Title: Chapter 7 Why Diversification Is a Good Idea
1Chapter 7Why Diversification Is a Good Idea
2-
- The most important lesson learned
- is an old truth ratified.
-
- - General Maxwell R. Thurman
3Outline
- 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
4Introduction
- Diversification of a portfolio is logically a
good idea - Virtually all stock portfolios seek to diversify
in one respect or another
5Carrying Your Eggs in More Than One Basket
- Investments in your own ego
- The concept of risk aversion revisited
- Multiple investment objectives
6Investments 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
7The 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
8The 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
9Multiple 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
10Role 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
11Variance of A Linear Combination
- One measure of risk is the variance of return
- The variance of an n-security portfolio is
12Variance of A Linear Combination (contd)
- The variance of a two-security portfolio is
13Variance 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 up any
return
Total Risk
Risk from A
Risk from B
Interactive Risk
14Variance of A Linear Combination (contd)
- 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
15Portfolio 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
16Portfolio 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 .10
Standard deviation .232 .136 .195
17Portfolio Programming in A Nutshell (contd)
- Example (contd)
- 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
18Portfolio 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?
19Portfolio Programming in A Nutshell (contd)
- Example (contd)
- Solution Two combinations achieve a 12 return
- 50 in B, 50 in C (.5)(14) (.5)(10) 12
- 20 in A, 80 in C (.2)(20) (.8)(10) 12
20Portfolio Programming in A Nutshell (contd)
- Example (contd)
- Solution (contd) Calculate the variance of the
B/C combination
21Portfolio Programming in A Nutshell (contd)
- Example (contd)
- Solution (contd) Calculate the variance of the
A/C combination
22Portfolio 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 portfolio variance. Thus, the
investor will likely prefer this combination to
the alternative of investing 20 in Stock A and
80 in Stock C.
23Concept of Dominance
- Dominance is a situation in which investors
universally prefer one alternative over another - All rational investors will clearly prefer one
alternative
24Concept 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
25Concept of Dominance (contd)
- Example (contd)
- In the previous example, the B/C combination
dominates the A/C combination
B/C combination dominates A/C
Expected Return
Risk
26Harry Markowitz Founder of Portfolio Theory
- Introduction
- Terminology
- Quadratic programming
27Introduction
- Harry Markowitzs 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
28Terminology
- Security Universe
- Efficient frontier
- Capital market line and the market portfolio
- Security market line
- Expansion of the SML to four quadrants
- Corner portfolio
29Security 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
30Efficient Frontier
- Construct a risk/return plot of all possible
portfolios - Those portfolios that are not dominated
constitute the efficient frontier
31Efficient Frontier (contd)
Expected Return
100 investment in security with highest E(R)
No points plot above the line
Points below the efficient frontier are dominated
All portfolios on the line are efficient
100 investment in minimum variance portfolio
Standard Deviation
32Efficient Frontier (contd)
- The farther you move to the left on the efficient
frontier, the greater the number of securities in
the portfolio
33Efficient 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
34Efficient Frontier (contd)
Expected Return
C
B
Rf
A
Standard Deviation
35Efficient 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
36Capital 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
37Capital Market Line and the Market Portfolio
(contd)
- 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
38Capital 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
39Capital Market Line and the Market Portfolio
(contd)
Expected Return
C
B
Rf
A
Standard Deviation
40Security 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 markets expected return
change
41Security Market Line (contd)
Expected Return
E(R)
Market Portfolio
Rf
1.0
Beta
42Expansion 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
43Security Market Line (contd)
Expected Return
Securities with Negative Expected Returns
Beta
44Corner 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
45Quadratic Programming
- The Markowitz algorithm is an application of
quadratic programming - The objective function involves portfolio
variance - Quadratic programming is very similar to linear
programming
46 Markowitz Quadratic
Programming Problem
47Lessons from Evans and Archer
- Introduction
- Methodology
- Results
- Implications
- Words of caution
48Introduction
- Evans and Archers 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
49Methodology
- 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
50Results
- Definitions
- General results
- Strength in numbers
- Biggest benefits come first
- Superfluous diversification
51Definitions
- 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
52Definitions (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
53General Results
Portfolio Variance
Number of Securities
54Strength 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
55Biggest 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
56Superfluous 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
57Implications
- 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
58Implications (contd)
- Owning all possible securities would require high
commission costs - It is difficult to follow every stock
59Words 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 Markowitzs efficient diversification
60Diversification 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
61Capital Asset Pricing Model
- Introduction
- Systematic and unsystematic risk
- Fundamental risk/return relationship revisited
62Introduction
- 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
63Systematic 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)
64Fundamental Risk/Return Relationship Revisited
- CAPM
- SML and CAPM
- Market model versus CAPM
- Note on the CAPM assumptions
- Stationarity of beta
65CAPM
- The more risk you carry, the greater the expected
return
66CAPM (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
67CAPM (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
68CAPM (contd)
- CAPM assumptions (contd)
- Investors look only one period ahead
- Everyone is equally adept at analyzing securities
and interpreting the news
69SML 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
70Market 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
71Market Model Versus CAPM (contd)
72Note 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
73Stationarity 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
74Equity 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
75Using A Scatter Diagram to Measure Beta
- Correlation of returns
- Linear regression and beta
- Importance of logarithms
- Statistical significance
76Correlation 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
77Linear 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
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
78Importance 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
79Statistical 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
80Arbitrage Pricing Theory
- APT background
- The APT model
- Comparison of the CAPM and the APT
81APT 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
82APT 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
of a stock
83The APT Model
- General representation of the APT model
84Comparison 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 SP 500 index is
used - APT requires specification of the relevant
macroeconomic factors
85Comparison 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
the market and vice versa