Question

You are planning to buy a house worth $500,000 today. You have $100,000 savings for the down payment. A 30-year fixed-rate mortgage is 6.00% APR.

a. Suppose if you take the 30-year loan, how much is your monthly payment?

b. Show the payment for interest and principal separately for the first three months.

c. Ten years later, the 30-year fixed-rate mortgage rate reduces to 3% APR. So you decide to refinance the debt with a new 30-year fixed-rate loan. How much is the new monthly payment?

Answer 1

a) Cost of house today = $500000

Down payment made = $100000

Loan taken on balance amount i.e. = $ 500000-100000 = $ 400000

We have to calculate monthly emi

n = 30 years * 12 = 360 months

r = 6%/12 = 0.5%

PV = $ 400000

PV = PMT [1 - (1+r)^-n]/r

$400000 = PMT [1-(1+0.005)^-360]/0.005

$400000 = PMT [1-0.1660]/0.005

$400000 = PMT [0.8340]/0.005

$400000 = PMT *166.7916

PMT = $2398.2023

montlhy installments would be = $ 2398

b) Emi as calculated above.

Interest is calculated as Loan amount outstanding * interest rate  

Eg : 400000*0.5% =$2000

$399602*0.5% = $1998

Principal amount is calculates as EMI amount - interest amount

Balance outstanding is calculated as Balance loan amount - principal amount

Month	EMI	Principal amount	interest amount	Balance outstanding
1	2398	$398	$2000	$399,602
2	2398	$400	$1,998	$399,202
3	2398	$402	$1,996	$398,800

c) After 10 years, monthly payment would be in case of refinancing if APR reduces to 3%

Loan balance outstanding = $334,743 ( as calculated from amoritisation schedule with formullas mentioned in part b)

r = 3%/12 = 0.25%

n = 30 years * 12 = 360months

PV = PMT [1 - (1+r)^-n]/r

$ 334,743 = PMT[1-(1+0.0025)^-360]/0.0025

$334743 = PMT[1-0.4070]/0.0025

$334743 = PMT * 237.2

PMT = $ 1411.2268

So, after refinancing at new apr of 3% , monthly repayments would be $1411