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Sample Actuarial Problems

Apply your math skills to actuarial exam questions.

Actuaries earn professional credentials by passing a series of examinations. This online exam is designed to give you an idea of the types of questions you might encounter on the preliminary actuarial examinations administered by the Casualty Actuarial Society and Society of Actuaries. The sample problems are actual questions from prior exams, but they do not cover all the topics or all levels of difficulty.

Answer the five multiple choice questions below, then click submit to see your results.

1

An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders who have only a homeowners policy will renew next year. The company estimates that 80% of policyholders who have both an auto and a homeowners policy will renew at least one of those policies next year.

Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowners policy, and 15% of policyholders have both an auto and a homeowners policy.

Using the company's estimates, calculate the percentage of policyholders that will renew at least one policy next year.

2

An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company's employees that choose coverages A, B, and C are 1?4 , 1?3 and 5?12 respectively.

Determine the probability that a randomly chosen employee will choose no supplementary coverage.

3

Let T1 be the time between a car accident and reporting a claim to the insurance company. Let T2 be the time between the report of the claim and payment of the claim. The joint density function of T1 and T2, f(t1, t2), is constant over the region 0 < t1 < 6, 0< t2 < 6, t1 + t2 < 10, and zero otherwise. Determine E[T1 + T2], the expected time between a car accident and payment of the claim.

4

An auto insurance company insures an automobile worth 15,000 for one year under a policy with a 1,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function

What is the expected claim payment?

5

The future lifetimes (in months) of two components of a machine have the following joint density function:

What is the probability that both components are still functioning 20 months from now?