Question

1. When you purchased your car, you took out a five-year annual-payment loan with an interest rate of 6.5% per year. The annual payment on the car is $4,500. You have just made a payment and have now decided to pay off the loan by repaying the outstanding balance. What is the payoff amount for the following scenarios?
a. You have owned the car for the one year (so there are four year left on the loan)?
answer: $15,416.09

b. You have owned the car for four years (so there is one year left on the loan)?
answer: $4,225.35

2. You have a loan outstanding. It requires making eight annual payments of $7,000 each at the end of the next eight years. Your bank has offered to restruture the loan so that instead of making the eight payments as originally agreed, you will make only one final payment in eight years. If the interest rate on the loan is 1%, what final payment will the bank require you to make so that it is indiferent to the two forms of payment?
answer: $58,000

3. You have an investment account that started with $3,000 10 years ago and which now has grown to $11,000.
a. What annual rate of return have you earned (you have made no additional contributions to the account)?
answer: 13.87%

b. If the investment account earns 15% per year from now on, what will the account's value be 10 years from now?
answer: $44,501.14

I got wrong and got answer but i tried to do myself, and yet i could not get same answer.

Can anyone show me step by step?

Also method of use finance calculator will be fine too.

Thank you.

Answer 1

1) To find the periodic payments (annual in our case) on any loan, we use the following formula -

Annual payments = Loan Amount / PVIFA (rate, n)

PVIFA here is the present value factor annuity, r is the periodic (annual) rate of interest, n is the no. of periods remaining

Now, we know the annual payments, so we can compute the remaining loan amount / outstanding balance as -

Loan amount = Annual payments x PVIFA (rate, n)

a) Outstanding balance = $4500 x PVIFA (6.5% , 4) = $4500 x 3.42579860158 = $15,416.09

b) Outstanding balance = $4500 x PVIFA (6.5%, 1) = $4500 x 0.93896713615 = $4,225.35

2) This is question is a form of ordinary annuity and we require to compute the future value of the annuity, which can be computed as follows -

where, FV = future value, A = periodic payments, r = rate of interest, n = no. of periods remaining

3) a) So, we are given the present value and the future value of the same investment. Now, present value is computed as follows -

PV = FV x PVIF(r, n)

or, $3000 = $11000 x PVIF (r, 10)

or, PVIF (r, 10) = 0.27273

We refer this value in the Present value interest factor table in year 10 -

At 13% = 0.29459

At 14% = 0.26974

Our value lies between the two, so we need to interpolate -

Difference required (from 13%) = 0.29459 - 0.27273 = 0.02186

Total difference (between 13% and 14%) = 0.29459 - 0.26974 = 0.02485

Rate = Lower rate + Difference in rates x (Difference required / Total difference)

or, Rate = 13% + 1% x (0.02186 / 0.02485) = 13.87%

b) Future value is computed as follows -

FV = PV x (1 + r)ⁿ = $11000 x (1 + 0.15)¹⁰ = $44,501.14