Homework04
Ebuchka
2019/03/28
firm_data1 = read.csv('3firmExample_data3.csv')
str(firm_data1)## 'data.frame': 59 obs. of 4 variables:
## $ date : Factor w/ 59 levels "1995/10/1","1995/11/1",..: 4 5 6 7 8 9 10 1 2 3 ...
## $ Nordstrom: num -0.03615 -0.0568 0.07821 -0.00302 -0.02757 ...
## $ Starbucks: num 0.00521 -0.02105 0.21244 0.2036 0.04797 ...
## $ Microsoft: num 0.1213 0.13923 0.03529 0.06501 0.00138 ...firm_data1$date## [1] 1995/3/1 1995/4/1 1995/5/1 1995/6/1 1995/7/1 1995/8/1 1995/9/1
## [8] 1995/10/1 1995/11/1 1995/12/1 1996/1/1 1996/2/1 1996/3/1 1996/4/1
## [15] 1996/5/1 1996/6/1 1996/7/1 1996/8/1 1996/9/1 1996/10/1 1996/11/1
## [22] 1996/12/1 1997/1/1 1997/2/1 1997/3/1 1997/4/1 1997/5/1 1997/6/1
## [29] 1997/7/1 1997/8/1 1997/9/1 1997/10/1 1997/11/1 1997/12/1 1998/1/1
## [36] 1998/2/1 1998/3/1 1998/4/1 1998/5/1 1998/6/1 1998/7/1 1998/8/1
## [43] 1998/9/1 1998/10/1 1998/11/1 1998/12/1 1999/1/1 1999/2/1 1999/3/1
## [50] 1999/4/1 1999/5/1 1999/6/1 1999/7/1 1999/8/1 1999/9/1 1999/10/1
## [57] 1999/11/1 1999/12/1 2000/1/1
## 59 Levels: 1995/10/1 1995/11/1 1995/12/1 1995/3/1 1995/4/1 ... 2000/1/1library(xts)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(PerformanceAnalytics)##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
##
## legenddate1 = as.Date(firm_data1[,1], "%Y/%m/%d")
#convert firm_data1 into time series data: xts
firm_data1.xts = as.xts(firm_data1[,-1], order.by = date1)
firm.data1<-coredata(firm_data1.xts)
summary(firm.data1)## Nordstrom Starbucks Microsoft
## Min. :-0.212880 Min. :-0.47970 Min. :-0.17634
## 1st Qu.:-0.057395 1st Qu.:-0.01734 1st Qu.:-0.01826
## Median : 0.004950 Median : 0.04064 Median : 0.03848
## Mean : 0.001545 Mean : 0.02846 Mean : 0.04271
## 3rd Qu.: 0.064980 3rd Qu.: 0.09416 3rd Qu.: 0.11174
## Max. : 0.312480 Max. : 0.27967 Max. : 0.28153skewness(firm.data1)## Nordstrom Starbucks Microsoft
## Skewness 0.2422885 -0.8882285 0.1712068rbind(apply(firm.data1, 2, summary),
apply(firm.data1, 2, skewness),
apply(firm.data1, 2, kurtosis))## Nordstrom Starbucks Microsoft
## Min. -0.212880000 -0.47970000 -0.17634000
## 1st Qu. -0.057395000 -0.01734000 -0.01826500
## Median 0.004950000 0.04064000 0.03848000
## Mean 0.001545085 0.02846068 0.04271153
## 3rd Qu. 0.064980000 0.09415500 0.11173500
## Max. 0.312480000 0.27967000 0.28153000
## 0.242288454 -0.88822851 0.17120676
## 0.351952075 1.85144933 -0.08728437library(plyr)
library(quantmod)## Loading required package: TTR## Version 0.4-0 included new data defaults. See ?getSymbols.tickers<-c("JWN", "SBUX", "MSFT")
data.env<-new.env()l_ply(tickers, function(sym) try(getSymbols(sym, env=data.env), silent=T))## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.stocks <- tickers[tickers %in% ls(data.env)]data <- xts()for(i in seq_along(stocks)) {
symbol <- stocks[i]
data <- merge(data, Ad(get(symbol, envir=data.env)))
}
head(data)## JWN.Adjusted SBUX.Adjusted MSFT.Adjusted
