Investments, tenth edition

 Active Portfolio 

 Zero-Net-Investment, 

Arbitrage (Zero-Beta) 

Portfolio  

Treynor-Black Procedure 

 Alpha (%)   Residual SD   Beta    w  in Active   Info Ratio    w (beta  5  0)  

 w (beta) 

 Sharpe 

Ratio 

 Incremental 

Sharpe Ratio 

 1  

4  

0.5  

2  

0.25  

3.57  

1.28  

0.43  

0.18 

 1  

4  

2  

1  

0.25  

3.57  

1.00  

0.43  

0.18 

 1  

3  

0.5  

2  

0.33  

6.35  

1.52  

0.48  

0.15 

 1  

3  

2  

1  

0.33  

6.35  

1.00  

0.48  

0.15 

 1  

2  

0.5  

2  

0.50  

14.29  

1.75  

0.61  

0.11 

 1  

2  

2  

1  

0.50  

14.29  

1.00  

0.61  

0.11 

 3  

4  

0.5  

2  

0.75  

10.71  

1.69  

0.83  

0.08 

 3  

4  

2  

1  

0.75  

10.71  

1.00  

0.83  

0.08 

 3  

3  

0.5  

2  

1.00  

19.05  

1.81  

1.06  

0.06 

 3  

3  

2  

1  

1.00  

19.05  

1.00  

1.06  

0.06 

 3  

2  

0.5  

2  

1.50  

42.86  

1.91  

1.54  

0.04 

 3  

2  

2  

1  

1.50  

42.86  

1.00  

1.54  

0.04 

 Table 10.2 

 Performance of APT vs. Index Model when diversification of residual SD is incomplete 

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338 

P A R T   I I I

  Equilibrium in Capital Markets

    10.4 

A Multifactor APT 

  We have assumed so far that only one systematic factor affects stock returns. This simpli-

fying assumption is in fact too simplistic. We’ve noted that it is easy to think of several 

factors driven by the business cycle that might affect stock returns: interest rate fluctua-

tions, inflation rates, and so on. Presumably, exposure to any of these factors will affect a 

stock’s 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 model expressed in Equation 10.1 to a 

two-factor  model:   

 

R

i

5 E(R

i

)

1 b

i1

F

1

1 b

i2

F

2

1 e

i

 

 (10.10)  

In Example 10.2, factor 1 was the departure of GDP growth from expectations, and factor 2 

was the unanticipated change in interest rates. Each factor has 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,  e  

 i  

 , also has zero expected 

value. Extending such a two-factor model to any number of factors is straightforward. 

 We can now generalize the simple APT to a more general multifactor version. But first 

we must introduce the concept of a    factor  portfolio,    which is a well-diversified portfolio 

constructed to have a beta of 1 on one of the factors and a beta of zero on any other factor. 

We can think of a factor portfolio as a  tracking portfolio.  That is, the returns on such a port-

folio track the evolution of particular sources of macroeconomic risk but are uncorrelated 

with other sources of risk. It is possible to form such factor portfolios 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.  

The multidimensional SML predicts that exposure to each risk factor contributes to the 

 security’s total risk premium by an amount equal to the factor beta times the risk premium 

of the factor portfolio tracking that source of risk. We illustrate with an example.

by the APT). The last set of columns shows the T-B position in the active portfolio that 

maximizes the Sharpe ratio of the overall risky portfolio. The final column shows the 

increment to the Sharpe ratio of the T-B portfolio relative to the APT portfolio. 

 Keep in mind that even when the two models call for a similar weight in the active 

portfolio (compare w in Active for the APT model to w(beta) for the T-B model), they 

nevertheless prescribe a different overall risky portfolio. The APT assumes zero invest-

ment beyond what is necessary to hedge out the market risk of the active portfolio. In 

contrast, the T-B procedure chooses a mix of active and index portfolios to maximize the 

Sharpe ratio. With identical investment in the active portfolio, the T-B portfolio can still 

include additional investment in the index portfolio. 

