You plan to borrow $35,000 at a 7.5% annual interest rate. The terms require you to amortize the loan with 7 equal end-of-year payments. How much interest would you be paying in Year 2?
The correct answer and explanation is :
To calculate the interest paid in Year 2 on a loan of $35,000 at a 7.5% annual interest rate with 7 equal end-of-year payments, we first need to determine the annual payment and then compute the interest portion for Year 2.
Step 1: Calculate the annual payment
We use the formula for the amortization of a loan, which is given by the following equation for an annuity:
[
PMT = \frac{P \times r}{1 – (1 + r)^{-n}}
]
Where:
- ( PMT ) is the annual payment,
- ( P ) is the principal loan amount ($35,000),
- ( r ) is the annual interest rate (7.5% or 0.075),
- ( n ) is the number of payments (7 years).
Plugging the values into the formula:
[
PMT = \frac{35,000 \times 0.075}{1 – (1 + 0.075)^{-7}} = \frac{2,625}{1 – (1.075)^{-7}} \approx \frac{2,625}{1 – 0.533} \approx \frac{2,625}{0.467} \approx 5,616.85
]
So, the annual payment is approximately $5,616.85.
Step 2: Determine the interest in Year 2
In an amortized loan, each payment consists of both principal repayment and interest. The interest portion of the payment in any given year is calculated as:
[
\text{Interest for Year 2} = \text{Remaining Balance at the end of Year 1} \times \text{Interest Rate}
]
At the end of Year 1, the remaining balance after the first payment can be found by subtracting the principal portion of the first payment from the original loan.
The first year’s interest is:
[
\text{Interest in Year 1} = 35,000 \times 0.075 = 2,625
]
The principal repayment in Year 1 is the total annual payment minus the interest:
[
\text{Principal in Year 1} = 5,616.85 – 2,625 = 2,991.85
]
At the end of Year 1, the remaining loan balance is:
[
\text{Remaining Balance at the end of Year 1} = 35,000 – 2,991.85 = 32,008.15
]
Now, for Year 2, the interest is:
[
\text{Interest for Year 2} = 32,008.15 \times 0.075 = 2,400.61
]
Final Answer:
The interest paid in Year 2 is approximately $2,400.61.
Explanation:
In amortized loans, each payment consists of both principal and interest. In the early years of the loan, a larger portion of the payment goes toward interest, while in later years, more of the payment is applied to reducing the principal balance. By following the amortization schedule, we can break down each year’s interest and principal payments accurately. In this case, the interest in Year 2 is lower than in Year 1 because the principal balance has decreased after the first payment.
