Ch04 Capm and Apt

1

Chapter 4

CAPM & APT

Asst. Prof. Dr. Mete Feridun

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Capital Market Theory: An Overview

Capital market theory extends portfolio theory and develops a model for pricing all risky assets

Capital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset

3

Capital Asset Pricing Model (CAPM)

The asset pricing models aim to use the concepts of portfolio valuation and market equilibrium in order to determine the market price for risk and appropriate measure of risk for a single asset.

Capital Asset Pricing Model (CAPM) has an observation that the returns on a financial asset increase with the risk. CAPM concerns two types of risk namely unsystematic and systematic risks. The central principle of the CAPM is that, systematic risk, as measured by beta, is the only factor affecting the level of return.

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

developed independently by Sharpe (1964), Lintner (1965) and Mossin (1966) as a financial model of the relation of risk to expected return for the practical world of finance.

CAPM originally depends on the mean variance theory which was demonstrated by Markowitz’s portfolio selection model (1952) aiming higher average returns with lower risk.

5

Capital Asset Pricing Model (CAPM)

Equilibrium model that underlies all modern financial theory

Derived using principles of diversification with simplified assumptions

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development5

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7

Capital Asset Pricing Model Introduction Systematic and unsystematic risk Fundamental risk/return relationship

revisited

8

Introduction The Capital Asset Pricing Model (CAPM)

is a theoretical description of the way in which the market prices investment assets• The CAPM is a positive theory

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Systematic and Unsystematic Risk

Unsystematic risk can be diversified and is irrelevant

Systematic risk cannot be diversified and is relevant• Measured by beta

– Beta determines the level of expected return on a security or portfolio (SML)

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Fundamental Risk/Return Relationship Revisited

CAPM SML and CAPM Market model versus CAPM Note on the CAPM assumptions Stationarity of beta

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CAPM The more risk you carry, the greater the

expected return:

( ) ( )

where ( ) expected return on security

risk-free rate of interest

beta of Security

( ) expected return on the market

i f i m f

i

f

i

m

E R R E R R

E R i

R

i

E R

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CAPM (cont’d) The CAPM deals with expectations about

the future

Excess returns on a particular stock are directly related to:• The beta of the stock• The expected excess return on the market

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CAPM (cont’d) CAPM assumptions:

• Variance of return and mean return are all investors care about

• Investors are price takers– They cannot influence the market individually

• All investors have equal and costless access to information

• There are no taxes or commission costs

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CAPM (cont’d) CAPM assumptions (cont’d):

• Investors look only one period ahead

• Everyone is equally adept at analyzing securities and interpreting the news

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SML and CAPM If you show the security market

line with excess returns on the vertical axis, the equation of the SML is the CAPM • The intercept is zero

• The slope of the line is beta

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Market Model Versus CAPM The market model is an ex post model

• It describes past price behavior

The CAPM is an ex ante model• It predicts what a value should be

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Market Model Versus CAPM (cont’d)

The market model is:

( )

where return on Security in period

intercept

beta for Security

return on the market in period

error term on Security in period

it i i mt it

it

i

i

mt

it

R R e

R i t

i

R t

e i t

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Note on the CAPM Assumptions

Several assumptions are unrealistic:• People pay taxes and commissions

• Many people look ahead more than one period

• Not all investors forecast the same distribution

Theory is useful to the extent that it helps us learn more about the way the world acts• Empirical testing shows that the CAPM works

reasonably well

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Stationarity of Beta Beta is not stationary

• Evidence that weekly betas are less than monthly betas, especially for high-beta stocks

• Evidence that the stationarity of beta increases as the estimation period increases

The informed investment manager knows that betas change

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Equity Risk Premium Equity risk premium refers to the

difference in the average return between stocks and some measure of the risk-free rate• The equity risk premium in the CAPM is the

excess expected return on the market

• Some researchers are proposing that the size of the equity risk premium is shrinking

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Using A Scatter Diagram to Measure Beta

Correlation of returns Linear regression and beta Importance of logarithms Statistical significance

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Correlation of Returns Much of the daily news is of a general

economic nature and affects all securities• Stock prices often move as a group

• Some stock routinely move more than the others regardless of whether the market advances or declines

– Some stocks are more sensitive to changes in economic conditions

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Linear Regression and Beta To obtain beta with a linear regression:

• Plot a stock’s return against the market return

• Use Excel to run a linear regression and obtain the coefficients

– The coefficient for the market return is the beta statistic

– The intercept is the trend in the security price returns that is inexplicable by finance theory

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Importance of Logarithms Taking the logarithm of returns reduces the

impact of outliers• Outliers distort the general relationship

• Using logarithms will have more effect the more outliers there are

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Statistical Significance Published betas are not always useful

numbers• Individual securities have substantial

unsystematic risk and will behave differently than beta predicts

• Portfolio betas are more useful since some unsystematic risk is diversified away

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CAPM Assumptions

Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs Information is costless and available to all

investors Investors are rational mean-variance optimizers Homogeneous expectations

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Assumptions Asset markets are frictionless and information

liquidity is high. All investors are price takers; so that, they are not able

to influence the market price by their actions. All investors have homogenous expectations about

asset returns and what the uncertain future holds for them.

