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immun.gms : Financial Optimization: Risk Management

Several different models have been developed during the seventies
to manage financial risks. Immunization, dedication and combination
models of various flavors are presented. The presentation of the
models follows closely chapter one of the book Financial Optimization.

The models are designed to capture the essential features using a small
data set derived from U.S.Treasury Quotes for 6/23/89 from the Wall Street
Journal.

The family of models presented in this file are

IMMUN1  Dollar Duration Immunization Model
IMMUN2  Gain Oriented Immunization Model
IMMUN3  Dollar Convexity Immunization Model
FACTOR1 Factor Immunization Model
FACTOR2 Factor Immunization Model
DEDIC   Bond Dedication Model
COMB1   Combination Matching Model
COMB2   Horizon Matching Model
COMB3   Factor Horizon Matching Model
SIGMA   Minimum Cashflow Variance Model

All share the same liability and security data. Common formulations
are exploited to better show the relationship between these models
Reference:
Small Model of Types: QCP lp
$Title Financial Optimization: Risk Management (IMMUN,SEQ=110) $Ontext Several different models have been developed during the seventies to manage financial risks. Immunization, dedication and combination models of various flavors are presented. The presentation of the models follows closely chapter one of the book Financial Optimization. The models are designed to capture the essential features using a small data set derived from U.S.Treasury Quotes for 6/23/89 from the Wall Street Journal. The family of models presented in this file are IMMUN1 Dollar Duration Immunization Model IMMUN2 Gain Oriented Immunization Model IMMUN3 Dollar Convexity Immunization Model FACTOR1 Factor Immunization Model FACTOR2 Factor Immunization Model DEDIC Bond Dedication Model COMB1 Combination Matching Model COMB2 Horizon Matching Model COMB3 Factor Horizon Matching Model SIGMA Minimum Cashflow Variance Model All share the same liability and security data. Common formulations are exploited to better show the relationship between these models 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 $Stitle Liability Information Set tl liability time / tl0*tl5 / Table liab(tl,*) liability data month day year liability liabrate tl0 6 23 1989 0 0.092370 tl1 9 1 1989 50000 0.092281 tl2 6 15 1990 42000 0.090367 tl3 12 1 1990 40000 0.088643 tl4 12 1 1991 40000 0.085649 tl5 6 1 1993 45000 0.086548 ; Scalar basedate earliest liability date; basedate = smin(tl, jdate(liab(tl,'year'),liab(tl,'month'),liab(tl,'day'))); liab(tl,'days') = jdate(liab(tl,'year'),liab(tl,'month'),liab(tl,'day')) - basedate; liab(tl,'term') = liab(tl,'days')/365; Parameters rl(tl) interest rate of liability pl present value of liability kl dollar duration of liability ql dollar convexity of liability; rl(tl) = liab(tl,"liabrate"); pl = sum( tl, liab(tl,'liability') * exp(-rl(tl)*liab(tl,'term')) ); kl = -sum( tl, liab(tl,'liability') * exp(-rl(tl)*liab(tl,'term')) * liab(tl,'term') ); ql = sum( tl, liab(tl,'liability') * exp(-rl(tl)*liab(tl,'term')) * sqr(liab(tl,'term')) ); * note: (1) liability interest rate fitted from cubic spline prototype. * (2) continuous compounding used