Handout on Risk Aversion
For an individual with a utility of consumption function denoted )(CU that exhibits
positive but diminishing marginal utility, a measure of risk aversion commonly used in
Financial Economics is something called Relative Risk Aversion (RRA), which is defined
as follows:
RRA = )(
)(
CU
CU
C′
′
′
−.
Note that because the second derivative in the numerator is negative in sign, RRA is a
positive valued number. If we compare values of RRA for 2 individuals, the one with the
higher RRA is deemed to be more averse to risk that the other. At this point the measure
is quite abstract but we'll try to give it more substance in what follows. To do so it is
useful to introduce a class of utility functions that exhibit Constant Relative Risk
Aversion (CRRA) – which is to say that the risk aversion measure RRA has the same value
irrespective of the level of consumption.
A CRRA utility function is of the form
γ
γ
−
=
−
1
)(
1
C
CU ,
where γ is a parameter with any value γ > 0, except for γ = 1, in which case the function
takes the form )ln()( CCU =.
For each member of this class of utility functions one can apply the preceding definition
of Relative Risk Aversion to deduce that RRA = γ, irrespective of the level of
consumption. (In the ln(C) case, RRA = 1). The parameter γ is often referred to as the
coefficient of relative risk aversion.
If 2 individuals have different CRRA utility functions, the one with the higher value of γ
is deemed to be the more risk averse.
Now let's give this some substance by considering how individuals with CRRA utility
functions that have different values of γ would evaluate the following risky situation:
An individual's wealth will equal either 50,000 or 100,000 each with probability ½ so that
expected wealth is E[W] = 75,000.
The following table shows the certainty equivalent wealth WCE associated with various
values for the coefficient or relative risk aversion γ.
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Value of γ 1 2 5 10 30
Value of WCE 70,711 66,667 58,566 53,991 51,209
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