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Unformatted text preview: What Practitioners Need to Know. .. by Bob Hagin
LTCB—MAS Investment Management In addition to being relevant to practitioners, articles in Financial Analysts Ioumal are
required to meet high standards of academic excellence. As a byproduct, readers are
frequently faced with terms such as “standard error of the estimate” or “ttest" that—even
though possibly studied years ago—are certainly not everyday terms in an investment
practitioner’s world. We introduce here a new column designed to provide short ”English—
language” explanations of ”What Practitioners Need to Know. ” For our ﬁrst topic, we have selected ttests. . . . About tTeSts Suppose an investor has outperformed the av—
erage active manager by 2 per cent each year for
the past four years. What can we infer about this
investor’s skill? There is an old anecdote, “If you place an
infinite number of monkeys in front of an infi
nite number of typewriters, one of them will
type the full text of King Lear.” Similarly, there is
some probability that beating the average man
ager by 2 per cent per year for four years could
happen by pure chance. Thus we know enough
to be cautious about immediately inferring that this investor has skill (just as we would likely
not bestow the name ”Shakespeare" on the monkey that, by pure chance, typed King Lear). What we need in a situation such as this is a
way to estimate the likelihood that what we
observe (in this case, an investor who has out
performed the average manager by an average
of 2 per cent per year for four years) has not
occurred by chance. Intuitively, we know that
our conﬁdence in saying that someone has used
skill to outperform the other managers by 2 per
cent per year over the past four years depends
on the distribution of annual active returns. If, on one hand, the returns of active manag ers vary so widely that, on average, two~thirds
of all managers fall within a range of plus or
minus 10 per cent of the average manager, we
would not be very confident that someone who was above the mean by 2 per cent did so by skill.
On the other hand, if the returns of active managers are so tightly clustered around the
mean that, on average, twothirds of all manag
ers fall within a range of plus or minus 1 per cent of the average return, we would certainly
have more confidence that a manager who beat
the average by 2 per cent per year for four years
did so because of skill. Statistical tests, such as the ttest, allow us to quantify these intuitions. Normal Distributions
Many statistical tests assume that data are nor mally distributed. One of the most important
features of a normal distribution is that it is
completely described by its mean and standard
deviation. .
In brief, standard deviation measures varia
tion around an average. Say the average of 30
monthly rates of return is 15 per cent, and the
average of the returns’ deviations from this
average is 10 per cent. Conceptually, the returns between 5 per cent (15—10) and 25 per cent (15 +
10) fall within one standard deviation of the average return. Similarly, the returns between
—5 per cent [15 — (2 x 10)] and 35 per cent [15
+ (2 X 10)] fall within two standard deviations of
the average. In a normally distributed sample, approxi
mately 68 per cent of the values are within one
standard deviation of the mean, approximately
95 per cent of the values are within two stan
dard deviations of the mean, and more than 99
per cent of the values are within three standard deviations of the mean.
How can we use this knowledge to gauge our conﬁdence in an investor’s skill? Consider a
broad population of stocks with a normal distri
bution of returns. The return on a portfolio of FINANCIAL ANALYSIS JOURNAL I MAYJUNE 1990 El 17 stocks selected from this population is not likely
to have the same average return as the popula
tion itself. After all, the portfolio holds a smaller
number of stocks, and the weights of the stocks
in the portfolio may differ from their weights in
the overall population. We can, however, determine the range of
probable mean returns for a sample of stocks
drawn randomly from the overall population.
We noted earlier that 95 per cent of normally
distributed observations fall within a range of
plus or minus two standard deviations from the
observations’ mean. There is thus only a 5 per
cent chance that the mean of a sample portfolio
drawn randomly from a normally distributed
population will fall outside a band defined by
plus two and minus two standard deviations
from the population mean. Such bands can be used to define conﬁdence
internals—intervals within which we are likely
(with 68 per cent probability, or 95 per cent
probability, or 99 per cent probability) to ﬁnd
the means of random samples drawn from a
normally distributed population. The calcula
tion of confidence intervals allows us to quantify
our conﬁdence that what we observe are true
differences and not merely observations that are
likely to occur by mere chance. If we simplify our problem to the case of an
investor whose performance for one year was 2
per cent above the mean performance of a
population of similar managers, we merely have
to determine where this 2 per cent lies in rela
tion to the returns within a given conﬁdence
interval. If 2 per cent lies within the confidence
interval, we would conclude that this level of
return could easily be the result of chance. If 2
per cent lies outside this confidence interval, we
would conclude that this level of return is not
likely to be the result of chance. In our simpliﬁed case, in which we know the standard deviation of the population, we can
standardize our sample mean (which in this case
is only one measurement) by calculating a z
score. This can be done by dividing the differ
ence between the sample mean and the popu—
lation mean by the population’s standard
deviation: Sample Mean  Population Mean
2 = .
