Chapter 4 CAPM

1
Chapter 4CAPM APTAsst. 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

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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 Markowitzs
  portfolio selection model (1952) aiming higher
  average returns with lower risk.

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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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Capital Asset Pricing Model

• Introduction
• Systematic and unsystematic risk
• Fundamental risk/return relationship revisited

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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

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CAPM (contd)

• 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 (contd)

• 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 (contd)

• CAPM assumptions (contd)
• 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 (contd)

• The market model is

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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 stocks 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 investors risk-return
  utility function.

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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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.

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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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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Assumptions of Capital Market Theory

• 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 stocks 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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Capital Market Line
E(r)
CML
M
E(rM)
rf
s
sm
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Security Market Line
E(r)
SML
E(rM)
rf
ß
ß
1.0
M
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Capital Market Line

• CML E(rp) rF ?sp
•  
• E(rp) Expected return on portfolio
• rF Return on the risk free asset
• ? Market price risk
• sp Market portfolio risk

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Slope and Market Risk Premium

• M Market portfolio rf Risk free
  rate E(rM) - rf Market risk premium E(rM) -
  rf Market price of risk
• Slope of the CAPM

s
M
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SML Relationships

• b COV(ri,rm) / sm2
• Slope SML E(rm) - rf
• market risk premium
• SML rf bE(rm) - rf

SML E(rS)rF ?sSpS,M
(sSpS,M) is the market price of risk
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Expected Return and Risk on Individual Securities

• The risk premium on individual securities is a
  function of the individual securitys
  contribution to the risk of the market portfolio
• Individual securitys 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
• bx 1.25
• E(rx) .03 1.25(.08) .13 or 13
• by .6
• E(ry) .03 .6(.08) .078 or 7.8

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Graph of Sample Calculations
E(r)
SML
Rx13
.08
Rm11
Ry7.8
3
ß
1.0
1.25
.6
ß
ß
ß
m
y
x
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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 Asset

• Covariance between two sets of returns is

Because the returns for the risk free asset are
certain,
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

This is a linear relationship
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Combining a Risk-Free Asset with a Risky
Portfolio

• Standard deviation
• The expected variance for a two-asset portfolio
  is

Substituting the risk-free asset for Security 1,
and the risky asset for Security 2, this formula
would become
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
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Combining a Risk-Free Asset with a Risky
Portfolio

• Given the variance formula

the standard deviation is
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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Combining a Risk-Free Asset with a Risky
Portfolio

• 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.

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Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
D
M
C
B
A
RFR
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Risk-Return Possibilities with Leverage

• To 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
CML
Borrowing
Lending
M
RFR
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An Exercise to Produce the Efficient Frontier
Using Three Assets

• Risk, Return and Portfolio Theory

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An Exercise using T-bills, Stocks and Bonds
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Achievable PortfoliosResults Using only Three
Asset Classes
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.
The plotted points are attainable portfolio
combinations.
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Achievable Two-Security PortfoliosModern
Portfolio Theory
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
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Efficient FrontierThe Two-Asset Portfolio
Combinations
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 Portfolio

• Because the market is in equilibrium, all assets
  are included in this portfolio in proportion to
  their market value

68
The Market Portfolio

• Because 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

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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

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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

Unsystematic (diversifiable) Risk
Total Risk
Standard Deviation of the Market Portfolio
(systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
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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 CML

• Together, this means the only important
  consideration is the assets covariance with the
  market portfolio

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A Risk Measure for the CML

• Because all individual risky assets are part of
  the M portfolio, an assets rate of return in
  relation to the return for the M portfolio may be
  described using the following linear model

where 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
79
Variance of Returns for a Risky Asset
80
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 assets 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

83
Graph of Security Market Line (SML)
Exhibit 8.5
SML
RFR
84
The Security Market Line (SML)

• The equation for the risk-return line is

We then define as beta
85
Graph of SML with Normalized Systematic Risk
Exhibit 8.6
SML
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)

87
Determining the Expected Rate of Return for a
Risky Asset

• Assume RFR 6 (0.06)
• RM 12 (0.12)
• Implied market risk premium 6 (0.06)

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
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Determining the Expected Rate of Return for a
Risky Asset

• 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
Exhibit 8.7
91
Comparison of Required Rate of Return to
Estimated Rate of Return
Exhibit 8.8
92
Plot of Estimated Returnson SML Graph
Exhibit 8.9
.22 .20 .18 .16 .14 .12 Rm .10 .08 .06 .04 .02
C
SML
A
E
B
D
.20 .40 .60 .80
1.20 1.40 1.60 1.80
-.40 -.20
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 assets characteristic line with the model
  portfolio

where Ri,t the rate of return for asset i
during period t RM,t the rate of return for the
market portfolio M during t
94
Scatter Plot of Rates of Return
Exhibit 8.10
The characteristic line is the regression line of
the best fit through a scatter plot of rates of
return
Ri
RM
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 firms size declines

