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Bootstrap.lst
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GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 1
Bootstrapping the yield curve
C o m p i l a t i o n
2
3 * Bootstrap.gms: Bootstrapping the yield curve.
4 * Consiglio, Nielsen and Zenios.
5 * PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 4.2.
3
6 * Last modified: Apr 2008.
7
8
9 SET Time Time periods /2016 * 2039/;
10
11 ALIAS (Time, t, t1, t2);
12
13 SCALARS
14 Now Current year;
15
16 Now = 2016;
17
18 PARAMETER
19 tau(t) Time in years;
20
21 * Note: time starts from 0
22
23 tau(t) = ORD(t)-1;
24
25 SET
26 Bonds Bonds universe
27 /DK0009923641, DK0009923484, DK0009921942,
28 DK0009922403, DK0009922676, DK0009922916,
29 DK0009918138, DK0009923567, DK0009923054,
30 DK0009923138, DK0009922320/;
31
32
33 ALIAS(Bonds, i);
34
35 PARAMETERS
36 Price(i) Bond prices
37 Coupon(i) Coupons
38 Maturity(i) Maturities
39 F(t,i) Cashflows;
40
41 * Bond data. Prices, coupons and maturities from the Danish market
42
INCLUDE C:\Users\khoongwh\Documents\gamsdir\projdir\Bonds.inc
44 TABLE BondData(i,*)
45 Price Maturity Coupon
46 DK0009923641 102.35 2020 0.25
47 DK0009923484 101.02 2018 0.25
48 DK0009921942 100.61 2017 4
49 DK0009922403 109.89 2019 4
50 DK0009922676 113.95 2021 3
51 DK0009922916 107.63 2023 0.1
52 DK0009918138 149.49 2024 7
53 DK0009923567 99.71 2027 0.5
54 DK0009923054 109.46 2023 1.5
55 DK0009923138 112.39 2025 1.75
56 DK0009922320 168.26 2039 4.5;
57
58 * Copy/transform data. Note division by 100 to get unit data, and
59 * subtraction of "Now" from Maturity date (so consistent with tau):
60
61 Price(i) = BondData(i,"Price")/100;
62 Coupon(i) = BondData(i,"Coupon")/100;
63 Maturity(i) = BondData(i,"Maturity") - Now;
64
65 * Calculate the ex-coupon cashflow of Bond i in year t:
66
67 F(t,i) = 1$(tau(t) = Maturity(i))
68 + coupon(i) $ (tau(t) <= Maturity(i) AND tau(t) > 0);
69
70 VARIABLES
71 r(t) Spot rates
72 SumOfSquareDev Sum of square deviations;
73
74 EQUATION
75 ObjDef Objective function definition;
76
77 ObjDef.. SumOfSquareDev =e= SUM(i, sqr(Price(i) - SUM(t, F(t,i) * exp(-r(
t) * tau(t)))));
78
79 OPTION SOLVEOPT = REPLACE;
80
81 MODEL Bootstrap /ObjDef/;
82
83 SOLVE Bootstrap MINIMIZING SumOfSquareDev USING NLP;
84
85 FILE TermStructureHandle /"TermStructure1.csv"/;
86
87 TermStructureHandle.pc = 5;
88
89 PUT TermStructureHandle;
90
91 * Write the heading
92
93 PUT "Bootstrap"/;
94 PUT "Time (in years)","Spot Rates","Sc1_Spot_Rates","Sc2_Spot_Rates","Sc3_
Spot_Rates","Sc4_Spot_Rates","Sc5_Spot_Rates","Sc6_Spot_Rates","Discount F
actors","Forward Rates"/;
95
96 * Calculate and store in a file the results: spot rates and discount facto
rs
97
98 PARAMETER
99 Forward(t) One-period forward rates
100 Discount(t) Discount factors;
101
102 Forward(t) = r.l(t) $ ( tau(t) = 0 ) + ((tau(t) * r.l(t) - tau(t-1) * r.l(
t-1)) / (tau(t) - tau(t-1))) $ (tau(t) > 0 );
103 Discount(t) = exp(-r.l(t) * tau(t));
104
105 LOOP (t$(tau(t) > 0),
106
107 PUT tau(t),r.l(t):6:5,(r.l(t)+0.04):6:5,(r.l(t)+0.01):6:5,r.l(t):6:5,
(r.l(t)-0.01):6:5,(r.l(t)+ ( 0.02 * ( (CARD(t)-1) - tau(t))/(CARD(t)-2) )
):6:8,(r.l(t)+ ( 0.02 * (tau(t)-1)/(CARD(t)-2) ) ):6:5,Discount(t):6:5,For
ward(t):6:5/;
108
109 );
110
111
112 * Also calculate the yield-to-maturity y(i) of each bond.
