your annual interest rate is 18% = .18
your monthly interest rate is .18/12 = .015
P looks like it's the principal amount.
R looks like it might be the payment per month or the revenue per month, depending on how you want to look at it.
The first formula is which looks like the Future Value of a Present amount formula.
r is the annual interest rate.
m is the number of months in a year.
n is the total number of months.
P is the Present Amount.
What's missing is the F which, in this case, means Future Value of a Present Amount.
The complete formula is
The second formula is which looks like the future value of a revenue formula.
R is the monthly revenue.
r is the annual interest rate.
m is the number of months in a year.
n is the total number of months.
What's missing is the F, which in this case means Future Value of a monthly revenue amount.
The complete formula would show up as:
F =
What I believe the formulas are trying to tell you is that the future value of a present amount is equal to the future value of a series of revenue.
Without getting into the equations, I'll show you what I think they mean.
You start with $10,000 Principal.
The annual interest rate is 18% which is equivalent to .18
the monthly interest rate is .18/12 = .015
The number of months is 48.
The future value of this amount is equal to $20,434.78
If, instead of taking $10,000 and putting it into the account up front, you generate a stream of revenue equal to $293.75 per month, you will wind up with the same amount of money.
What the formula appears to be telling you is that the future value of the present amount of $10,000 is equivalent to the Future Value of the monthly revenue.
The future Value of the monthly revenue of $293.75 per month for 48 months at .015% interest year is equal to $20,434.78.
Whether you invest $10,000 for 48 months or you invest $293.75 per month for 48 months, you will wind up with the same amount of money at the end of the 48 month period.
A look at the monthly cash flows might show this up clearer.
First Cash flow is investing the money for 48 months.
You put $10,000 in your account and wait 48 months and then receive the value of that money at the end of the 48 months.
Looks like this:
YEAR BALANCE
0 $10,000.00
1 $10,150.00
2 $10,302.25
3 $10,456.78
4 $10,613.64
5 $10,772.84
6 $10,934.43
7 $11,098.45
8 $11,264.93
9 $11,433.90
10 $11,605.41
11 $11,779.49
12 $11,956.18
13 $12,135.52
14 $12,317.56
15 $12,502.32
16 $12,689.86
17 $12,880.20
18 $13,073.41
19 $13,269.51
20 $13,468.55
21 $13,670.58
22 $13,875.64
23 $14,083.77
24 $14,295.03
25 $14,509.45
26 $14,727.10
27 $14,948.00
28 $15,172.22
29 $15,399.81
30 $15,630.80
31 $15,865.26
32 $16,103.24
33 $16,344.79
34 $16,589.96
35 $16,838.81
36 $17,091.40
37 $17,347.77
38 $17,607.98
39 $17,872.10
40 $18,140.18
41 $18,412.29
42 $18,688.47
43 $18,968.80
44 $19,253.33
45 $19,542.13
46 $19,835.26
47 $20,132.79
48 $20,434.78
The money was invested in some account earning 1.5% per month for 48 months.
At the end of 48 months, the account was worth $20,434.78.
Second Cash Flow is investing money at $293.75 per month for 48 months.
Looks like this:
YEAR REVENUE BALANCE
0
1 $293.75 $293.75
2 $293.75 $591.91
3 $293.75 $894.53
4 $293.75 $1,201.70
5 $293.75 $1,513.48
6 $293.75 $1,829.93
7 $293.75 $2,151.13
8 $293.75 $2,477.15
9 $293.75 $2,808.05
10 $293.75 $3,143.92
11 $293.75 $3,484.83
12 $293.75 $3,830.86
13 $293.75 $4,182.07
14 $293.75 $4,538.55
15 $293.75 $4,900.38
16 $293.75 $5,267.63
17 $293.75 $5,640.40
18 $293.75 $6,018.75
19 $293.75 $6,402.79
20 $293.75 $6,792.58
21 $293.75 $7,188.22
22 $293.75 $7,589.79
23 $293.75 $7,997.39
24 $293.75 $8,411.10
25 $293.75 $8,831.01
26 $293.75 $9,257.23
27 $293.75 $9,689.84
28 $293.75 $10,128.93
29 $293.75 $10,574.62
30 $293.75 $11,026.99
31 $293.75 $11,486.14
32 $293.75 $11,952.18
33 $293.75 $12,425.22
34 $293.75 $12,905.35
35 $293.75 $13,392.68
36 $293.75 $13,887.32
37 $293.75 $14,389.38
38 $293.75 $14,898.97
39 $293.75 $15,416.20
40 $293.75 $15,941.19
41 $293.75 $16,474.06
42 $293.75 $17,014.92
43 $293.75 $17,563.90
44 $293.75 $18,121.10
45 $293.75 $18,686.67
46 $293.75 $19,260.72
47 $293.75 $19,843.38
48 $293.75 $20,434.78
You have the same amount of money at the end of 48 months whether you invested $10,000 up front or you invested $293.75 a month for 48 months.
What is not stated is that, in order to get the equivalent, you have to know hoe much revenue per month is equivalent to $10,000 investment up front.
The formula to find that is the Payment from a Present Value formula.
I used that formula to generate the revenue of $293.75 per month.
What you have is the following:
A principal of $10,000
Use the Future Value of a Present Amount formula to find the value of this investment at the end of 48 months at an interest rate of .015 per month.
Use the Payment on a Present Amount formula to determine what the monthly payments would be for this $10,000 investment.
Use the Future Value on a Payment formula to determine how much this series of payments is to the lender at the end of 48 months.
I started with $10,000.
I got monthly payments of $293.75 per month from that.
I got a future value of payments of $20,434.78 from that.
I then calculated the future value of the 10,000 initial investment from which I also got $20,434.78.
The formula of Payment on a Present Value calculates the monthly amount from the perspective of money that is borrowed.
You go to the bank and borrow $10,000 to buy a car.
You pay a monthly payment of $293.75.
The formula of Future Value of a Payment calculates the future value of those payments from the perspective of the person who offers you the loan.
I lend you $10,000 and you pay me $293.75 for 48 months, at the end of which I have $20,434.78, assuming I reinvest the interest I earn each month at the same interest rate.
The payment represents $10,000 that you borrow today (Present Value of a Payment).
The same payment represents $20,434.75 that I earn because I lent you $10,000 up front (Future Value of a Payment).
If I know what the Payment is, I can calculate both the Present Value and the Future Value from that same payment.
If I know the Present Value, I can calculate the payment from that using the Payment on a Present Value formula.
If I know the Future Value, I can calculate the payment from that using the Payment on a Future Value formula.
A summary of the formulas used can be found at this link:
http://www.algebra.com/algebra/homework/Finance/FINANCIAL-FORMULAS-101.lesson