{"id":3077,"date":"2015-03-24T19:04:44","date_gmt":"2015-03-24T22:04:44","guid":{"rendered":"https:\/\/www.antonioguilherme.web.br.com\/blog\/?page_id=3077"},"modified":"2023-05-06T14:45:22","modified_gmt":"2023-05-06T17:45:22","slug":"matematica-financeira","status":"publish","type":"page","link":"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/","title":{"rendered":"Matem\u00e1tica Financeira"},"content":{"rendered":"<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\"><strong><em><span style=\"font-family: verdana, geneva, sans-serif;\">&#8220;Finan\u00e7as consiste na arte de passar dinheiro de m\u00e3o em m\u00e3o at\u00e9 ele desaparecer&#8221;<\/span><\/em><\/strong><\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif; font-size: 14pt;\">Robert Sarnoff, foi Presidente da RCA<\/span><\/p>\n<p><a id=\"Inicio\"><\/a><span style=\"font-family: verdana, geneva, sans-serif;\">Independentemente da profiss\u00e3o, a matem\u00e1tica financeira se tornou indispens\u00e1vel \u00e0 vida cotidiana. O milion\u00e1rio a utiliza para preservar sua fortuna, e todos devem aplic\u00e1-la para construir seu patrim\u00f4nio. <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por raz\u00f5es obscuras, seu ensino se encontra restrito aos cursos relacionados \u00e0 \u00e1rea financeira, apesar de ser indispens\u00e1vel na engenharia.<\/span><a id=\"\u00cdndice\"><\/a><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Os engenheiros necessitam conhec\u00ea-la para criar solu\u00e7\u00f5es economicamente vi\u00e1veis, de modo que as empresas possam pagar seus sal\u00e1rios.<\/span><\/p>\n<h2 id='\u00edndice'  id=\"boomdevs_1\"><span style=\"font-family: verdana, geneva, sans-serif;\">\u00cdndice<\/span><\/h2>\n<ol style=\"list-style-type: upper-roman;\">\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Tempo_dinheiro\">Tempo \u00e9 dinheiro<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Valor_futuro\">Valor Futuro<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Valor_presente\">Valor Presente<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Pagamentos_constantes\">Pagamentos Constantes<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Amortiza\u00e7\u00f5es_constantes\">Amortiza\u00e7\u00f5es Constantes<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Infla\u00e7\u00e3o\">Infla\u00e7\u00e3o<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Viabilidade_projetos\">Viabilidade de Projetos<\/a><\/span>\n<ol style=\"list-style-type: lower-roman;\">\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#VPL\">Valor Presente L\u00edquido<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#TIR\">Taxa Interna de Retorno<\/a><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#TR\">Tempo de Retorno<\/a><\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<h2 id='tempo-\u00e9-dinheiro'  id=\"boomdevs_2\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Tempo_dinheiro\"><\/a>Tempo \u00e9 Dinheiro<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Tempo \u00e9 dinheiro, mas como medi-lo?<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A varia\u00e7\u00e3o do valor econ\u00f4mico das coisas<span id='easy-footnote-1-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-1-3077' title='A unidade de medi\u00e7\u00e3o do valor econ\u00f4mico \u00e9 o dinheiro'><sup>1<\/sup><\/a><\/span> com o tempo constitui a base da matem\u00e1tica financeira. A Figura 1 apresenta o conceito de valor futuro. Quanto maior o tempo, maior a mudan\u00e7a do valor futuro, que pode ser positiva ou negativa. Mas, como medir a <em>velocidade<\/em> desta varia\u00e7\u00e3o?<\/span><\/p>\n<figure id=\"attachment_3080\" aria-describedby=\"caption-attachment-3080\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf.png\"><img decoding=\"async\" class=\"wp-image-3080\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf-1024x768.png\" alt=\"Valor Futuro\" width=\"800\" height=\"600\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf-1024x768.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf-300x225.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf-624x468.png 624w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf.png 1920w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/a><figcaption id=\"caption-attachment-3080\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Figura 1. Valor Futuro<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A <em>velocidade<\/em> da varia\u00e7\u00e3o de valor pode ser expressa em $\/ano ou em $\/m\u00eas, onde $ representa qualquer unidade monet\u00e1ria<span id='easy-footnote-2-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-2-3077' title=' Reais, D\u00f3lares, Euros, etc.. '><sup>2<\/sup><\/a><\/span>. No entanto, esta velocidade possui um valor relativo diferente dependendo da compara\u00e7\u00e3o ser feita com o valor inicial ou final.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por isso, normaliza-se esta <em>velocidade<\/em> tendo como base o valor existente no instante inicial. A escolha do ponto inicial depende da situa\u00e7\u00e3o espec\u00edfica e n\u00e3o precisa necessariamente ser o momento presente. <span id='easy-footnote-3-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-3-3077' title=' pode ser o instante inicial de um projeto futuro'><sup>3<\/sup><\/a><\/span> Por exemplo, considerando a Figura 1, se a <em>velocidade<\/em> da varia\u00e7\u00e3o de valor fosse de R$100,00 por ano, a <em>velocidade normalizada<\/em> seria de 10% ao ano. Portanto, o conceito de <em>velocidade de varia\u00e7\u00e3o de valor<\/em> tamb\u00e9m depende da unidade de tempo considerada.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Utilizando este conceito, o valor futuro equivale ao valor inicial mais o incremento relativo \u00e0 velocidade de varia\u00e7\u00e3o.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Denomina-se a velocidade de crescimento de valor de <strong>taxa de desconto<\/strong>. Em opera\u00e7\u00f5es financeiras, denomina-se esta taxa de <strong><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/blog\/historia-dos-juros\/\">taxa de juros<\/a><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/blog\/historia-dos-juros\/\">.<\/a> <\/strong>Juros existem apenas em opera\u00e7\u00f5es de empr\u00e9stimo, para investimentos denominamos esta velocidade de <strong>taxa de retorno<\/strong>. Por isso, utilizaremos a denomina\u00e7\u00e3o de taxa de desconto para designar juros ou taxa de retorno indistintamente no desenvolvimento dos conceitos de matem\u00e1tica financeira. A matem\u00e1tica financeira independe da nomenclatura, mas a escolha correta da taxa na resolu\u00e7\u00e3o de quest\u00f5es pr\u00e1ticas requer experi\u00eancia e compreens\u00e3o do problema espec\u00edfico.<\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h2 id='valor-futuro'  id=\"boomdevs_3\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Valor_futuro\"><\/a>Valor Futuro<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O Valor Futuro da Figura 1 \u00e9 igual ao valor presente mais o aumento provocado pela taxa de desconto conforme a equa\u00e7\u00e3o abaixo:<a id=\"Eq1\"><\/a><\/span><\/p>\n<figure id=\"attachment_3104\" aria-describedby=\"caption-attachment-3104\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_2.png\"><img decoding=\"async\" class=\"wp-image-3104 size-medium\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_2-300x53.png\" alt=\"vf_2\" width=\"300\" height=\"53\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_2-300x53.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_2.png 398w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-3104\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 1<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>f<\/sub> \u00e9 o valor futuro;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>p<\/sub> \u00e9 o valor presente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<hr \/>\n<h5 id='exemplos'  id=\"boomdevs_4\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">1. Voc\u00ea emprestou R$ 100 e recebeu R$ 110 no m\u00eas seguinte. Qual a taxa de desconto desta opera\u00e7\u00e3o? Qual a taxa de retorno e qual a taxa de juros? <span id='easy-footnote-4-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-4-3077' title='Resposta: 10%. A taxa de juros e retorno s\u00e3o iguais, por\u00e9m denomina-se taxa de retorno para quem emprestou e taxa de juros para quem tomou o empr\u00e9stimo. '><sup>4<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">2. Voc\u00ea emprestou R$ 100 e recebeu R$ 90 depois de 1 ano. Qual a taxa de juros deste empr\u00e9stimo? <span id='easy-footnote-5-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-5-3077' title='Resposta: -10%'><sup>5<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">3. Voc\u00ea emprestou R$ 100 e recebeu R$ R$101 no dia seguinte. Qual a taxa de juros deste empr\u00e9stimo? <span id='easy-footnote-6-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-6-3077' title='Resposta: 1%'><sup>6<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">4. Qual a maior taxa de juros dos exemplos anteriores? <span id='easy-footnote-7-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-7-3077' title=' Resposta: 1% do exemplo 3'><sup>7<\/sup><\/a><\/span><\/span><\/p>\n<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Quando a frequ\u00eancia de capitaliza\u00e7\u00e3o da taxa de desconto \u00e9 diferente da base de tempo do fluxo financeiro, surge a quest\u00e3o da capitaliza\u00e7\u00e3o da taxa de desconto. A Figura 2 mostra este conceito.<\/span><\/p>\n<figure id=\"attachment_3120\" aria-describedby=\"caption-attachment-3120\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf8.png\"><img decoding=\"async\" class=\"wp-image-3120 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf8.png\" alt=\"vf8\" width=\"600\" height=\"401\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf8.png 600w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf8-300x201.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-3120\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Figura 2. Valor Futuro ao longo do tempo<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Considerando a taxa de desconto constante<span id='easy-footnote-8-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-8-3077' title=' esta condi\u00e7\u00e3o \u00e9 extremamente importante. Se ela n\u00e3o for verdadeira, a equa\u00e7\u00e3o n\u00e3o pode ser utilizada. '><sup>8<\/sup><\/a><\/span>, calcula-se o valor futuro 2 a partir da <a href=\"#Eq1\">equa\u00e7\u00e3o 1<\/a> da seguinte maneira:<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4.png\"><img decoding=\"async\" class=\"aligncenter wp-image-3108\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4-1024x62.png\" alt=\"vf_4\" width=\"800\" height=\"49\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4-1024x62.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4-300x18.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4-624x38.png 624w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_4.png 1300w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/a>Analogamente, escreve-se o valor futuro no per\u00edodo n da seguinte maneira:<a id=\"Eq2\"><\/a><\/span><\/p>\n<figure id=\"attachment_3110\" aria-describedby=\"caption-attachment-3110\" style=\"width: 221px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_5.png\"><img decoding=\"async\" class=\"wp-image-3110\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_5-300x54.png\" alt=\"Equa\u00e7\u00e3o 2\" width=\"221\" height=\"40\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_5-300x54.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_5.png 387w\" sizes=\"(max-width: 221px) 100vw, 221px\" \/><\/a><figcaption id=\"caption-attachment-3110\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 2. Valor Futuro<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 um n\u00famero inteiro que representa o per\u00edodo futuro.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p align=\"justify\"><span style=\"font-family: verdana, geneva, sans-serif;\">Calcula-se a quantidade total de juros paga de acordo com a equa\u00e7\u00e3o abaixo:<\/span><\/p>\n<figure id=\"attachment_3113\" aria-describedby=\"caption-attachment-3113\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6.png\"><img decoding=\"async\" class=\"wp-image-3113\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6-1024x117.png\" alt=\"Equa\u00e7\u00e3o 3\" width=\"800\" height=\"92\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6-1024x117.