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Required Growth Probability:
An Objective Approach to Fundamental Security Analysis
Dan W. French
And
David Javakhadze
Contact:
Dan W. French
Department of Finance
Robert J. Trulaske, Sr. College of Business
University of Missouri
403 Cornell Hall
Columbia, MO 65211
573-882-3739
[email protected]
ABSTRACT
We argue that it is possible to construct an objective set of decision criteria for security selection
that can enable an active fund manager to employ fundamental analysis and outperform. We
consider an approach that uses probability-based security analysis and find considerable
evidence that the approach produces abnormal returns. By studying a probability-based approach
to security selection we are able explore an area of asset pricing that has received little attention.
JEL Classification: G10, G11, G14
Draft Date: February 28, 2013
Required Growth Probability:
An Objective Approach to Fundamental Security Analysis
1. Introduction
Every day there are thousands of security analysts and portfolio managers who use publicly
available financial data in an effort to identify overvalued and undervalued securities and even to
rank such securities by their extent of their over or undervaluation. It is analysts’ belief in the
superiority of their judgment (relative to the other analysts) that motivates their work. In order to
justify the efforts they spend on valuation, they all must accept that pricing inefficiencies exist
even though observed prices reflect the collective expectations of the market (i.e. other analysts).
They therefore believe that their valuations are more precise as a reflection of “true” values than
are valuations of the “average” analyst. In other words, they believe that they can correctly
identify over and undervalued securities more often than would occur by chance alone and
construct portfolios based on these valuations that would yield superior risk-adjusted
performance.
This paper analyzes the efficacy of a different approach to stock selection. In this method, they
analyst uses the market price to infer the market’s collective expectation for a firm’s future
revenue growth. The analyst then uses financial data to estimate probability that the firm can
actually meet this expectation. The process involves reversing the traditional discounted cash
flow (DCF) approach to back out the firm’s implied revenue growth and fitting the firm’s
historical growth distribution. Then, firms more likely to achieve their required revenue growth
(and portfolios constructed using this criterion) should exhibit superior performance.
2
This alternative approach relies on fewer subjective and potentially biased judgments on the part
of analysts because it avoids making assumptions and forecasts about future growth as does the
valuation approach. It simply solves for the growth rate implied by the current price. The
subjective nature of this alternative approach could come into play when the analyst estimates
the probability of the firm actually achieving the required growth rate. However, an objective
estimate of the probability can result from fitting a probability distribution using observed
patterns of historical firm revenue growth. We call this measure of the probability that a firm will
achieve the growth rate needed to justify its current market price the Required Growth
Probability (RGP). It is the thesis of this paper that implementing a portfolio construction
strategy using RGPs will be more effective at achieving superior performance than traditional
valuation measures because RGP relies less on subjective estimates and therefore less influenced
by analyst bias.
Our findings suggest the RGP approach can in fact be fruitful to an active fund manager and
more generally suggest that disciplined, objective stock selection is a superior approach to
traditional fundamental analysis and the subjectivity it typically entails. That is, an alternative
strategy that is based on conventional fundamental analysis yet applied with less subjectivity can
result in superior performance. The abnormal performance we observe appears to be strongest
among growth stocks but exists throughout the growth-value spectrum. It is also robust to size
and momentum effects.
The organization of the paper is as follows. First we review the theory that motivates our interest
in this approach to investing. Second we describe our proprietary dataset and the methodology
used to generate the probability measure that will be the focus of the remainder of the paper.
3
Section 4 contains our empirical work which begins with preliminary tests of outperformance
and concludes with tests that explain the outperformance we observe. Section 5 discusses the
findings and concludes.
2. Theoretical Motivation
Financial literature contains tests of a variety of fundamental investment analysis techniques
beginning with studies based on price/earnings (P/E) multiples. For example, Whitbeck and
Kisor (1963) devise a model for estimating fundamental P/E multiple for a firm’s stock and
demonstrate a strategy that would have yielded high investment returns. Basu (1977) finds that
strategies using merely price multiples to select stocks appeared to capture abnormal returns.
Similar strategies such as the use of book-to-market ratio, first identified by Rosenberg, Reid and
Lanstein (1984) and later codified by Fama and French (1992) would ultimately prove to be of
seminal importance to the asset pricing theory.
In practice, however, most fundamental analysis is done quite subjectively. In most cases this is
necessarily so. Financial statements contain overwhelming amounts of information that make it
difficult for an analyst or fund manager to distill into a single decision criterion. Yet a single
decision criterion is often just what an active fund manager needs in order to select within a large
universe of stocks.
One study that summarizes multiple fundamental data items is that of Piotroski (2000), who
argues using that many fundamental indicators used in conjunction may result in abnormal
returns while using any one in isolation may not. The approach we study also summarizes much
fundamental data, but additionally results in a single decision criterion. In this way, the approach
4
is in the same vein as Ou and Penman (1989) who summarize a variety of financial statement
data items into a single probability measure that predicts an increase or decrease in earnings.
Yet in general the use of probabilities is conspicuously absent in the field of asset pricing,
despite the fact that probability as a means to evaluate risk is pervasive in the financial services
industry, from measuring credit default risk to insurance underwriting risk. Moreover,
probability is simple metric that could give a clear signal to a fund manager making stock
selection decisions. By studying a probability-based approach to investing we are able explore an
area of asset pricing that has received little attention.
