Exhibit — Case 1:21-md-02989-CMA (Document 585-12)

Summary

This exhibit (Document 585-12) in Case 1:21-md-02989-CMA, entered on the FLSD docket 06/30/2023, reproduces a paper by Richard Roll of the Anderson Graduate School of Management, University of California, Los Angeles, from The Journal of Finance, Volume 43, Issue 3 (July 1988), 541-566. The abstract states that even with hindsight the ability to explain stock price changes is modest and that there is little relation between explanatory power and either the firm's size or its industry. The paper reports that with all explanatory factors included, less than forty percent of the monthly return volatility in the typical stock can be explained, for a sample of the largest firms in the U.S. market. Its references cite works by Lawrence Harris, John Lintner, Jan Mossin, Stephen A. Ross, Myron Scholes, Joseph Williams and William F. Sharpe.

Summary drafted by a model from the document's text below and checked by script against that text before publication. It is a navigation aid, not a reading of what the document proves. Where AI is used

Full text

Text recognized from a scanned document; may contain errors. Pages 9–11, 16, 18–20, 22, 23, and 26 of 29 are not shown because the text could not be recognized reliably.

Case 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 1 of
29

EAHIBIT
45
Case 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 2 of
29

Richard Roll

The Journal of Finance, Volume 43, Issue 3, Papers and Proceedings of the
Forty-Seventh Annual Meeting of the American Finance Association, Chicago, Illinois,
December 28-30, 1987 (Jul., 1988), 541-566.

Stable URL:

Your use of the ISTOR archive indicates your acceptance of ISTOR’s Terms and Conditions of Use, available at
http://www_jstor.org/about/terms.html. JSTOR’s Terms and Conditions of Use provides, in part, that unless you
have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and
you may use content in the JSTOR archive only for your personal, non-commercial use.

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

The Journal of Finance is published by American Finance Association. Please contact the publisher for further
permissions regarding the use of this work. Publisher contact information may be obtained at

The Journal of Finance
©1988 American Finance Association

ISTOR and the JSTOR logo are trademarks of JSTOR, and are Registered in the U.S. Patent and Trademark Office.
For more information on JSTOR contact jstor-info@umich.edu.

©2002 JSTOR

Mon Jun 3 16:06:56 2002
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 3.
29

THE JOURNAL OF FINANCE ¢ VOL. XLIIL NO. 2 « JULY 1988

R 2
RICHARD ROLL*

ABSTRACT

Even with hindsight, the ability to explain stock price changes is modest. R’s were
calculated for the returns of Large stocks as explained by systematic economic influences,
by the returns on other stocks in the same industry, and by public firm-specific news
events. The average adjusted R? is only about .35 with monthly data and .20 with daily
data. There is little relation between explanatory power and either the firm's size or its
industry. There is little impravement in R? from eliminating aé/ dates surrounding news
reports in the financial press. However, the sample kurtosis is quite different when such
news events are eliminated, thereby revealing a mixture of return distributions. Non-
news dates also indicate the presence of a distributional mixture, perhaps due to traders
acting on private information.

THE MATURITY OF A science is often gauged by its success in predicting important
phenomena. Astronomy, the oldest science, is able to predict the positions of
planets and the reappearance of comets with a high degree of accuracy. Astro-
nomical phenomena are extraordinarily regular, and they permit the construction
of forecasting models with only trivial prediction errors.

Financial science too can boast about very high explanatory power if the
phenomenon is artfully selected. For instance, the daily change in an option’s
price can be accurately “predicted” by the concurrent change in the associated
stock price. Similarly, a good predictor of tomorrow’s asset price is today’s price.
But both of these examples are contrived; they are analogous to a meteorologist
who might claim a high degree of predictive power because the weather can he
forecast fairly accurately over the next hour!

The immaturity of our science is illustrated by the conspicuous lack of predic-
tive content about some of its most intensely interesting phenomena, particularly
changes in asset prices. General stock price movements are notoriously unpre-
dictable and financial economists have even developed a coherent theory (the
theory of efficient markets) to explain why they should be unpredictable.

