The derivation of fixed interest annuity formulas is based on compound interest and the concept of time value of money, allowing to express the total value of possibly very long time series of cashflows in terms of simple analytical formulas.
After a short recap of time value of money with present and future value of cash flows, this introduction defines annuities as series of constant cashflows. It subsequently shows in two intuitive ways how infinite series of constant cashflows, so-called perpetuities, have finite present values if interest rates are positive. Finally, common formulas for deferred and advance annuities will be derived by factoring in said perpetuities.
Compound Interest and Time Value
Future and present values of monetary balances follow from 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
Annuities are series of constant payments at regular time intervals and finding a price for such streams of cashflows means to convert them into an equivalent single payment at a specific point in time. But even though the payments comprising an annuity are nominally all same, their present or future values are all different. This is because in the concept of time value, money becomes less valuable the further its receipt is deferred into the future. As a consequence, we cannot simply add up all the nominal values of annuity cashflows in order to calculate the annuity’s current price.
The current price, or present value is calculated with equation (2), where the FV(t) are the annuity cashflows of constant value C. For simplicity, let’s assume annual compounding equivalent to a compounding frequency of m=1. An annuity paying an amount of C twice, once at the end of this year, and once at the end of next year, will then have the following value PV2:
\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}With a growing number of payments C the calculation of the sum of fractions of powers of (1+r) becomes increasingly impracticable, motivating the search for an analytical shortcut. In order to find one, the next section examines what happens when the number of cashflows C grows infinitely large.
Pricing a Perpetuity
Pricing a perpetuity means 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}So according to equation (3), an infinite number of fractions would need to be computed and summed up for PV∞. But fortunately, it is not necessary to attempt this computationally impossible task because it turns out that the problem has trivial solutions.
Logical Derivation of Perpetuity Valuation
Imagine you could find a bank in these troubled times which you would want to forever trust with your money. A bank which, even better, could plausibly promise to pay interest on your deposits into the far future. Delightedly, you rush to deposit and never touch again an amount of $1. In exchange, the bank pays annual interest of C=$1*r. Now, how would you value the right to annually receive $1*r in all eternity? But the question already has an answer, since the price you paid was the $1 that you deposited. Therefore, PV∞ in equation (3) will be $1:
\begin{align}
\$1 &= \$1 \cdot r ~\sum_{n=1}^{\infty}\frac{1}{(1+r)^n}
\end{align}All that’s left to find a solution is to divide both sides of equation(4) by ($1*r). Accordingly, a $1 perpetuity has a value of 1/r.
\begin{align}
\frac{1}{r} = \sum_{n=1}^{\infty}\frac{1}{(1+r)^n}
\end{align}Mathematical Derivation of Perpetuity Valuation
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 until infinity. Therefore:
(1+r)~PV_{\infty} = C + PV_{\infty}Subtracting PV∞ on both sides and dividing by r yields the same valuation formula for a perpetuity that we arrived at before:
\begin{align}
PV_{\infty} = \frac{C}{r}
\end{align}Note, however, that it’s impossible to price a perpetuity in a zero or negative interest economy, hinting at the absurdity of modern monetary policies.
Deducing the Value of Annuities from Perpetuities
It is now possible to deduce the value of a finite term annuity of N cashflows by factoring in a perpetuity, because the difference between PVN in equation (7) and PV∞ in equation (3) is equally a perpetuity.
\begin{align}
PV_{N} =C ~ \sum_{n=1}^{N}\frac{1}{(1+r)^n}
\end{align}More specifically, what separates PVN from PV∞ is a perpetuity starting to pay from the end of period (N+1):
PV_N + C~\sum_{n=(N+1)}^{\infty}\frac{1}{(1+r)^n} = PV_{\infty}But the value of a perpetuity starting to pay from the end of period (N+1) can equally be expressed in terms of a perpetuity PV∞ starting to pay at the end of the current year:
\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}Substituting into equation (7) gives:
\begin{aligned}
PV_{\infty}= &~C~\sum_{n=1}^{N}\frac{1}{(1+r)^n}\\
&+\frac{1}{(1+r)^N}~PV_{\infty}\\
\end{aligned}But in the above equation the sum from n=1 to N is obviously same as the annuity PVN:
\begin{aligned}
PV_{\infty}= &~PV_N+ \frac{1}{(1+r)^N}~PV_{\infty}\\
\end{aligned}We can now express 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) to obtain the present value formula for deferred annuities:
\begin{align}
PV_{N}= C \cdot \frac{1 - (1+r)^{-N}}{r}
\end{align}Next, we get the future value FVN of an annuity by computing the future value of the result of the formula for the present value PVN:
FV_{N}= PV_{N} \cdot (1+r)^{N}
Computing the future value of equation (9) then produces the formula for the future value of deferred annuities:
\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. Suppose an annuity had an annual payment frequency of m. Because of payments in time increments of 1/m this will slice the period interest in equations (9) and (10) to r=R/m, where R is an annual interest rate introduced merely for the purpose of substitution.
\begin{align}
PV_{N}&= m \cdot C\frac{1 - (1+\frac{R}{m})^{-N}}{R}
\end{align}\begin{align}
FV_{N}&= m \cdot 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{annual~interest ~ rate}\\
m &&~:~ & \mathrm{compounding}\\
&& ~~& \mathrm{frequency}\\
N &&~:~ & \mathrm{number ~ of ~ total}\\
&& ~~& \mathrm{payments}\\
\end{alignat*}Equations (11) and (12) still neglect the duration of annuities. In the simplified derivation with a compounding frequency m=1, annuities paid for N years. But to receive payments over the course of K years, we can substitute K=m*N.
\begin{align}
PV_{K}&= m~C\frac{1 - (1+\frac{R}{m})^{-N
\cdot m}}{R}
\end{align}\begin{align}
FV_{K}&= m \cdot C\frac{(1+\frac{R}{m})^{N \cdot m} -1}{R}
\end{align}Equations (13) and (14) are the commonly known formulas for the prices of deferred annuities placing payments at the ends of periods. For advance annuities paying at the start of periods, formulas need to be extended by a one period compounding step:
\begin{align}
PV_{aK} = (1 + \frac{R}{m}) \cdot PV_K
\end{align}\begin{align}
FV_{aK} = (1 + \frac{R}{m}) \cdot FV_K
\end{align}\begin{alignat*}{2}
PV_{aK} &&~:~ &\mathrm{advance ~ annuity}\\
&& & \mathrm{present ~ value}\\
FV_{aK} &&~:~ &\mathrm{advance ~ annuity}\\
&& & \mathrm{future ~ value}\\
\end{alignat*}References
Annuity Formulas, Wikipedia.org
