Problem 43 - Applied Calculus

Question

You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for $$\$ 375 .$$ The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the extended warranty are $$\$ 150$$ at the end of the first year of the extension and $$\$ 250$$ at the end of the second year of the extension. The interest rate is \(5 \%\) per year, compounded annually. Should you buy the extended warranty? Explain.

Step-by-Step Solution

Answer

Do not buy the extended warranty.

Step 1: Calculate Present Value of Year 1 Expense

The expense of \$150 incurred at the end of the first year should be calculated in present value terms using the formula for Present Value (PV):\[ PV = \frac{FV}{(1 + r)^n} \]where \( \text{FV} = 150 \) is the future value, \( r = 0.05 \) is the interest rate, and \( n = 1 \) is the number of years.Apply the formula:\[ PV_{1} = \frac{150}{(1 + 0.05)^1} = \frac{150}{1.05} \approx 142.86 \]

Step 2: Recall relevant definitions and theorems

State the key definitions and theorems to be used.

Step 3: Verify the required algebraic properties

Check closure, identity, inverses, and other required axioms.

Step 4: Construct the argument or example

Build the complete argument or construct the required example.

Step 5: State the conclusion

Summarize the result.

Step 6: Conclude with the answer

Do not buy the extended warranty.

Key Concepts

extended warranty

When considering an extended warranty, you are essentially weighing the protection and benefits it offers against its cost. In this scenario, the extended warranty for your car costs $375 and promises coverage for two additional years after the initial warranty expires.

This means for two years, any expenses that arise, which would typically be out-of-pocket costs for repairs, might instead be covered under the warranty. Understanding the potential expenses during this period helps in deciding whether the warranty is worth the investment. Here, you're anticipating costs of $150 and $250 in the two years after the initial warranty expires. If these costs exceed the price of the warranty when adjusted to their present value, then the warranty is a good investment.

To decide, you need to calculate these future expenses' present value, a vital financial analysis tool.

compounded interest

Compounded interest is a critical concept when calculating the present value of future expenses. It affects how money grows over time or, in reverse, how future money is valued today. In essence, compounding refers to earning "interest on interest," which can significantly impact the value over time.

In the problem, the interest rate is compounded annually at 5%. This rate influences the discounting of future expenses to present value. The formula for compounded interest follows:

• Future Value (FV) = Present Value (PV) × (1 + interest rate)

When dealing with future expenses, we use this concept inversely to find out what those future expenses are worth today. It's about calculating today's value of a payment made in the future. An understanding of the workings of compounded interest is essential when performing these calculations, ensuring we make sound financial decisions.

future value calculation

Calculating the future value is central to understanding how much money will be worth at a future date, given an interest rate. Future value helps determine how much an investment made today will be worth in the future.

In our context, the expenses of $150 and $250 represent future costs that the extended warranty would cover. Calculating their future value in context doesn't change them from $150 and $250, but understanding future value helps reverse-calculate their present value. The formula we use, solving for present value, is essentially rearranging the future value calculation formula:

• Present Value (PV) = Future Value (FV) / (1 + interest rate)^n

Here, you adapt the formula to discount future amounts, seeing if the sum of their present values outweighs or seems lesser than the warranty's cost. This reverse approach, discounting back from future value to present value, ensures you pay fair value depending on expectations from the warranty.

annual interest rate

The annual interest rate, stated here as 5%, is pivotal when calculating both present and future values of amounts. This rate influences how money is perceived and valued over time.

A higher interest rate would increase the discounting effect, lowering the present value of future expenses, and vice versa. It reflects the earnings capability of consideration, helping individuals make repayment or investment decisions. Essentially, the interest rates show how much more future money is worth relative to current money.

In this scenario, the annual interest rate helps adjust the future expenses of $150 and $250 to their present-day value. By applying the rate of 5%, the scenario accurately evaluates whether the cost of the warranty is justified based on anticipated future repairs, ensuring sound financial judgment when making the final decision.