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 leverage
hypothesis,” which underlies most theoretical models of the value premium. We also
show that tests that rely on direct inference using the level of operating leverage are
biased against rejecting alternative hypotheses, due both to empirical issues associated
with measuring operating leverage and theoretical issues associated with equilibrium
behavior. We consequently generate indirect implications of the hypotheses, which
do not require direct observation of operating leverage and account for equilibrium
effects. Specifically, the model predicts that the relationship between expected returns and book-to-market is weak and non-monotonic across industries, but strong
and monotonic within industries. The data support these predictions. The model also
suggests a measure of book-to-market that controls for industry. HML constructed
using this measure over the sample from June 1973 to January 2007 has a Sharpe ratio
that exceeds the ex post optimal combination of the three Fama-French factors, and an
information 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 support from the Center for the Research in Securities Prices at the University of Chicago Graduate School of
Business 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’ explanation 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 leverage 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 revenues (unobservable). Theoretically, simple predictions of the theory derived in partial
equilibrium models are not robust to equilibrium analysis. We consequently derive additional, 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 enhancing 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 operating leverage. Sagi and Seasholes show theoretically that operating leverage reduces asset return autocorrelation, 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 value
firms 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 because 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 consist disproportionately of growth options load negatively on HML. HML loadings consequently 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 does not generate any systematic variation in bookto-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 understates 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 associated 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 operating leverage, are plagued by measurement issues. The level of operating leverage depends,
2
theoretically, on the capitalized values of all future costs and revenues. While market values 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 leverage. 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 operating leverage is itself unobservable. 2 In particular, the model predicts that the relationship between expected returns and industry book-to-market is weak and non-monotonic,
but that the relationship between expected returns and book-to-market within 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 bookto-market, however, generates no significant variation in returns, despite generating significant variation in book-to-market and more variation in HML loadings than the intraindustry 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 (respectively, growth) firms in growth industries, despite large differences in these firms’ book-tomarket 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 linking it to the operating leverage mechanism. Aguerrevere (2006) argues that operating leverage model and
equilibrium effects together imply that competitive industries should be riskier than concentrated industries
in recessions, but less risky in expansions.
3
This investigation also supports another prediction of the model, regarding the interaction 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 between 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 leverage 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 factor”) 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 countercyclical price of risk.
4
The remainder of the paper is organized as follows. Section 2 discusses the basic intuition behind the operating leverage hypothesis in a model-free context, derives some basic
predictions of the hypothesis, and provides empirical support for these predictions. Sections 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 fundamental 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 momentum 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 VAi C VGi where i denotes the firm and the
subscript A 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.,
ˇ
i
D
i
VAi
VG
i
i
ˇA C
ˇG
.
i
V
Vi
(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, VAi D VRi
VCi . 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,
ˇAi
D
i
ˇR
VCi
C
VAi
i
ˇR
ˇCi .
(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 significantly impact the riskiness of deployed capital requires both “high operating leverage” and
i 4
“limited operational flexibility,” i.e., VCi =VAi 0 and ˇCi ˇR
.
All firms basically satisfy the “high” operating leverage conditions, with VCi =VAi 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 crosssection. We will exploit this variation in our empirical tests, and look at differences in the
return characteristics of relatively levered and unlevered firms.
The second condition, “limited” operational flexibility, is less obvious. It does not typically hold, for example, in standard quadratic adjustment cost models. Nevertheless, both
the data and introspection seem to suggest that the existence of limit operational flexibility 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 ag3
Growth options here represent the value of future investment opportunities. Under this definition, a firm
that is actively trying to develop a drug from a molecule on which it has a patent is not really a high growth
options firm– a firm that owns a patent on a similar molecule but is deferring active drug development is. It
seems reasonable that the high growth option firm is less risky. Bad news about the class of molecule related
to the firms’ patents will hurt the value of the firm that just owns the patent, but may devastate the firm that is
incurring costs actively developing the drug.
4
The asymmetric adjustment cost function in Zhang (2005), with a quadratic adjustment penalty that is
significantly higher for disinvestment than investment, is essentially designed to generate this limited operational 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 simple 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
ˇ
i
D
VAi
Vi
i
VC
i
i
ˇR C
ˇR
VAi
ˇCi
C
VGi
Vi
i
.
ˇG
(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 proportional 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
i
ˇ i D BM i OLi ˇR
ˇCi
i
ˇG
i
ˇR
i
C ˇG
(4)
5
In general, the value of deployed capital may be either greater or less than its book value. For example, if
deployed capital generates economic rents, then its market value can exceed its book value by the capitalized
value of these rents. Conversely, if investment in not costlessly reversible, then if demand falls the market
value of capital can fall below its book value.
7
where BM i is firm i’s book-to-market and OLi 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 assetsin-place and growth options.
If book-to-market and operating costs-to-book both correlate with a firm’s risk exposure, 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 preferred 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 timeseries 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-tomarket (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 portfolio 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 investment
tax 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, or
else carrying value (item 130). We also follow Fama and French in reducing shareholder equity by postretirement benefit assets (item 330) if available, in order to neutralize discretionary differences in accounting
methods firms choose to employ under the Financial Accounting Standards Board’s statement regarding employers’ accounting for postretirement benefits other than pensions (FASB 106). Results are not sensitive to
this adjustment.
7
This seems to be the primary reason Fama and French build HML using book-to-market constructed as
June book equity divided by the previous December’s market equity. Using market equity from the June date
of portfolio formation biases the value portfolio toward recent losers and the growth portfolio toward recent
winners, resulting in a value factor that is short momentum, yielding a lower factor Sharpe ratio.
