Historical Yield Curve Scenarios Generation without Resorting to Variance Reduction Techniques

Transcription

1 Working Paper Series National Centre of Copetence in Research Financial Valuation and Risk Manageent Working Paper No. 136 Historical Yield Curve Scenarios Generation without Resorting to Variance Reduction Techniques Francesco Audrino Fabio Trojani First version: February 2003 Current version: June 2003 This research has been carried out within the NCCR FINRISK project on Interest Rate and Volatility Risk.

2 Historical Yield Curve Scenarios Generation without Resorting to Variance Reduction Techniques Francesco Audrino and Fabio Trojani Institute of Finance University of Southern Switzerland First Version: February 2003 This version: June 2003 Corresponding author. Address: USI, Institute of Finance, Via Buffi 13, Centrocivico, CH-6900 Lugano, Switzerland. E-ail: Address: USI, Institute of Finance, Via Buffi 13, Centrocivico, CH-6900 Lugano, Switzerland. E-ail: Financial support by the National Centre of Copetence in Research Financial Valuation and Risk Manageent (NCCR FINRISK) is gratefully acknowledged. 1

3 Abstract We propose a ultivariate nonparaetric technique for generating reliable scenarios and confidence intervals for the ter structure of interest rates fro historical data. The approach is based on a functional gradient descent (FGD) estiation of the conditional ean vector and the conditional volatility atrix of a ultivariate interest rate series. The ethodology is coputationally feasible in large diensions and avoids the use of variance reduction techniques like for instance principle coponents analysis. Moreover, it can account for a non-linear tie series dependence of interest rates at all available aturities. Based on the estiated FGD ters structure dynaics we apply filtered historical siulation to copute out-of-saple ter structure scenarios and confidence intervals. We apply our ethodology to daily USD bond data and back-test its out-of-saple accuracy for forecasting horizons fro 1 to 10 days. When copared with soe further scenario generating technologies based on principal coponents, a ultivariate CCC-GARCH odel, or the exponential soothing volatility forecasting technique used by the RiskMetrics TM approach, we find epirical evidence of a clearly higher predictive potential of FGD-based scenarios generating techniques. Specifically, at forecasting horizons of one day FGD provided accurate ultivariate VaR coputations for ties to aturity between one onth and thirty years. For longer horizons (i.e. ten days) accurate VaR predictions are obtained for ties to aturity between roughly one and thirty years. Key words: Conditional ean and volatility estiation; Filtered Historical Siulation; Functional Gradient Descent; Ter structure; Multivariate CCC-GARCH odels 2

4 1 Introduction The quality and the effectiveness of interest rate risk anageent depends on the ability to generate relevant forward looking ter structure scenarios that properly represent the future. Based on such scenarios, future distributions of interest rate dependent portfolio exposures and associated risk easures like VaR can be ultiately derived fro the future distributions of the underlying future interest rates. One broadly used approach to the estiation of interest rate scenarios and associated risk easures is based on the historical/monte Carlo siulation of the standardized residuals in a ter structure odel based on state dependent conditional eans and/or volatilities; see Barone- Adesi et al. (1998) and Barone-Adesi et al. (1999), (2002), for an introduction to the filtered historical siulation ethod and Jashidian and Zhu (1997) and Reiers and Zerbs (1999) for the Monte Carlo ethod applied to generating ter structure scenarios. While in a pure Monte Carlo setting paraetric assuptions on the conditional distribution of standardized residuals have to be introduced, the historical siulation ethod is nonparaetric and can incorporate a quite broad variety of historical distributional patterns. Since we do not want to rely on paraetric assuptions on the distribution of standardized interest rate residuals we use in the following this ethod to copute out-of-saple interest rate scenarios and interval estiates. A necessary ingredient of the filtered historical siulation ethod is the estiation of a dynaic odel for the conditional eans and/or volatilities of the joint interest rate dynaics. Conditionally on such a odel estiate, standardized residuals can be then coputed and bootstrapped to generate out-of-saple confidence intervals for either soe interest rates at different aturities or the prices of soe interest rate dependent securities. The estiation of a dynaic odel for the conditional eans and/or volatilities of the joint interest rate dynaics is a challenging task because ter structures are typically high diensional objects. Moreover, in any relevant applications it can be necessary to odel not only the ter structure dynaics but also the ones of further risk factors like for instance exchange rates. For these reasons any authors have often applied soe for of variance reduction technique like principal coponent analysis to reduce the estiation proble to an acceptable diension. Soe exaples of such types of ethodologies are presented and discussed in Engle et al. (1990), Loretan (1997), Rodrigues (1997) and Alexander (2001). An even sipler approach to this estiation proble is adopted by RiskMetrics TM which uses an exponential soothing volatility estiator to estiate conditional volatilities. 3

