Optimization and portfolio selection 173
the mean, the standard deviation, and the extreme value of annual returns. You
are probably already familiar with the mean and standard deviation. The mean
is a measure of the central tendency and the standard deviation a measure of the
spread of the curve. These two parameters are enough to describe a normal and
the standard lognormal. But the three-parameter version of the lognormal uses
another parameter. A lognormal curve has either a largest value or a smallest
value. This third parameter, the extreme value, allows us to shift and flip the
distribution.
Now we must find a way of estimating these parameters from a sample.
We choose to solve this problem by using the sample mean and sample stand-
ard deviation to estimate the mean and standard deviation of the underlying
lognormal. Our estimate of the extreme value was selected on the basis of sim-
ulations. These simulations showed that only a rough estimate of the extreme
value is required to obtain a reasonable lognormal fit. We estimate it as
follows: First, calculate the minimum and the maximum of the sample and
take the one closest to the mean. The extreme value is obtained from this value
by moving it four standard deviations further from the mean. For example, if
the mean, standard deviation, minimum, and maximum of a sample are 12%,
8%, 15%, and 70%, respectively, then the extreme value is ( 15%) — (4)
(8%) 47%, since the minimum is closer to the mean than the maximum.
Consider the following table with statistics computed from a lognormal fit:
Parameters and statistics Example 1 Example 2 Example 3
Mean 12% 12% 12%
Standard deviation 22% 22% 22%
Extreme value 50% 74% 50%
DTR 7.5% 7.5% 13.5%
Upside probability 51.9% 74% 40.4%
Downside deviation 10.5% 15.5% 14.3%
Upside potential 10.6% 11.2% 7.8%
Upside potential ratio 1.01 0.73 0.55
The first two examples differ only in extreme values. One is a maximum
and the other a minimum. Please notice how different these two examples are
even though they have the same mean and standard deviation. These differ-
ences cannot be captured by the normal curve. The third example is similar to
the first but with a higher DTR.
The three basic parameters are estimated from a sample obtained from a
bootstrap procedure on historical returns.
Mean sample mean
SD sample standard deviation
Tau τ extreme value computed as described above