Question

Rachael has a 100,000, 30 year, fixed mortgage with a 12% nominal interest rate convertible monthly.

She has made payments at the end of each month for ten years.

Now she will begin making twice the payment each month.

How many years will she be able to take off the original 30 years assuming a balloon payment for the final fractional payment?

A. 12

B. 13

C. 14

D. 15

E. 16

Answer 1

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PV = Loan Amount = 100,000

n = 30 * 12 = 360 months

r = monthly interest rate = 12%/2 = 1%

Monthly loan payment = [r*PV] / [1 - (1+r)^-n]

= [1% * 100,000] / [1 - (1+1%)^-360]

= 1000 / 0.972183311

= 1,028.6126

P = Monthly loan payment = 1,028.61

x = payments made = 10*12 = 120 months

Outstanding loan balance = P * [1 - (1+r)^-(n-x)] / r

= 1,028.61 * [1 - (1+1%)^-(360-120)] / 1%

= 1,028.61 * 0.908194164 / 0.01

= 93,417.7599

PV1 = Outstanding loan balance = 93,417.76

P1 = new monthly payment = 2*1028.61 = 2,057.22

PV1 = P1 * [1 - (1+r)^-n] / r

93,417.76 = 2,057.22 * [1 - (1+1%)^-n] / 1%

[1 - (1.01)^-n] = 0.454097082

(1.01)^-n = 0.545902918

(1.01)^n = 1.83182754

n = log (1.83182754) / log(1.01)

n = 0.262884584 / 0.00432137378

n = 60.8335677 months

n = 5 yeras

Therefore, it will take of 15 years take off original 30 year loan assuming balloon payment for final fractional payment