Operating Leverage�

Robert Novy-Marx|

University of Chicago and NBER

This Draft: May 18, 2007

Abstract

This paper presents the first direct empirical evidence for the “operating leveragehypothesis,” which underlies most theoretical models of the value premium. We alsoshow that tests that rely on direct inference using the level of operating leverage arebiased against rejecting alternative hypotheses, due both to empirical issues associatedwith measuring operating leverage and theoretical issues associated with equilibriumbehavior. We consequently generate indirect implications of the hypotheses, whichdo not require direct observation of operating leverage and account for equilibriumeffects. Specifically, the model predicts that the relationship between expected re-turns and book-to-market is weak and non-monotonic across industries, but strongand monotonic within industries. The data support these predictions. The model alsosuggests a measure of book-to-market that controls for industry. HML constructedusing this measure over the sample from June 1973 to January 2007 has a Sharpe ratiothat exceeds the ex post optimal combination of the three Fama-French factors, and aninformation ratio relative to these factors that exceeds that of momentum.

Keywords:Real options, value premium, factor models, asset pricing.

JEL Classification:G12, E22.

�I would like to thank John Cochrane, Stuart Currey, Peter DeMarzo, Andrea Frazzini, Toby Moskowitz,Milena Novy-Marx, Josh Rauh, Amir Sufi, and Luke Taylor for discussions and comments. Financial sup-port from the Center for the Research in Securities Prices at the University of Chicago Graduate School ofBusiness is gratefully acknowledged.

|University of Chicago Graduate School of Business, 5807 S Woodlawn Avenue, Chicago, IL 60637.Email: rnm@ChicagoGSB.edu.

1 Introduction

Theoretical models that generate a value premium generally rely on the “operating leverage

hypothesis,” introduced to the real options literature by Carlson, Fisher and Giammarino

(2004). This hypothesis, expounded in some form at least as early as Lev (1974), states that

variable (i.e.,flow) production costs play much the same role as debt servicing in levering

the exposure of a firms’ assets to underlying economic risks. Operating leverage is critical

to models that generate a value premium, because absent operating leverage growth options

are riskier than deployed capital.

While the operating leverage plays a critical role in these theories, there exists little

supporting empirical evidence.1 We provide direct empirical evidence for the operating

leverage hypothesis, and show that this evidence is consistent with the hypothesis’ expla-

nation of the value premium. We also show that the relative paucity of empirical support

should be expected, both on practical and theoretical grounds. Practically, operating lever-

age is largely unobservable, depending not, as is often assumed, on the level of a firm’s

costs and revenues (observable), but on the capitalized values of all future costs and rev-

enues (unobservable). Theoretically, simple predictions of the theory derived in partial

equilibrium models are not robust to equilibrium analysis. We consequently derive addi-

tional, indirect implications of the theory, which do not require direct observation of the

level of operating leverage and account for equilibrium effects. The data strongly support

these predictions, providing further evidence for the operating leverage hypothesis and en-

hancing our understanding of factors that explain variation in stock returns across firms.

In our direct tests, we show that firms with “levered” assets earn significantly higher

average returns than firms with “unlevered” assets, where these characterizations refer to

the level of operating (not financial) leverage. The return difference between levered and

1 Sagi and Seasholes (2007) and Gourio (2005) both provide indirect evidence for the importance of oper-ating leverage. Sagi and Seasholes show theoretically that operating leverage reduces asset return autocorre-lation, and identify firm-specific attributes that improve the empirical performance of momentum strategies.Gourio presents evidence that operating income is more sensitive to gross domestic product shocks for valuefirms than for growth firms.

1

unlevered firms is explained by Fama and French’s (1993) three factor model. The three

factor model, and HML in particular, do a good job pricing portfolios sorted on operating

leverage.

This is consistent with a growth options / operating leverage explanation of the value

premium. In operating leverage models, value firms generate higher average returns be-

cause they consist mainly of assets-in-place, which are riskier than growth options. Firms

that consist disproportionately of assets-in-place load positively on HML, which covaries

with the return spread between deployed capital and growth options, while firms that con-

sist disproportionately of growth options load negatively on HML. HML loadings conse-

quently help explain differences in average returns between value firms and growth firms.

If HML covaries with return differences between assets-in-place and growth options,

however, then firms with particularly risky assets-in-place should also load more heavily on

HML, even after controlling for book-to-market. HML loadings should therefore positively

correlate with operating leverage, which increases the riskiness of deployed capital. We

observe each of these effects in the data.

Moreover, the three factor model prices the operating leverage portfolios well, despite

the fact that the operating leverage sort doesnotgenerate any systematic variation in book-

to-market. This seems to contradict Daniel and Titman’s (1997) contention that HML has

no power explaining returns after controlling for book-to-market as a characteristic.

While we find direct evidence for the operating leverage hypothesis, inference based

on the observed relationship between expected returns and operating leverage likely under-

states the role operating leverage plays in generating the cross section of expected returns.

While higher variable costs result in effectively more levered assets, they should also be as-

sociated with more flexibility on the cost side. When capital costs are small relative to flow

costs associated with productions, firms should be more willing to shut down unprofitable

production, even if it entails the loss of capital.

Furthermore, direct implications,i.e., those which rely directly on measures of operat-

ing leverage, are plagued by measurement issues. The level of operating leverage depends,

2

theoretically, on thecapitalizedvalues of all future costs and revenues. While market val-

ues provide a good proxy for their difference, finding proxies for the two individually is

problematic.

These facts lead us to consider alternative, indirect implications of operating lever-

age. Using an equilibrium model, based on the dynamic oligopoly model of Novy-Marx

(2007a), we derive implications of operating leverage that should be observable even if op-

erating leverage is itself unobservable.2 In particular, the model predicts that the relation-

ship between expected returns and industry book-to-market is weak and non-monotonic,

but that the relationship between expected returns and book-to-marketwithin industries

is strong and monotonic. This provides a theoretical basis for Cohen and Polk’s (1998)

contention that the value premium is largely an intra-industry phenomena.

Empirical investigation conducted here largely supports this prediction. Sorting firms

on the basis of intra-industry book-to-market generates significant variation in returns,

which is explained by the three-factor model. Sorting firms on the basis of industry book-

to-market, however, generates no significant variation in returns, despite generating sig-

nificant variation in book-to-market and more variation in HML loadings than the intra-

industry sort. The three-factor model consequently severely misprices the inter-industry

value-growth spread.

These results hold across both industry and intra-industry book-to-market quintiles.

Value firms in value industries earn significantly higher returns than growth firms in value

industries, and value firms in growth industries earn significantly higher returns than growth

firms in growth industries. The converse is false. The returns to value (respectively,

growth) firms in value industries are indistinguishable from the returns to value (respec-

tively, growth) firms in growth industries, despite large differences in these firms’ book-to-

market ratios and HML loadings.

2 Other equilibrium models employing operating leverage include Zhang (2005) and Aguerrevere (2006).Zhang focuses on the role costly reversibility plays in generating a value premium, without explicitly link-ing it to the operating leverage mechanism. Aguerrevere (2006) argues that operating leverage model andequilibrium effects together imply that competitive industries should be riskier than concentrated industriesin recessions, but less risky in expansions.

3

This investigation also supports another prediction of the model, regarding the inter-

action of value and size effects. Value firms are systematically smaller than growth firms

in the same industry, both in the data and the model. Firms in value industries, however,

are roughly the same size as firms in growth industries. It appears that the book-to-market

effect is concentrated in small firms at least partly because small size helps distinguish be-

tween inefficient value firms more exposed to priced risks, and more efficient, less exposed

firms in value industries.

A simple, alternative sorting methodology suggested by the model, which controls for

cross-industry variation in book-to-market, yields higher value-growth spreads than sorting

on book-to-market directly, despite generating much less variation in the book-to-market

characteristic. Using this sorting procedure, and following the methodology that Fama and

French (1993) use to construct HML, we construct an alternative value factor. This factor

prices HML, but carries a significantly positive three-factor alpha. The realized Sharpe

ratio on the factor exceeds that on the ex post tangency portfolio of the three Fama-French

factors over the June 1973 to January 2007 sample period. Its information ratio relative to

the Fama-French three over that same period exceeds that of momentum.

Following the same methodology, we also construct two additional factors. The first is

based on the operating leverage measure employed in our direct test of the operating lever-

age hypothesis. The second, which relies on the work of Sagi and Seasholes (2007) relating

return auto-correlation to firm-specific attributes in a real options framework, is an industry

relative momentum factor, similar to but distinct from that employed by Asness, Porter and

Stevens (2000). Both these factors also add significantly to the investment opportunity set.

Finally, while the model employed in this paper contains only a single risk factor, we

show that HML nevertheless arises naturally as a distinct, second factor (or “pseudo fac-

tor”) that, in conjunction with the market, helps explain the cross-section of returns. This

factor can be uncorrelated with the market, in which case we see very little dispersion in

market betas across firms. A high unconditional HML Sharpe ratio requires a counter-

cyclical price of risk.

4

The remainder of the paper is organized as follows. Section 2 discusses the basic intu-

ition behind the operating leverage hypothesis in a model-free context, derives some basic

predictions of the hypothesis, and provides empirical support for these predictions. Sec-

tions 3 and 4 present the formal, dynamic model of operating leverage, and discuss firms’

equilibrium behavior. Sections 5 and 6 relate firm values and expected returns to funda-

mental firm characteristics. Section 7 derives and tests an indirect implication of the model,

that the relationship between expected returns and book-to-market should be weak and

non-monotonic across industries, but strong and monotonic within industries. In section 8

we construct an alternative value factor, and consider the efficiency (or near-efficiency) of

HML. Section 9 takes an asset pricing perspective, considering how the inclusion of our

alternative value factor, as well as an operating leverage factor and an industry-relative mo-

mentum factor, alter the investment opportunity set. Section 10 discusses the role of HML

in a one-factor model. Section 11 concludes.

2 Basic Predictions

As with any real options model, a firm’s value consists of two pieces: the value of currently

deployed asset and growth options,V i D V iA

C V iG

where i denotes the firm and the

subscriptA and G signify assets-in-place and growth options, respectively. The firm’s

expected excess returns depend on its exposure to the underlying risk factors. This exposure

is a value weighted sum of the loadings of the firm’s assets-in-place and the firm’s growth

options on these risks,i.e.,

ˇi D�

V iA

V i

�ˇi

A C�

V iG

V i

�ˇi

G . (1)

Just as the value of equity equals the value of assets minus the value of debt, the value

of deployed assets consists of the capitalized value of the revenues they generate minus

the capitalized cost of operating the assets,V iA

D V iR � V i

C. The exposure of the assets to

the underlying risks is then a value weighted average of the exposures of the capitalized

5

revenues and the capitalized operating costs,

ˇiA D ˇi

R C�

V iC

V iA

��ˇi

R � ˇiC

�. (2)

While growth options are almost always riskier than revenues from deployed capital

in real options models, the presences of operating costs allow for deployed assets that are

riskier than growth options.3 This is the operating leverage hypothesis of Carlson, Fisher

and Giammarino (2004) and Sagi and Seasholes (2007). For operating leverage to signifi-

cantly impact the riskiness of deployed capital requires both “high operating leverage” and

“limited operational flexibility,” i.e.,V iC=V

iA � 0 andˇi

C � ˇiR.4

All firms basically satisfy the “high” operating leverage conditions, withV iC=V i

A�

0. Operating leverage is, to first order approximation, the inverse of a firm’s operating

margins, and thus is generally closer to ten than zero. While all firms satisfy the “high”

operating leverage criterion, the level of operating leverage varies a great deal in the cross-

section. We will exploit this variation in our empirical tests, and look at differences in the

return characteristics ofrelatively levered and unlevered firms.

The second condition, “limited” operational flexibility, is less obvious. It does not typ-

ically hold, for example, in standard quadratic adjustment cost models. Nevertheless, both

the data and introspection seem to suggest that the existence of limit operational flexibil-

ity is not unreasonable. In response to negative shocks firms’ revenues typically fall more

quickly than they can reduce costs; prices are more responsive than firms’ capital stocks.

A more subtle issue concerns the relationship between the level of operating leverage

and the degree of operational inflexibility. High operating cost firms should be more ag-

3 Growth options here represent the value of future investment opportunities. Under this definition, a firmthat is actively trying to develop a drug from a molecule on which it has a patent is not really a high growthoptions firm– a firm that owns a patent on a similar molecule but is deferring active drug development is. Itseems reasonable that the high growth option firm is less risky. Bad news about the class of molecule relatedto the firms’ patents will hurt the value of the firm that just owns the patent, but may devastate the firm that isincurring costs actively developing the drug.

4 The asymmetric adjustment cost function in Zhang (2005), with a quadratic adjustment penalty that issignificantly higher for disinvestment than investment, is essentially designed to generate this limited opera-tional flexibility.

6

gressive in cutting costs in bad times, so we should expect operational flexibility to be

positively correlated with operating leverage. This makes the work of an econometrician

looking for direct evidence of the operating leverage hypothesis more difficult. While sim-

ple theory suggests that expected returns should be increasing in operating leverage, the fact

that they should also be increasing in operational inflexibility, which is difficult to observe

and negatively correlated with the level of operating leverage, makes direct inference on the

expected return / operating leverage relationship difficult. Tests that attempt to relate the

level of operating leverage directly to expected returns will be biased against rejecting the

null hypothesis that expected returns are uncorrelated with the level of operating leverage.

