cant improvement over that obtained by hedging using an equally weighted combination

of assets representing the market portfolio. These results suggest that the cointegration

approach to hedging asset-speci¬c risk deserves to be taken seriously.

Our study also provides some evidence to support the possibility of creating statistical

arbitrage strategies based on cointegration effects within this set of equities. Evidence for

the presence of mean-reverting components in the asset price dynamics is re¬‚ected in the

downward-sloping variance ratio pro¬les shown in Figure 2.20.

Both the mean and median pro¬les show a signi¬cant decline in normalised vari-

ance, with the 30-day variance just under 70% of that which would be expected from

66 Applied Quantitative Methods for Trading and Investment

1.2

1

0.8

0.6

Mean Median

0.4

0.2

0

1 6 11 16 21 26

Figure 2.20 Average (mean and median) variance ratio pro¬les for the residual dynamics,

calculated across the full set of 50 equities

extrapolating the 1-day variance. This evidence in itself is clearly not conclusive, especially

considering the non-synchronous nature of our data sampling due to the different closing

times in the various markets. However, this result, together with previous studies (e.g.

Burgess, 1999) seems to strongly suggest that reverting components exist in the asset-

speci¬c components of price volatility which may in principle be open to exploitation

using suitable statistical arbitrage strategies.

2.7 DISCUSSION AND CONCLUSIONS

The major objective of our methodology is to construct combinations of time series which

are either decorrelated with major sources of economic risk and/or contain a deterministic

(and hence potentially predictable) component in the dynamics.

Given a particular asset there are two main methods for constructing combinations of

one or more assets with similar net exposures to market-wide risk factors. The cointe-

gration tools described in this chapter represent an “implicit” approach to factor mod-

elling in which neither the factors nor the sensitivities are estimated explicitly, but are

instead addressed from the perspective of the impact which they cause on sets of related

asset prices.

The standard approach to estimating explicit factor models is to postulate a set of

¬nancial and economic variables as the risk factors and to use regression-based techniques

to estimate the sensitivity coef¬cients βi,j (e.g. the sensitivity of a particular stock to, say,

long-term interest rates). Evidence for mean-reverting dynamics in the residuals of factor

models was reported by Jacobs and Levy (1988). However, the task of constructing such

Cointegration to Hedge and Trade International Equities 67

models is a dif¬cult problem in itself, and the interested reader is directed to a recent

extensive study by Bentz (1999).

A second approach which is sometimes referred to as “statistical” factor modelling

is to perform multivariate analysis of asset price returns and reconstruct the unobserved

risk factors as linear combinations of observed asset returns. The technique of “principal

components analysis” (PCA) is a natural tool in this case as it generates sequences of

linear combinations (factors) which account for the greatest possible amount of the total

variance, subject to the constraint of being orthogonal to the previous factors in the

sequence. Some examples of applications of PCA to statistical arbitrage modelling are

described in Burgess (1996), Schreiner (1998), Towers (1998), Tjangdjaja et al. (1998)

and Towers and Burgess (1998).

Still within the area of statistical factor modelling, a generalisation of PCA which

has recently attracted much attention in engineering disciplines is so-called “independent

components analysis” (ICA), which is based upon algorithms developed for the blind

separation of signals (Bell and Sejnowski, 1995; Amari et al., 1996). In ICA the orthog-

onality condition of PCA is strengthened to one of complete statistical independence,

thus taking into account higher moments. Whilst ICA techniques offer exciting poten-

tial in computational ¬nance, particularly in that they account not only for expectations

(“returns”) but also for variances (“risk”), applications in this domain are still rare (see

Moody and Wu (1997), Back and Weigend (1998)).

The explicit and statistical factor modelling approaches are those which are currently

most used in practice, with explicit factor models typically used for hedging and risk

control and statistical factor models typically used from a statistical arbitrage perspec-

tive. However we believe that the implicit factor modelling offered by the cointegration

approach presents a viable alternative to these more established techniques, both for hedg-

ing and for trading. By dealing with asset prices directly, cointegration modelling presents

both technical and conceptual advantages over the other methods. The technical motiva-

tion for the cointegration approach is that it avoids the need for explicit estimation of risk

factors and factor sensitivities, thus eliminating a potential source of estimation error from

the modelling process as a whole. The conceptual bene¬t of the cointegration approach

is that it can be viewed as a sophisticated version of “relative value” analysis. The syn-

thetic assets constructed by the cointegration procedure can be viewed from a hedging

perspective as “tracking baskets” and from a statistical arbitrage perspective as “synthetic

pairs” and thus are consistent with the way in which many traders naturally consider

relationships between sets of assets. The cointegration approach also has its weaknesses,

for instance there are many different methods for constructing cointegration relationships

and it is not clear which works best in practice. Furthermore, in some cases it is a positive

bene¬t to explicitly model market-wide risk factors, particularly if they are felt to contain

predictable components. Thus, whilst there are both pros and cons to the cointegration

approach, we feel that in a practical discipline such as investment ¬nance it is important

to take a “broad church” view and apply whichever set of tools works best in a given

situation. The aim of this chapter has been to provide a demonstration of and a motivation

for the relatively unused cointegration approach, in order to add another set of tools to

the ¬nancial toolbox.

In conclusion, this chapter has examined the application of the econometric concept

of cointegration as a tool for hedging and trading international equities. Section 2.2

introduced cointegration within the perspective of different frameworks for analysing

68 Applied Quantitative Methods for Trading and Investment

and modelling ¬nancial time series. Sections 2.3 and 2.4 described two complementary

perspectives in which cointegration is viewed as a basis for implicitly hedging unknown

risk factors and also as a basis for suggesting possible opportunities for statistical arbi-

trage. Section 2.5 provided a controlled simulation in which the cointegration approach

was demonstrated upon arti¬cial time series with known properties. Finally, the spread-

sheet analysis in Section 2.6 demonstrates an application of these tools to a real-world

problem. The data used for this demonstration consisted of the daily closing prices of

the 50 equities which constituted the STOXX 50 index as of 4 July 2002, analysed over

a period from 14 September 1998 to 3 July 2002. We have noted that the use of daily

closing prices will introduce some spurious effects due to the non-synchronous closing

times of the markets on which these equities trade, so the speci¬c results themselves can

only be taken as indicative of a more realistic study. This caveat admitted, the results

nevertheless serve both to illustrate the use of the tools and also to suggest the potential

bene¬ts which may be gained from their intelligent application to the tasks of hedging

and trading international equities.

REFERENCES

Amari, S., A. Cichocki and H. H. Yang (1996), “A New Learning Algorithm for Blind Signal

Separation”, in D. S. Touretzky et al. (eds), Advances in Neural Information Processing Systems

8, MIT Press, Cambridge, MA, pp. 757“763.

Back, A. D. and A. S. Weigend (1998), “Discovering Structure in Finance using Independent Com-

ponent Analysis”, in A-P. N. Refenes et al. (eds), Decision Technologies for Computational

Finance, Kluwer Academic Publishers, Dordrecht, pp. 309“322.

