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
ỵ
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.
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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
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