CVaR : Conditional Value at Risk models.

Description

```CVaR.gms: Conditional Value at Risk models.
Consiglio, Nielsen and Zenios.
PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 5.5
```

Category : GAMS FIN library

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

``````\$TITLE Conditional Value at Risk models

* CVaR.gms: Conditional Value at Risk models.
* Consiglio, Nielsen and Zenios.
* PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 5.5

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

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

SCALARS
Budget        Nominal investment budget
alpha         Confidence level
MU_TARGET     Target portfolio return
MU_STEP       Target return step
MIN_MU        Minimum return in universe
MAX_MU        Maximum return in universe
RISK_TARGET   Bound on CVaR (risk)
LossFlag      Flag selecting the type of loss;

Budget = 100.0;
alpha  = 0.99;

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;

PARAMETER
TargetIndex(l)   Target index returns;

* To test the model with a market index, uncomment the following two lines.
* Note that, this index can be used only with WorldIndexes.inc.

*\$INCLUDE "Index.inc";

*TargetIndex(l) = Index(l);

POSITIVE VARIABLES
x(i)            Holdings of assets in monetary units (not proportions)
VaRDev(l)       Measures of the deviations from the VaR;

VARIABLES
VaR             Value-at-Risk
z               Objective function value
Losses(l)       Measures of the losses;

EQUATIONS
BudgetCon        Equation defining the budget contraint
ReturnCon        Equation defining the portfolio return constraint
CVaRCon          Equation defining the CVaR allowed
ObjDefCVaR       Objective function definition for CVaR minimization
ObjDefReturn     Objective function definition for return mazimization
LossDef(l)       Equations defining the losses
VaRDevCon(l)     Equations defining the VaR deviation constraints;

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

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

CVaRCon ..           VaR + SUM(l, pr(l) * VaRDev(l)) / (1 - alpha) =L= RISK_TARGET;

VaRDevCon(l) ..      VaRDev(l) =G= Losses(l) - VaR;

LossDef(l)..         Losses(l) =E= (Budget - SUM(i, P(i,l) * x(i)))\$(LossFlag = 1) +
(TargetIndex(l) * Budget - SUM(i, P(i,l) * x(i)))\$(LossFlag = 2) +
(SUM(i, EP(i) * x(i)) - SUM(i, P(i,l) * x(i)))\$(LossFlag = 3);

ObjDefCVaR ..        z =E= VaR + SUM(l, pr(l) * VaRDev(l)) / (1 - alpha);

ObjDefReturn ..      z =E= SUM(i, EP(i) * x(i));

MODEL MinCVaR  'PFO Model 5.5.1' /BudgetCon, ReturnCon, LossDef, VaRDevCon, ObjDefCVaR/;

MODEL MaxReturn 'PFO Model 5.5.2' /BudgetCon, CVaRCon, LossDef, VaRDevCon, ObjDefReturn/;

FILE FrontierHandle /"CVaRFrontiers.csv"/;

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

PUT FrontierHandle;

PUT "Status","VaR","CVaR","Mean";

LOOP(i, PUT i.tl);

PUT /;

LossFlag = 2;

* Comment the following line if you want to
* track the market index.

TargetIndex(l) = 1.01;

FOR (MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,

SOLVE MinCVaR MINIMIZING z USING LP;

PUT MinCVaR.MODELSTAT:0:0,VaR.l:6:5,z.l:6:5,(MU_TARGET * Budget):8:3;

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

PUT /;
);
``````
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