While we will address this issue in greater detail, we hold off doing so until after we have

presented a formal model of operating leverage. For the remainder of this section, in which

we consider general implications, derived from what amount to accounting identities, we

will proceed as if operating leverage and operational inflexibility are uncorrelated.

Combining equations (1) and (2) gives

ˇi D�

V iA

V i

��ˇi

R C�

V iC

V iA

��ˇi

R � ˇiC

��C�

V iG

V i

�ˇi

G . (3)

We can make this equation friendlier by 1) identifying assets-in-place with book assets,

2) ignoring financing issues, and 3) assuming that the capitalized cost of operating is pro-

portional to annual operating costs.5 These heroic assumptions are not meant to be taken

seriously, but are made in the interest of eliciting the basic intuition in the simplest possible

framework, and to enable us to derive some basic empirical predictions.

Under these assumptions, we can rewrite the previous equation as

ˇi D BM i�OLi

�ˇi

R � ˇiC

���ˇi

G � ˇiR

��C ˇi

G (4)

5 In general, the value of deployed capital may be either greater or less than its book value. For example, ifdeployed capital generates economic rents, then its market value can exceed its book value by the capitalizedvalue of these rents. Conversely, if investment in not costlessly reversible, then if demand falls the marketvalue of capital can fall below its book value.

7

whereBM i is firm i’s book-to-market andOLi is the firm’s annual operating costs divided

by book assets times an arbitrary scale constant.

In the previous equation, book-to-market multiplies the difference in the risk factor

loadings on deployed assets and growth options. Under the operating leverage hypothesis,

high book-to-market firms earn higher returns because they are relatively more exposed to

assets-in-place, and assets-in-place are riskier than growth options.

Under this hypothesis firms should also earn higher returns, even after controlling for

the firm’s loading on this return difference, if the difference in the expected returns to

deployed capital and growth options is itself larger. That is, given two firms with the same

book-to-market ratios, and consequently the same relative loadings on assets-in-place and

growth options, the firm with the riskier assets-in-place should have the higher returns.

In the previous equation, operating leverage (OLi) is correlated with the extent to which

assets-in-place load on the risk factor, suggesting that high operating leverage firms should

earn higher returns. They should also load on HML, to the extent that HML picks up the

riskiness of assets-in-place when measuring the difference between the riskiness of assets-

in-place and growth options.

If book-to-market and operating costs-to-book both correlate with a firm’s risk expo-

sure, then we might expect their product, operating costs-to-market, to correlate even more

strongly with this risk exposure.

To test these predictions, we sort stocks into quintile portfolios based on book operating

leverage (OL) and market operating leverage (OLX), defined as annual operating costs

divided by book assets, and annual operating costs divided by market assets, respectively,

where operating costs is Compustat annual data item 41 (Costs of Goods Sold) plus item

189 (Selling, General, and Administrative Expenses), book assets is item 6 (Assets), and

market assets is book assets plus market equity (market capitalization, from CRSP) minus

book equity.6 We form portfolios in June of each year, using accounting data we are certain

6 For Book equity we employ a tiered definition largely consistent with that used by Fama and French(1993) to construct HML. Book equity is defined as shareholder equity, plus deferred taxes and minus pre-ferred stock if these are available. Shareholder equity is as given in Compustat (annual item 216) if available,or else common equity plus the carrying value of preferred stock (item 60 + item 130) if available, or else

8

was available at the time of portfolio formation. Market equity is lagged six months (i.e.,

we use prices from the previous December), to avoid taking unintentional positions in

momentum.7 Sorts are based on New York Stock Exchange (NYSE) break points. We

exclude banks and financial firms (i.e.,firms with a one-digit SIC code of six).

Table 1 provides return characteristics of portfolios sorted on book operating leverage

(Panel A) and market operating leverage (Panel B). The portfolios’ value-weighted average

monthly excess returns are given, together with the factor loadings and alphas from time-

series regressions of the portfolios’ returns on the Fama-French factors. Test-statistics are

provided in brackets.

The table also provides portfolio summary statistics. The right hand columns show

each portfolio’s time-series average operating leverage (OL, asset weighted) and book-to-

market (BM, value weighted) of the portfolios, as well as the average market equity (ME)

and number of firms (n) in the portfolio.

In panel A, we see that sorting on book operating leverage produces significant variation

in returns and HML loadings, without generating any systematic variation in the portfolios’

book-to-markets. The operating leverage sort does generate variation on average firm size,

but less size variation than is generated sorting on book-to-market. The levered portfo-

lio yields 41 basis points per month higher returns than the unlevered portfolio, and this

difference is statistically significant (test-statistic of 2.33), even though expected negative

correlation between the level of operating leverage and operational inflexibility should bias

the operating leverage sort against generating significant variation in expected returns. The

total assets minus stockholders’ equity (item 6 - item 216). Deferred taxes is deferred taxes and investmenttax credits (item 35) if available, or else deferred taxes and/or investment tax credit (item 74 and/or item 208).Prefered stock is redemption value (item 56) if available, or else liquidating value (item 10) if available, orelse carrying value (item 130). We also follow Fama and French in reducing shareholder equity by postre-tirement benefit assets (item 330) if available, in order to neutralize discretionary differences in accountingmethods firms choose to employ under the Financial Accounting Standards Board’s statement regarding em-ployers’ accounting for postretirement benefits other than pensions (FASB 106). Results are not sensitive tothis adjustment.

7 This seems to be the primary reason Fama and French build HML using book-to-market constructed asJune book equity divided by the previous December’s market equity. Using market equity from the June dateof portfolio formation biases the value portfolio toward recent losers and the growth portfolio toward recentwinners, resulting in a value factor that is short momentum, yielding a lower factor Sharpe ratio.

9

TABLE 1EXCESS RETURNS, THREE-FACTOR ALPHAS AND FACTOR LOADINGS, AND

CHARACTERISTICS OFPORTFOLIOS SORTED ON BOOK OPERATING LEVERAGE

AND MARKET OPERATING LEVERAGE, JUNE 1973 - JANUARY 2007

PANEL A: BOOK OPERATING LEVERAGE

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML OL BM ME n

Low 0.394 0.051 0.998 -0.081 -0.373 0.43 0.55 1535 685[1.47] [0.59] [47.83] [-2.98] [-11.92]

2 0.528 0.035 1.017 -0.058 -0.093 0.80 0.58 1521 581[2.16] [0.51] [63.08] [-2.79] [-3.87]

3 0.641 0.089 1.009 0.041 -0.019 1.06 0.58 1024 579[2.64] [1.24] [58.87] [1.82] [-0.74]

4 0.599 -0.019 1.006 0.226 0.017 1.37 0.66 615 669[2.35] [-0.21] [45.64] [7.89] [0.51]

High 0.803 0.157 1.043 0.138 0.082 2.48 0.54 525 762

Ope

ratin

glev

erag

equi

ntile

s

[3.01] [1.25] [34.81] [3.55] [1.84]

H - L 0.409 0.106 0.045 0.219 0.455[2.33] [0.62] [1.10] [4.13] [7.46]

Sharpe ratio (annual) of the high-minus-low strategy: 0.402

PANEL B: MARKET OPERATING LEVERAGE

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML OL BM ME n

Low 0.351 0.098 0.956 -0.131 -0.485 0.46 0.35 1975 789[1.35] [1.79] [73.16] [-7.69] [-24.75]

2 0.578 0.008 1.038 -0.032 0.027 0.79 0.55 1287 573[2.38] [0.12] [61.28] [-1.47] [1.06]

3 0.793 0.038 1.059 0.131 0.295 1.00 0.75 1004 519[3.23] [0.41] [47.59] [4.51] [8.84]

4 0.795 -0.050 1.044 0.204 0.460 1.29 0.88 571 577[3.35] [-0.58] [50.54] [7.59] [14.87]

High 0.985 0.079 1.025 0.394 0.498 2.17 1.04 246 806

Ope

ratin

glev

erag

equi

ntile

s

[3.94] [0.76] [41.44] [12.26] [13.42]

H - L 0.634 -0.019 0.069 0.525 0.982[3.48] [-0.17] [2.53] [14.76] [23.95]

Sharpe ratio (annual) of the high-minus-low strategy: 0.601

Source:Compustat and CRSP.The table shows the monthly value-weighted average excess returns to portfolios sorted on

operating leverage and market operating leverage, results of time-series regressions of these port-folios’ returns on the Fama-French factors, with test-statistics, and time-series average portfoliocharacteristics. Operating leverage and market operating leverage are cost of goods sold plus sell-ing, general and administrative expenses (Compustat annual data item 41 plus item 189), scaledrespectively by the book or market value of assets (item 6, or item 6 plus market equity minusbook equity).

10

annual Sharpe ratio of the levered-minus-unlevered strategy is 0.40, roughly three quarters

that of the high-minus-low book-to-market strategy over the same period (0.53).

The levered portfolio loads more heavily on HML than the unlevered portfolio, despite

the fact that it has essentially the same book-to-market as the unlevered portfolio. The

HML loading of the levered-minus-unlevered portfolio, 0.455, is less than half the loading

generated by a direct sort on book-to-market, but is highly significant (test-statistic equal to

7.46), and high for portfolios that do not exhibit variation in book-to-market as a character-

istic. In fact, the spread is roughly on par with that generated by sorting on pre-formation

HML slopes directly (Davis, Fama and French (2000)).

The Fama-French factors price the portfolios well. The observed three factor model

root mean squared pricing error is considerably lower than the observed market model root

mean squared pricing error, 8.6 as opposed to 14.6 basis points per month. It also performs

better statistically than the market model. While a GRS test (Gibbons, Ross and Shanken

(1989)) fails to reject the hypothesis that the true market model pricing errors on the five

portfolios are jointly zero (F5,397 D 1.47, p-value = 19.9%), the market model has trouble

with the extreme portfolios, mispricing the levered-minus-unlevered portfolio by 43.9 basis

points per month. This mispricing is statistically significant, with a test-statistic of 2.49.

The three factor model performs better. A GRS test similarly fails to reject the hypothesis

that the three factor model pricing errors are jointly zero (F5,395 D 1.08, p-value = 36.9%),

but the model does a much better job pricing the extreme portfolios. The observed three

factor mispricing of the levered-minus-unlevered portfolio is a statistically insignificant

10.6 basis points per month (test-statistic equal to 0.62).

These results are at odds with the Daniel and Titman (1997) result that HML has no

explanatory power in asset pricing tests after controlling for characteristics. The Fama-

French factors price the extreme operating leverage portfolios, despite the fact that these

portfolios do not exhibit systematic variation in book-to-market.

In panel B, we see that sorting on market operating leverage generates a significant

return spread and a high high-minus-low strategy Sharpe ratio. This is not in itself sur-

11

prising; the numerator, operating costs, is highly correlated with book equity, while the

denominator, market value of assets, is highly correlated with market equity. What is sur-

prising, however, is that the sort on market operating leverage generates a higher return

spread and high-minus-low strategy Sharpe ratio than the book-to-market sort (63.4 basis

points per month versus 56.8 basis points per month, and 0.601 versus 0.537, respectively),

despite generating only half as much variation in the book-to-market characteristic (a ratio

of extreme portfolios’ book-to-markets of three (1.04/0.35), as opposed to more than six

(1.72/0.27)).

Again, the Fama-French factors price the portfolios well. The observed three factor

model root mean squared pricing error is much smaller than the observed market model root

mean squared pricing error, 6.3 as opposed to 29.1 basis points per month. It also performs

better statistically than the market model. While a GRS test strongly rejects the hypothesis

that the market model pricing errors on the portfolios are jointly zero (F5,397 D 3.24, p-

value = 0.70%), the test again fails to reject the same hypothesis for the three factor model

(F5,395 D 1.07, p-value = 37.9%). The market model again has a particularly hard time

with the extreme portfolios, mispricing the high-minus-low portfolio by 71.1 basis points

per month, which is highly significant (test-statistic equal to 3.93). In contrast, the three

factor model does a spectacular job pricing the extreme portfolios, mispricing the levered-

minus-unlevered portfolio by a statistically insignificant -1.9 basis points per month, (test-

statistic equal to -0.17).

We can further investigate the interaction of operating leverage and book-to-market

by performing independent double sorts on the two characteristics. The double quintile

sort produces 25 portfolios that, because operating leverage and book-to-market are almost

uncorrelated, each contain roughly the same number of firms.

