURE

Risk

ER

RF

A

T

Clearly RF withT (the tangentportfolio) offersa series of portfolio

combinationsthat dominatethose producedby RF and A .

Further, they

dominate all butone portfolio onthe efficientfrontier!


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The New Efficient Frontier The New Efficient Frontier Lending PortfoliosLending Portfolios

9 - 3 FIGURE

Risk

ER

RF

A

T

Portfoliosbetween RF and T arelendingportfolios,

because theyare achieved byinvesting in theTangentPortfolio andlending funds tothe government(purchasing aT-bill, the RF ).

Lending Portfolios


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The New Efficient Frontier The New Efficient Frontier Borrowing PortfoliosBorrowing Portfolios

9 - 3 FIGURE

Risk

ER

RF

A

T

The line can beextended to risklevels beyondT byborrowing at RF

and investing itin T. This is aleveredinvestment thatincreases bothrisk andexpected returnof the portfolio.

Lending Portfolios Borrowing Portfolios


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9 - 4 FIGURE

ER

RF

A2

T

A

B

B2

Capital Market Line

The New Efficient Frontier The New Efficient Frontier The New (Super) Efficient Frontier The New (Super) Efficient Frontier

The optimalrisky portfolio

(the market

portfolio M)

Clearly RF withT (the marketportfolio) offersa series of

portfoliocombinationsthat dominatethose producedby RF and A.

Further, theydominate all butone portfolio onthe efficientfrontier!

This is nowcalled the new(or super)efficient frontier of risky

portfolios.Investors canachieve anyone of theseportfoliocombinations byborrowing or investing in RFin combinationwith the marketportfolio.


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The New Efficient Frontier The New Efficient Frontier The Implications Separation Theorem Market PortfolioThe Implications Separation Theorem Market Portfolio

All investors will only hold individually-determinedAll investors will only hold individually-determinedcombinations 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 theorem The separation theorem The investment decision (how to construct the portfolio of riskyThe investment decision (how to construct the portfolio of risky

assets) is separate from the financing decision (how muchassets) is separate from the financing decision (how muchshould 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 Condition The Equilibrium Condition The market portfolio must be the tangent portfolio T if everyoneThe market portfolio must be the tangent portfolio T if everyone

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


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ER

RF

M

CML

The New Efficient Frontier The New Efficient Frontier The Capital Market LineThe Capital Market Line

The optimalrisky portfolio

(the market

portfolio M)

The CML is thatset of superior portfoliocombinations

that areachievable inthe presence of the equilibriumcondition.


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The Capital Asset Pricing ModelThe Capital Asset Pricing Model

The Hypothesized Relationship The Hypothesized Relationshipbetween Risk and Returnbetween Risk and Return


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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 relationshipThere are no prices on the model, instead it hypothesizes the relationshipbetween 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.


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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 stocks intrinsicThe relevant risk in the dividend discount model to estimate a stocks intrinsic

(inherent economic worth) value.(inherent economic worth) value.

(As illustrated below)(As illustrated below)

Estimate InvestmentsRisk (Beta Coefficient)

Determine InvestmentsRequired Return

Estimate theInvestments IntrinsicValue

Compare to the actualstock price in themarket

2iM

i,M

COV = )( iM i RF ER RF k +=

g k D P c

= 10 Is the stockfairly priced?


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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 returns,All investors have identical expectations about expected returns,

standard deviations, and correlation coefficients for all securities.standard deviations, and correlation coefficients for all securities.2.2. All investors have the same one-period investment time horizon.All investors have the same one-period investment time 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 areThere 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 theThere are many investors, and no single investor can affect the

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

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


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Market Portfolio and Capital Market LineMarket Portfolio and Capital Market Line

The assumptions have the following The assumptions have the followingimplications:implications:1.1. The optimal risky portfolio is the one that isThe optimal risky portfolio is the one that is

tangent to the efficient frontier on a line that is drawntangent to the efficient frontier on a line that is drawnfrom RF. This portfolio will be the same for allfrom RF. This portfolio will be the same for allinvestors.investors.

2.2. This optimal risky portfolio will be theThis optimal risky portfolio will be the market market portfolio portfolio (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)


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The Capital Market LineThe Capital Market Line

9 - 5 FIGURE

ER

RF

MER M

M

P M

M P

RF ER RF k

+=

CML

The CML is that

set of achievableportfolio

combinationsthat are possiblewhen investing

in only twoassets (the

market portfolioand the risk-free

asset (RF).

