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Topic 5 Arbitrage Pricing Theory (APT) and

Multifactor Models of Risk and Return

- Multifactor models
- Arbitrage opportunities and profits
- The APT A single factor model
- Well-diversified portfolios
- Betas and expected returns
- The security market line
- Individual assets and the APT
- The APT and the CAPM
- A multifactor APT

Multifactor Models

- The index model gave us a way of decomposing

stock variability into market or systematic risk,

due largely to macroeconomic events, versus

firm-specific effects that can be diversified in

large portfolios. - In the index model, the return on the market

portfolio summarized the broad impact of macro

factors.

However, sometimes, rather than using a

market proxy, it is more useful to focus directly

on the ultimate sources of risk. This can

be useful in risk assessment when measuring ones

exposures to particular sources of uncertainty.

Factor models are tools that allow us to

describe and quantify the different factors that

affect the rate of return on a security during

any time period.

Factor models of security returns

A single-factor model

- Under a single-factor model, uncertainty in asset

returns has two sources a common or

macroeconomic factor, and firm-specific events. - The common factor is constructed to have

zero expected value, since we use it to measure

new information concerning the macro-economy

which, by definition, has zero expected value.

Let E(ri) expected return on stock i F

deviation of the common factor from its expected

value ßi sensitivity of firm

i to the common factor ei firm-specific

disturbance A single-factor model

The actual return on firm i will equal

its initially expected return plus a (zero

expected value) random amount attributable to

unanticipated economywide events, plus another

(zero expected value) random amount attributable

to firm-specific events.

The nonsystematic components of returns (the

eis) are assumed to be uncorrelated among

themselves and uncorrelated with the factor

F. Example Suppose that the macro factor,

F, is taken to be news about the state of the

business cycle, measured by the unexpected

percentage change in gross domestic product

(GDP), and that the consensus is that GDP will

increase by 4 this year. Suppose also

that a stocks ? value is 1.2.

If GDP increases by only 3, then the value

of F would be -1, representing a 1

disappointment in actual growth versus expected

growth. Given the stocks beta value, this

disappointment would translate into a return on

the stock that is 1.2 lower than previously

expected. This macro surprise, together

with the firm-specific disturbance (ei) determine

the total departure of the stocks return from

its originally expected value.

A two-factor model

- The systematic or macro factor summarized by the

market return arises from a number of sources

(e.g., uncertainty about the business cycle,

interest rates, inflation, etc.) - When we estimate a single-index regression,

we implicitly impose an incorrect assumption that

each stock has the same relative sensitivity to

each risk factor. - If stocks actually differ in their betas

relative to the various macroeconomic factors,

then lumping all systematic sources of risk into

one variable such as the return on the market

index will ignore the nuances that better explain

individual-stock returns.

- Example
- Suppose the 2 most important macroeconomic

sources of risk are uncertainties surrounding the

state of the business cycle, news of which we

will again measure by unanticipated growth in

GDP, and changes in interest rates (IR). - The return on any stock will respond both to

sources of macro risk as well as to its own

firm-specific influences. - ? A two-factor model describing the rate of

return on stock i in some time period

The 2 macro factors on the right-hand side

of the equation comprise the systematic factors

in the economy. Both of these macro

factors have zero expectation they represent

changes in these variables that have not already

been anticipated. The coefficients of each

factor measure the sensitivity of share returns

to that factor and are called factor

sensitivities, factor loadings, or factor betas.

ei reflects firm-specific influences.

- Now, consider 2 stocks, A and B.
- Stock A has a low sensitivity to GDP risk

(i.e. a low GDP beta) and has a relatively high

sensitivity to interest rates (i.e. a high

interest rate beta) . - Conversely, stock B is very sensitive to

economic activity, but it is not very sensitive

to interest rates (i.e. has a high GDP beta and a

small interest rate beta). - When GDP grows, As and Bs stock prices

will rise. When interest rates rise, As and Bs

stock prices will fall.

Suppose that on a particular day, a news

item suggests that the economy will expand.

GDP is expected to increase, but so are

interest rates. Is the macro news on

this day good or bad? For stock A, this

is bad news, since its dominant sensitivity is to

interest rates. But, for stock B, which

responds more to GDP, this is good news.

Clearly, a one-factor or single-index model

cannot capture such differential responses to

varying sources of macroeconomic uncertainty.

Of course, once a two-factor model can better

explain stock returns, it is easy to see that

models with even more factorsmultifactor

modelscan provide even better descriptions of

returns. However, there are many possible

sets of macro factors that might be considered.

