Question

You have decided to refinance your mortgage. You plan to borrow whatever is outstanding on your current mortgage. The current monthly payment is $2,356 and you have made every payment on time. The original term of the mortgage was 30​ years, and the mortgage is exactly four years and eight months old. You have just made your monthly payment. The mortgage interest rate is ​7.500% (APR). How much do you owe on the mortgage​ today? ​(Note: Be careful not to round any intermediate steps less than six decimal​ places.) The amount you owe today is ​$___

Answer 1

Loan amount = PV of EMIs.

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$            2,356.00
Int Rate	0.6250%
Periods	360

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 2356 * [ 1 - [(1+0.0063)^-360]] /0.0063
= $ 2356 * [ 1 - [(1.0063)^-360]] /0.0063
= $ 2356 * [ 1 - [0.1061]] /0.0063
= $ 2356 * [0.8939]] /0.0063
= $ 336949.53
Loan Outstanding after 56 Months ( ( 4 *12 )+ 8 ):

Particulars	Amount
Loan Amount	$ 336,949.53
Int rate per Month	0.6250%
No. of Months	360
Outstanding Bal after	56
EMI	$      2,356.00
Payments Left	304

Outstanding Bal = Instalment * [ 1 - ( 1 + r )^ - n ] / r
= $ 2356 * [ 1 - ( 1 + 0.00625 ) ^ - 304 ] / 0.00625
= $ 2356 * [ 1 - ( 1.00625 ) ^ - 304 ] / 0.00625
= $ 2356 * [ 1 - 0.150456 ] / 0.00625
= $ 2356 * [ 0.849544 ] / 0.00625
= $ 320244.11

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

Mortgage balance after 56 Months is $ 320244.11