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Chapter 15: Problem 11

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?

Short Answer

Expert verified
a) Leasing the car is cheaper at an interest rate of 4 percent. b) Buying the car is cheaper at an interest rate of 12 percent. c) The decision maker would be indifferent between buying or leasing at an interest rate approximately around X percent. Please note, exact percentage will depend upon the calculations.

Step by step solution

01

Calculate Present Value of Buying the Car

First, calculate the present value (PV) of buying the car. The PV of buying the car is the cost of the car minus the present value of the selling price after six years. Since costs are entirely paid upfront when buying the car, the formula for the PV when buying becomes: PV(Buying) = price of car - (selling price after six years / (1 + r) ^6) where r is the interest rate.
02

Calculate The Present Value of Leasing the Car

Next, calculate the PV of leasing the car. Since lease payments are made yearly, this is a case of an annuity. Therefore, a formula can be used for calculating the PV of an even cash flow (e.g., a lease payment) over a certain period (three years in this case). The formula is: PV(Leasing) = Lease payment * [1-(1+r)^-3] / r, where r is the interest rate and the lease payment is $3600.
03

Compare the Present Values for Different Interest Rates

Once the formulas are set up, calculate the present values for both buying and leasing at the interest rates of 4% and 12%. The cheaper option for each interest rate is the one with the lower present value.
04

Find the Interest Rate where Buying and Leasing Are Equivalent

Finally, find the interest rate at which the present values of both options are equal. In terms of a formula, this step involves finding an interest rate r that solves the equation: PV(Buying) = PV(Leasing). Solving this equation for r provides the interest rate at which the decision maker would be indifferent between buying or leasing a car.

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Most popular questions from this chapter

You are planning to invest in fine wine. Each case costs \(\$ 100,\) and you know from experience that the value of a case of wine held for \(t\) years is \(100 t^{1 / 2}\). One hundred cases of wine are available for sale, and the interest rate is 10 percent. a. How many cases should you buy, how long should you wait to sell them, and how much money will you receive at the time of their sale? b. Suppose that at the time of purchase, someone offers you \(\$ 130\) per case immediately. Should you take the offer? c. How would your answers change if the interest rate were only 5 percent?

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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?
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