Answer to Question #121800 in Financial Math for Piya

Question

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?
B. Find the value of the monthly payment?
C. Find the loan outstanding after making 20 payments?
D. Find the principal repaid in the 21st payment?

II. Fill out the loan amortization schedule provided in the solution template for the first 5 loan payments. What do you notice about the composition of the payment amount? Ending Loan Balance with Working for each payment:

Answer 1

A. Find the value of the mortgage on their house?

Amount = $326,000

Deposit = $75,000

Balance = Amount – Deposit

Balance = $326,000 - $75,000 = $ 251,000

B. Find the value of the monthly payment?

T= 30 years

Tmonths = 30*12 = 360 months

J₁₂ = 9%

Interest rate per month = 9% / 12 = 0.75% per month

Monthly payments = Amount

PMT = 251,000 (0.75%) * (1.0075)³⁶⁰ / ( (1.0075)³⁶⁰ -1)

PMT = $ 2019.6

C. Find the loan outstanding after making 20 payments?

Balance Tmonths = 360 – 20 = 340 months

PMT = $2019.6

Amount outstanding = 2019.6 * 1- (1.0075)^-340 / (0.0075)

Amount outstanding = 2019.6 * (1- (1.0075)^-340) / (0.0075)

Amount outstanding = $ 248,053.2

D. Find the principal repaid in the 21st payment?

Principal = P₃₄₀-P₃₃₉

P₃₃₉ = 2019.6 * (1- (1.0075)^-339) / (0.0075)

P₃₃₉ = $ 247,893.95

The principal paid in the 21^st payment = $248,053.15 - $247,893.95

P₂₁ = $ 159.20

II. Loan Amortization Schedule

Composition of the payment amount

As the interest rate decreases over the mortgage repayment period, the principal amount on the other hand increases implying that interest rate and principal payment amounts are inversely correlated. As payment (principal) increases, the ending loan balance is decreased as the loan is retired through consistent payments made over the years. The interest and principal amounts decrease with a margin of $1.03 and $1.05. The final ending balance is on a reducing balance which at the end of the mortgage loan repayment will equal to zero.

Workings

Payment Number 1

Beginning Balance = $251,000

Interest rate per month = 9% / 12 = 0.75% per month

Interest = Beginning balance * Interest rate per month

Interest = 251,000 (0.75%) = $1882.50

Principal = PMT / (1.0075)³⁶⁰

Principal = 2019.6 / 14.730576 = $137.10

Ending Balance = Beginning balance – Principal

Ending Balance = $251,000 – $137.10 = $ 250,862.90  

Payment Number 2

Interest = Beginning balance * Interest rate per month

Interest = $250,862.90 * 0.75% = $ 1,881.47

Principal = PMT / (1.0075)³⁵⁹

Principal = 2019.6 / 14.730576 = $ 138.13

Ending Balance = Beginning balance – Principal

Ending Balance = $250,862.90 – 138.13 = $ 250,724.77   

Payment Number 3

Beginning Balance = $ 250,724.77

Interest = Beginning balance * Interest rate per month

Interest = $ 250,724.77 * 0.75% = $ 1,880.44   

Principal = PMT / (1.0075)³⁵⁸

Principal = 2019.6 / 14.730576 = $ 139.17

Ending Balance = Beginning balance – Principal

Ending Balance = $250,724.77 – $ 139.17 = $250,585.60

Payment Number 4

Beginning Balance = $250,585.60

Interest = Beginning balance * Interest rate per month

Interest = $250,585.60 * 0.75% =$ 1,879.39

Principal = PMT / (1.0075)³⁵⁷

Principal = 2019.6 / 14.730576 = $ 140.21

Ending Balance = Beginning balance – Principal

Ending Balance = $250,724.77 – $ 140.21 = $ 250,445.39

Payment Number 5

Beginning Balance = $250,445.39

Interest = Beginning balance * Interest rate per month

Interest = $ 250,445.39 * 0.75% =$ 1,878.34

Principal = PMT / (1.0075)³⁵⁶

Principal = 2019.6 / 14.730576 = $ 141.26

Ending Balance = Beginning balance – Principal

Ending Balance = $250,445.39 – $ 141.26 = $ 250,304.13