A Noisy Principal Component Analysis for Forward Rate Curves
aa r X i v : . [ q -f i n . S T ] A ug A Noisy Principal Component Analysis for Forward Rate Curves
M´arcio Poletti Laurini a, ∗ , Alberto Ohashi b a FEA-RP USP - 14040-905, Ribeiro Preto, SP, Brasil b Mathematics Dept., Universidade Federal da Para´ıba, 13560-970, Jo˜ao Pessoa, PB, Brasil
Abstract
Principal Component Analysis (PCA) is the most common nonparametric method for estimatingthe volatility structure of Gaussian interest rate models. One major difficulty in the estimationof these models is the fact that forward rate curves are not directly observable from the marketso that non-trivial observational errors arise in any statistical analysis. In this work, we pointout that the classical PCA analysis is not suitable for estimating factors of forward rate curvesdue to the presence of measurement errors induced by market microstructure effects and numericalinterpolation. Our analysis indicates that the PCA based on the long-run covariance matrix iscapable to extract the true covariance structure of the forward rate curves in the presence ofobservational errors. Moreover, it provides a significant reduction in the pricing errors due to noisydata typically founded in forward rate curves.
Keywords:
Finance, Pricing, Principal component analysis, term-structure of interest rates, HJMmodels.
1. Introduction
The term-structure of interest rates is a high-dimensional object which has been the subjectof much research in the finance literature. It is the natural starting point for pricing fixed-incomesecurities and other financial assets. In particular, the identification of factors capable to explainits movements plays a crucial role in modeling complex interest rate derivative products. Sincethe seminal works of Steeley (1990), Stambaugh (1988) and Litterman and Scheinkman (1991),it is well-known that most of the covariance yield curve structure can be summarized by just afew unobservable factors. This stylized fact is fundamentally based on the Principal ComponentAnalysis (henceforth abbreviated by PCA) based on sample covariance matrices. In this case, asmall number of eigenvectors summarizes the whole second moment structure of the yield curves. ✩ We would like to thank Josef Teichmann for stimulating discussions on the topic of this paper. The secondauthor was supported by CNPq grant 308742. ∗ Tel.: +55-16-33290867
URL: [email protected] (M´arcio Poletti Laurini), [email protected] (Alberto Ohashi)
Preprint submitted to Arxiv October 18, 2018 he interest rate markets can be summarized by two fundamental high dimensional objects: theyield x y t ( x ) and forward rate curves x r t ( x ); t ≥ y t ( x ) = 1 x Z x r t ( z ) dz ; 0 ≤ t < ∞ , x ≥ . (1.1)See e.g. Filipovic (2009) for more details. In particular, the underlying covariance structure ofyield and forward rate curves play a major role in the statistical analysis of the term-structure ofinterest rate. See e.g. Rebonato (2002), Schmidt (2011) and other references therein. For instance,forward rate curves play a central role in pricing and hedging interest rate derivatives by means ofthe classical methodology proposed by Heath et al. (1992). Their contribution can be summarizedby the representation of the forward rate curve dynamics in terms of a stochastic partial differentialequation dr t ( x ) = (cid:16) ∂r t ( x ) ∂x + α HJM ( t, r t ( x )) (cid:17) dt + d X j =1 σ j ( t, r t ( x )) dB jt ; r ( x ) = ξ ( x ) , ≤ t < ∞ , x ≥ α HJM is the so-called (HJM) drift condition which is fully determined by the volatilitystructure σ = ( σ , . . . , σ d ), ( B , . . . , B d ) is a d -dimensional Brownian motion and x ξ ( x ) is agiven initial forward rate curve. Then, the initial forward rate curve ξ and the volatility structure σ fully determine the no-arbitrage dynamics of the model (1.2). See e.g Filipovic (2009) for furtherdetails.Plenty of spot interest rate data (and hence yield curve data) are available in fixed incomemarkets. However, due to the absence of explicit forward rate markets, implied forward rate curveshave to be estimated from interest rates based on other financial instruments. This already presentsa major difficulty in implementing derivative pricing models based on the classical Heath-Jarrow-Morton methodology.The most common non-parametric procedure for estimating the covariance structure of forwardrate curves is the PCA methodology. Basically, three common strategies are very popular in thePCA estimation of the forward rate curves: (A) One postulates the existence of a finite-dimensionalparameterized family of smooth curves G = { G ( z ; x ); z ∈ Z ⊂ R N , x ≥ } and a Z -valued stateprocess Y such that y t ( x ) = G ( Y t ; x ) for x ≥ , ≤ t < ∞ . (1.3)By interpolating the available yield data based on G , then one extracts the associated forwardrate curve by means of any numerical scheme to recover r t ( x ) = y t ( x ) − x ∂y t ( x ) ∂x . The PCA2s then applied on this estimated forward rate curves, as discussed in e.g. Jarrow (2002) andLord and Pelsser (2007). (B) Instead of (1.3), one shall use a non-parametric polynomial splinesmethod to interpolate the yield data and do the same step of (A) . See e.g. Vasicek and Fong (1982),Barzanti and Corradi (1998) and Chiu et al. (2008) for further details. Alternatively, one can useproxies to construct the forward rate curve jointly with a given interpolating family of smoothcurves G . See e.g. Bhar et al. (2002)), Alexander and Lvov (2003) and Gauthier and Simonato(2012) for further details.For a given initial forward rate curve, the fundamental object which encodes the whole dynamicsof (1.2) is volatility. In particular, due to closed form expressions for