113 * This is done by solving a Constrained Nonlinear System, CNS:
114
115 POSITIVE VARIABLES
116 y(i) Yield-to-Maturity of the bonds;
117
118 EQUATION
119 YieldDef(i) Equations defining the yield-to-maturity;
120
121 YieldDef(i) .. Price(i) =E= SUM(t, F(t,i) * exp(-y(i) * tau(t)));
122
123 MODEL FindYTM /YieldDef/;
124
125 * Solve as a square system for the yields-to-maturity
126
127 SOLVE FindYTM USING CNS;
128
129 PARAMETERS
130 PriceErrors(i) Price errors;
131
132 PriceErrors(i) = Price(i) - SUM(t, F(t,i) * exp(-r.l(t) * tau(t)));
133
134 * Write the heading
135
136 PUT "Bond","Yield-To-Maturity","Price error"/;
137
138 LOOP (i,
139
140 PUT i.tl,y.l(i):6:5,PriceErrors(i):6:5/;
141
142 );
143
144 * Model with positive forward constraints
145
146 EQUATION
147 PosForwardCon(t) Equations to constraint forward rates to be positiv
e;
148
149 PosForwardCon(t)$(tau(t) > 0).. tau(t) * r(t) =g= tau(t-1) * r(t-1);
150
151 MODEL BootstrapPosForward /ObjDef,PosForwardCon/;
152
153 SOLVE BootstrapPosForward MINIMIZING SumOfSquareDev USING NLP;
154
155 Forward(t) = r.l(t) $ ( tau(t) = 0 ) + ((tau(t) * r.l(t) - tau(t-1) * r.l(
t-1)) / (tau(t) - tau(t-1))) $ (tau(t) > 0 );
156 Discount(t) = exp(-r.l(t) * tau(t));
157
158 PUT "Positive Forwards Bootstrap"/;
159 PUT "Time (in years)","Spot Rates","Discount Factors","Forward Rates"/;
160
161 LOOP (t$(tau(t) > 0),
162
163 PUT tau(t),r.l(t):6:5,Discount(t):6:5,Forward(t):6:5/;
164
165 );
166
167 * Solve as a square system for the yields-to-maturity
168
169 SOLVE FindYTM using CNS;
170
171 PriceErrors(i) = Price(i) - SUM(t, F(t,i) * exp(-r.l(t) * tau(t)));
172
173 * Write the heading
174
175 PUT "Bond","Yield-To-Maturity","Price error"/;
176
177 LOOP (i,
178
179 PUT i.tl,y.l(i):6:5,PriceErrors(i):6:5/;
180
181 );
182
183 SCALARS
184 totalError
185 lambda;
186
187 VARIABLES
188 WeightedSumOfSquares
189
190 POSITIVE VARIABLES
191 ForwardRates(t) One-period forward rates;
192
193 EQUATIONS
194 WeightedObjFun
195 ForwardDef(t) Equations defining the forward rates;
196
197
198 WeightedObjFun.. WeightedSumOfSquares =e= lambda * SUM(i, sqr(Price(i) -
SUM(t, F(t,i) * exp(-r(t) * tau(t))))) + (1-lambda) *
199 SUM(t$(tau(t) > 0), sqr( Forwar
dRates(t) - ForwardRates(t-1)));
200
201 * Recall that the first forward rate, F(0,1), coincides with the one perio
d spot rate
202
203 ForwardDef(t).. ForwardRates(t) =E= r(t) $ ( tau(t) = 0 ) +
204 ((tau(t) * r(t) - tau(t-1) * r(t-1))
/ (tau(t) - tau(t-1))) $ (tau(t) > 0 );
205
206
207 MODEL BootstrapSmooth /WeightedObjFun,ForwardDef/;
208
209 PUT "Smooth Bootstrap"/;
210
211 FOR( lambda = 1.0 DOWNTO 0.0 BY 0.1,
212
213 SOLVE BootstrapSmooth MINIMIZING WeightedSumOfSquares USING NLP;
214
215 PUT "Lambda","Average Error"/;
216
217 totalError = SUM(i, sqr(Price(i) - SUM(t, F(t,i) * exp(-r.l(t) * tau(t))