96
The Effect of the Market Proxy

• The market portfolio of all risky assets must be
  represented in computing an assets
  characteristic line
• Standard Poors 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 SP 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 firms 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 securitys 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 (contd)

• 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
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118
Arbitrage Pricing Theory - APT

• Three 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

119
Assumptions of CAPMThat Were Not Required by APT

• APT 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

Ri
121
Arbitrage Pricing Theory (APT)

• For i 1 to N where
• return on asset i during a specified time
  period
• expected return for asset i

Ri Ei
122
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 is returns to movements in a
  common factor

Ri Ei bik
123
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 is returns to movements in a
  common factor
• a common factor with a zero mean that
  influences the returns on all assets

Ri Ei bik
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 is 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 is return that, by
  assumption, is completely diversifiable in large
  portfolios and has a mean of zero

Ri Ei bik
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 is 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 is return that, by
  assumption, is completely diversifiable in large
  portfolios and has a mean of zero
• number of assets

Ri Ei bik
N
126
Arbitrage Pricing Theory (APT)

• Multiple factors expected to have an
  impact on all assets

127
Arbitrage Pricing Theory (APT)

• Multiple factors expected to have an impact on
  all assets
• Inflation

128
Arbitrage Pricing Theory (APT)

• Multiple factors expected to have an impact on
  all assets
• Inflation
• Growth in GNP

129
Arbitrage Pricing Theory (APT)

• Multiple factors expected to have an impact on
  all assets
• Inflation
• Growth in GNP
• Major political upheavals

130
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

131
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.

132
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

133
Arbitrage Pricing Theory (APT)

• Bik determine how each asset reacts to this
  common factor
• Each asset may be affected by growth in GNP, but
  the effects will differ
• In 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
Arbitrage Pricing Theory (APT)

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

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
136
The Model of APT

• k
• Ri E( Ri ) ? dj ßij ei
  
• j1
• where, 
• R i The single period expected rate on
  security i , i 1,2.,n
• dj The zero mean j factor common to the
  all assets under consideration
• ßij The sensitivity of security is
  returns to the fluctuations in the j th common
  factor portfolio
• ei A random of i th security that
  constructed to have a mean of zero.

137
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

138
Arbitrage Portfolio

• Mean Stan. Correlation
• Return Dev. Of Returns
• Portfolio
• A,B,C 25.83 6.40 0.94
• D 22.25 8.58

139
Arbitrage Action and Returns
E. Ret.
P
D
St.Dev.
Short 3 shares of D and buy 1 of A, B C to form
P You earn a higher rate on the investment than
you pay on the short sale
140
The APT Model

• General representation of the APT model

141
Comparison of the CAPM and the APT

• The CAPMs market portfolio is difficult to
  construct
• Theoretically all assets should be included (real
  estate, gold, etc.)
• Practically, a proxy like the SP 500 index is
  used
• APT requires specification of the relevant
  macroeconomic factors

142
Comparison of the CAPM and the APT (contd)

• 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

143
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

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 boj0) is 3 percent
144
Example of Two Stocks and a Two-Factor Model

• the response of asset X to changes in the rate
  of inflation is 0.50

the response of asset Y to changes in the rate
of inflation is 2.00
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
145
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

146
Roll-Ross Study

• The methodology used in the study is as follows
• Estimate the expected returns and the factor
  coefficients from time-series data on individual
  asset returns
• 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 effect
• Reinganum - 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

149
Shankens 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

151
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.

152
Answer-market risk

• A) E(rm-rf)As2m
• Market Risk Premium 2(0.20)20.08
• B) E(rm) rf Eq. Risk prem
• 0.050.080.13 or 13
• C) Market Risk Premium 3(0.20)20.12
• D) E(rm) rf Eq. Risk prem
• 0.050.120.17 or 17

153
Example-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)/ s2m 0.085/0.2022.1
• B) E(rm)-rf As2m 3.5(0.20)20.14 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.

156
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 portfolios 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) ßy 1.25, ßx 1.15
• ßpwy ßy wx ßx
• 0.75(1.25)0.25(1.15)1.225
• B) E(rp)-rfßpE(rm)-rf
• 1.225(8)9.8

159
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)rfßE(rm)-rf
• 61.2(14-6)15.6

E(r)
Stock
SML
17
15.6
a
17-15.61.4
14
M
6
ß
1.0
1.2
161
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 rf5.
• 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?

162
Answer-SML

• A and B) aE(r)-rfßE(rm)-rf
• aXYZ 12-51.011-51
• UNDERVALUED
• aABC 13-51.511-5 -1
• OVERVALUED

163
Answer-SML-C
E(r)
Stock
SML
14
13
a
13-14-1
ABC
12
a
13-121
xyz
5
ß
1.0
1.5
164
Answer-SML

• D) E(r)rfßE(rm)-rf
• 81.316-818.4
• 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

166
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

167
Answer-SML

• A) ßA2-32/5-202 ßB3.5-14/5-200.7
• B) E(rA)rfßE(rm)-rf
• 0.5(232)17
• 0.5(3.514)8.75