## 2007-01-03 08:00:00 35.71681 14.13297 22.47883
## 2007-01-04 08:00:00 36.32145 14.14900 22.44119
## 2007-01-05 08:00:00 35.64033 14.08886 22.31321
## 2007-01-08 08:00:00 35.82799 14.03674 22.53152
## 2007-01-09 08:00:00 36.21721 13.97660 22.55411
## 2007-01-10 08:00:00 36.39095 13.93249 22.32826str(data)## An 'xts' object on 2007-01-03 08:00:00/2019-03-27 08:00:00 containing:
## Data: num [1:3079, 1:3] 35.7 36.3 35.6 35.8 36.2 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:3] "JWN.Adjusted" "SBUX.Adjusted" "MSFT.Adjusted"
## Indexed by objects of class: [POSIXct,POSIXt] TZ:
## xts Attributes:
## NULLdata<-xts(coredata(data), order.by = as.Date(index(data), tz=""))
head(data)## JWN.Adjusted SBUX.Adjusted MSFT.Adjusted
## 2007-01-03 35.71681 14.13297 22.47883
## 2007-01-04 36.32145 14.14900 22.44119
## 2007-01-05 35.64033 14.08886 22.31321
## 2007-01-08 35.82799 14.03674 22.53152
## 2007-01-09 36.21721 13.97660 22.55411
## 2007-01-10 36.39095 13.93249 22.32826tail(data)## JWN.Adjusted SBUX.Adjusted MSFT.Adjusted
## 2019-03-20 43.45 71.63 117.52
## 2019-03-21 43.76 72.26 120.22
## 2019-03-22 42.93 71.96 117.05
## 2019-03-25 43.54 72.30 117.66
## 2019-03-26 43.73 72.96 117.91
## 2019-03-27 44.65 72.74 116.77library(fBasics)## Loading required package: timeDate##
## Attaching package: 'timeDate'## The following objects are masked from 'package:PerformanceAnalytics':
##
## kurtosis, skewness## Loading required package: timeSeries##
## Attaching package: 'timeSeries'## The following object is masked from 'package:zoo':
##
## time<-##
## Attaching package: 'fBasics'## The following object is masked from 'package:TTR':
##
## volatilitySigma = cov(firm_data1[,2:4])
std = sqrt(diag(Sigma))
ones = rep(1,3)
one.vec = matrix(ones, ncol=1)
a = inv(Sigma)%*%one.vec
b = t(one.vec)%*%a
mvp.w =a / as.numeric(b)
mvp.w## [,1]
## Nordstrom 0.3635998
## Starbucks 0.1936537
## Microsoft 0.4427465mvp.ret<-sum((mvp.w)*colMeans(firm_data1[,2:4]))
mvp.ret## [1] 0.02498369mu<-0.06/12
return <- firm_data1[,2:4]
Ax <- rbind(2*cov(return), colMeans(return), rep(1, ncol(return)))
Ax <- cbind(Ax, rbind(t(tail(Ax, 2)), matrix(0, 2, 2)))
b0 <- c(rep(0, ncol(return)), mu, 1)
out<-solve(Ax, b0)
wgt<-out[1:3]
wgt## Nordstrom Starbucks Microsoft
## 0.875635380 0.116816458 0.007548163sum(wgt)## [1] 1ret.out<-sum(wgt*colMeans(return))
ret.out.annual<-ret.out*12
ret.out.annual## [1] 0.06std.out<-sqrt(t(wgt)%*%cov(return)%*%wgt)
std.out.annual<-std.out*sqrt(12)
std.out.annual## [,1]
## [1,] 0.335302return = firm_data1[,2:4]mu=0.06/12
minvariance <- function(return, mu) {
#return <- log(tail(assets, -1) / head(assets, -1))
Ax <- rbind(2*cov(return), colMeans(return), rep(1, ncol(return)))
Ax <- cbind(Ax, rbind(t(tail(Ax, 2)), matrix(0, 2, 2)))
b0 <- c(rep(0, ncol(return)), mu, 1)
zx<-solve(Ax, b0)
weight<-zx[1:ncol(return)]
ret.out<-sum(weight*colMeans(return))
std.out<-sqrt(t(wgt)%*%cov(return)%*%wgt)
list(weight=weight, rtn=ret.out, sd=std.out)
}
minvariance(return, mu)## $weight
## Nordstrom Starbucks Microsoft
## 0.875635380 0.116816458 0.007548163
##
## $rtn
## [1] 0.005
##
## $sd
## [,1]
## [1,] 0.09679334frontier <- function(return){
#return <- log(tail(assets, -1) / head(assets, -1))
n = ncol(return)
