Problem 5 - Calculus and its applications

Question

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 500 ; r=4.1 \% ; t=1 \mathrm{yr},\) compounded monthly

Step-by-Step Solution

Answer

A monthly payment of approximately \$42.58 is needed to amortize the loan.

Step 1: Identify Known Variables

We need to find the periodic payment amount \(p\) required to amortize a loan. We are given that the loan principal \(P = \$500\), the annual interest rate \(r = 4.1\%\), and the time \(t = 1\) year. The compounding is monthly, so there are \(n = 12\) periods per year.

Step 2: Calculate Monthly Interest Rate

To find the monthly interest rate, divide the annual rate by the number of compounding periods per year. The monthly rate \(i\) is given by \(i = \frac{r}{n} = \frac{0.041}{12} = 0.0034167\).

Step 3: Determine Total Number of Payments

Multiply the number of compounding periods per year \(n\) by the number of years \(t\) to obtain the total number of payments \(nt = 12 \times 1 = 12\).

Step 4: Apply the Amortization Formula

Use the amortization formula to calculate the monthly payment \(p\):\[p = \frac{P \cdot i}{1 - (1 + i)^{-nt}}\]Substitute the known values into the formula:\[p = \frac{500 \times 0.0034167}{1 - (1 + 0.0034167)^{-12}}\]

Step 5: Perform the Calculation

First, calculate \((1 + i)^{-nt}\):\((1 + 0.0034167)^{-12} \approx 0.959896\).Then substitute back into the formula:\[p = \frac{500 \times 0.0034167}{1 - 0.959896} = \frac{1.70835}{0.040104} \approx 42.58\]

Step 6: Review the Solution

The calculated monthly payment \(p\) is approximately \\(42.58. This is the amount needed to completely pay off the \\)500 loan over 1 year with the given interest rate and monthly compounding.

Key Concepts

Compounding Periods

When you hear the term "compounding periods," it refers to how often the interest is applied to the principal balance. In this exercise, the compounding is done monthly, which means the interest is compounded 12 times a year. This frequency of compounding can affect the total amount of interest paid over the life of a loan. The more often interest is compounded, the more interest you'll pay in total. Monthly compounding is common for various types of loans and investments, allowing for a more accurate reflection of the accumulating interest over the shorter period. It's important to identify how many times your interest compounds within a given time frame to accurately calculate other factors like the monthly interest rate and total payments.

Monthly Interest Rate

The monthly interest rate is crucial when calculating the payments needed to amortize a loan, especially when compounding is monthly. To find this rate, simply divide the annual interest rate by the number of compounding periods in a year. For instance, if the annual interest rate is 4.1%, and interest is compounded monthly, you divide 4.1% by 12 (the number of months in a year).

The formula for this is:

• \(i = \frac{r}{n}\)

Where:

• \(i\) is the monthly interest rate,
• \(r\) is the annual interest rate,
• \(n\) is the number of compounding periods per year.

This results in a monthly rate and ensures that the monthly payment covers not only the interest but also a part of the principal amount over the course of the loan.Understanding this concept will help you see how much you're paying in interest each month.

Amortization Formula

The amortization formula is used to calculate the payment amount necessary to pay off a loan over its term, which includes both principal and interest. This formula considers the loan amount, interest rate, and number of payments. The key here is that the payments spread out evenly over the loan term, covering both interest and part of the principal.

To calculate the monthly payment \(p\), we use:

• \[p = \frac{P \cdot i}{1 - (1 + i)^{-nt}}\]

Where:

• \(P\) is the principal amount of the loan,
• \(i\) is the monthly interest rate,
• \(n\) is the number of compounding periods per year,
• \(t\) is the total number of years.

The denominator \((1 - (1 + i)^{-nt})\) reflects how quickly the loan is paid off. Each monthly payment calculates a small reduction in principal and payment of interest accumulated. Using this formula provides a structured and straightforward way to determine consistent monthly payments, making financial planning easier. Understanding this calculation is critical to effectively manage loans.