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Uncertainty
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Hal Varian
January 1992
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Risk Premium
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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:
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u[wealth_,rho_] := (wealth^(1-rho))/(1-rho)
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Call the risk premium pi. To find it, we need to solve the following equation:
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rho=3
loss=500
FindRoot[u[5000-pi,rho] == .5*u[5000,rho]
+ .5*u[5000-loss,rho],{pi,500}]
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Let's graph the risk premium as a function of the loss.
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rho=3
plotrho3=Plot[5000 - (.5*5000^(1-rho) +
.5*(5000-loss)^(1-rho))^(1/(1-rho)),
{loss,0,2000}]
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If the consumer is more risk averse the risk premium increases more rapidly
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rho=7
plotrho7=Plot[5000 - (.5*5000^(1-rho) +
.5*(5000-loss)^(1-rho))^(1/(1-rho)),
{loss,0,2000}]
Show[{plotrho3,plotrho7}]
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A simple portfolio problem
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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
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eu[a_] := .5*u[wealth+.7*a,rho] +
.5u[wealth-.3*a,rho]
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Fix wealth at 1000, say, and rho at 3, and plot this as a function of a.
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wealth=1000
rho=5
Plot[Evaluate[eu[a]],{a,1,600}]
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It looks like this consumer would like to invest about $200. Let's check:
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FindMinimum[-eu[a],{a,200}]
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So the optimal investment is about $175.
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Mean-Variance Utility
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Let's check the calculation in the text for Normally distributed wealth and constant absolute risk averse utility function. [To be completed...]
^*)