Problem 7 - 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. \(P=\$ 150,000 ; r=5.15 \% ; t=30 \mathrm{yr},\) compounded semiannually

Step-by-Step Solution

Answer

The periodic payment amount needed is approximately \( \$50,021.45 \).

Step 1: Understand the Loan Parameters

The given loan amount, or principal, is \( P = \$150,000 \). The annual interest rate is \( r = 5.15\% = 0.0515 \), the term of the loan is \( t = 30 \) years, and since the interest is compounded semiannually, \( n = 2 \). Our goal is to find the periodic payment amount, \( p \).

Step 2: Calculate the Number of Payments

Since the loan is compounded semiannually, the number of compounding periods per year is \( n = 2 \). Over 30 years, the total number of payments is \( N = n \times t = 2 \times 30 = 60 \). Therefore, there will be 60 payments.

Step 3: Calculate the Periodic Interest Rate

The periodic interest rate is the annual rate divided by the number of compounding periods per year: \( i = \frac{r}{n} = \frac{0.0515}{2} = 0.02575 \).

Step 4: Use the Amortization Formula

The formula for the periodic payment \( p \) in an amortizing loan is given by:\[ p = \frac{Pi(1+i)^N}{(1+i)^N-1} \]Substitute the known values:\[ p = \frac{150,000 \times 0.02575 \times (1 + 0.02575)^{60}}{(1 + 0.02575)^{60} - 1} \]

Step 5: Compute the Payment Amount

First, calculate \((1+i)^{60}\):\((1+0.02575)^{60} \approx 4.4321\)Then, substitute back into the payment formula:\[ p = \frac{150,000 \times 0.02575 \times 4.4321}{4.4321-1} \approx \frac{171,607.5}{3.4321} \approx 50,021.45 \]Thus, the periodic payment \( p \) is approximately \( \$50,021.45 \).

Key Concepts

Loan payment calculation

Calculating the payment amount for a loan involves several essential steps. The main objective is to determine the periodic payment, often abbreviated as \( p \), that you will need to make to eventually pay off a loan. Here’s what you need to consider:

• The principal \( P \), which is the initial loan amount. In our example, it is \( \$150,000 \).
• The annual interest rate \( r \), which in this case is \( 5.15\% \). Remember to convert this percentage to a decimal form: \( 0.0515 \).
• The term \( t \), which is the number of years over which the loan will be paid. For this example, it's \( 30 \) years.
• The number of payment periods per year \( n \). Here, it's semi-annual, so \( n = 2 \).

Together, these parameters allow us to form the framework for loan payment calculation. The result is a single, periodic payment amount that simplifies the complexity of compound interest into manageable, regular payments.

Interest compounding

Interest compounding refers to the process of calculating interest on both the initial principal and the accumulated interest from prior periods. It’s one of the key reasons why the total amount paid back on a loan can seem higher than the original borrowing amount. Here’s how it works:

• In our example of semiannual compounding, interest is calculated and added to the principal every 6 months.
• The interest rate per period \( i \) is found by dividing the annual interest rate \( r \) by the number of periods \( n \): \( i = \frac{r}{n} \). If \( r = 0.0515 \) and \( n = 2 \), then \( i = 0.02575 \).
• This periodic interest significantly influences the overall cost of the loan because each period builds upon the previous one, expanding the balance subject to interest calculation.

Therefore, understanding compounding is crucial because it affects both the payment amount and the total interest paid over the life of the loan.

Amortization formula

The amortization formula is a mathematical expression used to calculate the periodic payment required to pay off a loan over a specified period. For loans like mortgages, this formula is essential to determining how much you will pay each period. The formula is:
\[p = \frac{Pi(1+i)^N}{(1+i)^N-1}\]
Where:

• \( p \) is the periodic payment amount.
• \( P \) is the principal or the initial loan amount.
• \( i \) is the periodic interest rate found using \( \frac{r}{n} \).
• \( N \) is the total number of payments (i.e., the total periods, \( n \times t \)), which in our example is \( 60 \).

Using this formula allows borrowers to find the consistent payment amount which will completely pay off both the principal and all interest by the end of the loan term. This way, you ensure that by the time you make the last payment, your debt balance has been fully amortized. The simplicity of a flat payment schedule relies heavily on this reliable, mathematical method.