9
Operating leverage quintiles
TABLE 1
E XCESS R ETURNS , T HREE -FACTOR A LPHAS AND FACTOR L OADINGS , AND
C HARACTERISTICS OF P ORTFOLIOS S ORTED ON B OOK O PERATING L EVERAGE
AND M ARKET O PERATING L EVERAGE , J UNE 1973 - J ANUARY 2007
Low
2
3
4
High
H-L
re
0.394
[1.47]
0.528
[2.16]
0.641
[2.64]
0.599
[2.35]
0.803
[3.01]
0.409
[2.33]
PANEL A: B OOK O PERATING L EVERAGE
FF3 alphas and factor loadings
characteristics
˛
MKT
SMB
HML
OL BM ME
n
0.051
0.998
-0.081
-0.373
0.43 0.55 1535 685
[0.59] [47.83] [-2.98] [-11.92]
0.035
1.017
-0.058
-0.093
0.80 0.58 1521 581
[0.51] [63.08] [-2.79] [-3.87]
0.089
1.009
0.041
-0.019 1.06 0.58 1024 579
[1.24] [58.87] [1.82]
[-0.74]
-0.019
1.006
0.226
0.017
1.37 0.66 615 669
[-0.21] [45.64] [7.89]
[0.51]
0.157
1.043
0.138
0.082
2.48 0.54 525 762
[1.25] [34.81] [3.55]
[1.84]
0.106
0.045
0.219
0.455
[0.62]
[1.10]
[4.13]
[7.46]
Operating leverage quintiles
Sharpe ratio (annual) of the high-minus-low strategy: 0.402
Low
2
3
4
High
H-L
re
0.351
[1.35]
0.578
[2.38]
0.793
[3.23]
0.795
[3.35]
0.985
[3.94]
0.634
[3.48]
PANEL B: M ARKET O PERATING L EVERAGE
FF3 alphas and factor loadings
characteristics
˛
MKT
SMB
HML
OL BM ME
0.098
0.956
-0.131
-0.485
0.46 0.35 1975
[1.79] [73.16] [-7.69] [-24.75]
0.008
1.038
-0.032
0.027
0.79 0.55 1287
[0.12] [61.28] [-1.47]
[1.06]
0.038
1.059
0.131
0.295
1.00 0.75 1004
[0.41] [47.59] [4.51]
[8.84]
-0.050
1.044
0.204
0.460
1.29 0.88 571
[-0.58] [50.54] [7.59]
[14.87]
0.079
1.025
0.394
0.498
2.17 1.04 246
[0.76] [41.44] [12.26] [13.42]
-0.019
0.069
0.525
0.982
[-0.17] [2.53] [14.76] [23.95]
n
789
573
519
577
806
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 portfolios’ returns on the Fama-French factors, with test-statistics, and time-series average portfolio
characteristics. Operating leverage and market operating leverage are cost of goods sold plus selling, general and administrative expenses (Compustat annual data item 41 plus item 189), scaled
respectively by the book or market value of assets (item 6, or item 6 plus market equity minus
book 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 characteristic. 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 FamaFrench 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 surprising, 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, pvalue = 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 leveredminus-unlevered portfolio by a statistically insignificant -1.9 basis points per month, (teststatistic 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 difference is only significant in the third and fourth book-to-market quintiles. The FamaFrench three factor model again does a better job pricing the levered-minus-unlevered
12
Book-to-market quintiles
TABLE 2
E XCESS R ETURNS , BOOK - TO -M ARKET, AND T HREE -FACTOR A LPHAS AND
FACTOR L OADINGS FOR P ORTFOLIOS D OUBLE S ORTED ON O PERATING L EVERAGE
AND B OOK - TO -M ARKET, J UNE 1973 - J ANUARY 2007
Monthly excess returns and BMs for
Return characteristics of
portfolios sorted on operating leverage
the levered-minus-unlevered
portfolios
and book-to-market
Operating leverage quintiles
FF3 alphas and loadings
L
2
3
4
H
re
˛
ˇmk t 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
(0.54)
0.637
(0.55)
0.770
(0.55)
0.691
(0.55)
0.858
(0.55)
0.316 0.044 0.016 0.241 0.411
[1.50] [0.21] [0.31] [3.66] [5.43]
3
0.474
(0.79)
0.727
(0.79)
0.821
(0.79)
0.835
(0.55)
1.036
(0.79)
0.562 0.240 0.042 0.353 0.421
[2.49] [1.08] [0.78] [5.07] [5.24]
4
0.507
(1.07)
0.851
(1.08)
1.121
(1.07)
0.841
(1.08)
1.001
(1.07)
0.494 0.264 -0.069 0.565 0.234
[2.18] [1.21] [-1.32] [8.29] [2.98]
H
0.908
(1.77)
0.534
[2.08]
1.001
(1.84)
0.604
[2.45]
1.141
(1.76)
0.741
[2.88]
0.974 1.070
(1.79) (1.85)
0.660 0.408
[2.51] [1.64]
0.162 0.071 -0.223 0.638 0.074
[0.68] [0.32] [-4.13] [9.11] [0.92]
high-minus-low portfolios
FF3 alphas and loadings
re
˛
-0.330 -0.199 -0.119 -0.117 -0.236
[-1.68] [-1.00] [-0.61] [-0.58] [-1.24]
ˇmk t
0.258
[5.48]
0.220
[4.63]
0.190
[4.04]
0.065 -0.041
[1.34] [-0.89]
ˇsmb
0.366
[5.98]
1.298
[18.39]
0.578
[9.35]
1.092
[15.34]
0.633
[10.35]
1.214
[17.23]
0.555
[8.87]
1.229
[17.05]
ˇhml
0.902
[15.22]
0.870
[12.73]
Source: Compustat and CRSP.
