权重约束下的投资组合优化
Portfolio Optimisation under weight constraints
在 Whosebug 的贡献者的大力帮助下，我设法组合了一个函数来导出 2 资产组合的权重，从而使夏普比率最大化。不允许卖空，权重总和为 1。我现在想做的是限制资产 A 不超过或低于用户定义权重的 10%。例如，我想限制资产 A 的权重不低于 54% 或高于 66%（即 60% +/- 10%）。所以在下面的例子中，我最终得到的权重是 (0.54,0.66) 而不是不受约束的 (0.243,0.7570) 。我认为这可以通过调整 bVect 来完成，但我不太确定如何去做。任何帮助，将不胜感激。
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)$solution
rp <- as.numeric(t(minw) %*% ret_A_B)
sp <- sqrt(t(minw) %*% covmat %*% minw)
wp <- t(matrix(minw))
sret <- sort(seq(t(minw) %*% ret_A_B,max(ret_A_B),length.out=100))
aMatt <- cbind(ret_A_B,aMat)
for (ri in sret[-1]){
bVect <- c(ri,bVec)
result <- tryCatch({solve.QP(covmat, zeros, aMatt, bVect, meq = 2,factorized = FALSE)},
warning = function(w){ return(NULL) } , error = function(w){ return(NULL)}, finally = {} )
if (!is.null(result)){
wp <- rbind(wp,as.vector(result$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$solution))) }
}
HR_weights <- wp[which.max(rp/sp),]
as.matrix(HR_weights)
}
HR_solve(asset_A,asset_B)
[,1]
[1,] 0.2429662
[2,] 0.7570338
好的，我找到了一种方法...如果您认为有更优雅的方法，请告诉我...
require(quadprog)
HR_solve <- function(asset_A,asset_B,mean_A,range_A) {
vol_A <- sd(asset_A)
vol_B <- sd(asset_B)
cor_AB <- cor(cbind(asset_A,asset_B),method="pearson")
ret_A_B <- as.matrix(c(mean(asset_A),mean(asset_B)))
vol_AB <- c(vol_A,vol_B)
covmat <- diag(as.vector(vol_AB))%*%cor_AB%*%diag(as.vector(vol_AB))
bVec <- c(1,0,0)
aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)))
zeros <- array(0, dim = c(nrow(covmat),1))
minw <- solve.QP(covmat, zeros, aMat, bVec, meq = 1 ,factorized = FALSE)$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=1000))
min_A <- mean_A * (1-range_A)
max_A <- mean_A * (1+range_A)
aMatt <- cbind(ret_A_B,aMat,-diag(2))
bVec <- c(1,min_A,0,-max_A,-1)
for (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)}, finally = {} )
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)
}
只需更改 aMat 和 bVec:
# sset A to be no less than 54% or more than 66%
aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)),c(1,0),c(-1,0))
bVec <- c(1,0,0,.54,-.66)
我想你应该看看下面的link。
http://economistatlarge.com/portfolio-theory/r-optimized-portfolio/r-code-graph-efficient-frontier
