financial economics: arbitrage pricing theory, part i

Overview and ComparisonsThe Factor Model

The Assumptions of APT

Financial Economics: Arbitrage Pricing

Theory, Part I

Shuoxun Hellen Zhang

WISE & SOE

XIAMEN UNIVERSITY

May, 2015

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Outline

1 Overview and Comparisons

2 The Factor ModelSingle-Factor ModelUsing the Market Model

3 The Assumptions of APTWell diversified portfoliosNo Arbitrage Condition

Same Beta: Proposition 1Different beta: Proposition 2

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Overview

The Arbitrage Pricing Theory (APT) was developed byStephen Ross (US, b.1944) in the mid-1970s.

Stephen Ross, “The Arbitrage Theory of Capital AssetPricing,” Journal of Economic Theory Vol.13 (December1976): pp.341-360.

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If we are willing to adopt the assumptions of the CAPM, in thisframework security prices only depend on one factor, namely the marketportfolios returns.

What we can observe from the data is that covariances between securitiestend to be positive if the same economic factors affect them. Forexample, stocks in one country are likely to all be affected by thebusiness cycle, interest rates, changes in technologies, movements in thecost of labor and raw materials...

The idea behind factor models is to summarize all relevant economic

variables in a set of indicators that are the main factors explaining the

security markets returns. The assumption is that few such factors are

sufficient to characterize the aggregate component of securities returns,

and any remaining uncertainty in security returns can be attributed to

firm-specific (or idiosyncratic) shocks.

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APT and the CAPM

The APT bears a close resemblance to the CAPM.

In fact, a special case of the APT implies a relationshipbetween expected returns on arbitrary, but well-diversified,portfolios and the return on the market portfolio that isidentical to the relationship implied by the CAPM.

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APT and the CAPM

But the APT allows for a more flexible, or general, depictionof aggregate risk than does the CAPM.

And, as its name suggests,the APT is a no-arbitrage theory ofasset pricing, that does not require the strong assumptionsimposed by the CAPM as an equilibrium theory.

The weakness of the APT—the cost, if you will, of its addedflexibility and generality—is that it applies only to“well-diversified portfolios” and “most” individual securities:there is no guarantee that it applies to all individual assets.

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Single-Factor ModelUsing the Market Model

The Single-Factor Model

The simplest version of the APT is also the one that can bestbe thought of as the “no-arbitrage” variant of the CAPM.

There is single factor that is sufficient to capture theaggregate component of the ex-post return on individual assetj as follows:

rj = E (rj) + βj f + εj

where E (f ) = 0, E (εj) = 0, Cov(f , εj) = 0 andCov(εj , εk) = 0 for all j = 1, 2, ..., J and k = 1, 2, ..., J withk 6= j .

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The Single-Factor Model

rj = E (rj) + βj f + εj

Deviations from the expected return on security are explainedby

an unexpected movement in the common factor f(aggregate/systemic shock)

εj idiosyncratic (or asset-specific) shock

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The Market Model

If we assert that the common component of unanticipatedmovements in individual security returns is explained byunanticipated movements in the return on a broad index ofsecurities, such as the S&P500 index, then we have the marketmodel, which is a variant of the single-factor model accordingto which the random return rj on each asset j = 1, 2, ..., J isrelated to the random return rM on the market portfolio via

rj = αj + βj [rM − E (rM)] + εj

where E (εj) = 0, Cov(rM , εj) = 0 and Cov(εj , εk) = 0 for allj = 1, 2, ..., J and k = 1, 2, ..., J with k 6= j

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where E (εj) = 0, Cov(rM , εj) = 0 and Cov(εj , εk) = 0 for allj = 1, 2, ..., J and k = 1, 2, ..., J with k 6= j .

The market model is called a single-factor model since itbreaks the random return on each asset down into twouncorrelated or “orthogonal” components:

βj [rM − E (rM)] which depends on the model’s singleaggregate variable or “factor,” in this case the marketreturn.

εj which is a purely idiosyncratic effect.

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The parameter’s αj and βj can be estimated through astatistical regression of each asset’s return on the marketreturn, imposing the additional statistical assumption thatthe regression error εj is uncorrelated across assets as wellas with the market portfolio.

As we will see, it is partly this added assumption that εj ispurely idiosyncratic that gives the APT its “bite.”

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Using the Market Model



Before moving on to see how the APT makes use of themarket model, let’s take note of two other purposes that themarket model can serve.

