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 …