5 This paper proposes a ultivariate nonparaetric technique based on Functional Gradient Descent (FGD, Audrino and Bühlann (2003)) for estiating out-of-saple scenarios and confidence intervals of the ter structure of interest rates fro historical data. The ethodology is coputationally feasible in large diensions and avoids the use of variance reduction techniques like for instance principle coponents analysis. This allows us to estiate jointly the whole ter structure dynaics, fro the very short aturity segents (i.e. the overnight aturity) up to its very long end (i.e. 10 to 30 years aturity rates) 1. Moreover, the non paraetric nature of our approach allows us to account for non-linear dependencies of conditional eans and volatilities on potentially all available interest rate aturities, a feature that is crucial - as we show below - in order to produce an accurate forecasting power also for interest rates in the very short aturity spectru or for prediction horizons longer than one day. Based on the estiated FGD ter structure dynaics we apply filtered historical siulation to copute non paraetric out-of-saple ter structure scenarios and confidence intervals for forecasting horizons fro one to ten days. We apply our scenarios generating ethodology to daily USD bond data and back-test its out-of-saple forecasting accuracy, relative to soe further historical siulation techniques based on principal coponent analysis, a ultivariate AR-CCC-GARCH (Bollerslev (1990)) ter structure dynaics, or the exponential soothing volatility forecasting technique used by the RiskMetrics TM approach. Based on several perforance easures, our results produce epirical evidence of a clearly higher predictive potential of FGD-based scenarios generating techniques. The paper is organizes as follows. Section 2 presents the odel dynaics underlying our approach and the FGD estiation procedure necessary to estiate it. A short description of our filtered historical siulation procedure is also included. Section 3 introduces our application to a tie series of daily USD ter structures and presents the results of our out-of-saple backtesting analysis. Section 4 concludes and suarizes. 2 The yield curve scenarios generating ethodology This section introduces first our ultivariate odel for the conditional ean and volatilities of the joint ter structure dynaics. In a second step, the FGD estiation procedure is presented theoretically, together with a coputationally feasible algorith that can be applied to estiate 1 Moreover, the incorporation of a possibly high nuber of further risk factors, like for instance exchange rates, to forecast interest rates can be easily accoplished when using FGD. 4

6 the odel. Finally, the filtered historical siulation approach relevant for our setting is briefly reviewed. 2.1 The general odel We consider a ultivariate tie series R = {r t } t Z, r t = (r t,t+t1,.., r t+td ), of spot interest rates for a given set of fixed ties to aturity T 1 <... < T d. Therefore, r t is the yield curve at tie t. Denote further by X = {x t } t Z, x t = r t r t 1, the corresponding tie series of interest rate changes 2. It is assued that R is a strictly stationary process. Denoting by F t 1 the inforation available up to tie t 1, we odel the dynaics of the conditional ean µ t = E (x t F t 1 ) and the conditional variance V t = Cov(x t F t 1 ) of yield curve changes x t by odelling explicitly the joint interest rate dynaics for all available aturities. No variance reduction technique is used in the whole procedure. The basic idea is to extend the classical constant conditional correlation (CCC) GARCH odel firstly introduced by Bollerslev (1990) to take into account possible nonparaetric nonlinearities in the functional for of µ t and V t. We thus consider a tie series process of the for x t = µ t + Σ t z t, (2.1) where the following assuptions are introduced: (A1) (Innovations) {z t } t Z is a sequence of i.i.d. ultivariate innovations with zero ean and covariance atrix Cov(z t F t 1 ) = I d. (A2) (CCC construction) The conditional covariance atrix V t = Σ t Σ t is alost surely positive definite for all t. A typical eleent of V t is given as v t,ij = ρ t,ij (v t,ii v t,jj ) 1/2, where i, j = 1,.., d. The paraeter ρ t,ij = Corr(x t,ti, x t,tj F t 1 ) is the conditional correlation between the single series coponents i and j of the process X. It is assued in the sequel that ρ t,ij is tie invariant, i.e. ρ t,ij = ρ ij for soe scalars 1 ρ ij 1. Recall that by construction we have ρ ii = 1. 2 We odel interest rate changes rather than levels in order to lower both the siultanous correlations and the autocorrelations of the interest rates series under scrutiny. This produces filtered standardized residuals having better statistical properties for the historical siulation of out-of-saple yield curve scenarios. 5

