In: Economics
Suppose you make some income when healthy, IH = $2000, and none when sick, IS = 0, and are considering the following an insurance contract with premium, r = 540, and insurance payout when sick, q = $1800.
a. What probability of sickness would make the insurance contract actuarially fair? What would the probability of sickness need to be for the insurer to make positive profits in expectation? Explain/show your work.
b. Is this potential contract an offer of full insurance or partial insurance? Explain/show your work.
c. What is your expected income if you purchase this contract and your probability of sickness is 0.2?
d. Assume the individual’s utility over income is U(I) = √ I and has a probability of sickness, p = 0.2. Calculate your expected utility E[U(I)] (a) with the contract and (b) without the contract.
e. Is this individual risk averse? Explain. (2 points) f. Should the individual purchase this contract? Explain.
a)
Let probability of sickness=p
In case of actuarially fair premium, Premium=probability of sickness*Payout=p*1800
Since premium is $540 put actuarially fair premium=540
So, p*1800=540
p=540/1800=0.30
An insurer will make the expected positive profits if probability sickness is less than the probability calculated in the case of actuarially fair premium. So,
p should be less than 0.30
b)
In case of sickness loss of income is $2000, while insurance covers only $1800. It means it is a partial insurance.
c)
Probability of sickness=p=0.20
Income in case of sickness=(0+1800-540) =$1260
Probability of no sickness=1-p=1-0.20=0.80
Income in case of no sickness=(2000-540) =$1460
Expected income=0.20*1260+0.80*1460=$1420
d)
Expected utility with contract
Probability of sickness=p=0.20
Income in case of sickness=(0+1800-540) =$1260
Utility, U(1260)=12601/2=35.49648
Probability of no sickness=1-p=1-0.20=0.80
Income in case of no sickness=(2000-540) =$1460
Utility, U(1460)=14601/2=38.20995
Expected Utility=0.20*35.49648+0.80*38.20995=37.66726
Expected utility without contract
Probability of sickness=p=0.20
Income in case of sickness=0
Utility, U(0)=01/2=0
Probability of no sickness=1-p=1-0.20=0.80
Income in case of no sickness=$2000
Utility, U(2000)=20001/2=44.72136
Expected Utility=0.20*0+0.80*44.72136=35.77709
e)
Expected income in case of no insurance=0.2*0+0.80*2000=$1600
Expected Utility in case of no insurance=35.77709 (calculated in above parts)
Expected income in case of insurance=$1420 (calculated in above parts)
Expected Utility in case of insurance=37.66726 (calculated in above parts)
We can observe that utility is higher in case of lower income risk despite lower expected income. We can say that person is risk averse.
Expected Utility has improved in case of insurance. He should buy the insurance.