However, many financial economists seem to helieve that, with hindsight, they
could explain most agsset-price movements with authenticated information. The
prevailing paradigm about stock price changes ascribes them to (1) unpredictable
movements in pervasive economic factors,’ (2) unpredictable changes in the
firm’s market environment, i.e., industry information, and (3) unpredictable

* Anderson Graduate School] af Management, University of California, Los Angeles.

‘There is a controversy within the paradigm ahout the identity and number of pervasive, or
“systematic,” economic factors, but almost everyone would agree that there is at least one, and some
researchers have uncovered evidence that there are several.

541
> 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page
29

Richard Roll
President of the American Finance Association
1987
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 5 ¢
29

542 The Journal of Finance

events apecific to the firm itself. By observing and measuring these influences ex
post, one could presumably “explain” stock price movements with an astronomical
degree of accuracy. The R? should be close to 1.0.

This paper offers a simple empirical investigation of the paradigm. Is it really
true that we can explain the actual price movements of individual common stocks
by broad economic influences, industry influences, and specific news events about
the firm? The paper is organized by these categories of information: the second
section examines pervasive factors, the third section industry factors, and the
fourth section firm-specific factors.

The results are not very gratifying. With all explanatory factors included, less
than forty percent of the monthly return volatility in the typical stock can be
explained, and this is for a sample of the largest firms in the U.S. market. The
explanatory success for daily return data is even less. The paucity of explanatory
power represents a significant challenge to our science. We ought to discover
either (a) measureable influences that will explain the remaining sixty percent,
or (b) a coherent reason why it should forever remain unexplained.

I. The Data

All data in the paper refer to equities of corporations traded on the New York
and American Stock Exchanges. Monthly returns are used in the first two sections
below and daily returns are used in the fourth section. The monthly returns
cover a five-year period, September 1982 through August 1987,” and they derive
from two sources: through December 1986, the source was the CRSP monthly
stock returns file, and from January through August 1987 the source was IDC.'
The daily data covered the 1982-86 calendar years inclusive and were obtained
from the CRSP daily returns file.

Data pertaining to news events were obtained from the Dow-Jones News
Retrieval System. This data source contains every mention of most publicly held
companies in a number of different publications and news services. The system
was used here to retrieve every news item about a company that appeared either
in the Wall Street Journal or on the Dow-Jones news wire (the Broad Tape)
during the sample period. These news items included stories exclusively about a
company and also stories of a more general nature in which the company was
mentioned.

II. The R? of Systematic Factors

Systematic, or non-diversifiable, factors play a major role in the most widely
studied theories of asset pricing: the Capital Asset Pricing Model (CAPM)* and
the Arbitrage Pricing Theory (APT).° For our purposes here, the principal

? This was the most recent five-year period available at the time the calculatians were made.

7 CRSP is the acronym for the University of Chicago’s Center for Research in Securities Prices,
and IDC is the acronym for Interactive Data Corporation.

* See Sharpe [11], Lintner [5], and Mossin [7].

5 See Ross [9].
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 6.
29

R? 543

distinction between the two theories is the number of non-diversifiable factors.
The “market model” version of the CAPM implies a single pervasive market
factor, while the APT allows more than one (though the APT is also consistent
with just one).

Several factors may turn out to explain a larger proportion of intertemporal
return volatility than a single factor, but this finding alone would not constitute
evidence that a multiple-factor theory is better. That conclusion would also
require an empirical finding that additional factors are indeed pervasive, non-
diversifiable, and most important, that they are associated with additional risk
premia. Such interesting and difficult questions are ignored in this paper.

Instead, the paper merely reports on the cross-sectional distribution of R,’
adjusted for degrees of freedom, from using a single factor and from using multiple
factors in regressions of the type

where r;, is the total return on stock j in period ¢, f,, is systematic factor i in
period t, the a’s and b’s are estimated regression coefficients, and e;, is the
“unexplained” return. The adjusted R? from such a regression is defined as

where T is the time-series sample size and s(x) is the sample standard deviation
of x.