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6-300x34.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6-624x72.png 624w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_6.png 1300w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/a><figcaption id=\"caption-attachment-3113\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 3. Juros<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que os juros crescem exponencialmente com o n\u00famero de per\u00edodos de capitaliza\u00e7\u00e3o.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A quantidade de juros depende da frequ\u00eancia de capitaliza\u00e7\u00e3o. Na pr\u00e1tica, o mercado utiliza frequ\u00eancias de capitaliza\u00e7\u00e3o mensais, trimestrais, semestrais e anuais. Pa\u00edses com taxas de juros ou infla\u00e7\u00e3o elevadas empregam frequ\u00eancias de capitaliza\u00e7\u00e3o mensal, enquanto economias est\u00e1veis operam com frequ\u00eancias semestrais ou anuais. O ajuste da frequ\u00eancia de capitaliza\u00e7\u00e3o afeta a taxa de desconto real e o c\u00e1lculo do valor futuro da seguinte maneira:<a id=\"Eq4\"><\/a><\/span><\/p>\n<figure id=\"attachment_5960\" aria-describedby=\"caption-attachment-5960\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_20.png\"><img decoding=\"async\" class=\"wp-image-5960 size-medium\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_20-300x87.png\" alt=\"Equa\u00e7\u00e3o\" width=\"300\" height=\"87\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_20-300x87.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_20.png 514w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-5960\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 4. Valor Futuro<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">m \u00e9 a frequ\u00eancia de capitaliza\u00e7\u00e3o por ano;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o n\u00famero de anos;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto nominal anual.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Normalmente, anunciam-se as taxas como um percentual ao ano &#8211; % a.a. Por\u00e9m, essa informa\u00e7\u00e3o encontra-se incompleta porque n\u00e3o expressa a frequ\u00eancia de capitaliza\u00e7\u00e3o.\u00a0\u00a0<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A equa\u00e7\u00e3o 5 mostra como ajustar as taxas anuais de acordo com a frequ\u00eancia de capitaliza\u00e7\u00e3o.<a id=\"Eq5\"><\/a><\/span><\/p>\n<figure id=\"attachment_40221\" aria-describedby=\"caption-attachment-40221\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado.png\"><img decoding=\"async\" class=\"wp-image-40221\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado.png\" alt=\"\" width=\"800\" height=\"75\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado.png 1145w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado-300x28.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado-1024x96.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/i_capitalizado-768x72.png 768w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/a><figcaption id=\"caption-attachment-40221\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 5. Taxas anuais com diversos per\u00edodos de capitaliza\u00e7\u00e3o<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i% <sub>a.a<\/sub>\u00a0 \u00e9 a taxa anual capitalizada anualmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i% <sub>a.s<\/sub> \u00e9 a taxa anual capitalizada semestralmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i% <sub>a.m<\/sub> \u00e9 a taxa anual capitalizada mensalmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i% <sub>a.d <\/sub> \u00e9 a taxa anual capitalizada diariamente.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Em decorr\u00eancia das elevadas taxas de juros e\/ou infla\u00e7\u00e3o, as taxas de juros capitalizadas mensalmente podem ser expressas ao m\u00eas. Quanto isto ocorre, deve-se utilizar as equa\u00e7\u00f5es abaixo:<\/span><\/p>\n<figure id=\"attachment_6045\" aria-describedby=\"caption-attachment-6045\" style=\"width: 364px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_22_1.png\"><img decoding=\"async\" class=\"wp-image-6045 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_22_1.png\" alt=\"\" width=\"364\" height=\"344\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_22_1.png 364w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_22_1-300x284.png 300w\" sizes=\"(max-width: 364px) 100vw, 364px\" \/><\/a><figcaption id=\"caption-attachment-6045\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 6<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>aa<\/sub> \u00e9 a taxa de desconto anual capitalizada anualmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>mm<\/sub> \u00e9 a taxa de desconto mensal capitalizada mensalmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>s<\/sub><sub>s<\/sub> \u00e9 a taxa de desconto semestral capitalizada semestralmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>tt<\/sub> \u00e9 a taxa de desconto trimestral capitalizada trimestralmente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>dm<\/sub> \u00e9 a taxa de desconto di\u00e1ria capitalizada mensalmente;<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Como o tempo existe de forma cont\u00ednua, a capitaliza\u00e7\u00e3o dos juros tamb\u00e9m pode ocorrer continuamente.\u00a0 A partir da <a href=\"#Eq4\">equa\u00e7\u00e3o 4<\/a> , calcula-se a taxa de desconto com capitaliza\u00e7\u00e3o cont\u00ednua da seguinte maneira:<a id=\"Eq7\"><\/a><\/span><\/p>\n<figure id=\"attachment_5989\" aria-describedby=\"caption-attachment-5989\" style=\"width: 275px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/i_continuo.png\"><img decoding=\"async\" class=\"wp-image-5989\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/i_continuo.png\" alt=\"\" width=\"275\" height=\"70\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/i_continuo.png 361w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/i_continuo-300x76.png 300w\" sizes=\"(max-width: 275px) 100vw, 275px\" \/><\/a><figcaption id=\"caption-attachment-5989\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 7<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por exemplo, uma taxa de desconto de 12% ao ano capitalizada mensalmente equivale, baseado na <a href=\"#Eq5\">equa\u00e7\u00e3o 5<\/a>, a 12,68 % aa. Por\u00e9m, se a capitaliza\u00e7\u00e3o fosse cont\u00ednua, esta taxa passaria para 12,74% de acordo com a equa\u00e7\u00e3o 7.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-1'  id=\"boomdevs_5\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">5. Qual a capitaliza\u00e7\u00e3o do exemplo 1? <span id='easy-footnote-9-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-9-3077' title='Resposta: mensal'><sup>9<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">6. Qual a capitaliza\u00e7\u00e3o do exemplo 3? <span id='easy-footnote-10-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-10-3077' title='Resposta : di\u00e1ria'><sup>10<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">7. Qual a taxa de desconto do exemplo 3 em base mensal? <span id='easy-footnote-11-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-11-3077' title='Resposta: i=(1+0,01)&lt;sup&gt;30&lt;\/sup&gt;-1= 34,8 % a.m.'><sup>11<\/sup><\/a><\/span><\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<hr \/>\n<h2 id='valor-presente'  id=\"boomdevs_6\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Valor_presente\"><\/a>Valor Presente<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Calcula-se o valor presente a partir da <a href=\"#Eq2\">equa\u00e7\u00e3o 2<\/a> da seguinte maneira:<a id=\"Eq8\"><\/a><\/span><\/p>\n<figure id=\"attachment_3115\" aria-describedby=\"caption-attachment-3115\" style=\"width: 353px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_7.png\"><img decoding=\"async\" class=\"wp-image-3115 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_7.png\" alt=\"Equa\u00e7\u00e3o\" width=\"353\" height=\"138\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_7.png 353w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_7-300x117.png 300w\" sizes=\"(max-width: 353px) 100vw, 353px\" \/><\/a><figcaption id=\"caption-attachment-3115\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 8. Valor Presente<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Dependendo do caso, calcula-se o valor presente a partir da <a href=\"#Eq4\">equa\u00e7\u00e3o 4<\/a>.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-2'  id=\"boomdevs_7\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">8. Qual o valor futuro de $ 1 000,00 depois de 6 meses com uma taxa de 1% a.m capitalizada mensalmente? <span id='easy-footnote-12-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-12-3077' title=' Resposta: $ 1061,52. Observe que este valor \u00e9 maior do que 6 vezes $10.'><sup>12<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">9. Qual o valor presente de $ 1 000,00 daqui a 6 meses com uma taxa de 1% a.m capitalizada mensalmente? <span id='easy-footnote-13-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-13-3077' title=' Resposta: $ 942,05'><sup>13<\/sup><\/a><\/span><\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<hr \/>\n<h2 id='sistema-de-pagamentos-constantes'  id=\"boomdevs_8\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Pagamentos_constantes\"><\/a>Sistema de Pagamentos Constantes<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O sistema de presta\u00e7\u00f5es, pagamentos, ou recebimentos peri\u00f3dicos acontece frequentemente e denomina-se <strong>Tabela Price<\/strong>. Neste sistema, conforme mostra a Figura 3, pagamentos iguais ocorrem com periodicidade constante durante n per\u00edodos.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Como calcular o valor futuro e o valor presente deste sistema de maneira mais simples do que pela aplica\u00e7\u00e3o direta das <a href=\"#Eq2\">equa\u00e7\u00f5es 2<\/a> e 8?<\/span><\/p>\n<figure id=\"attachment_40224\" aria-describedby=\"caption-attachment-40224\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/vf91.png\"><img decoding=\"async\" class=\"wp-image-40224 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/vf91.png\" alt=\"\" usemap=\"#imgmap2016626172610\" width=\"600\" height=\"450\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/vf91.png 600w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2021\/04\/vf91-300x225.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-40224\" class=\"wp-caption-text\">Figura 3. Sistema de pagamentos constantes<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Utilizando a equa\u00e7\u00e3o 2, calcula-se o valor futuro deste fluxo financeiro atrav\u00e9s da seguinte s\u00e9rie:<\/span><\/p>\n<figure id=\"attachment_3130\" aria-describedby=\"caption-attachment-3130\" style=\"width: 975px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_10.png\"><img decoding=\"async\" class=\"wp-image-3130 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_10.png\" alt=\"Equa\u00e7\u00e3o\" width=\"975\" height=\"87\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_10.png 975w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_10-300x27.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_10-624x56.png 624w\" sizes=\"(max-width: 975px) 100vw, 975px\" \/><\/a><figcaption id=\"caption-attachment-3130\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 9<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">P<sub>mt<\/sub> \u00e9 o pagamento ou recebimento constante. <span id='easy-footnote-14-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-14-3077' title=' PMT significa payments e foi mantido neste texto porque a maioria das calculadoras financeiras utiliza esta nomenclatura.'><sup>14<\/sup><\/a><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Multiplicando ambos os lados da equa\u00e7\u00e3o por (1+i), teremos que:<\/span><\/p>\n<figure id=\"attachment_3132\" aria-describedby=\"caption-attachment-3132\" style=\"width: 1020px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_11.png\"><img decoding=\"async\" class=\"wp-image-3132 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_11.png\" alt=\"Equa\u00e7\u00e3o\" width=\"1020\" height=\"79\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_11.png 1020w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_11-300x23.