What Practitioners Need
to
Know
.
. .
About Utility
Mark Kiitzinan
Windham
Capital
Manage-
ment
An important axiom
of modern
financial theory
is that rational
investors seek
to maximize ex-
pected utility. Many
financial
ana-
lysts, however, find the concept of
utility somewhat nebulous. This
column discusses
the origin of
utility theory
as well as its appli-
cation within the context
of finan-
cial analysis.
In
his classic paper, "Exposition
of a New Theory
on the Measure-
ment
of Risk," first published in
1738, Daniel Bernoulli proposed
the following:
"the determination
of
the value of an item must not
be based
on its price, but rather
on
the utility it yields. The price
of the item
is dependent only on
the thing itself
and is equal for
everyone;
the utility, however, is
dependent
on the particular cir-
cumstances
of the person making
the estimate. Thus there
is no
doubt that
a gain of one thousand
ducats
is more significant to a
pauper than
to a rich man though
both gain
the same amount."^
Bernoulli's insight that
the utility
of a gain depends
on one's wealth
may seem rather obvious
and
perhaps even pedestrian,
yet it
has profound implications
for the
theory
of risk. Figure A helps us
to visualize Bernoulli's notion
of
utility, which economists today
call diminishing marginal utility.
The horizontal axis represents
wealth, while the vertical axis rep-
resents utility.
The relation be-
tween wealth
and utility is mea-
sured
by the curved line. Utility
clearly increases with wealth,
be-
cause
the curve has a positive
slope.
The positive slope simply
indicates that
we prefer more
Figure A
Diminishing Marginal Lltihty
Figure B
Changes in Utility versus
Changes in Wealth
Wealth
wealth
to less wealth. This as-
sumption seldom invites dispute.
As wealth increases,
the incre-
ments
to Utility become progres-
sively smaller,
as the concave
shape
of the curve reveals. This
concavity indicates that we derive
less and less satisfaction with each
subsequent unit
of incremental
wealth. Technically, Bernoulli's
notion
of utility implies that its
first derivative with respect
to
wealth
is positive, while its sec-
ond derivative
is negative.
A negative second derivative
im-
plies that
we would experience
greater
disxiiiXiVf
from a decline
in
wealth than
the utility we would
derive from
an equal increase in
wealth. This tradeoff
is apparent
in Figure
B, which shows the
changes
in utility associated with
an equal increase
and decrease in
wealth.
According
to Bernoulli, the pre-
cise change
in utility associated
with
a change in wealth equals
the logarithm of the sum
of initial
wealth plus
the increment to
wealth, divided
by initial wealth.
For example,
the increase in util-
ity associated with
an increase in
Initial Wealth
wealth from $100 dollars
to $150
equals 0.405465,
as follows:
0.405465
/
=
ln
\
100-H50
100
The next
$50 increment to
wealth, however, yields
a smaller
increment
to utility:
0.287682
Risk Aversion
From Bernoulli's assumption
of
diminishing marginal utility,