A car dealership borrowed $1,000,000 from its local Bank to finance its inventory of 2016 new cars. The terms of the loan were as follows: 3% compounded continuously for first 3 years. 4% compounded continuously for the next 2 two years. 5% compounded continuously for the last 5 years. The dealership, however, wished to pay EVEN monthly payments to the Bank for the life of the loan, or for 10 years. What is the dealership monthly payment? And what is the effective annual interest rate cost to the dealership? Thanks a lot for any help.
The easiest way to tackle this interesting problem is to first convert all interest rates from continuous compound to monthly compound as follows:
3% comp. cont.=3.00375312695 comp. monthly.
4% ,,,,,,,,,,,,,,,,,, =4.00667408025 ,,,,,,,,,,,,,,,,,,,,,,,,,
5%,,,,,,,,,,,,,,,,,,,,=5.01043114934,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Next, we project the FV of the $1,000,0000 for the next 10 years at the above rates using this common TVM formula:FV=PV[1 + R]^N. After we do that for the above 3 terms and their respective interest rates, we get this FV for the $1,000,000=$1,521,961.56. From these two amounts and using the above TVM formula, we can easily find the averaged interest rate. We find that it comes to=4.20735858258 comp. monthly, which is equivalent to effective annual rate of=4.29%.
Now, will use this common TVM formula to find the monthly payment of the dealership:PMT=PV. R.{[1 + R]^N/ [1 + R]^N - 1}=PMT NEEDED TO PAY OFF A LOAN OF $1.
When we plug all the numbers into this formula, we find that the even monthly payments comes to=$10,223.36. And that is it.