The marketportfolio is theoptimal riskyportfolio, it

contains all riskysecurities andlies tangent (T)on the efficient

frontier.

The CML hasstandard

deviation of portfolio returns

as theindependent

variable.


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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 expectedThe slope of the CML is the incremental expectedreturn divided by the incremental risk.return divided by the incremental risk.

This is calledThis is called the market price for risk. Or the 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

CMLtheof Slope M

M

=[9-4]


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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 aSolving for the expected return on a portfolio in the presence of aRF asset and given theRF asset and given the market price for risk :market price for risk :

Where:Where: ERER MM = expected return on the market portfolio M= expected return on the market portfolio M

MM = the standard deviation of returns on the market portfolio= the standard deviation of returns on the market portfolio PP = the standard deviation of returns on the efficient portfolio being= the standard deviation of returns on the efficient portfolio being

consideredconsidered

)(

- RF ERRF R E P M

M P +=[9-5]


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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 aIn an efficient capital market investors will require areturn on a portfolio that compensates them for thereturn 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 alongThis means that portfolios should offer returns alongthe CML.the CML.


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

A is anundervaluedportfolio. Expectedreturn is greater than the requiredreturn.

Demand for Portfolio A willincrease driving upthe price, andtherefore theexpected return willfall until expectedequals required(market equilibriumcondition isachieved.)

Requiredreturn on A

Expectedreturn on A

B is a portfolio thatoffers and expectedreturn equal to therequired return.

9 - 6 FIGURE

ER

RF

B

C

A

CML

C is an overvaluedportfolio. Expectedreturn is less thanthe required return.

Selling pressure

will cause the priceto fall and the yieldto rise untilexpected equalsthe required return.

RequiredReturn on C

ExpectedReturn on C


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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-adjustedWilliam Sharpe identified a ratio that can be used to assess the risk-adjustedperformance of managed funds (such as mutual funds and pension plans).performance of managed funds (such as mutual funds and 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 wellSharpe ratio is a measure of portfolio performance that describes how wellan assets returns compensate investors for the risk taken.an assets returns compensate investors for the risk taken.

Its value is the premium earned over the RF divided by portfolio riskso it isIts value is the premium earned over the RF divided by portfolio riskso it ismeasuring valued added per unit of risk.measuring valued added per unit of risk. Sharpe ratios are calculated ex post (after-the-fact) and are used to rankSharpe ratios are calculated ex post (after-the-fact) and are used to rank

portfolios or assess the effectiveness of the portfolio manager in addingportfolios or assess the effectiveness of the portfolio manager in addingvalue to the portfolio over and above a benchmark.value to the portfolio over and above a benchmark.

RF-ER

ratioSharpe P

P

=[9-6]


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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 Financehowever, the recent announcement of FinanceMinister Flaherty and the subsequent drop in IncomeMinister Flaherty and the subsequent drop in IncomeTrust values has done much to eliminate thisTrust values has done much to eliminate thishistorical performance.historical performance.


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Income Trust Estimated ValuesIncome Trust Estimated Values

Return P Sharpe

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 TruCost of Income Trusts," Canadian Investment Review 19, no. 5 (Spring2006), Table 3, p. 15.


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CAPM and Market RiskCAPM and Market Risk

The Capital Asset Pricing Model The Capital Asset Pricing Model


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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 areare

caused by two different factors:caused by two different factors:

Non-diversifiable risk (system wide changes in the economy andNon-diversifiable risk (system wide changes in the economy andmarkets that affect all securities in varying degrees)markets that affect all securities in varying degrees) Diversifiable risk (company-specific factors that affect the returnsDiversifiable risk (company-specific factors that affect the returns

of only one security)of only one security) Figure 9 7 illustrates what happens to portfolio riskFigure 9 7 illustrates what happens to portfolio risk

as the portfolio is first invested in only oneas the portfolio is first invested in only oneinvestment, and then slowly invested, naively, in moreinvestment, and then slowly invested, naively, in moreand more securities.and more securities.


