Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?

Question

BlackTom AI · Accepted Answer

Present value of an ordinary annuity (monthly payments at end of each month) is used: PV = PMT * [1 - (1 + r)^(-n)] / r.

1) Relevant knowledge points and concepts
- PMT (payment) = $440 per month.
- Annual nominal interest rate = 7.8% compounded monthly → monthly rate r = 0.078/12 = 0.0065 (0.65% per month).
- Number of payments n = 5 years × 12 months = 60.
- Use the present-value-of-an-annuity formula because Todd makes a uniform series of monthly payments and wants the equivalent lump-sum today.

2) Complete reasoning process step by step
- Compute the monthly interest rate and number of periods.
- Compute the discount factor (1 + r)^(-n).
- Compute the annuity present-value factor [1 - (1 + r)^(-n)]/r.
- Multiply that factor by the monthly payment 440 to get the maximum present value (most he can afford today).

3) Calculations with intermediate steps
- r = 0.078/12 = 0.0065
- n = 5 × 12 = 60
- (1 + r)^(-n) = (1.0065)^(-60) ≈ 0.67791581
- Numerator: 1 - (1.0065)^(-60) ≈ 1 - 0.67791581 = 0.32208419
- Annuity factor: 0.32208419 / 0.0065 ≈ 49.55156818
- PV = 440 × 49.55156818 ≈ 21,802.89

4) Why the correct answer is correct
The formula directly converts the known monthly payment stream into its present value using the exact monthly interest rate and number of months. The computed annuity factor (≈49.551568) accurately discounts each of the 60 monthly payments at the monthly rate 0.65%, and multiplying by $440 gives $21,802.89 — the largest lump-sum purchase price affordable given those payments and interest terms.

5) Why the other options are incorrect
- $20,712.74 and $21,504.61 are slightly lower; they result from either incorrect rounding, wrong discount factor, or using a different rate/term (e.g., mistaking annual for monthly compounding).
- $32,161.83 is much larger and would correspond to a drastically lower discount rate or treating the payments as lasting far longer (or computing a future value rather than present value). None of these match the correct monthly-compounded PV calculation.

6) Conclusion and confirmation
Using the present-value-of-an-annuity formula with r = 0.0065 and n = 60 gives PV ≈ $21,802.89, which matches the provided correct answer.

Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?Single choice

Similar Questions

The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?

You are scheduled to receive annual payments of $10,100 for each of the next 23 years. Your discount rate is10 percent. What is the difference in the present value if you receive these payments at the beginning ofeach year rather than at the end of each year?

An annuity pays $100 per year for 20 years, with the first payment starting 2 years from now. What is the present value (PV) of this annuity (rounded to the closest dollar), given that the discount rate is 5%?[Fill in the blank]

What is the value today of $26,839 payments per year, at a discount rate of 5% p.a. if the first payment is received eight years from now and the final payment is received thirty years from now? Please write the answer in two decimal places.

Peter paid annual instalments of $1,000 on 15 March 2010 for 5 years. The effective annual interest rate is 6%. Calculate the value of the loan on 15 March 2005.

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler