What Determines Compound Growth — and What Follows From It

The Formula and Its Inputs

• The formula: A = P(1 + r/n)^(nt), where A is the ending amount, P the principal, r the annual rate as a decimal, n the number of compounding periods per year, and t the number of years.
• Principal (P): Scales the result proportionally. Doubling the starting amount doubles the ending amount, holding everything else constant.
• Rate (r): Enters the base of the exponent, so its effect is not proportional — a difference of one percentage point produces a gap that widens the longer the term runs.
• Time (t): Enters the exponent, which is why compounding is described as accelerating. Each additional period earns a return on all previously accumulated interest as well as on the principal.
• Frequency (n): More frequent compounding raises the effective yield, but with diminishing effect — the limit as n rises without bound is continuous compounding, A = Pe^(rt).
• Additional contributions: Periodic deposits are not in the basic formula; a future-value-of-an-annuity calculation adds them, and each one compounds only over its own remaining time.

What Reduces the Realised Result

• Fees: A fee expressed as a percentage of the balance reduces the effective rate directly, and because the rate sits in the exponent base, the shortfall compounds along with everything else.
• Taxes: Interest taxed as it is earned reduces the amount available to compound in the next period, which is why tax timing changes the trajectory and not merely the final deduction.
• Inflation: The real value of a balance grows at approximately the nominal rate minus the inflation rate, so a nominal gain and a real loss can occur simultaneously.
• Withdrawals: Removing an amount removes not only that amount but all the growth it would have generated over the remaining term.
• Interruptions in contribution: Because early contributions compound for the longest, a gap early in a long horizon has a larger arithmetic effect than an equivalent gap late in it.

The Same Arithmetic Applied to Debt

• Symmetry: Compounding does not care which side of the ledger it is on. A balance that is owed and not paid grows by the same formula that a balance that is held and not withdrawn grows by.
• Daily compounding on revolving credit: Credit card issuers commonly compute interest on the average daily balance and compound daily, which makes the effective annual cost slightly higher than the stated APR.
• Minimum payments: When a payment covers little more than accrued interest, the principal falls slowly and the repayment term extends dramatically, which is why disclosure rules require a minimum-payment payoff estimate on US credit card statements.
• Capitalisation: When unpaid accrued interest is added to principal — as can occur with some student loans after a deferment — subsequent interest is charged on the larger figure.
• Negative amortisation: If a scheduled payment is smaller than the interest accruing, the balance rises even though payments are being made.

Doubling Time and the Rule of 72

The Rule of 72 states that the number of periods required for a balance to double is approximately 72 divided by the periodic rate expressed as a percentage. At 6 percent the estimate is 12 years; at 8 percent, 9 years; at 12 percent, 6 years. The exact figure is the natural logarithm of 2, about 0.693, divided by the rate, and the reason 72 rather than 69 is used is that it divides evenly by many small integers and compensates slightly for discrete rather than continuous compounding. The approximation is most accurate for rates roughly between 6 and 10 percent and drifts at the extremes. The same rule applied to an inflation rate estimates how long the price level takes to double, which is a standard way of making an inflation figure concrete.