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The CAPM and Market RiskThe CAPM and Market RiskPortfolio Risk and DiversificationPortfolio Risk and Diversification

9 - 7 FIGURE

Number of Securities

Total Risk ()

Unique (Non-systematic) Risk

Market (Systematic) Risk

Market or systematicrisk is riskthat cannot

be eliminatedfrom the

portfolio byinvesting theportfolio into

more and

differentsecurities.


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Relevant RiskRelevant RiskDrawing a Conclusion from Figure 9 - 7Drawing a Conclusion from Figure 9 - 7

Figure 9 7 demonstrates that an individual securitiesFigure 9 7 demonstrates that an individual securitiesvolatility 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 isWhen combined into portfolios, company-specific risk isdiversified away.diversified away. Since all investors are diversified then in an efficientSince all investors are diversified then in an efficient

market, no-one would be willing to pay a premium formarket, no-one would be willing to pay a premium forcompany-specific risk.company-specific risk.

Relevant risk to diversified investors then is systematicRelevant risk to diversified investors then is systematicrisk.risk.

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


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Measuring Systematic RiskMeasuring Systematic Risk

The Beta CoefficientThe Beta Coefficient The Capital Asset Pricing Model The Capital Asset Pricing Model

(CAPM)(CAPM)


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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 numberAs a coefficient the beta is a pure numberand has no units of measure.and has no units of measure.


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


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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 8Market Returns (%)

The slope of the regression

line is beta.

The line of best fit isknown in

finance as thecharacteristic

line.

The plottedpoints are the

coincidentrates of returnearned on the

investmentand the marketportfolio over past periods.


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The Formula for the Beta CoefficientThe Formula for the Beta Coefficient

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

,2iM

iM i

M

i,M

COV

==[9-7]


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

The beta of the market portfolio is ALWAYS = 1.0 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 The beta of a security compares the volatility of its returns to the volatility of themarket returns:market returns:

ss = 1.0= 1.0 -- the security has the same volatility as the market as athe security has the same volatility as the market as awholewhole

ss > 1.0> 1.0 -- aggressive investment with volatility of returns greateraggressive investment with volatility of returns greaterthan the marketthan the market

ss < 1.0< 1.0 -- defensive investment with volatility of returns less thandefensive investment with volatility of returns less thanthe marketthe market

ss < 0.0< 0.0 -- an investment with returns that are negatively correlatedan investment with returns that are negatively correlatedwith the returns of the marketwith 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


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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 - P rec ious Metals & Minerals 2.52Imperial Oil Ltd. Energy - Oil & Gas: Integrated Oils 0.80

Table 9-2 Canadian BETAS

Source: Res earch Insight, Compus tat North Ame rican databas e, June 2006.


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The Beta of a PortfolioThe Beta of a Portfolio

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

Weights of individual assets are found by dividing theWeights of individual assets are found by dividing thevalue of the investment by the value of the totalvalue of the investment by the value of the totalportfolio.portfolio.

... nn B B A A P www +++=[9-8]


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The Security Market LineThe Security Market Line



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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 (theThe SML is the hypothesized relationship between return (thedependent 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:k k ii = the required return on security i = the required return on security i

ERER MM RF = market premium for risk RF = market premium for risk ii = 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)

)( iM i RF ER RF k +=[9-9]


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

9 - 9 FIGURE

M = 1

ER

RF

MER M

iM i RF ER RF k )( +=

The SML isused topredict

requiredreturns for individual

securities

The SMLuses thebeta

coefficient asthe measureof relevant

risk.


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9 - 10 FIGURE

A

ER

RF

B

A

B

SML

The CAPM and Market RiskThe CAPM and Market RiskThe SML and Security ValuationThe SML and Security Valuation

iM i RF ER RF k )( +=Required returnsare forecast usingthis equation.

You can see thatthe required returnon any security isa function of itssystematic risk ( )and market

factors ( RF and market premium for risk)

A is anundervaluedsecurity becauseits expected returnis greater than therequired return.

Investors willflock to A and bidup the price

causing expectedreturn to fall till itequals therequired return.

RequiredReturn A

ExpectedReturn A

Similarly, B is anovervaluedsecurity.

Investors will sellto lock in gains,but the sellingpressure willcause the marketprice to fall,

causing theexpected return torise until it equalsthe requiredreturn.


