Description
MeanVarMip.gms: Mean-variance model with diversification constraints. Consiglio, Nielsen and Zenios. PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 3.4 Last modified: Apr 2008.
Category : GAMS FIN library
Mainfile : MeanVarMip.gms includes : Estimate.gdx
$TITLE Mean-variance model with diversification constraints
* MeanVarMip.gms: Mean-variance model with diversification constraints.
* Consiglio, Nielsen and Zenios.
* PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 3.4
* Last modified: Apr 2008.
SET Assets;
ALIAS(Assets,i,j);
PARAMETERS
RiskFreeRate
ExpectedReturns(i) Expected returns
VarCov(i,j) Variance-Covariance matrix ;
* Read from Estimate.gdx the data needed to run the mean-variance model
$GDXIN Estimate
$LOAD Assets=subset RiskFreeRate=MeanRiskFreeReturn VarCov ExpectedReturns
$GDXIN
* Risk attitude: 0 is risk-neutral, 1 is very risk-averse.;
SCALAR
StockMax Maximum number of stocks /3/
lambda Risk attitude;
VARIABLES
x(i) Holdings of assets;
PARAMETER
xlow(i) lower bound for active variables ;
* In case short sales are allowed these bounds must be set properly.
xlow(i) = 0.0;
x.UP(i) = 1.0;
BINARY VARIABLE
Y(i) Indicator variable for assets included in the portfolio;
VARIABLES
PortVariance Portfolio variance
PortReturn Portfolio return
z Objective function value;
EQUATIONS
ReturnDef Equation defining the portfolio return
VarDef Equation defining the portfolio variance
NormalCon Equation defining the normalization contraint
LimitCon Constraint defining the maximum number of assets allowed
UpBounds(i) Upper bounds for each variable
LoBounds(i) Lower bounds for each variable
ObjDef Objective function definition;
ReturnDef .. PortReturn =E= SUM(i, ExpectedReturns(i)*x(i));
VarDef .. PortVariance =E= SUM((i,j), x(i)*VarCov(i,j)*x(j));
LimitCon .. SUM(i, Y(i)) =L= StockMax;
UpBounds(i).. x(i) =L= x.UP(i)* Y(i);
LoBounds(i).. x(i) =G= xlow(i)* Y(i);
NormalCon .. SUM(i, x(i)) =E= 1;
ObjDef .. z =E= (1-lambda) * PortReturn - lambda * PortVariance;
MODEL MeanVarMip /ReturnDef, VarDef, LimitCon, UpBounds, LoBounds, NormalCon, ObjDef/;
OPTION MINLP = SBB, optcr = 0;
FILE FrontierHandle /"MeanVarianceMIP.csv"/;
FrontierHandle.pc = 5;
FrontierHandle.pw = 1048;
PUT FrontierHandle;
PUT "Lambda","z","Variance","ExpReturn";
LOOP (i, PUT i.tl);
PUT /;
FOR (lambda = 0 TO 1 BY 0.1,
SOLVE MeanVarMip MAXIMIZING z USING MINLP;
PUT lambda:6:5, z.l:6:5, PortVariance.l:6:5, PortReturn.l:6:5;
LOOP (i, PUT x.l(i):6:5 );
PUT /;
)
PUTCLOSE;
***** Transaction Cost *****
* In this section the MeanVar.gms model is modified by imposing transaction
* costs. We consider a more realistic setting with fixed and proportional costs.
SCALARS
FlatCost Flat transaction cost / 0.001 /
PropCost Proportional transaction cost / 0.005 /;
POSITIVE VARIABLES
x_0(i) Holdings for the flat cost regime
x_1(i) Holdings for the linear cost regime;
* Amount at which is possible to make transactions at the flat fee.
