Answer to Question #120135 in Financial Math for Reynold

Question

I. A family buys a house worth $326,000. They pay $75,000 deposit and take a mortgage for the balance at J12=9% p.a. to be amortized over 30 years with monthly payments.

A. Find the value of the mortgage on their house? (1 mark)
B. Find the value of the monthly payment? (3 marks)
C. Find the loan outstanding after making 20 payments? (4 marks)
D. Find the principal repaid in the 21st payment? (5 marks)

Answer 1

cost of the house : $ 326,000

Initial deposit : $ 75,000

mortgage value : $(326,000 - 75,000) = $ 251,000

$\bold{Answer : A }$ $ $\bold{251,000}$

Now Amortizing period = 360 months (30 years)

rate of interest is 9%.

and per month rate of interest is (9/12)= 0.75%.

B.

To calculate per month payment we will use the following formula :

$$P_n=x*\frac{(1-(1+r)\raisebox{0.25em}{-n})}{r}$$

x: monthly payment

P: principal amount.

r: rate of interest converted to per month equivalent.

n: number of periods.

Now here

P = $251,000

r = 9/12=0.75% = 0.0075

n= 30 years = 360 months.

$\therefore$ $251000=(x)\frac{1-(1+0.0075)\raisebox{0.25em}{-360}}{0.0075}$

or $x= \frac{251000*0.0075}{1-(1.0075)\raisebox{0.15em}{-360}}$

$= \frac{1882.5}{0.932113}$

$= 2019.59$

$\bold{Answer: B}$ monthly payment is $ 2019.6

C.

After making 20 payments ,number of months left to be paid is (360-20)=340

so we need to calculate the outstanding balance of left 340 months when we know family is paying $2019.6 every month @ 9% annually.

$$P_n=x*\frac{(1-(1+r)\raisebox{0.25em}{-n})}{r}$$

where x is the monthly payemnt ,here x = $2019.6

r = rate of interest ,here r= 0.0075

and n = number of months left ,here n = 340

so,

$P_n=2019.6\frac{(1-(1+0.0075)\raisebox{0.25em}{-340})}{0.0075}$

$= 2019.6 * \frac{0.921171}{0.0075}$

$= 2019.6*122.8228$

$= 248052.92$

Answer : C the balance outstanding after making 20 payments is $ 248052.92

D.

Principal repaid in the 21st payment:

This can be calculated simply by applying P₃₄₀ $-$ P₃₃₉

Now

P₃₃₉ $=$ $2019.6 * \frac{1-(1+0.0075)\raisebox{0.15em}{-339}}{0.0075}$

$= 2019.6*$ $\frac{1-0.0794}{0.0075}$

$= 2019.6*\frac{0.9206}{0.0075}$

$= 2019.6*122.747$

$= 247899.84$

so principal paid in 21st payment is

P₃₄₀ $-$ P₃₃₉ $= 248052.92-247899.84$

$= 153.08$

Answer : D: $ 153.08