Chapter # 07: The Arbitrage Pricing Chapter # 07: The Arbitrage Pricing TheoryTheory

Chapter Outline:Chapter Outline:

Concept and understanding of Arbitrage Pricing Theory Concept and understanding of Arbitrage Pricing Theory

(APT) ;(APT) ;

Application & Benefit of APTApplication & Benefit of APT

Assumptions of APTAssumptions of APT

Significance of APTSignificance of APT

Comparison with CAPM and Arbitrage Pricing TheoryComparison with CAPM and Arbitrage Pricing Theory

Single-Factor & APT ModelSingle-Factor & APT Model

APT with an infinite number of securities;APT with an infinite number of securities;

APT with a finite number of securities;APT with a finite number of securities;

Empirical tests of APTEmpirical tests of APT;;

7.1 Concept and understanding of APT7.1 Concept and understanding of APT

Arbitrage pricing theory (APT) is a valuation Arbitrage pricing theory (APT) is a valuation model. Compared to model. Compared to CAPM, it uses fewer , it uses fewer assumptions but is harder to use. The basis assumptions but is harder to use. The basis of arbitrage pricing theory is the idea that of arbitrage pricing theory is the idea that the price of a security is driven by a the price of a security is driven by a number of factors. These can be divided number of factors. These can be divided into two groups: into two groups: macro factors and macro factors and company specific factors.company specific factors. Stephen Ross Stephen Ross

developed the theory in 1976.developed the theory in 1976.

Contd.Contd.

The APTThe APT (Ross, 1976) (Ross, 1976) holds that return of a holds that return of a

stock depends on several economic and stock depends on several economic and

industry factors rather than assuming only the industry factors rather than assuming only the

market factor like CAPMmarket factor like CAPM. . In the APT, sensitivity In the APT, sensitivity

to any of the macro economic factors are to any of the macro economic factors are

represented by their respective beta represented by their respective beta

coefficientcoefficient. . The APT assumes that there exist The APT assumes that there exist

a linear relationship between return of the a linear relationship between return of the

risky assets and macro economic variables.risky assets and macro economic variables.

Chen, Ross and Roll (1986) identified five Chen, Ross and Roll (1986) identified five

factors model to analyze stock returnfactors model to analyze stock return..

Contd.Contd.

Ross stated the return on a stock must Ross stated the return on a stock must follow a very simple relationship that is follow a very simple relationship that is described by the following formula:described by the following formula:

Expected ReturnExpected Return = rf + b1 x (factor 1) + = rf + b1 x (factor 1) + b2 x (factor 2)... + bn x (factor n)b2 x (factor 2)... + bn x (factor n)

Where:Where:

rf =rf = the risk free interest rate, which is the the risk free interest rate, which is the

interest rate the investor would expect to interest rate the investor would expect to

receive from a risk-free investment.  receive from a risk-free investment. 

Typically, U.S. Treasury Bills are used for Typically, U.S. Treasury Bills are used for

U.S. dollar calculations, while German U.S. dollar calculations, while German

Government bills are used for the EuroGovernment bills are used for the Euro b =b = the sensitivity of the stock or security the sensitivity of the stock or security

to each factorto each factor factor factor = the risk premium associated with = the risk premium associated with

each entityeach entity

Contd.Contd.

APT, describes a mechanism used by APT, describes a mechanism used by

investors to identify an asset,investors to identify an asset, such as, a such as, a

share of common stock, which is share of common stock, which is

incorrectly priced. Investors can incorrectly priced. Investors can

subsequently bring the price of the subsequently bring the price of the

security back into alignment / security back into alignment /

configuration with its actual value. configuration with its actual value.

This theory predicts a relationship between This theory predicts a relationship between

the returns of a portfolio and the returns of the returns of a portfolio and the returns of

a single asset through a linear combination a single asset through a linear combination

of many independent macro-economic of many independent macro-economic

variables.variables.

APT uses the risky asset's expected return APT uses the risky asset's expected return

and the risk premium of a number of macro-and the risk premium of a number of macro-

economic factorseconomic factors..

7.2 Application & Benefit of APT7.2 Application & Benefit of APT Arbitrageurs use the APTArbitrageurs use the APT model for making profit by taking model for making profit by taking

advantage of mispriced securities. A mispriced security will have advantage of mispriced securities. A mispriced security will have

a price that differs from the theoretical price predicted by the a price that differs from the theoretical price predicted by the

model. model. APT, describes a mechanism used by investors to identify APT, describes a mechanism used by investors to identify

an asset,an asset, such as, a share of common stock, which is incorrectly such as, a share of common stock, which is incorrectly

priced. Investors can subsequently bring the price of the priced. Investors can subsequently bring the price of the

security back into alignment / configuration with its actual value. security back into alignment / configuration with its actual value.

