The derivation of fixed interest annuity formulas is based on compound interest and the concept of time value of money. Mathematically the pricing of time series of cashflows, each discounted at a different factor, reduces to neat analytical solutions. A key insight underlying the approach is that infinite series of payments have finite values if interest rates are positive.
The fact that simple annuities cannot be priced if interest rates are zero or negative may serve as a reminder of the insanity of modern monetary theory.
Compound Interest and Time Value
Future and present values of monetary balances follow compound interest. This insight is the essence of the concept of time value of money.
\begin{align}
FV(t) = PV \cdot (1 + \frac{r}{m})^{m \cdot t}
\end{align}\begin{aligned}
FV &: \mathrm{future ~ value}\\
PV &: \mathrm{present ~ value}\\
r &: \mathrm{interest ~ rate}\\
m &: \mathrm{compounding ~ frequency}\\
t &: \mathrm{time}
\end{aligned}According to Equation (1), a current amount of money having the present value PV will have the higher future value FV(t) at a future time t.
\begin{align}
PV = FV(t) ~ (1 + \frac{r}{m})^{-m \cdot t}
\end{align}Conversely, rearranging Equation (1) to Equation (2), an amount FV(t) available at a future time t will have a lesser present value PV.
Annuities as Series of Constant Payments
For the purpose of this introduction, annuities are series of constant payments at regular time intervals. The derivation of annuity valuation formulas then requires calculating a price for all of those payments at a single point in time. Let this point be the present moment. Furthermore, assume that FV(t) equals a constant amount C as in an annuity. In addition, set the compounding frequency m equal to the number of payments per year. Then Equation (2) determines the value of every single payment at present time. All that remains to be done is adding up the resulting terms, each discounted to present value by a different factor.
Additionally, the subsequent explanations assume one payment per year setting m=1. Since it’s possible to later on substitute back an interest rate covering year fractions, this will not result in a loss of generality.
Now consider an annuity paying C dollars once per year over the course of two years:
\begin{aligned}
PV_2 &= C \cdot (\frac{1}{1+r}+\frac{1}{(1+r)^2})\\
&= C \cdot \sum_{n=1}^2\frac{1}{(1+r)^n}
\end{aligned}While in the above situation it’s still easy to compute the present value, the aim is to derive a formula for an unlimited number of payments. Clearly, computing a possibly infinite number of powers of (1+r) is computationally expensive. However, it turns out that the problem simplifies by first finding the value of a perpetuity.
Pricing a Perpetuity
Pricing a perpetuity is finding a value for an investment paying a constant amount of C in all eternity.
\begin{align}
PV_{\infty} =C ~ \sum_{n=1}^{\infty}\frac{1}{(1+r)^n}
\end{align}Finding the value PV∞ looks frightening but is actually trivial. Imagine you could find a bank in these troubled times that you would want to trust with your money in all eternity. And that, even better, could plausibly promise paying interest r on your deposits during all that time. Well, you rush to deposit $1 to never touch that amount again. In exchange, the bank annually pays you the interest C=$1*r. Now, what would you pay for the right to receive $1*r in all eternity? But the question already has an answer, since $1 were deposited and the bank gets to keep the amount for ever. So, the below equation captures the hypothetical scenario:
\begin{align}
\$1 &= \$1 \cdot r ~\sum_{n=1}^{\infty}\frac{1}{(1+r)^n}
\end{align}Dividing both sides of Equation (3) by $1*r then solves for the perpetuity. It’s value is simply 1/r.
\begin{align}
\frac{1}{r} = \sum_{n=1}^{\infty}\frac{1}{(1+r)^n}
\end{align}Alternative Derivation of Perpetuity
Alternatively to the previous method, there is also a simple mathematical transform to price a perpetuity. Multiply both sides of Equation (3) by (1+r) to get:
\begin{aligned}
&(1+r)~PV_{\infty}\\
&= (1+r)~C ~ \sum_{n=1}^{\infty}\frac{1}{(1+r)^n}\\
&= C ~ (1 + \sum_{n=1}^{\infty}\frac{1}{(1+r)^n})\\
&= C + C~ \sum_{n=1}^{\infty}\frac{1}{(1+r)^n})\\
&= C + PV_{\infty}
\end{aligned}The third line of above equations has the infinite sum multiplied by (1+r), which cancels the denominator of the first term. Afterwards, the infinite series still contains powers of 1/(1+r) starting from one. And clearly it still holds terms until infinity. Therefore:
(1+r)~PV_{\infty} = C + PV_{\infty}Subtracting PV∞ on both sides and dividing by r again arrives at the simple valuation formula for a perpetuity:
\begin{align}
PV_{\infty} = \frac{C}{r}
\end{align}However, I have not yet heard a convincing economic argument as to how Equation (6) should work in a zero or negative interest economy.
