After this page you can
Value a level perpetuity, a perpetuity-due, an arithmetically increasing annuity and a geometrically growing perpetuity, and say when a growing perpetuity has no finite value
A stream that never ends, and still has a price
The first time a student meets a perpetuity the instinct is that it must be worth an unlimited amount. It is not, and the reason is the whole idea of discounting: each payment is worth less than the one before, the amounts shrink geometrically, and a geometric series with a ratio below one has a finite sum.
For ₹1 at the end of every year for ever, at an effective rate i:
There is a way of seeing this that is worth more than the algebra. A sum of 1 ÷ i invested at rate i earns exactly 1 in interest each year. Spend the interest, leave the capital alone, and you have paid 1 a year for ever without ever touching the principal. So 1 ÷ i is not a limit arrived at by summing a series; it is the amount of capital whose income is the payment.
Paid at the start of each year instead, every payment arrives a year earlier, so the value is a year's interest larger:
The second form is the easier one to remember and to check: a perpetuity-due is a perpetuity-immediate plus one payment today.
Payments that increase
Two kinds of growth appear, and they behave quite differently.
Arithmetic: 1, 2, 3, 4, …
The payment rises by a fixed amount each year. Over n years the present value is written (Ia)n, and
Run it to infinity and it stays finite, which surprises people:
The payments grow without limit and the discounting still wins, because an arithmetic increase cannot outrun a geometric discount. At 8% the factor is 168.75 — large, and finite.
Geometric: payments growing at a fixed percentage
Here the payment is multiplied by (1 + g) each year, and the value is remarkably simple:
This is the one to look at closely, because of what happens to the denominator.
The mechanism: growth against the discount rate
The geometric perpetuity converges only when g < i. The payments are growing at g and being discounted at i, so the terms shrink only if the discounting is the stronger of the two.
If g ≥ i the series diverges and the value is infinite. That is not a failure of the arithmetic; it is the arithmetic reporting that the assumption is impossible. A payment stream growing for ever at a rate above the return the market offers would be worth more than all the money there is, so some part of the assumption — the growth, the term, or the rate — is wrong.
And notice how violent the formula becomes just before it breaks. At i = 8%, growth of 3% gives a divisor of 5% and a value of 20 times the first payment; growth of 7% gives a divisor of 1% and a value of 100 times. A one-point change in an assumption has quintupled the answer. Any valuation resting on this formula is therefore extremely sensitive to g, and an answer that does not say so is incomplete.
Why an actuary is made to learn this
Three live uses, each with the same shape.
Pensions in payment increase. A pension rising with inflation for the rest of a life is a growing annuity, and the gap between the discount rate and the increase rate is what the liability is most sensitive to.
Share valuation. A share paying a dividend that grows for ever is exactly the geometric perpetuity, and the formula above is the one used to price it.
Endowments and trusts. A fund intended to pay out for ever without eroding its capital is the 1 ÷ i argument made at the start of this page.
Questions students ask
Do perpetuities actually exist?
A few historical government stocks were issued with no redemption date, and some preference shares are irredeemable. But the formula earns its place mostly as an approximation and a bound: a 50-year annuity at a realistic rate is worth very nearly the perpetuity, so 1 ÷ i is a quick upper bound for any long level stream, and comparing it with the annuity value tells you how much the final years are actually contributing.
Why is (Ia)∞ finite when the payments grow for ever?
Because arithmetic growth is linear and discounting is exponential, and exponential decay beats linear growth eventually — the terms t vt rise for a while and then fall away to nothing. Geometric growth is the case where the two are the same kind of thing, which is why that one can diverge and this one cannot.
Can I value a growing annuity without a new formula?
Yes, and it is the better method when the growth is geometric. Discount at the net rate (1 + i) ÷ (1 + g) − 1 and treat the payments as level. That is the same arithmetic as converting a money rate to a real rate using the inflation page's exact formula, which is not a coincidence: growth in the payments and inflation in the currency are the same operation seen from different sides.
A worked example
Invented examples, built for this page, all at an effective rate of 8%, so v = 1 ÷ 1.08 = 0.925925926.
