Simple vs Compound Interest and APR vs APY Explained With Numbers

Short answer: simple interest is calculated on the original amount only. Compound interest is calculated on the original amount plus the interest already added, so it grows faster. APR is the quoted yearly rate; APY is the yearly rate after compounding is included, so APY is higher than APR whenever interest compounds more than once a year.

What is the difference between simple and compound interest?

With simple interest, interest = principal x rate x time. Nothing is added to the base. With compound interest, each period's interest is added to the balance and then earns interest itself.

Worked example: 10,000 at 5% a year for 10 years, computed with Mathify's repository code.

• Simple interest: 5,000.00 of interest, for a final amount of 15,000.00.
• Compounded annually: 16,288.95 (interest 6,288.95).
• Compounded quarterly: 16,436.19.
• Compounded monthly: 16,470.09 (interest 6,470.09).
• Compounded daily (365 periods a year): 16,486.65.

The big jump is from simple to compound. Moving from annual to daily compounding adds 197.70 over ten years here. The simple interest calculator and the compound interest calculator show these with your own inputs, and the daily compound calculator covers the daily case. For an in-depth walk-through, see the compound interest guide.

What do APR and APY mean?

APR (annual percentage rate) is the yearly rate quoted before compounding within the year. APY (annual percentage yield) is the rate you actually earn or pay over a year once compounding is counted.

The conversion is APY = (1 + APR / n) to the power n, minus 1, where n is the number of compounding periods a year. We computed:

• 5% APR compounded monthly: APY of 5.1162%, which is 0.1162 percentage points above the APR.
• 5% APR compounded annually: APY of 5.0000%. Annual compounding makes them equal.
• 5% APR compounded daily: APY of 5.1267%.
• 3.5% APR monthly: APY of 3.5567%.
• 24% APR monthly: APY of 26.8242%, a gap of 2.82 percentage points.

The APY vs APR calculator does this conversion. Separate pages cover APR and APY.

Why does it matter for loans and savings?

For savings, compare APY, because it includes compounding and puts accounts on the same footing. For borrowing, a higher compounding frequency raises what you owe over a year. Take 1,000 at 24% APR on a balance that compounds monthly with no payments for one year:

• Simple interest would give 1,240.00.
• Monthly compounding gives 1,268.24, which is the 26.82% APY applied to 1,000.

That is only an illustration of the mathematics. Real credit products have payments, minimums and fees, so use the credit card payoff calculator for a real balance.

One caution on definitions: in lending, rules about what an APR includes (such as fees) vary by country and product. Mathify's APR vs APY page treats APR as the nominal rate you type in and does not try to add fees. Check your lender's disclosure for the legal figure.

How much does compounding frequency matter at different rates?

At low rates, very little. At high rates, noticeably more. APY for the same APR under different compounding, computed with the repository's formula:

• 4.25% APR compounded monthly: 4.3338% APY, a gap of 0.0838 percentage points.
• 5% APR: 5.1162% monthly and 5.1267% daily, a difference of about 0.01 percentage points between the two.
• 24% APR: 26.8242% monthly and 27.1149% daily, a difference of about 0.29 percentage points.

So switching from monthly to daily compounding rarely changes a savings decision, while the gap between simple interest and compound interest over many years is large. The longer the term and the higher the rate, the more compounding matters. Mathify's daily compounding treats a year as 365 periods.

How do you pick the right calculator?

Ask what is changing over time:

1. A single sum earning interest with no deposits: use the simple or compound interest calculator.
2. You have a quoted APR and want the yearly effect: use APR vs APY.
3. You pay a loan back in instalments: use the EMI calculator or loan calculator, because interest is charged on a shrinking balance.
4. You want growth after inflation: the inflation calculator shows how prices change over the same period.

Does the time unit change simple interest?

Yes, if the day count differs. For 1,000 at 5% simple interest:

• 6 months gives 25.00.
• 180 days gives 24.66, because Mathify divides days by 365 to get a fraction of a year.

Banks and bond markets use different day-count conventions, so two lenders can show slightly different interest for the same dates. If the figures matter, ask which convention applies.

Common mistakes

• Comparing a savings APR with another account's APY. Compare like with like.
• Assuming the "rate" on a loan is simple interest when it compounds.
• Ignoring compounding frequency. Monthly and daily compounding differ.
• Mixing up the periodic rate and the yearly rate. A monthly rate of 2% is not an APR of 2%.
• Applying the formula with the rate as a whole number (5) instead of a decimal (0.05).

Key takeaways

• Simple: 10,000 at 5% for 10 years gives 15,000. Compounded monthly it gives 16,470.09.
• APY is at least APR, and equals it only with one compounding period a year.
• More frequent compounding raises the result, with diminishing returns.
• Check which rate a quote uses before comparing two products.

This article is general information about how the arithmetic works, not financial advice. For a time-to-double rule of thumb, see the Rule of 72 accuracy table.

FAQ

Is compound interest always better?

It is better when you earn it and worse when you owe it. The same compounding that grows savings also increases a debt that is not being paid down.

How do I convert APR to APY?

Divide the APR by the number of compounding periods, add 1, raise it to the power of the number of periods, and subtract 1. At 5% compounded monthly that gives 5.1162%.

Is APY or APR higher?

APY is higher or equal. It is equal only when interest compounds once a year, as with 5% annual compounding above.

How often is interest compounded?

It depends on the product and is stated in its terms. Common choices are daily, monthly, quarterly and annually.

Which one should I compare on a savings account?

Compare APY, because it already includes compounding. Also check whether the rate is fixed or variable and whether fees apply.