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CFA Institute Sorting out Risks Using Known APT Factors Author(s): Michael A. Berry, Edwin Burmeister, Marjorie B. McElroy Source: Financial Analysts Journal, Vol. 44, No. 2 (Mar. - Apr., 1988), pp. 29-42 Published by: CFA Institute Stable URL: http://www.jstor.org/stable/4479100 Accessed: 23/10/2009 04:12 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=cfa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. CFA Institute is collaborating with JSTOR to digitize, preserve and extend access to Financial Analysts Journal. http://www.jstor.org

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Page 1: Berry Burmeister Mcelroy 1988

CFA Institute

Sorting out Risks Using Known APT FactorsAuthor(s): Michael A. Berry, Edwin Burmeister, Marjorie B. McElroySource: Financial Analysts Journal, Vol. 44, No. 2 (Mar. - Apr., 1988), pp. 29-42Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4479100Accessed: 23/10/2009 04:12

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=cfa.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

CFA Institute is collaborating with JSTOR to digitize, preserve and extend access to Financial AnalystsJournal.

http://www.jstor.org

Page 2: Berry Burmeister Mcelroy 1988

by Michael A. Berry, Edwin Burmeister and Marjorie B. McElroy

Sorting Out Risks Using Known APT

Arbitrage Pricing Theory (APT) differs from the Capital Asset Pricing Model (CAPM) in hypothesizing that actual and expected security returns are sensitive, not to just one type of nondiversifiable risk (i.e., beta or market risk), but to a variety of different types of risk. Prior testing has shown the CAPM to be inferior to an APT model that incorporates unanticipated changes in five macroeconomic variables-default risk, the term structure of interest rates, inflation or deflation, the long-run expected growth rate of profits for the economy and residual market risk.

It is possible to estimate the sensitivities of individual securities or portfolios to these five risk factors. Such measurements allow one to explore variations in sensitivities to different types of risk across both equity market sectors and industries. The resulting risk exposure profiles do not depend on any particular market index.

APT risk profiles may be used for either active or passive portfolio management. Passive managers wishing to reduce their portfolios' systematic risk characteristics might consider a technique termed "risk sterilization"; here assets with different risk profiles are combined so as to negate, or sterilize, exposure to selected riskfactors. Alternatively, active managers can attempt to achieve excess returns by constructing portfolios in accordance with their forecasts of risk factor realizations.

T HE CAPITAL ASSET PRICING Model (CAPM) predicts that only one type of nondiversifiable risk influences expected

security returns-namely, "market risk." In contrast, Arbitrage Pricing Theory (APT) explic- itly recognizes that a variety of risks may affect expected returns.1 In particular, five different

types of risk factors have been shown to have significant influence on expected returns:

* risk of changes in default premiums, * risk that the term structure of interest rates

may change, * risk of unanticipated inflation or deflation, * risk that the long-run expected growth rate

of profits for the economy will change, and * residual market risk, or any remaining risk

needed to explain a market index such as the S&P 500.

This article examines the importance of these five different types of risk factors.2 By defini- tion, a risk factor is an element of surprise- such as unanticipated inflation, which arises be- cause the actual value of inflation may depart from its expected value.3 We discuss below how to measure the five risk factors and how expo- sure to them varies across different industries. The large differences in risk exposure profiles

1. Footnotes appear at end of article.

Michael Berry is Assistant Professor of Business Adminis- tration at the Darden School of the University of Virginia. Edwin Burmeister is Commonwealth Professor of Econom- ics at the University of Virginia. Marjorie McElroy is Professor of Economics at Duke University.

The authors are grateful for a research grant from the Institute of Quantitative Research in Finance, and Profes- sors Burmeister and McElroy acknowledge support from the National Science Foundation (SES-8618403). The authors thank Richard W. McEnally and Stephen A. Ross for their valuable comments, and Mark T. Finn, President of Delta Financial, Inc., for his investment management insights.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 D 29

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across economic sectors and industries suggest several strategies for managing risks so as to achieve superior portfolio returns.

Arbitrage Pricing Theory APT assumes that arbitrage profit opportunities are quickly eliminated through competitive forces. That is, an investor cannot earn a posi- tive expected rate of return on any combination of assets without incurring some risk and with- out making some net investment. (This assump- tion is essentially an equilibrium condition for capital markets analogous to "supply equals demand.")

APT models the discrepancy between the actual (realized) return on any asset and its expected return as a linear function of the realizations of relevant risk factors, plus returns resulting from asset-specific events. The expect- ed return for any asset, in turn, is equal to the sum of the quantities of different types of risk inherent in that asset, times their respective risk "prices."4 (The measurement of risk quantities and prices is discussed in more detail later.)

Previous technical work on APT has indicated that the five types of risk identified at the outset of this article are highly significant for determin- ing realized and expected returns.5 In other words, the five different types of risk have non- zero APT prices; they are relevant risks, influ- encing equilibrium returns. The present article extends the previous work by identifying and measuring, over various economic sectors and industries, the risk exposures associated with these relevant risks. The implications for practi- tioners are twofold: (1) Practitioners can use APT risk exposure profiles to manage their own risk exposures and (2) they can do so without having to implement the full, and complex, APT machinery.

Three Common Questions About APT and Risk Factors Do APT risk factors help explain stock returns any

better than a model using only a market index such as the S&P 500? Rigorous statistical testing has shown that there is virtually zero probability that the five risk factors identified above add no new information over and above that already embodied in the S&P 500.6 The APT model with these five risk factors is vastly superior to both the market model and the CAPM for explaining stock returns.

What kinds of variables qualify as legitimate risk

factors in an APT framework? Economic variables that are legitimate risk factors must possess three important properties:

(1) At the beginning of every period, the factor must be completely unpredictable to the market.

(2) Each APT factor must have a pervasive influence on stock returns.

(3) Relevant factors must influence expected return; i.e., they must have non-zero prices.

Property (1) means that, for the market as a whole, a risk factor cannot be forecast either from its own past value or from any other publicly available information. Thus, at the start of every time period (day, week, month or other period), the expected value of the factor is zero. For example, the rate of inflation is not a legiti- mate APT risk factor because it is partially predictable. Unexpected inflation, however, is a legitimate factor; unexpected inflation by defini- tion cannot be predicted because it is the differ- ence between the actual rate of inflation over the period and the rate that had been expected at the beginning of the period. Similarly, the growth rate of GNP is not a legitimate factor because its value in a given period can be predicted partially from realizations in prior periods; the unpredictable portion, however, could be used as a factor. (Of course, good forecasters on average may be able to predict the realizations of certain factors better than the market as a whole; if so, provided they pursue an active APT portfolio strategy, they will be rewarded by earning larger-than-average re- turns.)