Part 1: four perpetuities on the same ₹10,000
Payments Factor Value ₹10,000 at each year end, for ever 1 ÷ 0.08 = 12.50 ₹1,25,000 ₹10,000 at each year start, for ever 1.08 ÷ 0.08 = 13.50 ₹1,35,000 ₹10,000, 20,000, 30,000, … for ever 1.08 ÷ 0.082 = 168.75 ₹16,87,500 ₹10,000 growing at 3% a year, for ever 1 ÷ (0.08 − 0.03) = 20.00 ₹2,00,000 Check row two against row one. A perpetuity-due is a perpetuity-immediate plus one payment today:
1,25,000 + 10,000 = ₹1,35,000. It is also the immediate value accumulated one year:1,25,000 × 1.08 = ₹1,35,000. Two routes, same figure.Check row one by the capital argument. ₹1,25,000 at 8% earns
1,25,000 × 0.08 = ₹10,000a year. Spend exactly that and the capital is untouched for ever, which is what the perpetuity promises.Check row four from first principles, without the shortcut. The payments are
10,000 × 1.03t−1at timet, so the present value is a geometric series with first term10,000 vand ratio1.03 v:10,000 v ÷ (1 − 1.03 v) = 10,000 × 0.925925926 ÷ (1 − 0.953703704) = 9,259.26 ÷ 0.046296296 = ₹2,00,000which is the
1 ÷ (i − g)answer exactly. The shortcut is the series, not an approximation to it.Part 1 continued: how fragile row four is
Change only the growth assumption and nothing else:
Growth g Divisor i − g Value 0% 0.08 ₹1,25,000 3% 0.05 ₹2,00,000 5% 0.03 ₹3,33,333 7% 0.01 ₹10,00,000 8% 0 no finite value Moving the growth assumption from 5% to 7% — two percentage points, well inside the range of honest disagreement — triples the answer. At 0% growth the row agrees with the level perpetuity in the first table, which is the check that the formula is being applied correctly. At
g = ievery payment has the same present value and there are infinitely many of them, so the sum does not exist; the formula's divisor of zero is reporting that, not failing.Part 2: an increasing annuity over five years, two ways
Payments of ₹10,000, ₹20,000, ₹30,000, ₹40,000 and ₹50,000 at the ends of years 1 to 5, at 8%.
By formula. First the ingredients:
v5 = 0.680583197 | a5 = (1 − 0.680583197) ÷ 0.08 = 3.99271004 | ä5 = 3.99271004 × 1.08 = 4.31212684(Ia)5 = ( 4.31212684 − 5 × 0.680583197 ) ÷ 0.08 = 0.90921086 ÷ 0.08 = 11.3651357The payments are ₹10,000 times 1, 2, 3, 4, 5, so the present value is
10,000 × 11.3651357 = ₹1,13,651.36.By adding up the five terms. No formula, just five discounted payments:
Time Payment Discount factor Present value 1 ₹10,000 0.925925926 ₹9,259.26 2 ₹20,000 0.857338820 ₹17,146.78 3 ₹30,000 0.793832241 ₹23,814.97 4 ₹40,000 0.735029853 ₹29,401.19 5 ₹50,000 0.680583197 ₹34,029.16 Total ₹1,13,651.36 The two routes agree to the paisa. The second is the definition and the first is a closed form for it, and over five payments the long way is quick enough to be worth doing as a check — which is exactly how to catch a misremembered
(Ia)formula under exam conditions.One observation worth taking from the table. The
v5 = 0.680583197that appears in the last row is the same quantity that appears twice in the formula, once insidea5and once in then vnterm. Computing it once, carefully, and reusing it is both faster and safer than recomputing a rounded version at each step.
Where this fits in CM1
This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CM1 topics explained
- Simple and compound interest, and the principle that rules out one of them
- Present value, the discount factor, and why d is not i
- Nominal rates payable p times a year, and the loan that costs more than it says
- Force of interest explained, and how it relates to i, v and d
- When the interest rate changes: steps, and a force that varies
- Continuous payment streams, and why they are worth more than they look
- Annuity functions in CM1: immediate, due, deferred and temporary
- Money rates, real rates and inflation: the exact link, not the subtraction
- Real yields from an inflation index, and index-linked payments
- Loan schedules: the prospective and retrospective methods
- Life table functions explained: l(x), d(x), q(x) and p(x)
- Net premiums and the equivalence principle, worked from first principles
- Reserves in CM1: prospective, retrospective, and why they agree
- Thiele's differential equation, explained term by term
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.