Mathematics and Finance Summer School at CMI

Chennai Mathematical Institute

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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.optim in tseries package.

• 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")