portfolio management - chapter 7
TRANSCRIPT
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Chapter 7
Why Diversification Is a Good Idea
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Prof. Rushen Chahal
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The most important lesson learned
is an old truth ratified.
- General Maxwell R. Thurman
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Outline
Introduction
Carrying your eggs in more than one basket
Role of uncorrelated securities
Lessons from Evans and Archer
Diversification and beta
Capital asset pricing model
Equity risk premium Using a scatter diagram to measure beta
Arbitrage pricing theory
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Introduction
Diversification of a portfolio is logically a good
idea
Virtually all stock portfolios seek to diversify in
one respect or another
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Carrying Your Eggs in More Than
One Basket Investments in your own ego
The concept of risk aversion revisited
Multiple investment objectives
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Investments in Your Own Ego
Never put a large percentage of investment
funds into a single security
I
f the security appreciates, the ego is stroked andthis may plant a speculative seed
If the security never moves, the ego views this as
neutral rather than an opportunity cost
If the security declines, your ego has a verydifficult time letting go
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The Concept of
Risk Aversion Revisited Diversification is logical
If you drop the basket, all eggs break
Diversification is mathematically sound
Most people are risk averse
People take risks only if they believe they will be
rewarded for taking them
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The Concept of Risk
Aversion Revisited (contd) Diversification is more important now
Journal of Finance article shows that volatility of
individual firms has increased
Investors need more stocks to adequately diversify
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Multiple Investment Objectives
Multiple objectives justify carrying your eggs
in more than one basket
Some people find mutual funds unexciting
Many investors hold their investment funds in
more than one account so that they can play
with part of the total
E.g., a retirement account and a separate brokerageaccount for trading individual securities
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Role of Uncorrelated Securities
Variance of a linear combination: the practical
meaning
Portfolio programming in a nutshell Concept of dominance
Harry Markowitz: the founder of portfolio
theory
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Variance of A Linear Combination
One measure of risk is the variance of return
The variance of an n-security portfolio is:
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2
1 1
where proportion of total investment in Security
correlation coefficient between
Security and Security
n n
p i j ij i j
i j
i
ij
x x
x i
i j
W V W W
V
! !
!
!!
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Variance of A Linear Combination
(contd) The variance of a two-security portfolio is:
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2 2 2 2 2 2 p A A B B A B AB A B x x x xW W W V W W!
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Variance of A Linear Combination
(contd) Return variance is a securitys total risk
Most investors want portfolio variance to be
as low as possible without having to give upany return
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2 2 2 2 2
2 p A A B B A B AB A B x x x xW W W V W W! Total Risk Risk from A Risk from B Interactive Risk
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Variance of A Linear Combination
(contd) If two securities have low correlation, the
interactive risk will be small
If two securities are uncorrelated, theinteractive risk drops out
If two securities are negatively correlated,
interactive risk would be negative and would
reduce total risk
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Portfolio Programming
in A Nutshell Various portfolio combinations may result in a
given return
The investor wants to choose the portfolio
combination that provides the least amount of
variance
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Portfolio Programming
in A Nutshell (contd)Example
Assume the following statistics for Stocks A, B, and C:
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Stock A Stock B Stock C
Expected return .20 .14 .10Standard deviation .232 .136 .195
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Portfolio Programming
in A Nutshell (contd)Example (contd)
The correlation coefficients between the three stocks are:
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Stock A Stock B Stock C
Stock A 1.000Stock B 0.286 1.000
Stock C 0.132 -0.605 1.000
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Portfolio Programming
in A Nutshell (contd)Example (contd)
An investor seeks a portfolio return of 12%.
Which combinations of the three stocks accomplish this
objective? Which of those combinations achieves the least
amount of risk?
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution: Two combinations achieve a 12% return:
1) 50% in B, 50% in C: (.5)(14%) + (.5)(10%) = 12%
2) 20% in A, 80% in C: (.2)(20%) + (.8)(10%) = 12%
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Calculate the variance of the B/C
combination:
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2 2 2 2 2
2 2
2
(.50) (.0185) (.50) (.0380)
2(.50)(.50)( .605)(.136)(.195)
.0046 .0095 .0080
.0061
p A A B B A B AB A B x x x xW W W V W W!
!
!
!
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Calculate the variance of the A/C
combination:
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2 2 2 2 2
2 2
2
(.20) (.0538) (.80) (.0380)
2(.20)(.80)(.132)(.232)(.195)
.0022 .0243 .0019
.0284
p A A B B A B AB A B x x x xW W W V W W!
!
!
!