我想你会从中学到很多东西。我将 post 代码放在这里，以防将来 link 关闭。
# Economist at Large
# Modern Portfolio Theory
# Use solve.QP to solve for efficient frontier
# Last Edited 5/3/13
# This file uses the solve.QP function in the quadprog package to solve for the
# efficient frontier.
# Since the efficient frontier is a parabolic function, we can find the solution
# that minimizes portfolio variance and then vary the risk premium to find
# points along the efficient frontier. Then simply find the portfolio with the
# largest Sharpe ratio (expected return / sd) to identify the most
# efficient portfolio
library(stockPortfolio) # Base package for retrieving returns
library(ggplot2) # Used to graph efficient frontier
library(reshape2) # Used to melt the data
library(quadprog) #Needed for solve.QP
# Create the 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, from earliest start date possible (where all stocks have
# data) through most 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 be a m x n matrix with one column per security
# short argument is whether short-selling is allowed; default is no (short
# selling prohibited)max.allocation is the maximum % allowed for any one
# security (reduces concentration) risk.premium.up is the upper limit of the
# risk premium modeled (see for loop below) and risk.increment is the
# increment (by) value used in the for loop
covariance <- cov(returns)
print(covariance)
n <- ncol(covariance)
# Create initial Amat and bvec assuming only equality constraint
# (short-selling is allowed, no allocation constraints)
Amat <- matrix (1, nrow=n)
bvec <- 1
meq <- 1
# Then modify the Amat and bvec if short-selling is prohibited
if(short=="no"){
Amat <- cbind(1, diag(n))
bvec <- c(bvec, rep(0, n))
}
# And modify Amat and bvec if a max allocation (concentration) is specified
if(!is.null(max.allocation)){
if(max.allocation > 1 | max.allocation <0){
stop("max.allocation must be greater than 0 and less than 1")
}
if(max.allocation * n < 1){
stop("Need to set max.allocation higher; not enough assets to add to 1")
}
Amat <- cbind(Amat, -diag(n))
bvec <- c(bvec, rep(-max.allocation, n))
}
# Calculate the number of loops
loops <- risk.premium.up / risk.increment + 1
loop <- 1
# Initialize a matrix to contain allocation and statistics
# This is not necessary, but speeds up processing and uses less memory
eff <- matrix(nrow=loops, ncol=n+3)
# Now I need to give the matrix column names
colnames(eff) <- c(colnames(returns), "Std.Dev", "Exp.Return", "sharpe")
# Loop through the quadratic program solver
for (i in seq(from=0, to=risk.premium.up, by=risk.increment)){
dvec <- colMeans(returns) * i # This moves the solution along the 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 the eff.frontier function based on no short and 50% alloc. restrictions
eff <- eff.frontier(returns=returns$R, short="no", max.allocation=.50,
risk.premium.up=1, risk.increment=.001)
# Find the optimal portfolio
eff.optimal.point <- eff[eff$sharpe==max(eff$sharpe),]
# graph efficient frontier
# Start with color scheme
ealred <- "#7D110C"
ealtan <- "#CDC4B6"
eallighttan <- "#F7F6F0"
ealdark <- "#423C30"
ggplot(eff, aes(x=Std.Dev, y=Exp.Return)) + geom_point(alpha=.1, color=ealdark) +
geom_point(data=eff.optimal.point, aes(x=Std.Dev, y=Exp.Return, label=sharpe),
color=ealred, size=5) +
annotate(geom="text", x=eff.optimal.point$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")
在 Whosebug 的贡献者的大力帮助下，我设法组合了一个函数来导出 2 资产组合的权重，从而使夏普比率最大化。不允许卖空，权重总和为 1。我现在想做的是限制资产 A 不超过或低于用户定义权重的 10%。例如，我想限制资产 A 的权重不低于 54% 或高于 66%（即 60% +/- 10%）。所以在下面的例子中，我最终得到的权重是 (0.54,0.66) 而不是不受约束的 (0.243,0.7570) 。我认为这可以通过调整 bVect 来完成，但我不太确定如何去做。任何帮助，将不胜感激。
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)$solution
rp <- as.numeric(t(minw) %*% ret_A_B)
sp <- sqrt(t(minw) %*% covmat %*% minw)
wp <- t(matrix(minw))
sret <- sort(seq(t(minw) %*% ret_A_B,max(ret_A_B),length.out=100))
aMatt <- cbind(ret_A_B,aMat)
for (ri in sret[-1]){
bVect <- c(ri,bVec)
result <- tryCatch({solve.QP(covmat, zeros, aMatt, bVect, meq = 2,factorized = FALSE)},
warning = function(w){ return(NULL) } , error = function(w){ return(NULL)}, finally = {} )
if (!is.null(result)){
wp <- rbind(wp,as.vector(result$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$solution))) }
}
HR_weights <- wp[which.max(rp/sp),]
as.matrix(HR_weights)
}
HR_solve(asset_A,asset_B)
[,1]
[1,] 0.2429662
[2,] 0.7570338
好的，我找到了一种方法...如果您认为有更优雅的方法，请告诉我...
