Compound interest, and why frequency matters

Two different mechanisms, not two words for the same thing

Simple interest is calculated on the amount you originally deposited, every period, for the whole term. The interest it produces never itself earns anything. Compound interest is calculated on the current balance, which includes interest already credited, so each period starts from a slightly larger base than the one before. That single change turns a straight line into a curve.

Simple interest
  I = P × r × t
  Total = P + I

Compound interest
  A = P × (1 + r ÷ n)^(n × t)
  Interest = A − P

  P = principal          r = nominal annual rate as a decimal
  n = compounding periods per year          t = years

Put $10,000 in at 5% for ten years. Simple interest pays $10,000 × 0.05 × 10 = $5,000, so the balance is $15,000, and it pays exactly $500 in the tenth year just as it did in the first. Compounded annually the same deposit becomes $10,000 × 1.05^10 = $16,288.95, because the tenth year earns 5% on about $15,513 rather than on $10,000. The gap of $1,288.95 is interest that the interest earned.

The distinction is not academic, because both appear in real products. Term deposits that pay interest out to a separate account at maturity behave simply. Some fixed-term loans, and most of the arithmetic on a simple interest calculator, use the simple form. Savings accounts that credit interest into the same balance compound.

What compounding frequency is actually worth

This is the part that is oversold. Increasing n means the interest is credited more often, so it starts earning sooner, and the balance is genuinely higher. But the effect has a hard ceiling — continuous compounding — and most of the available gain is captured by the very first step away from annual.

Annual to monthly is worth $181.14 over the decade. Monthly to daily is worth $16.55 — about one and a half dollars a year on a $10,000 balance. The theoretical maximum, compounding continuously, is $16,487.21, which daily compounding gets within 56 cents of. So an account advertising daily compounding at 4.9% is worse than one compounding monthly at 5.0%, and no amount of frequency closes a gap in the underlying rate.

Nominal rate, effective annual rate, and how to convert

A rate quoted as “5% per annum, compounded monthly” is a nominal rate. It is not what you earn in a year, because it is shorthand for 0.05 ÷ 12 applied twelve times. The effective annual rate is the single figure that would produce the same year-end balance with one compounding period, and it is the only fair basis for comparing two accounts.

Effective annual rate from a nominal rate
  EAR = (1 + r ÷ n)^n − 1

  5% compounded monthly: (1 + 0.05 ÷ 12)^12 − 1 = 5.1162%
  5% compounded daily:   (1 + 0.05 ÷ 365)^365 − 1 = 5.1268%

Nominal rate from an effective annual rate
  r = ((1 + EAR)^(1 ÷ n) − 1) × n

  5% EAR with monthly compounding: 4.8889% nominal

The two directions matter in different places. Going nominal to effective lets you compare a savings account quoted monthly against a term deposit quoted annually. Going the other way explains why a 5% effective return needs only a 4.8889% nominal rate: compounding does part of the work, so less rate is required. Note also that the effective rate rises with n but converges quickly, which is the same conclusion as the table above expressed as a percentage rather than as dollars.

Regular deposits do more than frequency ever will

Chasing compounding frequency is optimising the smallest term in the equation. What actually builds a savings balance is the stream of deposits going into it, and that has its own formula — the future value of a level series, which grows each deposit for however many periods remain after it is made.

Future value of regular deposits
  FV = PMT × ((1 + i)^k − 1) ÷ i

  PMT = deposit each period    i = rate per period    k = number of deposits

Combined with an opening balance
  FV = P × (1 + i)^k + PMT × ((1 + i)^k − 1) ÷ i

The timing of deposits inside a period is a second-order effect of the same kind. Depositing at the start of each month rather than the end gives every deposit one extra period of growth, which on this plan is worth $323.50 over the ten years — more than the frequency change, and still small beside a $50 increase in the deposit itself. The savings goal calculator works this in reverse: it solves for the deposit needed to reach a target by a date.

Starting earlier beats a better rate, and the rule of 72

Because growth is exponential in time and only linear in the rate, an extra few years at the front of a savings plan is usually worth more than a rate improvement you might chase by switching products. Saving $5,000 a year for 30 years at 5% reaches $332,194. The same $5,000 a year for 25 years at 6% — a full percentage point better, but starting five years later — reaches $274,323. The later start costs about $58,000 despite the better rate.

The rule of 72 is the mental shortcut for the same idea. Divide 72 by the rate expressed as a whole number and you get a rough doubling time.

Use it to sanity-check a projection rather than to produce one. If a calculator tells you a balance at 6% will triple in twelve years, the rule of 72 says twelve years is roughly one doubling, and you should go back and check your inputs.

Tax, inflation, and compounding in the wrong direction

Every figure above is a gross nominal figure, and two things sit between it and what you actually end up with. Interest credited to a savings account is generally assessable income in the year it is paid, so it is taxed at your marginal rate — a nominal 5% is a smaller number once tax is taken, and how much smaller depends on your circumstances, which is a matter for the ATO or a registered tax agent rather than for this page. Inflation then reduces what the remaining balance can buy. A return that is positive in dollars can still be negative in purchasing power, and that is the number that decides whether you are better off.

The same mechanism, working against you

Compounding is indifferent to whose side it is on. Interest charged on a debt and not repaid is added to the balance, and the larger balance then attracts more interest, which is the entire reason a revolving credit balance is so hard to clear at the minimum payment. A home loan is the useful inverse: because interest is charged on the outstanding balance, every dollar of principal you remove cancels all the future interest that dollar would ever have generated. That is why the timing of extra repayments matters so much, and it is covered in how mortgage interest is charged and repaid.

What the OzCalc interest calculators do and do not model

The compound interest calculator applies the two formulas above directly: it grows the opening balance at (1 + r ÷ n) for n × t periods, and grows a level deposit stream alongside it. Deposits are treated as arriving at the end of each period, which is the conservative assumption. The simple interest calculator is the non-compounding counterpart, and the savings goal calculator rearranges the same annuity formula to solve for a deposit or a timeframe.

• The rate is constant for the whole term. Real savings rates move, and bonus rates often depend on monthly conditions being met or apply only up to a balance cap.
• No tax is deducted from interest, and no inflation adjustment is applied, so every projection is a gross nominal figure.
• No account fees or withdrawal penalties are modelled.
• Deposits are level and never missed. A stream that stops for six months is materially different from the projection.
• Nothing here is a forecast of investment returns. A fixed rate applied to a balance is a reasonable model of a savings account and a poor model of a market-linked investment, which does not deliver the same return every year.

For the official version and the surrounding guidance, Moneysmart publishes both an explanation of compound interest and a compound interest calculator, along with a savings goals calculator. If two calculators disagree, the usual culprit is the compounding frequency or whether deposits are assumed at the start or the end of each period.

Common questions