7 (A3) (Functional for for conditional variances) The conditional variances are given by a nonparaetric functional for given by v t,ii = σ 2 t,i = Var(x t,ti F t 1 ) = F i ({r t j,tk : j = 1, 2,... ; k = 1,..., d}) where F i is a function taking values in R +. (A4) (Functional for for conditional eans) The conditional ean µ t is given by a nonparaetric functional for given by µ t = (µ t,1,..., µ t,d ), µ t,i = G i ({r t j,tk : j = 1, 2,... ; k = 1,..., d}) where G i is a function taking values in R. Assuption (A1) is standard, for instance when working with ultivariate tie series odels of the GARCH faily. For estiations purposes a specific pseudo log likelihood for z t (for instance a ultivariate noral one) is introduced; see Section 2.2 below. By Assuption (A2) the conditional covariance atrix V t is of the for V t = Σ t Σ t = D t RD t, where D t = diag(σ t,1,..., σ t,d ) and R = [ρ ij ] d i,j=1 is a atrix of constant correlations. The nonparaetric functional fors (A3)-(A4) perit a rich specification of conditional eans, variances and (indirectly) conditional covariances. For instance, cross-dependencies across the different interest rates can be odelled. Siilarly, a ean reversion or a nonlinearity in conditional eans can be easily accounted for, as well as functional fors for conditional volatilities that are dependent on the level of current and past interest rates. As a consequence, several odels in the literature are special cases of the above setting. The standard paraetric CCC-GARCH odel is encopassed by (2.1). Siilarly, ultivariate AR-CCC-GARCH odels where the conditional eans µ t,i incorporate ean reversion in the standard way are special cases of the above setting. Finally, also volatility odels where volatility includes asyetric responses to past shocks are a special case of the above specification. Estiation of the above ultivariate odel in its full generality is a very challenging task, because of the curse of diensionality proble arising when the diension d is not a very low one. A coputationally feasible but still very generaly version of the above odel can be estiated 6

8 by the Functional Gradient Descent (FGD) technique (Friedan et al., 2000, and Friedan, 2001). Applications of this ethodology to the estiation of ultivariate equity dynaics (see Audrino and Barone-Adesi, 2002, and Audrino and Bühlann, 2003) have deonstrated that FGD is a powerful ethodology which allows to construct accurate estiates and predictions for the ultivariate conditional ean µ t and volatility atrix V t also for very large diensions d. In this paper we apply a version of the FGD technique to estiate the joint dynaics of the whole ter structure, fro the very short aturity segents (i.e. the overnight aturity) up to its very long end (i.e. 10 to 30 years aturity rates). Unlike several studies on the estiation and the prediction of the yield curve, this approach avoids relying on any for of variance reduction technique, like for instance Principal Coponents or Factor Analysis (PCA and FA, respectively). This has several advantages. First, we do not need to rely in our approach on restrictive assuptions necessary to apply consistently PCA or FA in a general tie series context based on stochastic conditional eans and volatilities (see for instance Mardia (1971) for an exposition of PCA and FA). Second, we can estiate the joint ter structure dynaics also over its very short ter aturity spectru where the high variability of short ter interest rates can ake the application of variance reduction techniques cubersoe. Third, the joint ter structure dynaics estiated by FGD are directly interpretable in ters of observable interest rate variables and can be naturally related to the prices of further interest rates derivatives, as for instance forward rates. By contrast, in PCA or FA the estiated factors are typically interpreted ex post as soe abstract shift-, slope- or curvature factors in the spot yield curve. These factors cannot be however naturally reconverted into forward rate factors without introducing iplicitly strong restrictions in the estiated forward curve dynaics (see for instance Lekkos (2000)) for a discussion of this point). The next section introduces the FGD odelling approach for estiating the conditional ean µ t and the conditional atrix function V t in a version of the general odel (2.1) based on Assuptions (A1)-(A4). 2.2 Conditional ean and volatility estiation using Functional Gradient Descent The ain idea of our FGD approach is to copute estiates Ĝi( ) and F i ( ) for the nonparaetric functions G i ( ) and F i ( ), i = 1,.., d, which iniize a joint negative pseudo log likelihood 7

