An animator needs a laptop for audio/video editing, and notices that he can pay $1500 for a Dell XPS laptop, or lease from the manufacturer for monthly payments of $65 each for three years. The designer can borrow at an interest rate of 12% APR compounded monthly. What is the cost of leasing the laptop over buying it outright?
Group of answer choices

Leasing costs $428 more than buying.

Leasing costs $365 more than buying.

Leasing costs $512 more than buying.

Leasing costs $457 more than buying.

To solve this problem, we need to compare the cost of leasing the laptop with the cost of buying it outright. Here's how we do it step by step:

### 1. **Cost of Leasing**:
The designer is leasing the laptop for 36 months (3 years) at $65 per month. The total cost of leasing is:

\[
\text{Total leasing cost} = 65 \times 36 = 2340 \, \text{dollars}.
\]

### 2. **Cost of Buying**:
If the designer were to buy the laptop outright, the cost is $1500.

### 3. **Cost of Borrowing**:
The cost of borrowing must be considered because the designer can borrow at an APR of 12% compounded monthly.

To compute the present value (PV) of the lease payments (the equivalent cost of leasing in terms of buying the laptop outright), we use the formula for the present value of an annuity:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $ P = 65 $ is the monthly payment,
- $ r = \frac{12\%}{12} = 0.01 $ is the monthly interest rate (12% APR compounded monthly),
- $ n = 36 $ is the total number of months.

Let's calculate this to find the present value of the lease payments.

\[
PV = 65 \times \frac{1 - (1 + 0.01)^{-36}}{0.01}
\]

Now I'll compute this value.The present value of the lease payments, when borrowing at an interest rate of 12% APR compounded monthly, is approximately $1,957.

### 4. **Cost Comparison**:
Now, we compare the cost of buying outright versus leasing.

- **Cost of buying outright**: $1500
- **Present value of leasing**: $1957

Thus, the cost difference between leasing and buying is:

\[
\text{Cost difference} = 1957 - 1500 = 457 \, \text{dollars}.
\]

### Conclusion:
Leasing the laptop costs **$457 more than buying** outright. Therefore, the correct answer is:

**Leasing costs $457 more than buying.**

---

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does the interest rate affect the cost difference between leasing and buying?
2. What would the cost difference be if the lease term was 2 years instead of 3?
3. How would a higher APR change the present value of the lease payments?
4. What if the monthly payment were lower—how would that impact the comparison?
5. Can you use the present value formula to evaluate other financing options?

**Tip:** When comparing leasing and buying, always account for the interest rate to understand the true cost of financing.

An animator needs a laptop for audio/video editing, and notices that he can pay $1500 for a Dell XPS laptop, or lease from the manufacturer for monthly payments of $65 each for three years. The designer can borrow at an interest rate of 12% APR compounded monthly. What is the cost of leasing the laptop over buying it outright?

Compare the cost of leasing a Dell XPS laptop with buying it outright. Using finance concepts such as the present value of an annuity and interest rates, this problem walks through the detailed steps to determine the cost difference. Ideal for high school and undergraduate students studying finance.

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