In this part of our portfolio optimization we explore the efficient frontier, EF, and the minimum variance locus, MVL, of mean-CVaR portfolios. We proceed in the same way as for mean-variance portfolios: We select the two assets which lead to the smallest and largest returns and divide their range into equidistant parts which determine the target returns for which we try to find the efficient portfolios. We compute the global minimum risk portfolio and start from the closest returns to this point in both directions of the EF and the MVL. Note that only in the case of the long-only portfolio constraints do we reach both ends of the EF and the MVL. Usually, constraints will shorten the EF and MVL, and may even happen, that the constraints were so strong that do not find any solution at all.

The data employed in the study consists of daily closing prices of kSE-30. The data set encompassed the trading days from 4th August, 2008 to 2nd Febuary 2015, 2009. The data is collected from the historical data available on the website of Pakistan stock Exchange.

In the following we compute and compare long-only constraint efficient frontiers of mean-CVaR portfolio. Going forward we will compare unlimited short, box, and group constrained efficient frontiers of mean-CVaR portfolios.

The long-only mean-variance portfolios. In this case all the weights are bounded between zero and one.

## 
## Title:
##  CVaR Portfolio Frontier 
##  Estimator:         covEstimator 
##  Solver:            solveRglpk.CVAR 
##  Optimize:          minRisk 
##  Constraints:       LongOnly 
##  Portfolio Points:  5 of 5 
##  VaR Alpha:         0.05 
## 
## Portfolio Weights:
##     DGKC  ENGRO   FCCL   FFBL    FFC    HBL   HCAR   HUBC   JSCL  KAPCO
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000
## 2 0.0000 0.0000 0.0000 0.0000 0.0252 0.0000 0.0000 0.0000 0.5201 0.0664
## 3 0.0000 0.0000 0.0000 0.0000 0.1443 0.0431 0.0000 0.0000 0.1476 0.3226
## 4 0.0000 0.0000 0.0000 0.1187 0.0578 0.0473 0.0744 0.1984 0.0000 0.2687
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000
##      KEL   LUCK   MARI    MCB   MLCF    NBP    NML   OGDC   PAEL   PIOC
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 2 0.0000 0.0000 0.0000 0.0000 0.0000 0.1585 0.0000 0.0000 0.0000 0.0000
## 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0897 0.0000 0.0000 0.0013 0.0000
## 4 0.0000 0.0412 0.0091 0.0000 0.0000 0.0000 0.0000 0.0185 0.0000 0.0381
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
##      POL    PPL    PSO    PTC  SEARL    TRG    UBL
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 2 0.0000 0.1355 0.0000 0.0944 0.0000 0.0000 0.0000
## 3 0.0000 0.1064 0.0000 0.1449 0.0000 0.0000 0.0000
## 4 0.0000 0.0320 0.0000 0.0000 0.0694 0.0000 0.0264
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 
## Covariance Risk Budgets:
##     DGKC  ENGRO   FCCL   FFBL    FFC    HBL   HCAR   HUBC   JSCL  KAPCO
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000
## 2 0.0000 0.0000 0.0000 0.0000 0.0051 0.0000 0.0000 0.0000 0.8587 0.0086
## 3 0.0000 0.0000 0.0000 0.0000 0.1135 0.0303 0.0000 0.0000 0.3441 0.1997
## 4 0.0000 0.0000 0.0000 0.1274 0.0508 0.0427 0.0919 0.2126 0.0000 0.2433
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000
##      KEL   LUCK   MARI    MCB   MLCF    NBP    NML   OGDC   PAEL   PIOC
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0630 0.0000 0.0000 0.0000 0.0000
## 3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0874 0.0000 0.0000 0.0009 0.0000
## 4 0.0000 0.0458 0.0084 0.0000 0.0000 0.0000 0.0000 0.0161 0.0000 0.0460
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
##      POL    PPL    PSO    PTC  SEARL    TRG    UBL
## 1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 2 0.0000 0.0334 0.0000 0.0312 0.0000 0.0000 0.0000
## 3 0.0000 0.0794 0.0000 0.1447 0.0000 0.0000 0.0000
## 4 0.0000 0.0287 0.0000 0.0000 0.0608 0.0000 0.0254
## 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 
## Target Returns and Risks:
##      mean     Cov    CVaR     VaR
## 1 -0.1417  4.3654  8.1617  5.1264
## 2 -0.0784  2.5584  5.4619  3.4576
## 3 -0.0151  1.3261  3.3478  1.9356
## 4  0.0482  1.0755  2.6980  1.5903
## 5  0.1115  2.6869  5.7461  4.4269
## 
## Description:
##  Tue Jan 17 11:53:33 2017 by user: azam.yahya

The printout lists the weights, the covariance risk budgets and the target return and risk values along the minimum variance locus and the efficient frontier starting with the portfolio with the lowest return and ending with the portfolio with the highest achievable return at the end of the efficient frontier. To shorten the output, we have lowered the number of frontier points to 5 points.

However, to plot the efficient frontier we repeat the optimization with 25 points at the frontier as shown below. It shows for 25 equidistant return points the minimum variance locus and the efficient frontier. Added are the risk-return points for the individual assets and the equal weights portfolio. The line through the origin is the tangency line for a zero risk-free rate. The curved line with the maximum the tangency point is the Sharpe ration along the frontier.

Next, the plot below shows the results for the weights, the weighted returns and the covariance risk budgets along the minimum variance locus and the efficient frontier. It shows for the weights, weighted returns, and covariance risk budgets 25 equidistant return points, along the minimum variance locus and the efficient frontier. Note that the strong separation line marks the position between the minimum variance locus and the efficient frontier. Target returns are increasing from left to right, whereas target risks, are increasing to the left and right with respect to the separation line.