9 λ under soe further constraint on the functional for of Ĝi( ) and F i ( ). More specifically, given an initial estiate Ĝi0( ) and F i0 ( ), i = 1,.., d - coputed for instance fro a paraetric AR-CCC-GARCH odel - the estiates Ĝi( ) and F i ( ) are obtained as additive nonparaetric expansions around Ĝi0( ) and F i0 ( ). Such nonparaetric expansions are based on soe siple estiates of the gradient of the loss function λ in a neighborhood of the initial estiates Ĝi0( ) and F i0 ( ). These siple estiates are obtained fro a base learner S least squares fitting 3. Fro the siple estiates of the gradient of the loss function λ, FGD deterines Ĝi( ) and F i ( ) as additive nonparaetric expansions of Ĝi0( ) and F i0 ( ) which iniize the joint negative pseudo log likelihood λ. Therefore, our FGD approach ais at producing estiates which iprove locally the pseudo log like likelihood of soe initial estiates Ĝi0( ) and F i0 ( ) by eans on nonparaetric additive expansions Ĝi( ) and F i ( ). Conditionally on the first p observations, the negative pseudo log likelihood iplied by a Gaussian distribution of z t in (2.1) is given by: = n t=p+1 n t=p+1 ( ) log (2π) d/2 det(v t ) 1/2 exp( ξt T Vt 1 ξ t /2) ( log(det(d t )) + 1 ) 2 (D 1 t ξ t ) R 1 (Dt 1 ξ t ) + n d log(2π)/2 + n log(det(r))/2 (2.2) where ξ t = x t µ t, D t is a diagonal atrix with eleents v t,ii and n = n p. Therefore, a natural conditional loss function for our FGD estiation procedure is given by the functional for λ R (x, G, F) = log(det(d(f)) (D(F) 1 (x G)) R 1 (D(F) 1 (x G)) log(det(r)) + d 2 log(2π), D(F) = diag( F 1,..., F d ), x G = (x 1 G 1,..., x d G d ), (2.3) where the ters d log(2π)/2 and log(det(r))/2 are constants that do not affect the optiization. As highlighted by the subscript R, the value of the loss function λ R depends on the unknown correlation atrix R. At any step of our FGD optiization procedure, the updated optial values of R, G, F will be constructed by a two step procedure. For a given initial correlation 3 Well known exaples of base learners are regression trees, projection pursuit, neural nets or splines; see also Friedan et al. (2000), Friedan (2001), Audrino and Barone-Adesi (2002), Audrino and Bühlann (2003) and Bühlann and Yu (2003) for ore details. 8

10 atrix R, updated estiates for all G i s and F i s are obtained by iniizing λ R with respect to G, F. In a second step, given the updated estiates Ĝ and F the correlation atrix is updated using the epirical oents of the resulting standardized ultivariate residuals. Therefore, given estiates Ĝ = (Ĝ1,..., Ĝd) and ˆF = ( ˆF 1,..., ˆF d ), we copute the standardized residuals ˆε t,i = ( x t,i Ĝi(r t 1,...) ) / ˆF i (r t 1,...) 1/2, t = p + 1,..., n to obtain the epirical correlation atrix n ˆR = (n p) 1 ˆε tˆε T t, ˆε t = (ˆε t,1,..., ˆε t,d ), (2.4) t=p+1 as an updated estiate of R. The optiization of λ R with respect to G, F is perfored by calculating the partial derivatives of the loss function λ R with respect to all G i s and F i s. In our setting, they are given for any i = 1,..., d, by and λ R (x, G, F) G i = λ R (x, G, F) = 1 ( 1 F i 2 F i d γ ij (x j G j ) F 1/2 i F 1/2, (2.5) j j=1 d γ ij (x i G i )(x j G j ) j=1 F 3/2 i F 1/2 j ), (2.6) respectively, where [γ ij ] d i,j=1 = R 1. This step of the optiization suggests the nae Functional Gradient Descent. Indeed, given initial estiates Ĝi0( ) and F i0 ( ), i = 1,.., d, the above gradients are used by the FGD ethodology to define a set of siple additive expansions of the functions Ĝi0( ) and F i0 ( ) which iprove the optiization criterion precisely in the directions of steepest descent of the loss function λ R. Since these expansions define a nonparaetric estiate of G and F, the resulting optiization is a functional one. Details on the FGD algorith used in the paper are presented below. In Step 2 of the algorith the above gradients are fitted by eans of a base learner S. In Step 3 and 4, the estiated gradients are used to define a set of additive expansions Ĝi0( ) and F i0 ( ) which iprove the optiization criterion precisely in the directions of steepest descent of λ. Algorith: Estiating conditional eans and volatilities Step 1 (initialization). Choose appropriate starting function Ĝi,0( ) and ˆF i,0 ( ) and define for 9

11 i = 1,.., d and t = p + 1,.., n: Ĝ i,0 (t) = Ĝi,0(r t 1, r t 2,...) ˆF i,0 (t) = ˆF i,0 (r t 1, r t 2,...). Copute ˆR 0 as in (2.4) using Ĝ0 and ˆF 0. Set = 1. Natural starting functions in our application can be obtained by eans of univariate AR-GARCH estiates for the single coponents, i = 1,..., d, of the process X. Step 2 (projection of coponent gradients to base learner). For every coponent i = 1,..., d, perfor the following steps. (I) (ean) Copute the negative gradient U t,i = λ (x ˆR 1 t, G, ˆF 1 (t)) G G= Ĝ 1 (t), t = p + 1,..., n. i This is explicitly given in (2.5). Then, fit the negative gradient vector U i = (U p+1,i,..., U n,i ) with a base learner S, using always the first p tie-lagged predictor variables (i.e. (r t 1,.., r t p ) is the predictor for U t,i ): ĝ,i ( ) = S X (U i )( ), r t 1 t p = where S X (U i )(x) denotes the predicted value at x fro the base learner S using the response vector U i and a predictor variable X (say). (II) (volatility) Copute the negative gradient W t,i = λ (x ˆR 1 t, Ĝ 1(t), F) F F=ˆF 1 (t), t = p + 1,..., n. i This is explicitly given in (2.6). Then, analogously to (I) fit the negative gradient vector W i = (W p+1,i,..., W n,i ) with the base learner S, using again the first p tie-lagged predictor variables ˆf,i ( ) = S X (W i )( ). Step 3 (line search). Perfor a one-diensional optiization for the step-length, ŵ (e),i ŵ (vol),i = argin w = argin w n t=p+1 n t=p+1 λ ˆR 1 (x t, Ĝ 1(t) + wĝ,i (r t 1 t p ), ˆF 1 (t)), λ ˆR 1 (x t, Ĝ 1(t), ˆF 1 (t) + w ˆf,i (r t 1 t p )), 10

