Interest Calculator — Simple vs Compound

In 2026, a borrower comparing two $20,000 personal loan offers - one quoted at 8% simple and another at 8% compound annual - needs to see that the compound version costs $1,386 more over 5 years. The widget shows this gap on a single split graph: blue straight line for simple, green exponential curve for compound, shaded green between them. The mathematical truth becomes visually obvious in seconds.

Simple interest was the only form of interest used in European banking from antiquity through the 13th century. Roman law (lex Genucia, 342 BC) capped interest at 8.33% per year simple (unciarium fenus). The Catholic Church's 1311 Council of Vienne formally banned compound interest (anatocismus) as usury - a ban that persisted in most of Europe until the Reformation. This is why early European bond markets used coupon payments (simple interest paid each year) rather than zero-coupon compound instruments.

Leonardo Fibonacci's 1202 Liber Abaci contained the first written compound-interest tables in Western Europe - though Fibonacci was careful to frame them as mercantile rather than usurious calculations. His chapter on multi-period investment math is essentially the modern (1+r)^t formula applied to grain trade and currency exchange. Italian double-entry bookkeeping developed by Luca Pacioli in 1494 standardized the recording of compound interest in business ledgers.

Jacob Bernoulli's 1683 study of compound interest discovered the mathematical constant e ≈ 2.71828 as the limit of (1 + 1/n)^n. Bernoulli also proved the inequality (1+r)^t ≥ 1 + rt for r ≥ 0 and t ≥ 1 - which is the mathematical reason compound interest always equals or exceeds simple interest at the same nominal rate. The widget's green gap is a visual proof of Bernoulli's inequality.

Modern US law tightly regulates the disclosure of interest. The Truth in Lending Act (TILA, 1968) and its Federal Reserve Regulation Z (12 CFR 1026) require lenders to disclose APR - the nominal annual rate - on every consumer loan. APR can be either simple or compound depending on how the loan amortizes; for most amortizing loans (mortgages, auto), APR is the compound rate. For some short-term loans (payday, family AFR-rate notes), simple interest is still used and must be disclosed as such.

The IRS Applicable Federal Rate (AFR) for intra-family loans uses simple-interest calculation by convention. The Internal Revenue Code Section 1274 specifies the AFR for tax purposes; the Section 7872 below-market loan rules treat any spread between the actual rate and AFR as either interest income (to the lender) or imputed gift. Family loans below $10,000 with simple-interest rates at AFR are typically the safest documentation - which is the basis for the widget's Family Loan preset.

For investors, the choice between simple and compound interest is almost always made by the institution issuing the instrument. Savings bonds, CDs, and money-market accounts use compound; short-term Treasury bills use simple (discount-yield basis); intra-family loans use simple. Knowing which math applies - and seeing the gap visualized - is critical for both lenders and borrowers to evaluate offers correctly. The widget's split graph is the fastest way to see both curves side-by-side and decide.