Lecture 6

Arbitrage Pricing Theory and Multifactor Models of Risk and

Return

30055 Financial Economics Prof. Petrova

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Index models decompose stock variability into market and firm-specific The return on the market portfolio summarizes the broad

impact of macro factors

Sometimes rather than using a market proxy is useful to focus directly on the ultimate sources of risk Useful in risk assessments measuring risk exposure

Factor models allow to describe and quantify different factors that affect the rate of return on a security during any period

Index vs. Factor Models

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Single Factor Model

Returns on a security come from two sources: Common macro

Constructed to have exp. value of 0. F is the deviation of the common factor from its expected

value

Firm specific events Possible common macro-economic factors

Gross Domestic Product Growth Interest Rates

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Single Factor Model Equation

Ri = Excess return on security

i= Factor sensitivity or factor loading or factor beta

F = Surprise in macro-economic factor

(F could be positive or negative but has expected value of zero)

ei = Firm specific events (zero expected value)

ei assumed uncorrelated among themselves and with F

= () + +

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Suppose that F is the news about the state of the business cycle, measured as the unexpected change in GDP and that the consensus is that GDP will increase by 5% next year. Suppose that stock has a F sensitivity of 1.2. If GDP increases by 3% what would be the effect on the return on the stock?

Factor Model Example

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Factors model decomposition to systematic and non-systematic compelling, but confining to one systematic risk to a single factor is not

Number of systematic risk sources GDP Interest rates Term structure Inflation Employment Exchange rates

Single-index models assume that all securities have the same sensitivity to the various macro-factors

Multifactor models

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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 (Two-factor) Model Equation

Ri = Excess return for security i GDP = Factor sensitivity for GDP IR = Factor sensitivity for Interest Rate ei = Firm specific events

( ) iiIRiGDPii eIRGDPRER +++=

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Interpretation

The expected return on a security is the sum of: 1.The risk-free rate 2.The sensitivity to GDP times the risk

premium for bearing GDP risk 3.The sensitivity to interest rate risk times the

risk premium for bearing interest rate risk

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Consider two firms Firm A - Regulated power utility in a residential

area Firm B - An airline

What are the sensitivities of A and B to: GDP IR

Is a macro news that the economy will expand bad or good for A & B?

Two-factor Model Example

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Consider United Airlines We estimate a two-factor model and find that: r=.12+1.2GDP - .3IR +e What is the interpretation of this model?

Two-factor Model Example

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Multifactor model a description of the factors that affect security returns No theory in the equation Where does E(r) come from?

One difference b/n single and multiple-factor

economy is that a factor risk premium can be negative

E.g. security with a positive IR beta hedges the value of portfolio against IR risk

A Multifactor SML

( ) IRIRGDPGDPfi RPRPrrE ++=

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Suppose rf=3% RPGDP=7% RPIR=2% What is E(r)?

United Example Continued

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Arbitrage Pricing Theory

APT developed by Stephen Ross in 1976 Security returns are described by factor models There is a large number of securities and sufficient to diversify

firm-specific risk Well-functioning markets do not allow for persistent arbitrage

opportunities

Arbitrage occurs if there is a zero investment portfolio with a sure profit.

Since no investment is required, investors can create large positions to obtain large profits.

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Arbitrage Pricing Theory

Regardless of wealth or risk aversion, investors will want an infinite position in the risk-free arbitrage portfolio. In efficient markets, profitable arbitrage opportunities will quickly disappear.

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If two assets re equivalent in all economically relevant aspects they should have the same price Enforced by arbitrageurs

Arbitrageur investor looking for mispriced securities

Risk arbitrage vs. pure arbitrage Derivative vs. primitive securities

Derivative securities condition of no-arbitrage leads to exact pricing

Primitive securities use diversification arguments

The Law of One Price

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APT & Well-Diversified Portfolios

RP = E (RP) + bPF + eP F = some factor For a well-diversified portfolio, eP

approaches zero as the number of securities in the portfolio increases

and their associated weights decrease If the securities in the portfolio are equally

weighted: 2 = 22() = 1/n(2())

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RP = E (RP) + bPF Non-factor risk is diversified away and only

factor risk commands a risk premium in market equilibrium

APT & Well-Diversified Portfolios

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Figure 10.1 Returns as a Function of the Systematic Factor

rp=10% +1.0*F

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Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

What is the arbitrage opportunity here? Form an arbitrage strategy and find the riskless payoff.

rp=10% +1.0*F

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Consider a single factor APT. Portfolio A has a beta of 1.0 and an expected return of 16%. Portfolio B has a beta of 0.8 and an expected return of 12%. The risk-free rate of return is 6%. If you wanted to take advantage of an arbitrage opportunity, you should take a short position in portfolio __________ and a long position in portfolio _______.

Example

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Consider the one-factor APT. The variance of returns on the factor portfolio is 6%. The beta of a well-diversified portfolio on the factor is 1.1. The variance of returns on the well-diversified portfolio is approximately

Example

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Figure 10.3 An Arbitrage Opportunity Portfolio with Different Betas

D is composed of .5 in A and .5 in the risk-free asset

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No-Arbitrage Equation of APT

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the APT, the CAPM and the Index Model

Assumes a well-diversified portfolio, but residual risk is still a factor.

Does not assume investors are mean-variance optimizers.

Uses an observable, market index

Reveals arbitrage opportunities

APT CAPM Model is based on an

inherently unobservable market portfolio.

Rests on mean-variance efficiency.

The actions of many small investors restore CAPM equilibrium.

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Multifactor APT

Use of more than a single systematic factor Requires formation of factor portfolios What factors?

Factors that are important to performance of the general economy

What about firm characteristics?

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Two-Factor Model

The multifactor APT is similar to the one-

factor case.

= () + 11 + 22 +

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Two-Factor Model

Track with diversified factor portfolios: beta=1 for one of the factors and 0 for

all other factors.

The factor portfolios track a particular source of macroeconomic risk, but are uncorrelated with other sources of risk.

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Consider the multifactor APT with two factors. Stock A has an expected return of 17.6%, a beta of 1.45 on factor 1, and a beta of .86 on factor 2. The risk premium on the factor 1 portfolio is 3.2%. The risk-free rate of return is 5%. What is the risk-premium on factor 2 if no arbitrage opportunities exist?

Example

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Fama-French Three-Factor Model

SMB = Small Minus Big (firm size) HML = High Minus Low (book-to-market ratio) Are these firm characteristics correlated with

actual (but currently unknown) systematic risk factors?

ittiHMLtiSMBMtiMiit eHMLSMBRR ++++=

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The Multifactor CAPM and the APT

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

Lecture 6Index vs. Factor ModelsSingle Factor ModelSingle Factor Model EquationFactor Model ExampleMultifactor modelsMultifactor ModelsMultifactor (Two-factor) Model EquationInterpretationTwo-factor Model ExampleTwo-factor Model ExampleA Multifactor SMLUnited Example ContinuedArbitrage Pricing TheoryArbitrage Pricing TheoryThe Law of One PriceAPT & Well-Diversified PortfoliosAPT & Well-Diversified PortfoliosFigure 10.1 Returns as a Function of the Systematic FactorFigure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage OpportunityExampleExampleFigure 10.3 An Arbitrage OpportunityPortfolio with Different BetasNo-Arbitrage Equation of APTthe APT, the CAPM and the Index ModelMultifactor APTTwo-Factor ModelTwo-Factor ModelExampleFama-French Three-Factor ModelThe Multifactor CAPM and the APT