Problem 46 - 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; its cost exceeds the present value of the covered expenses.

Step 1: Calculate the Present Value of Expenses

First calculate the present value (PV) of the expenses that would be covered by the extended warranty. The expenses are \(150 at the end of the first year and \)250 at the end of the second year. The present value is calculated by discounting these amounts back to their present cost using the given interest rate of 5%. The formula for present value is: \[ PV = \frac{C}{(1 + r)^n} \]where \( C \) is the cash flow, \( r \) is the interest rate, and \( n \) is the number of periods. For \(150 in Year 1: \[ PV_1 = \frac{150}{(1 + 0.05)^1} = \frac{150}{1.05} \approx 142.86 \]For \)250 in Year 2: \[ PV_2 = \frac{250}{(1 + 0.05)^2} = \frac{250}{1.1025} \approx 226.76 \]Add the present values of both years:\[ PV_{total} = PV_1 + PV_2 \approx 142.86 + 226.76 = 369.62 \]

Step 2: Compare Present Value with Warranty Cost

Now compare the total present value of expenses, which is $369.62, with the cost of the extended warranty, which is $375. Since the present value of the expenses ($369.62) is less than the cost of the extended warranty ($375), purchasing the warranty is not cost-effective.

Key Concepts

Extended Warranty

When buying a car, an extended warranty can provide additional protection beyond the standard warranty period. It covers potential repair costs that may occur after the manufacturer's warranty expires. The concept here is to decide whether the peace of mind offered by the extended warranty is worth the extra cost. With an extended warranty, you may cover costly repairs or replacements, but it requires an up-front payment. In this scenario, the extended warranty costs $375 and covers repair expenses estimated at $150 after the first year and $250 after the second year. The decision to buy depends on whether these future repair costs exceed the price of the warranty when considering interest on money or Present Value.

Interest Rate

Interest rates are a crucial factor in financial decisions. They represent the cost of borrowing money or the reward for saving. In this case, an interest rate of 5% per year compounded annually means any future cash flows need to be adjusted to their equivalent value today, which is known as the Present Value. Calculating present value helps in understanding whether the current cost of an investment, such as an extended warranty, is justified by future savings or returns. Interest rate impacts how much future values decline in today's terms, guiding decisions about spending, investments, or saving for future expenses.

Discounting Cash Flows

Discounting cash flows is a method used to calculate the present value of future cash flows. It helps determine what future amounts of money are worth in today's dollars. When considering expenses that will be incurred in the future, like those covered by an extended warranty, discounting reflects these costs accurately at today's value after accounting for the time value of money. For example, the costs of $150 at the end of one year and $250 at the end of two years are discounted using the provided 5% interest rate. The calculations show their present values as approximately $142.86 and $226.76 respectively. These figures help compare the true cost of the future expenses to the cost of the extended warranty.

Cost-Benefit Analysis

Cost-benefit analysis is a process used to evaluate the total expected costs versus the benefits of one or more actions. This analysis helps in making informed decisions by comparing financial outlays and expected returns or savings. In the car warranty example, you're essentially comparing the cost of the extended warranty ($375) with the present value of anticipated expenses ($369.62) that would be covered by the warranty. Since the calculated total expenses in today's terms are less than the cost of the warranty, it suggests that without extraordinary risks, the additional warranty purchase may not be financially justifiable. Making a decision like this involves understanding the expected value of what you receive against what you pay.