997, chap. 11) or Backus et al. (1998) for graduate textbook presentations of afﬁnemodels.5These models differ basically due to the fact that the latter allowed for the volatility of the short-term interestrate to be stochastic. In order to incorporate the supply side of ﬁnancial assets, Cox et al. (1985b) derived ageneral equilibrium model that yields a closed-form expression for asset prices.6One of the ﬁrst 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 identifythe factors with the consumption marginal rate of substitution.
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Applied Quantitative Methods for Trading and Investment141210
%
8642
1Y6Y
Figure 3.3
2Y7Y
3Y8Y
4Y10Y
19971998
199119921993199419951996
1987198819891990
198119821983198419851986
1980
197719781979
1976
1975
197219731974
0
5Y
German yield curve: 1986–1998
Most papers on afﬁne models have focused on the US term structure. The pronouncedhump-shape of the US yield curve and the empirical work pioneered by Litterman andScheinkman (1991) have led to the conclusion that three factors are required to explainthe movements of the whole term structure of interest rates. These factors are usuallyidentiﬁed as the level, the slope and the curvature of the term structure. Most studieshave concluded that the level is the most important factor in explaining interest ratevariations 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 haveused three-factor models with stochastic volatility in order to ﬁt the term structure ofinterest rates (see, for instance, Balduzzi et al. (1996) and Gong and Remolona (1997a)).However, stochastic volatility models pose admissibility problems, as the factors determining 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 oftenvery difﬁcult to estimate. In fact, frequently small deviations of the parameters from theestimated values generate widely different and implausible term structures.Alternatively a constant volatility or Gaussian model can be ﬁtted. In the German case,Figure 3.1 suggests that this may be a reasonable choice, though in this chart end-monthdata is presented. These models overcome the empirical problems posed by stochasticvolatility models and are also capable of reproducing a wide variety of shapes of theyield curve, though they face some shortcomings regarding the limiting properties of theinstantaneous forward rate.7Furthermore, some term structures may have properties identiﬁable with a smallernumber of factors. For instance, according to Buhler et al. (1999), principal component7
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 Germanterm structure of interest rates consistently from 1970 up to 1999. Consequently, a decisionon the number of factors is also required.The ﬂexibility of afﬁne models allows for considering observable as well as nonobservable factors, such as macroeconomic variables. The advantage of estimating thesemodels using only latent or non-observable factors is that it avoids making ex-ante restrictions on the behaviour of the factors determining the yield curve. In this chapter, we willstart by incorporating only latent factors.In order to ﬁt afﬁne models to a cross-section/time-series of interest rates three practicalquestions have to be answered: (i) how many factors should be considered; (ii) what are theproperties of the factors; and (iii) how can they be identiﬁed. A sensible way to proceedabout the ﬁrst question is by performing forward rate regressions in line with Backuset al. (1997), to assess the adequacy of Gaussian models.Regarding the second question, two- and three-factor models were estimated.8 Giventhat the two-factor Gaussian model can be speciﬁed in such a way that the short-terminterest rate is equal to the sum of a constant with the two latent factors, the usualconjecture is that these factors reﬂect the real interest rate and the expected inﬂationrate.9 A third factor may be included reﬂecting potential international inﬂuences on thedomestic 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 isassessed in three different ways: the ﬁrst two are based on the explicit identiﬁcation ofone of the factors with the inﬂation rate and relating the other factor to the ex-ante realinterest rate, in line with Zin (1997). Within this framework, a ﬁrst exercise consistedin identifying one factor with the inﬂation rate process, modelled as an AR(1) process,following Fung et al. (1999). The second factor was left unconstrained and assumed toreﬂect the “real” determinants of the term structure of interest rates, such as the outputgap or the real interest rate.A second exercise was performed based on the assumption that the inﬂation is givenby two factors, one of them being a common factor with the term structure. The jointfactor can be taken as a proxy for core inﬂation. It can be shown that this assumptionprovides the general case of a joint model developed in Fung et al. (1999), where thefactor loadings are not restricted to the values of the parameters of the interest rateloadings. Consequently, the second factor of the term structure is left unconstrained andshould reﬂect the “real” determinants of the term structure of interest rates, such as theoutput gap or the real interest rate, while the second factor of inﬂation should reﬂectnon-core inﬂation movements.10 The third way chosen to analyse the identiﬁcation issue8The three-factor model was run only for the shorter database, given that the longer database does not includeinformation on the money market rates. Therefore, it is reasonable to assume that two factors are enough toexplain the behaviour of the yield curve in the range of maturities included in that database.9This corresponds to the Fisher hypothesis. The two-factor model is also consistent with the idea that theBundesbank followed a type of Taylor rule in setting ofﬁcial interest rates as recently documented by Claridaet al. (1998) and Clarida and Gertler (1997). In accordance with such a rule the short-term interest rate wasadjusted in response to the deviation of inﬂation and output from their targets.10In Remolona et al. (1998), inﬂation-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 ona nominal basis, following Campbell et al. (1997). The best-known examples of inﬂation-linked securities arethe inﬂation-indexed government bonds that exist only in a few countries, namely in the UK, the USA andFrance (see, for instance, Deacon and Derry (1994) and Gong and Remolona (1997b)).
