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. 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

a. For 4 percent interest rate, the better option is to lease or buy based on NPCs. b. For 12 percent interest rate, the better option is to lease or buy based on NPCs. c. The indifference interest rate is solved based on equating NPCs.

Step-by-step solution

Calculate the NPC for buying and selling the car

The NPC for buying and selling the car is the cost of buying the car now minus the present value of the money obtained from selling the car after six years. The formula to calculate the present value is \( PV = FV / (1+r)^n \), where \( FV \) is the future value of money (the selling price), \( r \) is the interest rate, and \( n \) is the number of periods (years). Hence, calculate the NPC for buying and selling the car when \( r = 4\% \) and \( r=12\% \) respectively.

Calculate the NPC for leasing the car

To calculate the NPC for leasing, one should sum up the present values of lease payment for each year. Similar to step 1, use the formula \( PV = FV / (1+r)^n \), where \( FV \) is the lease payment per year, \( r \) is the interest rate, and \( n \) is the year. With the interest rates \( r = 4\% \) and \( r=12\% \), calculate the NPCs.

Comparison of options

Based on the NPCs calculated in steps 1 and 2, compare the two options (buying vs leasing) for each of the two interest rates (4% & 12%). If the NPC of buying and selling the car is less than leasing, then buying is the better option, else leasing is the better option.

Finding the indifference interest rate

The interest rate at which one would be indifferent between buying and leasing the car, is when the NPCs of both options are equal. This requires setting up an equation where the NPC of buying and selling the car equals the NPC for leasing the car, and solving for the interest rate, \( r \). Since it's not an easy algebraic equation, a numeric method or trial-error process could be used to find the approximate interest rate.

Key concepts

Lease vs Buy Decision

Deciding whether to lease or buy a car involves understanding what each option entails.
Each choice has its costs and benefits, and analyzing these will help you make an informed decision.
The key here is to compare the net present costs (NPC) of both options. **Buying a Car:**
- **Upfront Cost:** You pay for the car upfront, which means you need a significant amount of cash or financing. - **Resale Value:** You can sell the car after some years and retrieve part of your investment. - **Maintenance Costs:** Ownership means you are responsible for repairs and maintenance. **Leasing a Car:**
- **Monthly Payments:** A lease involves paying monthly or yearly installments. - **Maintenance:** Some lease agreements cover maintenance costs. - **No Resale Value:** You return the car after the lease period with no resale income. The decision largely hangs on personal financial situations and preferences. Some prefer ownership and the eventual resale value, while others appreciate the lower upfront cost of leasing.

Interest Rate Impact

Interest rates play a crucial role in determining the cost-effectiveness of leasing versus buying a car.
The interest rate affects the present value of future costs, shaping the final NPC calculation. **Low Interest Rates (e.g., 4%):**
- **Buying Advantage:** With lower rates, buying typically presents a lower NPC compared to leasing.
- The cost to borrow money is less, making the upfront payment less costly over time. - **Leasing Cost:** The effect of interest on lease payments is relatively minor, but still adds up over the lease period. **High Interest Rates (e.g., 12%):**
- **Leasing Advantage:** Higher rates increase the cost of borrowing, - thus raising the effective cost of buying a car. - **Reduced Buying Appeal:** The future resale value is heavily discounted, diminishing the total benefit of ownership. In a nutshell, when interest rates rise, leasing often becomes the more attractive option, while lower rates may favor buying outright.

Present Value Calculation

Understanding the concept of present value is essential in analyzing the lease versus buy decision.
Present value helps us determine the value of future payments or receipts in today's terms.**Formula for Present Value:**
\[PV = \frac{FV}{(1+r)^n}\]
- **FV:** Future value of the payment or receipt.
- **r:** Interest rate expressed as a decimal.- **n:** Number of periods until the payment or receipt.**Application in Lease vs Buy:**
- When buying, you calculate the present value of the car's resale value to know its worth today.
- When leasing, you calculate the present value of each lease payment to determine the total cost in today's dollars.By comparing these present values, you can decide which option has the lower net cost. This aids in realizing the effective burden each option puts on your finances, enabling a clearer, data-driven decision-making process.