Property (2) means that firm-specific events do not constitute legitimate APT factors. An investor might earn excess returns if he or she is able to identify firms with favorable firm-specif- ic events (such as the development of a profit- able new product), but this fact is not relevant for APT-based portfolio management strategies in which distinctly different types of economy- wide risks are managed in particular ways and firm-specific risks are diversified away.

Property (3) is an empirical issue that can only be answered by careful econometric work, such as that cited above.

How does an investor know if he or she has a correct set of APT factors, or if there are missing factors? First, there is no one "correct" set of factors; there are many equivalent sets of correct

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 D 30

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factors, all of which give rise to equivalent empirical results.7 Intuitively, a factor such as "unexpected change in the money supply" might work, as well as the factor "unexpected inflation"; a set of factors with "unexpected change in the money supply" substituted for "unexpected inflation" would give equivalent results. The choice of a set of "correct" factors can be made on empirical grounds: The factors should adequately explain asset returns; they should pass the statistical tests necessary to qualify as legitimate APT factors; the actual asset returns should exhibit plausible sensitiv- ities to the realizations of these factors; and the factors should have non-zero APT prices. Ex- tensive testing demonstrates that, by these cri- teria, our five factors work very well indeed. Another equivalent set of factors would do as well, but it would not do better.

Second, the worry over possible missing fac- tors is substantially resolved by using a residual market factor.8 The residual market factor is that part of the S&P 500 return not explained by the other four factors. Any missing factor is embod- ied in this residual market factor in exactly the same manner that all factors are embodied in the market return for the market model or the CAPM. Moreover, our work suggests that no important factors are missing, although the search continues.

Equations of APT APT is based on the premise that the discrepan- cy between the actual (realized) return on an asset and its expected return is equal to the sum of the quantities of different types of risk inher- ent in that asset multiplied by the realizations (actual end-of-period values) of the correspond- ing risk factors, plus an asset-specific error term. This premise is expressed by Equation (1):

ri- Eri = bil x f1 + bi2 X f2+ bi3 x f3+ b4 X f4 + bi5 x f5+ ei.

The left-hand side of Equation (1) is the discrepancy between the actual and expected return for the ith asset. Here "ri" denotes actual return and "Eri" expected return. The factor realizations for the five different types of risk are denoted by fl, f2, f3, f4 and f5. Later, we will describe how these factor realizations are mea- sured; for now, it is important to remember that, at the beginning of each period, the ex- pected value of every factor is zero. The coeffi- cients b11, bi2, bi3, bi4 and bi5 denote the quanti-

ties of the five different types of risk inherent in the ith asset. These quantities are also called sensitivity coefficients or factor loadings. Thus, for example, b1l x f1 is that part of the discrepan- cy between the actual and expected rate of return for the ith asset that is due to a non-zero realization for the first risk factor, f1. Finally, ei denotes the asset-specific error term for the ith asset; it also has a beginning-of-period expected value of zero.

Equation (1) holds for every time period.9 Note that the beginning-of-period expectation of the right-hand side of Equation (1) is zero. Thus taking beginning-of-period expectations of both sides of the equation yields the logically necessary conclusion that, at the beginning of the period, investors expect the actual return at the end of the period to be Eri.

APT predicts that the expected rate of return on the ith asset equals the riskless rate of return plus the sum of the quantities of different types of risk inherent in the ith asset times their respective prices. In this study, the riskless rate of return is measured by the 30-day Treasury bill rate, which we denote by TB. The prices of the five different types of risk are denoted by P1, P2, P3, P4 and P5. The expected rate of return for the ith asset may thus be written as follows:

Eri = TB + bil x P1 + bi2x P2 + bi3 X P3 + bi4 X P4 + b5 x P5. (2)

Over time periods where the means of the factor realizations are zero, this expected rate of return equals the mean of realized returns plus the mean of asset-specific errors.

Substituting the value of Eri from Equation (2) into Equation (1) gives the full APT. Econo- metric estimation of this nonlinear model has shown that the five risk prices are highly signifi- cant, with t-statistics of 4.27, 4.76, 1.83, 2.21 and 3.21, respectively, and are thus relevant for determining expected (equilibrium) returns.10

Suppose, for example, there is a portfolio (call it portfolio "q") that has one unit of type-1 (default) risk and no other type of risk associat- ed with it. In this case we have

bql = 1 and bq2 = bq3 = bq4 = bq5 = 0.

From Equation (2) we see that the expected rate of return on this special portfolio is

Erq = TB + P1 or P1 = Erq- TB.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 D 31

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This last relation holds generally: The price of any particular risk is equal to the expected excess rate of return on a special portfolio that has one unit of exposure to this risk and zero exposures to all other types of risk.

Practitioners could develop numerous invest- ment strategies using the expected rates of return predicted by APT. There are, however, simpler but important APT-based investment strategies that involve only the management of different types of relevant economy-wide risks in well-diversified portfolios where firm-specific influences on returns have been diversified away. No matter what valuation approach an investor employs to pick individual stocks, these picks will exhibit our five different types of risk. If a portfolio manager ignores these risks, erratic return performance will result in times of large factor realizations.

Measurement of Factors The first two risk factor measures are con-

structed from time series of returns on portfo- lios of corporate bonds, government bonds and Treasury bills. These total monthly returns are denoted by CB, GB and TB, respectively.11 The return on Treasury bills, TB, is especially impor- tant. We take TB as the measure of the riskless rate of return because it is known at the begin- ning of each month and is free of default risk.12

The first risk factor measures any unusual spread between the total monthly returns on government and corporate bonds:

f,= GB - CB + C,

where C is a constant chosen to make the beginning-of-month expected value of f, equal to zero.13 This factor measure reflects a default premium; the government and corporate bond portfolios both have 20-year maturities, but government bonds are essentially free of default risk. When fl is positive, the spread between government and corporate bond returns ex- ceeds its long-run average.

The second risk factor is the spread between the total monthly returns on government bonds and Treasury bills:

f2= GB - TB1,

where TB1 is the return on Treasury bills for the next month, which is first learned at the end of the current month. This second factor measures the slope of a total returns curve; it is related to changes in the term structure of interest rates,

because GB is the return on a 20-year portfolio while TB1 is the return on a 30-day portfolio.

