In this chapter we have discussed an early part of our PhD study which was presented in the 18th International AFIR Colloquium in Rome (2008) and submitted to the Annals of Actuarial Science (February 2010) as a joint paper. We have revisited the Wilkie investment model by re-estimating the parameters on updated data to 2009. We have considered models for retail prices, including an ARCH model, wage inflation, share dividend yields, share dividends and prices, long-term interest rates, short-term interest rates and index-linked bond yields. We have also recursively estimated the parameters on incrementally larger data sets and displayed those recursive estimates using graphical representation in order to analyse their stability.
The updated parameters of the retail prices model have not changed significantly. Because of low and stable inflation during last 15 years, the mean level of inflation
QM U and the standard deviation QSD have decreased slightly. The model still does
not satisfy the normality assumption and especially the two parameters QM U and
QSD are not stable over time.
Although the ARCH model satisfies the normality assumption for the 1923-1994 data, its performance gets worse on the updated data and the residuals are not normally distributed any more. The parameters have not changed significantly. It has been seen that the suggested ARCH model is a useful description for the periods that include the 1960s and 1970s.
The parameters in the wages models have not changed significantly for the updated
data. The parameter estimates over different sub-periods are quite stable exceptW SD.
The share dividend yield model is still satisfactory and the parameters are relatively stable over time.
The performance of the share dividend model is almost the same but its parameters
intervals are highly unstable and change greatly as the sub-periods change.
We modified the long-term interest rate model to apply it to the updated data
by introducing a fixed parameter called CM IN which is equal to 0.5%. In order
to avoid negative real interest rates, we used the modified model for 1923-2009 to
estimate the parameters. The value of CM U decreased, and CY and CSD increased
significantly in the model with updated data. The residuals of the modified model are not normally distributed according to the Jarque-Bera test statistic, and except for
CY, the parameters are not stable either.
The short-term interest rates model is the best model among them all. It satisfies all the diagnostic tests and fits the data better over the interval 1923-2009. Moreover, its parameters are quite stable.
We have also re-estimated the parameters for the index-linked bond yields for the period 1923-2009 but could not study the stability of the parameters due to lack of data.
Finally, we have presented the comments on the Wilkie model and discussed some Wilkie type stochastic investment models briefly.
Chapter 2
A Descriptive Yield Curve Model
for the UK Term Structures: The
Cairns Model
2.1
Introduction
Descriptive model can be defined as a model which takes a snapshot of the bond market as it is today. The aim is to get a good description of todays prices: that is, of the rates of interest which are implicit in todays prices (Cairns, 2004b).
A descriptive model, on its own, gives no indication of how the term structure might change in the future. It is known that there is randomness in the future but this sort of model does not describe this feature.
Cairns (2004b) summarizes the number of uses descriptive models have as below:
• They can be used to assess which bonds are over- or under-priced (so called
cheap/dear analysis)
• They give a broad picture of market rates of interest which are implied by market prices.
• They can assist in the analysis of monetary policy.
• They can be used in the construction of yield indices.
• Finally descriptive models provide sufficient information to get a precise mar- ket value of a non-profit insurance portfolio or to price, for example, annuity contracts.
Van Wijck (2006) discusses the methodology and the applications of the descriptive yield curve models in details.
In this chapter, we discuss the Cairns model as a descriptive parametric model to fit the daily nominal spot rates (January 1979-December 2009), implied inflation spot rates (January 1985-December 2009) and real spot rates (January 1985-December 2009) published on the Bank of England’s web page by changing the fixed parameters (exponential rates in the model) to find the best set of parameters for each data set. We try three fixed parameter sets which have been suggested by Cairns (1998) and Cairns and Pritchard (2001) and then we use the least squares method with a penalty function to find the optimized set of parameters for each set of yield curve data. We compare the root mean squared errors obtained by using the four parameter sets for each yield curve to decide which set of parameters fit each yield curve data best. Once
we decide these exponential rate parameters (C parameter sets), we analytically solve
the equations in Cairns model as described in the following sections and fit these four different models to the data. We estimate the remaining time dependent parameters
(b parameters) and find the fitted values for each day. We compare these models by
examining the root mean squared errors, fitted values for some specific dates and fitted values for short, medium and long term maturities for each yield curve to choose the
best set of C parameters. The overall aim of this chapter is to fill in the gaps in the
nominal, implied inflation and real yield curve data provided by the Bank of England by fitting the Cairns model.
Section 2.2 introduces the yield curve terminology by giving some basic definitions and the data and the methodology used by the Bank of England to construct UK yield curves. Section 2.3 presents the Cairns model and the least squares method
used, including the penalty function, to estimate the optimised C parameter set. In Section 2.4, we explore the data by looking at some descriptive statistics and estimate
the time dependentb parameters for eachC parameter set. We also discuss the nature
of the b parameters by considering the simultaneous correlations between them. We
plot and interpret the standard errors (root mean squared errors) of the residuals and also the ratios of these errors to decide the best fit in terms of the least squares method in Section 2.5. We compare how well each model fits some specific dates in Section 2.6. Similarly, we examine the short term, medium term and long term fit of the models by considering particular maturities in Section 2.7. Finally, Section 2.8 summarizes this chapter.