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Trading Off Smoothness vs. the Length of the Forward Rate Curve

Trading Off Smoothness vs. the Length of the Forward Rate Curve

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EXHIBIT 5.17 Ranking by Smoothness of Forward Rate Curve



Rank



Example



Smoothing Technique



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23



Example H-Qf1c

Example H-Qf1a

Example F

Example F

Example F

Example G-4Cfb

Example G-3Cfb

Example G-3Cfe

Example E

Example E

Example E

Example G-3Cfa

Example G-4Cfc

Example G-3Cfc

Example G-3Cfd

Example G-4Cfa

Example H-Qf1b

Example C

Example D

Example D

Example D

Example A

Example B



Quartic Forward Rate with f 00 (0)5x1, f 00 (T)5x2 and f 0 (T)50 to Max f Smoothness

Quartic Forward Rate with f 00 (0)50, f 00 (T)50 and f 0 (T)50

Cubic y, Max f Smoothness

Cubic y, Max y Smoothness

Cubic y, y 0 (T)¼0 and y 00 (0)¼0

Cubic Forward Rate with f 0 (0)¼x1 and f 0 (T)¼x2 to Max f Smoothness

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼x2 to Max f Smoothness

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼0 to Max f Smoothness

Quadratic f, f 0 (T)¼0

Quadratic f, Min f Length

Quadratic f, Min y Length

Cubic Forward Rate with f 00 (0)¼0 and f 0 (T)¼0

Cubic Forward Rate with f 0 (0)¼x1 and f 0 (T)¼x2 to Min f Length

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼x2 to Min f Length

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼0 to Min f Length

Cubic Forward Rate with f 0 (0)¼0 and f 0 (T)¼0

Quartic Forward Rate with f 00 (0)5x1, f 00 (T)5x2 and f 0 (T)50 to Min Length of f

Linear forwards

Quadratic y, Min y Length

Quadratic y, Min f Length

Quadratic y, y0 (T)¼0

Yield step function

Linear yields



Length of

Yield Curve

Fitted



Length of

Forward

Curve

Fitted



Smoothness

of Yield

Curve Fitted



Smoothness

of Forward

Curve Fitted



11.58

11.51

11.75

11.73

11.64

14.15

14.15

13.59

11.07

11.07

11.07

15.36

11.38

11.38

11.40

19.83

11.23

12.15

11.46

11.56

12.00

13.12

10.99



21.01

19.91

24.94

22.62

23.14

48.89

48.89

40.52

15.28

15.26

15.28

52.23

15.79

15.79

15.87

78.05

15.52

31.59

18.70

17.21

20.48

13.12

16.87



505.76

495.93

510.10

473.48

475.92

687.21

687.21

690.46

1,061.01

1,061.76

1,062.55

597.41

4,111.96

4,111.96

4,200.81

2,128.84

5,267.60

619.35

3,346.49

3,780.67

4,409.57

123,120.00

1,776.50



4,287.35

4,309.97

4,564.23

4,811.18

5,038.87

5,931.25

5,931.25

6,179.14

7,767.53

7,779.04

7,791.63

8,118.72

24,073.20

24,073.23

24,615.22

25,398.14

25,629.13

27,834.72

35,901.19

43,108.27

56,443.05

123,120.00

319,346.00



113



EXHIBIT 5.18 Ranking by Length of Forward Rate Curve



Rank



Example

Description



Smoothing Technique



1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23



Example A

Example E

Example E

Example E

Example H-Qf1b

Example G-3Cfc

Example G-4Cfc

Example G-3Cfd

Example B

Example D

Example D

Example H-Qf1a

Example D

Example H-Qf1c

Example F

Example F

Example F

Example C

Example G-3Cfe

Example G-3Cfb

Example G-4Cfb

Example G-3Cfa

Example G-4Cfa



Yield step function

Quadratic f, Min f Length

Quadratic f, f 0 (T)¼0

Quadratic f, Min y Length

Quartic Forward Rate with f 00 (0)5x1, f 00 (T)5x2 and f 0 (T)50 to Min Length of f

