Question 286379
Loan is $100 million to be repaid at the end of 4 years.
Monthly deposits are made.
Interest rate is 22% per year compounded monthly.


This is a Payment for a Future Value type problem.


Using a financial calculator, you would enter:


Number of Time periods = 4 * 12 = 48
Interest Rate per Time Period = 22% / 100% = .22 / 12 = .0183333333 
Future Amount = $100 million
Payments are made at the end of each month.
You are looking for the payment amount.


Your answer would be (using the financial calculator)"


Payment is $1,317,274.45 per month.


Sinking Fund Balance for the first 3 months would be:


(1+i) = 1.0183333333 per month
p = $1,317,274.45 per month


End of month 1 = $1,317,274.456
End of month 2 = $1,317,274.456 * (1+i) = $1,341,424.482 + p = $2,658,698.932
End of month 3 = $2,658,698.932 * (1+i) = $2,707,441.746 + p = $4,024,716.196
.....
End of month 47 = $96,906,113.47 * (1+i) = $98,682,725.55 + p = $100,000,000.00


If you do not have a financial calculator, you can get an online version that will help you do the same problem.


As a last resort, you can use the formula given in the following lesson on Payment for a Future Value.


<a href = "http://www.algebra.com/algebra/homework/Finance/PMTFV.lesson" target = "_blank">PAYMENT FOR A FUTURE VALUE</a>


The equation to use from that lesson is:


PAYMENT FOR A FUTURE VALUE EQUATION


{{{ PMT(FV) = (FV / (((((1+i)^n-1))/i))) }}}


PMT = Payment per Time Period
FV = Future Value
i = Interest Rate per Time Period
n = Number of Time Periods


FV = $100,000,000
i = .22 / 12 = .01833333333
n = 12*4 = 48


Intermediate calculations would be:


(1.01833333333)^48 - 1 = 1.391762614

 
Equation would become:


PMT = 100,000,000 / (1.391762614/.01833333333) which would become:


PMT = $1,317,274.45


An online financial calculator that will also calculate this for you can be found at the following link:


<a href = "http://www.arachnoid.com/lutusp/finance.html" target = "_blank">http://www.arachnoid.com/lutusp/finance.html</a>


You enter the information that you have and you click on the value you are looking for.


For example:


I entered values for:

Future Value.
Interest Rate per time period.
Number of time periods.
Present Value (I set it equal to 0).


I set the payment to be made  at the end of the time period.


I then clicked on the button for Payment.


I got the same result that I gave you using both my own financial calculator plus solving the equation from the lesson.


One last item.


All equations used here assume re-investment of the interest earned each month.


That re-investment is assumed to be at the same interest rate used for calculating the payment.


I used Excel to calculate the money accruing in the sinking fund for every time period, not just the first 3.


The results are shown below:


<pre>
Time Period	
      Present Value of Loan	
                         Payment	
                                         Interest Rate	
                                                        Sinking Fund Balance
0     $41,810,169.363		         0.018333333	$0.000
1	                 1,317,274.450		        $1,317,274.450
2                        1,317,274.450		        $2,658,698.932
3		         1,317,274.450		        $4,024,716.196
4		         1,317,274.450		        $5,415,777.110
5		         1,317,274.450		        $6,832,340.807
6		         1,317,274.450		        $8,274,874.839
7		         1,317,274.450		        $9,743,855.328
8		         1,317,274.450		        $11,239,767.126
9		         1,317,274.450		        $12,763,103.974
10		         1,317,274.450		        $14,314,368.663
11		         1,317,274.450		        $15,894,073.206
12		         1,317,274.450		        $17,502,738.998
13		         1,317,274.450		        $19,140,896.997
14		         1,317,274.450		        $20,809,087.892
15		         1,317,274.450		        $22,507,862.287
16		         1,317,274.450		        $24,237,780.879
17		         1,317,274.450		        $25,999,414.646
18		         1,317,274.450		        $27,793,345.031
19		         1,317,274.450		        $29,620,164.140
20		         1,317,274.450		        $31,480,474.933
21		         1,317,274.450		        $33,374,891.424
22		         1,317,274.450		        $35,304,038.884
23		         1,317,274.450		        $37,268,554.047
24		         1,317,274.450		        $39,269,085.321
25		         1,317,274.450		        $41,306,293.002
26		         1,317,274.450		        $43,380,849.491
27		         1,317,274.450		        $45,493,439.515
28		         1,317,274.450		        $47,644,760.357
29		         1,317,274.450		        $49,835,522.080
30		         1,317,274.450		        $52,066,447.769
31		         1,317,274.450		        $54,338,273.761
32		         1,317,274.450		        $56,651,749.897
33		         1,317,274.450		        $59,007,639.762
34		         1,317,274.450		        $61,406,720.942
35		         1,317,274.450		        $63,849,785.276
36		         1,317,274.450	 	        $66,337,639.123
37		         1,317,274.450		        $68,871,103.624
38		         1,317,274.450		        $71,451,014.974
39		         1,317,274.450		        $74,078,224.699
40		         1,317,274.450		        $76,753,599.935
41		         1,317,274.450		        $79,478,023.717
42		         1,317,274.450		        $82,252,395.269
43		         1,317,274.450		        $85,077,630.299
44		         1,317,274.450		        $87,954,661.305
45		         1,317,274.450		        $90,884,437.879
46		         1,317,274.450		        $93,867,927.024
47		         1,317,274.450		        $96,906,113.470
48		         1,317,274.450		        $100,000,000.000
</pre>