require(quadprog)
HR_solve <- function(asset_A,asset_B,mean_A,range_A) {
vol_A <- sd(asset_A)
vol_B <- sd(asset_B)
cor_AB <- cor(cbind(asset_A,asset_B),method="pearson")
ret_A_B <- as.matrix(c(mean(asset_A),mean(asset_B)))
vol_AB <- c(vol_A,vol_B)
covmat <- diag(as.vector(vol_AB))%*%cor_AB%*%diag(as.vector(vol_AB))
bVec <- c(1,0,0)
aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)))
zeros <- array(0, dim = c(nrow(covmat),1))
minw <- solve.QP(covmat, zeros, aMat, bVec, meq = 1 ,factorized = FALSE)$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=1000))
min_A <- mean_A * (1-range_A)
max_A <- mean_A * (1+range_A)
aMatt <- cbind(ret_A_B,aMat,-diag(2))
bVec <- c(1,min_A,0,-max_A,-1)
for (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)}, finally = {} )
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)
}
只需更改 aMat 和 bVec:
# sset A to be no less than 54% or more than 66%
aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)),c(1,0),c(-1,0))
bVec <- c(1,0,0,.54,-.66)
我想你应该看看下面的link。
http://economistatlarge.com/portfolio-theory/r-optimized-portfolio/r-code-graph-efficient-frontier
我想你会从中学到很多东西。我将 post 代码放在这里，以防将来 link 关闭。
# Economist at Large
# Modern Portfolio Theory
# Use solve.QP to solve for efficient frontier
# Last Edited 5/3/13
# This file uses the solve.QP function in the quadprog package to solve for the
# efficient frontier.
# Since the efficient frontier is a parabolic function, we can find the solution
# that minimizes portfolio variance and then vary the risk premium to find
# points along the efficient frontier. Then simply find the portfolio with the
# largest Sharpe ratio (expected return / sd) to identify the most
# efficient portfolio
library(stockPortfolio) # Base package for retrieving returns
library(ggplot2) # Used to graph efficient frontier
library(reshape2) # Used to melt the data
library(quadprog) #Needed for solve.QP
# Create the 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, from earliest start date possible (where all stocks have
# data) through most 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 be a m x n matrix with one column per security
# short argument is whether short-selling is allowed; default is no (short
# selling prohibited)max.allocation is the maximum % allowed for any one
# security (reduces concentration) risk.premium.up is the upper limit of the
# risk premium modeled (see for loop below) and risk.increment is the
# increment (by) value used in the for loop
covariance <- cov(returns)
print(covariance)
n <- ncol(covariance)
# Create initial Amat and bvec assuming only equality constraint
# (short-selling is allowed, no allocation constraints)
Amat <- matrix (1, nrow=n)
bvec <- 1
meq <- 1
# Then modify the Amat and bvec if short-selling is prohibited
if(short=="no"){
Amat <- cbind(1, diag(n))
bvec <- c(bvec, rep(0, n))
}
# And modify Amat and bvec if a max allocation (concentration) is specified
if(!is.null(max.allocation)){
if(max.allocation > 1 | max.allocation <0){
stop("max.allocation must be greater than 0 and less than 1")
}
if(max.allocation * n < 1){
stop("Need to set max.allocation higher; not enough assets to add to 1")
}
Amat <- cbind(Amat, -diag(n))
bvec <- c(bvec, rep(-max.allocation, n))
}
# Calculate the number of loops
loops <- risk.premium.up / risk.increment + 1
loop <- 1
# Initialize a matrix to contain allocation and statistics
# This is not necessary, but speeds up processing and uses less memory
eff <- matrix(nrow=loops, ncol=n+3)
# Now I need to give the matrix column names
colnames(eff) <- c(colnames(returns), "Std.Dev", "Exp.Return", "sharpe")
# Loop through the quadratic program solver
for (i in seq(from=0, to=risk.premium.up, by=risk.increment)){
dvec <- colMeans(returns) * i # This moves the solution along the 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 the eff.frontier function based on no short and 50% alloc. restrictions
eff <- eff.frontier(returns=returns$R, short="no", max.allocation=.50,
risk.premium.up=1, risk.increment=.001)
# Find the optimal portfolio
eff.optimal.point <- eff[eff$sharpe==max(eff$sharpe),]
# graph efficient frontier
# Start with color scheme
ealred <- "#7D110C"
ealtan <- "#CDC4B6"
eallighttan <- "#F7F6F0"
ealdark <- "#423C30"
ggplot(eff, aes(x=Std.Dev, y=Exp.Return)) + geom_point(alpha=.1, color=ealdark) +
geom_point(data=eff.optimal.point, aes(x=Std.Dev, y=Exp.Return, label=sharpe),
color=ealred, size=5) +
annotate(geom="text", x=eff.optimal.point$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")