Annuities: The Intuition First
Imagine you're saving for retirement. You decide to put ₹10,000 into a fixed deposit every year for 20 years. At the end of each year, the bank gives you 8% interest on whatever is in your account. What will your total savings be at the end of 20 years?
That's an annuity — a series of equal payments made at regular intervals. The key idea is that each payment earns interest for a different length of time. The first ₹10,000 earns interest for 20 years, the second for 19 years, and the last one earns interest for just 1 year (or zero, depending on when you count).
An annuity is not a single lump sum. It's a stream of identical cash flows spaced equally in time. The "value" of an annuity is what that entire stream is worth at a particular point in time, given a certain interest rate.
Two Kinds of Annuity Value
There are two questions you can ask:
- Future Value (FV) — "If I deposit ₹P every year for n years at r% interest, how much will I have at the end?"
- Present Value (PV) — "If someone promises to pay me ₹P every year for n years, and I can earn r% elsewhere, what is that promise worth right now?"
Both are built from the same core idea: each payment is a separate compound-interest problem, and you add them up.
The Precise Mathematics
Let:
- P = the regular payment (the annuity amount)
- r = interest rate per period (as a decimal; e.g., 8% = 0.08)
- n = number of payments
Future Value of an Ordinary Annuity
"Ordinary" means payments happen at the end of each period. The first payment earns interest for n−1 periods, the second for n−2, and the last earns no interest at all.
FV=P(1+r)n−1+P(1+r)n−2+⋯+P(1+r)+P
This is a geometric series. Factor out P and sum it:
FV=P⋅r(1+r)n−1
FVordinary=P⋅r(1+r)n−1
Present Value of an Ordinary Annuity
Here we discount each future payment back to today. The first payment (one period away) is discounted by (1+r)1, the second by (1+r)2, and so on.
PV=(1+r)P+(1+r)2P+⋯+(1+r)nP
Again, a geometric series:
PV=P⋅r1−(1+r)−n
PVordinary=P⋅r1−(1+r)−n
A Concrete Example
You want to know how much ₹10,000 saved every year for 20 years at 8% will grow to.
P=10000, r=0.08, n=20
FV=10000⋅0.08(1.08)20−1
(1.08)20≈4.66096
FV=10000⋅0.084.66096−1=10000⋅0.083.66096=10000⋅45.762=457,620
So you'd have about ₹4,57,620 after 20 years.
The fraction r(1+r)n−1 is called the annuity future value factor. For 8% and 20 years, it's about 45.762. You can find these factors in tables or calculate them directly.
What About "Annuity Due"?
If payments happen at the beginning of each period (like rent paid on the 1st of the month), it's an annuity due. Each payment earns one extra period of interest compared to an ordinary annuity.
FVdue=FVordinary×(1+r)
PVdue=PVordinary×(1+r)
The Core Insight
An annuity formula is just a shortcut for adding up a geometric series. You could compute each payment's value separately and sum them — the formula just does it in one step. The logic is always: each payment is a separate compound-interest calculation, and the total is their sum.
A common mistake is to treat the annuity as a single lump sum and apply the compound interest formula directly. That would give the wrong answer because it assumes all the money is deposited at the start. The annuity formula correctly accounts for the fact that later payments earn less interest.