1. Chris owns a $350,000 house and has 2% chance of experiencing a fire in any given year. Assume that only one fire per year can occur and that is a fire occurs, the house is completely destroyed.
Suppose that Chris purchases a full contract from Allstate Insurance Company for an actuarially fair premium. This contract would pay all losses due to the fire. Assume that Chris’s contract is the only contract that Allstate Insurance Company sold.
a.) What is the probability distribution of total losses for Allstate Insurance Company if they sell a contract to Chris?
b.) What is the actuarially fair premium Allstate Insurance Company will charge Chris in the coming year?
c.) What is the amount of risk Allstate Insurance Company faces if they have Chris as their only customer?
2.) Kate, who owns the same type of house and faces the same probability distribution of losses as Chris, also purchases full insurance for an actuarially fair premium from Allstate Insurance Company. We assume that the two houses are independent of each other. In other words, 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 Allstate Insurance Company if they sell contracts to both Chris and Kate?
b.) What is the expected loss or expected payout for Allstate Insurance Company if they sell contracts to both Chris and Kate?
c.) What is the amount of risk Allstate Insurance Company faces if they sell contracts to both Chris and Kate?
d.) Briefly explain the benefit(s) to Allstate Insurance Company as the number of insurance contracts sold increases?
3. Now suppose Matt owns a $500,00 house and has a 3% chance of experiencing a fire in any given year. Assume as before that the fire will result in a total loss. Suppose the Allstate Insurance Company offers Chris and Matt the same insurance contract and charges them the same premium. In other words, they put Chris Matt into the same risk pool.
a.) What is the probability distribution of total losses for Allstate Insurance Company if they sell contracts to Chris and Matt?
b.) What premium must Allstate Insurance Company charge each of Chris and Matt if they want to ‘break even’
c.) Will Chris purchase this contract if he is charged the ‘break-even’ premium? Will Matt purchase this contract if he is charged the ‘break-even’ premium? Briefly explain your season.
d.) What is the amount of risk Allstate Insurance Company faces if they sell contracts to both Chris and Matt?
e.) What requirement of a pooling arrangement is illustrated in this case?
4.) Compare the situation in question 2 and 3 above. In particular, examine the results you obtain in 1( c ), 2 ( c ) and 3 (d ). Explain carefully the tradeoff’ that is illustrated.
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