Accumulated interest rate

At interest rates they are percentages that express a compensation that must be paid to the person who lends or invests a sum of money.

Over time, these rates can vary, either with increases or decreases. Thus, considering the variation in interest rates, we can obtain the so-called accumulated interest rate over a period of time.

The accumulated interest rate can be obtained from a formula, which will be presented below. It is important to emphasize that this formula can also be used to calculate other types of accumulated fees, such as the rate of inflation.

Accumulated interest rate formula

Consider $\dpi{120} \mathrm{n}$ interest rates, $\dpi{120} \mathrm{i_1}$ the first rate, $\dpi{120} \mathrm{i_2}$ the second rate, and so on until $\dpi{120} \mathrm{i_n}$ , the last rate. THE formula for calculating the accumulated interest rate é:

$\dpi{120} \mathbf{i_{cumulative} = [(1+ i_1)\times (1+i_2)\times ...\times (i+i_n) - 1]\times 100}$

Example 1: O Broad Consumer Price Index (IPCA) is an index used to measure inflation in Brazil. Based on the IPCA for the months of a year and the above formula, we can obtain the accumulated IPCA.

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Month	IPCA (%)	IPCA/100
January	0,32	0,0032
February	0,43	0,0043
March	0,75	0,0075
April	0,57	0,0057
May	0,13	0,0013
June	0,01	0,0001
July	0,19	0,0019
August	0,11	0,0011
September	-0,04	-0,0004
October	0,1	0,001
November	0,51	0,0051
December	1,15	0,0115

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To use the formula, we must divide the rates (%) by 100, getting numbers in decimal form. Therefore, we are going to use the IPCA/100 values presented in the third column of the table above.

$\dpi{100} \small \mathbf{i_{a} = [(1.0032)\times (1.0043)\times (1.0075) \times... \times (1.0011) \times (0.9996) \times (1.001) \times (1.0051) \times (1.0115) - 1]\times 100}$