)))
218
219 PUT lambda:3:2,totalError:6:5/
220
221 PUT "Time (in years)","Spot Rates","Sc1_Spot_Rates","Sc2_Spot_Rates","Sc
3_Spot_Rates","Sc4_Spot_Rates","Sc5_Spot_Rates","Sc6_Spot_Rates","Discount
Factors","Forward Rates"/;
222
223 * Calculate and store in a file the results: spot rates and discount facto
rs
224 Discount(t) = exp(-r.l(t) * tau(t));
225
226 LOOP (t$(tau(t) > 0),
227
228 PUT tau(t),r.l(t):6:5,(r.l(t)+0.04):6:5,(r.l(t)+0.01):6:5,r.l(t):6:5,(
r.l(t)-0.01):6:5,(r.l(t)+ ( 0.02 * ( (CARD(t)-1) - tau(t))/(CARD(t)-2) ) )
:6:8,(r.l(t)+ ( 0.02 * (tau(t)-1)/(CARD(t)-2) ) ):6:5,Discount(t):6:5,Forw
ard(t):6:5/;
229
230 );
231
232 );
233
234 * Write spot rates to SpotRates.inc and
235 * yield-to-maturity to YieldRates.inc
236 * These data will be used by Immunization.gms .
237
238 * A smoothing parameter equal to 0.5 guarantees a
239 * reasonable term structure.
240
241
242
243 lambda = 0.5
244
245 SOLVE BootstrapSmooth MINIMIZING WeightedSumOfSquares USING NLP;
246
247 FILE SpotRatesHandle /"SpotRates.inc"/;
248
249 PUT SpotRatesHandle;
250
251 LOOP (t$(tau(t) > 0),
252
253 PUT t.tl:0:0,",",r.l(t):6:5/;
254
255 );
256
257 PUTCLOSE SpotRatesHandle;
258
259 FILE YieldRatesHandle /"YieldRates.inc"/;
260
261 PUT YieldRatesHandle;
262
263 LOOP (i,
264
265 PUT i.tl:0:0,",",y.l(i):6:5/;
266
267 );
268
269
270 DISPLAY r.l,y.l;
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 2
Bootstrapping the yield curve
Include File Summary
SEQ GLOBAL TYPE PARENT LOCAL FILENAME
1 1 INPUT 0 0 C:\Users\khoongwh\Desktop\Wei Hao\NUS\
Applied Mathematics Major\Modules\Year
3 Semester 1 (On Exchange)\42123 Optim
ization in Finance (ST4245 Statistical
Methods for Finance)\Projects\Bootstra
p.gms
2 43 INCLUDE 1 43 .C:\Users\khoongwh\Documents\gamsdir\p
rojdir\Bonds.inc
COMPILATION TIME = 0.000 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 3
Bootstrapping the yield curve
Equation Listing SOLVE Bootstrap Using NLP From line 83
---- ObjDef =E= Objective function definition
ObjDef.. (0.1154949)*r(2017) + (0.0691657999999997)*r(2018)
+ (0.261626700000001)*r(2019) + (0.0652836000000013)*r(2020)
+ (0.321942)*r(2021) + (0.2565504)*r(2022) - (0.525291200000002)*r(2023)
+ (1.38228)*r(2024) + (0.906039000000002)*r(2025) + (0.32295)*r(2026)
+ (1.629045)*r(2027) + (0.380592)*r(2028) + (0.412308)*r(2029)
+ (0.444024)*r(2030) + (0.47574)*r(2031) + (0.507456)*r(2032)
+ (0.539172)*r(2033) + (0.570888)*r(2034) + (0.602604)*r(2035)
+ (0.63432)*r(2036) + (0.666036000000001)*r(2037) + (0.697752)*r(2038)
+ (16.939868)*r(2039) + SumOfSquareDev =E= 0 ;
(LHS = -0.13972975, INFES = 0.13972975 ****)
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 4
Bootstrapping the yield curve
Column Listing SOLVE Bootstrap Using NLP From line 83
---- r Spot rates
r(2017)
(.LO, .L, .UP, .M = -INF, 0, +INF, 0)
