Is there an R function how to minimize the Tracking Error by finding optimal weights of portfolio assets

Question

Let´s say I have a portfolio of in total n assets and their returns over x periods and als the return of a benchmark over the same periods. My objective is to find a vector of weights w such that

w∗=arg min TE(w)

where TE(w) is the tracking error defined as follows:

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

In short, I want to replicate the return of a benchmark using a portfolio of n assets over x periods. The only constraints should be that the sum of all assets is equal to one and that no asset has a weight higher than 0.4.

Does anyone have an idea how I could solve this problem with R?

Answer 1

There are several options. Here is one with constrOptim from stats:

#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