MATH SOLVE

8 months ago

Q:
# Artemis took out a 30-year loan from her bank for $190,000 at an APR of9.6%, compounded monthly. If her bank charges a prepayment fee of 6months' interest on 80% of the balance, what prepayment fee would Artemisbe charged for paying off her loan 16 years early?OA. $5822.42OB. $7390.60O C. $6182.58O D. $7382.56

Accepted Solution

A:

The prepayment fee of $6182.58 would be charged to Artemis for paying off her loan 16 years early.Answer: Option CStep-by-step explanation:30 year loan at 9.6% interest yields. Number of month = 30 (12) = 360 months
Annual percent interest of [tex]\frac{9.6 \%}{12}[/tex] = monthly percent interest of .8%The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows [tex]P M T= \frac{P \times r}{1-(1+r)^{n}}[/tex]With r and n adjusted for periodicity, where
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the interest rate (also known as the discount rate)
n = the number of periods in which payments will be made
[tex]P M T= \frac{190000 \times 0.008}{1-(1+0.008)^{-360}} = \frac{1520}{1-(1.008)^{-360}} = \frac{1520}{1-0.0567}=\frac{1520}{0.9432}[/tex] PMT = $1611.50 per monthHer loan 16 year early. It means [tex]\text { Worth of loan after } 14 \text { year } = 190000 \times(1.008)^{168} = 3.814 \times 190000=\$ 724641.16[/tex]Worth of monthly payments for 14 year [tex] = \frac{1611.50 \times\left\{(1.008)^{168}-1\right)}{0.008} = \frac{1611.50 \times(3.81-1)}{0.008} = \frac{4.534 .6}{0.008} = \$ 566825.15[/tex]Amount still owed after 14 year = difference of the above two
=$724641.16 - $566825.18
=$157816.01
Prepayment fee = [tex](0.8 \times 157816.02) \times\left((1.008)^{6}-1\right)[/tex] = 126252.82 (1.049-1) = 126252.82 (1.0489-1) = $6182.63