meanvarx.gms : Financial Optimization: Risk Management

**Description**

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 a nonlinear mixed-integer problem. Two important modeling tricks are demonstrated: (1) use of only the triangular part of the Q matrix, and (2) introduction of the marginal variance to improve computational performance of large QP problems.

**Reference**

- 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.

**Small Model of Type :** MINLP

**Category :** GAMS Model library

**Main file :** meanvarx.gms

$Title Financial Optimization: Risk Management (MEANVARX,SEQ=113) $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 a nonlinear mixed-integer problem. Two important modeling tricks are demonstrated: (1) use of only the triangular part of the Q matrix, and (2) introduction of the marginal variance to improve computational performance of large QP problems. 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. $Offtext $Eolcom ! Set i securities / cn, fr, gr, jp, sw, uk, us /; alias (i,j) ; Parameter mu(i) expected return of security / cn 0.1287 fr 0.1096 gr 0.0501 jp 0.1524 sw 0.0763 uk 0.1854 us 0.0620 / Table q(i,j) covariance matrix cn fr gr jp sw uk us cn 42.18 fr 20.18 70.89 gr 10.88 21.58 25.51 jp 5.30 15.41 9.60 22.33 sw 12.32 23.24 22.63 10.32 30.01 uk 23.84 23.80 13.22 10.46 16.36 42.23 us 17.41 12.62 4.70 1.00 7.20 9.90 16.42 ; * we will continue to use only the lower triangle of the q-matrix * and adjust the off diagonal entries to give the correct results. q(i,j) = 2*q(j,i) ; q(i,i) = q(i,i)/2; Scalars tau bounding parameter on turnover of current holdings lambda return versus variance component tradeoff parameter ; Set pd portfolio data labels / old current holdings fraction of the portfolio umin minimum increase of holdings fraction of security i umax maximum increase of holdings fraction of security i lmin minimum decrease of holdings fraction of security i lmax maximum decrease of holdings fraction of security i / Table bdata(i,pd) portfolio data and trading restrictions * - increase - - decrease - old umin umax lmin lmax cn 0.2 0.03 0.11 0.02 0.30 fr 0.2 0.04 0.10 0.02 0.15 gr 0.0 0.04 0.07 0.04 0.10 jp 0.0 0.03 0.11 0.04 0.10 sw 0.2 0.03 0.20 0.04 0.10 uk 0.2 0.03 0.10 0.04 0.15 us 0.2 0.03 0.10 0.04 0.30 ; bdata(i,'lmax') = min(bdata(i,'old'),bdata(i,'lmax')); ! tighten bound Abort$(abs(sum(i, bdata(i,'old'))-1) >= 1e5) 'error in bdata', bdata; Variables omega objective variable definition for minlp x(i) fraction of portfolio of current holdings of i xi(i) fraction of portfolio increase xd(i) fraction of portfolio decrease mvar(i) marginal variance y(i) binary switch for increasing current holdings of i z(i) binary switch for decreasing current holdings of i Binary Variables y,z; positive variables x, xi, xd; Equations 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 mvardef(i) marginal variance definition obj objective function objx objective function; 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 ; mvardef(i).. mvar(i) =e= sum(j, q(i,j)*x(j)) ; obj.. omega =e= sum((i,j), x(i)*q(i,j)*x(j)) - lambda*sum(i, mu(i)*x(i)); objx.. omega =e= sum(i, x(i)*mvar(i)) - lambda*sum(i, mu(i)*x(i)); Models mean / budget, xdef, turnover, maxinc, mininc, maxdec, mindec, binsum, obj / marg / budget, xdef, turnover, maxinc, mininc, maxdec, mindec, binsum, mvardef, objx /; lambda = 0.5 ; tau = .3 ; Solve mean minimizing omega using minlp ; Solve marg minimizing omega using minlp ; Parameter report summary report; report(i,'old') = bdata(i,'old'); report(i,'inc') = xi.l(i); report(i,'dec') = xd.l(i); report(i,'new') = x.l(i); Display report;