.95
0.90
0.90
Beta
1.10
0.85
0.850
12
24
36
48
60
72
84
96
108
120
Time to settlement (in months)
Figure 3.7 Test of the expectations theory of interest rates. Forward regression coefﬁcient in1986–1998Table 3.3 Parameter estimates in the forward regressionsMβσ (β)ασ (α)
30.9530.029−0.0030.007
120.9650.0220.0050.019
240.9750.0150.0200.025
360.9800.0130.0290.030
480.9810.0120.0340.033
600.9810.0120.0380.036
840.9800.0130.0430.040
1200.9790.0140.0460.043
of the expectations theory of the term structure, in line with equations (3.33) and (3.34).In Table 3.3, the parameter estimates are exhibited.The pure expectations theory is easily rejected: as shown in Figure 3.6 and Table 3.3,average one-period short-term forward rates vary with maturity, which contradicts equation (3.33) with n = 0.40 However, as shown in Figure 3.7, forward regressions generate slope coefﬁcients close to one for all maturities, with relatively small standarderrors,41 suggesting that the assumption of constant term premiums might be a reasonableapproximation.The results42 of the term structure estimation are shown in Figure 3.8a and b (averageyield curves), Figure 3.9a and b (volatility curves), Figure 3.10a and b (term premiumcurves), Figure 3.11a and b (one-period forward curves), Figure 3.12a and b (expectedshort-term interest rate curves), Figure 3.13a and b (time-series yields), Figure 3.14a and b(time-series inﬂation), Figure 3.15a and b (factor loadings) and Figure 3.16a–f (time-seriesfactors). The parameters and their standard errors are reported in Tables 3.4 and 3.5.4340In the steady state, it is expected that short-term interest rates remain constant. Thus the one-period forwardcurve should be ﬂat under the absence of risk premium.41Newey–West standard errors.42Many of the charts which follow need to be viewed in colour to be fully understood. Each of the followingﬁgures are included as an Excel ﬁle with suitably titled ﬁlenames on the CD-Rom.43The inﬂation rate used in the identiﬁed models was obtained from Datastream.
Modelling the Term Structure of Interest Rates
109
8.0
7.5
%
7.0
6.5
6.0
5.5
5.00
12
24
Observed
36
486072Term to maturity (in months)
2f non-id.
84
3f non-id.
96
108
2f-ident.
120
2f-ident.2
(a)8.07.5
%
7.06.56.05.55.00
12
24
36
48
60
72
84
96
108
120
Term to maturity (in months)Observed
2f non-id.
2f ident.2
2f ident.
(b)
Figure 3.8
Average yield curves: (a) 1986–1998 and (b) 1972–1998
According to Figures 3.8a and b, 3.9a and b, all estimates reproduce very closely both theaverage yield and the volatility curves, though worse results are obtained for 1986–1998when the second factor is identiﬁed with inﬂation as in equation (3.55) or equation (3.56).44This result must be associated with the link between the behaviour of the second factor andthe inﬂation, which implies a worse ﬁtting for the long-term interest rates.44
The results with the alternative identiﬁed model (in equation (3.56)) are labelled “2f-ident.2”.
110
Applied Quantitative Methods for Trading and Investment
Standard deviation of yields
3.02.52.01.51.00.50.00
12
24
36
48
60
72
84
96
108
120
Term to maturity (in months)Observed
2f non-id.
3f non-id.
2f ident.
2f ident.2
(a)
Standard deviation of yields
3.0
2.5
2.0
1.5
1.0
0.50
12
24
36
48
60
72
84
96
108
120
Term to maturity (in months)Observed
2f non-id.
2f ident.
2f ident.2
(b)
Figure 3.9
Volatility curves: (a) 1986–1998 and (b) 1972–1998
Concerning the term premium and the one-period forward, Figures 3.10a and b, 3.11aand b show again that the results obtained differ in the shorter sample, depending on usingidentiﬁed or non-identiﬁed models. The estimates obtained with the identiﬁed model inequation (3.55) point to lower risk premium in the maturities between 2 and 7 years,while the risk premium is higher at both ends of the curve. This model still presents anestimate for the 8-year risk premium around 1.7 in both samples, though it converges
Modelling the Term Structure of Interest Rates
111
2.5
2.0
%
1.5
1.00.5
0.0−0.5
0
12
24
36
48
60
72
84
96
108
120
Term to maturity (in months)2f non-id.
3f non-id.
2f ident.
2f ident.2
(a)2.5
2.0
%
1.5
1.0
0.5
0.0
−0.50
12
24
36
48
60
72
84
96
108
Term to maturity (in months)2f non-id.
