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108 Fooled by Compounding Winter 2012
Fooled by CompoundingR. DaviD McLean
R. DaviD McL ean
is an associate professor of
finance at the Universityof Alberta in Edmonton,
AB, Canada, and a vis-
iting assistant professor
of finance at MIT in
Cambridge, MA.
Compounding can make things
appear to be larger than they really
are. This effect can arise when
returns resulting from an event
are compounded over a long holding period. In
this setting it is not uncommon for authors to
describe returns resulting from compounding
as a return caused by the individual event.
This mistake results in exaggerating the sig-
nif icance of the event. In this article, I review
several examples of this common mistake,
which are found in a popular book on rare
events, newspaper articles, investment advi-
sors’ research reports, and finance journalarticles. I also describe alternative methods of
return measurement that are not affected by
compounding and show that these methods
can lead to different inferences than measures
that include compounding.
For an understanding of how com-
pounding can distort things, consider the fol-
lowing example. A portfolio normally yields
a return of 1% per month. In one month the
portfolio has an abnormal event, which yields
a monthly return that is greater than 1%. A
benchmark portfolio has a return of 1% inevery month. An analyst wants to commu-
nicate the significance of the event. To do so
she calculates the buy-and-hold return of the
portfolio and compares it to that of the bench-
mark over holding periods of 1, 5, 10, and 50
years. The event occurs in month t , and the
returns are measured beginning in month t
through the subsequent holding periods.
Over the 1-year holding period, the port-
folio has a buy-and-hold return of 13.80%, while
the benchmark has a buy-and hold return of
12.68%. The analyst reports an abnormal buy-
and-hold return of 13.80% − 12.68% = 1.12%.
The analyst repeats this exercise and computes
abnormal buy-and hold-returns of 1.80%,
3.27%, and 387.71% over the 5-, 10-, and 50-
year holding periods, respectively. Hence, for
the same event, we have four different abnormal
returns, which range from 1.80% to 387.71%.
Which abnormal return accurately describes themagnitude of the return in the event month?
The point that I strive to make in this
article is that none of the previous returns
accurately describes the abnormal return in
the event month. In the preceding example,
the return in the event month is 2%, so the
abnormal return in the event month is simply
2% − 1% = 1%. If the analyst wants to com-
municate the size of the event, she can simply
state that the return in the event month is 1%
larger than the returns in the other months.
The problem with the buy-and-hold returns isthat they ref lect both the event and the com-
pounding, and therefore do not reveal how
important the event was.
Compounding in this setting multiplies
the abnormal return in the event month by
the compound factor from the other months.
To see this, let E equal the return in the event
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110 Fooled by Compounding Winter 2012
By removing the ten biggest one-day moves from
the U.S. stock market over the past f ifty years we
see a huge difference in returns—and yet conven-
tional finance sees these one day jumps as mere
anomalies.
The issue here is to attr ibute the total dif ference in
buy-and-hold returns to “one-day jumps” that occurred
on the 10 largest days. When the 10 largest days are
removed from the total buy-and-hold return of the S&P
500 over a 50-year period, not only are the returns of
the 10 days removed, but so is the compounding of those
days’ returns over the 12,829 other days in the holding
period, and this is what really matters. If half of thereturns from the S&P 500 over the last 50 years were
due to jumps, then these 10 jumps should be visible in
e x h i b i t 1
ivst th S&p 500 vr th lst 50 yrs
This exhibit plots the value of $1 that is invested in the S&P 500 during the period 1955 to 2005. In Panel A, the value is also computed
excluding the 10 days with the highest returns. In Panel B, the value is also computed excluding both the 10 days with the highest returns
and the 10 days with the lowest returns.
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the Journal oF portFoliomanagement 111Winter 2012
Exhibit 1, which plots this investment. Do we observe
10 jumps in the line that plots the return of the S&P
500 in Exhibit 1?
In March 2006, the Financial Times ran a series of
articles called “Mastering Uncertainty.” Benoit Man-
delbrot and Nassim Taleb wrote an article for this seriestitled “A Focus on the Exceptions That Prove the Rule.”1
As in The Black Swan, the point of the article is largely to
show that just a few outliers account for the bulk of many
things (e.g., book sales, internet searches, and wealth).
Mandelbrot and Taleb [2006] applied this framework to
the stock market, and refer to a graph similar to that in
Exhibit 1, stating the following:
Taken together, these facts should be enough to
demonstrate that it is the so-called outlier and not
the regular that we need to model. For instance, a
very small number of days account for the bulk of
the stock market changes: just 10 trading days repre-
sent 63 per cent of the returns of the past 50 years.
