Question 286378: You enter into a contract that requires you to pay $100 000 at the end of every four months(three times per year) for four years.At the end of the fourth year you will also be required to pay a lump sum of $500 000.The interest rate is 24% per annum compounded monthly.How much have you borrowed?How much do you still owe after two years
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! You pay $100,000 at the end of every 4 months.
You do that for 4 years.
At the end of the 4th year you are paying a lump sum of $500,000.
Interest rate is 24% per year compounded monthly.
How much have you borrowed?
How much do you still owe after 2 years.
The easiest way to solve part of this problem is to simulate what happens in a spreadsheet or some such other mechanized program.
That's the part for the remaining balance after 2 years.
I used Microsoft Excel Spreadsheet.
The answer I got is as follows:
Time Period
Present Value of Loan
Payment
Interest Rate
Borrower Balance
Lender Balance
0 $937,471.542 0.02 $937,471.542 $0.000
1 $956,220.973 $0.000
2 $975,345.392 $0.000
3 $994,852.300 $0.000
4 $100,000.000 $914,749.346 $100,000.000
5 $933,044.333 $102,000.000
6 $951,705.220 $104,040.000
7 $970,739.324 $106,120.800
8 $100,000.000 $890,154.110 $208,243.216
9 $907,957.193 $212,408.080
10 $926,116.337 $216,656.242
11 $944,638.663 $220,989.367
12 $100,000.000 $863,531.437 $325,409.154
13 $880,802.065 $331,917.337
14 $898,418.107 $338,555.684
15 $916,386.469 $345,326.798
16 $100,000.000 $834,714.198 $452,233.334
17 $851,408.482 $461,278.000
18 $868,436.652 $470,503.560
19 $885,805.385 $479,913.631
20 $100,000.000 $803,521.492 $589,511.904
21 $819,591.922 $601,302.142
22 $835,983.761 $613,328.185
23 $852,703.436 $625,594.749
24 $100,000.000 $769,757.505 $738,106.644
25 $785,152.655 $752,868.777
26 $800,855.708 $767,926.152
27 $816,872.822 $783,284.675
28 $100,000.000 $733,210.278 $898,950.369
29 $747,874.484 $916,929.376
30 $762,831.974 $935,267.964
31 $778,088.613 $953,973.323
32 $100,000.000 $693,650.385 $1,073,052.789
33 $707,523.393 $1,094,513.845
34 $721,673.861 $1,116,404.122
35 $736,107.338 $1,138,732.204
36 $100,000.000 $650,829.485 $1,261,506.848
37 $663,846.075 $1,286,736.985
38 $677,122.996 $1,312,471.725
39 $690,665.456 $1,338,721.160
40 $100,000.000 $604,478.765 $1,465,495.583
41 $616,568.341 $1,494,805.494
42 $628,899.707 $1,524,701.604
43 $641,477.701 $1,555,195.636
44 $100,000.000 $554,307.256 $1,686,299.549
45 $565,393.401 $1,720,025.540
46 $576,701.269 $1,754,426.051
47 $588,235.294 $1,789,514.572
48 $600,000.000 $0.000 $2,425,304.863
I calculated the Present Value of the loan as follows:
Annual Interest Rate of 24% compounded monthly is equivalent to a monthly interest rate of 2%.
The Effective Annual Interest Rate of a monthly 2% interest rate is equal to 1.02^12 = a factor of 1.268241795 which is equivalent to an Effective Annual Interest Rate of 26.8241795%.
To get an equivalent interest rate every 4 months, you need to solve the equation of:
(1+x)^3 = 1.268241795.
Take the cube root of both sides of this equation to get:
1+x =  which equals 1.08243216.
Subtract 1 from both sides of this equation to get x = .08243216 which is equal to an interest rate of 8.243216% every 4 months.
What you have so far is payments every 4 months at an interest rate of 8.243216% every 4 months.
Now that the payment schedule and the compound interest rate schedule is synchronized, you can use the financial calculators to determine the Present Value of the $100,000 payments every 4 months for the number of time periods required by the loan.
The number of time periods required by the loan is 4 years multiplied by 3 time periods per year to get a total of 12 time periods, each of which is 4 months long.
The inputs to the financial calculator are:
Present Value = 0
Future Value = 0
Payment = $100,000
Interest Rate = 8.243216%
Number of Time Periods is equal to 12.
The calculator tells you that:
Present Value of the loan payments = $744,202.7377
Future Value of the loan payments = $1,925,304.863.
$744,202.7377 is what you owe today.
$1,925,304.863 is what the lender receives by the end of the loan period.
That includes principal of the loan, interest you paid on the loan, additional interest that the lender received by re-investing the interest received from the loan.
On top of that you have to pay the lender $500,000 at the end of the loan period.
The total the lender receives is $1,925,304.863 + $500,000 = $2,425,304.863.
The present value of $2,425,304.863 for 48 time periods at 2% per month is equal to $937,471.5419.
The present value of $2,425,304.863 for 12 time periods at 8.243216% every 4 months is also equal to $937,471.5419.
The interest rates are equivalent and yield the same Present Value as they should.
The Present Value of the loan, taking into account the payments you make plus the $500,000 you give at the end of the loan period is equal to $937,471.5419.
That's the amount of the loan.
The remaining balance of the loan after 2 years was more difficult to determine using just formulas, so I resorted to the Excel Spreadsheet which provides you with the remaining balance on the loan after 2 years.
That remaining balance is equal to $769,757.505
That is the amount that you still owe after 2 years.
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