Future value

From Wikipedia, the free encyclopedia

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):

$FV = PV \cdot (1+rt)$

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:

$FV = PV \cdot (1+i)^n$

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:

$i_2=\left[\left(1+\frac{i_1}{n_1}\right)^\frac{n_1}{n_2}-1\right]{\times}n_2$

where i₁ is the periodic interest rate with compounding frequency n₁ and i₂ is the periodic interest rate with compounding frequency n₂.

If the compounding frequency is annual, n will be 1, and to get the annual interest rate (which may be referred to as the effective interest rate, or the Annual percentage rate), the formula can be simplified to:

$r = ( 1 + { i \over n } )^n - 1$

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: 1.05².) 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):

$FV_{annuity} = {(1+r)^n-1 \over r} \cdot {(payment\ amount)}$