Chapter 7
Why Diversification Is a Good Idea
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
• 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
 One
measure of risk is the variance of
return
 The variance of an n-security portfolio is:
n
n
   xi x j ij i j
2
p
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 (cont’d)
 The
variance of a two-security portfolio is:
  x   x   2 xA xB  AB A B
2
p
2
A
2
A
2
B
2
B
12
Variance of A Linear
Combination (cont’d)
 Return
 2p
variance is a security’s total risk
 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 (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%.
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% 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% in Stock B and 50% in
Stock C achieves an expected return of 12% with the
lower portfolio variance. Thus, the investor will likely
prefer this combination to the alternative of investing
20% in Stock A and 80% 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
Risk
0.02
0.025
0.03
25
Harry Markowitz: 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 important 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 are 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 below the efficient
frontier are dominated
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 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
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 assets in the proportion of its
market value to the aggregate market value of all
assets
– It is the only risky assets risk-averse investors will
hold
36
Capital Market Line and the
Market Portfolio (cont’d)
 Implication
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
• Shows how naïve diversification reduces the
dispersion of returns in a stock portfolio
– Naïve diversification refers to the selection of
portfolio components randomly
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
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 portfolio 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 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 beta
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

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
 Importance of logarithms
 Statistical significance
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 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
76
Linear Regression and Beta
 To
obtain beta with a linear regression:
• Plot a stock’s 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
77
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
78
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
79
Arbitrage Pricing Theory
 APT
background
 The APT model
 Comparison of the CAPM and the APT
80
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
81
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
82
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
83
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
84
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
85