Portfolio Optimization
Markowitz’s Portfolio Optimization (1952) minimizes the portfolio variance for a given level of expected return, i.e., \[ \min_{\omega}\omega^T\Sigma\omega \] such that \[ \omega^T\mu=\mu_0, \] where \(\mu_0\) is the expected level of return.
Note that you need to provide \(\Sigma\).
However, in reality we do not know what is true \(\Sigma\).
So we should estimate the \(\Sigma\).
The popular estimator for \(\Sigma\) is \[ S=\frac{1}{n-1}\sum_{i=1}^n(r_i-\bar{r})(r_i-\bar{r})^T. \]
You can use other estimators (like Bayes estimator) of \(\Sigma\). This will provide a different frontier.
You can do it easily using
portfolio.optimintseriespackage.Suppose you are considering the global portfolio with passive investment strategy, where you want to invest in the ETF of FTSE, DAX, SMI and CAC.
Your annualized expected return is 12.5%.
Consider annualized risk-free rate of return as 3%.
Index_Value<-as.matrix(EuStockMarkets)
## log-return in percentage terms
r<-diff(log(Index_Value))*100
## expected return
expected_return <- 12.5/252
## portfolio covariance
Sigma<-cov(r)
library(tseries)
port_optim<-portfolio.optim(r
,pm=expected_return
,covmat=Sigma
,rf=3/252)
## optimized portfolio weights
weight<-port_optim$pw*100
names(weight)<-colnames(EuStockMarkets)
weight## DAX SMI CAC FTSE
## 0.000000 16.560381 2.717417 80.722202## expected return
port_optim$pm## [1] 0.04960317## volatility at optimized weights
port_optim$ps## [1] 0.7633583### Plot the efficient frontier
er<-seq(0.045,0.075,0.001)
frontier<-matrix(NA,nrow=length(er),ncol=2)
for(i in 1:length(er)){
port_optim<-portfolio.optim(r
,pm=er[i]
,covmat=Sigma
,rf=3/252)
frontier[i,]<-c(port_optim$ps,port_optim$pm)
}
plot(frontier,col="red",type = "l"
,xlab="volatility"
,ylab = "expected return")