Question

What us the payoff on a mortgage loan of a $200,000 with a contract intrest rate of 4 percent annual rate with monthly compounding after five years (60 payments have been made)? the monthly payments of $954.83.

$200,000

$165,210

$180,895

$177,345

$140,000

Answer 1

Loan shall be equal to PV of EMIs made.

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
PV Annuity	$    200,000.00
Int Rate	0.333%
Cash Flow	$            954.83

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
$ 200000 = $ 954.83 * [ 1 - [ ( 1 + 0.00333 ) ^ - n ] ] /0.00333
0.6982 = [ 1 - [ ( 1.00333 ) ^ - n ] ]
[ ( 1.00333 ) ^ - n ] = 1 - 0.6982
[ ( 1.00333 ) ^ - n ] = 0.3018

Take Log on both sides
Log [ ( 1.00333 ) ^ - n ] = Log ( 0.3018)
Log( a^ b ) = b * Log (a )

-n * Log ( 1.00333 ) = Log ( 0.3018)
-n * 0.00144 = -0.52028
-n = -0.52028 / 0.00144
n = 0.52028 / 0.00144
n = 361.31
Original Loan period is 360 Months ( Difference is due to rounding off problem).

Loan balance after 60 Months:

Particulars	Amount
Loan Amount	$ 200,000.00
Int rate per Month	0.3333%
No. of Months	360
Outstanding Bal after	60
EMI	$         954.83
Payments Left	300

Outstanding Bal = Instalment * [ 1 - ( 1 + r )^ - n ] / r
= $ 954.83 * [ 1 - ( 1 + 0.003333 ) ^ - 300 ] / 0.003333
= $ 954.83 * [ 1 - ( 1.003333 ) ^ - 300 ] / 0.003333
= $ 954.83 * [ 1 - 0.368492 ] / 0.003333
= $ 954.83 * [ 0.631508 ] / 0.003333
= $ 180912.93

Answer C is correct. Difference is due to rounding off Problem.

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