MIPC
Erik De Luca
2023-10-15
knitr::optsxx_chunk$set(echo = TRUE)
library(quantmod)
library(highcharter)
library(PerformanceAnalytics)
library(tidyverse)
library(dplyr)
library(data.table)
library(DEoptim)
library(timetk)
library(plotly)
library(yahoofinancer)
library(RColorBrewer)
theme_set(theme_bw())
pal = brewer.pal(8, "Dark2")
conflicted::conflict_prefer("select", "dplyr")1 Importazione dati
Costruisco una tabella con i tickers, le categorie e i pesi dei titoli che comporranno il nostro portafoglio.
detailPortfolio =
tibble(
tickerList = c(
"SPG", # REITS
"OHI",
"CCOM.TO", # Commodities
"DBC",
"FCX",
"KSM-F6.TA",
"0P0001MN8G.F", # CAT BOND
"CDZ.TO", # dividend
"NOBL",
"NSRGY",
"CNI",
"WFAFY",
"UU.L",
"KO",
"NVS",
"NVDA", # nvidia no dividendi
# "SHY", # short term bond -- in veritร sono etf che riproducono l'andamento
# "VGSH",
"SPTS",
"IBGS.AS",
"6C=F", # cash -- ho messo un future sui dollari canadesi per rappresentare la liquiditร
# "XT2D.L", # swap -- non sapevo cosa mettere
"XGIU.MI" # Inflation linked
),
category = c(
rep("REITS",2),
rep("Commodities",4),
rep("CAT BOND",1),
rep("Dividends",9),
rep("Short term bonds",2),
rep("Cash",1),
rep("Inflation linked",1)
),
weight = c(
.08,
.06,
.046,
# comodities
.009,
.01,
.057,
.07,
# cat bond
.02,
# div
.06,
.02,
.01,
.017,
.005,
.005,
.05,
.023,
# .07,
# .07,
.05,
.128,
.15,
# .00,
.13
)
)
detailPortfolio1.1 Indici a base fissa
Importo i dati da Yahoo Finance degli ultimi 5 anni e costruisco il portafoglio senza i pesi di ciascun titolo.
stockData = lapply(detailPortfolio$tickerList,
getSymbols,
src = "yahoo",
from = as.Date("2018-09-30"),
to = as.Date("2023-09-29"),
auto.assign = F
)
# aggiusto i ticket che hanno cambiato nome durante l'importazione
detailPortfolio$tickerList[detailPortfolio$tickerList == "KSM-F6.TA"] = "KSM.F6.TA"
detailPortfolio$tickerList[detailPortfolio$tickerList == "6C=F"] = "X6C"
# compatto le diverse liste
nominalPortfolio = do.call(merge,stockData) %>%
na.omit
# il CAT BOND รจ privo di volumme
nominalPortfolio$X0P0001MN8G.F.Volume = 1
for(i in 1:nrow(detailPortfolio))
{
columnSelect = (!names(nominalPortfolio) %like% "Volume") & names(nominalPortfolio) %like% detailPortfolio$tickerList[i]
nominalPortfolio[,columnSelect] = nominalPortfolio[,columnSelect] / rep(coredata(nominalPortfolio[1,columnSelect])[1],5)
}
as_tibble(nominalPortfolio)Di seguito il grafico dellโandamento degli indici a base fissa dei titoli che compongono il portafoglio.
grafico = highchart(type = "stock")
for(i in 1:nrow(detailPortfolio))
grafico = hc_add_series(grafico,
Cl(nominalPortfolio[,names(nominalPortfolio) %like% detailPortfolio$tickerList[i]]),
name = detailPortfolio$tickerList[i])
grafico1.2 Andamento titoli nel portafoglio
Aggiungo i pesi iniziali del portafoglio.
portfolio = nominalPortfolio
for(i in 1:nrow(detailPortfolio))
{
columnSelect = (!names(portfolio) %like% "Volume") & names(portfolio) %like% detailPortfolio$tickerList[i]
portfolio[,columnSelect] = coredata(nominalPortfolio[,columnSelect]) * detailPortfolio$weight[i]
}
as_tibble(portfolio)Nel grafico seguente sono presenti i diversi titoli che compongono il portafoglio con il loro peso.
grafico = highchart(type = "stock")
for(i in 1:nrow(detailPortfolio))
grafico = hc_add_series(grafico,
Cl(portfolio[,names(portfolio) %like% detailPortfolio$tickerList[i]]),
name = detailPortfolio$tickerList[i])
grafico1.3 Creazione del portafoglio
Creo il portafoglio sommando gli indici dei titoli moltiplicati per i loro pesi. Cosรฌ da ottenere lโandamento complessivo del portafoglio.
