Chapter 6 Why Diversification Is a Good Idea

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Chapter 6Why Diversification Is a Good Idea
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Introduction

• Diversification 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 Basket

• Investments in your own ego
• The concept of risk aversion revisited
• Multiple investment objectives

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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 (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

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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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Lessons from Evans and Archer

• Introduction
• Methodology
• Results
• Implications
• Words of caution

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Introduction

• 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

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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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Results

• Definitions
• General results
• Strength in numbers
• Biggest benefits come first
• Superfluous diversification

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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

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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
Portfolio Variance
Number of Securities
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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 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 portfolio may be outweighed by the
  transaction costs

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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 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 proper
  diversification
• Although naïve diversification reduces risk, it
  can also reduce return
• Unlike Markowitzs efficient diversification

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Markowitzs Contribution

• 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

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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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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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Markowitz Quadratic
Programming Problem
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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 (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

B/C combination dominates A/C
Expected Return
Risk
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Terminology

• Security Universe
• Efficient frontier
• Capital market line and the market portfolio
• Security market line
• Expansion of the SML to four quadrants
• Corner portfolio

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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 (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
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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-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

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Efficient Frontier (contd)
Expected Return
C
B
Rf
A
Standard Deviation
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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
  efficient frontier
• Follows the curve from point B to point C

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Theorem

• For any constant Rf on the returns axis, the
  weights of the tangency portfolio B are

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Example with Rf0 and Rf6.5
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Graphically
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What is the zero-beta portfolio?

• The zero beta portfolio P0 is the portfolio
  determined by the intersection of the frontier
  with a horizontal line originating from the
  constant Rf selected.
• Property whatever Rf we choose, we always have
  Cov(B,P0)0
• (Notice, however, that the location of B and P0
  will depend on the value selected for Rf)

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• Note that the last proposition is true even if
  the risk-free rate (i.e. a riskless security)
  doesnt exist in the economy.
• The way the tangency portfolio B was determined
  also remains valid even if there is no riskless
  rate in the economy.
• All one has to do is replace Rf by a chosen
  constant c. The mathematics of the last
  propositions will remain valid.

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Fisher Black zero beta CAPM (1972)

• For a chosen constant c on the vertical axis of
  returns, the tangency portfolio B can be
  computed, and for ANY portfolio x we have a
  linear relationship if we regress the returns of
  x on the returns of B
• Moreover, c is the expected rate of return of a
  portfolio P0 whose covariance with B is zero.

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Fisher Black zero beta CAPM (Contd)

• The name zero beta stems from the fact that the
  covariance between P0 and B is zero, since a zero
  covariance implies that the beta of P0 with
  respect to B is zero too.
• If a riskless asset DOES exist in the economy,
  however, we can replace the constant c in Blacks
  zero beta CAPM by Rf and the portfolio B is the
  market portfolio.

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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 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

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Capital 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

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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 greater
  than the market portfolio by constructing a
  borrowing portfolio

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Capital Market Line and the Market Portfolio
(contd)
Expected Return
C
B
Rf
A
Standard Deviation
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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 markets expected return
  change

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Security Market Line (contd)
Expected Return
E(R)
Market Portfolio
Rf
1.0
Beta
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• Notice that we obtained very poor results. The
  R-squared is only 27.93 !
• However, the math of the CAPM is undoubtedly
  true.
• How then can CAPM fail in the real world?
• Possible explanations are that true asset returns
  distributions are unobservable, individuals have
  non-homogenous expectations, the market portfolio
  is unobservable, the riskless rate is ambiguous,
  markets are not friction-free.

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Using Artificial Market Portfolio
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• We obtained a perfect 100 R-squared this time !
• The reason is that when portfolio returns are
  regressed on their betas with respect to an
  efficient portfolio, an exact linear relationship
  holds.

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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 (contd)
Expected Return
Securities with Negative Expected Returns
Beta
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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

• Introduction
• Systematic and unsystematic risk
• Fundamental risk/return relationship revisited

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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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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 risk you carry, the greater the expected
  return

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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 beta

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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
• 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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Using A Scatter Diagram to Measure Beta

• Correlation of returns
• Linear regression and beta
• Importance of logarithms
• Statistical significance

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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 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

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Linear 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

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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

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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

• 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

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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 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

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APT
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Replicating the Randomness

• Lets try to replicate the random component of
  security A by forming a portfolio with the
  following weights

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Key Point in Reasoning

• Since we were able to match the random components
  exactly, the only terms that differ at this point
  are the fixed components.
• But if one fixed component is larger than the
  other, arbitrage profits are possible by
  investing in the highest yielding security
  (either A or the portfolio of factors) while
  short-selling the other (being long in one and
  short in the other will assure an exact
  cancellation of the random terms).

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• Therefore the fixed components MUST BE THE SAME
  for security A and the portfolio of factors
  created, otherwise unlimited profits would be
  possible.
• So we have

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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 SP 500 index is
  used
• APT requires specification of the relevant
  macroeconomic factors

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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 will
  result from factor sensitivities that move with
  the market and vice versa