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Chapter 2: Problem 24

Annuity Values. You want to buy a new car, but you can make an initial payment of only \(\$ 2,000\) and can afford monthly payments of at most \(\$ 400\) a. If the APR on auto loans is 12 percent and you finance the purchase over 48 months, what is the maximum price you can pay for the car? b. How much can you afford if you finance the purchase over 60 months?

Short Answer

Expert verified
a) The maximum price for the car that can be paid if the purchase is financed over 48 months is \$26,374.59. b) If the purchase is financed over 60 months, the maximum affordable amount is \$32,889.85.

Step by step solution

01

Calculate monthly interest rate and specify monthly payment and the number of months

Firstly, convert the annual percentage rate into a monthly rate by dividing it by 12. So, the monthly rate, \(r = 12\% / 12 = 0.01\). The annuity payment per time period (P) is $400, which is given. And the total number of periods (n) would be different for part a and b. For part a, n = 48 months and for part b, n = 60 months.
02

Calculate the future value of annuity for 48 months

Substitute the aforementioned values into the given formula for the future value of an annuity: \(FV_1 = 400 \times \[((1 + 0.01)^{48} - 1) / 0.01] = \$24,374.59\). The maximum price of the car can be calculated by adding the initial payment. So, Maximum Price P1 = FV1 + initial payment = \$24,374.59 + $2,000 = \$26,374.59.
03

Calculate the future Value of annuity for 60 months

Continuing with the same method, only altering the total periods, n to 60, \(FV_2 = 400 \times \[((1 + 0.01)^{60} - 1) / 0.01]\) = \$30,889.85. The maximum affordable price for 60 months can be obtained similarly. So, Maximum Price P2 = FV2 + initial payment = \$30,889.85 + $2,000 = \$32,889.85.

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Key Concepts

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

APR (Annual Percentage Rate)
When you're looking to finance a car, one of the first things you'll encounter is the APR, or Annual Percentage Rate. The APR represents the yearly interest generated by a sum that's charged for borrowing or earned through an investment. It doesn't just include interest, but it can also list fees or other costs, depending on your agreement. Understanding APR is crucial because it directly affects your monthly payments over the life of the loan. A higher APR means that you'll pay more over time. When you see a 12% APR, like in this exercise, it denotes that on an annual basis, you'll pay 12% of the loan balance as interest. However, this needs to be converted to a monthly interest rate to make it relevant for monthly car payments.
Future Value of Annuity
The Future Value (FV) of an annuity is a key piece of determining how much you can afford when financing a car purchase. An annuity is a series of equal payments made at regular intervals, and the future value is the total value of these payments at a specific point in the future. In the context of the exercise, the 'future value of annuity' calculation lets us figure out how much money we will have paid over the term of the loan, considering interest. It's computed using the formula: \[FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right)\]where:
  • \(P\) is the payment per period (\$400)
  • \(r\) is the monthly interest rate
  • \(n\) is the number of payment periods
By calculating the future value, you can understand how much you'll effectively pay over either a 48 or 60-month financing plan.
Monthly Interest Rate
The monthly interest rate is essential when breaking down an APR into smaller, understandable portions for monthly payments. Since the APR is an annual rate, we need to convert this to a monthly rate to match the frequency of car loan payments. To find the monthly interest rate from the annual rate, you simply divide the APR by 12. For example, with an APR of 12%, the monthly interest rate is 1% (or 0.01 when expressed as a decimal). This monthly rate is then used to calculate other aspects of the loan, such as the future value of annuities and the total interest accrued over the term. Thus, understanding how to derive and utilize the monthly interest rate is vital to accurately determining your loan obligations.

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Most popular questions from this chapter

Growth of Funds. If you earn 8 percent per year on your bank account, how long will it take an account with \(\$ 100\) to double to \(\$ 200 ?\)

Real versus Nominal Dollars. Your consulting firm will produce cash flows of \(\$ 100,000\) this year, and you expect cash flow to keep pace with any increase in the general level of prices. The interest rate currently is 8 percent, and you anticipate inflation of about 2 percent. a. What is the present value of your firm's cash flows for Years 1 through 5? b. How would your answer to (a) change if you anticipated no growth in cash flow?

Calculating Interest Rate. In a discount interest loan, you pay the interest payment up front. For example, if a 1 -year loan is stated as \(\$ 10,000\) and the interest rate is 10 percent the borrower "pays" \(.10 \times \$ 10,000=\$ 1,000\) immediately, thereby receiving net funds of \(\$ 9,000\) and repaying \(\$ 10,000\) in a year a. What is the effective interest rate on this loan? b. If you call the discount \(d\) (for cxample, \(d=10 \%\) using our numbers), express the effective annual rate on the loan as a function of \(d\) c. Why is the effective annual rate always greater than the stated rate \(d\) ?

Annuity Valuc. The \(\$ 40\) million lottery payment that you just won actually pays \(\$ 2\) million per year for 20 years. If the discount rate is 10 percent, and the first payment comes in 1 year, what is the present value of the winnings? What if the first payment comes immediately?

Valuing Delayed Annuities. Suppose that you will receive annual payments of \(\$ 10,000\) for a period of 10 years. The first payment will be made 4 years from now. If the interest rate is 6 percent, what is the present valuc of this stream of payments?
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