2f ident.
2f ident.2
(b)
Figure 3.10
Term premium curves: (a) 1986–1998 and (b) 1972–1998
120
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Applied Quantitative Methods for Trading and Investment9.08.58.0
%
7.57.06.56.05.55.00
12
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36
48
60
72
84
96
108
120
Time to settlement (in months)2f non-id.
3f non-id.
2f ident.
2f ident.2
(a)9.08.58.0
%
7.57.06.56.05.55.00
12
24
36
48
60
72
84
96
108
Time to settlement (in months)2f non-id.
2f ident.
2f ident.2
(b)
Figure 3.11 Average forward curves: (a) 1986–1998 and (b) 1972–1998
120
Modelling the Term Structure of Interest Rates
113
monotonically to a positive value, around 4% in maturities higher than 20 years.45 In thenon-identiﬁed models, the term premium rises rapidly up to 1.5 in the 5-year maturity,converging to around 1.6 and 1.25 in the 10-year maturity, respectively using the twoand the three-factor model.46Accordingly, the expected short-term interest rates exhibit differences between the identiﬁed and the non-identiﬁed models, as well as between the samples (Figure 3.12a andb). The most remarkable feature is the positive slope in the identiﬁed models and usingthe shorter sample, namely with the model in (3.56). In the longer sample, the short-terminterest rate is roughly constant in all models, as the shape of the one-period forward andthe term premium curves are similar.Regarding the time-series results, Figure 3.13a and b show the observed and the estimated yields for several maturities considered, while in Figure 3.14a and b the observedand the estimated inﬂation are shown (according to the models in (3.55) and (3.56)).It can be concluded that all estimates reproduce very closely both the yields across thematurity spectrum and the inﬂation.47 However, the ﬁt for the yields is poorer in twosubperiods: the ﬁrst in Spring 1994, where predicted short-term rates are above actualrates and predicted long-term rates are below actual rates;48 the second in 1998, whenlong bond yields fell as a consequence of the Russian and Asian crises, whilst short ratesremained stable. Therefore, the estimated long-term interest rates remained higher thanthe actual rates.The identiﬁed model in (3.56) provides more accurate estimates for the inﬂation, as ituses a speciﬁc factor for this variable, instead of the inﬂation being explained only bythe second factor of the term structure.Focusing on the parameter estimates (Table 3.4), the factors are very persistent, asthe estimated ϕ’s are close to one and exhibit low volatility.49 It is the second factorthat contributes positively to the term premium (σ2 λ2 ) and exhibits higher persistency.The market price of risk of this factor becomes higher when it is identiﬁed with inﬂation. Conversely, its volatility decreases. All variables are statistically signiﬁcant, as theparameters evidence very low standard deviations (Table 3.5). This result conﬁrms thegeneral assertion about the high sensitivity of the results to the parameter estimates.50As was seen in the previous charts, the results obtained with the three-factor model andusing the shorter sample exhibit some differences to those obtained with the two-factor45Notice that the identiﬁed model in (3.56) provides lower estimates for the risk premium as the market priceof risk of the ﬁrst factor is higher, giving a lower contribution for the risk premium.46The term premium estimate for the 10-year maturity using the two-factor non-identiﬁed model is in line withother estimates obtained for different term structures. See, e.g., De Jong (1997), where an estimate of 1.65 forthe 10-year term premium in the US term structure is presented.47The quality of the ﬁt for the yields contrasts sharply with the results found for the US by Gong and Remolona(1997c). In fact, in the latter it was concluded that at least two different two-factor heteroskedastic models areneeded to model the whole US term structures of yields and volatilities: one to ﬁt the medium/short end of thecurves and another to ﬁt the medium/long terms. The lowest time-series correlation coefﬁcients between theobserved and the estimated ﬁgures is 0.87, for the 10-year maturity, and the cross-section correlation coefﬁcientsare, in most days of the sample, above 0.9.48The surprising behaviour of bond yields during 1994 is discussed in detail in Campbell (1995) with referenceto the US market.49Nevertheless, given that the standard deviations are low, the unit root hypothesis is rejected and, consequently,the factors are stationary. The comments of Jerome Henry on this issue are acknowledged.50Very low standard deviations for the parameters have usually been obtained in former Kalman ﬁlter estimatesof term structure models, as in Babbs and Nowman (1998), Gong and Remolona (1997a), Remolona et al.(1998), and Geyer and Pichler (1996).
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Applied Quantitative Methods for Trading and Investment7.06.86.66.4
%
6.26.05.85.65.45.25.00
12
24
36
48
60
72
84
96
108
120
Horizon (in months)2f non-id.
