In order to derive the formula for future value with an annuity, we need to first understand the formula for finding the future value without an annuity, as well as the formula for geometric series.
An annuity is a reoccurring amount, such as a payment or reoccurring investment. Imagine if you decided to save $2,000 per year for the next 30 years. If you are obtaining interest on your invested money, each annuity deposit of $2,000 will receive one year less interest than the previous deposit. By the end of the 30 years, your last deposit of $2,000 will receive no interest, while the first deposit of $2,000 may have quadrupled depending on the interest rate.
How much money would you have total? In order to manually figure out the total amount, you would have to solve for the future value for each annuity, based on the number of years it was invested, and add them all together (which would be very time-consuming).
However, if we know the annuity is a fixed amount (e.g. $2,000) then we can derive a formula for the future value of our investment with this fixed annuity. To begin, let's discuss the basic formula for determining future value.
In order to determine the future value of an investment, we assume a yearly percentage increase and multiply the original amount by that percentage each year. If we invested $2,000 one time, and we wanted to know what the future value of that investment will be in 5 years (assuming that it will grow by 5% each year), we would have:
$2,000 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05
We multiply by 1.05 because we want 100% of the value plus the additional 5%, which as decimals is 1.00 + 0.05 = 1.05. Whenever we repeatedly multiply the same number, we use powers to represent the amount of times this multiplication occurs. Therefore, this formula can be simplified into $2,000(1.05)^5:
Or we can also rewrite it generically as FV = PV(1 + r)^n:
In this future value formula, FV represents future value, PV represents present value (the $2,000), r represents the percentage rate as a decimal (5% = 0.05, so 1 + 0.05 = 1.05), and n represents the number of periods (the 5).
Note: We will use r to represent interest rate instead of i, in order to avoid confusion, because i represents imaginary numbers in algebra. The n number of periods could be years, or if interest is accrued semi-annually, monthly, or daily then n would represent the number of those periods.
Now we can begin to derive the formula for future value with a fixed annuity.
Once we add a fixed annuity to the formula, we have to modify our equation. Remember that the equation without an annuity was simply PV(1 + r)^n:
However, now we have to do this formula for each individual annuity and add them together. Imagine if we invest $2,000 each year for 5 years. The first $2,000 was invested at the end of the year and gains interest for 4 years FV = PV(1 + r)^4:
The second $2,000 was invested at the end of the second year and gains interest for 3 years FV = PV(1 + r)^3:
Continuing this for each annuity, we have to add them all together to get the total future value
FV = PV(1 + r)^4 + PV(1 + r)^3 + PV(1 + r)^2 + PV(1 + r)^1 + PV(1 + r)^0:
Notice that when investing the $2,000 for 5 years, it actually only accrued interest for 4 years, because usually annuity payments are considered to be invested at the end of the year (the final formula changes slightly if it is invested at the beginning of the year). Therefore, the first power can be generalized to (n – 1) since 5 – 1 = 4. The second annuity payment can be generalized to (n – 2) since 5 – 2 = 3, and so on.
If we generalize the equation for n periods, then it produces this formula
FV = PV(1 + r)^n-1 + PV(1 + r)^n-2 + PV(1 + r)^n-3 … + PV(1 + r)^0:
This type of formula is a geometric series, which is any series where each unit is transformed by multiplication or division. In this situation, each unit of (1 + r) is being transformed by a different power, which is multiplication. The generic formula for a geometric series is s = x(1 - y^n) / (1 – y):
S represents "geometric series" and y represents a constant. At the end of this article, we will replace s with FV, and we will replace x with PMT since this number represents the annuity payment.
The only difference between our equation for future value and this generic simplified geometric equation is that we need to replace the y with the constant that is present in the annuity formula.
In our formula, the constant is (1 + r). We can derive this constant, also called the common ratio, by dividing each term by the following term. This division leaves (1 + r) left over. For example if you divide PV(1 + r)^n-1 by PV(1 + r)^n-2, then the PV and all but one (1 + r) are canceled out. Using real numbers, if we invest for 5 years then PV(1 + r)^4 divided by PV(1 + r)^3 would simplify to (1 + r).
Now, to derive the formula for the future value with an annuity, we must insert our constant into the y position of the generic geometric equation, which gives us s = x[1 – (1 + r)^n / 1 – (1 + r)]:
We can immediately simplify the denominator by distributing the negative sign, which transforms the bottom from (1 – (1 + r)) into (1 – 1 – r). Then, since 1 – 1 = 0, we are left with – r in the denominator, giving us s = x(1 – (1 + r)^n) / (– r):
Next, we will want to get rid of the negative sign in the denominator, which we can accomplish by multiplying the top and bottom by (– 1). Remember that multiplying anything over itself is the same as multiplying by 1. (– 1)/(– 1) simplifies to 1. This results in the formula becoming s = x(– 1 + (1 + r)^n) / (r):
Finally, we can rearrange the numerator and replace s and x for FV and PMT in order to reveal the familiar formula seen in textbooks FV = PMT[(1 + r)^n – 1 / r]:
Normally an annuity is considered to be paid or invested at the end of the year. However, when an annuity payment is invested at the beginning of the year it is called an annuity due. This changes our formula by multiplying everything by (1 + r) to give us the formula FV = PMT{[(1 + r)^n – 1] / r} * (1 + r)
Multiplying our formula by (1 + r) basically adds one year of interest earned to each annuity payment, because all of the payments are multiplied collectively by (1 + r). Remember that $2,000(1.05)^4 was $2,000 * 1.05 * 1.05 * 1.05 * 1.05, so by multiplying all these units collectively by the additional (1 + r), it essentially transforms (1.05)^4 into (1.05)^5 while simultaneously doing the same for (1.05)^3, (1.05)^2, and so on.
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