带约束的R中的长短期投资组合优化
我想解决一个相当常见(且简单)的优化问题,尽管似乎没有关于这方面的帖子:多头/空头市场中性最小方差优化。“R伪码”中的优化形式: 因此,我的问题是,如何在R中的solve.QP中的长/短优化中实现abs约束?带约束的R中的长短期投资组合优化,r,mathematical-optimization,portfolio,quantitative-finance,quadprog,R,Mathematical Optimization,Portfolio,Quantitative Finance,Quadprog,我想解决一个相当常见(且简单)的优化问题,尽管似乎没有关于这方面的帖子:多头/空头市场中性最小方差优化。“R伪码”中的优化形式: 因此,我的问题是,如何在R中的solve.QP中的长/短优化中实现abs约束? 作为进一步说明,本文展示了如何在Matlab中实现这一点,但是这在R中的solve.QP中似乎不起作用。来自优秀的系统投资者工具箱,请参见130/30优化()谢谢@Osssan;那里有很棒的代码,尽管我希望知道为什么我的方法不起作用;他的代码也解决了风险/回报的权衡问题,而不是我试图解决的
作为进一步说明,本文展示了如何在Matlab中实现这一点,但是这在R中的solve.QP中似乎不起作用。来自优秀的系统投资者工具箱,请参见130/30优化()谢谢@Osssan;那里有很棒的代码,尽管我希望知道为什么我的方法不起作用;他的代码也解决了风险/回报的权衡问题,而不是我试图解决的优化问题(以最小的预期回报);我不知道这对我所遇到的错误是否有意义。是的,我知道这可能不会直接回答你的问题,但可能函数
min.risk.portfolio
,min.te.portfolio
等可能会让你对你的具体问题有一些见解。我知道这很旧,但我现在刚刚找到这篇文章。你的股票回报在哪里?我看到了numStocks,但我不知道你从哪里得到的。我敢打赌,你有一个或多个安全性的坏数据。请看下面的链接。按原样运行它。当你觉得舒服的时候,把那些股票代码换成你真正想分析的。
min (t(h) %*% D %*% h ) s.t. # minimize portfolio variance, h weights, D covar matrix
sum(h) == 0 # market neutral; weights sum to 0
sum(abs(h)) == 1 # book-size/fully-invested; abs(weights) sum to 1
h %*% e >= threshold # the portfolio expected return is > some threshold
h <= maxPos # each long position is less than some maxPos
h >= -maxPos # each short position is greater than -maxPos
portSolveMinVol <- function(er,targetR,factorVols,factorCorrel,idioVol) {
require(quadprog)
# min ( -d'b + 1/2 b'Db)
# A'b >= b_0
# b = weights --> what we are solving for
# D = covariance matrix
# d = we can set this to zero as we have no linear term in the objective function
# set up the A matrix with all the constraints
# weights sum to 0
# abs weights sum to 1
# max pos < x, greater than -x
# return > some thresh
numStocks <- length(er) # er is the expected return vector
numAbs <- numStocks # this is redundant but I do this to make the code easier to read
VCV <- factorVols %*% t(factorVols) * factorCorrel # factor covariance matrix
S <- matrix(0,ncol=numStocks,nrow=numStocks)
diag(S) <- idioVol * idioVol # stock specific covariance (i.e., 0's except for diagonal)
common <- factorBetas %*% VCV %*% t(factorBetas) # stock common risk covar matrix
# need to fill in the Dmat b/c of the abs constraint
Dmat <- matrix(0,ncol=numStocks+numAbs,nrow=numStocks+numAbs)
Dmat[1:numStocks,1:numStocks] <- (common + S) # full covariance matrix
dvec <- rep(0,numStocks + numAbs) # ignored but solve.QP wants it
# A'b >= b_0
Amat <- matrix(0,nrow= 3,ncol=numStocks + numAbs)
Amat[1,] <- c(rep(1,numStocks),rep(0,numAbs)) # sum weights equal zero
Amat[2,] <- c(rep(0,numStocks),rep(1,numAbs)) # sum abs weights equal 1
Amat[3,] <- c(er,rep(0,numAbs)) # expected return >= threshold
# add contraints on min and max pos size
maxpos <- matrix(0,nrow=numStocks,ncol=numStocks + numAbs)
minpos <- matrix(0,nrow=numStocks,ncol=numStocks + numAbs)
for(i in 1:numStocks) {
maxpos[i,i] = -1 # neg and neg b/c of >= format of contraints
minpos[i,i] = 1 # pos and neg b/c of >= format of contraints
}
absmaxpos <- matrix(0,nrow=numStocks,ncol=numStocks + numAbs)
absminpos <- matrix(0,nrow=numStocks,ncol=numStocks + numAbs)
# add contraints on the sum(abs(wi)) = 1 and each
for(i in 1:numStocks) {
absmaxpos[i,i] = 1
absmaxpos[i,i+numAbs] = -1
absminpos[i,i] = 1
absminpos[i,i+numAbs] = 1
}
# Set up the Amat
Amat <- rbind(Amat,maxpos,minpos,absmaxpos,absminpos)
# set up the rhs
bvec <- c(0, # sum of weights
1, # sum of abs weights
0.005, # min expected return
rep(-0.025,numStocks), # max pos
rep(-0.025,numStocks), # min pos
rep(0,numAbs), # abs long dummy var
rep(0,numAbs)) # abs short dummy var
# meq is the number of first constraints that are equality
res <- solve.QP(Dmat, dvec, t(Amat), bvec, meq=2)
res
}
set.seed(1)
nStocks <- 100
nBetas <- 5
er <-rnorm(nStocks,mean=0.0012,0.0075)
factorVols <- 0.08 + runif(nBetas,0,0.15)
factorCorrel <- matrix(rep(0,nBetas*nBetas),nrow=nBetas,ncol=nBetas)
for(i in 1:(nBetas)) {
for(j in 1:(nBetas)) {
factorCorrel[i,j] = rnorm(1,mean=0.2,sd=0.05)
factorCorrel[j,i] = factorCorel[i,j]
}
}
diag(factorCorrel) <- 1
idioVol <- abs(rnorm(nStocks,mean=0.01,sd=0.05))
res <- portSolveMinVol(er,0.005,factorVols,factorCorrel,idioVol)
Error in solve.QP(Dmat, dvec, t(Amat), bvec, meq = 2) : matrix D in
quadratic function is not positive definite!