(0.1155) ObjDef
r(2018)
(.LO, .L, .UP, .M = -INF, 0, +INF, 0)
(0.0692) ObjDef
r(2019)
(.LO, .L, .UP, .M = -INF, 0, +INF, 0)
(0.2616) ObjDef
REMAINING 20 ENTRIES SKIPPED
---- SumOfSquareDev Sum of square deviations
SumOfSquareDev
(.LO, .L, .UP, .M = -INF, 0, +INF, 0)
1 ObjDef
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 5
Bootstrapping the yield curve
Model Statistics SOLVE Bootstrap Using NLP From line 83
MODEL STATISTICS
BLOCKS OF EQUATIONS 1 SINGLE EQUATIONS 1
BLOCKS OF VARIABLES 2 SINGLE VARIABLES 24
NON ZERO ELEMENTS 24 NON LINEAR N-Z 23
DERIVATIVE POOL 20 CONSTANT POOL 67
CODE LENGTH 356
GENERATION TIME = 0.000 SECONDS 4 MB 24.9.1 r63795 WEX-WEI
EXECUTION TIME = 0.000 SECONDS 4 MB 24.9.1 r63795 WEX-WEI
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 6
Bootstrapping the yield curve
Solution Report SOLVE Bootstrap Using NLP From line 83
S O L V E S U M M A R Y
MODEL Bootstrap OBJECTIVE SumOfSquareDev
TYPE NLP DIRECTION MINIMIZE
SOLVER CONOPT FROM LINE 83
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 2 Locally Optimal
**** OBJECTIVE VALUE 0.0022
RESOURCE USAGE, LIMIT 0.031 1000.000
ITERATION COUNT, LIMIT 37 2000000000
EVALUATION ERRORS 0 0
**** Reading with SOLVEOPT=REPLACE (0) ****
CONOPT 3 24.9.1 r63795 Released Aug 30, 2017 WEI x86 64bit/MS Windows
C O N O P T 3 version 3.17E
Copyright (C) ARKI Consulting and Development A/S
Bagsvaerdvej 246 A
DK-2880 Bagsvaerd, Denmark
The model has 24 variables and 1 constraints
with 24 Jacobian elements, 23 of which are nonlinear.
The Hessian of the Lagrangian has 23 elements on the diagonal,
253 elements below the diagonal, and 23 nonlinear variables.
Pre-triangular equations: 0
Post-triangular equations: 1
** Optimal solution. Reduced gradient less than tolerance.
CONOPT time Total 0.031 seconds
of which: Function evaluations 0.001 = 3.2%
1st Derivative evaluations 0.000 = 0.0%
2nd Derivative evaluations 0.001 = 3.2%
Directional 2nd Derivative 0.001 = 3.2%
LOWER LEVEL UPPER MARGINAL
---- EQU ObjDef . . . 1.000
ObjDef Objective function definition
---- VAR r Spot rates
LOWER LEVEL UPPER MARGINAL
2017 -INF 0.034 +INF EPS
2018 -INF -0.002 +INF 1.8162E-9
2019 -INF 0.007 +INF 3.6811E-9
2020 -INF -0.003 +INF 5.3130E-9
2021 -INF 0.002 +INF 1.0419E-8
2022 -INF 3.542 +INF EPS
2023 -INF -0.005 +INF 5.2872E-8
2024 -INF -6.277E-4 +INF 3.8153E-8
2025 -INF 0.002 +INF 5.5020E-8
2026 -INF 0.002 +INF EPS
2027 -INF 0.005 +INF 7.6606E-8
2028 -INF 0.003 +INF EPS
2029 -INF 0.003 +INF EPS
2030 -INF 0.003 +INF EPS
2031 -INF 0.003 +INF EPS
2032 -INF 0.003 +INF EPS
2033 -INF 0.004 +INF EPS
2034 -INF 0.004 +INF EPS
2035 -INF 0.004 +INF EPS
2036 -INF 0.004 +INF EPS
2037 -INF 0.004 +INF EPS
2038 -INF 0.005 +INF EPS
2039 -INF 0.013 +INF -4.150E-9
LOWER LEVEL UPPER MARGINAL
---- VAR SumOfSqua~ -INF 0.002 +INF .