columnNames = c("Open", "High", "Low", "Close", "Volume", "Adjusted")
myPortfolio = matrix(NA, nrow(portfolio),ncol = length(columnNames))
for(i in 1:length(columnNames))
{
columnSelect = names(portfolio) %like% columnNames[i]
myPortfolio[,i] = sapply(1:nrow(portfolio), function(r) sum(coredata(portfolio[r,columnSelect])))
}
colnames(myPortfolio) = paste("Portfolio", columnNames, sep = ".")
myPortfolio = xts(myPortfolio, order.by = index(portfolio))
as_tibble(myPortfolio)p = myPortfolio %>%
fortify() %>%
ggplot(aes(x = Index, y = Portfolio.Open)) +
geom_smooth(method = "gam",
formula = formula(y ~ s(x)),
fill = pal[5],
aes(color = pal[5]),
alpha = .3) +
geom_line(color = pal[3]) +
geom_pointrange(aes(ymin = Portfolio.Low,
ymax = Portfolio.High),
alpha = 0.22,
fill = 'turquoise1',
size = .1) +
# geom_col(aes(y = Portfolio.Volume),inherit.aes = F) +
xlab("") +
ylab("") +
scale_y_continuous(labels = function(x) scales::percent(x-1)) +
scale_color_manual(values = pal[5], labels = "Prediction of portfolio trends using splines") +
theme(legend.position = "bottom",
legend.title = element_blank())
p
# ggplotly(p)2 Decomposizione serie storica
Decompongo la serie storica per visualizzare le sue differenti componenti.
pfDecomposto = decompose(ts(myPortfolio$Portfolio.Open %>% as.vector(),
start = c(2022, 9, 29),
# end = c(2023, 09, 28),
frequency = 7))
# plot(pfDecomposto)
p = tibble(Dates = seq(as.Date("2022-09-29"),
length = length(pfDecomposto$x),
by = "days"),
Serie = pfDecomposto$x %>% coredata(),
Seasonal = pfDecomposto$seasonal %>% coredata(),
Trend = pfDecomposto$trend %>% coredata(),
Random = pfDecomposto$random %>% coredata()) %>%
gather(key = "Type", -Dates, value = "y") %>%
ggplot(aes(x = Dates, y = y)) +
geom_line() +
facet_grid(rows = vars(Type),
scales = "free_y") +
ylab("") +
scale_x_date(date_breaks = "1 month",
date_labels = "%b")
ggplotly(p, dynamicTicks = TRUE) %>%
# rangeslider()
plotly::layout(hovermode = "x")3 Ottimizzazione
Lโattuale portafoglio ha questi indici, il nostro obiettivo ora รจ quello di ottimizzare il portafoglio per massimizzare lโindice sharpe rato, ovvero massimizzare il rendimento del portafoglio e al contempo minimizzare il suo rischio.
nominalPortfolioAdj = nominalPortfolio[,names(nominalPortfolio) %like% "Adjusted"] %>%
CalculateReturns() %>%
to.yearly.contributions() %>%
na.omit()
weight = detailPortfolio$weight
nominalPortfolioAdj = nominalPortfolioAdj[,names(nominalPortfolioAdj) != "Portfolio Return"]
mean_ret = colMeans(nominalPortfolioAdj)
cov_mat = nominalPortfolio[,names(nominalPortfolio) %like% "Adjusted"] %>%
CalculateReturns() %>%
to.quarterly.contributions() %>%
na.omit() %>%
cov()
# solo quando i volumi non sono degeneri
return3mesi = nominalPortfolio[,!names(nominalPortfolio) %like% "Volume"] %>%
CalculateReturns() %>%
to.period.contributions("quarters") %>%
na.omit()
var3m = VaR(R = return3mesi[,names(return3mesi) %like% "Adjusted"],
method = "historical",
portfolio_method = "component",
weights = weight)
port_risk = var3m$hVaR
cov_mat = cov_mat[rownames(cov_mat) != "Portfolio Return",
colnames(cov_mat) != "Portfolio Return"]
port_returns = sum(mean_ret * weight)
port_risk = sqrt(t(weight) %*% (cov_mat %*% weight))
sharpe_ratio = port_returns/port_risk
tibble("Return" = port_returns,
"Risk" = port_risk,
"VaR a 3 mesi" = var3m$hVaR,
"Sharpe ratio" = sharpe_ratio)Verrร fatta una simulazione tramite metodo MonteCarlo dove si ripeterร lโesperimento estraendo casualmente i pesi, cercando cosรฌ la miglior combinazione di portafoglio. In questo caso lโesperimento verrร ripetuto 5000 volte ma i pesi del portafoglio non saranno completamente casuali, varieranno attorno ai valori da noi preimpostati.