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Alternative Pricing ModelsAlternative Pricing Models



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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 anteEx 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 returnsBeta 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 haveBecause 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)


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

A pricing model that uses three factors to relateA pricing model that uses three factors to relateexpected 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 firms common equity (MVE)The market value of a firms common equity (MVE)3.3. Ratio of a firms book equity value to its market value of equity.Ratio of a firms book equity value to its market value of equity.

(BE/MVE)(BE/MVE)

This model has become popular, and many think itThis model has become popular, and many think itdoes a better job than the CAPM in explaining exdoes a better job than the CAPM in explaining exante stock returns.ante stock returns.


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

A pricing model that uses multiple factors to relate expectedA pricing model that uses multiple factors to relate expectedreturns to risk by assuming that asset returns are linearly relatedreturns to risk by assuming that asset returns are linearly relatedto a set of indexes, which proxy risk factors that influenceto a set of indexes, which proxy risk factors that influencesecurity returns.security returns.

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

Underlying factors represent broad economic forces which areUnderlying factors represent broad economic forces which areinherently unpredictable.inherently unpredictable.

... 11110 niniii

F b F b F ba ER ++++=[9-10]


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

Where:Where: ER ER i i = the expected return on security i= the expected return on security i aa 0 0 = the expected return on a security with zero systematic risk= the expected return on a security with zero systematic risk bb i i = the sensitivity of security i to a given risk factor = the sensitivity of security i to a given risk factor F F i i = the risk premium for a given risk factor = the risk premium for a given risk factor

The model demonstrates that a securitys risk is based on its sensitivity to broadThe model demonstrates that a securitys risk is based on its sensitivity to broadeconomic forces.economic forces.

... 11110 niniii

F b F b F ba ER ++++=[9-10]


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

Underlying factors represent broad economic forcesUnderlying factors represent broad economic forceswhich 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 inflation

2.2. 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 isnt forecast, cant be usedClearly, something that isnt forecast, cant be usedto price securities todaythey can only be used toto price securities todaythey can only be used toexplain prices after the fact.explain prices after the fact.


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Concept Review QuestionsConcept Review Questions

The Capital Asset Pricing Model The Capital Asset Pricing Model

C R i Q i 1


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Concept Review Question 1Concept Review Question 1Risk AversionRisk Aversion

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


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

APPENDIX 1APPENDIX 1

C l l ti g B t C ffi i t U i g E A tCalculating a Beta Coefficient Using Ex Ante


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Calculating a Beta Coefficient Using Ex AnteCalculating a Beta Coefficient Using Ex AnteReturnsReturns

Ex Ante means forecastEx 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 AmericanA good example of this is Air Canada or AmericanAirlines, before and after September 11, 2001. AfterAirlines, before and after September 11, 2001. Afterthe World Trade Centre terrorist attacks, athe World Trade Centre terrorist attacks, afundamental 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 inestimating future returns.estimating future returns.


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Appendix 1 AgendaAppendix 1 Agenda

The beta coefficient The 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


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The Beta CoefficientThe Beta Coefficient

Under the theory of the Capital Asset Pricing ModelUnder the theory of the Capital Asset Pricing Modeltotal 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 theSystematic risk is the only relevant risk to the

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

Systematic Risk Unsystematic Risk

Total Risk of the Investment

Th B C ffi iTh B C ffi i


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The Beta CoefficientThe Beta CoefficientThe FormulaThe Formula

ReturnsMarkettheof Variance

markettheandreturnsi''stock betweenReturnsof CovarianceBeta =

,2iM

iM i

M

i,M

COV

==[9-7]


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The Term Relevant RiskThe 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 riskIt 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 highlyIt 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 instantlyIn a highly efficient market, the prices of all the securities adjust instantlyto cause the expected return of the investment to equal the requiredto 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 inherentWhen 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 appropriatelyIn this pe rfect world, the R(r) then will justly and appropriately

compensate the investor only for the risk that they perceive ascompen sate the investor only for the risk that they perceive asrelevantrelevant

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 riskoccurs when R-square is high, and the beta coefficient is greater than 1.0


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The Proportion of Total Risk that is SystematicThe Proportion of Total Risk that is Systematic

Every investment in the financial markets vary withEvery investment in the financial markets vary withrespect to the percentage of total risk that isrespect to the percentage of total risk that issystematic.systematic.

Some stocks have virtuallySome stocks have virtually no systematicno sy stematic risk.risk.

