Question

Jimmy paid off a mortgage by paying $750 per month for 14 years. What was the original amount of the mortgage if the interest rate charged was 5.50% compounded semi-annually?

Answer 1

Monthly Payment = $750

Interest rate = 5.50% compounded semi-annually.

Calculating Annual Percentage Rate(APR) compounded monthly from compounded semi-annually using EAR formula:-

$(1+\frac{r_1}{m_1})^{m_1}-1 = (1+\frac{r_2}{m_2})^{m_2}-1$

where, r1 = Interest rate compounded semi-annually = 5.50%

m1 = no of times compounding in a year = 2

r2 = interest rate compounded monthly

m2 = no of times = 12

12 0.055 (1+ -12-1=(1+ 2 12 -1

$(1+0.0275)^{2}= (1+\frac{r_2}{12})^{12}$

$1.05575625= (1+\frac{r_2}{12})^{12}$

Taking 12-root on both sides,\

1.00453168 = 1+r2/12

r2 = 5.4380%

So, APR of monthly compounding is 5.4380%

Now, Calculating the Original Loan amount:-

$Monthly Payment = P*r*\frac{(1+r)^{n}}{(1+r)^{n}-1}$

Where, P = Loan AMount

r = Periodic Interest rate = 5.4380%/12 = 0.453166%

n= no of periods = 14yrs*12 =168

Monthly payemnt = $750

$750 = P*0.00453166*\frac{(1+0.00453166)^{168}}{(1+0.00453166)^{168}-1}$

$750 = P*0.00453166*\frac{2.13742144119}{2.13742144119-1}$

P = $68,075.64

So, Original amount of Mortgage is $68,075.64