Notations

 

 

= instruments in the portfolio;

= index of instrument in the portfolio;

= number of scenarios;

=  index of scenarios;

= vector of weights of instruments in the portfolio, ;

= rate of return of instrument i under scenario ;

= random return vector;

= loss function;

= average rate of return of the i-th instrument over set of scenarios ;

= target level of rate of return of the portfolio;

=  bound on the portfolio CVaR;

=  penalty coefficient;

=  confidence level in CVaR.

 

 

 

Optimization Problem 1

 

minimizing portfolio CVaR

 

(CS.1)

 

subject to

 

budget  constraint

(CS.2)

 

constraint on the portfolio mean return

 

(CS.3)

no short constraints

(CS.4)

 

The constraint (CS.3) may be infeasible. To prevent possible infeasibility we have added to the performance function a compensation variable assuring feasibility.

 

Optimization Problem 2

 

minimizing weighted average of  risk and penalty

(CS.5)

 

subject to

 

budget  constraint

(CS.6)

 

constraint on  portfolio mean return combined with  penalty variable

(CS.7)

no short constraints and non-negativity constraint on penalty variable

(CS.8)

 

Optimization Problem 3

 

maximizing portfolio mean return

 

(CS.9)

 

subject to

 

CVaR constraint

 

(CS.10)

 

budget  constraint

 

(CS.11)

 

no short constraints

 

(CS.12)

 

© 2006-2016 American Optimal Decisions, Inc.