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Loan amortization Jan sold her house on December 31 and took a \(\$ 10,000\) mortgage as part of the payment. The 10 -year mortgage has a 10 percent nominal interest rate, but it calls for semiannual payments beginning next June \(30 .\) Next year, Jan must report on Schedule \(B\) of her IRS Form 1040 the amount of interest that was included in the 2 payments she received during the year. a. What is the dollar amount of each payment Jan receives? b. How much interest was included in the first payment? How much repayment of principal? How do these values change for the second payment? c. How much interest must Jan report on Schedule B for the first year? Will her interest income be the same next year? d. If the payments are constant, why does the amount of interest income change over time?

Step by step solution

01

Determine Payment Amount

To determine the fixed payment amount for each semiannual period, we use the formula for an amortizing loan payment:\[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \]where \( P = \\(10,000 \) is the loan principal, \( r = \frac{0.10}{2} = 0.05 \) is the semiannual interest rate, and \( n = 2 \times 10 = 20 \) is the total number of payments.Plug these values into the formula:\[ PMT = \frac{10,000 \cdot 0.05}{1 - (1 + 0.05)^{-20}} \]This calculates to approximately \( PMT \approx \\)804.62 \).

02

Calculate Interest and Principal for First Payment

For the first payment, the interest portion can be calculated as:\[ \text{Interest} = P \cdot r = 10,000 \cdot 0.05 = \\(500 \]The principal repayment is simply the payment minus the interest:\[ \text{Principal Repayment} = PMT - \text{Interest} = 804.62 - 500 = \\)304.62 \].

03

Calculate Interest and Principal for Second Payment

Reduce the principal by the amount of the first payment's principal repayment:\[ \text{New Principal} = 10,000 - 304.62 = 9,695.38 \]Calculate the interest for the second payment as:\[ \text{Interest for Second Payment} = 9,695.38 \cdot 0.05 = \\(484.77 \]Calculate the principal repayment for the second payment:\[ \text{Principal Repayment for Second Payment} = 804.62 - 484.77 = \\)319.85 \]

04

Total Interest Reported for First Year

Add the interest portions of the first and second payments to determine the total interest for the year:\[ \text{Total Interest for Year} = 500 + 484.77 = \$984.77 \].

05

Explanation of Changing Interest Income

The interest income decreases every payment period because each payment reduces the principal, and the interest is calculated on the remaining principal balance. As the principal diminishes, the interest portion of each payment decreases, while the principal repayment portion increases.

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Key Concepts

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

Amortizing Loan Payment

An amortizing loan is a type of loan where each payment goes towards both interest and principal repayment. This ensures that the loan will be fully paid off by the end of the term.
The payments are usually equal amounts over each period, which makes budgeting easier for both lenders and borrowers.

• Each payment remains constant.
• As time progresses, the portion of interest decreases and the portion of principal repayment increases.

The formula used to calculate the periodic payment is:\[ PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}} \]where \( P \) is the initial loan principal, \( r \) is the interest rate per period, and \( n \) is the total number of payments.
Using this formula provides a structured way to determine how much should be paid in each period.

Interest Income

Interest income is the amount earned from the interest portion of received loan payments. In the context of loan amortization, the interest income decreases over time.
This is because interest is calculated based on the remaining principal amount, which reduces as each payment is made.

• Initially, a larger portion of a payment goes towards interest.
• As the principal decreases, interest computations yield smaller amounts.

For Jan's situation, the interest income for the first year can be calculated by summing the interest portions from the payments she received.
This amount must be reported on tax filings, as it's considered taxable income.

Principal Repayment

Principal repayment refers to the part of the payment that reduces the outstanding balance of the loan.
In each successive payment of an amortizing loan, the portion dedicated to repaying the principal increases.

• Initially, principal repayment is less due to higher interest amounts.
• Over time, as the principal is paid down, the interest reduces, allowing more payment towards principal.

This shift happens naturally in an amortizing loan structure and ensures that the principal will be entirely repaid by the end of the loan term.
The movement from interest reduction to principal repayment is what leads to a gradual decrease in interest income.

Mortgage Calculation

Mortgage calculation involves determining the appropriate payment amount for a given loan, considering interest rate, period, and total loan amount.
Using the amortizing loan payment formula ensures the calculation is accurate and balanced over the term.

• The calculations involve principal payoff schedules.
• Interest and principal changes are predictable and systematic for each period.

For Jan's particular mortgage, we calculated semiannual payments which include both the interest and principal repayment.
By understanding the mortgage calculations, one can better manage financial expectations and investment returns over time.