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼x2 to Min f Length

Cubic Forward Rate with f 0 (0)¼x1 and f 0 (T)¼x2 to Min f Length

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼0 to Min f Length

Linear yields

Quadratic y, Min f Length

Quadratic y, Min y Length

Quartic Forward Rate with f 00 (0)50, f 00 (T)50 and f 0 (T)50

Quadratic y, y0 (T)¼0

Quartic Forward Rate with f 00 (0)5x1, f 00 (T)5x2 and f 0 (T)50 to Max f Smoothness

Cubic y, Max y Smoothness

Cubic y, y0 (T)¼0 and y00 (0)¼0

Cubic y, Max f Smoothness

Linear forwards

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼0 to Max f Smoothness

Cubic Forward Rate with f 00 (0)¼x1 and f 0 (T)¼x2 to Max f Smoothness

Cubic Forward Rate with f 0 (0)¼x1 and f 0 (T)¼x2 to Max f Smoothness

Cubic Forward Rate with f 00 (0)¼0 and f 0 (T)¼0

Cubic Forward Rate with f 0 (0)¼0 and f 0 (T)¼0



114



Length of

Yield Curve

Fitted



Length of

Forward

Curve

Fitted



Smoothness

of Yield

Curve Fitted



Smoothness

of Forward

Curve Fitted



13.12

11.07

11.07

11.07

11.23

11.38

11.38

11.40

10.99

11.56

11.46

11.51

12.00

11.58

11.73

11.64

11.75

12.15

13.59

14.15

14.15

15.36

19.83



13.12

15.26

15.28

15.28

15.52

15.79

15.79

15.87

16.87

17.21

18.70

19.91

20.48

21.01

22.62

23.14

24.94

31.59

40.52

48.89

48.89

52.23

78.05



123,120.00

1,061.76

1,061.01

1,062.55

5,267.60

4,111.96

4,111.96

4,200.81

1,776.50

3,780.67

3,346.49

495.93

4,409.57

505.76

473.48

475.92

510.10

619.35

690.46

687.21

687.21

597.41

2,128.84



123,120.00

7,779.04

7,767.53

7,791.63

25,629.13

24,073.23

24,073.20

24,615.22

319,346.00

43,108.27

35,901.19

4,309.97

56,443.05

4,287.35

4,811.18

5,038.87

4,564.23

27,834.72

6,179.14

5,931.25

5,931.25

8,118.72

25,398.14



115



Yield Curve Smoothing

70,000.00



Smoothness of Forward Rate Curve



60,000.00



50,000.00



40,000.00



30,000.00



20,000.00



10,000.00



0.00

0.00



10.00



20.00



30.00



40.00



50.00



60.00



70.00



80.00



90.00



Length of Forward Rate Curve



EXHIBIT 5.19 Trade-Off between Smoothness and Length of Forward Rate Curve for

Various Smoothing Techniques

EXHIBIT 5.20 Best Techniques: Smoothness Under 7,000, Length Under 30, Ranked by

Length of Forward Curve

Example

Description



Smoothing Technique



Example H-Qf1a Quartic Forward Rate

with f 00 (0)¼0, f 00 (T)¼0

and f 0 (T)¼0

Example H-Qf1c Quartic Forward Rate

with f 00 (0)¼x1, f 00 (T)¼

x2 and f 0 (T)¼0 to Max

f Smoothness

Example F

Cubic y, Max y

Smoothness

Example F

Cubic y, y0 (T)¼0 and

y00 (0)¼0

Example F

Cubic y, Max f

Smoothness



Length of

Length of Smoothness of Smoothness of

Yield Curve

Forward

Yield Curve Forward Curve

Fitted

Curve Fitted

Fitted

Fitted



11.51



19.91



495.93



4,309.97



11.58



21.01



505.76



4,287.35



11.73



22.62



473.48



4,811.18



11.64



23.14



475.92



5,038.87



11.75



24.94



510.10



4,564.23



It is interesting to see that the best five consist of two quartic forward rate

smoothing approaches, ranked first and second by smoothness, and three cubic yield