derivative prices and hedging,it is common (see e.g. Rutkowski (1996), Jarrow (2002) and Falini (2010)) to assume that thevolatility structure is deterministic. In this case, the stochastic dynamics of forward rates is givenby a Gaussian HJM model: dr t ( x ) = (cid:16) ∂r t ( x ) ∂x + α HJM ( x ) (cid:17) dt + d X j =1 σ j ( x ) dB jt ; r ( x ) = ξ ( x ) , ≤ t < ∞ , x ≥ . (1.4)The most common alternative to estimate the underlying volatility structure is to use PCAmethodology (see e.g. Jarrow (2002) Filipovic (2009), Schmidt (2011) and other references therein)based on the static covariance matrix of a given sample ( r t ( x ) , . . . , r t ( x n )). The PCA methodologyprovides the following estimator for the volatility structureˆ σ i = ˆ ϕ i q ˆ λ i ; i = 1 , . . . , ˆ m, (1.5)where ˆ m is the estimated number of principal components of the forward rate curves and theestimated eigenvalues and eigenvectors of the associated static covariance matrix are given by ˆ λ i and ˆ ϕ i , respectively. One fundamental assumption behind (1.5) and, more generally, on the use ofPCA in forward rate curves is the following one: Assumption (I)
There is no observational errors in forward rate curves.At this stage, a natural question is the validity of assumption (I) in the term-structure of interestrates. In fact, we shall compare the existing literature of principal components between yield andforward rate curves to see some evidence of violation of assumption (I) . In one hand, the linearityof the relation (1.1) strongly suggests that dimension of the forward rate and yield curves must beidentical (See Proposition 2.1). On the other hand, apparently, distinct results in the literature havebeen reported on the spectral structure of the forward rate and yield curves. Akahori et al. (2006)and Liu (2010) report a remarkable difference in the estimated number of factors between forwardrate and yield curves and they suggest that a possible explanation for this would be the violation3f the random walk hypothesis. The same type of behavior was reported by Lekkos (2000) whoargues that averaging the forward rates over time to maturities would induce a strong dependenceon the yield data. He argues that PCA method artificially estimates a small number of principalcomponents for yield curves. Alexander and Lvov (2003) study statistical properties of the UKLibor rates. They show that the strategy (A) induces equivalent loading factor structures betweenyield and implied forward rate curves. Lord and Pelsser (2007) report a visible difference in thePCA of forward and yield curves by using estimated Svensson curves for the Euro market.Essentially, the existing literature restricts the discussion into two lines: (i) More factors areneeded to account the correlation in forward rates curves. An averaging effect would be the reasonfor an artificial dependence on the yield curves. (ii) One way to remedy this pattern is to firstsmooth the yield data by means of a parametric or non-parametric form and then to calculate theimplied forward rates. One should notice that one important assumption behind (i) and (ii) is (I) .More importantly, the current literature only suggests ad hoc methods based on (A-B) .In this article, we take a rather different strategy. Throughout this article, we assume that theobserved curve time series, which we denote by X ( · ) , . . . , X n ( · ), they are subject to errors in thesense that X t ( u ) = r t ( u ) + ε t ( u ); u ≥ , t = 0 , , . . . . (1.6)The possible existence of the noise term ε in (1.6) reflects the classical interpolation procedures (A-B) when extracting forward rate curves from yield data. It can also be induced by observationalerrors due to market microstructure effects. The existence of an underlying bid and ask bondprice structure contaminates the yield and forward rate curves (see e.g Mizrach and Neely (2011)and Goyenko et al. (2011)). These noisy discrete data are smoothed to provide “observed” curves x X t ( x ) where both r t ( · ) and ε t ( · ) are unobservable. We investigate in detail the existence andthe impact of observational errors { ε t ( x ); t ∈ N , x ∈ R + } in (1.6) in the classical PCA methodology.We show that market microstructure effects and common interpolation procedures (A-B) inducenoisy forward rate curves which cause severe bias in the PCA methodology. The starting point ofour analysis is the fact that the ranks of the covariance operators of the forward rate and yield curvesare identical (see Proposition 2.1). In addition, we show that PCA based on the so-called long-run covariance matrix (henceforth abbreviated by LRCM) significantly improves the estimation ofthe covariance structure of forward rate curves. The impact of noisy data in pricing interest ratederivatives is also discussed.This article is organized as follows. In Section 2, we report an elementary result about theequivalence of ranks between the covariance operators of the forward rate and yield curves. InSection 3, we describe some alternatives of estimating covariance structures in the presence ofobservational errors, the so-called LRCM estimators. Section 4 presents a detailed simulation4nalysis reporting the performance of the PCA based on LRCM estimators, as well as, the roleplayed by measurement errors in the PCA methodology applied to the term-structure of interestrate. In order to compare the simulation results with a real data set, Section 5 provides an empiricalanalysis on the number of principal components for US and UK term-structure on interest-rates. InSection 6, we analyse the impact of neglecting observational errors in pricing interest rate derivativesin light of the PCA methodology. Section 7 presents the final remarks.
