From 608f6d04c64186a08494b9094ab9c2c59b09e97b Mon Sep 17 00:00:00 2001 From: Geisyane Date: Thu, 8 Apr 2021 10:46:55 -0300 Subject: [PATCH] =?UTF-8?q?Primeira=20vers=C3=A3o=20do=20post?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../index.pt.md | 314 ++++++++++++++++++ .../FE.png | Bin 0 -> 9203 bytes .../Rplot02.png | Bin 0 -> 4709 bytes .../Rplot03.png | Bin 0 -> 5515 bytes .../calculator-1680905_1920.jpg | Bin 0 -> 284750 bytes 5 files changed, 314 insertions(+) create mode 100644 content/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r/index.pt.md create mode 100644 static/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/FE.png create mode 100644 static/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/Rplot02.png create mode 100644 static/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/Rplot03.png create mode 100644 static/img/fronteira_eficiente/calculator-1680905_1920.jpg diff --git a/content/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r/index.pt.md b/content/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r/index.pt.md new file mode 100644 index 0000000..d6c6848 --- /dev/null +++ b/content/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r/index.pt.md @@ -0,0 +1,314 @@ +--- +title: Como obter a fronteira eficiente usando o R +author: Geisyane Karina Gonzaga Kath +date: '2021-04-08' +slug: [] +categories: + - tutoriais + - economia + - finanças +tags: + - investimentos + - carteira otima + - fronteira eficiente + - markowitz +description: '' +images: + - 'img/fronteira_eficiente/calculator-1680905_1920.jpg' +--- +### O QUE É FRONTEIRA EFICIENTE + +Uma carteira de ativos é dada por um conjunto de ativos (ou ainda por um único ativo) cuja soma da proporção a ser investida em cada ativo é 1. Assim, é possível compreender a definição de fronteira eficiente que foi utilizada por Markowitz: dado um conjunto de ativos, há inúmeras possibilidades de combinações dos mesmos formando carteiras, por isso, para representá-las em um diagrama de forma eficiente e de fácil visualização, basta considerar que se duas carteiras possuem o mesmo risco e uma delas tem o maior retorno, a de menor retorno é desprezada. Do mesmo modo, se duas carteiras possuem o mesmo retorno esperado e uma delas possui o menor risco, esta será a escolhida. Ou seja, a fronteira eficiente é formada apenas pelas carteiras que seriam de interesse de investidores. + + +No livro Moderna Teoria de Carteiras e Análise de Investimentos (2012), capítulo 5, os autores utilizam a imagem a seguir para representar a infinidade de carteiras possíveis. + +|![](/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/FE.png) | +|:--:| +| *Fonte: Elton, et. al (2012)* | + + +Perceba que, seguindo o conceito anteriormente apresentado de fronteira eficiente, a carteira B é preferível no lugar de A, visto que se trata de correr o mesmo risco e obter um possível retorno maior. Da mesma forma, a carteira E é preferível em relação a F, pois embora o retorno esperado seja o mesmo, o risco é menor. Assim, a fronteira eficiente é aquela que vai da carteira de menor risco até a que possui o maior retorno, observando as regras mencionadas: trata-se, no caso da figura, do segmento CEB. + +### SEM VENDAS A DESCOBERTO E SEM RETORNO À TAXA LIVRE DE RISCO + +Para representar a fronteira eficiente no R, primeiramente é necessário instalar os seguintes pacotes: + +``` +install.packages('quadprog') +install.packages('PerformanceAnalytics') +install.packages("IntroCompFinR", repos="http://R-Forge.R-project.org") +install.packages('BatchGetSymbols') + +``` + +Depois, é preciso baixar os dados de interesse, que são os retornos dos ativos em determinado período. Para isso, + + +``` + +library(quadprog) +library(PerformanceAnalytics) +library(IntroCompFinR) +library(BatchGetSymbols) + +# Selecione os ativos que deseja trabalhar + +papeis <- c('MGLU3.SA', 'PETR4.SA', 'ITUB4.SA', 'WEGE3.SA') + +# Adicione uma data inicial e final para os retornos que quer