The third and fourth risk factors are con- structed from the GNP accounts. The third risk factor is unexpected deflation, measured as fol- lows:

f3 -the rate of inflation expected at the beginning of the month minus the actual rate of inflation realized at the end of the month. 14

The fourth factor uses the growth rate in real final sales as a proxy for the long-run growth rate in profits for the economy. First, a series for monthly real final sales (excluding services) is obtained through detailed calculations based on monthly and quarterly data available in the GNP accounts. Then the expected growth rate in real final sales is calculated from its lagged values and from real disposable income. Finally, for every month the fourth factor is defined as follows:

f4= the long-run growth rate in real final sales (profits) expected at the beginning of the month minus the long-run growth rate in real final sales (profits) expected at the end of that month.

A positive realization for the factor f4 thus means that investors revised downward their expectation for the long-run growth rate of real final sales and profits.

The fifth and final factor is the residual mar- ket factor, defined as follows:

f5= that part of the S&P 500 return not explained by f1, f2, f3 and f4.

By definition, stock returns are positively corre- lated with the realization of f5.

Extensive statistical tests reveal that none of these five monthly factors (f1, f2, f3, f4 or f5) can be predicted either from past values or from any other publicly available information.15 Howev- er, we would not claim unpredictability for all these factors if they are measured on a daily or weekly basis. And, of course, it is possible that some of these factors can be forecast on a monthly basis using private information.

Quantities of Different Types of Risk in the S&P 500 In order to estimate the quantities of the first

four different types of risk inherent in the excess return on the S&P 500 (denoted by rm - TB), we estimated an ordinary-least-squares regression

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 D 32

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for the following equation:

rm - TB = a + bm1 x f1 + bm2 x f2 + bm3 X f3 +

bm4 x f4 + u, (3)

where u is an error term. The following results were obtained for a monthly sample beginning in January 1972 and ending in December 1982:

rm- TB = 0.0022 - 1.33 x f1 + 0.56 x f2 +

(0.62) (-3.94) (4.96) 2.29 x f3 - 0.93 x f4. (4) (1.99) (-2.27)

Thus, for example, the quantity of type-2, or term-structure, risk inherent in the market as measured by the S&P 500 is estimated to be:

bn,2= 0.56,

which indicates that a realization for f2 of 1 per cent per month, when all other factor realiza- tions are zero (f1 = f3 = f4 = 0), will raise the excess rate of return on the S&P 500 by 0.56 per cent per month. Analogously, the other quanti-

ties of different types of risk in the S&P 500 are estimated to be:

bin1 = -1. 33, bin3= 2.29, bn4 = -0.93.

The numbers in parentheses below Equation (4) are t-statistics and are used to calculate the probability that any particular type of risk factor is not significant for explaining the S&P 500 return.16 These probabilities are 0.0001, 0.0001, 0.049 and 0.025, respectively, for the first four types of risk; all are within the standard statisti- cal significance level of 0.05. Thus our first four risk factors have significant explanatory power, and together they account for approximately one-quarter of the variation in the S&P 500 return.

These estimated coefficients, which measure the quantities of the four different types of risk, also help to predict the effect of various scenari- os on the excess return for the S&P 500. For example, what happens to the excess return on the S&P 500-that is, the return on the S&P 500

Table I Quantities of Different Types of Risk for Seven Economic Sectorsa

Sector Type-i Risk Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name (default) (term (inflation or (unexpected (residual (adjusted (Durbin-

structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Cyclical -1.63 0.55 2.84 -1.04 1.14 0.77 1.67

(-6.93)b (6.97) (3.55) (-3.64) (18.47) [4]c [6] [4] [2] [3]

Growth -2.08 0.58 3.16 -0.92 1.28 0.84 1.94 (-9.80) (8.21) (4.38) (-3.57) (23.05)

[2] [4] [3] [3] [2] Stable -1.40 0.68 2.31 -0.22 0.74 0.73 1.81

(-7.09) (10.25) (3.43) (-0.93) (14.20) [5] [3] [5] [7] [6]

Oil -0.63 0.31 2.19 -0.83 1.14 0.50 1.79 (-1.62) (2.42) (1.65) (-1.75) (11.12)

[7] [7] [6] [4] [4] Utility -1.06 0.72 1.54 0.23 0.62 0.67 1.84

(-4.93) (10.02) (2.11) (0.87) (11.03) [6] [2] [7] [6] [7]

Transportation -2.07 0.58 4.45 -1.13 1.37 0.66 2.01 (-5.65) (4.75) (3.57) (-2.53) (14.24)

[3] [4] [1] [1] [1] Financials -2.48 1.00 3.20 -0.56 0.99 0.72 1.85

(-8.44) (10.21) (3.21) (-1.57) (12.86) [1] [1] [2] [5] [5]

a. An equally weighted portfolio was formed for the stocks in each sector, then sector total monthly returns were calculated from the data tapes produced by the Center for Research in Security Prices at The University of Chicago. The growth, cyclical, stable and oil sector definitions were taken from research performed by Farrell in which specific industry effects were identified; the utility, transportation and financial sectors were taken from the Standard & Poor's 1982 manual. The growth sector is defined to be "represented by companies expected to show an above-average rate of secular expansion." Cyclical stocks are defined as those with an above-average exposure to the vagaries of the economy. Stable firms are those whose earnings power is less affected than the average in the economy. (See. JL. Farrell, Jr., "Homogeneous Stock Groupings," Financial Analysts Journal, May/June 1975.)

b. Numbers in parentheses are t-statistics. c. The numbers in square brackets report the rank order of the (absolute value of) the corresponding type of risk exposure across the seven

economic sectors. Thus, for example, financials ranked 1st in exposure to type-1 and type-2 risks.

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 C 33

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Figure A Risk Exposures for Economic Sectors, 1972-1983

4.50

4.00 -

3.50 -

3.00-

2.50-

2.00-

Eh ^bA~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1.50

0.50-

0.00 -I Cyc. Grow. Stable Oil Utility Trans. Fin. S&P

E Inflation Default LII Market U Profits [ Term Structure

in excess of the return on Treasury bills mea- sured by TB-if there is an unexpected inflation of 0.5 per cent per month (6.17 per cent at a compound annual rate) and nothing else changes? To answer this question, we calculate the following:

bm3 X f3 = (2.29) x (-0.005) = -0.0115.

We thus predict that, in this scenario, the excess return on the S&P 500 would fall by 1.15 per cent per month (14.6 per cent at a com- pound annual rate), assuminmg that the realiza- tions of the other three factors (fl, f2 and f4) are all zero. It is important to remember that, in this example, the inflation of 0.5 per cent per month must be unexpected; inflation that is fully antici- pated is presumed to be already capitalized into the beginning-of-month returns.

The influence of the other risk factors on realized returns can be calculated in a similar way, as can the influence of more complicated combinations.