- Two principles guide us when we specify a

reasonable list of factors - We want to limit ourselves to macro factors with

considerable ability to explain security returns.

- If our model calls for hundreds of

explanatory variables, it does little to simplify

our description of security returns. - We wish to choose factors that seem likely to be

important risk factors (i.e. factors that concern

investors sufficiently that they will demand

meaningful risk premiums to bear exposure to

those sources of risk).

A multifactor security market line

- The Security Market Line of the CAPM
- Securities will be priced to give investors

an expected return comprised of 2 components the

risk-free rate, which is compensation for the

time value of money, and a risk premium,

determined by multiplying a benchmark risk

premium (i.e., the risk premium offered by the

market portfolio, RPM) times the relative measure

of risk (i.e., beta)

We can think of beta as measuring the

exposure of a stock or portfolio to marketwide or

macroeconomic risk factors. Thus, one

interpretation of the SML is that investors are

rewarded with a higher expected return for their

exposure to macro risk, based on both the

sensitivity to that risk (beta) as well as the

compensation for bearing each unit of that source

of risk (i.e., the risk premium, RPM). but are

not rewarded for exposure to firm-specific

uncertainty (the residual term ei). How

might this single-factor view of the world

generalize once we recognize the presence of

multiple sources of systematic risk?

A multifactor index model gives rise to a

multifactor security market line in which the

risk premium is determined by the exposure to

each systematic risk factor, and by a risk

premium associated with each of those factors.

Example (two-factor economy) The

expected rate of return on a security is the sum

of (1) the risk-free rate of return (2) the

sensitivity to GDP risk (the GDP beta) times the

risk premium for GDP risk and (3) the

sensitivity to interest rate risk (the interest

rate beta) times the risk premium for interest

rate risk.

One difference between a single and multiple

factor economy is that a factor risk premium can

be negative. For example, a security with a

positive interest rate beta performs better when

rates increase, and thus would hedge the value of

a portfolio against interest rate risk.

Investors might well accept a lower rate of

return, that is, a negative risk premium, as the

cost of this hedging attribute.

Arbitrage Opportunities and Profits

- An arbitrage opportunity arises when an investor

can construct a zero investment portfolio that

will yield a sure profit. - To construct a zero investment portfolio,

one has to be able to sell short at least one

asset and use the proceeds to purchase (go long

on) one or more assets. - Clearly, any investor would like to take as

large a position as possible in an arbitrage

portfolio.

- An obvious case of an arbitrage opportunity

arises when the law of one price is violated. - When an asset is trading at different prices

in two markets (and the price differential

exceeds transaction costs), a simultaneous trade

in the two markets can produce a sure profit (the

net price differential) without any investment. - One simply sells short the asset in the

high-priced market and buys it in the low-priced

market. - The net proceeds are positive, and there is

no risk because the long and short positions

offset each other.

- Another example
- Imagine that 4 stocks are traded in an

economy with only 4 distinct, possible

scenarios. - The rates of return of the 4 stocks for each

inflation-interest rate scenario

The current prices of the 4 stocks and rate

of return statistics

Consider an equally weighted portfolio of

the first three stocks (A, B, and C), and

contrast its possible future rates of return with

those of D ? The equally weighted

portfolio will outperform D in all scenarios.

The rate of return statistics of the 2

alternatives are ?Since the equally

weighted portfolio will fare better under any

circumstances, investors will take a short

position in D and use the proceeds to purchase

the equally weighted portfolio. Suppose we

sell short 300,000 shares of D and use the 3

million proceeds to buy 100,000 shares each of A,

B, and C.

- ?The dollar profits in each of the 4 scenarios

will be - The net investment is zero.
- Yet, our portfolio yields a positive profit

in any scenario.

?Investors will want to take an infinite

position in such a portfolio because larger

positions entail no risk of losses, yet yield

evergrowing profits. In principle, even a

single investor would take such large positions

that the market would react to the buying and

selling pressure The price of D has to come down

and/or the prices of A, B, and C have to go up.

The arbitrage opportunity will then be

eliminated. That is, market prices will

move to rule out arbitrage opportunities.

Violation of this restriction would indicate the

grossest form of market irrationality.