Bell, A. and T. Sejnowski (1995), “An Information-maximisation Approach to Blind Separation

and Blind Deconvolution”, Neural Computation, 7(6), 1129“1159.

Bentz, Y. (1999), Identifying and Modelling Conditional Factor Sensitivities: An Application to

Equity Investment Management, unpublished PhD thesis, London Business School.

Burgess, A. N. (1996), “Statistical Yield Curve Arbitrage in Eurodollar Futures using Neural Net-

works”, in A-P. N. Refenes et al. (eds), Neural Networks in Financial Engineering, World Scien-

ti¬c, Singapore, pp. 98“110.

Burgess, A. N. (1999), A Computational Methodology for Modelling the Dynamics of Statistical

Arbitrage, unpublished PhD thesis, London Business School. (http://cocreativity.net/papers.html).

Cochrane, J. H. (1988), “How Big is the Random Walk in GNP?”, Journal of Political Economy,

96, 5, 893“920.

Engle, R. F. and C. W. J. Granger (1987), “Cointegration and Error-correction: Representation,

Estimation and Testing”, Econometrica, 55, 251“276.

Granger, C. W. J. (1983), “Cointegrated Variables and Error-correcting Models”, UCSD Discussion

Paper.

Hoerl, A. E. and R. W. Kennard (1970a), “Ridge Regression: Biased Estimation for Nonorthogonal

Problems”, Technometrics, 12, 55“67.

Hoerl, A. E. and R. W. Kennard (1970b), “Ridge Regression: Application to Nonorthogonal Prob-

lems”, Technometrics, 12, 69“82.

Hull, J. C. (1993), Options, Futures and Other Derivative Securities, Prentice Hall, Englewood

Cliffs, NJ.

Jacobs, B. I. and K. N. Levy (1988), “Disentangling Equity Return Regularities: New Insights and

Investment Opportunities”, Financial Analysts Journal, May“June 1988, 18“43.

Lo, A. W. and A. C. MacKinlay (1988), “Stock Market Prices Do Not Follow Random Walks:

Evidence from a Simple Speci¬cation Test”, The Review of Financial Studies, 1988, 1, 1, 41“66.

Moody, J. E. and L. Wu (1997), “What is the ˜True Price™? “ State Space Models for High-

frequency FX Data”, in A. S. Weigend et al. (eds), Decision Technologies for Financial En-

gineering, World Scienti¬c, Singapore, pp. 346“358.

Cointegration to Hedge and Trade International Equities 69

Ross, S. A. (1976), “The Arbitrage Pricing Theory of Capital Asset Pricing”, Journal of Economic

Theory, 13, 341“360.

Schreiner, P. (1998), Statistical Arbitrage in Euromark Futures using Intraday Data, unpublished

MSc thesis, Department of Mathematics, King™s College, London.

Sharpe, W. F. (1964), “Capital Asset Prices: A Theory of Market Equilibrium”, Journal of Finance,

19, 425“442.

Tjangdjaja, J., P. Lajbcygier and N. Burgess (1998), “Statistical Arbitrage Using Principal Com-

ponent Analysis For Term Structure of Interest Rates”, in L. Xu et al. (eds), Intelligent Data

Engineering and Learning, Springer-Verlag, Singapore, pp. 43“53.

Towers, N. (1998), Statistical Fixed Income Arbitrage, Deliverable Report D4.6, ESPRIT project

“High performance Arbitrage detection and Trading” (HAT), Decision Technology Centre,

London Business School.

Towers, N. (2000), Decision Technologies for Trading Predictability in Financial Markets, unpub-

lished PhD thesis, London Business School.

Towers, N. and A. N. Burgess (1998), “Optimisation of Trading Strategies using Parametrised

Decision Rules”, in L. Xu et al. (eds), Intelligent Data Engineering and Learning, Springer-

Verlag, Singapore, pp. 163“170.

3

Modelling the Term Structure of Interest

Rates: An Application of Gaussian Af¬ne

Models to the German Yield Curve—

NUNO CASSOLA AND JORGE BARROS LU´S

I

ABSTRACT

This chapter shows that a two-factor constant volatility model describes quite well the

time series and the cross sectional behaviour of the German yield curve between 1972

and 1998. The empirical analysis supports the expectations theory with constant term

premiums. Thus, an average term premium structure can be calculated and short-term

interest rate expectations can be derived from the adjusted forward rate curve. The non-

observable factors that explain the German yield curve are extracted using a Kalman

¬lter technique. Following the conjecture that these factors capture, respectively, the

expected short-term real interest rate and the expected in¬‚ation rate, alternative methods

of identifying one of the factors with the in¬‚ation process are discussed. The ¬rst factor

(real interest rate) is more important for explaining interest rate movements at the short-

end of the yield curve. The second factor (in¬‚ation) carries greater weight in explaining

the longer end of the yield curve. These ¬ndings are of interest to risk managers for

analysing the shape of the yield curve under different scenarios and to policy makers

for assessing the impact of ¬scal and monetary policies. The Matlab codes necessary to

reproduce the results are presented in great detail.

3.1 INTRODUCTION

The identi¬cation of the factors that determine the time-series and cross-section behaviour

of the term structure of interest rates is a recurrent topic in the ¬nance literature. Even

though the interest rates for different maturities typically exhibit high correlations, these

are far from being perfectly correlated. The main conclusion from the literature is that

the yield curve is determined by different factors, usually described by its level, slope

and curvature. Nevertheless, the identi¬cation of the yield curve factors is a controversial

subject that has several empirical and practical implications, namely for the management

of portfolios of ¬xed income securities, in¬‚uencing the investment and hedging strategies

within the context of portfolios of ¬xed income securities.

—

This chapter is based on Cassola and Lu´s (2003) and on the PhD thesis of the second author (Lu´s, 2001).

± ±

Applied Quantitative Methods for Trading and Investment. Edited by C.L. Dunis, J. Laws and P. Na¨m

±

™ 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5

72 Applied Quantitative Methods for Trading and Investment

Moreover, a better knowledge of these factors would facilitate the task of risk managers,

in particular for Value-at-Risk (VaR) purposes. In fact by simulating different paths for

the factors the behaviour of the yield curve under different scenarios could be traced and

alternative empirical distributions for interest rates could be retrieved (see e.g. Bolder

(2001)). The analysis of the determining factors of the yield curve is also relevant for

policy makers, namely in assessing the impact of monetary and ¬scal policies (see, for

instance, Fleming and Remolona (1998) and Bliss (1997)).

