Future Value of an Annuity
1. Formula and Definition
The equation below calculates the future value of a stream of equal payments made at regular intervals over a specified period of time at a given rate. This value is referred to as the future value (FV) of an annuity.
In plain terms, the FV of an annuity equation calculates how much a stream of payments will be worth at a specified time in the future. The FV is an accumulated value in that it represents the accumulation of both payments made or borrowed and interest earned or charged.
In practice the FV of an annuity equation is used to calculate the accumulated growth of a series of payments such as deposits to a savings account or contributions to a retirement plan. However the calculation applies to liabilities as well as assets. So in addition to computing the growth of a savings plan, the equation can also calculate the accumulation of borrowings against a line of credit.
By rearrangement of the equation above it is possible to solve for the payment amount (PMT ) necessary to accumulate a desired FV. While the FV tells us what the ending accumulated balance will be, the PMT amount tells us how much we must save or borrow each period in order to achieve the specified FV. For example, if we want to accumulate $5,000 in 3 years, the PMT calculation tells us how much we must set aside each period in order to achieve this goal.
Note the distinction between the FV of an annuity and the PV of an annuity. The FV of an annuity equation answers the question "What will it be worth then?" while the PV of an annuity equation answers the question "What is it worth now (or some time prior to then)?". The PV of an annuity is discussed separately here.
The formula above assumes an ordinary annuity. one in which each payment is made at the end of the compounding period. An annuity-due is one in which the payments are made at the beginning of the compounding period. See Annuity-Due for more information on the distinction between an annuity-due and an ordinary annuity. This distinction is further illustrated in example problems #7 and #32.
2. Future Value of an Annuity Illustrated
The following simplified example illustrates the basic operation of the FV of an annuity formula.
What is the accumulated value of a $25 payment to be made at the end of each of the next three years if the prevailing rate of interest is 9% compounded annually? Or, put another way, "How much will I have at the end of three years if I save $25 a year at 9%?"
This problem is represented graphically in the diagram below.
In this situation the inputs to the FV of an annuity equation are as follows. PMT = 25.00
i = 0.09
n = 3
. and plugging these values into the equation.
. tells us that the accumulated value at the end of three years is $81.95 (rounded).
According to our calculation, the FV ($81.95 rounded) is greater than the sum of the three $25 payments ($75.00). The difference between the two values is interest. Each payment of $25 except the last one starts earning interest as soon as it is made. This interest is added to the accumulated balance each period.
So the first payment earns interest for two years.
. and the second payment earns interest for one year.
. but the third payment is made at the very end of the term and earns no interest at all.
The mechanics of the FV calculation are illustrated below.
. so at 9%, three payments of $25 plus accumulated interest will be equal to $81.95 in three years.
How much interest is in the accumulated balance?
81.95 - 75.00 = 6.95
Additional problems illustrating the calculation of the FV of an annuity can be viewed here under Application #5 "Find FV Annuity."
3. Solving for Other Variables
While the basic FV of an annuity formula presented above allows us to calculate FV. we often need to calculate one of the other variables in the equation such as the number of compounding periods (n ), the payment amount (PMT ), or the interest rate (i ). These calculations are illustrated below. Calculating the PV of an annuity (the current value of a series of periodic payments) is discussed separately here.
a. Number of compounding periods (n)
Solving for n is a simple matter of algebraic rearrangement of the basic FV of an annuity formula for which the following algebraic identity is helpful.
. and rearranging the FV of an annuity equation to solve for n as we did for "y" above, we get this.
. and using the values for the other variables from our earlier example, we calculate the number of compounding periods (n ) as.
b. Payment amount (PMT)
Rearranging the basic FV of an annuity formula to solve for PMT is a little easier than it was for n. What we end up with looks like this.
. and again using the values for the other variables from our original example, we calculate PMT as.
c. Interest rate (i)
This is the tough one.
Unfortunately there is no easy way to isolate the interest rate (i ) variable in the basic FV of an annuity equation.
What we end up with is a value referred to as kFV defined below.
. and since we know that FV is 81.95 and PMT is 25.00, then kFV must be equal to 3.2780.
We know that n = 3 so we keep substituting different values for i into the right hand side of the equation until we get close to the value of 3.2780.
This process of iteration (getting successively closer to the desired value for kFV ) continues until an acceptably accurate value for i is found.
Alternatively, we can find two reasonably accurate values that bracket our desired kFV and then calculate an i based on interpolation.
This entire process is illustrated more clearly in example problem #38.
For practical purposes i is typically computed using a calculator or computer program rather than through manual iteration. In Excel the RATE function is used for this purpose. The built-in TVOM functions of the HP-12C make it easy to calculate i for an annuity. However, if we have to code the calculation of i in a financial application then we're basically stuck with iteration.
4. Compounding Frequency
The equation for the FV of an annuity presented at the top of this page assumes annual compounding.
But what if instead of annual payments of $25, we were dealing with semiannual payments of $12.50. In total the annual amount of the payment is the same (12.50 x 2 = 25.00). However, the FV is not.