The approach we study makes important use of the discounted cash flow analysis (DCF). As an
expansion on the dividend discount model introduced by Gordon (1959), DCF analysis holds that
the value of an asset is equal to the present value of future cash flows. While widely used by
practitioners, the model also has support in the literature (see Myers (1974), Kaplan and Ruback
(1995, 1996), Gilson, Hotchkiss and Ruback (2000), Inselbag and Kaufold (1997) among
others). Typically such models make projections about a firm’s future free cash flow and then
discount these cash flows at the firm’s cost of capital. Common DCF models have two stages,
often ten years of cash flows in the first stage and a second stage representing the terminal
growth rate in years 11 and thereafter.
3.
Data and Empirical Approach
The method we use to generate the probabilities is based loosely on ideas formalized by
Rappaport and Mauboussin (2001) who argue that measuring price-implied market expectations
is key to generating abnormal returns because the stock price itself is the most unambiguous
5
source of firm-specific information. Accordingly, the method uses the stock price as the input
into a reversed discounted cash flow valuation. By reversing this process the analyst must solve
for implied revenue growth rather than intrinsic value.
To perform the reversed DCF, the analyst first calculates enterprise value, a metric often used by
practitioners in place of market cap due to its ability to capture all sources of financing. The
method is not assumption-free however, but the assumptions required to complete the process are
made in accordance with a set of predetermined guidelines and applied uniformly. For instance,
in order to solve for the first stage free cash flow growth rate, the analyst must make assumptions
about the long-term terminal growth rate and the allocation of free cash flow within the first
stage. The discount rate he uses is the cost of capital estimated under a general version of the
CAPM (Sharpe (1964), Lintner (1965) and Black (1972)). With this information, he can
calculate the first stage FCF growth rate and, by deduction, the first stage revenue growth rate.
The next step in the process is to determine the probability that the required revenue growth will
be met. To calculate this, the firm’s history of revenue growth over the previous eight quarters is
fit to a log-normal distribution, overweighting the most recent quarters under the theory that the
most recent observations should have the most predictive content. To eliminate the effect of
seasonality, growth rates are calculated on a quarter-on-quarter basis as
(1)
where
and
is the growth rate in quarter t,
is the revenue in the current quarter t
is the revenue in the same quarter of the previous year.
6
The use of the log-normal distribution is this case is logical for several reasons. First, growth
defined this way is a non-negative variable and heavily right-skewed. Also, the assumption of
log normality has been used in the accounting literature for a variety of fundamental variables
(see Hilliard and Leitch (1975)).
After transforming the data, the weighted mean and variance can be used to fully describe the
cumulative distribution function of the normal distribution. The mean is weighted under the
assumption that more recent observations have more predictive content. We define
as follows: we simply integrate the probability density function of the normal distribution at the
log-transformed required revenue growth. More formally,
√
where
∫
(
) )
(2)
equals the natural log of the required revenue growth rate,
mean and
is the sample weighted
is the sample standard deviation. The integral is subtracted from one because we are
concerned with the probability that the realized growth rate is greater than or equal to .
Our proprietary dataset is provided by Transparent Value, LLC, an asset manager utilizing such a
probability-based strategy with more than $500 million under management. That data includes
the required revenue growth calculated by the firm using its set of guidelines as well as the
probabilities that result. We have independently replicated the probabilities using the required
revenue growth numbers and found them to be nearly identical.
The sample we analyze includes the required revenue growth and probabilities of stocks over the
period 1999 to 2011. This sample includes both real-time calculated probabilities beginning in
2008 as well as back-calculated probabilities over the period 1999 through 2007. Descriptive
7
statistics of the sample can be found in Table 1. Due to the nature of our data source, the firms in
our sample are generally large and thus tend to have lower betas.
4.
Empirical Analysis
A. Fama-Macbeth Regressions
Our initial examination of the abnormal return generating potential of the required growth
probability is a stepwise Fama Macbeth (1973) approach to return analysis using the three
factors of Fama French (1992) plus the Carhart (1997) momentum factor. We perform 156 crosssectional regressions of return in month t on size (market capitalization), book-to-market, prior
return and probability. Prior return is defined as one plus the six month return in months j-6
through j-2. The regressions are performed over the period June 1999 through December 2011.
The average coefficients of the times series are shown in Table 2. In the first column we add
(
) to the market model and find that it is statistically significant in terms of
(
explaining returns. Columns (2) and (3) show
) added to the three-factor and
(
four-factor models. The coefficient estimates of
)in these regressions are all
highly significant.
The results of Fama Macbeth analysis indicate that the probability has explanatory power. The
coefficient on probability is positive, as expected, and statistically significant under all model
specifications. This provides compelling preliminary evidence that the probability measure we
test may in fact have the potential to contribute to abnormal returns.
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B. Performance of Probability-based Portfolios
Having established preliminary evidence that probability has explanatory power, it is important
that we begin to assess the actual potential profitability of strategy that uses
. To do
this we sort all stocks in to quintiles and test the absolute performance. We form the quintile
portfolios monthly, and calculate the mean raw return to each quintile over our entire sample
period. The results, shown in Panel A of Table 3, reveal that performance increases
monotonically by quintile, from 0.63% in the (equally-weighted) low probability quintile to
0.99% in the (equally-weighted) high probability quintile, suggesting a long/short strategy could
generate as much as 0.36% per month. Results using value-weighted quintiles are similar.
