1. Rainie owns a $200,000 house and has an 8% chance of experiencing a fire in any given year. Assume that only one fire per year can occur and that if a fire occurs, the house is completely destroyed. Suppose that Rainie purchases a full insurance contract from Lemonade Insurance Company for an actuarially fair premium. This contract would pay all losses due to the fire. Assume that Rainie's contract is the only insurance contract Lemonade Insurance Company sold. a. What is the probability distribution of total losses for Lemonade Insurance Company if they sell a contract to Rainie? (2 points) b. What is the actuarially fair premium [AFP] Lemonade Insurance Company will charge Rainie in the coming year? (1 point) C. What is the amount of risk Lemonade Insurance Company faces if they have Rainie as their only customer? (2 points) 2. Cat, who owns the same type of house and faces the same probability distribution of losses as Rainie, also purchases full insurance for an actuarially fair premium from Lemonade 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 Lemonade Insurance Company if they sell contracts to both Rainie and Cat? (2 points) b. What is the expected loss or expected payout for Lemonade Insurance. Company if they sell contracts to both Rainie and Cat? (1 point) C. What is the amount of risk Lemonade Insurance Company faces if they sell contracts to both Rainie and Cat? (2 points) d. Briefly explain the benefit(s) to Lemonade Insurance Company as the number of insurance contracts sold increases? (2 points) 3. Now suppose Ben owns a $600,000 house and has an 8% chance of experiencing a fire in any given year. Assume as before that the fire will result in a total loss. Suppose the Lemonade Insurance Company offers Rainie and Ben the same insurance contract and charges them the same premium. In other words, they put Rainie and Ben into the same risk pool. a. What is the probability distribution of total losses for Lemonade Insurance Company if they sell contracts to Rainie and Ben? (2 points) b. What premium must Lemonade Insurance Company charge each of Rainie and Ben if they want to break even'? (2 points) c. Will Rainie purchase this contract if she is charged the break-even' premium? Will Ben purchase this contact if he is charged the 'break- even' premium? Briefly explain your reason. (2 points) d. What is the amount of risk Lemonade Insurance Company faces if they sell contracts to both Rainie and Ben? (2 points) BONUS: 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. (4 points)
