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Faculty Working Papers
College of Commerce and Business Administration
University of Illinois at Urbana-Champalgn
FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
August 1, 1980
AN INVESTOR LOSS FUNCTION FOR EARNINGS FORECASTS
WITH AN EMPIRICAL APPLICATION
James C. McKeown, Professor, Department of
Accountancy
William S. Hopwood, Assistant Professor,
Department of Accountancy
;;694
Summary
This paper deals with an investor loss function for earnings
forecasts. Specifically we. develop a theoretical framework that
measures loss in the context of the problem of resource allocation
under uncertainty. The framework maps a set of forecasts into an
expected return. This expected return is compared to that return
which could be expected given perfect knowledge of future earnings
and the resulting difference measures the investor's loss. Some
empirical results are given.
This paper deals with an investor loss function for earnings fore-
casts. Specifically we develop a th oretical frar.ework that measures loss
in the context of the problem of resource allocation under uncertainty.
The framework naps a set of forecasts into an expected return. This ex-
pected return is compared to that return which could be expected given
perfect knowledge of future earnings and the resulting difference mea-
sures the investor's loss.
The need for a loss function arises from the need of investors to
assess the value of alternative forecasts. In addition empirical re-
search studies have often compared various forecast models. The typical
approach has been to compare forecast accuracy or dispersion. This
approach, however, is limited. For example Foster [1977, p. 10] stated
"it is important to recognize that our measures of dispersion are essen-
tially surrogate criteria for evaluating alternative forecasting models.
A more complete analysis would specify the loss function in a specific
decision context." In addition Gonedes et al [1976, p. 94] vrote, "There
is a more fundamental deficiency in these prediction performance studies...
Specifically they are not based upon any explicit theoretical structure
that connects their frameworks to resource allocation under uncertainty."
The purpose of the present study is to develop a theoretical frame-
work to overcome these limitations. This framework takes the form of an
investor loss function (henceforth ILF) . A secondary purpose is to em-
pirically apply the ELF for purposes of comparing forecasts generated
from several models that have recently been employed in the literature.
The paper is in four parts. Part one discusses some of the issues
associated with an ILF, part two presents an operational form of the
-2-
ILF, part three is an empirical application and part four contains sum-
mary and conclusions.
BACKGROUND ISSUES
An important result of research on the information content of ac-
counting earnings is that ex ante knowledge of annual earnings can provide
an opportunity for an investor to earn an abnormal return. For example,
Ball and Brown [1968] reported that if one were to know the sign of the
unexpected annual earnings change 12 months in advance it would be possible
to earn a 7% abnormal return via a simple trading rule. These findings
pose some interesting questions with respect to earnings forecasts.
1) If the market is efficient, then we wouldn't expect a forecast
model to enable an individual to earn an abnormal return via
utilizing only publicly available information.
2) If the expectation in (1) is correct, then from the individual
investor's standpoint the problem of comparing various sets of
forecasts might become irrelevant. One could expect to earn
no better than the risk conditioned rate of return since the
risk-return relationship should not depend on the portfolio
selection process.
3) Looking at (2) from another side, we would not expect a return
less than the risk conditioned return via the same reasoning.
Given the above reasoning, it might seem that investors would only
be concerned with constructing minimum variance portfolios and would not
be concerned with using earnings forecasts in their decision models. Re-
search, however, points to the exact opposite. For example, Nordby [1973]
found that 99% of responding analysts claimed that ':hey use earnings
forecasts in their decision making process.
What then accounts for the extensive use of forecasts in practice?
The answer seemingly must be one of two things:
-3-
(1) Some individuals do earn an abnormal return through use of
forecasts and the market is not efficient.
(2) Some individuals think that they can earn an abnormal return,
but the market is efficient and they are possibly irrationally
allocating resources for purposes of obtaining forecasts.
Research and reasoning can be used to support both (1) and (2). A
large amount of research has been conducted which favors efficiency. On
the other hand there is evidence which is not consistent with efficiency.
A good example of this is the Value Line Investment survey. Black [1973]
found reasonably strong evidence that the survey is able to predict returns
of securities in a way that cannot be accounted for by differences in risk.
(Note that the survey relies heavily on a determination of "earnings
momentum".) In addition, Joy et al [1977, p. 207] presented evidence that,
"...the information contained in quarterly earnings was not fully impounded
into stock prices at the time of announcement."