In this chapter, the German yield curve is analysed, given its relevance in the inter-

national bond markets and its informational content about future macroeconomic devel-

opments in one of the major European Union economies. Two databases are used in this

chapter, the ¬les “datgerse.txt” and “datgersb.txt”, which are both included on the CD-

Rom. The ¬rst database comprises monthly averages of nine daily spot rates for maturities

of 1 and 3 months and 1, 2, 3, 4, 5, 7 and 10 years, between January 1986 and December

1998. The spot rates were estimated using the Nelson and Siegel (1987) and Svensson

(1994) smoothing techniques from raw market data on euro“Deutschemark short-term

interest rates and par yields of German government bonds.1 The smoothed estimates were

used to construct one-period forward rates and perform forward rate regressions.2

As illustrated in Figure 3.1, the German end-month interest rates ¬‚uctuated markedly in

the period from January 1986 to December 1998 (between 3% and 10%). These interest

rate moves corresponded to signi¬cant shifts in the shape of the yield curve (Figure 3.2).

In fact, during this period the yield curve moved from a ¬‚at pattern to a positively sloped

curve. The sharp deceleration of the German economic growth between 1992 and 1993 was

accompanied by a negatively sloped yield curve. Finally, at the end of the sample period,

the yield curve returned to an almost null slope, achieving the lowest levels in the sample.

14

12

10

8

%

6

4

2

0

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1m 3m 1Y 2Y 3Y

4Y 5Y 7Y 10Y

Figure 3.1 German yield curve: 1986“1998

1

A TSP 4.3 routine was written for the estimation of spot rates. A likelihood ratio test was adopted for

choosing between the Nelson and Siegel (1987) and the Svensson (1994) methods, in each sample day, as

described in Cassola and Lu´s (1996).

±

2

On this issue the research assistance of F´ tima Silva, then af¬liated with the Research Department of the

a

Banco de Portugal, is acknowledged.

Modelling the Term Structure of Interest Rates 73

12

10

8

%

6

4

2

0

0

1

2

3

4

5

6

7

8

9

10

Term to Maturity (Years)

Jan - 86 May-88 Aug- 92 Jul - 96 Dec -98

Figure 3.2 German yield curves

The second database for Germany covers a longer period, between September 1972

and December 1998.3 However this sample includes only spot rates for annual maturities

between 1 and 10 years, excluding the 9-year maturity (Figure 3.3). For the overlapping

period, interest rates in the two data sets are very similar for equivalent maturities, which

suggests robustness of market data and smoothing techniques.

The analysis assumes that bond yields are functions of several macroeconomic and

¬nancial variables, observable or latent. In af¬ne models, parameters are linear in both

the maturity of the assets and the factors, which makes these models easier to implement.4

Additionally, compared to principal component analysis, af¬ne models have the advantage

of allowing for correlation among the factors, providing several outputs that are relevant

for ¬nancial market participants and policy makers, such as forward rates and the structure

of term premia.

Af¬ne models are the result of four decades of evolution in asset pricing theory. In

the context of yield curve modelling these can be seen as developments of the one-factor

models by Vasicek (1977) and Cox et al. (1985a), where the short-term interest rate was

the single factor.5 Multifactor models were developed because of the discrepancy between

the yield curve implied by the theory and the observed time-series properties of bond

yields, namely the fact that the observed curves are substantially more concave than

implied by the theory (see, for instance, Backus et al. (1998)).6

3

We are grateful to Manfred Kremer, then af¬liated with the Research Department of the Bundesbank, for

providing the data.

4

See Campbell et al. (1997, chap. 11) or Backus et al. (1998) for graduate textbook presentations of af¬ne

models.

5

These models differ basically due to the fact that the latter allowed for the volatility of the short-term interest

rate to be stochastic. In order to incorporate the supply side of ¬nancial assets, Cox et al. (1985b) derived a

general equilibrium model that yields a closed-form expression for asset prices.

6

One of the ¬rst multifactor models was developed by Ross (1976), with the Arbitrage Pricing Theory (APT).

Contrary to the consumption CAPM (CCAPM) developed by Breeden (1979), the APT does not try to identify

the factors with the consumption marginal rate of substitution.

74 Applied Quantitative Methods for Trading and Investment

14

12

10

8

%

6

4

2

0

1972

1973

1974

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1Y 2Y 3Y 4Y 5Y

6Y 7Y 8Y 10Y

Figure 3.3 German yield curve: 1986“1998

Most papers on af¬ne models have focused on the US term structure. The pronounced

hump-shape of the US yield curve and the empirical work pioneered by Litterman and

Scheinkman (1991) have led to the conclusion that three factors are required to explain

the movements of the whole term structure of interest rates. These factors are usually

identi¬ed as the level, the slope and the curvature of the term structure. Most studies

have concluded that the level is the most important factor in explaining interest rate

variations over time.

Moreover, given the apparent stochastic properties of the volatility of interest rates,

Gaussian or constant volatility models are often rejected. Therefore, several papers have

used three-factor models with stochastic volatility in order to ¬t the term structure of

interest rates (see, for instance, Balduzzi et al. (1996) and Gong and Remolona (1997a)).

However, stochastic volatility models pose admissibility problems, as the factors deter-

mining the volatility of interest rates enter “square rooted” and thus must be positive.

In addition, the parameters of a three-factor model with stochastic volatility are often

very dif¬cult to estimate. In fact, frequently small deviations of the parameters from the

estimated values generate widely different and implausible term structures.

Alternatively a constant volatility or Gaussian model can be ¬tted. In the German case,

Figure 3.1 suggests that this may be a reasonable choice, though in this chart end-month

data is presented. These models overcome the empirical problems posed by stochastic

volatility models and are also capable of reproducing a wide variety of shapes of the

yield curve, though they face some shortcomings regarding the limiting properties of the

instantaneous forward rate.7

Furthermore, some term structures may have properties identi¬able with a smaller

number of factors. For instance, according to Buhler et al. (1999), principal component

7

See, e.g., Campbell et al. (1997), p. 433, on the limitations of a one-factor homoskedastic model.

Modelling the Term Structure of Interest Rates 75

analysis reveals that two factors explain more than 95% of the variation in the German

term structure of interest rates consistently from 1970 up to 1999. Consequently, a decision

on the number of factors is also required.

The ¬‚exibility of af¬ne models allows for considering observable as well as non-

observable factors, such as macroeconomic variables. The advantage of estimating these

models using only latent or non-observable factors is that it avoids making ex-ante restric-

tions on the behaviour of the factors determining the yield curve. In this chapter, we will

start by incorporating only latent factors.

In order to ¬t af¬ne models to a cross-section/time-series of interest rates three practical

questions have to be answered: (i) how many factors should be considered; (ii) what are the

properties of the factors; and (iii) how can they be identi¬ed. A sensible way to proceed

about the ¬rst question is by performing forward rate regressions in line with Backus

et al. (1997), to assess the adequacy of Gaussian models.

Regarding the second question, two- and three-factor models were estimated.8 Given

that the two-factor Gaussian model can be speci¬ed in such a way that the short-term

interest rate is equal to the sum of a constant with the two latent factors, the usual

conjecture is that these factors re¬‚ect the real interest rate and the expected in¬‚ation

rate.9 A third factor may be included re¬‚ecting potential international in¬‚uences on the

domestic yield curve. Considering the fact that the two- and three-factor models are nested,

a chi-square test can be used to select the best model.

