CARA: vNM utility with constant absolute risk aversion

This is a classic graduate/advanced undergraduate level problem in microeconomics/finance. It asks to calculate utility function that corresponds to a constant Arrow-Pratt measure of absolute risk-aversion.

Here's a problem from Mas-Colell, Whinston and Green textbook on Microeconomic Theory:

Given that the Arrow-Pratt measure of absolute risk-aversion is a constant, derive the corresponding form of the von Neumann-Morgenstern utility function.

Solution Preview

Please see the attached file for detailed solutions. The text below is contained within the attached Word file along with the formulas:

Problem: Given that the Arrow-Pratt measure of absolute risk-aversion is a constant, derive the corresponding form of the von Neumann-Morgenstern utility funciton.

First of all let's be clear regarding the definitions:

Expected utility is the sum of utility in each outcome weighted by the probability.

Arrow-Pratt measure of absolute risk-aversion - the measure of how much an individual dislikes risk at different levels of wealth. For example, if you ...

Solution Summary

This solution shows how to find the von Neumann-Morgenstern utility functions that displays constant measure of absolute risk-aversion (Arrow-Pratt measure) - CARA.

All of the mathematics required for this problem are presented in the beginning along with the necessary definitions. Note that the properties of integrals are stated without proof, so a student without a background in calculus will need to refer to the text to see the proof. Once the properties are given, the solution explains how to apply them correctly to arrive at the final solution. All the equations are beautifully typed using Microsoft equation editor.

An expected utility maximizing individual has utility of eno-of-period wealth given by
u(W)= W^(1-y)-1, if y is not equal to 1
ln(W), if y=1
1. Show that this individual has constant relative riskaversion and decreasing absoluteriskaversion.
2. Consider the special case where y=2. Suppose that this individu

** Please see the attached file for fully formatted problem description **
---
Define RiskAversion. Consider a risk-averse von Neumann-Morgenstern individual having wealth w who must decide whether to accept or decline a simple gamble offering a chance of winning or losing a small amount of wealth h with probabilities p an

Consider a risk averse agent. He faces a health risk of D(he/she has to go to the hospital and pay D, and he/she will be fine) with probability k. He/she can buy insurance at price q (that is he/she can buy at cost q a contract that pays 1 dollar if he/she has to go to the hospital). How many units of the contract will the agent

The risk-free rate is 5% and two stocks have the following expected returns and standard deviations:
STOCK A
E(ra) = 10%
sigma(a)=20%
STOCK B
E(rb) = 15%
sigma(b) = 27%
An investor with a degree of riskaversion of 3 and a utility function of the form U = E(r) - 1/2Asigma^2 would find that:
a) only stock A i

What types of utility curves (increasing, decreasing, or constant marginal utility of wealth) are generally associated with each of the following attitudes toward risk:
(a) risk-averse,
(b) risk-neutral, and
(c) risk-seeking?