 To obtain the Sharpe ratio of the risky portfolio, we need the Sharpe ratio of the 

index portfolio. As an estimate, we use the average return and standard deviation of 

the broad market index (NYSE  1  AMEX  1  NASDAQ) over the period 1926–2012. The 

top row (over the column titles) of  Table 10.2  shows an annual Sharpe ratio of 0.35. The 

rows of the table are ordered by the information ratio of the active portfolio. 

  Table 10.2  shows that the T-B procedure noticeably improves the Sharpe ratio beyond 

the information ratio of the APT (for which the IR is also the Sharpe ratio). However, as 

the information ratio of the active portfolio increases, the difference in the T-B and APT 

active portfolio positions declines, as does the difference between their Sharpe ratios. 

Put differently, the higher the information ratio, the closer we are to a risk-free arbitrage 

opportunity, and the closer are the prescriptions of the APT and T-B models. 

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  C H A P T E R  

1 0

  Arbitrage Pricing Theory and Multifactor Models of Risk and Return 

339

  To generalize the argument in Example 10.4, note that the factor exposures of any port-

folio,  P,  are given by its betas,  b  

 P 1

  and  b  

 P 2

 . A competing portfolio,  Q,  can be formed by 

investing in factor portfolios with the following weights:  b  

 P 1

  in the first factor portfolio, 

 b  

 P 2

  in the second factor portfolio, and 1  2   b  

 P 1

   2   b  

 P 2

  in T-bills. By construction, portfolio 

 Q  will have betas equal to those of portfolio  P  and expected return of   

 E (r

Q

)

5 b

P1

E (r

1

)

1 b

P2

E (r

2

)

1 (1 2 b

P1

2 b

P2

)r

f

 

 

5 r

f

1 b

P1

3E(r

1

)

2 r

f

4 1 b

P2

3E(r

2

)

2 r

f

4

 

 (10.11)   

 Using the numbers in Example 10.4:   

E(r

Q

)

5 4 1 .5 3 (10 2 4) 1 .75 3 (12 2 4) 5 13%  

 Suppose that the two factor portfolios, portfolios 1 and 2, have expected returns 

 E ( r  

1

 )  5  10% and  E ( r  

2

 )  5  12%. Suppose further that the risk-free rate is 4%. The risk 

premium on the first factor portfolio is 10%  2  4%  5  6%, whereas that on the second 

factor portfolio is 12%  2  4%  5  8%. 

 Now consider a well-diversified portfolio, portfolio  A,  with beta on the first factor, 

 b  

 A 1

   5  .5, and beta on the second factor,  b  

 A 2

   5  .75. The multifactor APT states that the 

overall risk premium on this portfolio must equal the sum of the risk premiums required 

as compensation for each source of systematic risk. The risk premium attributable to risk 

factor 1 should be the portfolio’s exposure to factor 1,  b  

 A 1

 , multiplied by the risk pre-

mium earned on the first factor portfolio,  E ( r  

1

 )  2   r  

 f 

 . Therefore, the portion of portfolio 

 A ’s risk premium that is compensation for its exposure to the first factor is  b  

 A 1

 [ E ( r  

1

 )  2   r  

 f 

 ]  5  

.5(10%   2   4%)   5   3%, whereas the risk premium attributable to risk factor 2 is 

 b  

 A 2

 [ E ( r  

2

 )  2   r  

 f 

 ]  5  .75(12%  2  4%)  5  6%. The total risk premium on the portfolio should 

be 3%  1  6%  5  9% and the total return on the portfolio should be 4%  1  9%  5  13%. 

 Example  10.4 

Multifactor SML 

 Suppose that the expected return on portfolio  A  from Example 10.4 were 12% rather 

than 13%. This return would give rise to an arbitrage opportunity. Form a portfolio from 

the factor portfolios with the same betas as portfolio  A.  This requires weights of .5 on 

the first factor portfolio, .75 on the second factor portfolio, and  2 .25 on the risk-free 

asset. This portfolio has exactly the same factor betas as portfolio  A:  It has a beta of .5 on 

the first factor because of its .5 weight on the first factor portfolio, and a beta of .75 

on the second factor. (The weight of  2 .25 on risk-free T-bills does not affect the sensitiv-

ity to either factor.) 