The table shows value-weighted average excess monthly returns and book-to-market ratios (in parentheses) of portfolios double sorted on operating leverage and 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].
portfolios, despite the fact that the levered and unlevered portfolios do not differ in bookto-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 monotonic, 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 investment 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 of n competitive firms, which are assumed
to maximize the expected present value of risk-adjusted cash flows discounted at the constant risk-free rate r . 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 good S ti D K it =ci where K it is the
8
In the case of complete irreversibility (i.e., ˛ D 0) we still allow for the free disposal of capital.
14
firm’s capital stock and ci is the firm’s capital requirement per unit of production (i.e.,
the firm’s inverse capital productivity). Aggregate production is then S t D
K t D (K 1t , K 2t , ..., K nt )0 and
K t , where
D (c1 1 , c2 1 , ..., cn 1 ) denote the vectors of firms’ capital
stocks and firms’ capital productivities, respectively. Aggregate capital employed in the
industry is K t D 1 K t where 1 D (1, 1, ..., 1) is the n-vector of ones.
In the goods market firms face a stochastic level of iso-elastic demand, with price elasticity 1= > 1=n.9 Operating revenue generated by each unit of capital employed by firm
i is therefore P t =ci , where
Pt D
Xt
St
,
(5)
where we assume that the multiplicative demand shock X t is a geometric Brownian process
under the risk-neutral measure,
dX t D X X t dt C X X t dB t
where X < r and X are known constants, and B t 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
K it
R (K t , X t ) D
ci
i
Xt
Kt
K it .
(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 rate ı. The change in a firm’s capital
stock can therefore be written as dK it D
9
ıK it C d U ti
dLit , where U ti (respectively, Lit )
This condition assures that no firm can increase its own revenues by decreasing output.
15
denotes firm i’s gross cumulative investment (respectively, disinvestment) up to time t .
4
The Optimal Investment Strategy
The value of a firm’s investment depends on the price of the industry good, and consequently 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 exhibiting 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 firm i is given by
V i (K t , X t ) D
max
Z
e
Et
i
fdUtCs
,dLitCs g
1
rs
ˇ
i
i
i
i
Ri (K tCs , X tCs )ds dU tCs C ˛dL tCs ˇfdU tCs , dL tCs g
(7)
0
where fdU t i , dL t i g is used to denote other firms’ investment/disinvestment at time t , 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 consequently internalize more of the negative price externality associated with their own investment, 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 extremely
interesting, 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 a Q-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 efficient,” in that its capital requirement per unit of production is not too high, satisfying
ci < cmax where c 1
n
Pn
j D1 cj
c
1
(8)
n
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,”
j
S ti =S t D (cmax ci ) = cmax cj , which requires that firms’ capital stocks satisfy
K0i D
for each i, where c 2 1
n
cci
1
c2
1
ci2
n
c2
n
!
K0
n
(9)
Pn
2
iD1 cj .
Then all firms will invest in new capital, in proportion to their existing capital, whenever
11
This first condition
is satisfied trivially
set of firms’ unit costs c1 c2 ... cM ,
n
o if, given the order
Pi
ci
1
we let n max i 2 f1, ..., M gj ci < 1 where c i D i j D1 cj and restrict attention to the first n firms.
i
17
the goods price P t reaches PU , and disinvest whenever P t reaches PL , where
D
r Cı
PU D
(1 C )C
(1 L) ˘(
PL D
(˛ C )C
,
(1 L) ˘()
(10)
1)
(11)
is the capitalized flow costs associated with operating a unit of capital in perpe-
tuity, C K t =S t is industry average operating costs per unit of production,
Pn
C D Pn
j D1
j D1
K jt
K jt =cj
n
D
c
1
n
c2 c
,
(12)
L D H is the “pseudo”-Lerner index, 12 where 1= is the price elasticity and H Pn
j
2
j D1 (S t =S t ) is the Herfindahl index associated with the equilibrium organization,
n
X
H D
Pn
iD1
and ˘(
1
!2
K it =ci
D
j
j D1
K t =cj
1
1
1
n
C
c
,
) and ˘() are the perpetuity factors for the equilibrium price process at the
investment and disinvestment thresholds, respectively,
˘(x) D
1
2
2(r Cı)
y 0 (1) y 0 (x)
y(x)
x (13)
q
1 2 2(r Cı)
1
1
where y(x) D x
x , and ˇp D
C 2
, ˇn D
2
2
2
2
2
2
q
2
X
1 2 2(r Cı)
1
C 2 , D r Cı , D X C ı C ( 1) 2 and D X , and D
2
2
ˇp
ˇn
12
In the standard Cournot model the market Lerner index (fraction by which output-weighted average
marginal cost falls below price in the goods market) is equal to H . In the economy in this paper, in which
capital 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 power
index as L D (P FMC )=P where FMC is the “full marginal cost” of production, which includes the
costs of operating as well as purchasing capital, then L D L. More generally, L will lie in a region that
includes L.
18
PU =PL > 1 is uniquely determined by
˛C
˘()
D
˘( 1 )
1C
5
.
(14)
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(P t ) C i
(q(P t ) C
„
„ƒ‚…
shadow
price of
capital
where i D
cmax
ci
ƒ‚
) C an
…
Pt
PL
ˇn
C ap
„
Pt
PU
ƒ‚
capitalized
rents to
deployed capital
ˇp !
