Question

A consumer faces the following decision: She can buy a computer for \(\$ 1000\) and \(\$ 10\) per month for Internet access for three years, or she can receive a \(\$ 400\) rebate on the computer (so that its cost is \(\$ 600\) ) but agree to pay \(\$ 25\) per month for three years for Internet access. For simplification, assume that the consumer pays the access fees yearly (i.e., \(\$ 10\) per month \(=\$ 120\) per year). a. What should the consumer do if the interest rate is 3 percent? b. What if the interest rate is 17 percent? c. At what interest rate will the consumer be indifferent between the two options?

Short answer

a) The consumer should choose the option with the lowest present value at an interest rate of 3%. \nb) The consumer should choose the option with the lowest present value at an interest rate of 17%. \nc) The interest rate at which the consumer would be indifferent between the two that can be found by setting the two formulas for \(PV_1\) and \(PV_2\) equal to each other and solving for \(r\).

Step-by-step solution

Calculate total costs for both options for interest rate of 3%

Firstly, we need to calculate the total cost for a period of three years for both scenarios, taking the present value of all costs into account with an interest rate of 3%. Remember that the present value \(PV\) of a future amount \(FV\) can be calculated using the formula \( PV = \frac{FV}{(1 + r)^n} \) where \(r\) is the interest rate and \(n\) is the number of periods. Scenario 1: \(PV_{1} = 1000 + \frac{120}{(1.03)} + \frac{120}{(1.03)^2} + \frac{120}{(1.03)^3}\). Scenario 2: \(PV_{2} = 600 + \frac{300}{(1.03)} + \frac{300}{(1.03)^2} + \frac{300}{(1.03)^3}\)

Compare the total costs for 3%

Once we have the present values for both scenarios, we compare them to see which one is smaller. The one with the smaller present value would be the favorable choice for the consumer at an interest rate of 3%.

Repeat step 1 and 2 for interest rate of 17%

For this part of the problem, we need to perform the same calculations as in steps 1 and 2 but using an interest rate of 17%.

Calculate the interest rate of indifferences

For the final part of the problem, we need to set the present value of both scenarios equal to each other and solve for the unknown interest rate. Setting \(PV_1 = PV_2\), we insert the formulas from step 1, but in this case we don't know the interest rate \(r\). Solving this equation gives the interest rate at which the consumer is indifferent between the two options. This requires knowledge of algebra and possibly iterative methods to solve.

Key concepts

Understanding Present Value

Present value is a fundamental concept in finance that helps us compare different cash flows occurring over time. Essentially, it allows us to determine how much a future sum of money is worth today. This is important because money today is typically worth more than the same amount in the future due to its potential earning ability.
For the exercise at hand, present value helps us decide which option is cheaper by comparing costs now instead of in the future. By calculating the present value of payments for internet access over three years, and combining them with the initial computer cost, we get the total present value for each scenario.
The formula to calculate present value is:

• \( PV = \frac{FV}{(1 + r)^n} \)

Where:

• \( PV \) is the present value.
• \( FV \) is the future value of money.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( n \) is the number of periods, such as years.

By applying this formula to each payment, we discount future costs to make a fair comparison between the two options. An accurate present value calculation is essential in consumer decision-making as it provides a clear perspective of the total cost over time.

Making Interest Rate Comparisons

Interest rate comparison is crucial when evaluating financial decisions involving future payments. Interest rates affect the present value calculations, which in turn influence the consumer's decision.
In our exercise, an interest rate change significantly alters the outcomes. A lower rate makes future money worth more today, meaning future costs weigh heavily when calculating present value. In contrast, a higher rate diminishes the value of future cash, making the upfront cost more notable.
To find the best option, we evaluate both options at the specified interest rates of 3% and 17%.

• At 3%, the scenario with lower monthly charges has a lower total present value, as future savings on internet fees become relatively more significant.
• At 17%, the weight of future savings decreases, and the upfront rebate plays a bigger role in reducing total costs.

This variation shows why interest rate comparison is vital. Consumers must consider expected interest rates and opt for the option that minimizes their present value cost under that rate.

Comprehensive Cost Analysis

Cost analysis allows consumers to evaluate each scenario comprehensively, factoring in all potential payments and savings in the decision-making process. It doesn’t just consider the sticker price but integrates ongoing costs, such as internet fees, and how these vary over time with interest rates.
In this exercise, a comprehensive cost analysis involves comparing the total cost of two options, incorporating:

• The upfront price of the computer (either \\(1000 or \\)600).
• The ongoing internet fees (either \\(120 annually or \\)300 annually).

By calculating the present value for each payment within these elements, we perform a comprehensive cost analysis. It results in a total that accurately reflects the burden of option on the consumer's current financial resources.
Finally, by solving for the interest rate that makes both options equal, we determine the break-even point or rate of indifference. This rate is where the consumer would have no preference between options because they result in equivalent present value costs.