What is an annuity?
An annuity is a list of equal payments made at equal time gaps. Rent, a monthly savings plan and a loan EMI are all annuities.
Key features
- Payment (PMT): the fixed amount each time.
- Payment interval: the time between payments (month, quarter, year).
- Term: the total time from first to last payment.
- Interest rate: the yearly rate, split per period: i = rate ÷ payments per year.
- Number of payments: n = years × payments per year.
In an ordinary simple annuity, payments are made at the end of each period, and interest is compounded at the same frequency as the payments (monthly payments, monthly compounding).
Future value and present value
Future value FV = PMT × ((1 + i)^n − 1) ÷ i: how much a savings annuity grows to.
Present value PV = PMT × (1 − (1 + i)^−n) ÷ i: how much money today equals all the future payments. A loan is a present value.
Changing the conditions of an annuity
Change one thing at a time and watch the result (try it in the 3D):
- Higher interest rate → bigger payment for the same loan, more total interest. For savings, a higher rate makes the future value bigger.
- Longer term → smaller payment, but many more payments, so much more total interest.
- More frequent payments (weekly instead of monthly) → the loan balance drops sooner, so slightly less interest.
- Bigger regular payment → the loan ends years earlier and total interest falls sharply.
- Starting a savings annuity earlier → compound interest has more time to work, so the future value grows a lot.
A spreadsheet or a financial calculator (TVM solver) can redo the sum quickly, so you can compare choices.
Mortgages
A mortgage is a loan to buy land or a home. The property is the security: if the borrower stops paying, the lender can take it.
- Down payment: the money you pay yourself at the start. Price − down payment = loan (principal).
- Amortization period: the full time to pay off the loan, often 20–30 years.
- Term: in some countries the rate is fixed only for a shorter term (say 5 years), then renewed.
- Fixed or variable rate: a fixed rate stays the same; a variable rate moves with the market.
- Payment frequency: monthly, bi-weekly (26 a year) or weekly.
- Extra costs: fees, insurance and property tax.
Building an amortization table
An amortization table shows each payment row by row. For each row:
- Interest = balance × i.
- Principal paid = payment − interest.
- New balance = old balance − principal paid.
Example: loan 100,000 at 6% a year, monthly, i = 0.005, payment 644.30.
| # | Payment | Interest | Principal | Balance |
|---|---|---|---|---|
| 0 | 100,000.00 | |||
| 1 | 644.30 | 500.00 | 144.30 | 99,855.70 |
| 2 | 644.30 | 499.28 | 145.02 | 99,710.68 |
| 3 | 644.30 | 498.55 | 145.75 | 99,564.93 |
The interest column falls a little each row and the principal column rises. The last payment may be a few cents different to make the balance exactly zero.
Try it
In the 3D, set rate 6%, 25 years, 12 payments a year. Note the total interest. Now change only the years to 15. Predict first: will total interest go up or down? Then check. Next, keep 25 years and switch to 26 payments a year.
At home: make a 4-row amortization table in a spreadsheet with the three rules above and check it against the table on this page.
Key formulas and definitions
- i = annual rate ÷ payments per year, n = years × payments per year
- FV = PMT × [(1 + i)^n − 1] ÷ i
- PV = PMT × [1 − (1 + i)^−n] ÷ i
- PMT = PV × i ÷ [1 − (1 + i)^−n]
- Interest in a period = balance × i; principal paid = PMT − interest
- Total interest = PMT × n − loan
Worked examples
1. You save 100 at the end of every month for 2 years at 6% a year compounded monthly. Find the future value.
i = 0.06 ÷ 12 = 0.005, n = 24. FV = 100 × (1.005^24 − 1) ÷ 0.005 = 100 × (1.12716 − 1) ÷ 0.005 = 100 × 25.432 = 2,543.20.
2. What one amount today equals 500 a year for 5 years at 4% a year (end of year payments)?
i = 0.04, n = 5. PV = 500 × (1 − 1.04^−5) ÷ 0.04 = 500 × (1 − 0.82193) ÷ 0.04 = 500 × 4.4518 = 2,225.91.
3. Find the monthly payment on a 100,000 mortgage at 6% a year over 25 years.
i = 0.005, n = 300. 1.005^−300 = 0.22396. PMT = 100,000 × 0.005 ÷ (1 − 0.22396) = 500 ÷ 0.77604 = 644.30.
4. For that mortgage, find the total paid and the total interest.
Total paid = 644.30 × 300 = 193,290. Total interest = 193,290 − 100,000 = 93,290. Almost as much as the loan!
5. Row 1 of the amortization table: balance 100,000, i = 0.005, payment 644.30. Find interest, principal and new balance.
Interest = 100,000 × 0.005 = 500.00. Principal = 644.30 − 500.00 = 144.30. New balance = 100,000 − 144.30 = 99,855.70.
6. India example: a ₹30,00,000 home loan at 9% a year for 20 years, monthly EMI. Find the EMI.
i = 0.09 ÷ 12 = 0.0075, n = 240. 1.0075^−240 = 0.16641. EMI = 30,00,000 × 0.0075 ÷ (1 − 0.16641) = 22,500 ÷ 0.83359 ≈ ₹26,992.
7. Same 100,000 at 6%, but over 15 years. Compare total interest with 25 years.
n = 180, 1.005^−180 = 0.40748. PMT = 500 ÷ 0.59252 = 843.86. Total = 843.86 × 180 = 151,895; interest ≈ 51,895. That saves about 41,400 compared with 25 years, for about 200 more per month.
Common mistakes
- Using the yearly rate as i. For monthly payments at 6% a year, i = 0.005, not 0.06.
- Using years as n. For 25 years of monthly payments, n = 300.
- Thinking each payment cuts the loan by the full payment. Only the principal part reduces the balance.
- Thinking a longer term is cheaper because the payment is smaller. The total interest is much higher.