ppt efficient frontier

7/28/2019 PPT Efficient Frontier

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Chapter 7

Why Diversif ication I s a Good I dea


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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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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

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 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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Role of Uncorrelated SecuritiesVariance 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

CombinationOne 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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Combination (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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Combination (contd)Return variance is a securitys total r isk

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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Combination (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%.

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/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% in Stock B and 50% inStock 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 DominanceDominanceis 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

Exp

ec

tedRe

turn

B/C combination

dominates A/C


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Harry Markowitz: Founder of

Portfolio TheoryIntroduction

Terminology

Quadratic programming


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Introduction Harry Markowitzs Portfolio SelectionJournal

of Finance article (1952) set the stage for modernportfolio theory

The first major publication indicating the important 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 are required


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TerminologySecurity 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 UniverseThe secur ity universeis 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 FrontierConstruct a risk/return plot of all possible

portfolios

Those portfolios that are not dominatedconstitute the eff icient 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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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 variance

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 portfol io

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

It is the only risky assets risk-averse investors willhold


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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 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 portfol io

Investors can achieve portfolio returns

greater than the market portfolio by

constructing a borrowing portfol io


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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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Expansion of the SML to

Four QuadrantsThere are securities with negative betas and

negative expected returns

A reason for purchasing these securities is theirrisk-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 PortfolioA corner portfol iooccurs 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

ofquadratic 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 ofFinance article

Very consequential research regarding portfolioconstruction

Shows how nave diversif icationreduces the

dispersion of returns in a stock portfolioNave 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 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

Systematic r iskis the risk that remains after

no further diversification benefits can be

achievedUnsystematic r iskis 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

Number of Securities

Portfolio Variance


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Strength in Numbers

Portfolio variance (total risk) declines as thenumber of securities included in theportfolio increases

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

Risk-averse investors should always diversifyto 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

Superf luous diversif icationrefers 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 portfolio may be outweighed by thetransaction 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 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 notmean 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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Capital Asset Pricing Model

Introduction

Systematic and unsystematic risk

Fundamental risk/return relationshiprevisited


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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 betaBeta determines the level of expected return on a

security or portfolio (SML)

i /


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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 (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 t 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

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

that betas change


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Equity Risk Premium

Equi ty r isk 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

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

Importance of logarithmsStatistical 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 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 Excel to run a linear regression and obtainthe 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

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 pr icing 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 notcapture 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 FR 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 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

ppt efficient frontier

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