Initiation steps:

1. Verify that MATLAB version is 7.2 or higher.

2. Install Portfolio Safeguard (PSG).

3. Verify paths in MATLAB to the folder containing PSG libraries '...\Aorda\PSG\MATLAB'  and '...\Aorda\PSG\MATLAB\Tools'.

 

As example, we consider Conditional Value-at-Risk minimization, and show main steps in PSG MATLAB environment:

 

 

Conditional Value-at-Risk minimization. Mathematical formulation

 This example demonstrates CVaR (see PSG Function cvar_risk) minimization with average (expected) return (see PSG Function avg_g) and budget constraints.

General mathematical formulation of the minimization problem:

                             

subject to

portfolio return constraint:

budget constraint:

bounds on decision variables:

 

where

= vector of decision variables;

= portfolio return of instruments;

= Conditional Value-at-Risk for return;

= average return.

 

PSG functions in CVaR minimization problem:

1. Conditional Value-at-Risk in PSG is named  "CVaR Risk for Loss" (see cvar_risk). Function call :

cvar_risk(α, matrix)

where

= confidence level,

matrix = matrix of scenarios (see Matrix of Scenarios) with rows containing  returns of instruments,  benchmarks and probabilities.

Inputs for cvar_risk:

;

 

 

 

 

2. Average ( Expected value) of random linear gains in PSG is called Average Gain  (see avg_g). Function call:

avg_g(matrix)

where

matrix = matrix of scenarios (see Matrix of Scenarios) with rows containing  returns of instruments,  benchmarks and probabilities.

Inputs for avg_g:

H, c, p are the same as in cvar_risk function.

 

 There are three possibilities for optimizing this problem in PSG MATLAB environment: PSG Toolbox GUI, Special PSG MATLAB Subroutines and Programmatical PSG.

 

Example 1. Conditional Value-at-Risk Minimization. Toolbox

Quick Start with PSG Toolbox

See also:

 

Example 2. Conditional Value-at-Risk Minimization. Special Subroutines

Firstly, CVaR is minimized with PSG Subroutine riskconstrprog (see riskconstrprog).

 

Remarks

 

Define input data in MATLAB:

H = [1,4,8,3; 7,5,4,6; 2,8,1,0; 0,3,4,9];

c = [0.2; 0.11; 0.6; 0.1];

p=[0.25,0.25,0.25,0.25]; %Can be set p=[], because probabilities are equal.

Aeq = [1, 1, 1, 1];

beq = 1;

r = -4.5;

lb = [0; 0; 0; 0];

 

Run subroutine (see riskconstrprog function):

[xout, fval, status, output] = riskconstrprog('cvar_risk', 'avg', 0.95, H, c, p, [], H, c, p, [], r, [], [], Aeq, beq, lb);

 

Output:

Optimal point:

xout =

        0

   0.6000

   0.1900

   0.2100

Objective value on the optimal point:

fval =

  -4.3500

Status of the solved optimization problem:

status =

optimal

Full report for solution of the optimization problem:

output =

                 xout: [4x1 double]

                 fval: -4.3500

               frval1: -4.3500

                fdval: []

               frval2: -4.5000

                fAval: []

                rAval: []

              fAeqval: 1.0000

              rAeqval: 4.4409e-016

    data_loading_time: 0.0100

   preprocessing_time: 0

         solving_time: 0

MATLAB code for CVaR minimization using riskconstrprog is in: ./Aorda/PSG/MATLAB/Examples/Problems/Example_CVaR_riskconstrprog.m.

 

Example 3. Conditional Value-at-Risk Minimization.  Programmatical PSG

Define input data in MATLAB:

H=[1,4,8,3; 7,5,4,6; 2,8,1,0; 0,3,4,9];

c=[0.2; 0.11; 0.6; 0.1];

p=[];

Aeq=[1, 1, 1, 1];

lb=[0; 0; 0; 0];

vars={'x1', 'x2', 'x3', 'x4'};

 

Pack data to PSG Data Objects and save them to array of structures toolboxstruc_arr:

 

Pack input data to PSG structure containing matrix headers and data (see PSG Data Objects in MATLAB).

Pack matrices to the elements of the array "toolboxstruc_arr" (see function tbpsg_matrix_pack):

toolboxstruc_arr(1) = tbpsg_matrix_pack('matrix_scenarios',H,vars,c,p);

toolboxstruc_arr(2) = tbpsg_matrix_pack('matrix_budget',Aeq,vars);

 

Pack point to the elements of the array "toolboxstruc_arr" (see function tbpsg_point_pack):

toolboxstruc_arr(3) = point_pack('point_lowerbounds',lb,vars);

 

Create string with Problem Statement (see Problem Statement in MATLAB):

problem_statement = sprintf('%s\n',...

'minimize',...

'cvar_risk(0.95, matrix_scenarios)',...

'Constraint: >= 4.5',...

'avg_g(matrix_scenarios)',...

'Constraint: == 1',...

'linear(matrix_budget)',...

'Box: >= point_lowerbounds');

 

Solve problem with tbpsg_run:

[solution_str, outargstruc_arr] = mpsg_solver(problem_statement,toolboxstruc_arr)

 

You will get the following output (see PSG Solution in MATLAB):

 Full report for the solution of the optimization problem: (Full report with the optimization problem solution)

solution_str =

 

Problem: solution_status = optimal

Timing: Data_loading_time = 0.18, Preprocessing_time = 0.00, Solving_time = 0.00

Variables: optimal_point = point_problem_1

Objective:   = -4.350000000000

Constraint: constraint_1 = 4.500000000000e+000, [0.000000000000e+000]

Constraint: constraint_2 = 1.000000000000e+000, [-4.440892098501e-016]

Function: avg_g(matrix_scenarios) = 4.500000000000e+000

Function: cvar_risk(0.95,matrix_scenarios) = -4.350000000000e+000

Function: linear(matrix_budget) = 1.000000000000e+000

 

 Optimal point in PSG point format (see PSG Point in MATLAB):

outargstruc_arr =

 

   object_name: 'point_problem'

     variables: 'x1        x2        x3        x4'

          type: 'point'

        values: [0 0.6000 0.1900 0.2100]

 

Convert PSG solution to structure (see tbpsg_solution_struct):

>> output_structure = tbpsg_solution_struct(solution_str, outargstruc_arr)

 

output_structure =

 

                    status: {'optimal'}

                      time: [0.2200 0 0]

                 objective: -4.3500

             function_name: {3x1 cell}

            function_value: [3x1 double]

      vector_function_name: {}

     vector_function_value: {}

        optimal_point_data: [0 0.6000 0.1900 0.2100]

        optimal_point_vars: {'x1'  'x2'  'x3'  'x4'}

     point_constraint_data: [4.5000 1.0000]

     point_constraint_vars: {'constraint_1'  'constraint_2'}

          point_slack_data: [0 -4.4409e-016]

           point_dual_data: [3.7037 -22.1111]

   vector_constraints_data: {}

         vector_slack_data: {}

          vector_dual_data: {}

               matrix_data: {}

               matrix_vars: {}

               vector_data: {}

 

MATLAB code for CVaR minimization is in: ./Aorda/PSG/MATLAB/Examples/Problems/Example_CVaR_General.m.