The APT A Single Factor Model

- Recall A single factor model

where E(ri) expected return on

stock i F deviation of the common factor

from its expected value ßi

sensitivity of firm i to the common factor

ei firm-specific disturbance

Risk of a portfolio of securities

- Construct an n-asset portfolio with weights wi (

). - The rate of return on this portfolio
- where is the weighted

average of the ?i of the n securities. - ? The portfolio nonsystematic component (which is

uncorrelated with F) - which is a weighted average of the ei of the

n securities.

- We can divide the variance of this portfolio into

systematic and nonsystematic sources. - The portfolio variance is
- where variance of the factor F
- nonsystematic risk of

the portfolio. - Note

Well-diversified portfolios

- If the portfolio were equally weighted (wi

1/n), then the nonsystematic variance would be - where average nonsystematic

variance. - When the portfolio gets large in the sense

that n is large and the portfolio remains equally

weighted across all n securities, the

nonsystematic variance approaches 0.

The set of portfolios for which the

nonsystematic variance approaches 0 as n gets

large consists of more portfolios than just the

equally weighted portfolio. Any portfolio

for which each wi approaches 0 as n gets large

will satisfy the condition that the portfolio

nonsystematic risk will approach 0 as n gets

large. ?Define a well-diversified portfolio as

one that is diversified over a large enough

number of securities with proportions wi, each

small enough that for practical purposes the

nonsystematic variance is negligible.

Because the expected value of eP is 0, if

its variance also is 0, we can conclude that any

realized value of eP will be virtually 0. ? For

a well-diversified portfolio Note Large

(mostly institutional) investors can hold

portfolios of hundreds and even thousands of

securities thus the concept of well-diversified

portfolios clearly is operational in contemporary

financial markets. Well-diversified portfolios,

however, are not necessarily equally weighted.

Betas and expected returns

- Consider a well-diversified portfolio A with

E(rA) 10 and ?A 1. - ? The return on this portfolio
- The well-diversified portfolios return is

determined completely by the systematic factor.

- Now consider another well-diversified portfolio

B, with an expected return of 8 and ?B also

equal to 1.0. - Could portfolios A and B coexist with the

return pattern depicted?

Clearly not No matter what the systematic

factor turns out to be, portfolio A outperforms

portfolio B, leading to an arbitrage

opportunity. If you sell short 1 million

of B and buy 1 million of A, a 0 net investment

strategy, your riskless payoff would be 20,000,

as follows

You should pursue it on an infinitely large

scale until the return discrepancy between the

two portfolios disappears. ? Well-diversified

portfolios with equal betas must have equal

expected returns in market equilibrium, or

arbitrage opportunities exist.

- What about portfolios with different betas?
- ? Their risk premiums must be proportional to

beta. - e.g. Suppose that the risk-free rate is 4

and that well-diversified portfolio, C, with a

beta of 0.5, has an expected return of 6. - Portfolio C plots below the line from the

risk-free asset to portfolio A.

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Consider a new portfolio, D, composed of

half of portfolio A and half of the risk-free

asset. Portfolio Ds beta will be 0.5 ? 1

0.5 ? 0 0.5, and its expected return will be

0.5 ? 10 0.5 ? 4 7. Now, portfolio D

has an equal beta but a greater expected return

than portfolio C. From our analysis, we

know that this constitutes an arbitrage

opportunity.

Conclusion To preclude arbitrage

opportunities, the expected return on all

well-diversified portfolios must lie on the

straight line from the risk-free asset.

The equation of this line will dictate the

expected return on all well-diversified

portfolios. Note that risk premiums are

indeed proportional to portfolio betas.

- Formally
- Suppose that 2 well-diversified portfolios

(U V) are combined into a zero-beta portfolio,

Z, by choosing the weights shown below - Notes

Portfolio Z is riskless It has no

diversifiable risk because it is well

diversified, and no exposure to the systematic

factor because its beta is zero. To rule

out arbitrage, then, it must earn only the

risk-free rate. ? ?

(i.e. risk premiums are proportional to betas)

The security market line

- Now, consider the market portfolio as a

well-diversified portfolio, and measure the

systematic factor as the unexpected return on the

market portfolio. - Because the market portfolio must be on the

straight line from the risk-free asset and the

beta of the market portfolio is 1, we can

determine the equation describing that line - (i.e. the SML relation of the CAPM)

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- We have used the no-arbitrage condition to obtain

an expected return-beta relationship identical to

that of the CAPM, without the restrictive

assumptions of the CAPM. - This suggests that despite its restrictive

assumptions the main conclusion of the CAPM (i.e.

the SML expected return-beta relationship) should

be at least approximately valid.