After this analysis, however, it remains possible that the high versus low quintile results are due
to exposures to known risk factors, suggesting that further testing is needed to determine how
robust they are to multifactor asset pricing models. To test the robustness of the results, we
regress the returns of each quintile on the Fama French 3 factors and Carhart factor. 1
Panel B of Table 3 displays these results. It does not appear that the results shown in Panel A are
due to exposure to known risk factors. Specifically, the alphas (using both equal-weighted and
value-weighted portfolios) are positive and significantly different from zero for the highest
probability portfolio while not significantly different from zero for the lowest probability
portfolios. The result is a high-low difference in alpha of 0.608% per month using equallyweighted portfolios and 0.682% per month using value-weighted portfolios.
Does the unusual nature of our sample bias our results due to exposure to small or large stocks?
To examine this we sort all stocks in to quintiles based on both probability and size. If the size
1
Data used in this analysis are taken from Ken French’s website.
9
effect is affecting our results, we would expect to see higher high-probability-minus-lowprobability differences among small stocks.
The size effect does not appear pronounced. Returns consistently increase with probability
quintile. Further, as shown in Table 4, there appears to be little disparity in results among the
various size quintiles. Surprisingly, the largest difference actually occurs among large stocks,
contrary to what we would expect to see due to the size effect. The value-weighted quintile
results shown in Panel B of Table 4 reinforce this finding. The large value-weighted size
quintile, for which the largest firms in the universe tend to dominate results, has the highest highlow difference in performance.
From this analysis we conclude that the size effect is not driving the results. Yet we must still
establish that size quintiles are not exposed to known risk factors that might impact our findings.
Table 5 shows the results of regressions under the three and four factor models. We perform five
regressions, one for each size quintile, where the dependent variable is the return of the high
probability quintile minus the return of the low probability quintile. The alphas in all regressions
are positive and significant, indicating that, regardless of size, the high probability minus low
probability approach has alpha-producing potential.
C. Performance of Probability Factor
How does our
measure perform when treated as a factor itself? To answer this, we
form a high-minus-low probability factor in the same way as the HML factor returns of the Fama
French 3-factor model. Specifically, we form six value-weighted portfolios by sorting size into
two portfolios and
in to three portfolios. The
10
factor return equals the
average return of the two high
portfolios minus the average return of the two low
portfolios. We can then regress our factor, High-Minus-Low-Probability (HMLP),
on the Fama French three factors and Carhart factor.
First we examine the nature of HMLP by calculating summary statistics. Table 6, Panel A,
displays the mean and variance of HMLP as compared to the four other factors. While the mean
return to HMLP over our sample period is not as high as SMB, its Sharpe Ratio is higher than all
other factors. For a factor to capture the fundamental riskiness of a stock, it must sometimes have
negative returns despite its positive mean return. The Sharpe Ratio of HMLP, being higher than
that of other factors due to the overall lower volatility of HMLP, suggests that HMLP is
capturing abnormal returns that are less fully explained by risk and instead may be a real source
of abnormal returns.
In Panel B of Table 6 we show the correlations of HMLP with the four factors. HMLP is
correlated with SMB, HML and MOM, yet the correlation with HML is negative. The
correlation of HMLP with SMB is understandable, as our sample consists of mainly large stocks.
As Table 9, Panel A shows, around 95% for all the stocks classified in Fama-French 25 book-tomarket portfolios as a small, we are unable to estimate
large stocks have greater
. Also, as Panel B shows,
then then small stocks. Similarly, the correlation with
MOM is also expected to the extent that HMLP has a positive mean return and probabilities in
consecutive quarters are correlated. However the strong negative correlation with HML is
noteworthy. It may mean that HMLP is positive when HML is negative, suggesting that our
probability measure is best at identifying outperforming stocks with growth outperforms value.
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Alternatively, it may simply indicate that probability works best among growth stocks regardless
of the direction of HML.
The results of the factor regressions, shown in Table 7, indicate that neither the three-factor nor
four-factor model can explain the returns to a high
minus low
portfolio. Consistent with the correlation analysis, HMLP loads very heavily negatively on
HML.
The HMLP/HML relation warrants further investigation. In Table 8 we examine the performance
of quintiles based on probability and book-to-market. For robustness, we also use alternative
measures of value, market leverage and prior 36 month return. From this analysis we find that
the high-minus-low
difference is not larger among growth stocks as we would
expect if probability works when growth outperforms value. Rather, we see no discernible trend
in the high-minus-low difference across value/growth quintiles, confirming that the HMLP effect
is robust to the value/growth effect.
Fama and French (1996) divide all stocks in to three portfolios based on book-to-market ratio:
value, neutral and growth. In Figure 1 we chart monthly proportion (percent) of the value,
growth and neutral stocks in the top 30% and bottom 30%
high (low)
buckets. That is, among
stocks we identify percent of the value stocks, percent of growth stocks
and percent of neutral stocks in each month and plot it for the period 1999-2011.Growth stocks
tend to occur more frequently among stocks in the top HMLP portfolio, indicating that the
highest probabilities occur among growth stocks. If high probability stocks outperform low
probability stocks, this provides evidence that the negative relation of HMLP with HML is due to
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the probability measure’s ability to identify mispriced growth stocks with slightly greater
frequency than value stocks.