12 where Ĝ 1(t) + wĝ,i ( ) and ˆF 1 (t) + w ˆf,i ( ) are defined as the functions which are constructed by adding in the i th coponent only. This can be expressed ore explicitly using the functional for (2.3) 4. Step 4 (up-date). Select the best coponent for the conditional ean and volatility, respectively, as i (e) i (vol) = argin i = argin i n t=p+1 n t=p+1 λ ˆR 1 (x t, Ĝ 1(t) + ŵ (e),i ĝ,i (r t 1 t p ), ˆF 1 (t)) λ ˆR 1 (x t, Ĝ 1(t), ˆF 1 (t) + ŵ (vol),i ˆf,i (r t 1 t p )). If the iproveent in iniizing the epirical criterion (2.2) for the coponent i (e) conditional ean is larger than the one for the coponent i (vol) then up-date as Ĝ ( ) = Ĝ 1( ) + ŵ (e) ˆF ( ) = ˆF 1 ( ),i (e) ĝ (e),i ( ), in the in the conditional variance, and set j = 1. Else, up-date as Ĝ ( ) = Ĝ 1( ), ˆF ( ) = ˆF 1 ( ) + ŵ (vol),i (vol) ˆf (vol),i ( ) and set j = 2. Then, copute the new estiate ˆR according to (2.4) using Ĝ and ˆF. Step 5 (iteration). Increase by one and iterate Steps 2 4 up to an optial level = M. More details on the deterination of M are given in Reark 4 below. The resulting functions ĜM, ˆF M are our FGD estiates for conditional eans and volatilities. More forally, they are given by: Ĝ M ( ) = Ĝ0( ) + ˆF M ( ) = ˆF 0 ( ) + M ŵ (e),i (e) =1 M =1 ŵ (vol),i (vol) ĝ (e),i ( )I {j =1} ˆf (vol),i ( )I {j =2}. 4 The line search guarantees that the negative log-likelihood is onotonically decreasing in the nuber of iteration steps. 11

13 Reark 1. The base learner S in Step 2 deterines the FGD estiates ĜM( ) and ˆF M ( ) via the predicted values of the gradient of the objective function λ. The base learner should be a weak one - not involving a too large nuber of paraeters to be estiated - in order to avoid an iediate overfitted estiate at the first iteration of the algorith. The coplexity of the FGD estiates ĜM( ) and ˆF M ( ) is increased by adding further nonparaetric ters at every step of the above iterations. We use decision trees as base learners, because particularly in high diensions they are able to perfor a very effective variable selection by selecting only a few explanatory variables as predictors. This is not an exclusive choice: further base learners could be applied and copared based on soe for of cross-validation. Reark 2. As entioned, it is desirable to use sufficiently weak base learners in the above FGD algorith. A siple effective way to reduce the coplexity of a base learner is via shrinkage towards zero. In this case, the up-date Step 4 of the FGD algorith can be replaced by an updating step given by: Ĝ ( ) = Ĝ 1( ) + ν ŵ (e),i (e) ˆF ( ) = ˆF 1 ( ) + ν ŵ (vol),i (vol) ĝ (e),i ( ) or ˆf (vol),i ( ), (2.7) where ν [0, 1] is a shrinkage factor. This reduces the variance of the base learner by the factor ν 2. Reark 3. The initialization Step 1 in the above algorith is crucial, since FDG ais at iproving locally by eans of nonparaetric additive expansions the pseudo log likelihood criterion of an initial odel estiate. Therefore, it is iportant to start fro initial good estiates, in order to obtain a satisfactory perforance. In our application we ake use of the fit of a diagonal VAR(p i )-CCC-GARCH(1,1) odel 5 to initialize the FGD algorith by eans of functions G i,0, F i,0, i = 1,.., d, given by G i,0 (r t 1, r t 2,...) = µ t,i = φ k,i x t k,ti, p i k=1 F i,0 (r t 1, r t 2,...) = σ 2 t,i = α 0,i + α 1,i (x t 1,Ti µ t 1,i ) 2 + β i σ 2 t 1,i, where the autoregressive paraeter p i is selected in order to optiize the Akaike s Inforation Criterion (AIC) for each individual series i. We estiate by pseudo axiu likelihood this odel for each of the d individual series, thereby neglecting in the first step the structure of the 5 See Bollerslev (1990) for ore details. 12