All investors are risk averse and they operate in the market rationally and perceive utility in terms of expected return.

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Assumptions (cont.) All investors are operating in perfect markets which

enables them to operate without tax payments on returns and without cost of transactions entailed in trading securities.

All securities are highly divisible for instance they can be traded in small parcels (Elton and Gruber, 1995, p.294).

All investors can lend and borrow unlimited amount of funds at the risk-free rate of return.

All investors have single period investment time horizon in means of different expectations from their investments leads them to operate for short or long term returns from their investments.

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Assumptions of Capital Market Theory

1. All investors are Markowitz efficient investors who want to target points on the efficient frontier. • The exact location on the efficient frontier and,

therefore, the specific portfolio selected, will depend on the individual investor’s risk-return utility function.

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2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). • Clearly it is always possible to lend money at

the nominal risk-free rate by buying risk-free securities such as government T-bills. It is not always possible to borrow at this risk-free rate, but we will see that assuming a higher borrowing rate does not change the general results.

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3. All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.• Again, this assumption can be relaxed. As long

as the differences in expectations are not vast, their effects are minor.

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4. All investors have the same one-period time horizon such as one-month, six months, or one year. • The model will be developed for a single

hypothetical period, and its results could be affected by a different assumption. A difference in the time horizon would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.

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5. All investments are infinitely divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio. • This assumption allows us to discuss

investment alternatives as continuous curves. Changing it would have little impact on the theory.

34


6. There are no taxes or transaction costs involved in buying or selling assets. • This is a reasonable assumption in many

instances. Neither pension funds nor religious groups have to pay taxes, and the transaction costs for most financial institutions are less than 1 percent on most financial instruments. Again, relaxing this assumption modifies the results, but does not change the basic thrust.

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7. There is no inflation or any change in interest rates, or inflation is fully anticipated.• This is a reasonable initial assumption, and it

can be modified.

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8. Capital markets are in equilibrium.• This means that we begin with all investments

properly priced in line with their risk levels.

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Some of these assumptions are unrealistic Relaxing many of these assumptions would

have only minor influence on the model and would not change its main implications or conclusions.

A theory should be judged on how well it explains and helps predict behavior, not on its assumptions.

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Resulting Equilibrium Conditions

All investors will hold the same portfolio for risky assets – market portfolio

Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Risk premium on the market depends on the average risk aversion of all market participants

Risk premium on an individual security is a function of its covariance with the market

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Capital Market Line

If a fully diversified investor is able to invest in the market portfolio and lend or borrow at the risk free rate of return, the alternative risk and return relationships can be generally placed around a market line which is called the Capital Market Line (CML).

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Security Market Line

The SML shows the relationship between risk measured by beta and expected return. The model states that the stock’s expected return is equal to the risk-free rate plus a risk premium obtained by the price of the risk multiplied by the quantity of the risk.

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E(r)E(r)

E(rE(rMM))

rrff

MMCMLCML

mm

Capital Market Line

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E(r)E(r)

E(rE(rMM))

rrff

SMLSML

MMßßßß = 1.0= 1.0

Security Market Line

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Capital Market Line

CML: E(rp)= rF+ λσp

E(rp): Expected return on portfolio

rF : Return on the risk free asset λ : Market price risk σp : Market portfolio risk

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M = Market portfoliorf = Risk free rate

E(rM) - rf = Market risk premium

E(rM) - rf = Market price of risk

= Slope of the CAPM

Slope and Market Risk Premium

MM

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SML Relationships

= [COV(ri,rm)] / m

2

Slope SML = E(rm) - rf

= market risk premium

SML = rf + [E(rm) - rf]

(σSpS,M) is the market price of risk

SML: E(rS)=rF+ λσSpS,M

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Expected Return and Risk on Individual Securities

The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

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Exercise

If E(rm) - rf = .08 and rf = .03

Calculate exp. ret. based on betas given below:

x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6

E(ry) = .03 + .6(.08) = .078 or 7.8%

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E(r)E(r)

RRxx=13%=13%

SMLSML

mm

ßß

ßß1.01.0

RRmm=11%=11%RRyy=7.8%=7.8%

3%3%

xxßß1.251.25

yyßß.6.6

.08.08

Graph of Sample Calculations

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Disequilibrium Example

Suppose a security with a beta of 1.25 is offering expected return of 15%

According to SML, it should be 13% So the security is underpriced: offering too

high of a rate of return for its level of risk

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Risk-Free Asset An asset with zero standard deviation Zero correlation with all other risky assets Provides the risk-free rate of return (RFR) Will lie on the vertical axis of a portfolio

graph

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Risk-Free AssetCovariance between two sets of returns is

n

1ijjiiij )]/nE(R-)][RE(R-[RCov

Because the returns for the risk free asset are certain,

0RF Thus Ri = E(Ri), and Ri - E(Ri) = 0

Consequently, the covariance of the risk-free asset with any risky asset or portfolio will always equal zero. Similarly the correlation between any risky asset and the risk-free asset would be zero.