spline approaches. Of this best five group, the quartic forward rate approach, optimized for smoothness, was the smoothest. The quartic forward rate approach, with all

derivatives in constraints 23 to 25 set to zero, was the shortest. We now turn to a more

comprehensive approach to performance measurement of smoothing techniques.



116



RISK MANAGEMENT TECHNIQUES FOR INTEREST RATE ANALYTICS



THE SHIMKO TEST FOR MEASURING ACCURACY

OF SMOOTHING TECHNIQUES

In 1994, Adams and van Deventer (1994) published “Fitting Yield Curves and

Forward Rate Curves with Maximum Smoothness.” In 1993, our thoughtful friend

David Shimko responded to an early draft of the Adams and van Deventer paper by

saying, “I don’t care about a mathematical proof of ‘best,’ I want something that

would have best estimated a data point that I intentionally leave out of the smoothing

process—this to me is proof of which technique is most realistic.” A statistician

would add, “And I want something that is most realistic on a very large sample of

data.” We agreed that Shimko’s suggestion was the ultimate proof of the accuracy

and realism of any smoothing technique. The common academic practice of using

one set of fake data and then judging which technique “looks good” or “looks bad”

is as ridiculous as it is common. Therefore, we atone for the same sin, which we have

used in the first 10 installments of this series, by explaining how to perform the

Shimko test as in Adams and van Deventer (1994).

The Shimko test works as follows. First, we assemble a large data set, which

in the Adams and van Deventer case was 660 days of swap data. Next, we select one

of the maturities in that data set and leave it out of the smoothing process. For

purposes of this example, say we leave out the seven-year maturity because that

leaves a wide five-year gap in the swap data to be filled by the smoothing technique.

We smooth the 660 yield curves one by one. Using the smoothing results and the

zero-coupon bond yields associated with the 14 semiannual payment dates of the

seven-year interest rate swap, we calculate the seven-year swap rate implied by the

smoothing process. We have 660 observations of this estimated seven-year swap rate,

and we compare it to the actual seven-year swap rates that we left out of the

smoothing process. The best smoothing technique is the one that most accurately

estimates the omitted data point over the full sample.

This test can be performed on any of the maturities that were inputs to the

smoothing process, and we strongly recommend that all maturities be used one at a

time. Because this test suggested by Shimko is a powerful test applicable to any contending smoothing techniques, we strongly recommend that no assertion of superior

performance be made without applying the Shimko test on a large amount of real data.4



SMOOTHING YIELD CURVES USING COUPON-BEARING BOND PRICES

AS INPUTS

For expositional purposes, we have assumed in this chapter that the raw inputs to

this process are zero-coupon bond yields. When, instead, the inputs are the prices and

terms on coupon-bearing bonds, the analysis changes in a minor way, which we

illustrate in Chapter 17. The initial zero-coupon bond yields are guessed, and an

iteration of zero-coupon bond yields is performed that minimizes the sum of squared

bond–pricing errors. The Dickler, Jarrow, and van Deventer analysis discussed in

Chapter 3 was done in that manner using Kamakura Risk Manager version 8.0.

Kamakura Corporation produces a weekly forecast of implied forward rates derived

in this manner. The implied forward U.S. Treasury yields as of May 3, 2012, are

shown in the graph in Exhibit 5.21.