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_11-624x48.png 624w\" sizes=\"(max-width: 1020px) 100vw, 1020px\" \/><\/a><figcaption id=\"caption-attachment-3132\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 10<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Subtraindo a equa\u00e7\u00e3o 10 da equa\u00e7\u00e3o 9, obtemos:<\/span><\/p>\n<figure id=\"attachment_3134\" aria-describedby=\"caption-attachment-3134\" style=\"width: 1328px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12.png\"><img decoding=\"async\" class=\"wp-image-3134 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12.png\" alt=\"vf_12\" width=\"1328\" height=\"129\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12.png 1328w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12-300x29.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12-1024x99.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vf_12-624x61.png 624w\" sizes=\"(max-width: 1328px) 100vw, 1328px\" \/><\/a><figcaption id=\"caption-attachment-3134\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 11<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por sua vez, calcula-se o valor presente, aplicando a equa\u00e7\u00e3o 8 na equa\u00e7\u00e3o 11 da seguinte maneira:<a id=\"Eq12\"><\/a><\/span><\/p>\n<figure id=\"attachment_5872\" aria-describedby=\"caption-attachment-5872\" style=\"width: 1070px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp.png\"><img decoding=\"async\" class=\"wp-image-5872 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp.png\" alt=\"equa\u00e7\u00e3o\" width=\"1070\" height=\"135\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp.png 1070w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp-300x38.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp-768x97.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vp-1024x129.png 1024w\" sizes=\"(max-width: 1070px) 100vw, 1070px\" \/><\/a><figcaption id=\"caption-attachment-5872\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 12<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Esta equa\u00e7\u00e3o possui quatro vari\u00e1veis; V<sub>p<\/sub>, P<sub>mt<\/sub>, i e n, e deve-se conhecer tr\u00eas delas para calcular o valor da restante. <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Quando se conhece a taxa de desconto (i) e o n\u00famero de per\u00edodos (n), basta aplicar diretamente a equa\u00e7\u00e3o 12 para calcular uma das outras duas vari\u00e1veis (V<sub>p<\/sub> ou P<sub>mt<\/sub>).<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Contudo, quando n\u00famero de per\u00edodo se torna inc\u00f3gnita, precisamos manipular algebricamente a equa\u00e7\u00e3o para achar a solu\u00e7\u00e3o anal\u00edtica. Para isso, aplicamos a seguinte mudan\u00e7a de vari\u00e1veis:<\/span><\/p>\n<figure id=\"attachment_5851\" aria-describedby=\"caption-attachment-5851\" style=\"width: 294px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_1.png\"><img decoding=\"async\" class=\"wp-image-5851 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_1.png\" alt=\"financeira_1\" width=\"294\" height=\"65\" \/><\/a><figcaption id=\"caption-attachment-5851\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 13<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Substituindo esta nova vari\u00e1vel na equa\u00e7\u00e3o 10, teremos que:<\/span><\/p>\n<figure id=\"attachment_5856\" aria-describedby=\"caption-attachment-5856\" style=\"width: 695px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_2-1.png\"><img decoding=\"async\" class=\"wp-image-5856 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_2-1.png\" alt=\"financeira_2\" width=\"695\" height=\"434\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_2-1.png 695w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_2-1-300x187.png 300w\" sizes=\"(max-width: 695px) 100vw, 695px\" \/><\/a><figcaption id=\"caption-attachment-5856\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 14<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Substituindo a equa\u00e7\u00e3o 13 na equa\u00e7\u00e3o 14, teremos que:<\/span><\/p>\n<figure id=\"attachment_5866\" aria-describedby=\"caption-attachment-5866\" style=\"width: 905px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_5.png\"><img decoding=\"async\" class=\"wp-image-5866 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_5.png\" alt=\"Equa\u00e7\u00e3o\" width=\"905\" height=\"190\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_5.png 905w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_5-300x63.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_5-768x161.png 768w\" sizes=\"(max-width: 905px) 100vw, 905px\" \/><\/a><figcaption id=\"caption-attachment-5866\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 15. N\u00famero de Per\u00edodos de Pagamentos Constantes<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A partir desta equa\u00e7\u00e3o, observa-se que o valor do <strong>pagamento peri\u00f3dico (P<sub>mt<\/sub>) deve ser maior que o valor presente multiplicado pela taxa i (V<sub>p<\/sub>.i) para que exista solu\u00e7\u00e3o<\/strong>. <span id='easy-footnote-15-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-15-3077' title='demonstre que isto \u00e9 sempre verdade. Dica: Perpetuidade'><sup>15<\/sup><\/a><\/span> Em outras palavras, V<sub>p<\/sub>i\/P<sub>mt<\/sub> deve ser menor do que 1.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Finalmente, para calcular o valor de i, manipulamos a equa\u00e7\u00e3o 12 e criamos a seguinte fun\u00e7\u00e3o:<\/span><\/p>\n<figure id=\"attachment_5880\" aria-describedby=\"caption-attachment-5880\" style=\"width: 773px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png\"><img decoding=\"async\" class=\"wp-image-5880 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png\" alt=\"financeira_7\" width=\"773\" height=\"135\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7.png 773w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7-300x52.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/financeira_7-768x134.png 768w\" sizes=\"(max-width: 773px) 100vw, 773px\" \/><\/a><figcaption id=\"caption-attachment-5880\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 16<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Esta equa\u00e7\u00e3o n\u00e3o possui solu\u00e7\u00e3o anal\u00edtica, mas a disponibilidade de calculadoras e planilhas eletr\u00f4nicas permite resolver este problema rapidamente utilizando <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/\">t\u00e9cnicas num\u00e9ricas de solu\u00e7\u00e3o de equa\u00e7\u00f5es n\u00e3o <\/a>lineares. <span id='easy-footnote-16-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-16-3077' title=' HP 12 \u00e9 a calculadora financeira mais utilizada em problemas financeiros, mas a HP 50g possui todas as fun\u00e7\u00f5es financeiras e de engenharia. '><sup>16<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A Figura 4 apresenta o valor presente normalizado com rela\u00e7\u00e3o ao pagamento<span id='easy-footnote-17-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-17-3077' title='considerando P&lt;sub&gt;mt&lt;\/sub&gt; igual a 1 na equa\u00e7\u00e3o 10'><sup>17<\/sup><\/a><\/span> para diversos valores de i e n.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que o valor presente dividido pelo pagamento diminui com o aumento do n\u00famero de presta\u00e7\u00f5es e com aumento da taxa de juros. Al\u00e9m disso, ele converge para o valor de 1\/i quando o n\u00famero de per\u00edodos tende a infinito. <span id='easy-footnote-18-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-18-3077' title=' demonstre fazendo o limite de n tendendo a infinito da equa\u00e7\u00e3o 10'><sup>18<\/sup><\/a><\/span> Este valor, denominado perpetuidade, representa o valor presente de um pagamento perp\u00e9tuo com taxa i.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por outro lado, o valor presente tende a zero quando a taxa i tende a infinito, demonstrando que quanto maior a taxa, menos valem as receitas futuras.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Al\u00e9m disso, a Figura 4 permite resolver a equa\u00e7\u00e3o 16 graficamente. Determina-se a taxa i pela interse\u00e7\u00e3o da reta paralela ao eixo i no valor de V<sub>p<\/sub>\/P<sub>mt\u00a0<\/sub> com a curva do n\u00famero de per\u00edodos desejado.<\/span><\/p>\n<p>[em manuten\u00e7\u00e3o]<\/p>\n<p style=\"text-align: center;\"><span style=\"font-family: verdana, geneva, sans-serif; font-size: 14pt;\">Figura 4. Valor presente normalizado da Tabela Price<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Calcula-se o total de juros pagos neste sistema da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_5756\" aria-describedby=\"caption-attachment-5756\" style=\"width: 1160px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price.png\"><img decoding=\"async\" class=\"wp-image-5756 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price.png\" alt=\"equa\u00e7\u00e3o\" width=\"1160\" height=\"149\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price.png 1160w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price-300x39.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price-768x99.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/juros_price-1024x132.png 1024w\" sizes=\"(max-width: 1160px) 100vw, 1160px\" \/><\/a><figcaption id=\"caption-attachment-5756\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">equa\u00e7\u00e3o 17<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que o total de juros pagos cresce exponencialmente com o n\u00famero de per\u00edodos. Por isso, deve-se evitar financiamentos de longo prazo com este tipo de pagamento.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-3'  id=\"boomdevs_9\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">10. Considere uma compra de $ 1 000 parcelada em 5 vezes iguais com juros de 1% ao m\u00eas com capitaliza\u00e7\u00e3o mensal. Determine o valor da presta\u00e7\u00e3o e o total de juros pagos. <span id='easy-footnote-19-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-19-3077' title='Resposta: Tabela abaixo.'><sup>19<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><h2 id=\"tablepress-42-name\" class=\"tablepress-table-name tablepress-table-name-id-42\">Tabela Price<\/h2>\n<span id=\"tablepress-42-description\" class=\"tablepress-table-description tablepress-table-description-id-42\">Tabela 1 - Capital $ 1 000,00 e juros 1% a.m<\/span>\n\n<table id=\"tablepress-42\" class=\"tablepress tablepress-id-42\" aria-labelledby=\"tablepress-42-name\" aria-describedby=\"tablepress-42-description\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">M\u00eas<\/th><th class=\"column-2\">Saldo Devedor <br \/>\nInicio m\u00eas<\/th><th class=\"column-3\">Pagamento<\/th><th class=\"column-4\">Amortiza\u00e7\u00e3o<\/th><th class=\"column-5\">Juros<\/th><th class=\"column-6\">Saldo devedor<br \/>\nFinal m\u00eas<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">0<\/td><td class=\"column-2\">$1 000,00<\/td><td class=\"column-3\">-<\/td><td class=\"column-4\">-<\/td><td class=\"column-5\">$10,00<\/td><td class=\"column-6\">$1 010,00<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">1<\/td><td class=\"column-2\">$1 010,00<br \/>\n<\/td><td class=\"column-3\">$206,04<\/td><td class=\"column-4\">$196,00<\/td><td class=\"column-5\">$8,04<\/td><td class=\"column-6\">$812,00<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">2<\/td><td class=\"column-2\">$812,00<\/td><td class=\"column-3\">$206,04<\/td><td class=\"column-4\">$197,96<\/td><td class=\"column-5\">$6,06<\/td><td class=\"column-6\">$612,02<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">3<\/td><td class=\"column-2\">$612,02<\/td><td class=\"column-3\">$206,04<\/td><td class=\"column-4\">$199,98<\/td><td class=\"column-5\">$4,06<\/td><td class=\"column-6\">$410,04<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">4<\/td><td class=\"column-2\">$410,04<\/td><td class=\"column-3\">$206,04<\/td><td class=\"column-4\">$201,98<\/td><td class=\"column-5\">$2,04<\/td><td class=\"column-6\">$206,04<\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td class=\"column-1\">5<\/td><td class=\"column-2\">$206,04<\/td><td class=\"column-3\">$206,04<\/td><td class=\"column-4\">$204,00<br \/>\n<br \/>\n<\/td><td class=\"column-5\">-<\/td><td class=\"column-6\">-<\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-1\">Total<\/td><td class=\"column-2\">-<\/td><td class=\"column-3\">$1030,2<\/td><td class=\"column-4\">$1000<\/td><td class=\"column-5\">$30,2<\/td><td class=\"column-6\">-<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-42 from cache --><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">11. Seu cart\u00e3o de cr\u00e9dito prop\u00f4s financiar um saldo devedor de $ 1 000,00 em 24 presta\u00e7\u00f5es mensais de $ 99,84. Determine a taxa de juros mensais e capitaliza\u00e7\u00e3o mensal cobrada. <span id='easy-footnote-20-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-20-3077' title='Resposta: 8,61% a.m.'