Inspection of Table 2 reveals that high operating leverage portfolios yield higher returns

than the low operating leverage portfolios across book-to-market quintiles, though this dif-

ference is only significant in the third and fourth book-to-market quintiles. The Fama-

French three factor model again does a better job pricing the levered-minus-unlevered

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TABLE 2EXCESSRETURNS, BOOK-TO-MARKET, AND THREE-FACTOR ALPHAS AND

FACTOR LOADINGS FORPORTFOLIOS DOUBLE SORTED ON OPERATING LEVERAGE

AND BOOK-TO-MARKET, JUNE 1973 - JANUARY 2007

Monthly excess returns and BMs for Return characteristics ofportfolios sorted on operating leverage the levered-minus-unlevered

and book-to-market portfoliosOperating leverage quintiles FF3 alphas and loadings

L 2 3 4 H re ˛ ˇmkt

ˇsmb

ˇhml

L 0.374 0.397 0.400 0.314 0.662 0.288 -0.022 0.076 0.102 0.502(0.25) (0.27) (0.26) (0.28) (0.26) [1.36] [-0.11] [1.52] [1.57] [6.69]

2 0.542 0.637 0.770 0.691 0.858 0.316 0.044 0.016 0.241 0.411(0.54) (0.55) (0.55) (0.55) (0.55) [1.50] [0.21] [0.31] [3.66] [5.43]

3 0.474 0.727 0.821 0.835 1.036 0.562 0.240 0.042 0.353 0.421(0.79) (0.79) (0.79) (0.55) (0.79) [2.49] [1.08] [0.78] [5.07] [5.24]

4 0.507 0.851 1.121 0.841 1.001 0.494 0.264 -0.069 0.565 0.234(1.07) (1.08) (1.07) (1.08) (1.07) [2.18] [1.21] [-1.32] [8.29] [2.98]

H 0.908 1.001 1.141 0.974 1.070 0.162 0.071 -0.223 0.638 0.074Boo

k-to

-mar

ketq

uint

iles

(1.77) (1.84) (1.76) (1.79) (1.85) [0.68] [0.32] [-4.13] [9.11] [0.92]re 0.534 0.604 0.741 0.660 0.408

[2.08] [2.45] [2.88] [2.51] [1.64]

˛ -0.330 -0.199 -0.119 -0.117 -0.236[-1.68] [-1.00] [-0.61] [-0.58] [-1.24]

ˇmkt

0.258 0.220 0.190 0.065 -0.041[5.48] [4.63] [4.04] [1.34] [-0.89]

ˇsmb

0.366 0.578 0.633 0.555 0.902[5.98] [9.35] [10.35] [8.87] [15.22]

ˇhml

1.298 1.092 1.214 1.229 0.870high

-min

us-lo

wpo

rtfo

lios

FF

3al

phas

and

load

ings

[18.39] [15.34] [17.23] [17.05] [12.73]

Source:Compustat and CRSP.The table shows value-weighted average excess monthly returns and book-to-market ratios (in paren-

theses) of portfolios double sorted on operating leverage and book-to-market, and results of time-seriesregressions of both sorts’ high-minus-low portfolios’ returns on the Fama-French factors, with test-statistics[in square brackets].

portfolios, despite the fact that the levered and unlevered portfolios do not differ in book-

to-markets. While a GRS test on the five levered-minus-unlevered portfolios cannot reject

the null hypothesis that the pricing errors on the portfolios are jointly zero for either model

(F5,397 D 1.64, p-value = 14.7% for the market model;F5,395 D 0.49, p-value = 78.2% for

the three factor model), the observed root mean squared pricing error for the three factor

13

model is less than half that for the market model, 16.4 versus 41.9 basis points per month.

While value firms provide higher returns than growth firms across operating leverage

quintiles, the return spread between value and growth firms is hump-shaped, not mono-

tonic, in operating leverage. This is at odds with a naive interpretation of equation 4, one

that assumes that the relative risk factor loadings of deployed capital and growth options

are independent of the level of operating leverage. It is, however, consistent with a more

sophisticated interpretation, one that recognizes that higher operating costs may influence

firms to reduce production sooner in the face of falling demand, resulting in higher cost

betas for more levered firms.

This view needs to be made explicit, of course. The next section presents a formal,

dynamic model of operating leverage, which will be used to generate additional testable

implications. The presentation of the model, and the analysis of firms’ equilibrium invest-

ment behavior, is unavoidably somewhat technical. Readers interested only in the model’s

predictions, and empirical tests of these predictions, may wish to skip sections 3 and 4.

3 The Economy

We employ the industry equilibrium model of Novy-Marx (2007a), extended to include a

counter-cyclic risk price. The industry consists ofn competitive firms, which are assumed

to maximize the expected present value of risk-adjusted cash flows discounted at the con-

stant risk-free rater . These firms employ capital, which may be brought into the industry

at a price that we will normalize to one and may be sold outside the industry at a price

˛ < 1, in conjunction with non-capital inputs to produce a flow of a non-storable good or

service.8

A firm can produce a flow of the industry good proportional to the level of capital it

employs, and proportional to its firm-specific production efficiency. That is, at any time

a firm “i” can produce a quantity (“supply”) of the goodS it D Ki

t=ci whereKit is the

8In the case of complete irreversibility (i.e.,˛ D 0) we still allow for the free disposal of capital.

14

firm’s capital stock andci is the firm’s capital requirement per unit of production (i.e.,

the firm’s inverse capital productivity). Aggregate production is thenSt D ��� K t , where

K t D (K1t , K2

t , ..., Knt )0 and��� D (c�1

1 , c�12 , ..., c�1

n ) denote the vectors of firms’ capital

stocks and firms’ capital productivities, respectively. Aggregate capital employed in the

industry isKt D 111111111K t where111111111 D (1, 1, ..., 1) is then-vector of ones.

In the goods market firms face a stochastic level of iso-elastic demand, with price elas-

ticity 1= > 1=n.9 Operating revenue generated by each unit of capital employed by firm

i is thereforePt=ci , where

Pt D�

Xt

St

�

, (5)

where we assume that the multiplicative demand shockXt is a geometric Brownian process

under the risk-neutral measure,

dXt D �X Xtdt C �X Xt dBt

where�X < r and�X are known constants, andBt is a standard Wiener process.

Production also entails a non-discretionary operating cost, assumed to be proportional

to the level of capital employed, with a unit cost per period of�. Firm i’s operating profits

may consequently be written, in terms of primatives and the control, as

Ri(K t , Xt ) DKi

t

ci

�Xt

��� K t

�

� Kit�. (6)

Finally, each firm’s capital stock changes over time due to investment, disinvestment

and depreciation. At any time firms may acquire and deploy new capital within the industry,

at a unit price equal to one, or sell capital that will be redeployed outside the industry, at

the unit price < 1. Capital depreciates at a constant rateı. The change in a firm’s capital

stock can therefore be written asdKit D �ıKi

t C dU it � dLi

t , whereU it (respectively,Li

t )

9 This condition assures that no firm can increase its own revenues by decreasing output.

15

denotes firmi’s gross cumulative investment (respectively, disinvestment) up to timet .

4 The Optimal Investment Strategy

The value of a firm’s investment depends on the price of the industry good, and conse-

quently on the aggregate level of capital employed in the industry. As a consequence, the

value of a firm depends not only on how it invests, but also on how other firms invest.

Moreover, because each firm’s investment itself affects prices, any given firm’s investment

strategy affects the investment strategy employed by other firms. We are interested in ex-

hibiting a simple equilibrium, and will therefore restrict our attention to Nash-Cournot

strategies.10

4.1 The Firm’s Optimization Problem

Firms are assumed to maximize discounted cash flows, so the value of firmi is given by

V i(K t , Xt ) D (7)

maxfdU i

tCs ,dLitCs g

Et

�Z 1

0

e�rs�Ri(K tCs, XtCs )ds�dU i

tCs C ˛dLitCs

�ˇfdU �i

tCs , dL�itCsg

�

wherefdU �it , dL�i

t g is used to denote other firms’ investment/disinvestment at timet , and

the expectation is with respect to the risk-neutral measure.

4.2 Equilibrium

Firms’ equilibrium behavior, provided in the following proposition, represents a Cournot

outcome. More efficient firms invest more, so have larger market shares. They conse-

quently internalize more of the negative price externality associated with their own invest-

ment, and this offsets their greater production efficiency, resulting in marginal valuations

10 Consideration of collusive strategies, such as that employed by Green and Porter (1984), while extremelyinteresting, is beyond the scope of this analysis.

16

of capital that equate across firms. Firms all invest and disinvest at the same times, when

the shadow price of capital equals the purchase and sale prices of capital, respectively. The

details of this equilibrium investment behavior are provided in the proposition. In an effort

to avoid excessive digression, proofs of all propositions are left for the appendix. For a

more detailed exposition of the strategy, the economic intuition underlying firms’ behavior,

or aQ-theoretic characterization of the strategy, please see Novy-Marx (2007a).

Proposition 4.1.Suppose that the current state of the economy satisfies the following “long

history” conditions:

1. The long run participation constraint: each firm’s production is “sufficiently effi-

cient,” in that its capital requirement per unit of production is not too high, satisfying

ci < cmax �c

1 �

n

(8)

wherec � 1n

PnjD1 cj is the equal-weighted industry average capital requirement

per unit of production.11

2. The long run industry organization: firms produce in proportion to their “cost wedges,”

S it =S

jt D (cmax � ci) =

�cmax � cj

�, which requires that firms’ capital stocks satisfy

Ki0 D

cci �

�1 �

n

�c2

i

c2 ��1 �

n

�c2

!K0

n(9)

for eachi, wherec2 � 1n

PniD1 c2

j .

Then all firms will invest in new capital, in proportion to their existing capital, whenever

11 This first condition is satisfied trivially if, given the order set of firms’ unit costsc1 � c2 � ... � cM ,

we letn � maxni 2 f1, ..., Mgj ci < ci

1� i

owhereci D 1

i

PijD1 cj and restrict attention to the firstn firms.

17

the goods pricePt reachesPU , and disinvest wheneverPt reachesPL, where

PU D(1 C )C

(1 � L)˘(��1)(10)

PL D(˛ C )C

(1 � L)˘(�), (11)

D �

rCıis the capitalized flow costs associated with operating a unit of capital in perpe-

tuity, �C � �Kt=St is industry average operating costs per unit of production,

C DPn

jD1 KjtPn

jD1 Kjt =cj

D n

�c �

�1 �

n

�c2

c

�, (12)

L D H is the “pseudo”-Lerner index,12 where1= is the price elasticity andH �Pn

jD1(Sjt =St)

2 is the Herfindahl index associated with the equilibrium organization,

H DnX

iD1

Ki

t=ciPnjD1 K

jt =cj

!2

D 1

�1 �

�1 �

n

�Cc

�,

and˘(��1) and˘(�) are the perpetuity factors for the equilibrium price process at the

investment and disinvestment thresholds, respectively,

˘(x) D�1 � �2

2(rCı)

�y0(1)�y0(x)

y(x)

�x�� (13)

wherey(x) D x p � xˇn , and p Dq�

�

�2 � 12

�2C2(rCı)

�2 ��

�

�2 � 12

�, ˇn D �

��

�2 � 12

��q�

�

�2 � 12

�2C2(rCı)

�2 , � D 1rCı��

, � D ��X C ı C ( � 1)

�2X

2

�and� D �X , and� D

12 In the standard Cournot model the market Lerner index (fraction by which output-weighted averagemarginal cost falls below price in the goods market) is equal to H . In the economy in this paper, in whichcapital is costly, long-lived, and only partially reversible, the market power index is generally not equal to H . In the case of fully reversible capital, and if we follow Pindyck (1987) and calculate the market powerindex asL� D (P � FMC )=P whereFMC is the “full marginal cost” of production, which includes thecosts of operating as well as purchasing capital, thenL� D L. More generally,L� will lie in a region thatincludesL.

18

PU =PL > 1 is uniquely determined by

˘(�)

�˘(��1)D

˛ C

1 C . (14)

5 Valuation

Because investment and disinvestment occur when the marginal value of deployed capital

equals the purchase and sale prices of capital, respectively, the value of a firm’s currently

deployed capital is the capitalized value of the profits it is expected to generate. This fact,

in conjunction with standard technical conditions (value matching and smooth pasting at

the investment and disinvestment thresholds), allows us to explicitly characterize the value

of a firm as a function of prices in the goods market. This characterization is provided in

the following proposition.

Proposition 5.1. Average-Q for firm i, as a function of the price of the industry good, is

given by

Qit D q(Pt) C �i

(q(Pt ) C ) C an

�Pt

PL

�ˇn

C ap

�Pt

PU

�p

!(15)

„ƒ‚…shadowprice ofcapital

„ ƒ‚ …capitalized

rents todeployed capital

„ ƒ‚ …expansion

and contractionoptions

where�i D cmaxci

� 1 is firm i’s “excess productivity,” the shadow price of capital is given

by

q(Pt) D�

1�L

C

�Pt�(Pt) � (16)

where

P�(P) D �P C�

y�

PPU

�

y�

PLPU

���˘�

PU

PL

�� �

�PL C

�y

�P

PL

�

y�

PUPL

���˘�

PL

PU

�� �

�PU (17)

19

for y(x) D x p � xˇn, andan andap are given by

an D �(1 C ) � � p(˛ C )

( ˇn � 1) y (�)(18)

ap D(˛ C ) � ��ˇn(1 C )�

p � 1�

y (��1). (19)

Figure 1 depicts the relationship between firms’ values and prices in the goods market

provided in the previous proposition. A high cost (marginal) producer has an average

valuation equal to the industry’s shadow price of capital. This firm invests when its average-

Q, which equals its marginal-q, equals one, the purchase price of capital (right hand edge

of figure 1), and disinvest when its average-Q equals the sale price of capital (left hand edge

of figure 1). More efficient producers, which capture rents on both their current production

and the production of future capital deployments, have richer valuations. They invest and

disinvest at the same critical goods-price levels, however, because they internalize more of

the price externality associated with investment, due to their greater market shares. This

reduces an efficient firm’s marginal valuation of capital to the point that it equates with the

marginal valuation of less efficient firms.