117



Yield Curve Smoothing



4.500

4.000

3.500



2.500

Series 121

Series 109

Series 97

Series 85

Series 73

Series 61

Series 49

Series 37

Series 25

Series 13

Series 1



2.000

1.500

1.000

0.500



1

M

3 ont

M h

6 ont Bill

M h R

on Bi at

th ll R e

Bi at

ll e

R

a

1 te

Y

2 ear

Y

3 ears

Y

4 ear

Y s

e

5 ars

Y

6 ear

Y s

e

7 ars

Y

e

8 ars

Y

9 ear

Y s

10 ear

s

11 Yea

Y rs

12 ea

Y rs

13 ea

Y rs

1 4 ea

Y rs

1 5 ea

Y rs

16 ea

Y rs

1 7 ea

Y rs

18 ea

Y rs

19 ear

Y s

20 ear

Y s

ea

rs



0.000



Months Forward



Percent



3.000



U.S. Treasury Maturity



EXHIBIT 5.21 Kamakura Corporation, 10-Year Forecast of U.S. Treasury Yield Curve

Implied by Forward Rates Using Maximum Smoothness Forward Rate Smoothing



We now turn to interest rate simulation using the smoothed yield curves that we

have generated in this chapter.



APPENDIX: PROOF OF THE MAXIMUM SMOOTHNESS FORWARD

RATE THEOREM

Schwartz (1989) demonstrates that cubic splines produce the maximum smoothness

discount functions or yield curves if the spline is applied to discount bond prices or

yields respectively. In this appendix, we derive by a similar argument the functional

form that produces the forward rate curve with maximum smoothness. Let f(t) be the

current forward rate function, so that

0

1

Zt

PðtÞ ¼ exp@À f ðsÞdsA

ð5:A1Þ

0



is the price of a discount bond maturing at time t. The maximum smoothness term

structure is a function f with a continuous derivative that satisfies the optimization

problem



RISK MANAGEMENT TECHNIQUES FOR INTEREST RATE ANALYTICS



118



ZT

min



f 00 ðsÞds

2



ð5:A2Þ



0



subject to the constraints

Zti

f sịds ẳ log Pi , for i ẳ 1, 2, : : : , m:



5:A3ị



0



Here the Pi ẳ P(ti), for i ¼ 1, 2, . . . , m are given prices of discount bonds with

maturities 0 , t1 , t2 , . . . , tm , T.

Integrating twice by parts we get the following identity:

Zt



1

f sịds ẳ

2



0



Zt



1

t sị2 f 00 sịds ỵ tf 0ị ỵ t2 f 0 0ị

2



5:A4ị



0



Put

gtị ẳ f 00 tị, 0 # t # T



5:A5ị



and define the step function

utị ẳ 1 for t $ 0

¼ 0 for t , 0:

The optimization problem can then be written as

ZT

min



g2 ðsÞds



ð5:A6Þ



0



subject to

1

2



ZT



1

ðti À sÞ2 uðti À sịgsịds ẳ log Pi ti f 0ị ti2 f 0 0ị

2



5:A7ị



0



for I ẳ 1, 2, . . . , m. Let λi for i ¼ 1, 2, . . . , m be the Lagrange multipliers corresponding to the constraints (5.A7). The objective then becomes

ZT

g2 sịds



min Zẵg ẳ

0



m

X



0



1



i @

2

iẳ1



ZT

0



1

1

ti sị2 uti sịgsịds ỵ logPi ỵ ti f 0ị ỵ ti2 f 0 0ịA 5:A8ị

2



119



Yield Curve Smoothing



According to the calculus of variations, if the function g is a solution to equation

(5.A8), then

d

Zẵg ỵ hẳ0 ẳ 0

d



5:A9ị



for any function h(t) identically equal to w00 (t) where w(t) is any twice differentiable

function defined on [0, T] with w0 (0) ẳ w(0) ẳ 02. We get

d

Zẵg ỵ hẳ0 ẳ 2

d



ZT "