Risks by Sectors and Industries If all portfolios had the same risk profile, there would be no benefit to using a portfolio man- agement strategy that takes into account differ- ent types of risk. In fact, however, different stocks offer very different profiles of risk expo- sure. For example, unexpected inflation would have only a minor effect on profits (hence on returns) in those industries where it is easy to pass on inflationary costs in the form of higher product prices. In other industries, however, unexpected inflation could have devastating ef- fects on profits.

Table I reports the quantities of the five differ- ent types of risk exposure for seven sectors of the economy, as well as their rank orderings. Figure A illustrates how risk profiles differ across the seven economic sectors and provides the risk profile of the S&P 500 for comparison. These risks were calculated by running ordi- nary-least-squares regressions analogous to Equation (4), where the left-hand-side variables

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 C] 34

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were returns on equally weighted portfolios representing each sector. (The sample period was again January 1972 to December 1982.)

Wide differences in the risk exposures are evident. For example, the financial, growth and transportation sectors are especially sensitive to default (type-1) risk. Why? Because firms in these sectors are highly levered, any unantici- pated widening of the spread between govern- ment and corporate bond returns increases these firms' appropriate discount rate.

We also see that the utility sector is relatively insensitive to both type-3 (unexpected inflation) and type-4 (unexpected change in growth rate of profits) risks. Utilities can pass through their cost increases more readily than firms in other sectors because utility prices are highly regulat- ed to keep profits a constant share of the capital base. Regulation cushions the effects on utilities of unexpected inflation, and from Figure A and Table I we see that the utility sector has the lowest exposure to type-3 risk. Similarly, be- cause profits are regulated, the utility sector has no risk exposure to unexpected changes in the growth rate of profits; for the utility sector, the estimated quantity of type-4 risk is not signifi- cantly different from zero.

At the other extreme, the cyclical, growth and transportation sectors exhibit significant sensi-

tivities to both type-3 and type-4 risks. We might expect growth companies to be hurt by unexpected decreases in profit growth. These firms tend to use sophisticated capital-intensive technology and to grow by making highly le- vered investments in plant and equipment; un- expected declines in profit growth postpone the day when these investments produce positive cash flows. Moreover, growth companies tend to lack diversification. The sensitivity of cyclical and transportation sectors to unexpected changes in general prosperity is well known, and is reflected in the size (and the statistical significance) of our estimated quantities of type- 3 and type-4 risks for these sectors.

Many other interesting patterns of risk expo- sure can be picked out from Table I. Suffice it to say that our APT risk profiles for these sectors correspond closely to intuition regarding the distribution of different types of risk in the economy. Our work quantifies this intuition.

Industry Differences Table II reports the risk exposures for 82

different industry classifications, as well as the rank orderings of these risk exposures. As re- quired by the sector definitions, the risk sensi- tivities of the industries in Table II cluster around their respective sector sensitivities in

Table 11 Quantities of Different Types of Risk for 82 Industries

Industry Type-I Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name Risk (term (inflation or (unexpected (residual (adjusted (Durbin-

(default) structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Aerospace -0.75 0.41 2.31 -1.18 1.32 0.49 2.04

(-1.63)a (2.65) (1.47) (-2.10) (10.90) [73]b [73] [61] [24] [23]

Auto, OEM -1.56 0.40 4.76 -0.59 1.16 0.57 1.71 (-4.22) (3.23) (3.78) (-1.31) (11.91)

[43] [75] [21] [62] [42] Auto Replace- -1.86 0.42 4.15 -1.15 1.04 0.60 2.03

ment Parts (-5.62) (3.83) (3.70) (-2.86) (11.97) [28] [68] [26] [26] [57]

Auto, Truck Mfg. -0.43 0.52 -1.30 -1.31 1.01 0.35 1.87 (-0.89) (3.20) (-0.79) (-2.23) (7.98)

[78] [54] [81] [20] [59] Automobile -1.94 0.45 2.92 -1.47 0.92 0.24 1.88

(-3.12) (2.20) (1.38) (-1.94) (5.63) [26] [64] [45] [17] [68]

Beverages, -1.10 0.41 1.09 -0.86 0.77 0.15 2.01 Brewers (-1.77) (1.99) (0.52) (-1.14) (4.71)

[64] [70] [76] [44] [74] Beverages, -1.26 0.81 2.46 -0.54 0.95 0.48 2.12

Distillers (-3.08) (5.93) (1.77) (-1.08) (8.87) [59] [7] [57] [66] [65]

Beverages, -2.13 0.68 5.32 -0.57 1.12 0.57 1.94 Soft Drinks (-5.37) (5.13) (3.94) (-1.17) (10.73)

[16] [24] [12] [64] [45]

FINANCIAL ANALYSTS JOURNAL / MARCH-APRIL 1988 U 35

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Table II- Continued

Industry Type-1 Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name Risk (term (inflation or (unexpected (residual (adjusted (Durbin-

(default) structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Building -2.41 0.64 2.82 -0.76 1.09 0.41 2.17 Materials, (-4.62) (3.68) (1.59) (-1.19) (7.91)

Cement [9] [33] [46] [51] [51] Building -2.22 0.65 6.29 -0.88 1.57 0.52 2.00 Materials, (-3.90) (3.41) (3.25) (-1.27) (10.47)

A/C& [15] [28] [8] [42] [10] Plumbing

Building Materials, -2.47 0.99 2.52 -0.64 0.90 0.43 2.24 Roof & Wall (-4.91) (5.91) (1.47) (-1.04) (6.81)

[7] [2] [55] [60] [70] Chemicals, -1.98 0.54 3.84 -1.05 1.04 0.66 1.97

Major (-6.70) (5.43) (3.82) (-2.92) (13.44) [24] [50] [33] [32] [56]

Chemicals, -0.90 0.58 3.84 -1.05 1.04 0.66 1.97 Misc. (-2.12) (4.15) (0.84) (-2.38) (10.59)

[70] [42] [75] [22] [37] Coal, -3.04 0.98 2.07 -0.94 1.40 0.37 1.59

Bituminous (-4.10) (3.96) (0.82) (-1.04) (7.17) [2] [3] [66] [37] [17]

Conglomerates -1.51 0.58 3.71 -1.13 1.36 0.73 1.84 (-5.06) (5.86) (3.65) (-3.09) (17.30)

[46] [43] [36] [29] [20] Containers, -0.98 0.56 3.00 -0.48 0.96 0.64 1.74

Metal & (-3.60) (6.13) (3.24) (-1.45) (13.45) Glass [66] [47] [44] [67] [64]

Containers, -0.96 0.55 5.03 -1.51 1.19 0.35 2.06 Paper (-1.64) (2.82) (2.52) (-2.11) (7.70)

