IntroductionOften, in medium- and long-term operations, for methodological or accounting reasons, loan operations are analyzed period by period, with regard to the payment of interest and the actual repayment of the borrowed capital, also called the principal.AmortizationLet’s begin by defining some elements for this study. First, let’s define the outstanding balance (or state of the debt) recursively.Consider the time instants 0, 1, 2, …, n, in the unit expressed by the interest rate (in everything that follows, we will assume the compound capitalization regime). Let P be the value of the principal (or initial capital borrowed). The outstanding balance at time zero (0), indicated by S0, is the principal P itself, and the outstanding balance at time t is equal to the outstanding balance at the previous time (t − 1), plus the interest produced by it, minus the payment made at time t.We will use the following notation:- St: outstanding balance at time t- St−1: outstanding balance at time (t − 1)- i: interest rate- Rt: payment made at time t- Jt: interest in the period from (t − 1) to tThus, symbolically, we will have:\[ S_t = S_{t-1} + J_t - R_t \]If the interest produced in each period is paid at the end of the period and if we call the difference between Rt and Jt amortization at time t (indicated by At), we will have:\[ A_t = R_t - J_t \]Amortization is how much is actually being paid toward the value of your loan: the installment amount minus the interest.\[ S_t = S_{t-1} - A_t \]In other words, the principal, at its present value, will be the sum of the amortizations:\[ VP = A_1 + A_2 + A_3 + ... + A_t \]Thus, there are countless sequences of amortizations whose sum is the principal.It should be noted that the term installment is used to represent the payment plus taxes and other charges. Disregarding these taxes and charges, the installment is reduced to the payment Rt, which is equal to the sum of the amortization and the interest in each period.Finally, we call a schedule a demonstrative table in which, at each time instant, the interest, amortization, outstanding balance, installment, taxes, and other charges appear.ExamplesA loan of R\$ 50,000.00 must be repaid in four semiannual installments at an interest rate of 5% per semester, with interest paid semiannually. Obtain the schedule, knowing that the amortizations are semiannual, with the following values:- A1 = 5,000- A3 = 10,000- A2 = 15,000- A4 = 20,000SemesterOutstanding 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 semiannual installments at an interest rate of 5% per semester, with interest paid semiannually. Obtain the schedule, knowing that the semiannual amortizations are equal.- A1 = A2 = A3 = A4 = 50,000/4 = 12,500SemesterOutstanding 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 semiannual installments at a rate of 5% per semester, with interest paid semiannually. Obtain the schedule, knowing that:- A1 = 0- A3 = 0- A2 = 0- A4 = 50,000SemesterOutstanding balanceAmortizationInterestInstallment050,000---150,00002,5002,500250,00002,5002,500350,00002,5002,5004-50,0002,50052,500Total-50,00010,00060,000Constant Amortization System (SAC)Among the countless ways that exist to amortize the principal, the constant amortization system (SAC) is one of the most widely used in practice. This system consists of making all amortization installments equal. Thus, considering a principal P to be amortized in n installments A1, A2, A3, …, At, and assuming payment of interest in all periods, we will have:\[ A_1 = A_2 = A_3 = A_4 = \frac{P}{t} = A_5 = ... = A_t \]The value of the installments is given by:\[ R_1 = A_1 + J_1 = A_1 + P*i \]\[ R_2 = A_2 + J_2 = A_2 + (P - A_1)*i \]\[ R_3 = A_3 + J_3 = A_3 + (P - A_2 - A_1)*i \]\[ R_4 = A_4 + J_4 = A_4 + (P - A_3 - A_2 - A_1) · i \]\[ R_t = A_4 + J_4 = A_{t-1} + (P - (t - 1) · A_t) · i \]Since A is constant:\[ R_t = A + (P - (t - 1) · A) · i \]ExamplesA loan of 800 thousand dollars must be repaid in five semiannual installments through SAC at a rate of 4% per semester. Obtain the schedule.SemesterOutstanding balanceAmortizationInterestInstallment0800---164016032192248016025.60185.60332016019.20179.20416016012.80172.805-1606.40166.4Total-80096896Exercises1) A loan of R$ 21,000.00 must be paid in six semiannual installments at a rate of 8% per semester. Obtain the schedule, knowing that the semiannual amortizations have the following values: 1,000, 2,000, 3,000, 4,000, 5,000, and 6,000.2) A bank grants a company a credit of R$ 120,000.00 to be repaid through SAC in six quarterly installments. Obtain the schedule, knowing that the interest rate is 5% per quarter.3) A loan of R$ 40,000.00 must be repaid through SAC in 40 monthly installments. Knowing that the interest rate is 2% per month, obtain the amortization, interest, installment, and outstanding balance corresponding to the 21st month.4) A loan of 250 thousand dollars must be repaid through SAC in 50 monthly installments, with an interest rate charged of 2% per month. Find: the value of the second installment and the value of the 37th installment.Answer Key3) 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.
Sign in and join the conversation
— Comments 0
, Reactions 1
Be the first to comment