Annuity vs Perpetuity: What Changes When the Payments Never Stop

An annuity pays a fixed amount for a set number of periods and then stops; a perpetuity has no last payment. Rs 2,04,75,00,000 a year for five years at 12.00 per cent is worth Rs 7,38,07,79,274 today. The same amount growing 5.00 per cent a year for ever, on Gordon's formula, is worth Rs 29,25,00,00,000, or 3.96 times as much.

Everything in this guide turns on one thing: where the stream stops. A stream that stops has a countable number of terms, and its value is simply their sum. Every term is smaller than the one before it by more than growth adds back, and an endless run of shrinking terms settles on a total. So a stream that does not stop still has a finite value. The settling is the whole of the perpetuity formula, and the settling is also where the formula's one hard limit comes from. The formula works only while the rate used for discounting stays above the rate the payments grow at.

What is an annuity, and what is a perpetuity?

The comparison is worthless until both sides are properly on the table. Take the two objects one at a time.

An annuity is a level payment repeated a stated number of times and then finished. Three things have to be true for the word to apply. The amount is the same every period. The gap between payments is the same every time. And there is a last one, known in advance. A shop lease with nine years left to run pays rent on those terms. A four year college fee, paid once a year, has the same shape. So does the repayment on a household's five year vehicle loan: the same instalment, the same interval, and a final instalment everybody can point to on a calendar. Change any of the three and the stream is no longer an annuity. A list of cash flows is perfectly fine to value, but it has to be valued term by term rather than through a single multiplier.

A perpetuity is the same idea with the last payment removed. The amount arrives at the same interval, for ever, and nobody has written down a date on which it stops. A stream with no last payment sounds like a mathematical curiosity, and it is often treated as one. The shape is an ordinary one. A plot of agricultural land let to a tenant produces rent with no end date attached to it. A temple trust set up to pay for a daily meal is built to run without one. And, most importantly for anything in valuation, an ordinary operating business has no end date either. The shape therefore turns up in every model that has to say what a company is worth after the years somebody bothered to forecast.

Notice what is NOT part of either definition. Neither one says anything about certainty. Neither one says anything about risk. An annuity is not safer than a perpetuity and a perpetuity is not more speculative than an annuity. The only thing separating them is whether a last payment exists.

How can a stream with no last payment be worth a finite amount?

The endless stream is the sticking point for almost everybody meeting the idea, and the sticking point deserves to be seen rather than asserted. Infinitely many payments sounds like infinite money. The sum is not infinite. Each payment is pulled back to today through a discount factorThe single multiplier that shrinks one future rupee back to today, once the year it lands in and the rate are both fixed. that gets harsher every year further out.

The arithmetic is easiest to watch on the numbers used throughout. Sankalp Industrial Systems Limited, an invented manufacturer of industrial valves and precision castings, is assumed to produce Rs 2,04,75,00,000 of cash in the first year of the stream, with that amount rising 5.00 per cent a year afterwards, and the assumed rate for pulling it back is 12.00 per cent. Year one contributes Rs 2,04,75,00,000 divided by 1.12, or Rs 1,82,81,25,000. Year two's payment is 5.00 per cent larger but gets divided by 1.12 twice, so what reaches today is 1.05 over 1.12 of the previous year's contribution. The ratio of 1.05 to 1.12 is 0.9375 exactly, so every year contributes 93.75 per cent of what the year before contributed, for ever.

A run of numbers where each is 93.75 per cent of the last does not grow without limit. The running total piles up towards a ceiling and stops moving. The first twelve terms already come to Rs 15,76,71,66,010. Continued for ever, the pile reaches Rs 29,25,00,00,000 and never passes it. The formula that jumps straight to that ceiling is the perpetuity formula, and the arithmetic behind it is nothing more exotic than the ratio 0.9375 doing its work an unlimited number of times.

Try it out

An endless stream of payments carries a finite value. Which of these is the reason?