x_0.UP(i) = 0.1;
EQUATIONS
HoldingCon(i) Constraint defining the holdings
ReturnDefWithCost Equation defining the portfolio return with cost
FlatCostBounds(i) Upper bounds for flat transaction fee
LinCostBounds(i) Upper bonds for linear transaction fee;
HoldingCon(i).. x(i) =e= x_0(i) + x_1(i);
ReturnDefWithCost.. PortReturn =e= SUM(i, ( ExpectedReturns(i)*x_0(i) - FlatCost*Y(i) ) ) +
SUM(i, (ExpectedReturns(i) - PropCost)*x_1(i));
FlatCostBounds(i).. x_0(i) =l= x_0.UP(i) * Y(i);
LinCostBounds(i).. x_1(i) =l= Y(i);
MODEL MeanVarWithCost /ReturnDefWithCost, VarDef,HoldingCon, NormalCon, FlatCostBounds, LinCostBounds, ObjDef/;
OPTION MINLP = SBB, optcr = 0;
FILE FrontierHandleTwo /"MeanVarianceWithCost.csv"/;
FrontierHandleTwo.pc = 5;
FrontierHandleTwo.pw = 1048;
PUT FrontierHandleTwo;
PUT "Lambda","z","Variance","ExpReturn";
LOOP (i, PUT i.tl);
LOOP (i, PUT i.tl);
PUT /;
FOR (lambda = 0 TO 1 BY 0.1,
SOLVE MeanVarWithCost MAXIMIZING z USING MINLP;
PUT lambda:6:5, z.l:6:5, PortVariance.l:6:5, PortReturn.l:6:5;
LOOP (i, PUT x_0.l(i):6:5 );
LOOP (i, PUT x_1.l(i):6:5 );
PUT /;
)
PUTCLOSE;
***** Portfolio Revision *****
* In this section the MeanVar.gms model is modified by imposing zero-or-range
* variable to cope with portfolio revision.
SET Bound /Lower, Upper/;
PARAMETER
BuyLimits(Bound,i)
SellLimits(Bound,i)
InitHold(i) Current holdings;
* We set the curret holding to the optimal unconstrained mean-variance portfolio
* with lambda = 0.5
InitHold('Cash_EU') = 0.3686;
InitHold('YRS_1_3') = 0.3597;
InitHold('EMU') = 0.0;
InitHold('EU_EX')= 0.0;
InitHold('PACIFIC')= 0.0;
InitHold('EMERGT') = 0.0591;
InitHold('NOR_AM') = 0.2126;
InitHold('ITMHIST') = 0.0;
BuyLimits('Lower',i) = InitHold(i) * 0.9;
BuyLimits('Upper',i) = InitHold(i) * 1.10;
SellLimits('Lower',i) = InitHold(i) * 0.75;
SellLimits('Upper',i) = InitHold(i) * 1.25;
POSITIVE VARIABLES
buy(i) Amount to be purchased
sell(i) Amount to be sold;
BINARY VARIABLE
Yb(i) Indicator variable for assets to be purchased
Ys(i) Indicator variable for assets to be sold;
EQUATIONS
BuyTurnover
LoBuyLimits(i)
UpBuyLimits(i)
UpSellLimits(i)
LoSellLimits(i)
BinBuyLimits(i)
BinSellLimits(i)
InventoryCon(i) Inventory constraints;
InventoryCon(i).. x(i) - buy(i) + sell(i) =e= InitHold(i);
UpBuyLimits(i).. InitHold(i) + buy(i) =l= BuyLimits('Upper',i);
LoBuyLimits(i).. InitHold(i) + buy(i) =g= BuyLimits('Lower',i);
UpSellLimits(i).. InitHold(i) - sell(i) =l= SellLimits('Upper',i);
LoSellLimits(i).. InitHold(i) - sell(i) =g= SellLimits('Lower',i);
BinBuyLimits(i).. buy(i) =l= Yb(i);
BinSellLimits(i).. sell(i) =l= Ys(i);
BuyTurnover.. SUM(i, buy(i) ) =l= 0.05;
MODEL MeanVarRevision /NormalCon, HoldingCon, InventoryCon, ReturnDef,
UpBuyLimits, LoBuyLimits, UpSellLimits, LoSellLimits,
BinBuyLimits, BinSellLimits, BuyTurnover,
VarDef, ObjDef/;
OPTION MINLP = SBB, optcr = 0;
FILE FrontierHandleThree /"MeanVarianceRevision.csv"/;
FrontierHandleThree.pc = 5;
FrontierHandleThree.pw = 1048;
PUT FrontierHandleThree;
PUT "Model status","Lambda","z","Variance","ExpReturn";
LOOP (i, PUT i.tl);
LOOP (i, PUT i.tl);
LOOP (i, PUT i.tl);
PUT /;
FOR (lambda = 0 TO 1 BY 0.1,
SOLVE MeanVarRevision MAXIMIZING z USING MINLP;
PUT MeanVarRevision.modelstat:0:0,lambda:6:5, z.l:6:5, PortVariance.l:6:5, PortReturn.l:6:5;
LOOP (i, PUT x.l(i):6:5 );
LOOP (i, PUT buy.l(i):6:5 );
LOOP (i, PUT sell.l(i):6:5 );
PUT /;
)
PUTCLOSE;