F:V

Amortized Loans Homework Problems

1. Your bank pays 8% annual interest, compounded monthly. Suppose you deposit $100 each month for 18 years.

a) What is the total amount you deposited?

b) What is the future value of the account (at the end of the 18 years)?

Answers:

a) $21,600

b) This is a problem that, strictly speaking, we shouldn't ask until after we have covered annuities.

The future value is given by

(1 + i)n - 1 A

i

where

n

=

18 ? 12

=

216,

i

=

.08 12

,

and

the

montly

payment

A

=

100.

Plug

in

those

numbers

and

get, rounded to the nearest penny,

F.V. = $48008.61

2. In the previous problem you had hoped to have a future value of $96, 019. What would the monthly deposit have to be? Answer: $200.00

3. Sandy, who is about to graduate is considering taking out a loan at 6% to buy her first car, an Oldsmobile Alero. She plans to finance the entire $17, 785 at 6% for 36 months. What would be the monthly payment? How much total interest will she pay? Answer: $541 $1691

4. See the previous problem. Oldsmobile has a special program for graduating seniors: 3.9% annual interest. Sandy decided to buy the Alero under this program. What is her monthly payment? How much interest does she pay? Answer: $524 $1079

5. What is the future value of $541 deposited monthly, at an annual rate of 6% (compounded monthly) at the end of three years? Answer: $21280.83

6. Suppose you borrow 150, 000 dollars. You have arranged for a 15 year (amortized) loan with an interest rate of 8% (compounded monthly). a) What will your monthly payment be? b) After 10 years, what will the balance (amount owed) be? c) After those 10 years, how much interest will you have paid? d) After 15 years, when the loan expires, what will be your total interest paid?

e) You decide to send an extra $200 per month to pay off the loan earlier. When will the loan be paid off now? f) By sending an extra $200 per month, what will be your total interest paid? Answer: a) With P = 150, 000 and i = .08/12 and n = 15 ? 12 = 180,

P i(1 + i)n A = (1 + i)n - 1 = 1433.48 to the nearest penny. b) After n payments, the amount owed is P (1 + i)n - A (1 + i)n - 1

i so with A = 1433.48 and n = 12 ? 10 = 120, we get

$70, 696.90 to the nearest penny. c) The principal paid up to this point would be P - the amount owed. So the balance is

150000 - 70696.90 = $79, 303.10

The total amount of money paid up to this point would be n ? A = 120 ? 1433.48 = $172053.60.

If you stored the monthly amount in your calculator, so you didn't round first, your answer could be slightly different.

The interest paid up to this point would be (total paid) - (the principal paid).

172053.60 - 79303.10 = $92, 750.50 d) The total paid after 15 years is 180 ? 1433.48 = 258026.40. The principal is 150, 000, so the interest paid is

258026.40 - 150000 = $108, 026.40 . Again, if you stored your numbers without rounding, your numbers will be slightly different. If you use the Excel program, you would get $258, 026.06 because there is no rounding in the computations until the final answer is displayed.

e) After n payments, the amount owed is

P (1 + i)n - A (1 + i)n - 1 i

This time we know A = 1433.48 + 200 = 1633.48 and we need to know the value of n, the number of payments, which makes the amount owed 0. You can use logarithms and solve for n:

n

=

ln(

A A-P

i

)

=

142.56

months

ln(1 + i)

rounding to the nearest hundredth.

An alternative is to plot the function 150000(1+.08/12)x-1633.48((1+.08/12)x-1)/(.08/12) as a function of x and find where it is zero.

f) With this value of n, the total paid is

142.56 ? 1633.48 = $232867.32 so the total interest paid under this scheme is

232867.32 - 150000 = $82, 867.32

Depending when you did your rounding, you could have slightly different numbers.

7. This question concerns a loan of $100, 000 at 8% per year, compounded monthly. You are making a regular payment of $750 per month. a) Fill in the entries in the schedule for the first 3 months:

Payment/.Month # 0 1 2 3

Interest 0

Principal 0

Balance 100,000

b) What will be the balance after the 120th payment? c) After how many payments will the loan be repaid? d) Approximately how much interest will you have paid on this loan? Answer: b) $84, 754.50 c) 331 d) $148, 000.

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