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Chapter 15: Problem 12

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

Expert verified
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

01

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: \(PV_{2} = 600 + \frac{300}{(1.03)} + \frac{300}{(1.03)^2} + \frac{300}{(1.03)^3}\)
02

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%.
03

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%.
04

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.

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Most popular questions from this chapter

You are offered the choice of two payment streams: (a) \(\$ 150\) paid one year from now and \(\$ 150\) paid two years from now; (b) \(\$ 130\) paid one year from now and \(\$ 160\) paid two years from now. Which payment stream would you prefer if the interest rate is 5 percent? If it is 15 percent?

Suppose the interest rate is 10 percent. If \(\$ 100\) is invested at this rate today, how much will it be worth after one year? After two years? After five years? What is the value today of \(\$ 100\) paid one year from now? Paid two years from now? Paid five years from now?

A bond has two years to mature. It makes a coupon payment of \(\$ 100\) after one year and both a coupon payment of \(\$ 100\) and a principal repayment of \(\$ 1000\) after two years. The bond is selling for \(\$ 966 .\) What is its effective yield?

Suppose you can buy a new Toyota Corolla for \(\$ 20,000\) and sell it for \(\$ 12,000\) after six years. Alternatively, you can lease the car for \(\$ 300\) per month for three years and return it at the end of the three years. For simplification, assume that lease payments are made yearly instead of monthly- \(i . e .,\) that they are \(\$ 3600\) per year for each of three years. a. If the interest rate, \(r,\) is 4 percent, is it better to lease or buy the car? b. Which is better if the interest rate is 12 percent? c. At what interest rate would you be indifferent between buying and leasing the car?

Suppose the interest rate is 10 percent. What is the value of a coupon bond that pays \(\$ 80\) per year for each of the next five years and then makes a principal repayment of \(\$ 1000\) in the sixth year? Repeat for an interest rate of 15 percent.
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