set.seed(1)
num_port = 5000
# Creating a matrix to store the weights
all_wts = matrix(nrow = num_port,
ncol = nrow(detailPortfolio))
# Creating an empty vector to store
# Portfolio returns
port_returns = vector('numeric', length = num_port)
# Creating an empty vector to store
# Portfolio Standard deviation
port_risk = vector('numeric', length = num_port)
# Creating an empty vector to store
# Portfolio Sharpe Ratio
sharpe_ratio = vector('numeric', length = num_port)
for (i in seq_along(port_returns)) {
precisione = 0.9
wts = sapply(1:length(weight), function(i) runif(1,precisione * weight[i], (2 - precisione) * weight[i]))
# wts = runif(length(tickerList))
wts = wts/sum(wts)
# Storing weight in the matrix
all_wts[i,] = wts
# Portfolio returns
port_ret = sum(wts * mean_ret)
# port_ret <- ((port_ret + 1)^252) - 1
# Storing Portfolio Returns values
port_returns[i] = port_ret
# Creating and storing portfolio risk
port_sd = sqrt(t(wts) %*% (cov_mat %*% wts))
# Piรน preciso ma ci mette troppo
# port_sd = VaR(
# R = return3mesi[, names(return3mesi) %like% "Adjusted"],
# method = "historical",
# portfolio_method = "component",
# weights = wts
# )$hVaR
port_risk[i] = port_sd
# Creating and storing Portfolio Sharpe Ratios
# Assuming 0% Risk free rate
sr = port_ret/port_sd
sharpe_ratio[i] = sr
}3.1 Pesi del portafoglio
Di seguito vengono mostrati i pesi assegnati dal portafoglio con sharpe ratio maggiore.
# Storing the values in the table
portfolio_values = tibble(Return = port_returns,
Risk = port_risk,
SharpeRatio = sharpe_ratio)
# Converting matrix to a tibble and changing column names
all_wts = all_wts %>%
data.frame() %>%
tibble
colnames(all_wts) = detailPortfolio$tickerList
# Combing all the values together
portfolio_values = tibble(cbind(all_wts, portfolio_values))
colnames(portfolio_values)[1:nrow(detailPortfolio)] = detailPortfolio$tickerList
min_var = portfolio_values[which.min(portfolio_values$Risk),]
max_sr = portfolio_values[which.max(portfolio_values$SharpeRatio),]
# weightOLD = weight
weight = max_sr[,1:nrow(detailPortfolio)] %>%
as.numeric() %>%
round(4)
# con l'arrotondamento potrebbe non fare 1 e lo calibro con il primo titolo, ciรฒ non influenzerร significativamente sullo scostamento del portafoglio
weight[1] = weight[1] + 1 - sum(weight)
max_sr %>%
t() %>%
data.frame()p = max_sr %>%
gather(detailPortfolio$tickerList, key = Asset,
value = Weights) %>%
cbind(Category = factor(detailPortfolio$category)) %>%
mutate(Asset = Asset %>%
as.factor() %>%
fct_reorder(Weights),
Percentage = str_c(round(Weights*100,2), "%")) %>%
ggplot(aes(x = Asset,
y = Weights,
fill = Category,
label = Percentage)) +
geom_bar(stat = 'identity') +
geom_label(nudge_y = .01, size = 3) +
theme_minimal() +
labs(x = 'Tickers',
y = 'Weights',
title = "Weights of the portfolio tangent to the efficient frontier") +
scale_y_continuous(labels = scales::percent) +
guides(fill = guide_legend(override.aes = aes(label = ""))) +
theme(legend.position = "bottom") +
coord_flip()
ggplotly(p)3.2 Frontiera efficiente
Il grafico sottostante mostra i valori dei portafogli creati durante il processo di ottimizzazione.
