Answered

Stephanie has a homeowners insurance policy for her $355,000 home with an annual premium of $0.42 per $100 of value and a deductible of $500. Under this policy, in the event of a major mishap, Stephanie would have a total annual out-of-pocket expense of [($355,000 / 100) · $0.42] + $500 = $1,991.



Stephanie would like to lower her premium by increasing her deductible. If Stephanie wants to increase her deductible to $1,000, what annual premium would result in an annual out-of-pocket expense that is about the same as her current plan?


A$0.16 per $100 of value
B$0.28 per $100 of value
C$0.35 per $100 of value
D$0.46 per $100 of value

Verified Answer :

  • Define the total cost formula as \(\text{Total} = \left(\frac{V}{100} \times R\right) + D\).
  • Substitute the target total expense \(\$1,991\), new deductible \(\$1,000\), and home value \(\$355,000\) into the formula.
  • Calculate the new premium rate \(R_{new} = \frac{1,991 - 1,000}{3,550} \approx 0.279\).
  • Match the result to the provided options to conclude that \(\$0.28\) per \(\$100\) is the best choice, which corresponds to option \(\boxed{B}\).

Explanation

  1. Analyzing the Cost Components

To find the appropriate new premium rate for Stephanie's homeowners insurance, we first need to understand the relationship between the total annual out-of-pocket expense, the property value, the premium rate, and the deductible. The total cost is given by the sum of the annual premium and the insurance deductible. Since the premium is calculated per \$\$100\$ of the home's value, the annual premium is defined as \(\left(\frac{\text{Home Value}}{100}\right) \times \text{Premium Rate}\). Therefore, the total out-of-pocket expense can be expressed as:
\[\text{Total Expense} = \left(\frac{V}{100} \times R\right) + D\]
where \(V\) is the home value, \(R\) is the premium rate per \(\$100\), and \(D\) is the deductible. We are given that for a home value \(V = \$355,000\), a deductible \(D = \$500\), and a premium rate \(R = \$0.42\), the total expense is \(\$1,991\).

  1. Solving for the New Premium Rate

Stephanie wants to increase her deductible \(D_{new}\) to \(\$1,000\) while maintaining the same total out-of-pocket expense of \(\$1,991\). We set up the equation for the new plan using the same formula:
\[1,991 = \left(\frac{355,000}{100} \times R_{new}\right) + 1,000\]
This simplifies to:
\[1,991 = 3,550 \times R_{new} + 1,000\]
To find \(R_{new}\), we subtract the new deductible from the total expense:
\[3,550 \times R_{new} = 1,991 - 1,000\]
\[3,550 \times R_{new} = 991\]
Solving for \(R_{new}\) gives:
\[R_{new} = \frac{991}{3,550} \approx 0.27915\]
Rounding this to two decimal places, we find that the required premium rate is approximately \(\$0.28\) per \(\$100\) of home value.

  1. Conclusion

Comparing our result to the given options:

A\(\$0.16\) per \(\$100\) of value
B\(\$0.28\) per \(\$100\) of value
C\(\$0.35\) per \(\$100\) of value
D\(\$0.46\) per \(\$100\) of value

Our calculated value of \(\$0.28\) per \(\$100\) matches option (b). Thus, the correct answer is B.

Examples

Insurance companies often adjust premium rates when policyholders choose higher deductibles. This problem demonstrates how to calculate the necessary premium rate to maintain a consistent total cost, which is a common task in financial planning and risk management. By understanding these algebraic relationships, individuals can better manage their budget and choose the insurance plan that best balances their immediate costs and long-term financial security.

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