RMI 2101

Summer 2017

Homework Assignment 9

Show ALL of your work in order to receive full credit for an answer.

1. Aron owns a $10,000 home and has a 10% chance of experiencing a fire in any

given year. Assume that only one fire per year can occur and that if a fire occurs,

her home is completely destroyed.

Suppose that Aron purchases a full insurance contract from Cooper Insurance for

an actuarially fair premium. This contract would pay all losses due to the fire.

Assume that Aron’s contract is the only insurance contract Cooper Insurance

sold.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell a contract to Aron? [1 point]

b. What is the actuarially fair premium [AFP] Cooper Insurance will charge

Aron in the coming year? [1 point]

c. What is the amount of risk Cooper Insurance faces if they have Aron as

their only customer? [2 points]

2. Barbara, who owns the same type of house and faces the same probability

distribution of losses as Aron, also purchase full insurance for an actuarial fair

premium from Cooper Insurance. We assume that the two houses are

independent of each other….meaning that if one house has a fire, this has no

impact on the probability of the other house having a fire.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to both Aron and Barbara? [2 points]

b. What is the expected loss or expected payout for Cooper Insurance if they

sell contracts to both Aron and Barbara? [1 point]

c. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Barbara? [2 points]

d. Briefly explain the benefits to Cooper Insurance as the number of

insurance contracts sold increases? [2 points]

3. Now suppose Carlo owns a $20,000 home and has a 10% chance of experiencing

a fire in any given year. Assume as before that the fire will result in a total loss.

Suppose Cooper Insurance offers Aron and Carlo the same insurance contract

and charges them the same premium….in other words, they put Aron and Carlo

into the same risk pool.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to Aron and Carlo? [2 points]

b. What premium must Cooper Insurance charge each of Aron and Carlo if

they want to ‘break-even’? [1 point]

c. Will Aron purchase this contract if he is charged the ‘break-even’

premium? Will Carlo purchase this contact if he is charged the ‘breakeven’

premium? Briefly explain your reason. [2 points]

d. Explain what requirement of a pooling arrangement is illustrated in this

case? [2 points]

e. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Carlo? [2 points]

4. Compare the situation in question 2 and question 3 above. In particular, examine

the results you obtain in 2 (c) versus 3 (e). Explain carefully the ‘tradeoff’ that is

illustrated. [4 points]

RMI 2101

Summer 2017

Homework Assignment 9

Show ALL of your work in order to receive full credit for an answer.

1. Aron owns a $10,000 home and has a 10% chance of experiencing a fire in any

given year. Assume that only one fire per year can occur and that if a fire occurs,

her home is completely destroyed.

Suppose that Aron purchases a full insurance contract from Cooper Insurance for

an actuarially fair premium. This contract would pay all losses due to the fire.

Assume that Aron’s contract is the only insurance contract Cooper Insurance

sold.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell a contract to Aron? [1 point]

b. What is the actuarially fair premium [AFP] Cooper Insurance will charge

Aron in the coming year? [1 point]

c. What is the amount of risk Cooper Insurance faces if they have Aron as

their only customer? [2 points]

2. Barbara, who owns the same type of house and faces the same probability

distribution of losses as Aron, also purchase full insurance for an actuarial fair

premium from Cooper Insurance. We assume that the two houses are

independent of each other….meaning that if one house has a fire, this has no

impact on the probability of the other house having a fire.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to both Aron and Barbara? [2 points]

b. What is the expected loss or expected payout for Cooper Insurance if they

sell contracts to both Aron and Barbara? [1 point]

c. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Barbara? [2 points]

d. Briefly explain the benefits to Cooper Insurance as the number of

insurance contracts sold increases? [2 points]

3. Now suppose Carlo owns a $20,000 home and has a 10% chance of experiencing

a fire in any given year. Assume as before that the fire will result in a total loss.

Suppose Cooper Insurance offers Aron and Carlo the same insurance contract

and charges them the same premium….in other words, they put Aron and Carlo

into the same risk pool.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to Aron and Carlo? [2 points]

b. What premium must Cooper Insurance charge each of Aron and Carlo if

they want to ‘break-even’? [1 point]

c. Will Aron purchase this contract if he is charged the ‘break-even’

premium? Will Carlo purchase this contact if he is charged the ‘breakeven’

premium? Briefly explain your reason. [2 points]

d. Explain what requirement of a pooling arrangement is illustrated in this

case? [2 points]

e. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Carlo? [2 points]

4. Compare the situation in question 2 and question 3 above. In particular, examine

the results you obtain in 2 (c) versus 3 (e). Explain carefully the ‘tradeoff’ that is

illustrated. [4 points]

RMI 2101

Summer 2017

Homework Assignment 9

Show ALL of your work in order to receive full credit for an answer.

