1. Jack owns a $500,000 house and has a 4% 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 Jack 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 Jack'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 Jack? (1 point)

b. What is the actuarially fair premium [AFP] Lemonade Insurance Company will charge Jack in the coming year? (1 point)

c. What is the amount of risk Lemonade Insurance Company faces if they have Jack as their only customer? (2 points)

2. Paul, who owns the same type of house and faces the same probability distribution of losses as Jack, 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 Jack and Paul? (1 point)

b. What is the expected loss or expected payout for Lemonade Insurance Company if they sell contracts to both Jack and Paul? (1 point)

c. What is the amount of risk Lemonade Insurance Company faces if they sell contracts to both Jack and Paul? (2 points)

d. Briefly explain the benefit(s) to Lemonade Insurance Company as the number of insurance contracts sold increases? (1 point)