Markowitz Asset Allocation Dana Uhlig Tue Jun 26 13:27:

Transcription

1 Markowitz Asset Allocation Dana Uhlig Tue Jun 26 13:27: library(quantmod) ## Loading required package: xts ## Loading required package: zoo ## ## Attaching package: 'zoo' ## The following objects are masked from 'package:base': ## ## as.date, as.date.numeric ## Loading required package: TTR ## Version included new data defaults. See?getSymbols. ## Learn from a quantmod author: = ls()) #directory for graphics if (!dir.exists("./pics-out")) dir.create(path="pics-out") titles = c("dax","rwe","gold","oil","$/ ") all = c( ,12.870, , 34.9, , , , , 33.08, , , , , 33.77, , , , , 37.04, , , , , 43.71, , , , , 45.81, , , , , 46.04, , , , , 39.78, , , , , 43.35, , , , , 46.15, , , , , 44.95, , , , , 47.72, , , , , 52.69, ) S = matrix(all,ncol=5,byrow=true) nr.assets = dim(s)[2] n = dim(s)[1] colnames(s) = titles time = seq(from=as.date(' '),by="month",length.out=n) (S.xts = as.xts(s,frequency = 12, order.by = time)) 1

2 ## ## ## ## ## ## ## ## ## ## ## ## ## ############# help functions ############################################# #normalising function for rescaling assets (S0=1) normalise_series = function(xdat) xdat / coredata(xdat)[1] normalise_assetmatrix = function(assets){ n = dim(assets)[1] #sample size nr = dim(assets)[2] #assets } S.scaled = assets for (k in (1:nr)){ S.scaled[,k] = normalise_series(assets[,k]) } return(s.scaled) ######################## (S.scaled = normalise_assetmatrix(s.xts)) ## ## ## ## ## ## ## ## ## ## ## ## ## ### chart matplot(time,s.scaled,type='l',xaxt="n",col=1:nr.assets,xlab="",ylab="scaled prices",lwd=3) axis(side=1, at=time, labels = format(time,"%b %y")) legend("topleft", titles, col=1:nr.assets, lty=1:nr.assets,bty="n",lwd=1) 2

3 scaled prices DAX RWE gold oil $/ Jan 16 Mar 16 May 16 Jul 16 Sep 16 Nov 16 Jan 17 ### using ts / quantmod charts all.cols = topo.colors(nr.assets, alpha = 1) chart_series(s.scaled[,1], lwd=2, name="") 3

4 / Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Jan 2017 add_ta(s.scaled[,2], on = 1, col = all.cols[2], lty = 2,lwd=2) 4

5 / Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Jan 2017 add_ta(s.scaled[,3], on = 1, col = all.cols[3], lty = 3,lwd=2) 5

6 / Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Jan 2017 add_ta(s.scaled[,4], on = 1, col = all.cols[4], lty = 4,lwd=2) 6

7 / Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Jan 2017 add_ta(s.scaled[,5], on = 1, col = all.cols[5], lty = 5,lwd=2) legend("topleft", titles, col=all.cols, lty=1:nr.assets,bty="n",lwd=1) 7

8 DAX RWE / gold oil $/ Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Jan 2016 Feb 2016 Mar 2016 Apr 2016 May 2016 Jun 2016 Jul 2016 Aug 2016 Sep 2016 Oct 2016 Nov 2016 Dec 2016 Jan 2017 #################################################################################################### #alternative individual computation DAX = as.xts(s[,1],frequency = 12, order.by = time) RWE = as.xts(s[,2],frequency = 12, order.by = time) GOLD = as.xts(s[,3],frequency = 12, order.by = time) OIL = as.xts(s[,4],frequency = 12, order.by = time) USEUR = as.xts(s[,5],frequency = 12, order.by = time) DAX.scaled = normalise_series(dax) RWE.scaled = normalise_series(rwe) GOLD.scaled = normalise_series(gold) OIL.scaled = normalise_series(oil) USEUR.scaled = normalise_series(useur) ############################################################################## #matrix of annulized returns (Xi = 12*diff(log(S))) ## [1,] ## [2,] ## [3,]

