An insurance company has a portfolio of 100 policyholders. 25 of the policyholders have loss distributions...

An insurance company has a portfolio of 100 policyholders. 25 of the policyholders have loss distributions with mean 50and variance 10, and 75 of the policyholders have loss distributions with mean 10and variance 30. Assuming the insurance company charges all policyholders the same premium, what premium should the company charge so that there is a 95% chance that aggregate claims will be less than the total premium collected?

Expert Solution

Expected loss of all policyholders = 25 * 50 + 75 * 10 = 2000

Variance of loss of all policyholders = 25 * 10 + 75 * 30 = 2500

Standard deviation of loss of all policyholders = = 50

By Central limit theorem, we can assume that Loss of all policyholders X ~ N(2000 , 50)

Let k be the total premium charged to all 100 policyholders such that,

P(X < k) = 0.95

=> k =

=> k = 2000 + 1.645 * 50 = $2082.25

Premium per policyholders = 2082.25 / 100 = $20.82

The company should charge $20.82 per policyholder so that there is a 95% chance that aggregate claims will be less than the total premium collected.

orchestra answered 2 years ago

An auto insurance company concludes that 30% of policyholders with only collision coverage will have a...

An auto insurance company concludes that 30% of policyholders with only collision coverage will have a claim next year, 40% of policyholders with only comprehensive coverage will have a claim next year and 50% of policyholders with both collision and comprehensive coverage will have a claim next year. Records show 60% of policyholders have collision coverage 70% have comprehensive coverage and all policyholders have at least one of these coverages. Calculate the percentage of policyholders expected to have an accident...

An insurance company’s projected loss ratio is 73%, its commission payments and dividends to policyholders is...

An insurance company’s projected loss ratio is 73%, its commission payments and dividends to policyholders is 12%, and its yield on investments is 6%. If it has a positive (profitable) operating ratio, what can you say about its loss adjustment expense ratio?

An insurance company has 20, 000 policyholders. The amount claimed yearly by a policyholder has an...

An insurance company has 20, 000 policyholders. The amount claimed yearly by a policyholder has an expected value of $150 and a standard deviation of $750. Use the central limit theorem (CLT) to approximate the probability that the the total amount claimed in the coming year will be larger than 3.3 million dollars.

A fire insurance company has 3,840 policyholders. The probability that a policyholder files at least one...

A fire insurance company has 3,840 policyholders. The probability that a policyholder files at least one claim in a given year is 1/1,200. Approximate the probability that more than 6 policyholders file at least one claim in a given year.

A health insurance company wants to know the percentage of its policyholders, who have tried alternative...

A health insurance company wants to know the percentage of its policyholders, who have tried alternative (e.g. herbal) treatments. A random sample of 25 of the company’s policyholders were asked whether or not (Yes or No) they have ever tried such treatment. Below are their responses: Yes Yes No Yes Yes Yes Yes No Yes Yes Yes No No No No No No Yes No No Yes Yes No Yes No a. What is the point estimate of the corresponding...

An insurance policy has an ordinary deductible of 100 and a maximum covered loss of 10,000....

An insurance policy has an ordinary deductible of 100 and a maximum covered loss of 10,000. You observe the following six payments: 100 200 500 600 1000 5000 In addition, there are three payments at the limit. You ﬁt a single parameter Pareto distribution with θ = 50 to the ground up distribution using maximum likelihood. Determine the estimated parameterα.

Accounting Ethics: A health insurance company sold insurance policies that allow policyholders to obtain services from...

Accounting Ethics: A health insurance company sold insurance policies that allow policyholders to obtain services from selected hospitals. The policies fully disclosed the names of these hospitals. However, the insurance company did not mention that the hospitals did not have the capability to perform certain costly procedures, such as heart bypass operations. As a result, if a policyholder one day developed a severe heart blockage, the patient would not have insurance coverage to pay for such a heart bypass procedure....

In early March an insurance company portfolio manager has a stock portfolio of $500 million with...

In early March an insurance company portfolio manager has a stock portfolio of $500 million with a portfolio beta of 1.20. The portfolio manager plans on selling the stocks in the portfolio in June to be able to pay out funds to policyholders for annuities, and is worried about a fall in the stock market. In April, the CME Group S&P 500 mini= Futures contract index is 2545 for a June futures contract ($50 multiplier for this contract). Suppose in...

Diagrams of the normal distribution are almost mandatory A health insurance company charges policyholders a $2450...

Diagrams of the normal distribution are almost mandatory A health insurance company charges policyholders a $2450 annual premium for health insurance for hospitalization. The company estimates that each time a patient is hospitalized costs the company $3600. Furthermore, they have estimated that 85% of patients will not be hospitalized, 10% will be hospitalized once a year, and no one will be hospitalized more than twice. (a) Find the insurance company’s expected profit per policyholder. (b) What is the expected profit...

A large automobile insurance company selected samples of single and married male policyholders and recorded the...

A large automobile insurance company selected samples of single and married male policyholders and recorded the number who made an insurance claim over the preceding three-year period. Single Policyholders Married Policyholders n1 = 900 n2 = 400 number making claims = 144 number making claims = 28 (a) Use α = 0.05. Test to determine whether the claim rates differ between single and married male policyholders. State the null and alternative hypotheses. (Let p1 = claim rate for single male...

Question