Regarding the third issue, the link between the two factors and observable variables is

assessed in three different ways: the ¬rst two are based on the explicit identi¬cation of

one of the factors with the in¬‚ation rate and relating the other factor to the ex-ante real

interest rate, in line with Zin (1997). Within this framework, a ¬rst exercise consisted

in identifying one factor with the in¬‚ation rate process, modelled as an AR(1) process,

following Fung et al. (1999). The second factor was left unconstrained and assumed to

re¬‚ect the “real” determinants of the term structure of interest rates, such as the output

gap or the real interest rate.

A second exercise was performed based on the assumption that the in¬‚ation is given

by two factors, one of them being a common factor with the term structure. The joint

factor can be taken as a proxy for core in¬‚ation. It can be shown that this assumption

provides the general case of a joint model developed in Fung et al. (1999), where the

factor loadings are not restricted to the values of the parameters of the interest rate

loadings. Consequently, the second factor of the term structure is left unconstrained and

should re¬‚ect the “real” determinants of the term structure of interest rates, such as the

output gap or the real interest rate, while the second factor of in¬‚ation should re¬‚ect

non-core in¬‚ation movements.10 The third way chosen to analyse the identi¬cation issue

8

The three-factor model was run only for the shorter database, given that the longer database does not include

information on the money market rates. Therefore, it is reasonable to assume that two factors are enough to

explain the behaviour of the yield curve in the range of maturities included in that database.

9

This corresponds to the Fisher hypothesis. The two-factor model is also consistent with the idea that the

Bundesbank followed a type of Taylor rule in setting of¬cial interest rates as recently documented by Clarida

et al. (1998) and Clarida and Gertler (1997). In accordance with such a rule the short-term interest rate was

adjusted in response to the deviation of in¬‚ation and output from their targets.

10

In Remolona et al. (1998), in¬‚ation-indexed bonds issued by the UK government are used for this purpose.

However, these securities do not exist in Germany. Consequently, the analysis in this chapter is conducted on

a nominal basis, following Campbell et al. (1997). The best-known examples of in¬‚ation-linked securities are

the in¬‚ation-indexed government bonds that exist only in a few countries, namely in the UK, the USA and

France (see, for instance, Deacon and Derry (1994) and Gong and Remolona (1997b)).

76 Applied Quantitative Methods for Trading and Investment

is based on econometric evidence about the leading indicator properties of one factor for

in¬‚ation developments in Germany.

The latent factors are estimated by a Kalman ¬lter, while a maximum likelihood pro-

cedure is used to estimate the time-constant parameters of the yield curve, following

the pioneering work by Chen and Scott (1993a,b).11 This econometric technique allows

a wide range of model speci¬cations (including observable and non-observable factors),

which compute the optimal estimate for the state variables at a given moment using all the

past information available. In Figure 3.4 the reader ¬nds a brief summary of the exercise

performed in this chapter.

Two Matlab routines, supplied on the accompanying CD-Rom, were used for each of

the three model speci¬cations, one for the Kalman ¬lter and the other for the maximum

likelihood estimation. The leading properties of the factor were assessed using a RATS 4.0

standard program.12 The results obtained in this chapter illustrate that both two- and three-

factor models ¬t quite well the yield and the volatility curves, also providing reasonable

estimates for the one-period forward and term premium curves.13

Assessment of the properties of the

factors' volatility

Forward regressions

(equation (3.34))

Estimation of the two-and three-factor

models

Kalman filter

(equations (3.52) to (3.54))

Choice between two-and three-factor

models

Chi-square test

Identification of the two-factor model

with macroeconomic variables

Leading indicator

One-and two-factor models

properties for inflation

for inflation (equations(3.55) and (3.57))

Conclusions for yield curve modelling,

asset management, Value-at-Risk and

monetary policy

Figure 3.4 Yield curve af¬ne models using latent factors

11

The model is derived in discrete time, as in this way it matches the frequency of the data, allows the

identi¬cation of the factors with observable macroeconomic variables and avoids the problem of estimating a

continuous-time model with discrete-time data (see, for example, A¨t-Sahalia (1996)).

±

12

The chapter does not present the routine to assess the leading indicator properties, given that a routine adapted

from Doan (1995) was used.

13

The Matlab codes were written based upon codes made available by Mike Wickens and Eli Remolona.

Initially, the two-factor model was run only with the equations for the yields, disregarding the volatilities. The

Modelling the Term Structure of Interest Rates 77

However, two periods of poorer model performance are identi¬ed, both related to world-

wide gyrations in bond markets “ Spring 1994 and 1998 “ which were characterised by

sharp changes in long-term interest rates while short-term rates remained stable. Thus, it

seems that the third factor fails to capture the potential external in¬‚uences on German

interest rates. In addition, the two-factor model ¬ts quite well the yield and the volatility

curve, providing a good ¬t of the time series of bond yields and more reasonable estimates

for the one-period forward and term premium curves than the three-factor model.

The remainder of the chapter is structured as follows. In Section 3.2 some background

on asset pricing is presented. In Section 3.3 the theoretical framework of Duf¬e and Kan

(1996) (DK hereafter) af¬ne models is explained. In Section 3.4 a test of the expecta-

tions theory is developed that will be used to empirically motivate the Gaussian model.

In Section 3.5 we discuss alternative ways of identifying the factors in the model. The

econometric methodology is presented in Section 3.6. Section 3.7 includes the presen-

tation of the data and the results of the estimation. The main conclusions are stated in

Section 3.8.

3.2 BACKGROUND ISSUES ON ASSET PRICING

A fundamental result in modern asset pricing theory is that the price of any ¬nancial

asset corresponds to the present value of its expected future cash-¬‚ows, this present value

being obtained by applying a positive stochastic discount factor (hereafter sdf, denoted

by Mt ). If the future cash-¬‚ows correspond only to the ¬nancial asset price in the next

period we have:

Pt = Et [Pt+1 Mt+1 ] (3.1)

Another fundamental result in ¬nance theory is that asset prices and returns are related

to their risk, which is the ability of the asset to offer higher cash-¬‚ows when they are

more needed. In fact, the more an asset helps to smooth income ¬‚uctuations, the less

risky it is and the higher will be its demand for insuring against “bad times”. Employing

some simple algebra in equation (3.1), the following equation illustrates this link between

asset prices and their risk:

1

Pt = Et [Pt+1 ] + Covt [Pt+1 , Mt+1 ] (3.2)

f

1 + it+1

This result shows that the asset price is the discounted expected value of its future

payoff, adjusted by the covariance of its payoff with the sdf and where the discount

factor is the inverse of the return on a risk-free asset. The covariance term consists of a

risk factor and it is positive for assets that pay higher returns when they are more needed.