9 ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [12,] #script table 3.2 annulized returns Xi * 100 ## [1,] ## [2,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [12,] #factor 12 for annualized returns: 1/as.numeric(time[n]-time[1])*as.numeric(time[2]-time[1]) ## [1] /12 ## [1] /as.numeric(time[n]-time[1])*as.numeric(diff(time)) ## [1] ## [7] /12 ## [1] #same as S.scaled after removing first line (NA)! 12*diff(log(S.scaled))[-1,] ## ## ## ## ## ## ## ## ## ##

10 ## ## #or individual monthlyreturn(dax,type = 'log') ## monthly.returns ## ## ## ## ## ## ## ## ## ## ## ## ## monthlyreturn(dax,type = 'arithmetic') ## monthly.returns ## ## ## ## ## ## ## ## ## ## ## ## ## DAX.ret.month = monthlyreturn(dax,type='log')[-1] #same as for scaled prices, since log(s_t / S_{t-1})= log((s_t/s_0) / (S_{t-1}/S_0)) norm(monthlyreturn(dax.scaled) - monthlyreturn(dax)) ## [1] e-16 #caution! There are some small differnces (due to time scaling????) cbind(dax.ret.month * 12,Xi[,1]) ## monthly.returns..2 ## ## ## ## ## ## ## ## ##

11 ## ## ## cbind(xi[,1], 12*DAX.ret.month,as.numeric(time[n]-time[1])*(1/as.numeric(diff(time)))*DAX.ret.month) ##..1 monthly.returns monthly.returns.1 ## ## ## ## ## ## ## ## ## ## ## ## norm(xi[,1] - 12*DAX.ret.month) ## [1] e-14 norm(xi[,1] - as.numeric(time[n]-time[1])*(1/as.numeric(diff(time)))*dax.ret.month) ## [1] RWE.ret.month = monthlyreturn(rwe,type = 'log')[-1] GOLD.ret.month =monthlyreturn(gold,type = 'log')[-1] OIL.ret.month = monthlyreturn(oil,type = 'log')[-1] USEUR.ret.month = monthlyreturn(useur,type = 'log')[-1] (Xi.alternative = cbind(dax.ret.month, RWE.ret.month, GOLD.ret.month, OIL.ret.month, USEUR.ret.month)*12 ## monthly.returns monthly.returns.1 monthly.returns.2 ## ## ## ## ## ## ## ## ## ## ## ## ## monthly.returns.3 monthly.returns.4 ## ## ## ## ## ## ## ##

12 ## ## ## ## Xi ## [1,] ## [2,] ## [3,] ## [4,] ## [5,] ## [6,] ## [7,] ## [8,] ## [9,] ## [10,] ## [11,] ## [12,] ######################################################################################################## #estimation of covariance matrix and expected returns #first moment: expected annulized returns r=colmeans(xi) #table 3.3 first line - mean r*100 ## #alternative computation apply(x=xi,margin = 2,FUN = mean) ## r ## #second moment: variance #unbiased variance estimator apply(x=xi, MARGIN = 2, FUN = var) ## #biased variance estimator (table 3.3 second line) apply(x=xi, MARGIN = 2, FUN = var) * (n-1)/n ## #covariance, unbiased (better estimator than biased version used in script!) (C.unbiased = cov(xi)) 12

13 ## DAX ## RWE ## gold ## oil ## $/\ #diagonal = variances apply(x=xi, MARGIN = 2, FUN = var) ## diag(c.unbiased) ## #biased covariance = version script Pichler (C.biased = C.unbiased*(n-1)/n) ## DAX ## RWE ## gold ## oil ## $/\ #variances on diagonal apply(x=xi, MARGIN = 2, FUN = var)*(n-1)/n ## diag(c.biased) ## ######################################################################## # CAPM Markowitz: min( portfoliovar(x)) such that: expected portfolio mean(x) >= mu and sum(x)<=1 #see Theorem 3.12 (1. step: we compute the efficient portfolios for a given expected portfolio return (s # and later we write a function for doing this (step 2) C = C.biased #because this version is used in the lecture notes #(even using the unbiased estimator is better!!!) #step 1: C.inv = solve(c) #table 3.4 (a) and 3.4 (b) round(c*100,2) ## DAX ## RWE ## gold ## oil ## $/\

