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Matemática Financeira - Modelos Genéricos de Anuidades

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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 - Modelos Genéricos de Anuidades

  1. 1. Washington Franco Mathias José Maria Gomes Matemática Financeira Com + de 600 exercícios resolvidos e propostos 5ª Edição
  2. 2. Capítulo 6 MODELOS GENÉRICOS DE ANUIDADES Mathias Gomes
  3. 3. Modelos Genéricos de Anuidades EXEMPLO Anuidades Diferidas: são aquelas em que os ter- mos são exigíveis pelo menos a partir do segundo período. Carência: é o intervalo de tempo do diferimento. Calcula-se o Valor Atual pelo modelo básico mais um cálculo de equivalência para a data focal zero. O cálculo do Montante é feito pelo Modelo Básico. Mathias Gomes
  4. 4. Exemplo Uma pessoa vai receber dezesseis prestações mensais iguais a $ 400,00, com um diferimento de 15 meses: 400 400 400 400 (Meses) 0 1 2 ... 15 16 17 ... 30 31 ... 39 40 Sendo a taxa de juros igual a 2% a.m., pergunta-se: a) Qual o valor atual das prestações na data zero ? b) Qual o montante na data focal 40 ? Mathias Gomes
  5. 5. Exemplo Resolução: a) Valor do principal na data focal zero: P0 P15 400 400 400 400 (Meses) 0 1 2 ... 15 16 17 ... 30 31 ... 39 40 Procedemos de duas etapas: I) Calculamos o principal na data focal 15, segundo o modelo básico: Mathias Gomes
  6. 6. Exemplo P15 = R.a¬ 2 16 P15 = 400,00 x13,577709 P15 = $5.431,08 II) A seguir, achamos o valor atual na data focal zero e à taxa de 2% a.m.: P15 P0 = (1,02)15 5.431,08 P0 = ≅ $4.035,38 1,345868 Mathias Gomes
  7. 7. Exemplo b) Montante na data focal 40: O montante na data focal 40 pode ser obtido diretamente do valor atual (P0): P 40 = P 0(1 + i ) 40 P 40 = 4.035,38.(1,02) 40 P 40 = 4.035,38 x 2,208040 P 40 = $8.910,28 Mathias Gomes
  8. 8. Modelo Genéricos de Anuidades EXEMPLO Anuidade em que o período dos termos não coincide com aquele a que se refere a taxa. • Supondo os termos constantes e periódicos; • Calcula-se a taxa equivalente ao período dos termos e recai-se no modelo básico. Mathias Gomes
  9. 9. Exemplo Um aparelho de som estereofônico é vendido em 5 prestações de $ 2.000,00 a serem pagas a cada 2 meses. Sendo a taxa de juros cobrada de 3% a.m., qual o valor do aparelho à vista ? Se Se o mesmo aparelho pudesse ser pago em uma única vez após 10 meses, qual a quantia que a loja cobraria, admitida a mesma taxa de juros ? Resolução: Graficamente, a situação é a seguinte: P0 2.000 2.000 2.000 2.000 2.000 (Meses) 0 1 2 3 4 5 6 7 8 9 10 Mathias Gomes
  10. 10. Exemplo Como a taxa de juros está referida em termos mensais e as prestações estão referidas a bimestres, calculamos a taxa bi- mestral equivalente: 1 + i’ = (1 + 0,03)2 i’ = 1,0609 -1 i’ = 0,0609 a.b. ou i’ = 6,09% a.b. Agora temos uma anuidade nas mesmas condições do modelo básico, pois a taxa se refere ao mesmo intervalo de tempo dos termos e podemos calcular: a) Preço à vista: P 0 = R.a¬i n Mathias Gomes
  11. 11. Exemplo onde: R = 2.000,00 por bimestre i = i’ = 6,09% a .b. n = 5 bimestres Portanto: (1, 0609 ) 5 − 1 a ¬6,09 5 = (1, 0609 ) 5 . 0 , 0609 a ¬6,09 4 , 202070 5 = E o valor atual: P 0 = 2 . 000 , 00 x 4 , 202070 P 0 = $ 8 . 404 ,14 Concluímos que o preço do aparelho à vista é $ 8.404,14. Mathias Gomes
  12. 12. Exemplo b) Preço após 10 meses: O montante na data focal 10 pode ser obtido por capitalização direta: P10 = P0 (1 + i’)5 P10 = 8.404,14 (1,0609)5 P10 = 8.404,14 x 1,343916 P10 = $ 11.294,46 Mathias Gomes
  13. 13. Modelo Genéricos de Anuidades EXEMPLO Anuidade com termos constantes, segundo o modelo básico, mais parcelas intermediárias iguais. • Uniformiza-se a anuidade, de modo que todos os termos sejam iguais entre si e com taxas de juros referida ao período dos termos. Esta é uma anuidade tipo modelo básico. • Por diferença calcula-se o valor das parcelas in- termediárias. Recai-se no modelo básico. Mathias Gomes
  14. 14. Exemplo Um carro é vendido em oito prestações mensais. As prestações de ordem ímpar são iguais a $ 1.000,00, enquanto que as de or- dem par são iguais a $ 2.000,00. Considerando-se a taxa de ju- ros de 2% a.m., qual é o preço à vista ? Resolução: Graficamente, tem-se: 2.000 2.000 2.000 2.000 1.000 1.000 1.000 1.000 (Meses) 0 1 2 3 4 5 6 7 8 Mathias Gomes
