Introduction> Deferred (spelled with an i in Portuguese) means postponed or delayed.There are situations in which the buyer only begins to pay after a grace period. We will analyze these types of situations.Deferred uniform sequencesGiven a sequence: 0, 1, 2, m, m + 1, m + 2... Where the first installment begins to be paid at m + 1.To calculate the present value (PV) of the deferred uniform sequence, we can proceed as follows:1. We calculate the present value of the uniform sequence at date m (Vm), that is, one period before the start of the uniform sequence. To do this, it is enough to note that:\[ V_m = R [\frac{(1 + i)^n - 1}{(1 + i)^n · i}]\]2. We calculate the capital PV, which invested at date 0 produces an amount equal to Vm at date m. That is:\[ PV = \frac{V_m}{(1+i)^m}\]ExamplesA plot of land is sold for R$ 50,000 in cash or in six equal monthly installments, with the first due three months after the purchase. If the financing interest rate is 2% per month, what is the value of each installment?- m = 2- i = 0.02- PV = 50,000- R = ?\[ PV = \frac{V_m}{(1+i)^m}\]\[ 50,000 = \frac{V_2}{(1+0.02)^2} => V_2 = 52,020\]\[ V_m = R [\frac{(1 + i)^n - 1}{(1 + i)^n · i}]\]\[ V_2 = R [\frac{(1 + 0.02)^6 - 1}{(1 + 0.02)^6 · 0.02}]\]\[ 52,020 = R · 5.601431\]\[ R = 9,286.91\]Amount in uniform sequencesWe have seen the present value of a uniform sequence. Now we will see the value of its amount.We call the amount of the sequence, at date n, the sum of the amounts of each capital R, invested from the date considered until date n.Thus, denoting the amount by M, we will have:\[ M = R + R · (1+i)^{1} + R · (1+i)^{2} + ... +R · (1+i)^{t} \]Thus, the first term is R and this geometric progression grows by (1 + i).Given that the formula for a geometric progression is:\[ S = \frac{a_1 · (q^n - 1)}{q - 1} \]Thus:\[ M = \frac{R · ((1+i)^t - 1)}{1 + i - 1} \]\[ M = R[\frac{(1+i)^t - 1}{i}] \]ExamplesAn investor invests R$ 2,000.00 monthly in an investment fund that remunerates investments at a compound interest rate of 1% per month. If the investor makes seven investments, what is the amount at the time of the last deposit?We have:- R = 2,000- i = 1% per month- t = 7\[ M = R[\frac{(1+i)^t - 1}{i}] \]\[ M = 2,000[\frac{(1+0.01)^7 - 1}{0.01}] = 14,427.07 \]An executive, thinking about his future retirement, decides to make 240 monthly deposits of R$ 700.00 each into a fund that yields 0.4% per month (real rate). His goal is to generate an amount that allows him to withdraw x reais per month for 360 months until his balance is depleted. Find the value of x assuming that the 1st withdrawal is made one month after the last deposit. Assume that all monetary values are given in real terms relative to the date of the beginning of the deposits.- Amount of the deposits immediately after the 240th deposit:\[ M = 700[\frac{(1+0.004)^{240} - 1}{0.004}] =281,172.52 \]- The amount above is the present value of the withdrawals that begin one month later. Therefore,\[ 281,172.52 = R[\frac{(1+0.004)^{360} - 1}{(0.004)^{360} · 0.004}] => R = 1,475.21 \]Check the spreadsheet below and copy it to run your own retirement-related experiments.https://docs.google.com/spreadsheets/d/1xYi51tznE1N46asp-zQE7YbpWJmR6Fv5dCEwaDbTju8/edit?usp=sharingExercises1) A person deposits R$ 3,500.00 monthly for seven months in a fund that remunerates their deposits at a rate of 0.9% per month. What is the amount immediately after the last deposit?2) How much should a person deposit monthly for 15 months in an investment fund that yields 1.8% per month so that, immediately after the last deposit, they have an amount of R$ 60,000.00?3) A company must pay a note of R$ 50,000.00 one year from now. How much should it invest monthly, starting today, if the deposits are equal and remunerated at 0.85% per month, so that, one month after the last deposit, the balance is sufficient to pay the note?4) How much should I deposit monthly in an investment fund that pays 0.8% per month in interest so that, at the end of the 18th deposit, I have an amount of R$ 2,000,000.00?5) An executive, anticipating a supplement to his retirement income, decides to make 180 monthly deposits of R reais each on dates 1, 2, 3, …, 180, aiming at withdrawals of R$ 1,500.00 per month on dates 181, 182, …, 420. Find the value of R, assuming that the invested money yields 0.5% per month.6) Suppose you retire at age 70 with savings of R$ 800,000.00 invested in a fund. How much could you withdraw from this fund and spend per year for the next 25 years, assuming these expenses are constant in real terms? Assume a real rate of 4% per year of return on the fund and the first withdrawal one year after retirement.7) Following the previous exercise. How much should you invest per year (constant deposit in real terms) in this fund from age 35 to 70 (36 deposits) to achieve savings of R$ 800,000.00?8) A sofa set is sold for R$ 6,000.00 in cash or in four equal monthly installments, with the first due three months after the purchase. What is the value of each installment if the financing interest rate is 2.8% per month?9) The sale of a motorcycle is advertised in ten equal monthly installments of R$ 2,000.00 each, with the first due two months after the purchase. What is the cash price if the financing rate is 3.5% per month?10) A sound system is sold for R$ 2,300.00 cash. In installments, it is sold in six equal monthly payments, with the customer given a two-month grace period, that is, the first payment is due only three months after the purchase. The interest rate charged by the store is 2% per month. Find the amount of each payment.Answer Key1) R$ 25,171.512) R$ 3,519.953) R$ 3,975.444) R$ 103,746.145) 719.94 reais6) R$ 51,209.577) R$ 10,309.508) R$ 1,697.679) R$ 16,070.7310) R$ 427.20
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