As we know that value, as defined by book-to-market ratio, tends to outperform growth, and that
high probability stocks tend to outperform low probability stocks, the negative relation suggests
that much of the return to HMLP comes from growth stocks rather than value stocks. If a growth
stock outperforms it is likely that there is a divergence from the standard efficient pricing of
stocks because ordinarily we expect growth stocks to underperform. Thus, the negative relation
we see is evidence that the abnormal returns our probability measure appears to generate may be
the result of behavioral biases. That is to say, the ability of our
measure to identify
outperforming growth stocks (and as Table 9 Panel B shows growth stocks on average have
greater
than value stocks) is evidence that prices are more likely to be irrational due
to behavioral biases. The coincidence of this phenomenon with the ability of our probability
measure to produce abnormal returns indicates HMLP, or probability more generally, may work
due to its ability to identify when prices are more likely to be inefficient due to behavioral biases.
D. Performance over Different Periods
A potential concern with our data is the relatively short time series that we have available. While
the data cover only the period 1999 through 2011, several distinct market and economic cycles
occurred during this period. We therefore test how robust the probability measure is as a stock
selection tool to different environments by calculating the alphas generated by regressing the
returns of each probability quintile on the four-factor model. The difference in alphas between
the high quintile and low quintile tells us how a theoretical long/short trading strategy might
perform in each period.
13
Table 10 presents these results. In general we conclude the
stock selection system
works well in both bull and bear markets, recession and expansion. That is to say, our more
general results reported in previous sections are not driven by any one period in our time series.
5.
Discussion
What explains the surprisingly robust ability of the required revenue growth probability measure
that we study to generate abnormal returns? Our findings are interesting primarily because only
public information is used in the calculation of the probability. However, in contrast to more
typical fundamental analysis, the probability measure is far more objective and applied in a
uniform fashion without human judgment or subjectivity. This suggests that the abnormal returns
are at least in part the result of biases by the balance of fundamental analysts and investors –
those who ultimately set market prices. In general, our results suggest the probability approach
picks up on systematic behavioral biases among market participants.
Lakonishok, Shliefer and Vishny (1994) as well as La Porta, Lakonishok, Shleifer and Vishny
(1997) questioned the nature of value strategies in a rational pricing framework. They concluded
that value stocks outperform growth not because they are truly riskier, but because they exploit
the behavioral shortcomings of other investors. What the authors term “glamour” stocks tend to
have prices that imply growth that the firm ultimately cannot deliver because investors have
extrapolated past growth rates too far in to the future. Another way of saying this is that market
expectations for glamour stocks tend to be too optimistic. That prices could be set by
systematically biased market participants became very clear just a few years later. Indeed,
14
subsequent work such as that of Chan, Karceski and Lakonishok (2000) found enormous
divergences between “value” metrics and operating performance. They concluded that the latenineties outperformance of growth over value was unsubstantiated by performance and instead
the result of irrational pricing.
Chan, Karceski and Lakonishok (2000) refer to the divergence between value and performance
as a disruption, and clearly in the late nineties it was. However, smaller and less systematic
disruptions can occur among subsets of stocks. The evidence in this paper suggests that the
probability of meeting required revenue growth can help identify and avoid “glamour” stocks by
measuring the implied market expectations and objectively balancing these expectations against
performance. While value stocks may still outperform growth in general, this approach appears
to be able to identify which value stocks have overly optimistic implied expectations and thus
can lead to outperformance among such growth stocks.
The objectivity of the approach is crucial. For a behavioral bias to be exploited it is very
important that the analyst not commit the same behavioral bias himself. The fashion in which our
probability measure is calculated suggests this does not happen. The stock price is the most
unambiguous piece of firm-specific information, so using this as the input rather than subjective
or even arbitrary projections about future performance mitigates analyst input.
In summary, the probability measure we test in this paper appears to have alpha-producing
potential that arises from the ability for the measure to identify and exploit the irrational pricing
of securities by market participants. The probability seems to work best among growth stocks for
which investors have driven prices to irrationally high levels, yet we find outperformance among
value stocks as well. We interpret this as evidence that removing the subjectivity from the
15
analysis process and distilling the investment decision to a single criterion may enable active
managers to identify mispricings.
16
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17
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18
Table 1: Sample Description
This table shows descriptive statistics. Panel A shows statistics for our sample. Panel B shows
the same statistics for the universe of stocks in the Compustat database. Variable definition is as
follows: CHat – cash scaled by total assets, CAPXat – capital expenditures scaled by total assets,
PPESALES – property, plant and equipment scaled by total assets, EBITSALE – earnings
before interest and taxes scaled by total assets, DEBTAT – debt scaled by total assets, Q –
Tobin’s Q defined as total assets plus market value of equity minus book value of equity, scaled
by total assets, ROE – returns on equity, ROA – return on assets, BM – book-to-market ratio,
Mkval – market value of equity, BETA – ninety-day beta.
Panel A: Sample Descriptive Statistics
Variable
Mkval
BETA
PPESALES
EBITSALE
DEBTAT
Q
ROE
ROA
CHat
CAPXat
Mean
8572.17
0.87
1.73
-0.78
0.43
2.15
-0.07
0.01
0.11
0.06
Median
2166.87
1.07
0.21
0.10
0.42
1.59
0.11
0.05
0.06
0.04
Std Dev
25906.38
23.55
90.48
39.87
0.22
2.00
9.39
0.32
0.12
0.06
Maximum
508329.45
275.61
10189.91
4.34
3.56
78.56
103.97
2.17
0.95
0.90
Minimum
0.94
-2376.78
-0.26
-4347.58
0.00
0.31
-790.61
-19.45
0.00
-0.03
Std Dev
11334.770
290.359
124.537
136.820
103.634
1071.360
70.386
353.405
0.192
1.078
Maximum
508329.45
50761.88
19269.94
4037.11
25968.97
222021.00
7770.330
2369.430
1.000
341.000
Minimum
0.000
-35936.12
-646.44
-30175.70
0.00
0.03
-23155.88
-130077.00
-0.53
-2.09
Panel B: Compustat Descriptive Statistics
Variable
Mkval
BETA
PPESALES
EBITSALE
DEBTAT
Q
ROE
ROA
Chat
CAPXat
Mean
1867.200
2.830
3.935
5.405
2.867
0.755
0.086
2.508
0.133
0.073
Median
106.068
0.791
0.208
0.053
0.429
1.381
0.073
0.015
0.052
0.037
19
Table 2: Fama-MacBeth Regressions
This table presents the time-series average coefficients for 156 cross-sectional regressions for
each month from June 1999 to December 2011. The dependent variable is the stock return for
month t. A firm’s size is its market capitalization (price times shares outstanding). The book-tomarket ratio (BE/ME) is the prior year’s book value of equity divided by the firm’s market value.
