The prepayment fee of $6182.58 would be charged to Artemis for paying off her loan 16 years early.
Answer: Option C
<u>Step-by-step explanation:</u>
30 year loan at 9.6% interest yields.
Number of month = 30 (12) = 360 months
Annual percent interest of
= monthly percent interest of .8%
The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows
![P M T= \frac{P \times r}{1-(1+r)^{n}}](https://tex.z-dn.net/?f=P%20M%20T%3D%20%5Cfrac%7BP%20%5Ctimes%20r%7D%7B1-%281%2Br%29%5E%7Bn%7D%7D)
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
![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}](https://tex.z-dn.net/?f=P%20M%20T%3D%20%5Cfrac%7B190000%20%5Ctimes%200.008%7D%7B1-%281%2B0.008%29%5E%7B-360%7D%7D%20%3D%20%5Cfrac%7B1520%7D%7B1-%281.008%29%5E%7B-360%7D%7D%20%3D%20%5Cfrac%7B1520%7D%7B1-0.0567%7D%3D%5Cfrac%7B1520%7D%7B0.9432%7D)
PMT = $1611.50 per month
Her loan 16 year early. It means
![\text { Worth of loan after } 14 \text { year } = 190000 \times(1.008)^{168} = 3.814 \times 190000=\$ 724641.16](https://tex.z-dn.net/?f=%5Ctext%20%7B%20Worth%20of%20loan%20after%20%7D%2014%20%5Ctext%20%7B%20year%20%7D%20%3D%20190000%20%5Ctimes%281.008%29%5E%7B168%7D%20%3D%203.814%20%5Ctimes%20190000%3D%5C%24%20724641.16)
Worth of monthly payments for 14 year
![= \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](https://tex.z-dn.net/?f=%20%3D%20%5Cfrac%7B1611.50%20%5Ctimes%5Cleft%5C%7B%281.008%29%5E%7B168%7D-1%5Cright%29%7D%7B0.008%7D%20%3D%20%5Cfrac%7B1611.50%20%5Ctimes%283.81-1%29%7D%7B0.008%7D%20%3D%20%5Cfrac%7B4.534%20.6%7D%7B0.008%7D%20%3D%20%5C%24%20566825.15)
Amount still owed after 14 year = difference of the above two
=$724641.16 - $566825.18
=$157816.01
Prepayment fee = ![(0.8 \times 157816.02) \times\left((1.008)^{6}-1\right)](https://tex.z-dn.net/?f=%280.8%20%5Ctimes%20157816.02%29%20%5Ctimes%5Cleft%28%281.008%29%5E%7B6%7D-1%5Cright%29)
= 126252.82 (1.049-1) = 126252.82 (1.0489-1) = $6182.63