Uncertainty

Hal Varian
January 1992

Risk Premium

A consumer has a constant relative risk aversion utility function with risk aversion of 3.  He currently has monthly income of 5,000 and has a 50 percent chance of incurring a loss of 500.  How much would he be willing to pay to avoid this loss?

First we define the CRRA utility function:

u[wealth_,rho_] := (wealth^(1-rho))/(1-rho)

Call the risk premium pi.  To find it, we need to solve the following equation:

rho=3
loss=500
FindRoot[u[5000-pi,rho] == .5*u[5000,rho] 
+ .5*u[5000-loss,rho],{pi,500}]

Let's graph the risk premium as a function of the loss.

rho=3
plotrho3=Plot[5000 - (.5*5000^(1-rho) + 
.5*(5000-loss)^(1-rho))^(1/(1-rho)),
{loss,0,2000}]

If the consumer is more risk averse the risk premium increases more rapidly

rho=7
plotrho7=Plot[5000 - (.5*5000^(1-rho) + 
.5*(5000-loss)^(1-rho))^(1/(1-rho)),
{loss,0,2000}]
Show[{plotrho3,plotrho7}]

A simple portfolio problem

Let's calculate how much a consumer would invest in a risky asset.
Again, we assume a CRRA utility function and suppose that the risk asset has a 50 percent chance of a rate of return of 70% and a 50  percent chance of -30%.  Utility as a function of the investment a is

eu[a_] := .5*u[wealth+.7*a,rho] + 
.5u[wealth-.3*a,rho]

Fix wealth at 1000, say, and rho at 3, and plot this as a function of a.

wealth=1000
rho=5
Plot[Evaluate[eu[a]],{a,1,600}]

It looks like this consumer would like to invest about $200.  Let's check:

FindMinimum[-eu[a],{a,200}]

So the optimal investment is about $175.

Mean-Variance Utility

Let's check the calculation in the text for Normally distributed wealth and  constant absolute risk averse utility function.    [To be completed...]