><sup>20<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">12. Qual a taxa de juros equivalente que seu cart\u00e3o cobra ao ano com capitaliza\u00e7\u00e3o anual? <span id='easy-footnote-21-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-21-3077' title='Resposta: 169% a.a.'><sup>21<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">13. Determinada loja oferece um produto em 10 parcelas mensais de R$ 500,00 no carn\u00ea sem juros. \u00c9 melhor comprar com o financiamento do cart\u00e3o de cr\u00e9dito do exemplo anterior ou diretamente pela loja? <span id='easy-footnote-22-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-22-3077' title='Resposta: pela loja. '><sup>22<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">14. Considerando que voc\u00ea possui R$ 5.000,00 aplicados na caderneta de poupan\u00e7a rendendo 0,5% ao m\u00eas com capitaliza\u00e7\u00e3o mensal, calcule o menor desconto que voc\u00ea deve aceitar para comprar o produto do exemplo anterior \u00e0 vista. <span id='easy-footnote-23-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-23-3077' title='Resposta:R$ 4.865,21 ou 2,7% de desconto. '><sup>23<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">15. Considerando que a taxa de lucro da loja \u00e9 de 20% ao ano, determine o maior desconto que o comerciante deve aceitar para vender o produto \u00e0 vista. <span id='easy-footnote-24-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-24-3077' title=' Resposta: R$ 4 603,54 ou 8%.'><sup>24<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">16. Por que o maior desconto aceit\u00e1vel pelo comerciante \u00e9 maior do que o menor desconto aceit\u00e1vel pelo comprador? Resposta: a taxa de lucro da loja \u00e9 maior do que a taxa do rendimento da poupan\u00e7a do consumidor. <span id='easy-footnote-25-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-25-3077' title=' Por isso, devemos negociar sempre em nossas compras e vendas. '><sup>25<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">17. Estime o valor presente de $ 100,00 depositados mensalmente numa aplica\u00e7\u00e3o que rende 10% ao ano com capitaliza\u00e7\u00e3o mensal. <span id='easy-footnote-26-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-26-3077' title='Resposta: $ 3 138,42'><sup>26<\/sup><\/a><\/span><\/span><\/p>\n<hr \/>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h2 id='sistema-de-amortiza\u00e7\u00f5es-constantes'  id=\"boomdevs_10\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Amortiza\u00e7\u00f5es_constantes\"><\/a>Sistema de Amortiza\u00e7\u00f5es Constantes<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Utiliza-se o sistema de presta\u00e7\u00f5es constantes em opera\u00e7\u00f5es financeiras para o consumidor e financiamento de bens de consumo de valores reduzidos devido ao valor constante das presta\u00e7\u00f5es. <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Em opera\u00e7\u00f5es de alto valor e longo prazo <span id='easy-footnote-27-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-27-3077' title=' financiamento imobili\u00e1rio e investimentos na \u00e1rea de energia'><sup>27<\/sup><\/a><\/span> utiliza-se o sistema de amortiza\u00e7\u00f5es constantes &#8211; <strong>sistema SAC<\/strong>, conforme a Tabela 2.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que as presta\u00e7\u00f5es mensais decrescem, mas os pagamentos iniciais s\u00e3o maiores do que os pagamentos do sistema de pagamentos constantes. Contudo, o total de juros pagos \u00e9 menor. Por isso, este sistema de pagamento tornou-se o preferido em empr\u00e9stimos de longo prazo.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A amortiza\u00e7\u00e3o se resume ao valor presente dividido pelo n\u00famero de pagamentos e os juros s\u00e3o dados por:<\/span><\/p>\n<figure id=\"attachment_5798\" aria-describedby=\"caption-attachment-5798\" style=\"width: 619px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_1.png\"><img decoding=\"async\" class=\"wp-image-5798 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_1.png\" alt=\"\" width=\"619\" height=\"334\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_1.png 619w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_1-300x162.png 300w\" sizes=\"(max-width: 619px) 100vw, 619px\" \/><\/a><figcaption id=\"caption-attachment-5798\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 18<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">A<sub>k<\/sub> \u00e9 a amortiza\u00e7\u00e3o no per\u00edodo de tempo k;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>p<\/sub> \u00e9 o valor presente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o n\u00famero total de per\u00edodos de tempo;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de juros ou retorno;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">k \u00e9 um dos per\u00edodos de tempo.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A Equa\u00e7\u00e3o 19 fornece o valor total de juros pagos pelo sistema SAC.<\/span><\/p>\n<figure id=\"attachment_5799\" aria-describedby=\"caption-attachment-5799\" style=\"width: 964px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_2.png\"><img decoding=\"async\" class=\"wp-image-5799 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_2.png\" alt=\"sac_2\" width=\"964\" height=\"171\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_2.png 964w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_2-300x53.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/sac_2-768x136.png 768w\" sizes=\"(max-width: 964px) 100vw, 964px\" \/><\/a><figcaption id=\"caption-attachment-5799\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 19<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Comparando as tabelas 1 e 2, observa-se que o total de juros pagos \u00e9 menor, mas os pagamentos durante a primeira metade do financiamento s\u00e3o maiores do que o financiamento com a Tabela Price.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><h2 id=\"tablepress-43-name\" class=\"tablepress-table-name tablepress-table-name-id-43\">Sistema SAC<\/h2>\n<span id=\"tablepress-43-description\" class=\"tablepress-table-description tablepress-table-description-id-43\">Tabela 2 - Capital $ 1 000,00 e taxa de juros de 1% a.m<\/span>\n\n<table id=\"tablepress-43\" class=\"tablepress tablepress-id-43\" aria-labelledby=\"tablepress-43-name\" aria-describedby=\"tablepress-43-description\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">M\u00eas<\/th><th class=\"column-2\">Saldo Devedor<br \/>\nIn\u00edcio m\u00eas<\/th><th class=\"column-3\">Pagamento<\/th><th class=\"column-4\">Amortiza\u00e7\u00e3o<\/th><th class=\"column-5\">Juros<\/th><th class=\"column-6\">Saldo Devedor<br \/>\nFinal m\u00eas<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">0<\/td><td class=\"column-2\">$1 000,00<\/td><td class=\"column-3\">-<\/td><td class=\"column-4\">-<\/td><td class=\"column-5\">$10,00<\/td><td class=\"column-6\">$1 010,00<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">1<\/td><td class=\"column-2\">$1 010,00<\/td><td class=\"column-3\">$210<\/td><td class=\"column-4\">$200,00<\/td><td class=\"column-5\">$8,00<\/td><td class=\"column-6\">$808,00<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">2<\/td><td class=\"column-2\">$808,00<\/td><td class=\"column-3\">$208,00<\/td><td class=\"column-4\">$200,00<\/td><td class=\"column-5\">$6,00<\/td><td class=\"column-6\">$606,00<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">3<\/td><td class=\"column-2\">$606,00<\/td><td class=\"column-3\">$206,00<\/td><td class=\"column-4\">$200,00<\/td><td class=\"column-5\">$4,00<\/td><td class=\"column-6\">$404,00<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">4<\/td><td class=\"column-2\">$404,00<\/td><td class=\"column-3\">$204,00<\/td><td class=\"column-4\">$200,00<\/td><td class=\"column-5\">$2,00<\/td><td class=\"column-6\">$202,00<\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td class=\"column-1\">5<\/td><td class=\"column-2\">$202,00<\/td><td class=\"column-3\">$202,00<\/td><td class=\"column-4\">$200,00<\/td><td class=\"column-5\">-<\/td><td class=\"column-6\">-<\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-1\">Total<\/td><td class=\"column-2\">-<\/td><td class=\"column-3\">$1 030,00<\/td><td class=\"column-4\">$1 000,00<\/td><td class=\"column-5\">$30,00<\/td><td class=\"column-6\">-<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-43 from cache --><\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h2 id='infla\u00e7\u00e3o'  id=\"boomdevs_11\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Infla\u00e7\u00e3o\"><\/a>Infla\u00e7\u00e3o<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">At\u00e9 agora, consideramos a moeda $ sem infla\u00e7\u00e3o. Contudo, a infla\u00e7\u00e3o e as varia\u00e7\u00f5es de pre\u00e7os ocorrem ao longo do tempo e independem da moeda escolhida. A quest\u00e3o reside em como tratar este fen\u00f4meno na an\u00e1lise financeira.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A legisla\u00e7\u00e3o brasileira vigente n\u00e3o permite o reajuste de pre\u00e7os de contratos em intervalos inferiores a 12 meses, mas aceita reajuste de pre\u00e7os em base anual de acordo com \u00edndices gerais de pre\u00e7os, tais como <a href=\"http:\/\/www.ibge.gov.br\/home\/estatistica\/indicadores\/precos\/inpc_ipca\/defaultseriesHist.shtm\" target=\"_blank\" rel=\"noopener\">IPCA<\/a> e <a href=\"https:\/\/www3.bcb.gov.br\/sgspub\/localizarseries\/localizarSeries.do?method=prepararTelaLocalizarSeries\" target=\"_blank\" rel=\"noopener\">IGP-M<\/a>.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por outro lado, contratos internacionais apresentam cl\u00e1usulas de reajuste de pre\u00e7o sempre que a infla\u00e7\u00e3o ultrapassa determinado valor pr\u00e9 acordado. Este sistema \u00e9 conhecido como gatilho inflacion\u00e1rio.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Existem duas formas de considerar a infla\u00e7\u00e3o na an\u00e1lise financeira; o modelo p\u00f3s-fixado e o modelo prefixado. Como a infla\u00e7\u00e3o futura \u00e9 desconhecida, sua inclus\u00e3o nos modelos requer alguns truques.<\/span><\/p>\n<h5 id='modelo-p\u00f3s-fixado'  id=\"boomdevs_12\"><span style=\"font-family: verdana, geneva, sans-serif;\">Modelo P\u00f3s-fixado<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Neste modelo, considera-se o fluxo de caixa em moeda constante e com taxa de juros sem infla\u00e7\u00e3o, conforme as an\u00e1lises anteriores. Para isso, converte-se a moeda do fluxo em moeda indexada de acordo com algum \u00edndice escolhido. Contudo, o problema deste m\u00e9todo consiste no desconhecimento dos \u00edndices inflacion\u00e1rios futuros e, por isso, denomina-se de p\u00f3s-fixado.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Para contornar este problema, pode-se utilizar uma taxa de infla\u00e7\u00e3o estimada para corrigir a taxa nominal de retorno da seguinte maneira<span id='easy-footnote-28-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-28-3077\" title=\" as taxas devem estar na mesma base de tempo\"><sup>28<\/sup><\/a><\/span>:<\/span><\/p>\n<figure id=\"attachment_5829\" aria-describedby=\"caption-attachment-5829\" style=\"width: 1025px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2.png\"><img decoding=\"async\" class=\"wp-image-5829 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2.png\" alt=\"inflacao_2\" width=\"1025\" height=\"65\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2.png 1025w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2-300x19.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2-768x49.