By going short an over priced security,By going short an over priced security, while concurrently while concurrently

going long the portfolio, the APT calculations were based on, going long the portfolio, the APT calculations were based on, the the

arbitrageur is in a position to make a theoretically risk-free arbitrageur is in a position to make a theoretically risk-free

profit.profit.

ContdContd

APT is based on the law of one price which APT is based on the law of one price which sates that two otherwise identical assets sates that two otherwise identical assets cannot sell at different prices. cannot sell at different prices. APT APT assumes that assets return are linearly assumes that assets return are linearly related to a set of indexes where each related to a set of indexes where each index represents a factor that influence s index represents a factor that influence s the return on an asset.the return on an asset.

7.3 Essence / Spirit of the 7.3 Essence / Spirit of the Arbitrage Pricing TheoryArbitrage Pricing Theory

Given the impossibility of empirically verifying Given the impossibility of empirically verifying the CAPM, an alternative model of asset the CAPM, an alternative model of asset pricing called the Arbitrage Pricing Theory pricing called the Arbitrage Pricing Theory (APT) has been introduced by Ross , 1976.(APT) has been introduced by Ross , 1976.

A security’s expected return and risk A security’s expected return and risk are directly related to its sensitivities to are directly related to its sensitivities to changes in one or more factors (e.g., changes in one or more factors (e.g., inflation, interest rates, productivity, inflation, interest rates, productivity, etc.)etc.)

7.3 Assumptions of APT7.3 Assumptions of APT

Unlike CAPM, APT assumesUnlike CAPM, APT assumes , , Investors have homogenous beliefs;Investors have homogenous beliefs;

Investors are risk averse utility maximizesInvestors are risk averse utility maximizes Markets are perfectMarkets are perfect

7.4 Calculating Asset’s Return with APT7.4 Calculating Asset’s Return with APT

As per APT, formula for calculating asset’s expected rate of As per APT, formula for calculating asset’s expected rate of

return is:return is:

E(rj) = rf + bj1RP1 + bj2RP2 + bj3RP3 + bj4RP4 + ... + bjnRPnE(rj) = rf + bj1RP1 + bj2RP2 + bj3RP3 + bj4RP4 + ... + bjnRPn

where:where:

E(rj) = the asset's expected rate of returnE(rj) = the asset's expected rate of return

rf = the risk-free raterf = the risk-free rate

bj = the sensitivity of the asset's return to the particular factorbj = the sensitivity of the asset's return to the particular factor

RP = the risk premium associated with the particular factorRP = the risk premium associated with the particular factor

The general idea behind APTThe general idea behind APT is that two things can is that two things can

explain the expected return on a financial asset: explain the expected return on a financial asset:

11) macroeconomic/security-specific influences and ) macroeconomic/security-specific influences and

2) the asset's sensitivity to those influences. This 2) the asset's sensitivity to those influences. This

relationship takes the form of the relationship takes the form of the

linear linear regression formula above formula above..

There are an infinite number of security-specific influences for There are an infinite number of security-specific influences for

any given security including any given security including inflation, production measures, , production measures,

investor confidence, exchange rates, market indices or investor confidence, exchange rates, market indices or

changes in interest rates. It is up to the changes in interest rates. It is up to the analyst to decide which  to decide which

influences are relevant to the asset being analyzed.influences are relevant to the asset being analyzed.

7.4 Significance of APT7.4 Significance of APT

The APT was a revolutionary model because it The APT was a revolutionary model because it

allows the user to adapt the model to the allows the user to adapt the model to the

security being analyzed. With other pricing security being analyzed. With other pricing

models, it helps the user decide whether a models, it helps the user decide whether a

security is undervalued or overvalued and so security is undervalued or overvalued and so

he or she can profit from this information. APT he or she can profit from this information. APT

is also very useful for building portfolios is also very useful for building portfolios

because it allows managers to test whether because it allows managers to test whether

their portfolios are exposed to certain factors.their portfolios are exposed to certain factors.

However, APT may be more customizable than However, APT may be more customizable than

CAPM,CAPM, but it is also more difficult to apply but it is also more difficult to apply

because determining which factors influence because determining which factors influence

a stock or portfolio takes a considerable amount a stock or portfolio takes a considerable amount

of research. of research.