6 Expected Returns

Equation (15), which specifies average-Q as a function of firm and industry characteristics,

can also be used to calculate the sensitivity of firm value to demand, providing a means to

study the relationship between market-to-book and expected returns. This relationship has

been studied extensively in the finance literature, in which it is now common practice to

include Fama and French’s (1993) adjustment to expected returns to account for the “value

premium.”

The following proposition relates firms’ risk factor loadings, and consequently their

expected rates of return, to prices in the goods market. In conjunction with the previous

20

0.8 0.9 1 1.1 1.2 1.3P

0

0.5

1

1.5

2

2.5

3

3.5

Q

Figure 1: Tobin’s Q in the cross-sectionThe figure depicts average-Q for three firms in an industry, as a function of the price ofthe industry good. The bottom curve (dotted line) shows a high cost (marginal) producer(ci D cmax), which has an average-Q equal to the industry’s shadow price of capital. Themiddle curve (dashed line) shows the average firm in the industry (ci D C ). The top curve(solid line) shows a low cost producer (ci D (C =cmax)C ). Other parameters arer D 0.05,� D 0.03, � D 0.20, ı D 0.02, C D 1, � D 1, D 1, H D 0.02, and˛ D 0.25.

low cost producer

average cost producer

high cost producer

proposition this will allow us to explicitly relate firms’ expected returns to their book-to-

markets.

Proposition 6.1. The expected excess rate of return to firmi is ˇit�t where�t is the time-t

price of exposure to the priced risk factor (X ) and

ˇit D

�Qi

t

��1

�Pt

ci

C C iˇn

�Pt

PL

�ˇn

ˇn C C i

p

�Pt

PU

�p

p

!(20)

21

where

C iˇn

D� ˇn

�cmax

ci

�� 1

�an �

��PL

ci

���ˇP � �

y (�)

�(21)

C i

pD

� p

�cmax

ci

�� 1

�ap �

��PU

ci

����1 � ��ˇn

y (��1)

�. (22)

Figure 2 depicts the relationship between risk-factor loadings and prices in the goods

market provided in the previous proposition.13 In normal times inefficient producers are

more exposed to the underlying risks in the economy, because the exposure of their rev-

enues to the risk factor is levered more by their high production costs. In good times,

however, they are relatively insulated from these risks, which are largely absorbed by the

capacity response resulting from firms’ competitive investment decisions. At these times,

efficient producers still benefit from positive economic shocks. In response to these shocks

they expand, buying capital at a price that is lower than its average value to the firm, and

capturing the expected surplus.

6.1 Unconditional Expected Returns

While equation (20) specifies firms’ conditional expected rates of returns, it is now simple

to characterize firms’ unconditional expected rates of return. A firm’s unconditional ex-

pected excess rate of return is the average price of risk scaled by the firm’s exposure to the

risk factor. This is given explicitly in the following proposition.

13 This figure contains the intuition behind the results of Kogan (2004), Zhang (2005) and Aguerrevere(2006). Kogan considers a perfectly competitive economy, comprised completely of marginal producers.Firms are more exposed to fundamental risks, and consequently expect higher, more volatile returns whenprices in the goods market, and firms’ values relative to book capital, are low. Zhang shows that both high op-erating costs and operating inflexibility are required to generate a value premium, and these are the necessaryconditions for generating significant variation in the exposures of high and low cost producers to fundamentalrisks. Aguerrevere argues that competitive industries should be riskier than concentrated industries in “bad”times, but less risky in good times. Competitive industries “look like” high cost producers, deriving most oftheir value from assets-in-place and little from future investment opportunities, and assets-in-place are highlyexposed to fundamental risks in normal times but insensitive to these risks in expansions.

22

0.8 0.9 1 1.1 1.2 1.3P

1

2

3

4

Β

Figure 2: Risk factor loadings in the cross-sectionThe figure depicts risk factor loadings for three firms in an industry, as a function of pricesin the goods market. The strongly arching dotted line shows a high cost (marginal producer,ci D cmax), the dashed line the average firm in the industry (ci D C ), and the relatively flatsolid line shows a low cost producer (ci D (C =cmax)C ). Other parameters arer D 0.05,� D 0.03, � D 0.20, ı D 0.02, C D 1, � D 1, D 1, H D 0.02, and˛ D 0.25.

high cost producer

average cost producer

low cost producer

Proposition 6.2. The unconditional expected rate of return to firmi is

rei D

Z PU

PL

ˇi(p)�(p)d�(p) (23)

whereˇi(Pt) is given in Proposition 6.1 andd�(p), the stationary density for the risk-

neutral price process, is given by

d�(p) D �

p��1

P�

U � P�

L

!dp (24)

for � D 2�

�2 � 1.

23

7 Indirect Inference

While the results of section 2 provide direct evidence relating the level of operating leverage

to the cross section of returns, analysis of the formal model presented in section 3 reveals

limitations to direct inference on operating leverage. In particular, the model doesnot

make strong predictions regarding expected returns and the level of operating leverage.

While high operating leverage is associated with higher (more levered) exposures to the

risk factor through the firm’s revenues, high operating leverage is also associated with more

operational flexibility, as high operating costs increase the effective reversibility of capital.

This increased operational flexibility largely offsets the impact of increased leverage.

Moreover, the level of operating leverage, which depends on thecapitalizedvalue of all

future costs and revenues, is basically unobservable. While market values provide a good

proxy for the difference in the capitalized values of costs and revenues, it is difficult to find

good proxies for these individually. Cross-industry differences in accounting practices, and

the prevalence of leases, add further noise to accounting variables that might conceivably

be related to operating leverage. Attenuation bias arising from noise is observed operating

leverage reduces the power of tests that employ the measure. These facts provide another

incentive to develop additional implications of the model.

This leads us to look for alternative predictions. These predictions should account for

the fact that the degree of operational inflexibility should be negatively correlated with

the level of operating leverage. We prefer implications that are testable even if operating

leverage is completely unobservable.

7.1 Expected returns and book-to-market in and across industries

Figure 3, below, depicts the model-implied relationship between expected returns and

book-to-market, both within and across industries. The bold, hump-shaped curve shows

the relationship between expected value-weighted average industry returns and industry

book-to-market,i.e., the expected return / book-to-market relationshipacrossindustries.

24

Low levels of operating leverage are associated with both high book-to-markets and low

expected returns. Book-to-markets in low operating leverage industries are low because

few rents accrue to non-capital factors of production. Expected returns in these industries

are low because deployed capital is relatively unlevered and thus less risky. Increasing op-

erating leverage is associated with higher book-to-markets, as more of the industry’s value

accrues to factors that do not contribute to book value. At low levels of operating lever-

age, increasing operating leverage is also associated with higher expected returns, because

it increases the riskiness of deployed capital. At high levels of operating leverage, how-

ever, increasing operating leverage is associated with lower expected returns, as the role

of operating leverage in increasing the effective reversibility of capital begins to dominate

the leverage effect. The net result is a weak, non-monotonic inter-industry relationship

between book-to-market and expected returns.

The upward sloping lines in Figure 3 show the relationship between expected returns

and book-to-marketswithin industries. The top, solid line depicts a labor intensive (high

operating leverage) industry, the middle, dashed line an average industry, and the bottom,

dotted line a capital intensive (low operating leverage) industry. Within industries the rela-

tionship between expected returns and operating leverage is strong and monotonic. Within

industries, inefficient, high book-to-market firms earn higher returns than more efficient,

lower book-to-market firms.

The figure suggests our next set of empirical tests. While the model predicts that book-

to-market and expected returns are strongly correlatedwithin an industry, it predicts that

the relationship between book-to-market and expected returns is weak, and non-monotonic,

acrossindustries.

In order to test this prediction, we perform separate sorts based on intra-industry book-

to-market and industry book-to-market. The first sort is used to identify value (inefficient)

and growth (efficient) firms across industries, while the second sort is used to generate

value and growth industries.

The intra-industry sort each year assigns each stock to a portfolio based on the firm’s

25

0.25 0.5 0.75 1 1.25 1.5 1.75BM

0.06

0.08

0.1

0.12

0.14

E@reD

Figure 3: BM / expected return relationship, in and across industriesThe figure depicts the unconditional relationship between expected excess returns and book-to-market in three different industries, and across industries. The top curve (solid line)shows the expected return / book-to-market relationship in a highly levered industry (� D2.5), the middle curve (dashed line) shows a moderately levered industry (� D 1), whilethe bottom curve (dotted line) shows a relatively unlevered industry (� D 0.4). The boldline depicts the relationship between expected excess industry returns and industry book-to-market. Other parameters arer D 0.05, � D 0.03, � D 0.20, ı D 0.02, D 1, H D 0.02,˛ D 0.25, and� D 0.05.

highleverageindustry

averageleverageindustry

lowleverageindustry

industry average (value-weighted)

book-to-market ratio relative to other firms in the same industry. For example, a firm is

assigned to the value portfolio if it has a book-to-market higher than eighty percent of

NYSE firms in the same industry. Each quintile portfolio consequently contains roughly

twenty percent of the firms in each industry.

The industry sort each year assigns each stock to a quintile portfolio based on the book-

to-market of the firm’s industry (total industry book value divided by total industry market

value). The industries we employ are the Fama-French 49. The middle portfolio contains

only nine industries.

Table 3 provides average excess returns and results of time series regressions of the

portfolios’ returns on the Fama-French factors. Panel A shows that the intra-industry book-

26

to-market sort generates a significant return spread, and a high-minus-low strategy Sharpe

ratio higher than that generated by the straight book-to-market sort (0.583 vs. 0.537).

The Fama-French factors accurately price these portfolios. While the observed market

model root mean squared pricing error on the five intra-industry book-to-market portfolios

is 24.8 basis points per month, the observed three factor model root mean squared pricing

error is only 3.5 basis points per month. GRS tests strongly reject the null hypothesis that

the market model pricing errors are jointly zero (F5,397 D 4.35, p-value = 0.073%), but

fail to reject the same null hypothesis for the three factor model (F5,395 D 0.47, p-value

= 80.0%). The market model performs particularly poorly on the value-growth spread,

mispricing the high-minus-low portfolio by 61.9 basis points per month, with a test-statistic

of 4.25, while the three factor alpha is only 1.9 basis points per month, with a test-statistic

of 0.19.

These inter-industry results, presented in Panel B, contrast strongly with the intra-

industry results presented in Panel A. Value industries do not provide significantly higher

returns than growth industries, despite having significantly higher book-to-market ratios

and HML loadings. The Sharpe ratio on the strategy that buys value industries and sells

growth industries is only 0.08.

The three factor model also performs worse than the market model in explaining the

portfolio returns. In section 2 we saw HML accurately pricing the operating leverage

portfolios that exhibited no variation in the book-to-market characteristic; here we see

HML mispricing portfolios constructed by sorting on book-to-market. While the observed

root mean squared three factor model pricing error is as small as the observed root mean

squared market model pricing error, 16.7 versus 17.0 basis points per month, GRS tests

strongly reject the null hypothesis that the Fama-French pricing errors do not differ from

zero (F5,395 D 4.44, p-value = 0.061%), while failing to reject the same null hypothesis

for market model (F5,397 D 1.60, p-value = 16.0%). The three factor model performs

particularly poorly at pricing the value-growth spread. The three factor alpha on the value-

minus-growth strategy is -49.9 basis points per month, with a test-statistic of -4.64, while

27

TABLE 3EXCESS RETURNS, THREE-FACTOR ALPHAS AND FACTOR LOADINGS,

AND CHARACTERISTICS OFPORTFOLIOS SORTED ON BOOK-TO-MARKET

WITHIN AND ACROSSINDUSTRIES, JUNE 1973 - JANUARY 2007

PANEL A: BOOK-TO-MARKET WITHIN INDUSTRY

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML BM ME n

Low 0.427 0.031 1.039 -0.098 -0.300 0.31 1945 962[1.64] [0.64] [87.83] [-6.34] [-16.96]

2 0.550 0.006 0.983 -0.089 0.068 0.53 1632 701[2.47] [0.11] [70.16] [-4.89] [3.23]

3 0.655 0.046 0.966 -0.047 0.200 0.72 1104 721[3.08] [0.81] [70.65] [-2.67] [9.76]

4 0.735 0.022 1.009 0.049 0.314 0.99 771 807[3.33] [0.37] [70.39] [2.65] [14.60]

High 0.938 0.049 1.049 0.201 0.546 1.49 318 1168

Intr

a-in

dust

ryB

Mqu

intil

es

[4.09] [0.78] [70.08] [10.30] [24.33]

High-Low 0.511 0.017 0.011 0.298 0.846[3.38] [0.19] [0.49] [10.50] [25.87]

Sharpe ratio (annual) of the high-minus-low strategy: 0.583

PANEL B: INDUSTRY BOOK-TO-MARKET

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML BM ME n

Low 0.487 0.252 0.950 -0.126 -0.517 0.32 1205 1004[1.82] [3.11] [48.89] [-4.99] [-17.76]

2 0.511 -0.104 1.082 -0.003 0.054 0.52 925 972[1.97] [-1.07] [46.44] [-0.08] [1.55]

3 0.608 -0.036 1.058 -0.017 0.152 0.65 840 852[2.46] [-0.39] [47.32] [-0.59] [4.53]

4 0.784 0.042 1.007 -0.100 0.462 0.81 1137 793[3.45] [0.40] [40.00] [-3.07] [12.25]

High 0.579 -0.248 0.987 -0.008 0.607 1.10 1140 738Indu

stry

BM

quin

tiles

[2.73] [-3.31] [55.07] [-0.36] [22.58]

High-Low 0.091 -0.499 0.037 0.118 1.124[0.47] [-4.64] [1.43] [3.51] [29.03]

Sharpe ratio (annual) of the high-minus-low strategy: 0.081

Source:Compustat and CRSP.The table shows the value-weighted average excess returns to quintile portfolios sorted on

industry book-to-market and intra-industry book-to-market, results of time-series regressions ofthese portfolios’ returns on the Fama-French factors, with test-statistics, and time-series averageportfolio characteristics.

