An annuity is a stream of equal cash flows C at regular intervals for n periods. Salaries, EMIs, SIPs, bond coupons, lease rentals, pension payouts, toll-road concession receipts: half of finance is annuities. You could discount each cash flow separately. A closed form exists, though. Derive it once and it's yours for good.
How do you derive the annuity formula?
Derivation (ordinary annuity, payments at period-end). PV = C/(1+r) + C/(1+r)² + … + C/(1+r)^n, a geometric series with first term a = C/(1+r) and common ratio x = 1/(1+r). You can use the standard trick. Multiply the sum S by x and subtract it from S. Everything telescopes:
S − xS = a − a·x^n ⟹ S = a(1 − x^n)/(1 − x)
Substitute back and simplify by multiplying top and bottom by (1+r):
PV_annuity = C × [1 − (1+r)^(−n)] / r
The bracket divided by r is the annuity factor, the value today of ₹1 per period for n periods. You should be able to reproduce two sanity checks. As n grows without limit, the factor tends to 1/r, which is the perpetuity coming up next. As r falls to 0, it tends to n, because with no time value n rupees are just n rupees.
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Future value of an annuity, and annuity due vs ordinary
Future value is what the stream grows to by time n, as with a SIP:
FV_annuity = C × [(1+r)^n − 1] / r
Ordinary vs due. An ordinary annuity pays at period-end (loan EMIs in effect, bond coupons). An annuity due pays at period-start (rent, insurance premiums, school fees, most SIP setups). Every cash flow arrives one period earlier, so:
PV_due = PV_ordinary × (1+r) and FV_due = FV_ordinary × (1+r)
It's one factor of (1+r), applied to the whole stream. Take ₹50,000 a year for 20 years at 7%. As an ordinary annuity it's worth 50,000 × [1 − 1.07^(−20)]/0.07 = 50,000 × 10.5940 = ₹5,29,701, and as a due it's worth × 1.07 = ₹5,66,780. Get the timing wrong and you misprice the stream by a full period's interest, which here is ₹37,079.
How do you calculate an EMI?
The EMI: solving the annuity for C. A loan is an annuity in reverse. You get PV from the bank today and repay it as n equal instalments whose present value, at the loan's periodic rate, equals the principal. Rearrange and you get:
EMI = P × r / [1 − (1+r)^(−n)], with r the monthly rate and n the number of months.
Worked build (India): ₹80 lakh home loan, 8.5% p.a., 20 years. 8.5% is a typical recent level for prime borrowers. Indian home-loan rates float with the RBI repo rate, so check current rates on any large lender's site.
- r = 0.085/12 = 0.0070833; n = 240
- Annuity factor = [1 − (1.0070833)^(−240)] / 0.0070833 = 115.2308
- EMI = 80,00,000 / 115.2308 = ₹69,425.86 ≈ ₹69,426
Over 240 months you pay ₹1.67 crore in all, and total interest comes to ₹86.62 lakh, more than the house loan itself. There's nothing sinister in that. It's simply what renting ₹80 lakh for two decades costs at 8.5%.
What does an amortisation schedule show?
The amortization schedule shows where each EMI goes. Each month, interest is the opening balance times the monthly rate. Subtract it from the EMI and what's left is principal repaid, which is how far the balance steps down. Here are selected rows, in ₹:
| Month | Opening balance | Interest | Principal | Closing balance |
|---|---|---|---|---|
| 1 | 80,00,000 | 56,667 | 12,759 | 79,87,241 |
| 2 | 79,87,241 | 56,576 | 12,850 | 79,74,391 |
| 12 | 78,54,571 | 55,637 | 13,789 | 78,40,782 |
| 60 | 70,69,525 | 50,076 | 19,350 | 70,50,175 |
| 120 | 56,29,059 | 39,873 | 29,553 | 55,99,506 |
| 143 | 48,93,648 | 34,663 | 34,763 | 48,58,886 |
| 180 | 34,29,035 | 24,289 | 45,137 | 33,83,898 |
| 240 | 68,938 | 488 | 68,938 | 0 |
Read the shape, because you'll see it in every levered structure, from home loans to LBO debt waterfalls. Early payments are almost all interest, and in month 1 the split is 82% interest. Each month the principal component grows by a factor of (1+r), because each ₹1 of principal repaid stops accruing interest and frees that much more of the next EMI for principal. Not until month 143, year 12 of 20, does principal exceed interest within an EMI. After five years of faithful payments, the borrower still owes ₹70.5 lakh of the original ₹80 lakh. Five years of EMIs, and the debt has barely moved.
Prepayment arithmetic. Prepay ₹5 lakh at the end of year 5 and the balance drops to ₹65.5 lakh. Keep the same EMI and solve the annuity for n. The remaining term falls from 180 to ~156 months, so you finish two years earlier and save ~₹11.4 lakh of interest. A prepayment earns the loan rate, risk-free, tax quirks aside. So "prepay the 8.5% loan or invest?" is a pure opportunity-cost question, the very kind you're learning to answer. RBI bars prepayment penalties on floating-rate loans to individuals, but check your own loan's terms.
What is a growing annuity?
The growing annuity, cash flows growing at g for n periods, is the bridge to DCF. Its derivation runs the same geometric-series trick with ratio (1+g)/(1+r):
PV = [C₁/(r − g)] × [1 − ((1+g)/(1+r))^n] (for r ≠ g, with C₁ the first cash flow, at t=1)
Say the first cash flow is ₹10 crore, growing 8%/yr for 5 years, with r = 12%. Then PV = [10/0.04] × [1 − (1.08/1.12)^5] = 250 × 0.16626 = ₹41.57 crore. Check it by brute force: discount all five flows, 10/1.12 + 10.8/1.12² + …, and you land on 41.57 again. This formula is the explicit forecast period of every DCF you'll ever build.