(15)
…
expansion
and contraction
options
1 is firm i’s “excess productivity,” the shadow price of capital is given
by
q(P t ) D
1 L
C
P t (P t )
(16)
where
P
P
U
P
y PL
U
P (P ) D P C
P y P
PL C PUL ˘ PPUL
y
U
˘ P
PL
y
19
PL
PU
(17)
for y(x) D x ˇp
x ˇn , and an and ap are given by
(1 C ) ˇp (˛ C )
(ˇn 1) y ()
(˛ C ) ˇn (1 C )
.
D
ˇp 1 y ( 1 )
an D
(18)
ap
(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 averageQ, 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
Q
low cost producer
3.5
3
average cost producer
2.5
2
1.5
high cost producer
1
0.5
P
0
0.8
0.9
1
1.1
1.2
1.3
Figure 1: Tobin’s Q in the cross-section
The figure depicts average-Q for three firms in an industry, as a function of the price of
the 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. The
middle 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 are r 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.
proposition this will allow us to explicitly relate firms’ expected returns to their book-tomarkets.
Proposition 6.1. The expected excess rate of return to firm i is ˇ it t where t is the time-t
price of exposure to the priced risk factor ( X ) and
ˇ it D
i
Qt
1
P t
C Cˇi n
ci
Pt
PL
21
ˇn
ˇn C Cˇip
Pt
PU
!
ˇp
ˇp
(20)
where
Cˇi n
Cˇip
cmax
D
ˇn
ci
cmax
D
ˇp
ci
1 an
1 ap
ˇP
PL
ci
y ()
1
ˇn
PU
.
ci
y ( 1 )
(21)
(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 revenues 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 expected 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 when
prices in the goods market, and firms’ values relative to book capital, are low. Zhang shows that both high operating costs and operating inflexibility are required to generate a value premium, and these are the necessary
conditions for generating significant variation in the exposures of high and low cost producers to fundamental
risks. 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 of
their value from assets-in-place and little from future investment opportunities, and assets-in-place are highly
exposed to fundamental risks in normal times but insensitive to these risks in expansions.
22
Β
high cost producer
4
3
average cost producer
2
low cost producer
1
P
0.8
0.9
1
1.1
1.2
1.3
Figure 2: Risk factor loadings in the cross-section
The figure depicts risk factor loadings for three firms in an industry, as a function of prices
in 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 flat
solid line shows a low cost producer (ci D (C =cmax )C ). Other parameters are r 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.
Proposition 6.2. The unconditional expected rate of return to firm i is
Z
rie
PU
ˇ i (p)(p)d(p)
D
(23)
PL
where ˇ i (P t ) is given in Proposition 6.1 and d(p), the stationary density for the riskneutral price process, is given by
!
d(p) D p
1
PL
PU
for D
2
2
1.
23
dp
(24)
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 does not
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 the capitalized value 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 relationship across industries.
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 operating 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 leverage, increasing operating leverage is also associated with higher expected returns, because
it increases the riskiness of deployed capital. At high levels of operating leverage, however, 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-markets within 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 relationship 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 bookto-market and expected returns are strongly correlated within an industry, it predicts that
the relationship between book-to-market and expected returns is weak, and non-monotonic,
across industries.
In order to test this prediction, we perform separate sorts based on intra-industry bookto-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
E@re D
high
leverage
industry
0.14
0.12
average
leverage
industry
0.1
low
leverage
industry
0.08
0.06
industry average (value-weighted)
BM
0.25
0.5
0.75
1
1.25
1.5
1.75
Figure 3: BM / expected return relationship, in and across industries
The figure depicts the unconditional relationship between expected excess returns and bookto-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 ( D
2.5), the middle curve (dashed line) shows a moderately levered industry ( D 1), while
the bottom curve (dotted line) shows a relatively unlevered industry ( D 0.4). The bold
line depicts the relationship between expected excess industry returns and industry book-tomarket. Other parameters are r 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.
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 bookto-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 intraindustry 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 valueminus-growth strategy is -49.9 basis points per month, with a test-statistic of -4.64, while
27
Intra-industry BM quintiles
TABLE 3
E XCESS R ETURNS , T HREE -FACTOR A LPHAS AND FACTOR L OADINGS ,
AND C HARACTERISTICS OF P ORTFOLIOS S ORTED ON B OOK - TO -M ARKET
W ITHIN AND ACROSS I NDUSTRIES , J UNE 1973 - J ANUARY 2007
PANEL A: B OOK - TO -M ARKET W ITHIN I NDUSTRY
FF3 alphas and factor loadings
characteristics
re
˛
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
[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
Industry BM quintiles
Low
2
3
4
High
High-Low
PANEL B: I NDUSTRY B OOK - TO -M ARKET
FF3 alphas and factor loadings
re
˛
MKT
SMB
HML
0.487 0.252
0.950
-0.126
-0.517
[1.82] [3.11] [48.89] [-4.99] [-17.76]
0.511 -0.104 1.082
-0.003
0.054
[1.97] [-1.07] [46.44] [-0.08]
[1.55]
0.608 -0.036 1.058
-0.017
0.152
[2.46] [-0.39] [47.32] [-0.59]
[4.53]
0.784 0.042
1.007
-0.100
0.462
[3.45] [0.40] [40.00] [-3.07] [12.25]
0.579 -0.248 0.987
-0.008
0.607
[2.73] [-3.31] [55.07] [-0.36] [22.58]
0.091 -0.499 0.037
0.118
1.124
[0.47] [-4.64] [1.43]
[3.51]
[29.03]
characteristics
BM ME
n
0.32 1205 1004
0.52
925
972
0.65
840
852
0.81
1137
793
1.10
1140
738
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 of
these portfolios’ returns on the Fama-French factors, with test-statistics, and time-series average
portfolio 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 intraindustry 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 generate 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-tomarket 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, improving 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 errors 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 threefactor root mean squared pricing error is 52.9 basis points per month, versus 12.8 basis