3f non-id.
2f ident.
2f ident.2
(a)7.06.86.66.4
%
6.26.05.85.65.45.25.00
12
24
36
48
60
72
84
96
108
120
Horizon (in months)2f non-id.
2f ident.
2f ident.
(b)
Figure 3.12
Average expected short-term interest rates: (a) 1986–1998 and (b) 1972–1998
Modelling the Term Structure of Interest Rates
115
(i) 1-month
Observed3f non-id.2f ident.2
2f non-id.2f ident.
Observed3f non-id.2f ident.2
2f non-id.2f ident.
(v) 7-year
2f non-id.2f ident.
Observed3f non-id.2f ident.2
1998
1997
1996
1994
1993
1995
1998
1997
1996
1995
1994
1993
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1998
01997
2
01996
4
2
1995
6
4
1994
8
6
1993
10
8
1992
12
10
1991
12
1990
2f non-id.2f ident.
(vi) 10-year14
1989
1992
Observed3f non-id.2f ident.2
14
1988
1991
1990
1989
1988
1987
1986
1998
1997
1996
1995
1994
01993
2
01992
4
2
1991
6
4
1990
8
6
1989
10
8
1988
12
10
1987
12
1986
14
1987
2f non-id.2f ident.
(iv) 5-year
(iii) 3-year
1986
1992
Observed3f non-id.2f ident.2
14
Observed3f non-id.2f ident.2
1991
1990
1989
1988
1986
1998
1997
1996
1995
1994
01993
2
01992
4
2
1991
6
4
1990
8
6
1989
10
8
1988
12
10
1987
12
1986
14
1987
(ii) 1-year
14
2f non-id.2f ident.
(a)
Figure 3.13 Time-series yield estimation results: (a) 1986–1998 and (b) 1972–1998
model. In order to opt between the two- and the three-factor model, a decision must bemade based on the estimation accuracy of the models. Consequently, a hypothesis test isperformed, considering that both models are nested and the two-factor model correspondsto the three-factor version imposing ϕ3 = σ3 = 0.This test consists of computing the usual likelihood ratio test l = −2(ln v − ln v ∗ ) ∼2χ (q), v and v* being the sum of the likelihood functions respectively of the two- and
116
Applied Quantitative Methods for Trading and Investment(i) 1-year
(ii) 3-year14
12
12
10
10
8
8
6
6
4
4
2
2
0
0
Observed2f ident.
197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
14
Observed2f ident.
2f non-id.2f ident.2
(iv) 7-year
(iii) 5-year14
12
12
10
10
8
8
6
6
4
4
2
2
0
0197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
14
Observed2f ident.
Observed2f ident.
2f non-id.2f ident.2
(v) 8-year
2f non-id.2f ident.2
(vi) 10-year
12
12
10
10
8
8
6
6
4
4
2
2
0
0197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
197219731974197519761977197819791980198119821983198419851986198719881989199019911992199319941995199619971998
14
14
Observed2f ident.
2f non-id.2f ident.2
Observed2f ident.
2f non-id.2f ident.2
2f non-id.2f ident.2
(b)
Figure 3.13 (continued )
the three-factor models and q the number of restrictions (in this case two restrictions areimposed). The values for ln v and ln v* were −306.6472 and −304.5599, implying thatl = 4.1746. As χ 2 (2)0.95 = 5.991, the null hypothesis is not rejected and the two-factormodel is chosen. This conclusion is also suggested by the results already presented,due to the fact that the three-factor model does not seem to increase signiﬁcantly the
Modelling the Term Structure of Interest Rates
117
6543
%
210−1−2
Jan-97
Jan-98Sep-96
Jan-96
Jan-95
Jan-94
Jan-93
Estimated
Sep-94
Observed
Jan-92
Jan-91
Jan-90
Jan-89
Jan-88
Jan-87
−4
Jan-86
−3
Estimated 2
(a)5432
0−1−2−3
Observed
Estimated
Estimated 2
(b)
Figure 3.14
Observed and estimated yearly inﬂation: (a) 1986–1998 and (b) 1972–1998
Sep-98
Sep-92
Sep-90
Sep-88
Sep-86
Sep-84
Sep-82
Sep-80
Sep-78
Sep-76
−5
Sep-74
−4Sep-72
%
1
–––6.89340
––0.00931
1972–19982f non-id.2f ident.2f ident.2
––0.99748
–––0.99773
ϕ1π
σ1π
1986–19982f non-id.3f non-id.2f ident.2f ident.2
0.001270.001590.00141
0.005080.005470.00526
1972–19982f non-id.2f ident.2f ident.2
0.000780.001140.001560.00154
0.004890.004940.004980.00566
σ1
1986–19982f non-id.3f non-id.2f ident.2f ident.2
δ
––0.08405
–––1.72529
λ1π σ1π
0.962830.992250.95572