 Now invest $1 in portfolio  Q  and sell (short) $1 in portfolio  A.  Your net investment is 

zero, but your expected dollar profit is positive and equal to   

$1 3 E(r

Q

) 2 $1 3 E (r

A

) 5 $1 3 .13 2 $1 3 .12 5 $.01

 Moreover, your net position is riskless. Your exposure to each risk factor cancels out 

because you are long $1 in portfolio  Q  and short $1 in portfolio  A,  and both of these 

well-diversified portfolios have exactly the same factor betas. Thus, if portfolio  

A ’s 

expected return differs from that of portfolio  Q ’s, you can earn positive risk-free profits 

on a zero-net-investment position. This is an arbitrage opportunity. 

 Example  10.5 

Mispricing and Arbitrage 

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340 

P A R T   I I I

  Equilibrium in Capital Markets

  Because  portfolio   Q  in Example 10.5 has precisely the same exposures as portfolio  A   to 

the two sources of risk, their expected returns also ought to be equal. So portfolio  A  also ought 

to have an expected return of 13%. If it does not, then there will be an arbitrage opportunity.  

7

  

  

 

We conclude that any well-diversified portfolio with betas  

b  

 P 1

 

 and  

b  

 P 2

   must 

have the return given in Equation 10.11 if arbitrage opportunities are to be precluded. 

Equation 10.11 simply generalizes the one-factor SML. 

 Finally, the extension of the multifactor SML of Equation 10.11 to individual assets 

is precisely the same as for the one-factor APT. Equation 10.11 cannot be satisfied by 

every well-diversified portfolio unless it is satisfied approximately by individual securities. 

Equation 10.11 thus represents the multifactor SML for an economy with multiple sources 

of risk. 

 We pointed out earlier that one application of the CAPM is to provide “fair” rates 

of return for regulated utilities. The multifactor APT can be used to the same ends. The 

nearby box summarizes a study in which the 

APT was applied to find the cost of capital 

for regulated electric companies. Notice that 

empirical estimates for interest rate and infla-

tion risk premiums in the box are negative, as 

we argued was reasonable in our discussion 

of Example 10.2. 

 

   

 Using the factor portfolios of Example 10.4, find the equilib-

rium rate of return on a portfolio with  b  

1

   5  .2 and  b  

2

   5  1.4. 

 CONCEPT CHECK 

10.3 

  

7

 The risk premium on portfolio  A  is 9% (more than the historical risk premium of the S&P 500) despite the fact 

that its betas, which are both below 1, might  seem  defensive. This highlights another distinction between multi-

factor and single-factor models. Whereas a beta greater than 1 in a single-factor market is aggressive, we cannot 

say in advance what would be aggressive or defensive in a multifactor economy where risk premiums depend on 

the sum of the contributions of several factors. 

  

8

 Eugene F. Fama and Kenneth R. French, “Multifactor Explanations of Asset Pricing Anomalies,”  Journal of 

Finance  51 (1996), pp. 55–84. 

    10.5 

The Fama-French (FF) Three-Factor Model 

  The currently dominant approach to specifying factors as candidates for relevant sources 

of systematic risk uses firm characteristics that seem on empirical grounds to proxy for 

exposure to systematic risk. The factors chosen are variables that on past evidence seem to 

predict average returns well and therefore may be capturing risk premiums. One example 

of this approach is the Fama and French three-factor model and its variants, which have 

come to dominate empirical research and industry applications:  

8

  

  

   

 

R

it

5 a

i

1 b

iM

R

Mt

1 b

iSMB

SMB

t

1 b

iHML

HML

t

1 e

it

 

 (10.12)  

where

   SMB   5   Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the 

return on a portfolio of large stocks.  

  HML   5   High Minus Low, i.e., the return of a portfolio of stocks with a high 

book-to-market ratio in excess of the return on a portfolio of stocks with a 

low  book-to-market  ratio.    

 Note that in this model the market index does play a role and is expected to capture sys-

tematic risk originating from macroeconomic factors. 

 These two firm-characteristic variables are chosen because of long-standing observa-

tions that corporate capitalization (firm size) and book-to-market ratio predict deviations 

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341