2. Rank equivalence in covariance operators for forward rate and yield curves
In this section, we give a simple remark showing the number of principal components in forwardrate curves should be exactly the same of the yield curves under some mild conditions. In spite ofits simplicity, it is the starting point to investigate the violation of assumption (I) and it also givesa comparison criteria for our statistical analysis.Let { P ( t, T ); ( t, T ) ∈ ∆ } be the term-structure of bond prices where ∆ := { ( t, T ); 0 ≤ t ≤ T < ∞} . Let s ( t, T ) := − log P ( t, T ) T − t ; ( t, T ) ∈ ∆ , (2.1)be the spot-interest rate prevailing at time t for maturity T and let y t ( x ) := s ( t, t + x ) be thecorrespondent yield curve at time indexed by the time to maturity x = T − t .The forward rate prevailing at time t for maturity T is f ( t, T ) := − ∂ logP ( t, T ) ∂T ; ( t, T ) ∈ ∆ , and the forward rate curve is r t ( x ) := f ( t, t + x ) for x = T − t . For simplicity of exposition, weanalyze the PCA methodology based on a space E of curves so that forward rate and yield dataare interpreted as sample curves over time.In the sequel, we consider a discrete-time setup t ∈ N := { , , , . . . } and we assume that y t and r t are discrete-time weakly stationary E -valued process. That is, there exist functions( µ y ( · ) , µ r ( · ) , Q r ( · , · ) , Q y ( · , · )) such that the following identities hold for every tµ r ( u ) = E r t ( u ) , µ y ( u ) = E y t ( u ) ,Q y ( u, v ) = Cov ( y t ( u ) , y t ( v )) , Q r ( u, v ) = Cov ( r t ( u ) , r t ( v )); u, v ∈ K. (2.2)Otherwise, we assume that the first difference process satisfies such properties. The set E isa separable Hilbert space of functions from a bounded set K := [0 , x ⋆ ) ⊂ R + to R such that J, T : E → E defined by 5 J x ( f ) := 1 x Z x f ( y ) dy ; f T x ( f ) := x ddx f ( x ) , (2.3)are bounded linear operators. In the sequel, we denote E equipped with an inner product h· , ·i / = k · k . Assuming that the discrete-time processes r and y are square-integrable, the covarianceoperators induced by the kernels in (2.2) admit spectral decompositions over E as follows Q y ( · ) = ∞ X k =1 λ k ( y ) h· , ϕ k ( y ) i ϕ k ( y ) , Q r ( · ) = ∞ X k =1 λ k ( r ) h· , ϕ k ( r ) i ϕ k ( r ) , where ( ϕ k ( r )) ∞ k =1 and ( ϕ k ( y )) ∞ k =1 are orthonormal bases for E with eigenvalues ( λ k ( r )) ∞ k =1 and( λ k ( y )) ∞ k =1 , respectively. The number of principal components of the forward and yield curves arethe number of non-zero eigenvalues of Q r and Q y , respectively. We assume that λ ( r ) > . . . >λ p ( r ) > λ p + i = 0; λ ( y ) > . . . > λ q ( y ) > λ q + i = 0 for every i ≥ max ( p, q ) < ∞ so that E k r t − µ r k = p X i =1 λ i ( r ) , E k y t − µ y k = q X i =1 λ i ( y ); t ∈ N . The explained variance associated with the k -th principal component for r and y can be expressed,respectively, by λ k ( r ) P pi =1 λ i ( r ) , λ k ( y ) P qi =1 λ i ( y ) . In the sequel, we denote Λ := I d − T where I d is the identity operator and T is given by (2.3).For a given bounded linear operator G , the correspondent self-adjoint operator will be denoted by G ∗ . The following simple remark shows how the covariance operators of the forward rate and yieldcurves are related to each other. Proposition 2.1.
Let y and r be square-integrable weakly-stationary E -valued discrete-time pro-cesses. Assume that Q r and Q y are finite-rank operators and E is a Hilbert space realizing (2.3).Then Q r = Λ Q y Λ ∗ and Q y = JQ r J ∗ . In particular, Rank Q r = Rank Q y . Proof:
We recall that Q r and Q y are the unique self-adjoint, non-negative and bounded operatorssuch that h Q r f, g i = E h r t − µ r , f ih r t − µ r , g i (2.4) h Q y f, g i = E h y t − µ y , f ih y t − µ y , g i ; f, g ∈ E, t ∈ N . (2.5)Moreover, we shall write y t ( x ) = J x ( r t ) and r t ( x ) = Λ x ( y t ). This fact, together with relations (2.4)and (2.5) and the continuity assumptions on Λ and J allow us to conclude that Q r = Λ Q y Λ ∗ and Q y = JQ r J ∗ . Now let us define the following finite-rank, non-negative and self-adjoint operators6 Q r := Q y Λ ∗ Λ , ¯ Q y := Q r J ∗ J. By the very definition, ¯ Q y and Q y share the same non-zero eigenvalues and therefore, Rank Q y = Rank ¯ Q y ≤ Rank Q r J ∗ ≤ Rank Q r . The same argument applies to ¯ Q r and Q r so that Rank Q r ≤ Rank Q y . This concludes the proof.Despite the simplicity of Proposition 2.1, it provides an important information on dimensionreducing techniques for forward rate and yield curves based on PCA: The effective number ofprincipal components should be the same for forward rates and yield curves. Proposition 2.1 contradicts the heuristic argument given by Lekkos (2000) who argues that theidentity y = J ( r ) would smooth the spectral structure of the yield curve covariance operator. Italso shed some light on the results reported by many authors who compare the PCA for forwardand yield curves. In one hand, Akahori and Liu (2011), Liu (2010), Lord and Pelsser (2007) andLekkos (2000) report substantial differences between the correlation structure of forward and yieldcurves. On the other hand, Alexander and Lvov (2003) report a very stable estimation by usingdiscretely compounded forward rates and the parametric Svensson family to extract the forwardrates. Their empirical results together with Proposition 2.1 might suggest that in order to extractthe spectral properties of forward rates from the observed bond prices, a reasonable choice of aparametric family of smooth curves is a good starting point. However, one should notice that thisprocedure may potentially introduce an a priory loading factor structure on forward rates. Aspointed out by Lord and Pelsser (2007), nonparametric procedures based on splines and bootstraptechniques might introduce a non-negligible noise on the forward curve.Proposition 2.1 implies that a remarkable difference in the number of principal componentsbetween forward and yield curves is a strong evidence for the presence of unobserved noise in thedata. If this is the case, then unobserved errors might induce a nontrivial bias in the statisticalanalysis of the forward rate curves. This remark will be the starting point for our analysis in theremainder of this article.
