IntroductionUnder simple interest, the interest of each period is always calculated on the initial capital. Compound interest is different: the interest of each period is added to the amount of the previous period, and this new amount starts earning interest in the following period. This is the famous interest on interest.The compound interest amount formulaConsider a capital C invested at an interest rate i per period. Let us calculate the amount (M) obtained at compound interest after t periods.Amount at period zero:\[ M_0 = C \]Amount after 1 period:\[ M_1 = C + C \cdot i = C \cdot (1 + i) \]Amount after 2 periods (interest is now applied to M1):\[ M_2 = C \cdot (1 + i) + i \cdot C \cdot (1 + i) = C \cdot (1 + i) \cdot (1 + i) = C \cdot (1 + i)^{2} \]Amount after 3 periods:\[ M_3 = C \cdot (1 + i)^{2} + i \cdot C \cdot (1 + i)^{2} = C \cdot (1 + i)^{3} \]Following the pattern, the amount after t periods is given by:\[ M_t = C \cdot (1 + i)^{t} \]As with simple interest, the rate i and the term t must be in the same unit of time.ExamplesA capital of R$ 6,000.00 (Brazilian reais) was invested at compound interest for three months, at a rate of 2% p.m. What are the amount and the total interest earned?\[ C = 6{,}000;\ i = 2\% \text{ p.m.};\ t = 3 \]\[ M_t = C \cdot (1 + i)^{t} = 6{,}000 \cdot (1 + 0.02)^{3} = 6{,}367.25 \]\[ J = 6{,}367.25 - 6{,}000 = 367.25 \]At simple interest, the interest would be 6,000 · 0.02 · 3 = 360.00. The difference of R$ 7.25 is the interest on interest.What capital, invested at compound interest at 2.5% p.m., produces an amount of R$ 3,500.00 after one year?\[ M = 3{,}500;\ i = 2.5\% \text{ p.m.};\ t = 12 \]\[ 3{,}500 = C \cdot (1 + 0.025)^{12} \]\[ 3{,}500 = C \cdot 1.3449 \]\[ C = \frac{3{,}500}{1.3449} = 2{,}602.42 \]How much interest does a capital of R$ 10,000 earn when invested for 2 years at compound interest of 10% p.a.?Answer: R$ 2,100 — M = 10,000 · 1.10² = 12,100, and the interest is 12,100 - 10,000 = 2,100. R$ 2,000 would be simple interest; the extra R$ 100 is the second year's interest on the R$ 1,000 of interest from the first year. R$ 12,100 is the amount, not the interest.Financial calculatorsFor quick and accurate calculations, you can use financial calculators (such as the HP 12C) or the financial functions of spreadsheets, designed for this kind of problem. In them:PV (present value) is the capital;FV (future value) is the amount;i is the interest rate per period;n is the number of periods, our t.In most calculators, PV and FV appear with opposite signs, because they follow the cash flow convention: money going out is represented by a negative number, and money coming in by a positive number. Someone investing R$ 1,000 enters PV = -1,000 and gets a positive FV.A calculator is a great ally, not only because it makes the arithmetic easier, but also because it keeps numbers with all their decimal places. In the previous example, the factor (1.025)12 = 1.344889 was rounded to 4 places (1.3449), which led to the result of R$ 2,602.42. Without rounding the factor, the correct result is R$ 2,602.45.Equivalent ratesAs with simple interest, equivalent rates are those that generate the same amount when applied to the same capital for the same term. If t1 and t2 are the numbers of periods of each rate within that term:\[ C \cdot (1 + i_1)^{t_1} = C \cdot (1 + i_2)^{t_2} \]\[ (1 + i_1)^{t_1} = (1 + i_2)^{t_2} \]Or:\[ 1 + i_1 = (1 + i_2)^{\frac{t_2}{t_1}} \]\[ i_1 = (1 + i_2)^{\frac{t_2}{t_1}} - 1 \]Unlike simple interest, here it is not enough to divide or multiply the rate: 1% p.m. at compound interest is not equivalent to 12% p.a., but to (1.01)12 - 1 = 12.68% p.a.ExamplesAt compound interest, what quarterly rate is equivalent to 15% p.a.?A year has 4 quarters:\[ t_1 = 4;\ i_1 = ? \]\[ t_2 = 1;\ i_2 = 15\% \]\[ (1 + i_1)^{4} = (1 + 0.15)^{1} \]\[ 1 + i_1 = (1 + 0.15)^{\frac{1}{4}} = 1.0356 \]\[ i_1 = 1.0356 - 1 = 0.0356 = 3.56\% \text{ per quarter} \]At compound interest, what monthly rate is equivalent to 8% per quarter?A quarter has 3 months:\[ t_1 = 3;\ i_1 = ? \]\[ t_2 = 1;\ i_2 = 8\% \]\[ i_1 = (1 + 0.08)^{\frac{1}{3}} - 1 = 0.0260 = 2.60\% \text{ p.m.