Population Standard Deviation Assume that the average active return from the
population of investment managers is zero, the standard deviation is 6 per cent and our inves
tor’s ”sample mean" is 2 per cent. We thus
obtain a zscore of 0.67—(20)/6. This means that
(given our simplifying assumptions) an active
return of 2 per cent falls only 0.67 standard
deviation above the mean—well short of the
two standard deviations that contain 95 per cent
of the observations. Thus our investor’s 2 per
cent active return is within the 95 per cent
conﬁdence interval and likely the result of
chance, not skill. Our problem becomes slightly more compli cated when we (more realistically) measure our
manager over more than one year. Technically, instead of determining the probability of one
observation, we need to compare the means of
two distributions (the mean of our population
and the mean of our manager). To complicate
matters further, in practice we usually need to
make inferences from a relatively small sample
Size. This was the problem that W. S. Gosset faced
around the turn of the century. If we think of
the dreary working conditions portrayed so well
by Charles Dickens, it is easy to imagine Gos
set’s problem. Gosset—a chemist at the Guih
ness Breweries—was asked to make inferences
about the quality of various brews. But Gosset
had two problems. First, quite understandably, Guinness was
unwilling to supply Gosset with a large number
of samples. But this limitation on sample size
spurred Gosset to an important discovery. He
found that, when working with small samples,
errors were introduced unless the normal distri
bution was replaced with a distribution that had
more variability (and a higher probability of
large deviations). Having discovered something of great impor
tance to the scientific community, Gosset faced a second problem: Guinness prohibited him
from using his name to publish the results of his onthejob discovery. Undaunted, and believing
in the importance of.his discovery, Gosset pub—
lished his findings anonymously under the pen
name ”Student." Statisticians have ever since
been introduced to "Student’s t" or, as it has
come to be known, the "ttest." The ttest is
especially important for ﬁnancial researchers
who—working with annual returns—share
Gosset’s problem of being forced to work with
small samples. FINANCIAL ANALYSIS IOURNAL / MAYJUNE 1990 U 18 The tTest The ingredients of a ttest are quite intuitive.
First, we expect to have more conﬁdence in
statistics derived from large samples than in
statistics derived from small samples. We might
expect, however, that an increase in sample size
of 10 will have a larger effect on our conﬁdence
when the sample goes from 10 to 20 than when
it goes from 90 to 100. We will skip the mathe
matics, but it turns out that the confidence
interval around the mean is governed by the
square root of the sample size. In the case of the
95 per cent conﬁdence interval deﬁned by plus
or minus two standard deviations from the
mean: 95% Conﬁdence Interval _ 2 X Standard Deviation of Sample Square Root of Sample Size The ”2" in front of the ”standard deviation of
sample” refers to the two standard deviations
associated with the 95 per cent conﬁdence inter
val. Note that an increase in sample size from
four to nine has the same relative effect as an increase from nine to 16.
Let’s take an easy example. Say that the marketrelative return of all managers has a
mean of zero, our sample size is four, and its
standard deviation is 6 per cent (which is typi
cal). Plugging these values into the foregoing
equation, we have: 2x6%
2 95% Confidence Interval = 2% d: = +8to —4. Thus, if we calculated the active return above or
below the mean return for a fouryear random
sample, and repeated the experiment 100 times,
we would expect 95 per cent of these returns to
be within the 95 per cent conﬁdence interval.
Note that, although the conﬁdence interval will
bob around for each sample, 95 per cent of the
intervals so formed will capture the true mean
of the population. What would happen if the standard deviation
or the size of the sample changed? If we cut
standard deviation in half—to 3 per cent—the
conﬁdence interval would also be halved; it would then range from minus 1 to plus 5 per
cent [2% :t (2 x 3%/2)]. Similarly, if we in
creased the sample size from four to 16 years,
the conﬁdence interval would be reduced to from minus 1 to plus 5 per cent [2% 1 (2 X 6%/4)]. How conﬁdent can we be that an observed
value is not a chance deviation from the mean?