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Applied Quantitative Methods for Trading and Investment
is based on econometric evidence about the leading indicator properties of one factor forinﬂation developments in Germany.The latent factors are estimated by a Kalman ﬁlter, while a maximum likelihood procedure is used to estimate the time-constant parameters of the yield curve, followingthe pioneering work by Chen and Scott (1993a,b).11 This econometric technique allowsa wide range of model speciﬁcations (including observable and non-observable factors),which compute the optimal estimate for the state variables at a given moment using all thepast information available. In Figure 3.4 the reader ﬁnds a brief summary of the exerciseperformed in this chapter.Two Matlab routines, supplied on the accompanying CD-Rom, were used for each ofthe three model speciﬁcations, one for the Kalman ﬁlter and the other for the maximumlikelihood estimation. The leading properties of the factor were assessed using a RATS 4.0standard program.12 The results obtained in this chapter illustrate that both two- and threefactor models ﬁt quite well the yield and the volatility curves, also providing reasonableestimates for the one-period forward and term premium curves.13Assessment of the properties of thefactors' volatility
Forward regressions(equation (3.34))
Estimation of the two-and three-factormodels
Kalman filter(equations (3.52) to (3.54))
Choice between two-and three-factormodels
Chi-square test
Identification of the two-factor modelwith macroeconomic variables
One-and two-factor modelsfor inflation (equations(3.55) and (3.57))
Leading indicatorproperties for inflation
Conclusions for yield curve modelling,asset management, Value-at-Risk andmonetary policy
Figure 3.4 Yield curve afﬁne models using latent factors11The model is derived in discrete time, as in this way it matches the frequency of the data, allows theidentiﬁcation of the factors with observable macroeconomic variables and avoids the problem of estimating acontinuous-time model with discrete-time data (see, for example, Aı̈t-Sahalia (1996)).12The chapter does not present the routine to assess the leading indicator properties, given that a routine adaptedfrom Doan (1995) was used.13The 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 identiﬁed, both related to worldwide gyrations in bond markets – Spring 1994 and 1998 – which were characterised bysharp changes in long-term interest rates while short-term rates remained stable. Thus, itseems that the third factor fails to capture the potential external inﬂuences on Germaninterest rates. In addition, the two-factor model ﬁts quite well the yield and the volatilitycurve, providing a good ﬁt of the time series of bond yields and more reasonable estimatesfor 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 backgroundon asset pricing is presented. In Section 3.3 the theoretical framework of Dufﬁe and Kan(1996) (DK hereafter) afﬁne models is explained. In Section 3.4 a test of the expectations 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. Theeconometric methodology is presented in Section 3.6. Section 3.7 includes the presentation of the data and the results of the estimation. The main conclusions are stated inSection 3.8.
3.2 BACKGROUND ISSUES ON ASSET PRICINGA fundamental result in modern asset pricing theory is that the price of any ﬁnancialasset corresponds to the present value of its expected future cash-ﬂows, this present valuebeing obtained by applying a positive stochastic discount factor (hereafter sdf, denotedby Mt ). If the future cash-ﬂows correspond only to the ﬁnancial asset price in the nextperiod we have:(3.1)Pt = Et [Pt+1 Mt+1 ]Another fundamental result in ﬁnance theory is that asset prices and returns are relatedto their risk, which is the ability of the asset to offer higher cash-ﬂows when they aremore needed. In fact, the more an asset helps to smooth income ﬂuctuations, the lessrisky it is and the higher will be its demand for insuring against “bad times”. Employingsome simple algebra in equation (3.1), the following equation illustrates this link betweenasset prices and their risk:Pt = Et [Pt+1 ]
1f
1 + it+1
+ Covt [Pt+1 , Mt+1 ]
(3.2)
This result shows that the asset price is the discounted expected value of its futurepayoff, adjusted by the covariance of its payoff with the sdf and where the discountfactor is the inverse of the return on a risk-free asset. The covariance term consists of arisk 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:f
Et [it+1 ] = it+1 −
Covt [Mt+1 , it+1 ]Et [Mt+1 ]
(3.3)
estimates obtained for the volatilities proved unreasonable. Consequently, we opted for including the volatilityequations in the Kalman ﬁlter. This procedure can be considered as corresponding to estimating the modelimposing restrictions on the parameters in order to get well-behaved volatility curves.
