Portfolio_Optimization
Neha Awasthi
2022-10-30
Introduction
Portfolio optimization is the process of selecting the best portfolio, out of the set of all feasible portfolios for our desired outcome. We typically maximize return and minimizes financial risk during the consideration. Sharpe Ratio is a measure of calculating risk adjusted return. Higher the Sharpe Ratio, more is the excess returns over that of a risk free investment, relative to the increased risk of the investment.
In this project, we are going to use Monte-Carlo simulation method to optimize portfolios using R. We write a function that takes Tickers, beginning date, end date and risk-free rate as input and gives us a feasible set. We optimize it using weight simulation and find an efficient frontier and then try to find an optimal portfolio.
Let’s load all the libraries needed first
library(quantmod)## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numeric## Loading required package: TTR## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoolibrary(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:xts':
##
## first, last## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionlibrary(ggplot2)Let’s write an R function that a vector of TICKERs, a begin date, an end date, (annualized) risk-free rate as arguments and produce an list containing the vector of stock means, the covariance matrix and relevant information (weights, mean, sigma, SR) for each simulated portfolio.
myMeanVarPort <- function(ticks,begin_date,end_date,rf_rate){
#Create a portfolio for the required tickers and given dates
retout <- NULL
retout <- xts(retout)
for(i in 1:length(ticks)){
prices = getSymbols(ticks[i], auto.assign = F)
returns <- periodReturn(prices, period = "monthly",
type = "arithmetic")
retout <- merge.xts(retout, returns)
}
colnames(retout) <- ticks
dates <- paste(begin_date, end_date, sep='/')
retout = retout[dates]
# Calculate mean
meanret <- colMeans(retout,na.rm = T)
# Get Covariance matrix
covar <- var(retout)
meanret <- as.matrix(meanret)
# Simulate weights for the portfolios for 100N portfolios
set.seed(12)
niter <- 100*length(ticks)
randomnums <- data.frame(replicate(length(ticks), runif(niter, 1, 10)))
wt_sim <- randomnums / rowSums(randomnums)
## initialize weight and Results matrices
weight <- matrix(data = NA, nrow = length(ticks), ncol = 1)
Results <- matrix(data = NA, nrow = niter, ncol = (length(ticks)+3))
#run the simulations.
# loop: each i is a portfolio
for (i in 1:niter){
# inner loop places weights into Results
for (k in 1:length(ticks)) {
Results[i,k] = weight[k,1] = wt_sim[i,k]
}
Results[i,length(ticks)+1] <- t(weight) %*% meanret #portfolio mean
Results[i,length(ticks)+2] <- sqrt(t(weight) %*% covar %*% weight) #portfolio sigma
Results[i,length(ticks)+3] <- (Results[i,length(ticks)+1] - rf_rate)/Results[i,length(ticks)+2] #portfolio SR ratio
}
#Calculate Portfolio Mean return, risk and Sharpe ratio
mretp <- t(weight) %*% meanret
sretp <- sqrt(t(weight) %*% covar %*% weight)
shrretp <- (mretp - rf_rate)/sretp
# Store the output in the list containing the vector of stock means, the covariance matrix, weights, mean, sigma, SR for each simulated portfolio
out_lst <-list()
out_lst[["Stock_means"]] <- meanret
out_lst[["Covariance_Matrix"]] <- covar