28

the market model alpha is 24.9 basis points per month and insignificant (test-statistic equal

to 1.36).

It is also interesting to note the manner in which firm size varies across book-to-market

portfolios for the two different sorts. While size is strongly negatively correlated with intra-

industry book-to-market (as it is with straight book-to-market), it is essentially uncorrelated

with industry book-to-market. That is, while value firms within an industry tend to be

significantly smaller than growth firms in the same industry, firms in value industries are

roughly as large as firms in growth industries.

This result is consistent with the model presented in this paper, and helps explain why

the value effect is concentrated in small firms. Size helps distinguish value firms that gen-

erate higher returns because they are truly more exposed to risk from value firms that are

average producers, with average risk exposures, in high book-to-market industries.

It is useful, as a robustness check, to consider the interaction between industry book-to-

market and intra-industry book-to-market. Table 4 shows the results for portfolios sorted

independently across the two variables. Each portfolio has roughly the same number of

firms, by construction. The results across both industry and intra-industry book-to-market

quintiles are consistent with those of Table 3. While value firms generate higher returns

than growth firms across industry book-to-market quintiles, value (growth) firms in value

industries do not produce higher returns than value (growth) firms in growth industries.

The three factor model helps price the intra-industry high-minus-low portfolios, im-

proving the observed root mean squared pricing error relative to the market model, 31.4

versus 66.0 basis points per month. GRS tests reject the hypothesis that these pricing er-

rors are jointly zero for both models, though this rejection is less emphatic for the three

factor model (F5,395 D 4.35, p-value = 0.072% for the three factor model;F5,397 D 9.04,

p-value = 0.000% for the market model).

In contrast, the three factor model performs worse than the market model, or no model

at all, in explaining the industry high-minus-low portfolio returns. The observed three-

factor root mean squared pricing error is 52.9 basis points per month, versus 12.8 basis

29

TABLE 4EXCESSRETURNS, BOOK-TO-MARKET RATIOS, AND THREE-FACTOR ALPHAS

AND FACTOR LOADINGS FORPORTFOLIOS DOUBLE SORTED ON BOOK-TO-MARKET

WITHIN AND ACROSSINDUSTRIES, JUNE 1973 - JANUARY 2007

Monthly excess returns and BMs for Return characteristics ofportfolios sorted on industry BM the value industries-minus-

and within industry BM growth industries portfoliosIndustry BM quintiles FF3 alphas and loadings

L 2 3 4 H re ˛ ˇmkt

ˇsmb

ˇhml

L 0.396 0.362 0.548 0.473 0.489 0.093 -0.506 0.079 0.251 1.016(0.19) (0.28) (0.33) (0.41) (0.54) [0.44] [-3.12] [2.04] [4.96] [17.42]

2 0.511 0.426 0.622 0.822 0.565 0.054 -0.446 0.015 0.144 0.944(0.29) (0.45) (0.54) (0.67) (0.84) [0.28] [-3.12] [0.43] [3.23] [18.36]

3 0.676 0.505 0.679 0.961 0.548 -0.128 -0.607 0.053 -0.053 0.971(0.41) (0.62) (0.71) (0.67) (1.06) [-0.61] [-3.84] [1.39] [-1.07] [17.10]

4 0.739 1.088 0.773 0.906 0.501 -0.239 -0.608 0.032 -0.437 0.988(0.63) (0.84) (0.93) (1.06) (1.31) [-0.98] [-3.59] [0.78] [-8.27] [16.23]

H 1.006 1.104 0.795 1.333 0.782 -0.224 -0.455 0.048 -0.555 0.748

Intr

a-in

dust

ryB

Mqu

intil

es

(1.12) (1.33) (1.41) (1.54) (1.91) [-1.00] [-2.80] [1.24] [-10.96] [12.83]re 0.609 0.742 0.247 0.860 0.293

[2.74] [4.05] [1.53] [5.43] [1.73]

˛ -0.112 0.304 -0.048 0.618 -0.060[-0.72] [2.03] [-0.31] [4.18] [-0.41]

ˇmkt

0.075 -0.041 0.021 -0.028 0.043[2.01] [-1.14] [0.58] [-0.78] [1.22]

ˇsmb

0.881 0.521 0.327 0.129 0.075[18.29] [11.16] [6.85] [2.79] [1.63]

ˇhml

0.911 0.660 0.403 0.462 0.644

Intr

a-in

dust

ryH

-Lpo

rtfo

lios

FF

3al

phas

and

load

ings

[16.42] [12.28] [7.32] [8.69] [12.10]

Source:Compustat and CRSP.The table shows value-weighted average excess returns and book-to-market ratios (in parentheses) of

portfolios double sorted on intra-industry book-to-market and industry book-to-market, and results of time-series regressions of both sorts’ high-minus-low portfolios’ returns on the Fama-French factors, with test-statistics [in square brackets].

points per month for the market model. GRS tests reject the hypothesis that the three factor

pricing errors on these portfolios are jointly zero (F5,395 D 6.34, p-value = 0.001%), but

fail to reject the same hypothesis for the market model (F5,397 D 0.48, p-value = 79.0%).

Interestingly, the dispersion in HML loadings across industries exceeds those within

industries despite the facts that 1) the dispersion in book-to-market within industries is

30

approximately twice that observed across industries, and 2) the intra-industry variation

in book-to-market is strongly associated with differences in expected returns while the

variation in book-to-market across industries is not.14

These results suggests that HML has both a priced and an unpriced component. The

priced component appears related to variation in firms’ efficiencies, identifiable as differ-

ences in book-to-market ratios within industries. The unpriced component appears related

to industry variation, which affects book-to-market ratios but is largely unrelated to differ-

ence in expected returns.

7.2 Industry-relative book-to-market

While it appears that HML has a priced component related to variation in intra-industry

book-to-market ratios and an unpriced component related to variation in industry book-

to-markets, the intra-industry sort used in Tables 3 and 4 is not the most effective way to

isolate the priced component. Firms do not earn significantly higher returns because they

have higher book-to-markets than most other firms in their industry; firms earn significantly

higher returns because they are significantly more exposed to the priced risk factor in the

economy. This exposure is associated withsignificantlyhigher book-to-market than the

industry average. That is, the cardinal ranking of a firm’s book-to-market within its industry

is not important,per se.

The model suggests that book-to-marketrelative to the book-to-market of other firms

in the industry will better identify those firms that load heavily on priced risk factors. This

motivates a simple, alternative univariate sorting methodology, whereby firms are assigned

to portfolios on the basis of their industry-relative book-to-markets,BM i=BMi, where

BM i is the book-to-market of firmi andBMi

is the book-to-market of firmi’s indus-14 This fact essentially guarantees the inefficiency of HML. The construction of HML ensures that the

factor covaries positively with the returns to a portfolio long value industries and short growth industries. Thisvariation, which can be hedged, is unpriced absent systematic variation in expected returns across industries,tautologically.

31

try.15,16 In an average year this sorting procedure assigns 53.5 percent of stocks to the same

quintile portfolio as the book-to-market sort. It assigns 32.6 percent of stocks to portfolios

one different in cardinal ranking from their assignment under the book-to-market sort, 11.3

percent to portfolios two different, and 2.5 percent to portfolios three different. In only 0.12

percent of cases does the procedure classify growth stocks as intra-industry value stocks or

value stocks as intra-industry growth stocks.

Note that this sorting procedure does not guarantee industries equal representation in

the portfolios; industries with high cross-sectional variation in book-to-market will be over-

represented in both the value and growth portfolios, while industries with little variation on

book-to-market will be overrepresented in the neutral portfolio. Unlike the straight book-

to-market sorting procedure, however, the relative book-to-market procedure does not bias

the value (growth) portfolio towards high (low) book-to-market industries.

Table 5 provides the average excess returns to quintile sorted portfolios, and results

of time-series regressions of the portfolios’ returns on the Fama-French factors. We also

report time-series average portfolio characteristics. Results for the standard book-to-market

sort are also provided, in Panel B, for comparison.

The Sharpe ratio of the high-minus-low strategy for the sort on industry-relative book-

to-market is significantly higher than for the straight book-to-market sort, 0.74 versus 0.54.

Approximately one quarter of this difference is due to a greater return spread between the

15 That is,BMi �

Pallj 11indj Dindi

bej =P

allj 11indj Dindimej , where11indj Dindi

is an indicator that takes thevalue one if firmsi andj are in the same industry and zero otherwise, andbej andmej are firmj ’s bookand market equities, respectively.

16 Theory supports scaling by the industry book-to-market,i.e., the value, not equal, weighted averagebook-to-market of firms in the industry. Using an equal-weighted average, like that employed by Asness,Porter and Stevens (2000) in their investigation of industry-relative characteristics, bias inefficient producerswith high expected returns towards the neutral portfolio. In the extreme, imagine an industry that consists ofa single, efficient oligopolistic firm and a large number of inefficient, marginal producers. Scaling the indi-vidual firms’ book-to-markets by the value-weighted industry average results in marginal producers havingthe maximum possible industry-relative book-to-market, while scaling by the equal weighted average resultsin these firms having an industry-relative book-to-market of one (the expected average across industries).Empirical tests confirm that scaling by industry book-to-market more effective, in a Sharpe ratio sense, thanscaling by equal-weighted industry average book-to-market. Scaling book-to-market by the equal-weightedindustry average does improve the Sharpe ratio of the high-minus-low quintile strategy, but only one third asmuch as the value weighted scaling procedure.

32

TABLE 5EXCESS RETURNS, THREE-FACTOR ALPHAS AND FACTOR LOADINGS, AND

CHARACTERISTICS OFPORTFOLIOS SORTED ON BOOK-TO-MARKET AND

INDUSTRY-RELATIVE BOOK-TO-MARKET, JUNE 1973 - JANUARY 2007

PANEL A: BOOK-TO-MARKET RELATIVE TO INDUSTRY BOOK-TO-MARKET

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML BM ME n

Low 0.386 -0.016 1.049 -0.098 -0.298 0.30 1810 944[1.46] [-0.28] [78.13] [-5.62] [-14.81]

2 0.500 0.009 0.948 -0.144 0.030 0.54 2011 668[2.34] [0.19] [85.94] [-10.06] [1.84]

3 0.721 0.105 0.997 -0.059 0.188 0.78 1359 699[3.30] [1.96] [77.98] [-3.56] [9.80]

4 0.790 0.054 1.035 0.191 0.248 1.02 662 854[3.36] [1.01] [80.67] [11.43] [12.89]

High 1.006 0.088 1.094 0.505 0.381 1.45 223 1219Rel

ative

BM

quin

tiles

[3.80] [1.35] [70.06] [24.88] [16.28]

H - L 0.620 0.104 0.045 0.603 0.679[4.28] [1.20] [2.17] [22.28] [21.75]

Sharpe ratio (annual) of the high-minus-low strategy: 0.738

PANEL B: BOOK-TO-MARKET

FF3 alphas and factor loadings characteristicsre ˛ MKT SMB HML BM ME n

Low 0.392 0.084 0.988 -0.130 -0.406 0.27 1912 1091[1.52] [1.63] [79.95] [-8.11] [-21.90]

2 0.650 0.024 1.056 -0.074 0.150 0.55 1304 768[2.74] [0.35] [64.39] [-3.46] [6.11]

3 0.714 -0.002 1.021 -0.015 0.342 0.79 906 744[3.20] [-0.03] [60.32] [-0.67] [13.49]

4 0.782 -0.047 0.993 0.045 0.573 1.08 685 783[3.72] [-0.79] [70.56] [2.48] [27.19]

High 0.960 -0.070 1.080 0.216 0.797 1.72 399 998

BM

quin

tiles

[4.13] [-1.18] [75.74] [11.65] [37.29]

H - L 0.568 -0.154 0.092 0.346 1.203[3.11] [-2.16] [5.39] [15.60] [46.98]

Sharpe ratio (annual) of the high-minus-low strategy: 0.537

Source:Compustat and CRSP.The table shows value-weighted average excess returns to portfolios sorted on book-to-

market scaled by industry book-to-market (Panel A) and portfolios sorted on book-to-market(Panel B), results of time-series regressions of these portfolios’ returns on the Fama-Frenchfactors, with test-statistics, and time-series average portfolio characteristics.