Such stocks are not influenced by the health of the economy inSuch stocks are not influenced by the health of the economy ingeneraltheir financial results are predominantly influenced bygeneraltheir financial results are predominantly influenced bycompany-specific factors.company-specific factors.

An example is cigarette companiespeople consume cigarettesAn example is cigarette companiespeople consume cigarettesbecause they are addictedso it doesnt matter whether thebecause they are addictedso it doesnt matter whether theeconomy is healthy or notthey just continue to smoke.economy is healthy or notthey just continue to smoke.

Some stocks have a high proportion of their total riskSome stocks have a high proportion of their total riskthat 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.


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E t R t D tE t R t D t


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Ex ante Return DataEx ante Return DataA SampleA 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:

PossibleFuture State

of theEconomy Probability

PossibleReturns onthe Stock

PossibleReturns onthe Market

Boom 25.0% 28.0% 20.0%Normal 50.0% 17.0% 11.0%

Recession 25.0% -14.0% -4.0%

By observationyou can see therange is muchgreater for thestock than themarket and they

move in thesame direction.

Since the betarelates the stock

returns to themarket returns,the greater rangeof stock returnschanging in thesame direction asthe market

indicates the betawill be greater than 1 and will bepositive.(Positivelycorrelated to themarket returns.)


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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 outcomesThis means that we have considered all of the possible outcomesin this discrete probability distributionin this discrete probability distribution

PossibleFuture State


PossibleReturns onthe Stock

PossibleReturns onthe Market

Boom 25.0% 28.0% 20.0%Normal 50.0% 17.0% 11.0%

Recession 25.0% -14.0% -4.0%100.0%

MeasuringMeasuring Expected Return on the StockExpected Return on the Stock


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MeasuringMeasuring 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 fromThe expected return is weighted average returns fromthe given ex ante datathe given ex ante data

(1) (2) (3) (4)Possible

Future Stateof the

Economy Probability

PossibleReturns onthe Stock (4) = (2)*(3)

Boom 25.0% 28.0% 0.07Normal 50.0% 17.0% 0.085Recess ion 25.0% -14.0% -0.035

Expected return on the Stock = 12.0%

MeasuringMeasuring Expected Return on the MarketExpected Return on the Market


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MeasuringMeasuring 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 fromThe expected return is weighted average returns fromthe given ex ante datathe given ex ante data

(1) (2) (3) (4)Possible

Future Stateof the

Economy Probability

PossibleReturns onthe Market (4) = (2)*(3)

Boom 25.0% 20.0% 0.05Normal 50.0% 11.0% 0.055Recession 25.0% -4.0% -0.01

Expected return on the Marke t = 9.5%

Measuring Variances Standard Deviations ofMeasuring Variances Standard Deviations of


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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, squareUsing the expected return, calculate the deviations away from the mean, squarethose 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 meantheir 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 Stateof the

Economy Probability

PossibleReturns onthe Stock (4) = (2)*(3) Deviations

SquaredDeviations

Weightedand

SquaredDeviations

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 re turn (stock) = 12.0% Variance (stock)= 0.02455Standard Deviation (stock) = 15.67%

Measuring Variances Standard Deviations ofMeasuring Variances Standard Deviations of


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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 market Now do this for the possible returns on the market

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

Future State


Possible

Returns onthe Market (4) = (2)*(3) Deviations

SquaredDeviations

Weightedand

SquaredDeviations

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 re turn (market) = 9.5% Variance (market) = 0.007425Standard Deviation (market)= 8.62%


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CovarianceCovariance

From Chapter 8 you know the formula for the covarianceFrom Chapter 8 you know the formula for the covariancebetween the returns on the stock and the returns on thebetween the returns on the stock and the returns on themarket 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 Bi B

n

iii A AB k k k k COV

==[8-12]

ff


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Correlation CoefficientCorrelation Coefficient

Correlation is covariance normalized by the product of the standard deviationsCorrelation is covariance normalized by the product of the standard deviationsof both securities. It is a relative measure of co-movement of returns on aof both securities. It is a relative measure of co-movement of returns on ascale from -1 to +1.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 stockand 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 -1. 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+1 is perfect positive correlation (there is no diversification potential when combiningthese two securities together in a two-asset portfolio.)these two 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 twosecurities in a two-asset portfolio that will eliminate all portfolio risk)securities in a two-asset portfolio that will eliminate all portfolio risk)

B A

AB AB

COV

=[8-13]

Measuring CovarianceMeasuring Covariance


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Measuring CovarianceMeasuring Covariancefrom Ex Ante Return Datafrom Ex Ante Return Data

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

together with the probability of occurrencethen add the productstogether with the probability of occurrencethen add the productsup.up.

