Transcript Example3_4

Example 3.4
Interpretation of the Standard
Deviation: Rules of Thumb
DOW.XLS

This file contains monthly closing prices for the Dow
Jones Index from January 1947 through January
1993.

The monthly returns from the index are also shown
starting with February 1947. Each return is the
monthly percentage change (expressed) as a
decimal) in the index.
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How well do the rules of thumb work for these data?
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Rules of Thumb
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Many data sets follow “rules of thumb”.
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Approximately 68% of the observations are within
one standard deviation of the mean.
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Approximately 95% of the observations are within two
standard deviations of the mean.
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Approximately 99.7% - almost all - of the
observations are within three standard deviations of
the mean.
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Index Time Series Plot
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A time series plot of the index show that the index
has been increasingly fairly steadily over the period.
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Whenever a series indicates a clear trend such as
the index does, most of the measures we have been
discussing are less relevant.
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For example, the mean of the index for this period
has at most historical interest. We are probably more
interested in predicting the future of the Dow, and the
historical mean has little relevance for predicting the
future.
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Time Series Plot of Dow Closing
Index
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Time Series Plot of Dow Returns
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Return Time Series Plot
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A time series plot of the returns show no obvious
trend over the period.
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The measures we have been discussing are relevant
in discussing the series of returns, which fluctuate
around a stable mean.
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We first calculate the mean and standard deviation of
the returns by using the Excel functions AVERAGE
and STDEV in cells B4 and B5. See the table on the
next slide.
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Rules of Thumb for Dow Jones
Data
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Returns -- continued
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The average return is 0.59% and the standard
deviation of about 3.37%.
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Therefore, the rules of thumb (if they apply) imply, for
example, that about 2/3 of all returns are within the
interval 0.59% + 3.37%, that is from -2.78% to 3.95%.
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In order to determine if the rules of thumb apply to
these returns, we can use a frequency table.
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Creating the Frequency Table
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We first enter the upper limits of the suitable
categories in the range A8:A15.
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Any categories can be chosen but it is convenient to
choose categories in which each breakpoint is one
standard deviation higher than the previous one with
the open-ended categories on either end are “more
than 3 standard deviations from the mean”.
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Next we use the FREQUENCY function to fill in
column C. “=FREQUENCY(Returns,Bins)”
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Frequency Table continued
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Finally, we use the frequencies in column C to
calculate the actual percentage of return within k
standard deviations of the mean for k=1, k=2 and k=3
and we compare these with the percentages from the
rules of thumb.
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The agreement between these percentages is not
perfect - there are a few more observations within
one standard deviation of the mean than the rule of
thumb predicts - but in general the rules of thumb
work quite well.
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