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Combining a Risk-Free Asset with a Risky Portfolio

Expected return

the weighted average of the two returns

))E(RW-(1(RFR)W)E(R iRFRFport

This is a linear relationship

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Standard deviation

The expected variance for a two-asset portfolio is

211,22122

22

21

21

2port rww2ww)E(

Substituting the risk-free asset for Security 1, and the risky asset for Security 2, this formula would become

iRFiRF iRF,RFRF22

RF22

RF2port )rw-(1w2)w1(w)E(

Since we know that the variance of the risk-free asset is zero and the correlation between the risk-free asset and any risky asset i is zero we can adjust the formula

22RF

2port )w1()E( i

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Given the variance formula22

RF2port )w1()E( i

22RFport )w1()E( i the standard deviation is

i)w1( RF

Therefore, the standard deviation of a portfolio that combines the risk-free asset with risky assets is the linear proportion of the standard deviation of the risky asset portfolio.

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Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

56

Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

RFR

M

C

AB

D

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Risk-Return Possibilities with LeverageTo attain a higher expected return than is available at point M (in exchange for accepting higher risk)

Either invest along the efficient frontier beyond point M, such as point D

Or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M

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Capital Market Line - CML A line used in the capital asset pricing

model to illustrate the rates of return for efficient portfolios depending on the risk-free rate of return and the level of risk (standard deviation) for a particular portfolio.

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Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

)E( port

)E(R port

RFR

M

CML

Borrowing

Lending

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An Exercise to Produce the Efficient Frontier Using Three

Assets

Risk, Return and Portfolio Theory

61

An Exercise using T-bills, Stocks and Bonds

Base Data: Stocks T-bills BondsExpected Return(%) 12.73383 6.151702 7.0078723

Standard Deviation (%) 0.168 0.042 0.102

Correlation Coefficient Matrix:Stocks 1 -0.216 0.048T-bills -0.216 1 0.380Bonds 0.048 0.380 1

Portfolio Combinations:

Combination Stocks T-bills BondsExpected Return Variance

Standard Deviation

1 100.0% 0.0% 0.0% 12.7 0.0283 16.8%2 90.0% 10.0% 0.0% 12.1 0.0226 15.0%3 80.0% 20.0% 0.0% 11.4 0.0177 13.3%4 70.0% 30.0% 0.0% 10.8 0.0134 11.6%5 60.0% 40.0% 0.0% 10.1 0.0097 9.9%6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%7 40.0% 60.0% 0.0% 8.8 0.0044 6.6%8 30.0% 70.0% 0.0% 8.1 0.0028 5.3%9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%10 10.0% 90.0% 0.0% 6.8 0.0014 3.8%

Weights Portfolio

Historical averages for

returns and risk for three asset

classes

Historical correlation coefficients

between the asset classes

Portfolio characteristics for each combination of securities

Each achievable portfolio combination is plotted on expected return, risk (σ) space, found on the following slide.

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Achievable PortfoliosResults Using only Three Asset Classes

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.0 5.0 10.0 15.0 20.0

Standard Deviation of the Portfolio (%)

Po

rtfo

lio E

xpec

ted

Ret

urn

(%

) Efficient Set

Minimum Variance

Portfolio

The plotted points are attainable portfolio

combinations.

The efficient set is that set of achievable portfolio

combinations that offer the highest rate of return for a

given level of risk. The solid blue line indicates the efficient

set.

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Achievable Two-Security Portfolios

Modern Portfolio Theory8 - 9 FIGURE

Ex

pe

cte

d R

etu

rn %

Standard Deviation (%)

13

12

11

10

9

8

7

6

0 10 20 30 40 50 60

This line represents the set of portfolio combinations that are achievable by varying relative weights and using two non-correlated securities.

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Efficient FrontierThe 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 %


A

E

B

C

D

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Efficient FrontierThe Two-Asset Portfolio Combinations

8 - 10 FIGURE

Ex

pe

cte

d R

etu

rn %


A

E

B

C

D

Rational, risk averse investors will only want to hold portfolios such as B.

The actual choice will depend on her/his risk preferences.