14 correlation atrix R. This causes soe statistical loss in efficiency but has the advantage that the odel estiation is fast and therefore coputable also in very high diensions d. Reark 4. The stopping criterion in Step 4,5, is iportant. It can be viewed as a regularization device which is very effective when fitting a coplex odel. We deterine the stopping criterion by eans of a cross validation schee: for a give saple size n, we split the (in-saple) estiation period into two subsaples, the first of saple size 0.7 n (used as training set) and the second of saple size 0.3 n (used as test set). The optial value M to stop the algorith is then chosen as the one which optiizes the cross-validated log-likelihood 6. Our FGD procedure, connected with tree-structured base learners, provides a coputationally feasible and siple ethod aiing at iproving the pseudo log likelihood criterion, given a set of initial odel estiates. FGD perfors a one-diensional sequence of estiated predictions which are optiized by selecting the optial stopping value M with the above cross validation procedure. One could alternatively try to estiate predictions for the conditional ean µ t and the covariance atrix V t with soe ore coplex ultivariate GARCH odel. However, this becoes rapidly an intractable odel-selection and estiation proble in large diensions d. As entioned, in applications to the estiation of the ter structure dynaics this proble has been often circuvented by resorting to soe variance reduction techniques such as PCA or FA. Based on the FGD estiates for the ultivariate conditional ean vector µ t and for the covariance atrix V t, we apply in the next sections a filtered historical siulation procedure to generate out-of-saple scenarios for the ter structure of interest rates. Such an historical siulation procedure is briefly reviewed in the next section. 2.3 Siulation of future yield curve scenarios We generate future scenarios for the tie series R of interest rate levels. To this end, we apply a ultivariate version of the filtered historical siulation procedure proposed first by Barone-Adesi et al. (1998). Our historical siulation is based on a odel-based bootstrap of ultivariate filtered historical residuals, iplied by an FGD estiation of the ter structure dynaics. Using the bootstrapped residuals, we construct out of saple scenarios for the ter 6 This cross-validation schee has been shown to work well in epirical applications of FGD; see again Audrino and Barone-Adesi (2002) and Audrino and Bühlann (2003). 13

15 structure. The FGD odel estiate is used as the filter for the estiation of standardized ultivariate residuals. More details on the coplete siulation ethodology are as follows. In a first step, we filter the ultivariate standardized innovations z t with our odel (2.1): z t = (Σ t ) 1 (x t µ t ), V t = Σ t Σ T t = D t RD t, t = 1,..., n, where the individual conditional ean functions µ t,i = G i ( ) and (squared) volatility functions σt,i 2 = F i( ), i = 1,..., d are estiated by eans of our FGD technique, as described in detail by the algorith of Section 2.2. Under Assuption (A1), the standardized ultivariate innovations are i.i.d. and can be therefore bootstrapped. The historical standardized residuals are drawn randoly (with replaceent) and are used to generate pathways for future interest rate changes (and, consequently, for future interest rate levels). Hence, we apply a odel-based bootstrap (Efron and Tibshirani, 1993) where fro an i.i.d. resapling of the standardized ultivariate residuals z t we recursively generate a tie series of interest rates using the structure and the fitted paraeters of the estiated optial odel (2.1). Specifically, we draw randoly dates with corresponding standardized innovations z 1, z 2,..., z x, (2.8) where x is the tie horizon at which we want to generate future scenarios (in general fro 1 up to 10 days). We then construct for each tie to aturity T i pathways for future conditional eans and (squared) volatilities and interest rate levels, fro tie n + 1 up to tie n + x (say), based on the odel structure (2.1). More forally we copute the entities µ t+b,i = Ĝi({rt+b s,k ; s = 1, 2,..., p, k = 1,..., d}), v t+b,ii = ( σ t+b,i )2 = F i ({rt+b s,k ; s = 1, 2,..., p, k = 1,..., d}), v t+b,ij = ρ ij v t+b,ii v t+b,jj, x t+b,t i = µ t+b,i + ( Σ t+bẑ b ) i, r t+b,t i = r t+b 1,T i + x t+b,t i, b = 1,..., x, i, j = 1,..., d, (2.9) where all quantities denoted by are based on the odel structure estiated by eans of the FGD algorith in section

