Efficient equity portfolios using mean variance optimisation in R
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The document outlines the methodology for constructing efficient equity portfolios using mean-variance optimization and the R package 'fportfolio'. It details data manipulation, portfolio optimization steps, risk-free rate settings, as well as plotting techniques to visualize the efficient frontier. The analysis includes performance metrics such as Sharpe ratios and constraints on asset weights.
![Efficient Equity Portfolios using Mean
Variance Optimization
Gregg Barrett
======================== Efficient Equity Portfolios ========================
Data
setwd("~/R/datasets")
dat = read.csv("Stock_Bond.csv", header = T)
prices = cbind(dat$GM_AC, dat$F_AC, dat$CAT_AC, dat$UTX_AC,
dat$MRK_AC, dat$IBM_AC)
n = dim(prices)[1]
returns = 100 * (prices[2:n, ] / prices[1:(n-1), ] - 1)
pairs(returns)
mean_vect = colMeans(returns)
cov_mat = cov(returns)
sd_vect = sqrt(diag(cov_mat))](https://image.slidesharecdn.com/efficientequityportfolios-160531115335/85/Efficient-equity-portfolios-using-mean-variance-optimisation-in-R-1-320.jpg)
![Function
library(quadprog)
## Warning: package 'quadprog' was built under R version 3.2.3
eff_front_fn = function(returns, muP, mu_free, lower_limit_weight = -Inf, up
per_limit_weight = +Inf){
n_stocks = dim(returns)[2]
mean_vect = apply(returns, 2, mean)
cov_mat = cov(returns)
sd_vect = sqrt(diag(cov_mat))
Amat = cbind(rep(1,n_stocks),mean_vect)
bvec = c(1,NaN)
if( is.finite(lower_limit_weight) ){
Amat = cbind(Amat, diag(1, nrow = n_stocks))
bvec = c(bvec, lower_limit_weight * rep(1, n_stocks))
}
if( is.finite(upper_limit_weight) ){
Amat = cbind(Amat, -diag(1, nrow = n_stocks))
bvec = c(bvec, -upper_limit_weight * rep(1, n_stocks))
}
sdP = muP
weights = matrix(0, nrow = length(muP), ncol = n_stocks)
for( i in 1:length(muP) ){
bvec[2] = +muP[i]
result = solve.QP(Dmat = 2*cov_mat, dvec = rep(0, n_stocks), Amat = Ama
t, bvec = bvec, meq=2)
sdP[i] = sqrt(result$value)
weights[i,] = result$solution
}
sharpe = ( muP - mu_free ) / sdP
ind_ms = (sharpe == max(sharpe))
ind_mv = (sdP == min(sdP))
list( muP = muP, sdP = sdP, weights = weights, sharpe = sharpe, max_sharp
e = ind_ms, min_variance = ind_mv )
}](https://image.slidesharecdn.com/efficientequityportfolios-160531115335/85/Efficient-equity-portfolios-using-mean-variance-optimisation-in-R-2-320.jpg)
![Setting the risk free rate, the target
porfolio return and constraints
mu_free = 3.0/365
muP = seq(min(mean_vect), max(mean_vect), length.out=300)
mvo = eff_front_fn(returns, muP, mu_free = mu_free, lower_limit_weight = -0.