for liabilities. Display pl, kl, ql; $Stitle security information Sets i securities / bond-1*bond-8 / ts time points / 89-07, 89-08, 90-02, 90-08, 91-02 91-08, 92-02, 92-08, 93-02, 93-08 / Table stime(ts,*) time information for securities month day year 89-07 7 15 1989 89-08 8 15 1989 90-02 2 15 1990 90-08 8 15 1990 91-02 2 15 1991 91-08 8 15 1991 92-02 2 15 1992 92-08 8 15 1992 93-02 2 15 1993 93-08 8 15 1993 ; stime(ts,'days') = jdate(stime(ts,'year'),stime(ts,'month'),stime(ts,'day')) - basedate; stime(ts,'term') = stime(ts,'days') / 365; Table sdata(*,i) security data bond-1 bond-2 bond-3 bond-4 89-07 103.8125 107.2500 89-08 7.4375 3.6250 90-02 7.4375 3.6250 90-08 7.4375 3.6250 91-02 7.4375 3.6250 91-08 107.4375 3.6250 92-02 3.6250 92-08 103.6250 yield 8.35 8.56 8.08 8.29 price 99.9063 100.2812 113.0625 97.1563 accr 3.3491 6.3688 5.2597 2.5635 + bond-5 bond-6 bond-7 bond-8 89-08 4.1250 4.3125 4.375 5.9375 90-02 4.1250 4.3125 4.375 5.9375 90-08 4.1250 4.3125 4.375 5.9375 91-02 4.1250 4.3125 4.375 5.9375 91-08 4.1250 4.3125 4.375 5.9375 92-02 4.1250 4.3125 4.375 5.9375 92-08 104.1250 4.3125 4.375 5.9375 93-02 4.3125 4.375 5.9375 93-08 104.3125 104.375 105.9375 yield 8.37 8.35 8.35 8.36 price 99.6563 100.9375 101.3750 112.0625 accr 2.9171 3.0497 3.0939 4.1989 ; * note: (3) accr is interest accrued in the current coupon period * to date. Parameters r(i) current yield of security cf(ts,i) cash flow of security p(i) present value of security (current price) pv(i) present value of cashflows of security k(i) dollar duration of security q(i) dollar convexity of security; cf(ts,i) = sdata(ts,i) * 10; r(i) = sdata('yield',i) / 100; p(i) = (sdata('price',i) + sdata('accr',i)) * 10; pv(i) = sum( ts, cf(ts,i) * (1+r(i)/2)**(-2*stime(ts,'term')-1) ); k(i) = -sum( ts, cf(ts,i) * stime(ts,'term') * (1+r(i)/2)**(-2*stime(ts,'term')-1) ); q(i) = sum( ts, cf(ts,i) * (stime(ts,'term')+1) * stime(ts,'term') * (1+r(i)/2)**(-2*stime(ts,'term')-2) ); * note: (4) dollar amounts scaled by 10 to indicate par value. * (5) current price used to indicate present value. * (6) dollar duration and dollar convexity uses semiannual * compounding. Display k, q; $Stitle immunization model equations Variables omega objective value x(i) security ; Positive Variable x; Equations objdef1 objective definition immun1 objdef2 objective definition immun2 objdef3 objective definition immun3 pvm present value match ddm dollar duration match dcm dollar convexity constraint; objdef1.. omega =e= sum(i, k(i)*r(i)*x(i)); objdef2.. omega =e= sum(i, (p(i)-pv(i))*x(i)); objdef3.. omega =e= sum(i, q(i)*x(i)); pvm.. sum(i, p(i)*x(i)) =e= pl; ddm.. sum(i, k(i)*x(i)) =e= kl; dcm.. sum(i, q(i)*x(i)) =g= ql; Model immun1 dollar duration immunization model / objdef1,pvm,ddm / immun2 gain oriented immunization model / objdef2,pvm,ddm / immun3 dollar convexity immunization model / objdef3,pvm,ddm,dcm/; Solve immun1 maximizing omega using lp ; Solve immun2 minimizing omega using lp ; Solve immun3 minimizing omega using lp ; $Stitle factor loading model Set tf factor dates in years from 6-23-89 / tf0 * tf10 / j factors / shift, tilt, curve / ja(j) active factors ; * shape risk factors * shift parallel shifts to term structure curve * tilt changes in steepness of term structure curve * curve