Arbitage Pricing Model

Chapter 9

AN INTRODUCTION TO ASSET

PRICING MODELS

Chapter 9 Questions

What are the assumptions of the capital asset pricing model?

What is a risk-free asset and what are its risk-return characteristics?

What is the covariance and correlation between the risk-free asset and a risky asset or portfolio of risky assets?

What is the expected return when we combine the risk-free assets and a portfolio of risky assets?

Chapter 9 Questions

What is the standard deviation when we combine the risk-free asset and a portfolio of risky assets?

When you combine the risk-free asset and a portfolio of risky assets on the Markowitz efficient frontier, what does the set of possible portfolios look like?

Given the initial set of portfolio possibilities with a risk-free asset, what happens when you add financial leverage (that is, borrow)?

Chapter 9 Questions

What is the market portfolio, what assets are included in this portfolio, and what are the relative weights for the alternative assets included?

What is the capital market line (CML)?

What do we mean by complete diversification?

How do we measure diversification for an individual portfolio?

What are systematic and unsystematic risk?

Chapter 9 Questions

Given the capital market line (CML), what is

the separation theorem?

Given the CML, what is the relevant risk

measure for an individual risky asset?

What is the security market line (SML) and

how does it differ from the CML?

What is beta and why is it referred to as a

standardized measure of systematic risk?

Chapter 9 Questions

How can we use the SML to determine the expected (required) rate of return for a risky asset?

Using the SML, what do we mean by an undervalued and overvalued security, and how do we determine whether an asset is undervalued or overvalued?

What is an asset’s characteristic line and how do we compute the characteristic line for an asset?

Chapter 9 Questions

What is the impact on the characteristic

line when we compute it using different

return intervals (such as weekly versus

monthly) and when we employ different

proxies (that is, benchmarks) for the

market portfolio (for example, the S&P

500 versus a global stock index)?

Chapter 9 Questions

What is the arbitrage pricing theory

(APT) and how does it differ from the

capital asset pricing model (CAPM) in

terms of assumptions?

How does the APT differ from the

CAPM in terms of risk measure?

Capital Market Theory:

An Overview

Capital market theory extends portfolio theory

and seeks to develops a model for pricing all

risky assets based on their relevant risks

Asset Pricing Models

Capital asset pricing model (CAPM) allows for the

calculation of the required rate of return for any

risky asset based on the security’s beta

Arbitrage Pricing Theory (APT) is a multi-factor

model for determining the required rate of return

Assumptions of

Capital Market Theory

All investors are Markowitz efficient investors who invest on the efficient frontier.

Investors can borrow or lend any amount of money at the risk-free rate of return (RFR).

Investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.

All investors have the same one-period time horizon such as one-month, six months, or one year.

Assumptions of


All investments are infinitely divisible, which

means that it is possible to buy or sell

fractional shares of any asset or portfolio.

There are no taxes or transaction costs

involved in buying or selling assets.

There is no inflation or any change in interest

rates, or inflation is fully anticipated.

Capital markets are in equilibrium.

Making Assumptions

Some of these assumptions are clearly

unrealistic

Relaxing many of these assumptions would

have only minor influence on the model and

would not change its main implications or

conclusions.

The primary way to judge a theory is on how

well it explains and helps predict behavior,

not on its assumptions.


and a Risk-Free Asset

Perhaps surprisingly, there are rather large implications for capital market theory when a risk-free asset exists.

What is a risk-free asset? An asset with zero variance

Provides the risk-free rate of return (RFR)

It will be an “intercept” value on a portfolio graph between expected return and standard deviation.

Since it has zero variance, it will also have zero correlation with all other risky assets

Risk-Free Asset

Covariance between two sets of returns isn

1i

jjiiij )]/nE(R-)][RE(R-[RCov

Because the returns for the risk free asset are certain,

0RFThus Ri = E(Ri), and Ri - E(Ri) = 0

Consequently, the covariance of the risk-free asset with any

risky asset or portfolio will always equal zero. Similarly the

correlation between any risky asset and the risk-free asset

would be zero.

Combining a Risk-Free

Asset with a Portfolio

Expected return is the weighted average

of the two returns

))E(RW-(1(RFR)W)E(R iRFRFport

This is a linear relationship



Standard deviation: The expected variance for a

two-asset portfolio is

211,221

2

2

2

2

2

1

2

1

2

port rww2ww)E(Substituting the risk-free asset for Security 1, and the risky

asset for Security 2, this formula would become

iRFiRF iRF,RFRF

22

RF

22

RF

2

port )rw-(1w2)w1(w)E(

Since we know that the variance of the risk-free asset is

zero and the correlation between the risk-free asset and any

risky asset i is zero we can adjust the formula22

RF

2

port )w1()E( i



Given the variance formula22

RF

2

port )w1()E( i

22

RFport )w1()E( ithe standard deviation is

i)w1( RF

Therefore, the standard deviation of a portfolio that

combines the risk-free asset with risky assets is the

linear proportion of the standard deviation of the risky

asset portfolio.



