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Chapter 4 CAPM

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Title: Chapter 4 CAPM


1
Chapter 4CAPM APTAsst. Prof. Dr. Mete
Feridun
2
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.

4
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.

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

6
(No Transcript)
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

9
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)

10
Fundamental Risk/Return Relationship Revisited
  • CAPM
  • SML and CAPM
  • Market model versus CAPM
  • Note on the CAPM assumptions
  • Stationarity of beta

11
CAPM
  • The more risk you carry, the greater the expected
    return

12
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

13
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

14
CAPM (contd)
  • CAPM assumptions (contd)
  • Investors look only one period ahead
  • Everyone is equally adept at analyzing securities
    and interpreting the news

15
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

16
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

17
Market Model Versus CAPM (contd)
  • The market model is

18
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

19
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

20
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

21
Using A Scatter Diagram to Measure Beta
  • Correlation of returns
  • Linear regression and beta
  • Importance of logarithms
  • Statistical significance

22
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

23
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

24
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

25
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

26
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

27
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.

28
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.

29
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.

30
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.

31
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.

32
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.

33
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.

34
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.

35
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.

36
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.

37
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.

38
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

39
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).

40
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.

41
Capital Market Line
E(r)
CML
M
E(rM)
rf
s
sm
42
Security Market Line
E(r)
SML
E(rM)
rf
ß
ß
1.0
M
43
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

44
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
45
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
46
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

47
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

48
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
49
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

50
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

51
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.
52
Combining a Risk-Free Asset with a Risky
Portfolio
  • Expected return
  • the weighted average of the two returns

This is a linear relationship
53
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
54
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.
55
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.

56
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
D
M
C
B
A
RFR
57
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

58
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.

59
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient
Frontier
CML
Borrowing
Lending
M
RFR
60
An Exercise to Produce the Efficient Frontier
Using Three Assets
  • Risk, Return and Portfolio Theory

61
An Exercise using T-bills, Stocks and Bonds
62
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.
63
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.
64
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
65
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.
66
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

67
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

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

71
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

Unsystematic (diversifiable) Risk
Total Risk
Standard Deviation of the Market Portfolio
(systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
75
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

77
A Risk Measure for the CML
  • Together, this means the only important
    consideration is the assets covariance with the
    market portfolio

78
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
88
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
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