out_lst[["Portfolio_Weight"]] <- weight
out_lst[["Portfolio_Mean"]] <- mretp
out_lst[["Portfolio_Sigma"]] <- sretp
out_lst[["Portfolio_Sharpe_ratio"]] <- shrretp
out_lst[["Results"]] <- Results
return(out_lst)
}Show results for the given inputs
ticks <- c('GE', 'XOM', 'GBX', 'SBUX', 'PFE', 'HMC', 'NVDA')
begin_date <- 20140101
end_date <- 20171231
rf_rate <- 0.02
output <- myMeanVarPort(ticks,begin_date,end_date,rf_rate)The vector of stock means
output[["Stock_means"]]## [,1]
## GE -0.008371841
## XOM -0.003140133
## GBX 0.015364882
## SBUX 0.009024095
## PFE 0.004415191
## HMC -0.002464943
## NVDA 0.058165669Covariance Matrix:
output[["Covariance_Matrix"]]## GE XOM GBX SBUX PFE
## GE 0.0029425540 0.0010346975 1.211210e-03 8.072761e-04 0.0006465268
## XOM 0.0010346975 0.0016851744 1.625325e-03 2.360760e-04 0.0005344605
## GBX 0.0012112097 0.0016253253 1.036163e-02 6.399308e-05 0.0015767495
## SBUX 0.0008072761 0.0002360760 6.399308e-05 2.115326e-03 0.0005236355
## PFE 0.0006465268 0.0005344605 1.576749e-03 5.236355e-04 0.0018874080
## HMC 0.0006216792 0.0002043222 2.127414e-03 6.802793e-04 0.0008354260
## NVDA 0.0013543229 0.0010916138 3.182648e-03 3.671076e-04 0.0003674847
## HMC NVDA
## GE 0.0006216792 0.0013543229
## XOM 0.0002043222 0.0010916138
## GBX 0.0021274140 0.0031826485
## SBUX 0.0006802793 0.0003671076
## PFE 0.0008354260 0.0003674847
## HMC 0.0031606549 0.0011638388
## NVDA 0.0011638388 0.0111256402Weight:
output[["Portfolio_Weight"]]## [,1]
## [1,] 0.13229426
## [2,] 0.23842279
## [3,] 0.04149393
## [4,] 0.13445616
## [5,] 0.21049501
## [6,] 0.18606403
## [7,] 0.05677381Portfolio Mean:
output[["Portfolio_Mean"]]## [,1]
## [1,] 0.003767694Portfolio Sigma:
output[["Portfolio_Sigma"]]## [,1]
## [1,] 0.03192729Portfolio Sharpe Ratio:
output[["Portfolio_Sharpe_ratio"]]## [,1]
## [1,] -0.5084148Let’s check the data type of output of our function
class(output)## [1] "list"Let’s show the result as a data frame that includes portfolio mean, sigma and Sharpe ratio and the plot the feasible set
Results <- output[["Results"]]
colnames(Results) <- c(ticks, "PortMean", "PortSigma", "PortSharpeRatio")
Results <- as.data.frame(Results)
head(Results)## GE XOM GBX SBUX PFE HMC NVDA
## 1 0.04591352 0.12500521 0.11204062 0.14103162 0.2409166 0.20463972 0.1304527
## 2 0.17032874 0.09825230 0.16522497 0.17753802 0.1241027 0.19185299 0.0727003
## 3 0.22918285 0.03894855 0.24130303 0.05836888 0.1482585 0.11881530 0.1651229
## 4 0.09390609 0.21051359 0.07800949 0.03641164 0.2498903 0.08883252 0.2424364
## 5 0.07590857 0.19030177 0.25140423 0.10023674 0.1593137 0.08202033 0.1408146
## 6 0.03183638 0.12317640 0.22950334 0.17884515 0.1092784 0.11006486 0.2172955
## PortMean PortSigma PortSharpeRatio
## 1 0.010364394 0.03713120 -0.25950161
## 2 0.006709984 0.03799179 -0.34981282
## 3 0.012159533 0.04662683 -0.16815356
## 4 0.015065803 0.04154418 -0.11876987
## 5 0.012226077 0.04463796 -0.17415499
## 6 0.017337207 0.04636429 -0.05743198ggplot(data = Results , aes(x = PortSigma, y = PortMean, color = PortSharpeRatio)) +
geom_point(pch = 10, size =3)
Let’s print the mean returns, sigma and sharpe ration for the
portfolio
weight<- output[["Portfolio_Weight"]]
covar <- output[["Covariance_Matrix"]]
