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Chapter 6
Why Diversif ication I s a Good I dea
Portfolio Construction, Management, & Protection, 4e, Robert A. Strong
Copyright 2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
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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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IntroductionDiversification 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 BasketInvestments in Your Own Ego
The Concept of Risk Aversion Revisited
Multiple Investment Objectives
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Investments in Your Own EgoNever put a large percentage of investment
funds into a single security
If the security appreciates, the ego is strokedand this 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 very
difficult time letting go
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The Concept of
Risk Aversion RevisitedDiversification 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 willbe rewarded for taking them
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The Concept of Risk
Aversion Revisited (contd)Diversification is more important now
AJournal of Financearticle shows that
volatility of individual firms has increased
Investors need more stocks to adequately diversify
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Multiple Investment ObjectivesMultiple 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 brokerage
account for trading individual securities
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Variance of a Linear Combination:
The Practical MeaningOne measure of risk is the variance of
return
The variance of an n-security portfolio is:
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
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Variance of a Linear Combination:
The Practical Meaning (contd)The variance of a two-security portfolio is:
2 2 2 2 2 2p A A B B A B AB A Bx x x x
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Variance of a Linear Combination:
The Practical Meaning (contd)Return variance is a securitys total risk
Most investors want portfolio variance to be
as low as possible without having to give up
any return
2 2 2 2 2
2p A A B B A B AB A Bx x x x
Total Risk Risk from A Risk from B Interactive Risk
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Variance of a Linear Combination:
The Practical Meaning (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 andwould reduce total risk
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Portfolio Programming
in a NutshellVarious 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:
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:
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 percent.
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 percent 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/Ccombination:
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 Bx x x x
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Portfolio Programming
in a Nutshell (contd)Example (contd)
Solution (contd):Calculate the variance of the A/Ccombination:
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 Bx x x x
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Portfolio Programming
in a Nutshell (contd)Example (contd)
Solution (contd):Investing 50 percent in Stock B and 50percent in Stock C achieves an expected return of 12
percent with the lower portfolio variance. Thus, the
investor will likely prefer this combination to the
alternative of investing 20 percent in Stock A and 80
percent in Stock C.
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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:
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
Exp
ec
tedRe
turn
B/C combination
dominates A/C
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Harry Markowitz: The
Founder of Portfolio TheoryIntroduction
Terminology
Quadratic Programming
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Introduction Harry Markowitzs Portfolio SelectionJournal
of Financearticle (1952) set the stage for modernportfolio theory
The first major publication indicating the importance ofsecurity return correlation in the construction of stock
portfolios
Markowitz showed that for a given level of expectedreturn and for a given security universe, knowledge ofthe covariance and correlation matrices is required
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TerminologySecurity Universe
Efficient Frontier
Capital Market Line and the MarketPortfolio
Security Market Line
Expansion of the SML to Four Quadrants
Corner Portfolio
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Security UniverseThe security universe is the collection of all
possible investments
For some institutions, only certain investmentsmay 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 (contd)
Standard Deviation
Expected Return100% Investment in Security
with Highest E(R)
100% Investment in MinimumVariance Portfolio
Points plotting below the
efficient frontier are dominated
by other portfolios
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-freerate/stock return combination are simply a
weighted average of the two expected returns
and variances
The risk-free rate has a variance of zero
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Efficient Frontier (contd)
Standard Deviation
Expected Return
Rf
A
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 efficientfrontier
Follows the curve from point B to point C
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Capital Market Line and the
Market PortfolioThe tangent line passing from the risk-free
rate through point B is the capital marketline (CML)
When the security universe includes all possibleinvestments, point B is the market portfolio
It contains every risky asset in the proportion of itsmarket value to the aggregate market value of allassets
It is the only risky asset risk-averse investors willhold
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Capital Market Line and the
Market Portfolio (contd)Implications 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 near
the 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
greaterthan the market portfolio by
constructing a borrowing portfolio
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Capital Market Line and the
Market Portfolio (contd)
Standard Deviation
Expected Return
Rf
A
B
C
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Security Market LineThe graphical relationship between
expected return and beta is the securitymarket line (SML)
The slope of the SML is the market price ofrisk
The slope of the SML changes periodically asthe risk-free rate and the markets expectedreturn change
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Security Market Line (contd)
Beta
Expected Return
Rf
Market Portfolio
1.0
E(R)
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Security Market Line (contd)
Beta
Expected Return
Securities with NegativeExpected Returns
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Corner PortfolioA corner portfoliooccurs 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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Quadratic ProgrammingThe Markowitz algorithm is an application
of quadratic programming
The objective function involves portfoliovariance
Quadratic programming is very similar to linear
programming
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Markowitz Quadratic
Programming Problem
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Lessons from
Evans and ArcherIntroduction
Methodology
Results
Implications
Words of Caution
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Introduction
Evans and Archers 1968Journal ofFinancearticle
Very consequential research regarding portfolioconstruction
Shows how nave diversificationreduces the
dispersion of returns in a stock portfolioNave diversification refers to the selection ofportfolio components randomly without any serioussecurity analysis
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Methodology
Used computer simulations:
Measured the average variance of portfolios of
different sizes, up to portfolios with dozens ofcomponents
Purpose was to investigate the effects of
portfolio size on portfolio risk when securities
are randomly selected
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Results
Definitions
General Results
Strength in Numbers
Biggest Benefits Come First
Superfluous Diversification
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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
Number of Securities
Portfolio Variance
Source: Adapted by Edwin J. Elton and Martin J. Gruber, Risk Production and Portfolio Size: An Analytical Solution, Journal of Business,
October 1977, 415437.
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Superfluous Diversification
Superfluous diversificationrefers to theaddition of unnecessary components to analready well-diversified portfolio
Deals with the diminishing marginal benefits ofadditional portfolio components
The benefits of additional diversification inlarge portfolios may be outweighed by thetransaction costs
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Implications
Very effective diversification occurs evenwhen the investor owns only a smallfraction of the total number of availablesecurities
Institutional investors may not be able to avoidsuperfluous diversification due to the dollar size
of their portfoliosMutual funds are prohibited from holding more than5 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 properdiversification
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 not mean to reduce beta
Investors differ in the extent to which they willtake risk, so they choose securities with
different betas
e.g., an aggressive investor could choose a portfolio
with a beta of 2.0e.g., a conservative investor could choose a portfolio
with a beta of 0.5
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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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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 (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 the expected market risk
premium
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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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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 ofreturns for the market
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 premiumrefers to thedifference in the average return betweenstocks 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 ofthe equity risk premium is shrinking
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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 stocks routinely move more than theothers regardless of whether the market
advances or declinesSome stocks are more sensitive to changes ineconomic 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 Microsoft Excel to run a linear regressionand obtain the coefficients
The coefficient for the market return is the betastatistic
The intercept is the trend in the security pricereturns that is inexplicable by finance theory
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Importance of Logarithms
Introduction
Statistical Significance
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Introduction
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 substantialunsystematic 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 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
Another alternative = Fama & Frenchs 3-Factor Model
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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) doesnot capture all the information relevant to the
price of a stock
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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 relevantmacroeconomic factors
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 willresult from factor sensitivities that move with
the market and vice versa APT can be viewed as a more general and more flexible
version of CAPM
Instead of having one all-encompassing measure of systematic
risk, APT breaks down the measure of systematic risk into a
variety of different variables that either drive or reflect
differences in systematic risk