You have just purchased a home and taken out a $$460,000$ mortgage. The mortgage has a $30-$year term with monthly payments and an APR of $6\mathrm{.}08\%.$

a$.$ How much will you pay in interest, and how much will you pay in principal, during the first year?

b$.$ How much will you pay in interest, and how much will you pay in principal, during the $20$th year $($i$.$e$.,$ between $19$ and $20$ years from now$)?$

Your mortgage has $26$ years left, and has an APR of $7\mathrm{.}449\%$ with monthly payments of $$1449.$

a$.$ What is the outstanding balance?

b$.$ Suppose you cannot make the mortgage payment and you are in danger of losing your house to foreclosure. The bank has offered to renegotiate your loan. The bank expects to get $$149,638$ for the house if it forecloses. They will lower your payment as long as they will receive at least this amount $($in present value terms$).$ If current $26-$year mortgage interest rates have dropped to $4\mathrm{.}842\%($APR$),$ what is the lowest monthly payment you could make for the remaining life of your loan that would be attractive to the bank?

X $*18.$ You have an outstanding student loan with required payments of $$600$ per month for the next four years. The interest rate on the loan is $10\%$ APR $($monthly$).$ You are considering making an extra payment of $$175$ today $($that is$,$ you will pay an extra $$175$ that you are not required to pay$).$ If you are required to continue to make payments of $$600$ per month until the loan is paid off, what is the amount of your final payment? What effective rate of return $($expressed as an APR with monthly compounding$)$ have you earned on the $$175?$

${}^{}$ X$*19.$ Consider again the setting of Problem $18.$ Now that you realize your best investment is to prepay your student loan, you decide to prepay as much as you can each month. Looking at your budget, you can afford to pay an extra $$250$ per month in addition to your required monthly payments of $$600,$ or $$850$ in total each month. How long will it take you to pay off the loan?

$*20.$ Oppenheimer Bank is offering a $30-$year mortgage with an APR of $5\mathrm{.}25\%$ based on monthly compounding. With this mortgage your monthly payments would be $$2,000$ per month. In addition, Oppenheimer Bank offers you the following deal: Instead of making the monthly payment of $$2,000$ every month, you can make half the payment every two weeks $($so that you will make $\frac{52}{2}=26$ payments per year$).$ With this plan, how long will it take to pay off the mortgage if the EAR of the loan is unchanged?

${}^{}$ X$*21.$ Your friend tells you he has a very simple trick for shortening the time it takes to repay your mortgage by one$-$third: Use your holiday bonus to make an extra payment on January $1$ of each year $($that is$,$ pay your monthly payment due on that day twice$).$ Assume that the mortgage has an original term of $30$ years and an APR of $12\%.$

a$.$ If you take out your mortgage on January $1($so that your first payment is due on February $1),$ and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage?

b$.$ If you take out your mortgage on July $1($so the first payment is on August $1),$ and you make the extra payment each January, in how many months will you pay off your mortgage?

c$.$ How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan?

$\frac{}{}$ X$22.$ You need a new car and the dealer has offered you a price of $$20,000,$ with the following payment options: $($a$)$ pay cash and receive a $$2000$ rebate, or $($b$)$ pay a $$5000$ down payment and finance the rest with a $0\%$ APR loan over $30$ months. But having just quit your job and started an MBA program, you are in debt and you expect to be in debt for at least the next $2\frac{1}{2}2$ years. You plan to use credit cards to pay your expenses; luckily you have one with a low $($fixed$)$ rate of $14\mathrm{.}51\%$ APR. Which payment option is best for you?

$23.$ The mortgage on your house is five years old. It required monthly payments of $$1390,$ had an original term of $30$ years, and had an interest rate of $10\%($APR$).$ In the intervening five years, interest rates ha