[68] [48] [18] [12] [34] Cosmetics -1.52 0.69 2.71 -0.36 1.12 0.70 2.19

(-5.31) (7.24) (2.79) (-1.04) (14.88) [45] [20] [50] [71] [46]

Drugs, -0.98 0.51 2.23 -0.83 0.95 0.48 2.01 Ethical (-2.72) (4.27) (1.82) (-1.89) (10.06)

[67] [55] [64] [47] [66] Drugs, -1.82 0.70 2.55 -0.45 1.16 0.64 1.79

Medical & (-5.39) (6.20) (2.21) (-1.08) (13.07) Hospital [32] [17] [54] [68] [41] Supply

Drugs, -1.48 0.68 1.59 -0.77 1.00 0.51 2.20 Proprietary (-3.87) (5.35) (1.22) (-1.65) (9.92)

[48] [23] [72] [50] [62] Eating Places -2.71 0.47 6.64 -0.69 1.36 0.59 1.62

(-6.06) (3.17) (4.37) (-1.26) (11.59) [4] [59] [6] [55] [18]

Electrical & -1.57 0.55 3.62 -1.19 1.21 0.63 2.04 Electronic, (-4.58) (4.78) (3.10) (-2.84) (13.42) Major [41] [49] [37] [23] [31]

Electrical -2.23 0.56 4.22 -0.91 1.10 0.72 1.63 Equipment (-8.09) (6.13) (4.51) (-2.71) (15.19)

[14] [45] [24] [41] [49] Electrical & -1.78 0.62 4.20 -0.58 1.08 0.58 1.63

Household (-4.86) (5.08) (3.37) (-1.29) (11.21) Appliances [33] [38] [25] [63] [52]

Electronics, -1.86 0.70 5.07 -1.47 1.60 0.72 2.09 Diversified (-5.03) (5.66) (4.04) (-3.27) (16.48)

[29] [19] [16] [16] [6] Electronics, -1.63 0.47 5.32 -1.23 1.34 0.64 1.98

Instruments (-4.41) (3.82) (4.23) (-2.73) (13.74) [40] [60] [13] [21] [22]

Electronics, -2.49 0.35 3.47 -1.17 1.63 0.58 1.81 Semiconductors (-4.98) (2.07) (2.04) (-1.93) (12.37) and [6] [78] [39] [25] [4] Computers

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Table II- Continued

Industry Type-I Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name Risk (term (inflation or (unexpected (residual (adjusted (Durbin-

(default) structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Food, -1.20 0.51 2.57 -0.24 0.68 0.37 2.13 Canned Goods (-3.28) (4.19) (2.06) (-0.53) (7.08)

[60] [56] [53] [75] [76] Food, -1.31 0.29 2.61 -0.03 0.62 0.26 2.06

Confec- (-3.18) (2.15) (1.87) (-0.05) (5.69) tionary [54] [81] [51] [77] [80]

Food, Corn & 0.04 0.75 -2.79 -0.35 1.22 0.27 2.03 Soybean (0.05) (3.18) (-1.16) (-0.40) (6.55) Refiners [82] [12] [82] [72] [28]

Food, Dairy -1.54 0.65 3.07 -0.20 0.65 0.51 1.97 Products (-4.99) (6.34) (2.92) (0.53) (7.94)

[44] [26] [42] [80] [79] Food, Meat -0.25 0.85 2.44 -2.52 1.03 0.24 2.05

Packers (-0.35) (3.51) (0.99) (-2.85) (5.44) [79] [5] [58] [2] [58]

Food, Sugar -0.94 0.67 -0.56 0.12 0.73 0.16 2.24 Refiners (-1.46) (3.10) (-0.25) (0.16) (4.28)

[69] [25] [79] [79] [75] Food, -1.30 0.69 2.76 -0.40 0.79 0.62 1.76

Packaged (-4.90) (7.79) (3.06) (-1.25) (11.38) [55] [22] [48] [69] [72]

Forest -2.28 0.78 3.54 -0.94 1.36 0.61 1.46 Products (-5.42) (5.52) (2.47) (-1.84) (12.29)

[12] [10] [38] [36] [19] Home -2.07 0.70 3.89 -0.98 1.07 0.40 1.77

Furnishings (-3.91) (3.96) (2.16) (-1.51) (7.67) [20] [16] [32] [34] [53]

Hotels & -2.13 0.76 6.62 -1.50 1.60 0.40 1.86 Motels (-2.91) (3.12) (2.67) (-1.69) (8.31)

[16] [11] [7] [141 [7] Leisure Time -1.14 0.45 3.99 -0.04 1.60 0.44 2.04

Products (-1.83) (2.16) (1.89) (0.05) (9.78) [62] [65] [31] [78] [5]

Machine -1.45 0.30 4.08 -1.85 1.23 0.34 2.05 Tools (-2.39) (1.51) (1.98) (-2.50) (7.73)

[49] [80] [29] [5] [26] Machine -2.07 0.73 3.02 -0.84 1.27 0.58 2.25

Tools, (-5.00) (5.26) (2.15) (-1.67) (11.68) Hand [21] [14] [43] [45] [25]

Machinery, -1.76 0.53 3.75 -1.08 1.09 0.61 1.79 Construction & (-5.27) (4.78) (3.31) (-2.66) (12.43) Material [35] [51] [35] [31] [50] Handling

Machinery, -1.09 0.64 2.07 -2.10 1.00 0.33 1.75 Agricultural (-1.99) (3.52) (1.10) (-3.15) (6.96)

[65] [29] [65] [4] [60] Machinery, -1.35 0.44 3.82 -1.01 1.14 0.67 1.70

Industrial (-4.63) (4.51) (3.85) (-2.84) (14.80) [53] [67] [34] [33] [44]

Machinery, -2.24 0.56 1.01 -1.47 1.29 0.57 1.71 Specialty (-5.38) (4.02) (0.71) (-2.89) (11.79)

[13] [46] [77] [15] [24] Metals, -1.56 0.24 2.30 -1.38 1.17 0.38 2.08

Aluminum (-2.93) (1.36) (1.32) (-2.15) (8.49) [42] [82] [59] [18] [38]

Metals, -0.14 0.41 2.59 -2.24 1.50 0.46 2.13 Copper (-2.44) (2.50) (1.76) (-2.08) (8.51)

[81] [69] [52] [3] [13] Metals, -1.76 0.36 3.30 -0.95 1.14 0.30 2.00

Steel (-2.84) (1.75) (1.57) (-1.25) (7.00) Products [34] [77] [41] [35] [43]

Metals & -0.81 0.64 2.23 -1.36 1.18 0.34 1.65 Mining, Misc. (-1.37) (3.27) (1.11) (-1.89) (7.62)