It is not the purpose of the present paper to take a position on effi-
ciency or lack of efficiency in the market. However it must be noted
that the theory of market efficiency does not explain the empirically
observed behavior of users of earnings forecasts. An alternative frsme-
2
work which might explain such behavior is that of Grossman and Stiglitz
[1976] who pointed out that costless information is not only sufficient
for market efficiency but necessary as well. Their alternative is sum-
marized (p. 218):
In the structure we have developed, the market never
fully adjusts. Prices never fully reflect all the
information possessed by the informed individuals.
Capital markets are not efficient, but the difference
is just enough to provide the revenue required to com-
pensate the informed for purchasing the information.
Note that this framework allows for the possibility that there is a value
of gathering information and therefore a corresponding loss for gathering
-4-
information which is less than optimal. (It is this loss that is the
focus of the present study.) We define optimal information to be that
information which produces the maximum possible net revenue associated
with its use, where net revenue is computed as the difference between
the revenue associated with the use of the information and the acquisi-
tion cost of that information.
If the market is assumed to be efficient and the market's earnings
expectation model is assumed to be optimal, then one might gauge the use-
fulness of a given forecast method by measuring the abnormal returns
associated with an investment strategy based on an ex ante knowledge of
the forecast error (i.e., unexpected earnings) of the given model. This
statement is elaborated on by Brown and Kennelly [1972, p. 404]:
This experimental design permits a direct comparison
between alternative forecasting rules. .. .The. . .conten-
tion is based on the hypothesis (and evidence) that the
stock market is "both efficient and unbiased in that,
if information is useful in forming capital asset
prices, then the market will adjust asset prices to
that information quickly and without leaving any
opportunity for further abnormal gain" (Ball and Brown,
1968). There is, then a presumption that the con-
sensus of the market reflects, at any point, an esti-
mate of future EPS which is the best possible from
generally available data. Since the abnormal rate
of return measures the extent to which the market has
reacted to errors in its previous expectations, the
abnormal rate of return can be used to assess the
predictive accuracy of any device which attempts to
forecast a number that is relevant to investors. To
our knowledge, Ball and Brown (1968) were the first
to make use of this fact.
This basic type of reasoning can be used to derive several types
of empirical tests all based on different sets of assumptions as listed
below.
-5-
Category Assumptions Test
1 A) Efficient rcarket Compare prediction models
B) Earnings has infor- on ability to approximate
nation content market prediction model
(assumed to be optimal as
defined above)
2 A) Efficient market
B) Given prediction Information content
model approximates
market model
3 A) Efficient market Joint test of
information content
and prediction model
Most of the accounting research has fallen in one of the above
three categories. For example Fester's [1977] study on quarterly
accounting data falls into category 1 while Ball and Brown [1968] and
Brown and Kennelly [1972] among many others fall into category 2. It
can be argued also that categories 1 and 2 are subsumed under category 3
which reduces to the first two cases if the assumptions are correct.
In the present study we develop a methodology (ILF) for comparing pre-
diction methods which dees not rely on any of the Table 1 assumptions.
Instead we assume that there exists some market expectation (prediction)
for earnings (not necessarily optimal as defined above) which has been
impounded in the market equilibrium (price). The error in this market
earnings expectation has not been impounded by the market. An investor
who had knowledge of this (ncn- impounded) information would be able to
use it to formulate an investment strategy which would produce an abnor-
mal return (as measured by the market model). Therefore we define the
investor's loss (due to inaccurate earnings predictions) to be the
difference between the abnormal return he/she could earn given a
-6-
strategy based on correct forecasts of earnings Cand thereby of the
non- impounded information) and the return he/she could earn given a
strategy based on less than perfect forecasts. This definition is
operationalized below and discussed in the context of decision theory.
It is also applied to a comparison of several forecasts models found
in the literature.
It should be emphasized that in developing the loss function we are
not concerned with market efficiency per se but rather an individual in-
vestor's perceptions with respect to market efficiency. In particular we
assume that the investor believes that there is a possibility of earning
a return higher than predicted by the Sharp-Lintner [1964, 1965] capital
asset pricing model. As pointed out previously, unless this assumption
holds, there is no private value of earnings forecasts bcsed on publicly
available information. If there is no private value, then from the in-
dividual's standpoint the process of comparing forecasts is dubious.