For the CAPM single-factor market model (k = 1) and the APT five-factor
market model (k = 5), Figure 1 presents the cross-sectional frequency distribution
of R? for the 2030 individual stocks listed on the NYSE and AMEX aa of
September 1982, and for which there were at least thirty monthly observations
through August 1987. For the CAPM, /f,,, was defined as the equal-weighted index
of all stocks in the sample available in month ¢. For the APT, f, through f; were
factor scores obtained from a large-scale factor analysis using all stocks that were
continuously listed during the five-year period. The first APT factor, f,, is very
highly correlated with the CAPM’s single factor, and indeed the APT f, is
tantamount to an equal-weighted market index. By construction, the second
through the fifth APT factors are uncorrelated with f, and also with each other.

As Figure 1 reveals quite clearly, the entire distribution of R’ is displaced to a
somewhat higher level for the multiple-factor model (APT), relative to the single-
factor model (CAPM). The mean R’s were, respectively, .179 for the CAPM and
.244 for the APT. Of the 2030 stocks, 1571 (77.4 percent) had a higher R? with
the multiple-factor model. The second panel of Figure 1 shows the cross-sectional
frequency distribution of the difference between the APT and CAPM R’s for
individual firms.

The average explanatory power of well-accepted market models is quite modest,
but there are some firms with impressive #’s (in either a single- or a multiple-
factor regression), and perhaps a study of these firms would help us understand
why the explanatory power is rather limited for the average firm.

* Every 2? reported. in the paper is adjusted for degrees of freedom, even if this is not explicitly
stated.
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 7.
29

544 The Journal of Finance

Adjusted R~Square For Market Models

September, 1982 - August, 1967

14 : %

12
11
10
a

7 SC? 7
A
x

Frequency, t%)

x
x

Cx

7
6
§
4
a
2
1
a

‘a

.05 6.16.15 0.2 0.25 0.3 0.35 0.4045 05 0.45 2.6 0.66 0.7 0,75

KS
LX
x

Lass Than

R-Square Difference, APT vs. CAPM

Saptember, 1962- August, 19487

Frequency, (%)

MSS EAA AQAA Y

U qu T tT T
Less Than -0.16-0.1-0.95 0 0.05 0.4 0.15 0.2 0.26 2.3 0.35 04 0.46 0.6 0.55
APT R-Square - CAPM A-Squarea

Figure 1.

A. R? and Firm Size: The Portfolio- Diversification Effect

Firm size is a possible explanation of why R’s differ; larger firms generally
have many divisions and often operate in more than a single industry and market.
Thus, they superficially resemble diversified portfolios of smaller firms, and it is
well known that diversified portfolios have high R’s, at least with respect to the
single CAPM factor and with respect to the broad market APT factor, f,.

This possibility is supported to a modest extent by the results depicted in
Figure 2, a cross-sectional scatter diagram of CAPM R? against the natural
logarithm of firm size.’ There is a discernible positive cross-sectional correlation.

? Firm size is defined as market price times the number of outstanding shares on the deginning
date of the sample period, i.e., August 31, 1982,
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 8
29

R? 545

CAPM Adjusted A-Square vs. Size

NYSE and AMEX Stacks. Seplember, 1962 - August, 1967

6.8

Average R? = 1786
0.7 4

Adjusied R-Square

Cross-Sectional Regresston:

R?2 = 6224 + 0337 Ln{Size} R?= 198
—6.2 T T T T T T T

fos 2 4 6 8 10

4
4

Figure 2.

A simple linear regression between R? and In(Size) indicates a statistically
significant connection (but this statistical significance is suspect because the
observations are probably not cross-sectionally independent). When firms are
relatively small, say below $10 million, few R*s exceed 0.3, while when firms are
relatively large,.say above $150 million, there are many firms with R’s above 0.4
and a few even above 0.5. Based on this evidence taken in isolation, diversification
may indeed be an explanation for why larger firms display a greater degree of
explanatory power.