0



#

m

1X

2

gsị ỵ

i ti sị uti sị hsịds

4 iẳ1



In order that this integral is zero for any function h, we must have

gtị ỵ



m

1X

i ti tị2 uti tị ¼ 0

4 i¼1



ð5:A10Þ



for all t between 0 and T. This means that

gtị ẳ 12ei t2 ỵ 6di t ỵ 2ci for tiÀ1 , t # ti , i ¼ 1, 2, : : : , m ỵ 1,



5:A11ị



where

ei ẳ

di ¼



m

1 X

λj

48 j¼1



m

1 X

λj tj

12 j¼1



ci ¼ À



ð5:A12Þ



m

1X

λj t2

8 j¼i j



and we define t0 ẳ 0, tm ỵ 1 ẳ T. Moreover, equation (5.A10) implies that g and g0

(and therefore f 00 and f000 ) are continuous. From equation (5.A4) we get

f tị ẳ ei t4 ỵ di t3 ỵ ci t2 ỵ bi t ỵ ai , ti1 , t # ti , i ¼ 1, 2, : : : , m ỵ 1



5:A13ị



Continuity of f, f 0 , f 00 and f 000 then implies that

ei ti4 ỵ di t3 ỵ ci ti2 ỵ bi ti ỵ ai ẳ ei ỵ1 ti4 ỵ diỵ1 ti3 ỵ ciỵ1 ti2 þ biþ1 ti þ aiþ1 , i

¼ 1, 2, : : : , m



5:A14ị



4ei ti3 ỵ 3di ti2 ỵ 2ci ti ỵ bi ẳ 4eiỵ1 ti3 ỵ 3diỵ1 ti2 ỵ 2ciỵ1 ti ỵ biỵ1 , i ẳ 1, 2, :::, m

12ei ti2 ỵ 6di ti ỵ 2ci ẳ 12eiỵ1 ti2 þ 6diþ1 ti þ 2ciþ1

24ei ti þ 6di ¼ 24eiþ1 ti ỵ 6diỵ1



5:A15ị



RISK MANAGEMENT TECHNIQUES FOR INTEREST RATE ANALYTICS



120



The constraints (5.A3) become

1

1

1

1

5

4

3

2

ị ỵ di ti4 ti1

ị ỵ ci ti3 ti1

ị ỵ bi ti2 ti1

ị ỵ ai ti ti1 ị

ei ti5 ti1

5

4

3

2

!

Pi

, i ẳ 1, 2, :::, m

ẳ log

Pi1



5:A16ị



where we define P0 ẳ 1. This proves the theorem.



NOTES

1. These techniques are reviewed in great detail, with worked examples, in a series of blogs on

the Kamakura Corporation website at www.kamakuraco.com:



Basic Building Blocks of Yield Curve Smoothing, Part 1, November 2, 2009.

www.kamakuraco.com/Blog/tabid/231/EntryId/150/Basic-Building-Blocks-of-YieldCurve-Smoothing-Part-1.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 2: A Menu of Alternatives,

November 17, 2009. www.kamakuraco.com/Blog/tabid/231/EntryId/152/BasicBuilding-Blocks-of-Yield-Curve-Smoothing-Part-2-A-Menu-of-Alternatives.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 3: Stepwise Constant Yields and

Forwards versus Nelson-Siegel,” November 18, 2009. www.kamakuraco.com

/Blog/tabid/231/EntryId/156/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-3Stepwise-Constant-Yields-and-Forwards-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 4: Linear Yields and Forwards

versus Nelson-Siegel, November 20, 2009. www.kamakuraco.com/Blog/tabid/231

/EntryId/157/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-4-Linear-Yieldsand-Forwards-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 5: Linear Forward Rates and Related

Yields versus Nelson-Siegel, November 30, 2009. www.kamakuraco.com/Blog

/tabid/231/EntryId/158/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-5-LinearForward-Rates-and-Related-Yields-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 6: Quadratic Yield Splines and

Related Forwards versus Nelson-Siegel, December 3, 2009. www.kamakuraco.com

/Blog/tabid/231/EntryId/159/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-6

