IntroductionIn this lesson we continue studying loan amortization. After the constant amortization system (SAC, Sistema de Amortização Constante), we will look at two other widely used systems: the French system, known in Brazil as the Price table (Tabela Price), in which the installments are equal, and the American system, in which the principal is repaid all at once at the end.French system (Price table)According to Poitras (2000), the interest tables underlying the French system were published by the Flemish mathematician and engineer Simon Stevin in the 16th century. In the 18th century, the Welsh philosopher and mathematician Richard Price used calculations of this kind in his studies of pensions and life insurance in England, which is why the system became known in Brazil as the Price System or Price table.In this system, the installments are equal and periodic from the moment they start being paid.So, considering a principal P to be paid at instants 1, 2, 3, …, n, at an interest rate i (in the time unit of the payments), the constant installments form a uniform sequence, whose installment we denote by R. Recalling the present value formula:\[ P = R \left[\frac{(1 + i)^n - 1}{(1 + i)^n \cdot i}\right] \]In each period, interest is calculated on the previous outstanding balance, and the amortization is what remains of the installment after the interest is paid. Since the balance falls every period, the interest falls and the amortization grows, while the installment stays the same.To calculate the outstanding balance at a given instant, just calculate the present value of the installments still to be paid. Bringing them to present value removes the interest embedded in them, and what remains is exactly the balance still to be amortized, that is, the outstanding balance.ExamplesA company took out a loan of R$ 800,000.00 to buy a building. The loan must be repaid under the French system in five half-yearly installments at 4% per half-year. Build the schedule.The installment is:\[ R = \frac{800{,}000}{4.451822} = 179{,}701.69 \]Half-yearOutstanding balanceAmortizationInterestInstallment0800,000.00---1652,298.31147,701.6932,000.00179,701.692498,688.55153,609.7626,091.93179,701.693338,934.40159,754.1519,947.54179,701.694172,790.09166,144.3113,557.38179,701.695-172,790.096,911.60179,701.69Total-800,000.0098,508.45898,508.45In a loan of R$ 100,000.00 to be repaid under the French system over 40 months at 3% per month, what is the outstanding balance in the 25th month? (Assume this month's installment has been paid.)First, the installment:\[ 100{,}000 = R \left[\frac{(1 + 0.03)^{40} - 1}{(1 + 0.03)^{40} \cdot 0.03}\right] \]\[ 100{,}000 = R \cdot 23.114772 \]\[ R = \frac{100{,}000}{23.114772} = 4{,}326.24 \]After the 25th installment is paid, 15 remain. The outstanding balance is the present value of these 15 installments:\[ S_{25} = 4{,}326.24 \left[\frac{(1 + 0.03)^{15} - 1}{(1 + 0.03)^{15} \cdot 0.03}\right] = 51{,}646.37 \]A loan of R$ 10,000.00 will be repaid under the Price table in 4 monthly installments at 2% per month. The installment is R$ 2,626.24. How much is amortized in the 1st installment?Answer: R$ 2,426.24 — the 1st month's interest is 2% of 10,000 = 200, and the amortization is the installment minus the interest: 2,626.24 − 200 = 2,426.24. R$ 2,500.00 would be the SAC amortization (10,000 / 4), and R$ 200.00 is the interest.In a Price table loan, 25 of the 40 monthly installments of R$ 4,326.24, at 3% per month, have been paid. How is the outstanding balance calculated?Answer: Present value, at 3% per month, of the 15 remaining installments — bringing the installments to present value removes the interest embedded in them. 15 × 4,326.24 includes future interest that has not yet accrued, and equal amortizations belong to SAC, not to the Price table.American systemIn this system, the principal is repaid all at once at the end of the loan term. Interest is usually paid periodically; it can also be capitalized and paid all at once together with the principal (it all depends on the agreement between the parties).ExampleFor a loan of 800 thousand dollars, a client proposes to repay the principal two and a half years from now, paying only the interest every half-year at 4% per half-year. Build the schedule.Half-yearOutstanding balanceAmortizationInterestInstallment0800,000---1800,000-32,00032,0002800,000-32,00032,0003800,000-32,00032,0004800,000-32,00032,0005-800,00032,000832,000Total-800,000160,000960,000Comparing the three systemsFor the same loan of 800 thousand at 4% per half-year over five half-years, total interest was 96 thousand under SAC (previous lesson), 98.5 thousand under the French system and 160 thousand under the American system. This does not mean one system is always "more expensive": at the same rate, all of them have the same present value, equal to the principal. More interest is paid when the money stays borrowed for longer. That is why the choice depends on the situation: the American system suits those who will only have the money to repay at the end (and need low payments until then); the Price table, those who want equal, predictable installments; SAC, those who can afford larger installments at the beginning to reduce the debt faster.A loan of R$ 200,000.00 under the American system, for 3 years at 8% per year, with interest paid annually. What is the last payment and the total interest?Answer: R$ 216,000.00 and R$ 48,000.00 — annual interest is 8% of 200,000 = 16,000; in the last year the principal plus that interest is paid, and in total it is 3 × 16,000 = 48,000. R$ 251,942.40 is what would be paid if interest were capitalized and paid only at the end.For the same principal, rate and term, the American system generates more total interest than SAC. Why?Answer: Because the outstanding balance does not decrease over the term, and interest is charged on the full principal in every period — the rate is the same in both; what changes is how long the money stays borrowed. Equal installments belong to the Price table, not to the American system.Exercises1) A bank grants a company a credit of R$ 60,000.00, to be repaid under the Price system over 20 quarters at 6% per quarter. Build the schedule up to the third quarter.2) When buying an apartment, Mr. Almeida received a loan of R$ 280,000.00, to be repaid under the Price system over 240 months at 0.9% per month. What is the outstanding balance after the 51st installment is paid?3) Mr. Moura received financing of R$ 500,000.00 to buy a house, under the Price system at 1.5% per month, to be repaid over 180 months. What is the state of the debt after the 64th installment is paid? (ignore cents)4) The amount of R$ 1,500,000.00 is financed at 10% per year, to be amortized under the American system over three years: interest is paid annually and the principal is repaid all at once at the end of the third year. Build the schedule.5) A person took out a loan of R$ 120,000.00 at compound interest of 2% per month, to be repaid in 10 equal installments. How much interest is paid in the 8th installment?6) (BNDES) Choose the correct option. The French Amortization System is characterized by:a) Increasing interest and increasing installment.b) Decreasing installment and increasing interest.c) Decreasing amortization and interest.d) Increasing interest and amortization.e) Increasing amortization and constant installment.7) The amount of 3,600 UR (units of reference) can be financed under SAC or under the French system. Given an interest rate of 1% per month and 180 monthly installments, find:a) The installment under the French system.b) The first three installments under SAC.c) The month from which the SAC installment becomes lower than the French system installment.Answers1) Installment: R$ 5,231.07. Quarter 1: interest R$ 3,600.00, amortization R$ 1,631.07, balance R$ 58,368.93. Quarter 2: interest R$ 3,502.14, amortization R$ 1,728.94, balance R$ 56,639.99. Quarter 3: interest R$ 3,398.40, amortization R$ 1,832.67, balance R$ 54,807.31.2) R$ 258,624.973) R$ 441,361.004) Years 1 and 2: interest and installment of R$ 150,000.00, balance of R$ 1,500,000.00. Year 3: interest of R$ 150,000.00, amortization of R$ 1,500,000.00, installment of R$ 1,650,000.00. Total interest: R$ 450,000.00.5) R$ 770.536) E7) a) 43.21 UR; b) 56.00 UR; 55.80 UR; 55.60 UR; c) 65th month.
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