However, some doubt about this conclusion can be derived from considering
Figure 3, which presents the cross-sectional relation between the five-factor
(APT) R? and In(Size). There is a much less perceptible positive relation here,
though the keen-eyed observer may still detect a small one. Aside from a higher
R? level in general, the biggest difference between the multiple-factor and the
single-factor scatter is a significant improvement. in the R’s for small firms. This
may suggest that higher explanatory power of market models for large firms is
due less to general diversification than to large firms being less susceptible, for
some ag yet unknown reason, to systematic risks that are not general market
risks.

To investigate the diversification element in more detail, all of the firms in the
sample were sorted by size on the beginning date of the period and divided into
two groups, one group comprised of the decile of largest firms and the second
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 12
29

R? 549

size-matched portfolio, there is likely to be significant diversification across
industries. To the extent that the larger firms are not similarly diversified by
having divisions in several industries, their R’*s might naturally be lower.

To check this out, the four-digit SIC code enregistered in the CRSP file was
used to categorize stocks into industries and then size-matched portfolios were
constructed for as many large firms as possible within each industry. To be
specific, the following procedure was employed:

1. All firms were sorted by four-digit SIC code.

2. For a given four-digit code, the largst firm was selected.

3. From among the smaller firms in the same four-digit industry, a portfolio
was constructed, if possible, hy randomly choosing firms, without replace-
ment, until an aggregate market capitalization equal to that of the large
firm had been obtained.*

4. For the next largest firm in the industry, another size-matched portfolio
was constructed, and so on until as many portfolios as possible were
constructed from the smaller firms in a‘ given four-digit industry, each
portfolio matched in size to one of the larger firms in the same industry.

Table I presents the resulting industries included, and the number of portfolios
in each industry. There were ninety-six large firms, and thus ninety-six size- and
industry-matched portfolios, in sixty-six different industries.

The sample of large firms, their industry codes, and the size- and industry-
matched portfolios are presented in Table II, which also gives single-factor and
multiple-factor R’s and a few other pertinent statistics. For the ninety-six large
firms, the average CAPM R? was .2394 and the APT R? was .2872; the corre-
sponding portfolios had higher R’s on average, .3614 and .4426, respectively. On
average, 6.2 small firms in the same industry were required to construct a portfolio
with that same aggregate size as the large firm.

Figure 6 presents evidence on the critical question as to whether explanatory
power is improved significantly by controlling for industry. For both the CAPM
in the upper panel, and the APT in the lower panel, there appears to be only a
very slight cross-sectional relation between the R? of a large individual firm and
an equal-size portfolio of smaller firms within the same industry. The scatter is
almost, as disperse as that observed in Figure 5 when there was no control on
industry when forming the equal-size portfolio.

One might have anticipated significant differences across industries in the
ability of pervasive economic factors to explain returns. Yet, the lack of inter-
industry correlation between the R’s of large firms and the R’s of size- and
industry-matched portfolios of smaller firms suggests that the explanatory power
of pervasive factors is not very different from one industry to another.

To measure the extra explanatory power of industry events in addition to the
explanatory power of general pervasive economic factors, further regressions were
performed with each large firm in the sample using that firm’s equal-sized
industry-matched portfolio return as a supplemental regressor; i.e., the following

° The same method was employed in Section II to form size-matched portfolios without regard to
industry.
ise 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 13
29

550 The Journal of Finance

Table I

Four-Digit Industries with Smaller Firm Portfolios
Matched in Aggregate Size to the Largest Firms Within the Same

Industry
Portfolios Number of Frequency

per Industry Industries (%)

1 62 78.79

2 8 12.12

a 2 3.03

4 1 1.52

5 1 1.52

6 1 1.52

7 1 L.52

No. of Industries: 66
No. of Portfolios: 96
Number Number, Number
of of af
Industry Portfolios Industry Portfolios Industry Portfolios