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/inflacao_2-1024x65.png 1024w\" sizes=\"(max-width: 1025px) 100vw, 1025px\" \/><\/a><figcaption id=\"caption-attachment-5829\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 20<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>r<\/sub> \u00e9 a taxa de desconto aparente;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto real;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i<sub>i<\/sub> \u00e9 a taxa de infla\u00e7\u00e3o.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se, a partir da equa\u00e7\u00e3o 20, que para taxas de infla\u00e7\u00e3o baixas o produto das taxas pode ser desprezado e a taxa aparente ser\u00e1 dada aproximadamente pela soma da taxa real com a taxa da infla\u00e7\u00e3o.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-4'  id=\"boomdevs_13\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">19. Considere uma aplica\u00e7\u00e3o de $ 1 000 em caderneta de poupan\u00e7a rendendo TR mais 0,5 % a.m. Determine a rentabilidade total sabendo que a TR valorizou 0,1771 % a.m. <span id='easy-footnote-29-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-29-3077\" title=\"Resposta: ( 1+0,1771) x (1+0,5)=0,6779 a.m.\"><sup>29<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">20. Qual a rentabilidade anual desta aplica\u00e7\u00e3o? <span id='easy-footnote-30-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-30-3077\" title=\"Resposta: 8,44% a.a.\"><sup>30<\/sup><\/a><\/span><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">21. Considerando a taxa de infla\u00e7\u00e3o anual igual a 9,8% a.a, determine o rendimento real da aplica\u00e7\u00e3o anterior.<span id='easy-footnote-31-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-31-3077\" title=\" Resposta: -1,2% a.a pelo c\u00e1lculo exato e -1,36% a.a pelo c\u00e1lculo aproximado. Portanto, infla\u00e7\u00e3o de 9,8% a.a n\u00e3o pode ser considerada baixa.\"><sup>31<\/sup><\/a><\/span><\/span><\/p>\n<hr \/>\n<h5 id='modelo-pr\u00e9-fixado'  id=\"boomdevs_14\"><span style=\"font-family: verdana, geneva, sans-serif;\">Modelo Pr\u00e9 Fixado<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">No modelo pr\u00e9 fixado ajustam-se os valores do fluxo de caixa de acordo com a infla\u00e7\u00e3o estimada para cada intervalo de tempo e a taxa de juros utilizada dever\u00e1 ser a aparente, que considera a taxa de infla\u00e7\u00e3o.<\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h2 id='an\u00e1lise-de-viabilidade-de-projetos'  id=\"boomdevs_15\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"Viabilidade_projetos\"><\/a>An\u00e1lise de Viabilidade de Projetos<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A an\u00e1lise de viabilidade de projetos tem como objetivo escolher projetos e decidir investimentos. Por exemplo:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">\u00c9 melhor construir uma hidrel\u00e9trica ou uma termel\u00e9trica?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Qual o financiamento m\u00e1ximo que podemos fazer para um determinado investidor?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Dado o valor do investimento necess\u00e1rio para a constru\u00e7\u00e3o de uma usina, qual deve ser o pre\u00e7o da energia vendida?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Dado o pre\u00e7o da energia vendida e o valor do investimento, qual o retorno do projeto?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Qual o pre\u00e7o m\u00ednimo aceit\u00e1vel num leil\u00e3o de energia?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Qual o pre\u00e7o m\u00e1ximo aceit\u00e1vel num leil\u00e3o de energia?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Qual o impacto do atraso da obra na viabilidade do projeto?<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Qual a taxa de retorno da fotovoltaica domiciliar?<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Para responder a estas e outras perguntas, deve-se detalhar inicialmente o fluxo de caixa do projeto. O fluxo de caixa representa todas as receitas e despesas do projeto durante toda sua vida \u00fatil em fun\u00e7\u00e3o do tempo. A partir do fluxo de caixa, realizam-se diversas an\u00e1lises utilizando os conceitos de matem\u00e1tica financeira apresentados anteriormente.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A Figura 5 apresenta o fluxo de caixa de um projeto fict\u00edcio. Por conven\u00e7\u00e3o, despesas aparecem como vetores negativos e receitas como vetores positivos. Quando as receitas ou despesas s\u00e3o iguais, os vetores aparecem como linhas pontilhadas.<\/span><\/p>\n<figure id=\"attachment_22823\" aria-describedby=\"caption-attachment-22823\" style=\"width: 2117px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1.png\"><img decoding=\"async\" class=\"wp-image-22823 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1.png\" alt=\"\" width=\"2117\" height=\"1412\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1.png 1920w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-300x200.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-1024x683.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-768x512.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-1536x1024.png 1536w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-2048x1366.png 2048w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-1200x800.png 1200w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_1-1980x1321.png 1980w\" sizes=\"(max-width: 2117px) 100vw, 2117px\" \/><\/a><figcaption id=\"caption-attachment-22823\" class=\"wp-caption-text\">Figura 5. Fluxo de caixa<\/figcaption><\/figure>\n<hr \/>\n<h5 id='exemplo'  id=\"boomdevs_16\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplo<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">18. Esboce o fluxo de caixa de um projeto de energia solar com as seguintes caracter\u00edsticas:<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><span id=\"tablepress-41-description\" class=\"tablepress-table-description tablepress-table-description-id-41\">Projeto Fotovoltaico<\/span>\n\n<table id=\"tablepress-41\" class=\"tablepress tablepress-id-41\" aria-describedby=\"tablepress-41-description\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">Descri\u00e7\u00e3o<\/th><th class=\"column-2\">Valor<\/th><th class=\"column-3\">Observa\u00e7\u00e3o<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">Equipamentos<\/td><td class=\"column-2\">$ 6.000,00<\/td><td class=\"column-3\">Investimento inicial<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">Baterias<\/td><td class=\"column-2\">$ 3.600,00<\/td><td class=\"column-3\">Investimento Inicial com troca a cada 5 anos<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">Troca Inversor<\/td><td class=\"column-2\">$ 1.300,00<\/td><td class=\"column-3\">a cada 10 anos<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">Custo opera\u00e7\u00e3o e manuten\u00e7\u00e3o<\/td><td class=\"column-2\">$ 60 por ano<\/td><td class=\"column-3\"><\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">Vida \u00fatil<\/td><td class=\"column-2\">20 anos<\/td><td class=\"column-3\"><\/td>\n<\/tr>\n<tr class=\"row-7\">\n\t<td class=\"column-1\">Taxa de desconto<\/td><td class=\"column-2\">5% a.a.<\/td><td class=\"column-3\"><\/td>\n<\/tr>\n<tr class=\"row-8\">\n\t<td class=\"column-1\">Energia Gerada<\/td><td class=\"column-2\">81 kWh\/m\u00eas<\/td><td class=\"column-3\">m\u00e1xima, 5 horas de sol pleno por dia<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-41 from cache --><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>Resposta:<\/strong><\/span><\/p>\n<figure id=\"attachment_22825\" aria-describedby=\"caption-attachment-22825\" style=\"width: 2117px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1.png\"><img decoding=\"async\" class=\"wp-image-22825 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1.png\" alt=\"\" width=\"2117\" height=\"1412\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1.png 2117w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-300x200.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-1024x683.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-768x512.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-1536x1024.png 1536w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-2048x1366.png 2048w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-1200x800.png 1200w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_1_1-1980x1321.png 1980w\" sizes=\"(max-width: 2117px) 100vw, 2117px\" \/><\/a><figcaption id=\"caption-attachment-22825\" class=\"wp-caption-text\">Figura 6. Fluxo de caixa do exemplo<\/figcaption><\/figure>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h3 id='valor-presente-l\u00edquido'  id=\"boomdevs_17\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"VPL\"><\/a>Valor Presente L\u00edquido<\/span><\/h3>\n<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O<strong> Valor Presente L\u00edquido &#8211; VPL &#8211;<\/strong> de um fluxo de caixa equivale ao valor monet\u00e1rio de todo o fluxo trazido ao instante de tempo inicial do fluxo de caixa. O VPL corresponde \u00e0 soma das receitas e despesas l\u00edquidas do fluxo de caixa descontadas para o instante de refer\u00eancia com uma determinada <strong>taxa de desconto<\/strong>. Portanto, o VPL depende da taxa de desconto. Normalmente, como existem diversas receitas e despesas no mesmo per\u00edodo de tempo, a diferen\u00e7a entre receitas e despesas em determinado instante de tempo determina o valor da receita l\u00edquida daquele momento. A Equa\u00e7\u00e3o 21 descreve matematicamente o procedimento de c\u00e1lculo do VPL.<a id=\"Eq21\"><\/a><\/span><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_5943\" aria-describedby=\"caption-attachment-5943\" style=\"width: 764px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl.png\"><img decoding=\"async\" class=\"wp-image-5943 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl.png\" alt=\"\" width=\"764\" height=\"319\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl.png 764w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl-300x125.png 300w\" sizes=\"(max-width: 764px) 100vw, 764px\" \/><\/a><figcaption id=\"caption-attachment-5943\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 21. Valor presente l\u00edquido<\/span><\/figcaption><\/figure>\n<p style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>PL<\/sub> \u00e9 o valor presente l\u00edquido [$];<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">F<sub>n<\/sub> \u00e9 a receita l\u00edquida no instante n [$];<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">x \u00e9 uma vari\u00e1vel auxiliar;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o per\u00edodo;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">N \u00e9 o n\u00famero m\u00e1ximo de per\u00edodos.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Calcula-se VPL atrav\u00e9s da equa\u00e7\u00e3o 21 manualmente com uma calculadora ou digitalmente com uma planilha eletr\u00f4nica, mas a grande quest\u00e3o consiste na escolha da taxa de desconto.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>A taxa de desconto consiste na taxa de retorno desejada pelo investidor<\/strong>. Ela deve ser:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">maior do que a taxa de juros dos empr\u00e9stimos banc\u00e1rios;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">maior do que a maior taxa de retorno dos investimentos de risco equivalente do investidor;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">maior do que zero;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">vari\u00e1vel e n\u00e3o uma constante determinada pelo governo.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>VPLs negativo <\/strong>significam que o projeto n\u00e3o consegue pagar seus custos e tornaram-se <strong>invi\u00e1veis<\/strong> com a taxa de desconto escolhida.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por outro lado, <strong>VPLs positivos<\/strong> significam que o projeto paga seus custos com alguma folga e possuem <strong>viabilidade<\/strong>\u00a0com a taxa de desconto escolhida.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">No caso particular de VPL igual a zero, o projeto encontra-se no limiar da viabilidade porque n\u00e3o agrega nem destr\u00f3i valor para o investidor.