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Washington Franco Mathias
        José Maria Gomes


Matemática
 Financeira
        Com + de 600 exercícios
          reso...
Capítulo 4


          EQUIVALÊNCIA
               DE
            CAPITAIS
Mathias
Gomes
Equivalência de
          Capitais                                       EXEMPLO


           • Data Focal: é a data que s...
Exemplo
          Certa pessoa tem um nota promissória a receber com valor no-
          minal de $ 15.000,00, que vencerá...
Exemplo
      Sejam:         x = quantia que possui na data zero.
                     y = quantia que possuirá na data 12...
Exemplo
          c) Daqui a 2 anos:
                 z = 20.000 (1,02)24 + 15.000
                 z = 32.168,74 + 15.000...
Exemplo
          Considere-se o exercício resolvido anteriormente. As expres-
          sões de primeiro grau em x, y e z...
Exemplo

                 Podemos concluir que, usando a taxa de juros compos-
          tos a que se referem as aplicaçõe...
Exemplo
                 Consideremos os valores nominais seguintes:

                     Capital ($)    Datas de vencime...
Exemplo
                        C1       1.100,00
              V1 =             =          = 1.000,00
                   ...
Exemplo
          Logo, podemos concluir que:

                 V1 = V2 = V3 = V4 = V5

          Como os capitais são equ...
Valor Atual de um Conjunto
          de Capitais              EXEMPLO


           Conjunto de Capitais: é uma carteira de...
Exemplo
                 Consideremos os valores nominais seguintes:

                   Capital ($)    Data de Vencimento...
Exemplo
                Temos, então:


                            1.000       2.000      5.000
                        V...
Conjuntos Equivalentes
          de Capitais                                                                       EXEMPLO...
Exemplo
                 Verificar se os conjuntos de valores nominais, referidos
          à data zero, são equivalentes ...
Exemplo
                  V 1 = 1.000,00 + 2.000,00 + 1.500,00 + 500,00
                  V 1 = 5.000,00
          b) 2º c...
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Matemática Financeira - Equivalência de Capitais

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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 - Equivalência de Capitais