1041 1 4641 1 4931 2
1211 1 3561 1 4941 1
1311 5 3569 1 5065 1
1382 1 3574 1 5311 2
2043 1 3612 1 6411 2
2211 1 4629 1 §912 1
2461 1 3662 a 6944 1
2621 1 a679 1 6025 2
2711 1 3694 1 6331 1
2819 1 3721 1 6411 1
2834 2 3732 1 6552 2
2869 1 3792 1 6711 q
2899 1 3822 1 6723 4
2911 2 3825 1 6792 1
3011 1 3841 1 6799 a
3069 1 4011 1 7011 1
3079 1 4511 2 7311 1
3312 1 4832 1 7379 1
3429 1 4899 1 FAGi 1
3494 1 4911 6 7392 1
3519 1 4923 1 8062 1
36334 1 4924 1 8911 1

regression was computed for each of the ninety-six large firms,

where j = 1, ..., 96, the f’s are the systematic factors used previously (k = 1 for
the CAPM and k = 5 for the APT), and /;,, is the return on the industry-matched
portfolio of smaller firms. Table III gives the average increase in R’ from adding
the industry portfolio return, /;. On average, the improvement is slightly greater
for the CAPM than for the APT, which may indicate that the non-general-
market, higher order APT factors, f2,..., fs, have a differential impact across
industries.
551

29

ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 14
R?

QOUBLIE A qo ‘ON
29

The Journal of Finance

ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 15
552
The Journal of Finance

29

ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 17
554
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 21
29

558 The Journal of Finance

attributes the remaining volatility to unigue information about the firm. With
hindsight, such information should be discoverable by tracing its appearance in
the financial press, an implicit presumption being that anything insignificant
enough to be ignored by the media is probably also immaterial in its impact on
market prices. .

It should, therefore, be possible to increase greatly the R* of pervasive factors
by considering only periods when there is no reported news about the firm or
about its industry. When there is no firm-specific news, all of the observed
changes in prices would presumably be explained by pervasive factors. We can
investigate this part of the paradigm by expurgating those data observations that
coincide with news stories in the financial press and then running the same
regressions as before on the remaining information-cleansed observations.

In the U.S. market, there are two prominent sources of financial news, the
Dow-Jones news service, or “Broad Tape,” which contains a real-time record of
major news developments of all kinds, and The Wall Street Journal, which
records not only major developments but significant analyses of these events by
staff writers and others. Given the available resources, it was not possible to
collect news dates from these sources for every listed firm. Thus, for ease of
comparison with the previous sections of the paper, these news events were
collected for the ninety-six large firms whose industry influences were analyzed
in Section III.

Daily data were used in the analysis of this section because of the high
frequency with which large firms are mentioned in the financial press. For many
large firms, few months pass without a mention; there are not enough news-free
months to constitute an adequate sample of observations. In contrast, consistent
daily mention is unusual, so a large number of no-news daily observations for
most firms can be collected even if a few days are excluded around the publication
date of each news item. On average over all ninety-six stocks, 23.7 percent of the
daily observations were excluded by being either the day of a news event or the
preceding day; this means that about 965 of the 1264 trading days in the 1982-
86 calendar years are available for the non-news analysis.

It is probably important to expurgate the date of the news publication and at
least one day before publication. For items that appear only in the Wall Street
Journal and not on the Broad Tape, the news might actually have been publicly
available on the previous day. Even for items on the Broad Tape, insiders
sometimes receive forewarning, and the price may move in response to their
trades. We report below on the consequences of widening the window of excluded
dates around each news appearance.

Figure 10 presents the resulting R*s contrasting two effects, the CAPM single-
factor model veraus the APT multiple-factor model and the presence or absence
of firm-specific news. The top panel shows R’*s with all available daily observa-
tions; the bottom panel shows R’s for regressions excluding the day of a news
appearance and the previous trading day. The 45-degree line lies below all
observations, which means that every stock had a higher R* in the multiple-
factor daily regressions.’°

© This contrasts with only about three quarters of the stocks with higher multiple-factor R*s using
monthly data.
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 24 c
29

R? 561

increase in R-Square

From Eliminating News Oate and Prior Qay

0.16
a
0.14 4 " © «
0.1 ~ .
o 5A ao ht o a a
3. a ® of a
0.08 4 7
o APT
-6.1 T T T T T “ T T T
a 20 40 60 30

Percentage of Days with News

Figure E2,

non-news-date returns had heen easier to explain with the systematic factors,
the depicted relation would have been stronger.

