Compound vs simple interest
With simple interest, you earn a fixed amount each period based only on your original deposit. Put $10,000 in an account paying 7% simple interest and you collect $700 every year — no more, no less — because the calculation always ignores the interest you have already been paid.
With compound interest, the interest you earn is added back to your balance, and the next period's interest is calculated on that larger total. Year one still earns $700, but year two earns 7% of $10,700, year three earns 7% of $11,449, and so on. Each year the base grows, so each year's interest grows too. This "interest earning interest" is the entire idea — small at first, then increasingly dramatic.
The compound interest formula
The standard formula for a lump sum left to compound is:
A is the final amount, P is the principal (your starting balance), r is the annual interest rate as a decimal (7% = 0.07), n is the number of times interest compounds per year, and t is the number of years.The two pieces that trip people up are r/n and nt. Dividing the annual rate by n gives the rate applied per compounding period; multiplying years by n gives the total number of periods. So for monthly compounding you use n = 12, for daily you use n = 365, and for annual you use n = 1.
Why compounding frequency matters
Because interest starts earning its own interest sooner, compounding more often produces a slightly larger result for the same rate. Take $10,000 at 7% for 30 years:
- Annually (n = 1): about $76,123
- Monthly (n = 12): about $81,165
- Daily (n = 365): about $81,600
Notice the jump from annual to monthly is meaningful, but the jump from monthly to daily is tiny. There is a ceiling — even continuous compounding tops out near $81,660. The lesson: frequency helps, but it is a refinement, not the main event. The rate and the number of years do the heavy lifting.
A worked example: $10,000 at 7% for 30 years
Let's follow that $10,000 year by year, compounded annually, and compare it against the same deposit earning simple interest. Under simple interest you would add a flat $700 each year, ending at $31,000 ($10,000 principal plus $21,000 of interest). Under compounding, the balance accelerates:
| Year | Compounded balance | Simple-interest balance |
|---|---|---|
| 0 | $10,000 | $10,000 |
| 5 | $14,026 | $13,500 |
| 10 | $19,672 | $17,000 |
| 20 | $38,697 | $24,000 |
| 30 | $76,123 | $31,000 |
At year 5 the two are nearly level. By year 30 the compounded balance is almost two and a half times the simple-interest one — a difference of roughly $45,000 from the exact same deposit and rate. Every cent of that gap is interest that was reinvested and then went on to earn more interest. You can run your own figures with the compound interest calculator or project a lump sum forward with the future value calculator.
The Rule of 72
You don't always need a calculator to gauge compounding. The Rule of 72 estimates how many years it takes money to double: simply divide 72 by the interest rate written as a whole number.
- At 6%: 72 ÷ 6 = 12 years to double
- At 8%: 72 ÷ 8 = 9 years
- At 12%: 72 ÷ 12 = 6 years
It is an approximation, most accurate for rates between about 4% and 12%, but it is excellent for quick mental math. Our 7% example doubles in roughly 72 ÷ 7 ≈ 10.3 years — which is exactly why $10,000 reached about $19,672 by year 10 and then roughly doubled again to near $39,000 by year 20.
Why starting early beats investing more later
Time is the single most powerful lever in compounding, because the last doublings are the largest. In the table above, the balance grew about $4,000 in the first five years but nearly $37,000 in the final five — all from money that was already in the account. The dollars you invest earliest get the most years to compound, so they do the most work.
Consider two savers. Ana invests $5,000 a year from age 25 to 35 (ten years, $50,000 total) and then stops, leaving it to grow. Ben waits until 35 and invests $5,000 a year all the way to 65 (thirty years, $150,000 total). At a 7% return, Ana often ends up with more at 65 despite contributing a third of what Ben did — because her early dollars had an extra decade to compound. Starting sooner usually beats contributing more later. The savings goal calculator can show how much earlier contributions shrink the amount you ultimately need to set aside.
APR vs APY (nominal vs effective rate)
When you compare accounts, watch which rate is quoted. The APR (annual percentage rate) is the nominal rate — the headline figure before compounding is folded in. The APY (annual percentage yield), also called the effective rate, already includes the within-year compounding. A 7% nominal rate compounded monthly works out to an APY of about 7.23%, because the monthly compounding adds a little extra. APY is always equal to or higher than APR, so when shopping for a savings account or comparing returns, line them up by APY for a fair comparison. The APY calculator converts a nominal rate into its effective yield.
The takeaway
Compound interest is not magic — it is just interest applied to a growing balance, again and again. The three things that decide your outcome are the rate, the frequency, and above all the time you give it. Save early, let it compound, and compare offers by their effective yield. The same math that quietly builds wealth is the math working against you on debt, so respect it in both directions.
Frequently asked questions
- What is the difference between compound and simple interest?
- Simple interest is calculated only on your original principal, so you earn the same amount every period. Compound interest is calculated on the principal plus all previously earned interest, so each period you earn interest on a larger and larger base. Over short spans the gap is small, but over decades compounding pulls far ahead.
- Does it matter how often interest compounds?
- Yes, but less than people expect. More frequent compounding (monthly or daily instead of annually) raises the final amount because interest starts earning interest sooner. At 7% on $10,000 over 30 years, annual compounding gives about $76,123 while daily compounding gives roughly $81,600 — a real but modest difference compared to the effect of the rate and time.
- What is the Rule of 72?
- The Rule of 72 is a shortcut for estimating how long an investment takes to double: divide 72 by the annual interest rate as a percentage. At 8%, money doubles in about 72 ÷ 8 = 9 years. It is an approximation that works best for rates between roughly 4% and 12%, but it is accurate enough for quick mental math.
- What is the difference between APR and APY?
- APR (annual percentage rate) is the nominal yearly rate before compounding is taken into account. APY (annual percentage yield), also called the effective rate, includes the effect of compounding within the year. APY is always equal to or higher than the APR, so comparing two accounts by APY gives a true apples-to-apples picture of what you will actually earn.
This guide is general educational information, not financial advice.