16 The epirical distribution of siulated odel-based interest rate levels at the chosen future tie point n+x for each series i = 1,..., d, is obtained by replicating the above procedure a large nuber of ties, e.g ties. Confidence bounds for the ter structure of interest rates at the future tie point n + x for a confidence level q are finally estiated by the lower and upper 1 q 2 -quantiles of the siulated epirical distribution of interest rates. We focus on confidence levels q = 0.90, 0.95, Epirical Results In this section we back-test on real data our scenario generation technique based on FGD for forecasting horizons x = 1, 3, 5, 10 days and for three different confidence levels q = 0.90, 0.95, We copare the perforance of our approach with an historical siulation procedure based on (i) the industry standard benchark 7 used by RiskMetrics TM and (ii) a standard ultivariate AR-CCC-GARCH. The second coparison is particularly useful, because it highlights the exact contribution of the FGD technique in enhancing the accuracy of VaR predictions for the yield curve relatively to a standard ultivariate GARCH odel. For copleteness, we also coputed in our study soe historical siulation scenarios based on a three factor decoposition of the yield curve dynaics. However, the back-testing results of this procedure were very poor and are therefore oitted. 3.1 Data We consider ultivariate tie series for the yield curves of daily interest rate levels r t,ti at twelve different aturities T i. For the lowest aturity segents, i.e. overnight, 1 week, 2 weeks, 1 onth, 2 onths, 3 onths, 6 onths and 1 year, we ake use of Euro dollar interest rates. For the higher aturities, i.e. 2 years, 5 years, 10 years and 30 years, we ake use of interest rates of US governent bonds. The data span the tie period between January 7 RiskMetrics TM uses an EWMA conditional variance estiator of the for V t = (1 λ)ξ t 1 ξt 1 T + λv t 1, λ = 0.94, (3.1) where V 0 can be fixed to be the saple covariance atrix or soe presaple data selection to begin the soother. This odel is extreely easy to estiate since it contains only one paraeter of interest. One obvious drawback is that it forces all assets to have the sae soothing coefficient λ = 0.94, irrespectively of the specific dynaic features of a given interest rate. 15

17 1, 1996 and Septeber 30, 2002, for a total of 1760 trading days, and have been downloaded fro Data Strea International. We split our saple in a back-testing period used to test the predictive accuracy of our FGD ethodology and an in-saple estiation period used to initialize the odel paraeter estiates. The back-testing period goes fro January 3, 2000 to Septeber 30, 2002, for a total of 716 trading days. In our back-testing exercise the odel paraeters are re-estiated every 20 working days, as new data becoe available for prediction purposes, using all ultivariate past observations in the estiation of the odel dynaics. The updated conditional ean and volatility dynaics are then used to copute out of saple VaR predictions based on historical siulation for the whole back-testing period. Table 1 presents suary statistics of the tie series of interest rate changes in our saple, for each aturity. Figure 1 plots the yield curves in our saple as a function of tie and aturity. TABLE 1 AND FIGURE 1 ABOUT HERE. Table 1 shows that the saple eans of all interest rate changes in our saple are negative, highlighting the fact that in our back-testing period the Fed reduced several ties the target interest rate. This effect is ore pronounced for interest rates up to 2 years ties to aturity and is clearly visible in Figure 1. In particular, we can expect a back-test based on such a tie span to be a quite hard test to pass for a VaR prediction odel. Finally, the volatilities for interest rates up to 1 onth tie to aturity tend to be larger than the ones of rates corresponding to further tie to aturities. The Ljung-Box statistics LB(20) testing for autocorrelations in the level of interest rate changes up to the 20 th order are strongly significant for aturities up to 1 year, showing evidence of soe autocorrelation at shorter ties to aturity for the euro bonds interest rates in our saple. For higher ties to aturity they are not significant at the 5% confidence level. The LB(20) statistics for testing the null hypothesis of no autocorrelation in the absolute interest rate changes are all highly significant, supporting a volatility clustering hypothesis. Finally, when analyzing the saple correlations between interest rates of different aturities (not reported here) we observe that, as expected, the tie series of interest rate changes of different ties to aturities are positively correlated, with higher correlations for the longer ties to aturity; for exaple, the saple correlations of interest rate changes at 3 and 6 onths and at 2 and 5 years are 0.73 and 0.91, respectively. 16

