Portfolio Management - Chapter 7

8/3/2019 Portfolio Management - Chapter 7

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Chapter 7

Why Diversification Is a Good Idea

Prof. Rushen Chahal 1

Prof. Rushen Chahal


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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

I

f the security appreciates, the ego is stroked andthis 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 verydifficult 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 brokerageaccount 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 j

i j

i

ij

x x

x i

i j

W V W W

V

! !

!

!!


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(contd) The variance of a two-security portfolio is:


2 2 2 2 2 2 p A A B B A B AB A B x x x xW W W V W W!


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(contd) Return variance is a securitys total risk

Most investors want portfolio variance to be

as low as possible without having to give upany return


2 2 2 2 2

2 p A A B B A B AB A B x x x xW W W V W W! Total Risk Risk from A Risk from B Interactive Risk


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(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 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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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%.

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% 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/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 B x x x xW W W V W W!

!

!

!


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Solution (contd): 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 B x x x xW W W V W W!

!

!

!


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Solution (contd): Investing 50% in Stock B and 50% in Stock C

achieves an expected return of 12% with the lower portfoliovariance. 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 (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

ExpectedReturn

B/C combination

dominates A/C


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Harry Markowitz: Founder of

P

ortfolio Theory Introduction

Terminology

Quadratic programming



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Introduction

Harry Markowitzs Portfolio Selection Journal ofFinance article (1952) set the stage for modernportfolio theory

The first major publication indicating the important ofsecurity return correlation in the construction of stockportfolios

Markowitz showed that for a given level of expected return

and for a given security universe, knowledge of thecovariance 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 (contd)


Standard Deviation

Expected Return100% investment in security

with highest E(R)

100% investment in minimumvariance 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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The farther you move to the left on the

efficient frontier, the greater the number of

securities in the portfolio



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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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Standard Deviation

Expected Return

RfA

B

C


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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 possibleinvestments, point B is the market portfolio

It contains every risky assets in the proportion of itsmarket value to the aggregate market value of all assets

It is the only risky assets risk-averse investors will hold



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Market Portfolio (contd) Implication 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 nearthe 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 greater

than the market portfolio by constructing a

borrowing portfolio



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Market Portfolio (contd)


Standard Deviation

Expected Return

RfA

B

C


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Security Market Line

The graphical relationship between expectedreturn 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 therisk-free rate and the markets expected return

change



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Security Market Line (contd)


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 (contd)


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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QuadraticProgramming

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 Archers 1968 Journal of Financearticle

Very consequential research regarding portfolio

construction

Shows how nave diversification reduces thedispersion of returns in a stock portfolio

Nave diversification refers to the selection of portfoliocomponents 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 randomlyselected



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Results

Definitions

General results

Strength in numbers Biggest benefits come first

Superfluous diversification



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Definitions

Systematic riskis the risk that remains after

no further diversification benefits can be

achieved

Unsystematic riskis 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 isdiversifiable



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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


NumberofSecurities

Portfolio Variance


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Strength in Numbers

Portfolio variance (total risk) declines as thenumber of securities included in the portfolioincreases

On average, a randomly selected ten-securityportfolio will have less risk than a randomlyselected three-security portfolio

Risk-averse investors should always diversify toeliminate 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 portfolioprovides only modest additional benefits



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Superfluous Diversification

Superfluous diversification refers to theaddition of unnecessary components to analready well-diversified portfolio

Deals with the diminishing marginal benefits ofadditional portfolio components

The benefits of additional diversification in large

portfolio may be outweighed by the transactioncosts



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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 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 notmean 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 apositive 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

F

F

! -

!

!

!

!


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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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Market Model Versus CAPM

The market model is an ex postmodel

It describes past price behavior

The CAPM is an ex ante model

It predicts what a value should be


M k M d l


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Market Model

Versus CAPM (contd)

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

E F

E

F

!

!

!

!

!!

N t th


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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 thatbetas change



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Equity Risk Premium

Equity risk premium refers to the difference inthe average return between stocks and somemeasure of the risk-free rate

The equity risk premium in the CAPM is the excessexpected return on the market

Some researchers are proposing that the size of

the equity risk premium is shrinking


U i A S tt Di t


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Using A Scatter Diagram to

Measure Beta

Correlation of returns

Linear regression and beta

I

mportance of logarithms Statistical significance



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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 stock routinely move more than the othersregardless of whether the market advances ordeclines

Some stocks are more sensitive to changes in economicconditions



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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 thecoefficients

The coefficient for the market return is the betastatistic

The intercept is the trend in the security price returnsthat 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 anumber of distinct factors determine themarket return

Roll and Ross state that a securitys long-runreturn 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 ofa 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

! !

!

!!

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 relevant

macroeconomic 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 will

result from factor sensitivities that move with themarket and vice versa

Portfolio Management - Chapter 7

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