IntroductionIn medium- and long-term operations, for methodological or accounting reasons, loans are usually analyzed period by period: how much was paid in interest and how much of the borrowed capital, also called the principal, was paid back.Amortization and outstanding balanceLet us start by defining the elements of this study. The first is the outstanding balance (or state of the debt), which we define recursively.Consider the instants 0, 1, 2, …, n, in the unit of time of the interest rate (throughout, the regime is compound interest). Let P be the principal, that is, the amount borrowed. The outstanding balance at instant 0, denoted S0, is the principal P itself. The outstanding balance at instant t is the balance at the previous instant (t − 1), plus the interest it produced, minus the payment made at instant t.Notation:St: outstanding balance at instant tSt−1: outstanding balance at instant t − 1i: interest rateRt: payment made at instant tJt: interest for the period from t − 1 to t, that is, Jt = St−1 · iSymbolically:\[ S_t = S_{t-1} + J_t - R_t \]If the interest of each period is paid at the end of the period, we call amortization at instant t (denoted At) the difference between the payment and the interest:\[ A_t = R_t - J_t \]Amortization is how much of the loan you actually paid back in that period: the amount paid minus the interest. Substituting into the first equation:\[ S_t = S_{t-1} - A_t \]In other words, in each period the outstanding balance falls by exactly the amount amortized. Since the debt ends at zero, the sum of all amortizations equals the principal:\[ P = A_1 + A_2 + A_3 + ... + A_n \]There are, therefore, countless possible sequences of amortizations: any one whose sum is the principal. Each amortization system is a rule for choosing that sequence.In practice, the word installment is often used for the payment plus taxes and other charges (insurance, fees). Leaving those charges aside, the installment reduces to the payment Rt, which is the sum of the amortization and the interest of the period.Finally, we call a schedule (amortization table) the table that shows, at each instant, the interest, the amortization, the outstanding balance and the installment (and, when applicable, taxes and charges).ExamplesA loan of R\$ 50,000.00 must be repaid in four half-yearly installments at an interest rate of 5% per half-year, with interest paid every half-year. Build the schedule, given that the half-yearly amortizations are:A1 = 5,000A2 = 10,000A3 = 15,000A4 = 20,000The interest of each half-year is 5% of the outstanding balance of the previous half-year; the installment is the amortization plus the interest.Half-yearOutstanding balanceAmortizationInterestInstallment050,000---145,0005,0002,5007,500235,00010,0002,25012,250320,00015,0001,75016,7504-20,0001,00021,000Total-50,0007,50057,500A loan of R\$ 50,000.00 must be repaid in four half-yearly installments at an interest rate of 5% per half-year, with interest paid every half-year. Build the schedule, given that the half-yearly amortizations are equal.A1 = A2 = A3 = A4 = 50,000 / 4 = 12,500Half-yearOutstanding balanceAmortizationInterestInstallment050,000---137,50012,5002,50015,000225,00012,5001,87514,375312,50012,5001,25013,7504-12,50062513,125Total-50,0006,25056,250A loan of R\$ 50,000.00 must be repaid in four half-yearly installments at 5% per half-year, with interest paid every half-year. Build the schedule, given that:A1 = 0A2 = 0A3 = 0A4 = 50,000Half-yearOutstanding balanceAmortizationInterestInstallment050,000---150,00002,5002,500250,00002,5002,500350,00002,5002,5004-50,0002,50052,500Total-50,00010,00060,000Compare the three examples: the principal and the rate are the same, but total interest ranges from R$ 6,250 to R$ 10,000. The more slowly the loan is amortized, the longer the outstanding balance stays high and the more interest it generates.A loan of R$ 10,000.00 charges 3% per month. In the first month, the borrower amortizes R$ 4,000.00. How much is paid that month, and what is the outstanding balance after the payment?Answer: Pays R$ 4,300.00; balance of R$ 6,000.00 — the month's interest is 3% of 10,000 = 300, and the payment is amortization plus interest. The balance falls only by the amount amortized, 10,000 − 4,000 = 6,000; the interest paid does not reduce the debt.Constant amortization system (SAC)Among the countless ways of amortizing the principal, the constant amortization system — known in Brazil as SAC (Sistema de Amortização Constante) — is one of the most used in practice; in Brazil it is common in home financing. In it, all amortizations are equal. Considering a principal P to be amortized in n installments, with interest paid every period:\[ A_1 = A_2 = ... = A_n = A = \frac{P}{n} \]The installments are:\[ R_1 = A + J_1 = A + P \cdot i \]\[ R_2 = A + J_2 = A + (P - A) \cdot i \]\[ R_3 = A + J_3 = A + (P - 2A) \cdot i \]and, in general, since t − 1 amortizations have already been paid before the t-th payment:\[ R_t = A + J_t = A + (P - (t - 1) \cdot A) \cdot i \]Note that the outstanding balance falls by A every period, so the interest falls by A · i every period, and so do the installments: in SAC they decrease, forming an arithmetic progression.ExampleA loan of 800 thousand dollars must be repaid in five half-yearly installments under SAC at 4% per half-year. Build the schedule.The amortization is 800 / 5 = 160 thousand dollars per half-year. Values in the table are in thousands of dollars:Half-yearOutstanding balanceAmortizationInterestInstallment0800---164016032192248016025.60185.60332016019.20179.20416016012.80172.805-1606.40166.40Total-80096896A loan of R$ 60,000.00 will be repaid under SAC in 12 monthly installments at 2% per month. What is the 5th installment?Answer: R$ 5,800.00 — the amortization is 60,000 / 12 = 5,000; before the 5th installment, 4 amortizations have been paid, so the balance is 40,000 and the interest is 800. R$ 6,200.00 is the 1st installment, and R$ 5,900.00 comes from subtracting only 3 amortizations.In the same loan (R$ 60,000.00, SAC, 12 months, 2% per month), by how much does the installment fall from one month to the next?Answer: R$ 100.00 — each month the balance falls by one amortization (5,000), and the interest falls by 2% of that, 100. A fixed installment is a feature of the French system (Price table), not of SAC.Exercises1) A loan of R$ 21,000.00 must be repaid in six half-yearly installments at 8% per half-year. Build the schedule, given that the half-yearly amortizations are: 1,000, 2,000, 3,000, 4,000, 5,000 and 6,000.2) A bank lends a company R$ 120,000.00 to be repaid under SAC in six quarterly installments. Build the schedule, given that the interest rate is 5% per quarter.3) A loan of R$ 40,000.00 must be repaid under SAC in 40 monthly installments. Given that the interest rate is 2% per month, find the amortization, the interest, the installment and the outstanding balance for the 21st month.4) A loan of 250 thousand dollars must be repaid under SAC in 50 monthly installments, at an interest rate of 2% per month. Find the value of the 2nd installment and of the 37th installment.Answers1) Installments: R$ 2,680.00; R$ 3,600.00; R$ 4,440.00; R$ 5,200.00; R$ 5,880.00; R$ 6,480.00. Total interest: R$ 7,280.00.2) Amortization of R$ 20,000.00 per quarter; interest of R$ 6,000.00, 5,000.00, 4,000.00, 3,000.00, 2,000.00 and 1,000.00; installments of R$ 26,000.00, 25,000.00, 24,000.00, 23,000.00, 22,000.00 and 21,000.00.3) Amortization: R$ 1,000.00; Interest: R$ 400.00; Installment: R$ 1,400.00; Outstanding balance: R$ 19,000.00.4) 9,900 dollars and 6,400 dollars.
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