33

high and low portfolios, and three quarters to the fact that returns to the strategy are less

volatile.

Despite the large difference in the Sharpe ratios of the two strategies, the Fama-French

factors price the industry-relative book-to-market sorted portfolios well. The three factor

root mean squared pricing error of the five portfolios is only 6.6 basis points per month,

compared to 26.1 basis points per month for the market model. GRS tests reject that the

market model pricing errors are jointly zero (F5,397 D 5.88, p-value = 0.003%), but fail

to reject the same hypothesis for the three factor pricing errors (F5,395 D 1.29, p-value =

26.9%). The market model alpha on the high-minus-low strategy is 65.1 basis points per

month, with a test-statistic of 4.47, while the three factor alpha is only 10.4 basis points per

month and insignificant (test-statistic equal to 1.20).

Sorting on industry-relative book-to-market generates less spread in book-to-market,

as it must, because some high (low) book-to-market firms are average firms in high (low)

book-to-market industries. It produces greater variation, however, in average firm size;

firms with high book-to-market relative to their industries tend to be smaller. This is con-

sistent both with our model and the results presented in Table 3. The value effect seems

to be concentrated in small firms, at least in part, because size helps distinguish between

firms that have high book-to-markets because they are less efficient, and consequently more

exposed to economic risks, and firms that have high book-to-markets because they are in

high book-to-market industries.

The factor loadings of the high-minus-low strategy are consistent with the magnitude

on the variation in characteristics generated by the sort. The strategy load less on HML,

but more on SMB, than the corresponding strategy using a straight book-to-market sort.

While the high-minus-low strategy using industry-relative book-to-market loads on both

HML and SMB, the strategy’s high Sharpe ratio is not simply due to a fortunate rotation

of the Fama-French factors. We demonstrate this explicitly in the next section, noting here

only that while the three factor alpha on the strategy is not significantly higher than zero, it

is significantly higher than the alpha on the high-minus-low strategy for the straight book-

34

to-market sort (a difference of 25.8 basis points per month with a test-statistic of 3.05).

8 Efficiency of HML

The previous results suggest HML is not mean-variance efficient, and provides guidance

for constructing alternative value factors that may be closer to the efficient frontier. In this

section we construct two alternative factors. The first is based on a simple univariate sorting

procedure that is no more complicated than the construction of HML. The second uses a

somewhat more complicated two-stage procedure, which attempts to strip the unpriced

component out of HML to produce a cleaner exposure to the priced component. Both

alternative factors carry Sharpe ratios twice that of HML.

Our first, preferred, procedure employs the identical methodology Fama and French use

to construct HML, except that instead of book-to-market we sort on industry-relative book-

to-market, as we did to produce the portfolios tested in Panel A of Table 5. That is, we

begin by constructing six value weighted portfolios using the intersection of two size port-

folios (stocks above and below the size of the NYSE median) and three book-to-market

portfolios (firm book-to-market divided by industry book-to-market below the 30th per-

centile of NYSE industry-relative book-to-market, between the 30th and 70th percentiles,

and above the 70th percentile). The factor, HML�, is then constructed as 1/2 (small value -

small growth + large value - large growth). This industry relative factor HML� has a Sharpe

ratio twice that of HML (1.09 versus 0.54), due both to its higher average returns (57.6 ver-

sus 47.9 basis points per month) and lower standard deviation (1.83 versus 3.08 percent

per month). HML� prices HML well, but has a significant positive alpha relative to the

Fama-French factors (26.7 basis points per month with a test-statistic of 4.73). Its monthly

correlation with HML is 68.6 percent (for comparison, HML replicated in the same sample

is 99.3 percent correlated with the Fama and French monthly HML series).

Our second, more complicated strategy, is similar to that employed by Cohen and Polk

(1998). This strategy constructs separate intra-industry (priced) and inter-industry (un-

35

priced) factors, then uses the unpriced factor to remove as much of the unpriced vari-

ation from the priced factor as possible. To do so, we again employ the methodology

of Fama and French to construct two alternative versions of HML, one based on intra-

industry book-to-market (HMLN) and one based on industry book-to-market (HMLX).

These employ the same definitions used in the sorts of Tables 3 and 4. Our second factor,

HML C, is the part of intra-industry HML that is orthogonal to inter-industry HML,i.e.,

HML C D HMLN � ˇHMLX. The strategy is basically long a dollar of intra-industry

value stocks, short a dollar of intra-industry growth stocks, short 50 cents of value in-

dustries, and long 50 cents of growth industries. The Sharpe ratio of this orthogonalized

intra-industry value factor, HMLC, is again twice that of HML (1.13). The high Sharpe

ratio here is driven exclusively by the low volatility of the strategy (it returns 43.9 basis

points per month, with a standard deviation of only 1.34 percent per month). It has a sig-

nificant positive alpha relative to the Fama-French factors (30.7 basis points per month with

a test-statistic of 5.30), and is relatively weakly correlated with HML (32.4 percent at the

monthly frequency). HMLC has a significantly higher Sharpe ratio, and significantly lower

correlation with HML, than the factors produced by Cohen and Polk.

Figure 4 shows the time series of returns to the three strategies. The figure depicts trail-

ing one year average monthly returns to HML (blue), HML� (green) and HMLC (red). The

fact that HML is more volatile than HML�, and much more volatile than HMLC, is readily

apparent in the figure. The trailing one year average returns to these alternative factors

follow roughly the same basic trends as HML, exhibiting significantly higher correlations

with HML at the annual frequency (81.4 percent for HML�, and 49.7 percent for HMLC).

While HML� prices HML, either alone or in conjunction with MKT and SMB, it has a

significant positive alpha of 26.7 basis points per month with respect to the Fama-French

factors (test-statistic equal to 4.73). Moreover, it does a “better” job than HML pricing the

25 intra-industry book-to-market / industry book-to-market sorted portfolios of Table 4.

Using HML� instead of HML in conjunction with MKT and SMB yields an observed root

mean squared pricing error for the 25 portfolios of 21.1 basis points per month, compared

36

Jan 76 Jan 81 Jan 86 Jan 91 Jan 96 Jan 01 Jan 06−4

−3

−2

−1

0

1

2

3

4

5A

vera

ge m

onth

ly re

turn

s (%

)

HML

HML*

HML+

Figure 4: Average Monthly Returns to Value-Minus-Growth StrategiesThe figure shows one year trailing average monthly returns to three different value-minus-growth strategies. The blue (darkest) path is Fama and French’s HML. The green (nextdarkest) path (HML�) results from replicating the Fama-French procedure for constructingHML using industry-relative book-to-market. The red (lightest) path (HMLC) is the part ofHML constructed using intra-industry book-to-market not explained by HML constructedusing inter-industry book-to-market.

to 21.7 basis points per month for the Fama-French model and 32.1 basis points per month

for the market model. GRS tests reject the hypothesis that the pricing errors are jointly

zero for all three models, but the rejection is least emphatic for the model that includes

HML � (F25,375 D 1.95 for a p-value = 0.457%, compared toF25,375 D 2.36 for a p-value

= 0.031% for the Fama-French model, andF25,377 D 2.76 for a p-value = 0.002% for the

market model).

More surprising, HML� also does a “better” job pricing the 25 Fama-French book-to-

37

market / size portfolios. Using HML� instead of HML in conjunction with MKT and SMB

yields an observed root mean squared pricing error for these 25 portfolios of 14.7 basis

points per month, compared to 15.3 basis points per month for the Fama-French model

and 41.7 basis points per month for the market model. While GRS tests reject that the

pricing errors are jointly zero for all three models, the rejection is again least emphatic

for the model that includes HML� (F25,375 D 2.17 for a p-value = 0.115%, compared to

F25,375 D 2.70 for a p-value = 0.003% for the Fama-French model, andF25,377 D 3.70 for

a p-value = 0.000% for the market model).17

9 Investment Perspective

As noted previously, HML� seems to both price HML and have a significant positive alpha

relative to the Fama-French factors. In this section we will consider in greater detail how

the inclusion of this factor in the set of available assets affects the investment opportunity

set.

We also consider the impact of including a factor based on operating leverage, “levered-

minus-unlevered” (LMU). This factor is again constructed following the basic Fama-French

methodology, as 1/2 (small levered - small unlevered + large levered - large unlevered),

where the leverage tertiles are based on firms’ operating leverage within their own indus-

tries. This factor generates an average excess return of 32.6 basis points per month with

a standard deviation of 1.78 percent per month, giving is an annual Sharpe ratio of 0.63.

LMU is correlated 51.5 percent with HML, -40.8 percent with the market, and -22.8 percent

with SMB over the sample period of June 1973 to January 2007.

Finally, inspired by both Sagi and Seasholes (2007), who demonstrate how momentum

arises naturally in real options models, and the success of our own industry-relative value

factor, we additionally consider an industry relative momentum factor, UMD�. This factor

17 The average R-squared for the 25 portfolios is 87.6% for the model that includes HML�, compared to91.2% for the Fama-French model (and 71.9% for the market model). The difference is largely driven by thelarge, high book-to-market portfolios; HML� does not covary as strongly with the returns to big value stocksas does HML.

38

is constructed along the lines of UMD (or HML), as 1/2 (small winners - small losers +

large winners - large losers), except that “winners” and “losers” are defined by a stocks

performance relative to its own industry. That is, the tertile sort on returns over the first

eleven months of the previous year is based onr i � r i, as opposed tor i , wherer i is the

value weighted return to firmi’s industry.18 Asness, Porter and Stevens (2000) consider a

similar measure, which employs equal rather than value weighted industry returns, in both

Fama-MacBeth tests and to produce quintile sorted portfolios.

Table 6 reports alphas from time-series regressions of the two value factors, HML and

HML �, the two momentum factors, UMD and UMD�, and the operating leverage factor,

LMU, on various subsets of these five factors plus MKT and SMB. These “unexplained”

factor returns are reported both over the whole sample and over subsamples.

Panels A and B examine the relation of Fama and French’s value factor, HML, and our

industry relative value factor, HML�, to the covariance structure of asset returns. Panel A

shows that HML appears to be within the span of Fama and French’s MKT and SMB and

the industry relative value factor HML�. Panel B shows that the converse is false; HML�

resides outside the span of the Fama-French factors. The high Sharpe ratio associated with

high-minus-low relative book-to-market trading strategies in not due to a rotation of the

Fama-French factors. In fact, looking at Panel C it appears that HML� is as far outside the

span of the Fama-French factors as UMD.

Panels C and D examine the relationship of the momentum factor, UMD, and the in-

dustry relative momentum factor, UMD�, to the covariance structure of asset returns. Panel

C shows that while UMD appears to be outside the span of the Fama and French factors,

it appears to reside within the span of these factors plus UMD�. Again, the converse is

false. Panel D shows that UMD� resides outside the span of the Fama-French factors plus

UMD. This result is consistent with Asness, Porter and Stevens (2000), who find, contrary

to Moskowitz and Grinblatt (1999), that industry-relative momentum has predictive power

18 That is,r i �P

allj 11indj Dindimej r j =

Pallj 11indj Dindi

mej , where11indj Dindiis an indicator that takes the

value one if firmsi andj are in the same industry and zero otherwise, andmej is the market capitalizationof firm j

39

TABLE 6SPANNING TESTS: ˛S (BASIS POINTS PER MONTH)

FROM REGRESSIONS OF THEFORM yt D ˛ C ˇ 0xt C �t

Sub-periodsJune 1973 - June 1973 - April 1990 -

Independent Variables (x) January 2007 March 1990 January 2007

Panel A: HML as the dependent variable (y)

MKT, SMB, HML* -4.8 -12.4 2.4[-0.53] [-1.57] [0.16]

Panel B: HML* as the dependent variable (y)

MKT, SMB, HML 26.7 18.9 31.1[4.73] [3.27] [3.73]

Panel C: UMD as the dependent variable (y)

MKT, SMB, HML 96.0 92.7 111.1[4.43] [3.57] [3.25]

MKT, SMB, HML, UMD* -9.4 -15.7 -0.5[-0.96] [-1.32] [-0.03]

Panel D: UMD* as the dependent variable (y)

MKT, SMB, HML, UMD 21.3 26.3 16.7[3.18] [3.15] [1.59]

MKT, SMB, HML*, UMD 0.300 0.287 0.300[4.35] [3.36] [2.75]

Panel E: LMU as the dependent variable (y)

MKT, SMB, HML 26.8 26.8 36.1[3.47] [3.31] [2.98]

MKT, SMB, HML, UMD 18.9 26.4 24.9[2.46] [3.15] [2.08]

MKT, SMB, HML*, UMD* 17.8 25.8 22.5[2.09] [2.94] [1.74]

Source:Compustat and CRSP.The table reports intercepts from time-series regressions on various subsets of the Fama-

French factors (MKT, SMB and HML), momentum (UMD), industry-relative value andmomentum factors (HML� and UMD�), and the operating leverage factor (LMU). Resultsare given in basis points per month. Test-statistics are provided in brackets.

40

for a firm’s stock return beyond that captured by momentum.19

Panel E shows that the levered-minus-unlevered factor adds significant information to

the Fama-French factors. This information, however, is partly contained in UMD and

partly contained in HML�. The operating leverage factor consequently contributes only

marginally to MKT, SMB and the industry relative factors HML� and UMD�, and this

contribution is statistically insignificant in the second half of our sample.

