In this section the Kalman filter algorithm described in Section (1.1.3) is applied to the Canadian zero-coupon data for the period 1991 to 2018 using spot-rate data for only the third day of the month. We obtain our interest rates using real world data as discussed earlier. Tables 3.3 to 3.5 present estimates for the one factor, two factor and, three factor models respectively. The standard deviations for these parameters are presented in brackets
in the tables.
κ, the mean reversion speed, characterizes the mean half-life for the model. The mean half-life represents the expected time for the process to return halfway to its long term mean, and is computed as half life = ln(2)/κ. The mean half-life for the one factor model is 1732.87yrs. For the two factor model it is 990.2yrs for the first state variable and 34.5yrs for the second state variable. The mean half-life for the second state variable is lower than that of the first. For the three factor model the mean half-lives are 1386.3yrs, 33yrs and 10yrs for the first, second and third state variables respectively. The mean half-life decreases with increasing number of factors. The one factor model has the biggest mean half-life and this indicates weaker mean reversion. According to Chen and Scott [1993], mean reversion is useful for capturing underlying trends. Slow mean reversion in particular is useful for predicting long-term trends such as long-term bond prices. These trends tend to be sensitive to the κ and λ parameters.
Table 3.3 Estimates From The One Factor Model
Parameter Estimate κ 0.0004 Log L= 29776.8 (2.97e−5) θ 0.8027 BIC= -59511.42 (4.66e−5) σ 0.0014 (9.7e−5) λ -0.0001 (2.04e−5)
Table 3.4 Estimates From The Two Factor Model
Parameter State State
Variable 1 Variable 2 κ 0.0007 0.0201 Log L= 33741.2 (4.201e−4) (4.5e−5) θ 0.6031 0.1002 BIC= -67384.48 (1.807e−1) (1.81e−1) σ 0.0045 0.0016 (1.19e−4) (2.72e−5) λ -0.0001 -0.0001 (4.85e−2) (2.45e−2)
and the BIC. The log likelihoods for the one-factor model, two-factor model and three-factor model are 29776.8, 33741.2 and 35074.59 respectively. The log likelihood increases when increasing number of factors with that of the three factor being the highest. The BICs for the one-factor, two-factor and three-factor models are -59511.42, -67384.48 and -70017.19 respectively. The BIC decreases by 12% from the one factor model to the two factor model and by 3.8% from the two factor to the three factor model. The results from the log likelihood and the BIC indicate that the three-factor model provides the best fit for the term structure. The standard deviations of the measurement errors shown in Appendix A also show that that the results from the three-factor model are more precise than those from the one-factor and two-factor model. The measurement errors decrease with increasing number of factors for most of the bonds.
Using the estimated parameters we computed the estimated model spot rates and com- pare these to the actual (observed) spot rates. The unobservable factors used for the esti- mation were obtained from the filtered estimates based on information available at time t. Seven days were selected out of the sample. Table 4.6 shows the root mean square errors
Table 3.5 Estimates From The Three Factor Model
Parameter State State State
Variable 1 Variable 2 Variable 3
κ 0.0005 0.0209 0.0649 Log L= 35074.59
θ 0.0649 0.1697 0.7280 BIC=-70017.19
σ 0.00152 0.00706 0.00786
λ -0.0001 -0.0001 0.1315
for the one-factor, two-factor and three-factor models for the days selected and Figures 3.6 to 3.12 show the plots of the actual spot rates compared to the estimated spot rates. In Table 4.6 we observe that the sum of squared errors decreases with an increasing number of parameters for all the days selected. From Figures 3.6 to 3.12 we observe that the one-factor model performs poorly for all the days selected. It fits the data poorly at the extremes of the yield curve and only passes through a few points in the middle of the curve. The two-factor and three-factor models are very close in fit for most of the days except for 2018- 10-03 and 2002-06-03 when the three-factor model outperforms the two-factor model. The three-factor model in that case is able to capture the curvature of the yield curve and this is consistent with the results obtained by Litterman and Scheinkman [1991]–i.e. that the first factor captures the level of the curve, the second factor the steepness, and the third factor the curvature. The average spot rates for each bond for the one-factor, two-factor and three-factor models are shown in Figure 3.5. In this plot we observe that the two-factor and three-factor models perform similarly with both fitting the short end of the curve much better than the long end.
Table 3.6 RMSE for The Selected Days (Bps)
1997-03-03 2002-06-03 2004-03-03 2007-08-03 2013-09-03 2016-05-03 2018-10-03
One Factor 0.884 0.403 0.527 0.799 0.174 0.133 0.598
Two Factor 0.245 0.392 0.121 0.047 0.087 0.272 0.212
Three Factor 0.059 0.055 0.139 0.035 0.092 0.094 0.073
In Figures 3.13 and 3.14 we plotted the estimated time series for the one-factor, two-factor and three-factor models over the sample period and compared them to the actual rates for the 3-month spot rate representing the short end and the 30-year rate representing the long end. The fit of the estimated curve to the data gets better with increasing number of factors for both the short end rate and the long end rate. The three-factor model outperforms the two-factor model and the one-factor model for the short end of the yield curve.
In all the cases presented above the three factor model fits the general shape of the yield as well as the slope. This indicates that increasing the number of factors increases the abil- ity of the model to capture the shape of the yield curve. Also three factors are sufficient to capture the level, the slope and curvature of the yield curve.