You take out a 25-year $210,000 mortgage loan with an APR of 12% and monthly payments. In 16 years, you decide to sell your house and pay off the mortgage. What is the principal balance on the loan? (Round the monthly loan payment to 2 decimal places when computing the answer. Round your answer to 2 decimal places.)
The Correct Answer and Explanation is:
To find the principal balance on the mortgage after 16 years, we need to calculate the monthly payment first and then determine how much principal remains after 16 years of payments.
Step 1: Find the Monthly Payment
The formula to calculate the monthly mortgage payment (PMT) for a fixed-rate mortgage is given by the formula:PMT=P×r(1+r)n(1+r)n−1PMT = P \times \frac{r(1 + r)^n}{(1 + r)^n – 1}PMT=P×(1+r)n−1r(1+r)n
Where:
- PPP is the loan amount ($210,000)
- rrr is the monthly interest rate (annual rate divided by 12 months)
- nnn is the total number of payments (loan term in years multiplied by 12)
Here, the annual interest rate is 12%, so the monthly rate r=12%12=1%=0.01r = \frac{12\%}{12} = 1\% = 0.01r=1212%=1%=0.01. The loan term is 25 years, so n=25×12=300n = 25 \times 12 = 300n=25×12=300 months.
Plugging in the values:PMT=210,000×0.01(1+0.01)300(1+0.01)300−1PMT = 210,000 \times \frac{0.01(1 + 0.01)^{300}}{(1 + 0.01)^{300} – 1}PMT=210,000×(1+0.01)300−10.01(1+0.01)300
Using a calculator:PMT=210,000×0.01(1.01)300(1.01)300−1≈210,000×0.01×20.9619.96PMT = 210,000 \times \frac{0.01(1.01)^{300}}{(1.01)^{300} – 1} \approx 210,000 \times \frac{0.01 \times 20.96}{19.96}PMT=210,000×(1.01)300−10.01(1.01)300≈210,000×19.960.01×20.96PMT≈210,000×0.209619.96≈210,000×0.0105≈2,205.08PMT \approx 210,000 \times \frac{0.2096}{19.96} \approx 210,000 \times 0.0105 \approx 2,205.08PMT≈210,000×19.960.2096≈210,000×0.0105≈2,205.08
Thus, the monthly payment is approximately $2,205.08.
Step 2: Calculate the Remaining Principal After 16 Years
Now, we need to find the remaining principal after 16 years, which is the remaining balance on the loan after 192 payments (16 years × 12 months). The formula for the remaining balance after a certain number of payments is:B=P×(1+r)n−(1+r)t(1+r)n−1B = P \times \frac{(1 + r)^n – (1 + r)^t}{(1 + r)^n – 1}B=P×(1+r)n−1(1+r)n−(1+r)t
Where:
- BBB is the remaining balance
- PPP is the loan amount ($210,000)
- rrr is the monthly interest rate (0.01)
- nnn is the total number of payments (300 months)
- ttt is the number of payments made (192 months)
Plugging in the values:B=210,000×(1.01)300−(1.01)192(1.01)300−1B = 210,000 \times \frac{(1.01)^{300} – (1.01)^{192}}{(1.01)^{300} – 1}B=210,000×(1.01)300−1(1.01)300−(1.01)192
Using a calculator:B≈210,000×20.96−6.1419.96≈210,000×14.8219.96≈210,000×0.7422≈155,862.98B \approx 210,000 \times \frac{20.96 – 6.14}{19.96} \approx 210,000 \times \frac{14.82}{19.96} \approx 210,000 \times 0.7422 \approx 155,862.98B≈210,000×19.9620.96−6.14≈210,000×19.9614.82≈210,000×0.7422≈155,862.98
Thus, the remaining principal balance after 16 years is approximately $155,862.98.
Conclusion
After 16 years, the principal balance on the mortgage loan is $155,862.98.