Our investigation would not be complete without examining the risk-reward trade-offs

available to investors who can take positions in these factors, all of which represent viable

trading strategies. Table 7 reports the maximum ex post Sharpe ratio and the weights in the

corresponding tangency portfolio for various subsets of the seven considered factors over

the entire sample period, of June 1973 to January 2007.

Here it is again immediately apparent that the high Sharpe ratio associated with the

industry-relative book-to-market high-minus-low strategy is not simply due to a rotation in

the Fama-French factors. The realized Sharpe ratio on HML� alone exceeds that achieved

with the ex post optimal combination of the Fama-French factors (1.09 vs. 0.99).20 More-

over, allowing an investor to trade in the market as well as HML� significantly improves the

investment opportunity set. The ex post Sharpe ratio attainable using only HML� and MKT

exceeds that attainable using the three Fama-French factors plus UMD (1.29 vs. 1.27).

Adding the operating leverage factor to the Fama-French three improves the risk-return

trade-off by roughly two-thirds of the improvement wrought by the inclusion of the mo-

mentum factor (0.99 to 1.17, as opposed to 1.27). This difference in performance between

the leverage factor and momentum is even smaller in practice, because the LMU turns over

significantly less frequently than UMD, generating less trading costs. The weight on LMU

in the optimal portfolio that includes the Fama-French three and the leverage factor is large

19 Moskowitz and Grinblatt do not skip a month between ranking and portfolio formation, and their resultreflects strong industry momentum, together with short term reversals on individual stocks, at extremely shorthorizons. They also use a courser industry definition, employing only twenty industries.

20 While HML� does not simply result from a rotation of the Fama-French factors, it does largely subsumeSMB. Tangency portfolios that include both HML� and SMB typically take small short positions in SMB thathave little effect on the portfolio’s Sharpe ratio.

41

TABLE 7EX POST MEAN-VARIANCE EFFICIENT PORTFOLIOS:

WEIGHTS (%) AND SHARPE RATIOS (ANNUAL ),JUNE 1973 - JANUARY 2007

INCLUDED ASSETS

FF3 + MOMENTUM INDUSTRY RELATIVE

MKT SMB HML UMD HML* UMD* LMU Sharpe Ratio

Panel A: Strategies without momentum

100.00 0.41100.0 0.54

100.0 1.0927.59 19.37 53.04 0.9918.55 81.45 1.2920.71 13.45 23.23 42.61 1.1717.27 49.30 33.43 1.46

Panel B: Strategies that include momentum

22.45 12.91 42.63 22.01 1.2719.92 15.96 32.92 31.20 1.3915.48 17.73 66.79 1.5415.51 56.99 27.50 1.7215.52 46.70 21.94 15.84 1.77

Source:Compustat and CRSP.The table shows portfolio weights (in percents) and realized Sharpe ratios (annual) for

various subsets of the three Fama-French factors and momentum (MKT, SMB, HML andUMD) and our industry-relative factors based on value, momentum and operating leveragestrategies (HML�, HML� and LMU).

(42.6 percent), though this partly reflects the low volatility of the levered-minus-unlevered

strategy. On a volatility-weighted basis the optimal portfolio loads roughly equally on

HML and LMU. The leverage factor also generates an improvement of similar magnitude

when added to HML� and MKT, increasing the ex post Sharpe ratio from 1.29 to 1.46.

Allowing the inclusion of momentum strategies yields similar results. In conjunction

with the Fama-French three, industry-relative UMD� yields higher ex post Sharpe ratios

than UMD (1.39 vs. 1.27). UMD�’s outperformance of UMD is even more marked in con-

junction with strategies that include HML�: the optimal combination of MKT, HML� and

42

UMD generated a Sharpe ratio of “only” 1.54 over the period, while the optimal combina-

tion of MKT, HML � and UMD� generated a Sharpe ratio of 1.72. Allowing for trading in

LMU further increases this Sharpe ratio to 1.77.

10 The Role of HML

While the model presented in this paper is a one factor model, HML nevertheless arises

as a distinct second factor (or “pseudo-factor”) that, in conjunction with the market, helps

explain firms’ returns. That is, while the returns to both the market and HML are ultimately

driven by the single demand factor (X ), HML adds significant power to the market factor

for explaining the cross-section of assets returns.

HML helps explain firms’ returns for two reasons. First, HML helps identify firms that

have higher unconditional loadings on the risk factor. While inefficient firms have higher

unconditional loadings on the underlying risk factor, they are less exposed to this factor

in good times. A firm’s loading on the market factor in the time series is not sufficient to

determine the firm’s unconditional loading on the factor. The firm’s loading on HML is

correlated with the firm’s unconditional loading, and consequently helps predict returns.

If the price of risk is counter-cyclic, then HML additionally helps identify firms that

load higher on the risk factor when its price is high. A firm’s HML loading essentially proxy

for the counter-cyclicality of it’s risk exposure. If the price of risk is itself counter-cyclic,

then not only do inefficient firms load more heavily on the risk factor unconditionally, they

load disproportionately heavily when it has unusually high expected returns. Indeed HML

only has a high Sharpe ratio, in the model, if the price of risk is counter-cyclic.

While a single underlying factor drives both the market and HML, these derived factors

need not be strongly correlated. They may, in fact, be negatively correlated. The condi-

tional correlation between HML and the market depends on the state of the economy, with

HML and the market covarying positively in “recessions,” but negatively in “expansions.”

Figure 2 provides the intuition for this prediction. Below the critical threshold at which

43

deployed capital and growth options are equally exposed to the underlying risk factor, and

all the lines in Figure 2 intersect, both the market and HML load positively on the risk

factor, so covary positively. In “good times,” however, value firms are less exposed to the

risk factor than are growth firms, so HML is short the factor. At these times HML and

the market, which is always long the factor, covary negatively. The sign of the uncondi-

tional correlation between HML and the market is ambiguous; it can be either positive or

negative, depending on the parameters describing the economy.

In simulations HML has significant incremental power in explaining returns. In fact, in

these simulations HML and the market together explain almost all cross-sectional variation

in returns.21 With parameters that yield low unconditional correlations between HML and

the market we also observe very little dispersion in market betas, even when value firms

have significantly higher expected returns than growth firms.

A tension exists, between having a low correlation between HML and the market on

the one hand, and a high unconditional value premium on the other. Low correlations

between HML and the market require that the economy is often in a “good” state, because

it is at these times that HML and the market covary negatively.22 These “good” states

of the world are also, however, the times at which efficient firms are more exposed than

inefficient firm to the underlying risk factor. Low correlations between HML and the market

are therefore generally associated with less value-growth dispersion in unconditional risk

factor loadings. Nevertheless, employing a counter-cyclic risk premium it is possible to

pick parameters that yield uncorrelated value and market factors and a Sharpe ratio on the

value factor that exceeds that on the market.23

21 This is not particularly surprising, as the dependent variables have no real independent variation in themodel.

22 More precisely, it requires that the stationary distribution of the risk-neutral price process is skewedtowards the investment trigger. This occurs if the drift in the goods market price is “high” relative to itsvolatility.

23 Absent the counter-cyclic risk premium, HML exhibits a Sharpe ratio significantly below the market’sin long history simulations (10,000 years), though realized HML Sharpe ratios in 30 year subsamples exhibita great deal of variation and often exceed that on the market.

44

11 Conclusion

This paper provides direct empirical evidence, previously absent in the literature, for the

“operating leverage hypothesis,” underlying most theoretical explanations of the value pre-

mium. We also identify the reason that direct evidence has been elusive: difficulties, both

practical and theoretical, associated with testing the hypothesis directly.

We provide additional indirect support for the theory by deriving and testing implica-

tions that 1) do not require direct observation of operating leverage, and 2) account for

equilibrium effects. We demonstrate that, consistent with the predictions of our dynamic

equilibrium model, expected returns and book-to-market are strongly correlated within in-

dustries, but almost uncorrelated across industries.

Our results have important implications for investors. Investment strategies suggested

by the model significantly improve the investment opportunity set relative to the three factor

model and momentum. Over the sample from June 1973 to January 2007, the Sharpe ratio

available to investors with access to just three assets– the market and value and momentum

factors constructed using a procedure suggested by our model– exceeded that which could

have been achieved using the three Fama-French factors and momentum by over 35 percent.

Finally, while not emphasized in this paper, the model makes theoretical contributions,

and provides a rich, tractable environment for generating further empirical predictions. The

model has implications for the interpretation of investment-cashflow sensitivity regressions,

which we explore in Novy-Marx (2007b). The model makes additional, unexplored pre-

dictions relating the cross-section of asset returns to industry organization. It also suggests

a role for HML, as a “pseudo”-factor that helps identify stocks that both 1) load heavily on

risk factors unconditionally, and 2) load disproportionately heavily on risk factors when the

price of risk is high. This factor plays an important role, adding explanatory power beyond

that provided by the market, even in a model with a single priced risk factor in which assets

exhibit little variation in market loadings.

45

A Appendix: Proofs of Propositions

Proof of Proposition 4.1

Lemma A.1. LetEx [f (Xt)] � E [f (Xt ) jX0 D x] andE [�(!)I A] � E [�(!)11A(!)] for 11A(!) D1 if ! 2 A and 11A(!) D 0 otherwise. Now supposeX (1,v)

t is a drifted geometric Brownian process

between an upper reflecting barrier atv and lower reflecting barrier at 1, and letTv D minft >0jX (1,v)

t D vg andT1 D minft > 0jX (1,v)t D 1g denote the first passage times to the upper and

lower barriers, respectively. Then the state prices for hitting the lower barrier before the upper

barrier, and for hitting the upper barrier before the lower barrier, are given by

Eu[e�(rCı)T1 I T1 < Tv ] Dy (u=v)

y (1=v)(25)

Eu[e�(rCı)Tv I Tv < T1] Dy (u)

y (v)(26)

wherey(x) D x p � xˇn , and p andˇn are given in Proposition 5.1.

Proof of lemma: The state prices, discounting atr C ı, for the first passage of the process to the

upper and lower barriers may be written as

Euhe�(rCı)Tv

iD Eu

he�(rCı)Tv I Tv < T1

iC Eu

he�(rCı)T1 I T1 < Tv

iE1he�(rCı)Tv

i(27)

Euhe�(rCı)T1

iD Eu

he�(rCı)T1 I T1 < Tv

iC Eu

he�(rCı)Tv I Tv < T1

iEvhe�(rCı)T1

i.(28)

Solving the preceding equations forEuhe�(rCı)Tv I Tv < T1

iandEu

he�(rCı)T1 I T1 < Tv

iusing

Euhe�(rCı)Tv

iD�

uv

�p andEu

he�(rCı)T1

iD uˇn yields the lemma. �

Lemma A.2. The perpetuity factor for a geometric Brownian process currently atx with reflecting

barriers at a � x andb � x, which we will denote�b

a(x), is a homogeneous degree-zero function

of a, b, andx jointly, and

u�v

1(u) D u� C

�y (u=v)

y (1=v)

�(˘(v) � �) C

�y (u)

y (v)

��˘(v�1) � �

�(29)

wherey(x) D x p � xˇn and� ,˘(x), p , andˇn are given in Proposition 5.1.

Proof of lemma: SupposeX (1,v)0 D u 2 [1, v], whereX

(1,v)t is a geometric Brownian process

between an upper reflecting barrier atv and lower reflecting barrier at1. Then the value of the cash

46

flow e�ıtX(1,v)t discounted atr and starting att D 0 is

u�v

1(u) D Eu

�Z 1

0

e�(rCı)tX(1,v)t dt

�

D Eu

"Z T1_Tv

0

e�(rCı)tX(1,v)t dt

#C E1

�Z 1

T1

e�(rCı)tX(1,v)t dt I T1 < Tv

�

CEv

�Z 1

Tv

e�(rCı)tX(1,v)t dt I Tv < T1

�(30)

D�u � Eu[e�(rCı)T1 I T1 < Tv ] � Eu[e�(rCı)Tv I Tv < T1] v

��

C Eu[e�(rCı)T1 I T1 < Tv]˘(v) C Eu[e�(rCı)Tv I Tv < T1] v˘(v�1)

where� D 1rCı��

is the perpetuity factor for a geometric Brownian process discounted atr C ı,

and

˘(v) � E1

�Z 1

0

e�(rCı)tX(1,v)t dt

�

˘(v�1) � v�1Ev

�Z 1

0

e�(rCı)tX(1,v)t dt

�

are the perpetuity factors for the reflected process when it is at the lower and upper barriers, respec-

tively.

Then defining

�v

1(u) � Eu[e�(rCı)T1 I T1 < Tv ]

�v

1(u) � Eu[e�(rCı)Tv I Tv < T1]

completes the proof of the proposition, except for the explicit functional form for˘(v) and˘(v�1).