p = portfolio_values %>%
ggplot(aes(x = Risk, y = Return, color = SharpeRatio)) +
geom_point() +
theme_classic() +
scale_y_continuous(labels = scales::percent) +
scale_x_continuous(labels = scales::percent) +
labs(x = 'Annual risk',
y = 'Annual return',
title = "Portfolio optimization and efficient frontier") +
geom_point(aes(x = Risk,
y = Return),
data = max_sr,
color = 'darkred')
ggplotly(p)3.3 Portafoglio ottimizzato
# ricreo il portafoglio con i nuovi pesi
portfolio = nominalPortfolio
for(i in 1:nrow(detailPortfolio))
{
columnSelect = (!names(portfolio) %like% "Volume") & names(portfolio) %like% detailPortfolio$tickerList[i]
portfolio[,columnSelect] = coredata(nominalPortfolio[,columnSelect]) * weight[i]
}
columnNames = c("Open", "High", "Low", "Close", "Volume", "Adjusted")
myPortfolio = matrix(NA, nrow(portfolio),ncol = length(columnNames))
for(i in 1:length(columnNames))
{
columnSelect = names(portfolio) %like% columnNames[i]
myPortfolio[,i] = sapply(1:nrow(portfolio), function(r) sum(coredata(portfolio[r,columnSelect])))
}
colnames(myPortfolio) = paste("Portfolio", columnNames, sep = ".")
myPortfolio = xts(myPortfolio, order.by = index(portfolio))myPortfolio %>%
fortify() %>%
ggplot(aes(x = Index, y = Portfolio.Open)) +
geom_smooth(method = "gam",
formula = formula(y ~ s(x)),
fill = pal[5],
aes(color = pal[5]),
alpha = .3) +
geom_line(color = pal[3]) +
geom_pointrange(aes(ymin = Portfolio.Low,
ymax = Portfolio.High),
alpha = 0.22,
fill = 'turquoise1',
size = .1) +
# geom_col(aes(y = Portfolio.Volume),inherit.aes = F) +
xlab("") +
ylab("") +
scale_y_continuous(labels = function(x) scales::percent(x-1)) +
scale_color_manual(values = pal[5], labels = "Prediction of portfolio trends using splines") +
theme(legend.position = "bottom",
legend.title = element_blank())
4 VaR
Nel seguente grafico sono rappresentati i diversi titoli con il loro rendimento, la loro varianza sulle ascisse e invece sono colorati in base al loro coefficiente di variazione. Questo grafico aiuta nella scelta dei pesi iniziali (pre ottimizzazione) dato che รจ facile visualizzare quelli che rendono meglio.
returnTicker = xts()
for(i in 1:nrow(detailPortfolio))
returnTicker = cbind(returnTicker,dailyReturn(Cl(nominalPortfolio[,names(portfolio) %like% detailPortfolio$tickerList[i]])))
colnames(returnTicker) = detailPortfolio$tickerList
returnTickerIndici = returnTicker %>%
as.tibble() %>%
summarise_all(sum) %>%
pivot_longer(1:nrow(detailPortfolio), names_to = "Titoli", values_to = "Rendimento") %>%
add_column(returnTicker %>%
as.tibble() %>%
summarise_all(sd) %>%
pivot_longer(1:nrow(detailPortfolio), names_to = "Titoli", values_to = "Varianza") %>%
dplyr::select(Varianza)
) %>%
mutate(Variazione = ifelse(round(Rendimento, 2) != 0,
Varianza/abs(Rendimento),
1))
hValMin = 1.8 # giocando con questo parametro si cambia l'asse delle y modificando la distanza dei punti estremi ai punti centrali
hValMax = 1
p = returnTickerIndici %>%
mutate(quantili = punif(Rendimento,
min = hValMin * min(Rendimento), # se non metto hvalmin il min di Rendimento vale 0 e di conseguenza il log tende a meno infinito
max = hValMax * max(Rendimento),
log.p = T)) %>%
ggplot(aes(y = quantili,
x = Varianza,
color = Variazione,
label = Titoli,
z = Rendimento # serve solo per l'etichetta nel grafico interattivo
)) +
geom_point(size = 1.5) +
scale_color_distiller(palette = "RdYlGn", direction = -1) +
scale_x_log10(labels = scales::percent_format(accuracy = .2),
breaks = scales::breaks_log(n = 10, base = 10)) +
scale_y_continuous(labels = function(x) scales::percent(qunif(x,
min = hValMin * min(returnTickerIndici$Rendimento),
max = hValMax * max(returnTickerIndici$Rendimento),
log.p = T), # associo i valori originali invertendo la funzione di ripartizione