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Portfolio Programming
in A Nutshell (contd)Example (contd)
Solution (contd): Investing 50% in Stock B and 50% in Stock C
achieves an expected return of 12% with the lower portfoliovariance. Thus, the investor will likely prefer this combination
to the alternative of investing 20% in Stock A and 80% in Stock
C.
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Concept of Dominance
Dominance is a situation in which investors
universally prefer one alternative over another
All rational investors will clearly prefer one
alternative
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Concept of Dominance (contd)
A portfolio dominates all others if:
For its level of expected return, there is no other
portfolio with less risk
For its level of risk, there is no other portfolio with
a higher expected return
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Concept of Dominance (contd)
Example (contd)In the previous example, the B/C combination dominates the A/C combination:
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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.005 0.01 0.015 0.02 0.025 0.03
Risk
ExpectedReturn
B/C combination
dominates A/C
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Harry Markowitz: Founder of
P
ortfolio Theory Introduction
Terminology
Quadratic programming
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Introduction
Harry Markowitzs Portfolio Selection Journal ofFinance article (1952) set the stage for modernportfolio theory
The first major publication indicating the important ofsecurity return correlation in the construction of stockportfolios
Markowitz showed that for a given level of expected return
and for a given security universe, knowledge of thecovariance and correlation matrices are required
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Terminology
Security Universe
Efficient frontier
Capital market line and the market portfolio Security market line
Expansion of the SML to four quadrants
Corner portfolio
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Security Universe
The security universe is the collection of all
possible investments
For some institutions, only certain investments
may be eligible
E.g., the manager of a small cap stock mutual fund
would not include large cap stocks
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Efficient Frontier
Construct a risk/return plot of all possible
portfolios
Those portfolios that are not dominated
constitute the efficient frontier
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Efficient Frontier (contd)
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Standard Deviation
Expected Return100% investment in security
with highest E(R)
100% investment in minimumvariance portfolio
Points below the efficient
frontier are dominated
No points plot above
the line
All portfolios
on the line
are efficient
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Efficient Frontier (contd)
The farther you move to the left on the
efficient frontier, the greater the number of
securities in the portfolio
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Efficient Frontier (contd)
When a risk-free investment is available, the
shape of the efficient frontier changes
The expected return and variance of a risk-free
rate/stock return combination are simply a
weighted average of the two expected returns and
variance
The risk-free rate has a variance of zero
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Efficient Frontier (contd)
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Standard Deviation
Expected Return
RfA
B
C
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Efficient Frontier (contd)
The efficient frontier with a risk-free rate:
Extends from the risk-free rate to point B
The line is tangent to the risky securities efficient
frontier
Follows the curve from point B to point C
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Capital Market Line and the
Market Portfolio The tangent line passing from the risk-free
rate through point B is the capital market line(CML)
When the security universe includes all possibleinvestments, point B is the market portfolio
It contains every risky assets in the proportion of itsmarket value to the aggregate market value of all assets
It is the only risky assets risk-averse investors will hold
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Capital Market Line and the
Market Portfolio (contd) Implication for investors:
Regardless of the level of risk-aversion, allinvestors should hold only two securities:
The market portfolio The risk-free rate
Conservative investors will choose a point nearthe lower left of the CML
Growth-oriented investors will stay near themarket portfolio
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Capital Market Line and the
Market Portfolio (contd)
Any risky portfolio that is partially invested in
the risk-free asset is a lending portfolio
Investors can achieve portfolio returns greater
than the market portfolio by constructing a
borrowing portfolio
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Capital Market Line and the
Market Portfolio (contd)
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Standard Deviation
Expected Return
RfA
B
C
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Security Market Line
The graphical relationship between expectedreturn and beta is the security market line(SML)
The slope of the SML is the market price of risk
The slope of the SML changes periodically as therisk-free rate and the markets expected return
change
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Security Market Line (contd)
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Beta
Expected Return
Rf
Market Portfolio
1.0
E(R)
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Expansion of the SML to
Four Quadrants
There are securities with negative betas and
negative expected returns
A reason for purchasing these securities is their
risk-reduction potential
E.g., buy car insurance without expecting an accident
E.g., buy fire insurance without expecting a fire
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Security Market Line (contd)
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Beta
Expected Return
Securities with NegativeExpected Returns
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Corner Portfolio
A corner portfolio occurs every time a new
security enters an efficient portfolio or an old
security leaves
Moving along the risky efficient frontier from right
to left, securities are added and deleted until you
arrive at the minimum variance portfolio
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QuadraticProgramming
The Markowitz algorithm is an application of
quadratic programming
The objective function involves portfolio variance
Quadratic programming is very similar to linear
programming
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Markowitz Quadratic
Programming Problem
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Lessons from
Evans and Archer
Introduction
Methodology
Results Implications
Words of caution
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Introduction
Evans and Archers 1968 Journal of Financearticle
Very consequential research regarding portfolio
