Market model

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Chapter 7

Why Diversification Is a Good Idea

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The most important lesson learned is an old truth ratified.

- General Maxwell R. Thurman

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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

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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 (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

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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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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

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Variance of A Linear Combination

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 (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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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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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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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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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?

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Example (cont’d)

Solution: Two combinations achieve a 12% 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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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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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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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.

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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: Founder of Portfolio Theory

Introduction Terminology Quadratic programming

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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

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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 (cont’d)

Standard Deviation

Expected Return100% investment in security with highest E(R)

100% investment in minimum variance portfolio

Points below the efficient frontier are dominated

No points plot above the line

All portfolios on the line are efficient

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Efficient Frontier (cont’d) The farther you move to the left on the

efficient frontier, the greater the number of securities in the portfolio

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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 variance

– 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 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 (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

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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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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

Introduction Methodology Results Implications Words of caution

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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

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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 (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

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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 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 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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Fundamental Risk/Return Relationship Revisited

CAPM SML and CAPM Market model versus CAPM Note on the CAPM assumptions Stationarity of beta

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CAPM The more 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

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CAPM (cont’d) CAPM assumptions (cont’d):

• 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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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

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Market Model Versus CAPM (cont’d)

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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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 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

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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 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

Economy & Finance

Market model