no

2. Relating a Uniform Series to Its
Present & Future Equivalent Values
General cash-flow diagram involving a series of uniform (equal) receipts,
each of amount A, occurring at the end of each period for N periods with
interest at 𝑖% per period. Such a uniform series is often called an annuity.

3. Relating a Uniform Series to Its
Present & Future Equivalent Values
It should be noted that the formulas and tables to be presented are derived such that A occurs at the
end of each period, and thus,
1. 𝑃 (present equivalent value) occurs one interest period before the first A (uniform amount),
2. 𝐹 (future equivalent value) occurs at the same time as the last A, and N periods after P, and
3. 𝐴 (annual equivalent value) occurs at the end of periods 1 through N, inclusive.

4. Finding F when Given A
If a cash flow in the amount of 𝐴 dollars occurs at the end of each period for 𝑁 periods and 𝑖%
is theinterest (profit or growth) rate per period, the future equivalent value, 𝐹, at the end of the
Nth period is obtained by summing the future equivalents of each of the cash flows. Thus,
𝐹 = 𝐴
𝐹
𝑃
, 𝑖%, 𝑁 − 1 + 𝐴
𝐹
𝑃
, 𝑖%, 𝑁 − 2 + 𝐴
𝐹
𝑃
, 𝑖%, 𝑁 − 3 +··· + 𝐴
𝐹
𝑃
, 𝑖%, 1 + 𝐴
𝐹
𝑃
, 𝑖%, 0
𝐹 = 𝐴[ 1 + 𝑖 𝑁−1 + 1 + 𝑖 𝑁−2 + 1 + 𝑖 𝑁−3 +··· + 1 + 𝑖 1 + 1 + 𝑖 0]

5. Finding P when Given A
Put
in

6. Finding A when Given F
Finding A when Given P

7. Finding the Number of Cash Flows in
an Annuity Given 𝐴, 𝑃, 𝑎𝑛𝑑 𝑖
Sometimes we may have information about a present amount of money (𝑃), the
magnitude of an annuity (𝐴), and the interest rate (𝑖). The unknown factor in this case is
the number of cash flows in the annuity (N).

8. Finding the Interest Rate, i, Given
A, F, and N
Now let’s look at the situation in which you know the amount (𝐴) and duration (𝑁) of a
uniform payment series. You also know the desired future value of the series (𝐹). What
you don’t know is the interest rate that makes them equivalent. As was the case for an
unknown 𝑁, there is no single equation to determine 𝑖. However, we can use the known
relationships between 𝑖, 𝐴, 𝐹, 𝑎𝑛𝑑 𝑁 and the method of linear interpolation to
approximate the interest rate.

9. Finding the Interest Rate, i, Given
A, F, and N
Now let’s look at the situation in which you know the amount (𝐴) and duration (𝑁) of a
uniform payment series. You also know the desired future value of the series (𝐹). What
you don’t know is the interest rate that makes them equivalent. As was the case for an
unknown 𝑁, there is no single equation to determine 𝑖. However, we can use the known
relationships between 𝑖, 𝐴, 𝐹, 𝑎𝑛𝑑 𝑁 and the method of linear interpolation to
approximate the interest rate.