Because the payments themselves get smaller over timeBecause what each year contributes today shrinks faster than the payments growBecause the far years are treated as worth nothing and dropped

What is an annuity factor, and what is it actually counting?

Once a stream is accepted as having a value, the practical question is how to get at it without adding up terms one by one. Reaching that total in one step is what a factor does. A factor is the single number a level payment is multiplied by to give the present value of the entire run, and every shape in this guide has one.

The factor for a stream that stops

$$ A(n,r) \;=\; \frac{1-(1+r)^{-n}}{r} $$

A(n,r)	the annuity factor: what one rupee a year for n years is worth today
n	how many payments there are, counted from the first to the last
r	the rate per period used to pull each payment back, here the invented company's assumed 12.00 per cent

What it says in wordsOne, less what one rupee received n years out is worth today, all divided by the rate. A level payment multiplied by the result gives the value of every payment in the run.

At 12.00 per cent, five payments give one less 1.12 to the power minus five, all over 0.12. The factor is 3.604776. The assumed Rs 2,04,75,00,000 multiplied by that comes to Rs 7,38,07,79,274, written from here on as Rs 738.08 crore where the last two digits are not doing any work.

There is a reading of that factor worth carrying around. The factor is a number of years bought. Five payments at 12.00 per cent buy 3.604776 years of payment, not five, because the later ones arrive shrunken. And the factor climbs more slowly the further out the run goes. No intuition in this guide is more useful.

Payments in the run	Factor at 12.00 per cent	Value of Rs 2,04,75,00,000 a year	Extra years bought by the last stretch
1	0.892857	Rs 1,82,81,25,000	the first payment, almost all of it
3	2.401831	Rs 4,91,77,49,522	1.508974 across payments two and three
5	3.604776	Rs 7,38,07,79,274	1.202945 across payments four and five
10	5.650223	Rs 11,56,88,31,651	2.045447 across the next five
25	7.843139	Rs 16,05,88,27,332	2.192916 across the next fifteen
50	8.304498	Rs 17,00,34,60,655	0.461359 across the next twenty five
no last payment	8.333333	Rs 17,06,25,00,000	0.028835 across all remaining time

The last column, read downwards, carries the shape of the whole subject. Payments two and three together buy 1.508974 years. Payments twenty six to fifty, twenty five of them, buy 0.461359. And every payment after the fiftieth, an unlimited number of them, buys 0.028835 between them.

Why is the level perpetuity factor just one divided by the rate?

Look at the bottom row of that table again. The factor for a stream with no last payment is 8.333333, or one divided by 0.12. There is no other term in it. Since the stream has no number of years, no count of years appears. Nothing has to be raised to a count that was never fixed, so no power appears either.

The reason falls out of the annuity factor. As n is pushed further and further out, 1.12 to the power minus n gets closer and closer to nothing, so the numerator settles on one and the whole expression settles on one over the rate. A level perpetuity factor is the annuity factor with the only term that mentioned an ending taken out of it. Rs 2,04,75,00,000 a year for ever, level, at 12.00 per cent, is Rs 17,06,25,00,000. Because 0.12 divides cleanly, Rs 17,06,25,00,000 is exact rather than rounded.

The size of that factor is worth pausing on. A stream that runs for the rest of time, at this rate, is worth eight and a third years of it. Not fifty years of it, and certainly not an unbounded amount. The rate does all of the compressing. A perpetuity valued at a high rate therefore looks much less like magic than one valued at a low rate.

What does growth do to the factor, and what does Gordon's formula need to work?

Now let the payment rise. If it grows at a constant rate for ever, the factor becomes one divided by whatever is left when growth is taken out of the rate. Gordon's formula, set out by Myron Gordon in his 1959 paper on dividends, earnings and stock prices, is the version valuation work actually uses. A business that never grows again is a strange thing to assume.