SumOfSquareDev Sum of square deviations
**** REPORT SUMMARY : 0 NONOPT
0 INFEASIBLE
0 UNBOUNDED
0 ERRORS
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Bootstrapping the yield curve
Equation Listing SOLVE FindYTM Using CNS From line 127
---- YieldDef =E= Equations defining the yield-to-maturity
YieldDef(DK0009923641).. (4.025)*y(DK0009923641) =E= -1.0235 ;
(LHS = -1.01, INFES = 0.0134999999999998 ****)
YieldDef(DK0009923484).. (2.0075)*y(DK0009923484) =E= -1.0102 ;
(LHS = -1.005, INFES = 0.00520000000000009 ****)
YieldDef(DK0009921942).. (1.04)*y(DK0009921942) =E= -1.0061 ;
(LHS = -1.04, INFES = 0.0339 ****)
REMAINING 8 ENTRIES SKIPPED
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Bootstrapping the yield curve
Column Listing SOLVE FindYTM Using CNS From line 127
---- y Yield-to-Maturity of the bonds
y(DK0009923641)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(4.025) YieldDef(DK0009923641)
y(DK0009923484)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(2.0075) YieldDef(DK0009923484)
y(DK0009921942)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(1.04) YieldDef(DK0009921942)
REMAINING 8 ENTRIES SKIPPED
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Bootstrapping the yield curve
Model Statistics SOLVE FindYTM Using CNS From line 127
MODEL STATISTICS
BLOCKS OF EQUATIONS 1 SINGLE EQUATIONS 11
BLOCKS OF VARIABLES 1 SINGLE VARIABLES 11
NON ZERO ELEMENTS 11 NON LINEAR N-Z 11
DERIVATIVE POOL 20 CONSTANT POOL 56
CODE LENGTH 343
SINGLE =E|X= EQUS 11 SINGLE UNFIXED VARS 11
GENERATION TIME = 0.016 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
EXECUTION TIME = 0.016 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 10
Bootstrapping the yield curve
Solution Report SOLVE FindYTM Using CNS From line 127
S O L V E S U M M A R Y
MODEL FindYTM
TYPE CNS
SOLVER CONOPT FROM LINE 127
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 5 Locally Infeasible
RESOURCE USAGE, LIMIT 0.000 1000.000
ITERATION COUNT, LIMIT 0 2000000000
EVALUATION ERRORS 0 0
**** Reading with SOLVEOPT=REPLACE (0) ****
CONOPT 3 24.9.1 r63795 Released Aug 30, 2017 WEI x86 64bit/MS Windows
C O N O P T 3 version 3.17E
Copyright (C) ARKI Consulting and Development A/S
Bagsvaerdvej 246 A
DK-2880 Bagsvaerd, Denmark
** An equation in the pre-triangular part of the model cannot
be solved because the critical variable is at a bound.
Residual= 6.93000000E-02
Tolerance (Rtnwtr)= 2.00000000E-08
The critical pre-triangular constraints and the variables they are
solved for in solution order:
Equation YieldDef(DK0009922916) has been solved with respect to
variable y(DK0009922916). Value = 0
After the last operation the residual is still 0.0693000000000001
CONOPT time Total 0.000 seconds
of which: Function evaluations 0.000 = 0.0%
1st Derivative evaluations 0.000 = 0.0%
**** ERRORS/WARNINGS IN EQUATION YieldDef(DK0009922916)
1 error(s): Infeasibility in pre-triangular part of model.
**** ERRORS/WARNINGS IN VARIABLE y(DK0009922916)
1 error(s): Infeasibility in pre-triangular part of model.