It is easy to show that virtually all of what Taleb
[2007] and Mandelbrot and Taleb [2006] were refer-
ring to is not returns on the ten largest days, but rather the
compounding of those returns over thousands of other days.
Panel A of Exhibit 1 shows the total returns of an investor
who invested $1 in the S&P 500 in 1955 and held that
investment through 2005 (the data are from CRSP),
as reported by Taleb [2007] and Mandelbrot and Taleb[2006]. The investment is shown both with and without
the 10 largest-return days for the S&P 500. When the
10 largest days are included, the investor has terminal
wealth of $191. When the 10 largest days are excluded,
the investor’s terminal wealth is $112. Hence, excluding
the 10 largest days costs the investor $78.92, or 41% of
her total return over the 50-year holding period.
Panel A of Exhibit 2 displays the returns of the 10
highest-return days, which range from 4.77% to 8.81%.
If the investor were to invest only $1 on each of these 10
days, in isolation, the investor would earn $0.55 in dollar
returns. Of the $78.92 reduction in terminal value thatresults from excluding the 10 largest days, only $0.55, or
0.7%, comes from the returns that could be independently
achieved on those 10 days. The other $78.37, or 99.93%,
results from compounding, as shown in Exhibit 3.
To fur ther understand the problems of attr ibuting
half of the S&P 500’s returns to just 10 days, consider
Panel B of Exhibit 1. Like Panel A, Panel B of Exhibit 1
shows the total return of an investor who invested $1 in
the S&P 500 in 1955 and held it through 2005. Panel B
shows this investment with all days and compares it to an
investment that removes both the 10 largest-return and
10 worst-return days. The investment that excludes both
the 10 best and 10 worst days has a terminal value of $253,which is 32% greater than the investment that includes
all of the days ($191). Clearly, the 10 highest-return days
do not represent half of the market’s returns, because
we can make an investment that excludes them and get
a larger return. Why does excluding the 10 worst days
make such a big difference? Panel B in Exhibit 2 shows
that the 10 worst days have larger returns (in absolute
value) than the 10 best, so compounding these returns
over thousands of other days has a greater effect than
does compounding the 10 largest returns.
e x h i b i t 2
Th 10 Hhst- 10 lwst-Rtr dsfr th S&p 500, 1955–2005
This exhibit reports the 10 highest-return days (Panel A) and the
10 lowest-return days (Panel B) for the S&P 500 during the period
1955–2005. The far right column reports the simple interest that
would be earned from investing $1 on each of these days.
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112 Fooled by Compounding Winter 2012
For those who want to make statements regarding
the magnitude of high-return days, they can simply mea-
sure the returns on these days and compare them to the daily
mean return of the sample. With respect to the S&P 500,
its mean daily return over the period 1955–2005 is 0.045%,
and its standard deviation is 0.908%. Each of the returns on
the 10 largest days that are described in Exhibit 2 is more
than five standard deviations greater than the sample’s
mean. The returns on these days are certainly outliers, but
in isolation they do not account for nearly half of the total
returns that could be achieved by investing in the S&P
500 over this 50-year period.
ert th irtcf Hh-Rtr ds: mr es
Journalists also exaggerate the effects of high-returndays with compounding. As an example, an article that
appeared in The New York Times on October 8, 2008,
attributed the effects of compounding over a 40-year
period to returns that occurred over just 90 days:
From 1963 to 2004, the index of American stocks
he tested gained 10.84 percent annually in a
geometric average, which avoided overstating the
true performance. For people who missed the 90
biggest-gaining days in that period, however, the
annual return fell to just 3.2 percent. Less than
1 percent of the trading days accounted for 96
percent of the market gains.
Even professional investment advisors make this
mistake. The statement from The New York Times article
referred to two studies conducted by Towneley Capital
Management. Following is an excerpt from the intro-
duction of a study by Towneley [2005]2:
What surprised us, however, was the conclusion
that practically all of the market’s gains or losses over
several decades occurred during only a handful of
days or months. For example, in the original study,95% of market gains between 1963 and 1993 were
generated during a mere 1.2% of the trading days.
Many people have contacted us over the last
10 years asking for copies of the study. Recently,
however, we began receiving requests for an
updated version. Since, we also were curious to see
how the last decade might have changed result s,
e x h i b i t 3
Wh ec th 10 lrst ds mttrs
This exhibit breaks out the total difference between investing $1 in the S&P 500 during the period 1955–2005 and investing $1 in the
S&P 500 over the same period, but excluding the 10 largest-return days. The total difference is divided into simple interest and compounding
effects. Simple interest from the 10 largest days is the sum of the simple interest that could be earned by investing $1 on each of these days.