  15. 15. Exemplo a) Uniformizando a anuidade de modo a se ter 8 termos iguais a $ 1.000,00: P0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 (Meses) 0 1 2 3 4 5 6 7 8 O valor atual (P0’) dessa anuidade na data focal zero, é: Mathias Gomes
  16. 16. Exemplo P 0 ' = R.a¬2 8 P 0 ' = 1.000 x 7,325481 P 0 ' = $7.325,48 b) Considerando apenas a diferença entre a anuidade original e a anuidade uniformizada (item a), obtemos a anuidade consti- tuída pelas parcelas intermediárias: P0 1.000 1.000 1.000 1.000 (Meses) 0 1 2 3 4 5 6 7 8 Mathias Gomes
  17. 17. Exemplo Calculando-se a taxa equivalente bimestral, tem-se: 1 + i ' = (1,02 ) 2 i ' = 0,0404 a.b. = 4,04 % a.b. Podemos calcular agora: (1, 0404 ) 4 − 1 a ¬4,04 4 = 4 ≅ 3 , 626476 (1, 0404 ) . 0 , 0404 P 0 ' ' = 1 . 000 , 00 x 3 , 626476 P 0 ' ' = $ 3 . 626 , 48 Mathias Gomes
  18. 18. Exemplo Agora podemos determinar o preço do carro à vista: P 0 = P 0 '+ P 0 ' ' P 0 = 7.325,48 + 3.626 ,48 P 0 = $10 .951,96 Logo, o preço do carro à vista é $ 10.951,96. Mathias Gomes
  19. 19. Modelo Genéricos de Anuidades EXEMPLO Anuidade compostas por duas anuidades diferidas em seqüência. Calcula-se o Valor Atual em datas focais intermediárias e, em seguida, o valor atual na data focal zero. O montante pode ser obtido a partir do valor atual, fazendo-se a capitalização até a da- ta focal requerida. Mathias Gomes
  20. 20. Exemplo Uma pessoa comprou um gravador, para pagar em 7 presta- ções, do seguinte modo: - 3 prestações de $ 100,00 no 7°, 8° e 9° mês: - 4 prestações de $ 100,00 no 13°, 14°, 15° e 16° mês. A taxa de juros cobrada foi de 2% a.m. Pergunta-se o valor do gravador à vista. Resolução: O cálculo é o seguinte: P 6 = 100,00.a¬ 2 3 P 6 = 100,00.2,883883 P 6 = $288,39 Mathias Gomes
  21. 21. Exemplo E, para a segunda anuidade: P12 = 100 ,00 .a¬ 2 4 P12 = 100 ,00 x3,807729 P12 = $380 ,77 Portanto: P6 P12 P0 = + (1 + i ) 6 (1 + i )12 P 0 = 256 ,08 + 300 ,23 P 0 = $556 ,31 Portanto, o preço do gravador à vista é $ 556,31. Mathias Gomes
  22. 22. Modelo Genéricos de Anuidades EXEMPLO Anuidades Perpétuas • São as anuidades com duração ilimitada. • Só tem sentido calcular o valor atual, porque o montante será infinito. • O valor atual é dado por: R P= i P = valor atual R = termo constante i = taxa de juros Mathias Gomes
  23. 23. Exemplo Se um apartamento está rendendo um aluguel de $ 500,00 por mês e se a taxa da melhor aplicação no mercado financeiro é de 1% a.m., qual seria uma primeira estimativa do valor do i- móvel. Resolução: Admitindo-se as hipóteses de duração ilimitada do apartamento e de ser o aluguel constante, tem-se: R P= i 500,00 P= = $50.000,00 0,01 Ou seja, numa primeira aproximação, o imóvel seria avaliado em $ 50.000,00. Mathias Gomes
  24. 24. Modelo Genéricos de Anuidades EXEMPLO Anuidades Variáveis: são anuidades cujos ter- mos não são iguais entre si. Dada uma anuidade: - Temporária; - Imediata e postecipada; - Periódica. O Valor Atual é a soma dos valores atuais de cada um de seus termos. O Montante pode ser obtido pela soma do montante de cada um dos seus termos. Mathias Gomes
  25. 25. Exemplo Um terreno foi comprado para ser pago em 5 prestações tri- mestrais, com os seguintes valores: 1° trimestre: 20.000,00 2° trimestre: 5.000,00 3° trimestre: 10.000,00 4° trimestre: 3.000,00 5° trimestre: 30.000,00 Sendo a taxa de juros para aplicações financeiras vigente no mercado de 2,5% a.m., pergunta-se o valor do terreno à vis- ta. Resolução: Como a taxa de juros está referida ao mês e as prestações são trimestrais, calculamos a taxa trimestral equi- valente: Mathias Gomes
  26. 26. Exemplo 1 + i ' = (1 + i ) 3 1 + i ' = (1,025 ) 3 i ' = 0,07689 a.t. = 7,689 % a.t . Calculando-se o valor atual de cada prestação à taxa de juros i’, tem-se: 20.000 5.000 10.000 3.000 30.000 P0 = + + + + (1,07689) (1,07689) (1,07689) (1,07689) (1,07689)5 1 2 3 4 P0 = 18.572,00 + 4.311,49 + 8.007,30 + 2.230,67 + 20.714,03 P0 = $53.835,49 Portanto, o preço do terreno à vista é de, aproximadamente, $ 54.000,00 Mathias Gomes

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