Since both the expected return and the

standard deviation of return for such a

portfolio are linear combinations, a

graph of possible portfolio returns and

risks looks like a straight line between

the two assets.

Portfolio Possibilities Combining

the Risk-Free Asset and Risky

Portfolios on the Efficient

Frontier

)E( port

)E(R port

RFR

M

C

AB

D

Risk-Return Possibilities

with Leverage

To attain a higher expected return than is

available at point M (in exchange for

accepting higher risk)

Either invest along the efficient frontier

beyond point M, such as point D

Or, add leverage to the portfolio by borrowing

money at the risk-free rate and investing in

the risky portfolio at point M

Portfolio Possibilities Combining

the Risk-Free Asset and Risky

Portfolios on the Efficient

Frontier

)E( port

)E(R port

RFR

M

The Market Portfolio

Portfolio M lies at the point of tangency, so it has the highest portfolio possibility line

This line of tangency is called the Capital Market Line (CML)

Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML (the CML is a new efficient frontier) Therefore this portfolio must include all risky

assets (or else some assets would have no demand)

The Market Portfolio

Because the market is in equilibrium, all

assets are included in this portfolio in

proportion to their market value

Because it contains all risky assets, it is

a completely diversified portfolio, which

means that all the unique risk of

individual assets (unsystematic risk) is

diversified away

Systematic Risk

Only systematic risk remains in the market

portfolio

Systematic risk is the variability in all risky

assets caused by macroeconomic variables

Systematic risk can be measured by the

standard deviation of returns of the market

portfolio and can change over time

Factors Affecting

Systematic Risk

Systematic risk factors are those

macroeconomic variables that affect the

valuation of all risky assets

Variability in growth of the money supply

Interest rate volatility

Variability in aggregate industrial

production

How to Measure

Diversification

All portfolios on the CML are perfectly

positively correlated with each other and

with the completely diversified market

Portfolio M

A completely diversified portfolio would

have a correlation with the market

portfolio of +1.00

Diversifying Away

Unsystematic Risk

The purpose of diversification is to reduce the standard deviation of the total portfolio

As you add securities, you expect the average covariance for the portfolio to decline, but not to disappear since correlations are not perfectly negative.

About how many securities must you add to obtain a completely diversified portfolio? About 90% of the benefit after 12-18 stocks

Maximum benefit needs between 30 and 40

The Portfolio Standard

Deviation

Standard Deviation of Return

Number of Stocks in the Portfolio

Standard Deviation of

the Market Portfolio

(systematic risk)Systematic Risk

Total

Risk

Unsystematic

(diversifiable)

Risk

The CML and the

Separation Theorem

The CML leads all investors to invest in the M portfolio (The Investment Decision)

Individual investors should differ in position on the CML depending on risk preferences (which leads to the Financing Decision) Risk averse investors will lend part of the portfolio

at the risk-free rate and invest the remainder in the market portfolio (points left of M)

Aggressive investors would borrow funds at the RFR and invest everything in the market portfolio (points to the right of M)

A Risk Measure for the

CML

If…

the relevant risk in a portfolio is the average

covariance with all other assets in the portfolio,

and …

the only relevant portfolio is the market portfolio (M),

then it follows that …

the covariance with the market portfolio is the

relevant (systematic) risk of an asset.

A Risk Measure for the

CML

Because all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return for the M portfolio may be described using the following linear model:

Rit = ai +biRMt +

where:

Rit = return for asset i during period t

ai = constant term for asset i

bi = slope coefficient for asset i

RMt = return for the M portfolio during period t

=random error term

The Capital Asset

Pricing Model

The existence of a risk-free asset resulted in

deriving a capital market line (CML) that

became the relevant frontier

An asset’s covariance with the market

portfolio is the relevant risk measure

This can be used to determine an appropriate

required rate of return on a risky asset - the

capital asset pricing model (CAPM)

The Capital Asset

Pricing Model

CAPM indicates what should be the expected

or required rates of return on risky assets

This helps to value an asset by providing an

appropriate discount rate to use in dividend

valuation models

You can compare an expected rate of return

to the required rate of return implied by

CAPM - over/ under valued?