A. Estimating the News Process with a Simple Mixture Model

Even though the results are disappointing, there is some very slight evidence,
particularly from those firms that have experienced extraordinary events such aa
merger, that public news reduces the explanatory power of systematic factors.
Thus, it seems worth investigating the phenomenon a bit further, if only to help
in understanding why the power is so poor on average.

It ig natural to model the overall stochastic process of price changes with a
mixture of probability models, i.e., a different probability distribution during
news and non-news periods. This idea has been investigated previously by a
number of different authors, most of whom have presented relatively sophisti-
cated statistical models.’ The principal feature of mixed distributions empha-
sized by most authora involves the higher moments, particularly the fourth
moment or the sample kurtosis. Kurtosis can reveal something about the proha-
bility of information and the difference between the information-related distri-
bution and the non-information-related distribution of returns. Damodaran [1],
for instance, finds that kurtosis is closely correlated to such measures of infor-
mation as the number of analysts following a stock and the number of Wail
Street Journal stories, inter alia.

* One of the first, and still one of the most interesting models was by Press [8]. Press’ model
postulated a mixture of normal distributions; the number of distinct normals in the mixture was
itself a Poisson-distributed random variable. There are many papers in this tradition (¢.g., Epps and
Epps [2], Morgan [4], and Westerfield {12]). Harris [3], Damodaran [1], and Harris [4] are recent
contributions of the same genre.
ise 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 25
29

562 The Journal of Finance

A very simple model of this type can be constructed as follows. Assume that a
market-model residual, e,, can be decomposed as

Cp = Xi + Be ¥t,
where
x = background trading noise (no news),
y = news-related residual return,

#, = 1 if there is news on date t, otherwise 0.

Assume E(e) = E(x) = E(y) = E(xy) = E(xg) = E(gy) = 0.
Define
V; = variance of jf, (j = 4, x, ¥)
p = Prob{g, = 1}.
If x and y are normally distributed,
E{x + gy] = 3[Vi + pV; + 2pV.V,],
and the “kurtosis” is
K = E(e*)/(V2) = 3{(1 + [p(1 — p)/LV./V, + p]’} > 3.
Q = K/3 -1=[p(1 — p)JAR + pf, (1)

where R = V,/V, is the ratio of noise to news variances. Q@ can be directly

estimated, but there is no straightforward way to decompose the estimate of @

into its components R and p. However, provided that @ is positive, R can be

bounded. As it turns out, for every one of the ninety-six firms in our sample, the

sample value of Q.is positive (which is a result typical of daily stock-return data).
Solving equation (1) for p gives

p= {1 — 2RQ + [1 — 4RQ(L + R))}/2(Q + DI,
and, to obtain a non-imaginary p,
R+ RR’ < 1/(4Q),
which implies that the maximum value of R is given by
| Rox = {[1 + 1/Q]'” — 14/2. (2)

This is, therefore, the minimum ratio of news variance to noise variance. At this
minimum ratio of variances, the probability of news is given by

Dmax = [1 — 2Rmax@]/[2(Q + 1)]. (3)

The cross-sectional frequency distribution of the reciprocal of Raa, is given in
the top panel of Figure 13. The bottom panel gives the corresponding frequency
distribution for pmax. The average values of these statistics, along with several
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 27
29

564 The Journal of Finance

Table IV
Volatility, Kurtosis, Estimated Information Statistics, and R?
for Windows Around Public News Dates
(Average Values over Ninety-Six Industry-Matched Large Firms)

