Simple vs. Compound Interest: Definitions and Formulas
By Personal Finance DeskUpdated Nov 10, 2026
Key takeaways
- Simple interest: I = P × r × t. Interest depends only on the principal, the rate and the time.
- Compound interest: A = P(1 + r/n)^(nt), with total interest equal to A minus P. Interest is earned on earlier interest ("interest on interest").
- More frequent compounding (monthly instead of annually, for example) raises the total, though the gains shrink as frequency increases.
- The gap between the two methods is small over a year or two and grows quickly over decades.
- Borrowers pay less under simple interest. Savers and investors earn more under compound interest.
Simple interest is calculated only on the original principal, so the interest charged or earned is the same every period. Compound interest is calculated on the principal plus all interest already added, so each period’s interest is larger than the last. The simple interest formula is I = P × r × t. The compound interest formula is A = P(1 + r/n)^(nt). At the same rate and term (longer than one compounding period), compounding always produces more interest. That works in your favor on savings and investments and against you on debt.
Simple Interest Formula
The formula for simple interest is:
I = P × r × t
- I = interest
- P = principal (the amount borrowed or deposited)
- r = annual interest rate as a decimal (5% = 0.05)
- t = time in years
The total amount owed or held at the end is A = P(1 + r × t).
Example (hypothetical): You borrow $10,000 at 5% simple interest for 3 years. I = 10,000 × 0.05 × 3 = $1,500. You repay $11,500 in total.
Because the formula is linear, you can rearrange it to solve for any missing value:
- Rate: r = I ÷ (P × t)
- Time: t = I ÷ (P × r)
- Principal: P = I ÷ (r × t)
Time must be expressed in the same unit as the rate. For an annual rate over 9 months, t = 9/12 = 0.75. For a period counted in days, lenders divide by 365 or 360 depending on the day-count convention in the contract. A 360-day year produces slightly more interest than a 365-day year at the same stated rate.
Compound Interest Formula
The compound interest formula gives the final amount:
A = P(1 + r/n)^(nt)
- A = final amount (principal plus interest)
- P = initial principal
- r = annual interest rate as a decimal
- n = number of compounding periods per year
- t = number of years
Compound interest itself is CI = A - P, or CI = P[(1 + r/n)^(nt) - 1].
With annual compounding (n = 1), the formula simplifies to A = P(1 + r)^t. Here is $10,000 at 5% compounded annually for 3 years, year by year:
| Year | Opening balance | Interest at 5% | Closing balance |
|---|---|---|---|
| 1 | $10,000.00 | $500.00 | $10,500.00 |
| 2 | $10,500.00 | $525.00 | $11,025.00 |
| 3 | $11,025.00 | $551.25 | $11,576.25 |
| Total interest | $1,576.25 |
Under simple interest, the same deposit earns $500 each year, or $1,500 in total. Compounding adds $76.25 over three years because each year’s interest is calculated on a larger balance.
Simple vs. Compound Interest Over Time
The difference looks minor over three years. It becomes the main story over longer horizons. Same assumptions: $10,000 at 5%, compounding annually, no withdrawals or additional deposits.
| Years | Simple interest earned | Compound interest earned | Difference |
|---|---|---|---|
| 1 | $500.00 | $500.00 | $0.00 |
| 3 | $1,500.00 | $1,576.25 | $76.25 |
| 10 | $5,000.00 | $6,288.95 | $1,288.95 |
| 20 | $10,000.00 | $16,532.98 | $6,532.98 |
| 30 | $15,000.00 | $33,219.42 | $18,219.42 |
Simple interest grows in a straight line: each additional year adds the same $500. Compound interest grows exponentially: by year 30, the account is earning more than $1,500 a year in interest on a $10,000 starting balance. After one compounding period the two methods are identical, which is why short-term comparisons understate the effect.
Formula for the difference between compound and simple interest
To find how much more compound interest produces than simple interest at the same rate (annual compounding):
CI - SI = P[(1 + r)^t - 1 - r × t]
For exactly two years this reduces to a shortcut: CI - SI = P × r². For $10,000 at 5% over two years, the difference is 10,000 × 0.0025 = $25 (compound interest of $1,025 against simple interest of $1,000).