The Security Market

Line (SML)

The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)

This is shown as the risk measure

The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance

The Security Market

Line (SML)

)Cov(RFR-R

RFR)E(R Mi,2

M

Mi

RFR)-R(Cov

RFR M2

M

Mi,

2

M

Mi,CovWe then define as beta

RFR)-(RRFR)E(R Mi i

)( i

Graph of SML

)E(R i

)Beta(Cov 2Mim/0.1

mR

SML

0

Negative

Beta

RFR

Determining the

Expected Return

The expected rate of return of a risk

asset is determined by the RFR plus a

risk premium for the individual asset

The risk premium is determined by the

systematic risk of the asset (beta) and

the prevailing market risk premium

(RM-RFR)

RFR)-(RRFR)E(R Mi i

Determining the

Expected Return

In equilibrium, all assets and all portfolios of

assets should plot on the SML

The SML gives the market “going rate of return” or

what you should earn as a return for a security

Any security with an expected return that plots

above the SML is underpriced

Any security with an expected return that plots

below the SML is overpriced

Identifying Undervalued

and Overvalued Assets

Compare the required rate of return to

the expected rate of return for a specific

risky asset using the SML over a

specific investment horizon to determine

if it is an appropriate investment

Independent estimates of expected

return for the securities provide price

and dividend outlooks

Calculating Beta: The

Characteristic Line

The systematic risk input of an individual asset is derived

from a regression model, referred to as the asset’s

characteristic line with the model portfolio:

tM,iiti, RRwhere:

Ri,t = the rate of return for asset i during period t

RM,t = the rate of return for the market portfolio M during t

miii R-R

2M

Mi,Covi

error term random the

Issues in Beta

Estimation

The Impact of the Time Interval Number of observations and time interval used in

regression vary Value Line Investment Services (VL) uses weekly rates

of return over five years

Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years

There is no “correct” interval for analysis

Weak relationship between VL & ML betas due to difference in intervals used

Interval effect impacts smaller firms more

Issues in Beta

Estimation

The Effect of the Market Proxy A measure of the market portfolio is needed

S&P 500 Composite Index is most often used Includes a large proportion of the total market value of

U.S. stocks

Value weighted series

Weaknesses of Using S&P 500as the Market Proxy Includes only U.S. stocks

The theoretical market portfolio should include all types of assets from all around the world

Arbitrage Pricing Theory

(APT)

CAPM is criticized because of the

difficulties in selecting a proxy for the

market portfolio as a benchmark

An alternative pricing theory with fewer

assumptions was developed:


Assumptions of


Capital markets are perfectly

competitive

Investors always prefer more wealth to

less wealth with certainty

The stochastic process generating

asset returns can be presented as a k-

factor model

Assumptions of CAPM

Not Required by APT

APT does not assume

A market portfolio that contains all risky

assets, and is mean-variance efficient

Normally distributed security returns

Quadratic utility function for investors


For i = 1 to N where:

= return on asset i during a specified time period

= expected return for asset i

= reaction in asset i’s returns to movements in a common factor

= a common factor with a zero mean that influences the returns on all assets

= a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero

= number of assets

ikikiiiitt bbbER ...21

Ri

Ei

bik

k

i

N


Multiple factors expected to have an impact on all assets:

Inflation

Growth in GNP

Major political upheavals

Changes in interest rates

And many more….

Contrast with CAPM assumption that only beta is relevant


(APT)

bik measure how each asset (i) reacts to a common factor (k)

Each asset may be affected by a factor, but the effects will differ

In application of the theory, the factors are not identified

Similar to the CAPM, the unique effects are independent and will be diversified away in a large portfolio


(APT)

APT assumes that, in equilibrium, the

return on a zero-investment, zero-

systematic-risk portfolio is zero when

the unique effects are diversified away

The expected return on any asset i (Ei)

can be expressed as:


(APT)

where:

= the expected return on an asset with zero

systematic risk where

ikkiii bbbE ...22110

0

01 EEi

00 E

1= the risk premium related to each of the common

factors - for example the risk premium related to

interest rate risk

bik = the pricing relationship between the risk premium and

asset i - that is how responsive asset i is to this common

factor K

Example of Two Stocks

and a Two-Factor Model

= changes in the rate of inflation. The risk premium

related to this factor is 1 percent for every 1

percent change in the rate

1

)01.( 1

= percent growth in real GNP. The average risk premium

related to this factor is 2 percent for every 1 percent

change in the rate

= the rate of return on a zero-systematic-risk asset (zero

beta: boj=0) is 3 percent

2

)02.( 2

)03.( 3

3



= the response of asset X to changes in the rate of

inflation is 0.501xb

)50.( 1xb

= the response of asset Y to changes in the rate of inflation

is 2.00 )50.( 1yb1yb

= the response of asset X to changes in the growth rate of

real GNP is 1.50

= the response of asset Y to changes in the growth rate of

real GNP is 1.75

2xb

2yb

)50.1( 2xb

)75.1( 2yb



= .03 + (.01)bi1 + (.02)bi2

Ex = .03 + (.01)(0.50) + (.02)(1.50)

= .065 = 6.5%

Ey = .03 + (.01)(2.00) + (.02)(1.75)

= .085 = 8.5%

22110 iii bbE

Empirical Tests of the

APT

Studies by Roll and Ross and by Chen

support APT by explaining different rates of

return with some better results than CAPM

Reinganum’s study indicated that the APT

does not explain small-firm results

Dhrymes and Shanken question the

usefulness of APT because it was not

possible to identify the factors and therefore

may not be testable

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