-Quadratic-Yield-Splines-and-Related-Forwards-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 7: Quadratic Forward Rate Splines

and Related Yields versus Nelson-Siegel,” December 8, 2009. www.kamakuraco

.com/Blog/tabid/231/EntryId/161/Basic-Building-Blocks-of-Yield-Curve-Smoothing

-Part-7-Quadratic-Forward-Rate-Splines-and-Related-Yields-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 8: Cubic Yield Splines and Related

Forwards versus Nelson-Siegel, December 10, 2009. www.kamakuraco.com/Blog

/tabid/231/EntryId/162/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-8-CubicYield-Splines-and-Related-Forwards-versus-Nelson-Siegel.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 9: Cubic Forward Rate Splines and

Related Yields versus Nelson-Siegel, December 30, 2009. www.kamakuraco.com

/Blog/tabid/231/EntryId/167/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part

-9-Cubic-Forward-Rate-Splines-and-Related-Yields-versus-Nelson-Siegel-Updated

-January-6-2010.aspx.



Yield Curve Smoothing



121



Basic Building Blocks of Yield Curve Smoothing, Part 10: Maximum Smoothness

Forward Rates and Related Yields versus Nelson-Siegel, January 5, 2010. www

.kamakuraco.com/Blog/tabid/231/EntryId/168/Basic-Building-Blocks-of-Yield-CurveSmoothing-Part-10-Maximum-Smoothness-Forward-Rates-and-Related-Yields-versusNelson-Siegel-Revised-May-8-2012.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 11: The Shimko Test for Measuring

Accuracy of Smoothing Techniques,” January 13, 2010. www.kamakuraco.com/

Blog/tabid/231/EntryId/170/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part11-The-Shimko-Test-for-Measuring-Accuracy-of-Smoothing-Techniques.aspx.

Basic Building Blocks of Yield Curve Smoothing, Part 12: Smoothing with Bond Prices as

Inputs,” January 20, 2010. www.kamakuraco.com/Blog/tabid/231/EntryId/172

/Basic-Building-Blocks-of-Yield-Curve-Smoothing-Part-12-Smoothing-with-Bond-Prices

-as-Inputs.aspx.

2. www.wikipedia.com has generally been a very useful reference for the topics “spline

interpolation” and “spline.” We encourage interested readers to review the current version

of Wikipedia on these topics.

3. The authors wish to thank Keith Luna of Western Asset Management Company for

helpful comments on this section of Chapter 5.

4. As an example of the volumes of data that can be analyzed, Dickler, Jarrow, and van

Deventer present a history of forward rate curves, zero-coupon yield curves, and par

coupon bond yield curves for every business day (more than 12,000 business days) from

January 2, 1962, to August 22, 2011 in these volumes available on www.kamakuraco

.com:

Daniel T. Dickler, Robert A. Jarrow, and Donald R. van Deventer, “Inside the Kamakura

Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates,”

Kamakura Corporation memorandum, September 13, 2011. kamakuraco.com/Portals

/0/PDF/KamakuraYieldBook-ForwardRates-FinalVersion-20110913.pdf.

Daniel T. Dickler and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An

Analysis of 50 Years of Daily U.S. Treasury Forward Rates,” Kamakura blog, www

.kamakuraco.com, September 14, 2011. www.kamakuraco.com/Blog/tabid/231

/EntryId/333/Inside-the-Kamakura-Book-of-Yields-An-Analysis-of-50-Years-Of-Daily

-U-S-Treasury-Forward-Rates.aspx.

Daniel T. Dickler, Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura

Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Zero-coupon Bond

Yields,” Kamakura Corporation memorandum, September 26, 2011. kamakuraco.com/

Portals/0/PDF/KamakuraYieldBook-ZeroCouponBondYields-Final-20110930.pdf.

