Problem 8 - 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 payment amount is approximately \$5,108.46 per period.

Step 1: Understand the Amortization Formula

The formula to find the periodic payment for an amortizing loan is given by: \[ p = \frac{P \cdot r/n}{1 - (1 + r/n)^{-n \cdot t}} \] where \(P\) is the principal (loan amount), \(r\) is the annual interest rate (written as a decimal), \(n\) is the number of compounding periods per year, and \(t\) is the total number of years.

Step 2: Convert Interest Rate to Decimal

Convert the annual interest rate from a percentage to a decimal by dividing by 100. Thus, \(r = 5.15\% = 0.0515\).

Step 3: Identify Key Values

From the problem, identify the necessary values: \(P = 150,000\), \(r = 0.0515\), \(t = 30\), and since it is compounded semiannually, \(n = 2\).

Step 4: Substitute Values into Formula

Substitute the given values into the amortization formula: \[ p = \frac{150,000 \cdot 0.0515/2}{1 - (1 + 0.0515/2)^{-2 \cdot 30}} \] Simplify within the formula to calculate \(p\).

Step 5: Simplify the Denominator

First, calculate the denominator part: \((1 + 0.0515/2)^{-60}\). Compute \(0.0515/2 = 0.02575\), so the expression becomes \((1 + 0.02575)^{-60}\).

Step 6: Calculate Exponent

Calculate \((1 + 0.02575)^{-60}\) using a calculator. This gives the approximate result of \(0.2439\).

Step 7: Continue with Denominator Simplification

Subtract the above result from 1 to continue simplifying the denominator: \(1 - 0.2439 = 0.7561\).

Step 8: Compute the Numerator

Calculate the numerator: \(150,000 \cdot 0.02575 = 3,862.5\).

Step 9: Calculate Payment

Divide the numerator by the denominator to find \(p\): \(p = 3,862.5 / 0.7561 \approx 5,108.46\).

Step 10: Confirm the Payment Amount

The periodic payment amount \(p\) required to amortize the loan is approximately \$5,108.46.

Key Concepts

Compound Interest

Compound interest is a powerful concept in financial mathematics and plays a significant role in loan calculations like amortization. It refers to the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.
In simple terms, when you have compound interest, the interest you earned in one period gets added to your original amount, and then in the next period, interest is earned on this new total. This process repeats throughout the duration of the loan or investment.
Key Features of Compound Interest:

• The principle grows at an exponential rate because interest is applied to both the initial principal and accumulated interest.
• The frequency of compounding (e.g., annually, semiannually, quarterly) can significantly affect the total interest earned or paid.

In the given solution, the interest is compounded semiannually, meaning the amount grows more quickly than with simple interest due to the compounding effect twice a year. Understanding compound interest helps grasp why the final payment amounts in loans and investments can be quite different based on how often the interest compounds throughout the duration.

Loan Payment Calculation

Loan payment calculation is crucial in determining how much you need to pay periodically to amortize a loan. Amortization involves spreading the loan payments over a term so that the loan is fully paid off by the end.
This process uses a specific formula that considers the principal amount, interest rate, and the compounding periods.
In this exercise, the formula used is\[ p = \frac{P \cdot r/n}{1 - (1 + r/n)^{-n \cdot t}} \]Where:

• \(P\) = Principal (loan amount)
• \(r\) = Annual interest rate (decimal form)
• \(n\) = Number of compounding periods per year
• \(t\) = Total number of years

The goal is to find \(p\), the periodic payment needed to amortize the loan.
By understanding and using this formula, you can accurately calculate the periodic payments for any loan, considering the specific terms and interest rates involved. This way, you ensure that you pay off the loan entirely by the end of the term.

Financial Mathematics

Financial mathematics provides the tools and formulas necessary to solve problems related to loans and investments. It combines principles of mathematics and economics to handle monetary elements like interest, loans, and amortizations effectively.
Understanding financial mathematics is critical for making informed financial decisions.
Essential Concepts in Financial Mathematics:

• **Interest Rates:** These are critical as they determine how much you will pay over the life of a loan or gain on an investment.
• **Time Value of Money:** This concept explains how the present value of a sum of money differs from its future value due to interest.
• **Amortization:** This is the process of gradually paying off a debt over time through regular payments. Each payment covers both interest and principal.

In the context of this exercise, financial mathematics helps calculate the periodic payment necessary to amortize a loan, ensuring the borrower knows exactly what they need to pay to be debt-free at the end of the loan term. It empowers individuals with the knowledge to approach loans strategically, maximizing financial benefits.