Prior return is the raw buy-and-hold return from month j−6 to month j−2. (
) is
natural log
estimated by equation (2). T-statistics are in parenthesis. *** represents
a significant difference at the 1% level, ** at the 5% level, and * at the 10% level.
(
)
(
)
(
)
(
Model
(1)
(2)
(3)
Intercept
0.973*
(1.98)
0.091
(0.30)
3.851*
(1.98)
0.078
(0.26)
-0.132
(-1.17)
0.952***
(4.82)
0.124*
(1.73)
0.169***
(2.58)
3.653**
(2.14)
0.003
(0.01)
-0.131
(-1.28)
0.923***
(5.66)
0.513
(0.89)
0.187***
(3.06)
Beta
ln(Size)
Ln(BE/ME)
LN(1+prior)
ln(Probability)
20
)
Table 3: Performance of High minus Low Probability Portfolios
Panel A shows Equal-weighted and value-weighted returns for each
quintile. Panel
B shows four-factor regressions of monthly excess equally-weighted and value-weighted
portfolio returns. T-statistics are presented in brackets. *** represents a significant
difference at the 1% level, ** at the 5% level, and * at the 10% level.
Panel A: Monthly Returns for Probability Quintile
Quintile
1 (Low)
2
3
4
5 (High)
High-Low
Monthly EW Returns
0.63%
0.80%
0.83%
0.84%
0.99%
0.37%
Monthly VW Returns
0.16%
0.32%
0.42%
0.43%
0.66%
0.50%
Panel B: Monthly High minus Low Probability Returns as Dependent Variable
Equal-Weighted
Alpha
RMRF
SMB
HML
UMD
Adjusted R2
Low
2
Probability Quintiles
3
4
-0.020
(-0.18)
1.169
(44.95)***
0.402
(12.43)***
0.366
(11.33)***
-0.093
(-4.83)***
95.5%
0.224
(1.96)**
1.104
(41.49)***
0.427
(12.92)***
0.191
(5.77)****
-0.143
(-7.21)***
95.1%
0.316
(2.92)***
1.072
(42.53)***
0.410
(13.07)***
0.052
(1.67)*
-0.178
(-9.48)***
95.6%
Low
2
3
-0.277
(-1.83)*
1.108
(31.51)***
0.049
(1.12)
0.223
(5.11)***
-0.044
(-1.69)
89.5%
-0.024
(-0.23)
0.958
(38.00)***
-0.038
( -1.20)
0.074
(2.37)**
0.026
(1.39)
89.5%
0.083
(0.73)
0.936
( 35.08)***
-0.117
( -3.53)***
0.152
( 4.58)***
0.037
(1.88)*
90.3%
Value-Weighted
Alpha
RMRF
SMB
HML
UMD
Adjusted R2
0.309
(2.47)***
1.076
(36.91)***
0.475
(13.10)***
-0.002
( -0.07)
-0.175
(-8.08)***
64.5%
Probability Quintiles
4
21
0.125
( 1.08)
0.992
(36.68)***
-0.068
(-2.03)**
-0.029
( -0.88)
0.013
(0.65)
91.6%
High
High-Low
0.588
(3.31)***
1.148
( 27.80)***
0.423
( 8.24)***
-0.308
( -6.01)***
-0.221
( -7.21)
91.2%
0.608
(2.91)***
-0.021
( -0.44)
0.021
( 0.35)
-0.675
(-11.16)***
-0.128
( -3.54)
48.3%
High
High-Low
0.405
( 3.61)***
0.993
( 37.95)***
-0.034
( -1.05)
-0.192
( -5.91)***
-0.035
( -1.82)*
92.9%
0.682
(3.05)***
-0.115
( -2.22)**
-0.083
( 1.28)
-0.416
(-6.43)***
0.009
( 0.23)
21.8%
Table 4: Monthly Returns by Size and Probability Quintiles
This table shows monthly returns by size and probability quintiles for the period 1999-2011.
Portfolios are determined by using only NYSE cutoffs and are rebalanced monthly2. A firm’s
size is its market capitalization (stock price multiplied by shares outstanding).
is
defined by equation (2). *** represents a significant difference at the 1% level, ** at the 5%
level, and * at the 10% level.