Standard Deviations Kurtosis Minimum P{News}

Returns Residuais {1/Rmex) R?
Including All Daily Observations
CAPM = 23.454 21.488 17.725 20.457 0.14392 0.16282
APT 24.454 20.927 17.935 20.738 0.143857 0.20532
Exciuding One Day Before and the Day of the News
CAPM = 22.491 20.427 $.228 7.750 0.17594 0.17650
APT 22.491 19.843 §.256 7.785 0.17780 0.22102
Exciuding Two Days Before and the Day of the News
CAPM = 22.419 20,344 7.829 7.209 0.18150 0.17821
APT 22.419 19.738 7.800 7.164 0.18522 0.22367
Excluding Two Days Before Through One Day After the News
CAPM = 22.263 20.192 7.371 6.582 0.18162 0.17880
APT 22.263 19.557 7.312 6.502 0.18520 0.22548

estimate of news volatility, V,, several times the volatility of background trading
noise, V.. This is the case for both the complete sample including every daily
observation and the sample censored of news dates in the financial press.

Table IV presents sample averages for several variables and for three different
windows around news dates. All of the censored samples display much lower
kurtosis and much lower estimated news volatility than the full (uncensored)
sample, but widening the window to censor more days before and after the news-
publication date seems to have only a minor additional effect.

The results in Table [V indicate that the sample volatility for the total return
or for the residual return is almost as large in the censored samples as in the
complete sample. This is in striking contrast with kurtosis, and it reveals
immediately that the probability is rather small of observing the more volatile
news-related member of the mixture of distributions. In keeping with this
inference, the average value of py.» for the complete sample is about .14.

The average probability of “news” is even higher in the censored samples than
in the complete sample. This seems to imply that the financial press misses a
great deal of relevant information generated privately. However, the volatility
induced by private information is lower than that induced by the big newsworthy
events. Evidently, the very simple model presented here should be generalized at
least enough to subsume a three-distribution mixture, say y, for public news, x,
for noise, and an additional z, for private information.

B. Caveats

There are a few things that were considered but omitted from the paper because
of space or time limitations.
1) In daily regressions of stock returns on indicia (including APT factors),
ase 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 28
29

R? 565

there is a serious problem of coefficient bias hecause of non-synchronous trad-
ing.'* The regressions above were estimated also with leading and lagged values
of the explanatory variables, but this had very little impact on the R*s. Large
firms are less susceptible to this problem anyway, and apparently the problem is
more material for slope coefficients than for R’s.

2) The CAPM market index and the APT factors are composed of returns on
traded securities. A certain proportion of these securities would experience firm-
or industry-specific news on any given day; thus, the indexes and factors must
contain extra volatility arising from such unique events (relative to the volatility
induced by purely general economic influences unpolluted by specific firm news).
In principle, one could construct better factors by eliminating every stock with
news on each day. It seems doubtful that this would materially increase R’, but
one cannot. know for sure until it is tried.

3) Several authors have suggested that volatility of asset prices can be hetter
explained by psychological factors, fads, etc., than by information. The results
above are actually consistent with such a view. After all, the unexplained volatility
on non-news dates could conceivably indicate not that private information is
being incorporated into prices hut that mania is periodically gripping investors.
Perhaps the components of the distributional mixture consist of low-variance
background liquidity traders and high-variance traders striken by either panic or
euphoria. It would be nice to have a method for detecting the difference.

V. Summary and Conclusions

Most scholars and practitioners have resigned themselves to poor ex ante fore-
casting power for stock price changes. However, the current paradigm of financial
markets implies much better explanatory power ex post. With hindsight, stock
price changes should be explainable by general systematic influences, industry
influences, and events unique to the firm. This paper attempted an empirical
investigation of the paradigm.

Regressions of individual monthly stock returns on either a single market
index or on multiple factors produced explanatory power, as measured by the
average adjusted R?, in the neighborhood of 0.30. Adding an industry factor
increased the average R* to around 0.35.

For the decile of the largest AMEX and NYSE firms, portfolios of smaller
firms were constructed to match each large firm in aggregate size. These portfolios
had much higher R’s than their corresponding size-matched large individual
firms, thereby indicating that diversification by the large firm is not much of an
explanation for the slightly larger explanatory power. There was no perceptible
cross-sectional relation between the R? of the large firm and the R’ of its aggregate
size-matched portfolio.

