r - Portfolio Optimisation under weight constraints -

with lot of contributors stackoverflow have managed put function derive weights of 2-asset portfolio maximises sharpe ratio. no short sales allowed , sum of weights add 1. constrain asset not being more or less 10% user defined weight. example constrain weight of asset no less 54% or more 66% (i.e 60% +/- 10%). on below example end weights of (0.54,0.66) instead of unsconstrained (0.243,0.7570) .i assume can done tweaking bvect not sure how go it. appreciated.

asset_a <- c(0.034320510,-0.001209628,0.031900161,0.023163947,-0.001872938,-0.010322489,0.006090395,-0.003270854,0.017778990,0.017204915)   asset_b <- c(0.047103261,0.055175057,0.021019816,0.020602347,0.007281368,-0.006547404,0.019155238,0.005494798,0.025429958,0.014929124)       require(quadprog)       hr_solve   <- function(asset_a,asset_b) {      vol_a    <-  sd(asset_a)      vol_b   <-  sd(asset_b)      cor_ab  <-   cor(cbind(asset_a,asset_b),method="pearson")      ret_a_b      <- as.matrix(c(mean(asset_a),mean(asset_b)))        vol_ab <- c(vol_a,vol_b)       covmat <- diag(as.vector(vol_ab))%*%cor_ab%*%diag(as.vector(vol_ab))        amat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)))       bvec  <- c(1,0,0)       zeros <- array(0, dim = c(nrow(covmat),1))       minw <-  solve.qp(covmat, zeros, amat, bvec, meq = 1 ,factorized = false)$solution       rp <- as.numeric(t(minw) %*% ret_a_b)       sp <- sqrt(t(minw) %*% covmat %*% minw)       wp <- t(matrix(minw))        sret <- sort(seq(t(minw) %*% ret_a_b,max(ret_a_b),length.out=100))       amatt <- cbind(ret_a_b,amat)        (ri in sret[-1]){       bvect  <- c(ri,bvec)       result <-  trycatch({solve.qp(covmat, zeros, amatt, bvect, meq = 2,factorized = false)},                             warning = function(w){ return(null) } , error = function(w){ return(null)}, = {} )      if (!is.null(result)){      wp <-  rbind(wp,as.vector(result$solution))      rp <-c(rp,t(as.vector(result$solution) %*% ret_a_b))      sp <- c(sp,sqrt(t(as.vector(result$solution)) %*% covmat %*% as.vector(result$solution))) }       }        hr_weights <- wp[which.max(rp/sp),]       as.matrix(hr_weights)     }   hr_solve(asset_a,asset_b)             [,1] [1,] 0.2429662 [2,] 0.7570338 

i think should take @ link below.

http://economistatlarge.com/portfolio-theory/r-optimized-portfolio/r-code-graph-efficient-frontier

i think you'll learn lot that. i'll post code here, in case link gets shut down sometime in future.

# economist @ large # modern portfolio theory # use solve.qp solve efficient frontier # last edited 5/3/13  # file uses solve.qp function in quadprog package solve # efficient frontier. # since efficient frontier parabolic function, can find solution # minimizes portfolio variance , vary risk premium find # points along efficient frontier. find portfolio # largest sharpe ratio (expected return / sd) identify # efficient portfolio  library(stockportfolio) # base package retrieving returns library(ggplot2) # used graph efficient frontier library(reshape2) # used melt data library(quadprog) #needed solve.qp  # create portfolio using etfs, incl. hypothetical non-efficient allocation stocks <- c(  "vtsmx" = .0,  "spy" = .20,  "efa" = .10,  "iwm" = .10,  "vwo" = .30,  "lqd" = .20,  "hyg" = .10)  # retrieve returns, earliest start date possible (where stocks have # data) through recent date returns <- getreturns(names(stocks[-1]), freq="week") #currently, drop index  #### efficient frontier function #### eff.frontier <- function (returns, short="no", max.allocation=null,  risk.premium.up=.5, risk.increment=.005){  # return argument should m x n matrix 1 column per security  # short argument whether short-selling allowed; default no (short  # selling prohibited)max.allocation maximum % allowed 1  # security (reduces concentration) risk.premium.up upper limit of  # risk premium modeled (see loop below) , risk.increment  # increment (by) value used in loop   covariance <- cov(returns)  print(covariance)  n <- ncol(covariance)   # create initial amat , bvec assuming equality constraint  # (short-selling allowed, no allocation constraints)  amat <- matrix (1, nrow=n)  bvec <- 1  meq <- 1   # modify amat , bvec if short-selling prohibited  if(short=="no"){  amat <- cbind(1, diag(n))  bvec <- c(bvec, rep(0, n))  }   # , modify amat , bvec if max allocation (concentration) specified  if(!is.null(max.allocation)){  if(max.allocation > 1 | max.allocation <0){  stop("max.allocation must greater 0 , less 1")  }  if(max.allocation * n < 1){  stop("need set max.allocation higher; not enough assets add 1")  }  amat <- cbind(amat, -diag(n))  bvec <- c(bvec, rep(-max.allocation, n))  }   # calculate number of loops  loops <- risk.premium.up / risk.increment + 1  loop <- 1   # initialize matrix contain allocation , statistics  # not necessary, speeds processing , uses less memory  eff <- matrix(nrow=loops, ncol=n+3)  # need give matrix column names  colnames(eff) <- c(colnames(returns), "std.dev", "exp.return", "sharpe")   # loop through quadratic program solver  (i in seq(from=0, to=risk.premium.up, by=risk.increment)){  dvec <- colmeans(returns) * # moves solution along ef  sol <- solve.qp(covariance, dvec=dvec, amat=amat, bvec=bvec, meq=meq)  eff[loop,"std.dev"] <- sqrt(sum(sol$solution*colsums((covariance*sol$solution))))  eff[loop,"exp.return"] <- as.numeric(sol$solution %*% colmeans(returns))  eff[loop,"sharpe"] <- eff[loop,"exp.return"] / eff[loop,"std.dev"]  eff[loop,1:n] <- sol$solution  loop <- loop+1  }   return(as.data.frame(eff)) }  # run eff.frontier function based on no short , 50% alloc. restrictions eff <- eff.frontier(returns=returns$r, short="no", max.allocation=.50,  risk.premium.up=1, risk.increment=.001)  # find optimal portfolio eff.optimal.point <- eff[eff$sharpe==max(eff$sharpe),]  # graph efficient frontier # start color scheme ealred <- "#7d110c" ealtan <- "#cdc4b6" eallighttan <- "#f7f6f0" ealdark <- "#423c30"  ggplot(eff, aes(x=std.dev, y=exp.return)) + geom_point(alpha=.1, color=ealdark) +  geom_point(data=eff.optimal.point, aes(x=std.dev, y=exp.return, label=sharpe),  color=ealred, size=5) +  annotate(geom="text", x=eff.optimal.point$std.dev,  y=eff.optimal.point$exp.return,  label=paste("risk: ",  round(eff.optimal.point$std.dev*100, digits=3),"\nreturn: ",  round(eff.optimal.point$exp.return*100, digits=4),"%\nsharpe: ",  round(eff.optimal.point$sharpe*100, digits=2), "%", sep=""),  hjust=0, vjust=1.2) +  ggtitle("efficient frontier\nand optimal portfolio") +  labs(x="risk (standard deviation of portfolio)", y="return") +  theme(panel.background=element_rect(fill=eallighttan),  text=element_text(color=ealdark),  plot.title=element_text(size=24, color=ealred)) ggsave("efficient frontier.png")