---- EQU YieldDef Equations defining the yield-to-maturity
LOWER LEVEL UPPER
DK0009923641 -1.023 -1.010 -1.023 INFES
DK0009923484 -1.010 -1.005 -1.010 INFES
DK0009921942 -1.006 -1.040 -1.006 INFES
DK0009922403 -1.099 -1.120 -1.099 INFES
DK0009922676 -1.139 -1.150 -1.139 INFES
DK0009922916 -1.076 -1.007 -1.076 INFES
DK0009918138 -1.495 -1.496 -1.495 INFES
DK0009923567 -0.997 -0.999 -0.997 INFES
DK0009923054 -1.095 -1.095 -1.095 INFES
DK0009923138 -1.124 -1.124 -1.124 INFES
DK0009922320 -1.683 -1.716 -1.683 INFES
---- VAR y Yield-to-Maturity of the bonds
LOWER LEVEL UPPER
DK0009923641 . . +INF
DK0009923484 . . +INF
DK0009921942 . . +INF
DK0009922403 . . +INF
DK0009922676 . . +INF
DK0009922916 . . +INF
DK0009918138 . 0.006 +INF
DK0009923567 . 0.005 +INF
DK0009923054 . 0.001 +INF
DK0009923138 . 0.003 +INF
DK0009922320 . 0.010 +INF
**** REPORT SUMMARY : 11 INFEASIBLE (INFES)
SUM 0.190
MAX 0.069
MEAN 0.017
0 DEPENDENT
2 ERRORS ( ****)
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 11
Bootstrapping the yield curve
Equation Listing SOLVE BootstrapPosForward Using NLP From line 153
---- ObjDef =E= Objective function definition
ObjDef.. - (6.53824308999931E-10)*r(2017) - (1.81621976804926E-9)*r(2018)
- (3.68111682180938E-9)*r(2019) - (5.31297786057554E-9)*r(2020)
- (1.04185048800333E-8)*r(2021) + (3.22221479799363E-12)*r(2022)
- (5.28718838108151E-8)*r(2023) - (3.81533427151396E-8)*r(2024)
- (5.50196887970058E-8)*r(2025) - (2.55239482076374E-10)*r(2026)
- (7.66057518294002E-8)*r(2027) + (1.22364391227831E-10)*r(2028)
+ (1.31859671343748E-10)*r(2029) + (1.41193275694691E-10)*r(2030)
+ (1.50355329275226E-10)*r(2031) + (1.59336508270027E-10)*r(2032)
+ (1.68128072303257E-10)*r(2033) + (1.76721894941904E-10)*r(2034)
+ (1.85110492275041E-10)*r(2035) + (1.93287049390708E-10)*r(2036)
+ (2.01245444574638E-10)*r(2037) + (2.08980271060399E-10)*r(2038)
+ (4.14968560126741E-9)*r(2039) + SumOfSquareDev =E= 0 ; (LHS = 0)
---- PosForwardCon =G= Equations to constraint forward rates to be positive
PosForwardCon(2017).. r(2017) =G= 0 ; (LHS = 0.0335437568779823)
PosForwardCon(2018).. - r(2017) + 2*r(2018) =G= 0 ;
(LHS = -0.0383792223610765, INFES = 0.0383792223610765 ****)
PosForwardCon(2019).. - 2*r(2018) + 3*r(2019) =G= 0 ; (LHS = 0.02464443591395)
REMAINING 20 ENTRIES SKIPPED
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Bootstrapping the yield curve
Column Listing SOLVE BootstrapPosForward Using NLP From line 153
---- r Spot rates
r(2017)
(.LO, .L, .UP, .M = -INF, 0.0335437568779823, +INF, EPS)
(-6.53824E-10) ObjDef
1 PosForwardCon(2017)
-1 PosForwardCon(2018)
r(2018)
(.LO, .L, .UP, .M = -INF, -0.00241773274154711, +INF,
1.81621976839885E-9)
(-1.816220E-9) ObjDef
2 PosForwardCon(2018)
-2 PosForwardCon(2019)
r(2019)
(.LO, .L, .UP, .M = -INF, 0.00660299014361861, +INF,
3.68111682231755E-9)
(-3.681117E-9) ObjDef
3 PosForwardCon(2019)
-3 PosForwardCon(2020)
REMAINING 20 ENTRIES SKIPPED
---- SumOfSquareDev Sum of square deviations
SumOfSquareDev
(.LO, .L, .UP, .M = -INF, 0.00215287879316589, +INF, 0)
1 ObjDef
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Bootstrapping the yield curve
Model Statistics SOLVE BootstrapPosForward Using NLP From line 153
MODEL STATISTICS
BLOCKS OF EQUATIONS 2 SINGLE EQUATIONS 24
BLOCKS OF VARIABLES 2 SINGLE VARIABLES 24
NON ZERO ELEMENTS 69 NON LINEAR N-Z 23
DERIVATIVE POOL 20 CONSTANT POOL 67
CODE LENGTH 355
GENERATION TIME = 0.016 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
EXECUTION TIME = 0.016 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
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Bootstrapping the yield curve
Solution Report SOLVE BootstrapPosForward Using NLP From line 153
S O L V E S U M M A R Y
MODEL BootstrapPosForward OBJECTIVE SumOfSquareDev
TYPE NLP DIRECTION MINIMIZE
SOLVER CONOPT FROM LINE 153
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 2 Locally Optimal
**** OBJECTIVE VALUE 7.6437
RESOURCE USAGE, LIMIT 0.000 1000.000
ITERATION COUNT, LIMIT 5 2000000000
EVALUATION ERRORS 0 0
**** Reading with SOLVEOPT=REPLACE (0) ****
CONOPT 3 24.9.1 r63795 Released Aug 30, 2017 WEI x86 64bit/MS Windows
C O N O P T 3 version 3.17E
Copyright (C) ARKI Consulting and Development A/S
Bagsvaerdvej 246 A
DK-2880 Bagsvaerd, Denmark
The model has 24 variables and 24 constraints
with 69 Jacobian elements, 23 of which are nonlinear.