It is also pointed out that the loss function derived in this paper does
not depend on the need to compare forecast methods but simply specifies
the loss associated with different forecast sources. If all forecast
methods have equal loss, then there is no need to compare forecast
methods. This is an empirical question.
CFERATIONALIZATION OF THE ILF
In order to use the ILF, it is first necessary to operationally define
a market earnings expectation model. In this study we use the cross-sectional
model employed by Ball and Brown [1968] which regresses individual firms'
earnings changes on market earnings changes. I'e use this model since Ball
and Brown observed that ex ante knowledge of its residuals made it possible
for an investor to earn a 7% abnormal return.
-7-
To facilitate operationalization of the loss function we make the
following definitions:
(1) E(F |r )
0
(2) a - ;± + ;± z a +•
j=i
N
(3) E(F |* ) = a + b ( E E(F |r ) * = {A }, j#L
xc0 x j-1 J 0 J 0
(4) E[ln(l + R - R )] = B E(ln(l + R - R ))
it it l mt rt
Where:
a) (1) represents the investor's expectation of earnings change
for firm i and period t~. This expectation could be the re-
sult of intuition, statistical modeling or judgmental opinion
and is conditioned upon r, the set of prior earnings for firm i.
b) (2) is an empirical description of the relationship between
market earnings changes and firm i earnings changes, A. . The
coefficients a. and b are assumed to be estimated by the investor.
Note that the investor is assumed to use the same coefficients
in (3) . Also the subscript t denotes time previous to t
:■'
(In the empirical application, a. and b are estimated by
regression on previous years' data.)
c) (3) represents the investor's expectation of earnings changes
(A ) for firm i in period t. conditioned upon his expectation
itQ U
of the market earnings ($ ) in period tn. Since this depends
on E(F. |r. ) (which is ex ante), it is ex ante. The coeffi-
Ja0 1 a
cients a. and b. are from (2) and are taken as known in the
equation.
d) (4) is the log form Sharp-Lintner [1964, 1965] capital asset
pricing nodel where R. represents the return on asset i in
period t, R represents the market return in period t and Rc
mt t
is the risk free rate of return.
Given the above definitions we can define the investor's anticipation
of the individual components of change in earnings. It is this individual
-8-
component which Ball and Brown [1968] found to enable one to earn an
abnormal return when known in advance. We proceed to define the in-
vestor's anticipation of the individual component of changes in earnings
subtracting (3) from (1):
(5) ait ' E(Fit 'V - E(Fit 'V
0 0 0
When a is greater than zero, then the investor expects a positive
0
individual component of change in earnings. When a. is less than zero,
ato
a negative individual component of change is anticipated. This is con-
sistent with the Ball and Erown market conditioned definition of indi-
vidual component of change in earnings e>xept it is based en predicted
earnings as opposed to actual earnings.
Given that a. is positive (negative), the investor would be expected
to buy long (sell short) in asset i. Also to the degree that his expectations
are correct, he will earn an abnormal return, AEC,. Similarly let AR^ represent
the abnormal return assuming that the investor has perfect forecasts of
the future earnings (i.e., his predictions of all firms are perfectly
accurate). Then define the investors loss function (ILF) as
(6) ILF = ARpp - ARp
where ILF has the interpretation as being the loss incurred from not
having perfect forecasts. The minimum expected value is expected to be
0 in the case of having perfect forecasts, and equal to the abnormal
return of having perfect forecasts in the case of having useless fore-
casts.
-9-
A Further Interpretation
In terms of standard decision theory the ILF measures the cost, in
terms of return, of not having perfect forecasts. A larger abnormal
return earned means a smaller loss. This relationship is depicted in
Figure 1. In point A the loss is at a maximum and the abnormal return
[Figure 1 about here]
is at a minimum. Point B depicts the case of perfect forecasts.
Note that the loss depends strictly on the abnormal return. This
is important because abnormal returns are risk adjusted or independent
of the market. This means that forecasting methods can be preference
ranked based on the ILF without a separate consideration of risk.
