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Calculating Future Value of Insurance Savings. Beverly and Kyle Nelson currently insure their cars with separate companies, paying \(\$ 650\) and \(\$ 575\) a year. If they insured both cars with the same company, they would save 10 percent on the annual premiums. What would be the future value of the annual savings over 10 years based on an annual interest rate of 6 percent? (Obj. 6 )

Short Answer

Expert verified
The future value of the savings is \(\$1614.59\).

Step by step solution

01

Calculate Total Current Premium Cost

First, we need to find out the combined amount Beverly and Kyle are currently paying for their car insurance. Add the individual premiums: \(650 + 575 = 1225\).
02

Calculate Annual Savings

Next, we determine the savings if both cars were insured with the same company, which offers a 10% discount. The annual savings is 10% of the combined current premiums: \(0.10 \times 1225 = 122.50\).
03

Set Up Future Value Formula

In order to find the future value of the annual savings, we use the Future Value of an Annuity formula: \(FV = P \times \frac{(1 + r)^n - 1}{r}\), where \(P\) is the annual savings, \(r\) is the annual interest rate, and \(n\) is the number of years.
04

Insert Values into Future Value Formula

Plugging in the values: annual savings \(P = 122.50\), interest rate \(r = 0.06\), and number of years \(n = 10\), gives us: \[FV = 122.50 \times \frac{(1 + 0.06)^{10} - 1}{0.06}\].
05

Compute Future Value

Calculate \((1 + 0.06)^{10} = 1.790847\) and substitute back into the formula: \[(1.790847 - 1)/0.06 = 13.180783\]. Now, multiply by the annual savings: \(122.50 \times 13.180783 = 1614.59\). The future value of the savings over 10 years is \(\$1614.59\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Insurance Savings
Insurance savings refer to the reduction in yearly premiums when using strategies such as combining policies or finding more affordable coverage. In the example with Beverly and Kyle, they can reduce their overall insurance cost by opting for a joint insurance policy. By insuring their vehicles with the same provider, they are eligible for a 10% discount on their premiums.
This leads to an annual saving of $122.50. The logic here is to minimize expenses in one of the common financial obligations—car insurance.
Over time, these savings can accumulate to a notable amount, especially when augmented by smart investment strategies like reinvesting savings with compounding interest.
Annuity Formula
An annuity is a series of equal payments made at regular intervals. To calculate the future value of these payments, one uses the Future Value of an Annuity formula. For Beverly and Kyle, each annual saving due to their insurance discount works like an annuity payment.
The formula to find the future value (FV) is given by:
  • \[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
  • Where:
    • \( P \) is the annual payment (in this case, the savings).
    • \( r \) is the annual interest rate.
    • \( n \) is the number of periods (years).
Using this method helps to determine how a consistent saving grows over time with a specified interest rate, illustrating the power of compounding.
Interest Rate
The interest rate plays a crucial role in the calculation of future value for savings or investments. It determines how much the saved funds will grow each year. In the Nelsons' case, an interest rate of 6% is used to grow their annual saving over 10 years.
Interest rates can vary depending on economic conditions and the financial products available. Here, the rate is compounded annually, meaning the interest is calculated once per year and added to the principal amount, which then earns interest the following year.
This compounding effect means that even a small initial saving can grow significantly over time, especially when paired with a consistent savings strategy.
Financial Planning
Financial planning involves managing income, savings, and investments to achieve financial stability and goals. Savings from reduced insurance premiums can be a key component of financial planning.
For Beverly and Kyle, the decision to combine insurance policies and save annually is a strategic financial decision. By reinvesting these savings with a 6% interest rate, they can secure a future value of over $1614.
Key aspects of financial planning include assessing current financial situations, identifying financial goals, and developing a plan to achieve them.
  • Regular review and adjustment of the plan may be necessary to stay on track and account for changes in income or expenses.
  • Being mindful of opportunities for savings and investment is critical to growing personal wealth over time.
Through systematic planning, individuals like Beverly and Kyle can maximize their financial outcomes, ensuring a secure and prosperous future.

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Most popular questions from this chapter

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