$1.$ You take out a car loan of $$10,000$ at $12\%$ with monthly compounding. The loan is to be paid off in monthly installments over a $50-$month period.

$($a$)$ Construct the amortization schedule for the loan and provide the monthly payment, interest payment, principal payment and outstanding balance for the $15$th payment period the $25$th payment period, and the $45$th payment period $($outstanding balance for the end of the periods$).$

$($b$)$ After $15$ months you decide to pay off the entire loan. What is the balance that needs to be paid off?

$($c$)$ How much interest is saved by paying off the loan after $15$ months?

$($d$)$ Suppose instead of paying off the loan in $15$months, you increase your payments by a fixed amount, $$100$ per month $($beginning at the $15$th period$)$ for the remaining period of the loan. When is the loan paid off $($what payment period$)?$

$($e$)$ How much interest is saved in this scenario?

$($f$)$ Would it be more suitable to pay off the loan at the end of the $15$th month, or increase the payments by $$100$ beginning at the end of the $15$th month?

$2.$ Now consider that instead of an amortized loan, you repaid the loan in $(1)($a$)$ using a sinking fund approach $($where you can earn a $5\%$ monthly compounded rate for the sinking fund deposits, and the loan rate is the same as in $(1)($a$)).$

$($a$)$ Construct the sinking fund schedule for the loan and provide the sinking fund deposit, interest earned on the sinking fund, fund increase and fund balance for the $15$th period the $25$th period, and the $45$th period $($fund balance for the end of the periods$).$

$($b$)$ Compare the total interest paid $($interest paid on the loan$)$ for both the amortized loan $(1($a$))$ and sinking fund loan. Is one method financially superior to the other?

$($c$)$ What determines the more financially beneficial approach?

$($d$)$ What interest rate would equalize the two methods?