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Matemática Financeira - Juros Compostos

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Material de Apoio do livro Matemática Financeira, dos autores Washington Franco Matias e José Maria Gomes, da Editora Atlas.

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Matemática Financeira - Juros Compostos

  1. 1. Washington Franco Mathias José Maria Gomes Matemática Financeira Com + de 600 exercícios resolvidos e propostos 5ª Edição
  2. 2. Capítulo 3 JUROS COMPOSTOS Mathias Gomes
  3. 3. Juros Compostos Juros Simples: • Apenas o capital inicial rende juros; • O Juro é diretamente proporcional ao tempo e à taxa. Juros Compostos: • O Juro gerado pela aplicação, em um período, será incorporado; • No período seguinte, o capital mais o juro passa a ge- rar novos juros; • O regime de juros compostos é mais importante, por- que retrata melhor a realidade. Mathias Gomes
  4. 4. Diferença entre os regimes de capitalização Co= 1000,00 i= 20 % a.a. n= 4 anos n Juros Simples Juros Compostos Juro por Período Montante Juro por período Montante 1 1000 x 0,2 = 200 1200 1000 x 0,2 = 200 1200 2 1000 x 0,2 = 200 1400 1200 x 0,2 = 240 1440 3 1000 x 0,2 = 200 1600 1440 x 0,2 = 288 1728 4 1000 x 0,2 = 200 1800 1728 x 0,2 = 346 2074 Mathias Gomes
  5. 5. Montante EXEMPLO O cálculo do montante, em juros compostos é dado pela fórmula: C n = C o (1 + i ) n Cn = montante ao fim de “n” períodos Co = capital inicial n = número de períodos i = taxa de juros por período Mathias Gomes
  6. 6. Exemplo Uma pessoa toma $ 1.000,00 emprestado a juros de 2% a.m. pelo prazo de 10 meses com capitalização composta. Qual o montante a ser devolvido ? Resolução: C0 = 1.000 i = 2% a .m. n = 10 meses Temos: C n = C 0 (1 + i ) n C 10 = C 0 (1 + i )10 C 10 = 1.000 (1 + 0,02 )10 C 10 = 1.000 (1,02 )10 Mathias ∴ C 10 = $1.218,99 Gomes
  7. 7. Cálculo de Juro EXEMPLO O juro é dado pela fórmula seguinte: Jn =C.[( +i) −1 o 1 ] n Jn = juros após “n” períodos Co = capital inicial n = número de períodos i = taxa de juros por período Mathias Gomes
  8. 8. Exemplo Qual o juro pago no caso do empréstimo de $ 1.000,00 à taxa de juros compostos de 2% a.m. e pelo prazo de 10 meses ? Resolução: C0 = 1.000 i = 2% a .m. n = 10 meses Temos: Jn = [C 0 (1 + i ) n − 1] J 10 = 1.000[(1 + 0,02 )10 − 1] J 10 = 1.000[(1,02 )10 − 1] J 10 = 1.000[0,21899 ] Mathias ∴ J 10 = $218,99 Gomes
  9. 9. Valor Atual e Valor Nominal EXEMPLO • O Valor Atual corresponde ao valor da aplicação em uma data inferior à do vencimento. • O Valor Nominal é o valor do título na data do seu vencimento. N V= (1 + i ) n V = valor atual N = valor nominal i = taxa de juros n = número de períodos que antecede o vencimento do título Mathias Gomes
  10. 10. Exemplo a) Por quanto devo comprar um título, vencível daqui a 5 me- ses, com valor nominal de $ 1.131,40, se a taxa de juros com- postos corrente for de 2,5% a.m. ? Resolução: N=1.131,40 V n = 5 Meses Mathias Gomes
  11. 11. Exemplo N = 1.131,40 i = 2,5 % a.m. n = 5 meses N V = (1 + i ) n 1.131,40 1.131,40 V = 5 ≅ (1,025) 1,131408 V ≅ $1.000,00 Portanto, se comprar o título por $ 1.000,00, não esta- rei fazendo mau negócio. Mathias Gomes