Panel A: Equal-Weighted Monthly Returns
Probability Quintiles
Size Quintile
Low
2
3
4
High
Diff High-Low
Small
0.87%
1.57%
1.53%
1.39%
1.29%
0.42%
2
3
4
Large
0.75%
0.66%
0.61%
0.25%
0.86%
0.66%
0.69%
0.31%
0.86%
0.47%
0.77%
0.47%
0.91%
0.96%
0.61%
0.36%
1.10%
1.04%
0.92%
0.93%
0.35%
0.38%
0.31%
0.69%
Average
0.63%
0.82%
0.82%
0.85%
1.06%
0.43%
Panel B: Value-Weighted Monthly Returns
Probability Quintiles
Size Quintile
Low
2
3
4
High
High-Low
Small
2
3
4
0.93%
0.72%
0.65%
0.63%
1.40%
0.82%
0.66%
0.71%
1.32%
0.84%
0.48%
0.77%
1.25%
0.90%
0.98%
0.58%
1.16%
1.09%
1.04%
0.88%
0.22%
0.37%
0.39%
0.26%
Large
0.15%
0.20%
0.46%
0.30%
0.60%
0.44%
Average
0.62%
0.76%
0.77%
0.80%
0.95%
0.34%
2
Qualitatively similar results are obtained when portfolios are rebalanced quarterly.
22
Table 5: Four-factor Regressions on Monthly Size and Probability Quintiles (High minus
Low)
This table shows regression results of High minus Low
portfolio monthly returns
for each size quintile on Fama-French-Carhart factors for the period 1999-2011. Portfolios are
determined by using only NYSE cutoffs and are rebalanced monthly3.
is defined by
equation (2). *** represents a significant difference at the 1% level, ** at the 5% level, and * at
the 10% level.
Panel A: Monthly Probability Quintile (High-Low) Equal-Weighted Returns as the
Dependent Variable
Small
2
3
4
Large
0.652
0.645
0.512
0.484
0.865
Alpha
(3.32)**
(2.00)**
(1.97**)
(1.98)**
(1.91)*
*
-0.030
-0.015
0.054
0.032
-0.020
MKT
(-0.4)
(-0.2)
(0.89)
(0.54)
(-0.33)
0.002
-0.193
0.051
-0.013
0.021
SMB
(0.02)
(-2.04)**
(0.69)
(-0.17)
(0.27)
-0.589
-0.474
-0.463
-0.491
-0.595
HML
(-6.25)***
(-5.00)***
(-6.21)***
(-6.71)***
(-7.9)***
-0.152
-0.154
-0.058
-0.070
-0.018
MOM
(-2.68)***
(-2.71)***
(-1.29)
(-1.6)
(-0.4)
21.81%
14.85%
24.32%
24.40%
31.10%
Adjusted R2
Panel B: Monthly Probability Quintile (High-Low) Value-Weighted Returns as the
Dependent Variable
Small
2
3
4
Large
0.396
0.641
0.505
0.418
0.636
Alpha
(1.27)
(1.99)**
(1.99)**
(1.63)*
(2.23)**
0.054
-0.026
0.051
0.025
-0.083
MKT
(0.75)
(-0.34)
(0.87)
(0.42)
(-1.25)
0.012
-0.134
0.092
0.028
-0.095
SMB
(0.14)
(-1.44)
(1.25)
(0.38)
(-1.15)
-0.507
-0.474
-0.474
-0.523
-0.473
HML
(-5.64)***
(-5.08)***
(-6.47)***
(-7.02)***
(-5.73)***
-0.099
-0.165
-0.058
-0.063
0.034
MOM
(-1.84)*
(-2.95)**
(-1.32)
(-1.41)
(0.7)
19.9%
15.35%
27.39%
27.14%
17.78%
Adjusted R2
3
Qualitatively similar results are obtained when portfolios are rebalanced quarterly.
23
Table 6: Factor Summary Statistics and Correlations
This table shows monthly summary statistics and correlations of HMLP and Fama-FrenchCarhart four factors. HMLP factor is created to mimic return difference between high and low
probability portfolios. HMLP factor is created similar to Fama-French HML factor. HMLP is the
sample average of the value-weighted returns on the two high probability portfolios minus the
average value-weighted return of the two low probability portfolios. Returns are monthly (%). Pvalues in the Panel 2 are italicized.
Panel A: Descriptive Statistics
Quarterly Rebalancing
Annual Rebalancing
Variable
Mean
Std Dev
Sharpe Ratio
Mean
Std Dev
Mkt
Smb
Hml
Mom
HMLP
0.099
0.497
0.302
0.307
0.335
4.865
3.873
3.742
6.349
2.388
0.02
0.128
0.081
0.048
0.14
0.078
0.507
0.356
0.401
0.35
4.865
3.854
3.714
6.339
2.625
Panel B: Factor Pearson Correlations
Quarterly Rebalancing
Mkt
Smb
1
0.29484
Mkt
0.0002
0.29484
1
Smb
0.0002
-0.17011
-0.36651
Hml
0.0355
<.0001
-0.32736
0.13415
Mom
<.0001
0.0983
-0.16643
0.02174
HMLP
0.0398
0.7897
Annual Rebalancing
Mkt
Smb
1
0.29156
Mkt
0.0003
0.29156
1
Smb
0.0003
-0.15274
-0.38185
Hml
0.062
<.0001
-0.33344
0.14549
Mom
<.0001
0.0757
0.02431
0.22402
HMLP
0.7678
0.0059
Hml
-0.17011
0.0355
-0.36651
<.0001
1
-0.1579
0.0513
-0.13911
0.0863
Hml
-0.15274
0.062
-0.38185
<.0001
1
-0.14488
0.0769
-0.36922
<.0001
24
Mom
-0.32736
<.0001
0.13415
0.0983
-0.1579
0.0513
1
0.21513
0.076
Mom
-0.33344
<.0001
0.14549
0.0757
-0.14488
0.0769
1
0.21425
0.0085
Sharpe
Ratio
0.016
0.132
0.096
0.063
0.133
HMLP
-0.16643
0.0398
0.02174
0.7897
-0.13911
0.0863
0.21513
0.0076
1
HMLP
0.02431
0.7678
0.22402
0.0059
-0.36922
<.0001
0.21425
0.0085
1
Table 7: Fama-French-Carhart factors and HMLP
This table shows regression results of the HMLP factor on both Fama-French three and FamaFrench-Carhart four factors. HMLP factor is created to mimic return difference between high and
low probability portfolios. HMLP factor is created similarly to Fama-French HML factor. HMLP
is the sample average of the value-weighted returns on the two high probability portfolios minus
the average value-weighted return of the two low probability portfolios. In (1)-(2) probability
portfolios are rebalanced quarterly. In (3)-(4) probability portfolios are rebalanced monthly.
