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Chapter 6: Problem 57

You want to buy a new sports car from Muscle Motors for \(\$ 48,000 .\) The contract is in the form of a 48 -month annuity due at a 9.25 percent APR. What will your monthly payment be?

Short Answer

Expert verified
The monthly payment for the 48-month annuity due to purchase the sports car at a 9.25% APR will be approximately $1,236.96.

Step by step solution

01

Convert the APR to a monthly interest rate

To get the monthly interest rate, divide the annual percentage rate (APR) by 12 (the number of months in a year). In this case, the APR is 9.25%. \(i = \frac{0.0925}{12}\)
02

Calculate the present value of the annuity due

The formula for the present value of an annuity due is: \(PV_{AD} = PMT \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)\), Where: - \(PV_{AD}\) is the present value of the annuity due, - \(PMT\) is the monthly payment, - \(i\) is the monthly interest rate, and - \(n\) is the number of payments, which is 48 in this case. We know the present value of the annuity due is $48,000, the monthly interest rate from Step 1, and the number of payments is 48. So, we can plug these values into the formula: \(48,000 = PMT \times \frac{1 - (1 + i)^{-48}}{i} \times (1 + i)\)
03

Solve for the monthly payment

Now we need to solve the equation from Step 2 for PMT: \(PMT = \frac{48,000}{\frac{1 - (1 + i)^{-48}}{i} \times (1 + i)}\) Plug in the value from Step 1 for the monthly interest rate, \(i\), and calculate the value for PMT: \(PMT = \frac{48,000}{\frac{1 - (1 + \frac{0.0925}{12})^{-48}}{\frac{0.0925}{12}} \times (1 + \frac{0.0925}{12})}\) Solve for PMT: \(PMT \approx \$1,236.96\) The monthly payment for the 48-month annuity due will be approximately $1,236.96.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Present Value of an Annuity Due
Understanding the present value of an annuity due is crucial when planning to make an investment or purchase like a car where payments are made in advance. An annuity due is a series of equal payments made at the beginning of consecutive periods.

The key to calculating the present value of an annuity due is recognizing that each payment occurs one period earlier than it would in an ordinary annuity, so each payment has one extra period to earn interest. This gets reflected in the formula as multiplying by (1 + i), which accounts for the additional period.

Therefore, to find the present value, we use the formula:
\[PV_{AD} = PMT \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)\],
where PMT represents the regular payment amount. In financial calculations, this value represents the amount needed today to satisfy the payment obligations of the annuity given a specific interest rate.
Monthly Interest Rate Calculation
When dealing with loans or investments that compound monthly, it's necessary to break down the annual interest rate into a monthly one. This conversion is vital, as it reflects the more frequent compounding period and is used to determine the payment amounts and to calculate the present value of the annuity due.

To convert an annual percentage rate (APR) to a monthly interest rate, divide the APR by 12, because there are 12 months in a year. So, with an APR of 9.25%, the calculation is:
\[i = \frac{0.0925}{12}\].
Knowing the monthly interest rate allows us to plug it into various formulas to calculate payment schedules, total interest paid over time, and other important financial figures.
Annuity Due Payment Formula
Once you've established the present value of the annuity and the monthly interest rate, you can find the amount of each payment using the annuity due payment formula. This formula rearranges the present value formula to solve for the payment amount (PMT).

The annuity due payment formula is derived as:
\[PMT = \frac{PV_{AD}}{\frac{1 - (1 + i)^{-n}}{i} \times (1 + i)}\].
By using this formula, you're able to calculate the amount you'd need to pay at the beginning of each period to reach a specific financial goal, such as paying off a loan. For example, given a present value of $48,000, a monthly interest rate calculated from a 9.25% APR, and a total of 48 payments for a car loan, you can use the formula to determine the monthly payment required.

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Most popular questions from this chapter

This is a classic retirement problem. A time line will help in solving it. Your friend is celebrating her 35 th birthday today and wants to start saving for her anticipated retirement at age \(65 .\) She wants to be able to withdraw \(\$ 80,000\) from her savings account on each birthday for 15 years following her retirement; the first withdrawal will be on her 66th birthday. Your friend intends to invest her money in the local credit union, which of fers 9 percent interest per year. She wants to make equal annual payments on each birthday into the account established at the credit union for her retirement fund. a. If she starts making these deposits on her 36 th birthday and continues to make deposits until she is \(65 \text { (the last deposit will be on her } 65 \text { th birthday })\) what amount must she deposit annually to be able to make the desired withdrawals at retirement? b. Suppose your friend has just inherited a large sum of money. Rather than making equal annual payments, she has decided to make one lump-sum payment on her 35 th birthday to cover her retirement needs. What amount does she have to deposit? c. Suppose your friend's employer will contribute \(\$ 1,500\) to the account every year as part of the company's profit-sharing plan. In addition, your friend expects a \(\$ 30,000\) distribution from a family trust fund on her 55 th birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

You have your choice of two investment accounts. Investment A is a 10 -year annuity that features end-of-month \(\$ 1,000\) payments and has an interest rate of 11.5 percent compounded monthly. Investment \(\mathrm{B}\) is an 8 percent continuously compounded lump-sum investment, also good for 10 years. How much money would you need to invest in \(\mathrm{B}\) today for it to be worth as much as Investment A 10 years from now?

Prepare an amortization schedule for a five-year loan of \(\$ 20,000 .\) The interest rate is 12 percent per year, and the loan calls for equal annual payments. How much interest is paid in the third year? How much total interest is paid over the life of the loan?

Investment X offers to pay you \(\$ 3,000\) per year for eight years, whereas Investment Y offers to pay you \(\$ 5,000\) per year for four years. Which of these cash flow streams has the higher present value if the discount rate is 5 percent? If the discount rate is 22 percent?

You're trying to choose between two different investments, both of which have up-front costs of \(\$ 30,000\). Investment G returns \(\$ 55,000\) in six years. Investment H returns \(\$ 90,000\) in 11 years. Which of these investments has the higher return?
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