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Estimate of beta in CAPM

First, sincerj = αj + βj [rM − E (rM)] + εj

can be written as

rj = αj + βj rM + εj

whereαj = αj − βjE (rM)

the regressions from the market model yield slope coefficients

βj =Cov(rj , rM)

σ2M

that can be viewed as estimates of the CAPM beta for asset j .13 / 71




Ease of Computation

Second, since


impliesE (rj) = αj

σ2j = E ([rj − E (rj)]2)

= E ([βj [rM − E (rM)] + εj ]2)

= β2j E ([rM − E (rM)]2) + 2βjE ([rM − E (rM)]εj) + E (ε2

j )

= β2j σ

2M + 2βjCov(rM , εj) + σ2

εj

= β2j σ

2M + σ2

εj

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Ease of Computation


Similarly, the market model implies that for any two assets jand k 6= j

cov(rj , rk) = E ([rj − E (rj)][rk − E (rk)])

= E ([βj [ ˜rM − E ( ˜rM)] + εj ][βk [ ˜rM − E ( ˜rM)] + εk ])

= βjβkE ([ ˜rM − E ( ˜rM)]2) + βjE ([ ˜rM − E ( ˜rM)]εk)

+ βkE ([ ˜rM − E ( ˜rM)]εj) + E (εjεk)

= βjβkσ2M + βjcov( ˜rM , εk) + βkcov( ˜rM , εj) + cov(εj , εk))

= βjβkσ2M

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Ease of Computation

Recall one difficulty with MPT: it requires estimates of theJ(J + 1)/2 variances and covariances for J individual assetreturns.

But since the market model implies

σ2j = β2

j σ2M + σ2

εj

cov(rj , rk) = βjβkσ2M

it allows these variances and covariances to be estimatedbased on only 2J + 1 underlying “parameters”: the J βj ’s, theJ σ2

εj’s, and the 1 σ2

M .

With J = 100, J(J + 1)/2 = 5050 but 2J + 1 = 201. Thesaving are considerable!

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Well diversified portfoliosNo Arbitrage Condition

The market model


implies that the parameter αj measures the expected return onasset j :

E (rj) = αj

Recognizing this fact, let’s rewrite the equations for themarket model as

rj = E (rj) + βj [rM − E (rM)] + εj

for all j = 1, 2, ..., J . By itself, however, the market model tellsus nothing about how expected returns are determined or howthey are related across different assets.

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Assumptions

The APT starts by assuming that asset returns aregoverned by a factor model such as the market model.

To derive implications for expected returns, the APT makestwo additional assumptions:

There are enough individual assets to create manywell-diversified portfolios.

Investors act to eliminate all arbitrage opportunities acrossall well-diversified portfolios(No Arbitrage Conditions).

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Well diversified portfolios

Before moving all the way to well-diversified portfolios, let’sconsider an “only somewhat” diversified portfolio.

Consider, in particular, a portfolio consisting of only twoassets, j and k , both of which have returns described by themarket model.

Let w be the share of the portfolio allocated to asset j and1− w the share allocated to asset k .

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rj = E (rj) + βj [rM − E (rM)] + εj

rk = E (rk) + βk [rM − E (rM)] + εk

imply that the return rw on the portfolio is

rw = wE (rj) + (1− w)wE (rk)

+ wβj [rM − E (rM)] + (1− w)βk [rM − E (rM)]

+ wεj + (1− w)εk

= E (rw ) + βw [rM − E (rM)] + εw

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rw = wE (rj) + (1− w)wE (rk)


+ wεj + (1− w)εk

= E (rw ) + βw [rM − E (rM)] + εw

The first implication is that the expected return on theportfolio is just a weighted average of the expected returns onthe individual assets, with weights corresponding to those inthe portfolio itself:

E (rw ) = wE (rj) + (1− w)wE (rk)

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rw = wE (rj) + (1− w)wE (rk)


+ wεj + (1− w)εk

= E (rw ) + βw [rM − E (rM)] + εw

The second implication is that the portfolio’s beta is the sameweighted average of the betas of the individual assets:

βw = wβj + (1− w)βk

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rw = wE (rj) + (1− w)wE (rk)


+ wεj + (1− w)εk

= E (rw ) + βw [rM − E (rM)] + εw

The third implication is subtle but very important. We can seethat the idiosyncratic component of the portfolio’s return is aweighted average of the idiosyncratic components of theindividual asset returns:

εw = wεj + (1− w)εk

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εw = wεj + (1− w)εk

But the variance of the idiosyncratic component of theportfolio’s return is not a weighted average of the variances ofthe idiosyncratic components of the individual asset returns:

σ2εw = E (ε2

w ) = E ([wεj + (1− w)εk ]2)

= w 2E (ε2j ) + 2w(1− w)E (εjεk) + (1− w)2E (ε2

k)

= w 2σ2εj

+ (1− w)2σ2εk

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σ2εw = w2σ2

εj + (1− w)2σ2εk

For example, if the individual returns have idiosyncratic components withequal variances

σ2εj = σ2

εk= σ2

and the portfolio gives equal weight to the two assets w = 1− w = 1/2,then

σ2εw = (1/2)2σ2 + (1/2)2σ2 = (1/2)σ2

Even with just two assets, the portfolio’s idiosyncratic risk is cut in half.

Of course, this is just a special case of the gains from diversification

exploited by MPT. But the APT pushes the idea to its logical and

mathematical limits.

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Let’s head in the same direction, by considering a portfoliowith a large number N of individual assets.

Let wi ,i = 1, 2, ...,N , denote the share of the portfolioallocated to asset i .

And consider, in particular, the “equal weighted” case, where

wi = 1/N

for all i = 1, 2, ...,N .

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Since each of the individual asset returns are generated by the marketmodel, the return on this equal-weighted portfolio will be

rw = E (rw ) + βw [rw − E (rw )] + εw

where . . . the expected return on the equal-weighted portfolio is just

the average of the expected returns on the individual assets:

E (rw ) = ΣNi=1wiE (ri ) = (1/N)ΣN

i=1E (ri )

the equal-weighted portfolio’s beta is just the average of the individualassets’ betas:

βw = ΣNi=1wiβi = (1/N)ΣN

i=1βi

. . . and the idiosyncratic component of the return on theequal-weighted portfolio is an average of the idiosyncratic components ofthe individual asset returns:

εw = ΣNi=1wiεi = (1/N)ΣN

i=1εi

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εw = ΣNi=1wiεi = (1/N)ΣN

i=1εi

however, the variance of εw will not equal the average of thevariances of the individual εi ’s. In fact,

σ2εw = (1/N)2ΣN

i=1E (ε2i ) + (1/N)2ΣN

i=1Σh 6=iE (εiεh)

= (1/N)2ΣNi=1E (ε2

i )

= (1/N)[1

NΣN

i=1σ2εi

] = (1/N)σ2εi

where σ2εi

denotes the average the variances of the individualεi .

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σ2εw = (1/N)[

1

NΣN

i=1σ2εi

] = (1/N)σ2εi

Hence, the variance of εw is not the average of the variancesof the individual εi ’s.

The variance of εw is 1/N times the average of the variancesof the individual εi ’s. Hence, the amount of idiosyncratic riskin the portfolio’s return can be made arbitrarily small bymaking N, the number of assets included in the portfolio,sufficiently large.

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More generally, for portfolio’s with unequal weights

εw = ΣNi=1wiεi

implies

σ2εw = ΣN

i=1w 2i E (ε2

i ) + ΣNi=1Σh 6=iwiwhE (εiεh)

= ΣNi=1w 2

i E (ε2i )

= ΣNi=1w 2

i σ2εi

Any portfolio in which the share of each individual asset wi

becomes arbitrarily small as the number of assets N growslarger and larger will have negligible idiosyncratic risk.

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σ2εw = ΣN

i=1w 2i σ

2εi

From a practical perspective, this means: holding the dollarvalue of your portfolio fixed, buy some shares in more andmore individual companies by buying fewer and fewer shares ineach individual company.

This, specifically, is what is meant by a well-diversifiedportfolio: one in which the number of assets included issufficiently large to make the portfolio’s idiosyncratic riskvanish.

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In his 1976 paper, Stephen Ross notes that while theassumption

cov(εj , εk) = 0 for all j , k = 1, 2, ..., J with k 6= j

helps in making a portfolio’s idiosyncratic risk vanish as thenumber of assets included becomes larger, this can still happenwith positive correlations between individual assets’idiosyncratic returns provided the correlations are not too largeand the number of assets is sufficiently large.

