Answer to Question #121975 in Financial Math for Piya

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?
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?


III. Suppose that after making 50 payments, the interest rate changes to J2=9% p.a.:

A. Convert the interest rate J2=9% to J12 equivalent
B. Assuming that the family seeks to accept the change in interest rates, what would be their new payment based on the new interest rate?
C. Assuming that the family seeks to continue their initial monthly payment calculated in part I, how many full payments would be required to pay off the loan and what would be the final concluding smaller payment one period later?

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 21st payment = $248,053.15 - $247,893.95

P₂₁ = $ 159.20

III.

A. J1=(1+j²/2)²−1=(1+0.045)²=0.092025

J12=12((1+0.092025/1)^1/12-1)

i=0.088357=8.8357%

B. New payment

Tmonths = 30*12 = 360 months

No. of months/payments remaining after 50th payment = 360 – 50 = 310

J12 = 9% compounded semi-annually = 9%*2 = 18%

Interest rate per month = 18% / 12 = 1.5% per month

PMT = 251,000 * (1.5%) * (1.015) / ((1- (1.015)^-310)

PMT = $ 3,802.64  

C. No. of full payments:

I=7.5% =0.0075

No. of payments = 30*12 = 360 payments.

Final concluding smaller payment one period later

Principal = P₃₆₀-P₃₅₉

Full payment: P₃₆₀ = 251,000

P₃₅₉ = 2019.6 * (1- (1.0075)^-359) / (0.0075)

P₃₅₉ = $  250,862.55  

PMT = $ 251,000 - $ 250,862.55 

PMT = $ 137.45