The same result may be obtained for interest rates, instead of prices:

Covt [Mt+1 , it+1 ]

f

Et [it+1 ] = it+1 ’ (3.3)

Et [Mt+1 ]

estimates obtained for the volatilities proved unreasonable. Consequently, we opted for including the volatility

equations in the Kalman ¬lter. This procedure can be considered as corresponding to estimating the model

imposing restrictions on the parameters in order to get well-behaved volatility curves.

78 Applied Quantitative Methods for Trading and Investment

According to equation (3.3), the excess return of any asset over the risk-free asset

depends on the covariance of its rate of return with the stochastic discount factor. Thus,

an asset whose payoff has a negative correlation with the stochastic discount factor pays

a risk premium. With some additional self-explanatory algebra, the following result is

obtained:

Covt [Mt+1 , it+1 ] Vart [Mt+1 ]

f f

Et [it+1 ] = it+1 + ’ = it+1 + βit+1 ,Mt+1 »t (3.4)

Et [Mt+1 ]

Vart [Mt+1 ]

In equation (3.4), βit+1 ,Mt+1 is the coef¬cient of a regression of it+1 on Mt+1 , i.e., it

measures the correlation between the asset™s return and the sdf or the quantity of risk,

while »t = ’Vart [Mt+1 ]/Et [Mt+1 ] is the market price of risk. Therefore, as in the static

CAPM, the excess return over the risk-free interest rate of any ¬nancial asset depends on

the quantity of risk and its market price.

3.3 DUFFIE“KAN AFFINE MODELS

OF THE TERM STRUCTURE

Af¬ne models are built upon a log-linear relationship between asset prices and the sdf,

on the one side, and the factors or state variables, on the other side. These models were

originally developed by Duf¬e and Kan (1996), for the term structure of interest rates.

As referred to in Balduzzi et al. (1996), “Duf¬e and Kan (1996) show that a wide range

of choices of stochastic processes for interest rate factors yield bond pricing solutions of

a form now widely called exponential-af¬ne models”.

Let us start by writing equation (3.1) in logs:

pt = log(Et [Pt+1 Mt+1 ]) (3.5)

where lowercase letters denote the logs of the corresponding uppercase letters. With the

assumption of joint log-normality of bond prices and the nominal pricing kernel and using

the statistical result that if log X ∼ N (µ, σ 2 ) then log E(X) = µ + σ 2 /2 it is obtained

from equation (3.5) that:

pt = Et [mt+1 + pt+1 ] + 1 Vart [mt+1 + pt+1 ] (3.6)

2

Duf¬e and Kan (1996) de¬ne a general class of multifactor af¬ne models of the term

structure, where the log of the pricing kernel is a linear function of several factors stT =

(s1,t , . . . , sk,t ). DK models offer the advantage of nesting the most important term structure

models, from Vasicek (1977) and Cox et al. (1985a) one-factor models to three-factor

models like the one presented in Gong and Remolona (1997a). An additional feature of

these models is that they allow the estimation of the term structure simultaneously on

a cross-section and a time-series basis. Furthermore they provide a way of computing

and estimating simple closed-form expressions for the spot, forward, volatility and term

premium curves.

Expressed in discrete time, the discount factors in DK models are speci¬ed as:

’mt+1 = ξ + γ T st + »T V (st )1/2 µt+1 (3.7)

Modelling the Term Structure of Interest Rates 79

where V (st ) is the variance“covariance matrix of the random shocks to the sdf and is

de¬ned as a diagonal matrix with elements vi (zt ) = ±i + βiT st . Under certain conditions,

the volatility functions vi (st ) are positive;14 βi has non-negative elements and µt are the

independent shocks normally distributed as µt ∼ N (0, I ). Following equation (3.4), the

parameters in »T are the market prices of risk, as they govern the covariance between

the stochastic discount factor and the latent factors of the yield curve. Thus, the higher

these parameters are the higher is the covariance between the discount factor and the asset

return and the lower is its expected rate of return or the less risky the asset is (when the

covariance is negative).

The k-dimensional vector of factors st is de¬ned as follows:

st+1 = (I ’ )θ + st + V (st )1/2 µt+1 (3.8)

where has positive diagonal elements which ensure that the factors are stationary and

θ is the long-run mean of the factors. Asset prices are also log-linear functions of the

factors. Adding a second subscript in order to identify the term to maturity (denoted by

n), bond prices are given as follows:

’pn,t = An + Bn st

T

(3.9)

where An is a parameter and Bn a vector of parameters to be estimated. The parameters

in Bn are commonly known as the factor loadings, given that their values measure the

impact of a one-standard deviation shock to the factors on the log of asset prices.

In term structure models, the identi¬cation of the parameters is easier, considering

the restrictions imposed by the maturing bond price. In fact, when the term structure is

modelled using zero-coupon bonds paying one monetary unit, the log of the price of a

maturing bond must be zero. Consequently, from equation (3.6), the common normalisa-

tion A0 = B0 = 0 results. The following recursive restrictions between the parameters are

obtained by computing the moments in equation (3.6), using equations (3.7) and (3.9),

equating the independent terms and the terms in st in equation (3.8) respectively to An

and Bn in equation (3.9) and assuming p0,t = 0 and independent shocks:

k

1

An = An’1 + ξ + ’ )θ ’ (»i + Bi,n’1 )2 ±i

T

Bn’1 (I (3.10)

2 i=1

k

1

= (γ + )’ (»i + Bi,n’1 )2 βiT

T T T

Bn Bn’1 (3.11)

2 i=1

Our empirical analysis is based on interest rates of nominal zero-coupon bonds or spot

rates, which can easily be computed from bond prices as:

pn,t

yn,t = ’ (3.12)

n

Consequently, from equations (3.9) and (3.12), the yield curve is de¬ned as:

1

yn,t = (An + Bn st )

T

(3.13)

n

14

See Backus et al. (1998).

80 Applied Quantitative Methods for Trading and Investment

Using equations (3.10), (3.11) and (3.13), the short-term or one-period interest rate is:

k k

1 1

=ξ’ +γ’

»2 ±i T

»2 βiT st

y1,t (3.14)

i i

2 2

i=1 i=1

Correspondingly, the expected value of the short rate is:

k k

1 1

Et (y1,t+n ) = Et ξ’ +γ’

»2 ±i T

»2 βiT st+n

i i

2 2

i=1 i=1

k k

1 1

=ξ’ +γ’

»2 ±i T

»2 βiT Et (st+n )

i i

2 2

i=1 i=1

k k

1 1

=ξ’ +γ’ »2 βiT [(I ’ )θ +

n n

»2 ±i T

st ] (3.15)

i i

2 2

i=1 i=1

The volatility curve of the yields is derived from the variance“covariance matrix in

the speci¬cation of the factors. From equations (3.8) and (3.13), the volatility curve is

given by:

1T

Vart (yn,t+1 ) = 2 Bn V (st )Bn (3.16)

n

The instantaneous or one-period forward rate is the log of the inverse of the gross

return:

fn,t = pn,t ’ pn+1,t (3.17)

