Chapter 6
Why Diversification
Is a Good Idea
Portfolio Construction, Management, & Protection, 4e, Robert A. Strong
Copyright ©2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.
1
The most important lesson learned
is an old truth ratified.
General Maxwell R. Thurman
2
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
3
Introduction

Diversification of a portfolio is logically a
good idea

Virtually all stock portfolios seek to
diversify in one respect or another
4
Carrying Your Eggs in More
Than One Basket
Investments in Your Own Ego
 The Concept of Risk Aversion Revisited
 Multiple Investment Objectives

5
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
6
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
7
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
8
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
9
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

10
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:

n
n
 2p   xi x j ij i j
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
11
Variance of a Linear Combination:
The Practical Meaning (cont’d)

The variance of a two-security portfolio is:
  x   x   2xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
12
Variance of a Linear Combination:
The Practical Meaning (cont’d)

Return variance is a security’s total risk
 2p
 xA2 A2
Total Risk

 xB2 B2
Risk from A
 2 xA xB  AB A B
Risk from B
Interactive Risk
Most investors want portfolio variance to
be as low as possible without having to
give up any return
13
Variance of a Linear Combination:
The Practical Meaning (cont’d)
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

14
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
15
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
16
Portfolio Programming
in a Nutshell (cont’d)
Example (cont’d)
The correlation coefficients between the three stocks
are:
Stock A
Stock B
Stock A
1.000
Stock B
0.286
1.000
Stock C
0.132
–0.605
Stock C
1.000
17
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?
18
Portfolio Programming
in a Nutshell (cont’d)
Example (cont’d)
Solution: Two combinations achieve a 12 percent
return:
1)
2)
50% in B, 50% in C: (.5)(14%) + (.5)(10%) = 12%
20% in A, 80% in C: (.2)(20%) + (.8)(10%) = 12%
19
Portfolio Programming
in a Nutshell (cont’d)
Example (cont’d)
Solution (cont’d): Calculate the variance of the B/C
combination:
 2p  x A2 A2  xB2 B2  2 x A xB  AB A B
 (.50) 2 (.0185)  (.50) 2 (.0380)
 2(.50)(.50)( .605)(.136)(.195)
 .0046  .0095  .0080
 .0061
20
Portfolio Programming
in a Nutshell (cont’d)
Example (cont’d)
Solution (cont’d): Calculate the variance of the A/C
combination:
 2p  x A2 A2  xB2 B2  2 x A xB  AB A B
 (.20) 2 (.0538)  (.80) 2 (.0380)
 2(.20)(.80)(.132)(.232)(.195)
 .0022  .0243  .0019
 .0284
21
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.
22
Concept of Dominance

Dominance is a situation in which
investors universally prefer one alternative
over another
 All
rational investors will clearly prefer one
alternative
23
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
24
Concept of Dominance
(cont’d)
Example (cont’d)
In the previous example, the B/C combination dominates the A/C
combination:
0.14
Expected Return
0.12
0.1
B/C combination
dominates A/C
0.08
0.06
0.04
0.02
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Risk
25
Harry Markowitz: The Founder
of Portfolio Theory
Introduction
 Terminology
 Quadratic Programming

26
Introduction

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
27
Terminology
Security Universe
 Efficient Frontier
 Capital Market Line and the Market
Portfolio
 Security Market Line
 Expansion of the SML to Four Quadrants
 Corner Portfolio

28
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
29
Efficient Frontier

Construct a risk/return plot of all possible
portfolios
 Those
portfolios that are not dominated
constitute the efficient frontier
30
Efficient Frontier (cont’d)
Expected Return
No points plot above
the line
All portfolios
on the line
are efficient
100% Investment in Security
with Highest E(R)
Points plotting below the
efficient frontier are dominated
by other portfolios
100% Investment in Minimum
Variance Portfolio
Standard Deviation
31
Efficient Frontier (cont’d)

The farther you move to the left on the
efficient frontier, the greater the number of
securities in the portfolio
32
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 riskfree 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
33
Efficient Frontier (cont’d)
Expected Return
C
B
Rf
A
Standard Deviation
34
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
35
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

36
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
37
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
38
Capital Market Line and the
Market Portfolio (cont’d)
Expected Return
C
B
Rf
A
Standard Deviation
39
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
40
Security Market Line (cont’d)
Expected Return
E(R)
Market Portfolio
Rf
1.0
Beta
41
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
42
Security Market Line (cont’d)
Expected Return
Securities with Negative
Expected Returns
Beta
43
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
44
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
45
Markowitz Quadratic
Programming Problem
46
Lessons from
Evans and Archer
Introduction
 Methodology
 Results
 Implications
 Words of Caution

47
Introduction

Evans and Archer’s 1968 Journal of
Finance article
 Very
consequential research regarding
portfolio construction
how naïve diversification reduces
the dispersion of returns in a stock portfolio
 Shows

Naïve diversification refers to the selection of
portfolio components randomly without any serious
security analysis
48
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
49
Results
Definitions
 General Results
 Strength in Numbers
 Biggest Benefits Come First
 Superfluous Diversification

50
Definitions
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
51
Definitions (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
52
General Results
Portfolio Variance
Number of Securities
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.
53
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 risk as possible
54
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
55
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
56
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 firm’s equity shares
57
Implications (cont’d)

Owning all possible securities would
require high commission costs

It is difficult to follow every stock
58
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
59
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

60
Capital Asset Pricing Model
Introduction
 Systematic and Unsystematic Risk
 Fundamental Risk/Return Relationship
Revisited

61
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
62
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)
63
Fundamental Risk/Return
Relationship Revisited
CAPM
 SML and CAPM
 Market Model versus CAPM
 Note on the CAPM Assumptions
 Stationarity of Beta

64
CAPM

The more systematic risk you carry, the
greater the expected return:
E ( Ri )  R f   i  E ( Rm )  R f 
where E ( Ri )  expected return on security i
R f  risk-free rate of interest
 i  beta of Security i
E ( Rm )  expected return on the market
65
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
66
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
67
CAPM (cont’d)

CAPM assumptions (cont’d):
 Investors
look only one period ahead
 Everyone
is equally adept at analyzing
securities and interpreting the news
68
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
69
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
70
Market Model
versus CAPM (cont’d)

The market model is:
Rit   i   i ( Rmt )  eit
where Rit  return on Security i in period t
 i  intercept
 i  beta for Security i
Rmt  return on the market in period t
eit  error term on Security i in period t
71
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
72
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
73
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
74
Using A Scatter Diagram to
Measure Beta
Correlation of Returns
 Linear Regression and Beta

75
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
76
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

77
Importance of Logarithms
Introduction
 Statistical Significance

78
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
79
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
80
Arbitrage Pricing Theory
APT Background
 The APT Model
 Comparison of the CAPM and the APT

81
APT 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

82
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
83
The APT Model

General representation of the APT model:
RA  E ( RA )  b1 A F1  b2 A F2  b3 A F3  b4 A F4
where RA  actual return on Security A
E ( RA )  expected return on Security A
biA  sensitivity of Security A to factor i
Fi  unanticipated change in factor i
84
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
85
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
86