The difference due to compounding is the portion of the total difference that is not the result of the simple interest that could be earned byinvesting on these 10 days.
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the Journal oF portFoliomanagement 113Winter 2012
we asked Dr. Seyhun to revise the study, incor-
porating data through the year 2004. The results
were virtually unchanged: 96% of market gains
between 1963 and 2004 occurred during only
0.9% of the trading days.
It is not surprising that Towneley found similar
results in both samples, as compounding worked the same
from 1963 to 1993 as it did from 1993 to 2004. Towneley
is not the only major investment advisory firm to mistake
the effects of compounding for returns created over just a
few days. The following statement was obtained from the
website of John Hancock Investment Advisors [2008]:
Market upswings are as unpredictable as declines,
and history shows that a significant amount of
the long-term return available from investing in
stocks comes from gains made in a relatively small
number of trading days.
Finance professors and f inance journal editors also
can be confused by compounding. The following state-
ment was made by Estrada [2008]:
As these figures show, in all cases a very small
number of days accounts for the bulk of returns
delivered by emerging equity markets. Investors
in these markets do not obtain their long-term
returns smoothly and steadily over time but largelyas a result of booms and busts. A neglig ible pro-
portion of days determine a massive creation or
destruction of wealth.
In fact, the opposite is true. Investors do earn their
returns smoothly over time, as Exhibit 1 shows, and not
in a few boom and busts, as Estrada claimed. Estrada
[2009] made a similar misstatement in another article:
As these figures show, in all cases a very small
number of days account for the bulk of returns
delivered by equity markets. Investors do not obtain
their long-term returns smoothly and steadily over
time but largely as a result of booms and busts.
These examples show that the effects of compounding
exaggerate the size of high-return days. In each of these
examples, both the actual returns on high-return days and
the compounding of these returns over thousands of other
days are referred to as returns that occurred on just the
high-return days.
buy-and-Hold abnoRmalReTuRnS (bHaRs)
Compounding can distort inference in event studies
and mutual fund performance measurement. This section
describes several examples of these effects. I describe two
return-measurement methodologies that are not distorted
by compounding: cumulative abnormal returns (CARs)
and average abnormal returns (AARs).
evt Sts
As discussed earlier, compounding can also create
confusion in abnormal return measurement. This problem
can arise when buy-and-hold abnormal returns (BHARs)
are used to test whether a portfolio’s returns exceeds those
of its benchmark over long holding periods, as in the
example given at the beginning of this article. BHAR
is a comparison between the total buy-and-hold return
of a portfolio and that of its benchmark. BHAR, there-
fore, is affected by compounding. BHAR is computed
as follows:
Mitchell and Staf ford [2000] and Fama [1998] also
described problems that can ar ise with BHAR because
of compounding.3 Fama cited the results of Desai and
Jain [1997] and Ikenberry, Rankine, and Stice [1996]
as examples of how BHARs can distort inference;
both studies used BHARs to analyze abnormal returns
following stock splits. Ikenberry, Rankine, and Stice
reported that stock splits generate a one-year BHAR of
7.93%, while the BHARs in the second and third years
are zero. The BHAR over the entire period (three-year
BHAR) is 12.15%. There were no differences between
the portfolios in years two and three, yet the BHAR stil lgrows during these years because of compounding.
To see how compounding can distort inference
with BHAR in a generic setting, consider an example
similar to the one at the beginning of this article.
A firm has a return of 2% in the month following an
event, while the benchmark has a return of 1% in the
same month. Both the firm and its benchmark have
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114 Fooled by Compounding Winter 2012
returns of 1% in all other months. If we measure the
BHAR in just the month following the event, it is 1%
(1 × 1.02 − 1 × 1.01). If we measure the returns over one
year, the buy-and-hold return for the firm is 13.80% and
for the benchmark it is 12.68%; the BHAR is therefore
1.12%. When the measurement horizon is extended to5 years, the BHAR is 1.80%, and if extended to 10 years,
the BHAR is 3.27%. Hence, as the horizon grows, the
BHAR also grows, even though the returns are identical
after the first month. For this reason the 3-year BHAR
in Ikenberry, Rankine, and Stice [1996] exceeds the
1-year BHAR, even though the second- and third-years
BHARs are zero.
Why do people use BHARs? Barber and Lyon [1997]
pointed out that BHARs accurately measure the investor’s
experience. Indeed, it is true that if an investor were to
invest in the stock-split portfolio described by Ikenberry,
Rankine, and Stice [1996], then his terminal wealth would
be 12.15% higher three years later compared to holding the
benchmark during the same three-year period. It is also
true, however, that the investor could have switched to
the benchmark portfolio after the end of the first year and
achieved the same return at the end of the third year.