According to the de¬nition in equation (3.17), the price equation in (3.9) and the recur-

sive restrictions in (3.13) and (3.14), the one-period forward curve is:

fn,t = (An+1 + Bn+1 st ) ’ (An + Bn st )

T T

k

1

=ξ+ ’ )θ ’ (»i + Bi,n )2 ±i

T

Bn (I

2 i=1

k

1

+γ+ ’ I) ’ (»i + Bi,n )2 βiT st

T T

Bn ( (3.18)

2 i=1

The term premium is usually computed as the one-period log excess return of the

n-period bond over the short rate. Using equations (3.9)“(3.11) and (3.14), it is equal to:

= Et pn,t+1 ’ pn+1,t ’ y1,t

n,t

k k

2

Bi,n ±i

=’ »i Bi,n ±i + ’ (»i Bi,n + Bi,n )βiT st

2

(3.19)

2

i=1 i=1

From equations (3.15), (3.18) and (3.19), one can conclude that the forward rate is

equal to the expected future short-term interest rate plus the term premium and a constant

Modelling the Term Structure of Interest Rates 81

term that is related to the mean of the factors. Computing the term premium from the

basic pricing equation, the following result is obtained:15

T

Bn V (st )Bn

= ’» V (st )Bn ’

T

(3.20)

n,t

2

The ¬rst component in equation (3.20) is a pure risk premium, where » is the price

associated with the quantity of risk V (st )Bn . The parameters in » determine the signal

of the term premium. The second component is a Jensen inequality term. From equation

(3.20) one can conclude that at least one of the market prices of risk must be negative in

order to have a positive term premium.

The model estimated belongs to the class of Gaussian or constant volatility models. It

is a generalisation of the Vasicek (1977) one-factor model and a particular case of the DK

model, implying that some form of the expectations theory holds. As will be seen, this

model seems to be adequate to ¬t the German term structure, given that the expectations

theory is valid to a close approximation in this case. Following equation (3.7), the sdf in

a two- or a three-factor Gaussian model is written as:16

k

»2 2

’mt+1 = δ + σ + si,t + »i σi µi,t+1

i

(3.21)

2i

i=1

with k = 2 or 3. The factors are assumed to follow a ¬rst-order autoregressive order, with

zero mean:17

si,t+1 = •i si,t + σi µi,t+1 (3.22)

Within the DK framework, these models are characterised by:

θi = 0

= diag(•1 , •2 )

±i = σi2

βi = 0 (3.23)

k

»2 2

ξ =δ+ i

σ

2i

i=1

γi = 1

The recursive restrictions are:

k

1

An = An’1 + δ + [»2 σi2 ’ (»i σi + Bi,n’1 σi )2 ] (3.24)

i

2 i=1

Bi,n = (1 + Bi,n’1 •i ) (3.25)

Given that V (st ) was previously de¬ned as a diagonal matrix with elements vi (st ) = ±i + βiT st , equation

15

(3.20) corresponds to equation (3.19).

16

As will be seen later, this speci¬cation was chosen in order to write the short-term interest rate as the sum

of a constant (δ) with the factors.

17

This corresponds to considering the differences between the “true” factors and their means.

82 Applied Quantitative Methods for Trading and Investment

As can be seen from equation (3.24), in a homoskedastic model, the element on the

right-hand side of equation (3.10) related to the risk is zero. Thus, there are no interactions

between the risk and the factors in¬‚uencing the term structure, i.e., the term premium is

constant. Given equations (3.14), (3.24) and (3.25), the short-term interest rate is:18

k

y1,t = δ + si,t (3.26)

i=1

These models have the appealing feature that the short term is the sum of the factors.

Our conjecture is that the yield curve may be determined by two factors, one of them

being related to in¬‚ation and the other to a real factor, possibly the ex-ante real interest

rate. Following equations (3.24) and (3.25), the one-period forward rate is given by:

k k

2

1 ’ •in

1

=δ+ ’ »i σi + + [•in si,t ]

»2 σi2

fn,t σi (3.27)

1 ’ •i

i

2 i=1 i=1

This speci¬cation of the forward-rate curve accommodates very different shapes. How-

ever, the limiting forward rate cannot be simultaneously ¬nite and time-varying. In fact, if

•i < 1, the limiting value will not depend on the factors, corresponding to the following

expression:19

k

»i σi2 σi2

lim fn,t = δ + ’ ’ (3.28)

(1 ’ •i ) 2(1 ’ •i )2

n’∞

i=1

From equations (3.16) and (3.25), the volatility curve is:

k

1

Vart (yn,t+1 ) = 2 (Bi,n σi2 )

2

(3.29)

n i=1

Notice that as the factors have constant volatility, given by Vart (si,t+1 ) = σi2 , the

volatility of the yields does not depend on the level of the factors. Finally, the term

18

As referred to in Campbell et al. (1997), in a one-factor model setting, the Bi,n coef¬cients in a Gaussian

model measure the sensitivity of the log of bond prices to changes in short-term interest rate. This is different

from duration, as it does not correspond to the impact on bond prices of changes in the respective yields, but

instead in the short rate.

If •i = 1 interest rates are non-stationary. In that case, the limiting value of the instantaneous forward

19

yn,t = ’(pn,t /n) is time-varying but assumes in¬nite values. Effectively, (1 ’ •in )/(1 ’ •i ) = n in this case.

Thus, the expression for the instantaneous forward will be given by:

k k

1

fn,t = δ + ’n»i σi2 ’ n2 σi2 + si,t .

2

i=1 i=1

Accordingly, even if »i < 0, the forward rate curve may start by increasing, but at the longer end it will

decrease in¬nitely. Obviously, if »i > 0, the forward rate curve will decrease monotonously.

Modelling the Term Structure of Interest Rates 83

premium in these models will be:

= Et pn,t+1 ’ pn+1,t ’ y1,t

n,t

k 2

1 ’ •in

1

= ’ »i σi +

»2 σi2 σi

1 ’ •i

i

2 i=1

k

Bi,n σi2

2

= ’»i σi2 Bi,n ’ (3.30)

2

i=1

According to equations (3.27) and (3.30), the one-period forward rate in these Gaussian

models corresponds to the sum of the term premium with a constant and with the factors

weighted by the autoregressive parameters of the factors. Once again, the limiting case

is worth noting. When •i < 1, the limiting value of the risk premium differs from the

forward only by δ. Thus, within constant volatility models, the expected short-term rate

for a very distant settlement date is δ, i.e., the average short-term interest rate. From

equations (3.22) and (3.26), the expected value of the short rate for any future date t + n

can be computed:

k

Et (y1,t+n ) = δ + [•in si,t ] (3.31)

i=1

Comparing equations (3.27), (3.30) and (3.31) we conclude that the expectations theory

of the term structure holds with constant term premiums:20

fn,t = Et (y1,t+n ) + (3.32)

n

Thus, assessing the adequacy of Gaussian models corresponds to testing the validity of

the expectations theory of the term structure with constant term premiums.