  1. 1. Washington Franco Mathias José Maria Gomes Matemática Financeira Com + de 600 exercícios resolvidos e propostos 5ª Edição
  2. 2. Capítulo 4 EQUIVALÊNCIA DE CAPITAIS Mathias Gomes
  3. 3. Equivalência de Capitais EXEMPLO • Data Focal: é a data que se considera como base de comparação dos valores referidos a datas dife- rentes. • Equação de Valor: é a equação que torna possí- vel igualar capitais diferentes, referidos a datas di- ferentes, para uma mesma data focal, fixada uma certa taxa de juros. • Capitais equivalentes: dois ou mais capitais, com datas de vencimento determinadas, são equivalen- tes quando, levados para uma mesma data focal à mesma taxa de juros, tiverem valores iguais nesta data. Mathias Gomes
  4. 4. Exemplo Certa pessoa tem um nota promissória a receber com valor no- minal de $ 15.000,00, que vencerá em dois anos. Além disso, possui $ 20.000,00 hoje, que irá aplicar à taxa de 2% a.m., du- rante dois anos. Considerando que o custo de oportunidade do capital hoje, ou seja, a taxa de juros vigente no mercado, é de 2% a.m., pergunta-se: a) Quanto possui hoje ? b) Quanto possuirá daqui a um ano ? c) Quanto possuirá daqui a dois anos ? Resolução: Representamos o problema graficamente: Z Y 20.000 X (Meses) 0 12 24 Mathias Gomes
  5. 5. Exemplo Sejam: x = quantia que possui na data zero. y = quantia que possuirá na data 12 meses. z = quantia que possuirá na data 24 meses. Temos então: a) Hoje: x = 20.000 + 15.000/(1,02)24 = 20.000 + 9.325,82 Portanto: x = 29.325,82 b) Daqui a 1 ano: y = 20.000 (1,02)12 + 15.000/(1,02)12 y = 25.364,84 + 11.827,40 y = 37.192,24 Mathias Gomes
  6. 6. Exemplo c) Daqui a 2 anos: z = 20.000 (1,02)24 + 15.000 z = 32.168,74 + 15.000 z = 47.168,74 Assim, à taxa considerada, podemos dizer que a pessoa possui hoje $ 29.325,82. Ela possuirá $ 37.192,24 daqui a um ano e $ 47.168,74 daqui a 2 anos. Mathias Gomes
  7. 7. Exemplo Considere-se o exercício resolvido anteriormente. As expres- sões de primeiro grau em x, y e z são equações de valor. Assim, o valor y = 37.192,24, calculado na data focal 12, é composto de duas parcelas: $ 25.364,84, que é o mon- tante de $ 20.000,00 à taxa de juros compostos de 2% a.m. e $ 11.827,40, que é o valor atual ou valor presente de $ 15.000,00 à taxa de juros compostos de 2% a.m. O valor z = 47.168,74 foi obtido através de uma equa- ção de valor, com a qual passamos diretamente da data focal 0 para a data 24. Podemos pensar em uma nova equação de valor, em que usemos o valor y = 37.192,44 referido à data focal 12 e, daí, passemos à data focal 24. Nestas condições, temos: z´= 37.192,24 (1,02)12 z’ = 47.168,74 Mathias Gomes
  8. 8. Exemplo Podemos concluir que, usando a taxa de juros compos- tos a que se referem as aplicações de capital, as equações de valor em z e z’ dão resultados iguais. Logo, a solução deste problema de comparação de capitais no regime de juros com- postos não depende da data focal considerada. Mathias Gomes
  9. 9. Exemplo Consideremos os valores nominais seguintes: Capital ($) Datas de vencimento (anos) 1.100,00 1 1.210,00 2 1.331,00 3 1.464,10 4 1.610,51 5 Admitindo-se uma taxa de juros compostos de 10% a.a., verificar se os capitais são equivalentes na data focal zero. Resolução: Calculemos os valores atuais na data zero: Mathias Gomes
  10. 10. Exemplo C1 1.100,00 V1 = = = 1.000,00 (1 + i )1 (1,10)1 C2 1.210,00 V2 = = = 1.000,00 (1 + i ) 2 (1,10) 2 C3 1.331,00 V3 = = = 1.000,00 (1 + i ) 3 (1,10) 3 C4 1.464,10 V4 = = = 1.000,00 (1 + i ) 4 (1,10) 4 C5 1.610,51 V5 = = = 1.000,00 (1 + i ) 5 (1,10) 5 Mathias Gomes
  11. 11. Exemplo Logo, podemos concluir que: V1 = V2 = V3 = V4 = V5 Como os capitais são equivalentes a esta taxa de juros, isto quer dizer que o possuidor de dois ou mais destes capitais, ficará indiferente quanto aos valores nominais. Em outras pa- lavras, a pessoa fica indiferente a possuir $ 1.100,00 em 1 a- no ou $ 1.464,10 daqui a 4 anos, desde que a taxa de juros seja de 10% a.a. Mathias Gomes
  12. 12. Valor Atual de um Conjunto de Capitais EXEMPLO Conjunto de Capitais: é uma carteira de aplicações. Pode ser caracterizada pelo valor nominal do título e por sua data de vencimento. CAPITAL DATA DE VENCIMENTO C1 1 C2 2 Cn n Valor da carteira na data “zero” à taxa de juros “i”: c1 c2 cn V= + + ...+ (1+ i) (1+ i) 1 2 (1+ i)n Mathias Gomes
  13. 13. Exemplo Consideremos os valores nominais seguintes: Capital ($) Data de Vencimento (Mês) 1.000,00 6 2.000,00 12 5.000,00 15 Admitindo-se taxa de juros de 3% a.m., pergunta-se qual o valor atual deste conjunto na data focal zero. Resolução: A situação é a seguinte: V 5.000 2.000 1.000 (Meses) 0 6 12 15 Mathias Gomes
  14. 14. Exemplo Temos, então: 1.000 2.000 5.000 V = 6 + 12 + (1,03) (1,03) (1,03)15 V = 837,48 + 1.402,76 + 3.209,31 V = 5.449,55 Podemos concluir que $ 5.449,55 é o valor da carteira na data zero, à taxa de 3% a.m. Ou seja, se a pessoa vender a carteira hoje (data zero) por $ 5.449,55, o comprador esta- rá ganhando uma taxa de 3% a.m. Mathias Gomes
  15. 15. Conjuntos Equivalentes de Capitais EXEMPLO Dados dois conjuntos de valores nominais com seus prazos contados a partir da mesma data de ori- gem: 1º Conjunto 2º Conjunto Capital Data de Capital Data de vencimento vencimento C1 m1 C'1 m'1 C2 m2 C'2 m'2 ... ... ... ... Cn mn C'n m'n Dada uma taxa de juros “i”, os conjuntos são equivalentes em uma data focal se os seus valores a- tuais forem iguais: c1 c2 cn c '1 c' n + + ... + = + ... + (1 + i ) m 1 (1 + i ) m 2 (1 + i ) m n (1 + i ) m '1 (1 + i ) m '1 Mathias Gomes
  16. 16. Exemplo Verificar se os conjuntos de valores nominais, referidos à data zero, são equivalentes à taxa de juros de 10% a.a. 1º Conjunto 2º Conjunto Capital Data de Capital Data de ($) vencimento ($) vencimento 1.100,00 1º ano 2.200,00 1º ano 2.420,00 2º ano 1.210,00 2º ano 1.996,50 3º ano 665,5 3º ano 732,05 4º ano 2.196,15 4º ano Resolução: Para fazer a comparação dos valores atuais à taxa dada, escolhemos a data focal zero: a) 1º conjunto de capitais: 1.100 2.420 1.996,50 732,05 V1 = 1 + 2 + 3 + (1,1) (1,1) (1,1) (1,1) 4 Mathias Gomes
  17. 17. Exemplo V 1 = 1.000,00 + 2.000,00 + 1.500,00 + 500,00 V 1 = 5.000,00 b) 2º conjunto de capitais: 2.200 1.210 665,50 2196,15 V2 = 1 + 2 + 3 + (1,1) (1,1) (1,1) (1,1) 4 V 2 = 2.000,00 + 1.000,00 + 500,00 + 1.500,00 V 2 = 5.000,00 Como V1 = V2, concluímos que, à taxa de 10% a.a., os dois conjuntos são equivalentes. Isto quer dizer que uma pessoa ficará indiferente entre possuir uma carteira de títulos igual ao 1º ou ao 2º conjunto, desde que a taxa de juros vigen- tes no mercado seja 10% a.a. Mathias Gomes

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