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.