Description

```MAD.gms: Mean absolute deviation model.
Consiglio, Nielsen and Zenios.
PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 5.3
```

Category : GAMS FIN library

Mainfile : MAD.gms   includes :  Corporate.inc  WorldIndices.inc

``````\$title Mean absolute deviation model

* MAD.gms: Mean absolute deviation model.
* Consiglio, Nielsen and Zenios.
* PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 5.3

* Uncomment one of the following lines to include a data file

* \$include "Corporate.inc"
\$include "WorldIndices.inc"

SCALARS
Budget       Nominal investment budget
MU_TARGET    Target portfolio return
MU_STEP      Target return step
MIN_MU       Minimum return in universe
MAX_MU       Maximum return in universe;

Budget = 100.0;

PARAMETERS
pr(l)       Scenario probability
P(i,l)      Final values
EP(i)       Expected final values;

pr(l) = 1.0 / CARD(l);

P(i,l) = 1 + AssetReturns ( i, l );

EP(i) = SUM(l, pr(l) * P(i,l));

MIN_MU = SMIN(i, EP(i));
MAX_MU = SMAX(i, EP(i));

* Assume we want 20 portfolios in the frontier

MU_STEP = (MAX_MU - MIN_MU) / 20;

DISPLAY MAX_MU;

POSITIVE VARIABLES
x(i)            Holdings of assets in monetary units (not proportions)
y(l)            Measures of the absolute deviation;

VARIABLES
z           Objective function value;

EQUATIONS
BudgetCon       Equation defining the budget contraint
ReturnCon       Equation defining the portfolio return constraint
ObjDef          Objective function definition for MAD
yPosDef(l)      Equations defining the positive deviations
yNegDef(l)      Equations defining the negative deviations;

BudgetCon ..     SUM(i, x(i)) =E= Budget;

ReturnCon ..     SUM(i, EP(i) * x(i)) =G= MU_TARGET * Budget;

yPosDef(l) ..    y(l) =G= SUM(i, P(i,l) * x(i)) - SUM(i, EP(i) * x(i));

yNegDef(l) ..    y(l) =G= SUM(i, EP(i) * x(i)) - SUM(i, P(i,l) * x(i));

ObjDef    ..     z =E= SUM(l, pr(l) * y(l));

MODEL MeanAbsoluteDeviation 'PFO Model 5.3.1' /BudgetCon, ReturnCon, yPosDef, yNegDef, ObjDef/;

OPTION SOLVEOPT = REPLACE;

FrontierHandle.pc = 5;
FrontierHandle.pw = 1048;

PUT FrontierHandle;

FOR (MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,

SOLVE MeanAbsoluteDeviation MINIMIZING z USING LP;

PUT z.l:6:5,(MU_TARGET * Budget):8:3;

LOOP (i, PUT x.l(i):6:2);

PUT /;

);

* Compute variances and covariances
* for comparison between Mean Variance and Mean Absolute Deviation

ALIAS (i,i1,i2);

PARAMETERS
VP(i,i);

VP(i,i) = SUM(l, SQR(P(i,l) - EP(i))) / (CARD(l)- 1);

VP(i1,i2)\$(ORD(i1) > ORD(i2)) = SUM(l, (P(i1,l) - EP(i1))*(P(i2,l) - EP(i2))) / (CARD(l) - 1);

DISPLAY VP;

EQUATION
ObjDefMV        Objective function definition for Mean-Variance;

ObjDefMV ..      z =E= SUM((i1,i2), x(i1)* VP(i1,i2) * x(i2));

MODEL MeanVariance /BudgetCon, ReturnCon, ObjDefMV/;

PUT "SD","Mean"/;

FOR  (MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,

SOLVE MeanVariance MINIMIZING z USING NLP;

z.l = SQRT(z.l);

PUT z.l:6:5,(MU_TARGET * Budget):8:3;

LOOP (i, PUT x.l(i):6:2);

PUT /;

);

SCALARS
lambdaPos      Weight attached to positive deviations
lambdaNeg      Weight attached to negative deviations;

lambdaPos = 0.5;
lambdaNeg = 0.5;

EQUATIONS
yPosWeightDef(l)   Equations defining the positive deviations with weight attached
yNegWeightDef(l)   Equations defining the positive deviations with weight attached;

yPosWeightDef(l) ..    y(l) =G= lambdaPos * (SUM(i, P(i,l) * x(i)) - SUM(i, EP(i) * x(i)));

yNegWeightDef(l) ..    y(l) =G= lambdaNeg * (SUM(i, EP(i) * x(i)) - SUM(i, P(i,l) * x(i)));

MODEL MeanAbsoluteDeviationWeighted /BudgetCon, ReturnCon, yPosWeightDef, yNegWeightDef, ObjDef/;

FOR  (MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,

SOLVE MeanAbsoluteDeviationWeighted MINIMIZING z USING LP;

PUT z.l:6:5,(MU_TARGET * Budget):8:3;

LOOP (i, PUT x.l(i):6:2);

PUT /;

);

lambdaPos = 0.2;
lambdaNeg = 0.8;

FOR  (MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,

SOLVE MeanAbsoluteDeviationWeighted MINIMIZING z USING LP;

PUT z.l:6:5,(MU_TARGET * Budget):8:3;

LOOP (i, PUT x.l(i):6:2);

PUT /;

);

* Note that, the last two models will yield the same portfolios! See PFO Section 5.2.2 .
``````
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