Distribution:
,
Moment 44
Appendix
Normal Generating Function (MGF):
Constant Absolute Risk Aversion utility function
Maximization of
is equivalent to the minimization of
equivalent to the maximization of
, which is
45
Log-Normal distribution:
Y is distributed as a lognormal with parameters : and F if
is distributed as a
.
,
Remark: Let
(i)
(ii)
Constant Relative Risk Aversion utility function
Both expressions are used, but the first one is more common.
(>1 is the Arrow-Pratt relative risk aversion coefficient. Most evidence suggests that ( is low, between
1 and 3, and at most 10.
46
Notice that this expected utility is written as a function of the mean and variance of
, not in terms of the
mean and variance of Y. That is the reason why the expected utility appears to be an increasing function the
variance. As shown above, the variance of Y depends on both : and
the expected utility in portfolio analysis, one has to first transform Y to
of
, and use them as the estimates of : and
of
. To use this expression of
, estimate the mean and variance
.
It is useful find to the expected utility in terms of the mean and variance of Y. This will show the tradeoff
between the mean and variance more clearly. To do this, we need to find equations for : and
and
. From the equations of
and
in terms of
, we have
From the second equation, we have
Substitute this into the equation for : to get
Therefore,
Therefore, we can write the expected utility as
The second expression shows the tradeoff between the mean and the variance of Y. This equation can be used
once we estimate the mean and variance of Y (not lnY). Some authors use the mixture of these moments. One
example is to use
and
47
But this requires estimation of the moments of Y and also lnY.
Indifference Curves
Since the expected utility depends only on the mean and variance of W, we can find the equation for the
indifference curves in the space of
Rewriting the
and
.
equation, we derive
Rearranging the terms gives
and we have
Maximization of
Maximization of the expected utility is equivalent to
or
Certainty Equivalent
The certainty equivalent C is determined by
. Let
.