meanret<- output[["Stock_means"]]
mretp <- t(weight) %*% meanret
sretp <- sqrt(t(weight) %*% covar %*% weight)
shrretp <- (mretp - rf_rate)/sretp
cat("The portfolio mean return, sigma and sharpe ratio: ", mretp, sretp,shrretp)## The portfolio mean return, sigma and sharpe ratio: 0.003767694 0.03192729 -0.5084148Optimization
We optimize the portfolios by maximizing sharpe ratio
#Optimization
minmret = min(Results$PortMean)
maxmret = max(Results$PortMean)
seqmret = seq(round(minmret,3)-.001, maxmret+.001, .001)
optim <- Results %>% mutate(portnumber = index(Results)) %>%
mutate(ints = cut(PortMean ,breaks = seqmret),
lower = as.numeric( sub("\\((.+),.*", "\\1", ints) )) %>%
group_by(ints) %>%
summarise( lowerval = min(lower),
sig_optim = min(PortSigma),
ret_sigm_optim = PortMean[which.min(PortSigma)],
ret= PortMean,
retn_optim = PortMean[which.max(PortMean)],
sr_optim = PortSharpeRatio[which.max(PortSharpeRatio)],
numb = length(PortSigma),
portID=portnumber[which.max(PortMean)])## `summarise()` has grouped output by 'ints'. You can override using the
## `.groups` argument.head(optim)## # A tibble: 6 Ă— 9
## # Groups: ints [2]
## ints lowerval sig_optim ret_si…¹ ret retn_…² sr_op…³ numb portID
## <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int> <int>
## 1 (0.001,0.002] 0.001 0.0351 0.00185 0.00185 0.00185 -0.517 1 350
## 2 (0.002,0.003] 0.002 0.0328 0.00272 0.00276 0.00294 -0.488 6 216
## 3 (0.002,0.003] 0.002 0.0328 0.00272 0.00272 0.00294 -0.488 6 216
## 4 (0.002,0.003] 0.002 0.0328 0.00272 0.00272 0.00294 -0.488 6 216
## 5 (0.002,0.003] 0.002 0.0328 0.00272 0.00294 0.00294 -0.488 6 216
## 6 (0.002,0.003] 0.002 0.0328 0.00272 0.00232 0.00294 -0.488 6 216
## # … with abbreviated variable names ¹​ret_sigm_optim, ²​retn_optim, ³​sr_optimPlot the optimized potfolios
xcoord_minvar <- min(optim$sig_optim)
ycoord_minvar<- optim$ret[which.min(optim$sig_optim)]
ggplot(data = optim , aes(x = sig_optim, y = retn_optim, color = sr_optim)) +
geom_point(pch = 10, size = 3) +
annotate("segment", x = xcoord_minvar, y = ycoord_minvar,
xend = xcoord_minvar + .005, yend =ycoord_minvar ,
arrow=arrow(), color = "blue") +
annotate("text", x = .040, y = ycoord_minvar, label = 'Min Variance', color = 'red')+
annotate("segment", x = 0.0395, y = 0.018,
xend = 0.0395+ .005, yend =0.018 ,
arrow=arrow(), color = "blue") +
annotate("text", x = .050, y = 0.018, label = 'Optimal Portfolio', color = 'red')
Conclusion:
Monte-Carlo Simulation is one of the methods typically used to optimize portfolios. There is however no one correct method or result as the paramters may differ and it involves some speculation. That said, let’s looks at some advantages and disadvantages of this method:
Advantages:
The biggest advantage of this method is its ability to factor in a range of values for various inputs converting chances into choices. It can also be easily plotted for visual aid.
Disadvantages:
Since it depends on inputs it needs the assumptions to be fair Another disadvantage is that ot tends to underestimate the behaviorial aspect of finance and the irrationality demonstrated by participants.
For all its pros and cons, Monte-Carlo simulation remains one of the most widely used methods in finance for optimization and is a useful tool for advisors.