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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 upas 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 isalso called by a VBA macro, you will also need to establish a reference to theSolver add-in in the VBA editor:With a Visual Basic module active, click References on the Tools menu, andthen select the Solver.xla check box under Available References. If Solver.xla

doesn't appear under Available References, click Browse and open Solver.xla inthe \Office\Library subfolder.

Kurt Hess, kurthess@waikato.ac.nz 199512653.xls.ms_office Introduction 1/2/2014

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Illustration of Extended Nelson & Siegel Spot Rate Modelprogrammed by Kurt Hess May 2004, kurthess@waikato.ac.nz

Time to maturity m 3.0 30.00

Long-run levels of interest rates b0 5.0% 50

Short-run component b1 2.0% 120

Medium-term component b2 8.0% 80 determines magnitude and the direction of the hump

Decay parameter 1 t1 1.000 100 determines decay of short-term component, must be > 0

Decay parameter 2 t2 1.200 120 determines decay of medium-term component, must be > 0

Spot rate at time t rt,i 7.9141%

7.9141% with VBA Function

Components of N&S spot rate

Comp 1 5.000% b0

Comp 2 0.633% b1*((1-EXP(-m/tau1))/(m/tau1))

Comp 3 2.281% b2*((1-EXP(-m/tau2))/(m/tau2)-EXP(-m/tau2))

7.91%

0.0%

1.0%2.0%3.0%4.0%5.0%6.0%7.0%8.0%9.0%

10.0%

0 2 4 6 8 10 12 0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

0 2 4 6 8

2

2

1

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Kurt Hess, Waikato Management School Page 2 199512653.xls.ms_office Extended Nelson Siegel 1/2/2014

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Fitting Extended Nelson & Siegel Spot Rate with Solver programmed by Kurt Hess May 2004, kurthess@waikato.ac.nzTime to maturity m 3.0 30

Long-run levels of interest rates b0 7.30% 73.02474

Short-run component b1 -2.80% 71.97526

Medium-term component b2 -1.40% -14.0174 determines magnitude and the direction of the hump

Decay parameter 1 t1 0.412 41.19045 determines decay of short-term component, must be > 0

Decay parameter 2 t2 2.905 290.466 determines decay of medium-term component, must be > 0

Spot rate at time t rt,i 6.5429%

Objective Functions see formulasNon-weighted objective function x103 0.200134

Inverse duration weighted function x 105

0.028763

Initial Guess Values:

Bond DataShort-term rate 4.50%

Settlement date 14-Feb-99

Issuer Coupon Maturity Bid Ask Mid Clean Mid Dirty

o e

Price Duration Weights (wi) (cheap)/ rich

NZ Government 6.50% 15-Feb-00 100.563 100.583 100.57% 103.80% 103.573% 0.956271 0.361346082 0.23%

NZ Government 8.00% 15-Feb-01 102.786 102.854 102.82% 106.80% 106.742% 1.821322 0.189721979 0.06%

NZ Government 10.00% 15-Mar-02 108.406 108.526 108.47% 112.60% 113.399% 2.647526 0.130516077 (0.79%)

NZ Government 5.50% 15-Apr-03 96.673 96.827 96.75% 98.57% 97.483% 3.706899 0.093216656 1.08%

NZ Government 8.00% 15-Apr-04 105.034 105.234 105.13% 107.78% 108.039% 4.253648 0.081234908 (0.26%)

NZ Government 8.00% 15-Nov-06 106.518 106.809 106.66% 108.64% 108.878% 5.884208 0.058724089 (0.24%)

NZ Government 7.00% 15-Jul-09 100.549 100.903 100.73% 101.29% 101.176% 7.537708 0.045842151 0.11%

NZ Government 6.00% 15-Nov-11 91.666 92.049 91.86% 93.34% 93.343% 8.770603 0.039398058 (0.00%)

Total 1

6.54%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%7.0%

8.0%

9.0%

0 2 4 6 8 10

N&S Zero Rate

0.82178

0.0%

20.0%

40.0%

60.0%

80.0%

100.0%

120.0%

0 2 4 6

N&S D

Minimize

Minimize

Default Values Set Random ValuesStep through

optimization

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Formula Objective Function back to top Extended Nelson Siegel Model (parameters explained on top)

D: Duration

Pi: Price of bond i

^Pi: Model price of bond i

N: number of bonds in universe

Subject to:

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

Rate at the end of the estimation horizon must remain positive

Discount functions must be non-increasing

References:Nelson, C. R. & Siegel, A. F. (1987). Parsimonious modeling of yield curves, Journal of Business 60(4): 473489.as discussed inBliss, R. R. (1997). Testing Term Structure Estimation Methods. Advances in Futures and Options Research(9), 197-231.

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Kurt Hess, Waikato Management School Page 4 199512653.xls.ms_office Fitting Bond Universe 1/2/2014