Q = cov(return)
Ax <- rbind(2*cov(return), colMeans(return), rep(1, n))
Ax <- cbind(Ax, rbind(t(tail(Ax, 2)), matrix(0, 2, 2)))
r <- colMeans(return)
rbase <- seq(min(r), max(r), length = 100)
s <- sapply(rbase, function(x) {
b0 <- c(rep(0, ncol(return)), x, 1)
y <- head(solve(Ax, b0), n)
sqrt(y%*%Q%*%y)
})
plot(s, rbase, xlab = 'Std', ylab = 'Return')
}
frontier(return)
library(timeSeries)
library(PerformanceAnalytics)
#install.packages("rugarch", dependencies=TRUE)
#install.packages("PerformanceAnalytics", dependencies=TRUE)
#install.packages("fAssets", dependencies=TRUE)
#install.packages("fPortfolio",dependencies=TRUE)
library(fPortfolio)## Loading required package: fAssetsreturn = firm_data1[,2:4]ret.ts<- timeSeries(return, date1)
chart.CumReturns(ret.ts, legend.loc = 'topleft', main = '')
Spec = portfolioSpec()
setSolver(Spec) = "solveRshortExact"
setTargetReturn(Spec) = mean(colMeans(ret.ts))## or set your own target return
Spec##
## Model List:
## Type: MV
## Optimize: minRisk
## Estimator: covEstimator
## Params: alpha = 0.05
##
## Portfolio List:
## Portfolio Weights: NA
## Target Return: 0.0242391
## Target Risk: NA
## Risk-Free Rate: 0
## Number of Frontier Points: 50
##
## Optim List:
## Solver: solveRshortExact
## Objective: portfolioObjective portfolioReturn portfolioRisk
## Trace: FALSEsetEstimator(Spec)="covMcdEstimator"setEstimator(Spec)= "covOGKEstimator"setEstimator(Spec)= "shrinkEstimator"Spec##
## Model List:
## Type: MV
## Optimize: minRisk
## Estimator: shrinkEstimator
## Params: alpha = 0.05
##
## Portfolio List:
## Portfolio Weights: NA
## Target Return: 0.0242391
## Target Risk: NA
## Risk-Free Rate: 0
## Number of Frontier Points: 50
##
## Optim List:
## Solver: solveRshortExact
## Objective: portfolioObjective portfolioReturn portfolioRisk
## Trace: FALSEConstraints="Short"
efficientPortfolio(ret.ts, Spec, Constraints)##
## Title:
## MV Efficient Portfolio
## Estimator: shrinkEstimator
## Solver: solveRshortExact
## Optimize: minRisk
## Constraints: Short
##
## Portfolio Weights:
## Nordstrom Starbucks Microsoft
## 0.3780 0.2044 0.4176
##
## Covariance Risk Budgets:
## Nordstrom Starbucks Microsoft
## 0.3872 0.2126 0.4001
##
## Target Returns and Risks:
## mean mu Cov Sigma CVaR VaR
## 0.0242 0.0242 0.0734 0.0667 0.1324 0.1243
##
## Description:
## Thu Mar 28 05:03:04 2019 by user: NomiotangencyPortfolio(ret.ts, Spec, Constraints)##
## Title:
## MV Tangency Portfolio
## Estimator: shrinkEstimator
## Solver: solveRshortExact
## Optimize: minRisk
## Constraints: Short
##
## Portfolio Weights:
## Nordstrom Starbucks Microsoft
## -0.0147 0.2486 0.7661
##
## Covariance Risk Budgets:
## Nordstrom Starbucks Microsoft
## -0.0037 0.1924 0.8113
##
## Target Returns and Risks:
## mean mu Cov Sigma CVaR VaR
## 0.0398 0.0398 0.0870 0.0853 0.1485 0.1131
##
## Description:
## Thu Mar 28 05:03:05 2019 by user: NomiominvariancePortfolio(ret.ts, Spec, Constraints)## Warning in as.vector(invSigma %*% one)/(one %*% invSigma %*% one): Recycling array of length 1 in vector-array arithmetic is deprecated.
