IntroductionIn this lesson, we will continue discussing amortization according to the systems adopted in different countries or situations.French System (or Price system)According to Poitras (2000), the French system was developed by the Belgian mathematician and physicist Simon Stevin in the 16th century. However, it was used by the English economist and mathematician Richard Price in the 18th century in the English pension calculations of the time. Thus, it became known in Brazil as the Price System.In this system, payments are equal and periodic, from the moment they begin to be paid.Thus, considering a principal P to be paid at times 1, 2, 3,…, t, at an interest rate i (expressed in the time unit of the payment periodicity), the constant payments constitute a uniform sequence in which each installment is indicated by R. Remember:\[ V_m = R [\frac{(1 + i)^t - 1}{(1 + i)^t · i}]\]When we want to calculate the outstanding balance, at a given time, in the French system, the procedure consists of the following: we calculate the present value of the payments yet to fall due. With this, we eliminate the value of the interest contained in the payments. Thus, this present value corresponds to the balance to be amortized, that is, the outstanding balance.ExamplesA loan of R$ 800,000.00 was obtained by a company when purchasing a building. The loan must be repaid under the French system in five semiannual payments at a rate of 4% per semester. Prepare the schedule.SemesterOutstanding balanceAmortizationInterestPayment0800,000---1652,298.30147,701.7032,000179,701.702498,688.53153,609.7726,091.93179,701.703338,934.37159,754.1619,947.54179,701.704172,790.04166,144.3313,557.37179,701.705-172,790.046,911.60179,701.70Total-800,00098,508.44898,508.44In a loan of R$ 100,000.00 to be paid under the French system, over 40 months and at a rate of 3% per month, what is the outstanding balance in the 25th month? (Assume the payment for this month has been made.)\[ V_m = R [\frac{(1 + i)^t - 1}{(1 + i)^t · i}]\]\[ 100,000 = R [\frac{(1 + 0.03)^{40} - 1}{(1 + 0.03)^{40} · 0.03}]\]\[ 100,000 = R · 23.114772 \]\[ R = \frac{100,000}{23.114772} = 4,326.24 \]The outstanding balance in the 25th month is the present value of the uniform sequence of payments yet to fall due (15 payments):\[ V_m = R [\frac{(1 + i)^t - 1}{(1 + i)^t · i}]\]\[ V_m = 4,326.24 [\frac{(1 + 0.03)^{15} - 1}{(1 + 0.03)^{15} · 0.03}] = 51,646.37 \]American SystemUnder this system, the principal is paid in a single amount at the end of the loan period. In general, interest is paid periodically; however, it may eventually be capitalized and paid in a single amount together with the principal (it all depends on the agreement between the interested parties).ExamplesFor a loan of 800 thousand dollars, a client proposes to repay the principal two and a half years from now, paying only the interest semiannually at a rate of 4% per semester. Prepare the schedule.SemesterOutstanding balanceAmortizationInterestPayment0800,000---1800,000-32,00032,0002800,000-32,00032,0003800,000-32,00032,0004800,000-32,00032,0005-800,00032,000832,000Total-800,000160,000.00960,000.00Exercises1) A bank grants a credit of R$ 60,000.00 to a company, to be paid under the Price System over 20 quarters, at a rate of 6% per quarter. Prepare the schedule up to the third quarter.2) When buying an apartment, Mr. Almeida received a loan of R$ 280,000.00, which must be paid under the Price system over 240 months at a rate of 0.9% per month. What is the outstanding balance after the payment of the 51st installment?3) Mr. Moura received financing of R$ 500,000.00 for the purchase of a house, with the Price System adopted at a rate of 1.5% per month for payment over 180 months. What is the status of the debt after payment of the 64th installment? (ignore cents)4) The amount of R$ 1,500,000.00 is financed at a rate of 10% per year, to be amortized under the American system, with a three-year grace period on amortizations. Knowing that interest is paid annually, construct the schedule.5) A person obtained a loan of R$ 120,000.00 at a compound interest rate of 2% per month, which must be paid in 10 equal installments. What is the amount of interest to be paid in the 8th installment?6) (BNDES) Mark the correct option. The French Amortization System is characterized by:a) Increasing interest and increasing payment.b) Decreasing payment and increasing interest.c) Decreasing amortization and interest.d) Increasing interest and amortization.e) Increasing amortization and constant payment.7) The amount of 3,600 UR can be financed under the SAC system or the French system. Knowing that the interest rate is 1% per month and that 180 monthly installments will be paid, obtain:a) The payment amount under the French system.b) The values of the first three payments under the SAC system.c) The month from which the payment under SAC becomes lower than the payment under the French system.Answer Key2) R$ 258,624.973) R$ 441,370.005) R$ 770.006) E7) a) 43.21 UR; b) 56 UR; 55.80 UR; 55.60 UR; c) 65th month.
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