Question

Part Three: Short Answer (2 points each; 8 points total) Assume that Calista has preferences over money that may be described by the following utility function: U(SX) SQRT SX) Further, assume that a fair coin is tossed. If the coin comes up heads, then Calista will be paid $100, otherwise Calista will be paid $0. Answer each question and show all work. 1. What is the expected value of the coin toss? 2. What is the expected utility of the coin toss? 3. What is Calista's risk premium for this gamble (measured in utils)? 4. What is Calista's risk premium for this gamble (measured in dollars)?

Answer 1

U($X)=[1+SQRT($X)]

We know that if a fair coin is tossed,

Probability of a heads (p1)=0.5

Probability in case of not a heads (p2)=0.5

Payoff in case of heads=M1=$100

Payoff in case of not a heads=M2=$50

1)

Expected value of coin toss will be equal to the expected payoff of this game.

Expected value of coin toss=p1*M1+p2*M2=0.5*100+0.5*50=$75

Expected value of coin toss=$75

2)

Utility in case of heads (i.e. X=$100)= )=U1=[1+SQRT(100)]=11 utils

Utility in case of not a heads (i.e. X=$50)=U2=[1+SQRT(50)]=8.07 utils

Expected Utility of coin toss=p1*U1+p2*U2=0.5*11+0.5*8.07=9.535 utils

Expected Utility of coin toss=9.535 utils

3)

In this case expected payoff = $75

Now calculate the Utility for certain payoff $75 (i.e. X=$75)

U($X)=[1+SQRT($X)]

U($75)=[1+SQRT(75)]= 9.660 utils

We find that Calista has more utility for certain payoff of $75, against the expected utility as derived in part (b)

Risk premium for the gamble =Utility (at X= $75)-Expected utility of game

Risk Premium = 9.660-9.536 =0.124 utils

4)

We can calculate the risk premium by calculating first the amount which will equate Expected utility and utility at certain $75. We need to find the X for which U(X)=Expected Utility.

U(X)=1+SQRT(X)= 9.535

SQRT(X)=9.660-1=8.535

X=(8.535)^2=$72.85

Calista will get the same utility at $72.85 as the expected utility had in game. She can at most pay the difference between these dollar amounts as risk premium.

Risk Premium=75-72.85=$2.15