EMPIRICAL APPLICATION
Forecast >!odels
The empirical results of this study focus on the ability of several
statistical models to predict annual EPS from quarterly EPS. This purpose
has been suggested by the Financial Accounting Standards Board in the
discussion memorandum, Interim Financial Accounting and Reporting (.FASB,
1978). In addition there has been a considerable amount of research
done on the predictive ability of models using quarterly EPS (e.g.,
Lcrek, 1979; Foster, 1977; Brown and Rozeff, 1978).
We focus on several models that have been given considerable atten-
tion with respect to their ability to represent the time series of
quarterly EPS. These are
1) a seasonally and consecutively differenced first order and
seasonal moving average model [Griffin, 1977; Watts, 1975]
-10-
Figure 1
The Relationship Between Loss and Abnormal Return
Loss
Abnormal Return
-11-
2) a seasonally differenced first order autoregression model
3
with a constant drift term [Fester, 1977]
3) a seasonally differenced first order autoregressive and
seasonal coving average model [Brown and Rozeff, 1978]
4) individually identified and estimated Box-Jenkins models
for each firm.
la the Bos-Jenkins notation the first three are referred to as
(0,1,1) x (0,1,1), (1,0,0) x (0,1,0) and (1,0,0) x (0,1,1), respectively.
For the remainder of the study these are referred to as the GW, F, and
BR model's.
Population Studied
Data pertaining to 267 firms was obtained from the Ccmpustat quar-
terly and CRSP monthly tapes. For a firm to be included in the group,
it was required to have no missing EPS or returns data for the 64 con-
secutive quarters beginning with the first quarter of 1962. This pro-
vided a sample period frcm 1962 through 1977. The EPS number used was
primary earnings per share excluding extraordinary items and discontin-
ued operations, adjusted for capital changes. The return figure
selected from CRSP included both a dividend and price component.
Note that, unlike previous research, all firms which met the sur-
vivorship test were retained for analysis. We define this group to be
the population of interest and make no attempt to generalize to a larger
number of years or group of firms. To use statistical testing to make
inferences about a larger group of firms would be unwarranted because
there is no reason to believe that firms which fail to meet the survivor-
ship test are the same as those that do. In fact, a priori reasoning
-12-
indicates that firns meeting the test are very likely to be larger and
less risky than on the average. Also attempting to generalize across
all years would be unwarranted because structural changes in the economy
might produce a shifting in the relative performance of different fore-
cast methods. Even if this was not a problem, in order to generalize
to all years, it would be necessary to obtain a reasonably large random
sample of years. This is not possible because of limited data avail-
ability.
Since statistical testing is used for making inferences about a
larger population and under the circumstances we felt that such infer-
ences would be unwarranted, no statistical tests are presented in this
paper. Instead our goal is to present results for an entire population
therefore avoiding the need to make statistical generalizations. We feel
that this approach is useful because it presents results for a large
population which is of interest in its own right.
AcDlicaticn of the Forecasting Models
For purposes of assessing the ILF's of the 4 forecast methods, the
years of 1974 through 1977 wera used as holdout periods. Therefore the
267 series were each modeled 16 times, once for each method using pre
1974 data (48 quarters in the base period) and again for each method
(52 quarters in the base period) using all pre 1975 data, etc. The re-
sult was that each model made predictions for the 4 quarters in each of
the four hold out years. These quarterly forecasts were aggregated
within each year to form annual forecasts. These forecasts represent
F in (1) above.
Z0
-13-
Next the coefficients a, and b. in (3) were estimated for each
i i
firm. The procedure was done for each hold out year and was based on
all data prior to the holdout year. The market index was a weighted
average of all of the sample firms' EPS except the one for which the
model was being estimated. The residuals of the models were tested for
autocorrelation and the null hypothesis of no autocorrelation was re-
jected for only 8 firms which was attributed to chance.
The a and b. coefficients were then applied to compute the in-
vestor's anticipation of the market conditioned EPS in (3) and finally
the anticipation of the unexpected earnings change in (5).
Application of the Market Model
The equilibrium market model (4) was estimated for each firm and
for each of the four years. The estimation included data in the 5 years
preceeding the holdout year. The residuals from these models when
applied to the four holdout years constitute abnormal returns. The mar-
ket index used was the value weighted market index containing the
dividend-price returns of all firms as supplied on the CRSP tape.