} \]At compound interest, what annual rate is equivalent to 2% p.m.?Answer: 26.82% p.a. — (1.02)¹² - 1 = 1.2682 - 1 = 0.2682. The 24% comes from multiplying 2% by 12, which only works for simple interest; with compound interest, each month's interest also earns interest in the following months.Bank certificates of deposit and bank deposit receiptsCertificados de depósito bancário (CDBs, Brazilian bank certificates of deposit) are securities issued by banks to raise funds that they use for their lending. They are registered in the holder's name and can be transferred to another investor.Recibos de depósito bancário (RDBs, bank deposit receipts) are similar to CDBs, but cannot be transferred.These securities can pay a fixed (prefixed) rate, known on the day of the investment, or a floating (postfixed) rate, tied to an index such as the CDI (the Brazilian interbank deposit rate). In both cases, the income is subject to Brazilian income tax (IR, imposto de renda), withheld by the bank on redemption or at maturity. The tax rate follows the regressive table of Law 11,033/2004, in force since January 2005 and still valid in 2026: the longer the money stays invested, the lower the rate.Holding periodIncome tax rate on the returnUp to 180 days22.5%181 to 360 days20.0%361 to 720 days17.5%Over 720 days15.0%Redemptions made before 30 days also pay IOF (a Brazilian tax on financial transactions), which is levied on the return and decreases day by day until it reaches zero on the 30th day. In the examples below, the terms are 30 days or longer, so IOF does not enter the calculation.To measure the investor's effective gain, we calculate the interest rate of the transaction net of the income tax paid. This is the net rate; the rate advertised by financial institutions, which does not consider the tax, is the gross rate.ExamplesAn investor put R$ 20,000.00 in a 90-day prefixed CDB at a rate of 10% p.a. What are the amount, the income tax (22.5%), the net amount and the net rate for the period?Using the 360-day commercial year, 90 days correspond to 90/360 of the year (in practice, Brazilian banks usually count business days, on a 252-day basis, but the reasoning is the same):\[ M_{90} = 20{,}000 \cdot (1 + 0.10)^{\frac{90}{360}} = 20{,}482.27 \]\[ IR = 0.225 \cdot (20{,}482.27 - 20{,}000) = 108.51 \]\[ M_{net} = 20{,}482.27 - 108.51 = 20{,}373.76 \]\[ i_{net} = \frac{20{,}373.76}{20{,}000} - 1 = 0.0187 = 1.87\% \text{ for the period (90 days)} \]A CDB held for 2 years and 1 month earned R$ 4,000 of gross interest. How much net return does the investor receive under the regressive income tax table?Answer: R$ 3,400 — the term exceeds 720 days, so the tax rate is 15%: IR = 0.15 · 4,000 = 600 and the net return is 4,000 - 600 = 3,400. R$ 3,300 would use the 17.5% rate, which applies only up to 720 days.Exercises1) If you deposit $ 25,000 today in an interest-bearing account at 0.98% per month, for 12 months with monthly compounding, calculate how much will be redeemed at the end of one year.2) At an interest rate of 1.05% per thirty days, $ 20,000 was invested for two hundred and seventy days. Calculate the amount redeemed.3) The bank financed $ 20,000 for 12 months at an interest rate of 3.5% per month. Calculate the interest and the final payment considering compound interest.4) For your next trip, 7 months from now, you will need $ 5,000. Calculate how much you should invest today, considering an interest rate of 1.25% per month at compound interest.5) Earning an interest rate of 0.96% per month, $ 49,040.69 was redeemed after nine months. Calculate the amount invested.6) Calculate the monthly interest rate that triples a capital in three years.7) The ratio between the future value and the present value of a financial transaction is 1.16578. If the term of this transaction was 12 months, calculate the monthly interest rate at compound interest.8) An investment for 61 days was made at an interest rate of 32.8% per 365 days. Calculate the equivalent interest rate per 30 days.9) In a transaction, $ 8,000 was invested and, after 60 days, $ 8,240 was redeemed. Calculate the daily interest rate of the transaction.10) When investing some capital in a 30-day prefixed CDB, an investor wants to earn a net rate of 1% for the period. Given that the income tax is 22.5% of the interest, what gross rate should the investor accept?Answer key1) $ 28,103.762) $ 21,971.363) J = $ 10,221.37; M = $ 30,221.374) $ 4,583.585) $ 45,0006) 3.10% per month7) 1.29% per month8) 2.36% per 30 days9) 0.04927681% per day10) 1.29% for the period (30 days)References[1] http://www.bb.com.br/pbb/pagina-inicial/voce/produtos-e-servicos/investimentos/investimentos-de-baixo-risco-a-longo-prazo/cdb-di#/[2] BRAZIL. Law No. 11,033 of December 21, 2004, art. 1 (income tax rates on fixed-income investments).
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