3.4 A FORWARD RATE TEST

OF THE EXPECTATIONS THEORY

Equation (3.32) implies that forward rates are martingales if the term structure of risk

premium is ¬‚at. In fact, by the law of iterated expectations, it is obtained as:

fn,t = Et (Et+1 (y1,t+n ) + = Et (fn’1,t+1 ) + ( ’

n) n’1 ) (3.33)

n

As in the steady state it is expected that short-term interest rates remain constant,

according to equation (3.33) the one-period forward curve should be ¬‚at under the absence

of risk premium ( n = 0). As noted by Backus et al. (1998), equation (3.33) leads to a

forward regression test of the expectations theory of the term structure. The regression is:21

fn’1,t+1 ’ y1,t = constant + cn (fn,t ’ y1,t ) + residual (3.34)

20

Also known as the non-pure version of the expectations theory.

21

The term related to the slope of the term structure of risk premium is included in the constant, as it is

assumed to be time-constant.

84 Applied Quantitative Methods for Trading and Investment

with cn = 1. A rejection of this hypothesis can be taken as evidence that term premiums

vary with time, i.e., that the expectations theory does not hold.

It can be shown that the theoretical values of cn implied by the two-factor model

correspond to:

(B1 + Bn ’ Bn+1 )T 0 (B1 ’ T (Bn ’ Bn’1 ))

cn = (3.35)

(B1 + Bn ’ Bn+1 )T 0 (B1 + Bn ’ Bn+1 )

In the general case:

T

B1 0 B1

lim cn = T =1 as Bn’1 = Bn = Bn+1 when n ’ ∞

B1 0 B1

n’∞

3.5 IDENTIFICATION

Contrary to the pioneer interest rate models, such as Cox et al. (1985a), where the short-

term interest rate in¬‚uenced the whole term structure, the latent factor models do not

use explicit determinants of the yield curve. Therefore, as referred to in Backus et al.

(1997), the major outstanding issue in this context is the economic interpretation of the

interest rate behaviour approximated with af¬ne models, in terms of its monetary and

real economic factors. Our conjecture is that two factors seem to drive the German term

structure of interest rates: one factor related to the ex-ante real interest rate and a second

factor linked to in¬‚ation expectations.

As previously stated, the identi¬cation will thus be assessed in three different ways, the

¬rst being based on the model developed in Fung et al. (1999). As it is supposed that one

of the latent factors is related to in¬‚ation, one factor is identi¬ed with that variable, which

is assumed to be an AR(1) process. The in¬‚ation, denoted by πt , will thus be de¬ned as:22

(πt+1 ’ π) = ρ(πt ’ π) + ut+1 (3.36)

where π is the unconditional mean of the in¬‚ation rate and ρ is a parameter that measures

the rate of mean reversion. Considering that the short-term interest rate is a risk-free rate,

as there are no expectation revisions in a one-period investment, the hypothesis is that

it is the sum of the short-term real interest rate (r1,t ) with the one-period in¬‚ation rate

expectation:

y1,t ’ δ = (r1,t ’ r) + Et (πt+1 ’ π) (3.37)

where the long-run mean of the short-term interest rate is equal to the sum of the uncon-

ditional means of the real interest rate and the in¬‚ation rate (δ = r + π). The component

of the short-term rate related to the second factor may be considered as the one-period

in¬‚ation expectation:

s2,t = Et (πt+1 ’ π) = ρ(πt ’ π) = ρ πt

˜ (3.38)

˜

where πt denotes deviation of in¬‚ation from the steady-state value.

22

In¬‚ation is measured as the annual change in the log CPI. This measure is preferred because the quarter-

on-quarter annualised change in the log CPI is extremely volatile, often with negative readings. Our measure

should be interpreted as capturing the underlying or smoothed one-period in¬‚ation rate. In¬‚ation is measured

as year-on-year changes in the CPI in deviations from its mean, due to the de¬nition of the factors as having

zero mean.

Modelling the Term Structure of Interest Rates 85

From equations (3.36) and (3.38) the value of the second factor in t + 1 is:

s2,t+1 = Et+1 (πt+2 ) = ρ πt+1 = ρ(ρ πt + ut+1 ) = ρs2,t + ρut+1

˜ ˜ ˜ (3.39)

Comparing equations (3.22) and (3.39), we have the following identi¬cation between

the parameters of the second factor and the in¬‚ation process:

ρ = •2

(3.40)

ρut+1 = σ2 µ2,t+1

Using equations (3.38) to (3.40), the following relationship between the in¬‚ation and

the second factor is obtained:

1

πt = s2,t

˜ (3.41)

•2

An advantage of this identi¬cation procedure is that ex-ante real interest rates can

be derived. The major drawback of this technique is that it implies the second factor

to explain simultaneously the in¬‚ation as well as the long-term rates, which in some

periods may evidence signi¬cantly different volatilities. Consequently, during the periods

of higher volatility of the long-term rates, the estimated in¬‚ation tends to present a more

irregular behaviour than the true in¬‚ation.

Secondly, the procedure sketched above is based on the assumption in equation (3.36),

which is not necessarily the optimal model for forecasting in¬‚ation. In fact, this model is

too simple concerning its lag structure and also does not allow for the inclusion of other

macroeconomic information that market participants may use to form their expectations

of in¬‚ation. For example, information about developments in monetary aggregates, com-

modity prices, exchange rates, wages and unit labour costs, etc., may be used by market

participants to forecast in¬‚ation and, thus, may be re¬‚ected in the bond pricing process.

However, a more complex model would certainly not allow a simple identi¬cation of the

factor.

One way to overcome these problems is by using a joint model for the term structure

and the in¬‚ation, where the latter still shares a common factor with the interest rates but

is also determined by a second speci¬c factor.23 Thus, in¬‚ation can be modelled as a

function of two factors:

1

πt = (Aπ + Bπ sπ,t )

T

(3.42)

n

=( T

2t , s1π,t )

where and

πt

s1π,t+1 = •1π s1π,t + σ1π µ1π,t+1 (3.43)

Comparing to the model developed in Fung et al. (1999), this model offers the advan-

tage of not restricting the factor loadings to the values of the parameters of the term

structure loadings, as well as providing an additional factor to explain the in¬‚ation moves.

23

In this case, the loading of the second factor in the in¬‚ation equation is independent from the loading in term

structure equations, as no explicit relationship is assumed between the yields and the in¬‚ation and, consequently,

there are no recursive restrictions between the yields and the in¬‚ation.

86 Applied Quantitative Methods for Trading and Investment

Therefore, the second term structure factor does not have to be liable simultaneously for

the long-term interest rates and the in¬‚ation, avoiding more volatile estimates for the

in¬‚ation in periods of higher volatility of the long-term rates.

