alan.gms : A Quadratic Programming Model for Portfolio Analysis

**Description**

This is a mini mean-variance portfolio selection problem described in 'GAMS/MINOS: Three examples' by Alan S. Manne, Department of Operations Research, Stanford University, May 1986. Integer variables have been added to restrict the number of securities selected. The resulting MINLP problem is solved with different option settings to demonstrate some DICOPT features. Finally, the model is solved by complete enumeration using GAMS procedural facilities.

**Reference**

- Manne, A S, GAMS/MINOS: Three examples. Tech. rep., Department of Operations Research, Stanford University, 1986.

**Small Model of Types :** MINLP nlp rminlp

**Category :** GAMS Model library

**Main file :** alan.gms

```
$Title A Quadratic Programming Model for Portfolio Analysis (ALAN,SEQ=124)
$Ontext
This is a mini mean-variance portfolio selection problem described in
'GAMS/MINOS: Three examples' by Alan S. Manne, Department of Operations
Research, Stanford University, May 1986.
Integer variables have been added to restrict the number of securities
selected. The resulting MINLP problem is solved with different option
settings to demonstrate some DICOPT features. Finally, the model is
solved by complete enumeration using GAMS procedural facilities.
Manne, A S, GAMS/MINOS: Three examples. Tech. rep., Department of
Operations Research, Stanford University, 1986.
$Offtext
Set i securities /hardware, software, show-biz, t-bills/; alias (i,j)
Scalar target target mean annual return on portfolio (%) / 10 /
Parameters mean(i) mean annual returns on individual securities (%)
/ hardware 8
software 9
show-biz 12
t-bills 7 /
Table v(i,j) variance-covariance array (%-squared annual return)
hardware software show-biz t-bills
hardware 4 3 -1 0
software 3 6 1 0
show-biz -1 1 10 0
t-bills 0 0 0 0
Variables x(i) fraction of portfolio invested in asset i
variance variance of portfolio
Positive Variable x;
Equations fsum fractions must add to 1.0
dmean definition of mean return on portfolio
dvar definition of variance;
fsum.. sum(i, x(i)) =e= 1.0 ;
dmean.. sum(i, mean(i)*x(i)) =e= target;
dvar.. sum(i, x(i)*sum(j,v(i,j)*x(j))) =e= variance;
Model portfolio / fsum, dmean, dvar / ;
Solve portfolio using nlp minimizing variance;
* now allow only three assets in our portfolio
Scalar maxassets max assets in portfolio /3/;
Binary Variables active(i) indicator: if 1 then asset is in portfolio;
Equations setindic(i) if active is 0 then not in portfolio,
maxactive defines max number of assets in portfolio;
setindic(i).. x(i) =l= active(i);
maxactive.. sum(i, active(i)) =l= maxassets;
Model p1 / fsum, dmean, dvar, setindic, maxactive/ ;
Solve p1 using minlp minimizing variance;
* now we change the solution options for dicopt
File opt /dicopt.opt/; Putclose opt 'stop 1'; p1.optfile=1;
Option limcol=0,limrow=0;
Solve p1 using minlp minimizing variance;
if (p1.modelstat<>%modelstat.Optimal% and
p1.modelstat<>%modelstat.LocallyOptimal% and
p1.modelstat<>%modelstat.FeasibleSolution% and
p1.modelstat<>%modelstat.IntegerSolution%,
abort 'Could not solve p1 minimizing variance');
* just to be sure we also do complete enumeration and put results
* on a separate file.
Sets b / zero, one /
Parameter boole(b) / zero 0, one 1 /; alias (b,b1,b2,b3,b4);
File res /results.put/; put res;
Scalar min / 1.0e10 /; p1.solprint=%solprint.Summary%;p1.optfile=0;
Loop(i, put ' ',i.tl:4 ); put ' variance';
Loop((b1,b2,b3,b4),
active.fx('hardware') = boole(b1);
active.fx('software') = boole(b2);
active.fx('show-biz') = boole(b3);
active.fx('t-bills') = boole(b4);
Solve p1 minimizing variance using rminlp;
Put / boole(b1):5:0 boole(b2):5:0 boole(b3):5:0 boole(b4):5:0;
if(p1.solvestat <> %solvestat.NormalCompletion%,
Put ' *** failed solvestat=' p1.solvestat:0:0 ' modelstat=', p1.modelstat:0:0;
display 'Solver failed' , p1.solvestat, p1.modelstat;
else
If(p1.modelstat <= %modelstat.LocallyOptimal% or p1.modelstat = %modelstat.FeasibleSolution%,
put variance.l:15:5;
If(variance.l < min,
put ' *' ;
min = variance.l )
Else
Put ' infeas, etc' ) ) );
```