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Hence, the APT’s first assumption—that individual assetreturns are generated by a factor model — implies that thereturn on any portfolio is

rw = E (rw ) + βw [rM − E (rM)] + εw

And the APT’s second assumption—that there are a sufficientnumber of assets J to create well-diversified portfolios —implies that the return on any well-diversified portfolio is

rw = E (rw ) + βw [rM − E (rM)]

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We still have not said anything about what determines theexpected return E (rw ).

But we can now, based on the APT’s third assumption: thatinvestors act to eliminate arbitrage opportunities across allwell-diversified portfolios.

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No Arbitrage Condition

An arbitrage opportunity exists if an investor can earn profitswithout exposing him or herself to risk and making any netinvestment.

Example: IBM stock is sold for $ 60 on the NYSE andsimultaneously trades at $ 58 on NASDAQ. This offers arisk-free opportunity to earn money by at the same time buyingIBM stock on NASDAQ and selling IBM stock on the NYSE.

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No Arbitrage Condition

In a well functioning capital market such opportunities shouldnot exist since market participants would instantaneously takeadvantage of them and thus move prices so that theydisappear.

Law of one price: two assets which are equivalent in alleconomically relevant aspects must have the same marketprice.

No-arbitrage conditions such as this one underpin much ofmodern finance theory. Let us turn to the implications ofno-arbitrage in the context of the APT.

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Example

Suppose that we have the following situation: well diversifiedportfolios A and B have the same factor loading, i.e. βA = βB .However, E (rA) = 0.10 and E (rB) = 0.08. Is this consistentwith no arbitrage?

No, an arbitrage opportunity arises. A market participant canconstruct an arbitrage portfolio

position return on positionlong position in A E (rA) + βA[rM − E (rM)]short position in B −E (rB)− βB [rM − E (rM)]net proceeds E (rA)− E (rB)

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Example

The exposure to factor risk cancels out:βA[rM − E (rM)]− βB [rM − E (rM)] = 0. If you buy $1 millionof A and selling short $1 million of B would yield a profit of$20,000(=(0.1− 0.08) ∗ $1, 000, 000). Note that this involveszero net investment and is risk-free.

The no-arbitrage condition thus implies that well-diversifiedportfolios with equal factor loadings must have equal expectedreturns. This means that they plot on a straight line in theexpected return - beta diagram.

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Proposition 1

Our first no-arbitrage argument applies to well-diversifiedportfolios with the same betas.

Proposition (1)

The absence of arbitrage opportunities requires well-diversifiedportfolios with the same betas to have the same expectedreturns.

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Proof

To see why this proposition must be true, consider twowell-diversified portfolios, one with

r 1w = E (rw ) + βw [rM − E (rM)]

and the other with

r 2w = E (rw ) +4+ βw [rM − E (rM)]

where 4 > 0. These portfolios have the same beta, but thesecond has a higher expected return.

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Proof

r 1w = E (rw ) + βw [rM − E (rM)]

r 2w = E (rw ) +4+ βw [rM − E (rM)]

Now consider a strategy of taking a “long position” worth x inportfolio 2 and a “short position” worth −x in portfolio 1.

Note that the two portfolios may contain some of the sameindividual assets; hence, in practice, this strategy may involvetaking appropriate long and short positions in just some ofthose assets so as to capture the net differences between thetwo portfolios.

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Proof

r1w = E (rw ) + βw [ ˜rM − E ( ˜rM)]

r2w = E (rw ) +4+ βw [ ˜rM − E ( ˜rM)]

Now consider a strategy of taking a “long position” worth x in portfolio 2and a “short position” worth −x in portfolio 1.

This strategy is self-financing, in that it requires “no money down” att = 0.But the strategy yields a payoff at t = 1 of

x(1 + r2w )− x(1 + r1

w ) = xE (rw ) + x4+ xβw [ ˜rM − E ( ˜rM)]

− xE (rw )− xβw [ ˜rM − E ( ˜rM)]

= x4 > 0

Thus, 4 > 0 is inconsistent with the absence of arbitrage opportunities.

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Proof

So suppose instead that

r 1w = E (rw ) + βw [rM − E (rM)]

and the other with

r 2w = E (rw ) +4+ βw [rM − E (rM)]

where now 4 < 0. These portfolios have the same beta, butthe second has a lower expected return.

With 4 < 0, the profitable strategy involves taking a longposition worth x in portfolio 1 and a short position worth −xin portfolio 2.

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Proof

r1w = E (rw ) + βw [ ˜rM − E ( ˜rM)]

r2w = E (rw ) +4+ βw [ ˜rM − E ( ˜rM)]

With 4 < 0, the profitable strategy involves taking a long position worthx in portfolio 1 and a short position worth −x in portfolio 2.