scale = 1),
breaks = scales::pretty_breaks(10)) +
labs(x = "Variation", y = "Return", color = "Coefficient \nof variation") +
theme(legend.position = "right",
legend.title.align = 0)
ggplotly(p, tooltip = c("z", "x", "color", "label"))return3mesi = nominalPortfolio %>%
CalculateReturns %>%
to.period.contributions("quarters")
weight_max_sr = max_sr %>%
t() %>%
head(nrow(detailPortfolio)) %>%
as.vector()
VaR(return3mesi[,names(return3mesi) %like% "Open"],
method = "historical",
weights = weight_max_sr,
portfolio_method = "marginal") %>%
pivot_longer(1:length(weight_max_sr)+1, names_to = "Titoli", values_to = "VaR") %>%
mutate(VaR = round(VaR *100, 2)) %>%
ggplot(aes(x = Titoli, y = VaR, fill = VaR)) +
geom_col() +
geom_hline(aes(yintercept = PortfolioVaR), color = "orchid") +
coord_flip() +
scale_fill_distiller(palette = "RdYlGn", direction = 1) +
theme(legend.position = "none") +
scale_x_discrete(labels = function(x) str_remove(x,".Open")) +
labs(x = "") +
scale_y_continuous(labels = scales::percent_format(),
breaks = scales::pretty_breaks(8))
Nellโistogramma seguente รจ rappresentata la simulazionedi 1000000 campioni presi da una normale di media pari alla media del rendimento del portafoglio su base quadrimestrale e la varianza pari alla varianza del portafoglio su base quadrimestrale.
media = sapply(1:nrow(return3mesi), function(i) sum(return3mesi[i,names(nominalPortfolio) %like% "Adjusted"] * weight_max_sr)) %>% mean(na.rm = T)
varianza = sapply(1:nrow(return3mesi), function(i) sum(return3mesi[i,names(nominalPortfolio) %like% "Adjusted"] * weight_max_sr)) %>% sd(na.rm = T)
set.seed(1)
df = data.frame(x = rnorm(1E6, media, varianza))
ggplot(df, aes(x, ..density..)) +
geom_histogram(color = "violet",
fill = "orchid1",
alpha = .5,
bins = 30) +
geom_density(color = "aquamarine") +
geom_vline(xintercept = quantile(df$x, probs = .05),
color = "aquamarine2") +
annotate('text',
x = quantile(df$x, probs = .05),
y = 0.01,
color = "aquamarine4",
label = paste("VaR = ",df$x %>%
quantile(probs = .05) %>%
round(4))) +
xlab("")
5 Correlazione
Il correlogramma รจ utile per vedere la diversificazione del portafoglio.
correlazione = return3mesi[,names(return3mesi) %like% "Adjusted"] %>%
na.omit %>%
cor
colnames(correlazione) = stringr::str_remove(colnames(correlazione),".Adjusted")
rownames(correlazione) = stringr::str_remove(colnames(correlazione),".Adjusted")
# categories = levels(factor(detailPortfolio$category))
# pal[which(detailPortfolio$category[detailPortfolio$tickerList == "XGIU.MI"] == categories)]
# Funzione personalizzata per etichette colorate
color_labels = function(labels, colors) {
mapply(function(label, color) {
as.expression(bquote(bold(.(color)(.(label)))))
}, labels, colors, SIMPLIFY = FALSE)
}
p = correlazione %>%
reshape2::melt() %>%
ggplot(aes(x=Var1, y=Var2, fill = value, color = value)) +
geom_tile() +
scale_fill_distiller(name = "Correlation",
palette = "RdYlGn") +
# geom_label(aes(label = round(value,2)), size =2,label.size = 0) +
scale_x_discrete(limits = rev(rownames(correlazione))) +
# scale_color_identity() + # Imposta scale_color_identity per rimuovere il mapping del colore
guides(color = guide_legend(override.aes = list(color = NULL))) + # Imposta color su NULL per nasconderlo
theme(axis.title = element_blank(),
axis.text.x = element_text(angle = 30,vjust = .95, hjust = .95))
ggplotly(p)# map_chr(detailPortfolio$tickerList,~ palette.colors(length(levels(factor(detailPortfolio$category))),"Dark2")[as.numeric(labels(factor(detailPortfolio$category,ordered = T)[detailPortfolio$tickerList == .x]))])
#
# sapply(detailPortfolio$tickerList,function(x) as.numeric(labels(factor(detailPortfolio$category,ordered = T)[detailPortfolio$tickerList == x])))