Kurt Hess, kurthess@waikato.ac.nz document.xls Introduction 04/08/2023

IMPORTANT INSTALLATION INFORMATION

Programmed by and copyright Kurt Hess March 2004, kurthess@waikato.ac.nz

Illustration of Extended Nelson & Siegel Spot Rate Model

Fitting Extended Nelson & Siegel Spot Rate with Solver

The SOLVER macros in this workbook will only run if your Excel is set up as follows.

You must have SOLVER installed with your Excel.Go to Tools Menu and see whether item Solver appears there.If it does not, go to Tools - Add-ins and tick "Solver Add-in".

This 1st step will allow you to use SOLVER from Excel but because SOLVER is also called by a VBA macro, you will also need to establish a reference to the Solver add-in in the VBA editor:With a Visual Basic module active, click References on the Tools menu, and then select the Solver.xla check box under Available References. If Solver.xla doesn't appear under Available References, click Browse and open Solver.xla in the \Office\Library subfolder.

Kurt Hess, Waikato Management School Page 2 document.xls Nelson Siegel with Svensson 04/08/2023

Illustration of Extended Nelson & Siegel Spot Rate Model with Svensson 1994 Extensionprogrammed by Kurt Hess May 2004, kurthess@waikato.ac.nzTime to maturity m 3.3 33.00

Long-run levels of interest rates 5.2% 52

Short-run component 2.2% 122

Medium-term component 8.1% 81 determines magnitude and the direction of the hump

Decay parameter 1 1.050 105

Decay parameter 2 1.240 124

Svensson 1994 Extension 0.035 35

Spot rate at time t 8.9650%#VALUE! with VBA Function

Components of N&S spot rate

Comp 1 5.200%

Comp 2 0.670%

Comp 3 2.116%

Svensson (1994) Extension Comp 4 0.979%

b0

b1

b2

t1 determines decay of short-term component, must be > 0

t2 determines decay of medium-term component, must be > 0

b3

rt,i

b0

b1*((1-EXP(-m/tau1))/(m/tau1))

b2*((1-EXP(-m/tau1))/(m/tau1)-EXP(-m/tau1))

b3*((1-EXP(-m/tau2))/(m/tau2)-EXP(-m/tau2))

0 2 4 6 8 10 12

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

8.965%

0 2 4 6 8 10 12

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

rt , j (m ,Θ )=β0+β1(1−e(−m

τ 1)mτ1

)+ β2(1−e(−m

τ1)mτ1

−e(−mτ1)

+β31−e

m/ τ2

mτ2

−em / τ2)+εt , j

with ε t , j~N (0 , σ2 )Θ=( β0 , β1 , β2 , β3 , τ1 , τ2)

Kurt Hess, Waikato Management School Page 3 document.xls Fitting Bond Universe 04/08/2023

Fitting Nelson & Siegel / Svensson Spot Rate with Solver programmed by Kurt Hess May 2004, kurthess@waikato.ac.nzTime to maturity m 4.9 49

0

Long-run levels of interest rates 7.31% 73.121443

Short-run component -2.90% 71

Medium-term component 19.30% 193.04219 determines magnitude and the direction of the hump

Decay parameter 1 1.580 158

Decay parameter 2 1.773 177.28254

Svensson 1994 Extension -0.176245081 -176.2451 optional parameter proposed by Svensson (1994)

Spot rate at time t 6.6317%

Objective Functions

#NUM!

#NUM!

Initial Guess Values:

Bond DataShort-term rate 4.50%Settlement date 14-Feb-99

Issuer Coupon Maturity Bid Ask Mid Clean Mid Dirty DurationNZ Government 6.50% 15-Feb-00 101.563 100.583 101.07% #VALUE! #VALUE! 0.9564551 0.3613016268 #VALUE!NZ Government 8.00% 15-Feb-01 102.786 102.854 102.82% #VALUE! #VALUE! 1.8213216 0.189735185 #VALUE!NZ Government 10.00% 15-Mar-02 108.406 108.526 108.47% #VALUE! #VALUE! 2.6475262 0.1305251619 #VALUE!NZ Government 5.50% 15-Apr-03 96.673 96.827 96.75% #VALUE! #VALUE! 3.7068991 0.0932231445 #VALUE!NZ Government 8.00% 15-Apr-04 105.034 105.234 105.13% #VALUE! #VALUE! 4.2536484 0.0812405626 #VALUE!NZ Government 8.00% 15-Nov-06 106.518 106.809 106.66% #VALUE! #VALUE! 5.8842077 0.0587281769 #VALUE!NZ Government 7.00% 15-Jul-09 100.549 100.903 100.73% #VALUE! #VALUE! 7.5377078 0.0458453416 #VALUE!NZ Government 6.00% 15-Nov-11 91.666 92.049 91.86% #VALUE! #VALUE! 8.7706032 0.0394008005 #VALUE!

Total 1

b0

b1

b2

t1 determines decay of short-term component, must be > 0

t2 determines decay of medium-term component, must be > 0

b3

rt,i

see formulasNon-weighted objective function x103

Inverse duration weighted function x 105

Model Price Weights (wi) (cheap) / rich

0 1 2 3 4 5 6 7 8 9 10

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

6.632%

N&S Svensson Zero Rate

0 1 2 3 4 5 6 7 8 9 10

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

0.722562612446547

N&S Svensson Discount Factors

Minimize

Minimize

Default Values Set Random Values Step through optimization

Before using the minimization macros, you must establish a reference to the Solver add-in. With a Visual Basic module active, click References on the Tools menu, and then select the Solver.xla check box under Available References. If Solver.xla doesn't appear under Available References, click Browse and open Solver.xla in the \Office\Library subfolder.

Kurt Hess, Waikato Management School Page 4 document.xls Fitting Bond Universe 04/08/2023

Formula Objective Function Extended Nelson Siegel Model with Svensson (1994) beta3 extension (parameters explained on top)

D: DurationPi: Price of bond i

N: number of bonds in universe

Subject to:

Rate at the end of the estimation horizon must remain positive

Discount functions must be non-increasing

back to top

^Pi: Model price of bond i

Rate r at time 0 must remain positive (mmin is a value just slightly larger than 0)

min (∑i=1

N

(wi ε i )2)

w i=1/Di

∑ j=1

N1/D j

ε i=P̂i−Pi

0≤r (mmin )0≤r (m=∞ )exp (−r (mk )mk )≥exp (−r (mk+1 )mk+1 ) ∀mk<mmax

References:Nelson, C. R. & Siegel, A. F. (1987). Parsimonious modeling of yield curves, Journal of Business 60(4): 473—489.as discussed in Bliss, R. R. (1997). Testing Term Structure Estimation Methods. Advances in Futures and Options Research(9), 197-231.Svensson, L. (1994). Estimating and interpreting forward interest rates: Sweden 1992-4. Discussion paper, Centre for Economic Policy Research(1051).Anderson, N., Breedon, F., Deacon, M., Derry, A., & Murphy, G. (1996). Estimating and interpreting the yield curve. Chichester: John Wiley Series in Financial Economics and Quantitative Analysis. Chapter 2.4.6, pgs. 36-41.

rt , j (m ,Θ )=β0+β1(1−e(−m

τ 1)mτ1

)+ β2(1−e(−m

τ1)mτ1

−e(−mτ1)

+β31−e−

m /τ2

mτ2

−em / τ2)+εt , j

with ε t , j~N (0 , σ2 )Θ=( β0 , β1 , β2 , β3 , τ1 , τ2)