3. PCA based on LRCM and noisy data
One way to overcome the presence of observational errors in a PCA methodology is the so-called long-run covariance matrix (LRCM). We recall that the PCA method based on the usualstatic sample covariance matrix is only valid for independent and weakly stationary processes. In7he presence of some sort of temporal dependence caused by serial correlation or a contaminationprocess, the use of the static sample covariance matrix is not the correct one anymore.One alternative to correct those types of dependence is to use estimators based on the LRCM.In the sequel, we denote the process of interest as a vector w t , and assume weak stationarity andergodicity of w t . The LRCM of w is defined by V lr := lim n →∞ var ( √ n ¯ w ) = ∞ X j = −∞ γ ( j ) , where γ ( j ) := E h(cid:0) w t − E [ w t ] (cid:1)(cid:0) w t − j − E [ w t − j ] (cid:1) ⊤ i is the cross covariance in lag j , ¯ w is the samplemean and ⊤ denotes the transpose of a matrix. One important and standard case is γ ( j ) at j = 0.In this case, all the non-contemporaneous cross-variances are equal to zero and we retrieve the usualstatic covariance matrix V s := γ (0) whose the usual estimator will be denoted by ˆ V s .Although it is possible to estimate independently each covariance term γ ( j ) by the correspondentsample quantities ˆ γ ( j ), the natural estimator of the long run matrix ˆ V n := P n − j = − ( n − ˆ γ ( j ) is notconsistent because the number of parameters grows proportional to the sample size. In order toovercome this problem, a general non-parametric class of LRCM consistent estimators is introducedby Andrews (1991) ˆ V Alr := ( n − X j = − ( n − α ( j )ˆ γ ( j ) , (3.1)where α ( j ) is a sequence of weights of the form α ( j ) = K ( j/b ), where K ( · ) is a continuous symmetrickernel function such that K (0) = 1 and b a suitable bandwidth parameter such that b → ∞ as n → ∞ .Some optimal choices for the kernel function and the bandwidth parameter in (3.1) are discussedby Andrews (1991). The most important one is the bandwith parameter. Precise conditions forconsistency of the LRCM estimators of type (3.1) based on kernel methods require that the bandwithincreases slower than sample size.However it is well-known that this class of asymptotic estimators does not go well in finite sam-ples, in particular for processes with strong dependence and temporal heterogeneity (see e.g. M¨uller(2007)). As discussed by M¨uller (2007), statistical inference by using consistent LRCM estima-tors performs badly in small samples and also in the presence of dependence and contamina-tion/measument errors. To overcome these problems, Kiefer and Vogelsang (2002, 2005) introducea class of kernel-type estimators with a bandwidth rule given by a fixed portion of the sample size,known as fixed-b estimators. These LRCM estimators are not consistent due to a fixed bandwidthparameter. However, they explicitly incorporate parameter uncertainty and they present very goodfinite sample properties in hypothesis testing. See Kiefer and Vogelsang (2002, 2005) for details.8n particular, Kiefer and Vogelsang (2002) suggest the use of the whole sample as a possible band-width rule in the construction of the LRCM estimator (3.1). This strategy allows us to use all lagsof X in (1.6) in the estimation procedure. This will be particularly important for noisy data setswith contaminations induced by market microstructure effects and interpolation procedures (A-B) discussed in the Introduction.Let ˆ u t be the mean adjusted deviations of the series w t . The Kiefer and Vogelsang (2002)estimator is defined by ˆ V V Klr := T − T X i =1 T X j =1 ˆ u i (cid:18) − | i − j | T (cid:19) ˆ u j , (3.2)which corresponds to the use of a Bartlett kernel and bandwidth equal to the sample size in theLRCM estimation.In order to handle dependent error structures and outliers, M¨uller (2007) extends the resultsin Kiefer and Vogelsang (2005, 2002). He shows that the usual LRCM estimators are extremelyfragile in the presence of contamination and outliers. He suggests a class of LRCM estimators withbandwidths based on fixed portions of the sample size. The class of estimators proposed by M¨uller(2007) can be constructed in the same spirit of Kiefer and Vogelsang (2002, 2005) but with onefundamental difference: They are asymptotically robust to contaminations in the autocorrelationstructure, in particular to contamination by moving average process. In particular, he obtains aclass of estimators ˆ V UA ( p ) lr (see pages 1339-1342 in M¨uller (2007)) which trades optimally robustnessand efficiency. The ˆ V UA ( p ) lr estimator is defined byˆ V UA ( p ) lr := P Ti =1 P Tj =1 ˆ ξ i ˆ ξ j p , (3.3)where ˆ ξ t is the residual of the linear regression of ˆ u t against a p -dimensional series ˆ υ t ( l ), l = 1 , . . . , p where ˆ υ t ( l ) := p /T cos( lπ ( t − / /T . The parameter 1 ≤ p < ∞ controls the bias and efficiencyin the LRCM estimation procedure. M¨uller (2007) shows that this estimator has good propertiesw.r.t robustness and efficiency. In addition, they present well-known asymptotic distributions underthe null in hypothesis testing.In this article, we show that the LRCM estimators (3.2) and (3.3) allow us to filter the de-pendence structure generated by observational errors and market microstructure effects, typicallyfounded in forward rate markets.
