$title A Quadratic Programming Model for Portfolio Analysis (QALAN,SEQ=282)

$onText
This is the gamslib model ALAN expressed as a QCP and MIQCP

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.

Keywords: quadratic constraint programming, mixed integer quadratic constraint
          programming, portfolio optimization, complete enumeration, finance
$offText

Set i 'securities' / hardware, software, show-biz, t-bills /;

Alias (i,j);

Scalar target 'target mean annual return on portfolio            (%)' / 10 /;

Parameter 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;

Variable
   x(i)     'fraction of portfolio invested in asset i'
   variance 'variance of portfolio';

Positive Variable x;

Equation
   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 qcp minimizing variance;

* now allow only three assets in our portfolio
Scalar maxassets 'max assets in portfolio' / 3 /;

Binary Variable active(i) 'indicator: if 1 then asset is in portfolio';

Equation
   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 /;

option optCr = 1e-6;

solve p1 using miqcp minimizing variance;