Notations
= number of instruments in the portfolio;
= index of instruments in the portfolio;
= exposure to instrument
(in currency);
= vector of weights of instruments in the portfolio;
= rate of return of
-th instrument in time period
;
= hurdle rate of return;
= loss function (underperformance of the portfolio) at time
as a function of
(in currency);
= loss function (underperformance of the portfolio) at time
as a function of
(in fractions);
= loss function (underperformance of the portfolio) at time
as a function of
(in fractions);
= low partial moment (in currency) for loss function
with respect to benchmark 0;
= low partial moment (in currency) for loss function
with respect to benchmark 0;
= low partial moment (in fractions) for loss function
with respect to benchmark 0;
= upper partial moment (in currency) for loss function
with respect to benchmark 0;
= upper partial moment (in fractions) for loss function
with respect to benchmark 0;
= upper partial moment (in fractions) for loss function
with respect to benchmark 0;
= expected gain with respect to hurdle rate as a function of
(in currency);
= expected gain with respect to hurdle rate as a function of
(in currency);
= Omega function;
= number of strategies;
= set of managers using
-th strategy,
;
= lower bound in the constraint on allocations to strategies,
;
= upper bound in the constraint on allocations to strategies,
;
= lower bound in the box constraints for individual positions,
;
= upper bound in the box constraints for individual positions,
.
Optimization Problem
maximizing Omega function
(CS.1) |
subject to
budget constraint
(CS.2) |
constraints on allocations to strategies
(CS.3) |
box constraints for individual positions
(CS.4) |