4. Simulation study on the number of factors
In order to investigate the impact of observational errors in the classical PCA methodologyapplied to forward rate curves, we now provide a detailed simulation study as follows. In order to9tudy different types of data contamination in forward rate curves, we assume that both the yieldand forward rate curves are subject to errors X t = r t + ε t , Z t = y t + η t ; t = 0 , . . . , T, (4.1)where ε t and η t are the error components to be specified.In addition to the analysis of the curves (4.1), it is also important to consider the first-differenceof the time series ˜ X t := X t − X t − and ˜ Z t := Z t − Z t − ; t = 1 , . . . , T. We recall that a directapplication of the PCA to the processes (4.1) implicitly assumes that X and Z are already weaklystationary. If X and (or) Z are assumed to be non-stationary, it is necessary to apply the PCAdecomposition at the first-differences ˜ X and ˜ Z , which are possibly weakly stationary processes.There is also an intrinsic reason to study the first-difference of the time series. Due to the HJMrepresentation of the forward rate (see (1.2)), the PCA methodology must be computed in termsof the increments rather than the first levels. See Jarrow (2002) and Filipovic (2009) for furtherdetails.An obvious consequence of Proposition 2.1 is the following corollary. Corollary 4.1.
Assume that both r and y are independent weakly stationary processes. If η = ε =0 , then the number of principal components of X and Z must be identical. The number of principal components will be the key parameter of study in order to infer theperformance of the usual PCA methodology based on ˆ V s against the LRCM estimators definedin (3.1), (3.2) and (3.3). Of course, the same statement of Corollary 4.1 holds for the first-differenceprocesses ˜ X and ˜ Y .In order to investigate the impact of observational errors in the PCA methodology, we proceedthe analysis on two different prominent models: Gaussian HJM and Cox-Ingersoll-Ross models. Asfar as the Gaussian HJM model r t (see (1.4)) is concerned, the volatility parameter is calibratedbased on the classical interest rate curves studied by Diebold and Li (2006). It consists of zerocoupon (Treasury bond) bonds with maturities 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 90, 108and 120 months covering 1985-Jan to 2000-Dec. The Gaussian HJM simulation is based on thiscalibrated volatility parameter where the number of principal components is equal to 3 (three).The specification of the Cox-Ingersoll-Ross model (henceforth abbreviated by CIR model) isbased on Chen and Scott (2003). In this case, a three-factor CIR model is simulated by means ofa short rate process of the form short t = P i =1 Y it where each latent factor follows the stochasticdifferential equation dY it = κ i ( θ i − Y it ) dt + σ i q Y it dB it ; i = 1 , , . The parameters κ i , θ i and σ i used in the simulation study are chosen according to Chen and Scott102003) who estimate them based on weakly data (1980-1988) of the U.S Treasury market . Weperform 1,000 replications of these HJM and CIR data generating processes with sample size 1,000.We analyze the PCA methodology based on the standard sample covariance matrix estimatorˆ V s against the LRCM estimators ˆ V Alr , ˆ V V Klr and ˆ V UA ( p ) lr . Those estimators are applied to ( X, Z )and ( ˜ X, ˜ Z ). The ˆ V Alr estimator is specified with the quadratic-spectral kernel function and theoptimal bandwidth choice of Andrews (1991). The ˆ V V Klr estimator is specified with the Bartlettkernel function and the whole sample as a bandwidth rule. The ˆ V UA ( p ) lr estimator is specified with p = 4 components in the basis ˆ v t . See Section 3 for details.Figures 1-5 report the mean value of the cumulative R-Squared obtained in the Monte Carloexperiments with the PCA decompositions based on the static covariance matrix and the LRCMestimators. Figure 1 reports the PCA methodology applied to X = r + ε where ε = 0 and theassociated yield curve is computed via (1.1). Figure 2 reports the PCA methodology applied to Z = y + η where η = 0 and the associated forward rate curve is computed via y t ( x ) + x ∂y t ( x ) ∂x .Without the presence of observational errors, we notice that the number of principal componentsis correctly estimated for ( X, Z ) and ( ˜ X, ˜ Z ) by using any of the estimators. This result is notsurprising due to Proposition 2.1 and the fact that both r and y are weakly stationary processes.In the presence of observational errors the picture is rather different. We formulate the analysis based on two types of observational errors: Market microstructureeffects (MME) and interpolation error structure (IES) as described by (A-B) in the Introduction.In order to analyze the impact of MME in the PCA methodology, we introduce an additive errorstructure typically founded in interest rate markets due to transactions costs and liquidity premiumsin bond prices (see e.g Mizrach and Neely (2011) and Goyenko et al. (2011)). This phenomenonis directly observed by the existence of bid and ask prices. In the classical model of market mi-crostructure (e.g. Hasbrouck (1991)), the true price of the asset is within the range between theobserved bid and ask prices. In other words, the observed prices may be considered as the trueprice plus an additive measurement error.We are going to study the impact of the MME as follows. In the sequel, the forward rate process r in (4.2) and the yield process y in (4.3) are given, respectively, by the Gaussian HJM and CIRmodels as specified above. Figure 3 reports the PCA methodology applied to ( X t ( x ) = r t ( x ) + ε t ( x ); where ε is a Gaussian zero mean IID process with variance . Z t ( x ) = y t ( x ) + η t ( x ); where y is computed via formula (2 .