It can be virtually impossible to detect every influential It can be virtually impossible to detect every influential

factor much less determine how sensitive the security factor much less determine how sensitive the security

is to a particular factoris to a particular factor. But getting "close enough" is . But getting "close enough" is

often good enough; in fact studies find that four or five often good enough; in fact studies find that four or five

factors will usually explain most of a security's return: factors will usually explain most of a security's return:

surprises in inflation, GNP, investor confidence . surprises in inflation, GNP, investor confidence .

7.5 Difference between CAPM and Arbitrage 7.5 Difference between CAPM and Arbitrage

Pricing TheoryPricing Theory The Arbitrage Pricing Theory and the The Arbitrage Pricing Theory and the Capital Asset Capital Asset

Pricing ModelPricing Model (CAPM) are the two most influential  (CAPM) are the two most influential

theories on stock and asset pricing today. theories on stock and asset pricing today.

The difference between CAPM and arbitrage The difference between CAPM and arbitrage

pricing theory is that CAPM has a single non-pricing theory is that CAPM has a single non-

company factor and a single beta, company factor and a single beta, whereas whereas

arbitrage pricing theory separates out non-arbitrage pricing theory separates out non-

company factors into as many as proves company factors into as many as proves

necessarynecessary..

Arbitrage pricing theory does not rely Arbitrage pricing theory does not rely on measuring the performance of the on measuring the performance of the marketmarket. Instead, APT directly relates . Instead, APT directly relates the price of the security to the the price of the security to the fundamental factors driving it. fundamental factors driving it.

Each of these requires a separate Each of these requires a separate beta. The beta of each factor is the beta. The beta of each factor is the sensitivity of the price of the security sensitivity of the price of the security to that factor.to that factor.

Contd.Contd.

Intuitively, the APT makes a lot of sense Intuitively, the APT makes a lot of sense

because it removes the CAPM because it removes the CAPM

restrictions, and basically states "The restrictions, and basically states "The

expected return on an asset is a expected return on an asset is a

function of many factors as well as the function of many factors as well as the

sensitivity of the stock to these sensitivity of the stock to these

factorsfactors."  As these factors move, so ."  As these factors move, so

does the expected return on the stock, does the expected return on the stock,

and therefore its value to the investorand therefore its value to the investor. .

Contd.Contd.

In the CAPM theory, the expected return In the CAPM theory, the expected return

on a stock can be described by the on a stock can be described by the

movement of that stock relative to the rest movement of that stock relative to the rest

of the marketof the market.  The CAPM is really just a .  The CAPM is really just a

simplified version of the APT, whereby the simplified version of the APT, whereby the

only factor considered is the risk of a only factor considered is the risk of a

particular stock relative to the rest of the particular stock relative to the rest of the

market, as described by the market, as described by the stock's betastock's beta. .

Contd.Contd.

However, from a practical standpoint, However, from a practical standpoint,

CAPM remains the dominant pricing model CAPM remains the dominant pricing model

used todayused today.  When compared to the .  When compared to the

Arbitrage Pricing Theory, the Capital Asset Arbitrage Pricing Theory, the Capital Asset

Pricing Model is both elegant and Pricing Model is both elegant and

relatively simple to calculate. relatively simple to calculate.

Arbitrage Argument in APTArbitrage Argument in APT

1. 1. Arbitrage argument in APTArbitrage argument in APT When arbitrage opportunities exist, each investor wants When arbitrage opportunities exist, each investor wants

to take as large a position as possible.to take as large a position as possible. It will not take many investors to restore equilibrium.It will not take many investors to restore equilibrium. Implications derived from no-arbitrage argument is Implications derived from no-arbitrage argument is

stronger, because they do not depend on a large, well-stronger, because they do not depend on a large, well-educated investors. educated investors.

2. Arbitrage argument in APT2. Arbitrage argument in APT When arbitrage opportunities exist, each investor When arbitrage opportunities exist, each investor

wants to take as large a position as possible.wants to take as large a position as possible. It will not take many investors to restore It will not take many investors to restore

equilibrium.equilibrium. Implications derived from no-arbitrage argument is Implications derived from no-arbitrage argument is

stronger, because they do not depend on a large, stronger, because they do not depend on a large, well-educated investors. well-educated investors.

contd.contd. The problem with this is that the theory in itself The problem with this is that the theory in itself

provides no indication of what these factors are, so provides no indication of what these factors are, so

they need to be empirically determined. Obvious they need to be empirically determined. Obvious

factors include economic growth and interest factors include economic growth and interest

rates. For companies in some sectors other factors rates. For companies in some sectors other factors

are obviously relevant as well - such as consumer are obviously relevant as well - such as consumer

spending for retailers. spending for retailers.

