MODEL: ! (PortScenHedge);
! Scenario portfolio model, Hedge 1st asset;
! Minimize the variance;
SETS:
SCENE: PRB, R, DVU, DVL;
ASSET: X;
SXA( SCENE, ASSET): VE;
ENDSETS DATA:
SCENE= 1..12;
ASSET=
GMT, ATT, USX;
! Data based on original Markowitz example;
VE =
1.225 1.300 1.149
1.290 1.103 1.260
1.216 1.216 1.419
0.728 0.954 0.922
1.144 0.929 1.169
1.107 1.056 0.965
1.321 1.038 1.133
1.305 1.089 1.732
1.195 1.090 1.021
1.390 1.083 1.131
0.928 1.035 1.006
1.715 1.176 1.908;
! All scenarios happen to be equally likely;
PRB= .0833333 .0833333 .0833333 .0833333 .0833333
.0833333 .0833333 .0833333 .0833333 .0833333
.0833333 .0833333;
! The desired return;
TARGET = 1.15;
ENDDATA
! Minimize risk;
[OBJ] MIN = @SUM( SCENE: PRB * ( DVL + DVU) ^ 2);
!We are stuck with having asset 1 in the portfolio;
X( 1) = 1;
!Compute hedging portfolio value under each scenario;
@FOR( SCENE( S):
R( S)=
@SUM( ASSET( J)| J #GT# 1: VE( S, J) * X( J));
! Measure deviations hedge + benchmark from target;
DVU( S) - DVL( S) =
( R(S) + VE( S, 1))/ 2 - TARGET;
);
! Budget constraint(applies to remaining assets);
[BUDGET] @SUM( ASSET( J)| J #GT# 1: X( J)) = 1;
! Compute expected value of ending position;
[DEFAVG] AVG = @SUM( SCENE: PRB * R);
! Target ending value;
[RET] AVG > TARGET;
END
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