Question

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.

1. If the interest rate, r, is 4 percent, is it better to lease or buy the car?
2. Which is better if the interest rate is 12 percent?
3. At what interest rate would you be indifferent between buying and leasing the car?

Short answer

1. It is better to lease the car.
2. It is better to lease the car.
3. The interest rate will be nearly 3.8%.

Step-by-step solution

Explanation for part (a)

The NPV for buying a car is calculated below:

$$\begin{gathered} \text{r =}\frac{\text{4}}{\text{100}}\text{= 0.04} \\ \text{NPV = - 20,000 +}\frac{\text{12,000}}{{\left(\text{1 + 0.04}\right)}^{\text{6}}} \\ \text{= - \$ 10,516.23} \end{gathered}$$

The NPV for leasing the car is calculated below:

The payment needs to be made at the beginning of each year.

$$\begin{gathered} \text{NPV = - 3600 -}\frac{\text{3600}}{\text{1.04}}\text{-}\frac{\text{3600}}{{\left(\text{1.04}\right)}^{\text{2}}} \\ \text{= - 3600 - 3461.54 - 3328.40} \\ \text{= - \$ 10,389.94} \end{gathered}$$

Leasing the car is better than buying the car, as the NPV is higher, i.e., lower negative value.

Explanation for part (b)

The NPV for buying a car is calculated below:

$$\begin{gathered} \text{r =}\frac{\text{12}}{\text{100}}\text{= 0.12} \\ \text{NPV = - 20,000 +}\frac{\text{12,000}}{{\left(\text{1 + 0.12}\right)}^{\text{6}}} \\ \text{= - \$ 13,920.43} \end{gathered}$$

The NPV for leasing the car is calculated below:

The payment needs to be made at the beginning of each year.

$$\begin{gathered} \text{NPV = - 3600 -}\frac{\text{3600}}{\text{1.12}}\text{-}\frac{\text{3600}}{{\left(\text{1.12}\right)}^{\text{2}}} \\ \text{= - 3600 - 3214.29 - 2869.90} \\ \text{= - \$ 9,684.18} \end{gathered}$$

Leasing the car is better than buying the car, as the NPV is higher. i.e., lower negative value.

Explanation for part (c)

An investor will be indifferent between two options when the NPVs are equal. Thus, to be indifferent between buying and leasing the car, the NPVs must be equal. Hence, the NPVs are $\text{- 20,000 +}\frac{\text{12,000}}{{\left(\text{1 + r}\right)}^{\text{6}}}\text{= - 3600 -}\frac{\text{3600}}{\left(\text{1 + r}\right)}\text{-}\frac{\text{3600}}{{\left(\text{1 + r}\right)}^{\text{2}}}$.

The interest rate increases from 4% to 12%; thus lease become a better option. The NPVs are calculated below:

r	NPV Buy	NPV Lease
0.03	-9,950.19	-10,488.49
0.037	-10,350.41	-10,419.45
0.038	-10,406.06	-10,409.45
0.039	-10,461.33	-10,399.68
0.04	-10,516.23	-10,389.94

The interest rate will be 3.8%, where the NPVs are almost equal.