The Hessian of the Lagrangian has 23 elements on the diagonal,
253 elements below the diagonal, and 23 nonlinear variables.
Pre-triangular equations: 0
Post-triangular equations: 24
** Optimal solution. Reduced gradient less than tolerance.
CONOPT time Total 0.001 seconds
of which: Function evaluations 0.000 = 0.0%
1st Derivative evaluations 0.000 = 0.0%
LOWER LEVEL UPPER MARGINAL
---- EQU ObjDef . . . 1.000
ObjDef Objective function definition
---- EQU PosForwardCon Equations to constraint forward rates to be positive
LOWER LEVEL UPPER MARGINAL
2017 . . +INF 1.747
2018 . . +INF 1.449
2019 . . +INF 1.071
2020 . . +INF 0.745
2021 . . +INF 0.349
2022 . 21.242 +INF EPS
2023 . . +INF EPS
2024 . 0.032 +INF EPS
2025 . 0.021 +INF EPS
2026 . 0.006 +INF EPS
2027 . 0.032 +INF EPS
2028 . . +INF EPS
2029 . 0.005 +INF EPS
2030 . 0.006 +INF EPS
2031 . 0.006 +INF EPS
2032 . 0.007 +INF EPS
2033 . 0.007 +INF EPS
2034 . 0.007 +INF EPS
2035 . 0.008 +INF EPS
2036 . 0.008 +INF EPS
2037 . 0.008 +INF EPS
2038 . 0.009 +INF EPS
2039 . 0.201 +INF EPS
---- VAR r Spot rates
LOWER LEVEL UPPER MARGINAL
2017 -INF . +INF .
2018 -INF . +INF .
2019 -INF . +INF .
2020 -INF . +INF .
2021 -INF . +INF .
2022 -INF 3.540 +INF .
2023 -INF 3.035 +INF .
2024 -INF 2.659 +INF .
2025 -INF 2.366 +INF .
2026 -INF 2.130 +INF .
2027 -INF 1.939 +INF .
2028 -INF 1.778 +INF .
2029 -INF 1.641 +INF .
2030 -INF 1.525 +INF .
2031 -INF 1.423 +INF .
2032 -INF 1.335 +INF .
2033 -INF 1.257 +INF .
2034 -INF 1.187 +INF .
2035 -INF 1.125 +INF .
2036 -INF 1.069 +INF .
2037 -INF 1.019 +INF .
2038 -INF 0.973 +INF .
2039 -INF 0.939 +INF .
LOWER LEVEL UPPER MARGINAL
---- VAR SumOfSqua~ -INF 7.644 +INF .
