An insurance agent offers his clients auto insurance, homeowners insurance and renters insurance. The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive. The profile of the agent’s clients is as follows:

1. 17% of the clients have none of these three products.
2. 64% of the clients have auto insurance.
3. Twice as many of the clients have homeowners insurance as have renters insurance.
4. 35% of the clients have two of these three products.
5. 11% of the clients have homeowners insurance, but not auto insurance.

Calculate the percentage of the agent’s clients that have both auto and renters insurance.

• 7%
• 10%
• 16%
• 25%
• 28%

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

Let H be the percentage of clients with homeowners insurance and R be the percentage of clients with renters insurance. Because 36% of clients do not have auto insurance and none have both homeowners and renters insurance, we calculate that 8% (36% – 17% – 11%) must have renters insurance, but not auto insurance. (H – 11)% have both homeowners and auto insurance, (R – 8)% have both renters and auto insurance, and none have both homeowners and renters insurance, so (H + R – 19)% must equal 35%. Because H = 2R, R must be 18%, which implies that 10% have both renters and auto insurance.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.