Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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. $$ P=80,000, r=10.5, t=30, m=12 $$

Step-by-Step Solution

Answer

The periodic payment (R) required to amortize the $80,000 loan over 30 years with an interest rate of 10.5% compounded 12 times a year is approximately $733.76 per month.

Step 1: Calculate Monthly Interest Rate

We must first convert the annual interest rate (r) to a monthly interest rate (i). To do this, we will divide r by the number of times the interest is compounded in a year (m). \(i = \frac{r}{m}\) Now, we can plug in the given values of r and m. \(i =\frac{10.5}{12}\) Calculate the value of i: \(i ≈ 0.875\% \)

Step 2: Convert Interest Rate to Decimal Form

To make calculations easier, we will convert the monthly interest rate (i) from percentage to decimal form by dividing by 100. \(i_{decimal} = \frac{i}{100}\) Now, plug in the value of i. \(i_{decimal} ≈ \frac{0.875}{100}\) Calculate the value of \(i_{decimal}\). \(i_{decimal} ≈ 0.00875\)

Step 3: Calculate Total Number of Payments

Next, we need to find the total number of periodic payments required for the entire loan term. Multiply the term (t) in years by the number of payments made each year (m). \(n = t × m\) Now, plug in the given values of t and m. \(n = 30 × 12\) Calculate the value of n. \(n = 360\) The total number of payments is 360.

Step 4: Calculate Periodic Payment (R)

Using the annuity formula, we can now find the periodic payment (R) required to amortize the loan. \(R = P × \frac{i_{decimal}(1 +{i_{decimal}})^{n}}{(1 +{i_{decimal}})^{n} - 1}\) Now, plug in the values of P, \(i_{decimal}\), and n. \(R = 80,000 × \frac{(0.00875)(1 + 0.00875)^{360}}{(1 + 0.00875)^{360} - 1}\) Calculate the value of R. \(R ≈ 733.76\) The periodic payment (R) required to amortize the loan is approximately $733.76 per month.

Key Concepts

Periodic Payment Calculation

When you take out a loan, you'll typically repay it with regular payments over a certain period. This is called the loan amortization process. The periodic payment is how much you pay in each period, like every month or year, to gradually pay off your loan. To calculate this payment, several factors must be examined, including:

• The loan amount, known as the principal (P). In this example, it's $80,000.
• The interest rate, which affects how much extra you pay for borrowing the money.
• The number of periods (n) over which the loan is paid off, calculated as the number of years times how often payments are made each year.

It's important to accurately determine this periodic payment amount so you can plan your finances effectively and ensure the loan is paid off at the end of the term.

Interest Rate Conversion

Interest rates are often quoted annually, but loans can require payments at more frequent intervals. To calculate the periodic payment accurately, it's essential to convert the annual interest rate into a rate that's appropriate for the payment intervals, such as monthly.To do this, you'll need to:

• Divide the annual rate by the number of compounding periods per year. For example, if your loan is compounded monthly, you divide by 12. In our exercise, \(i = \frac{10.5}{12} \approx 0.875\%\).
• Convert this rate from a percentage to a decimal by dividing by 100, resulting in the monthly interest rate \(i_{decimal} \approx 0.00875\).

This converted rate is critical as it will be used in further calculations, ensuring a precise determination of each periodic payment amount.

Annuity Formula

The annuity formula is key to finding the periodic payment amount for loans that are amortized over time with equal payments. It helps us to figure out what constant payment will gradually pay down the loan entirely, including both principal and interest, by the end of the loan term.The formula used for calculating this periodic payment \(R\) is: \[R = P \times \frac{i_{decimal}(1 +{i_{decimal}})^{n}}{(1 +{i_{decimal}})^{n} - 1}\]Here's what each symbol represents:

• \(R\) is the periodic payment amount you're calculating.
• \(P\) is the principal, or the initial loan amount.
• \(i_{decimal}\) is the decimal version of the periodic interest rate.
• \(n\) is the total number of payments.

Plugging in these values from the exercise allows you to calculate the periodic payment. In this case, \(R \approx 733.76\), which means you need to pay \(733.76 each month to fully cover an \)80,000 loan over 30 years at 10.5% interest compounded monthly.