is defined by equation (2). *** represents a significant difference at the 1% level, **
at the 5% level, and * at the 10% level.
Alpha
MKT
SMB
HML
(1)
0.371
(1.93)**
-0.098
(-2.4)**
0.012
(0.23)
-0.106
(-1.94)**
MOM
Adjusted R2
3.8%
(2)
0.355
(1.85)*
-0.068
(-1.54)
-0.006
(-0.1)
-0.091
(-1.66)*
0.056
(1.69)*
4.9%
25
(3)
0.397
(1.95)**
-0.032
(0.75)
0.077
(1.33)
-0.237
(-4.05)***
13.0%
(4)
0.375
(1.85)*
0.003
(0.07)
0.054
(0.91)
-0.223
(-3.81)***
0.066
(1.89)*
14.53%
Table 8: Equal and Value-Weighted Monthly Returns across Measures of Value
This table shows returns by
and value/growth measures quintiles. Each year the
sample is sorted into quintiles on the basis of 3 alternative definitions of value (book-to-market,
market leverage and prior 36 month returns).
is defined by equation (2).
Panel A: Equal-Weighted Returns
Book-to-market Quintile
Growth
2
3
4
Value
Average
Low
-0.20%
0.36%
0.72%
1.13%
2.09%
0.82%
2
-0.02%
0.55%
0.69%
1.14%
1.86%
0.84%
Market Leverage Quintile
Growth
2
3
4
Value
Average
Low
-0.10%
0.13%
0.95%
1.16%
1.01%
0.63%
2
0.17%
0.59%
0.98%
1.04%
0.97%
0.75%
Prior 36 Month Return Quintile
Value
1
2
3
Growth
Average
Low
0.82%
0.41%
0.70%
0.50%
0.51%
0.59%
2
0.87%
0.76%
0.62%
0.40%
0.48%
0.63%
26
Probability Quintiles
3
4
High
0.00%
-0.08% -0.02%
0.44%
0.43%
0.98%
0.92%
0.87%
1.21%
1.43%
1.38%
1.46%
2.32%
1.98%
2.61%
1.02%
0.91%
1.25%
Probability Quintiles
3
4
High
-0.35% -0.26% -0.26%
0.69%
0.48%
1.15%
0.91%
1.00%
1.42%
1.29%
1.00%
1.41%
1.12%
1.61%
1.36%
0.73%
0.76%
1.02%
Probability Quintiles
3
4
High
1.34%
1.54%
1.21%
0.93%
0.94%
1.07%
1.04%
0.81%
0.98%
0.73%
0.66%
0.96%
0.67%
0.71%
1.01%
0.94%
0.93%
1.05%
High-Low
0.18%
0.62%
0.49%
0.34%
0.52%
0.43%
High-Low
-0.16%
1.03%
0.47%
0.25%
0.34%
0.39%
High-Low
0.19%
0.65%
0.28%
0.46%
0.70%
0.46%
Table 8 (Continued)
Panel B: Value-Weighted Returns
Book-to-market Quintile
Growth
2
3
4
Value
Average
Low
-0.40%
0.28%
0.48%
0.54%
1.10%
0.40%
2
0.03%
0.66%
0.62%
0.49%
1.23%
0.61%
Market Leverage Quintile
Growth
2
3
4
Value
Average
Low
-0.29%
-0.29%
0.82%
0.42%
0.51%
0.23%
2
0.21%
0.40%
0.30%
0.76%
0.40%
0.41%
Prior 36 Month Return Quintile
Value
1
2
3
Growth
Average
Low
0.33%
0.11%
0.40%
0.10%
0.12%
0.21%
2
0.34%
0.48%
0.35%
0.55%
-0.12%
0.32%
27
Probability Quintiles
3
4
High
0.24%
0.09%
0.25%
0.29%
0.58%
1.07%
0.36%
0.63%
0.90%
0.93%
0.95%
0.91%
1.36%
1.04%
1.79%
0.64%
0.66%
0.99%
Probability Quintiles
3
4
High
0.22%
-0.19% 0.17%
0.37%
0.34%
0.84%
0.59%
0.72%
0.75%
0.56%
0.71%
1.38%
0.78%
0.89%
0.93%
0.50%
0.49%
0.81%
Probability Quintiles
3
4
High
1.02%
0.91%
0.97%
0.62%
0.61%
0.76%
0.57%
0.45%
0.87%
0.46%
0.47%
0.63%
0.36%
0.44%
0.66%
0.60%
0.58%
0.78%
High-Low
0.65%
0.80%
0.42%
0.37%
0.69%
0.59%
High-Low
0.45%
1.13%
-0.07%
0.96%
0.42%
0.58%
High-Low
0.64%
0.65%
0.48%
0.53%
0.53%
0.57%
Table 9: Probabilities Across Fama-French 25 Size and Book-to-Market Potfolios
Panel A shows percentage of the firms in Fama-French 25 book-to-market portfolio for which
probabilities are NOT available. Panel B shows time-series mean value of probabilities for each
of the Fama-French 25 book-to-market portfolios.