Deducing the Value of Annuities from Perpetuities
It is now possible to deduce the value of a finite term annuity by factoring in a perpetuity:
\begin{align}
PV_{N} =C ~ \sum_{n=1}^{N}\frac{1}{(1+r)^n}
\end{align}To do so, simply add the missing terms of a perpetuity to both sides of Equation (7). Since the aim is to value an annuity of N terms, the missing terms run from N+1 to infinity. Consequently, the left hand side simplifies as follows:
PV_N + C~\sum_{n=(N+1)}^{\infty}\frac{1}{(1+r)^n} = PV_{\infty}For the right hand side there is another helpful transform of the added terms:
\begin{aligned}
&C\sum_{n=(N+1)}^{\infty}\frac{1}{(1+r)^n}\\
& = \frac{1}{(1+r)^N}~C~\sum_{n=1}^{\infty}\frac{1}{(1+r)^n}\\
& = \frac{1}{(1+r)^N}~PV_{\infty}
\end{aligned}Applying the two simplifications to Equation (7) results in:
\begin{aligned}
PV_{\infty}= &C~\sum_{n=1}^{N}\frac{1}{(1+r)^n}\\
&+\frac{1}{(1+r)^N}~PV_{\infty}
\end{aligned}In the above equation the sum from n=1 to N is obviously the annuity PVN. Therefore, it delivers the sought for solution for PVN in terms of PV∞:
\begin{align}
PV_{N}= PV_{\infty} (1 - \frac{1}{(1+r)^N})
\end{align}Finally, substitute Equation (6) into Equation (8) for the annuity value:
\begin{align}
PV_{N}= C \cdot \frac{1 - (1+r)^{-N}}{r}
\end{align}Equation (9) provides the present value of an annuity paying C over N periods in a financial environment of interest r. Next, the future value is found by letting the present value earn compound interest over N periods. Remember that throughout the derivation of Equation (9) the compounding frequence was kept at m=1. Therefore, an application of Equation (2) adds a term of (1+r) to the power of N for the future value:
FV_{N}= PV_{N} \cdot (1+r)^{N}
Consequently, Equation (10) has the solution for the future value of an annuity:
\begin{align}
FV_{N}= C \cdot \frac{(1+r)^{N} -1}{r}
\end{align}Now it’s time to generalize the formulas for compounding frequencies other than one. The derivation of Equation (9) and (10) equally works with time increments of 1/m and period interest r=R/m. This introduces a new variable R as the nominal interest rate and N periods now cover a time of N/m years. The result are generalized versions of the formulas for present and future values of annuities.
\begin{align}
PV_{N}&= m~C\frac{1 - (1+\frac{R}{m})^{-N}}{R}
\end{align}\begin{align}
FV_{N}&= m~C\frac{(1+\frac{R}{m})^{N} -1}{R}
\end{align}\begin{alignat*}{2}
PV_N &&~:~ & \mathrm{annuity ~ present ~ value}\\
FV_N &&~:~ & \mathrm{annuity ~ future ~ value}\\
R &&~:~ & \mathrm{interest ~ rate}\\
m &&~:~ & \mathrm{compounding}\\
&& ~~& \mathrm{frequency}\\
N &&~:~ & \mathrm{number ~ of ~ total}\\
&& ~~& \mathrm{payments}\\
\end{alignat*}Equations (11) and (12) compute the prices of deferred annuities placing payments at the ends of periods. As a final step in the derivation of annuity formulas the prices of advance annuities result from yet annother compounding step.
\begin{align}
PV_{aN} = (1 + \frac{R}{m}) \cdot PV_N
\end{align}\begin{align}
FV_{aN} = (1 + \frac{R}{m}) \cdot FV_N
\end{align}\begin{alignat*}{2}
PV_{aN} &&~:~ &\mathrm{advance ~ annuity}\\
&& & \mathrm{present ~ value}\\
FV_{aN} &&~:~ &\mathrm{advance ~ annuity}\\
&& & \mathrm{future ~ value}\\
\end{alignat*}References
Annuity Formulas, Wikipedia.org