[71] [32] [63] [19] [35]

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Table II- Continued

Industry Type-i Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name Risk (term (inflation or (unexpected (residual (adjusted (Durbin-

(default) structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Mobile -3.38 1.05 8.60 -0.93 1.93 0.54 2.12 Home (-5.29) (4.34) (3.49) (-1.06) (10.12) Builders [1] [1] [4] [39] [1]

Motion -1.71 0.59 2.73 -0.56 1.44 0.50 1.70 Pictures (-1.31) (3.42) (1.55) (-0.89) (10.66)

[37] [40] [49] [65] [16] Natural Gas -1.48 0.71 9.73 -1.11 1.56 0.35 1.87

Transmission (-1.80) (2.59) (3.48) (-1.11) (7.23) [47] [15] [1] [30] [11]

Office & -1.94 0.59 4.08 -1.67 1.53 0.75 1.85 Business (-0.95) (3.37) (0.77) (-2.40) (9.51) Equipment [25] [41] [30] [7] [12]

Oil & Gas -0.70 0.83 1.94 -1.84 1.85 0.43 1.77 Drilling (-0.95) (3.37) (0.77) (-2.04) (9.51)

[75] [6] [69] [6] [2] Oil, Crude -0.49 0.64 1.55 -0.94 1.34 0.46 1.59

Producers (-0.96) (3.79) (0.90) (-1.52) (10.01) [77] [30] [73] [38] [21]

Oil, -0.66 0.53 -0.74 -0.81 1.17 0.42 1.55 Integrated (-1.46) (3.38) (-0.46) (-1.41) (9.44) Domestic [76] [52] [80] [48] [39]

Oil, -0.73 0.40 2.80 -0.71 0.99 0.53 2.01 Integrated (-2.20) (3.69) (2.50) (-1.75) (11.44) Int'l [74] [74] [47] [52] [63]

Paper -1.85 0.63 1.97 -1.52 1.16 0.62 1.79 (-5.36) (5.51) (1.67) (-3.60) (12.82)

[30] [35] [68] [11] [40] Photo- -2.58 0.80 1.93 -1.60 1.20 0.49 1.75

graphic (-5.19) (4.83) (1.14) (-2.63) (9.18) [5] [9] [70] [9] [32]

Pollution -1.70 0.62 8.04 -1.56 1.57 0.49 1.90 Control (-2.83) (3.11) (3.94) (-2.13) (9.99)

[38] [37] [5] [10] [9] Publishing -1.37 0.56 4.97 -0.79 1.10 0.56 1.66

(-3.64) (4.48) (3.88) (-1.72) (11.10) [52] [44] [19] [49] [48]

Publishing, -1.44 0.48 5.30 -0.65 1.18 0.54 2.03 Newspapers (-3.52) (3.52) (3.81) (-1.30) (10.97)

[50] [58] [14] [58] [36] Radio-TV -1.39 0.47 4.42 -0.63 1.23 0.52 1.83

Broadcasters (-3.14) (3.19) (2.94) (-1.16) (10.59) [51] [61] [22] [82] [27]

Railroad -1.90 0.49 1.89 -0.61 1.19 0.54 2.08 Equipment (-4.73) (3.68) (1.39) (-1.24) (11.30)

[27] [57] [71] [61] [33] Retail Dept. -1.67 0.63 5.19 -0.69 0.92 0.41 2.08

Stores (-3.62) (4.10) (3.31) (-1.23) (7.59) [39] [36] [15] [54] [67]

Retail, -2.32 0.75 3.43 -0.33 0.85 0.44 2.02 Discount (-5.24) (5.07) (2.27) (-0.61) (7.27) Stores [11] [13] [40] [73] [71]

Retail -2.12 0.81 6.17 -0.66 1.21 0.56 1.81 Drug Stores (-4.79) (5.46) (4.10) (-1.21) (10.41)

[18] [8] [9] [57] [30] Retail -2.05 0.64 2.29 -0.31 0.68 0.41 2.04

Food Chains (-5.30) (4.96) (1.74) (-0.65) (6.66) [22] [34] [62] [1] [77]

Retail, -2.79 0.59 9.68 -1.51 1.57 0.47 2.01 Misc. (-4.17) (2.66) (4.25) (-1.85) (8.95)

[3] [39] [2] [13] [8] Service -2.00 0.69 9.01 -0.84 1.45 0.62 1.93

(-4.43) (4.59) (5.87) (-1.52) (12.21) [23] [21] [3] [36] [15]

Shoes -2.39 0.65 2.35 -0.70 0.91 0.22 2.08 (-3.37) (2.77) (0.98) (-0.81) (4.89)

[10] [27] [60] [53] [69]

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Table II- Continued

Industry Type-i Type-2 Risk Type-3 Risk Type-4 Risk Type-5 Risk R2 DW Name Risk (term (inflation or (unexpected (residual (adjusted (Durbin-

(default) structure) deflation) change in market R-squared) Watson growth rate risk) statistic) of profits)

Soaps -1.29 0.70 4.93 -0.88 0.78 0.60 1.89 (-4.49) (7.30) (5.05) (-2.51) (10.36)

[56] [18] [20] [43] [73] Telephone -0.78 0.38 1.28 0.36 0.35 0.30 2.06

(-2.95) (4.37) (1.42) (1.11) (5.07) [72] [76] [74] [81] [81]

Textile -1.74 0.45 5.80 -0.68 1.22 0.48 1.94 Apparel Mfg. (-3.66) (2.82) (3.60) (-1.17) (9.77)

[36] [66] [10] [56] [29] Textile -1.14 0.46 5.03 -0.65 1.00 0.46 2.08

Products (-2.78) (3.39) (3.61) (-1.29) (9.30) [63] [62] [17] [59] [61]

Tire & -1.85 0.46 2.47 -0.92 1.11 0.49 1.79 Rubber Goods (-4.44) (3.32) (1.74) (-1.80) (10.16)

[31] [63] [56] [40] [47] Tobacco -1.16 0.53 2.03 -0.30 0.67 0.51 2.10

(-4.25) (5.83) (2.20) (-0.91) (9.35) [61] [53] [67] [74] [78]

Toys -2.42 0.91 5.65 -1.65 1.80 0.33 2.11 (-2.58) (2.91) (1.77) (-1.45, (7.32)

[8] [4] [11] [8] [3] Wholesalers -2.09 0.64 4.24 -0.12 1.48 0.39 2.01

(-3.06) (2.81) (1.82) (-0.15) (8.20) [19] [31] [23] [76] [14]

a. Numbers in parentheses are t-statistics. b. The numbers in square brackets report the rank ordering of (the absolute value of) the corresponding type of risk exposure. Thus, for

example, the aerospace industry ranked 73rd out of 82 industries with respect to its exposure to type-1 and type-2 risks. At press time, three of the original 82 industries were dropped because the data were found to be problematic; this does not affect the reported rank orderings.

