Formula Calculation for Interest: Simple & Compound

This page explains the two standard formulas used to calculate interest: simple interest and compound interest. You enter the principal amount, the annual interest rate, and the time period, and the calculator returns the interest earned or owed, along with the total balance. The formulas below work for loans, savings accounts, and investments, and the worked example shows exactly how to apply them.

Formula

The simple interest formula is:

Simple Interest = P × r × t

where:

• P = principal (the initial amount of money, in dollars or any currency)
• r = annual interest rate (as a decimal, so 5% becomes 0.05)
• t = time the money is borrowed or invested, in years

The compound interest formula is:

Compound Interest = P × (1 + r/n)^(n×t) − P

where:

• P = principal
• r = annual interest rate (decimal)
• n = number of times interest is compounded per year (e.g., 12 for monthly, 1 for yearly)
• t = time in years

The total balance with compound interest is simply P × (1 + r/n)^(n×t). The “^” symbol means “raised to the power of.”

Worked Example

Using the default values from the calculator: P = $1,000, r = 5% (0.05 as a decimal), t = 3 years, and n = 12 (monthly compounding).

Simple interest:

1. Convert the rate: 5% ÷ 100 = 0.05
2. Multiply: $1,000 × 0.05 × 3 = $150
3. Total with simple interest: $1,000 + $150 = $1,150

Compound interest (monthly):

1. Divide the rate by compounding frequency: 0.05 ÷ 12 = 0.0041667
2. Add 1: 1 + 0.0041667 = 1.0041667
3. Multiply the exponent: n × t = 12 × 3 = 36
4. Raise to the power: 1.0041667^36 ≈ 1.161472
5. Multiply by principal: $1,000 × 1.161472 = $1,161.47
6. Subtract principal to get interest: $1,161.47 − $1,000 = $161.47

So with the same 5% rate, simple interest gives $150, while monthly compounding gives $161.47 – a difference of $11.47 because interest is earned on previously earned interest.

When to Use Each Formula

Simple interest is used for short-term loans, some auto loans, and certain bonds where interest is not reinvested. It is also the basis for most “flat rate” consumer loans. Compound interest is used for savings accounts, certificates of deposit, mortgages, credit cards, and most long-term investments. The key difference is that simple interest is calculated only on the original principal, while compound interest is calculated on the principal plus any interest already added.

For example, if you deposit $10,000 in a savings account that compounds monthly at 2% APY, after 10 years you will have about $12,210. If the bank used simple interest, you would have $12,000. The extra $210 comes from compounding.

How to Read the Result

The calculator shows two separate outputs: simple interest and compound interest. The “total” lines show the final balance (principal + interest). When comparing loan offers or investment options, always look at the total cost or total return, not just the interest rate. Also note that the compound interest formula assumes the rate stays constant and that no additional deposits or withdrawals are made. In reality, many accounts have variable rates and fees, so the actual result may differ.

For a savings account, the result tells you how much your money grows. For a loan, the result tells you how much extra you pay beyond the amount borrowed. A higher compounding frequency (e.g., daily vs. yearly) yields slightly more interest, but the effect is small for typical rates and time periods.

Common Mistakes and Misconceptions

• Using the percentage instead of the decimal: Always divide the percentage by 100 first. Forgetting this gives results 100 times too large.
• Mismatching time units: The formulas require time in years. If you have 6 months, use 0.5 years. If you have 18 months, use 1.5 years.
• Confusing APR and APY: APR (Annual Percentage Rate) is the nominal rate before compounding; APY (Annual Percentage Yield) includes the effect of compounding. The compound interest formula uses the nominal rate (r) and the compounding frequency (n) to get the effective yield.
• Forgetting to subtract the principal: The compound interest formula gives the total balance; to get the interest alone, subtract P.
• Rounding too early: Round only at the final step. Rounding intermediate values can change the final cents.

Limits of the Formula

These formulas assume a constant interest rate and no additional payments. Real-world loans often have fees, variable rates, or prepayment penalties. Savings accounts may have tiered rates (higher balances earn higher rates) or monthly maintenance fees that reduce the effective return. The formulas also do not account for inflation, taxes on interest income, or the time value of money beyond the stated rate. For complex financial products, consult the specific terms from your bank or lender.

Frequently Asked Questions

What is the difference between simple and compound interest?

Simple interest is calculated only on the original principal. Compound interest is calculated on the principal plus any interest already earned. Over time, compound interest grows faster because you earn “interest on interest.”

How do I calculate interest for less than a year?

Convert the time to years. For example, 6 months = 0.5 years, 90 days = 90/365 ≈ 0.2466 years. Then plug that decimal into the formula. For compound interest, the exponent n×t will be a fraction, which is fine.

Does compounding frequency matter?

Yes, but not dramatically for typical rates. The more frequently interest is compounded (e.g., daily vs. yearly), the slightly higher the total. For a $10,000 deposit at 5% for 10 years, yearly compounding gives $16,288.95, monthly gives $16,470.09, and daily gives $16,486.65. The difference between monthly and daily is only about $16.

What is the formula for compound interest when interest is compounded continuously?

The continuous compounding formula is A = P × e^(r×t), where e is approximately 2.71828. This is the mathematical limit of compounding as the frequency approaches infinity. It gives the maximum possible interest for a given nominal rate.

Can I use these formulas for credit card interest?

Credit cards typically use daily compounding on the average daily balance. The same compound interest formula applies, but you need to know the daily periodic rate (APR divided by 365) and the number of days in the billing cycle. Also, most cards have a grace period, so no interest is charged if you pay the full balance each month.

Why does my bank’s savings account show a different APY than the nominal rate?

APY (Annual Percentage Yield) is the effective annual rate after compounding. If a bank advertises a 5% APY, that means $100 grows to $105 in one year. The nominal rate might be slightly lower (e.g., 4.88% compounded daily) to achieve the same APY. Always compare APYs when comparing accounts.

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in p | Principal | $ | 1000 | 0
in r | Annual interest rate | % | 5 | 0
in t | Time | years | 3 | 0
in n | Compounding frequency (times per year) | - | 12 | 1
out si | Simple interest | $ | p * r / 100 * t | 2
out ci | Compound interest | $ | p * ((1 + r / 100 / n) ^ (n * t) - 1) | 2
out total_si | Total with simple interest | $ | p + p * r / 100 * t | 2
out total_ci | Total with compound interest | $ | p * (1 + r / 100 / n) ^ (n * t) | 2