7/29/2019 Ch 10 Mgmt 1362002

1/26

The Arbitrage Pricing Theory(Chapter 10)

Single-Factor APT Model

Multi-Factor APT Models

Arbitrage Opportunities

Disequilibrium in APT

Is APT Testable?

Consistency of APT and CAPM

7/29/2019 Ch 10 Mgmt 1362002

2/26

Essence of the Arbitrage Pricing Theory

Given the impossibility of empirically verifying

the CAPM, an alternative model of asset

pricing called the Arbitrage Pricing Theory(APT) has been introduced.

Essence of APT

A securitys expected return and risk are

directly related to its sensitivities to changesin one or more factors (e.g., inflation,

interest rates, productivity, etc.)

7/29/2019 Ch 10 Mgmt 1362002

3/26

Essence of the Arbitrage Pricing Theory(Continued)

In other words, security returns are generated by a

single-index (one factor) model:

where:

or, by a multi-index (multi-factor) model:

tj,t1,j1,jtj,IAr

(1)Factortorespectwith(j)securityofbeta

(t)periodin(1)FactorofValueI

j1,

t1,

tj,tn,jn,t2,j2,t1,j1,jtj, I+...IIAr

7/29/2019 Ch 10 Mgmt 1362002

4/26

Single-Factor APT Model(A Comparison With the CAPM)

CAPM (Zero Beta Version)Factor = Market Portfolio

Actual Returns:

Expected Returns:

APT (One Factor Version)Factor = Your Choice

Actual Returns:

Expected Returns:

tj,tM,jjtj, rAr

PremiumRiskMarket

jzMzj )]E(r)[E(r)E(r)E(r

tj,t1,j1,jtj, IAr

price.factor

denotes)(lambdatext,In the:Note*

*PriceFactor

j1,z1zj )]E(r)[E(I)E(r)E(r

7/29/2019 Ch 10 Mgmt 1362002

5/26

Single-Factor APT Model(A Comparison With the CAPM)

Continued

CAPM (Zero Beta Version)

Continued

Portfolio Variance:

APT (One Factor Version)

Continued

Portfolio Variance:

0),COV(sumingAs

)(x)(

xwhere

)()(r)(r

kj

m

1j

j22

jp2

m

1j

jjp

p2

M22

pp2

0),COV(sumingAs

)(x)(

xwhere

)()(I)(r

kj

m

1j

j22

jp2

m

1j

j1,jp1,

p2

122

p1,p2

7/29/2019 Ch 10 Mgmt 1362002

6/26

Multi-Factor APT Models

One Factor

Two Factors

)()(I)(r

)]E(r[E(I)E(r)E(r

IAr

p2

122

p1,p2

j1,z1zj

tj,t1,j1,jtj,

)()(I)(I)(r

)]E(r)[E(I)]E(r)[E(I)E(r)E(r

IIAr

p2

222

p2,122

p1,p2

j2,z2j1,z1zj

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

7/29/2019 Ch 10 Mgmt 1362002

7/26

Multi-Factor APT Models(Continued)

N Factors

)(+)(I+...)(I)(I)(r

)]E(r)[E(I+

....+

)]E(r)[E(I

)]E(r)[E(I)E(r)E(r

+I+...IIAr

p2

n22

pn,222

p2,122

p1,p2

jn,zn

j2,z2

j1,z1zj

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

7/29/2019 Ch 10 Mgmt 1362002

8/26

The Ideal APT Model

Ideally, you wish to have a model where all of

the covariances between the rates of return to

the securities are attributable to the effects ofthe factors. The covariances between the

residuals of the individual securities,

Cov(j, k), are assumed to be equal to zero.

7/29/2019 Ch 10 Mgmt 1362002

9/26

APT With an Unlimited Number ofSecurities

Given an infinite number of securities, if

security returns are generated by a process

equivalent to that of a linear single-factor ormulti-factor model, it is impossible to

construct two different portfolios, both having

zero variance (i.e., zero betas and zero

residual variance) with two different expectedrates of return. In other words, pure riskless

arbitrage opportunities are not available.

7/29/2019 Ch 10 Mgmt 1362002

10/26

Pure Riskless Arbitrage Opportunities(An Example)

Note: If two zero variance portfolios could be

constructed with two different expected rates

of return, we could sell short the one with thelower return, and invest the proceeds in the

one with the higher return, and make a pure

riskless profit with no capital commitment.

