MODEL: ! Markowitz Value at Risk Portfolio Model(PORTVAR);
! Find a portfolio that minimizes the loss threshold that
has a probability of .05 of being exceeded;
! Keywords: value at risk, Markowitz, portfolio, Second order cone, SOC;
SETS:
STOCKS: AMT, RET;
COVMAT(STOCKS, STOCKS): VARIANCE;
ENDSETS DATA:
STOCKS = ATT, GMC, USX;
STARTW = 1.0; ! How much we start with;
PROB = .05;! Risk threshold, must be < .05;
!Covariance matrix and expected returns,
see Markowitz(1959);
VARIANCE = .01080754 .01240721 .01307513
.01240721 .05839170 .05542639
.01307513 .05542639 .09422681 ;
RET = 1.0890833 1.213667 1.234583 ;
ENDDATA
!;
! Get the s.d. corresponding to this risk threshold;
PROB = @PSN( Z);
@FREE( Z); ! Generally, Z < 0;
! Maximize value not at risk, note Z < 0;
[VAR] MAX = ARET + Z * RISK;
! Use exactly 100% of the starting budget;
@SUM( STOCKS: AMT) = STARTW;
! Expected terminal value;
ARET = @SUM( STOCKS: AMT * RET) ;
! This is a second order cone constraint, and thus
tends to solve fast;
! In squared form;
@SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)) <= SD^2 ;
! In squared form using an "=" constraint. The "<=" relaxation is SOC;
! @SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)) = SD^2 ;
! In square root form;
! (@SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)))^0.5 <= SD;
! Measure risk by standard deviation;
RISK = SD;
! Measure risk by variance rather than SD. This is a "rotated" SOC;
! @SUM( COVMAT(I, J): AMT(I) * AMT(J) * VARIANCE(I, J)) <= RISK;
END
