Finance Formulas

Solve for n on Annuity - (FV) Calculator (Click Here or Scroll Down)

The formula for solving for number of periods (* n*) on an annuity shown above is used to calculate
the number of periods based on the future value, rate, and periodic cash flows.

The formula at the top of the page, solving for n, generally approaches the question "How long will it take to save
* $x* amount dollars by saving

Suppose that an individual receives an additional *$1000* pay or bonus semiannually. Suppose this individual would
like to find out how long until they save *$19600* by saving *$1,000* every half year at an effective rate of
*5%* every half year. **It is important to remember that the rate should match the frequency of the cash
flows/payments. For example, if cash flows are semi-annual, then the effective semi-annual rate is used. The term effective
implies that compounding is already adjusted for that period (see effective rate).

Using the formula at the top of the page to solve for the number of periods, *n*, for this example would show the
equation

The above equation can be reduced to

which results in 14 semi-annual periods.

*Note that this is for academic reasons only. Actual results may vary due to fees, exact cash flow dates, et cetera

**See section labeled Alternative Method for an additional way of solving this example.

Solving for the number of periods on an annuity requires first looking the future value of annuity formula.

The number of periods can be found by rearranging the above formula to solve for *n*. The first step would be to
multiply both sides by ** r/P**. Then we can add

From here it is seen that *(1+r)*^{n} is isolated on the right side of the equation. To
solve for the exponent * n* requires multiplying both sides by

Both sides of the equation above can be divided by ** ln(1+r)** which would result in the formula at
the top of the page, solving for

Another method of solving for the number of periods (** n**) on an annuity based on future value is to
use a future value of annuity (or increasing annuity) table. Solving for the number of periods can be achieved by dividing

For example, using the example provided in the preceding section, the future value of *$19600 *can be divided by
semi-annual cash flows of *$1,000* which results in *19.6*. Using the future value of annuity table, one can
see that at an effective rate of *5%*, *19.6* in the table matches up to *14* (semi-annual) periods.

__Formulas related to Annuity (FV) - Solve for n__- Number of Periods - PV & FV
- Future Value of Annuity
- Solve for n - Annuity (PV)