1. Aron owns a $10,000 home and has a 10% chance of experiencing a fire in any

given year. Assume that only one fire per year can occur and that if a fire occurs,

her home is completely destroyed.

Suppose that Aron purchases a full insurance contract from Cooper Insurance for

an actuarially fair premium. This contract would pay all losses due to the fire.

Assume that Aron’s contract is the only insurance contract Cooper Insurance

sold.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell a contract to Aron? [1 point]

b. What is the actuarially fair premium [AFP] Cooper Insurance will charge

Aron in the coming year? [1 point]

c. What is the amount of risk Cooper Insurance faces if they have Aron as

their only customer? [2 points]

2. Barbara, who owns the same type of house and faces the same probability

distribution of losses as Aron, also purchase full insurance for an actuarial fair

premium from Cooper Insurance. We assume that the two houses are

independent of each other….meaning that if one house has a fire, this has no

impact on the probability of the other house having a fire.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to both Aron and Barbara? [2 points]

b. What is the expected loss or expected payout for Cooper Insurance if they

sell contracts to both Aron and Barbara? [1 point]

c. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Barbara? [2 points]

d. Briefly explain the benefits to Cooper Insurance as the number of

insurance contracts sold increases? [2 points]

3. Now suppose Carlo owns a $20,000 home and has a 10% chance of experiencing

a fire in any given year. Assume as before that the fire will result in a total loss.

Suppose Cooper Insurance offers Aron and Carlo the same insurance contract

and charges them the same premium….in other words, they put Aron and Carlo

into the same risk pool.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to Aron and Carlo? [2 points]

b. What premium must Cooper Insurance charge each of Aron and Carlo if

they want to ‘break-even’? [1 point]

c. Will Aron purchase this contract if he is charged the ‘break-even’

premium? Will Carlo purchase this contact if he is charged the ‘breakeven’

premium? Briefly explain your reason. [2 points]

d. Explain what requirement of a pooling arrangement is illustrated in this

case? [2 points]

e. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Carlo? [2 points]

4. Compare the situation in question 2 and question 3 above. In particular, examine

the results you obtain in 2 (c) versus 3 (e). Explain carefully the ‘tradeoff’ that is

illustrated. [4 points]

RMI 2101

Summer 2017

Homework Assignment 9

1. Aron owns a $10,000 home and has a 10% chance of experiencing a fire in any

given year. Assume that only one fire per year can occur and that if a fire occurs,

her home is completely destroyed.

Suppose that Aron purchases a full insurance contract from Cooper Insurance for

an actuarially fair premium. This contract would pay all losses due to the fire.

Assume that Aron’s contract is the only insurance contract Cooper Insurance

sold.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell a contract to Aron? [1 point]

b. What is the actuarially fair premium [AFP] Cooper Insurance will charge

Aron in the coming year? [1 point]

c. What is the amount of risk Cooper Insurance faces if they have Aron as

their only customer? [2 points]

2. Barbara, who owns the same type of house and faces the same probability

distribution of losses as Aron, also purchase full insurance for an actuarial fair

premium from Cooper Insurance. We assume that the two houses are

independent of each other….meaning that if one house has a fire, this has no

impact on the probability of the other house having a fire.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to both Aron and Barbara? [2 points]

b. What is the expected loss or expected payout for Cooper Insurance if they

sell contracts to both Aron and Barbara? [1 point]

c. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Barbara? [2 points]

d. Briefly explain the benefits to Cooper Insurance as the number of

insurance contracts sold increases? [2 points]

3. Now suppose Carlo owns a $20,000 home and has a 10% chance of experiencing

a fire in any given year. Assume as before that the fire will result in a total loss.

Suppose Cooper Insurance offers Aron and Carlo the same insurance contract

and charges them the same premium….in other words, they put Aron and Carlo

into the same risk pool.

a. What is the probability distribution of total losses for Cooper Insurance

if they sell contracts to Aron and Carlo? [2 points]

b. What premium must Cooper Insurance charge each of Aron and Carlo if

they want to ‘break-even’? [1 point]

c. Will Aron purchase this contract if he is charged the ‘break-even’

premium? Will Carlo purchase this contact if he is charged the ‘breakeven’

premium? Briefly explain your reason. [2 points]

d. Explain what requirement of a pooling arrangement is illustrated in this

case? [2 points]

e. What is the amount of risk Cooper Insurance faces if they sell contracts to

both Aron and Carlo? [2 points]

4. Compare the situation in question 2 and question 3 above. In particular, examine

the results you obtain in 2 (c) versus 3 (e). Explain carefully the ‘tradeoff’ that is

illustrated. [4 points]