Efficient Frontier
Portfolio of two assets
\[ \mathbb{E}(R_P)=\omega_1\mathbb{E}(R_1)+\omega_2\mathbb{E}(R_2), \]
\(\mathbb{E}(R_P)\) is the expected return of the portfolio,
\(\omega_i\) is the weight of asset \(i\),
\(\mathbb{E}(R_i)\) is the expected return of asset \(i\).
Portfolio variance \[ \sigma_P^2=\omega_1^2\sigma_1^2+\omega_2^2\sigma_2^2+2\omega_1\omega_2\sigma_{1,2}, \]
\(\sigma_P^2\) is the portfolio variance
\(\sigma_i^2\) is the variance of assets \(i\)
\(\sigma_{1,2}\) is the covariance between asset 1 and 2
\(\omega_1+\omega_2=1\)
portfolio volatility or standard deviation is \[ \sigma_P=\bigg\{\omega_1^2\sigma_1^2+\omega_2^2\sigma_2^2+2\omega_1\omega_2\sigma_{1,2}\bigg\}^{\frac{1}{2}}. \]
Example
Two securities, say X and Y
\(\mathbb{E}(R_X)=5\%\) and \(\mathbb{E}(R_Y)=4\%\),
\(\sigma_X^2=9\%\), \(\sigma_Y^2=6\%\)
\(\sigma_{XY}=3\%\).
The following table presents the portfolio return and volatility for five different portfolio combinations.
| \(\omega_X\) | 100% | 80% | 60% | 40% | 20% | 0% |
| \(\omega_Y\) | 0% | 20% | 40% | 60% | 80% | 100% |
| —————- | ———- | ———– | ———- | ———– | ———- | ———- |
| \(\mathbb{E}(R_P)\) | 5% | 4.8% | 4.6% | 4.4% | 4.2% | 4% |
| \(\sigma_P^2\) | 9% | 6.96% | 5.64% | 5.04% | 5.16% | 6% |
| —————- | ———- | ———– | ———- | ———– | ———- | ———- |
- Though we consider six possible choices of \(\omega_X\); however an infinitely many choices are possible for two stocks.
Portfolio of \(N\) assets
- Expected portfolio return
\[ \mathbb{E}(R_P)=\omega^T\mu, \] where \(\omega^T=\{\omega_1,\omega_2,...,\omega_N\}\), \(\mu=\{\mu_1,\mu_2,...,\mu_N\}\), \(\mu_i=\mathbb{E}(R_i)\), \(i=1,2...,N\) and
Portfolio volatility as \[ \sigma_P=\sqrt{\omega^T\Sigma\omega}, \] where \[ \Sigma=\Bigg[\begin{array}{ccc} \sigma_1^2&...&\sigma_{1N}\\ \vdots & \ddots &\vdots\\ \sigma_{N1}&...&\sigma_1^2\\ \end{array}\Bigg] \] is the portfolio covariance matrix.
Calculate expected return and volatility for all possible portfolios that can be constructed by varying the portfolio weights of the assets.
The set of all possible portfolios, represented by their expected return and volatilities has the characteristic shape.
Considers 10000 portfolios, where portfolio weights are randomly simulated and corresponding portfolio return.
Consider the global portfolio with passive investment strategy, where you want to invest in the ETF of FTSE, DAX, SMI and CAC and use the data in
EuStockMarketsdataset.
Index_Value<-as.matrix(EuStockMarkets)
r<-diff(log(Index_Value))*100
no.of.portf<-10000
set.seed(1)
sigma<-mu<-rep(NA,no.of.portf)
for(i in 1:no.of.portf){
w <- sample(1:1000,4,replace=T)
w <- w/sum(w) ## weight for i-th portfolio
rp <- r%*%w ## returns of i-th portfolio
mu[i] <- mean(rp) ## mean return of i-th portfolio
sigma[i] <- sd(rp) ## volatility of i-th portfolio
}
##################################
plot(sigma,mu,xlab = "volatility"
,ylab="expected return",col="grey")
abline(h=0.065,col="red",lwd=2)
segments(0.8,0.04,0.8,0.065,col="blue",lwd=2)
segments(0.85,0.04,0.85,0.065,col="green",lwd=2)
arrows(0.775,0.07,0.8,0.065,col="black",lwd=2)
arrows(0.875,0.07,0.85,0.065,col="black",lwd=2)
text(0.77,0.071,"w1")
text(0.88,0.071,"w2")
points(0.779,0.065,col="blue",lwd=2)
text(0.779,0.066,"w")
points(0.81,0.07,col="blue",lwd=2)
text(0.81,0.0715,"v")
### Plot the efficient frontier
Sigma<-cov(r)
library(tseries)
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)
frontier[i,]<-c(port_optim$ps,port_optim$pm)
}
lines(frontier,col="red")
Portfolio
w1andw2has the same expected return but different portifolio volatility (aka standard deviation)Which portfolio you would choose?
———–Email: sourish [at] cmi.ac.in ——–