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

PossibleFuture

State of theEconomy Prob.

PossibleReturnson theStock

(4) =(2)*(3)

PossibleReturns onthe Market (6)=(2)*(5)

Deviationsfrom themean for the stock

Deviationsfrom themean 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(k stock ) = 12.0% E(k market ) = 9.5% Covariance = 0.01335

The Beta MeasuredThe Beta Measured


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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 theNow you can substitute the values for covariance and thevariance 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 aggressiveitsreturns 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%.

h f h k


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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 marketLet us assume we are comparing the possible marketreturns against itselfwhat will the beta be?returns against itselfwhat will the beta be?

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

PossibleFuture

State of theEconomy Prob.

PossibleReturnson theMarket

(4) =(2)*(3)

PossibleReturnson theMarket 6)=(2)*(5

Deviationsf rom themean for the stock

Deviationsfrom themean 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(k M) = 9.5% E(k M) = 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 !!!

0.1007425.007425.

Var CovBeta

M

M`M,===


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Using the Security Market LineUsing the Security Market Line

Expected versus Required ReturnExpected versus Required Return

How Do We use Expected and RequiredHow Do We use Expected and Required


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How Do We use Expected and Requiredow o We use pected a d equ edRates of Return?Rates of Return?

% Return

Risk-free Rate = 3%

BM= 1.0

E(kM)= 4.2%

Bs = 1.464

R(k s) = 4.76%

E(R s) = 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 theOnce you have estimated the expected and required rates of return, you can plot them on theSML and see if the stock is under or overpriced.SML and see if the stock is under or overpriced.

How Do We use Expected and RequiredHow Do We use Expected and Required


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How Do We use Expected and Requiredp qRates of Return?Rates of Return?

% Return

Risk-free Rate = 3%

BM=1.0

E(R M)= 4.2%

BS = 1.464

E(R s) = R(R s) 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.

U f h F BU f th F t B t


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Use of the Forecast BetaUse of the Forecast Beta

We can use the forecast beta, together with an estimate of theWe can use the forecast beta, together with an estimate of therisk-free rate and the market premium for risk to calculate therisk-free rate and the market premium for risk to calculate theinvestors required return on the stock using the CAPM:investors 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 Canadafree 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(k MM RF) RF)

RF]k [E RF M i += )(ReturnRequired

C l iC l i


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ConclusionsConclusions

Analysts can make estimates or forecasts for theAnalysts can make estimates or forecasts for thereturns 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 stocks betausing the CAPM but you will need the stocks betacoefficient, the expected return on the marketcoefficient, the expected return on the marketportfolio 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 DiscountModels to estimate the intrinsic value (inherentModels to estimate the intrinsic value (inherentworth) of the stock.worth) of the stock.


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Calculating the Beta using TrailingCalculating the Beta using Trailing

Holding Period ReturnsHolding Period Returns

APPENDIX 2APPENDIX 2

The Regression Approach to Measuring theThe Regression Approach to Measuring the


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g pp gg pp gBetaBeta

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 thestock is representative of the normal condition of the firm and itsrelationship 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 havedivested itself of a major subsidiary) there is good reason to believethat future returns will not reflect the pastand this approach to betaestimation SHOULD NOT be used.rather, use the ex ante approach.

Historical Beta EstimationHistorical Beta Estimation


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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

R e

t u r n s o n

S t o c

k

In this example, we have regressed the quarterly returns on the stock against thequarterly returns of a surrogate for the market (TSE 300 total return composite

index) and then using Excelused the charting feature to plot the historical points and add a regression trend line.

The regression line is a line of bestfit that describes the inherentrelationship between the returns on

the stock and the returns on themarket. The slope is the beta

coefficient.

The cloud of plotted pointsrepresents diversifiable or company

specific risk in the securities returns

that can be eliminated from a portfoliothrough diversification. Sincecompany-specific risk can be

eliminated, investors dont requirecompensation for it according to

Markowitz Portfolio Theory.