Compounding Periods
The value of n in the compound interest formula changes the result. Each compounding period uses a rate of r/n, and there are n × t periods in total. This table assumes $10,000 at a 10% annual rate for 10 years:
| Compounding frequency | Periods per year (n) | Rate per period (r/n) and total periods (nt) | Total interest |
|---|---|---|---|
| Annually | 1 | 10%, 10 | $15,937.42 |
| Semiannually | 2 | 5%, 20 | $16,532.98 |
| Quarterly | 4 | 2.5%, 40 | $16,850.64 |
| Monthly | 12 | 0.833%, 120 | $17,070.41 |
| Daily | 365 | 0.0274%, 3,650 | $17,179.10 |
For comparison, simple interest at 10% for 10 years yields $10,000.
Two patterns stand out. More frequent compounding always increases the total, and each step up adds less than the one before. Moving from annual to semiannual adds about $596; moving from monthly to daily adds about $109.
The upper limit is continuous compounding, where n approaches infinity and the formula becomes A = Pe^(rt), with e ≈ 2.71828. In the example above, continuous compounding produces about $17,182.82 in interest, only a few dollars more than daily compounding.
Effective annual rate, APR and APY
Because compounding frequency changes the outcome, a nominal rate alone does not tell you what you will earn or pay. The effective annual rate converts any compounding schedule into a single comparable figure:
EAR = (1 + r/n)^n - 1
A 10% rate compounded monthly has an effective annual rate of about 10.47%. In the U.S., deposit accounts quote an annual percentage yield (APY), which includes compounding, under the Truth in Savings Act. Loans quote an annual percentage rate (APR) under the Truth in Lending Act, which generally does not reflect intra-year compounding. When comparing a savings account to an investment, use APY. When comparing loans, compare APRs on the same basis and check how often interest is charged to the balance.
Related Compound Interest Concepts
Time value of money
A dollar today is worth more than a dollar in the future because today’s dollar can earn interest. Compound interest is the engine behind this idea. Rearranging the compound interest formula gives present value:
PV = FV ÷ (1 + r)^t
At 5%, $11,576.25 received in three years is worth $10,000 today. This discounting logic is the basis for discounted cash flow (DCF) analysis, bond pricing and retirement planning. Inflation works the same way in reverse, compounding the loss of purchasing power over time; historical rates are on our U.S. inflation rate by year page.
The Rule of 72
The Rule of 72 estimates how long it takes money to double with annual compounding: years to double ≈ 72 ÷ interest rate (with the rate as a whole number).
- At 6%: 72 ÷ 6 = 12 years (exact answer: about 11.9 years)
- At 8%: 72 ÷ 8 = 9 years (exact answer: about 9.0 years)
It is a mental shortcut, most accurate for rates in the single digits. If the rate is stated per period (per month, for example), the result is in periods. There is no equivalent doubling rule for simple interest because simple interest doubles the principal when r × t = 1: at 6%, that takes 16.7 years instead of roughly 12.
Compound Annual Growth Rate (CAGR)
CAGR answers the reverse question: given a starting value and an ending value, what steady annual rate of compounding would connect them?
CAGR = (Ending value ÷ Beginning value)^(1/t) - 1
Example: A portfolio grows from $10,000 to $16,000 over 5 years. CAGR = (16,000 ÷ 10,000)^(1/5) - 1 = 1.6^0.2 - 1 ≈ 9.86% per year.
A simple average would give 60% ÷ 5 = 12% per year, which overstates the result because it ignores compounding. If you solve this with a financial calculator or spreadsheet function, enter the starting value as negative (cash outflow) and the ending value as positive; the two must have opposite signs.
How CAGR is used in investing
CAGR is the standard way to express returns for stocks, mutual funds and portfolios over multi-year periods. It also allows comparisons with a benchmark: if an index returned 10% a year over five years and a fund manager returned 9% a year, the manager underperformed.