14 round(c.inv,2) ## DAX ## RWE ## gold ## oil ## $/\ #test C.inv %*% C ## DAX e e e e e-17 ## RWE e e e e e-17 ## gold e e e e e-17 ## oil e e e e e-17 ## $/\ e e e e e+00 C %*% C.inv ## DAX e e e e e-17 ## RWE e e e e e-15 ## gold e e e e e-16 ## oil e e e e e-16 ## $/\ e e e e e+00 #computation of efficient Markowirtz portfolio via formula (3.5) ones = rep(1,nr.assets) a = as.numeric(r%*% C.inv %*% r) b = as.numeric(r%*% C.inv %*% ones ) c = as.numeric(ones %*% C.inv %*% ones ) d = a*c-b*b mu = seq(from=0.00,to = 0.25,by=0.01) m = length(mu) xm = matrix(nrow = nr.assets, ncol=m) for (k in 1:m){ xm[,k]= a*c.inv%*%ones- b*c.inv%*%r+ mu[k]*(c*c.inv%*%r -b*c.inv%*%ones) } xm = xm/d row.names(xm) = titles colsums(xm) ## [1] xm<0 ## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] ## DAX FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## RWE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## gold FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 14

15 ## oil TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE ## $/\200 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## [,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] ## DAX FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## RWE FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE ## gold FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## oil FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## $/\200 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE ## [,23] [,24] [,25] [,26] ## DAX FALSE FALSE FALSE FALSE ## RWE TRUE TRUE TRUE TRUE ## gold FALSE FALSE FALSE FALSE ## oil FALSE FALSE FALSE FALSE ## $/\200 FALSE FALSE FALSE FALSE standarddev = sqrt((mu^2*c-2*mu*b+a)/d) plot(mu,standarddev,type="l",main="standard deviation of the optimal portfolio in dependence of mu") Standard deviation of the optimal portfolio in dependence of mu standarddev mu matplot(mu,t(xm),type='l',col=1:nr.assets,lty=1:nr.assets,ylab="x*",main="asset allocation") legend("topright", titles, col=1:nr.assets, lty=1:nr.assets,bty="n",lwd=1) 15

16 asset allocation x* DAX RWE gold oil $/ mu ######################################################################################################## # function for creating asset allocation plot (see script figure 3.4 and 3.6) graph.asset.allocation = function(portfolio,sd,mu, png.name="optimal-asset-allocation-mu.png",title="mar nr = dim(portfolio)[1] titles = row.names(portfolio) print(portfolio) print(titles) } par(mar=c(5,4,4,5)+.1) plot(mu,t(portfolio[1,]),ylim=c(min(portfolio),max(portfolio)),type='l',col=1,lty=1,ylab="x* asset all for (j in 2:nr) lines(mu,t(portfolio[j,]),type='l',col=j,lty=j) par(new=true) plot(mu, sd,type="l",col="darkgray",lwd=3,lty=6,xaxt="n",yaxt="n",xlab="",ylab="") axis(4) mtext("sd(x*(mu))",side=4,line=3,col="darkgray") legend("topleft", c(titles), col=c(1:nr), lty=1:nr,bty="n",lwd=rep(1,nr)) dev.copy(png,paste("pics-out/",png.name,sep="")) dev.off() graph.asset.allocation(portfolio=xm,sd=standarddev, mu=mu,png.name = 'test.png') ## [,1] [,2] [,3] [,4] [,5] ## DAX

17 ## RWE ## gold ## oil ## $/\ ## [,6] [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## [1] "DAX" "RWE" "gold" "oil" "$/\200" ## pdf ## 2 ######################################################################################################## #step 2: function for efficient Markowitz portfolio (Theorem 3.12) markowitz.portfolio = function(mu.ret, Cov.Matrix=diag(length(mu.ret)), mu.portfolio.min = 0){ if (missing(mu.ret)) stop("need vector of expected asset returns: mu.ret") nr = length(mu.ret) titles= names(mu.ret) if (sum(dim(cov.matrix)==nr)<2) stop("wrong dimensions") ones = rep(1,nr) Cov.inv = solve(cov.matrix) m = length(mu.portfolio.min) xm = matrix(nrow = nr, ncol=m) a = as.numeric(mu.ret%*% Cov.inv %*% mu.ret) b = as.numeric(mu.ret%*% Cov.inv %*% ones ) c = as.numeric(ones %*% Cov.inv %*% ones ) d = a*c-b*b for (k in 1:m){ xm[,k] = a*cov.inv%*%ones- b*cov.inv%*%mu.ret+ mu.portfolio.min[k] *(c*cov.inv%*%mu.ret -b*cov.inv%* } 17