- In contrast to the CAPM, the APT does not require

that the benchmark portfolio in the SML

relationship be the true market portfolio. - Any well-diversified portfolio lying on the

SML may serve as the benchmark portfolio. - Accordingly, the APT has more flexibility

than does the CAPM.

- The APT provides further justification for use of

the index model in the practical implementation

of the SML relationship. - Even if the index portfolio is not a precise

proxy for the true market portfolio, we now know

that if the index portfolio is sufficiently well

diversified, the SML relationship should still

hold true according to the APT.

Individual assets and the APT

- If arbitrage opportunities are to be ruled out,

each well-diversified portfolios expected excess

return must be proportional to its beta. - That is, for any two well-diversified

portfolios P Q - If this relationship is to be satisfied by

all well-diversified portfolios, it must be

satisfied by almost all individual securities. - Thus, the expected return-beta relationship

holds for all but possibly a small number of

individual securities.

- Recall that to qualify as well diversified, a

portfolio must have very small positions in all

securities. - If, for example, only one security violates

the expected return-beta relationship, then the

effect of this violation on a well-diversified

portfolio will be too small to be of importance

for any practical purpose, and meaningful

arbitrage opportunities will not arise. - But if many securities violate the expected

return-beta relationship, the relationship will

no longer hold for well-diversified portfolios,

and arbitrage opportunities will be available.

- Conclusion
- Imposing the no-arbitrage condition on a

single-factor security market implies maintenance

of the expected return-beta relationship for all

well-diversified portfolios and for all but

possibly a small number of individual securities.

The APT and the CAPM

- Similarities
- The APT serves many of the same functions as

the CAPM - The APT gives us a benchmark for rates of return

that can be used in capital budgeting, security

evaluation, or investment performance evaluation.

- The APT highlights the crucial distinction

between nondiversifiable risk (factor risk) that

requires a reward in the form of a risk premium

and diversifiable risk that does not.

- Dissimilarities
- Advantages of the APT
- The APT depends on the assumption that a rational

equilibrium in capital markets precludes

arbitrage opportunities. A violation of the

APTs pricing relationships will cause extremely

strong pressure to restore them even if only a

limited number of investors become aware of the

disequilibrium. - The CAPM relies on a number of restrictive

assumptions.

- The APT yields an expected return-beta

relationship using a well-diversified portfolio

that practically can be constructed from a large

number of securities. - In contrast, the CAPM is derived assuming an

inherently unobservable market portfolio. - Disadvantage of the APT
- The CAPM provides an unequivocal statement

on the expected return-beta relationship for all

assets, whereas the APT implies that this

relationship holds for all but perhaps a small

number of securities.

A Multifactor APT

- We have assumed so far that there is only one

systematic factor affecting security returns. - This simplifying assumption is in fact too

simplistic. - It is easy to think of several factors

driven by the business cycle that might affect

security returns interest rate fluctuations,

inflation rates, oil prices, etc. - Presumably, exposure to any of these factors

will affect a securitys risk and hence its

expected return. - We can derive a multifactor version of the

APT to accommodate these multiple sources of

risk.

- Suppose that we generalize the single-factor

factor model to a two-factor model - Factor 1 might be, for example, departures

of GDP growth from expectations, and factor 2

might be unanticipated inflation. - Each factor has a zero expected value

because each measures the surprise in the

systematic variable rather than the level of the

variable. - Similarly, the firm-specific component of

unexpected return ei also has zero expected

value.

- A factor portfolio
- A well-diversified portfolio constructed to

have a beta of 1 on one of the factors and a beta

of 0 on any other factor. - This is an easy restriction to satisfy,

because we have a large number of securities to

choose from, and a relatively small number of

factors. - Factor portfolios will serve as the

benchmark portfolios for a multifactor security

market line.

- Suppose that the two factor portfolios

(portfolios 1 2) have expected returns E(r1)

10 E(r2) 12. - Suppose further that the risk-free rate rf

is 4. - ?The risk premium on the first factor portfolio
- E(r1) - rf 10 - 4 6.
- The risk premium on the second factor

portfolio - E(r2) - rf 12 - 4 8.

- Now consider an arbitrary well-diversified

portfolio, portfolio A, with beta on the first

factor, ?A1 0.5, and beta on the second factor,

?A2 0.75. - The multifactor APT states that the overall

risk premium on this portfolio A must equal the

sum of the risk premiums required as compensation

to investors for each source of systematic risk. - The risk premium attributable to risk factor

1 - (As exposure to factor 1) ? (risk premium

earned on the first factor portfolio) - ?A1 ? E(r1) - rf 0.5 ? 6 3.