This strategy is self-financing at t = 0, but yields a payoff at t = 1 of

x(1 + r1w )− x(1 + r2

w ) = xE (rw ) + xβw [ ˜rM − E ( ˜rM)]

− xE (rw )− x4− xβw [ ˜rM − E ( ˜rM)]

= −x4 > 0

Thus, 4 < 0 is inconsistent with the absence of arbitrage opportunities.

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Proof

r 1w = E (rw ) + βw [rM − E (rM)]

r 2w = E (rw ) +4+ βw [rM − E (rM)]

Since 4 cannot be positive or negative, it must equal zero.

This proves the first proposition: in the absence of arbitrageopportunities, well-diversified portfolios with the same betasmust have the same expected returns.

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Example

Suppose that the risk-free rate is 4 percent and that we havethe following situation for well diversified portfolios A and C:

portfolio beta expected returnA 1 0.14B 0.5 0.08

Is this consistent with no arbitrage?

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Figure: Returns as a function of the systematic factor

The expected return on C means that there is an arbitrage opportunity.

We can construct a portfolio that has the same beta as C and a higher

return, which we have already seen violates no arbitrage.

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Figure: Returns as a function of the systematic factor

D is such an arbitrage portfolio, it is a portfolio invested halfin the risk-free asset (with β = 0) and half in the welldiversified portfolio A. Thus βD = 0.5 ∗ 0 + 0.5 ∗ 1 = 0.5 andE (rD) = 0.5rf + 0.5E (rA) = 0.09.

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Different Beta

Hence, no-arbitrage implies that well-diversified portfolios withdifferent factor loadings must have expected returnsproportional to their betas. This means that they plot on astraight line in the expected return - beta diagram.

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Proposition 2

Now that we’ve seen how the APT’s arguments work, we canextend the results to apply to well-diversified portfolios withdifferent betas.

Proposition (2)

The absence of arbitrage opportunities requires the expectedreturns on all well-diversified portfolios to satisfy

E (rw ) = λ0 + λ1βw

for constants λ0 and λ1 that are the same for all portfolios.

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Proposition 2

The APT implies that expected returns on well-diversifiedportfolios line up along E (rw ) = λ0 + λ1βw .

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Proposition 2

The graph of the APT relationship resembles the CAPM’ssecurity market line. This is not a coincidence. To see why,let’s start by interpreting λ0 and λ1.

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Proposition 2

To interpret λ0, consider a well-diversified portfolio, call itportfolio 1, with βw = 0, so that

r 1w = E (r 1

w ) + 0× [rM − E (rM)] = E (r 1w )

But if the return on portfolio 1 always equals its own expectedreturn, either:

Portfolio 1 is a well-diversified portfolio that forms a“synthetic” risk-free asset

or portfolio 1 consists entirely of risk-free assets.

Either way, its return must equal the risk-free rate:

r 1w = E (r 1

w ) = rf

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Proposition 2

But proposition 2 requires the expected return on portfolio 1to equal λ0 as well:

E (r 1w ) = λ0 + λ1 × 0 = λ0

Together withE (r 1

w ) = rf

this observation tells us that

λ0 = rf

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Proposition 2

Now we know that λ0 is given by the risk-free rate rf .

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Proposition 2

Next, consider a second well-diversified portfolio, call itportfolio 2, with βw = 1, so that

r 2w = E (rw ) + 1× [rM − E (rM)]

Portfolio 2 is well diversified and has the same beta as themarket portfolio, so it must also have the same expectedreturn as the market portfolio. Hence,

r 2w = E (r 2

w ) + [rM − E (rM)]

= E (rM) + [rM − E (rM)]

= rM

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Proposition 2

But if portfolio 2 is such that

r 2w = rM

then either

Portfolio 2 is a well-diversified portfolio that always hasthe same return as the market portfolio or

portfolio 2 actually is the market portfolio

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Proposition 2

In any case, proposition 2 requires the expected return onportfolio 2 to be given by

E (r 2w ) = λ0 + λ1 × 1 = λ0 + λ1

Together withr 2w = rM

andλ0 = rf

this observation tells us that

λ1 = E (rM)− rf

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Proposition 2

Since λ0 = rf and λ1 = E (rM)− rf , the lineE (rw ) = λ0 + λ1βw is the CAPM’s security market line.