Empirical Results: Losses
The loss for each forecast method was calculated by computing the
annual cumulative abnormal return (CAR) associated with each forecast
method and subtracting this from the CAR associated with an investment
strategy based on ex ante knowledge of the actual EPS. The CARs were
computed by assuming a long investment given a. in (5) was positive
"o
(henceforth CAR+) and a short investment given that a was negative
1C0
(henceforth CAR-) .
-14-
Table 1 gives the cumulative abnormal residuals for the 4 forecast
methods and actual EPS, for the 12 months prior to and including the
earnings announcement date. For example, long investments for the BR
method made 11 months prior to the annual announcement month and termi-
nating at the end of the announcement month show an abnormal return of
.00391 for 1976. Quick inspection reveals that only the GW (for the
year 1977) and the actuals demonstrate a strong apparent pattern of
abnormal return.
Table 2 presents the loss (as defined in equation 6) for each of
the four methods. Note that in all cases the loss is positive with the
smallest average loss being associated with the GW method and the larg-
est average loss being associated with the BJ method. Also note that
the GW method had the smallest loss in three out of the four years with
1975 being the exception. In addition, the percentage difference be-
tween the GW and ether methods was substantial for 1976 and 1977. This
can be seen by examining the ratio of the next smaller loss to the GW
for these two years. This ratio is 1.39 (.0501 * .0360) for 1976 and
1.99 (.0730 * .0367) for 1977. Also, with the exception of 1975, the
percentage differences between the F, ER and BJ methods were small.
Finally, note that no consistent pattern of rankings exists between these
three methods.
Empirical Results: Forecast Errors
The question is now examined as to whether forecast error measures
are consistent with those of the ILF. Therefore Table 3 presents the
average absolute percentage forecast errors for the four methods over
the hold out period. The results are presented for the quarterly and
annual forecasts.
-15-
[Table 3 about here]
For both the quarterly and annual forecasts the rankings based on
the five year averages are fairly consistent. In all cases the BR method
has the smallest error and the F method has the largest error. Also the
BJ method performs better than the GW in all cases except for the two
quarter ahead forecasts. These results are consistent with those of
Collins and Kopwood [1980] who found identical rankings for the annual
forecast generated prior to the first quarter of the year.
Table 4 presents the sane error analysis as Table 3 but based on
the mean ranks. This information is presented because it has the advan-"
tage of not depending on a particular choice of an error metric.
[Table 4 about here]
Note that, while the average quarterly rankings vary depending on
the forecast horizon, the rankings based on the annual error are identical
to those based on the absolute percentage error. The fact that the rank-
ings are the same is particularly relevant to the present study since the
losses in Table 2 are based on annual forecasts.
Discussion of the Empirical Results
One thing immediately noticeable about the results is that the ILF
and error analysis produced different rankings of the methods. The ILF
ranks were: GW=1, F=4, BR =2 and BJ = 3, and the error analysis
ranks were: GW = 3, F = 4, ER = 1, BJ = 2. This discrepancy can be
accounted for by the fact that for a given investment decision the size
of the forecast error need not necessarily be related to whether or not
the best Investment decision is made. For example, two different fore-
casts can be quite different in terms of accuracy but both can induce
-16-
the same investment decision. This can be seen from the situation where
the magnitude of the unexpected earnings implied from one forecast is
much larger than another but both induce a buy decision.
The net result is that the investment performance is determined by
the percent of time that the correct decision is made where each deci-
sion outcome is weighted by its investment return for that outcome.
These percentages are presented in Table 5. Note that these ranks are
consistent with those of the ILF.
[Table 5 about here]
Another way of looking at investment performance is by considering
the percentage of time that a given method results in the same decision
which would have been made had the forecast been perfectly accurate.
This approach will be exactly the same as that presented in Table 5 if
it is true that a perfect forecast will always lead to the correct in-
vestment decision. One reason for looking at investment performance in
this manner is that it is that it eliminates as noise the cases where
utilization of a perfect forecast leads to the wrong investment decision.
This method of describing investment performance will henceforth be
referred to as "weighted agreement with the actual" (WAA) and the first
method will be referred to as "weighted agreement wTith the market" (VAK) .
Table 6 presents the WAA results. Notice that the rankings are
again the same but the means are larger. Their increased magnitude is
expected since a positive abnormal return is produced any time that the
same decision is made as would be made with a perfect forecast.
