The compound interest formula
Compound interest is interest on interest. Each time interest is added, it joins the balance and earns interest itself. The balance after t years is:
- A = P × (1 + r ÷ n)^(n × t)
- P is the starting amount, r is the yearly rate as a decimal (5% is 0.05), n is how many times a year interest is compounded and t is the number of years.
Put 10,000 at 5% for 10 years. Compounded yearly it grows to 16,288.95. Monthly gives 16,470.09, daily gives 16,486.65 and continuous compounding, A = P × e^(r × t), gives 16,487.21. Compounding more often helps, but the gap between monthly and daily is only about 16 on 10,000 over a decade. The rate and the time matter far more than the frequency.
Adding regular deposits
Each deposit starts earning from the day it goes in, so the earliest ones grow the most. With deposits of D made c times a year, the calculator adds up every deposit's growth with the standard annuity formula: D × (u^m − 1) ÷ (u − 1), where u is the growth over one deposit period and m is the number of deposits. Interest is taken to grow smoothly between compounding dates, so a deposit made part way through a period earns its share of that period.
Saving 200 a month at 6% compounded monthly for 30 years gives 200,903.01. You paid in 72,000, so 128,903.01 is interest, more than you deposited. If each deposit goes in at the start of the month instead of the end, every one earns an extra month and the total is 201,907.52.
Deposits are counted for each full deposit period in the term. A part period at the end gets no deposit, and a term shorter than one deposit period has none.
Stated rate, effective rate and time
A stated yearly rate with frequent compounding earns a little more than the number suggests. The effective yearly rate is (1 + r ÷ n)^n − 1. A stated 5% compounded monthly is an effective 5.1162%, and the calculator shows it so you can compare accounts on equal terms.
Time does most of the work. 5,000 at 7% compounded yearly is 9,835.76 after 10 years, 19,348.42 after 20 and 38,061.28 after 30. Each extra decade multiplies the balance by about 1.97, which is why starting early beats saving more later.
The rule of 72 gives a quick estimate of the time to double: 72 divided by the rate. At 6% that is 12 years. The exact figure is 11.9 years with yearly compounding and 11.6 years with monthly compounding. The calculator shows the exact time to double for your own rate and compounding when you make no deposits.
What this calculator leaves out
- Changing rates. The rate is held steady for the whole term. Savings and investment rates move, and investment returns are not guaranteed.
- Fees, tax and inflation. None are taken off. When prices rise, 100 in 20 years buys less than 100 today.
- How an account really credits interest. Banks differ on which dates interest is added and how they treat deposits made between those dates. Your statement can differ by a little.
To see how much a loan costs when interest works against you, try the loan calculator. For interest that is never added back to the balance, see the simple interest calculator.