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• 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

Topic 5 (Ch. 10) Arbitrage Pricing Theory (APT)

and Multifactor Models of Risk and Return

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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.

Multifactor Models

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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 one’s 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.

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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.

Factor models of security returns

A single-factor model

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Let E(ri): expected return on stock i F: deviation of the common factor from its expected value βi: 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.

iiii eFrEr )(

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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 stock’s value is 1.2.

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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 stock’s 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 stock’s return from its originally expected value.

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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.

A two-factor model

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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:

iIRGDPii eIRGDPrEr )(

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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.

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• 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, A’s and B’s stock prices will rise. When interest rates rise, A’s and B’s stock prices will fall.

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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.

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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 factors—multifactor models—can provide even better descriptions of returns.

However, there are many possible sets of macro factors that might be considered.

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• 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).

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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):

MffMf RPrrrErrE ])([)(

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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?

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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.

IRIRGDPGDPf RPRPrrE )(

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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.

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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.

Arbitrage Opportunities and Profits

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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.

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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:

High Real Interest Rates Low Real Interest Rates High

Inflation Low

Inflation High

Inflation Low

Inflation Probability: 0.25 0.25 0.25 0.25 Stock:

A -20 20 40 60 B 0 70 30 -20 C 90 -20 -10 70 D 15 23 15 36

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The current prices of the 4 stocks and rate of return statistics:

Correlation Matrix

Stock

Current

Price

Expected Return

(%)

Standard Deviation

(%)

A

B

C

D A $10 25.00 29.58 1.00 -0.15 -0.29 0.68 B 10 20.00 33.91 -0.15 1.00 -0.87 -0.38 C 10 32.50 48.15 -0.29 -0.87 1.00 0.22 D 10 22.25 8.58 0.68 -0.38 0.22 1.00

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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.

High Real Interest Rates Low Real Interest Rates High

Inflation Low

Inflation High

Inflation Low

Inflation Equally weighted

portfolio (A, B, and C)

23.33

23.33

20.00

36.67 D 15.00 23.00 15.00 36.00

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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.

Mean Standard Deviation

Correlation

3-stock portfolio 25.83 6.40 0.94 D 22.25 8.58

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The dollar profits in each of the 4 scenarios will be:

Þ The net investment is zero.

Yet, our portfolio yields a positive profit in any scenario.

Dollar High Real Interest Rates Low Real Interest Rates Stock Investment High Inflation Low Inflation High Inflation Low Inflation A $1,000,000 $-200,000 $200,000 $400,000 $600,000 B 1,000,000 0 700,000 300,000 -200,000 C 1,000,000 900,000 -200,000 -100,000 700,000 D -3,000,000 -450,000 -690,000 -450,000 -1,080,000 Portfolio 0 $250,000 $10,000 $150,000 $20,000

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Investors will want to take an infinite position in such a portfolio because larger positions entail no risk of losses, yet yield ever growing 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.

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Recall: A single factor model

The APT: A Single Factor Model

iiii eFrEr )(

where E(ri): expected return on stock i F: deviation of the common factor from its expected value βi: sensitivity of firm i to the common factor ei: firm-specific disturbance

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Construct an n-asset portfolio with weights wi ( ).

Þ 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.

n

iiw

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PPPP eFrEr )(

iiP w

iiP ewe

Risk of a portfolio of securities

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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:

2F

)(2Pe

)()()( 222iiiiP ewewVariancee

)(2222pFpp e

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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.

)(2ie

)(1)(1

)()1

()1

,( 22

222i

iiiP e

nne

ne

nnwe

Well-diversified portfolios

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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.

)(2Pe

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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.

FrEr Ppp )(

222FPP FPP

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Consider a well-diversified portfolio A with E(rA) = 10% and A = 1.

The return on this portfolio:

The well-diversified portfolio’s return is determined completely by the systematic factor.

FFrE AA 0.1%10)(

Betas and expected returns

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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?

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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:

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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.

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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 D’s 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.

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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.

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Formally:

Suppose that 2 well-diversified portfolios (U & V) are combined into a zero-beta portfolio, Z, by choosing the weights shown below:

Notes: 1 VU ww

0

V

UV

UU

UV

VVVUUZ ww

Portfolio

Expected

Return

Beta

Portfolio

Weight

U E(rU) U

UV

V

V E(rV) V

UV

U

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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)V

fV

U

fU rrErrE

)()(

fVUV

UU

UV

V

VVUUZ

rrErE

rEwrEwrE

)()(

)()(

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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)

PfMfP rrErrE ])([)(

The security market line

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The security market line

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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.

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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.

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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.

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If arbitrage opportunities are to be ruled out, each well-diversified portfolio’s 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.

Q

fQ

P

fP rrErrE

)()(

Individual assets and the APT

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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.

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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.

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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.

The APT and the CAPM

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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 APT’s 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.

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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.

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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 security’s risk and hence its expected return.

We can derive a multifactor version of the APT to accommodate these multiple sources of risk.

A Multifactor APT

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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.

iiiii eFFrEr 2211)(

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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.

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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%.

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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:

= (A’s exposure to factor 1) (risk premium earned on the first factor portfolio)

= A1 [E(r1) - rf] = 0.5 6% = 3%.

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The risk premium attributable to risk factor 2:

= (A’s 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:

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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

Þ The sum of B’s weights:

0.5 + 0.75 + (-0.25) = 1.

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B’s beta on the first factor

= 0.5 1 + 0.75 0 + (-0.25) 0

= 0.5 (same as A1)

B’s beta on the second factor

= 0.5 0 + 0.75 1 + (-0.25) 0

= 0.75 (same as A2)

Þ B’s expected return

= 0.5 E(r1) + 0.75 E(r2) – 0.25 rf

= 0.5 10% + 0.75 12% - 0.25 4% = 13%

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Þ 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).

AB rr

%1

])([])([ 22112211 FFrEFFrE AAABBB

]75.05.0%12[]75.05.0%13[ 2121 FFFF

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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

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beta on the second factor

= P1 0 + P2 1 + (1 - P1 - P2 ) 0 = P2

Þ 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.

fpPPPP rrErErE )1()()()( 212211

])([])([ 2211 fPfPf rrErrEr

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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%)].

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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

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Þ 5-factor model of excess security returns during holding period t as a function of the macro indicators:

Þ 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.

ittiGBtiCGtiUItiEItiIPiit eGBCGUIEIIPR

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• 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.

ittiHMLtiSMBMtiMiit eHMLSMBRR

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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.