If we do not know the standard deviation of the
underlying population, or if we have a small
sample (less than 30), we can answer this ques
tion directly with a t—test. The tdistribution is more "spread out" than
the normal distribution. Its exact form is a
function of the sample size (degrees of free
dom)—the smaller the sample, the more spread
out the distribution becomes. The equation for t contains the three things
that intuitively affect our conﬁdence—the mean
of the active return (the larger the active return,
the more conﬁdence), the standard deviation of
the active return (the more variable the return,
the more likely we are to conclude that an
aboveaverage return occurred by chance) and
the number of years (the more years, the more
conﬁdence). Speciﬁcally: t Active Return — Average Return Standard Deviation of Active Return
Square Root of Number of Years A useful rule of thumb is that a tstatistic must
be at least 2.0 to be signiﬁcant. To be more
speciﬁc, the t—distribution (following Gosset’s
insight) depends on the sample size. By consult
ing a table of tstatistics for various sample sizes,
we find that with a sample size of four we need
a t—statistic of at least 2.35 to conclude that there
is only a 5 per cent chance that beating the
market by 2 per cent per year for four years was
an accident. Because the t—distribution becomes
less spread out as we increase sample size, the
t—statistic required for statistical signiﬁcance de
creases as sample size increases. When the
sample size reaches 30, the tdistribution is very
close to the normal distribution. The data in Table I show that when the
annual standard deviation of the active invest ment return is 6 per cent, we cannot reject the
hypothesis that our manager’ 5 average active return is zero until the manager outperforms the average manager by 2 per cent for 25 years. At
this point, the required t and the calculated t are approximately equal. FINANCIAL ANALYSIS JOURNAL l MAYJUNE 1990 Cl 19 Table I Effect of Sample Size on Required tStatistic m Sample Size Required t Calculated r Active Return / (Std. Deo. / m 4 2.35 0.67 2.0 / (6.0 I 2) 9 1.35 1.00 2.0 / (6.0 I 3)
16 1.75 1.33 2.0 / (6.0 / 4)
25 1.71 1.67 2.0 / (6.0 / 5)
36 1.69 2.00 2.0 / (6.0 / 6) The foregoing example illustrates two things.
First, it is important not to rely on intuition in
estimating confidence. In this example, as in
many studies contained in Financial Analysts
Journal, we need an objective estimate of the
likelihood that the results are sensible. t
tests—one of the major ways to estimate that
likelihood—tell us the probability of being able to rely on the data used to make the analysis. Second, the example was selected to illustrate
the differences between the level of conﬁdence
that would be acceptable to many practitioners
(beating the market by an average of 2 per cent
for four years) and the objective standards to
which Financial Analysts Iournal holds its researchers. Current Issues concluded from page 12. returns. Note, however, that the hierarchy of risk and
returns conformed to expectations until 1989. Both
observations can be helpful in understanding the
fixed income securities market. Footnotes 1. See E. Altman, ”The Anatomy of the HighYield
Bond Market,” Financial Analysts journal, July/
August 1987 and Default Risk, Mortality Rates, and
the Performance of Corporate Bonds (Charlottesville,
VA: Research Foundation of the Institute of Char tered Financial Analysts, 1990); and M. Blume and
D. Keim, ”LowerGrade Bonds: Their Risks and
Retums," Financial Analysts Iournal, July/August
1987. 2.‘ E. Altman and S. Nammacher, Investing in junk
Bonds: Inside the High Yield Debt Market (New York:
John Wiley, 1987). 3. R. Ambarish and M. Subrahmanyam, ”Defaults
and the Valuation of High Yield Bonds,” in E.
Altman, ed., The High Yield Debt Market: Investment Performance and Economic Impact (Homewood, IL:
Dow JonesIrwin, 1990) and R. Bookstaber and R.
Clarke, “Problems in Evaluating the Performance
of Portfolios with Options," Financial Analysts Jour
nal, January/February 1985. 4. J. Fons, “Default Risks and Duration Analysis,” in The High Yield Debt Market, op. cit. 5. We used the mortality methodology in E. Altman,
“Measuring Corporate Bond Mortality and Perfor
mance” (Working paper, Salomon Brothers Cen
ter, New York University, February 1988 and June
1989) and Journal of Finance, September 1989. This
was updated with data from Alhnan, ”Default
Risk,” op. cit. ' 6. See Altman, ”Measuring Corporate Bond Mortal
ity,” op. cit. and P. Asquisth, D. Mullins and E.
Wolff, ”Original Issue High Yield Bonds: Aging
Analysis of Defaults, Exchanges and Calls," Jour
nal of Finance, September 1989. 7. See Altman, ”Measuring Corporate Bond Mortal ity,” op. cit. FINANCIAL ANALYSIS JOURNAL / MAYJUNE 1990 El 20 Copyright© 2003 EBSCO Publishing ...
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