Notations

 

I =  number of instruments;

J = number of scenarios;

= decision vector defining positions in instruments;

= rate of return of the i-th instrument under scenario j;

= random vector of rates of returns of instruments, i=1,…,I;

= vector of rates of returns of instruments, i=1,…,I, under scenarios j;

= Loss under scenario j;

 

, where is CVaR Risk for Loss function;

 

;

;

 

=  upper bounds;

= current portfolio value;

= (random) rate of return of hedge fund i;

= market beta for hedge fund i;

= constant in the inequality assuring market-neutrally (“zero-beta”);

= penalty coefficient for Risk included in objective.

 

Optimization Problem 1

Maximizing sum of expected return and weighted risk

 

                          (CS.1)

subject to

budget constraint

                                                                         (CS.2)

market-neutrality constraint

                                                         (CS.3)

constraint on individual positions  

                                                         (CS.4)

 

Optimization Problem 2

Maximizing sum of expected return and weighted risk

 

                                  (CS.5)

subject to

budget constraint

                                                                         (CS.6)

market-neutrality constraint

                                                         (CS.7)

constraint on individual positions  

                                                         (CS.8)

 

Optimization Problem 3

Maximizing sum of expected return and weighted risk

 

                          (CS.9)

subject to

budget constraint

                                                                         (CS.10)

market-neutrality constraint

                                                         (CS.11)

constraint on individual positions  

                                                         (CS.12)