Derivations of Interest Formulas
Deriving financial formulas involves understanding the accumulation and discounting processes
based on the time value of money. Below are the derivations for each of the formulas:
3.3.1 Single-Payment Compound Amount
Formula:
[ FV = PV(1 + i)^n ]
Derivation:
1. Start with a present value ((PV)).
2. After one period, the value grows by the interest rate ((i)), so the value is (PV \times (1 + i)).
3. After two periods, the value grows again: ((PV \times (1 + i)) \times (1 + i) = PV \times (1 + i)^2).
4. Continuing this process for (n) periods gives (FV = PV \times (1 + i)^n).
3.3.2 Single-Payment Present Worth Amount
Formula:
[ PV = \frac{FV}{(1 + i)^n} ]
Derivation:
1. Start with a future value ((FV)) and discount it to the present using the interest rate ((i)) over (n)
periods.
2. The present value is obtained by dividing the future value by the accumulated factor: (PV =
\frac{FV}{(1 + i)^n}).
3.3.3 Equal-Payment Series Compound Amount
Formula:
[ FV = A \left[\frac{(1 + i)^n - 1}{i}\right] ]
Derivation:
1. Consider a series of equal payments ((A)) made at the end of each period.
2. The first payment accumulates interest for (n-1) periods, the second for (n-2), and so on until the
last payment which accumulates no interest.
3. Sum the future values of all payments:
[ FV = A \left[(1 + i)^{n-1} + (1 + i)^{n-2} + \ldots + (1 + i) + 1 \right] ]
4. This series is a geometric progression with a common ratio of ((1 + i)).