## Use c() or as.vector() instead.##
## Title:
## MV Minimum Variance Portfolio
## Estimator: shrinkEstimator
## Solver: solveRshortExact
## Optimize: minRisk
## Constraints: Short
##
## Portfolio Weights:
## Nordstrom Starbucks Microsoft
## 0.3750 0.2047 0.4203
##
## Covariance Risk Budgets:
## Nordstrom Starbucks Microsoft
## 0.3823 0.2131 0.4045
##
## Target Returns and Risks:
## mean mu Cov Sigma CVaR VaR
## 0.0244 0.0244 0.0734 0.0667 0.1324 0.1237
##
## Description:
## Thu Mar 28 05:03:05 2019 by user: NomioportfolioConstraints(ret.ts, Spec, constraints = "LongOnly")##
## Title:
## Portfolio Constraints
##
## Lower/Upper Bounds:
## Nordstrom Starbucks Microsoft
## Lower 0 0 0
## Upper 1 1 1
##
## Equal Matrix Constraints:
## ceq Nordstrom Starbucks Microsoft
## Return 0.0242391 0.001545085 0.02846068 0.04271153
## Budget -1.0000000 -1.000000000 -1.00000000 -1.00000000
##
## Cardinality Constraints:
## Nordstrom Starbucks Microsoft
## Lower 0 0 0
## Upper 1 1 1Frontier <- portfolioFrontier(as.timeSeries(ret.ts), Spec, constraints = "Short") #vs "LongOnly" "Short"## Warning in as.vector(invSigma %*% one)/(one %*% invSigma %*% one): Recycling array of length 1 in vector-array arithmetic is deprecated.
## Use c() or as.vector() instead.frontierPlot(Frontier, col = c("orange", "red"), pch = 19)
weightsPlot(Frontier)
Data<-ret.ts
nAssets = getNAssets(portfolioData(Data))
Weights <- rep(1/nAssets, times = nAssets)
covRisk(Data, Weights)## Cov
## 0.07651345varRisk(Data, Weights, alpha = 0.05)###VaR of equal weights portfolio## VaR.5%
## -0.1474633cvarRisk(Data, Weights, alpha = 0.05)### CVaR for equal weight portfolio## CVaR.5%
## -0.166304setRiskFreeRate(Spec)<-0.01/12
tgPortfolio <- tangencyPortfolio(ret.ts, Spec, constraints = "Short")
tgPortfolio##
## Title:
## MV Tangency Portfolio
## Estimator: shrinkEstimator
## Solver: solveRshortExact
## Optimize: minRisk
## Constraints: Short
##
## Portfolio Weights:
## Nordstrom Starbucks Microsoft
## -0.0285 0.2502 0.7783
##
## Covariance Risk Budgets:
## Nordstrom Starbucks Microsoft
## -0.0066 0.1898 0.8168
##
## Target Returns and Risks:
## mean mu Cov Sigma CVaR VaR
## 0.0403 0.0403 0.0879 0.0865 0.1496 0.1146
##
## Description:
## Thu Mar 28 05:03:09 2019 by user: Nomiorf = 0.01/12
mr = colMeans(firm_data1[,2:4])
mr.mtx = matrix(mr, ncol=1)
mr_rf = mr - rf
mr_rf = matrix(mr_rf, ncol=1)
mr_rf## [,1]
## [1,] 0.0007117514
## [2,] 0.0276273446
## [3,] 0.0418781921a1 = inv(Sigma)%*%mr_rf
b1 = t(one.vec)%*%a1
tp = a1 / as.numeric(b1)
tp## [,1]
## Nordstrom -0.2061492
## Starbucks 0.2791516
## Microsoft 0.9269976ret.tp = sum(mr.mtx*tp)
ret.tp## [1] 0.04721981std.tp = sqrt((t(tp)%*%Sigma)%*%tp)
std.tp## [,1]
## [1,] 0.1015897sharpe.tp = (ret.tp - rf)/std.tp
sharpe.tp## [,1]