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The Market Portfolio Because portfolio M lies at the point of

tangency, it has the highest portfolio possibility line

Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML

Therefore this portfolio must include ALL RISKY ASSETS

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The Market PortfolioBecause the market is in equilibrium, all assets are included in this portfolio in proportion to their market value

68

The Market PortfolioBecause it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away

69

Systematic Risk Only systematic risk remains in the market

portfolio Systematic risk is the variability in all risky

assets caused by macroeconomic variables Systematic risk can be measured by the

standard deviation of returns of the market portfolio and can change over time

70

Examples of Macroeconomic Factors Affecting Systematic Risk Variability in growth of money supply Interest rate volatility Variability in

• industrial production• corporate earnings• cash flow

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How to Measure Diversification All portfolios on the CML are perfectly

positively correlated with each other and with the completely diversified market Portfolio M

A completely diversified portfolio would have a correlation with the market portfolio of +1.00

72

Diversification and the Elimination of Unsystematic Risk The purpose of diversification is to reduce the

standard deviation of the total portfolio This assumes that imperfect correlations exist

among securities As you add securities, you expect the average

covariance for the portfolio to decline How many securities must you add to obtain a

completely diversified portfolio?

73

Diversification and the Elimination of Unsystematic Risk

Observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation

74

Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return

Standard Deviation of Return

Number of Stocks in the Portfolio

Standard Deviation of the Market Portfolio (systematic risk)

Systematic Risk

Total Risk

Unsystematic (diversifiable) Risk

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The CML and the Separation Theorem The CML leads all investors to invest in the M

portfolio Individual investors should differ in position on

the CML depending on risk preferences How an investor gets to a point on the CML is

based on financing decisions Risk averse investors will lend part of the

portfolio at the risk-free rate and invest the remainder in the market portfolio

76

A Risk Measure for the CML Covariance with the M portfolio is the

systematic risk of an asset The Markowitz portfolio model considers

the average covariance with all other assets in the portfolio

The only relevant portfolio is the M portfolio

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A Risk Measure for the CMLTogether, this means the only important consideration is the asset’s covariance with the market portfolio

78

A Risk Measure for the CMLBecause all individual risky assets are part of the M portfolio, an

asset’s rate of return in relation to the return for the M portfolio may be described using the following linear model:

Miiiit RbaRwhere:

Rit = return for asset i during period t

ai = constant term for asset i

bi = slope coefficient for asset i

RMt = return for the M portfolio during period t

= random error term

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Variance of Returns for a Risky Asset)Rba(Var)Var(R Miiiit

)(Var)Rb(Var)a(Var Miii )(Var)Rb(Var0 Mii

risk icunsystemator portfoliomarket the

torelatednot return residual theis )(Var

risk systematicor return market to

related varianceis )Rb(Var that Note Mii

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The Capital Asset Pricing Model: Expected Return and Risk

The existence of a risk-free asset resulted in deriving a capital market line (CML) that became the relevant frontier

An asset’s covariance with the market portfolio is the relevant risk measure

This can be used to determine an appropriate expected rate of return on a risky asset - the capital asset pricing model (CAPM)

81

The Capital Asset Pricing Model: Expected Return and Risk

CAPM indicates what should be the expected or required rates of return on risky assets

This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models

You can compare an estimated rate of return to the required rate of return implied by CAPM - over/under valued ?

82

The Security Market Line (SML) The relevant risk measure for an individual

risky asset is its covariance with the market portfolio (Covi,m)

This is shown as the risk measure The return for the market portfolio should

be consistent with its own risk, which is the covariance of the market with itself - or its variance: 2

m

83

Graph of Security Market Line (SML)

)E(R i

Exhibit 8.5

RFR

imCov2m

mR

SML

84

The Security Market Line (SML)The equation for the risk-return line is

)Cov(RFR-R

RFR)E(R Mi,2M

Mi

RFR)-R(Cov

RFR M2M

Mi,

2M

Mi,Cov

We then define as beta

RFR)-(RRFR)E(R Mi i

)( i

85

Graph of SML with Normalized Systematic Risk

)E(R i

Exhibit 8.6

)Beta(Cov 2Mim/0.1

mR

SML

0

Negative Beta

RFR

86

Determining the Expected Rate of Return for a Risky Asset

The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset

The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium (RM-RFR)

RFR)-(RRFR)E(R Mi i

87


Assume: RFR = 6% (0.06)

RM = 12% (0.12)

Implied market risk premium = 6% (0.06)

Stock Beta

A 0.70B 1.00C 1.15D 1.40E -0.30

RFR)-(RRFR)E(R Mi i

E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%

E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%

E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%

E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%

E(RE) = 0.06 + -0.30 (0.12-0.06) = 0.042 = 4.2%

88


In equilibrium, all assets and all portfolios of assets should plot on the SML

Any security with an estimated return that plots above the SML is underpriced

Any security with an estimated return that plots below the SML is overpriced

A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor

89

Identifying Undervalued and Overvalued Assets

Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment

Independent estimates of return for the securities provide price and dividend outlooks

90

Price, Dividend, and Rate of Return Estimates

Stock (Pi) Expected Price (Pt+1) (Dt+1) of Return (Percent)

