What Practitioners Need
to
Know
.
. .
About Lognormality
i
CC
g
u
I
cn
10
Mark Kritzman
Windbam
Capital
Manage-
ment
When reading
the financial litera-
ture
we often see statements
to
the effect that
a particular result
depends
on
the assumption that
retums
are
lognormally
distrib-
uted. What exactly
is
a lognormal
distribution, and why is
it relevant
to financial analysis?
In order to
address this question,
let us start
with
a review
of logarithms.
Logarithms
A logarithm
is simply
the power
to which
a base must be raised to
yield a particular value. For exam-
ple,
the exponent
2 is
the loga-
rithm of 16
to the base 4, because
4 squared equals
16. The loga-
rithm of 8 to the base 4 equals
1.5,
because
4 raised to the power 1.5
equals
8.
The choice
of a base depends on
the context
in which we use log-
arithms.
For simple mathematical
procedures,
it
is common to use
the base 10, which explains
why
logarithms
to the base 10 are
called common logs. The base
10
is popular because the logarithms
of 10,
100, 1000 and so on equal
1,
2, 3
respectively.
Why should
we care about loga-
rithms?
In the days prior to
pocket calculators
^ong before
my time), logarithms were useful
for performing complicated com-
putations. Financial analysts
would multiply large numbers
by
summing their logarithms,
and
they would divide them
by sub-
tracting their logarithms.
For ex-
ample, given
a base
of 4, we can
multiply 16 times
8 by raising the
number 4 to the
35 power, which
is the sum of the logarithms 2
and
1.5. Of course, you might argue
that
it would have been easier to
multiply large numbers direaly
than
to raise a base to a fractional
power.
In
the olden days, how-
ever,
an analyst would use a slide
rule, which
is
a ruler with a slid-
ing central strip marked with
log-
arithmic scales.
In most financial applications,
in-
stead of logarithms to the base 10,
we
use logarithms to the base
2.71828, which
is denoted by the
letter
e in honor of
the famous
Swiss mathematician Euler. These
logarithms, which
are called nat-
ural logs
and are abbreviated
as
ln, have
a special property.
Sup-
pose we invest $100
at the begin-
ning
of the year at an annual
interest rate of 10096. At the end
ofthe year we will receive $200—
our original principal
of 1100 and
another
$100
of interest. Now
suppose
our interest is com-
pounded semiannually. Our year-
end payment will equal $225.
By
the middle
of
the year we will
have earned
$50 of interest,
which
is then reinvested
to gen-
erate
the additional $25.
In general, we can use the follow-