[Table 6 about here]
Since Tables 5 and 6 explain the losses in terms of the combined
percentage of investment success and individual decision outcome weighted
-17-
by returns, it is possible to further explain these results by consider-
ing how much of each percentage is due to the percentage of success and
how much is due to the weighting. To this end, Table 7 presents the
percentage of time that each forecast method led to the same decision
as would have been made had the forecast been perfect. This is identi-
cal to WAA but the weighting has been eliminated. Note that the rank-
ings are not the same as in WAA in the case of the F, BR and BJ methods.
This implies that making the correct decision more often does not neces-
sarily imply a higher average return. Keep in mind however that is for
the case where a perfect forecast is assumed to always produce the cor-
rect investment decision.
[Table 7 about here]
Table 8 removes the weighting for WAM. The rankings are fairly
consistent with the exception of a tie between F and BR. However, this
is not surprising since on the WAM they were virtually identical.
[Table 8 about here]
In summary, the conflict in rankings between the ILF and error
analysis is explained by the WAM and AM which imply that the method with
smaller error does not correspondingly make the correct investment deci-
sion a larger percent of the time. The WAA and AA imply that, given
that perfect forecasts always lead to the optimal decision, it is possi-
ble for a method to fare better in terms of the number of times that it
leads to the correct decision but fare worse in terms of its total return.
Summary and Conclusions
Previous research involving comparisons among forecast methods has
typically relied on various error metrics. In the present study an
-18-
alternative approach has been taken, namely conparing forecast methods
based on the outcomes of investment decisions which depend on earnings
forecasts. In particular an investors loss was defined as: the dif-
ference between the abnormal return he/she could earn given a strategy
based on correct forecasts of earnings (and thereby of the non-impounded
information) and the re'.urn he/she could earn given a strategy based on
less than perfect forecasts.
Several forecast models were examined based on their observed loss.
The results indicated that the models studied by Griffin and Katts
((0,1,1) x (0,1,1)) performed better than those of Brown and Rozeff
((1,0,0) x (0,1,1)), Foster ((1,0,0) x (0,1,0)) and individually identi-
fied Box- Jenkins models.
Furthermore these same models were examined based upon an outlier
adjusted mean absolute percentage error metric and a mean rank criterion.
These rankings were found to be identical to previous research on a dif-
ferent sample [Collins and Hopwood, 1980] but were different than those
produced by the loss function. This was explained by showing that a
smaller forecast error did not lead to a corresponding increase in per-
centage of times that the correct investment decision was made.
-19-
FOOTNOTES
In the present study, we define unexpected earnings as the
unexpected difference betx;een an individual firm's earnings and the
market conditioned expectation of the same number. This definition is
the same as used by Ball and Brown [1968],
2
The reader is referred to the Grossman and Stiglitz [1976]
paper for details relating to the assumptions and logic of their
analysis.
3
In the present study, we exclude the constant term based on
the evidence provided by Brown and Rozeff [1978] that this term is not
significant.
4^
For 1974, there were only 4 years of data available for re-
gression estimation.
-20-
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-21-
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(The Financial Analysts Federation, 1973).
R. Watts, "The Time Series Behavior of Quarterly Earnings," Unpublished
Paper, Department of Commerce, University of Newcastle (April 1975).
B. J. Winer, Statistical Principles in Experimental Design (McGraw-Hill,
1971).
M/B/117
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Table 2
Losses Associated With Each of the
Four Forecast Methods*
GW
F
BR
BJ
1974
.0863
.0938
.0871
.0925
1975
.0748
.0473
.0488
.0701
1976
.0360
.0597
.0589
.0501
1977
.0367
.0774
.0780
.0730
Mean
.0585
.0696
.0682
.0714
Rank
1
3
2
4
*Based on a composite investment.