18 } if (length(titles)==0) titles=paste(rep("asset",nr),1:nr) #print(titles) row.names(xm) = titles standarddev = sqrt((mu.portfolio.min^2*c-2*mu.portfolio.min*b+a)/d) return.list = list(xm/d,standarddev) names(return.list) = c("efficient_portfolio","standarddev") return(return.list) (test = markowitz.portfolio(r,c,0.03)) ## $efficient_portfolio ## [,1] ## DAX ## RWE ## gold ## oil ## $/\ ## ## $standarddev ## [1] sum(test$efficient_portfolio) ## [1] 1 (test2 = markowitz.portfolio(r,c,mu)) ## $efficient_portfolio ## [,1] [,2] [,3] [,4] [,5] ## DAX ## RWE ## gold ## oil ## $/\ ## [,6] [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## [,22] [,23] [,24] [,25] [,26] 18

19 ## DAX ## RWE ## gold ## oil ## $/\ ## ## $standarddev ## [1] ## [8] ## [15] ## [22] colsums(test2$efficient_portfolio,) ## [1] row.names(test2$efficient_portfolio) ## [1] "DAX" "RWE" "gold" "oil" "$/\200" graph.asset.allocation(portfolio=test2$efficient_portfolio,sd=test2$standarddev, mu=mu, png.name = 'asse ## [,1] [,2] [,3] [,4] [,5] ## DAX ## RWE ## gold ## oil ## $/\ ## [,6] [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## [1] "DAX" "RWE" "gold" "oil" "$/\200" 19

20 x* asset allocation DAX RWE gold oil $/ sd(x*(mu)) mu ## pdf ## 2 ######################################################################################################## #theorem Markowitz with risk free asset - section 3.8, Theorem 3.19 markowitz.portfolio.cash = function(mu.ret, Cov.Matrix=diag(length(mu.ret)), mu.portfolio.min = 0, r0=0) if (missing(mu.ret)) stop("need vector of expected asset returns: mu.ret") titles= names(mu.ret) nr = length(mu.ret) if (sum(dim(cov.matrix)==nr)<2) stop("wrong dimensions") ones = rep(1,nr) Cov.inv = solve(cov.matrix) m = length(mu.portfolio.min) xm.r0 = matrix(nrow = nr+1, ncol=m) #first nr compontens usual assets x^star, nr+1 = cash(r0) x0^star a = as.numeric(mu.ret%*% Cov.inv %*% mu.ret) b = as.numeric(mu.ret%*% Cov.inv %*% ones ) c = as.numeric(ones %*% Cov.inv %*% ones ) d = a-2*b*r0+c*r0*r0 for (k in 1:m){ xm.r0[,k] = c((mu.portfolio.min[k]-r0)*(cov.inv%*%mu.ret - r0 * Cov.inv%*%ones),d-(b-r0*c)*(mu.portf 20

21 } } if (length(titles)==0) titles=paste(rep("asset",nr),1:nr) print(titles) row.names(xm.r0) = c(titles,"cash") standarddev = sqrt(((mu.portfolio.min-r0)^2/d)) return.list = list(xm.r0/d,standarddev) names(return.list) = c("efficient_portfolio","standarddev") return(return.list) r0 = 0.02 mu.new = seq(from=0.0, by=0.01, to=0.84) (portfolio.riskfree = markowitz.portfolio.cash(r,c,mu.new,r0)) ## [1] "DAX" "RWE" "gold" "oil" "$/\200" ## $efficient_portfolio ## [,1] [,2] [,3] [,4] [,5] [,6] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\

