1.mortgage has 25 years left, and has an APR of 7.625% 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 $150,000 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 25-year mortgage interest rates have dropped to 5% (APR), what is the lowest monthly payment you could make for the remaining life of your loan that would be attractive to the bank?

answer-

a. The monthly discount rate is

0.07625/12=0.635%

Present Value= (1449/0.00635){1-(1/1.00635)³⁰⁰} =194,024.13.

b. Here the present value is $150,000 and the monthly payment needs to be calculated.

r=5 /1200 0.004167

Payment=150000 0.004167/ (1-1/1.004167^25*12) =876.88

2.You have an outstanding student loan with required payments of $500 per month for the next four years. The interest rate on the loan is 9% APR (monthly). You are considering making an extra payment of $100 today (that is, you will pay an extra $100 that you are not required to pay). If you are required to continue to make payments of $500 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 $100?

answer-

We begin with the timeline of our required payments

0  1 2 47 48

–500 –500   –500 –500

(1) Let’s compute our remaining balance on the student loan. As we pointed out earlier, the remaining balance equals the present value of the remaining payments. The loan interest rate is 9% APR, or 9% / 12 = 0.75% per month, so the present value of the payments is 

PV=(500/0.0075)(1-1/1.0075⁴⁸) =$20, 092.39

Using the annuity spreadsheet to compute the present value, we get the same number:

N I PV PMT FV

48 0.75 % 20,092.39 –500 0

Thus, your remaining balance is $20,092.39.

If you prepay an extra $100 today, your will lower your remaining balance to $20,092.39 – 100 = $19,992.39. Though your balance is reduced, your required monthly payment does not change. Instead,you will pay off the loan faster; that is, it will reduce the payments you need to make at the very end of the loan. How much smaller will the final payment be? With the extra payment, the timeline changes:

0 1 2 47 48

19,992.39 -500 -500 -500 -(500-X)

That is, we will pay off by paying $500 per month for 47 months, and some smaller amount, $500 – X,in the last month. To solve for X, recall that the PV of the remaining cash flows equals the outstanding balance when the loan interest rate is used as the discount rate:

19, 992.39=(500/0.0075)(1-1/(1+ 0.0075)⁴⁸)-X/1.0075⁴⁸ 

Solving for X gives

19,992.39 =20,092.39-X/1.0075⁴⁸ 

X =$143.14

So the final payment will be lower by $143.14.

You can also use the annuity spreadsheet to determine this solution. If you prepay $100 today, and make payments of $500 for 48 months, then your final balance at the end will be a credit of $143.14:

N I PV PMT FV

48 0.75 % 19,992.39 -500 143.14

(2) The extra payment effectively lets us exchange $100 today for $143.14 in four years. We claimed that the return on this investment should be the loan interest rate. Let’s see if this is the case:

$100 ×( 1.0075)⁴⁸ = $143.14 , so it is.

Thus, you earn a 9% APR (the rate on the loan).

3.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 $500, or $750 in total each month. How long will it take you to pay off the loan?

answer-

The timeline in this case is:

0 1 2 N

20,092.39 -750 -750 -750

and we want to determine the number of monthly payments N that we will need to make. That is, we need to determine what length annuity with a monthly payment of $750 has the same present value as the loan balance, using the loan interest rate as the discount rate. As we did in Chapter 4, we set the outstanding balance equal to the present value of the loan payments and solve for N.

(750/0.0075)(1-1/1.0075^N) =20, 092.39

(1-1/1.0075^N)=(20, 092.39* 0.0075) /750=0.200924

1/1.0075^N=1- 0.200924 =0.799076

1.0075^N  =1.25145

N=Log(1.25145)/Log(1.0075)=30.02

We can also use the annuity spreadsheet to solve for N.

N I PV PMT FV

30.02 0.75 % 20,092.39 –750 0

So, by prepaying the loan, we will pay off the loan in about 30 months or 2 ½ years, rather than the four years originally scheduled. Because N of 30.02 is larger than 30, we could either increase the 30th payment by a small amount or make a very small 31st payment. We can use the annuity spreadsheet to determine the remaining balance after 30 payments.

N I PV PMT FV

30 0.75 % 20,092.39 –750 –13.86

If we make a final payment of $750.00 + $13.86 = $763.86, the loan will be paid off in 30 months.