A 25 27 0.50 10.0 %B 40 42 0.50 6.2C 33 39 1.00 21.2D 64 65 1.10 3.3E 50 54 0.00 8.0

Current Price Expected Dividend Expected Future Rate

Exhibit 8.7

91

Comparison of Required Rate of Return to Estimated Rate of Return

Stock Beta E(Ri) Estimated Return Minus E(Ri) Evaluation

A 0.70 10.2% 10.0 -0.2 Properly ValuedB 1.00 12.0% 6.2 -5.8 OvervaluedC 1.15 12.9% 21.2 8.3 UndervaluedD 1.40 14.4% 3.3 -11.1 OvervaluedE -0.30 4.2% 8.0 3.8 Undervalued

Required Return Estimated Return

Exhibit 8.8

92

Plot of Estimated Returnson SML Graph

Exhibit 8.9)E(R i

Beta0.1

mRSML

0 .20 .40 .60 .80 1.20 1.40 1.60 1.80-.40 -.20

.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02

AB

C

D

E

93

Calculating Systematic Risk: The Characteristic Line

The systematic risk input of an individual asset is derived from a regression model, referred to as the asset’s characteristic line with the model portfolio:

tM,iiti, RRwhere: Ri,t = the rate of return for asset i during period tRM,t = the rate of return for the market portfolio M during t

miii R-R

2M

Mi,Cov

i

error term random the

94

Scatter Plot of Rates of ReturnExhibit 8.10

RM

RiThe characteristic line is the regression line of the best fit through a scatter plot of rates of return

95

The Impact of the Time Interval Number of observations and time interval used in

regression vary Value Line Investment Services (VL) uses weekly

rates of return over five years Merrill Lynch, Pierce, Fenner & Smith (ML) uses

monthly return over five years There is no “correct” interval for analysis Weak relationship between VL & ML betas due to

difference in intervals used The return time interval makes a difference, and its

impact increases as the firm’s size declines

96

The Effect of the Market Proxy The market portfolio of all risky assets must

be represented in computing an asset’s characteristic line

Standard & Poor’s 500 Composite Index is most often used• Large proportion of the total market value of

U.S. stocks• Value weighted series

97

Weaknesses of Using S&P 500as the Market Proxy

• Includes only U.S. stocks • The theoretical market portfolio should include

U.S. and non-U.S. stocks and bonds, real estate, coins, stamps, art, antiques, and any other marketable risky asset from around the world

98

Relaxing the Assumptions Differential Borrowing and Lending Rates

• Heterogeneous Expectations and Planning Periods

Zero Beta Model• does not require a risk-free asset

Transaction Costs• with transactions costs, the SML will be a band

of securities, rather than a straight line

99

Relaxing the Assumptions Heterogeneous Expectations and Planning

Periods• will have an impact on the CML and SML

Taxes• could cause major differences in the CML and

SML among investors

100

Empirical Tests of the CAPM Stability of Beta

• betas for individual stocks are not stable, but portfolio betas are reasonably stable. Further, the larger the portfolio of stocks and longer the period, the more stable the beta of the portfolio

Comparability of Published Estimates of Beta

• differences exist. Hence, consider the return interval used and the firm’s relative size

101

Relationship Between Systematic Risk and Return

Effect of Skewness on Relationship• investors prefer stocks with high positive

skewness that provide an opportunity for very large returns

Effect of Size, P/E, and Leverage• size, and P/E have an inverse impact on returns

after considering the CAPM. Financial Leverage also helps explain cross-section of returns

102

Relationship Between Systematic Risk and Return

Effect of Book-to-Market Value• Fama and French questioned the relationship

between returns and beta in their seminal 1992 study. They found the BV/MV ratio to be a key determinant of returns

Summary of CAPM Risk-Return Empirical Results• the relationship between beta and rates of return

is a moot point

103

The Market Portfolio: Theory versus Practice

There is a controversy over the market portfolio. Hence, proxies are used

There is no unanimity about which proxy to use An incorrect market proxy will affect both the beta

risk measures and the position and slope of the SML that is used to evaluate portfolio performance

104

What is Next? Alternative asset pricing models

105

Summary

The dominant line is tangent to the efficient frontier• Referred to as the capital market line (CML)• All investors should target points along this line

depending on their risk preferences

106

Summary All investors want to invest in the risky

portfolio, so this market portfolio must contain all risky assets• The investment decision and financing decision

can be separated

• Everyone wants to invest in the market portfolio

• Investors finance based on risk preferences

107

Summary The relevant risk measure for an individual

risky asset is its systematic risk or covariance with the market portfolio• Once you have determined this Beta measure

and a security market line, you can determine the required return on a security based on its systematic risk

108

Summary

Assuming security markets are not always completely efficient, you can identify undervalued and overvalued securities by comparing your estimate of the rate of return on an investment to its required rate of return

109

Summary When we relax several of the major

assumptions of the CAPM, the required modifications are relatively minor and do not change the overall concept of the model.