## [1,] 0.456606globminSpec <- portfolioSpec()globminPortfolio <- minvariancePortfolio(as.timeSeries(ret.ts), spec = globminSpec, constraints = "LongOnly")
print(globminPortfolio)##
## Title:
## MV Minimum Variance Portfolio
## Estimator: covEstimator
## Solver: solveRquadprog
## Optimize: minRisk
## Constraints: LongOnly
##
## Portfolio Weights:
## Nordstrom Starbucks Microsoft
## 0.3636 0.1937 0.4427
##
## Covariance Risk Budgets:
## Nordstrom Starbucks Microsoft
## 0.3636 0.1937 0.4427
##
## Target Returns and Risks:
## mean Cov CVaR VaR
## 0.0250 0.0733 0.1288 0.1183
##
## Description:
## Thu Mar 28 05:03:11 2019 by user: Nomiocol = rampPalette(ncol(ret.ts), “purple2green”) weights <- 100 * as.vector(getWeights(globminPortfolio)) names <- as.vector(names(ret.ts)) barplot(height = weights, names.arg = names, horiz = TRUE, las = 1, col = col) title(main = “Weights of Global Min Variance Portfolio”, xlab = “Weights %”) ```
library(quadprog)
mu = apply(firm_data1[,2:4], 2, mean)
Amat = cbind(rep(1,3),mu) # set the constraints matrix
muP = seq(.01,.08,length=300) # set of 300 possible given returnssdP = muP # set up storage for standard deviations of portfolio returns
weights = matrix(0,nrow=300,ncol=3) # storage for portfolio weights
i=1
for (i in 1:length(muP)) # find the optimal portfolios for each target expected return
{
bvec = c(1,muP[i]) # constraint vector
result = solve.QP(Dmat=2*Sigma,dvec=rep(0,3),Amat=Amat,bvec=bvec,meq=2)
sdP[i] = sqrt(result$value)
weights[i,] = result$solution
}pdf("3firmExample.pdf",width=6,height=5)
plot(sdP,muP,type="l",xlim=c(0,0.25),ylim=c(0,0.08),lty=1)mufree = 0.01/12 # input value of risk-free interest ratesharpe =( muP-mufree)/sdP # compute Sharpe ratiosind = (sharpe == max(sharpe)) # Find maximum Sharpe ratiooptions(digits=3)
weights[ind,] # Find tangency portfolio# show line of optimal portfolios## [1] -0.206 0.279 0.927ind2 = (sdP == min(sdP)) # find the minimum variance portfolioind3 = (muP > muP[ind2])Amat1 = cbind(rep(1,3),mu, diag(1,nrow=3)) # set the constraints matrix
t(Amat1)## Nordstrom Starbucks Microsoft
## 1.00000 1.0000 1.0000
## mu 0.00155 0.0285 0.0427
## 1.00000 0.0000 0.0000
## 0.00000 1.0000 0.0000
## 0.00000 0.0000 1.0000muP1 = seq(min(mu)+.0001,max(mu)-.0001,length=300)sdP1 = muP1 # set up storage for standard deviations of portfolio returns
weights1 = matrix(0,nrow=300,ncol=3) # storage for portfolio weights
i=1
for (i in 1:length(muP1)) # find the optimal portfolios for each target expected return
{
bvec1 = c(1,muP1[i], rep(0,3)) # constraint vector
result = solve.QP(Dmat=2*Sigma,dvec=rep(0,3),Amat=Amat1,bvec=bvec1,meq=2)
sdP1[i] = sqrt(result$value)
weights1[i,] = result$solution
}
pdf("3firmExample_noshort.ps",width=6,height=5)
