From one payment to a whole stream
The last two guides taught you to move a *single* sum across time: discount it back to find a present value, or roll it forward to find an accumulated value. But almost nothing in an actuary's world arrives as one lonely payment. A loan is repaid in monthly instalments. A pension drips out a cheque every year for decades. A bond pays coupons twice a year and the face value at the end. Each of these is a *stream* of payments, and we need a way to put one honest price tag on the entire stream at once.
An annuity-certain is the simplest such stream: a fixed payment made at regular intervals for a fixed, *guaranteed* number of periods. The word "certain" is doing real work — it means the payments are promised regardless of whether anyone is alive to receive them, in contrast to the *life* annuities (which depend on survival) you will meet several rungs from now. Here there is no dice-rolling at all; only the arithmetic of interest. That makes the annuity-certain the perfect training ground.
The trick is no trick at all — it is the equation of value from the previous guide, applied many times over. Discount each payment back to today, then add up the pile. Because the payments are equal and evenly spaced, that sum collapses into a neat shorthand. The whole subject of annuity-certains is really just *bookkeeping for the equation of value*, organised so you never have to discount fifty payments one by one.
Immediate vs due: end of period or start?
There is exactly one decision that splits annuities-certain into two families, and beginners trip over it constantly: *when within each period does the payment land?* If each payment arrives at the end of its period, it is an annuity-immediate — confusingly named, since "immediate" here means the opposite of immediate. Think of a loan you took out today and start repaying one month *later*: the first instalment is at the end of period one. Most loans, bonds, and ordinary debts work this way.
If instead each payment arrives at the start of its period, it is an annuity-due. Rent is the classic example: you pay for the month *before* you live in it, not after. So is an insurance premium — the cover only switches on once you have paid. The two families describe the *same* set of payments shifted by a single period, and that one-period shift is the only difference between them.
Present value and accumulated value, in words
Every annuity-certain has two natural valuation points. Its present value is what the whole future stream is worth *today*: discount each payment back to time zero and sum them. Its accumulated value is what the stream will have grown into by the *end*, once each payment has earned interest from its arrival until the final date. The two are the same stream viewed from opposite ends of the timeline, and they differ only by accumulating the present value forward over the full term.
Here is the picture in plain words, no algebra. The present value of a level annuity-immediate is: *the payment amount, times a factor that says "how many of today's dollars buys one dollar at the end of each of the next n periods."* That factor is bigger when the term n is longer (more payments to value) and smaller when the interest rate i is higher (future dollars are discounted harder). Push n to a very long horizon and the factor keeps growing but ever more slowly, because distant payments are squashed almost to nothing by discounting — a fact we are about to exploit.
A tiny worked example: a 5-year stream
Let's value a real stream by hand. Suppose you are promised $1,000 at the end of each of the next 5 years, and the effective annual interest rate is 5%. This is a level annuity-immediate. We do exactly what the equation of value demands: discount each $1,000 back to today using the discount factor 1 / (1.05) raised to the right power, then add the five present values together.
Effective annual rate i = 5% -> discount factor v = 1/1.05 = 0.952381
Year Payment Discounted to today
1 1,000 1,000 / 1.05^1 = 952.38
2 1,000 1,000 / 1.05^2 = 907.03
3 1,000 1,000 / 1.05^3 = 863.84
4 1,000 1,000 / 1.05^4 = 822.70
5 1,000 1,000 / 1.05^5 = 783.53
---------
Present value (annuity-immediate) = 4,329.48
Simple (wrong) sum of payments = 5,000.00 <- ignores time value
Annuity-DUE (paid at start of each year) = 4,329.48 * 1.05 = 4,545.95
Accumulated value at end of year 5 = 4,329.48 * 1.05^5 = 5,525.63Read the bottom three lines carefully, because they tie the whole guide together. The stream of five $1,000 payments is worth $4,329.48 today as an annuity-immediate — not the naive $5,000. Slide every payment to the *start* of its year and it becomes an annuity-due worth exactly $4,329.48 × 1.05 = $4,545.95, the single (1 + i) bump we promised. And if instead you let the whole stream sit and earn interest until the end of year 5, its accumulated value is $5,525.63 — the same stream, just valued at the far end of the timeline.
Perpetuities and deferred annuities
Now stretch the term to infinity. A perpetuity pays a fixed amount at every period *forever*, with no end date. It sounds as though it must cost an infinite amount — but it does not, and that surprise is one of the most beautiful results in interest theory. Because each successive payment is discounted a little harder than the last, the far-off payments contribute almost nothing, and the infinite sum settles to a finite, tidy number: the present value of a perpetuity-immediate is simply the payment divided by the interest rate. A $1,000-a-year perpetuity at 5% is worth exactly $1,000 / 0.05 = $20,000 today.
The mirror image of an infinitely long annuity is one that simply hasn't started yet. A deferred annuity is an ordinary annuity-certain whose payments begin not now but after a waiting period — say, an income stream you buy today that starts paying in 10 years. Valuing it needs no new machinery at all: work out the present value as if the annuity began at the end of the deferral, then discount that single lump back across the deferral period. Two steps you already own, stacked one on top of the other.
These two ideas are not academic curiosities. A perpetuity is the textbook model for an endowed scholarship or a consol bond, and it is the limiting case that pins down the value of very long pensions. A deferred annuity is exactly what a 35-year-old buys for retirement income decades away — and it is the gateway to the *life* annuities and pensions later in the ladder, where the deferral period is real human years and the payments are no longer certain but contingent on survival. We have not even touched payments that grow or shrink over time — the varying annuities — but every one of them is built from the same single move: discount, then add.
What you carry forward
You can now take any level stream of guaranteed payments and collapse it into a single honest number — at the front of the timeline (present value) or at the back (accumulated value), paid at the end of each period (immediate) or the start (due), running for a fixed term, forever (perpetuity), or starting later (deferred). Underneath all of it sits one move you have used since the first guide of this rung: discount with the time value of money, then add. The next guides put this engine to work on the things people actually borrow and buy — loans and their amortisation schedules, and the yields that tell you what a stream of payments truly earns.
One honest caveat before you go: every number in this guide assumed a single, constant interest rate that never changes over the whole term. That keeps the arithmetic clean and is a fine first model, but real rates move year to year and differ by maturity. When that matters, an actuary discounts each payment at its *own* rate drawn from a yield curve — a refinement you will meet in the term-structure guide, not a contradiction of anything here. A model is a careful map, never the territory.