110

Summary Betas of individual stocks are not stable

while portfolio betas are stable There is a controversy about the

relationship between beta and rate of return on stocks

Changing the proxy for the market portfolio results in significant differences in betas, SMLs, and expected returns

111

Arbitrage Pricing Theory APT background The APT model Comparison of the CAPM and the APT

112

Arbitrage Pricing Theory

Arbitrage Pricing Theory was developed by Stephen Ross (1976). His theory begins with an analysis of how investors construct efficient portfolios and offers a new approach for explaining the asset prices and states that the return on any risky asset is a linear combination of various macroeconomic factors that are not explained by this theory namely.

Similar to CAPM it assumes that investors are fully diversified and the systematic risk is an influencing factor in the long run. However, unlike CAPM model APT specifies a simple linear relationship between asset returns and the associated factors because each share or portfolio may have a different set of risk factors and a different degree of sensitivity to each of them.

113

APT Background Arbitrage pricing theory (APT) states that a

number of distinct factors determine the market return• Roll and Ross state that a security’s long-run

return is a function of changes in:– Inflation– Industrial production– Risk premiums– The slope of the term structure of interest rates

114

APT Background (cont’d) Not all analysts are concerned with the

same set of economic information• A single market measure such as beta does not

capture all the information relevant to the price of a stock

115

Arbitrage Pricing Theory (APT) CAPM is criticized because of the

difficulties in selecting a proxy for the market portfolio as a benchmark

An alternative pricing theory with fewer assumptions was developed:

Arbitrage Pricing Theory

116

The Assumptions of APT Capital asset returns’ properties are consistent

with a linear structure of the factors. The returns can be described by a factor model.

Either there are no arbitrage opportunities in the capital markets or the markets have perfect competition.

The number of the economic securities are either inestimable or so large that the law of large number can be applied that makes it possible to form portfolios that diversify the firm specific risk of individual stocks.

Lastly, the number of the factors can be estimated by the investor or known in advance (K. C. John Wei, 1988)

117

118

Arbitrage Pricing Theory - APTThree major assumptions:

1. Capital markets are perfectly competitive

2. Investors always prefer more wealth to less wealth with certainty

3. The stochastic process generating asset returns can be expressed as a linear function of a set of K factors or indexes

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Assumptions of CAPMThat Were Not Required by APTAPT does not assume A market portfolio that contains all risky

assets, and is mean-variance efficient Normally distributed security returns Quadratic utility function

120

Arbitrage Pricing Theory (APT)

For i = 1 to N where:

= return on asset i during a specified time period

ikikiiiitt bbbER ...21

Ri

121




= expected return for asset i


Ri

Ei

122





= reaction in asset i’s returns to movements in a common factor


Ri

Ei

bik

123





= reaction in asset i’s returns to movements in a common factor

= a common factor with a zero mean that influences the returns on all assets


Ri

Ei

bik

k

124

Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period= expected return for asset i= reaction in asset i’s returns to movements in a common

factor= a common factor with a zero mean that influences the

returns on all assets= a unique effect on asset i’s return that, by assumption, is

completely diversifiable in large portfolios and has a mean of zero


Ri

Ei

bik

ki

125

Arbitrage Pricing Theory (APT) For i = 1 to N where: = return on asset i during a specified time period= expected return for asset i= reaction in asset i’s returns to movements in a common

factor= a common factor with a zero mean that influences the

returns on all assets= a unique effect on asset i’s return that, by assumption, is

completely diversifiable in large portfolios and has a mean of zero

= number of assets


Ri

Ei

bik

ki

N

126

Arbitrage Pricing Theory (APT) Multiple factors expected to have an

impact on all assets:k

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Arbitrage Pricing Theory (APT)Multiple factors expected to have an impact

on all assets:• Inflation

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on all assets:• Inflation• Growth in GNP

129


on all assets:• Inflation• Growth in GNP• Major political upheavals

130


on all assets:• Inflation• Growth in GNP• Major political upheavals• Changes in interest rates

131


on all assets:• Inflation• Growth in GNP• Major political upheavals• Changes in interest rates• And many more….

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Arbitrage Pricing Theory (APT)Multiple factors expected to have an impact on all

assets:• Inflation• Growth in GNP• Major political upheavals• Changes in interest rates• And many more….