<\/span><\/p>\n<p>&nbsp;<\/p>\n<hr \/>\n<h5 id='exemplos-5'  id=\"boomdevs_18\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">22. Considerando a receita anual do exemplo 18 igual a $ 2 000 por ano, determine o VPL do projeto.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>Resposta<\/strong>:<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Normalmente, utiliza-se uma planilha eletr\u00f4nica e aplica-se a fun\u00e7\u00e3o financeira de valor presente.<span id='easy-footnote-32-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-32-3077\" title=\" chamada de VPL ou PV\"><sup>32<\/sup><\/a><\/span> Por\u00e9m, neste exemplo, resolveremos manualmente com uma calculadora para fins did\u00e1ticos. Para facilitar os c\u00e1lculos, decompomos o fluxo de caixa em partes mais f\u00e1ceis de calcular. <span id='easy-footnote-33-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-33-3077\" title=\" A linearidade da fun\u00e7\u00e3o VPL permite este artif\u00edcio\"><sup>33<\/sup><\/a><\/span> A arte de resolver este tipo de problema consiste em enxergar como separar o fluxo original no menor n\u00famero de fluxos de f\u00e1cil solu\u00e7\u00e3o.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>Primeira parte &#8211; Parcela constante do fluxo de caixa<br \/>\n<\/strong><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Como a receita e a despesa de O&amp;M<span id='easy-footnote-34-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-34-3077\" title=\" Opera\u00e7\u00e3o e Manuten\u00e7\u00e3o\"><sup>34<\/sup><\/a><\/span> s\u00e3o constantes ao longo do projeto, podemos utilizar a <a href=\"#Eq21\">equa\u00e7\u00e3o 21<\/a> fazendo:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i=5% ao ano com capitaliza\u00e7\u00e3o anual;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n=20;<\/span><\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Utilizando estes dados, o valor presente normalizado (V<sub>p<\/sub>\/PMT) ser\u00e1 igual a 12,46. Como a receita l\u00edquida desta parcela \u00e9 igual a $1 940 (2000-60), o valor presente l\u00edquido desta primeira parte \u00e9 igual a $24 172,40. Desta forma, reduziu-se o fluxo de caixa conforme a figura 6.<\/span><\/p>\n<p><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1.png\"><img decoding=\"async\" class=\"aligncenter wp-image-22827 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1.png\" alt=\"\" width=\"2117\" height=\"1412\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1.png 1920w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-300x200.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-1024x683.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-768x512.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-1536x1024.png 1536w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-2048x1366.png 2048w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-1200x800.png 1200w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/fluxo_caixa_solar_ex_2_1-1980x1321.png 1980w\" sizes=\"(max-width: 2117px) 100vw, 2117px\" \/><\/a><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><strong>Segunda parte &#8211; Parcelas \u00fanicas<\/strong><\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Finalmente, para calcular o valor presente destas tr\u00eas despesas utiliza-se a <a href=\"#Eq8\">equa\u00e7\u00e3o 8<\/a>, da seguinte maneira:<\/span><\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Para cinco anos, teremos um valor presente de -$2 820;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Para dez anos, teremos um valor presente de -$3 008;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Para quinze anos, teremos um valor presente de -$1 732.<\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Portanto, o <strong>valor presente l\u00edquido<\/strong> <strong>&#8211; VPL-<\/strong> deste projeto fica igual a $7 012,40. Contudo, qual o significado deste n\u00famero? Se compararmos o VPL com o total investido no projeto ($ 21 700), o lucro foi de 32% ao longo de 20 anos. <span id='easy-footnote-35-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-35-3077\" title=\" esta taxa equivale a 1,4% a.a.\"><sup>35<\/sup><\/a><\/span> Ser\u00e1 que o investidor fez um bom neg\u00f3cio?<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">23. Suponha que a efici\u00eancia do sistema foi inferior ao esperado, que o tempo permanece nublado mais do que previsto, e que, em decorr\u00eancia desses erros de c\u00e1lculo, a receita anual ficou reduzida para $ 1 437,00. Qual o novo VPL do projeto? <span id='easy-footnote-36-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-36-3077\" title=\"Resposta: zero\"><sup>36<\/sup><\/a><\/span><\/span><\/p>\n<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A <a href=\"#Eq21\">equa\u00e7\u00e3o 21<\/a> n\u00e3o possui solu\u00e7\u00e3o anal\u00edtica e isto representa uma limita\u00e7\u00e3o na utiliza\u00e7\u00e3o do V<sub>PL<\/sub>. Por\u00e9m, considerando a <a href=\"#Eq7\">equa\u00e7\u00e3o 7<\/a> e o fluxo de caixa uma fun\u00e7\u00e3o cont\u00ednua no tempo, o V<sub>pl<\/sub> pode ser escrito da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_6008\" aria-describedby=\"caption-attachment-6008\" style=\"width: 391px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/lapla_fin_1.png\"><img decoding=\"async\" class=\"wp-image-6008 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/lapla_fin_1.png\" alt=\"\" width=\"391\" height=\"87\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/lapla_fin_1.png 391w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/lapla_fin_1-300x67.png 300w\" sizes=\"(max-width: 391px) 100vw, 391px\" \/><\/a><figcaption id=\"caption-attachment-6008\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 22<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">s \u00e9 a taxa de desconto de capitaliza\u00e7\u00e3o cont\u00ednua;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">V<sub>pl<\/sub>(s) o valor presente l\u00edquido em fun\u00e7\u00e3o da taxa de retorno s.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que a fun\u00e7\u00e3o V<sub>pl<\/sub>(s) \u00e9 a <a href=\"https:\/\/pt.wikipedia.org\/wiki\/Transformada_de_Laplace\" target=\"_blank\" rel=\"noopener\">transformada de Laplace<\/a> do fluxo de caixa, e as todas as propriedades desta transformada aplicam-se ao valor presente l\u00edquido.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-6'  id=\"boomdevs_19\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">24. Determine o valor presente de um fluxo de caixa constante e infinito.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Considerando o valor de uma s\u00e9rie geom\u00e9trica infinita igual a;<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/serie_geo_1.png\"><img decoding=\"async\" class=\"aligncenter wp-image-6016 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/serie_geo_1.png\" alt=\"\" width=\"374\" height=\"102\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/serie_geo_1.png 374w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/serie_geo_1-300x82.png 300w\" sizes=\"(max-width: 374px) 100vw, 374px\" \/><\/a>e aplicando na <a href=\"#Eq21\">equa\u00e7\u00e3o 21<\/a>, o valor presente l\u00edquido ser\u00e1 igual a:<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/perpetuo_1.png\"><img decoding=\"async\" class=\"aligncenter wp-image-6015 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/perpetuo_1.png\" alt=\"perpetuo_1\" width=\"777\" height=\"107\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/perpetuo_1.png 777w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/perpetuo_1-300x41.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/perpetuo_1-768x106.png 768w\" sizes=\"(max-width: 777px) 100vw, 777px\" \/><\/a>A utiliza\u00e7\u00e3o da transformada de Laplace da fun\u00e7\u00e3o constante<span id='easy-footnote-37-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-37-3077\" title=\" Murray \"><sup>37<\/sup><\/a><\/span> permite obter diretamente o mesmo resultado, conforme a equa\u00e7\u00e3o abaixo.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/Laplace_pmt.png\"><img decoding=\"async\" class=\"aligncenter wp-image-6017 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/Laplace_pmt.png\" alt=\"Laplace_pmt\" width=\"487\" height=\"87\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/Laplace_pmt.png 487w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/Laplace_pmt-300x54.png 300w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/a>25. Determine o efeito do atraso de n anos no in\u00edcio dos pagamentos no valor presente l\u00edquido de uma perpetuidade.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Utilizando a transformada de Laplace de um degrau atrasado<span id='easy-footnote-38-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-38-3077\" title=\" Murray\"><sup>38<\/sup><\/a><\/span>, teremos que:<\/span><\/p>\n<figure id=\"attachment_6020\" aria-describedby=\"caption-attachment-6020\" style=\"width: 544px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl_1.png\"><img decoding=\"async\" class=\"wp-image-6020 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl_1.png\" alt=\"vpl_1\" width=\"544\" height=\"82\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl_1.png 544w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/vpl_1-300x45.png 300w\" sizes=\"(max-width: 544px) 100vw, 544px\" \/><\/a><figcaption id=\"caption-attachment-6020\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 23<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">s \u00e9 a taxa de desconto com capitaliza\u00e7\u00e3o cont\u00ednua;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto com capitaliza\u00e7\u00e3o discreta;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o n\u00famero de anos de atraso.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Este exemplo serve para esclarecer que a taxa de desconto discreta (i) somente se iguala \u00e0 taxa cont\u00ednua (s) no caso da perpetuidade. Isto ocorre porque a taxa discreta tende para a taxa cont\u00ednua quando o n\u00famero de per\u00edodos tende para infinito. Por isso, a equa\u00e7\u00e3o abaixo fornece rela\u00e7\u00e3o de convers\u00e3o entre a taxa cont\u00ednua (s), utilizada na transformada de Laplace, e a taxa discreta i para os outros casos em que esta hip\u00f3tese n\u00e3o se aplica.<\/span><\/p>\n<figure id=\"attachment_6021\" aria-describedby=\"caption-attachment-6021\" style=\"width: 187px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/s_i.png\"><img decoding=\"async\" class=\"wp-image-6021 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/s_i.png\" alt=\"s_i\" width=\"187\" height=\"39\" \/><\/a><figcaption id=\"caption-attachment-6021\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 24<\/span><\/figcaption><\/figure>\n<p style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">26. Determine o Valor Presente L\u00edquido de uma s\u00e9rie de pagamentos constantes com dura\u00e7\u00e3o finita.<\/span><\/p>\n<p style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">Conforme a figura abaixo, o fluxo de caixa discreto pode ser representado pela diferen\u00e7a de dois fluxos de pagamentos perp\u00e9tuos defasados no tempo.<\/span><\/p>\n<p><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_1.png\"><img decoding=\"async\" class=\"aligncenter wp-image-22829 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_1.png\" alt=\"\" width=\"600\" height=\"400\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_1.png 600w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_1-300x200.png 300w\" sizes=\"(max-width: 600px) 100vw, 600px\" \/><\/a><\/p>\n<p style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">A transformada de Laplace da fun\u00e7\u00e3o no tempo da figura 7 ser\u00e1 dada, de acordo com JEFFREY, por:<\/span><\/p>\n<figure id=\"attachment_6055\" aria-describedby=\"caption-attachment-6055\" style=\"width: 446px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_1.png\"><img decoding=\"async\" class=\"wp-image-6055 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_1.png\" alt=\"Equa\u00e7\u00e3o\" width=\"446\" height=\"75\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_1.png 446w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_1-300x50.png 300w\" sizes=\"(max-width: 446px) 100vw, 446px\" \/><\/a><figcaption id=\"caption-attachment-6055\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 25<\/span><\/figcaption><\/figure>\n<p style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">Aplicando as considera\u00e7\u00f5es do exemplo anterior na equa\u00e7\u00e3o 25, o valor presente em fun\u00e7\u00e3o da taxa de desconto discreta ser\u00e1 dado por:<\/span><\/p>\n<figure id=\"attachment_6056\" aria-describedby=\"caption-attachment-6056\" style=\"width: 573px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_2.png\"><img decoding=\"async\" class=\"wp-image-6056 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_2.png\" alt=\"Equa\u00e7\u00e3o\" width=\"573\" height=\"113\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_2.png 573w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/PMT_2-300x59.png 300w\" sizes=\"(max-width: 573px) 100vw, 573px\" \/><\/a><figcaption id=\"caption-attachment-6056\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 26<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">na \u00e9 o n\u00famero de per\u00edodos de capitaliza\u00e7\u00e3o referentes ao tempo a;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">nb \u00e9 o n\u00famero de per\u00edodos de capitaliza\u00e7\u00e3o referentes ao tempo b.