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 policy 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 policy and a homeowners policy.

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

• 20%
• 29%
• 41%
• 53%
• 70%

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

Solution: D

Let

[math]A[/math] = event that a policyholder has an auto policy

[math]H[/math] = event that a policyholder has a homeowners policy

Then based on the information given,

[[math]] \begin{align*} &\operatorname{P}( A ∩ H ) = 0.15 \\ &\operatorname{P}( A ∩ H^c ) = \operatorname{P}( A ) − \operatorname{P}( A ∩ H ) = 0.65 − 0.15 = 0.50 \\ &\operatorname{P}( A^c ∩ H ) = \operatorname{P}( H ) − \operatorname{P}( A ∩ H ) = 0.50 − 0.15 = 0.35 \\ \end{align*} [[/math]]

and the portion of policyholders that will renew at least one policy is given by

[[math]] 0.4 \operatorname{P}( A ∩ H^c ) + 0.6 \operatorname{P}( A^c ∩ H ) + 0.8 \operatorname{P}( A ∩ H ) = ( 0.4 )( 0.5 ) + ( 0.6 )( 0.35 ) + ( 0.8 )( 0.15 ) = 0.53 [[/math]]

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