The potentially large number of factors means more The potentially large number of factors means more

betas to be calculatedbetas to be calculated. There is also no guarantee . There is also no guarantee

that all the relevant factors have been identified. that all the relevant factors have been identified.

This added complexity is the reason arbitrage This added complexity is the reason arbitrage

pricing theory is far less widely used than CAPM. pricing theory is far less widely used than CAPM.

APT With an Unlimited Number of APT With an Unlimited Number of SecuritiesSecurities

Given an infinite number of securities, Given an infinite number of securities, if if

security returns are generated by a process security returns are generated by a process

equivalent to that of a linear single-factor or equivalent to that of a linear single-factor or

multi-factor model, it is impossible to multi-factor model, it is impossible to

construct two different portfolios, both construct two different portfolios, both

having zero variancehaving zero variance (i.e., zero betas (i.e., zero betas andand

zero residual variance) with two different zero residual variance) with two different

expected rates of return. expected rates of return. In other words, In other words,

pure riskless arbitrage opportunities are not pure riskless arbitrage opportunities are not

available.available.

APT With Limited Number of SecuritiesAPT With Limited Number of Securities

Several papers have placed further Several papers have placed further

restrictions on the model to ensure that the restrictions on the model to ensure that the

APT pricing equation holds strictly in a finite APT pricing equation holds strictly in a finite

economyeconomy. Connor (1984) shows that if an . Connor (1984) shows that if an

economy is insurable, i.e., through economy is insurable, i.e., through

exchange it is possible to form portfolios exchange it is possible to form portfolios

that are free of that are free of idiosyncraticidiosyncratic risk, then the risk, then the

APT pricing equation will be exactly linear in APT pricing equation will be exactly linear in

both the finite and the infinite economiesboth the finite and the infinite economies..

Wei (1988) provides Wei (1988) provides a synthesis of the CAPM and a synthesis of the CAPM and

the APT in a competitive equilibrium frameworkthe APT in a competitive equilibrium framework. He . He

shows that in a finite economy, a linear-beta shows that in a finite economy, a linear-beta

pricing equation will price all securities correctly pricing equation will price all securities correctly if if

the market portfolio is included as an additional the market portfolio is included as an additional

factorfactor..

To obtain an exact pricing equation in finite To obtain an exact pricing equation in finite

economies, no-arbitrage should be defined with economies, no-arbitrage should be defined with

respect to utility rather than the payoff. The precise respect to utility rather than the payoff. The precise

definition of no- arbitrage in a finite economy is as definition of no- arbitrage in a finite economy is as

follows: follows: An economy does not permit arbitrage if a An economy does not permit arbitrage if a

zero investment, zero factor risk portfolio does not zero investment, zero factor risk portfolio does not

change the expected utility of the investor.change the expected utility of the investor.

Pure Riskless Arbitrage Pure Riskless Arbitrage OpportunitiesOpportunities(An Example)(An Example)

If two zero variance portfolios could If two zero variance portfolios could

be constructed with two different be constructed with two different

expected rates of return, expected rates of return, we could we could

sell short the one with the lower sell short the one with the lower

return, and invest the proceeds in the return, and invest the proceeds in the

one with the higher return, and make one with the higher return, and make

a pure riskless profit with no capital a pure riskless profit with no capital

commitment.commitment.

Pure Riskless Arbitrage Pure Riskless Arbitrage OpportunitiesOpportunities

(An Example) - Continued(An Example) - Continued

0

0.25

-0.5 0 0.5 1 1.5

Expected Return (%)

Factor Beta

A

B

CD

E(rZ)1

E(rZ)2

““Approximately Linear” APT Approximately Linear” APT EquationsEquations

The APT equations are expressed as being The APT equations are expressed as being

“approximately linear.” That is, the absence of “approximately linear.” That is, the absence of

arbitrage opportunities does not ensure arbitrage opportunities does not ensure exactexact linear linear

pricing. There may be a few securities with expected pricing. There may be a few securities with expected

returns greater than, or less than, those specified by returns greater than, or less than, those specified by

the APT equation. However, because their number is the APT equation. However, because their number is

fewer than that required to drive residual variance of fewer than that required to drive residual variance of

the portfolio to zero, we no longer have a riskless the portfolio to zero, we no longer have a riskless

arbitrage opportunity, and no market pressure arbitrage opportunity, and no market pressure

forcing their expected returns to conform to the APT forcing their expected returns to conform to the APT

equation.equation.