construction
Shows how nave diversification reduces thedispersion of returns in a stock portfolio
Nave diversification refers to the selection of portfoliocomponents randomly
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Methodology
Used computer simulations:
Measured the average variance of portfolios of
different sizes, up to portfolios with dozens of
components
Purpose was to investigate the effects of portfolio
size on portfolio risk when securities are randomlyselected
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Results
Definitions
General results
Strength in numbers Biggest benefits come first
Superfluous diversification
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Definitions
Systematic riskis the risk that remains after
no further diversification benefits can be
achieved
Unsystematic riskis the part of total risk that
is unrelated to overall market movements and
can be diversified
Research indicates up to 75 percent of total risk isdiversifiable
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Definitions (contd)
Investors are rewarded only for systematic risk
Rational investors should always diversify
Explains why beta (a measure of systematic risk) is
important
Securities are priced on the basis of their beta
coefficients
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General Results
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NumberofSecurities
Portfolio Variance
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Strength in Numbers
Portfolio variance (total risk) declines as thenumber of securities included in the portfolioincreases
On average, a randomly selected ten-securityportfolio will have less risk than a randomlyselected three-security portfolio
Risk-averse investors should always diversify toeliminate as much risk as possible
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Biggest Benefits Come First
Increasing the number of portfolio
components provides diminishing benefits as
the number of components increases
Adding a security to a one-security portfolio
provides substantial risk reduction
Adding a security to a twenty-security portfolioprovides only modest additional benefits
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Superfluous Diversification
Superfluous diversification refers to theaddition of unnecessary components to analready well-diversified portfolio
Deals with the diminishing marginal benefits ofadditional portfolio components
The benefits of additional diversification in large
portfolio may be outweighed by the transactioncosts
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Implications
Very effective diversification occurs when the
investor owns only a small fraction of the total
number of available securities
Institutional investors may not be able to avoid
superfluous diversification due to the dollar size of
their portfolios
Mutual funds are prohibited from holding more than 5
percent of a firms equity shares
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Implications (contd)
Owning all possible securities would require
high commission costs
It is difficult to follow every stock
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Words of Caution
Selecting securities at random usually gives
good diversification, but not always
Industry effects may prevent proper
diversification
Although nave diversification reduces risk, it
can also reduce return
Unlike Markowitzs efficient diversification
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Diversification and Beta
Beta measures systematic risk
Diversification does notmean to reduce beta
Investors differ in the extent to which they will
take risk, so they choose securities with different
betas
E.g., an aggressive investor could choose a portfolio
with a beta of 2.0
E.g., a conservative investor could choose a portfolio
with a beta of 0.5
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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 apositive 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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( ) ( )
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
F
F
! -
!
!
!
!
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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 postmodel
It describes past price behavior
The CAPM is an ex ante model
It predicts what a value should be
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M k M d l
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Market Model
Versus CAPM (contd)
The market model is:
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( )
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
E F
E
F
!
!
!
!
!!
N t th
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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 thatbetas change
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Equity Risk Premium
Equity risk premium refers to the difference inthe average return between stocks and somemeasure of the risk-free rate
The equity risk premium in the CAPM is the excessexpected return on the market
Some researchers are proposing that the size of
the equity risk premium is shrinking
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U i A S tt Di t
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Using A Scatter Diagram to
Measure Beta
Correlation of returns
Linear regression and beta
I
mportance of logarithms Statistical significance
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Correlation of Returns
Much of the daily news is of a generaleconomic nature and affects all securities
Stock prices often move as a group
Some stock routinely move more than the othersregardless of whether the market advances ordeclines
Some stocks are more sensitive to changes in economicconditions
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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 thecoefficients
The coefficient for the market return is the betastatistic
The intercept is the trend in the security price returnsthat 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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Arbitrage Pricing Theory
APT background
The APT model
Comparison of the CAPM and the APT
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APT Background
Arbitrage pricing theory (APT) states that anumber of distinct factors determine themarket return
Roll and Ross state that a securitys long-runreturn is a function of changes in:
Inflation
Industrial production
Risk premiums The slope of the term structure of interest rates
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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 ofa stock
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The APT Model
General representation of the APT model:
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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
! !
!
!!
Comparison of the
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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 S&P 500 index is used
APT requires specification of the relevant
macroeconomic factors
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Comparison of the
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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 themarket and vice versa