Compound interest, contributions and inflation: the exponential your intuition reads as a line

Exponential, not a line

Under simple interest, interest is always charged on the initial capital and is the same every period: the balance grows like a straight line. Under compound interest, each period's interest joins the base of the next period, so you earn interest on the capital and interest on the interest already accrued. The base grows, and on a larger base the interest is larger, a loop that feeds itself. The result is not a line, it is an exponential. Intuition fails precisely here: because in the early years the two curves nearly touch, it is easy to conclude the difference is small. It is not; it just takes time to show up, and then it explodes.

Put the two formulas side by side on the same capital and rate and the difference stops being abstract. Take R$ 10,000 at 10% per year. Under simple interest you add R$ 1,000 every year, always: in 30 years the balance is R$ 40,000. Under compound interest each year earns 10% on a balance larger than the year before. The chart below traces both curves year by year.

Notice the shape: up to about five years the curves run almost together, which is deceptive. It is at the end that the exponential leaves the line behind. That is why time is the most powerful variable in the simulation, far more than the rate, and even more than the starting amount. A simulation combines five inputs, and each one moves the result in a different way.

Converting rates: the power, not the multiplication

The costliest everyday intuition error is converting rates by simple multiplication or division. If 1% per month were 12% per year, you could just multiply. But under compounding the next month earns on a balance that already absorbed the previous month's interest, so there is interest on the monthly interest itself, which multiplying by 12 ignores. The correct conversion uses a power: you compound the monthly rate twelve times.

Applying the formula, 1% per month becomes (1.01)^12 − 1 = 12.6825%, not 12%. The 0.68 percentage-point gap is precisely the interest that the monthly interest earned across the year, invisible to anyone who multiplies. And the error grows with the rate: 2% per month equals 26.82% per year, not 24%. The higher the rate, the more multiplying by 12 lies downward.

Worked example 1, the error of annualizing by 12. You invest R$ 10,000 at 1% per month for 10 years (120 months). The correct move is to compound the monthly rate: 10,000 × (1.01)^120 = R$ 33,003.87. If you treated 1% per month as "12% per year" and projected annually, you would get 10,000 × (1.12)^10 = R$ 31,058.48. The difference is R$ 1,945.39, nearly 6% of the total, and it exists only because the wrong version threw away the interest on the monthly interest for a decade.

Periodic contributions: the annuity formula

When there are monthly contributions, the balance stops being a single exponential and becomes the sum of many. The initial capital compounds over all n months. The first deposit compounds for n−1 months; the second, for n−2; and the last, made at the end, barely earns. Summing all those terms looks laborious, but they form a geometric progression, and the sum of a geometric progression has a closed form. That is where the future-value-of-an-annuity formula comes from.

The one-line derivation: sum the compounded PMT contributions, FV = PMT·[(1+i)^(n−1) + (1+i)^(n−2) + ... + (1+i) + 1]. The bracketed term is a geometric sum with ratio (1+i) and n terms, which equals ((1+i)^n − 1) / i. Substituting, you reach the closed form below. It holds for contributions made at the end of each period, the convention our calculator uses.

Worked example 2 (nominal part), R$ 1,000 of initial capital, R$ 500 contributed per month, 0.7% per month (equivalent to 8.73% per year), for 20 years (240 months). The initial capital becomes 1,000 × (1.007)^240 = R$ 5,334.24. The contribution stream becomes 500 × [(1.007)^240 − 1] / 0.007 = R$ 309,588.91. Summing, the nominal balance is R$ 314,923.15. Of that total, you deposited R$ 121,000 (the capital plus 240 contributions of R$ 500) and interest added R$ 193,923.15, more than you put in. Notice the imbalance: almost all the balance came from the contributions and from time, not from the initial capital. Check it in the calculator below.

The real return: stripping out tax and inflation

The nominal balance is a partial illusion: part of it the government takes in tax, and part inflation hollows out from within. On fixed income (CDB, Treasury bonds, LC), income tax is withheld at source upon redemption, at a rate that falls with the term, the regressive table of Law 11,033/2004. It applies only to the yield, never to the principal. Leaving the money in for over two years drops the bite from 22.5% to 15%.

Even before income tax, there is the IOF, a tax that only bites redemptions in the first 30 days. It follows a regressive table from Decree 6,306/2007: 96% of the yield on day one, falling day by day to 3% on day 29 and zeroing on day 30. In practice, anyone who does not redeem within the first month never pays IOF, which is why it barely appears in medium- and long-term simulations, but it punishes very short-term churn. After IOF (if any) and income tax, you are left with the nominal net return. Inflation still has to be stripped out to reach what matters.

The correct real-return calculation is the Fisher effect, formalized by Irving Fisher in "The Theory of Interest" (1930): you do not subtract inflation, you divide by it. The rough subtraction (real return ≈ nominal return − inflation) works as a mental estimate, but errs more and more as rates rise. With a 10% return and 6% inflation, subtraction gives 4%, but the exact figure is (1.10 ÷ 1.06) − 1 = 3.77%, a 0.23-point error. Raise it to a 30% return and 20% inflation and subtraction spits out 10%, when the real figure is (1.30 ÷ 1.20) − 1 = 8.33%: now the error is 1.67 points. Under high rates and high inflation, the approximation becomes a trap.

Worked example 2 (real and net part), go back to the R$ 314,923.15 balance from the contributions example. Interest was R$ 193,923.15; at 15% income tax (term over 720 days), the tax is R$ 29,088.47, and the nominal net balance drops to R$ 285,834.68. Now strip out inflation: at 4.5% per year for 20 years, prices multiply by (1.045)^20 = 2.4117. Dividing the net balance by that factor, today's real purchasing power is R$ 118,519.31. The balance looked like R$ 314k; after tax and inflation, it buys the equivalent of R$ 118.5k in today's money. It is this figure, not the nominal one, that answers "what is this really worth".

Useful shortcuts and common traps

The exponential yields a few mental shortcuts worth gold, as long as you know each one's range. The most famous is the rule of 72: to estimate in how many periods money doubles at a rate i (in %), divide 72 by i. At 8% per year, money doubles in about 72 ÷ 8 = 9 years. The rule does not come from nowhere: doubling requires (1+i)^n = 2, that is, n = ln(2)/ln(1+i), and ln(2) ≈ 0.693. For rates in the 6% to 10% band, the number 72 (a bit above 69.3) offsets the rounding and lands right on target.

The same ideas resurface in other financial calculations. The logic of summing a series of compounded installments is at the heart of a loan, see SAC versus Price. Converting between a percentage change and a multiplicative factor is the foundation of all of this, revisit it in percentage increase and discount. And when the topic is restating a value over time by an official index, the guide to monetary restatement by Selic and IPCA-E shows the same math applied to debts and judgments.