You take out a car loan of $

$1$

$0$

$,$

$0$

$0$

$0$

$$at

$1$

$2$

$\%$

$$with monthly compounding. The loan is to be paid off in monthly installments over a

$5$

$0$

$-$

month period.

$($

a

$)$

$$Construct the amortization schedule for the loan and provide the monthly payment, interest payment, principal payment and outstanding balance for the

$1$

$5$

th payment period the

$2$

$5$

th payment period, and the

$4$

$5$

th payment period

$($

outstanding balance for the end of the periods

$)$

$.$

$($

b

$)$

$$After

$1$

$5$

$$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

$1$

$5$

$$months$?$

$($

d

$)$

$$Suppose instead of paying off the loan in

$1$

$5$

months, you increase your payments by a fixed amount, $

$1$

$0$

$0$

$$per month

$($

beginning at the

$1$

$5$

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

$1$

$5$

th month, or increase the payments by $

$1$

$0$

$0$

$$beginning at the end of the

$1$

$5$

th month? 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?