In: Statistics and Probability
Is it true that for an expected utility maximizer to be a risk
averter, his/her utility function
should have a negative second derivative? Give an example. (Advice:
Look up the definition
of concavity.)
An important breakthrough in the analysis of decisions under risk was achieved when Daniel Bernoulli, a distinguished Swiss mathematician, wrote in St Petersburg in 1738 a paper in Latin entitled: “Specimen theoriae novae de mensura sortis,” or “Exposition of a new theory on the measurement of risk.”
Expected-utility (EU) theory has many proponents and many detractors. The researchers in both economics and finance have long considered the EU theory as an acceptable paradigm for decision making under uncertainty. Indeed, EU theory has a long and prominent place in the development of decision making under uncertainty. Even detractors of the theory use EU as a standard by which to compare alternative theories. Moreover, many of the models in which EU theory has been applied can be modified, often yielding better results. Whereas the current trend is to generalize the EU model, researchers often restrict EU criterion by considering a specific subset of utility functions. This is done to obtain tractable solutions to many problems. It is important to note the implications that derive from the choice of a particular utility function. Some results in the literature may be robust enough to apply for all risk-averse preferences, while others might be restricted to applying only for a narrow class of preferences.
Let us consider the following simple decision problem. An agent is offered a take-it-or-leave-it offer to accept lottery z(bar) with mean µ and variance ?2. Of course, the optimal decision is to accept the lottery if
Eu(w + z(bar)) u(w),
or, equivalently, if the certainty equivalent e of z(bar) is positive. In the following, we examine how this decision is affected by a change in the utility function. Notice at this stage that an increasing linear transformation of u has no effect on the decision maker’s choice, and on certainty equivalents. Indeed, consider a function v(·) such that
v(x) = a + bu(x) for all x,
for some pair of scalars a and b, where b > 0. Then, obviously
Ev(w + z(bar)) v(w)
yields exactly the same restrictions on the distribution of z(bar) as a condition. The neutrality of certainty equivalents to linear transformations of the utility function can be verified in the case of small risks by using the Arrow-Pratt approximation.
If v ? a + bu, it is obvious that
A(x) = ?v”(x) v’ (x) = ?bu”(x) bu’ (x) = ?u”(x) u’ (x) ,for all x.
Thus, risk premia for small risks are not affected by the linear transformation. Because the certainty equivalent equals the mean payoff of the risk minus the risk premium, the same neutrality property holds for certainty equivalents.
Risk aversion is a second-order phenomenon.
The general formula is
var(ax˜ + by)˜ = a2 var(x)˜ + b2 var(y)˜ + 2ab cov(x,˜ y).
For an expected utility maximizer with u: Z? R, the Arrow-Pratt measure of risk aversion is defined by,
?(z) = (-u”(z))/(u’(z))
if the DM is risk-averse and prefers more money to less, then ?(z) is positive (as the second derivative of a concave function is negative). Notice also, that, in some sense, if the utility function is ‘more’ concave then ?(z) will increase. And in fact, this intuition is correct in the sense that in many instances we can use the Arrow-Pratt measure to compare levels of risk aversion.
For example:
Another use of this measure is to compare the risk aversion of different decision makers:
Concavity
It is the property of curving a graph of the function upward or downward. The interval on which derivative of the function, f', increases or decreases can determine where the graph of f is curving upward or curving downward. The first derivative test tells us where a function is increasing or decreasing. And the second derivative gives the information that function is concave upward or concave downward.
A point P on the curve yy = f(x) is called the inflection point f is continues there and the curve changes from concave upward to concave downward or from concave down to concave upward at P.