consultar. +# Lembre que a data deve ser colocada na ordem ano-mês-dia. + +data.inicial <- as.Date('2019-06-17') +data.final <- as.Date('2020-07-31') + +# Especifique se deseja retornos diários("daily"), semanais("weekly"), mensais("monthly") ou anuais("yearly"). + +frequencia <- 'monthly' + +Retornos <- BatchGetSymbols(ticker = papeis, first.date = data.inicial, + last.date = data.final, freq.data = frequencia) + +# Aqui é importante observar a ordem que os dados foram gerados, para isso: + +Retornos$df.tickers$ticker + +> Retornos$df.tickers$ticker + [1] "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" + [8] "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" "ITUB4.SA" +[15] "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" +[22] "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" "MGLU3.SA" +[29] "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" +[36] "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" "PETR4.SA" +[43] "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" +[50] "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" "WEGE3.SA" + +# Observe que a ordem não é a do vetor informado em "papeis", mas sim ITUB4.SA, MGLU3.SA, PETR4.SA e WEGE3.SA + +Tabela_Retornos <- as.data.frame(Retornos[2]) +Tabela_Retornos + +``` + +Sendo de interesse apenas a coluna da tabela que contém os retornos ajustados pelo preço (df.tickers.ret.adjusted.prices), é importante extrair apenas os dados dessa coluna referente a cada um dos ativos. + + +``` +# Para obter apenas os dados da coluna de interesse + +dados<-Tabela_Retornos$df.tickers.ret.adjusted.prices +dados + +# Para colocar os dados em sequência e retirar os NA + +x <- seq_along(dados[!is.na(dados)]) +x + +# Para separar os dados em termos de cada ativo (ceiling é x/13 porque são 13 meses). +# Como são 4 ativos sendo analisados, serão 4 sep + +sep <- split(dados[!is.na(dados)], ceiling(x/13)) +sep + +# Para que os retornos fiquem em porcentagem: + +# ITUB4.SA +z1<-sep$'1'*100 +# MGLU3.SA +z2<-sep$'2'*100 +# PETR4.SA +z3<-sep$'3'*100 +# WEGE3.SA +z4<-sep$'4'*100 + +ativos<-c('ITUB4.SA','MGLU3.SA','PETR4.SA','WEGE3.SA') +retornos<-cbind(z1,z2,z3,z4) + +``` + +Para traçar a fronteira eficiente é preciso ter alguns insumos como o retorno médio de cada ativo e também a matriz de covariância dos mesmos, assim, + +``` +# Para calcular a média dos retornos de cada ativo: + +retorno_medio<- rbind(mean(retornos[,1]), mean(retornos[,2]), mean(retornos[,3]), mean(retornos[,4])) +retorno_medio + +# Para calcular o desvio padrão dos retornos de cada ativo: + +desvio_padrao<- rbind(sd(z1),sd(z2), sd(z3), sd(z4)) + +# Para nomear as linhas + +rownames(desvio_padrao)<-ativos +rownames(retorno_medio) <- ativos + +# Para obter a matriz de covariância dos ativos + +matriz_cov <- cov(retornos) + +rownames(matriz_cov) <- ativos +colnames(matriz_cov) <- ativos + +``` + +Para obter a carteira de menor risco e os pontos da fronteira eficiente: + + +``` +#Para que não haja vendas a descoberto: + +short<-FALSE + +# Cálculo da carteira com a menor risco possível + +carteira_min_risco <- globalMin.portfolio(retorno_medio, matriz_cov, shorts = short) + +# É possível visualizar o retorno esperado, o risco e a proporção a ser investida em cada ativo para a carteira de menor risco + +carteira_min_risco + +# Cálculo dos diferentes pontos da fronteira eficiente +#nport é o número de pontos com diferentes pesos a serem calculados pela função + +fronteira_eficiente <- efficient.frontier(retorno_medio, matriz_cov, nport = 1500, shorts = short) +fronteira_eficiente + +``` + +E para visualizar a fronteira eficiente: + + +``` + +# Visualização da Fronteira Eficiente +# xlim e ylim é a escala do gráfico, xaxs='i' e yaxs='i' é para que o eixo horizontal e vertical se iniciem em zero +#xlab e ylab é para renomear os eixos x e y respectivamente + +plot(fronteira_eficiente$sd,fronteira_eficiente$er, ylab= "Retorno esperado", xlab="Risco da carteira",col="blue", pch=16, xlim=c(0,17),ylim=c(0,14),xaxs='i',yaxs='i') + +# Observe