IntroductionUnlike simple interest, where the rate is always calculated based on the initial principal, in compound interest, the interest generated in each period is added to the amount from the previous period, causing this new amount to generate interest in the following period.Compound interest amount formulaLet a principal C and an interest rate i be considered, and calculate the amount obtained with compound interest after t time periods.Amount at period zero:\[ M_0 = C \]Amount after 1 period:\[ M_1 = C + Ci = C · (1 + i) \]Amount after 2 periods:\[ M_2 = C*(1 + i) + i · C · (1 + i) = C · (1 + i) · (1+i) \]Amount after 3 periods:\[ M_3 = C · (1 + i) · (1+i) + i · C · (1 + i) · (1+i) = C · (1 + i) · (1+i) · (1+i) \]In the end, it can be seen that the amount calculation can be reduced to a notation formula:\[ M_t = C · (1 + i)^{t} \]ExamplesA principal of R$ 6,000.00 was invested at compound interest for three months, at a rate of 2% per month. What are the amount and the total interest earned?\[ C = 6.000; i = 2\%a.m.; t = 3 \]\[ M_t = C · (1 + i)^{t} = 6.000 · (1 + 0,02)^{3} = 6.367,25 \]\[ J = 6.367,25 - 6.000 = 367,25 \]What principal, invested at compound interest at a rate of 2.5% per month, produces an amount of R$ 3,500.00 after one year?\[ M = 3.500; i = 2,5\%a.m. ; t = 12 \]\[ 3.500 = C · (1 + 0,025)^{12} \]\[ 3.500 = C · (1,3449) \]\[ C = \frac{3.500}{1,3449} = 2.602,42 \]Financial calculatorsFor quick and accurate calculations, financial calculators designed for this type of problem can be used. In them:- PV (present value) is the same as the principal;- FV (future value) is the same as the amount;- i represents the interest rate;- n represents t, the time;It is important to note that, in most calculators, the values of PV and FV appear with opposite signs. That is, in financial transactions, cash outflow is represented by negative numbers while inflow is represented by positive numbers.The calculator is a great financial friend, not only because it makes calculations easier, but also because it can store numbers in internal memory. In the previous example, the annual interest rate calculation was rounded to 4 decimal places, producing the result seen as 2.602,42, but, if it had been stored in memory, the final result would have been 2.602,45.Equivalent ratesSimilar to simple interest, equivalent rates are those that generate the same amount over equal periods and principals. Thus:\[ C · (1 + i_1)^{t_1} = C · (1 + i_2)^{t_2} \]\[ (1 + i_1)^{t_1} = (1 + i_2)^{t_2} \]Or:\[ 1 + i_1 = (1 + i_2)^{\frac{t_2}{t_1}} \]\[ i_1 = (1 + i_2)^{\frac{t_2}{t_1}} - 1 \]ExamplesIn compound interest, what is the equivalent quarterly rate to 15% p.a.?\[ t_1 = 4; i_1 = ? \]\[ t_2 = 1; i_2 = 15\% \]\[ (1 + i_1)^{4} = (1 + 0,15)^{1} \]\[ 1 + i_1 = (1 + 0,15)^{\frac{1}{4}} \]\[ i_1 = 1,0356 - 1 = 0,0356 = 3,56\% a.t. \]In compound interest, what is the equivalent monthly rate to 8% per quarter?\[ t_1 = 3; i_1 = ? \]\[ t_2 = 1; i_2 = 8\% \]\[ i_1 = (1 + 0,08)^{\frac{1}{3}} -1 = 0,026 = 2,6\% \]Bank deposit certificate and bank deposit receiptThe bank deposit certificates (CDBs) are securities issued by banks in general, intended to raise funds for their financing. Such securities are nominative and endorsable (can be transferred by endorsement).The bank deposit receipts (RDBs) are identical to CDBs, except that they are generally non-transferable.There are two forms of remuneration for these securities: fixed-rate and variable-rate. We will now look at the fixed-rate one, where its earnings are subject to taxation (income tax). Banco do Brasil makes it clear: [1]> The longer the amounts remain invested, the lower the tax rate charged will be. For earnings calculated up to 12/31/2004, the rate is 20%. Starting in January/2005, upon redemption or at maturity of the investment, income tax will be charged according to the period the funds remained invested, according to the table below:Holding PeriodRegressive RatesUp to 180 days22.5%From 181 to 360 days20.0%From 361 to 720 days17.5%Above 720 days15.0%To analyze the investor’s effective gain, the interest rate of the operation is calculated, taking into account the income tax paid. The rate thus obtained is called the net rate, whereas the one announced by financial institutions is called the gross rate because it does not consider tax.ExamplesAn investor invested R$ 20,000.00 in a 90-day fixed-rate CDB; the agreed rate was 10% p.a. What are the amount, income tax (22.5%), net amount, and net rate over the period considered?\[ M_{90} = 20.000 · (1 + 0,1)^{\frac{90}{360}} = 20.482,27 \]\[ IR = 0,225 · (20.482,27 - 20.000) = 108,51 \]\[ M_{líquido} = 20.482,27 - 108,51 = 20.373,76 \]\[ i_{líquido} = \frac{20.373,76}{20.000} - 1 = 0,0187 = 1,87\%a.t. \]Exercises1) If today you deposit $25,000 in an account paying 0.98% interest per month for 12 months with monthly compounding, calculate how much will be withdrawn at the end of one year.2) With an interest rate of 1.05% every thirty days, an amount of $20,000 was invested for a period of two hundred and seventy days. Calculate the amount withdrawn.3) The bank financed $20,000 for a 12-month period at an interest rate of 3.5% per month. Calculate the interest and the final payment amount considering compound interest.4) For your next trip in 7 months, it will be necessary to have $5,000 available. Calculate how much you should invest today considering an interest rate of 1.25% per month under compound interest.5) Receiving an interest rate of 0.96% per month, the amount of $49,040.69 was redeemed after a term of nine months. Calculate the invested amount.6) Calculate the monthly interest rate that allows a capital to triple in three years.7) The relationship between the present and future values of the financial operation is 1.16578. If the term of this operation was 12 months, calculate the monthly interest rate under compound interest.8) An investment for 61 days was made with an interest rate of 32.8% per 365 days. Calculate the equivalent interest rate for a 30-day period.9) In an operation, $8,000 was invested and after 60 days the amount of $8,240 was redeemed. Calculate the daily interest rate of the operation.10) By applying a certain capital in a 30-day pre-fixed CDB, an investor intends to earn a net rate of 1% over the period. Knowing that income tax is 22.5% of the interest, what gross rate should be accepted?Gabarito1) $28,103.762) $21,971.363) J = $10,221.37; M = $30,221.374) $4,583.585) $45,0006) 3.10%7) 1.29%8) 2.36%9) 0.04927681%10) 1.29% p.m.References[1] http://www.bb.com.br/pbb/pagina-inicial/voce/produtos-e-servicos/investimentos/investimentos-de-baixo-risco-a-longo-prazo/cdb-di#/
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