What is the probability distribution of total losses for Lemonade Insurance Company if they sell contracts to both Jessica and Rainie?

On-time Submission – Submit to Turnitin on Friday, November 11, 2022 no later than 11:59 PM

Late Submission – Submit to Turnitin on Saturday, November 12, 2022 no later than 11:59 PM

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1. Jessica owns a $450,000 house and has an 5% 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 Jessica 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 Jessica’s contract is the only insurance contract Lemonade Insurance Company sold.

1. What is the probability distribution of total losses for Lemonade Insurance Company if they sell a contract to Jessica? (2 points)

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

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

2. Rainie, who owns the same type of house and faces the same probability distribution of losses as Jessica, 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.

1. What is the probability distribution of total losses for Lemonade Insurance Company if they sell contracts to both Jessica and Rainie? (2 points)

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

1. What is the amount of risk Lemonade Insurance Company faces if they sell contracts to both Jessica and Rainie? (2 points)

1. 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 $850,000 house and has an 5% 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 Jessica and Ben the same insurance contract and charges them the same premium. In other words, they put Jessica and Ben into the same risk pool.
4. What is the probability distribution of total losses for Lemonade Insurance Company if they sell contracts to Jessica and Ben? (2 points)

1. What premium must Lemonade Insurance Company charge each of Jessica and Ben if they want to ‘break even’? (2 points)

1. Will Jessica 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)

1. What is the amount of risk Lemonade Insurance Company faces if they sell contracts to both Jessica and Ben? (2 points)

4. 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)