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Chapter 6
Why Diversification Is a Good Idea
Portfolio Construction, Management, & Protection, 5e, Robert A. StrongCopyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.
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The most important lesson learned is an old truth ratified.
General Maxwell R. Thurman
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Introduction Diversification of a portfolio is logically a
good idea• “Carrying Your Eggs in More than One Basket”
Virtually all stock portfolios seek to diversify in one respect or another
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Investments in Your Own Ego Never put a large percentage of investment
funds into a single security• If the security appreciates, the ego is stroked
and 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 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 (cont’d)
Diversification is more important now• A 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 brokerage account for trading individual securities
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Variance of a Linear Combination: The Practical Meaning
One 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 ji j
i
ij
x x
x i
i j
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Variance of a Linear Combination: The Practical Meaning (cont’d)
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 (cont’d)
Return variance is a security’s 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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Role of Uncorrelated Securities
If two securities have low correlation, the interactive risk will be small
If two securities are uncorrelated, the interactive 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 (cont’d)
Example
Assume the following statistics for Stocks A, B, and C:
Stock A Stock B Stock C
Expected return .20 .14 .10
Standard deviation .232 .136 .195
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Portfolio Programming in a Nutshell (cont’d)
Example (cont’d)
The correlation coefficients between the three stocks are:
Stock A Stock B Stock C
Stock A 1.000
Stock B 0.286 1.000
Stock C 0.132 –0.605 1.000
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Portfolio Programming in a Nutshell (cont’d)
Example (cont’d)
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 (cont’d)
Example (cont’d)
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 (cont’d)
Example (cont’d)
Solution (cont’d): Calculate the variance of the B/C combination:
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 (cont’d)
Example (cont’d)
Solution (cont’d): Calculate the variance of the A/C combination:
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 (cont’d)
Example (cont’d)
Solution (cont’d): Investing 50 percent in Stock B and 50 percent 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 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 (cont’d) 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 (cont’d)Example (cont’d)
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
ecte
d R
etu
rn
B/C combination dominates A/C
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Harry Markowitz: The Father of Portfolio Theory
Harry Markowitz’s “Portfolio Selection” Journal of Finance article (1952) set the stage for modern portfolio theory• The first major publication indicating the importance of security
return correlation in the construction of stock portfolios
• Markowitz showed that for a given level of expected return and for a given security universe, knowledge of the covariance and correlation matrices is required
• He identified the existence of an efficient frontier where portfolios dominated individual securities within the security universe
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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 (cont’d)
Standard Deviation
Expected Return100% Investment in Security with Highest E(R)
100% Investment in Minimum Variance 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 (cont’d) 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 variances
– The risk-free rate has a variance of zero
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Efficient Frontier (cont’d)
Standard Deviation
Expected Return
Rf A
B
C
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Efficient Frontier (cont’d) 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 possible
investments, point B is the market portfolio– It contains every risky asset in the proportion of its
market value to the aggregate market value of all assets– It is the only risky asset risk-averse investors will hold
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Capital Market Line and the Market Portfolio (cont’d)
Implications for investors:• Regardless of the level of risk-aversion, all
investors 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 the market portfolio
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Capital Market Line and the Market Portfolio (cont’d)
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 (cont’d)
Standard Deviation
Expected Return
Rf A
B
C
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Security Market Line The graphical relationship between
expected return 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 the risk-free rate and the market’s expected return change
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Security Market Line (cont’d)
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 (cont’d)
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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Quadratic Programming 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 Evans and Archer’s 1968 Journal of
Finance article• Very consequential research regarding portfolio
construction
• Shows how naïve diversification reduces the dispersion of returns in a stock portfolio
– Naïve diversification refers to the selection of portfolio components randomly without any serious security analysis
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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 randomly selected
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Two Types of Risk Systematic risk is the risk that remains
after no further diversification benefits can be achieved
Unsystematic risk is 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
is diversifiable
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Two Types of Risk (cont’d) 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, 415–437.
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Strength in Numbers Portfolio variance (total risk) declines as the
number of securities included in the portfolio increases• On average, a randomly selected ten-security portfolio
will have less risk than a randomly selected three-security portfolio
• Risk-averse investors should always diversify to eliminate as much unsystematic 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 portfolio provides only modest additional benefits
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Superfluous Diversification Superfluous diversification refers to the
addition of unnecessary components to an already well-diversified portfolio• Deals with the diminishing marginal benefits of
additional portfolio components
• The benefits of additional diversification in large portfolios may be outweighed by the transaction costs
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Implications of Evans and Archer 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 firm’s equity shares
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Implications (cont’d) 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 naïve diversification reduces risk,
it can also reduce return• Unlike Markowitz’s 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 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 The Capital Asset Pricing Model (CAPM)
is a theoretical description of the way in which the market prices investment assets
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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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CAPM The more systematic risk you carry, the
greater the expected return:
( ) ( )
where ( ) expected return on security
risk-free rate of interest
beta of Security
( ) expected return on the market
i f i m f
i
f
i
m
E R R E R R
E R i
R
i
E R
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CAPM (cont’d) The CAPM deals with expectations about
the future
Excess returns on a particular stock are directly related to:• The beta of the stock• The expected excess return on the market
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CAPM (cont’d) CAPM assumptions:
• Variance of return and mean return are all investors care about
• Investors are price takers; they cannot influence the market individually
• All investors have equal and costless access to information
• There are no taxes or commission costs• 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:
( )
where return on Security in period
intercept
beta for Security
return on the market in period
error term on Security in period
it i i mt it
it
i
i
mt
it
R R e
R i t
i
R t
e i t
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Market Model versus CAPM (cont’d)
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 of
returns 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 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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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 stocks routinely move more than the others regardless of whether the market advances or declines
– Some stocks are more sensitive to changes in economic conditions
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Linear Regression and Beta To obtain beta with a linear regression:
• Plot a stock’s return against the market return
• Use Microsoft 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 Logrithms 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 Background
Arbitrage pricing theory (APT) states that a number of distinct factors determine the market return• Roll and Ross state that a security’s long-run
return is a function of changes in:– Inflation– Industrial production– Risk premiums– The slope of the term structure of interest rates
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APT Background (cont’d) Not all analysts are concerned with the
same set of economic information• A single market measure (such as beta) does
not capture all the information relevant to the price of a stock
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The APT Model General representation of the APT model:
1 1 2 2 3 3 4 4( )
where actual return on Security
( ) expected return on Security
sensitivity of Security to factor
unanticipated change in factor
A A A A A A
A
A
iA
i
R E R b F b F b F b F
R A
E R A
b A i
F i
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Comparison of the CAPM and the APT
The CAPM’s market portfolio is difficult to construct:• Theoretically, all assets should be included (real estate,
gold, etc.)
• Practically, a proxy like the S&P 500 index is used
APT requires specification of the relevant macroeconomic factors
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Comparison of the CAPM and the APT (cont’d)
The CAPM and APT complement each other rather than compete• Both models predict that positive returns will
result from factor sensitivities that move with the market and vice versa
