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.
Future value (FV) using simple interest (i.e., without compounding)
where
- PV is the present value or principal
- t is the time in years
- r is the per annum interest rate
Simple interest is rarely used, as compounding is considered more meaningful.
Future value using compound interest
where
- PV is the present value or principal
- n is the number of compounding periods
- i is 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
- i1 is the periodic interest rate with compounding frequency n1
- 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:
where
- r is the annual rate
- i the periodic rate
- n the number of compounding periods per year
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):
where
- r is the interest rate
- n the number of periods
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