7/29/2019 Ch 10 Mgmt 1362002

11/26

Pure Riskless Arbitrage Opportunities(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

7/29/2019 Ch 10 Mgmt 1362002

12/26

Approximately Linear APT Equations

The APT equations are expressed as being

approximately linear. That is, the absence of

arbitrage opportunities does not ensure exact

linear pricing. There may be a few securities withexpected returns greater than, or less than, those

specified by the APT equation. However, because

their number is fewer than that required to drive

residual variance of the portfolio to zero, we no

longer have a riskless arbitrage opportunity, and

no market pressure forcing their expected returns

to conform to the APT equation.

7/29/2019 Ch 10 Mgmt 1362002

13/26

Disequilibrium Situation in APT:A One Factor Model Example

Portfolio (P) contains 1/2 of security (B) plus 1/2 of the

zero beta portfolio:

Portfolio (P) dominates security (A). (i.e., it has the same

beta, but more expected return).

1.0.5(0).5(2.0).5.5 Z1,B1,P1,

0

20

0 0.5 1 1.5 2 2.5

Expected Return (%) B

P

A

E(rP)

E(I1)E(rA)

E(rZ)

Beta

Equilibrium Line

7/29/2019 Ch 10 Mgmt 1362002

14/26

Disequilibrium Situation in APT:A One Factor Model Example

(Continued)

Arbitrage: Investors will sell security (A). Price

of security (A) will fall causing E(rA) to rise.

Investors will use proceeds of sale of security (A)to purchase security (B). Price of security (B)

will rise causing E(rB) to fall. Arbitrage

opportunities will no longer exist when all assets

lie on the same straight line.

7/29/2019 Ch 10 Mgmt 1362002

15/26

Anticipated Versus Unanticipated Events

Given a Single-Factor Model:

Substituting the right hand side of Equation #2 for Aj in

Equation #1:

#2}{Equation)E(I)E(rA

or

)E(IA)E(r

:thatstatecanwe,E(Since

#1}{EquationIAr

1j1,jj

1j1,jj

tj,t1,j1,jtj,

0)

ReturnReturnReturn

danticipateUndAnticipateActual

tj,1t1,j1,jtj,

tj,t1,j1,1j1,jtj,

)]E(I[I+)E(rrI)E(I)E(rr

7/29/2019 Ch 10 Mgmt 1362002

16/26

Anticipated Versus Unanticipated Events(Continued)

Note: If the actual factor value (I1,t) is exactly

equal to the expected factor value, E(I1), and the

residual (j,t) equals zero as expected, then allreturn would have been anticipated:

rj,t = E(rj)

If (I1,t) is not equal to E(I1), or (j,t) is not equal to

zero, then some unanticipated return (positive or

negative) will be received.

7/29/2019 Ch 10 Mgmt 1362002

17/26

Anticipated Versus Unanticipated Events(A Numerical Example)

Given:

Expected Return:

Anticipated Versus Unanticipated Return:

.5.01.15I.12)E(I.06)E(r j1,tj,t1,1Z

.09=

.06].5-[.12+.06=

)]E(r)[E(I)E(r)E(r j1,Z1Zj

cipatedUnantidAnticipate

.01+.015.09=

.01+.12]-.5[.15+.09=

)]E(I[I)E(rr tj,1t1,j1,jtj,

7/29/2019 Ch 10 Mgmt 1362002

18/26

Anticipated Versus Unanticipated Returns(A Graphical Display)

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21

rj,t = .115.105

E(rj) = .09

E(rZ) = .06

.03

E(rZ) E(I1)I1,t

7/29/2019 Ch 10 Mgmt 1362002

19/26

Consistency of the APT and the CAPM

Consider APT for a Two Factor Model:

In terms of the CAPM, we can treat each of the factors in

the same manner that individual securities are treated: (Seecharts above)

CAPM Equation:

