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Chapter 6

Why Diversif ication I s a Good I dea

Portfolio Construction, Management, & Protection, 4e, Robert A. Strong

Copyright 2006 by South-Western, a division of Thomson Business & Economics. All rights reserved.


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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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IntroductionDiversification 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 BasketInvestments in Your Own Ego

The Concept of Risk Aversion Revisited

Multiple Investment Objectives


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Investments in Your Own EgoNever put a large percentage of investment

funds into a single security

If the security appreciates, the ego is strokedand 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 RevisitedDiversification 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 willbe rewarded for taking them


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The Concept of Risk

Aversion Revisited (contd)Diversification is more important now

AJournal of Financearticle shows that

volatility of individual firms has increased

Investors need more stocks to adequately diversify


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Multiple Investment ObjectivesMultiple 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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Variance of a Linear Combination:

The Practical MeaningOne 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 j

i j

i

ij

x x

x i

i j


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The Practical Meaning (contd)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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The Practical Meaning (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

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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The Practical Meaning (contd)If two securities have low correlation, the

interactive risk will be small

If two securities are uncorrelated, theinteractive risk drops out

If two securities are negatively correlated,

interactive risk would be negative andwould reduce total risk


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Portfolio Programming

in a NutshellVarious 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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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 .10Standard deviation .232 .136 .195


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in a Nutshell (contd)Example (contd)

The correlation coefficients between the three stocks are:

Stock A Stock B Stock C

Stock A 1.000Stock B 0.286 1.000

Stock C 0.132 0.605 1.000


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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?


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Solution: Two combinations achieve a 12 percent 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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Solution (contd):Calculate the variance of the B/Ccombination:

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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Solution (contd):Calculate the variance of the A/Ccombination:

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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Solution (contd):Investing 50 percent in Stock B and 50percent 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.


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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:

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

ec

tedRe

turn

B/C combination

dominates A/C


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Harry Markowitz: The

Founder of Portfolio TheoryIntroduction

Terminology

Quadratic Programming


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Introduction Harry Markowitzs Portfolio SelectionJournal

of Financearticle (1952) set the stage for modernportfolio theory

The first major publication indicating the importance ofsecurity return correlation in the construction of stock

portfolios

Markowitz showed that for a given level of expectedreturn and for a given security universe, knowledge ofthe covariance and correlation matrices is required


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TerminologySecurity Universe

Efficient Frontier

Capital Market Line and the MarketPortfolio

Security Market Line

Expansion of the SML to Four Quadrants

Corner Portfolio


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Security UniverseThe security universe is the collection of all

possible investments

For some institutions, only certain investmentsmay 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 (contd)

Standard Deviation

Expected Return100% Investment in Security

with Highest E(R)

100% Investment in MinimumVariance Portfolio

Points plotting below the

efficient frontier are dominated

by other portfolios

No points plot above

the line

All portfolios

on the line

are efficient


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Efficient Frontier (contd)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 (contd)When a risk-free investment is available,

the shape of the efficient frontier changes

The expected return and variance of a risk-freerate/stock return combination are simply a

weighted average of the two expected returns

and variances

The risk-free rate has a variance of zero


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Efficient Frontier (contd)

Standard Deviation

Expected Return

Rf

A

B

C


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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 efficientfrontier

Follows the curve from point B to point C


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Capital Market Line and the

Market PortfolioThe tangent line passing from the risk-free

rate through point B is the capital marketline (CML)

When the security universe includes all possibleinvestments, point B is the market portfolio

It contains every risky asset in the proportion of itsmarket value to the aggregate market value of allassets

It is the only risky asset risk-averse investors willhold


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Market Portfolio (contd)Implications for investors:

Regardless of the level of risk-aversion, allinvestors 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 themarket portfolio


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Market Portfolio (contd)Any risky portfolio that is partially invested

in the risk-free asset is a lending portfolio

Investors can achieve portfolio returns

greaterthan the market portfolio by

constructing a borrowing portfolio


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Market Portfolio (contd)

Standard Deviation

Expected Return

Rf

A

B

C


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Security Market LineThe graphical relationship between

expected return and beta is the securitymarket line (SML)

The slope of the SML is the market price ofrisk

The slope of the SML changes periodically asthe risk-free rate and the markets expectedreturn change


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Security Market Line (contd)

Beta

Expected Return

Rf

Market Portfolio

1.0

E(R)


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Security Market Line (contd)

Beta

Expected Return

Securities with NegativeExpected Returns


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Corner PortfolioA corner portfoliooccurs 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 ProgrammingThe Markowitz algorithm is an application

of quadratic programming

The objective function involves portfoliovariance

Quadratic programming is very similar to linear

programming


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Markowitz Quadratic

Programming Problem


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Lessons from

Evans and ArcherIntroduction

Methodology

Results

Implications

Words of Caution


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Introduction

Evans and Archers 1968Journal ofFinancearticle

Very consequential research regarding portfolioconstruction

Shows how nave diversificationreduces the

dispersion of returns in a stock portfolioNave diversification refers to the selection ofportfolio components randomly without any serioussecurity analysis


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Methodology

Used computer simulations:

Measured the average variance of portfolios of

different sizes, up to portfolios with dozens ofcomponents

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 (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

Number of Securities

Portfolio Variance

Source: Adapted by Edwin J. Elton and Martin J. Gruber, Risk Production and Portfolio Size: An Analytical Solution, Journal of Business,

October 1977, 415437.


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Superfluous Diversification

Superfluous diversificationrefers to theaddition of unnecessary components to analready well-diversified portfolio

Deals with the diminishing marginal benefits ofadditional portfolio components

The benefits of additional diversification inlarge portfolios may be outweighed by thetransaction costs


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Implications

Very effective diversification occurs evenwhen the investor owns only a smallfraction of the total number of availablesecurities

Institutional investors may not be able to avoidsuperfluous diversification due to the dollar size

of their portfoliosMutual funds are prohibited from holding more than5 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 properdiversification

Although nave diversification reduces risk,

it can also reduce return Unlike Markowitzs 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 willtake risk, so they choose securities with

different betas

e.g., an aggressive investor could choose a portfolio

with a beta of 2.0e.g., a conservative investor could choose a portfolio

with a beta of 0.5


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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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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 (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 the expected market risk

premium


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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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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 ofreturns 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


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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 premiumrefers to thedifference in the average return betweenstocks 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 ofthe equity risk premium is shrinking


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Correlation of Returns

Much of the daily news is of a generaleconomic nature and affects all securities

Stock prices often move as a group

Some stocks routinely move more than theothers regardless of whether the market

advances or declinesSome stocks are more sensitive to changes ineconomic 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 Microsoft Excel to run a linear regressionand obtain the coefficients

The coefficient for the market return is the betastatistic

The intercept is the trend in the security pricereturns that is inexplicable by finance theory


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Importance of Logarithms

Introduction

Statistical Significance


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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


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Statistical Significance

Published betas are not always useful

numbers

Individual securities have substantialunsystematic 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

Another alternative = Fama & Frenchs 3-Factor Model


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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) doesnot capture all the information relevant to the

price of a stock


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Comparison of the


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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 S&P 500 index is used

APT requires specification of the relevantmacroeconomic factors

Comparison of the


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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 willresult from factor sensitivities that move with

the market and vice versa APT can be viewed as a more general and more flexible

version of CAPM

Instead of having one all-encompassing measure of systematic

risk, APT breaks down the measure of systematic risk into a

variety of different variables that either drive or reflect

differences in systematic risk

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