Question

It has been exactly 6 years since Alan bought his house for $375,000. Six years ago, he paid $75,000 as a down payment and took out a 35-year mortgage for the remaining amount with monthly payments and a quoted APR of 5.5% (with semi-annual compounding). Today, Alan has received a promotion to a higher paying managerial position at his firm. He has decided to pay part of his mortgage in order to reduce his monthly payments. Alan has already made his payment for this month, but he will now make a lumpsum payment of $75,000. What will be his new monthly payments (starting next month) for the rest of the mortgage term, assuming the interest rate stays the same?

Answer 1

EMI :
EMI or Instalment is sum of money due as one of several equal payments for loan/ Mortgage taken today, spread over an agreed period of time.

EMI = Loan / PVAF (r%, n)
PVAF = SUm [ PVF(r%, n) ]
PVF(r%, n) = 1 / ( 1 + r)^n
r = Int rate per period
n = No. of periods

How to calculate PVAF using Excel:
=PV(Rate,NPER,-1)
Rate = Disc Rate
NPER = No.of periods

Original Loan = Price - Down Payment

= $ 375000 - $ 75000

= $ 300000

Particulars	Amount
Loan Amount	$          300,000.00
Int rate per Month	0.4583%
No. of Months	420

EMI = Loan Amount / PVAF (r%, n)
Where r is Int rate per Month & n is No. of Months
= $ 300000 / PVAF (0.0046 , 420)
= $ 300000 / 186.2141
= $ 1611.05
Loan Outstanding after 6 Years:

Particulars	Amount
Loan Amount	$ 300,000.00
Int rate per Month	0.4583%
No. of Months	420
Outstanding Bal after	72
EMI	$      1,611.05
Payments Left	348

Outstanding Bal = Instalment * [ 1 - ( 1 + r )^ - n ] / r
= $ 1611.05 * [ 1 - ( 1 + 0.004583 ) ^ - 348 ] / 0.004583
= $ 1611.05 * [ 1 - ( 1.004583 ) ^ - 348 ] / 0.004583
= $ 1611.05 * [ 1 - 0.203649 ] / 0.004583
= $ 1611.05 * [ 0.796351 ] / 0.004583
= $ 279939.18

r = Int Rate per period
n = Balance No. of periods

New balance after paying $ 75000

= $ 279939.18 - $ 75000

= $ 204939.18

New EMI:

Particulars	Amount
Loan Amount	$          204,939.18
Int rate per Month	0.4583%
No. of Months	348

EMI = Loan Amount / PVAF (r%, n)
Where r is Int rate per Month & n is No. of Months
= $ 204939.18 / PVAF (0.0046 , 348)
= $ 204939.18 / 173.7492
= $ 1179.51