ABC Constrution Company borrowed $1,000,0000 from its local Bank for a term of 5 years. The interest charged is as follows: 1st. year=3%compounded monthly, 2nd.year=4%C.M., 3rd year=5%C.M., 4th. year=6% C.M., 5th. year=7%C.M. The Company wishes to pay the Bank in equal monthly payments for the entire 60 months. How would you calculate these equal payments? What is the total interest paid to the Bank over the 5-year period? Finally, what is the overall cost, one combined interest rate, to the Construction Company over the same period? Thank you for your help.
To solve this problem, at least 2 common TVM formulae will be used repeatedly:
1 - PV=P{[1 + R]^N - 1.[1 + R]^-N} R^-1=PV OF $1 PER PERIOD
This formula will find the PV of $1 per period, or $12 for each year at their respective interest rates.
Then these PV's will be brought forward to the present, discounting them each year at their respective rates using this 2nd and very common TVM formula:
2- PV=FV[1 + R]^-N=PV OF $1 IN THE FUTURE.
Then these PV's for the 5 years will be added together to give us the PV of all 60 payments.
Then we simply divide $1,000,000(I'm taking it as $1,000,000. If it is $10,000,000, then just multiply everything by 10).
Doing it by hand will involve about a dozen calculations. However, my computer is already programmed with all these financial formulae and will arrive at the answer very rapidly. Plugging in all numbers, the computer gives the accumulated PV as=54.0611975224. Now we simply divide $1,000,000 by this number and we get the even monthly payment of:$1,000,000/54.0611975224
=$18,497.56. Sixty payments of this amount will be made to pay off the entire $1,000,000 loan in 5 years.
Total interest paid will be: $18,497.56 x 60 - $1,000,000=$109,853.31.
The overall cost to the Company, or the combined interest rate, will be=4.17932032174%.
Finally, amortization of this loan confirms that all these numbers are accurate.
Once the overall interest is found, then the PV of all the 60 payments can be summed up on any good calculator, such as Wolfram/Alpha: Here is the same value summed up on W/A:
http://www.wolframalpha.com/input/?i=%E2%88%91%5B1+%2F+1.00348276693478333333333333333333%5E(n%2B1)%5D,+n%3D0+to+59
=54.06119752239511449364170823961036475493978759735782576366