  12. 12. Exemplo b) Uma pessoa possui uma letra de câmbio que vence daqui a 1 ano, com valor nominal de $ 1.344,89. Foi-lhe proposta a tro- ca daquele título por outro, vencível daqui a 3 meses e no valor de $ 1.080,00. Sabendo-se que a taxa corrente de mercado é de 2,5% a.m., pergunta-se se a troca proposta é vantajosa. Resolução: N=1.344,89 N*=1.080,00 0 3 12 Mathias Gomes
  13. 13. Exemplo O valor atual na data focal zero da letra de câmbio que vence em 12 meses é dado por: N 1344, 89 V1 = = (1 + i )12 (1, 025)12 1.344, 89 V1 = ≅ 1.000, 00 1, 344889 ∴ V 1 = $1.000, 00 Calculemos agora o valor atual na data zero, da letra que vence em 3 meses: N* 1080, 00 V2 = 3 = (1 + i ) (1, 025)3 Mathias Gomes
  14. 14. Exemplo 1.080,00 V2 = 1,076891 ∴ V 2 = $1.002,89 Comparando os dois valores atuais constatamos que: V 2 > V1 Ou seja, o título que vence em 3 meses tem um valor atual um pouco maior que o que vence em 12 meses. Portanto, a troca seria vantajosa. Mathias Gomes
  15. 15. Taxas Equivalentes EXEMPLO Duas taxas de juros são equivalentes se, consi- derados o mesmo prazo de aplicação e o mesmo capital, for indiferente aplicar em uma ou em ou- tra. iq = 1 + i − 1 q onde: iq = taxa referente a uma fração 1/q a que se refere a taxa “i”. i = taxa referente a um intervalo de tempo unitário Mathias Gomes
  16. 16. Exemplo a) Dada a taxa de juros de 9,2727% ao trimestre, determinar a taxa de juros compostos equivalente mensal. Resolução: iq = q 1 + i − 1 Sendo que: q = 3 meses i = 9,2727% a.t. Portanto: i 3 = 3 1 + 0,092727 − 1 i 3 = 3 1,092727 − 1 i 3 = 1,03 − 1 ∴ i 3 = 0,03a.m. ou i 3 = 3% a.m. Mathias Gomes
  17. 17. Exemplo b) Suponhamos que C0 = 1.000,00; iq = 2% a.m.; i = 26,824% a.a. e n = 1 ano. Verificar se i e iq são equivalentes. Resolução: Para verificar se as duas taxas são equivalentes, vamos aplicar o capital de $ 1.000,00 pelo mesmo prazo. Va- mos adotar 1 ano, que é o período de aplicação corresponden- te à taxa i. O montante à taxa i, é: C1 = 1.000(1,26824) C1 = $ 1.268,24 Calculando-se o montante em 12 meses para a taxa iq, tem-se: C1’ = 1.000(1,02)12 C1’ = 1.000(1,268242) Logo: C1’ = $ 1.268,24 Mathias Gomes
  18. 18. Exemplo Portanto, como C1 = C1’, podemos concluir que a taxa de 2% a.m. é equivalente à taxa de 26,824% ao ano. Note-se que esta taxa é maior que a taxa equivalente obtida a juros simples (ou seja: 2% x 12 meses = 24% ao ano). c) Se um capital de $ 1.000,00 puder ser aplicado às taxas de juros compostos de 10% ao ano ou de 33,1% ao triênio, deter- minar a melhor aplicação. Resolução: Para determinar qual a melhor aplicação, vamos a- plicar o capital disponível às duas taxas e por um mesmo prazo. Façamos a aplicação por 3 anos, que é o período da segunda ta- xa. Mathias Gomes
  19. 19. Exemplo Aplicando à taxa de 10% a.a. C3 = 1.000(1 + 0,10)3 C3 = 1.000(1,331) C3 = $ 1.331,00 Aplicando à taxa de 33,1% ao triênio, por um triênio: C1 = 1.000(1 + 0,331)1 C1 = 1.000(1,331) C1 = $ 1.331,00 É portanto, indiferente aplicar-se a qualquer das taxas; ou seja, as taxas são equivalentes. Mathias Gomes
  20. 20. Períodos Não-Inteiros Convenção Exponencial EXEMPLO Nesta convenção, os juros do período não- inteiro são calculados utilizando-se a taxa equiva- lente. n+ p / q Cn , p / q = Co(1 + i ) Co = Capital inicial n = número de períodos inteiros i = taxa de juros p/q = fração própria (p<q) de um período a que se refere a ta- xa “i” Cn,p/q = montante ao fim de (n+p/q) períodos Mathias Gomes