Table 3
Mean Absolute Percentage Forecast Errors
for the Four Methods*
1 quarter
ahead
2 quarters
ahead
Horizon
3 quarters
ahead
4 quarters
ahead
Annual
.3168049
.3131468
.3061375
.3112299
,3415264
,3495478
.3393090
,3848678
.3707929
.3946708
.3816767
.3S65412
.5296023
.5308849
.5381678
.5247461
.3030187
.3069276
.3082423
.3225231
.5466285
.5173653
.4795169
.5150834
.5507753
.5714754
,4954936
,4876365
.5682686
.54963 54
.4833038
.4882317
.6131159
.5739701
.5172408
.5441138
.5053421
.4871677
.4242318
.4296102
.3580204
.4006113
.3420058
.3310875
.3591631
,4279471
,3365913
,3659078
.3986770
.4062940
.3671510
.3940491
.5185392
.5335978
.4dl2312
.4635779
.3398120
.3441070
.3188220
.3034558
.3239681
.3249985
.3072657
.3128735
.3863555
.3890305
.3589815
.3675686
.3166372
.3067774
.3228528
,3333958
,3920255
,4139370
,3735617
,3929520
.3378085
.3605339
.3222657
.3465253
.4188867
,4277835
.38S5993
,4038368
.4160805
.4476179
.4003121
.4405104
.5193345
.5215177
.4842380
.4932370
.2949488
.307a918
.2880671
.3082472
.3607804
.3615235
.3348408
.3409591
*Errors larger than 3 were set equal to 3,
Table 4
Mean Ranks* of Forecast Errors Associated
with the Four Forecast Methods
Horizon
1 quarter
2 quarter
3 quarters
4 quarters
ahead
ahead
ahead
Year 1
ahead
Annual
GW
2.498127
2.310861
2.239700
2.344569
2.423221
F
2.513109
2.438202
2.576779
2.543071
2.438202
BR
2.329588
2.397004
2.584270
2.535581
2.468165
BJ
2.659176
2.853933
2.599251
Year 2
2.576779
2.670412
GW
2.647940
2.524345
2.704120
2.625468
2.726592
F
2.535581
2.749064
2.576779
2.610487
2.655431
BR
2.370787
2.397004
2.367041
2.292135
2.314607
BJ
2.445693
2.329588
2.352060
Year 3
2.471910
2.314607
GW
2.400749
2.325S43
2.479401
2.438202
2.468165
F
2.831461
2.831461
2.632959
2.692884
2.711610
BR
2.411985
2.333333
2.453184
2.516854
2.505618
BJ
2.355805
2.509363
2.434457
Year 4
2.352060
2.314 607
GW
2.539326
2.494382
2.434457
2.307116
2.483146
F
2.408240
2.340824
2.475655
2.565543
2.337079
BR
2.543071
2.486891
2.554307
2.441948
2.479401
BJ
2.509363
2.677903
2.535581
Average
2.685393
2.700375
GW
2.521536
2.413858
2.464419
2.428839
2.525281
F
2.572097
2.589888
2.565543
2.602996
2.535581
BR
2.413858
2.403558
2.489700
2.446629
2.439139
BJ
2.492509
2.592697
2.480337
2.521536
2.500000
Table 5
Weighted Percentage of Times that the Decision Leading
to a Positive Return Was Made
Overall
Year 1
Year 2
Year 3
Year 4
Average
Rank
GW
.4656
.4270
.4907
.6603
.5062
1
F
.4445
.5059
.4097
.5303
.4712
3
BR
.4633
.5012
.4179
.5189
.4746
2
BJ
.4484
.4416
.4436
.5453
.4675
4
Table 6
Weighted Percentage of Tizies that the Same Decision Was Made
as Would Have Been Made Had a Perfect Forecast Been Made
Overall
Year 1
Year 2
Year 3
Year 4
Average
Rank
GW
.5380
.5867
.6781
.6429
.6080
1
F
.2233
.5944
.6266
.5188
.5652
3
BR
.5430
.6138
.6020
.5607
.5792
2
BJ
.5054
.5496
.6422
.5523
.5603
4
Table 7
Percentage of Times That the Same Decision Was Made as Would
Have Been Had the Perfect Forecast Been Made
Overall
Year 1
Year 2
Year 3
Year 4
Average
Rank
GW
.5785
.5709
.6628
.6207
.6082
1
F
.5517
.5862
.5939
.5211
.5632
4
BR
.5900
.6015
.5824
.5862
.5900
2
BJ
.5479
.5709
.6360
.5824
.5825
3
Table 8
Percentage of Times That the Decision Leading to
a Positive Return Was Made
Overall
Year 1
Year 2
Year 3
Year 4
Average
Rank
GW
.4904
.4444
.4789
.6207
.5086
1
F
.4556
.4751
.4176
.5211
.4674
3.5
BR
.4789
.4904
.3985
.5019
.4674
3.5
BJ
.4751
.4521
.4598
.5519
.4847
2
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