Probability Solution

Solution 2:

Possessions value- $50000

Probability of accident=.1

This means that the expected value of loss is =10 % of 50000=$5000

 

  1. If the insurance fee(f) is $5500, he will not go for the insurance, since the cost of insurance is higher than the expected value of loss
  2. U=x ^.4

By the utility function , we can get the utility by applying the value

Merchant expected utility if un-insured is

(.9)*( 50000)^.4

To get that utlity with certainty , he would need the following risk free wealth

 

U(x)= x^.4

U(x)=  .9 * (50000)^.4

X= (.9)^.4 *50000

X=47936.58

So he will be willing to pay (1-.9^.4)*50000=2063.424$

Answer 3

Profit- 500000, prob=.9

Loss= 2500000, prob=.1

In linear case , expected point= 500000*.9-2500000*.1=200000

Hence it will go ahead with the project

U(x)=2x-.1×2+100

Current assets=3 million

In the first case, asset become, 3+.5=3.5, in the second case asset becomes, 3-2.5=.5,

The expected utility if uninsured is

=.9*(-.1*(3.5)^2 +2(3.5) +100)+.1*(-.1*(.5)^2+2(.5)+100)

=117.4

 

We need the asset level that corresponds to this value

2x-.1x^2+100=117.4

X is imaginary!

The utility function is unique because it is a quadratic equation, which may have imaginary roots

 

4

b.

Let the probability of success be p, and failure 1-p

Successà Asset=4mn

Failureà Asset=.5mn

U(x)=x^.5

Utility outcome= p(4)^.5+(1-p)(.5)^.5

U(x)=x-.1x^2

Utility outcome=p(4-.1(4)^2) + (1-p)(.5-.1(.5)^2)

U(x)=1-e^-x

Utility outcome=p (1-e^-4) +(1-p)(1-e^-.5)

U(x)= p(1-(3/4)^4)+(1-p)(1-(3/4)^.5)