  21. 21. Exemplo Um capital de $ 1.000,00 é emprestado à taxa de juros com- postos de 10% a.a., pelo prazo de 5 anos e 6 meses. Tendo por base a capitalização anual, qual será o montante ? Resolução: a) por etapas: 1ª etapa: calculamos o montante para os períodos inteiros: C5 = C0(1 + i)5 C5 = 1.000(1,10)5 C5 = 1.000(1,61051) C5 = $ 1.610,51 2ª etapa: como a taxa está em base anual (12 meses), te- mos: p = 6 meses p 1 q = 12 meses } ∴ = q 2 Mathias Gomes
  22. 22. Exemplo Portanto: C’n,p/q = Cn(1 + i)p/q C’5,1/2 = C5(1,10)1/2 C’5,1/2 = 1.610,51(1,048809) C’5,1/2 = $ 1.689,12 b) usando a fórmula: C’n,p/q = C0(1 + i)n+p/q C’5,1/2 = 1.000(1,10)5+1/2 C’5,1/2 = 1.000(1,10)5,5 C’5,1/2 = $ 1.689,12 Mathias Gomes
  23. 23. Taxa Efetiva e Nominal Diz-se que a taxa é nominal quando o pe- ríodo de capitalização não coincide com o período da taxa. i kn C nk = C o (1 + ) k e i k i f = (1 + ) − 1 k Mathias Gomes
  24. 24. Taxa Efetiva e Nominal EXEMPLO i = taxa nominal if = taxa efetiva k = número de capitalizações para 1 período da taxa efetiva n = número de períodos de capitalização da taxa no- minal C0 = Principal Cnk = Montante Mathias Gomes
  25. 25. Exemplo 1) Um banco faz empréstimos à taxa de 5% a.a., mas adotando a capitalização semestral dos juros. Qual seria o juro pago por um empréstimo de $ 10.000,00, feito por 1 ano ? Resolução: Adotando-se a convenção de que a taxa por perío- do de capitalização seja a taxa proporcional simples à taxa no- minal dada, tem-se: i 5 i = 5% a.a. i ' = = = 2,5%a.s. k 2 Onde k corresponde ao prazo de formação de juros, ou seja, é o número de vezes em que foi dividido o período corres- pondente à taxa dada. Nestas condições, o montante no primeiro semestre é dado por: Mathias Gomes
  26. 26. Exemplo C1 = C0 (1+i/k)1 C1 = 10.000 (1 + 0,025)1 = $ 10.250,00 E, no segundo semestre, tem-se: C2 = 10.250(1 + 0,025)1=$ 10.506,25 O montante que seria devido caso a capitalização fosse anual é dado por: C’ = C0(1 + i)1 C’ = 10.000(1 + 0,05) = $ 10.500,00 Constatamos que existe uma pequena diferença para mais no montante, quando o prazo de capitalização não coincide com o prazo da taxa. Mathias Gomes
  27. 27. Exemplo A taxa efetiva nesta operação, em que temos duas capitalizações, é dada por: if = 506,25/10.000,00 = 0,050625 a.a. ou if = 5,0625% a.a. E a taxa efetiva quando a capitalização é feita no período da taxa é: i’f = 500,00/10.000,00 = 0,05 a.a. ou i’f = 5% a.a. Mathias Gomes
  28. 28. Exemplo Observe-se que podemos obter o resultado diretamente, a- plicando os $ 10.000,00 em dois semestres: C2 = 10.000 (1,025)2 = 10.506,25 A taxa efetiva é dada por: 1 + if = (1,025)2 = 1,050625 if = 5,0625% a.a. Mathias Gomes
  29. 29. Exemplo 2) Um capital de $ 1.000,00 foi aplicado por 3 anos, à taxa de 10% a.a. com capitalização semestral. Calcular o montante e a taxa efetiva da operação. Resolução: i = 10% a.a. K=2 n = 3 anos Portanto: Cnk = C0 (1 + i/k)kn C6 = 1.000 (1 + 0,10/2)2.3 C6 = 1.000 (1 + 0,05)6 C6 = $ 1.340,10 A taxa efetiva é dada por: if = (1 + i/k)2 - 1 if = (1 + 0,05)2 - 1 if = 10,25% a.a. Mathias Gomes
  30. 30. Exemplo 3) Sabendo-se que uma taxa nominal de 12% a.a. é capitalizada trimestralmente, calcular a taxa efetiva. Resolução: Como em 1 ano existem 4 trimestres, temos k=4. Então: if = (1+i/k)k - 1 if = (1+0,12/4)4 - 1 if = (1,03)4 - 1 if =1,12551 - 1 if =0,12551 a.a. ou if = 12,551% a.a. Mathias Gomes

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