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Demonstre que as equa\u00e7\u00f5es 26 e <a href=\"#Eq12\">12<\/a> se igualam quando o tempo inicial (a) \u00e9 zero.<\/span><\/p>\n<hr \/>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Os exemplos anteriores mostraram, conforme era de se esperar, que os resultados obtidos com a utiliza\u00e7\u00e3o da Transformada de Laplace s\u00e3o equivalentes aos resultados encontrados com a matem\u00e1tica financeira convencional.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Contudo, a Transformada de Laplace permite resolver de forma mais simples problemas complicados para a matem\u00e1tica financeira convencional.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-7'  id=\"boomdevs_20\" style=\"text-align: left;\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">27. Projetos de gera\u00e7\u00e3o de energia entram em opera\u00e7\u00e3o de forma escalonada no tempo. Isto acarreta a venda crescente de energia ao longo do in\u00edcio do projeto, e este aumento afeta o valor presente l\u00edquido do empreendimento. Portanto, determine o valor presente l\u00edquido do fluxo de caixa da figura abaixo, que representa a receita com a venda de energia deste tipo de projeto.<\/span><\/p>\n<p><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1.png\"><img decoding=\"async\" class=\"aligncenter wp-image-22830 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1.png\" alt=\"\" width=\"2117\" height=\"1412\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1.png 1920w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-300x200.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-1024x683.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-768x512.png 768w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-1536x1024.png 1536w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-2048x1366.png 2048w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-1200x800.png 1200w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/11\/PMT_linear3_1-1980x1321.png 1980w\" sizes=\"(max-width: 2117px) 100vw, 2117px\" \/><\/a><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O valor presente l\u00edquido do fluxo de caixa da Figura 8, utilizando a Transformada de Laplace, ser\u00e1 dado por:<\/span><\/p>\n<figure id=\"attachment_6062\" aria-describedby=\"caption-attachment-6062\" style=\"width: 408px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_1.png\"><img decoding=\"async\" class=\"wp-image-6062 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_1.png\" alt=\"Equa\u00e7\u00e3o\" width=\"408\" height=\"77\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_1.png 408w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_1-300x57.png 300w\" sizes=\"(max-width: 408px) 100vw, 408px\" \/><\/a><figcaption id=\"caption-attachment-6062\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 27<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">No caso de capitaliza\u00e7\u00e3o discreta, o valor presente l\u00edquido ficar\u00e1 igual a:<\/span><\/p>\n<figure id=\"attachment_6066\" aria-describedby=\"caption-attachment-6066\" style=\"width: 480px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_2.png\"><img decoding=\"async\" class=\"wp-image-6066 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_2.png\" alt=\"pmt_linear_2\" width=\"480\" height=\"80\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_2.png 480w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/pmt_linear_2-300x50.png 300w\" sizes=\"(max-width: 480px) 100vw, 480px\" \/><\/a><figcaption id=\"caption-attachment-6066\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 28<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">P<sub>mt<\/sub> \u00e9 o pagamento constante;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">k \u00e9 o tempo a partir do qual o pagamento fica constante;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">s \u00e9 a taxa de desconto com capitaliza\u00e7\u00e3o cont\u00ednua;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">nk \u00e9 o per\u00edodo de tempo a partir do qual o pagamento fica constante;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto com capitaliza\u00e7\u00e3o discreta.<\/span><\/li>\n<\/ul>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<h3 id='taxa-interna-de-retorno'  id=\"boomdevs_21\"><span style=\"font-family: verdana, geneva, sans-serif;\">\u00a0<a id=\"TIR\"><\/a>Taxa Interna de Retorno<\/span><\/h3>\n<p align=\"justify\"><span style=\"font-family: verdana, geneva, sans-serif;\">O valor presente l\u00edquido permite analisar diversos projetos, mas depende da taxa de desconto escolhida. Por isso, criou-se o conceito de <strong>Taxa Interna de Retorno-TIR<\/strong>, que representa <span id='easy-footnote-39-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-39-3077\" title=\" algumas vezes chamada de taxa de retorno\"><sup>39<\/sup><\/a><\/span> <strong>a taxa de desconto que anula o valor presente do fluxo de caixa<\/strong>. Portanto, a taxa interna de retorno representa a taxa de desconto lim\u00edtrofe entre lucro e perda. Ou seja, para determinado fluxo de caixa e com esta taxa de desconto, a receita l\u00edquida do projeto permite apenas o pagamento do investimento.<\/span><\/p>\n<p align=\"justify\"><span style=\"font-family: verdana, geneva, sans-serif;\">A resolu\u00e7\u00e3o da equa\u00e7\u00e3o abaixo fornece o valor da TIR.<\/span><\/p>\n<figure id=\"attachment_3148\" aria-describedby=\"caption-attachment-3148\" style=\"width: 482px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png\"><img decoding=\"async\" class=\"wp-image-3148 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png\" alt=\"TIR\" width=\"482\" height=\"168\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir.png 482w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/tir-300x105.png 300w\" sizes=\"(max-width: 482px) 100vw, 482px\" \/><\/a><figcaption id=\"caption-attachment-3148\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 23. TIR<\/span><\/figcaption><\/figure>\n<p align=\"left\"><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">F<sub>n<\/sub> \u00e9 o fluxo de caixa no per\u00edodo n;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">TIR \u00e9 a taxa interna de retorno;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">n \u00e9 o per\u00edodo.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Infelizmente, esta equa\u00e7\u00e3o n\u00e3o possui solu\u00e7\u00e3o anal\u00edtica e sua solu\u00e7\u00e3o requer tamb\u00e9m <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/matematica\/metodo-de-newton-raphson\/\" target=\"_blank\" rel=\"noopener noreferrer\">t\u00e9cnicas de c\u00e1lculo num\u00e9rico<\/a>. Contudo, as modernas m\u00e1quinas de calcular e planilhas eletr\u00f4nicas<span id='easy-footnote-40-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-40-3077\" title=\" HP50g,\u00a0 HP12c e o Excel possuem fun\u00e7\u00e3o financeiras embutidas\"><sup>40<\/sup><\/a><\/span> permitem resolver esta quest\u00e3o rapidamente.<\/span><\/p>\n<hr \/>\n<h5 id='exemplos-8'  id=\"boomdevs_22\"><span style=\"font-family: verdana, geneva, sans-serif;\">Exemplos<\/span><\/h5>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">24. Qual a taxa interna de retorno do exemplo 18? <span id='easy-footnote-41-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-41-3077\" title=\" Utilizando a mesma planilha eletr\u00f4nica desenvolvida para resolver o exemplo 15 e variando a taxa de desconto, o VPL passa a ser negativo para i igual a 13,1%. Portanto, a TIR original do projeto \u00e9 aproximadamente 13%. Qual o significado deste valor? Depende do investidor. Se a rentabilidade do investimento mais rent\u00e1vel do investidor for menor do que 13%, investir neste projeto \u00e9 um bom neg\u00f3cio. Contudo, se o investidor possuir outros investimentos com maior rentabilidade, \u00e9 mais neg\u00f3cio investir mais no neg\u00f3cio mais lucrativo.\"><sup>41<\/sup><\/a><\/span><\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<hr \/>\n<h3 id='tempo-de-retorno'  id=\"boomdevs_23\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a id=\"TR\"><\/a>Tempo de Retorno<\/span><\/h3>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O tempo de retorno<span id='easy-footnote-42-3077' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/geracao-de-energia-eletrica\/aspectos-economicos\/matematica-financeira\/#easy-footnote-bottom-42-3077\" title=\"payback time\"><sup>42<\/sup><\/a><\/span> \u00e9 o n\u00famero de per\u00edodos necess\u00e1rio para que o valor presente do fluxo de caixa at\u00e9 aquele per\u00edodo seja igual ao investimento. Isto pode ser escrito da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_3151\" aria-describedby=\"caption-attachment-3151\" style=\"width: 420px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/payback_1.png\"><img decoding=\"async\" class=\"wp-image-3151 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/payback_1.png\" alt=\"Tempo de Retorno\" width=\"420\" height=\"180\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/payback_1.png 420w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2015\/03\/payback_1-300x129.png 300w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><\/a><figcaption id=\"caption-attachment-3151\" class=\"wp-caption-text\"><span style=\"font-family: verdana, geneva, sans-serif;\">Equa\u00e7\u00e3o 24. Tempo de Retorno<\/span><\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">N<sub>r<\/sub> \u00e9 o per\u00edodo\/tempo de retorno;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">i \u00e9 a taxa de desconto;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">I<sub>o<\/sub> \u00e9 o investimento inicial.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A equa\u00e7\u00e3o 24 \u00e9 semelhante \u00e0 equa\u00e7\u00e3o 23, mas a inc\u00f3gnita representa a \u00fanica diferen\u00e7a. Isto significa que os m\u00e9todos n\u00fameros utilizados no c\u00e1lculo da TIR tamb\u00e9m podem ser utilizados no c\u00e1lculo do tempo de retorno. Contudo, para fugir do c\u00e1lculo num\u00e9rico, considera-se a taxa de desconto igual a zero. Este resultado equivale ao menor tempo de retorno poss\u00edvel deste fluxo de caixa, e quanto maior a taxa de desconto maior o tempo de retorno.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Este indicador apresenta os seguintes problemas:<\/span><\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">N\u00e3o considera todo o fluxo de caixa porque o c\u00e1lculo termina no per\u00edodo de retorno.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">Assim como no caso da taxa interna de retorno, existe a possibilidade de diversas solu\u00e7\u00f5es para o problema.<\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por tudo isso, evita-se utilizar este m\u00e9todo na an\u00e1lise de projetos de elevado investimento e de longo prazo.<\/span><\/p>\n<p style=\"text-align: right;\"><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#\u00cdndice\">\u00cdndice<\/a><\/span><\/p>\n<p>&nbsp;<\/p>\n<h2 id='bibliografia'  id=\"boomdevs_24\"><span style=\"font-family: verdana, geneva, sans-serif;\">Bibliografia<\/span><\/h2>\n<ol>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">PUCCINI, A.L., <strong>Matem\u00e1tica Financeira Objetiva e Aplicada<\/strong>, Editora Saraiva, 1999.