29
Intra-industry H-L portfolios
FF3 alphas and loadings
Intra-industry BM quintiles
TABLE 4
E XCESS R ETURNS , BOOK - TO -M ARKET R ATIOS , AND T HREE -FACTOR A LPHAS
AND FACTOR L OADINGS FOR P ORTFOLIOS D OUBLE S ORTED ON B OOK - TO -M ARKET
W ITHIN AND ACROSS I NDUSTRIES , J UNE 1973 - J ANUARY 2007
Monthly excess returns and BMs for
Return characteristics of
portfolios sorted on industry BM
the value industries-minusgrowth industries portfolios
and within industry BM
Industry BM quintiles
FF3 alphas and loadings
L
2
3
4
H
re
˛
ˇmk t 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
(0.29) (0.45) (0.54) (0.67) (0.84)
0.054 -0.446 0.015 0.144 0.944
[0.28] [-3.12] [0.43] [3.23] [18.36]
3
0.676 0.505 0.679 0.961 0.548
(0.41) (0.62) (0.71) (0.67) (1.06)
-0.128 -0.607 0.053 -0.053 0.971
[-0.61] [-3.84] [1.39] [-1.07] [17.10]
4
0.739 1.088 0.773 0.906 0.501
(0.63) (0.84) (0.93) (1.06) (1.31)
-0.239 -0.608 0.032 -0.437 0.988
[-0.98] [-3.59] [0.78] [-8.27] [16.23]
H
1.006
(1.12)
0.609
[2.74]
-0.224 -0.455 0.048 -0.555 0.748
[-1.00] [-2.80] [1.24] [-10.96] [12.83]
re
1.104
(1.33)
0.742
[4.05]
0.795
(1.41)
0.247
[1.53]
1.333
(1.54)
0.860
[5.43]
0.782
(1.91)
0.293
[1.73]
˛
-0.112 0.304 -0.048 0.618 -0.060
[-0.72] [2.03] [-0.31] [4.18] [-0.41]
ˇmk t
0.075 -0.041 0.021 -0.028 0.043
[2.01] [-1.14] [0.58] [-0.78] [1.22]
ˇsmb
0.881
[18.29]
0.911
[16.42]
ˇhml
0.521
[11.16]
0.660
[12.28]
0.327
[6.85]
0.403
[7.32]
0.129
[2.79]
0.462
[8.69]
0.075
[1.63]
0.644
[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 timeseries regressions of both sorts’ high-minus-low portfolios’ returns on the Fama-French factors, with teststatistics [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 differences 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 difference 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 bookto-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 with significantly higher 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-market relative 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
i
to portfolios on the basis of their industry-relative book-to-markets, BM i =BM , where
i
BM i is the book-to-market of firm i and BM is the book-to-market of firm i’s indus14
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. This
variation, 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 overrepresented 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 bookto-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 bookto-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
P
P
i
That is, BM allj 11indj Dindi be j = allj 11indj Dindi me j , where 11indj Dindi is an indicator that takes the
value one if firms i and j are in the same industry and zero otherwise, and be j and me j are firm j ’s book
and market equities, respectively.
16
Theory supports scaling by the industry book-to-market, i.e., the value, not equal, weighted average
book-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 producers
with high expected returns towards the neutral portfolio. In the extreme, imagine an industry that consists of
a single, efficient oligopolistic firm and a large number of inefficient, marginal producers. Scaling the individual firms’ book-to-markets by the value-weighted industry average results in marginal producers having
the maximum possible industry-relative book-to-market, while scaling by the equal weighted average results
in 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, than
scaling by equal-weighted industry average book-to-market. Scaling book-to-market by the equal-weighted
industry average does improve the Sharpe ratio of the high-minus-low quintile strategy, but only one third as
much as the value weighted scaling procedure.
15
32
TABLE 5
E XCESS R ETURNS , T HREE -FACTOR A LPHAS AND FACTOR L OADINGS , AND
C HARACTERISTICS OF P ORTFOLIOS S ORTED ON B OOK - TO -M ARKET AND
I NDUSTRY-R ELATIVE B OOK - TO -M ARKET, J UNE 1973 - J ANUARY 2007
Relative BM quintiles
PANEL A: B OOK - TO -M ARKET R ELATIVE TO I NDUSTRY B OOK - TO -M ARKET
FF3 alphas and factor loadings
characteristics
re
˛
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 1219
[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
Low
BM quintiles
2
3
4
High
H-L
re
0.392
[1.52]
0.650
[2.74]
0.714
[3.20]
0.782
[3.72]
0.960
[4.13]
0.568
[3.11]
PANEL B: B OOK - TO -M ARKET
FF3 alphas and factor loadings
˛
MKT
SMB
HML
0.084
0.988
-0.130
-0.406
[1.63] [79.95] [-8.11] [-21.90]
0.024
1.056
-0.074
0.150
[0.35] [64.39] [-3.46]
[6.11]
-0.002 1.021
-0.015
0.342
[-0.03] [60.32] [-0.67]
[13.49]
-0.047 0.993
0.045
0.573
[-0.79] [70.56] [2.48]
[27.19]
-0.070 1.080
0.216
0.797
[-1.18] [75.74] [11.65] [37.29]
-0.154 0.092
0.346
1.203
[-2.16] [5.39] [15.60] [46.98]
characteristics
BM ME
n
0.27 1912 1091
0.55
1304
768
0.79
906
744
1.08
685
783
1.72
399
998
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-tomarket 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-French
factors, 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 consistent 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 bookto-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 portfolios (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 percentile 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 versus 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 variation 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 intraindustry 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,
HMLC , is the part of intra-industry HML that is orthogonal to inter-industry HML, i.e.,
HMLC D HMLN
ˇ HMLX. 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 industries, and long 50 cents of growth industries. The Sharpe ratio of this orthogonalized
intra-industry value factor, HML C , 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 significant 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). HML C 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 trailing one year average monthly returns to HML (blue), HML (green) and HML C (red). The
fact that HML is more volatile than HML , and much more volatile than HML C , 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 HML C ).
