是否有 R 函数如何通过找到投资组合资产的最佳权重来最小化跟踪误差

Question

假设我有一个投资组合，共有 n 个资产及其在 x 个时期内的 return 以及同期基准的 return。我的 objective 是找到一个权重向量 w 使得

w∗=arg 最小 TE(w)

其中 TE(w) 是跟踪误差，定义如下：

 TE = (sum((return.portfolio-return.benchmark)^2)/(x - 1))^0.5

简而言之，我想在 x 个时期内使用包含 n 个资产的投资组合来复制基准的 return。唯一的限制应该是所有资产的总和等于 1，并且没有资产的权重高于 0.4。

有人知道我如何用 R 解决这个问题吗？

Answer 1

有几个选项。这是来自 stats 的 constrOptim:

#some data
library(quantmod)
library(PerformanceAnalytics)

getReturns <- function(symbols, from='2000-01-01') {
  returns <- list()
  for (symbol in symbols) { #symbol=symbols[2]
    sym = getSymbols(symbol, from = from, adjustOHLC = TRUE, auto.assign = FALSE)
    return <- Return.calculate(Ad(sym))
    colnames(return) <- gsub("\.Adjusted", "", colnames(return))
    returns[[symbol]] <- return
  }
  returns <- do.call(cbind, returns)
  return(returns)
}

symbols <-  c("MSFT","AAPL","AMZN","CSCO","ADBE")
ret <- getReturns(symbols,from="2020-01-01")[-1,]

x = nrow(ret) # periods
N = ncol(ret) # assets

set.seed(1324)
Bench = rnorm(x,0.0015,0.03) #some benchmark
plot(ts(Bench))

# const opt
## needs a feasible starting value
#  A'b >= b0  x < 0.4 --> -x >= -.4 x> -0.4 or 0 if not short selling
# 
fTE <- function(b,rets,Ben) {
  Pret=apply(rets,1,function(rl)sum(rl*b))
  sd(Pret-Ben)
}
upblim=0.4
loblim=-0.4
Amat <- matrix(1, 1, N)
Amat <- rbind(Amat,-diag(N)) # upper limit
Amat <- rbind(Amat,diag(N)) # if lower limit
bvec       <- c(.9999999,rep(-upblim,N),rep(loblim,N))

# feasable sol
ws=c(3,4,5,6,1)
(ws=ws/sum(ws))
Amat %*% ws >= bvec

(sol <- constrOptim(ws, fTE, NULL, ui = Amat, ci = bvec,rets=ret,Ben=Bench))
sum(sol$par)
pie(abs(sol$par),labels = colnames(ret))

#> data.frame(Assets=colnames(ret),ws=sol$par)
#  Assets          ws
#1  MSFT -0.13233070
#2  AAPL  0.39999989
#3  AMZN  0.39999997
#4  CSCO  0.26382163
#5  ADBE  0.06850912

是否有 R 函数如何通过找到投资组合资产的最佳权重来最小化跟踪误差

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