Ch t i ti LiCharacteristic Line


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Characteristic LineCharacteristic Line

The characteristic line is a regression line that represents the The characteristic line is a regression line that represents therelationship between the returns on the stock and the returns on therelationship between the returns on the stock and the returns on themarket over a past period of time. (It will be used to forecast themarket over a past period of time. (It will be used to forecast thefuture, assuming the future will be similar to the past.)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 in The degree to which the characteristic line explains the variability inthe dependent variable (returns on the stock) is measured by thethe dependent variable (returns on the stock) is measured by thecoefficient of determination. (also known as thecoefficient of determination. (also known as the RR 22 (r-squared or(r-squared orcoefficient of determination)).coefficient of determination)).

If the coefficient of determination equals 1.00, this would mean thatIf the coefficient of determination equals 1.00, this would mean thatall of the points of observation would lie on the line. This wouldall of the points of observation would lie on the line. This would

mean that the characteristic line would explain 100% of themean that the characteristic line would explain 100% of thevariability of the dependent variable.variability of the dependent variable.

The The alphaalpha is the vertical intercept of the regression (characteristicis the vertical intercept of the regression (characteristicline). Many stock analysts search out stocks with high alphas.line). Many stock analysts search out stocks with high alphas.

Lo RLow R 22


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Low RLow R 22

An RAn R 22 that approaches 0.00 (or 0%) indicates that thethat approaches 0.00 (or 0%) indicates that thecharacteristic (regression) line explains virtually none of thecharacteristic (regression) line explains virtually none of thevariability 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 theIn this case, you should use other approaches to value the

stockdo not use the estimated beta coefficient.stockdo 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 Tobacco


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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 onthe Market %(S&P TSX)

Returns onImperialTobacco %

CharacteristicLine for ImperialTobacco

High alpha

R-square is very low 0.02

Beta is largely irrelevant


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Characteristic Line General MotorsCharacteristic Line General Motors


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Characteristic Line General MotorsC a acte st c e Ge e a oto sA Positive Beta with Predictive Power A Positive Beta with Predictive Power


Returns onGeneralMotors %

CharacteristicLine for GM

(high R 2)

Positive alpha

R-square isvery high 0.9

Beta is positiveand close to 1.0

An Unusual Characteristic LineAn Unusual Characteristic Line


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A Negative Beta with Predictive Power A Negative Beta with Predictive Power


Returns on aStock %

Characteristic Line for a stock that will provide excellent

portfolio diversification

(high R 2) Positive alpha

R-square isvery high

Beta is negative -1.0

Diversifiable RiskDiversifiable Risk


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

Volatility in a securitys returns caused by company-specificVolatility in a securitys returns caused by company-specificfactors (both positive and negative) such as:factors (both positive and negative) such as: a single company strikea single company strike a spectacular innovation discovered through the companys R&D programa spectacular innovation discovered through the companys R&D program equipment failure for that one companyequipment failure for that one company management competence or management incompetence for that particular firmmanagement competence or management incompetence for that particular firm a jet carrying the senior management team of the firm crashes (this could be either aa 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)positive or negative event, depending on the competence of 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 influences onlyObviously, diversifiable risk is that unique factor that influences onlythe one firm.the one firm.

OK lets go back and look at raw dataOK lets go back and look at raw data


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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 theEnding values for the total return composite index (surrogate for themarket)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. TSEEnding value of the market indicator series you plan to use (ie. TSE

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


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Demonstration Through ExampleDemonstration Through Example

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

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


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

Spreadsheet Data From YahooSpreadsheet Data From Yahoo


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Process:Process:

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

company you are interested in studying.company you are interested in studying. Use the historical quotes buttonand get 30 monthsUse the historical quotes buttonand get 30 months

of historical data.of historical data.

Use the download in spreadsheet format feature toUse the download in spreadsheet format feature tosave the data to your hard drive.save the data to your hard drive.

http://ca.finance.yahoo.com/http://ca.finance.yahoo.com/http://ca.finance.yahoo.com/


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ppAlcan 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



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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 onthe TSE in the

month innumbers of board lots

The day,month andyear

Opening price per share, thehighest price per share during themonth, the lowest price per shareachieved during the month and theclosing price per share at the endof the month



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From Yahoo, the only information you can use is theFrom Yahoo, the only information you can use is theclosing price per share and the date. Just

Documents

The Capital Asset Pricing Model (CAPM) Imp