SumOfSquareDev Sum of square deviations
**** REPORT SUMMARY : 0 NONOPT
0 INFEASIBLE
0 UNBOUNDED
0 ERRORS
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 15
Bootstrapping the yield curve
Equation Listing SOLVE FindYTM Using CNS From line 169
---- YieldDef =E= Equations defining the yield-to-maturity
YieldDef(DK0009923641).. (4.025)*y(DK0009923641) =E= -1.0235 ;
(LHS = -1.01, INFES = 0.0134999999999998 ****)
YieldDef(DK0009923484).. (2.0075)*y(DK0009923484) =E= -1.0102 ;
(LHS = -1.005, INFES = 0.00520000000000009 ****)
YieldDef(DK0009921942).. (1.04)*y(DK0009921942) =E= -1.0061 ;
(LHS = -1.04, INFES = 0.0339 ****)
REMAINING 8 ENTRIES SKIPPED
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Bootstrapping the yield curve
Column Listing SOLVE FindYTM Using CNS From line 169
---- y Yield-to-Maturity of the bonds
y(DK0009923641)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(4.025) YieldDef(DK0009923641)
y(DK0009923484)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(2.0075) YieldDef(DK0009923484)
y(DK0009921942)
(.LO, .L, .UP, .M = 0, 0, +INF, 0)
(1.04) YieldDef(DK0009921942)
REMAINING 8 ENTRIES SKIPPED
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Bootstrapping the yield curve
Model Statistics SOLVE FindYTM Using CNS From line 169
MODEL STATISTICS
BLOCKS OF EQUATIONS 1 SINGLE EQUATIONS 11
BLOCKS OF VARIABLES 1 SINGLE VARIABLES 11
NON ZERO ELEMENTS 11 NON LINEAR N-Z 11
DERIVATIVE POOL 20 CONSTANT POOL 56
CODE LENGTH 343
SINGLE =E|X= EQUS 11 SINGLE UNFIXED VARS 11
GENERATION TIME = 0.000 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
EXECUTION TIME = 0.031 SECONDS 3 MB 24.9.1 r63795 WEX-WEI
GAMS 24.9.1 r63795 Released Aug 30, 2017 WEX-WEI x86 64bit/MS Windows 10/26/17 02:25:11 Page 18
Bootstrapping the yield curve
Solution Report SOLVE FindYTM Using CNS From line 169
S O L V E S U M M A R Y
MODEL FindYTM
TYPE CNS
SOLVER CONOPT FROM LINE 169
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 5 Locally Infeasible
RESOURCE USAGE, LIMIT 0.000 1000.000
ITERATION COUNT, LIMIT 0 2000000000
EVALUATION ERRORS 0 0
**** Reading with SOLVEOPT=REPLACE (0) ****
CONOPT 3 24.9.1 r63795 Released Aug 30, 2017 WEI x86 64bit/MS Windows
C O N O P T 3 version 3.17E
Copyright (C) ARKI Consulting and Development A/S
Bagsvaerdvej 246 A
DK-2880 Bagsvaerd, Denmark
** An equation in the pre-triangular part of the model cannot
be solved because the critical variable is at a bound.
Residual= 6.93000000E-02
Tolerance (Rtnwtr)= 2.00000000E-08
The critical pre-triangular constraints and the variables they are
solved for in solution order:
Equation YieldDef(DK0009922916) has been solved with respect to
variable y(DK0009922916). Value = 0
After the last operation the residual is still 0.0693000000000001
CONOPT time Total 0.001 seconds
of which: Function evaluations 0.001 =100.0%
1st Derivative evaluations 0.000 = 0.0%
**** ERRORS/WARNINGS IN EQUATION YieldDef(DK0009922916)
1 error(s): Infeasibility in pre-triangular part of model.
**** ERRORS/WARNINGS IN VARIABLE y(DK0009922916)
1 error(s): Infeasibility in pre-triangular part of model.
---- EQU YieldDef Equations defining the yield-to-maturity
LOWER LEVEL UPPER
DK0009923641 -1.023 -1.010 -1.023 INFES
DK0009923484 -1.010 -1.005 -1.010 INFES
DK0009921942 -1.006 -1.040 -1.006 INFES
DK0009922403 -1.099 -1.120 -1.099 INFES
DK0009922676 -1.139 -1.150 -1.139 INFES
DK0009922916 -1.076 -1.007 -1.076 INFES
DK0009918138 -1.495 -1.495 -1.495 INFES
DK0009923567 -0.997 -0.997 -0.997 INFES
DK0009923054 -1.095 -1.095 -1.095 INFES
DK0009923138 -1.124 -1.124 -1.124 INFES
DK0009922320 -1.683 -1.683 -1.683 INFES
---- VAR y Yield-to-Maturity of the bonds
LOWER LEVEL UPPER
DK0009923641 . . +INF
DK0009923484 . . +INF
DK0009921942 . . +INF
DK0009922403 . . +INF
DK0009922676 . . +INF
DK0009922916 . . +INF
DK0009918138 . 0.006 +INF
DK0009923567 . 0.005 +INF
DK0009923054 . 0.001 +INF
DK0009923138 . 0.003 +INF
DK0009922320 . 0.011 +INF
**** REPORT SUMMARY : 11 INFEASIBLE (INFES)
SUM 0.154
MAX 0.069
MEAN 0.014
0 DEPENDENT
2 ERRORS ( ****)
1 PROJECTED