To obtain the explicit functional form for (v) and˘(v�1), note that the smooth pasting con-

dition implies

ddu

u�v

1(u)ˇˇuD1

D 0 (31)

ddu

u�v

1(u)ˇˇuDv

D 0, (32)

47

or

� C

�ˇnv p � pv

ˇn

�(˘(v) � �) C

�p � ˇn

�v�˘(v�1) � �

�

v p � vˇnD 0 (33)

� C

�ˇn � p

�v pCˇn�1 (˘(v) � �) C

�pv p � ˇnv

ˇn

� �˘(v�1) � �

�

v p � vˇnD 0. (34)

Solving the previous equations simultaneously yields the explicit values for˘(v) and˘(v�1). �

Proof of the proposition: The Bellman equation corresponding to firmi’s optimization problem

(equation (7)) is

rV i(K , X) D Ri(K , X) � ıK � rKV i(K , X)

C�X XV iX (K , X) C 1

2�2

X X 2V iXX (K , X). (35)

This equation essentially demands that the required return on the firm at each instant equals the

expected risk-adjusted return (cash flows and capital gains). It holds identically inKi , so taking

partial derivatives of the left and right hand sides with respect toKi yields

(r C ı)V iK i (K , X) D Ri

K i (K , X) � ıK � rK V iK i (K , X)

C�X XV iXK i (K , X) C 1

2�2

X X 2V iXXK i (K , X). (36)

Then using thatV i(K , X) is homogeneous degree one inK and inX , soqi(K , X) � V iKi

(K , X)

is homogeneous degree zero inK andX , and that� D ��X C ı C ( � 1)�2

X=2�

and� D �X ,

we can rewrite the previous equation as

(r C ı)qi(P) D�

1� n

c

�P C �Pq0

i(P) C 12�2P2q00

i (P) � �. (37)

Using the “myopic strategy” solution technique of Leahy (1993), we can “guess” that the firm’s

marginal valuation of capital is the product of 1) its marginal revenue products of capital and 2) the

unit value of revenues given the equilibrium price process. That is, we will guess thatqi(K t , Xt) �V i

Ki(K t , Xt) may be written as

qi(Kit , Pt ) D

�Ri

Ki(Ki

t , Pt ) C ���(Pt ) � (38)

whereRi(Kit , Pt ) D Ki

t

�Pt

ci� �

�is the firm’s unit profits and�(Pt ) D E

hR10

e�(rCı)s PtCs

Ptdsi

is the unit value of revenue for the geometric Brownian price process reflected above atPU and

below atPL, given explicitly in Lemma A.2, and D �rCı

is the capitalized unit cost of operating

48

capital in perpetuity.

The firm’s revenue depends on its capital stock directly, because it uses the capital stock to

produce the revenue generating good, and indirectly, because the price of the industry good depends,

partly, on the firm’s production. The firm’s marginal revenue product of capital, differentiating firm

revenueRi(K t , Xt ) D Kit Pt=ci with respect toKi , is

RiK i (K

it , Pt) C � D c�1

i Pt C c�1i Ki

tdPt

dK it

. (39)

Substituting the previous equation into equation (38), and using the facts thatdPt

dK it

D � Pt

ciSt,

which comes from differentiating the inverse demand functionPt D X t S

� t with respect toKi

t ,

andS it =St D

�c �

�1 �

n

�cj

�= c, which follows from the second condition of Proposition 4.1,

and the definitions ofL andC , we have that

q(Pt ) D�

1�L

C

�Pt �(Pt) � , (40)

where explicit dependence oni has been dropped because firms’ marginal valuations of capital

equate.

It is then simple to check thatq(P) satisfies the differential equation associated with the Bellman

equation. The marginal value of capital given in the previous equation satisfies equation (37) if and

only if

(r C ı)P�(P) D P C �P ddP

(P�(P)) C 12�2P2 d2

dP2 (P�(P)). (41)

This must hold for allP , so using the fact that

P�(P) D �P C aPˇn C bP p (42)

for somea andb (see, for example, equation (29)) and, matching terms of equalP -orders on the

left and right hand sides of equation (41), we then have that equation (37) holds if and only if

(r C ı � �)� D 1

(r C ı) � (� � �2

2)ˇn � �2

2ˇ2

n D 0

(r C ı) � (� � �2

2) p � �2

2ˇ2

p D 0,

which is easily verified.

The shadow price of capital,q(P), also satisfies the necessary boundary conditions. Evaluating

49

equation (40) atPU andPL yields

q(PU ) D 1 (43)

q(PL) D ˛. (44)

The smooth pasting condition at both boundaries,i.e., that q0i(PU ) D q0

i(PL) D 0, follows

immediately from equation (40) and the construction of�(P).

Finally, that firms invest/disinvest in proportion to their existing capital follows directly from

the fact that a firm internalizes more of the price externality that investment or disinvestment entails

when it has a bigger market share, soqi(P) is decreasing inKi=K, i.e., a firm’s value is strictly

convex in its capital share.

Proof of Proposition 5.1

The value of deployed capital is the expected discounted value of the operating profits it gener-

ates given the equilibrium goods price process,

OV i�Pt , Ki

t

�D

Ki

t

ci

!Pt�(Pt ) � Ki

t

��

r C ı

�. (45)

Substituting equation (40), the equilibrium condition on marginal-q, into the previous equation, we

get that a firm’s average-Q of assets-in-place is affine in its marginal-q, given by

OV it

Kit

D qt C�i (qt C ) (46)

where�i D cmaxci

� 1. The first term in the previous equation is just the shadow price of capital,

and is bounded betweenand one, while the second term represents the capitalized value of rents

expected to accrue to the deployed capital.

Total firm value also includes rents expected to accrue to future capital deployments, which will

be bought at a price below the value of the revenues it is expected to generate. It also accounts for

the costs associated with reducing capacity to support prices in “bad times,” when capital will be

sold at a price below the revenues it could have been expected to generate.

Firm i’s value satisfies the standard differential equation,�PVP C �2

2P2VPP D (r C ı)V ,

which implies

Qit D

OV it

Kit

C ain

�Pt

PL

�ˇn

C aip

�Pt

PU

�p

(47)

50

for someain and ai

p . This, taken with the differentiability of firm value at the investment and

disinvestment boundaries, implies the following proposition.

Capacity is insensitive to changes in the multiplicative demand shock away from the boundary,

so

dVi

dX

ˇˇX DX �

U

D Ki

�dP

dX

�d

dP

OV i

KiC ai

n

�P

PL

�ˇn

C aip

�P

PU

�p

! ˇˇX DX �

U

D Ki

X �

�ˇnai

n�ˇn C pai

p

�(48)

where we have used the facts that value of deployed capital is insensitive to changes inX at the

development boundary anddPdX

D P=X .

At the boundary, homogeneity of the value function impliesddX

Vi

Ki

ˇX DX C

UD 0, and the supply

response ensures the price never exceedsPU so d ln Ki

d ln X

ˇX DX

CU

D 1, so

d (Vi � Ki)

dX

ˇˇX DX C

U

D�

Vi

Ki� 1

�dKi

dX

ˇˇX DX C

U

DV �

i � Ki

X � . (49)

The value function is differentiable at the boundary,ddX

Vi

ˇX DX �

U

D ddX

(Vi � Ki)ˇX DX C

U

,

which, using the results of the previous two equations, yields

�ˇnai

n�ˇn C pai

p

�D Qi

U � 1, (50)

or, rearranging using the fact that at the investment boundaryOV iU=Ki D 1 C �i(1 C ) where

�i D cmax=ci � 1, that

( ˇn � 1) ain�

ˇn C� p � 1

�ai

p D �i(1 C ). (51)

A completely analogous calculation at the disinvestment boundary implies

( ˇn � 1) ain C

� p � 1

�ai

p�� p D �i(˛ C ). (52)

Solving the previous two equations simultaneously yields

ain

�i

D(1 C ) � � p(˛ C )

( ˇn � 1)��ˇn � � p

� (53)

aip

�iD

(˛ C ) � ��ˇn(1 C )� p � 1

� ��� p � ��ˇn

� . � (54)

51

Proof of Proposition 6.1

Proof of the proposition: Combining equations (15), (16), and (17), together with the fact that

˘(�)PL D (˛C )cmax and˘���1

�PU D (1 C )cmax, we have that a firm’s average-Q is given

by

Qit D

�Pt

ciC C i

ˇn

�Pt

PL

�ˇn

C C ip

�Pt

PU

�p

� . (55)

The proposition then follows directly.

Proof of Proposition 6.2

Lemma A.3. The stationary density for the risk-neutral price process is given by

d�P (p) D �

p��1

P�U

� P�L

!dp (56)

where� D 2��2 � 1.

Proof of lemma: SupposeXt is a geometric Brownian process with drift� and volatility� , and a

lower reflecting barrier atl and an upper reflecting barrier at1. Then

limt!1

t�1E�Z t

0

11[l,z]Xsds

�D P [X < z] (57)

where 11A(!) D 1 if ! 2 A and 11A(!) D 0 otherwise, andX is has the stationary distribution of

the processXt .

If Yt is a geometric Brownian process with the same drift and volatility, also reflected above at

1 but unreflected below, then by the Markovian nature of the processes

XtdD Yst

(58)

wheredD denotes “equal in distribution,” andst � minfsj

R s0

11[l,1]Ysds D tg. So

P [X < z] D P [Y < zjY > l ] (59)

DP [Y < z] � P [Y < l ]

1 � P [Y < l ]. (60)

Finally, using the fact that a Brownian process with positive drift reflected from above at zero

has a stationary distribution that is exponentially distributed, with exponent equal to its drift divided

52

by half its volatility squared, we have thatP [Y < y] D y� where� D (� � �2=2)=(�2=2).

Substituting this into the previous equation, and lettingXt D Pt=PU , l D PL=PU and z DPT =PU , yields the stationary distribution for the equilibrium price process,

P [P < p] Dp� � P

�L

P�U

� P�L

. (61)

Differentiating with respect top yields the lemma.

Proof of the proposition:The proposition follows directly from the preceeding lemma.

References

[1] Abel, Andrew B., and Janice C. Eberly, 1996, “Optimal Investment with Costly Reversibility,”Review of Economic Studies,Vol. 63, pp. 581-593.

[2] Aguerrevere, Felipe L., 2006, “Real Options, Product Market Competition, and Asset Re-turns,”mimeo.

[3] Asness, Cliff, R. Burt Porter and Ross Stevens, 2000, “Predicting Stock Returns UsingIndustry-relative Firm Characteristics,” AQR Capital Management working paper.

[4] Carlson, Murray, Adlai Fisher, and Ron Giammarino, 2004, “Corporate Investment and AssetPricing Dynamics: Implications for the Cross-section of Returns,”Journal of Finance56, pp.2577-2603.

[5] Cohen, Randolph B., and Christopher K. Polk, 1998, “The Impact of Industry Factors inAsset-Pricing Tests,”mimeo.

[6] Daniel, Kent, and Sheridan Titman, 1997, “Evidence on the Characteristics of Cross SectionalVariation in Stock Returns,”Journal of Finance52, 1-33.

[7] Davis, James L., Eugene F. Fama and Kenneth R. French, 2000, “Characteristics, Covariances,and Average Returns: 1929 to 1997,”Journal of Finance55, 389-406.

[8] Fama, Eugene F., and Kenneth R. French, 1992, “The Cross Section of Expected Stock Re-turns,”Journal of Finance47, pp. 427-465.

[9] Fama, Eugene F., and Kenneth R. French, 1993, “Common Risk Factors in the Returns onStocks and Bonds,”Journal of Financial Economics33, pp. 3-56.

[10] Gibbons, Michael R., Stephen A. Ross, and Jay Shanken, 1989, ”A Test of the Efficiency of aGiven Portfolio,”Econometrica57 (5), 1121-1152.

[11] Green, Edward J., and Robert H. Porter, 1984, ”Noncooperative Collusion under ImperfectPrice Information,”Econometrica52 (1), 87-99.

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[12] Grenadier, Steven R., 2002, “Option Exercise Games: An Application to the EquilibriumInvestment Strategy of Firms,”Review of Financial Studies15, pp. 691-721.

[13] Kogan, Leonid, 2004, “Asset Prices and Real Investment,”Journal of Financial Economics73, pp. 411-431.

[14] Leahy, John V., 1993, “Investment in Competitive Equilibrium: The Optimality of MyopicBehavior,”Quarterly Journal of Economics108, 1105-1133.

[15] Lev, Baruch, 1974, “On the Association Between Operating Leverage and Risk,”Journal ofFinancial and Quantitative Analysis9, 627-641.

[16] Moskowitz, Tobias J., and Mark Grinblatt, 1999, “Do Industries Explain Momentum?”Jour-nal of Finance54, 1249-1290.

[17] Novy-Marx, Robert, 2007a, “Real Options and Investment in a Dynamic Model of Oligopoly,”working paper.

[18] Novy-Marx, Robert, 2007b, “Investment-Cash Flow Sensitivity and the Value Premium,”working paper.

[19] Pastor,Lubos, and Robert F. Stambaugh, 2003, “Liquidity Risk and Expected Stock Returns,”Journal of Political Economics111, 642-685.

[20] Pindyck, Robert S., 1985, “The Measurement of Monopoly Power in Dynamic Markets,”Journal of Law and Economics28, 193-222.

[21] Sagi, Jacob S., and Mark S. Seasholes, 2007, “Firm-specific attributes and the cross-sectionof momentum,”Journal of Financial Economics,forthcoming.

[22] Zhang, Lu, 2005, “The Value Premium,”Journal of Finance60, pp. 67-103.

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