Empirical Tests of the APTEmpirical Tests of the APT

Currently, there is no conclusive evidence either Currently, there is no conclusive evidence either

supporting or contradicting APT. supporting or contradicting APT. Furthermore, the Furthermore, the

number of factors to be included in APT models has number of factors to be included in APT models has

varied considerably among studiesvaried considerably among studies. In one example, . In one example,

a study reported that most of the covariances a study reported that most of the covariances

between securities could be explained on the basis between securities could be explained on the basis

of unanticipated changes in four factors:of unanticipated changes in four factors: Difference between the yield on a long-term and Difference between the yield on a long-term and

a short-term treasury bond.a short-term treasury bond. Rate of inflationRate of inflation Difference between the yields on BB rated Difference between the yields on BB rated

corporate bonds and treasury bonds.corporate bonds and treasury bonds. Growth rate in industrial production.Growth rate in industrial production.

Is APT Testable?Is APT Testable?

Some question whether APT can ever be Some question whether APT can ever be

tested. The theory does not specify the tested. The theory does not specify the

“relevant” factor structure. “relevant” factor structure. If a study If a study

shows pricing to be consistent with some shows pricing to be consistent with some

set of “N” factors, this does not prove that set of “N” factors, this does not prove that

an “N” factor model would be relevant for an “N” factor model would be relevant for

other security samples as wellother security samples as well. If returns . If returns

are not explained by some “N” factor are not explained by some “N” factor

model, we cannot reject APT. Perhaps the model, we cannot reject APT. Perhaps the

choice of factors was wrong.choice of factors was wrong.

Using APT to Predict ReturnUsing APT to Predict Return

Haugen presents a test of the predictive Haugen presents a test of the predictive power of APT using the following factorspower of APT using the following factors:: Monthly return to U.S. T-BillsMonthly return to U.S. T-Bills Difference between the monthly returns on Difference between the monthly returns on

long-term and short-term U.S. Treasury bonds.long-term and short-term U.S. Treasury bonds. Difference between the monthly returns on Difference between the monthly returns on

long-term U.S. Treasury bonds and low-grade long-term U.S. Treasury bonds and low-grade corporate bonds with the same maturity.corporate bonds with the same maturity.

Monthly change in consumer price index.Monthly change in consumer price index. Monthly change in U.S. industrial production.Monthly change in U.S. industrial production. Dividend to price ratio of the S&P 500.Dividend to price ratio of the S&P 500.

Random Walk and Competing Random Walk and Competing TheoriesTheories

Generally, there are two competing Generally, there are two competing

approaches to predicting the movements of approaches to predicting the movements of

stocksstocks:  fundamental and technical :  fundamental and technical

analysis.  In the sections below, we'll be analysis.  In the sections below, we'll be

taking a closer look at these competing taking a closer look at these competing

theories, and how the random walk theories, and how the random walk

hypothesis aligns with eachhypothesis aligns with each. .

Fundamental TheoristsFundamental Theorists

Fundamental analysts believe the price of a stock is Fundamental analysts believe the price of a stock is

a function of its a function of its intrinsic valueintrinsic value, which depends , which depends

heavily on the future earnings potential for a heavily on the future earnings potential for a

company. By carefully studying fundamental company. By carefully studying fundamental

factors such as industry trends, economic news, factors such as industry trends, economic news,

and the company's and the company's earnings per shareearnings per share outlook,  outlook,

fundamental analyst can determine if the stock's fundamental analyst can determine if the stock's

price is above or below its intrinsic value.price is above or below its intrinsic value.

Comparing a stock's price to its intrinsic value Comparing a stock's price to its intrinsic value

allows the fundamental analyst to predict the allows the fundamental analyst to predict the

potential future direction of the stock's price. potential future direction of the stock's price.

Technical TheoristsTechnical Theorists

Market analysts that practice technical Market analysts that practice technical techniques believe that techniques believe that historical historical movements of a stock's price can be used movements of a stock's price can be used to predict future price direction. to predict future price direction.  Using Using methods such as charting, the technical methods such as charting, the technical analyst will examine the sequence of analyst will examine the sequence of upward and downward movements for a upward and downward movements for a stock.  stock.  These patterns of movements allow These patterns of movements allow the technical theorist to chart what they the technical theorist to chart what they believe will be future movements for the believe will be future movements for the stocks they are examining.stocks they are examining.

