Textbooks on economic analysis define a formula to calculate the equivalent future value *F* of a series of uniform payments *A* for *n* periods at interest rate *i*. It is assumed that the payments occur at the end of each period. Rent, insurance payments or deposits into a savings account occur are beginning-of-period cash flows. What formula should we use in these cases to calculate the equivalent future value?

We need to derive it. We start from an example cash flow:

Four uniform payments *A* are done at the beginning of 4 periods. The equivalent future value *F* of all the payments is calculated as the sum of the future value of each payment, using the formula *F* = *A*(1 + *i*)^{n}:

F = *A*(1 + i)^{4} + *A*(1 + i)^{3} + *A*(1 + i)^{2} + *A*(1 + i)

The general case for *n* years is:

F = *A*(1 + *i*)^{n} + *A*(1 + *i*)^{n-1} … + *A*(1 + *i*)^{2} + *A*(1 + *i*) (1)

We multiply each side by (1 + *i*) to obtain:

*F*(1 + *i*) = *A*(1 + *i*)^{n+1} + *A*(1 + *i*)^{n} … + *A*(1 + *i*)^{2} (2)

We subtract (1) from (2):

*Fi* = *A*(1 + *i*)^{n+1} - A(1 + *i*)

Solving for *F*:

*F* = *A* [ (1 + *i*)^{n+1} - (1 + *i*) ]/ *i*

We now have the general formula for beginning-of-period uniform series future value.

The idea of deriving the formula from a cash flow. came from the book *Engineering Economic Analysis* by Donald Newman, Ted Eschenbach and Jerome Lavelle, Ninth Edition, 2004, Chapter 4 *More interest Formulas, section Uniform Series Compoung Interest Formulas*, page 86.