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According to equation (3.3), the excess return of any asset over the risk-free assetdepends 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 paysa risk premium. With some additional self-explanatory algebra, the following result isobtained:Covt [Mt+1 , it+1 ]Vart [Mt+1 ]ff(3.4)−= it+1 + βit+1 ,Mt+1 λtEt [it+1 ] = it+1 +Vart [Mt+1 ]Et [Mt+1 ]In equation (3.4), βit+1 ,Mt+1 is the coefﬁcient of a regression of it+1 on Mt+1 , i.e., itmeasures the correlation between the asset’s return and the sdf or the quantity of risk,while λt = −Vart [Mt+1 ]/Et [Mt+1 ] is the market price of risk. Therefore, as in the staticCAPM, the excess return over the risk-free interest rate of any ﬁnancial asset depends onthe quantity of risk and its market price.
3.3 DUFFIE–KAN AFFINE MODELSOF THE TERM STRUCTUREAfﬁne 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 wereoriginally developed by Dufﬁe and Kan (1996), for the term structure of interest rates.As referred to in Balduzzi et al. (1996), “Dufﬁe and Kan (1996) show that a wide rangeof choices of stochastic processes for interest rate factors yield bond pricing solutions ofa form now widely called exponential-afﬁne 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 theassumption of joint log-normality of bond prices and the nominal pricing kernel and usingthe statistical result that if log X ∼ N (µ, σ 2 ) then log E(X) = µ + σ 2 /2 it is obtainedfrom equation (3.5) that:pt = Et [mt+1 + pt+1 ] + 12 Vart [mt+1 + pt+1 ]
(3.6)
Dufﬁe and Kan (1996) deﬁne a general class of multifactor afﬁne models of the termstructure, 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 structuremodels, from Vasicek (1977) and Cox et al. (1985a) one-factor models to three-factormodels like the one presented in Gong and Remolona (1997a). An additional feature ofthese models is that they allow the estimation of the term structure simultaneously ona cross-section and a time-series basis. Furthermore they provide a way of computingand estimating simple closed-form expressions for the spot, forward, volatility and termpremium curves.Expressed in discrete time, the discount factors in DK models are speciﬁed as:−mt+1 = ξ + γ T st + λT V (st )1/2 εt+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 isdeﬁned as a diagonal matrix with elements vi (zt ) = αi + βiT st . Under certain conditions,the volatility functions vi (st ) are positive;14 βi has non-negative elements and εt are theindependent shocks normally distributed as εt ∼ N (0, I ). Following equation (3.4), theparameters in λT are the market prices of risk, as they govern the covariance betweenthe stochastic discount factor and the latent factors of the yield curve. Thus, the higherthese parameters are the higher is the covariance between the discount factor and the assetreturn and the lower is its expected rate of return or the less risky the asset is (when thecovariance is negative).The k-dimensional vector of factors st is deﬁned as follows:st+1 = (I − )θ + st + V (st )1/2 εt+1
(3.8)
where  has positive diagonal elements which ensure that the factors are stationary andθ is the long-run mean of the factors. Asset prices are also log-linear functions of thefactors. Adding a second subscript in order to identify the term to maturity (denoted byn), bond prices are given as follows:−pn,t = An + BnT st
(3.9)
where An is a parameter and Bn a vector of parameters to be estimated. The parametersin Bn are commonly known as the factor loadings, given that their values measure theimpact of a one-standard deviation shock to the factors on the log of asset prices.In term structure models, the identiﬁcation of the parameters is easier, consideringthe restrictions imposed by the maturing bond price. In fact, when the term structure ismodelled using zero-coupon bonds paying one monetary unit, the log of the price of amaturing bond must be zero. Consequently, from equation (3.6), the common normalisation A0 = B0 = 0 results. The following recursive restrictions between the parameters areobtained 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 Anand Bn in equation (3.9) and assuming p0,t = 0 and independent shocks:1(λi + Bi,n−1 )2 αi2 i=1k
TAn = An−1 + ξ + Bn−1(I − )θ −
1(λi + Bi,n−1 )2 βiT2 i=1
(3.10)
k
TBnT = (γ T + Bn−1) −
(3.11)
Our empirical analysis is based on interest rates of nominal zero-coupon bonds or spotrates, which can easily be computed from bond prices as:yn,t = −
pn,tn
(3.12)
Consequently, from equations (3.9) and (3.12), the yield curve is deﬁned as:yn,t =14
See Backus et al. (1998).
