There are two directions in which the formula for discrete compounding naturally goes. One towards a limit of continuous compounding, and the other as a geometric series modelling the way that the quantity (for example, money) grows during the observation period.
The idea of $e$ was implicit by 1614 in Napier's work on natural logs. The use of $e$ as a constant was due to Euler in 1727 and he was explicitly setting it in the context of taking the compounding formula to the continuous limit. Just as the idea of $e$ was beginning to be born, in 1613 Richard Witt published the first book dedicated to the maths of (discrete) compounding.
A geometric series has as its sum: $a(\frac{1-r^n}{1-r})$. Notice here that $r$ is not a rate, but a growth multiplier - the equivalent in present value/future value analysis to $\frac{1}{(1+r)}$, where $r$ in that case is an actual interest rate. An annuity can be modelled with this formula as long as we set the initial payout amount per period to be $a$ and the common ratio to be a present value formula, being sensitive to compounding. The idea is to realise that an annuity describes a series of evenly spaced cash payments into the future. To work out how much each is worth, we'd like to present value them with a constant rate and a discounting period counter of $n$.
The annuity thus is a decent model for working our how valuable to you the coupon payments are. With computers these days doing the hard work of iterating the cash flows, the alternative is just to loop over all the coupon payments, and to make the last payment to include the principal. But in the days before computers, having mathematics do the hard work was a smart move.
So, for the $n$ periodic coupon payments $c$ on a bond, with the period return being the nominal discount rate $\frac{1}{(1+r)^n}$ , the series of payments $\frac{c}{(1+r)} + \frac{c}{(1+r)^2} + \frac{c}{(1+r)^3} \cdots \frac{c}{(1+r)^n}$ sums to $c(\frac{1-{\frac{1}{(1+r)}}^n}{1-\frac{1}{(1+r)}})$, or, bringing the $n$ down to the denominator, $c(\frac{1-{\frac{1}{(1+r)^n}}}{1-\frac{1}{(1+r)}})$
Finally, there's a limits exposition which shows that, in the case where $r<1$, as it usually is with discount factors, then the sum of a geometric series for an infinite number of steps is the surprising $\frac{a}{1-r}$. So far, again, $r$ is the discount factor and to make it into a rate you replace it with $(1+r)$ - the 1s cancel and you're left with $\frac{a}{r}$, where the $r$ is no longer a geometric series ratio, but a discount rate. This is the present value of a perpetuity.
Ground rent is an example of a perpetuity. If someone tells you the ground rent on a freehold is worth 15,000 for annual payments of 100, then they're telling you that the market rate to discount for that infinite flow of 100 payments, forever, is solved by setting $15000 = \frac{100}{r}$ which means $r=\frac{1}{150}$ or about 0.667% annualised.
Note how confusing it is to join together the maths of geometric series (a,r) with that for compounding (P,c,n,r) since in both of these worlds, by tradition, the choice of r can be semantically jarring, as they mean different things. In geometric series maths, it represents a multiplier, and in compounding it represents the percentage growth (1+r).
The idea of $e$ was implicit by 1614 in Napier's work on natural logs. The use of $e$ as a constant was due to Euler in 1727 and he was explicitly setting it in the context of taking the compounding formula to the continuous limit. Just as the idea of $e$ was beginning to be born, in 1613 Richard Witt published the first book dedicated to the maths of (discrete) compounding.
A geometric series has as its sum: $a(\frac{1-r^n}{1-r})$. Notice here that $r$ is not a rate, but a growth multiplier - the equivalent in present value/future value analysis to $\frac{1}{(1+r)}$, where $r$ in that case is an actual interest rate. An annuity can be modelled with this formula as long as we set the initial payout amount per period to be $a$ and the common ratio to be a present value formula, being sensitive to compounding. The idea is to realise that an annuity describes a series of evenly spaced cash payments into the future. To work out how much each is worth, we'd like to present value them with a constant rate and a discounting period counter of $n$.
The annuity thus is a decent model for working our how valuable to you the coupon payments are. With computers these days doing the hard work of iterating the cash flows, the alternative is just to loop over all the coupon payments, and to make the last payment to include the principal. But in the days before computers, having mathematics do the hard work was a smart move.
So, for the $n$ periodic coupon payments $c$ on a bond, with the period return being the nominal discount rate $\frac{1}{(1+r)^n}$ , the series of payments $\frac{c}{(1+r)} + \frac{c}{(1+r)^2} + \frac{c}{(1+r)^3} \cdots \frac{c}{(1+r)^n}$ sums to $c(\frac{1-{\frac{1}{(1+r)}}^n}{1-\frac{1}{(1+r)}})$, or, bringing the $n$ down to the denominator, $c(\frac{1-{\frac{1}{(1+r)^n}}}{1-\frac{1}{(1+r)}})$
Finally, there's a limits exposition which shows that, in the case where $r<1$, as it usually is with discount factors, then the sum of a geometric series for an infinite number of steps is the surprising $\frac{a}{1-r}$. So far, again, $r$ is the discount factor and to make it into a rate you replace it with $(1+r)$ - the 1s cancel and you're left with $\frac{a}{r}$, where the $r$ is no longer a geometric series ratio, but a discount rate. This is the present value of a perpetuity.
Ground rent is an example of a perpetuity. If someone tells you the ground rent on a freehold is worth 15,000 for annual payments of 100, then they're telling you that the market rate to discount for that infinite flow of 100 payments, forever, is solved by setting $15000 = \frac{100}{r}$ which means $r=\frac{1}{150}$ or about 0.667% annualised.
Note how confusing it is to join together the maths of geometric series (a,r) with that for compounding (P,c,n,r) since in both of these worlds, by tradition, the choice of r can be semantically jarring, as they mean different things. In geometric series maths, it represents a multiplier, and in compounding it represents the percentage growth (1+r).
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