Question

Suppose your income when healthy is IH = 5000 and income when sick is IS = 1000. You are considering purchasing an insurance contract with premium r = 700 and payout of q = 3500 when sick. Your utility over income is U(I) =√

(a) What probability of sickness would make the contract actuarially fair? What would the probability of sickness need to be for the insurer to make positive profits in expectation?

(b) Does this contract offer full or partial insurance? Explain.

(c) Suppose your probability of sickness is p = 0.2. Should you purchase this insurance contract? Explain and show your work

Answer 1

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Answer:

a)

Let probability of sickness=p

In case of actuarially fair premium, Premium=probability of sickness*Payout=p*3500

Since premium is $700 put actuarially fair premium=700

So, p*3500=700

p=700/3500=0.2

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 0.2

b)

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 4000 (5000-1000), while insurance covers only $3500. It means it is a partial insurance.

c)

Probability of sickness=p=0.20

Income in case of sickness=(1000+3500-700) =$3800

Probability of no sickness=1-p=1-0.20=0.80

Income in case of no sickness=(5000-700) =$4300

Expected income=0.20*3800+0.8*4300=$4200