Future value is the value of an asset at a specific date.
It 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, or more generally, rate of
return; it is the present value multiplied by the accumulation function.
The 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.
Overview Money value fluctuates over time: $100 has
a different value than $100 in five years.
This is because one can invest $100 today in a bank account or any other investment,
and that money will grow/shrink due to interest.
Also, if $100 today allows the purchase of an item, it is possible that $100 will not
be enough to purchase the same item in five years, because of inflation.
An investor who has some money has two options: to spend it right now or to invest it.
The financial compensation for saving it is that the money value will accrue through the
interests that he will receive from a borrower.
Therefore, to evaluate the real worthiness of an amount of money today after a given
period of time, economic agents compound the amount of money at a given interest rate.
Most actuarial calculations use the risk-free interest rate which corresponds the minimum
guaranteed rate provided the bank's saving account, for example.
If one wants to compare their change in purchasing power, then they should use the real interest
rate.
The operation of evaluating a present value into the future value is called capitalization.
The reverse operation which consists in evaluating the present value of a future amount of money
is called a discounting.
It follows that if one has to choose between receiving $100 today and $100 in one year,
the rational decision is to cash the $100 today.
If the money is to be received in one year and assuming the savings account interest
rate is 5%, the person has to be offered at least $105 in one year so that two options
are equivalent.
This is because if you cash $100 today and deposit in your savings account, you will
have $105 in one year.
Simple interest To determine future value using simple interest:
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.
Indeed, the Future Value in this case grows linearly: it doesn't take into account the
fact that the interest earned might be compounded itself and produce further interest.
Compound interest To determine future value using compound interest:
where PV is the present value, t is the number of compounding periods, and i is the interest
rate for that period.
Thus the future value increases exponentially with time when i is positive.
The growth rate is given by the period, and i, the interest rate for that period.
Alternatively the growth rate is expressed by the interest per unit time based on continuous
compounding.
For example, the following all represent the same growth rate:
3 % per half year 6.09 % per year
2.95588022 % per half year based on continuous compounding
5.91176045 % per year based on continuous compounding
Also the growth rate may be expressed in a percentage per period, with another period
as compounding basis; for the same growth rate we have:
6% per year with half a year as compounding basis
To convert an interest rate from one compounding basis to another compounding basis, the following
formula applies:
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, n2 will be 1, and to get the annual interest
rate, 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.
Problems become more complex as you account for more variables.
For example, when accounting for annuities, 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.
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, one must figure
out the true annual interest rate, which in this case would be 1.1025 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 ordinary annuity:
where r = interest rate; n = number of periods.
The simplest way to understand the above formula is to cognitively split the right side of
the equation into two parts, the payment amount, and the ratio of compounding over basic interest.
The ratio of compounding is composed of the aforementioned effective interest rate over
the basic interest rate.
This provides a ratio that increases the payment amount in terms present value.
See also Lifetime value
Present value Time value of money
References
External links calculate the different FV's with one's own
values