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
5
HML
HML*
+
HML
4
Average monthly returns (%)
3
2
1
0
−1
−2
−3
−4
Jan 76
Jan 81
Jan 86
Jan 91
Jan 96
Jan 01
Jan 06
Figure 4: Average Monthly Returns to Value-Minus-Growth Strategies
The figure shows one year trailing average monthly returns to three different value-minusgrowth strategies. The blue (darkest) path is Fama and French’s HML. The green (next
darkest) path (HML ) results from replicating the Fama-French procedure for constructing
HML using industry-relative book-to-market. The red (lightest) path (HML C ) is the part of
HML constructed using intra-industry book-to-market not explained by HML constructed
using 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 to F25,375 D 2.36 for a p-value
= 0.031% for the Fama-French model, and F25,377 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, and F25,377 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, “leveredminus-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 industries. 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
The average R-squared for the 25 portfolios is 87.6% for the model that includes HML , compared to
91.2% for the Fama-French model (and 71.9% for the market model). The difference is largely driven by the
large, high book-to-market portfolios; HML does not covary as strongly with the returns to big value stocks
as does HML.
17
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 on r i
r i , as opposed to r i , where r i is the
value weighted return to firm i’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 industry 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
P
P
That is, r i allj 11indj Dindi me j r j = allj 11indj Dindi me j , where 11indj Dindi is an indicator that takes the
value one if firms i and j are in the same industry and zero otherwise, and me j is the market capitalization
of firm j
18
39
TABLE 6
S PANNING T ESTS : ˛ S ( BASIS POINTS PER MONTH )
FROM R EGRESSIONS OF THE F ORM y t D ˛ C ˇ 0 x t C t
Sub-periods
June 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
[-0.53]
[-1.57]
2.4
[0.16]
Panel B: HML* as the dependent variable (y)
MKT, SMB, HML
26.7
18.9
[4.73]
[3.27]
31.1
[3.73]
Panel C: UMD as the dependent variable (y)
MKT, SMB, HML
96.0
92.7
[4.43]
[3.57]
MKT, SMB, HML, UMD*
-9.4
-15.7
[-0.96]
[-1.32]
111.1
[3.25]
-0.5
[-0.03]
Panel D: UMD* as the dependent variable (y)
MKT, SMB, HML, UMD
21.3
26.3
[3.18]
[3.15]
MKT, SMB, HML*, UMD
0.300
0.287
[4.35]
[3.36]
16.7
[1.59]
0.300
[2.75]
Panel E: LMU as the dependent variable (y)
MKT, SMB, HML
26.8
26.8
[3.47]
[3.31]
MKT, SMB, HML, UMD
18.9
26.4
[2.46]
[3.15]
MKT, SMB, HML*, UMD*
17.8
25.8
[2.09]
[2.94]
36.1
[2.98]
24.9
[2.08]
22.5
[1.74]
Source: Compustat and CRSP.
The table reports intercepts from time-series regressions on various subsets of the FamaFrench factors (MKT, SMB and HML), momentum (UMD), industry-relative value and
momentum factors (HML and UMD ), and the operating leverage factor (LMU). Results
are 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 Moreover, 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 momentum 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 result
reflects strong industry momentum, together with short term reversals on individual stocks, at extremely short
horizons. 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 subsume
SMB. Tangency portfolios that include both HML and SMB typically take small short positions in SMB that
have little effect on the portfolio’s Sharpe ratio.
41
TABLE 7
E X P OST M EAN -VARIANCE E FFICIENT P ORTFOLIOS :
W EIGHTS (%) AND S HARPE R ATIOS ( ANNUAL),
J UNE 1973 - JANUARY 2007
I NCLUDED A SSETS
FF3 + MOMENTUM
I NDUSTRY R ELATIVE
MKT SMB HML UMD HML* UMD* LMU Sharpe Ratio
Panel A: Strategies without momentum
100.00
100.0
100.0
27.59
18.55
20.71
17.27
19.37 53.04
81.45
13.45 23.23
42.61
33.43
49.30
0.41
0.54
1.09
0.99
1.29
1.17
1.46
Panel B: Strategies that include momentum
22.45
19.92
15.48
15.51
15.52
12.91 42.63
15.96 32.92
22.01
31.20
17.73
66.79
56.99
46.70
27.50
21.94
15.84
1.27
1.39
1.54
1.72
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 and
UMD) and our industry-relative factors based on value, momentum and operating leverage
strategies (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 conjunction 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 combination 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 conditional 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 unconditional 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 the
model.
22
More precisely, it requires that the stationary distribution of the risk-neutral price process is skewed
towards the investment trigger. This occurs if the drift in the goods market price is “high” relative to its
volatility.
