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Chapter 5: Problem 4

Find the periodic payment \(R\) required to amortize a loan of \(P\) dollars over \(t\) yr with interest charged at the rate of \(r \% /\) year compounded \(m\) times a year. \(P=16,000, r=9, t=4, m=12\)

Short Answer

Expert verified
The periodic payment (R) required to amortize the loan of \(16,000\) dollars over \(4\) years, with an interest rate of \(9\%\) compounded monthly, is approximately \(404.65\).

Step by step solution

01

Identify the given values

The exercise has given us the following values: 1. The loan amount (P): \(P=16,000\) 2. The interest rate (r): \(r=9\)% 3. The duration of the loan in years (t): \(t=4\) 4. The number of times interest is compounded per year (m): \(m=12\)
02

Convert the interest rate from percentage to decimal

To use the interest rate in our calculations, we need to convert it from a percentage to a decimal. To do this, we divide the interest rate by 100. \[ r = \frac{9}{100} = 0.09 \]
03

Calculate the annual interest rate in decimals

The annual interest rate 'i', compounded m times a year, can be calculated using the formula: \[ i = (1 + \frac{r}{m})^m - 1 \] Plugging in the given values, we get: \[ i = (1 + \frac{0.09}{12})^{12} - 1 \approx 0.093807 \]
04

Calculate the number of payments

The total number of payments, n, can be found by multiplying the number of years (t) by the number of compounding periods (m) per year. \[ n = t \times m = 4 \times 12 = 48 \]
05

Calculate the periodic payment (R)

Now we can use the annuity formula to find the periodic payment amount (R). The formula is as follows: \[ R = P \frac{i(1+i)^n}{(1+i)^n - 1} \] Plugging in the values for P, i, and n, we get: \[ R = 16,000 \times \frac{0.093807(1+0.093807)^{48}}{(1+0.093807)^{48} - 1} \] \[ R \approx 404.65 \] So, the periodic payment required to amortize the loan of \(16,000 over 4 years, with an interest rate of 9% compounded monthly, is approximately \)404.65.

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Key Concepts

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

Periodic Payment
When you take out a loan, you will often make regular payments to repay it. These are called periodic payments. The amount you pay each period is calculated so that after the agreed term, your loan is completely paid off.
Periodic payments are designed to cover both the interest and a portion of the loan principal.
  • The principal is the initial amount borrowed, in our example, $16,000.
  • The interest is the cost of borrowing money, influenced by the interest rate.
Periodic payments ensure that by the end of the loan term, the entire amount including interest, is repaid. Calculating periodic payments requires understanding the loan terms, such as the interest rate, loan duration, and compounding frequency.
Compounded Interest
Compounded interest is the process where the interest earned on a loan or deposit is reinvested to earn more interest.
This means that interest is calculated not only on the initial principal but also on the accumulated interest from previous periods.
  • The more frequently interest is compounded, the more interest will be paid over the life of the loan.
  • In our example, the interest is compounded monthly.
  • This complicates the calculation of the total interest paid because each month, on top of the original principal, the interest from previous months is also considered.
The formula for calculating the effect of compounded interest in one year is \[ i = \left(1 + \frac{r}{m}\right)^m - 1 \] where \( r \) is the annual interest rate and \( m \) is the number of compounding periods per year. Understanding compounded interest helps in managing loans effectively and anticipating the total interest paid.
Annuity Formula
The annuity formula is a crucial tool in financial mathematics for calculating periodic payments. It is used to amortize loans, meaning to schedule payments in such a way that both interest and principal are paid off after a specified period.
The formula for calculating periodic payment \( R \) for a loan is given by: \[ R = P \frac{i(1+i)^n}{(1+i)^n - 1} \]
  • \( P \) is the principal amount of the loan,
  • \( i \) is the interest rate per period, and
  • \( n \) is the total number of payments.
This formula allows you to determine how much you must pay regularly to fully repay your loan within the scheduled term. The annuity formula considers both the compounding of interest and the number of payments to ensure that the debtor is aware of the exact payment plan.
Financial Mathematics
Financial mathematics involves using mathematical formulas and theories to solve problems related to finance. It is vital in helping individuals and businesses make wise financial decisions.
Here are some of its components:
  • Understanding loans and interests, which are key to managing personal and corporate finances.
  • Utilizing formulas like the annuity formula to plan payments and investments accurately.
  • Assessing the long-term and short-term impacts of borrowing and investing decisions.
It helps in analyzing financial data and in making predictions about future trends and risks. With financial mathematics, someone can learn how to budget effectively, save on interests, and invest wisely for future needs.

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Most popular questions from this chapter

IRAs Martin has deposited \(\$ 375\) in his IRA at the end of each quarter for the past 20 yr. His investment has earned interest at the rate of \(8 \% /\) year compounded quarterly over this period. Now, at age 60 , he is considering retirement. What quarterly payment will he receive over the next 15 yr? (Assume that the money is earning interest at the same rate and that payments are made at the end of each quarter.) If he continues working and makes quarterly payments of the same amount in his IRA until age 65, what quarterly payment will he receive from his fund upon retirement over the following \(10 \mathrm{yr}\) ?

Auro FiNANCING Dan is contemplating trading in his car for a new one. He can afford a monthly payment of at most \(\$ 400 .\) If the prevailing interest rate is \(7.2 \% /\) year compounded monthly for a 48 -mo loan, what is the most expensive car that Dan can afford, assuming that he will receive \(\$ 8000\) for the trade-in?

Nina purchased a zero coupon bond for \(\$ 6724.53 .\) The bond matures in \(7 \mathrm{yr}\) and has a face value of \(\$ 10,000\). Find the effective annual rate of interest for the bond. Hint: Assume that the purchase price of the bond is the initial investment and that the face value of the bond is the accumulated amount.

Online retail sales stood at \(\$ 23.5\) billion for the year 2000 . For the next 2 yr, they grew by \(33.2 \%\) and \(27.8 \%\) per year, respectively. For the next \(6 \mathrm{yr}\), online retail sales were projected to grow at \(30.5 \%, 19.9 \%\), \(24.3 \%, 14.0 \%, 17.6 \%\), and \(10.5 \%\) per year, respectively. What were the projected online sales for 2008 ?

From age 25 to age 40 , Jessica deposited \(\$ 200\) at the end of each month into a tax-free retirement account. She made no withdrawals or further contributions until age \(65 .\) Alex made deposits of \(\$ 300\) into his tax- free retirement account from age 40 to age \(65 .\) If both accounts earned interest at the rate of \(5 \% /\) year compounded monthly, who ends up with a bigger nest egg upon reaching the age of 65 ?
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