An actuary at an insurance company estimates from exsiting data that on a $1000 policy an average 1 and 100 policyholders will file a 20000 claim. in average of 1 in 200 policyolders will file a 50000 claim in average 1 in 500 policyholders will file a 100000 claim.
a. what is the expected value to the company for each policy sold.
b. if the company sells 100000 policies can it expect a profit. explain your assumptions of this calculation.
To find the expected value for each policy sold, we need to calculate the expected claim amount for each claim type and then multiply it by the probability of that claim occurring.
a. Expected value calculation:
1. For the $20,000 claim:
- Probability of this claim occurring = 1 in 100 policyholders = 1/100 = 0.01
- Expected claim amount = $20,000
- Expected claim payout = Probability * Expected amount = 0.01 * $20,000 = $200
2. For the $50,000 claim:
- Probability of this claim occurring = 1 in 200 policyholders = 1/200 = 0.005
- Expected claim amount = $50,000
- Expected claim payout = Probability * Expected amount = 0.005 * $50,000 = $250
3. For the $100,000 claim:
- Probability of this claim occurring = 1 in 500 policyholders = 1/500 = 0.002
- Expected claim amount = $100,000
- Expected claim payout = Probability * Expected amount = 0.002 * $100,000 = $200
Now, we can calculate the overall expected value for each policy sold:
Expected value = Expected payout for all claim types
= Expected payout for $20,000 claim + Expected payout for $50,000 claim + Expected payout for $100,000 claim
= $200 + $250 + $200
= $650
Therefore, the expected value to the company for each policy sold is $650.
b. To determine if the company can expect a profit by selling 100,000 policies, we need to compare the expected value to the total revenue generated from selling those policies.
1. Revenue calculation:
- Number of policies sold = 100,000
- Revenue per policy = $1,000
- Total revenue = Number of policies sold * Revenue per policy = 100,000 * $1,000 = $100,000,000
2. Profit calculation:
- Profit = Total revenue - Total expected claim payout
- Total expected claim payout = Expected payout per policy * Number of policies sold
= $650 * 100,000
= $65,000,000
Now, let's compare the profit to zero to see if the company can expect a profit:
Profit = Total revenue - Total expected claim payout
= $100,000,000 - $65,000,000
= $35,000,000
Since the profit is positive ($35,000,000), the company can expect a profit from selling 100,000 policies.
Assumptions in this calculation:
1. The estimates made by the actuary are accurate and reliable.
2. The claim occurrences follow the stated probabilities for the different claim types.
3. The prices of the policies are set appropriately to cover the expected claim payouts and other expenses.
4. There are no significant external factors impacting claim occurrences or any other unexpected costs.