22 ## cash ## [,27] [,28] [,29] [,30] [,31] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,32] [,33] [,34] [,35] [,36] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,37] [,38] [,39] [,40] [,41] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,42] [,43] [,44] [,45] [,46] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,47] [,48] [,49] [,50] [,51] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,52] [,53] [,54] [,55] [,56] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,57] [,58] [,59] [,60] [,61] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,62] [,63] [,64] [,65] [,66] ## DAX ## RWE ## gold

23 ## oil ## $/\ ## cash ## [,67] [,68] [,69] [,70] [,71] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,72] [,73] [,74] [,75] [,76] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,77] [,78] [,79] [,80] [,81] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,82] [,83] [,84] [,85] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## ## $standarddev ## [1] ## [7] ## [13] ## [19] ## [25] ## [31] ## [37] ## [43] ## [49] ## [55] ## [61] ## [67] ## [73] ## [79] ## [85] colsums(portfolio.riskfree$efficient_portfolio) ## [1] ## [36] ## [71]

24 graph.asset.allocation(portfolio=portfolio.riskfree$efficient_portfolio,sd=portfolio.riskfree$standardde mu=mu.new, png.name = 'asset-allocation-with-cash.png') ## [,1] [,2] [,3] [,4] [,5] [,6] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,27] [,28] [,29] [,30] [,31] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,32] [,33] [,34] [,35] [,36] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,37] [,38] [,39] [,40] [,41] ## DAX

25 ## RWE ## gold ## oil ## $/\ ## cash ## [,42] [,43] [,44] [,45] [,46] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,47] [,48] [,49] [,50] [,51] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,52] [,53] [,54] [,55] [,56] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,57] [,58] [,59] [,60] [,61] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,62] [,63] [,64] [,65] [,66] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,67] [,68] [,69] [,70] [,71] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,72] [,73] [,74] [,75] [,76] ## DAX ## RWE ## gold ## oil ## $/\ ## cash

26 ## [,77] [,78] [,79] [,80] [,81] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,82] [,83] [,84] [,85] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [1] "DAX" "RWE" "gold" "oil" "$/\200" "cash" x* asset allocation DAX RWE gold oil $/ cash sd(x*(mu)) mu ## pdf ## 2 # # -> all functions exported to markowitz-functions.r -> including with source("markowitz-functions.r") #### same with unbiased covarince matrix (mw.portfolio = markowitz.portfolio(r,c.unbiased,mu)) ## $efficient_portfolio ## [,1] [,2] [,3] [,4] [,5] 26

27 ## DAX ## RWE ## gold ## oil ## $/\ ## [,6] [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## ## $standarddev ## [1] ## [8] ## [15] ## [22] colsums(mw.portfolio$efficient_portfolio,) ## [1] row.names(mw.portfolio$efficient_portfolio) ## [1] "DAX" "RWE" "gold" "oil" "$/\200" graph.asset.allocation(portfolio=mw.portfolio$efficient_portfolio, sd=mw.portfolio$standarddev, mu=mu, png.name = 'asset-allocation-fig-3-4-c-unbias ## [,1] [,2] [,3] [,4] [,5] ## DAX ## RWE ## gold ## oil ## $/\ ## [,6] [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE

28 ## gold ## oil ## $/\ ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## [1] "DAX" "RWE" "gold" "oil" "$/\200" x* asset allocation DAX RWE gold oil $/ sd(x*(mu)) mu ## pdf ## 2 28

29 (mw.riskfree = markowitz.portfolio.cash(r,c.unbiased,mu.new,r0)) ## [1] "DAX" "RWE" "gold" "oil" "$/\200" ## $efficient_portfolio ## [,1] [,2] [,3] [,4] [,5] [,6] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,7] [,8] [,9] [,10] [,11] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,12] [,13] [,14] [,15] [,16] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,17] [,18] [,19] [,20] [,21] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,22] [,23] [,24] [,25] [,26] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,27] [,28] [,29] [,30] [,31] ## DAX ## RWE ## gold ## oil ## $/\ ## cash ## [,32] [,33] [,34] [,35] [,36] ## DAX ## RWE ## gold ## oil ## $/\ ## cash