immun.gms : Financial Optimization: Risk Management

**Description**

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**

- 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 Types :** QCP lp

**Category :** GAMS Model library

**Main file :** immun.gms

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$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;
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