The factor for a stream that never stops

$$ P \;=\; \frac{C}{r} \qquad\qquad P_{g} \;=\; \frac{C}{r-g}, \quad g

C	the payment in the first period of the stream, not the payment before it
r	the rate per period, here the assumed 12.00 per cent
g	the constant rate the payment rises at afterwards, here an assumed 5.00 per cent
P, Pg	the value today of the level version and of the growing version

What it says in wordsDivide the first payment by the rate to value a stream that never changes, and by the rate less the growth to value one that rises at a steady pace. The second only means anything while growth stays underneath the rate.

With 0.12 less 0.05 in the denominator, the factor is one over 0.07, or 14.285714. Multiply by Rs 2,04,75,00,000 and the stream is worth Rs 29,25,00,00,000.

The condition attached to that formula is not decoration. Growth has to sit below the rate. If growth equals the rate the denominator is nothing at all and the expression means nothing. If growth exceeds the rate the formula returns a negative number. The arithmetic is saying politely that the terms are getting bigger rather than smaller and the pile never settles. An assumed growth rate creeping towards the discount rate is therefore a warning sign rather than an ambitious view: the arithmetic runs away before the assumption becomes interesting.

What do the three shapes do to one and the same cash flow?

Here is where the comparison earns its keep. Take one cash flow, hold the rate at 12.00 per cent, and change nothing except the shape assumed for it.

The cash flow belongs to Sankalp Industrial Systems Limited and is the free cash flow to the firmEverything a business throws off for every provider of its funding at once, counted after the tax bill and after the year's own capital spending. the invented company is assumed to produce in Year 6, being Rs 2,04,75,00,000. Where that figure comes from is settled separately and taken as given here. Discounting is at year end throughout, so a year's cash counts as landing on the last day it belongs to.

The three factors are 3.604776, 8.333333 and 14.285714, and the gap between the ends of that range is Rs 21,86,92,20,726. Say that slowly. The cash flow did not change. The rate did not change. Nobody revised a forecast, argued about a margin or looked at a competitor. All that happened was a decision about the shape, and the answer moved by more than four fifths of its own top value.

What does the choice of shape do to the finished valuation?

Factors in the abstract are easy to shrug at. Put them inside a model and they stop being abstract.

The invented company's discounted cash flow value is Rs 21,28,13,79,094. Two parts make it up. Rs 4,68,41,43,564 covers the five years somebody actually forecast, the explicit forecast periodThe stretch of years somebody has actually written out line by line, before the shorthand takes over., plus Rs 16,59,72,35,530 for everything afterwards, being the Rs 29,25,00,00,000 growing perpetuity pulled back five years at 0.567426856. The two together give an enterprise valueA price tag on the operating business itself, before anybody splits it between the lenders and the owners. of Rs 21,28,13,79,094.

Now swap the shape and change nothing else. Treat the period after Year 5 as a five year annuity instead of a perpetuity. The tail is then worth Rs 7,38,07,79,274 at Year 5 rather than Rs 29,25,00,00,000, and pulls back to Rs 4,18,80,52,376 today. The five forecast years are untouched. The whole company comes out at Rs 8,87,21,95,940. Set against the Rs 21,28,13,79,094 of a moment ago, that is 41.69 per cent.

Fully 58.31 per cent of the answer rides on a decision most people make in about four seconds, usually without noticing they made one.

How many years does it take before an annuity is practically a perpetuity?

Most readers expect the answer to be very large, and are surprised by how small it is.

Hold the payment level at Rs 2,04,75,00,000 and the rate at 12.00 per cent, and just extend the run. One payment gives 10.71 per cent of the perpetuity. Five payments give 43.26 per cent. Ten give 67.80 per cent. Twenty five payments give 94.12 per cent. Every payment from the twenty sixth to the end of time is worth under six per cent of the total between them. Fifty payments give 99.65 per cent. A hundred give 99.9988 per cent.

Think about what that does to an argument. Two people disagreeing about whether a business runs for twenty five more years or for ever are, at this rate and on a level stream, arguing about six per cent. Two people disagreeing about whether it runs for five years or for ever are arguing about more than half. The disagreement that matters sits early, not late.