The risk premium attributable to risk factor 2

(As exposure to factor 2) ? (risk premium

earned on the second factor portfolio) ?A2 ?

E(r2) - rf 0.75 ? 8 6. ? The total

expected return on the portfolio A

- Why the expected return on A must be 13?
- Suppose that the expected return on A were

12. - This return would give rise to an arbitrage

opportunity. - Form a portfolio B with the same betas as A
- weight on the first factor portfolio 0.5
- weight on the second factor portfolio

0.75 - weight on the risk-free asset -0.25
- The sum of Bs weights
- 0.5 0.75 (-0.25) 1.

- ?Bs beta on the first factor
- 0.5 ? 1 0.75 ? 0 (-0.25) ? 0
- 0.5 (same as ?A1)
- Bs beta on the second factor
- 0.5 ? 0 0.75 ? 1 (-0.25) ? 0
- 0.75 (same as ?A2)
- Bs expected return
- 0.5 ? E(r1) 0.75 ? E(r2) 0.25 ? rf
- 0.5 ? 10 0.75 ? 12 - 0.25 ? 4 13

- A long position in B and a short position in A

would yield an arbitrage profit. - The total return per dollar long or short in

each position would be - (i.e. a positive, risk-free return on a zero

net investment position).

- Generalization
- The factor exposure of any portfolio, P, is

given by its betas, ?P1 and ?P2. - Form a competing portfolio
- weight in the first factor portfolio ?P1

- weight in the second factor portfolio

?P2 - weight in T-bills 1 - ?P1 - ?P2
- This competing portfolio will have betas

equal to those of Portfolio P - beta on the first factor
- ?P1 ? 1 ?P2 ? 0 (1 - ?P1 - ?P2 ) ?

0 ?P1

- beta on the second factor
- ?P1 ? 0 ?P2 ? 1 (1 - ?P1 - ?P2 ) ? 0

?P2 - The expected return on this competing portfolio
- Any well-diversified portfolio with betas

?P1 and ?P2 must have return given in the above

equation if arbitrage opportunities are to be

precluded. - This establishes a multifactor version of

the APT.

Note The extension of the multifactor

SML to individual assets is precisely the same as

for the one-factor APT. If this

relationship is to be satisfied by all

well-diversified portfolios, it must be satisfied

by almost all individual securities. Thus,

the multifactor SML holds for all but possibly a

small number of individual securities.

Hence, the fair rate of return on any security

with ?1 0.5 and ?2 0.75 is 13 4 0.5 ?

(10 - 4) 0.75 ? (12 - 4).

- We discuss two examples of the multifactor

approach that are more well-known in the

literature - Example 1 5-factor model
- (Chen, Roll, and Ross, 1986)
- IP change in industrial production
- EI change in expected inflation
- UI change in unanticipated inflation
- CG excess return of long-term corporate

bonds over - long-term government bonds
- GB excess return of long-term government

bonds - over T-bills

- 5-factor model of excess security returns during

holding period t as a function of the macro

indicators - A multidimensional security characteristic line

with 5 factors. - As before, to estimate the betas of a given

security we can use regression analysis. Here,

however, because there is more than one factor,

we estimate a multiple regression of the excess

returns of the security in each period on the 5

macro factors. - The residual variance of the regression

estimates the firm-specific risk.

- Example 2 3-factor model (Fama French, 1996)
- An alternative approach to specifying macro

factors as candidates for relevant sources of

systematic risk uses firm characteristics that

seem on empirical grounds to represent exposure

to systematic risk. - SMB ( small minus big) the return of a

portfolio of small - stocks in excess of the return on

a portfolio of large stocks - HML ( high minus low) the return of a

portfolio of - stocks with high ratios of book

value to market value in - excess of the return on a

portfolio of stocks with low - book-to-market ratios.

- Notes
- In this model, the market index does play a role

and is expected to capture systematic risk

originating from macro factors. - These two firm-characteristic variables (SMB

HML) are chosen because of longstanding

observations that corporate capitalization (firm

size) and book-to-market ratio seem to be

predictive of average stock returns, and

therefore risk premiums. - Small firms are more sensitive to changes in

business conditions, and firms with high ratios

of book-to-market value are more likely to be in

financial distress.