1) and η t ( x ) = x R x ε t ( r ) dr. (4.2) See Chen and Scott (2003) for the expressions for the affine factors and bond prices in the multifactor CIR model. ( Z t ( x ) = y t ( x ) + η t ( x ); where η is a Gaussian zero mean IID process with variance . X t ( x ) = r t ( x ) + ε t ( x ); where ε t ( x ) = η t ( x ) + x ∂η t ∂x ( x ) , r t ( x ) = y t ( x ) + x ∂y t ∂x ( x ) . (4.3)We remarkably notice (see Figures 3 and 4) the same structure reported by the empirical analysisgiven in Liu (2010) when using the classical static covariance estimator ˆ V s in the PCA methodology.We observe a substantial difference between the number of factors capable to explain the covariancestructure of the yield and forward rate curves, specially when applied to the first-difference of thecurves.In Figures 3 and 4, the estimates based on the standard sample covariance matrix ˆ V s arehighly biased for the forward rate curves. This can also be observed in the first-difference of theyield curves. We remarkable notice that both ˆ V V Klr and ˆ V UA ( p ) lr estimate the correct number offactors, especially when computed on the first-difference of the forward rate curves in which thecontamination is more problematic. The ˆ V Alr estimator still indicate an excessive number of factorsfor the forward rate curves.Figure 4 reports one typical situation of MME founded in practice. One should notice thatthe presence of additive IID Gaussian measurement errors in the observed yield curve does notaffect the PCA estimation. Neither the usual sample static covariance matrix ˆ V s nor ˆ V Alr , ˆ V V Klr and ˆ V UA ( p ) lr are affected by the presence of this type of error in the yield curve. In contrast, theestimates based on ˆ V s for the forward rate and first-difference of the yield curves are clearly biased.The situation is even worst when dealing with forward rate first-differences.The explanation for the results reported in Figures 3 and 4 lies in the following argument.We readily see the existence of MME in bond markets impacts differently forward rate and yieldcurves. In (4.3), the MME introduces a moving average structure with negative persistence intothe forward rate curve due to the components ∂y t ∂x and ∂η t ∂x ( x ). One should notice that the existenceof a moving average error structure affects severally the usual estimators of the covariance matrix.See M¨uller (2007) and Vogelsang and Wagner (2013) for a discussion about this issue. In contrast,the MME in (4.2) only introduces an IID component into the observed yield curve process. ThisIID component does not affect the temporal dependence of the yield curve. The results reported inFigures 1-4 show that the PCA methodology applied to forward rate curves is a very delicate issuebecause it might be subject to non-negligible observational errors due to MME. The presence ofMME explains the large difference between the number of components indicated by the PCA methodfor yield and forward curves. This remark is particularly more evident when double differentiationof the underlying moving average process appears due to the first-difference of the observed forwardrate curves in (4.3). The main reason to use LRCM estimators instead of the usual ˆ V s is to handle12hese types of observational errors. In particular, our results strongly indicate the estimators ˆ V Klr and ˆ V UA ( p ) lr have successfully filtered the MME in the PCA methodology.Let us now treat the case of a non-additive error structure G ( y t , η t ) due to interpolation of theyield curve y t as described in (A-B) . Figure 5 reports the PCA methodology applied to ( Z t ( x ) = G ( y t ( x ) , η t ( x )); where G and η are generated by an IES induced by a cubic splines X t ( x ) = Z t ( x ) + x ∂Z t ∂x ( x ) , (4.4)where y is given by the CIR model as specified above. In this analysis, the error is generated by theclassical spline method of McCulloch (1975). We simulate bond prices generated by the CIR modeland we omit prices for 4 maturities randomly chosen, generating again 1,000 replications for thisexperiment. The omitted prices are then interpolated by cubic spline using the price curve, similarto the method of McCulloch (1975). Hence, observational errors for bond prices are then generatedby means of interpolation. From these prices, we generate yield and forward rate interpolatedcurves. The PCA method based on the estimators of Section 3 is then applied to ( Z, X, ˜ Z, ˜ X ) asspecified in (4.4).In Figure 5, the observational errors generated by the cubic splines do not significantly affectthe number of factors in yield curves neither in level nor first-difference. However, it has again anontrivial impact on the forward rate first-difference ˜ X . The number of factors of ˜ X computed bythe PCA based on ˆ V s and ˆ V Alr is not correctly estimated. The standard PCA method based on ˆ V s indicates the incorrect number of six factors which explain 99 % of total variation. Similar to theanalysis of MME given in Figures 3 and 4, the PCA based on ˆ V V Klr and ˆ V UA ( p ) lr again correctlyestimates the true number of three factors.The results obtained from these experiments show that the presence of observational errors gen-erated by MME or IES induce a significant bias in the classical PCA methodology applied to forwardrate curves. More importantly, our analysis shows that the LRCM estimators ( ˆ V V Klr , ˆ V UA ( p ) lr ) indi-cate the correct number of factors in the Monte Carlo experiments. The results of this section arerobust w.r.t to model specification and types of observational errors.
