chapter 10 slides fin 4352

8/3/2019 Chapter 10 Slides FIN 4352

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FIN 435 (Instructor- Saif Rahman)

Investment Analysis &

Portfolio Management

Chapter 10

Arbitrage Pricing Theory and MultifactorModels of Risk and Return


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Single Factor Model

Returns on a security come from two sources

Common macro-economic factor

Firm specific events

Possible common macro-economic factors

Gross Domestic Product Growth

Interest Rates


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Single Factor Model Equation

ri = Return for security I

= Factor sensitivity or factor loading or factor beta

F= Surprise in macro-economic factor

(F could be positive, negative or zero)

ei = Firm specific events

( )i i i i

r E r F e

i


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Multifactor Models

Use more than one factor in addition to market return

Examples include gross domestic product, expected

inflation, interest rates etc.

Estimate a beta or factor loading for each factor

using multiple regression.


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Multifactor Model Equation

ri = E(ri) + GDP GDP + IRIR + ei

ri = Return for security i

GDP= Factor sensitivity for GDP

IR = Factor sensitivity for Interest Rate

ei = Firm specific events

i

i

i

i


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Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment

portfolio with a sure profit. Risk-less profit with zero initial

outlay or investment.

Since no investment is required, an investor can create large

positions to secure large levels of profit. In efficient markets, profitable arbitrage opportunities will

quickly disappear.

Examplethe same product is being transacted in two shops.

The price in Shop A is Tk. 20 whereas in Shop B, the price isTk. 22. Assume buying and selling prices are same. What will

happen? How can you make risk-less profit with no initial outlay

or investment? How will this arbitrage opportunity disappear in

an efficient market?

The Law of One Price


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Arbitrage Price Theory

The Arbitrage Pricing Theory (APT) is a relatively new theoryof expected asset returns due to Ross (1976). The APT explicitly

accounts for multiple factors.

The APT requires three assumptions:1) Returns can be described by a factor model

2) There are no arbitrage opportunities

3) There are large numbers of securities that permit theformation of portfolios that diversify the firm-specific risk

of individual stocks


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If there are K factors, then the return generating process is:

ri = ai + i1F1 + i2F2 + . + iKFK+ ei

The expected returns of each security will be a function of its

factor s The model is derived by showing that for well diversified

portfolios, if the portfolios expected return (price) is not equal to

the expected return predicted by the portfolios s, then there will

be an arbitrage opportunity Note that fewer assumptions are necessary to derive the APT

(than are necessary to derive the CAPM)


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Current Expected Standard Corr.Stock Price$ Return% Dev.%

A 10 25.0 29.58 AB -0.15

B 10 20 33.91 BC -0.87C 10 32.5 48.15 AC -0.29

D 10 22.5 8.58

Arbitrage Example


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Mean S.D.

Portfolio

A,B,C 25.83 6.40

D 22.25 8.58

Arbitrage Portfolio


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Arbitrage Action and Returns

E. Ret.

St.Dev.

* P

* D

Short 3 shares of D and buy 1 of A, B & C to form P.You earn a higher rate on the investment than you pay on

the short sale.


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APT & Well-Diversified Portfolios

Based on the law of one price

Does not rely on mean-variance assumption (as theCAPM does)

It assumes that asset returns are linearly related to a setof indexes. Each index represents a factor thatinfluences the return on an asset.

rP = E (rP) + bPF + eP

F = some factor For a well-diversified portfolio:

eP approaches zero


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Comparing a Portfolio with an

Individual Security

F

E(r)%

Portfoli

o

F

E(r)%

Individual Security


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Disequilibrium Example

E(r)%

Beta for F

10

7

6

Risk Free 4

AD

C

.5 1.0


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Disequilibrium Example

Short Portfolio C

Use funds to construct an equivalent risk higher return

Portfolio D.

D is comprised of A & Risk-Free Asset

Arbitrage profit of 1%


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E(r)%

Beta (Market Index)

Risk Free

M

1.0

[E(rM) - rf]

Market RiskPremium

APT with Market Index Portfolio


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The CAPM is a special case of APT that would result if the singlecommon factor affecting all security returns was the return on the

market portfolio.

APT is more general, or robust, than the CAPM. It is based on less

restrictive assumptions.

APT does not identify either the number or the definition of the factors

affecting returns. These have to be empirically determined by fitting a

factor model to returns.

The CAPM is a well-specified model, where the parameters of the

model are spelled out up front. However, it relies on the MarketPortfolio, which is in principle non measureable. APT is more general

in that it gets to an expected return and beta relationship without the

assumption of the market portfolio.

APT can be extended to multifactor models.

APT and CAPM Compared


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In order to implement the APT we need to know what the factors

are! Here the theory gives no guidance. There is some evidence that

the following macroeconomic variables may be risk factors:

1)Changes in monthly GDP

2)Changes in the default risk premium

3)The slope of the yield curve

4)Unexpected changes in the price level5)Changes in expected inflation

Note that the difficulty of measuring expected inflation makes the

estimation of 4 & 5 difficult


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Fama-French Three-Factor Model

The factors chosen are variables that on past evidence seem to

predict average returns well and may capture the risk premiums

Where:

SMB = Small Minus Big, i.e., the return of a portfolio of small

stocks in excess of the return on a portfolio of large stocks

HML = High Minus Low, i.e., the return of a portfolio of stockswith a high book to-market ratio in excess of the return on a

portfolio of stocks with a low book-to-market ratio

it i iM Mt iSMB t iHML t it r R SMB HML e


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The Multifactor CAPM and the APM

A multi-index CAPM will inherit its risk factors

from sources of risk that a broad group of investors

deem important enough to hedge

The APT is largely silent on where to look for priced

sources of risk

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