1, upper_limit_weight = 0.5)
sdP = mvo$sdP
Plot
plot(sdP, muP, type = "l",
xlim = c(0, 1.1 * max(sd_vect)),
ylim = c(0, 1.1 * max(mean_vect)),
lty=3)
points(0, mu_free, cex = 4, pch = "*")
sharpe = ( muP - mu_free ) / sdP
ind_ms = (sharpe == max(sharpe))
print(mvo$weights[ind_ms,] )
## [1] -0.091164544 -0.002905131 0.335298962 0.383700258 0.319482622
## [6] 0.055587833
lines(c(0, 2), mu_free + c(0, 2) * (muP[ind_ms] - mu_free) / sdP[ind_ms],
lwd = 4, lty = 1, col = "blue")
points(sdP[ind_ms], muP[ind_ms], cex = 4, pch = "*")
ind_mv = (sdP == min(sdP))
points(sdP[ind_mv], muP[ind_mv], cex = 2, pch = "+")
ind3 = (muP > muP[ind_mv])
lines(sdP[ind3], muP[ind3], type = "l", xlim = c(0, 0.25),
ylim = c(0, 0.3), lwd = 3, col = "red")
text(sd_vect[1], mean_vect[1], "GM", cex = 1.15)
text(sd_vect[2], mean_vect[2], "F", cex = 1.15)
text(sd_vect[3], mean_vect[3], "CAT", cex = 1.15)
text(sd_vect[4], mean_vect[4], "UTX", cex = 1.15)
text(sd_vect[5], mean_vect[5], "MRK", cex = 1.15)
text(sd_vect[6], mean_vect[6], "IBM", cex = 1.15)](https://image.slidesharecdn.com/efficientequityportfolios-160531115335/85/Efficient-equity-portfolios-using-mean-variance-optimisation-in-R-3-320.jpg)
![##
##
## Rmetrics Package fBasics
## Analysing Markets and calculating Basic Statistics
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
## Loading required package: fAssets
## Warning: package 'fAssets' was built under R version 3.2.3
##
##
## Rmetrics Package fAssets
## Analysing and Modeling Financial Assets
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
##
##
## Rmetrics Package fPortfolio
## Portfolio Optimization
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
setwd("~/R/datasets")
dat = read.csv("Stock_Bond.csv", header = T)
prices = cbind(dat$GM_AC, dat$F_AC, dat$CAT_AC, dat$UTX_AC,
dat$MRK_AC, dat$IBM_AC)
n = dim(prices)[1]
returns = 100 * (prices[2:n, ] / prices[1:(n-1), ] - 1)
pairs(returns)](https://image.slidesharecdn.com/efficientequityportfolios-160531115335/85/Efficient-equity-portfolios-using-mean-variance-optimisation-in-R-5-320.jpg)
![mean_vect = colMeans(returns)
cov_mat = cov(returns)
sd_vect = sqrt(diag(cov_mat))
Constraints
returns3 = as.timeSeries(returns)
names(returns3) = c("GM", "F", "CAT", "UTX", "MRK", "IBM")
Constraints = c(
"minW[1:nAssets] = rep(-0.10, times = nAssets)",
"maxW[1:nAssets] = rep(0.50, times = nAssets)")
Spec
spec1 = portfolioSpec()
setRiskFreeRate(spec1) = 0.0082192](https://image.slidesharecdn.com/efficientequityportfolios-160531115335/85/Efficient-equity-portfolios-using-mean-variance-optimisation-in-R-6-320.jpg)
Efficient equity portfolios using mean variance optimisation in R
- 1. Efficient Equity Portfolios using Mean Variance Optimization Gregg Barrett ======================== Efficient Equity Portfolios ======================== Data setwd("~/R/datasets") dat = read.csv("Stock_Bond.csv", header = T) prices = cbind(dat$GM_AC, dat$F_AC, dat$CAT_AC, dat$UTX_AC, dat$MRK_AC, dat$IBM_AC) n = dim(prices)[1] returns = 100 * (prices[2:n, ] / prices[1:(n-1), ] - 1) pairs(returns) mean_vect = colMeans(returns) cov_mat = cov(returns) sd_vect = sqrt(diag(cov_mat))
- 2. Function library(quadprog) ## Warning: package 'quadprog' was built under R version 3.2.3 eff_front_fn = function(returns, muP, mu_free, lower_limit_weight = -Inf, up per_limit_weight = +Inf){ n_stocks = dim(returns)[2] mean_vect = apply(returns, 2, mean) cov_mat = cov(returns) sd_vect = sqrt(diag(cov_mat)) Amat = cbind(rep(1,n_stocks),mean_vect) bvec = c(1,NaN) if( is.finite(lower_limit_weight) ){ Amat = cbind(Amat, diag(1, nrow = n_stocks)) bvec = c(bvec, lower_limit_weight * rep(1, n_stocks)) } if( is.finite(upper_limit_weight) ){ Amat = cbind(Amat, -diag(1, nrow = n_stocks)) bvec = c(bvec, -upper_limit_weight * rep(1, n_stocks)) } sdP = muP weights = matrix(0, nrow = length(muP), ncol = n_stocks) for( i in 1:length(muP) ){ bvec[2] = +muP[i] result = solve.QP(Dmat = 2*cov_mat, dvec = rep(0, n_stocks), Amat = Ama t, bvec = bvec, meq=2) sdP[i] = sqrt(result$value) weights[i,] = result$solution } sharpe = ( muP - mu_free ) / sdP ind_ms = (sharpe == max(sharpe)) ind_mv = (sdP == min(sdP)) list( muP = muP, sdP = sdP, weights = weights, sharpe = sharpe, max_sharp e = ind_ms, min_variance = ind_mv ) }