How much of a growing perpetuity sits beyond the years anybody can picture?

The settling above was fast because the payment was level. Growth changes it, and changes it a lot.

Take the growing stream, worth Rs 29,25,00,00,000 in total. Its first five payments land in Years 6 to 10 of the invented company's timeline and are worth Rs 7,38,07,79,274 between them. Those five payments are 25.23 per cent of the whole, so 74.77 per cent of the value sits in Year 11 and beyond.

Three quarters of the value is in years nobody in the room can describe. Not because the arithmetic is doing anything strange, but because growth at 5.00 per cent partly offsets the discounting, so each year gives up only 6.25 per cent of the year before rather than 10.71 per cent. Slower shrinking means a longer tail, and a longer tail means more of the answer sits where nobody has an opinion worth defending.

What does a change in growth do to the value, and why is the movement lopsided?

Growth sits in the denominator, and a denominator that is being eaten into behaves very differently from one that is not. Hold the Year 6 cash flow still at Rs 2,04,75,00,000 and move only the growth assumption.

One percentage point up adds Rs 4,87,50,00,000 and one point down removes Rs 3,65,62,50,000, so the same size of change is worth a third more in one direction than the other. The reason is visible in the denominator column. Going up takes 0.07 to 0.06, a seventh of what was there. Going down takes 0.07 to 0.08, a seventh added to a bigger base. The further growth climbs towards the rate, the fiercer each further step becomes, and the curve turns almost vertical as the two meet.

Which four questions decide the shape of a stream?

All of the above is arithmetic, and arithmetic is the easy half. The hard half is deciding which shape a real stream has, and that decision is not made in a spreadsheet. The shape is settled by looking at the asset that produces the cash.

Four questions settle it, and they are worth asking out loud in that order. Does the stream have a stated end written into it somewhere. Does the asset producing the cash have one of its own, as a concessionA right to work a resource or run a service for a stated stretch of time, granted by whoever controls it. or a licence does. Is the asset assumed to be replaced or renewed when it runs out. Every vehicle in a fleet has a last year, and a fleet that is always being replaced still produces a stream with no end. And, last, has anybody actually written down what happens in the year after the model stops.

The fourth question is the one that does the work in practice. When a stream is genuinely finite, somebody usually knows and can name the date. When a stream has been treated as finite by accident, nobody ever decided anything, so nobody can name a date.

Who actually uses this, and what do they use it for?

The distinction is not a classroom device. Three different people reach for it in a normal working week, and each of them is asking a different question of the same arithmetic.

Three desks, one distinction, three different reasons

A lending officer uses it to size a facility against the stream that repays it. If the cash comes from an arrangement with a stated end, the loan cannot outlive it, and the annuity factor is the honest ceiling on what the stream supports. A household applying for a vehicle loan meets exactly the same test in miniature: the lender looks at how many salary months are actually in front of the borrower, not at the salary itself.

An analyst uses it to interrogate somebody else's model. The single most productive question to ask about a valuation is what the model assumes happens after its last forecast year, and the second is whether anybody wrote that assumption down. Where the tail was left as an annuity by default, the analyst has found more of the answer in one question than a week of margin debate would yield.

An infrastructure team uses it to keep two very different assets apart. A road operated under a fixed term arrangement and a warehouse held outright produce cash that looks identical on paper and is not the same object at all. One has a last payment. The other does not. Everything else in the appraisal follows from which of the two is on the table.

The everyday version is a household choosing between two plots of land. One is let on a nine year lease with no renewal and the other is let with no end date agreed. The rent cheque is the same size each month. At 12.00 per cent, the second plot is worth a little over one and a half times the first, and nothing about the two tenants, the two buildings or the two rent cheques explains the difference.

Which neighbouring subjects does this touch, and where does the arithmetic stop?

Two shapes and the arithmetic of each are the whole of the distinction, and the questions that sit next to it are listed below.

Sankalp Industrial Systems Limited, Sankalp Coatings Private Limited and Aruna Tooling Private Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.