One formula, four dressings
An annuity in CM1 is nothing more complicated than a series of level payments, and its present value is nothing more complicated than the sum of a geometric progression. The reason the topic feels bigger than that is the notation: four symbols, each a small variation on the same sum, and a question that uses the wrong one is wrong by exactly one year's interest.
So learn the one sum first. If ₹1 is paid at the end of each of the next n years, the present value is
That is the annuity-immediate. "Immediate" is an unhelpful word that means the opposite of what it sounds like: payments are made in arrear, at the end of each period. The closed form comes straight from the geometric sum, and it is worth deriving it once by hand so that the (1 − vn) on top and the bare i underneath stop being something to remember.
The other three
Annuity-due: än
Payments at the start of each year — times 0, 1, …, n−1. Every payment happens one year earlier than in the annuity-immediate, so every term is worth one year's interest more:
Two things fall out of that, and both are useful. The (1 + i) multiplier is why the mistake is always worth exactly one year's interest. And the denominator becoming d rather than i is not a coincidence: it is the same d = iv from the interest-theory topic, appearing because the payments have moved to the front of the year, which is precisely what a discount rate describes.
A second identity worth having: än = 1 + an−1. The payment at time 0 is undiscounted, and the rest form an (n−1)-year annuity-immediate. That is often the fastest route when a table of an values is to hand.
Deferred annuity: m|an
Nothing is paid for m years, then n payments begin. Discount the whole annuity back:
The second form is the better one to think with: a deferred annuity is a long annuity minus the bit at the front you do not get. It is also a free check on your arithmetic, because the two routes use different numbers and must give the same answer.
Temporary annuity, and the life versions
So far every payment is certain. The life-contingent versions replace "paid at time t" with "paid at time t if the life is still alive", which means weighting each term by a survival probability:
A temporary annuity cuts the sum off: ax:n runs t = 1 to n and nothing after. The structure is identical to the certain case; the only new ingredient is tpx, which comes out of a life table. Note that äx = 1 + ax exactly, with no survival factor on the 1, because the life is alive at the moment the policy starts — that is what "aged x" means.
Where marks are lost
Choosing immediate where the question says due, or the reverse. Read for the words: a premium is almost always paid in advance (annuity-due), a pension or a loan repayment almost always in arrear (annuity-immediate). When the wording is ambiguous, draw the timeline. Thirty seconds of timeline beats a confident wrong symbol.
Getting n wrong by one on a deferred annuity. "Payments for 10 years, starting in 5 years' time" has to be pinned down: is the first payment at time 5 or at time 6? Write out the first two and the last two payment times explicitly before you choose m and n. This is the error the am+n − am check is best at catching.
Using d in the annuity-immediate formula. an has i underneath and än has d. Swapping them produces an answer that is wrong by a factor of (1 + i)2 — large enough to notice if you sanity-check, invisible if you do not.
Two checks that take ten seconds
The bound. an can never exceed n, and it can never exceed 1/i (the value of a perpetuity). At 6%, no annuity-immediate, of any term, can be worth more than 16.667. An answer of 18 is wrong, and you know it without checking the working.
The ratio. än / an must equal 1 + i exactly. If you have calculated both and the ratio is not 1.06 at a 6% rate, one of them is wrong.
Questions students ask
Why is an annuity paid in arrear called "immediate"?
It is a historical term, not a description, and the best thing to do is stop trying to make it make sense. "Annuity-immediate" means payments at the ends of the periods. "Annuity-due" means payments at the starts. The dots over the a in the annuity-due symbol are the thing to read, not the English.
Do I need to memorise the closed forms?
You need an = (1 − vn)/i cold. Everything else is one step from it: multiply by (1 + i) for the due version, multiply by vm to defer, truncate the sum to make it temporary. Memorising four formulae separately is how you end up using a life annuity formula on a certain annuity.
What changes when the payments are not level?
The geometric sum stops collapsing so neatly and you get the increasing-annuity functions instead. The method does not change: write the payments on a timeline, multiply each by its discount factor, and sum. Every one of the increasing-annuity results in the syllabus is that sum done once and recorded so you do not have to repeat it.
A worked example
Take i = 6% and a payment of ₹25,000 a year for 10 years. Find the present value if the payments are in arrear, if they are in advance, and if they are deferred five years.
The building blocks
1.0610 = 1.7908477, so v10 = 0.5583948
a10 = (1 − 0.5583948) / 0.06 = 0.4416052 / 0.06 = 7.3600871
Payments in arrear
Payments in advance
PV = 25,000 × 7.8016923 = ₹1,95,042.31
Check it through d: d = 0.06/1.06 = 0.0566038, and 0.4416052 / 0.0566038 = 7.8016923. The same figure.
And check the difference. ₹1,95,042.31 − ₹1,84,002.18 = ₹11,040.13, which is 0.06 × 1,84,002.18 to the rupee — exactly one year's interest, as the (1 + i) multiplier says it must be. If that had not matched, one of the two answers would be wrong.
Deferred five years, then ten payments
5|a10 = 0.7472582 × 7.3600871 = 5.4998832
PV = 25,000 × 5.4998832 = ₹1,37,497.08
The second route, as a check
A ten-year annuity deferred five years is a fifteen-year annuity minus the first five years of it.
a15 = (1 − 0.4172651) / 0.06 = 9.7122490
a5 = (1 − 0.7472582) / 0.06 = 4.2123638
a15 − a5 = 9.7122490 − 4.2123638 = 5.4998852
5.4998832 against 5.4998852 — agreement to six decimal places, the last digit differing because v5 and v15 were each rounded to seven places before being used. That is rounding, not a mistake, and knowing the difference is most of what checking an answer is.
The bounds, as a final sanity check
A perpetuity at 6% is worth 1/0.06 = 16.6667. All three of 7.36, 7.80 and 5.50 sit below it and below their own term lengths, as they must.
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
- Perpetuities and increasing annuities: payments that never stop, or never stay level
- 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.