- 3. Setting the risk free rate, the target porfolio return and constraints mu_free = 3.0/365 muP = seq(min(mean_vect), max(mean_vect), length.out=300) mvo = eff_front_fn(returns, muP, mu_free = mu_free, lower_limit_weight = -0. 1, upper_limit_weight = 0.5) sdP = mvo$sdP Plot plot(sdP, muP, type = "l", xlim = c(0, 1.1 * max(sd_vect)), ylim = c(0, 1.1 * max(mean_vect)), lty=3) points(0, mu_free, cex = 4, pch = "*") sharpe = ( muP - mu_free ) / sdP ind_ms = (sharpe == max(sharpe)) print(mvo$weights[ind_ms,] ) ## [1] -0.091164544 -0.002905131 0.335298962 0.383700258 0.319482622 ## [6] 0.055587833 lines(c(0, 2), mu_free + c(0, 2) * (muP[ind_ms] - mu_free) / sdP[ind_ms], lwd = 4, lty = 1, col = "blue") points(sdP[ind_ms], muP[ind_ms], cex = 4, pch = "*") ind_mv = (sdP == min(sdP)) points(sdP[ind_mv], muP[ind_mv], cex = 2, pch = "+") ind3 = (muP > muP[ind_mv]) lines(sdP[ind3], muP[ind3], type = "l", xlim = c(0, 0.25), ylim = c(0, 0.3), lwd = 3, col = "red") text(sd_vect[1], mean_vect[1], "GM", cex = 1.15) text(sd_vect[2], mean_vect[2], "F", cex = 1.15) text(sd_vect[3], mean_vect[3], "CAT", cex = 1.15) text(sd_vect[4], mean_vect[4], "UTX", cex = 1.15) text(sd_vect[5], mean_vect[5], "MRK", cex = 1.15) text(sd_vect[6], mean_vect[6], "IBM", cex = 1.15)
- 4. ======== Efficient Equity Portfolios using the package “fPortfolio” from Rmetrics ======== Data library(fPortfolio) ## Warning: package 'fPortfolio' was built under R version 3.2.3 ## Loading required package: timeDate ## Warning: package 'timeDate' was built under R version 3.2.3 ## Loading required package: timeSeries ## Warning: package 'timeSeries' was built under R version 3.2.3 ## Loading required package: fBasics ## Warning: package 'fBasics' was built under R version 3.2.3
- 5. ## ## ## Rmetrics Package fBasics ## Analysing Markets and calculating Basic Statistics ## Copyright (C) 2005-2014 Rmetrics Association Zurich ## Educational Software for Financial Engineering and Computational Science ## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY. ## https://www.rmetrics.org --- Mail to: [email protected] ## Loading required package: fAssets ## Warning: package 'fAssets' was built under R version 3.2.3 ## ## ## Rmetrics Package fAssets ## Analysing and Modeling Financial Assets ## Copyright (C) 2005-2014 Rmetrics Association Zurich ## Educational Software for Financial Engineering and Computational Science ## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY. ## https://www.rmetrics.org --- Mail to: [email protected] ## ## ## Rmetrics Package fPortfolio ## Portfolio Optimization ## Copyright (C) 2005-2014 Rmetrics Association Zurich ## Educational Software for Financial Engineering and Computational Science ## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY. ## https://www.rmetrics.org --- Mail to: [email protected] setwd("~/R/datasets") dat = read.csv("Stock_Bond.csv", header = T) prices = cbind(dat$GM_AC, dat$F_AC, dat$CAT_AC, dat$UTX_AC, dat$MRK_AC, dat$IBM_AC) n = dim(prices)[1] returns = 100 * (prices[2:n, ] / prices[1:(n-1), ] - 1) pairs(returns)
- 6. mean_vect = colMeans(returns) cov_mat = cov(returns) sd_vect = sqrt(diag(cov_mat)) Constraints returns3 = as.timeSeries(returns) names(returns3) = c("GM", "F", "CAT", "UTX", "MRK", "IBM") Constraints = c( "minW[1:nAssets] = rep(-0.10, times = nAssets)", "maxW[1:nAssets] = rep(0.50, times = nAssets)") Spec spec1 = portfolioSpec() setRiskFreeRate(spec1) = 0.0082192
- 7. Model and plot mvo2 = portfolioFrontier(returns3, spec1, Constraints) frontierPlot(mvo2, frontier = c("both"), col = c("black", "grey"), labels = TRUE, return = c("mean"), risk = c("Sigma")) minvariancePoints(mvo2, col = "red") tangencyPoints(mvo2, col = "green") tangencyLines(mvo2, col = "purple") Examination of weights weightsSlider(mvo2)