23
Absent the counter-cyclic risk premium, HML exhibits a Sharpe ratio significantly below the market’s
in long history simulations (10,000 years), though realized HML Sharpe ratios in 30 year subsamples exhibit
a 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 premium. 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 implications 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 industries, 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 predictions 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. Let Ex [f (X t )] E [f (X t ) jX0 D x] and E [(!)I A] E [(!)11A (!)] for 11A (!) D
(1,v)
1 if ! 2 A and 11A(!) D 0 otherwise. Now suppose X t
is a drifted geometric Brownian process
between an upper reflecting barrier at v and lower reflecting barrier at 1, and let Tv D minft >
(1,v)
0jX t
(1,v)
D vg and T1 D minft > 0jX 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
where y(x) D x ˇp
Eu [e
(r Cı)T1
I T1 < T v ] D
Eu [e
(r Cı)Tv
I Tv < T 1 ] D
y (u=v)
y (1=v)
y (u)
y (v)
(25)
(26)
x ˇn , and ˇp and ˇn are given in Proposition 5.1.
Proof of lemma: The state prices, discounting at r C ı, for the first passage of the process to the
upper and lower barriers may be written as
h
Eu e
h
Eu e
i
h
D Eu e
i
h
(r Cı)T1
D Eu e
i
h
I Tv < T1 C Eu e
i
h
(r Cı)T1
I T1 < Tv C Eu e
(r Cı)Tv
(r Cı)Tv
i h
I T1 < Tv E1 e
i h
(r Cı)Tv
I Tv < T1 Ev e
(r Cı)T1
i
(r Cı)Tv
(27)
i
(r Cı)T1
.(28)
h
i
h
i
Solving the preceding equations for Eu e (r Cı)Tv I Tv < T1 and Eu e (r Cı)T1 I T1 < Tv using
h
i
h
i
ˇ
Eu e (r Cı)Tv D uv p and Eu e (r Cı)T1 D uˇn yields the lemma. Lemma A.2. The perpetuity factor for a geometric Brownian process currently at x with reflecting
b
barriers at a x and b x, which we will denote a(x), is a homogeneous degree-zero function
of a, b, and x jointly, and
v
u1 (u) D u C
where y(x) D x ˇp
y (u=v)
(˘(v)
y (1=v)
) C
y (u) ˘(v
y (v)
1
)
(29)
x ˇn and , ˘(x), ˇp , and ˇn are given in Proposition 5.1.
(1,v)
Proof of lemma: Suppose X0
(1,v)
D u 2 [1, v], where X t
is a geometric Brownian process
between an upper reflecting barrier at v and lower reflecting barrier at 1. Then the value of the cash
46
flow e
ıt X (1,v)
t
discounted at r and starting at t D 0 is
Z
v
u1 (u) D E
1
u
(r Cı)t
e
0
"Z
#
T1 _Tv
u
0
Z
Z
1
1
e
CE
(r Cı)t
(1,v)
Xt
dt I T1
e
(r Cı)t
(1,v)
Xt
dt I Tv
< T1
(30)
Tv
u
< Tv
1
CE
1
r Cı (1,v)
Xt
dt
T1
v
where D
(r Cı)t
e
D E
D
X t(1,v) dt
Eu [e
(r Cı)T1
I T1 < T v ]
Eu [e
(r Cı)Tv
C Eu [e
(r Cı)T1
I T1 < Tv ] ˘(v) C Eu [e
I Tv < T 1 ] v (r Cı)Tv
I Tv < T1 ] v˘(v
1
)
is the perpetuity factor for a geometric Brownian process discounted at r C ı,
and
Z
(r Cı)t
˘(v) E
0
˘(v
1
) v
1
1
e
Z 1
1 v
e
E
(1,v)
Xt
dt
(r Cı)t
X t(1,v) dt
0
are the perpetuity factors for the reflected process when it is at the lower and upper barriers, respectively.
Then defining
v
(r Cı)T1
I T1 < T v ]
v
(r Cı)Tv
I Tv < T 1 ]
1 (u) Eu [e
1(u) Eu [e
completes the proof of the proposition, except for the explicit functional form for ˘(v) and ˘(v
To obtain the explicit functional form for ˘(v) and ˘(v
1
1
).
), note that the smooth pasting con-
dition implies
d
du
d
du
ˇ
v
ˇ
u1 (u)ˇ
D 0
uD1
ˇ
v
ˇ
u1 (u)ˇ
D 0,
uDv
47
(31)
(32)
or
ˇn v ˇp
ˇp v ˇn (˘(v)
C
ˇn
ˇp v ˇp Cˇn
1
) C ˇp
v ˇp v ˇn
) C ˇp v ˇp
(˘(v)
C
v ˇp
ˇn v ˘(v
1)
ˇn v ˇ n
˘(v
1)
D 0
(33)
D 0.
(34)
v ˇn
Solving the previous equations simultaneously yields the explicit values for ˘(v) and ˘(v
1 ).
Proof of the proposition: The Bellman equation corresponding to firm i’s optimization problem
(equation (7)) is
rV i (K, X ) D Ri (K, X )
ı K rKV i (K, X )
CX X VXi (K, X ) C 12 X2 X 2 VXi X (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 in Ki , so taking
partial derivatives of the left and right hand sides with respect to Ki yields
i
(r C ı)VKi i (K, X ) D RK
i (K, X )
ı K rK VKi i (K, X )
i
1 2 2 i
CX X VXK
i (K, X ) C 2 X X VX XK i (K, X ).