When choosing a return-measurement method-
ology, it is important to consider the questions you are
trying to answer. In the preceding example, the questions
are, do stock splits create abnormal returns, and if so, how
big are they? BHAR does not give an accurate answer to
either question because it reports not only the abnormalreturns, but also the compounding of those returns over
the holding period.
mt Fs
The same type of inference problems with event
studies can also happen when analyzing mutual fund
returns. As an example, inference problems can occur
when investors compare charts that plot the dollar value
of an investment in a portfolio versus that in a bench-
mark portfolio. The findings in Evans [2010] highlight
the potential for such a problem.
Evans documented an incubation process in which
mutual fund companies privately start new funds and
then bring some of them public after an incubation
period. Only the funds with strong performance are
brought public, while those with weak performance
are terminated. The funds that are brought public have
abnormal returns of 3.5% per year during the incubation
period (Note that if a company starts 10 new funds, then
by luck we would expect 5 of the funds to beat the index
in the first year.). After the incubation period, however,
the average abnormal return for these funds is zero. Plot-
ting the growth of a dollar invested, or using BHAR tomeasure performance over the entire life of an incubated
fund, could make it appear as if the fund continued to
have abnormal performance after the incubation period,
when in fact there was none.
As an example, assume that a fund was incubated
for a year and had a return of 13.5% during that year.
Assume the fund’s benchmark had an average return
of 10% per year, and that after the fund was brought
public, it tied its benchmark over the next five years,
as is typical for incubated funds. $100 invested in the
benchmark would yield $110 after the first year and
$161.05 at the end of the fifth year. $100 invested in the
fund would yield $113.50 after the first year and $166.18
at the end of the fifth year. Hence, if the mutual fund
company plotted the growth of $100 over the entire
five-year period, it would appear that the fund created
$166.18 − $161.05 = $5.13 in value. But all that is hap-
pening is just the compounding of the first year’s abnormal
performance, which was $113.50 − $110.00 = $3.50, and
was never available to the investor:
Mutual Fund’s Five-Year
Terminal Value: $113.50×
1.104
=
$166.18Benchmark’s Five-Year
Terminal Value: $110.00 × 1.104 = $161.05
An investor in this case probably should not expect
to do any better with the fund than with the benchmark,
but compounding may fool her into believing otherwise.
The longer the holding period, the better the fund looks,
even though the fund is not creating any value after the
first year.
More generally, even if a manager does produce
abnormal performance, comparing the terminal value of
an investment in a fund to the terminal value of an iden-tical investment in the fund’s benchmark, as is commonly
done in the mutual fund industry, is an imprecise way
to measure a manager’s performance. This is because the
difference in terminal values contains both the abnormal
performance and the compounding of the abnormal per-
formance over the measurement period.
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the Journal oF portFoliomanagement 115Winter 2012
atrtv msrs f Rtrs
Two return methodologies that are not inf luenced by
compounding are cumulative abnormal returns (CARs)
and average abnormal returns (AARs), which are also
known as “calendar time” abnormal returns. Both of these methods, and especially AAR, are advocated
over BHAR by Fama [1998] and Mitchell and Stafford
[2000]. To compute CAR, we subtract the monthly
return of the benchmark from that of the portfolio, and
sum up the differences over the sample period,
To compute AAR, we subtract the monthly return
of the benchmark from that of the portfolio, and take
an average of the difference over the sample period,4
To see how these methods can yield a different
inference from BHAR, consider a mutual fund that has
a return of 2% in the first month and 1% thereafter, and
a benchmark portfolio that has a return of 1% per month
throughout the entire sample period. The appendix
reports the abnormal returns of the two funds with each of
these different measures over the various holding periods.
The AAR after one month is 1%. After one year, theAAR is 0.08% (0.01/12), and after five years, it is 0.01%
(0.01/60). Hence, as we increase the holding period, the
AAR in this example goes to zero, as it should, because
the manager only created value in the first month. In this
example, the outcome with AAR is the exact opposite of
the outcome with BHAR. The one-month BHAR is 1%,
the one-year BHAR is 1.12%, and the f ive-year BHAR
is 3.27%. The BHAR in this example will head toward
infinity as the holding period gets larger. The CAR in
this example is 1%, regardless of the holding period.