5. Empirical analysis for the numbers of factors
In this section, we compare the results described in Section 4 with a PCA application to twodistinct real data sets. The first one is given by Treasury bonds (zero coupon) with maturities 3,6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 90, 108 and 120 months (17 maturities), with monthlyobservations ranging from 1985-Jan to 2000-Dez. This data set is constructed based on the Fama-Bliss methodology (unsmoothed Fama-Bliss) and it was already used by Diebold and Li (2006).The second data set is the UK term-structure obtained from the Bank of England with maturities13 .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 1: Cumulative R obtained from the PCA decomposition for the level and first difference of the forwardand yield term structures. First experiment without observational errors, Gaussian HJM process. Mean values from1,000 Monte Carlo simulations. .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 2: Cumulative R obtained from the PCA decomposition for the level and first difference of the forward andyield term structures . Second experiment without observational errors, Cox-Ingersoll-Ross process. Mean valuesfrom 1,000 Monte Carlo simulations. .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 3: Cumulative R obtained from the PCA decomposition for the level and first difference of the forwardand yield term structures. First experiment with observational errors generated by market microstructure effects,Gaussian HJM process. Mean values from 1,000 Monte Carlo simulations. .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 4: Cumulative R obtained from the PCA decomposition for the level and first difference of the forwardand yield term structures. Second experiment with observational errors generated by market microstructure effects,Cox-Ingersoll-Ross process. Mean values from 1,000 Monte Carlo simulations. .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 5: Cumulative R obtained from the PCA decomposition for the level and first difference of the forward andyield term structures. Experiment with observational errors generated by cubic spline interpolation, Cox-Ingersoll-Ross process. Mean values from 1,000 Monte Carlo simulations. .40.60.81.0 1 2 3 4 5 6 7 8 9 1011121314151617 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 6: Cumulative R obtained from the PCA decomposition for the level and first difference of the forward andyield term structures - U.S. Treasury Curve - Fama-Bliss Database. .5 to 25 years (50 maturities) and daily data ranging from Jan/4/2005 to Feb/29/2012 summing1872 observations.In Figures 6 and 7, we report the cumulative R obtained from the PCA decomposition forthe forward rate and yield curves of the U. S. and U. K. markets, respectively. The usual staticcovariance matrix estimator ˆ V s applied to these two data sets shows the same behavior as describedby Liu (2010). We observe a large difference for the number of principal components between yieldand forward rate curves, both in the first-difference and level. In particular, we observe the sametype of behavior as described in Section 4. More importantly, we notice that both estimators ˆ V V Klr and ˆ V UA ( p ) lr indicate the same number of factors between yield and forward rate curves. This resultholds for both the first-difference and the level of the curves. More importantly, it corroborateswith our findings in Figure 4.According to Proposition 2.1, these results strongly suggest the presence of contamination errorsin the data. In the US data, the error contamination might be explained by the Fama-Bliss methodof construction of the forward rate curve (Fama and Bliss (1987)) which uses a piecewise constantfunction to approximate the discount factor. The Bank of England term structure is constructed19 .40.60.81.0 0 10 20 30 40 50 Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Forward Rate Difference
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Level
Factor R S qua r ed Method
StaticAndrewsV.KMüller
R Squared − Yield Rate Difference
Figure 7: Cumulative R obtained from the PCA decomposition for the level and first difference of the forward andyield term structures - U.K. Bank of England Term Structure Database.
20y means of a smooth cubic spline interpolation method for the yield and forward rate curves (seee.g Anderson and Sleath (2001) for details). Therefore, it is subject to observational errors similarto the Monte Carlo experiment (4.4) in Figure 5. The presence of MME in the US data might alsobe a source of observational errors as reported by Goyenko et al. (2011) and Mizrach and Neely(2011). The reader is urged to compare the Monte Carlo experiments (4.3) and (4.4) to Figures 6and 7, respectively. The empirical analysis of this section jointly with Section 4 and Proposition 2.1give strong support for the use of PCA based on the LRCM estimators described in Section 3.
6. Pricing interest rate derivatives
In Section 4, we study the impact of noisy data in the PCA methodology. In Section 5, wereport the existence of nontrivial measurement errors in forward rate markets. In this section, theprimary goal is to illustrate the importance of a correct spectral analysis in noisy forward ratecurves. The example we choose to illustrate this point is the pricing of interest rate derivativesin the presence of noisy data. We study the impact of observational errors in pricing interest ratederivatives by means of the PCA methodology. More precisely, we compare the pricing error of thePCA methodology based on ˆ V s against the LRCM estimators.Similar to Mercurio and Moraleda (2000), we consider a vanilla-type call option based on a zerocoupon bond with 10 years maturity, with time to maturities of .25, .5 and 1 years, and distinctstrike prices. The underlying data generating process is specified by a 2-factor Hull-White (g2++)model (see e.g pages 356-364 in Nawalkha et al. (2007)) which yields an analytic pricing formulafor vanilla options written on a zero coupon bond. This model is affine whose the correspondentforward rate volatility function is given by V ol [ df ( t, T )] = q υ e − κ ( T − t ) + υ e − κ ( T − t ) + 2 υ υ ρ e − ( κ + κ )( T − t ) ; 0 ≤ t ≤ T, where the parameters v , v , κ , κ , ρ ∈ R . The payoff of interest is given by V T := max (cid:8) P ( T , T ) − K, (cid:9) for some T < T , where T is the time of exercising the payoff V T . The Gaussian structure of the2-factor Hull-White model yields a closed form expression for the price C ( T , T ) of V T as follows C ( T , T ) = P (0 , T )Φ( d + ) − P (0 , T )Φ( d − ) , where Φ is the Gaussian cumulative distribution function, d ± := ln P (0 ,T ) KP (0 ,T ± v √ v and v is the integratedvolatility between [ T , T ] which follows equation (4.41) in Andersen and Piterbarg (2010), page(185). 21e construct the interest rate curves with maturities 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 40, 60,72, 90, 108 and 120 months with delta of order 1/252 (daily data) between each curve observation.In order to verify the robustness of our results, we perform a Monte Carlo study with two distinctset of parameters - the first one consist κ = . , κ = . , v = . , v = . ρ = − .