changes in overall curvature of the term structure curve Table a(tf,*) factor loading array for yearly time intervals term shift tilt curve tf0 0 42 -25 -6 tf1 1 43 -22 -4 tf2 2 44 -16 -2 tf3 3 45 -12 0 tf4 4 46 -6 4 tf5 5 47 0 8 tf6 6 47 2 8 tf7 7 46 4 6 tf8 8 45 6 4 tf9 9 44 8 2 tf10 10 43 10 0 ; Parameters sfac(ts,j) interpolated factor loadings on term of securities lfac(tl,j) interpolated factor loadings on term of liabilities f(i,j) factor sensitivities of securities fl(j) factor sensitivities of liabilities ; Loop( (ts,tf)$(a(tf,"term") eq trunc(stime(ts,"term"))), sfac(ts,j) = ( a(tf,j) + (a(tf+1,j)-a(tf,j)) * (stime(ts,"term")-a(tf,"term")) ) / 100 ); Loop( (tl,tf)$(a(tf,"term") eq trunc(liab(tl,"term"))), lfac(tl,j) = ( a(tf,j) + (a(tf+1,j)-a(tf,j)) * (liab(tl,"term")-a(tf,"term")) ) / 100 ); f(i,j) = - sum( ts, sfac(ts,j) * cf(ts,i) * stime(ts,"term") * (1+r(i)/2)**(-2*stime(ts,'term')-1) ); fl(j) = - sum( tl, lfac(tl,j) * liab(tl,"liability") * liab(tl,"term") * exp(-rl(tl)*liab(tl,"term")) ); Positive Variable dif(j) elastic difference in factor match (surplus); * note: nonzero solution values of dif(j) indicate infeasibilities. Equations objdef4 objective definition fm(j) factor match diffm(j) difference between factor match when infeasible ; objdef4.. omega =e= sum(i, k(i)*r(i)*x(i)) - sum(ja, dif(ja)) ; fm(ja).. sum(i, f(i,ja)*x(i)) =e= fl(ja) ; diffm(ja).. sum(i, f(i,ja)*x(i)) - fl(ja) =e= dif(ja) ; Model factor1 factor immunization model / objdef1, pvm, fm / factor2 factor immunization model / objdef4, pvm, diffm /; ja(j) = yes; Solve factor1 maximizing omega using lp; Solve factor2 maximizing omega using lp; $Stitle bond dedication model Parameters rr reinvestment rate del(tl) interval between liability payments d(tl,i) reinvested value of bond cashflow between liability dates; rr = .05; del(tl) = liab(tl,"term") - liab(tl-1,"term"); d(tl,i) = sum( ts$(liab(tl-1,"term") lt stime(ts,"term") and stime(ts,"term") le liab(tl,"term")), cf(ts,i) * (1+rr)**(liab(tl,"term")-stime(ts,"term")) ); Variable s(tl) cash holdings (surplus) at time tl l(tl) funded liability ; Positive Variable s, l; l.lo(tl) = liab(tl,"liability"); Equations objdef5 objective definition for cost cbal(tl) cashflow balance ; objdef5.. omega =e= sum(i, p(i)*x(i)) + s("tl0"); cbal(tl).. sum(i, d(tl,i)*x(i)) + s(tl-1)*(1+rr)**del(tl) =e= l(tl) + s(tl) ; Model dedic bond dedication model / objdef5, cbal /; Solve dedic minimizing omega using lp; $Stitle combination matching (horizon matching) l.lo(tl)$(liab(tl,"year") gt 1992) =0; Equation hm horizon matching ; hm.. kl*(sum(i, p(i)*x(i)) + s("tl0")) =e= pl*sum(i, k(i)*x(i)) ; Model comb1 combination matching model / objdef5, cbal, hm / comb2 horizon matching model / objdef5, cbal, ddm / comb3 factor horizon matching model / objdef5, cbal, fm / Solve comb1 minimizing omega using lp; Solve comb2 minimizing omega using lp; ja(j) = yes; Solve comb3 minimizing omega using lp; $Stitle minimum cashflow variance model Variable variance variance of cashflows ; omega.up = pl; l.lo(tl) = 0; Equation objdef6 cashflow variance ; objdef6.. variance =e= sum(tl, sqr( liab(tl,"liability")-l(tl)) ); Model sigma minimum cashflow variance model / objdef6, cbal, objdef2 /; Solve sigma minimizing variance using qcp;