Contrast with CAPM insistence that only beta is relevant

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Arbitrage Pricing Theory (APT)Bik determine how each asset reacts to this common

factorEach asset may be affected by growth in GNP, but

the effects will differIn application of the theory, the factors are not

identified

Similar to the CAPM, the unique effects are independent and will be diversified away in a large portfolio

134

Arbitrage Pricing Theory (APT) APT assumes that, in equilibrium, the

return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away

The expected return on any asset i (Ei) can be expressed as:

135


where:

= the expected return on an asset with zero systematic risk where

ikkiii bbbE ...22110

0

01 EEi

00 E1 = the risk premium related to each of the common

factors - for example the risk premium related to interest rate risk

bi = the pricing relationship between the risk premium and asset i - that is how responsive asset i is to this common factor K

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The Model of APT k

Ri= E( Ri )+ ∑ δj βij+ εi

j=1 where, R i : The single period expected rate on

security i , i =1,2….,n δj : The zero mean j factor common to the

all assets under consideration βij : The sensitivity of security i’s returns to

the fluctuations in the j th common factor portfolio

εi : A random of i th security that constructed to have a mean of zero.

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Arbitrage Pricing Theory-briefly

•Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit

Since no investment is required, an investor can create large positions to secure large levels of profit

In efficient markets, profitable arbitrage opportunities will quickly disappear

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Arbitrage Portfolio

Mean Stan. Correlation

Return Dev. Of Returns

Portfolio

A,B,C 25.83 6.40 0.94

D 22.25 8.58

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Arbitrage Action and Returns

E. Ret.E. Ret.

St.Dev.St.Dev.

* * PP* * DD

Short 3 shares of D and buy 1 of A, B & C to form Short 3 shares of D and buy 1 of A, B & C to form PP

You earn a higher rate on the investment than You earn a higher rate on the investment than you pay on the short saleyou pay on the short sale

140

The APT Model General representation of the APT model:

1 1 2 2 3 3 4 4( )

where actual return on Security

( ) expected return on Security

sensitivity of Security to factor

unanticipated change in factor

A A A A A A

A

A

iA

i

R E R b F b F b F b F

R A

E R A

b A i

F i

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Comparison of the CAPM and the APT

The CAPM’s market portfolio is difficult to construct:• Theoretically all assets should be included (real estate,

gold, etc.)

• Practically, a proxy like the S&P 500 index is used

APT requires specification of the relevant macroeconomic factors

142

Comparison of the CAPM and the APT (cont’d)

The CAPM and APT complement each other rather than compete• Both models predict that positive returns will

result from factor sensitivities that move with the market and vice versa

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Example of Two Stocks and a Two-Factor Model= changes in the rate of inflation. The risk premium

related to this factor is 1 percent for every 1 percent change in the rate

1)01.( 1

= percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate

= the rate of return on a zero-systematic-risk asset (zero beta: boj=0) is 3 percent

2)02.( 2

)03.( 3 3

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Example of Two Stocks and a Two-Factor Model= the response of asset X to changes in the rate of

inflation is 0.501xb

)50.( 1 xb

= the response of asset Y to changes in the rate of inflation is 2.00 )50.( 1 yb1yb

= the response of asset X to changes in the growth rate of real GNP is 1.50

= the response of asset Y to changes in the growth rate of real GNP is 1.75

2xb

2yb)50.1( 2 xb

)75.1( 2 yb

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Example of Two Stocks and a Two-Factor Model

= .03 + (.01)bi1 + (.02)bi2

Ex = .03 + (.01)(0.50) + (.02)(1.50)

= .065 = 6.5%

Ey = .03 + (.01)(2.00) + (.02)(1.75)

= .085 = 8.5%

22110 iii bbE

146

Roll-Ross StudyThe methodology used in the study is as follows:1. Estimate the expected returns and the factor

coefficients from time-series data on individual asset returns

2. Use these estimates to test the basic cross-sectional pricing conclusion implied by the APT

The authors concluded that the evidence generally supported the APT, but acknowledged that their tests were not conclusive

147

Extensions of the Roll-Ross Study

Cho, Elton, and Gruber examined the number of factors in the return-generating process that were priced

Dhrymes, Friend, and Gultekin (DFG) reexamined techniques and their limitations and found the number of factors varies with the size of the portfolio

148

The APT and Anomalies

Small-firm effectReinganum - results inconsistent with the APT

Chen - supported the APT model over CAPM January anomaly

Gultekin - APT not better than CAPM

Burmeister and McElroy - effect not captured by model, but still rejected CAPM in favor of APT

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Shanken’s Challenge to Testability of the APT

If returns are not explained by a model, it is not considered rejection of a model; however if the factors do explain returns, it is considered support

APT has no advantage because the factors need not be observable, so equivalent sets may conform to different factor structures

Empirical formulation of the APT may yield different implications regarding the expected returns for a given set of securities

Thus, the theory cannot explain differential returns between securities because it cannot identify the relevant factor structure that explains the differential returns

150

APT and CAPM Compared

APT applies to well diversified portfolios and not necessarily to individual stocks

With APT it is possible for some individual stocks to be mispriced - not lie on the SML

APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio

APT can be extended to multifactor models

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Example-market risk

Suppose the risk free rate is 5%, the average investor has a risk-aversion coefficient of A* is 2, and the st. dev. Of the market portfolio is 20%.