Efficient Market TheoristsEfficient Market Theorists The next theory we're going to talk about is the The next theory we're going to talk about is the

efficient market hypothesis (EMH).  Subscribers to efficient market hypothesis (EMH).  Subscribers to this theory believe the price of a stock reflects all this theory believe the price of a stock reflects all publically known information about a company.  publically known information about a company.  In fact, individuals subscribing to what is termed In fact, individuals subscribing to what is termed the "strong" EMH believe that stock prices also the "strong" EMH believe that stock prices also reflect what insiders know too.reflect what insiders know too.

Since public and private information concerning a Since public and private information concerning a company is instantly reflected in the market price company is instantly reflected in the market price of a stock, it's impossible for an investor to of a stock, it's impossible for an investor to achieve "excess" returns.  The principles of the achieve "excess" returns.  The principles of the random walk hypothesis are consistent with those random walk hypothesis are consistent with those of the efficient market hypothesis.of the efficient market hypothesis.

Random Walk TheoryRandom Walk Theory

Finally, Finally, the random walk hypothesis statesthe random walk hypothesis states that prices of stocks cannot be predicted.  that prices of stocks cannot be predicted.  The stock market is "informationally The stock market is "informationally efficient." The people buying and selling efficient." The people buying and selling stocks consist of a large number of rational stocks consist of a large number of rational investors with access to this information.  investors with access to this information.  While long term prices will reflect While long term prices will reflect performance of the company over time, performance of the company over time, short term movements in prices can best be short term movements in prices can best be

described as a random walkdescribed as a random walk..

Practical Implications RWHPractical Implications RWH The random walk hypothesis has some practical implications to The random walk hypothesis has some practical implications to

investors.  For example, since the short term movement of a investors.  For example, since the short term movement of a stock is random, there is no sense in worrying about timing the stock is random, there is no sense in worrying about timing the market.  A buy and hold strategy will be just as effective as any market.  A buy and hold strategy will be just as effective as any attempt to time the purchase and sale of securities.attempt to time the purchase and sale of securities.

When investors buy stocks, they usually do so because they When investors buy stocks, they usually do so because they believe the stock is worth more than they are paying.  In the believe the stock is worth more than they are paying.  In the same way, investors sell stocks when they believe the stock is same way, investors sell stocks when they believe the stock is worth less than the selling price.  If the efficient market theory worth less than the selling price.  If the efficient market theory and random walk hypothesis are true, then an investor's ability and random walk hypothesis are true, then an investor's ability to outperform the stock market is more luck than analytical to outperform the stock market is more luck than analytical skill.skill.

Arbitrage Pricing TheoryArbitrage Pricing Theory APT is introduced by Ross (1976).APT is introduced by Ross (1976). Like the CAPM, APT predicts the relationship Like the CAPM, APT predicts the relationship

between the risk and equilibrium expected returns between the risk and equilibrium expected returns on risky assets.on risky assets.

However, the APT relies on no-arbitrage condition However, the APT relies on no-arbitrage condition rather than the market portfolio.rather than the market portfolio.

To explain the APT, we begin with the concept of To explain the APT, we begin with the concept of Arbitrage, which is the exploitation of relative Arbitrage, which is the exploitation of relative mispricing among two or more securities to earn mispricing among two or more securities to earn risk-free profitsrisk-free profits

A riskless arbitrage opportunity arises if an investor A riskless arbitrage opportunity arises if an investor can construct a zero investment portfolio with a can construct a zero investment portfolio with a sure profitsure profit..

Arbitrage Pricing TheoryArbitrage Pricing Theory Since no investment is required, an investor can Since no investment is required, an investor can

create large positions to secure large levels of profit.create large positions to secure large levels of profit. One has to be able to sell short at least one asset and One has to be able to sell short at least one asset and

use the proceeds to purchase one or more assets.use the proceeds to purchase one or more assets. An obvious case of an arbitrage opportunity arises An obvious case of an arbitrage opportunity arises

in the violation of the law of one price: When an in the violation of the law of one price: When an asset is trading at different prices in two markets, asset is trading at different prices in two markets, sell short in the high priced market and buys it in the sell short in the high priced market and buys it in the low priced market.low priced market.