Table I. The analysis of risk exposure by indus- try, however, provides further insight into the reasons behind the factor sensitivities we ob- serve. For instance, the mobile home building industry is unique in that it ranks first in sensi- tivity to default risk (type-1), first in sensitivity to term-structure risk (type-2), fourth in sensi- tivity to unexpected inflation risk (type-3), and first in sensitivity to residual market risk (type- 5). The reasons this industry performed so poorly during the time period under consider- ation, when unexpected inflation and other unfavorable shocks often predominated, are ev- ident.

The industries that are most sensitive to unex- pected inflation risk include retailers, services, eating places, hotels and motels, drug stores, toys, and textile apparel manufacturers. For the most part, their products tend to be "luxuries," and the demand for "luxuries" plummets when consumer real incomes fall. These industries are thus not well insulated from unexpected drops in real income due to unexpected inflation. In contrast, the industries least sensitive to unex-

pected inflation tend to sell "necessities," the demands for which are relatively insensitive to declines in real income. These industries in- clude foods, cosmetics, tire and rubber goods, shoes, tobacco and breweries. Several indus- tries appear to exhibit no significant sensitivity to unexpected inflation risk. For instance, both corn and soybean refiners and sugar refiners have negative but insignificant sensitivity to unexpected deflation (f3). This is important, because the active and passive portfolio man- agement techniques we discuss below depend upon selection of assets with low or opposite sensitivities to the risk factors the portfolio manager wishes to control.

In sum, exposure to different types of risk varies considerably both across economic sec- tors and across industries.17 Are there some simple strategies for effectively managing these different types of risk?

A Risk-Sterilization Strategy For simplicity, assume that a portfolio manager has determined that he or she wishes to have

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exposure to only one type of risk-say, residual market risk. This objective can be attained by forming a portfolio of short and long positions such that the overall portfolio has zero quanti- ties of risk exposure to the first four risk factors. If short sales are precluded or limited in prac- tice, this objective may not be attainable.

A strategy that is always feasible, however, is to form a portfolio that has exposures to the first four different types of risk that are exactly proportional to the corresponding exposures for any particular market index the manager may select (say, the S&P 500) while selecting a de- sired exposure to residual market risk. (If this exposure to residual market risk is 1.0, the strategy is equivalent to holding a portfolio that mimics the S&P 500; otherwise it is not.) We call such a strategy risk sterilization.18

Briefly, if we multiply bi5 times Equation (3) and subtract the result from Equation (1), we find, after rearranging terms, that:

ri- Eri = ai + cil x fl + ci2 x f2 + cD X f3

+ c4 x f4+ bi5 x (rm - TB) + ei, (5)

where

Ci = bil - bi5 X bml,

c2 = bi - bi5 x b2,

and so on, and where ai is a constant. If

Cil= c = C3 = Ci4 = 0,

then Equation (5) simplifies to:

ri - Eri =ai + bi5 x (rm - TB) + ei.

That is, when all the ciis, as defined above, are zero, any discrepancy between the actual and expected rate of return for the ith asset is explained by only the excess return on the market, a constant and an asset-specific error term. For a large portfolio, this asset-specific error is diversified away.

Consider, then, a large diversified portfolio consisting only of stocks for which ci1 = ci2 = CD = ci4 = 0. Such a portfolio has a risk profile for the first four factors that is exactly proportional to the risk profile for the S&P 500. The same result can be obtained for any market index or any well-diversified portfolio the manager may select. Furthermore, by selecting appropriate weights for the stocks in this portfolio, the manager can achieve any desired exposure to the remaining type of risk-residual market risk, as measured by the S&P 500 in this exam-

ple. Further generalizations will suggest them-

selves to the reader. For example, a manager may design portfolio strategies to obtain various exposures to one particular type of risk or to particular combinations of different types of risk. Similarly, a pension fund manager can sterilize a portfolio so that its exposure to unex- pected inflation or deflation risk is exactly the same as the exposure of the S&P 500 to this risk, while at the same time selecting desired profiles of exposures to the remaining risks.

An Active APT Strategy An active APT investment strategy entails

being able to forecast the factor realizations accurately, at least on average. Suppose a man- ager believes that he or she can forecast (or can purchase a forecast of) unexpected inflation for the next month. Remember that the manager must accurately forecast unexpected inflation, not actual inflation. Thus, for example, a portfolio manager might forecast that the market as a whole expects inflation to be 4 per cent annual- ly, while it will actually be 6 per cent annually. In this case, the manager must forecast a realiza- tion for unexpected inflation of 2 per cent.

In fact, the portfolio manager "only" has to forecast correctly the signs of the factor realiza- tions. To illustrate this point, let's presume that a manager can accurately forecast the sign of unexpected inflation (which is the opposite sign of the realization for f3). Using the methods described above, he or she first selects a subset of stocks for which the exposures to the remain- ing types of risks (type-1, type-2 and type-4) are exactly proportional to the exposures of, say, the S&P 500 (so that ci1 = cC = cC4 0 O). Then, to bet on unexpected deflation in the next month-a positive realization of the factor f3- the manager chooses from the above subset of stocks a portfolio that has (1) the largest quanti- ties of type-3 risk (the largest bi3s) and (2) an overall portfolio exposure to market risk near 1.0. To bet on unexpected inflation next month-a negative realization of the factor f3- the manager follows exactly the same proce- dure, but selects stocks with the smallest quan- tities of type-3 risk (the smallest bi3s). If the forecasts are correct on average, this portfolio will outperform the S&P 500.

With accurate forecasts of unexpected infla- tion or deflation, the manager can outperform any index selected. The extent of the superior

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performance will be proportional to the size of the factor realization whose sign the manager forecast correctly, but which the market as a whole predicted to be zero.

Implications The use of APT as an investment management tool is in its infancy. But a reliable database for asset returns and risk factor realizations, plus the availability of computer power that was unthinkable even 10 years ago, means that practical APT-based strategies can now be im- plemented at low cost. We have described two simple portfolio management strategies-a pas- sive strategy that sterilizes portfolios from ex- cessive exposure to selected types of risk and an

active one in which a portfolio manager makes bets based on forecasts of risk factor realiza- tions. Other strategies, too numerous to dis- cuss, suggest themselves; for example, the APT methodology described here is also applicable to the risk management of fixed-income securi- ties.19 Moreover, once a set of relevant APT factors is known, portfolio managers need not invoke the full APT machinery to pursue these risk management strategies.