Daniel T. Dickler and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An

Analysis of 50 Years of Daily U.S. Treasury Zero-coupon Bond Yields,” Kamakura

blog, www.kamakuraco.com, September 26, 2011. www.kamakuraco.com/Blog

/tabid/231/EntryId/337/Inside-the-Kamakura-Book-of-Yields-An-Analysis-of-50-Years

-of-Daily-U-S-Treasury-Zero-Coupon-Bond-Yields.aspx.

Daniel T. Dickler, Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura

Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond

Yields,” Kamakura Corporation memorandum, October 5, 2011. kamakuraco.com

/Portals/0/PDF/KamakuraYieldBook-ParCouponBondYields-Final-20111005.pdf.



122



RISK MANAGEMENT TECHNIQUES FOR INTEREST RATE ANALYTICS

Daniel T. Dickler and Donald R. van Deventer, “Inside the Kamakura Book of Yields:

An Analysis of 50 Years of Daily U.S. Par Coupon Bond Yields,” Kamakura blog,

www.kamakuraco.com, October 6, 2011. www.kamakuraco.com/Blog/tabid/231/

EntryId/339/Inside-the-Kamakura-Book-of-Yields-An-Analysis-of-50-Years-of-DailyU-S-Treasury-Par-Coupon-Bond-Yields.aspx.



5. This proof was kindly provided by Oldrich Vasicek. We also appreciate the comments of

Volf Frishling, who pointed out an error in the proof in Adams and van Deventer (1994).

Robert Jarrow also made important contributions to this proof.

6. We are grateful to Robert Jarrow for his assistance on this point.



CHAPTER



6



Introduction to Heath, Jarrow, and

Morton Interest Rate Modeling



I



n Chapter 5, we mentioned the review by Dickler, Jarrow, and van Deventer of

50 years of daily U.S. Treasury interest rate movements. They concluded that U.S.

interest rates are driven by 5 to 10 random factors, a much larger number of factors

than major financial institutions typically use for interest rate risk management. In

a series of three papers, David Heath, Robert A. Jarrow, and Andrew Morton

(1990a, 1990b, 1992) introduced the Heath, Jarrow, and Morton (HJM) framework for modeling interest rates driven by a large number of random factors while

preserving the standard “no arbitrage” assumptions of modern finance. The HJM

framework is a very powerful general solution for modeling interest rates driven by

a large number of factors and for the valuation and simulation of cash flows on

securities of all types. The authors believe that an understanding of the HJM

approach is a fundamental requirement for a risk manager. In this chapter and the

following three chapters, we lay out four worked examples of the use of the HJM

approach with increasing realism. A full enterprise-wide risk management infrastructure relies on a sophisticated implementation of the HJM framework with the

5 to 10 driving risk factors and a very large number of time steps. Such an

implementation should make use of both the HJM implications for Monte Carlo

simulation and the “bushy tree” approach for valuing securities like callable bonds

and home mortgages. For expositional purposes, this and the following chapters

focus on the bushy tree approach, but we explain why an HJM Monte Carlo

simulation is essential for the simulation of cash flows, net income, and liquidity

risk in Chapter 10.

The authors wish to thank Robert A. Jarrow for his encouragement and advice

on this series of worked examples of the HJM approach. What follows is based

heavily on Jarrow’s classic book, Modeling Fixed Income Securities and Interest Rate

Options, 2nd ed. (2002), particularly Chapters 4, 6, 8, 9, and 15. In our Chapter 9,

we incorporate some modifications of Chapter 15 in Jarrow’s book that would have

been impossible without Jarrow’s advice and support.

In Chapters 6 through 9, we use data from the Federal Reserve statistical release

H15, published on April 1, 2011. U.S. Treasury yield curve data was smoothed using

Kamakura Corporation’s Kamakura Risk Manager version 7.3 to create zero-coupon

bonds via the maximum smoothness technique of Adams and van Deventer as

described in Chapter 5. Most of the applications of the HJM approach by financial

market participants have come in the form of the LIBOR market model and



123



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