In efficient markets, profitable arbitrage opportunity In efficient markets, profitable arbitrage opportunity will quickly disappear – Program trading and index will quickly disappear – Program trading and index arbitrage.arbitrage.

Arbitrage Example from Text Arbitrage Example from Text Current ExpectedCurrent Expected StandardStandard

Stock Stock Price$Price$ Return%Return% Dev. % Dev. %

A 10 25.0A 10 25.0 29.58 29.58

B 10 20.0 33.91B 10 20.0 33.91

C 10 32.5C 10 32.5 48.15 48.15

D 10 22.5D 10 22.5 8.58 8.58

MeanMean Stan.Dev.Stan.Dev. CorrelationCorrelation

Portfolio P(A,B,C)Portfolio P(A,B,C) 25.83 6.40 25.83 6.40 0.94 0.94

DD 22.25 8.58 22.25 8.58

Arbitrage Action and ReturnsArbitrage Action and Returns

Expected Expected ReturnReturn

Standard Standard DeviationDeviation

* P* P

* D* D

Short 3 shares of D and Buy 1 of A, B & C to form P Short 3 shares of D and Buy 1 of A, B & C to form P (Arbitrage Portfolio: Zero-investment Portfolio).(Arbitrage Portfolio: Zero-investment Portfolio).

You earn a higher rate on the investment than you You earn a higher rate on the investment than you pay on the short sale.pay on the short sale.

Arbitrage Pricing TheoryArbitrage Pricing Theory The critical property of an arbitrage portfolio is than The critical property of an arbitrage portfolio is than

any investor, regardless of risk aversion or wealth, any investor, regardless of risk aversion or wealth, will want to take an infinite position in it so that will want to take an infinite position in it so that profits will be driven to an infinite level.profits will be driven to an infinite level.

Because those large positions will force some prices Because those large positions will force some prices up and/or some down until the opportunity is up and/or some down until the opportunity is vanished, we can derive restrictions on security vanished, we can derive restrictions on security prices that satisfy that no arbitrage opportunities are prices that satisfy that no arbitrage opportunities are left in the market place.left in the market place.

There is an important distinction between arbitrage There is an important distinction between arbitrage and CAPM risk-versus-return dominance arguments and CAPM risk-versus-return dominance arguments in support of equilibrium price relationships.in support of equilibrium price relationships.

Arbitrage Pricing TheoryArbitrage Pricing Theory Risk-vs-Return Dominance argument in CAPMRisk-vs-Return Dominance argument in CAPM

It holds that when an equilibrium relationship is It holds that when an equilibrium relationship is violated, many investors will make limited portfolio violated, many investors will make limited portfolio changes, depending on wealth and risk-aversion.changes, depending on wealth and risk-aversion.

Aggregation of limited portfolio changes over many Aggregation of limited portfolio changes over many investors will restore the equilibrium price.investors will restore the equilibrium price.

Arbitrage argument in APTArbitrage argument in APT When arbitrage opportunities exist, each investor wants When arbitrage opportunities exist, each investor wants

to take as large a position as possible.to take as large a position as possible. It will not take many investors to restore equilibrium.It will not take many investors to restore equilibrium. Implications derived from no-arbitrage argument is Implications derived from no-arbitrage argument is

stronger, because they do not depend on a large, well-stronger, because they do not depend on a large, well-educated investors. educated investors.

Well-diversified Portfolio and APTWell-diversified Portfolio and APT APT: A theory of risk-return relationship derived APT: A theory of risk-return relationship derived

from no arbitrage conditions in large capital marketfrom no arbitrage conditions in large capital market.. APT posits a single-factor security market.APT posits a single-factor security market.

RRii = = ii + + iiRRMM + e, where R + e, where Rii = (r = (rii – r – rff)) Suppose we construct a well-diversified portfolio Suppose we construct a well-diversified portfolio

with a given beta – No firm-specific risk.with a given beta – No firm-specific risk.

RRpp = = pp + + ppRRMM

If the portfolio beta is zero, RIf the portfolio beta is zero, Rpp = = pp, implying a riskless , implying a riskless

excess return over risk-free rate.excess return over risk-free rate. This implies that This implies that pp should be zero, or else an should be zero, or else an

immediate arbitrage opportunity opens up (borrow at immediate arbitrage opportunity opens up (borrow at risk free rate and buy zero-beta portfolio).risk free rate and buy zero-beta portfolio).

Well-diversified Portfolio and APTWell-diversified Portfolio and APT Portfolio V (Portfolio V (vv and and vv) and Portfolio U () and Portfolio U (uu and and uu).).