que o eixo horizontal apresenta o desvio padrão e o eixo vertical apresenta o retorno esperado + +# Para visualizar a carteira de menor risco + +points(carteira_min_risco$sd, carteira_min_risco$er, col="green", pch=10, cex=2) +text(carteira_min_risco$sd, carteira_min_risco$er, labels="Risco Mínimo", pos=2) + +# Para visualizar os demais ativos (o desvio padrão e o retorno médio são respectivamente colocados) +# pos=2 o nome do ativo fica ao lado esquerdo do ponto +# pos=1 o nome do ativo fica logo abaixo do ponto +# pos=3 o nome do ativo fica acima do ponto + +points(9.9538,-1.2182, col='purple',pch=10, cex=2) +text(9.9538,-1.2182, "ITUB4", pos = 2) + +points(15.3075,10.4740, col='Yellow',pch=10, cex=2) +text(15.3075,10.4740, "MGLU3", pos = 3) + +points(16.9875,0.4360, col='gray',pch=10, cex=2) +text(16.9875,0.4360, "PETR4", pos = 2) + +points(13.7182,10.3032, col='black',pch=10, cex=2) +text(13.7182,10.3032, "WEGE3", pos = 1) + +``` + +![](/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/Rplot02.png) + +### SEM VENDAS A DESCOBERTO E COM RETORNO À TAXA LIVRE DE RISCO + +Aqui admite-se que o investidor tem a opção de investir em algum ativo com retorno livre de risco, para esse exemplo considere um retorno à taxa de juros de 0.11%. A forma de baixar os dados necessários é a mesma que no caso anterior e os mesmos dados foram utilizados para esse exemplo. + +``` +# Para um retorno à taxa livre de risco igual a 0.11% e sem vendas a descoberto +tx_livre_risco <- 0.11 +short <- FALSE + +# Carteira eficiente + +carteira_eficiente <-IntroCompFinR::tangency.portfolio(retorno_medio, matriz_cov, tx_livre_risco, shorts = short) + +# É possível visualizar o retorno esperado, o risco da carteira e também a proporção que deve ser investida em cada ativo para obtê-la + +carteira_eficiente + +# Carteira de menor risco + +carteira_min_risco <- globalMin.portfolio(retorno_medio, matriz_cov, shorts = short) + + +``` +Para encontrar a fronteira eficiente: + +``` + +# Pontos da Fronteira Eficiente + +fronteira_eficiente <- efficient.frontier(retorno_medio, matriz_cov, nport = 1500, shorts = FALSE) + +# Visualização da Fronteira Eficiente + +plot(fronteira_eficiente$sd,fronteira_eficiente$er, col="blue",ylab= "Retorno esperado", xlab="Risco da carteira", pch=16, xlim=c(0,18),ylim=c(0,15), xaxs='i',yaxs='i') + +points(carteira_min_risco$sd, carteira_min_risco$er, col="green", pch=10, cex=2) +text(carteira_min_risco$sd, carteira_min_risco$er, labels="Risco Mínimo", pos=2) + +points(carteira_eficiente$sd, carteira_eficiente$er, col="red", pch=10, cex=2) +text(carteira_eficiente$sd, carteira_eficiente$er, labels="Carteira Eficiente", pos=2) + +points(9.9538,-1.2182, col='purple',pch=10, cex=2) +text(9.9538,-1.2182, "ITUB4", pos = 2) + +points(15.3075,10.4740, col='Yellow',pch=10, cex=2) +text(15.3075,10.4740, "MGLU3", pos = 3) + +points(16.9875,0.4360, col='gray',pch=10, cex=2) +text(16.9875,0.4360, "PETR4", pos = 2) + +points(13.7182,10.3032, col='black',pch=10, cex=2) +text(13.7182,10.3032, "WEGE3", pos = 1) + +``` + +A linha que tangencia a carteira eficiente tem como coeficiente angular +$$ +\theta=\frac{\bar{R}_p-R_f}{\sigma_p} +$$ + +Em que + + +$\bar{R}_p$ é o retorno esperado da carteira ótima + + +$R_f$ é o retorno do ativo livre de risco + + +$\sigma_p$ é o risco da carteira ótima + + +Então, + + + +``` + +# Cálculo de theta + +theta <- (carteira_eficiente$er - tx_livre_risco)/carteira_eficiente$sd +theta + +# Adição da linha que tangencia a carteira eficiente e que tem como intercepto o retorno à taxa livre de risco + +abline(a = tx_livre_risco, b=theta, col="green") + + + +``` + + +![](/blog/2021-04-08-como-obter-a-fronteira-eficiente-usando-o-r_files/Rplot03.png) + +### REFERÊNCIA BIBLIOGRÁFICA + +ELTON, E. J.; GRUBER, M. J.; BROWN S. J.; GOETZMANN, W. N. Moderna Teoria de Carteiras e Análises de Investimentos. Tradução de Helga Hoffmann. 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