Notations

 

= instruments in the portfolio;

=  index of instrument in the portfolio;

=  number of sample-paths, where every sample-path is a joint trajectory of rates of returns of underlying instruments (we suppose that the joint trajectory corresponding to the k-th sample-path precedes the joint trajectory corresponding to the k+1-th sample-path) ;

=  number of scenarios (time moments) in each sample-path;

=  index of scenarios (time moment);

=  number of years in the time period ;

=  vector of positions (weights) of instruments in the portfolio, ;

= rate of return of the i-th instrument at moment j in k-th sample-path;

=  matrix of scenarios corresponding to the k-th sample path, k =1,...,K;

;

= matrix of scenarios for united single sample path;

;

if ;

= portfolio rate of return  at moment in -th sample-path;

= portfolio rate of return at the moment in the united single path;

=  portfolio uncompounded cumulative rate of return up to the moment in -th sample-path;

=  portfolio uncompounded cumulative rate of return up to the moment ;

= average over paths annualized portfolio rate of return over the time period ;

= average over paths annualized portfolio rate of return of the -th instrument in the portfolio over the time period ;

=  drawdown function at moment in -th sample-path;

= drawdown function at the moment in the united single path;

=  confidence level, ;

=  function having equally probable scenarios ,;

= CDaR Deviation Multiple with confidence level ;

= function having equally probable scenarios ;

=  CDaR Deviation with confidence level ;

= bound on portfolio risk;

= lower bound on positions;

= upper bound on positions.

 

Remark: J scenarios in K sample-paths are equally probable, i.e., every scenario probability equals .

 

 

Optimization Problem 1

 

maximizing average annualized portfolio return

subject to

constraint on  CDaR Deviation Multiple ( for multiple paths)

 

 

lower and upper bounds on weights

 

 

Optimization Problem 2

maximizing average annualized portfolio return

 

subject to

constraint on  CDaR Deviation (for united single path)

 

 

lower and upper bounds on weights

 

 

 

Initial Data

 

Number of sample-paths (matrices of scenarios), K = 11.

Number of instruments in the portfolio, I =  31.

Number of time moments in the path, J =  1175.

Lower bound on weights, .

Upper bound on weights,.

Confidence level, .

Each of the eleven sample-paths is represented by the corresponding matrix of scenarios of rates of returns of instruments in the portfolio: “matrix_1”, “matrix_2”, …, “matrix_11”.

United single sample path is represented by the matrix of scenarios "matrix_H" with 12925 scenarios.