To form zero-beta portfolio (V+U); buy Portfolio V and To form zero-beta portfolio (V+U); buy Portfolio V and sell Portfolio U with proportions of wsell Portfolio U with proportions of wvv = [- = [-uu/(/(vv--uu)], )], and wand wuu = [ = [vv/(/(vv--uu)] )]

Riskless portfolio, but non-zero excess return unless Riskless portfolio, but non-zero excess return unless vv and and uu equal zero. equal zero.

Beta(V+U) = Beta(V+U) = vv[-[-uu/(/(vv--uu)] + )] + uu[[vv/(/(vv--uu)] = 0)] = 0

R(V+U) = R(V+U) = vv[-[-uu/(/(vv--uu)] + )] + uu[[vv/(/(vv--uu)] )] 0 0 The alpha of any well-diversified portfolio must be The alpha of any well-diversified portfolio must be

zero, even if the beta is not zero.zero, even if the beta is not zero. (r(rpp – r – rff) = ) = pp(r(rMM– r– rff); E(r); E(rpp) = r) = rff + + pp[E(r[E(rMM) – r) – rff]: Same ]: Same

as CAPM without any assumption about either as CAPM without any assumption about either investor preferences or access to market portfolio.investor preferences or access to market portfolio.

APT and CAPM ComparedAPT and CAPM Compared APT applies only to well diversified portfolios and APT applies only to well diversified portfolios and

not necessarily to individual stocks in equilibriumnot necessarily to individual stocks in equilibrium.. However, APT relationship must However, APT relationship must almostalmost surely hold surely hold

true for individual securitiestrue for individual securities.. If APT relationship is violated by many individual If APT relationship is violated by many individual

assets, it would be virtually impossible for all well-assets, it would be virtually impossible for all well-diversified portfolios to satisfy the relationship.diversified portfolios to satisfy the relationship.

APT serves many of same functions as the CAPMAPT serves many of same functions as the CAPM.. APT is more general in that it gets to an expected APT is more general in that it gets to an expected

return and beta relationship without the assumption return and beta relationship without the assumption of the market portfolio.of the market portfolio.

APT can be extended to multifactor models.APT can be extended to multifactor models.

Multi-Factor APT ModelsMulti-Factor APT Models

One FactorOne Factor

Two FactorsTwo Factors

)(εσ)(Iσβ)(rσ

β)]E(r[E(I)E(r)E(r

εIβAr

p2

122

p1,p2

j1,z1zj

tj,t1,j1,jtj,

)(εσ)(Iσβ)(Iσβ)(rσ

β)]E(r)[E(Iβ)]E(r)[E(I)E(r)E(r

εIβIβAr

p2

222

p2,122

p1,p2

j2,z2j1,z1zj

tj,t2,j2,t1,j1,jtj,

Multi-Factor APT ModelsMulti-Factor APT Models(Continued)(Continued)

N FactorsN Factors

)(εσ + )(Iσβ + . . . )(Iσβ)(Iσβ)(rσ

β)]E(r)[E(I+

. . . . +

β)]E(r)[E(I

β)]E(r)[E(I)E(r)E(r

ε + Iβ +. . . IβIβAr

p2

n22

pn,222

p2,122

p1,p2

jn,zn

j2,z2

j1,z1zj

tj,tn,jn,t2,j2,t1,j1,jtj,

Multifactor Generalization of APTMultifactor Generalization of APT

Use a multifactor version of APT to Use a multifactor version of APT to accommodate multiple sources of riskaccommodate multiple sources of risk..

Generalize the single-factor model to a two-Generalize the single-factor model to a two-factor model: Rfactor model: Rii = = ii + + i1i1RRM1M1 + + i2i2RRM2M2 + e. + e.

Two-Factor APTTwo-Factor APT

E(rE(rpp) = ) =

rrf f + + p1p1 [E(r [E(rM1M1) – r) – rff] + ] + p2p2 [E(r [E(rM2M2) – r) – rff]]

SummarySummary The general idea behind APT is that The general idea behind APT is that

two things can explain the expected two things can explain the expected return on a financial assetreturn on a financial asset:: 1) 1) macroeconomic/security-specific macroeconomic/security-specific influences and 2) the asset's influences and 2) the asset's sensitivity to those influences. This sensitivity to those influences. This relationship takes the form of the relationship takes the form of the linear regression formula stated linear regression formula stated above. above.