I1,t

AI1

M,I1

rM,t

I2,t

AI2

0 0

M,I2

rM,t

j2,Z2j1,Z1Zj )]E(r)[E(I)]E(r)[E(I)E(r)E(r

jM,ZMZj )]E(r)[E(r)E(r)E(r

7/29/2019 Ch 10 Mgmt 1362002

20/26

Note that M,I1 and M,I2 are the CAPM (market) betas of

factors 1 and 2. Therefore, in terms of the CAPM, the

expected values of the factors are:

By substituting the right hand sides of Equations 1 and 2

for E(I1) and E(I2) in the APT equation, we get:

2Equation)]E(r)[E(r)E(r)E(I

1Equation)]E(r)[E(r)E(r)E(I

2IM,ZMZ2

1IM,ZMZ1

jM,ZMZj

j2,2IM,j1,1IM,jM,

j2,2IM,j1,1IM,ZMZj

j2,2IM,ZM

j1,1IM,ZMZj

)]E(r)[E(r)E(r)E(r:Since

:thatNote

))](E(r)[E(r)E(r)E(r

))]E(r)([E(r+

))]E(r)([E(r)E(r)E(r

7/29/2019 Ch 10 Mgmt 1362002

21/26

There are numerous securities that could have the same

CAPM beta (M,j), but have different APT betas relative to

the factors (1,j and 2,j).

Consistency of the APT and CAPM (an example)

Given: Factor 1 (Productivity) M,I1 = .5

Factor 2 (Inflation) M,I2 = 1.5

Security______

12

3456

1,j____

0.4

.81.21.62.0

2,j____.667.534

.400

.267

.1340

M,I11,j + M,I22,j = M,j___________________.5(0) + 1.5(.667) = 1.00.5(.4) + 1.5(.534) = 1.00

.5(.8) + 1.5(.400) = 1.00.5(1.2) + 1.5(.267) = 1.00.5(1.6) + 1.5(.134) = 1.00

.5(2.0) + 1.5(0) = 1.00

7/29/2019 Ch 10 Mgmt 1362002

22/26

Assuming the market is efficient, all of the

securities (1 through 6) will have equal returns on

the average over time since they have a CAPMbeta of 1.00. However, some would argue that it is

not necessarily true that a particular investor

would consider all securities with the same

expected return and CAPM beta equally desirable.

For example, different investors may have

different sensitivities to inflation.

Note: It is possible for both the CAPM and themultiple factor APT to be valid theories. The

problem is to prove it.

7/29/2019 Ch 10 Mgmt 1362002

23/26

Empirical Tests of the APT

Currently, there is no conclusive evidence eithersupporting or contradicting APT. Furthermore, the numberof factors to be included in APT models has varied

considerably among studies. In one example, a studyreported that most of the covariances between securitiescould be explained on the basis of unanticipated changes infour factors:

Difference between the yield on a long-term and a

short-term treasury bond. Rate of inflation

Difference between the yields on BB rated corporatebonds and treasury bonds.

Growth rate in industrial production.

7/29/2019 Ch 10 Mgmt 1362002

24/26

Is APT Testable?

Some question whether APT can ever betested. The theory does not specify therelevant factor structure. If a study showspricing to be consistent with some set ofN factors, this does not prove that an Nfactor model would be relevant for other

security samples as well. If returns are notexplained by some N factor model, wecannot reject APT. Perhaps the choice offactors was wrong.

7/29/2019 Ch 10 Mgmt 1362002

25/26

Using APT to Predict Return

Haugen presents a test of the predictive power of APT

using the following factors:

Monthly return to U.S. T-Bills

Difference between the monthly returns on long-term

and short-term U.S. Treasury bonds.

Difference between the monthly returns on long-term

U.S. Treasury bonds and low-grade corporate bonds

with the same maturity.

Monthly change in consumer price index.

Monthly change in U.S. industrial production.

Dividend to price ratio of the S&P 500.

7/29/2019 Ch 10 Mgmt 1362002

26/26

Haugen presents continued . . . Using data for 3000 stocks over the period 1980-1997,

he found that the APT did appear to have only limited

predictive power regarding returns.

He argues that the arbitrage process is extremely

difficult in practice. Since covariances (betas) must be

estimated, there is uncertainty regarding their values in

future periods. Therefore, truly risk-free portfolios

cannot be created using risky stocks. As a result, pureriskless arbitrage is not readily available limiting the

usefulness of APT models in predicting future stock

returns.