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.

a. To calculate the expected value for each policy sold, we need to multiply the probability of each claim amount by their respective claim amounts and sum them up.

For the $20,000 claim:
Probability = 1 in 100 policyholders = 1/100 = 0.01
Claim amount = $20,000

Expected value for $20,000 claim = Probability * Claim amount = 0.01 * $20,000 = $200

For the $50,000 claim:
Probability = 1 in 200 policyholders = 1/200 = 0.005
Claim amount = $50,000

Expected value for $50,000 claim = Probability * Claim amount = 0.005 * $50,000 = $250

For the $100,000 claim:
Probability = 1 in 500 policyholders = 1/500 = 0.002
Claim amount = $100,000

Expected value for $100,000 claim = Probability * Claim amount = 0.002 * $100,000 = $200

Adding up the expected values for each claim amount:
Expected value for each policy sold = $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 from selling 100,000 policies, we need to consider the premium charged for each policy and compare it to the expected value.

Let's assume the premium charged for each policy is $700. If the company sells 100,000 policies, the total premium income will be $700 * 100,000 = $70,000,000.

To check if the company can expect a profit, we need to subtract the total expected claim amount from the premium income:

Total expected claim amount = Expected value for each policy sold * Number of policies sold = $650 * 100,000 = $65,000,000

Profit = Premium income - Total expected claim amount = $70,000,000 - $65,000,000 = $5,000,000

Based on these assumptions, the company can expect a profit of $5,000,000 if they sell 100,000 policies.

a. To calculate the expected value of each policy sold, we need to multiply the probability of each claim amount by the corresponding claim amount and sum them up.

First, let's calculate the expected value for a $20,000 claim:
The average probability of filing a $20,000 claim is 1 in 100 policyholders.
So, the probability of NOT filing a $20,000 claim is 99 in 100 policyholders.
The expected value for a $20,000 claim is therefore: (1 / 100) * $20,000 = $200.

Next, let's calculate the expected value for a $50,000 claim:
The average probability of filing a $50,000 claim is 1 in 200 policyholders.
The probability of NOT filing a $50,000 claim is 199 in 200 policyholders.
The expected value for a $50,000 claim is therefore: (1 / 200) * $50,000 = $250.

Finally, let's calculate the expected value for a $100,000 claim:
The average probability of filing a $100,000 claim is 1 in 500 policyholders.
The probability of NOT filing a $100,000 claim is 499 in 500 policyholders.
The expected value for a $100,000 claim is therefore: (1 / 500) * $100,000 = $200.

Now, to find the overall expected value for each policy sold, we sum up the expected values for each claim amount:
Expected value = $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 when selling 100,000 policies, we need to compare the expected value to the company's costs and revenues associated with selling those policies.

Assumptions to consider in this calculation may include:
1. The cost of issuing and administering each policy.
2. The total premiums collected from selling 100,000 policies.
3. Any additional expenses or overhead costs.

If the expected value per policy (calculated above as $650) is greater than the total cost associated with issuing and administering each policy, including other expenses, the company can expect a profit.

It is important to note that the calculation of profit also depends on other factors, such as the accuracy of the actuary's estimates, the actual claims experience, and any other relevant costs or revenue sources for the company.