APT provides effective means for managing the different types of risk to which investors are exposed. Its use in the investment community is certain to increase as managers become more familiar with the new strategic investment op- portunities it offers.20 U

Footnotes

1. APT originated with the theoretical work of S.A. Ross, "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic Theory, December 1976. Less difficult expositions can be found in E.J. Elton and M.J Gruber, Modern Portfolio Theory and Investment Analysis, 2nd ed. (New York: John Wiley & Sons, 1984); T.E. Copeland and J.F. Weston, Financial Theory and Corporate Policy, 2nd ed. (Reading, MA: Addison-Wesley Publishing Company, 1984); B.G. Malkiel, A Random Walk Down Wall Street, 4th ed. (New York: W.W. Norton & Company, 1985); and R. Roll and S.A. Ross, "The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning," Financial Analysts Journal, MaylJune 1984.

2. The technical results reviewed in this article are contained in the following papers: E. Burmeister and K.D. Wall, "The Arbitrage Pricing Theory and Macroeconomic Factor Measures," The Finan- cial Review, February 1986; Burmeister, Wall and J.D. Hamilton, "Estimation of Unobserved Ex- pected Monthly Inflation Using Kalman Filter- ing," Journal of Business and Economic Statistics, April 1986; and M. McElroy and E. Burmeister, "Arbitrage Pricing Theory as a Restricted Nonlin- ear Multiple Regression Model: ITNLSUR Esti- mates," Journal of Business and Economic Statistics, January 1988. Questions, including requests for copies of these papers, should be directed to Edwin Burmeister, Department of Economics, 114 Rouss Hall, University of Virginia, Char- lottesville, VA 22901; telephone 804/924-3177.

3. The first papers to identify APT risk factors with plausible economic variables were Burmeister and Wall, "The Arbitrage Pricing Theory and Macroeconomic Factor Measures," op. cit; N.-F. Chen, Roll and Ross, "Economic Forces and the Stock Market," Journal of Business, July 1986; and

K.C. Chan, Chen and D.A. Hsieh, "An Explor- atory Investigation of the Firm Size Effect," jour- nal of Financial Economics, September 1985.

4. The price associated with an APT factor may be negative if investors want, perhaps for hedging purposes, to hold stocks whose returns increase when there is an unanticipated positive realiza- tion of that factor (and whose returns decrease when there is an unanticipated negative realiza- tion). This negative price reflects an attribute that investors find desirable. Similarly, the "quantity" of a particular type of risk inherent in an asset is negative if the return on that asset decreases (increases) when there is a positive (negative) realization of the corresponding risk factor. These technical issues are discussed in Burmeister and McElroy, "Joint Estimation of Factor Sensitivities and Risk Premia for the Arbitrage Pricing The- ory" (Paper prepared for the American Finance Association Meeting, Chicago, December 1987 and forthcoming in the Journal of Finance) and Burmeister and McElroy, "APT and Multifactor Asset Pricing Models with Measured and Unob- served Factors: Theoretical and Econometric Is- sues" (Paper prepared for the Southern Finance Association Meeting, Washington, D.C., Novem- ber 1987).

5. See McElroy and Burmeister, "Arbitrage Pricing Theory as a Restricted Nonlinear Multiple Re- gression Model," op. cit.

6. Ibid. 7. For more discussion of this, see McElroy and

Burmeister, "Arbitrage Pricing Theory as a Re- stricted Nonlinear Multiple Regression Model," op. cit. and especially Burmeister and McElroy, "Joint Estimation of Factor Sensitivities and Risk Premia," op. cit. We use the term "equivalent" to mean that the empirical results obtained using

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one set of factors can be derived exactly from an equivalent set of factors. Thus equivalent sets contain exactly the same information.

8. See Burmeister and Wall, "The Arbitrage Pricing Theory and Macroeconomic Factor Measures," op. cit. and McElroy and Burmeister, "Arbitrage Pricing Theory as a Restricted Nonlinear Multiple Regression Model," op. cit.

9. We use one month as the time period in this study. One month is the shortest time period for which it is possible to measure some of our factors. In addition, monthly data are free from many troubling anomalies present in daily or weekly data (such as "holiday effects," autocor- related returns, etc.).

10. See McElroy and Burmeister, "Arbitrage Pricing Theory as a Restricted Nonlinear Multiple Re- gression Model," op. cit. and Burmeister and McElroy, "Joint Estimation of Factor Sensitivities and Risk Premia," op. cit.

11. Data were obtained from the series constructed by Ibbotson Associates, Inc. For details, see R.G. Ibbotson and R.A. Sinquefield, Stocks, Bonds, Bills, and Inflation: The Past and the Future (Char- lottesville, VA: Financial Analysts Research Foundation, 1982).

12. The Ibbotson T-bill series serves this purpose well, because it is the one-month holding period return for a one-bill portfolio that is the shortest bill not less than 30 days in maturity.

13. The constant C was 0.5 per cent at a compound annual rate. It is crucial to remember that GB and CB measure total monthly returns, not yields to maturity.

14. The expected inflation series was estimated using the Kalman filtering methods of Burmeister, Wall and Hamilton ("Estimation of Unobserved Ex-

pected Monthly Inflation Using Kalman Filter- ing," op. cit.).

15. Some of these tests are reported in Burmeister, Wall and Hamilton, "Estimation of Unobserved Expected Monthly Inflation Using Kalman Filter- ing," op. cit.

16. The other standard ordinary-least-squares sum- mary statistics: R2 = 0.24; DW = 2.13 (rho = -0.064); and F = 10.1 with a probability value of 0.0001.

17. An analysis of mutual funds similar to the analy- sis of sectors and industries discussed here is presented in M.A. Berry, Burmeister and McEl- roy, "A Practical Perspective of Mutual Fund Risks: 1974-1982," Investment Management Review, March-April 1988.

18. The key to implementing risk sterilization is contained in McElroy and Burmeister, "Arbitrage Pricing Theory as a Restricted Nonlinear Multiple Regression Model," op. cit. This strategy can also be used to construct portfolios that track any diversified index; it could therefore be used in connection with index futures contracts.

19. An excellent non-technical introduction to the use of APT-based risk-management techniques for bond portfolios is contained in Ross, "Modify- ing Risk and Return in Managing Bond Portfo- lios," in The Revolution in Techniques for Managing Bond Portfolios (Charlottesville, VA: The Institute of Chartered Financial Analysts, 1983).

20. Personal computer software currently under de- velopment and testing will enable portfolio man- agers to achieve alternative investment objectives by implementing the strategies suggested here, as well as a variety of more sophisticated APT- based risk exposure optimization techniques.

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