plot(sdP1,muP1,type="l",xlim=c(0,0.25),ylim=c(0,0.08),lty=1)par(new=TRUE)## Warning in par(new = TRUE): calling par(new=TRUE) with no plotplot(sdP,muP,type="l",xlim=c(0,0.25),ylim=c(0,0.08),lty=1, col="green") # plot
mufree = 0.005 # input value of risk-free interest rate
sharpe =( muP-mufree)/sdP # compute Sharpe ratios
ind = (sharpe == max(sharpe)) # Find maximum Sharpe ratio
options(digits=3)
weights[ind,] # Find tangency portfolio# show line of optimal portfolios## [1] -0.326 0.297 1.029lines(c(0,2),mufree+c(0,2)*(muP[ind]-mufree)/sdP[ind],lwd=2,lty=3)
ind2 = (sdP == min(sdP)) # find the minimum variance portfolioind3 = (muP > muP[ind2])con = gzcon(url('https://github.com/systematicinvestor/SIT/raw/master/sit.gz', 'rb'))
source(con)
close(con)n <- dim(firm_data1.xts)[2]
constraints = new.constraints(n, lb = -Inf, ub = +Inf)constraints = add.constraints(rep(1, n), 1, type = '=', constraints)
ia <- create.historical.ia(firm_data1.xts, 12)weight <- min.risk.portfolio(ia, constraints)ia <- create.historical.ia(firm_data1.xts, 12)constraints = new.constraints(n, lb = 0, ub = 1)
constraints = add.constraints(diag(n), type='>=', b=0, constraints)
constraints = add.constraints(diag(n), type='<=', b=1, constraints)constraints = add.constraints(rep(1, n), 1, type = '=', constraints)weight <- min.risk.portfolio(ia, constraints)ifelse(!require(corpcor), install.packages("corpcor"), library(corpcor))## Loading required package: corpcor##
## Attaching package: 'corpcor'## The following object is masked _by_ '.GlobalEnv':
##
## cov.shrink## [1] "corpcor"ifelse(!require(lpSolve), install.packages("lpSolve"), library(lpSolve))## Loading required package: lpSolve## [1] "lpSolve"ef = portopt(ia, constraints, 50, 'Efficient Frontier') ##
## Attaching package: 'kernlab'## The following object is masked _by_ '.GlobalEnv':
##
## crossef## $weight
## Nordstrom Starbucks Microsoft
## [1,] 0.36360 0.1937 0.443
## [2,] 0.35417 0.1947 0.451
## [3,] 0.34475 0.1958 0.459
## [4,] 0.33533 0.1969 0.468
## [5,] 0.32590 0.1980 0.476
## [6,] 0.31648 0.1991 0.484
## [7,] 0.30705 0.2002 0.493
## [8,] 0.29763 0.2013 0.501
## [9,] 0.28820 0.2024 0.509
## [10,] 0.27878 0.2035 0.518
## [11,] 0.26935 0.2046 0.526
## [12,] 0.25993 0.2057 0.534
## [13,] 0.25050 0.2068 0.543
## [14,] 0.24108 0.2079 0.551
## [15,] 0.23165 0.2090 0.559
## [16,] 0.22223 0.2101 0.568
## [17,] 0.21280 0.2112 0.576
## [18,] 0.20338 0.2123 0.584
## [19,] 0.19395 0.2134 0.593
## [20,] 0.18453 0.2145 0.601
## [21,] 0.17510 0.2156 0.609
## [22,] 0.16568 0.2167 0.618
## [23,] 0.15625 0.2177 0.626
## [24,] 0.14683 0.2188 0.634
## [25,] 0.13740 0.2199 0.643
## [26,] 0.12798 0.2210 0.651
## [27,] 0.11855 0.2221 0.659
## [28,] 0.10913 0.2232 0.668
