# Future Value

**Future value** measures the nominal future sum of money that a given
sum of money is "worth" at a specified time in the future assuming a certain
interest rate; this value does not include corrections for inflation or other
factors that affect the true value of money in the future. This is used in
time value of money calculations.

To determine future value (FV) using
** simple interest** (i.e., without compounding):

where *PV* is the ** present value** or principal,

*t*is the time in years, and

*r*stands for the per annum interest rate. Simple interest is rarely used, as compounding is considered more meaningful.

To determine **future value** using compound interest:

where *PV* is the present value, *n* is the number of compounding
periods, and *i* stands for the interest rate per period.

In this usage, *i* is the interest rate per period, not the annual
interest rate. To convert an interest rate from one compounding basis to
another compounding basis (between different periodic interest rates), the
following formula applies:

where *i*_{1} is the periodic interest rate with compounding
frequency *n*_{1} and *i*_{2} is the periodic
interest rate with compounding frequency *n*_{2}.

If the compounding frequency is annual, *n* will be 1, and to get the
annual interest rate (which may be referred to as
** Annual percentage rate** or

**APR**), the formula can be simplified to:

where *r* is the annual rate, *i* the periodic rate, and *n*
the number of compounding periods per year.

For continuous compound interest, exponentials are used; see the time value of money article for more detail.

For example, what is the future value of 1 money unit in one year, given 10% interest? The number of time periods is 1, the discount rate is 0.10, the present value is 1 unit, and the answer is 1.10 units. Note that this does not mean that the holder of 1.00 unit will automatically have 1.10 units in one year, it means that having 1.00 unit now is the equivalent of having 1.10 units in one year.

These problems become more complex as you account for more variables. For
example, when accounting for annuities (annual payments), there is no simple
*PV* to plug into the equation. Either the *PV* must be calculated
first, or a more complex annuity equation must be used. Another complication
is when the interest rate is applied multiple times per period. For example,
suppose the 10% interest rate in the earlier example is compounded twice a
year (semi-annually). Compounding means that each successive application of
the interest rate applies to all of the previously accumulated amount, so
instead of getting 0.05 each 6 months, you have to figure out the true annual
interest rate, which in this case would be 1.1025 (you divide the 10% by two
to get 5%, then apply it twice: 1.05^{2}.) This 1.1025 represents the
original amount 1.00 plus 0.05 in 6 months to make a total of 1.05, and get
the same rate of interest on that 1.05 for the remaining 6 months of the year.
The second six month period returns more than the first six months because the
interest rate applies to the accumulated interest as well as the original
amount.

This formula gives the Future Value of an annuity (assuming compound interest):

where r = interest rate; n = number of periods.