IntroductionWe saw, in the previous lessons, how sets of values could be transformed into other equivalent ones, for comparison purposes, especially by transforming them into present values.In practice, it is common for these sets to have certain characteristics, such as periodicity, uniformity, growth (or decrease), according to certain mathematical laws. Such sets are called sequences of monetary values (the values may refer either to payments or to receipts).Uniform sequenceLet us consider the sequence of values y1, y2, y3,…, yt, respectively on dates 1, 2, 3, …, n (the unit of time may be month, semester, year, etc.). We say that this set constitutes a uniform sequence if all yt have the same value.By definition, the present value of this sequence on date 0 (that is, one period of time before the start of the sequence), at an interest rate i, in the considered unit of time, is:\[ V = \frac{R}{(1+i)^{1}} + \frac{R}{(1+i)^{2}} + ... +\frac{R}{(1+i)^{t}} \]\[ V = R [\frac{1}{(1+i)^{1}} + \frac{1}{(1+i)^{2}} + ... +\frac{1}{(1+i)^{t}}] \]The value in brackets could be calculated by adding term by term; however, for a high value of n, such a procedure would be time-consuming.Simplifications can be made if we note that the expression in brackets is the sum of the terms of a geometric progression (GP) whose first term (a1) is equal to \[ \frac{1}{(1+i)}\] and whose ratio is:\[ \frac{1}{(1+i)}\]Elementary mathematics teaches us that the sum of the first n terms of a geometric progression, in which the ratio is different from 1, is given by:\[ S = \frac{a_1 *(q^n -1)}{(q - 1)}\]In the end, considering the first installment and the aforementioned ratio:\[ S = \frac{(1 + i)^n - 1}{(1 + i)^n*i}\]And the present value:\[ V = R [\frac{(1 + i)^n - 1}{(1 + i)^n*i}]\]ExamplesA household appliance is sold on credit, in four equal monthly payments of R$ 550.00 each, with the first due one month after the purchase. If the store finances it at an interest rate of 5% per month, what is its cash price?n = 4i = 5% per monthR = 550\[ V = R [\frac{(1 + i)^n - 1}{(1 + i)^n*i}]\]\[ V = 550 [\frac{(1 + 0,05)^4 - 1}{(1 + 0,05)^4*0,05}] = 1.950,27 \]A used car is sold for cash for R$ 30,000.00, but it can be sold on credit in 12 equal monthly installments, with the first due one month after the purchase. Knowing that the financing interest rate is 2% per month, obtain the value of each installment.V = 30,000n = 12i = 2% per month\[ V = R [\frac{(1 + i)^n - 1}{(1 + i)^n*i}]\]\[ 30.000 = R [\frac{(1 + 0,02)^12 - 1}{(1 + 0,02)^12*0,02}] \]\[ 30.000 = R * 10,5753 \]\[ R = \frac{30.000}{10,5753} = 2.836,79 \]A plot of land is sold in four equal monthly installments of R$ 150,000.00 each, with the first given as a down payment. If the financing rate is 4% per month, what is the cash price?down payment = 150,000R = 150,000n = 3i = 4% per month\[ V = R_0 + R [\frac{(1 + i)^n - 1}{(1 + i)^n*i}]\]\[ V = 150.000 + 150.000 [\frac{(1 + 0,04)^3 - 1}{(1 + 0,04)^3*0,04}]\]\[ V = 150.000 + 150.000*2,775 = 566.263,65\]What is the internal rate of return of an investment of 5,000 reais to receive shortly afterward 12 monthly installments of 500 reais?investment = 5,000R = 500n = 12\[ V = -5000 + 500 [\frac{(1 + i)^n - 1}{(1 + i)^n*i}]\]Thus:IRRPresent value1%627.532%287.673%-22.99Assume that the IRR will be between 2% and 3%. By the midpoint, the IRR will be approximately 2.5%.Exercises1) Obtain the cash price of a car financed at a rate of 3% per month, with the number of installments equal to 10 and R$ 4,500.00 as the value of each monthly installment, with the first due one month after the purchase.2) A car is sold for cash for R$ 40,000.00 or on credit in three equal monthly installments, with no down payment. What is the value of each installment if the financing interest rate is 1.9% per month?3) A boat is sold for cash for R$ 16,000.00 or, alternatively, with a 20% down payment plus four equal monthly installments. What is the value of each installment if the interest rate is 2% per month?4) A motorcycle is sold in five monthly installments of R$ 2,000.00 each, with the first given as a down payment. What is the cash price if the financing interest rate is 2.5% per month?5) A microcomputer is sold for cash for R$ 2,500.00 or, alternatively, in four equal monthly installments, with the first given as a down payment. What is the value of each installment if the interest rate is 2.2% per month?6) An apartment, whose cash price is R$ 100,000.00, is sold on credit with a 30% down payment and the balance in 100 equal monthly installments later adjusted for inflation. What is the value of each installment before adjustment if the financing interest rate is 1% per month?7) The monthly rent of an apartment is R$ 2,000.00, adjustable every six months. If the tenant, upon signing the rental contract, wants to pay the rent for the first six months in advance, what amount should be paid if the interest rate is 2.3% per month?8) A certain company needs to make a decision: buy a machine for cash for R$ 980,000.00 or on credit in 12 postpaid monthly installments of R$ 99,630.00. Knowing that the investment rate fluctuates between 2% per month and 3% per month, what is the best alternative for the company?9) A bank grants a loan to a company in the amount of R$ 60,000.00 to be paid in four equal postpaid monthly installments of R$ 16,850.00 each. What is the financing interest rate?10) The cash price of a stereo system is R$ 820.00. In installments, payment may be made in five equal monthly deferred payments of R$ 198.00 each. What is the monthly interest rate of the financing?11) A large company takes out a loan from a certain bank at a rate of 3.5% per month. Knowing that the amount borrowed is R$ 10,000.00, to be paid in 24 equal monthly installments, with the first due one month after the loan, and knowing that the bank charges a credit opening fee equal to 2% of the amount borrowed, calculate the amount of each installment and the rate effectively paid by the company.Answer Key1) R$ 38,385.912) R$ 13,843.183) R$ 3,361.584) R$ 9,523.955) R$ 645.556) R$ 1,110.607) R$ 11,090.328) Cash payment.9) 4.82% per month10) 6.63% per month11) R$ 622.73; 3.69% per month
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