1) | F = | | A * (1 + r)n - 1 | +A * (1 + r)n - 2 | + . . . . |
+A * (1 + r)2 | +A * (1 + r)1 | +A |
This can be calculated by multiplying both sides by (1 + r) then subtracting the first formula from it. |
Only the end of the first formula and beginning of the new one will remain. |
2) | F * (1+r) = | A * (1 + r)n | +A * (1 + r)n - 1 | +A * (1 + r)n - 2 | + . . . . |
+A * (1 + r)2 | +A * (1 + r)1 |
1) | F = | | A * (1 + r)n - 1 | +A * (1 + r)n - 2 | + . . . . |
+A * (1 + r)2 | +A * (1 + r)1 | +A |
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| F * r = | A * (1 + r)n | | | | | |
- A |
Combining the A's on the right side: |
| F * r = | A * [(1 + r)n
- 1] |
Solving for A: |
| A = | F * r
|
| [(1 + r)n - 1] |
Since F = P * (1 + r)n, we can substitute it and A is terms of P, the value of the loan, |
r, the interest rate, and n, number of periods: |
| A = | P * r * (1 + r)n
|
| (1 + r)n - 1 |
The top and bottom of that can be divided by (1 + r)n to simplify it further: |
| A = | P * r |
| 1 - 1 / (1 + r)n |