Mathematics and Finance Summer School at CMI

Chennai Mathematical Institute

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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.

	X1	X2	X3	X4	X5	X6
W_x	100%	80%	60%	40%	20%	0%
W_y	0%	20%	40%	60%	80%	100%
Expected Return	5%	4.8%	4.6%	4.4%	4.2%	4%
Volatility	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 EuStockMarkets dataset.

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 w1 and w2 has the same expected return but different portifolio volatility (aka standard deviation)

• Which portfolio you would choose?

———–Email: sourish [at] cmi.ac.in ——–