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">BUSER, S.A., <strong>LaPlace Transforms as Present Value Rules: A Note<\/strong>, The Journal of Finance, Vol. XLI, n.1, mar\u00e7o de 1986.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">MURRAY, R. S., LIPSCHUTZ, S., LIU, J., <strong>Mathematical Handbook of Formulas and Tables<\/strong>, McGraw-Hill, 2009.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">JEFFREY,A., DAI, H.H., <strong>Handobook of Mathematical Formulas and Integrals<\/strong>, 4 edi\u00e7\u00e3o, Elsevier, 2008.<\/span><\/li>\n<\/ol>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\"><a href=\"#Inicio\">Voltar ao In\u00edcio<\/a><\/span><\/p>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; zoom: initial; place-content: initial; place-items: initial; place-self: initial; alignment-baseline: initial; animation: initial; 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padding-block: initial; padding: initial; padding-inline: initial; page: initial; page-orientation: initial; paint-order: initial; perspective: initial; perspective-origin: initial; pointer-events: initial; position: initial; quotes: initial; r: initial; resize: initial; ruby-position: initial; rx: initial; ry: initial; scroll-behavior: initial; scroll-margin-block: initial; scroll-margin: initial; scroll-margin-inline: initial; scroll-padding-block: initial; scroll-padding: initial; scroll-padding-inline: initial; scroll-snap-align: initial; scroll-snap-stop: initial; scroll-snap-type: initial; shape-image-threshold: initial; shape-margin: initial; shape-outside: initial; shape-rendering: initial; size: initial; speak: initial; stop-color: initial; stop-opacity: initial; stroke: initial; stroke-dasharray: initial; stroke-dashoffset: initial; stroke-linecap: initial; stroke-linejoin: initial; stroke-miterlimit: initial; stroke-opacity: initial; stroke-width: initial; tab-size: initial; 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text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; 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scroll-snap-align: initial; scroll-snap-stop: initial; scroll-snap-type: initial; shape-image-threshold: initial; shape-margin: initial; shape-outside: initial; shape-rendering: initial; size: initial; speak: initial; stop-color: initial; stop-opacity: initial; stroke: initial; stroke-dasharray: initial; stroke-dashoffset: initial; stroke-linecap: initial; stroke-linejoin: initial; stroke-miterlimit: initial; stroke-opacity: initial; stroke-width: initial; tab-size: initial; table-layout: initial; text-align: initial; text-align-last: initial; text-anchor: initial; text-combine-upright: initial; text-decoration: initial; text-decoration-skip-ink: initial; text-indent: initial; text-overflow: initial; text-shadow: initial; text-size-adjust: initial; text-transform: initial; text-underline-offset: initial; text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; zoom: initial; place-content: initial; place-items: initial; place-self: initial; alignment-baseline: initial; animation: initial; appearance: initial; aspect-ratio: initial; 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-webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; 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text-overflow: initial; text-shadow: initial; text-size-adjust: initial; text-transform: initial; text-underline-offset: initial; text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; 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-webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; zoom: initial; place-content: initial; place-items: initial; place-self: initial; alignment-baseline: initial; animation: initial; appearance: initial; aspect-ratio: initial; backdrop-filter: initial; backface-visibility: initial; background: initial; background-blend-mode: initial; baseline-shift: initial; block-size: initial; border-block: initial; border: initial; border-radius: initial; border-collapse: initial; border-end-end-radius: initial; border-end-start-radius: initial; border-inline: initial; border-start-end-radius: initial; border-start-start-radius: initial; inset: initial; box-shadow: initial; box-sizing: initial; break-after: initial; break-before: initial; break-inside: initial; buffered-rendering: initial; caption-side: initial; caret-color: initial; clear: initial; clip: initial; 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ry: initial; scroll-behavior: initial; scroll-margin-block: initial; scroll-margin: initial; scroll-margin-inline: initial; scroll-padding-block: initial; scroll-padding: initial; scroll-padding-inline: initial; scroll-snap-align: initial; scroll-snap-stop: initial; scroll-snap-type: initial; shape-image-threshold: initial; shape-margin: initial; shape-outside: initial; shape-rendering: initial; size: initial; speak: initial; stop-color: initial; stop-opacity: initial; stroke: initial; stroke-dasharray: initial; stroke-dashoffset: initial; stroke-linecap: initial; stroke-linejoin: initial; stroke-miterlimit: initial; stroke-opacity: initial; stroke-width: initial; tab-size: initial; table-layout: initial; text-align: initial; text-align-last: initial; text-anchor: initial; text-combine-upright: initial; text-decoration: initial; text-decoration-skip-ink: initial; text-indent: initial; text-overflow: initial; text-shadow: initial; text-size-adjust: initial; text-transform: initial; text-underline-offset: initial; text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; 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x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; zoom: initial; place-content: initial; place-items: initial; place-self: initial; alignment-baseline: initial; animation: initial; appearance: initial; aspect-ratio: initial; backdrop-filter: initial; backface-visibility: initial; background: initial; background-blend-mode: initial; baseline-shift: initial; block-size: initial; border-block: initial; border: initial; border-radius: initial; border-collapse: initial; border-end-end-radius: initial; border-end-start-radius: initial; border-inline: initial; border-start-end-radius: initial; 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text-align: initial; text-align-last: initial; text-anchor: initial; text-combine-upright: initial; text-decoration: initial; text-decoration-skip-ink: initial; text-indent: initial; text-overflow: initial; text-shadow: initial; text-size-adjust: initial; text-transform: initial; text-underline-offset: initial; text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; -webkit-print-color-adjust: initial; -webkit-rtl-ordering: initial; -webkit-ruby-position: initial; -webkit-tap-highlight-color: initial; -webkit-text-combine: initial; -webkit-text-decorations-in-effect: initial; -webkit-text-emphasis: initial; -webkit-text-emphasis-position: initial; -webkit-text-fill-color: initial; -webkit-text-security: initial; -webkit-text-stroke: initial; -webkit-transform-origin-x: initial; -webkit-transform-origin-y: initial; -webkit-transform-origin-z: initial; -webkit-user-drag: initial; -webkit-user-modify: initial; white-space: initial; widows: initial; width: initial; will-change: initial; word-break: initial; word-spacing: initial; x: initial; y: initial; z-index: initial;\"><\/div>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; 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position: initial; quotes: initial; r: initial; resize: initial; ruby-position: initial; rx: initial; ry: initial; scroll-behavior: initial; scroll-margin-block: initial; scroll-margin: initial; scroll-margin-inline: initial; scroll-padding-block: initial; scroll-padding: initial; scroll-padding-inline: initial; scroll-snap-align: initial; scroll-snap-stop: initial; scroll-snap-type: initial; shape-image-threshold: initial; shape-margin: initial; shape-outside: initial; shape-rendering: initial; size: initial; speak: initial; stop-color: initial; stop-opacity: initial; stroke: initial; stroke-dasharray: initial; stroke-dashoffset: initial; stroke-linecap: initial; stroke-linejoin: initial; stroke-miterlimit: initial; stroke-opacity: initial; stroke-width: initial; tab-size: initial; table-layout: initial; text-align: initial; text-align-last: initial; text-anchor: initial; text-combine-upright: initial; text-decoration: initial; text-decoration-skip-ink: initial; text-indent: initial; text-overflow: initial; text-shadow: initial; text-size-adjust: initial; text-transform: initial; text-underline-offset: initial; text-underline-position: initial; touch-action: initial; transform: initial; transform-box: initial; transform-origin: initial; transform-style: initial; transition: initial; user-select: initial; vector-effect: initial; vertical-align: initial; visibility: initial; -webkit-app-region: initial; border-spacing: initial; -webkit-border-image: initial; -webkit-box-align: initial; -webkit-box-decoration-break: initial; -webkit-box-direction: initial; -webkit-box-flex: initial; -webkit-box-ordinal-group: initial; -webkit-box-orient: initial; -webkit-box-pack: initial; -webkit-box-reflect: initial; -webkit-highlight: initial; -webkit-hyphenate-character: initial; -webkit-line-break: initial; -webkit-line-clamp: initial; -webkit-mask-box-image: initial; -webkit-mask: initial; -webkit-mask-composite: initial; -webkit-perspective-origin-x: initial; -webkit-perspective-origin-y: initial; 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Robert Sarnoff, foi Presidente da RCA Independentemente da profiss\u00e3o, a matem\u00e1tica financeira se tornou indispens\u00e1vel \u00e0 vida cotidiana. O milion\u00e1rio a utiliza para preservar sua fortuna, e todos devem aplic\u00e1-la para construir seu patrim\u00f4nio. Por raz\u00f5es obscuras, seu ensino se encontra [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":10897,"parent":3068,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"templates\/template-full-width.php","meta":{"_uag_custom_page_level_css":"","footnotes":""},"class_list":["post-3077","page","type-page","status-publish","has-post-thumbnail","hentry"],"uagb_featured_image_src":{"full":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2.png",1200,280,false],"thumbnail":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2-150x150.png",150,150,true],"medium":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2-300x70.png",300,70,true],"medium_large":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2-768x179.png",580,135,true],"large":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2-1024x239.png",580,135,true],"1536x1536":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2.png",1200,280,false],"2048x2048":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2.png",1200,280,false],"post-thumbnail":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2.png",1200,280,false],"twentytwenty-fullscreen":["https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/07\/matfin_2.png",1200,280,false]},"uagb_author_info":{"display_name":"admin","author_link":"https:\/\/www.antonioguilherme.web.br.com\/blog\/author\/admin\/"},"uagb_comment_info":0,"uagb_excerpt":"&#8220;Finan\u00e7as consiste na arte de passar dinheiro de m\u00e3o em m\u00e3o at\u00e9 ele desaparecer&#8221; Robert Sarnoff, foi Presidente da RCA Independentemente da profiss\u00e3o, a matem\u00e1tica financeira se tornou indispens\u00e1vel \u00e0 vida cotidiana. O milion\u00e1rio a utiliza para preservar sua fortuna, e todos devem aplic\u00e1-la para construir seu patrim\u00f4nio. Por raz\u00f5es obscuras, seu ensino se encontra&hellip;","_links":{"self":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/pages\/3077","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/comments?post=3077"}],"version-history":[{"count":298,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/pages\/3077\/revisions"}],"predecessor-version":[{"id":44168,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/pages\/3077\/revisions\/44168"}],"up":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/pages\/3068"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/media\/10897"}],"wp:attachment":[{"href":"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-json\/wp\/v2\/media?parent=3077"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}