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cmo.gms : Financial Optimization: Financial Engineering

Collateralized Mortgage Obligations (CMO) are used to restructure
the cashflows from underlying mortgage collateral into a set of
high quality bonds with different maturities. In the following
mixed-integer programming it is used to restructure a mortgage
pool into six normal tranches and one zero tranche.

The model is written in a general form to allow experimentation
with different periodicity. The following version is annual.
Reference:
Large Model of Type: MIP
$Title Financial Optimization: Financial Engineering (CMO,SEQ=114) $Ontext Collateralized Mortgage Obligations (CMO) are used to restructure the cashflows from underlying mortgage collateral into a set of high quality bonds with different maturities. In the following mixed-integer programming it is used to restructure a mortgage pool into six normal tranches and one zero tranche. The model is written in a general form to allow experimentation with different periodicity. The following version is annual. 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 Sets i tranches / n1*n6, m / n(i) normal tranches / n1*n6 / m(i) m tranches / m / tp time periods / 0*10 / ts(tp) settlement date / 0 / t(tp) payment periods / 1*10 / tl(tp) last payment / 10 / zpos(i,tp) possible payment periods by tranche; Alias (i,j), (tp, tau) ; * note: all rates are monthly rates x 12. for aggregation we need to * compute equivalent rates. * * (1+r/12)**12 == (1+rd)**d Scalars cg collateral annual gross coupon / .10 / nom nominal value of collateral / 100 / s servicing rate /.005 / psa pricing speed / 115 / d periodicity / 1 / age age of the collateral in years minn minimum number of normal tranches / 3 / minm minimum number of m tranches / 1 /; Table tranches(*,*) tranche data (annual coupon and yield) coupon yield low-wal up-wal n1 .085 .099 1.00 1.24 n2 .085 .0999 2.00 2.44 n3 .085 .0999 3.00 3.44 n4 .0875 .1006 4.00 4.44 n5 .0875 .1009 5.00 5.65 n6 .0875 .1017 7.00 7.94 n7 .0875 .1018 10.00 10.94 n8 .0875 .1025 15.00 20.00 m .0995 .105 r .1285 ; Parameters po(tp) outstanding collateral principal at zero prepayment a zero prepayment annuity bv(tp) temporary factor for bvf calculation bvfac(tp) temporary factor for bvf calculation - rev sum bvf(tp) bond value factor pe(tp) expected outstanding collateral principal b(tp) prepayment rate cflow(tp) expected collateral payments prin(tp) structuring principal payments from collateral sump sum of prin cd periodic gross coupon sd periodic servicing rate coupon(i) periodic coupon by tranche yield(i) periodic yield by tranche yr periodic residual yield wallo(i) lower weighted average life in periodicity walup(i) upper weighted average life in periodicity bign big number smalln small number cmax maximum cmo coupon tmax term to maturity of collateral rev(tp) reverse order addresses psum(tp) sum of principal payments on collateral till tp tpsum(tp) sum of time principal payment products walp(tp) wal on principal payments from collateral ; rev(tp) = card(tp) - 2*ord(tp) + 1 ; tmax = card(t) ; age = (360 - tmax*12/d)/12; b(t) = 0.06*(psa/100)*min((ord(t)+age*d)/(d*2.5),1) ; cd = (1+cg/12)**(12/d)-1 ; sd = (1+s/12)**(12/d)-1 ; yr = (1 + tranches("r","yield")/12)**(12/d)-1 ; coupon(i) = (1+tranches(i,"coupon")/12)**(12/d)-1; yield(i) = (1+tranches(i,"yield")/12)**(12/d)-1 ; wallo(n) = tranches(n,"low-wal")*d ; walup(n) = tranches(n,"up-wal")*d ; cmax = smax(i, coupon(i)); a = nom*cd/(1 - power(1+cd,-tmax)) ; po(tp) = nom*(1 - power(1+cd,ord(tp)-1-tmax))/(1 - power(1+cd,-tmax)); bv(tl) = 0; Loop(tp, bv(tp+(rev(tp)-1)) = (bv(tp+rev(tp))+(a-sd*po(tp+(rev(tp)-1))))/(1+cmax) ); bvf(tp)$po(tp) = min(bv(tp)/po(tp),1) ; bvf(tl) = 1; pe(ts) = nom ; Loop(t(tp), pe(tp) = pe(tp-1)*(1-b(tp))**(1/d)*po(tp)/po(tp-1)) ; cflow(t(tp)) = (1 + cd - sd)*pe(tp-1) - pe(tp) ; prin(t(tp)) = pe(tp-1)*bvf(tp-1) - pe(tp)*bvf(tp) ; sump = sum(tp, prin(tp)); walp(ts) = 0 ; tpsum(ts) = 0 ; psum(ts) = 0 ; Loop(t(tp), psum(tp) = psum(tp-1) + prin(tp) ; tpsum(tp) = tpsum(tp-1) + (ord(tp)-1)*prin(tp) ); walp(t) = tpsum(t)/psum(t) ; zpos(i,tp) = yes ; zpos(n,tp)$(walp(tp) gt walup(n)) = no ; Option age:6, tmax:6, cd:6, sd:6, cmax:6,a:6 ; Display age, tmax, cd, sd, cmax, a ; Display po, cflow, pe, bv, bvf, prin, walp, zpos ; bign = sump*.7; smalln = sump*.03; Variables x(i,tp) outstanding principal in each tranche p(i,tp) principal payments on tranches c(i,tp) cashflow in each tranche r(tp) residual payments tpp(i) product of time and principal payment z(i,tp) tranche utilization variable (1=usage) y(i,tp) upper triangular structure tin(i) tranche in the structure pv def pv tranches pvres def pv residuals proceeds gross proceeds ; Positive Variables x, c, r, y ; Binary Variables z, tin ; Equations obj objective function defpv def pv tranches defpvres def pv residuals pdef(i,tp) definition of principal payments cdef(i,tp) cashflow accounting retiren1(tp) retirement of normal tranches retire(i,tp) retirement of tranches retirem(m,tp) retirement of m tranche retirem1(m,tp) retirement of m tranche cbal(tp) cashflow balance tppdef(i) definition of tpp lowal(n) lower bound on weighted average life upwal(n) upper bound on weighted average life seq1(tp) sequencing constraint 1 seq2(i,tp) sequencing constraint 2 ydef(i,tp) definition of y tindef1(i) tranche in definition tindef2(i) tranche in definition mcon constraint on number of m tranches ncon constraint on number of tranches ; obj.. proceeds =e= sum(i, pv(i)) + pvres; defpv(i).. pv(i) =e= sum(zpos(i,t), c(i,t)*(1+yield(i))**(-ord(t))) ; defpvres.. pvres =e= sum(t, r(t)*(1+yr)**(-ord(t))) ; pdef(zpos(i,t(tp))).. p(i,tp) =e= x(i,tp-1) - x(i,tp) ; cdef(zpos(i,t(tp))).. c(i,tp) =e= coupon(i)*x(i,tp-1) + p(i,tp) ; retiren1(t(tp)).. sum(zpos(n,tp), p(n,tp)) =e= prin(tp) + sum(m, x(m,tp-1)*coupon(m) - c(m,tp)) ; retire(zpos(n,t)).. p(n,t) =l= cflow(t)*z(n,t) ; retirem(m,t).. c(m,t) =l= cflow(t)*z(m,t) ; retirem1(m,t).. p(m,t) =l= prin(t)*z(m,t) ; cbal(t).. sum(zpos(i,t), c(i,t)) + r(t) =e= cflow(t) ; tppdef(n).. tpp(n) =e= sum(zpos(n,t), ord(t)*p(n,t)) ; lowal(n).. wallo(n)*sum(ts, x(n,ts)) =l= tpp(n) ; upwal(n).. walup(n)*sum(ts, x(n,ts)) =g= tpp(n) ; seq1(t).. sum(zpos(i,t), z(i,t)) =e= 1 ; seq2(zpos(i,t)).. y(i,t) =g= y(i,t+1) ; ydef(zpos(i,t)).. y(i,t) =e= y(i-1,t) + z(i,t) ; tindef1(i).. sum(ts, x(i,ts)) =l= tin(i)*bign ; tindef2(i).. sum(ts, x(i,ts)) =g= tin(i)*smalln ; ncon.. sum(n, tin(n)) =g= minn ; mcon.. sum(m, tin(m)) =g= minm ; Model cmo structuring model / all /; p.lo(n,tp) = 0 ; x.fx(i,tl) = 0 ; x.fx(i,t)$(not zpos(i,t+1)) = 0 ; z.fx(i,tp)$(not zpos(i,tp)) = 0 ; z.fx(m,tp)$(not sum(n, zpos(n,tp))) = 1 ; Option optcr = 0.001 ; Parameter xx(tp,i) principals zs(t,i) binary time increments cs(t,i) cashflows ; Solve cmo maximizing proceeds using mip ; xx(tp,i) = x.l(i,tp) ; zs(t,i) = z.l(i,t) ; cs(t,i) = c.l(i,t) ; Display zs, xx, cs, r.l ; Parameter report; report(t,n) = z.l(n,t); report("value",i) = sum(ts, x.l(i,ts)); report("avl",n)$report("value",n) = tpp.l(n)/report("value",n)/d; report("pvalue",i) = pv.l(i); report("pvalue","resid") = pvres.l; report("pvalue","total") = proceeds.l; Display report;