0.950760.971010.981630.98169
ϕ1
––−2.65631
–––1.55172
λ2π σ1π
0.103402.307970.09198
––0.25683
–––0.00004
B1π
0.001650.000700.00160
0.001730.000760.001090.00111
σ2
B2π
0.991450.992590.99298
0.986100.975430.982160.98225
ϕ2
––0.06685
–––−0.05438
Parameter estimates
0.221921.139833.097983.20068
λ1 σ1
Table 3.4
–––
–0.001317––
σ3
−0.10560−5.29858−0.09869
−0.09978−1.71811−4.44754−4.45554
λ2 σ2
–––
–0.989882––
ϕ3
–––
–0.006502––
λ3 σ3
Modelling the Term Structure of Interest Rates
119
Table 3.5
Standard deviation estimates
δ
σ1
ϕ1
λ1 σ1
σ2
ϕ2
λ2 σ2
1986–19982f non-id.2f ident.2f ident.2
0.000090.000010.00001
0.000010.000010.00001
0.001420.000000.00000
0.006480.000000.00000
0.000030.000710.00071
0.001000.001000.00100
0.000840.000050.00005
1972–19982f non-id.2f ident.2f ident.2
0.000010.000000.00000
0.000010.000000.00000
0.000000.000000.00000
0.000000.000000.00000
0.002490.000000.00000
0.000840.000000.00000
0.000130.000000.00000
σ1π
ϕ1π
λ1π σ1π
λ2π σ1π
B1π
B2π
σ3
ϕ3
λ3 σ3
1986–19982f non-id.––––––2f ident.––––––2f ident.2 0.12343 0.00154 0.00653 0.00000 0.00000 0.00004
–––
–––
–––
1972–19982f non-id.––––––2f ident.––––––2f ident.2 0.00000 0.00000 0.00000 0.00000 0.00864 0.00025
–––
–––
–––
ﬁtting quality. Consequently, the factor analysis and the identiﬁcation are focused on thetwo-factor model.In what concerns to the factor loadings (Figure 3.15a and b), the ﬁrst factor – lesspersistent and volatile – is relatively more important for the short end of the curve, whilethe second assumes that role in the long end of the curve.51The results already presented illustrate two ways of assessing the identiﬁcation problem.A third one, as previously mentioned, is to analyse the correlation between the factorsand the variables with which they are supposed to be related – the real interest rate andthe expected inﬂation – including the analysis of the leading indicator properties of thesecond factor concerning the inﬂation rate.The comparison between the unobservable factors, on the one hand, and a proxy for thereal interest rate and the inﬂation rate, on the other hand, shows (in Figure 3.16a–d) thatthe correlation between the factors and the corresponding economic variables is high. Infact, using ex-post real interest rates as proxies for the ex-ante real rates, the correlationcoefﬁcients between one-month and three-month real rates, on the one side, and the ﬁrstfactor, on the other side, are around 0.75 and 0.6 respectively in the shorter and in thelonger sample.52 Using the identiﬁed models, that correlation coefﬁcient decreases to 0.6and 0.7 in the shorter sample, respectively using the models in (3.55) and (3.56), while51
Excluding the identiﬁed two-factor models for the shorter sample.This ﬁnding contrasts with the results in Gerlach (1995) for Germany, between January 1967 and January1995. However, it is in line with the conclusion of Mishkin (1990b) that US short rates have informationcontent regarding real interest rates.52
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0.80.60.40.20.00
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108
120
Term to maturity (in months)F1 non-id.
F2 non-id.
F1 ident.
F2 ident.
F1 ident.2
F2 ident.2
(a)1.21.0
%
0.80.60.40.20.00
12
24
36
48
60
72
84
96
Term to maturity (in months)F1 non-id.
F2 non-id.
F1 ident.
F2 ident.
F1 ident.2
F2 ident.2
(b)
Figure 3.15
Factor loadings in the two-factor models: (a) 1986–1998 and (b) 1972–1998
in the larger sample it increases to 0.75 using the model in (3.55) and slightly decreasesto 0.56 with the model in (3.56).The correlation coefﬁcients between the second factor and the inﬂation are close to 0.6and 0.7 in the same samples. When the identiﬁed model in (3.55) is used, the second factorbasically reproduces the inﬂ