Advances in Risk Management

RAYMOND THÉORET, PIERRE ROSTAN AND ABDELJALIL EL-MOUSSADEK 101

Maturity

Yield

0510

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

15 20 25 30

Evolved 2 Sig

Realized

Evolved 1 Sig

EKF 1 Sig

EKF 2 Sig

Figure 5.3 Interest-rate term structures forecasted versus realized on

23 October 2002

by computing the RMSE:

RMSE=

√

√

√√ 1

n

∑n

i= 1

(Forecasted yield−Realized yield)^2

Figure 5.4 shows the term structures of the RMSE obtained from different
methods. We observe that the EKF approach with Bollinger bands and anti-
thetic variable with±1 sigma performs best. Followed by the same approach
except that the volatility has been estimated by GARCH (1,1). The evolved
approach with±2 sigma whatever the method of volatility estimation (EKF
or GARCH) performs better than the naïve approach.^17
In addition, we observe that the RMSE term structures are downward-
sloping. Oneofthepossiblereasonsisthatthismethodprobablydoesnotuse
all the information about the factor values contained in the cross-sectional
dimension.
In order to measure the exact contribution of introducing the Bollinger
bands to the F&V and EKF models, we perform theF-ratio test on
the RMSE.
H 0 is rejected in both tests since Fstat>F(241,240,.05).We conclude that the
differences in RMSE are significant for the two levels of Sigma (One-sigma
and Two-sigma) used in the Bollinger bands technique compare to F&V
coupled to EKFwithoutBollinger bands. This result suggests that associating