Panel A: % of the firms for which probabilities are NOT available
Small
2
3
4
Big
Growth
96.07%
63.82%
26.44%
4.75%
0.68%
2
95.92%
65.89%
28.38%
2.55%
0.22%
3
97.57%
69.74%
31.78%
2.76%
0.53%
4
97.87%
71.88%
32.61%
2.64%
0.20%
Value
98.38%
66.06%
21.81%
3.14%
0.64%
Panel B: Probabilities across Size and Market-to-book Quintiles
Small
2
3
4
Big
Small-Big
Growth
0.515
0.537
0.558
0.547
0.523
-0.008
2
0.457
0.510
0.518
0.476
0.475
-0.018
3
0.451
0.461
0.482
0.476
0.481
-0.031
4
0.387
0.435
0.488
0.471
0.459
-0.072
28
Value
0.365
0.418
0.432
0.445
0.447
-0.081
Value-Growth
-0.149
-0.119
-0.126
-0.102
-0.076
Table 10: Performance in Market and Economic Cycles
This table presents the alphas from four-factor regressions of monthly excess equally weighted
and value-weighted
portfolio returns. The column Diff High-Low refers to the
difference in four-factor alphas between high and low portfolios. Portfolios are created based on
probabilities and are rebalanced monthly. T-statistics are presented in brackets. *** represents a
significant difference at the 1% level, ** at the 5% level, and * at the 10% level.
NBER Expansion
Alpha, Equally-weighted
Alpha, Value-weighted
Probability Quintile
3
4
0.306
0.354
(2.16)**
(1.95)*
-0.025
0.354
(-0.17)
(2.35)**
Low
-0.076
(-0.53)
-0.311
(-1.72)*
2
0.081
(0.57)
-0.077
(-0.51)
Low
-0.022
(-0.07)
0.227
( 0.46)
Probability Quintile
2
3
4
0.272
0.649
0.347
(1.18)
(2.58)*** (1.42)
-0.177
0.983
-0.268
( -0.63)
( 1.23)
( -0.83)
High
0.768
(2.96)***
0.418
(2.56)***
Diff High-Low
0.844
(2.81)***
0.728
(2.47)***
High
0.732
(1.96)*
0.273
( 1.08)
Diff High-Low
0.754
(1.46)
0.045
( 0.07)
High
0.489
(3.15)***
0.395
(3.17)***
Diff High-Low
0.543
(2.82)***
0.448
(1.80)***
High
1.716
(4.33)***
0.675
(2.47)**
Diff High-Low
1.671
(3.67)***
1.371
(2.9)***
NBER Recession
Alpha, Equally-weighted
Alpha, Value-weighted
Bull Market
Alpha, Equally-weighted
Alpha, Value-weighted
Probability Quintile
3
4
0.139
0.155
(1.16)
(1.12)
-0.089
0.073
(-0.7)
(0.6)
Low
-0.054
(-0.38)
-0.053
(-0.33)
2
-0.022
(-0.21)
-0.088
(-0.79)
Low
0.045
(0.21)
-0.696
(-2.12)**
Probability Quintile
2
3
4
0.843
0.963
0.877
(2.93)*** ( 4.28)*** ( 3.18)***
-0.397
0.714
0.308
(-1.69)*
(2.8)***
1.15
Bear Market
Alpha, Equally-weighted
Alpha, Value-weighted
29
Figure 1: This figure shows % of the value, growth and neutral firms in the top and bottom 30%
of the HMLP quintile. Time-series mean values are also reported.
Stocks in the top 30% of the HMLP
Jun-11
Dec-11
Dec-11
Dec-10
Jun-10
Dec-09
Jun-09
Dec-08
Jun-08
Dec-07
Jun-07
Jun-11
Value (top 30% in the FF HML)
Dec-06
Jun-06
Dec-05
Jun-05
Dec-04
Jun-04
Dec-03
Jun-03
Dec-02
Jun-02
Dec-01
Jun-01
Dec-00
Jun-00
Dec-99
Jun-99
80.00%
70.00%
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
Neutral (middle 40% in the FF HML)
Growth (bottom 30% in the FF HML)
Stocks in the bottom 30% of the HMLP
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
Value (top 30% in the FF HML)
Dec-10
Jun-10
Dec-09
Jun-09
Dec-08
Jun-08
Dec-07
Jun-07
Dec-06
Jun-06
Dec-05
Jun-05
Dec-04
Jun-04
Dec-03
Jun-03
Dec-02
Jun-02
Dec-01
Jun-01
Dec-00
Jun-00
Dec-99
Jun-99
0.00%
Neutral (middle 40% in the FF HML)
Growth (bottom 30% in the FF HML)
Time-series Average
Value
11.22%
21.93%
Stocks in the top 30% of the HMLP
Stocks in the bottom 30% of the HMLP
30
Neutral
32.01%
38.56%
Growth
56.77%
39.51%