A) Calculate the market risk premium. B) Find the expected rate of return on the market. C) Calculate the market risk premium as the risk-

aversion coefficient of A* increases from 2 to 3. D) Find the expected rate of return on the market

referring to part c.

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Answer-market risk

A) E(rm-rf)=A*σ2m

Market Risk Premium =2(0.20)2=0.08 B) E(rm) = rf +Eq. Risk prem

= 0.05+0.08=0.13 or 13%

C) Market Risk Premium =3(0.20)2=0.12 D) E(rm) = rf +Eq. Risk prem

= 0.05+0.12=0.17 or 17%

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Example-risk premiumExample-risk premium

Suppose an av. Excess return over Treasury bill of 8% with a st. dev. Of 20%.

A) Calculate coefficient of risk-aversion of the av. investor.

B) Calculate the market risk premium as the risk-aversion coefficient is 3.5

154

Answer-risk premium A) A*= E(rm-rf)/ σ2

m =0.085/0.202=2.1

B) E(rm)-rf =A*σ2m =3.5(0.20)2=0.14 or 14%

155

Example-Portfolio beta and risk premium

Consider the following portfolio:

A) Calculate the risk premium on each portfolio

B) Calculate the total portfolio if Market risk premium is 7.5%.

Asset BetaRisk

prem.Portfolio Weight

X 1.2 9% 0.5

Y 0.8 6 0.3

Z 0.0 0 0.2

Port. 0.84 1.0

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Answer-Portfolio beta and risk premium

A) (9%) (0.5)=4.5 (6%) (0.3)=1.8 =6.3% B) 0.84(7.5)=6.3%

157

Example-risk premium Suppose the risk premium of the market portfolio

is 8%, with a st. dev. Of 22%.

A) Calculate portfolio’s beta. B) Calculate the risk premium of the portfolio

referring to a portfolio invested 25% in x motor company with beta 0f 1.15 and 75% in y motor company with a beta of 1.25.

158

Answer-risk premium

A) βy= 1.25, βx= 1.15

βp=wy βy+ wx βx

=0.75(1.25)+0.25(1.15)=1.225

B) E(rp)-rf=βp[E(rm)-rf]

=1.225(8%)=9.8%

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Example-SML

Suppose the return on the market is expected to be 14%, a stock has a beta of 1.2, and the T-bill rate is 6%.

A) Calculate the expected return of the SML B) If the return is 17%, calculate alpha of the

stock

160

Answer-SML A) E(rp)=rf+β[E(rm)-rf]

=6+1.2(14-6)=15.6%E(r)E(r)

17%17% SMLSML

ßß1.01.0

15.6%15.6%

14%14%

6%6%

1.21.2

MM

StockStock

α=α= 17-15.6=1.417-15.6=1.4

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Example-SML Stock xyz has an expected return of 12% and risk of

beta is 1.5. Stock ABC is expected to return 13% with a beta of 1.5. The market expected return is 11% and rf=5%.

A) Based on CAPM, which stock is a better buy? B) What is the alpha of each stock? C) Plot the relevant SML of the two stocks D) rf is 8% ER on the market portfolio is 16%, and

estimated beta is 1.3, what is the required ROR on the project?

E) If the IRR of the project is 19%, what is the project alpha?

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Answer-SML A and B) α=E(r)-{rf+β[E(rm)-rf]}

αXYZ= 12-{5+1.0[11-5]}=1

• UNDERVALUED

αABC= 13-{5+1.5[11-5]}= -1

• OVERVALUED

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Answer-SML-CAnswer-SML-C

E(r)E(r)

14%14% SMLSML

ßß1.01.0

13%13%

12%12%

5%5%

1.51.5

xyzxyz

StockStock

α=α= 13-14=-113-14=-1

ααABCABC

=13-12=1=13-12=1

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Answer-SML

D) E(r)={rf+β[E(rm)-rf]}

= 8+1.3[16-8]=18.4% E) If the IRR of the project is 19%, it is

desirable. However, any project with an IRR by using similar beta is less than 18.4% should be rejected.

165

Example-SML

Consider the following table:

Market Return

Aggressive stock

Defensive stock

5% 2% 3.5%

20 32 14

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Example-SML cont..

A) What are the betas of the two stock? B) What is the E(ROR) on each stock if Market

return is equally likely to be 5% or 20%? C) If T-bill rate is 8% and Market return is equally

likely to be 5% or 20%, draw SML for the economy?

D) Plot the two securities on the SML graph and show the alphas

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Answer-SML

A) βA=2-32/5-20=2 βB=3.5-14/5-20=0.7

B) E(rA)={rf+β[E(rm)-rf]}

=0.5(2%+32%)=17% =0.5(3.5%+14%)=8.75%

Documents

Ch04 Capm and Apt