It is equally useful for planning. These hypothetical examples assume a constant annual return, with contributions made at the end of each year:
- Growth of a lump sum. $100,000 at 3% a year becomes about $180,611 after 20 years. At 6%, it becomes about $320,714.
- Saving for a target. To accumulate $50,000 for a down payment in 10 years, you would need to save about $4,165 a year at a 4% return, or about $3,975 a year at 5%.
- Starting early. To reach $1 million by age 65 at 6%, a 25-year-old needs to save about $6,462 a year. A 40-year-old needs about $18,227 a year, nearly three times as much, because 15 fewer years of compounding are available.
The third example is the core of most long-term wealth plans; our guide on how to become a millionaire builds on it. CAGR is a smoothed figure: real investment returns vary year to year, and CAGR does not show that volatility.
Simple and Compound Interest on Debt
Where simple interest applies
Many installment loans, including car loans and many mortgages and personal loans, calculate interest on the outstanding principal each period. This is sometimes called the reducing balance method. Each payment covers that period’s interest first, and the rest reduces principal. Because the interest is paid every period rather than added to the balance, it does not compound. The interest calculation is effectively simple interest applied to a shrinking balance.
That changes if interest goes unpaid. When missed interest is capitalized (added to the principal), future interest is charged on it, and the loan starts compounding. Deferred student loan interest can be capitalized in this way.
Paying early on a simple-interest loan reduces the principal sooner, which reduces every later interest charge. Splitting a monthly mortgage payment into half-payments every two weeks results in 26 half-payments, or 13 full payments, a year. If the extra amount is applied to principal and the loan has no prepayment restrictions, this can shorten the payoff period and lower total interest.
Where compounding works against you
Credit cards typically compound interest, often daily or monthly, on any balance carried past the grace period. Hypothetical example: a $25,000 balance at 20% compounded monthly, with no payments, accrues about $5,485 in interest over one year (roughly $457 a month on average). Under simple interest, the same year would cost $5,000. The effective annual rate at 20% compounded monthly is about 21.94%.
Interest calculation conventions are standardized within a country but can differ between countries, for example in how often mortgage interest is compounded. The loan agreement states the method that applies.
Which Is Better, Simple or Compound Interest?
It depends on which side of the transaction you are on:
- Borrowing: simple interest is better. You pay interest only on the principal (or on the remaining balance), never on past interest.
- Saving or investing: compound interest is better. Reinvested interest earns its own return, and the advantage grows with time, rate and compounding frequency.
When comparing products, the three inputs that matter most are the rate, the compounding frequency (captured in APY or the effective annual rate) and the time horizon. A slightly lower rate with more frequent compounding can beat a higher rate with annual compounding, so convert both to an effective annual rate before deciding.
FAQs
Is compound interest always more money than simple interest?
At the same positive rate, yes, once the term is longer than one compounding period. Over exactly one period the two are equal. For a fraction of a period, plugging a fractional exponent into the compound formula gives slightly less than simple interest, which is why lenders and banks usually apply simple interest to partial periods.
What financial products use simple interest?
Many car loans, personal loans and mortgages calculate interest on the outstanding principal, which is a simple-interest method. Some bonds also pay fixed coupons that are not reinvested, so the holder receives simple interest. Savings accounts, CDs and credit cards generally use compound interest.
How often does interest compound?
Common schedules are daily, monthly, quarterly, semiannually and annually. The account or loan terms state the frequency, and the APY on a deposit account reflects it.
Is compound interest taxable income?
In the U.S., interest from savings accounts, CDs and most bonds is generally taxable as ordinary income in the year it is credited, even if you leave it in the account to compound. Interest earned inside tax-advantaged accounts follows the rules of that account; for Roth accounts, see Roth IRA withdrawal rules.
The Bottom Line
The difference between simple and compound interest matters most to long-term savers and to anyone carrying a credit card balance. Compounding adds little over a year or two but can more than double the interest earned over 30 years, so starting early pays off. Before choosing a savings account or loan, convert each offer to an effective annual rate or APY and check how often interest is charged or credited.
