#' Turnover constrained portfolio optimization
#'
#' Calculate portfolio weights, variance, and mean return, given a set of
#' returns and a constraint on overall turnover
#'
#'
#' @param returns an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param mu.target target portfolio return
#' @param w.initial initial vector of portfolio weights. Length of the vector
#' must be equal to ncol(returns)
#' @param turnover constraint on turnover from intial weights
#' @param long.only optional long only constraint. Defaults to FALSE
#' @return returns a list with initial weights, buys, sells, and
#' the aggregate of all three. Also returns the portfolio's expected
#' return and variance
#' @author James Hobbs
#' @seealso \code{\link{TurnoverFrontier}}
#' @seealso \code{\link{solve.QP}}
#'
#' data(Returns)
#' opt <- TurnoverOpt(large.cap.returns,mu.target=0.01,
#' w.initial = rep(1/100,100),turnover=5)
#' opt$w.total
#' opt$port.var
#' opt$port.mu
#' @export
TurnoverOpt <- function(returns,mu.target = NULL,w.initial,turnover, long.only = FALSE){
nassets <- ncol(returns)
#using 3 sets of variabes...w.initial, w.buy, and w.sell
returns <- cbind(returns,returns,returns)
#The covariance matrix will be 3Nx3N rather than NxN
cov.mat <- cov(returns)
Dmat <- 2*cov.mat
#Make covariance positive definite
#This should barely change the covariance matrix, as
#the last few eigen values are very small negative numbers
Dmat <- make.positive.definite(Dmat)
mu <- apply(returns,2,mean)
dvec <- rep(0,nassets*3) #no linear part in this problem
#left hand side of constraints
constraint.sum <- c(rep(1,2*nassets),rep(1,nassets))
constraint.mu.target <- mu
constraint.weights.initial <- rbind(diag(nassets),matrix(0,ncol=nassets,nrow=nassets*2))
#Make both w_buy and w_sell negative, and check that it is > the negative turnover
constraint.turnover <- c(rep(0,nassets),rep(-1,nassets),rep(1,nassets))
constraint.weights.positive <-
rbind(matrix(0,ncol=2*nassets,nrow=nassets),diag(2*nassets))
temp.index <- (nassets*3-nassets+1):(nassets*3)
#need to flip sign for w_sell
constraint.weights.positive[temp.index,]<-
constraint.weights.positive[temp.index,]*-1
if(!is.null(mu.target)){
#put left hand side of constraints into constraint matrix
Amat <- cbind(constraint.sum, constraint.mu.target, constraint.weights.initial,
constraint.turnover, constraint.weights.positive)
#right hand side of constraints in this vector
bvec <- c(1,mu.target,w.initial,-turnover,rep(0,2*nassets))
n.eq <- 2+nassets
} else {
#min variance, no target mu
Amat <- cbind(constraint.sum, constraint.weights.initial,
constraint.turnover, constraint.weights.positive)
bvec <- c(1,w.initial,-turnover,rep(0,2*nassets))
n.eq <- 1 + nassets
}
#optional long only constraint
if(long.only == TRUE){
if ( length(w.initial[w.initial<0]) > 0 ){
stop("Long-Only specified but some initial weights are negative")
}
constraint.long.only <- rbind(diag(nassets),diag(nassets),diag(nassets))
Amat <- cbind(Amat, constraint.long.only)
bvec <- c(bvec,rep(0,nassets))
}
#Note that the first 5 constraints are equality constraints
#The rest are >= constraints, so if you want <= you have to flip
#signs as done above
solution <- solve.QP(Dmat,dvec,Amat,bvec,meq=(n.eq))
port.var <- solution$value
w.buy <- solution$solution[(nassets+1):(2*nassets)]
w.sell <- solution$solution[(2*nassets+1):(3*nassets)]
w.total <- w.initial + w.buy + w.sell
achieved.turnover <- sum(abs(w.buy),abs(w.sell))
port.mu <- w.total%*%(mu[1:nassets])
list(w.initial = w.initial, w.buy = w.buy,w.sell=w.sell,
w=w.total,achieved.turnover = achieved.turnover,
port.var=port.var,port.mu=port.mu)
}
#' Turnover constrained portfolio frontier
#'
#' Calculates an efficient frontier of portfolios with a
#' constraint on overall turnover
#'
#' @param returns an xts, vector, matrix, data frame, timeSeries or zoo object of
#' asset returns
#' @param minmu min feasible target portfolio return to use in optimization
#' @param maxmu max feasible target portfolio return to use in optimization
#' @param w.initial initial vector of portfolio weights. Length of the vector
#' must be equal to ncol(returns)
#' @param turnover constraint on turnover from intial weights
#' @param long.only optional long only constraint. Defaults to FALSE
#' @return returns a matrix, with the first column of mean return
#' second column of portfolio standard deviation, and subsequent columns of
#' asset weights
#' @author James Hobbs
#' @seealso \code{\link{TurnoverOpt}}
#'
#' data(Returns)
#' efront <- TurnoverFrontier(large.cap.returns,npoints=50,minmu=0.001,
#' maxmu=.05, w.initial=rep(1/100,100),turnover=5)
#' plot(x=efront[,"SD"],y=efront[,"MU"],type="l")
#' @export
TurnoverFrontier <- function(returns,npoints = 10, minmu, maxmu,
w.initial,turnover,long.only = FALSE)
{
p = ncol(returns)
efront = matrix(rep(0,npoints*(p+2)),ncol = p+2)
dimnames(efront)[[2]] = c("MU","SD",dimnames(returns)[[2]])
muvals = seq(minmu,maxmu,length.out = npoints)
for(i in 1:npoints) {
opt <- TurnoverOpt(returns,mu.target = muvals[i],w.initial,turnover,long.only)
efront[i,"MU"] <- opt$port.mu
efront[i,"SD"] <- sqrt(opt$port.var)
efront[i,3:ncol(efront)] <- opt$w.total
}
efront
}
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