3, and thesecond one κ = . , κ = . , v = . , v = . ρ = − .
3. The Monte Carlo experiments arebased on the observational process Z = y + η , where y is the correspondent yield curve processspecified by the above 2-factor Hull-White model and η is zero or an IID zero mean Gaussian noisewith variance . M SE ( P rice pca ( Z ) ( T , T ) , C ( T , T ))where P rice pca ( Z ) ( T , T ) is the price computed by means of the estimation of the integrated volatil-ity v based on PCA as a function of the sample covariance matrix ˆ V s and the LRCM ˆ V Alr , ˆ V V Klr andˆ V UA ( p ) lr , given, respectively, by formulas (3.1), (3.2) and (3.3).Table 1 reports the Mean Square Error pricing with the first set of parameters, η = 0 and basedon a sample size of 250 observations. The PCA pricing using the sample covariance matrix ˆ V s andthe estimators ˆ V Alr and ˆ V UA ( p ) lr presents similar results in terms of pricing errors, while the ˆ V V Klr estimator achieves the best results. This can be explained by the excellent finite sample propertiesof ˆ V V Klr compared with the other estimators. See Kiefer and Vogelsang (2005, 2002) for furtherdetails.The results reported in Tables 2 and 3 suggest that the pricing of interest rate derivatives basedon PCA with sample covariance matrix ˆ V s is sensitive to observational errors. Clearly, the pricingbased on the LRCM estimators ˆ V Alr , ˆ V V Klr and ˆ V UA ( p ) lr presents better performance. We clearlyobserve smaller Mean Squared Errors for all strikes and time to maturities. This restates thevalidity and relevance of the corrections w.r.t the estimation of the covariance matrix in the PCAmethodology.The estimator ˆ V V Klr presents the best performance among the methods discussed in Section 3.This result is particularly important because it is the simplest one to implement where no bandwidthparameter estimation is needed. These results strongly suggest that neglecting the underlyingdependence structure generated by observational errors may degenerate significantly the pricing ofinterest rate derivatives. We also present in Table 4 the experiment with the first parameter setand with contaminations errors, but now using a sample of 1,000 observations. The results aresimilar to Tables 2 and 3. The Monte Carlo experiment of this section is robust w.r.t sample sizeand parameters. 2225 Years to Mat. Strike 0.45 0.5 0.55 0.6Static ˆ V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr Table 1: MSE - Option Pricing on a Zero Coupon Bond - without measurement errors, first parameter set, samplesize 250.
7. Final remarks
In this article, we discuss the impact of the inherent presence of measurement errors in theclassical application of the PCA methodology in the estimation of forward rate curves. Our re-sults strongly suggest the classical PCA method based on the standard sample covariance matrixis not suitable for estimating factors of forward rate curves. The main reason is the appearanceof non-negligible observational errors in the implied forward rate curves. An alternative method-ology based on so-called long-run covariance matrix estimators seems to improve significantly thequality of the estimation of the principal components of forward rate curves. We illustrate theimportance of a correct spectral analysis in forward rate curves by presenting non-trivial effects ofnoisy data in pricing errors related to European call options. Lastly, the results of this paper yielda sound explanation for the remarkable difference between the estimated number of factors in yieldand forward rate curves reported in the literature (see e.g. Akahori et al. (2006), Akahori and Liu(2011), Lekkos (2000) and Liu (2010)).Our conclusion is supported by the following results presented in this paper. Proposition 2.1proves that the number of principal components of the observed forward rate and yield curvesmust be identical in the absence of measurement errors. Based on this fact, we perform a detailedsimulation analysis in three prominent interest rate models with a number of distinct parameters inthe presence of observational errors of various magnitudes and forms. Market microstructure effectsand interpolation errors due to extraction of forward rates from yield curves are carefully analyzed.The results clearly indicate a considerable bias in the classical PCA methodology applied to forward2325 Years to Mat. Strike 0.45 0.5 0.55 0.6Static ˆ V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr Table 2: MSE - Option Pricing on a Zero Coupon Bond - with measurement errors, first parameter set, sample size250 rate curves. In contrast, Monte Carlo experiments jointly with empirical analysis strongly suggestthat PCA based on long run covariance matrices is robust w.r.t measurement errors in forward ratecurves. The presence of observational errors in forward rate markets is validated by Proposition 2.1together with a detailed empirical analysis on the number of principal components in the UK andUS interest rate markets. Monte Carlo experiments reports that classical PCA based on samplecovariance matrices presents nontrivial pricing errors for European call options. In contrast, theuse of long run covariance matrices seems to correct this bias.
References
Akahori, J., Aoki, H., Nagata, Y., 2006. Generalizations of Ho-Lee’s binomial interest rate modelI: from one- to multi-factor. Asia-Pacific Financial Markets 13, 151–179.Akahori, J., Liu, N.-L., 2011. On a type-I error of a random walk hypothesis of interest rates.International Journal of Innovative Computing, Information and Control 7, 115–131.Alexander, C., Lvov, D., 2003. Statistical properties of forward LIBOR rates, working paper.Andersen, L. F. B., Piterbarg, V. V., 2010. Interest Rate Modeling, Volume I: Foundations andVanilla Models. Atlantic Financial Press.Anderson, N., Sleath, J., 2001. New estimates of the uk real and nominal yield curves. The Bankof England Working Paper Series 126, 1–41. 2425 Years to Mat. Strike 0.45 0.5 0.55 0.6Static ˆ V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr V s V Alr V V Klr V UA ( p ) lr Table 3: MSE - Option Pricing on a Zero Coupon Bond - with measurement errors, second parameter set, samplesize 250.