(36)
Then using that V i (K, X ) is homogeneous degree one in K and in X , so qi (K, X ) VKi i (K, X )
is homogeneous degree zero in K and X , and that D X C ı C ( 1)X2 =2 and D X ,
we can rewrite the previous equation as
(r C ı)qi (P ) D
n
1
c
P C P qi0 (P ) C 12 2 P 2 qi00 (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 that qi (K t , X t ) VKi i (K t , X t ) may be written as
qi (K it , P t ) D
where Ri (K it , P t ) D K it
Pt
ci
i
i
RK
(K
,
P
)
C
(P t )
t
t
i
hR
1
is the firm’s unit profits and (P t ) D E 0 e
(38)
i
(r Cı)s P tCs ds
Pt
is the unit value of revenue for the geometric Brownian price process reflected above at PU and
below at PL , given explicitly in Lemma A.2, and
48
D
r Cı
is the capitalized unit cost of operating
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
revenue Ri (K t , X t ) D K it P t =ci with respect to K i , is
i
i
RK
D ci 1 P t C ci 1 K it dPti .
i (K t , P t ) C (39)
dK t
Substituting the previous equation into equation (38), and using the facts that
dP t
dK ti
D
t
cP
,
i St
which comes from differentiating the inverse demand function P t D X t S t with respect to K it ,
and S ti =S t D c
1 n cj = c, which follows from the second condition of Proposition 4.1,
and the definitions of L and C , we have that
q(P t ) D
1 L
C
P t (P t )
,
(40)
where explicit dependence on i has been dropped because firms’ marginal valuations of capital
equate.
It is then simple to check that q(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
2
d
d
(r C ı)P (P ) D P C P dP
(P (P )) C 12 2 P 2 dP
2 (P (P )).
(41)
This must hold for all P , so using the fact that
P (P ) D P C aP ˇn C bP ˇp
(42)
for some a and b (see, for example, equation (29)) and, matching terms of equal P -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
(r C ı)
(
(r C ı)
(
2
2 )ˇn
2
2 )ˇp
)
2 2
2 ˇn
2 2
2 ˇp
D 1
D 0
D 0,
which is easily verified.
The shadow price of capital, q(P ), also satisfies the necessary boundary conditions. Evaluating
49
equation (40) at PU and PL yields
q(PU ) D 1
(43)
q(PL ) D ˛.
(44)
The smooth pasting condition at both boundaries, i.e., that qi0 (PU ) D qi0 (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, so qi (P ) is decreasing in K i =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 generates given the equilibrium goods price process,
!
V P t , K it
D
K it
ci
Oi
P t (P t )
K it
.
r Cı
(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
VO ti
K it
where i D
cmax
ci
D q t C i (q t C
)
(46)
1. The first term in the previous equation is just the shadow price of capital,
and is bounded between ˛ and 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, P VP C
2 2
P VPP
2
D (r C ı)V ,
which implies
Qit
D
P t ˇn
P t ˇp
VO ti
i
i
C an
C ap
PL
PU
K it
50
(47)
for some ain and api . 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
ˇ
dVi ˇˇ
dX ˇX DX
!ˇ
dP
P ˇn
P ˇp ˇˇ
d
VO i
i
i
D Ki
C an
C ap
ˇ
dX dP K i
PL
PU
X DXU
Ki
ˇn ain ˇn C ˇp api
D
X
U
(48)
where we have used the facts that value of deployed capital is insensitive to changes in X at the
development boundary and
dP
dX
D P =X .
ˇ
d Vi ˇ
At the boundary, homogeneity of the value function implies dX
Ki X DXUC D 0, and the supply
ˇ
d ln Ki ˇ
response ensures the price never exceeds PU so
C D 1, so
d ln X X DXU
ˇ
d (Vi Ki ) ˇˇ
ˇ
dX
X DX C
D
U
ˇ
dKi ˇˇ
1
dX ˇX DX C
Vi
Ki
D
Vi Ki
.
X
(49)
U
The value function is differentiable at the boundary,
d
dX
ˇ
Vi ˇX DX
D
U
d
dX
(Vi
ˇ
Ki ) ˇX DX C ,
U
which, using the results of the previous two equations, yields
ˇn ain ˇn C ˇp api
D QiU
1,
(50)
or, rearranging using the fact that at the investment boundary VOUi =K i D 1 C i (1 C
i D cmax =ci
) where
1, that
1) ain ˇn C ˇp
(ˇn
1 api
D i (1 C
).
(51)
A completely analogous calculation at the disinvestment boundary implies
(ˇn
1) ain C ˇp
1 api ˇp
D i (˛ C
).
(52)
Solving the previous two equations simultaneously yields
ain
i
api
i
D
(1 C ) ˇp (˛ C )
(ˇn 1) ˇn ˇp
D
(˛ C
ˇp
) 1 51
ˇn (1 C
ˇp
(53)
)
. ˇ
n
(54)
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
P t
C Cˇi n
ci
Pt
PL
ˇ n
C
Cˇip
Pt
PU
ˇp
.
(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
!
dP (p) D p
1
PL
PU
where D
2
2
dp
(56)
1.
Proof of lemma: Suppose X t is a geometric Brownian process with drift and volatility , and a
lower reflecting barrier at l and an upper reflecting barrier at 1. Then
Z
lim t
t!1
1
t
11[l,z] Xs ds
E
D P [X < z]
(57)
0
where 11A (!) D 1 if ! 2 A and 11A (!) D 0 otherwise, and X is has the stationary distribution of
the process X t .
If Y t 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
Xt
d
D Ys t
d
where D denotes “equal in distribution,” and s t minfsj
(58)
Rs
0
11[l,1] Ys ds D t g. So
P [X < z] D P [Y < zjY > l]
D
P [Y < z] P [Y < l]
.
1 P [Y < l]
(59)
(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 that P [Y < y] D y where D (
2 =2)=( 2 =2).
Substituting this into the previous equation, and letting X t D P t =PU , l D PL =PU and z D
PT =PU , yields the stationary distribution for the equilibrium price process,
P [P < p] D
p
PL
PL
PU
.
(61)
Differentiating with respect to p yields the lemma.
Proof of the proposition: The proposition follows directly from the preceeding lemma.
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