1(An + BnT st )n
(3.13)
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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:kk1 21 2 TTλ αi + γ −λ β st(3.14)y1,t = ξ −2 i=1 i2 i=1 i iCorrespondingly, the expected value of the short rate is:kk1 21 2 TTλ αi + γ −λ β st+nEt (y1,t+n ) = Et ξ −2 i=1 i2 i=1 i ikk1 21=ξ−λ αi + γ T −λ2 β T Et (st+n )2 i=1 i2 i=1 i ikk1 21 2 TT=ξ−λ αi + γ −λ β [(I − n )θ + n st ]2 i=1 i2 i=1 i i
(3.15)
The volatility curve of the yields is derived from the variance–covariance matrix inthe speciﬁcation of the factors. From equations (3.8) and (3.13), the volatility curve isgiven by:1Vart (yn,t+1 ) = 2 BnT V (st )Bn(3.16)nThe instantaneous or one-period forward rate is the log of the inverse of the grossreturn:fn,t = pn,t − pn+1,t(3.17)According to the deﬁnition in equation (3.17), the price equation in (3.9) and the recursive restrictions in (3.13) and (3.14), the one-period forward curve is:Tst ) − (An + BnT st )fn,t = (An+1 + Bn+1
1(λi + Bi,n )2 αi2 i=1k
= ξ + BnT (I − )θ −
k1+ γ T + BnT ( − I ) −(λi + Bi,n )2 βiT st2 i=1
(3.18)
The term premium is usually computed as the one-period log excess return of then-period bond over the short rate. Using equations (3.9)–(3.11) and (3.14), it is equal to:n,t = Et pn,t+1 − pn+1,t − y1,t
=−
ki=1

λi Bi,n αi +
2Bi,nαi
2
−
k
2(λi Bi,n + Bi,n)βiT st
(3.19)
i=1
From equations (3.15), (3.18) and (3.19), one can conclude that the forward rate isequal 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 thebasic pricing equation, the following result is obtained:15BnT V (st )BnTn,t = −λ V (st )Bn −2
(3.20)
The ﬁrst component in equation (3.20) is a pure risk premium, where λ is the priceassociated with the quantity of risk V (st )Bn . The parameters in λ determine the signalof 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 inorder to have a positive term premium.The model estimated belongs to the class of Gaussian or constant volatility models. Itis a generalisation of the Vasicek (1977) one-factor model and a particular case of the DKmodel, implying that some form of the expectations theory holds. As will be seen, thismodel seems to be adequate to ﬁt the German term structure, given that the expectationstheory is valid to a close approximation in this case. Following equation (3.7), the sdf ina two- or a three-factor Gaussian model is written as:16k  2λi 2−mt+1 = δ +(3.21)σ + si,t + λi σi εi,t+12 ii=1with k = 2 or 3. The factors are assumed to follow a ﬁrst-order autoregressive order, withzero mean:17si,t+1 = ϕi si,t + σi εi,t+1(3.22)Within the DK framework, these models are characterised by:θi = 0 = diag(ϕ1 , ϕ2 )αi = σi2βi = 0ξ =δ+
(3.23)kλ2i
2i=1
σi2
γi = 1The recursive restrictions are:1 2 2[λ σ − (λi σi + Bi,n−1 σi )2 ]2 i=1 i ik
An = An−1 + δ +
Bi,n = (1 + Bi,n−1 ϕi )
(3.24)(3.25)
15Given that V (st ) was previously deﬁned as a diagonal matrix with elements vi (st ) = αi + βiT st , equation(3.20) corresponds to equation (3.19).16As will be seen later, this speciﬁcation was chosen in order to write the short-term interest rate as the sumof a constant (δ) with the factors.17This corresponds to considering the differences between the “true” factors and their means.