Which is the best method to use? In the fund man-
ager example, AAR may provide the clearest picture if the question we want to answer is, should we expect the
manager to create value in the future? Over the entire
five-year period of this example, the manager only beat
the index in 1 of 60 months, and that is likely due to
luck. The AAR suggests that the fund’s returns are not
significantly different from the benchmark’s returns,
which appears to be correct. Note that the AAR of the
incubated fund described in the previous section would
also go to zero over a suff iciently long holding period.
ConCluSion
Accurate abnormal return measurement is crucialfor understanding the significance of high- and low-
return days, corporate events such as stock splits, and
mutual fund performance. In this article, I show that
compounding can distort inference in each of these
instances. This is true because total holding-period
returns contain not only the return from the event itself,
but also the return compounded over days that did not
contain the event.
I show that there is a tendency to confuse returns
from compounding as the return from an event. The
distorting effect of compounding is more pronounced
when long holding periods are used, because the effectsof compounding increase with time. An alternative
methodology, known as average abnormal returns
(AARs), or the calendar-time portfolio approach, is
described. This method of return measurement is not
affected by compounding, and therefore may provide
more honest appraisals of event significance and fund
manager abil ity.
Compounding does have a place in return mea-
surement, however: whether to compound depends on
the question that we are trying to answer. If an investor
wants to know what her wealth will be at retirement,then we need to compound the returns of her investment
over the savings period. If we are try ing to measure how
big the abnormal return from a particular event is, then
we should not be compounding the event’s return over
long holding periods.
a p p e n D i x
CompaRiSon oF abnoRmal ReTuRnmeaSuRemenT meTHodologieS
This exhibit displays the abnormal returns from a port-folio that had a return of 2% in an event month and 1% in all
other months. The benchmark portfolio had a return of 1%
in all months. Buy-and-hold abnormal return (BHAR) is the
difference between the buy-and-hold returns of the portfolio
and the benchmark,
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116 Fooled by Compounding Winter 2012
Cumulative abnormal return (CAR) is the sum of the
differences in monthly returns between the portfolio and
the benchmark,
Average abnormal return (AAR) is the average of the
difference in monthly returns between the portfolio and the
benchmark,
endnoTeS
The author is grateful to Claire Lang, Min Maung, Jay
Ritter, and Mengxin Zhao for helpful comments.1This art icle was reprinted in 2009.2The Towneley study is available at http://www.
towneley.com/pdf/MT%20Study%2004.pdf.3These articles also describe a number of statistical issues
that arise with BHAR.4Mutual fund alpha, a concept introduced by Jensen
[1968], is computed via an AAR methodology.
ReFeRenCeS
Barber, B., and J. Lyon. “Detecting Long-Horizon Abnormal
Stock Returns: The Empirical Power and Specification of
Test Statist ics.” Journal of Financ ial Economic s, 43 (1997),pp. 341-372.
Desai, H., and P. Jain. “Long-Run Common Stock Returns
Following Splits and Reverse Splits.” Journal of Business, 70
(1997), pp. 409-433.
Estrada, J. “Black Swans and Market Timing: How Not
to Generate Alpha.” The Journal of Investing , Vol. 17, No. 3
(2008), pp. 14-21.
——. “Invest ing in Emerging Markets: A Black Swan Per-
spective.” Corporate Finance Review , January/February 2009,
pp. 14-21.
Evans, R. “Mutual Fund Incubation.” Journal of Finance , 65
(2010), pp. 1581-1611.
Fama, E. “Market Efficiency, Long-Term Returns, and
Behavioral Finance.” Journal of Financial Economics, 49 (1998),
pp. 283-306.
Ikenberry, D., G. Rankine, and E. Stice. “What Do Stock
Splits Really Signal?” Journal of Financial and Quantitative Anal-
ysis, 31 (1996), pp. 357-377.
Jensen, M. “The Performance of Mutual Funds in the Period
1945–1964.” Journal of Finance , 23 (1968), pp. 389-416.
John Hancock Investment Advisors. “Saving for College in
a Volatile Market.” John Hancock Freedom 529 Market Volatility
Message , November 18, 2008. Available at www.johnhan-
cockfreedom529.com/public/site/page/0,,Market_Volatilty_
Message,00.shtm.
Mandelbrot, B., and N. Taleb. “A Focus on the Exceptions
That Prove the Rule.” Financial Times, March 23, 2006.
Mitchell, M., and E. Stafford. “Managerial Decisions and
Long-Term Stock-Price Performance.” Journal of Business, 73
(2000), pp. 287-329.
Taleb, N. The Black Swan: The Impact of the Highly Improbable .
New York, NY: Random House, 2007.
To order reprints of this article, please contact Dewey Palmieri at [email protected] or 212-224-3675.