Question 262175: Mr Patel intends to start a dry cleaning business and wishes to borrow money for this purpose.He feels that he will not be able to pay anything for the first three years,but therafter,he is prepared to pay back $4000.00 per year for five years.The bank agrees to advance him money at 18% interest per annum.How much will they be willing to advance him now under these conditions?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! doesn't pay a dime for the first 3 years.
pays back 4000 per year for the next five year.
bank agrees to advance him money at 18% per year for 5 years.
my interpretation of this problem is that the bank will lend him money at 18% per year for 5 years beginning at the end of year 3.
since he will be paying 4000 per year for the same 5 years, the present worth of those payments at 18% per year is equal to $12,508.68408.
the bank is willing to lend him $12,508.68408.
they give that money to him at the beginning of year 1 which is 3 years earlier than the year in which he begins making payments.
he pays then nothing for 3 years.
at the end of year 3 the balance in his account with them is still $12,508.68408.
he pays them $4,000 at the end of year 4, 5, 6, 7, 8.
at the end of year 8 his balance with them is 0.
annual compounding is assumed and end of year payments are assumed.
here's how it looks on a year by year basis.
beginning of year 1 balance = $12,508
end of year 1 balance = $12,508
end of year 2 balance = $12,508
end of year 3 balance = $12,508
end of year 4 balance = $12,508 * 1.18 = $14,759 - $4,000 = $10,759
end of year 5 balance = $10,759 * 1.18 = $12,696 - $4,000 = $8,696
end of year 6 balance = $8,696 * 1.18 = $10,261 - $4,000 = $6,261
end of year 7 balance = $6,261 * 1.18 = $7,388 - $4,000 = $3,388
end of year 8 balance = $3,388 * 1.18 = $3,998 - $4,000 = -$2
I truncated the displays to the nearest dollar but kept the full numbers in the calculator, so the answer is really $0 even though you see -$2.
The bank is making 18% on it's investment for the last 5 years, assuming they are re-investing the money they received each year as payment at the same rate of return.
this is the 5 years loan period that begins at the end of year 3.
At the end of the investment period they have earned $28,616.83904
$12,508.68408 compounded yearly at 18% interest rate = $28,616.83904 at the end of 5 years.
Since they gave the money 3 years earlier and didn't earn anything on it for the first 3 years, then their overall interest rate was equivalent to:
$12,508.68408 compounded yearly for 8 years to be equal to $28,616.83904.
Solving for the equivalent interest rate per year, we get an interest rate of 10.89864877% per year.
the formulas that you would use to solve this problem are:
PRESENT VALUE OF A PAYMENT
PV = present value
PMT = payment per time period
i = interest rate per time period
n = number of time periods
and:
FUTURE VALUE OF A PAYMENT
FV = future value
PMT = payment per time period
i = interest rate per time period
n = number of time periods
and:
FUTURE VALUE OF A PRESENT AMOUNT
FV = future value
PA = present amount
i = interest rate per time period
n = number of time periods
The first formula is used to find the present value of the payments for 5 years.
This assumes the loan started at the end of year 3 and goes for 5 years to end at the end of year 8.
That's all you really need to know to solve this problem.
Once you find the present value of the payments for 5 years, you have the amount the bank is willing to lend him.
The rest was just figuring out what annual interest rate the bank really made on the loan considering they gave him the money 3 years earlier than they would have.
The second formula is used to find the future value of the payments at the end of the 5 year loan period. This is what the bank made on the loan assuming they re-invested the money they received each year at the same interest rate of the loan.
The third formula is used to find the average interest rate of the investment over 8 years rather than 5.
You have the present value of $12,000 and the future value of $28,000 and you solve for the interest rate. the investment period is 8 years from the time they gave him the money to the time he made the final payment and the loan was terminated.
|
|
|