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Proposition 2

E (rw ) = λ0 + λ1βw = rf + βw [E (rM)− rf ]

This is the same relationship implied by the CAPM.

But the APT derives this relationship without assuming thatutility is quadratic and without assuming that returns arenormally distributed (although the returns do have to behavein accordance with the market model).

On the other hand, the APT holds only for well-diversifiedportfolios, not necessarily for individual assets. Hence, theAPT’s added generality comes at a cost, in terms of lesswidely-applicable results.

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Proposition 2

But to prove proposition 2, we still need to show that

E (rw ) = λ0 + λ1βw = rf + βw [E (rM)− rf ]

holds for all well-diversified portfolios, not just those withβw = 0 or βw = 1

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Proof

Consider, therefore, a third well-diversified portfolio, with

r 3w = E (r 3

w ) + βw [rM − E (rM)]

and βw different from zero and one. We need to show that

E (r 3w ) = λ0 + λ1βw

so, paralleling the argument from before, suppose instead that

E (r 3w ) = λ0 + λ1βw +4

where 4 > 0

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If E (r 3w ) lies above the line λ0 + λ1βw , then a strategy that

takes a long position in portfolio 3 and short positions inportfolios 1 and 2 will constitute an arbitrage opportunity.

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r 1w = λ0

r 2w = λ0 + λ1 + [rM − E (rM)]

r 3w = λ0 + λ1βw +4+ βw [rM − E (rM)]

The no-arbitrage argument requires two steps. First, form afourth well-diversified portfolio from the first two, by allocatingthe shares 1− βw to portfolio 1 and βw to portfolio 2. Thisportfolio has

r 4w = (1− βw )r 1

w + βw r 2w

= (1− βw )λ0 + βw (λ0 + λ1) + βw [rM − E (rM)]

= λ0 + λ1βw + βw [rM − E (rM)]

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Proposition 2

Second, observe that portfolio 4 has the same beta, but alower expected return, than portfolio 3. Proposition 1 impliesthat this is inconsistent with the absence of arbitrage.

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r 1w = λ0

r 2w = λ0 + λ1 + [rM − E (rM)]

r 3w = λ0 + λ1βw +4+ βw [rM − E (rM)]

r 4w = λ0 + λ1βw + βw [rM − E (rM)]

Since portfolios 3 and 4 have the same beta, a strategy thatallocates x to portfolio 3 and −x to portfolio 4 is self-financingat t = 0 and yields a payoff of x4 > 0 at t = 1. This confirmsthat 4 > 0 is inconsistent with the absence of arbitrage.

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Similarly, if

r 1w = λ0

r 2w = λ0 + λ1 + [rM − E (rM)]

But portfolio 3 has

r 3w = λ0 + λ1βw +4+ βw [rM − E (rM)]

with 4 < 0, we would begin by constructing portfolio 4 asbefore, with shares 1− βw allocated to portfolio 1 and βw toportfolio 2.

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r 1w = λ0

r 2w = λ0 + λ1 + [rM − E (rM)]

r 3w = λ0 + λ1βw +4+ βw [rM − E (rM)]

r 4w = λ0 + λ1βw + βw [rM − E (rM)]

Since portfolios 3 and 4 have the same beta, a strategy thatallocates −x to portfolio 3 and x to portfolio 4 isself-financing at t = 0 and yields a payoff of −x4 > 0 att = 1. This confirms that 4 < 0 is inconsistent with theabsence of arbitrage.

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With 4 < 0, portfolio 4 has the same beta, but a higherexpected return, than portfolio 3. Proposition 1 again impliesthat this is inconsistent with the absence of arbitrage.

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Hence, if

r 1w = λ0

r 2w = λ0 + λ1 + [rM − E (rM)]

r 3w = λ0 + λ1βw +4+ βw [rM − E (rM)]

then 4 = 0. All expected returns on well-diversified portfoliosmust satisfy

E (rw ) = λ0 + λ1βw

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According to the APT, all expected returns on well-diversifiedportfolios must satisfy

E (rw ) = λ0 + λ1βw

or equivalently,

E (rw ) = rf + βw [E (rM)− rf ]

One might argue, as well, that the APT must apply “most”individual securities, since if a large number of individual assetsviolated the APT relationship, it would be possible toconstruct a well-diversified portfolio of those assets andarbitrage away the higher or lower expected returns.

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financial economics: arbitrage pricing theory, part i

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