Also, portfolios were constructed to match a sample of large firms bath in size
and in industry. Again, there is little relation between size and explanatory power.
Perhaps most surprising, there was again virtually no cross-sectional relation
between the R? of a large firm and the R? of its size-matched and industry-

2 See, e.g., Scholes and Williams {10].
ise 1:21-md-02989-CMA Document 585-12 Entered on FLSD Docket 06/30/2023 Page 29
29

566 The Journal of Finance

matched companion portfolio. This indicates that explanatory power by system-
atic economic factors is not very different across industries.”

Daily data were employed here to investigate the incidence and impact of
unique news about. the firm. With daily data, the average R? of the same sample
of large firms dropped to around 0.20. Every mention of each firm in either the
Wall Street Journal or over the Dow-Jones Broad Tape was defined as an
information event. Regressions on systematic factors were conducted only with
non-information dates. Even with this information-censored data, the average
explanatory power was only marginally better. For example, using the multiple-
factor APT pervasive factors as regressors, the average R* increased from .205
using all days in calendar years 1982-86 to only .225 after excluding the infor-
mation event, two days before, and one day afterward.

There was, however, a dramatic decline in sample kurtosis from excluding
public news events. A simple mixture-of-distributions model (i-e., one distribution
for a non-news date and a second, higher variance distribution for a news date)
yielded two suggestive results: (1) the average minimum ratio of news variance
to background noise variance was around 20 for all sample dates but only around
7 for non-news dates, and (2) the estimated probability of news was modest but
material for both identifiable public news dates and for other dates. This seems
to imply the existence of either private information or else occasional frenzy
unrelated to concrete information.

13 Although explanatory power is not very industry related, the coefficients in the regressions are
very different in different industries.

REFERENCES

1. Aswath Damodaran. “Information Frequency: Measures and Implications.” Working paper,
Graduate School of Business Administration, New York University, September 1986.

2. T. W. Epps and M. L. Epps. “The Stochastic Dependence of Security Price Changes and
Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis.” Econome-
trica 44 (March 1976), 305-21.

3. Lawrence Harris. “Cross-Security Tests of the Mixture of Distributions Hypothesis.” Journal af
Financial and Quantitative Anatysts 21 (March 1986), 39-46.

. “Transaction Data Tests of the Mixture of Distributions Hypothesis.” Journal of Financial
and Quantitative Analysis 22 (June 1987), 127-41.

5. John Lintner. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock
Portfolios and Capital Budgets.” Review of Economics and Statistics 47 (February 1965), 13-
37.

6. Ieuan Morgan. “Stock Prices and Heteroskedasticity.” Journal of Business 49 (October 1976),
496-508.

7. Jan Mossin. “Equilibrium in a Capital Asset Market.” Econometrica 34 (October 1966), 768-83.

8. James S. Press. “A Compound Events Model for Security Prices.” Journal of Business 40 (July
1967), 317-35.

9. Stephen A. Ross. “The Arbitrage Theory of Capital Asset Pricing.” Journal af Economic Theory
13 (December 1976), 341-60.

10. Myron Scholes and Joseph Williams. “Estimating Betas from Nonsynchronous Data.” Journal
of Financial Economics 5 (December 1977), 309-27.

11. William F. Sharpe. “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of

Risk.” Journal of Finance 19 (September 1964), 425-42.

12. Randolph Westerfield. “The Distribution of Common Stock Price Changes: An Application of

Transactions Time and Suhordinated Stochastic Modeis.” Journal of Financial and Quantita-
tive Analysis 12 (December 1977), 743-65.


File and source

File

gov.uscourts.flsd.590042.585.12.pdf

Size

942,866 bytes

SHA-256

a403ac0007e3db715042b0b2b348c227773bc84b1599037a2551582329122229

Our copy

gov.uscourts.flsd.590042.585.12.pdf

Original

archive.org