Problem $1$: Discounted Cash Flow Valuation $(30$ points$)$

You are planning to buy a house. After subtracting the downpayment, closing costs, and

fees, you require a loan with a principal of $$350,000.$ Your bank offers a $30-$year loan

with a fully amortizing structure, monthly payments, and an effective annual rate $($EAR$)$

of $6\%.$ Assume the periodic payments of the loan will start on January $1,2025,$ and will

be paid on the first day of each month.

Part I

A$)$ For each period obtain the periodic payment, interest, principal payment, and

principal outstanding.

B$)$ Obtain the total amount paid

Part II

You know that any payment above the periodic payment given by the fully amortizing

loan will count towards repaying the principal, thus reducing the time needed to

repay the loan and the total amount paid. You plan to pay $$100$ above the required

amount each month.

C$)$ For each period until the loan has been completely repaid, obtain the periodic

payment, interest, principal payment, and principal outstanding

D$)$ How long does it take you to pay the loan?

E$)$ Obtain the total amount paid

Part III

Your new job awards a discretionary bonus based on performance at the end of

each year, and you are confident in your ability to obtain them. Because of this, you

plan to pay $($in addition to the extra $$100$ described in part II$)$ one full extra payment

each January, starting with January $1,2026.($periodic payments for each January,

starting from $2026$ will be $2$xA $+100)$

F$)$ For each period until the loan has been completely repaid, obtain the periodic

payment, interest, principal payment, and principal outstanding

G$)$ How long does it take you to pay the loan?

H$)$ Obtain the total amount paid