The reason the FV changes is that the compounding frequency (n ) has changed; it has increased from once a year to twice a year. The net effect of this changes is to increase the value of FV ; payments are now being made earlier (front-loaded) and since more cash is being made available sooner, the FV will be larger.
Q. How do we account for the non-annual payments in our calculation?
A. By a process referred to as synchronization.
The standard formula for the FV of an annuity assumes annual compounding. Synchronization involves modifying the values of i and n to take into account non-annual compounding.
In order to understand the process, we start by defining n (the number of compounding periods in the term of the annuity) as.
. where m is the number of compounding periods in a year and Y is the number of years in the term of the annuity. So when semiannual compounding is used with our example, n is calculated as.
This makes sense: we are compounding twice a year over three years so the number of compounding periods (n ) is six.
Now that we have adjusted n. we need to take care of i.
The 9% interest rate we were given in the original example was an annual nominal rate. This is typically how rates are specified in TVOM problems. A nominal rate is also referred to as an "applied" rate because it is the rate at which interest is applied to principal.
Since we increased our compounding frequency from annual to semiannual, the nominal interest rate also needs to change. On an annual basis we were charging 9%. We simply need to adjust this rate to make it proportional to the new compounding frequency.
So, since we doubled the compounding frequency, we must halve the interest rate.
More formally, we divide i by the number of compounding periods in the year (m ) and our applied interest rate becomes.
. and this also makes sense; we were applying 9% annually and now we are applying 4.5% semiannually.
What is potentially confusing is that even with non-annual compounding, we still reference an annual rate. So instead of "4.5% applied semiannually," we say "9% with semiannual compounding." It is understood that 9% is an annual rate and that only 4.5% is applied each semiannual compounding period.
We can now modify the basic FV of an annuity equation to incorporate the synchronization process as follows.
. and we calculate the FV under semiannual compounding as.
We can used the same modified equation to calculate the FV under other compounding frequencies. For example, under quarterly compounding the FV is.
. and summarizing the effect of increasing compounding frequencies. FVannual = 81.95
FVsemiannual = 83.96
FVquarterly = 85.01
. we can see that increasing n has the effect of increasing the FV of an annuity. This is because we are compounding more frequently so there is more interest being earned on interest and the FV grows larger.
5. Payment and Compounding Periods Do Not Coincide
One of the basic assumptions under TVOM theory is that of a "simple" annuity (see Assumptions and Definitions ). A simple annuity is one where the payments and compounding periods coincide.
For example, when we are compounding monthly . we should also be making payments monthly .
However, in real life it is not uncommon to find a situation where compounding is occurring more or less frequently than payments are being made.
There are two approaches for handling such situations: rate equivalence and deconstruction.
a. Rate Equivalence Approach
Under this approach we convert the rate used for compounding into an equivalent rate based on the payment frequency.
For example, consider our original example where we are making payments annually but compounding monthly.
What annual rate is equivalent to 9% compounded monthly?
The following equation takes the 9% annual rate and converts it to an annual effective rate under monthly compounding.
In other words, 9% compounded monthly is equivalent to 9.38% compounded annually.
Now we can perform our FV of an annuity calculation using the equivalent annual rate.
. and we find that the FV increases by $0.31 (82.26 - 81.95) using monthly rather than annual compounding.
b. Deconstruction Approach
As an alternative to the rate equivalence approach, we can compute the FV for each payment and the summation of all of these individual values will be the FV of the annuity.
Typically this approach is used when the payment amounts are not equal or the interval between payment dates varies. However, it can also be applied to standard annuities.
Using our original example, the FV of a series of three annual payments of $25 at 9% monthly compounding is computed as the sum of the FV s of three single sum payments of $25 each with terms of 2, 1 and 0 years.
Graphically this approach looks like this.
. and crunching the numbers.
. we find that our calculated FV of $82.26 is the same as it was under the rate equivalence approach.
6. Excel Spreadsheet
Excel provides us with two approaches to solve for the FV of an annuity.
a. The FV Function.
If all we want to know is the FV of an annuity, then we can use Excel's built-in FV Function as shown here.
. where the cell contents look like this.
The FV Function takes the following parameters.
Note that the function can be used for both single sum and annuity calculations depending on the parameters supplied.
b. The Accumulation Schedule.
A accumulation schedule can be constructed in Excel to compute the FV of the annuity without having to use any special functions. Such a s schedule looks like this.
. and the cell contents look like this.
Note that the schedule above relies entirely upon basic math operators; no special FV Function was used.
This approach is typical of how a programmer might solve the problem. Absent knowledge of a specific mathematical equation, a common operation (accruing interest on a cummulative balance in this case) is simply repeated over and over again to arrive at the solution.
One benefit of the discount schedule that we do not have with a direct FV calculation is that we can see the FV of the annuity for all of the payment frequencies. For example, we know that the value of the annuity after two payments is $52.25.
8. Programming Languages
Practical application of TVOM concepts often involves using a programming language to code the calculations.
Listed below are some very simple illustrations of how the standard TVOM equation for the future value of an annuity can be coded in four different programming languages: