$title Financial Optimization: Risk Management using MIQCP (QMEANVAR,SEQ=291)

$onText
Minimum and maximum trade constraints are added to the standard
mean-variance model. If it is not profitable to trade within these
ranges, no trade should take place. A turnover constraint is added
to improve stability of the solution to small changes in data. The
resulting model is now expressed as a MIQCP and a frontiers is
computed.

Dahl, H, Meeraus, A, and Zenios, S A, Some Financial Optimization
Models: Risk Management. In Zenios, S A, Ed, Financial Optimization.
Cambridge University Press, New York, NY, 1993.

Keywords: mixed integer quadratic constraint programming, financial optimization,
          risk management, finance
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$eolCom //

Set i 'securities' / cn, fr, gr, jp, sw, uk, us /;

Alias (i,j);

Parameter mu(i) 'expected return of security' / cn  0.1287, fr  0.1096
                                                gr  0.0501, jp  0.1524
                                                sw  0.0763, uk  0.1854
                                                us  0.0620             /;

Table q(i,j) 'covariance matrix'
          cn      fr      gr      jp      sw      uk      us
   cn  42.18
   fr  20.18   70.89
   gr  10.88   21.58   25.51
   jp   5.30   15.41    9.60   22.33
   sw  12.32   23.24   22.63   10.32   30.01
   uk  23.84   23.80   13.22   10.46   16.36   42.23
   us  17.41   12.62    4.70    1.00    7.20    9.90   16.42;

* we will continue to use only the lower triangle of the q-matrix
* and adjust the off diagonal entries to give the correct results.

q(i,j) = 2*q(j,i);
q(i,i) = q(i,i)/2;

Scalars tau 'bounding parameter on turnover of current holdings' / 0.3 /;

Set pd 'portfolio data labels'
       / old   'current holdings fraction of the portfolio'
         umin  'minimum increase of holdings fraction of security i'
         umax  'maximum increase of holdings fraction of security i'
         lmin  'minimum decrease of holdings fraction of security i'
         lmax  'maximum decrease of holdings fraction of security i' /

Table bdata(i,pd) 'portfolio data and trading restrictions'
*           - increase -   - decrease -
       old  umin    umax   lmin    lmax
   cn  0.2  0.03    0.11   0.02    0.30
   fr  0.2  0.04    0.10   0.02    0.15
   gr  0.0  0.04    0.07   0.04    0.10
   jp  0.0  0.03    0.11   0.04    0.10
   sw  0.2  0.03    0.20   0.04    0.10
   uk  0.2  0.03    0.10   0.04    0.15
   us  0.2  0.03    0.10   0.04    0.30;

bdata(i,'lmax') = min(bdata(i,'old'),bdata(i,'lmax')); // tighten bound

abort$(abs(sum(i, bdata(i,'old')) - 1) >= 1e5) 'error in bdata', bdata;

Variable
   var   'variance of portfolio'
   ret   'return of portfolio'
   x(i)  'fraction of portfolio of current holdings of i'
   xi(i) 'fraction of portfolio increase'
   xd(i) 'fraction of portfolio decrease'
   y(i)  'binary switch for increasing current holdings of i'
   z(i)  'binary switch for decreasing current holdings of i';

Binary   Variable y, z;
Positive Variable x, xi, xd;

Equation
   budget    'budget constraint'
   turnover  'restrict maximum turnover of portfolio'
   maxinc(i) 'bound of maximum lot increase of fraction of i'
   mininc(i) 'bound of minimum lot increase of fraction of i'
   maxdec(i) 'bound of maximum lot decrease of fraction of i'
   mindec(i) 'bound of minimum lot decrease of fraction of i'
   binsum(i) 'restrict use of binary variables'
   xdef(i)   'final portfolio definition'
   retdef    'return definition'
   vardef    'variance definition';

budget..    sum(i, x(i)) =e= 1;

xdef(i)..   x(i)  =e= bdata(i,'old') - xd(i) + xi(i);

maxinc(i).. xi(i) =l= bdata(i,'umax')*y(i);

mininc(i).. xi(i) =g= bdata(i,'umin')*y(i);

maxdec(i).. xd(i) =l= bdata(i,'lmax')*z(i);

mindec(i).. xd(i) =g= bdata(i,'lmin')*z(i);

binsum(i).. y(i) + z(i) =l= 1;

turnover..  sum(i, xi(i)) =l= tau;

retdef..    ret =e= sum(i, mu(i)*x(i));

vardef..    var =e= sum((i,j), x(i)*q(i,j)*x(j));

Model
   maxret / budget, xdef, turnover, maxinc, mininc, maxdec, mindec, binsum, retdef         /
   minvar / budget, xdef, turnover, maxinc, mininc, maxdec, mindec, binsum, retdef, vardef /;

Set
   p     'efficient frontier points' / oldr, minvar, p1*p4, maxret, oldv /
   pp(p) 'efficient frontier points' /               p1*p4               /;

Scalar
   rmin 'minimum variance'
   rmax 'maximum variance';

Parameter
   Report(p,i,*) 'portfolio details at different points'
   xres(*,p)     'summary report at different points';

option limCol = 0, limRow = 0, solPrint = off, optCr = 0;

loop(p('maxret'),
   solve maxret max ret using mip;
   rmax              = ret.l;
   xres(i,p)         = x.l(i);
   report(p,i,'inc') = xi.l(i);
   report(p,i,'dec') = xd.l(i);
);

xres(i,'oldr') = bdata(i,'old');
xres(i,'oldv') = bdata(i,'old');

loop(p('minvar'),
   solve minvar min var using miqcp;
   rmin              = ret.l;
   xres(i,p)         = x.l(i);
   report(p,i,'inc') = xi.l(i);
   report(p,i,'dec') = xd.l(i);
);

loop(p(pp),
   ret.fx = rmin + (rmax - rmin)/(card(pp) + 1)*ord(pp);
   solve minvar min var using miqcp;
   xres(i,p)         = x.l(i);
   report(p,i,'inc') = xi.l(i);
   report(p,i,'dec') = xd.l(i);
);

xres('mean',p) = sum(i, mu(i)*xres(i,p));
xres('var' ,p) = sum((i,j), xres(i,p)*q(i,j)*xres(j,p));
display xres, report;

$onText
Output data to Excel into sheet 'Results'
To create the two plots in Excel:

Plot 1: Allocation -
     1. Select data from C2:I8 and plot using an
           "Area Plot" (type: stacked area)

Plot 2: Variance vs. Mean Return -
     1. Create an XY Scatter Plot (type: 'scatter with data points connected by lines')
     2. Data:
            Series 1:
               'X-Values': =Results!$C$9:$H$9
               'Y-Values': =Results!$C$10:$H$10
            Series 2:
               'X-Values': =(Results!$B$9,Results!$I$9)
               'Y-Values': =(Results!$B$10,Results!$I$10)
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EmbeddedCode Connect:
- GAMSReader:
    symbols:
      - name: xres
- ExcelWriter:
    file: results.xlsx
    symbols:
      - name: xres
        range: Results!A1
endEmbeddedCode