Simple vs Compound Interest — Side-by-Side

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Learn: Simple vs Compound Interest

This section explains when each model is used, the exact formulas, and how banks in different regions refer to rates (UK: AER, US: APY, general: EAR). Everything here is simplified and for education only.

Quick definitions

  • Simple interest: interest grows linearly with time. Often used in short-term loans or straightforward agreements.
  • Compound interest: interest earns interest. Growth accelerates with more frequent compounding (monthly, daily, continuous).

Formulas at a glance

  • Simple total: \( A_{\text{simple}} = P(1 + r t) \)
  • Compound (discrete): \( A_{\text{comp}} = P\left(1 + \frac{r}{n}\right)^{n t} \)
  • Compound (continuous): \( A_{\text{cont}} = P\,e^{r t} \)
  • Effective Annual Rate (EAR): \( \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \) (continuous: \( e^{r} - 1 \))

Worked example (side-by-side)

Suppose P = £10,000, APR r = 5%, t = 3 years, compounding monthly (n = 12):

  • Simple: \( A = 10{,}000 \times (1 + 0.05 \times 3) = \) £11,500.00
  • Compound (monthly): \( A = 10{,}000 \times \left(1 + \frac{0.05}{12}\right)^{36} \approx \) £11,614.72
  • Extra from compounding vs simple over 3 years: ~£114.72
  • EAR for 5% nominal, monthly compounding: \( (1 + 0.05/12)^{12} - 1 \approx \) 5.116%

GEO notes (UK / US / EU naming)

  • UK: Savings are commonly advertised with AER (Annual Equivalent Rate) — an “effective” yearly rate including compounding. Mortgages often calculate interest daily but are paid monthly; lenders still quote a nominal APR.
  • US: Deposit accounts use APY (similar to EAR/AER). Loans/credit cards quote a nominal APR; the effective cost depends on compounding and fees.
  • General: When comparing offers, convert to an effective rate (EAR/APY/AER) so the compounding frequency is accounted for.

When each model makes sense

  • Use simple interest for short, fixed-term agreements where compounding isn’t stipulated (some notes, quick estimates, educational contexts).
  • Use compound interest for most savings accounts, investment growth, credit cards, and typical amortising loans/mortgages (compounding frequency matters).

Common pitfalls

  • Mixing units: ensure time \(t\) is in years (months ÷ 12; days ÷ 365).
  • Comparing nominal APRs only: two 5% APR offers can have different outcomes if one compounds monthly and another daily.
  • Ignoring fees/taxes/inflation: this calculator doesn’t model fees, tax treatment, or inflation. Real return ≈ nominal return − inflation (rough rule).
  • Assuming compounding you don’t have: read the product terms — some contracts are simple interest or compound at unusual intervals.

Disclaimer: Educational illustration only. Not financial advice. Always check your provider’s terms (compounding method, fees, and disclosure standard such as APR/APY/AER).

5 Fun Facts about Simple vs Compound Interest

Compounding beats higher simple rates

A 6% compounded rate can overtake an 8% simple rate after ~14 years—time and frequency can outrun a bigger headline rate.

Time advantage

Payments hide the clock

Many loans accrue daily but bill monthly—the compounding meter runs between statements even if you don’t see it.

Invisible compounding

Rule of 72 only works for compounding

72 ÷ rate% estimates years to double because returns are compounded; with simple interest, doubling happens strictly by elapsed time.

Quick doubles

Flat vs snowball

Simple interest traces a straight line; compound growth curves upward. Plotting both shows how quickly the gap widens.

Growth shape

Fees decide the winner

A tiny monthly fee on a compounded account can erase its edge over a fee-free simple rate—costs compound too.

Cost gravity

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