In order to overcome the problems detected in the ¬rst identi¬cation model, the leading

indicator properties of s2,t for in¬‚ation were also analysed, by setting up a VaR model

with p lags:24

xt = A1 xt’1 + · · · + Ap xt’p + µ + ut (3.44)

where xt is (2 — 1) and each of the Ai is a (2 — 2) matrix of parameters with generic

element denoted by [ak,j ] and ut ∼ IN(0, ). The vector xt is de¬ned as xt = (πt , s2,t )T .

i

Testing whether s2,t has leading indicator properties for in¬‚ation corresponds to testing

p

the hypothesis H0 : a1,2 = · · · = a1,2 = 0. This is a test for Granger causality, i.e., a test

1

of whether past values of the factor along with past values of in¬‚ation better “explain”

in¬‚ation than past values of in¬‚ation alone. This of course does not imply that bond yields

cause in¬‚ation. Instead it means that s2,t is possibly re¬‚ecting bond market™s expectations

as to where in¬‚ation might be headed.

In assessing the leading indicator properties of s2,t , the Granger causality test can be

supplemented with an impulse-response analysis. The vector MA(∞) representation of

the VaR is given by:

xt = µ + ut + + + ···

1 ut’1 2 ut’2 (3.45)

The matrix will then correspond to:

z

‚xt+z

= (3.46)

z

‚ut

that is, the row i, column j element of z identi¬es the consequences of a one-unit

increase in the j th variable™s innovation at date t (uj,t ) for the value of the ith variable

at time t + z(xi,t+z ), holding all other innovations at all dates constant.

A plot of the row i, column j element of z (‚xi,t+z /‚uj,t ), as a function of z is the

impulse-response function. It describes the response of (xi,t+z ) to a one-time impulse in

xj,t , with all other variables dated t or earlier held constant. Supposing that the date t

value of the ¬rst variable in the autoregression s2,t is higher than expected, so that u1,t is

positive, then we have:25

ˆ

‚ E(xi,t+z |s2,t , xt+1 , xt+2 , . . . , xt’p ) ‚xi,t+z

= (3.47)

‚s2,t ‚u1,t

Thus if s2,t is a leading indicator of in¬‚ation, a revision in market expectations of

ˆ

in¬‚ation ‚ E(xi,t+z |s2,t , xt’1 , xt’2 , . . . , xt’p ) should be captured by the marginal impact

of a shock to the innovation process in the equation for s2,t .

24

See, e.g., Hamilton (1994), chap. 11.

25

We use a Cholesky decomposition of the variance“covariance of the innovations to identify the shocks. We

order in¬‚ation ¬rst in the VaR to re¬‚ect the idea that whereas in¬‚ation does not respond, contemporaneously,

to shocks to expectations, these may be affected by contemporaneous information on in¬‚ation.

Modelling the Term Structure of Interest Rates 87

3.6 ECONOMETRIC METHODOLOGY AND APPLICATIONS

As the factors that determine the dynamics of the yield curve are non-observable and

the parameters are unknown, a Kalman ¬lter and a maximum likelihood procedure were

chosen for the estimation of the model.26 In a brief description, the Kalman ¬lter is an

algorithm that computes the optimal estimate for the state variables at a moment t using

the information available up to t ’ 1.

The starting point for the derivation of the Kalman ¬lter is to write the model in

state-space form, which includes an observation or measurement equation and a state or

transition equation, respectively:

= A · Xt + H · St + wt

Yt (3.48)

(r—n) (n—1) (r—k) (k—1)

(r—1) (r—1)

= C + F · St’1 + G · vt

St (3.49)

(k—k) (k—1)

(k—1) (k—k) (k—1)

(k—1)

where r is the number of variables to estimate, n is the number of observable exogenous

variables, k is the number of non-observable or latent exogenous variables (the factors),

and wt and vt are i.i.d. residuals, distributed as wt ∼ N (0, R) and vt ∼ N (0, Q).

The variance matrices are written as:

R = E(wt wt ) (3.50)

(r—r)

Q = E(vt+1 vt+1 ) (3.51)

(k—k)

As previously mentioned, two non-identi¬ed models were estimated, with two and three

factors, and using two different databases in the former case. Considering the shorter

database, the two-factor model was estimated using the ¬les on the CD-Rom “va2fgrse”

and “v2mlgrse”, containing respectively the Kalman ¬lter and the maximum likelihood

procedure, written in Matlab code. Conversely, for the three-factor model the ¬les used

were “va3fgrse” and “v3mlgrse”. The longer database was used only to estimate the

two-factor models, by the ¬les “va2fgrsb” and “v2mlgrsb”.

Though there are some packages containing automatic procedures to run the Kalman

¬lter (namely TSP), Matlab gives the chance to easily get, from the same ¬le, the econo-

metric results and the yield, forward, volatility and term premium curves, while imposing

the required restrictions on the parameters.

According to equation (3.13), the measurement or observation equation in our two- and

three-factor non-identi¬ed models may be written as:

®b

® ® ®

1,1 b2,1 · · · bk,1 ®

y1,t a1,t w1,t

y2,t a2,t s1,t

b1,2 b2,2 . bk,2 . w2,t

.

.

. = . + . + k = 2 or 3

. ° . » .

.

°.» °.» °. °.»

. .. .»

.

. . .

.

. . . sk,t

yl,t al,t wl,t

b1,l b2,l · · · bk,l

(3.52)

26

The maximum likelihood procedure is usually adopted when the parameters are unknown. Routines are

provided for these algorithms.

88 Applied Quantitative Methods for Trading and Investment

where y1,t , y2,t , . . . , yl,t are the l zero-coupon yields at time t with maturities j = 1, 2, . . . , u

periods and w1,t , w2,t , . . . , wl,t are the normally distributed i.i.d. errors, with null mean and

2

standard deviation equal to ej , of the measurement equation for each interest rate considered,

aj = Aj /j , bk,j = Bk,j /j .27

Following equation (3.22), the transition or state equation is:

® ® ® ® ®

s1,t+1 •1 s1,t σ1 v1,t+1

0 0 0 0

. . .

.. ..

° . »=° 0 0 »° . » + ° 0 0 »° . » (3.53)

. .

. . .

sk,t+1 •k sk,t σk vk,t+1

0 0 0 0

v1,t+1 , . . . , vk,t+1 being orthogonal shocks with null mean and variances equal to σ1 , . . . , σk2 ,

2

respectively. As in the estimation of the models, the information on the homoskedasticity

of yields is also exploited. The observation equation may be rewritten as follows:

®

® ® ®

· · · bk,1

b1,1

y1,t a1,t …1,t

a2,t .b

.

…2,t

b

y2,t

.

. 1,2 k,2

.

. . ®

. .

.

. ..

. . .

. .

. s1,t

.

.

al,t

b1,l · · · bk,l . + …l,t

yl,t

Vart (y1,t+1 ) = al+1,t + ° . » (3.54)

.

0 0 …l+1,t

0

Vart (y2,t+1 ) al+2,t sk,t …l+2,t

0 0

0

. .

.

» ° . » . .

.

° °.»