## [29,] 0.09970 0.2243 0.676
## [30,] 0.09028 0.2254 0.684
## [31,] 0.08085 0.2265 0.693
## [32,] 0.07143 0.2276 0.701
## [33,] 0.06200 0.2287 0.709
## [34,] 0.05258 0.2298 0.718
## [35,] 0.04315 0.2309 0.726
## [36,] 0.03373 0.2320 0.734
## [37,] 0.02430 0.2331 0.743
## [38,] 0.01488 0.2342 0.751
## [39,] 0.00546 0.2353 0.759
## [40,] 0.00000 0.2264 0.774
## [41,] 0.00000 0.2037 0.796
## [42,] 0.00000 0.1811 0.819
## [43,] 0.00000 0.1585 0.842
## [44,] 0.00000 0.1358 0.864
## [45,] 0.00000 0.1132 0.887
## [46,] 0.00000 0.0905 0.909
## [47,] 0.00000 0.0679 0.932
## [48,] 0.00000 0.0453 0.955
## [49,] 0.00000 0.0226 0.977
## [50,] 0.00000 0.0000 1.000
##
## $return
## [,1]
## [1,] 0.373
## [2,] 0.379
## [3,] 0.384
## [4,] 0.390
## [5,] 0.396
## [6,] 0.401
## [7,] 0.407
## [8,] 0.413
## [9,] 0.418
## [10,] 0.424
## [11,] 0.430
## [12,] 0.436
## [13,] 0.441
## [14,] 0.447
## [15,] 0.453
## [16,] 0.458
## [17,] 0.464
## [18,] 0.470
## [19,] 0.475
## [20,] 0.481
## [21,] 0.487
## [22,] 0.492
## [23,] 0.498
## [24,] 0.504
## [25,] 0.510
## [26,] 0.515
## [27,] 0.521
## [28,] 0.527
## [29,] 0.532
## [30,] 0.538
## [31,] 0.544
## [32,] 0.549
## [33,] 0.555
## [34,] 0.561
## [35,] 0.566
## [36,] 0.572
## [37,] 0.578
## [38,] 0.584
## [39,] 0.589
## [40,] 0.595
## [41,] 0.601
## [42,] 0.606
## [43,] 0.612
## [44,] 0.618
## [45,] 0.623
## [46,] 0.629
## [47,] 0.635
## [48,] 0.640
## [49,] 0.646
## [50,] 0.652
##
## $risk
## [1] 0.254 0.254 0.254 0.254 0.254 0.255 0.255 0.256 0.256 0.257 0.257
## [12] 0.258 0.259 0.259 0.260 0.261 0.262 0.263 0.264 0.266 0.267 0.268
## [23] 0.269 0.271 0.272 0.274 0.275 0.277 0.278 0.280 0.282 0.284 0.286
## [34] 0.287 0.289 0.291 0.293 0.295 0.298 0.300 0.302 0.306 0.309 0.314
## [45] 0.318 0.324 0.329 0.335 0.342 0.349
##
## $name
## [1] "Efficient Frontier"ifelse(!require(Ecdat), install.packages("Ecdat"), library(Ecdat))## Loading required package: Ecdat## Loading required package: Ecfun##
## Attaching package: 'Ecfun'## The following object is masked from 'package:base':
##
## sign##
## Attaching package: 'Ecdat'## The following object is masked from 'package:datasets':
##
## Orange## [1] "Ecdat"ifelse(!require(Ecfun), install.packages("Ecfun"), library(Ecfun))## [1] "Ecdat"ifelse(!require(quadprog), install.packages("quadprog"), library(quadprog))## [1] "Ecdat"data(CRSPday)class(CRSPday)## [1] "mts" "ts"CRSP.df = as.data.frame(CRSPday)
head(CRSP.df)## year month day ge ibm mobil crsp
## 1 1989 1 3 -0.01676 0.00000 -0.00275 -0.00762
## 2 1989 1 4 0.01705 0.00513 0.00551 0.01302
## 3 1989 1 5 -0.00279 -0.00204 0.00548 0.00281
## 4 1989 1 6 0.00000 -0.00613 0.00272 0.00306
## 5 1989 1 9 0.00000 0.00411 0.00543 0.00163
## 6 1989 1 10 -0.00560 -0.00717 0.00811 -0.00199