Question

Answer in details with steps

Jake faces a 60% chance of having a loss of $0 and a 40% chance of having a loss of $15.

Mark also faces a 20% chance of having a loss of $0 and an 80% chance of having a loss of $7.50.
Shannon faces a 75% chance of having a loss of $0 and a 25% chance of having a loss of $30.

1. What is the actuarially fair premium (AFP) for Jake and Mark each individually? (1 point)

2. What is the actuarially fair premium (AFP) for Shannon? (1 point)

3. If Shannon is added to the risk pool for insurance with Jake and Mark, what would the expected loss (P*) be? (1 point)

4. If the insurer charges all of them a price of $7, is that price a good deal for each of them? Why or why not? (hint - think about the AFP for each) (2 points)

Answer 1

Answer:-  

The expected loss of Jake = $ 0 x 0.60 + $ 15 x 0.40 = $ 0 + $ 6 = $ 6
The expected loss of Mark = $ 0 x 0.20 + $ 7.50 x 0.80 = $ 0 + $ 6 = $ 6
The expected loss of Shannon = $ 0 x 0.75 + $ 30 x 0.25 = $ 0 + $ 7.5 = $ 7.5

The Actuarially Fair Premium (AFP)  is the premium that allows one to exactly break even

a) Therefore the AFP for Jake and Mark individually is $ 6.

b) The AFP for Shannon is $ 7.5

c) If Shannon is added to the risk pool for insurance the total expected loss will be = $ 6 + $ 6 + $ 7.5 = $ 19.5
The average expected loss will be = $ 19.5 / 3 = $ 6.5

d) If the insurer charges all of them $ 7
For Jake the expected loss (AFP) is only $ 6 and he is paying $ 7 to the insurer therefore he is loosing $ 1 which is not a good deal for Jake.

Similarly for Mark also the expected loss (AFP) is $ 6 and is paying $ 7 and loosing $ 1 which is not a good deal.

In case of Shannon he is expected loss (AFP)   $ 7.5 and he is paying $ 7 to the insurer, and is is gaining the difference $ 0.5 and the insurer is loosing $ 0.5. Therefore it's a good deal for Shannon.