Question

Consider 3 individuals A, B & C with the following risk profiles (4 points)

Individual A: P(healthy) = 90% and I(healthy) = 1000 and I(sick) = 0

Individual B: P(healthy) = 95% and I(healthy) = 3000 and I(sick) = 500

Individual C: P(healthy) = 99% and I(healthy) = 4000 and I(sick) = 500

P() = probability

I() = Income

An insurance company has agreed to cover these individuals for $1000 each. They have to pay a premium of 100. True or False

1. Individual A has a fair and full insurance

2. Individual B has a fair and full insurance

An insurance company has agreed to cover these individuals for $2500 each. They have to pay a premium of 250. True or False

3. Individual B has a unfair but full insurance

4. Individual C has a unfair and full insurance

Answer 1

An insurance is actuarially fair, meaning that the premium is equal to expected claims: Premium = p ·A where p is the expected probability of a claim, and A is the amount that the insurance company will pay in the event of an accident.

In first case,the insurance company has agreed to cover these individuals for $1000 each. They have to pay a premium of 100.

Therefore,

Amount that the insurance company will pay in the event of an accident (A)= 1000

Premium=100

For Individual A: p= 0.1 as the probability of A to be healthy is 90/100=0.9 so probability of claim (p)= 1-0.9= 0.1

Since, Premium=p.A to be fair.

100=0.1*1000

Also the amount = income when healthy so it covers individual A fully as they don't earn anything when sick.

Hence, TRUE: Individual A has a fair and full insurance.

For Individual B: p= 0.05 as the probability of B to be healthy is 95/100=0.95 so probability of claim (p)= 1-0.95= 0.05

Since, Premium=p.A to be fair but 1000.05*1000=50 so its unfair for Individual B.

Also the amount income when healthy so it doesn't cover individual B fully as he earns only 500 when sick

Hence, FALSE: Individual B has a fair and full insurance.

In second case,the insurance company has agreed to cover these individuals for $2500 each. They have to pay a premium of 250.

Therefore,

Amount that the insurance company will pay in the event of an accident (A)= 2500

Premium=250

For Individual B: p= 0.05 as the probability of B to be healthy is 95/100=0.95 so probability of claim (p)= 1-0.95= 0.05

Since, Premium=p.A to be fair but 2500.05*2500 therefore it is unfair for Individual B.

Also the amount = income when healthy so it covers individual A fully as they don't earn anything when sick.

Hence, TRUE: Individual B has a unfair but full insurance.

For Individual C: p= 0.01 as the probability of B to be healthy is 99/100=0.99 so probability of claim (p)= 1-0.99= 0.01

Since, Premium=p.A to be fair but  2500.01*2500 therefore it is unfair for Individual C

Also the amount+income when sick income when healthy i.e. 2500+500 4000 ,so it does NOT cover individual C fully.

Hence, FALSE: Individual C has a unfair and full insurance.