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Applied Quantitative Methods for Trading and Investment
As can be seen from equation (3.24), in a homoskedastic model, the element on theright-hand side of equation (3.10) related to the risk is zero. Thus, there are no interactionsbetween the risk and the factors inﬂuencing the term structure, i.e., the term premium isconstant. Given equations (3.14), (3.24) and (3.25), the short-term interest rate is:18y1,t = δ +
k
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 thembeing related to inﬂation and the other to a real factor, possibly the ex-ante real interestrate. Following equations (3.24) and (3.25), the one-period forward rate is given by:2  kk1 − ϕin1 2 2fn,t = δ +λi σi − λi σi ++σi[ϕin si,t ]2 i=11 − ϕii=1
(3.27)
This speciﬁcation of the forward-rate curve accommodates very different shapes. However, the limiting forward rate cannot be simultaneously ﬁnite and time-varying. In fact, ifϕi < 1, the limiting value will not depend on the factors, corresponding to the followingexpression:19k λi σi2σi2lim fn,t = δ +−(3.28)−(1 − ϕi ) 2(1 − ϕi )2n→∞i=1From equations (3.16) and (3.25), the volatility curve is:1  2 2(B σ )n2 i=1 i,n ik
Vart (yn,t+1 ) =
(3.29)
Notice that as the factors have constant volatility, given by Vart (si,t+1 ) = σi2 , thevolatility of the yields does not depend on the level of the factors. Finally, the term18As referred to in Campbell et al. (1997), in a one-factor model setting, the Bi,n coefﬁcients in a Gaussianmodel measure the sensitivity of the log of bond prices to changes in short-term interest rate. This is differentfrom duration, as it does not correspond to the impact on bond prices of changes in the respective yields, butinstead in the short rate.19If ϕi = 1 interest rates are non-stationary. In that case, the limiting value of the instantaneous forwardyn,t = −(pn,t /n) is time-varying but assumes inﬁnite values. Effectively, (1 − ϕin )/(1 − ϕi ) = n in this case.Thus, the expression for the instantaneous forward will be given by:
fn,t = δ +
k i=1
1 2 2n σi +si,t .2i=1k
−nλi σi2 −
Accordingly, even if λi < 0, the forward rate curve may start by increasing, but at the longer end it willdecrease inﬁnitely. Obviously, if λi > 0, the forward rate curve will decrease monotonously.
Modelling the Term Structure of Interest Rates
83
premium in these models will be:n,t = Et pn,t+1 − pn+1,t − y1,t
2 k1 − ϕin1 2 2λ σ − λi σi +=σi2 i=1 i i1 − ϕik22σBi,ni−λi σi2 Bi,n −=2i=1
(3.30)
According to equations (3.27) and (3.30), the one-period forward rate in these Gaussianmodels corresponds to the sum of the term premium with a constant and with the factorsweighted by the autoregressive parameters of the factors. Once again, the limiting caseis worth noting. When ϕi < 1, the limiting value of the risk premium differs from theforward only by δ. Thus, within constant volatility models, the expected short-term ratefor a very distant settlement date is δ, i.e., the average short-term interest rate. Fromequations (3.22) and (3.26), the expected value of the short rate for any future date t + ncan be computed:kEt (y1,t+n ) = δ +[ϕin si,t ](3.31)i=1
Comparing equations (3.27), (3.30) and (3.31) we conclude that the expectations theoryof the term structure holds with constant term premiums:20fn,t = Et (y1,t+n ) +
(3.32)
n
Thus, assessing the adequacy of Gaussian models corresponds to testing the validity ofthe expectations theory of the term structure with constant term premiums.
3.4 A FORWARD RATE TESTOF THE EXPECTATIONS THEORYEquation (3.32) implies that forward rates are martingales if the term structure of riskpremium is ﬂat. In fact, by the law of iterated expectations, it is obtained as:fn,t = Et (Et+1 (y1,t+n ) +
n ) = Et (fn−1,t+1 ) + (
n−
n−1 )
(3.33)
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 ﬂat under the absenceof risk premium ( n = 0). As noted by Backus et al. (1998), equation (3.33) leads to aforward regression test of the expectations theory of the term structure. The regression is:21fn−1,t+1 − y1,t = constant + cn (fn,t − y1,t ) + residual20
(3.34)
Also known as the non-pure version of the expectations theory.The term related to the slope of the term structure of risk premium is included in the constant, as it isassumed to be time-constant.21
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Applied Quantitative Methods for Tr