18 Starting fro these suary statistics, it is reasonable to odel the joint yield curve dynaics based on soe ultivariate GARCH-type odel of the general for (2.1). The FGD technique of Section 2 is applied in the next sections to iprove directly on the VaR predictions of standard ultivariate AR-CCC-GARCH odels. In particular, using FGD we can also odel a possibly non-linear dependence between ultivariate interest rate series and do not have to resort to any variance reduction technique. We first discuss the back-testing results for one day prediction intervals and analyze in a second step the ones for longer forecasting horizons. 3.2 Back-testing one-day ahead confidence bounds We exaine and copare the out-of-saple perforance and the accuracy of one-day ahead confidence bounds for the yield curve, coputed by eans of three historical siulation-based procedures. The industry standard benchark used by RiskMetrics TM, one based on a standard ultivariate AR-CCC-GARCH odel dynaics and, finally, one which uses the above FGD technique to estiate the ter structure dynaics. For any available tie to aturity and any tie in the back-testing saple we copute by historical siulation confidence intervals on the value of the corresponding future interest rates. By plotting these confidence bounds as a function of tie to aturity this produces for each of the above ethodologies a set of out-ofsaple confidence envelopes for the whole yield curve at any relevant date. Soe exaples of such confidence envelopes are presented in Figure 2, where we plot the realized yield curves at soe given dates, together with the 95%-confidence envelopes obtained by eans of filtered historical siulation based on the RiskMetrics TM approach (the dotted lines in Figure 2) and the FGD technique (the dashed lines in Figure 2), respectively. FIGURE 2 ABOUT HERE. The ter structure realizations presented in Figure 2 suggest at first sight that both ethodologies yield reasonable confidence envelopes. In particular, in alost all graphs of Figure 2, the realized yield curves lie inside the corresponding 95%-confidence envelopes. A sall violation of the FGD-based envelope bounds is observed for instance in the ter structure on March 13, 2001, at weekly aturities. For the RiskMetrics TM approach one relatively large violation is observed on January 5, 2001, at the two onths aturity. Moreover, the FGD-based procedure 17

19 sees to be able to replicate better soe particular shapes of the epirical yield curves, especially at the shorter ties to aturity. Indeed, in soe cases the ter structure envelopes based on the RiskMetrics TM ethodology appear to be too sooth as a function of tie to aturity (see again for instance the graph in Figure 2 for the ter structure on January ). To copare ore consistently and ore precisely the effective perforance or the above VaR prediction ethodologies it is necessary to perfor soe ore foral statistical back-tests. To test the predictive perforance of FGD-based confidence envelopes of the yield curve we use two types of statistical tests, which are based on the frequency and the duration of yield curve envelope violations, i.e. the actual interest rate observations r t,ti that happen to fall outside the predicted confidence envelopes. The first type of tests we use are standard overall frequency tests and test the hypothesis that the expected nuber of violations is copatible with the given confidence interval. For exaple, for a 95%-confidence envelope and a saple of 1000 back-testing days, one should expect 50 violations at any give tie to aturity. In Table 2 we report the observed nuber of violations for one-day ahead confidence bound forecasts at each tie to aturity T i, fro 1 onth to 30 years, i.e. i = 4,.., 12. For shorter ties to aturity no one of the ethodologies under scrutiny could provide accurate VaR estiation procedures in the present setting. We report the observed nuber of violations at the confidence levels 0.9, 0.95, 0.99 for the FGD-based ethodology (CCC-FGD), the RiskMetrics TM approach (RM) and the historical siulation ethodology based on a standard ultivariate AR-CCC-GARCH odel dynaics (CCC). Under the null hypothesis, the observed nuber of violations is binoially distributed around its expected value and with a standard deviation ranging fro (for the 90%-confidence bounds) to (99%-confidence bounds). Back-testing results arked by one and two asterisks, respectively, denote a significant difference fro the expected nuber of violations under the null hypothesis at the 5% and the 1% significance level, respectively. TABLE 2 ABOUT HERE. Fro Table 2 the FGD-based historical siulation strategy is the one that produces the lowest nuber of null hypothesis rejections when using overall frequency tests. In particular, for the 95% and the 99%-confidence envelopes we reark that only in one case a significant difference fro the expected nuber of violations is observed. The RiskMetrics TM approach yields very often confidence intervals which are too tight and that are therefore often violated a significantly 18

20 larger nuber of ties than expected under the null hypothesis. Siilarly, also a standard CCC- GARCH-based historical siulation produces often too tight confidence intervals, especially for short and interediate tie to aturities. Based on the results of pure overall frequency tests we conclude that the joint non-linear dependence of the yield curve dynaics estiated by FGD iproves the accuracy of daily VaR confidence intervals coputed by historical siulation. A second type of tests that can be applied in our back-testing exercise are likelihood-ratio Weibull duration tests; see Christoffersen and Pelletier (2002). The basic idea of these tests relies on the fact that if a odel for constructing the VaR confidence intervals at a confidence level q is correctly specified, then the conditional expected duration between consecutive violations - i.e. the expected no-hit duration - is constant and equal to 1/q days. Such an hypothesis can be tested as follows 8. Let D j = t j t j i be the no-hit duration for tie t j, where t j denotes the day of violation nuber j. Then, under the null hypothesis that the odel is correctly specified, E(D j ) = 1/q days for any j = 1, 2,... This hypothesis can be tested together with the independence hypothesis on the process of no-hit durations against soe specific dependence alternative. To this end, we consider alternatives where the distribution of no-hit durations is a Weibull distribution with density given by f W (D; a, b) = a b bd b 1 exp ( (ad) b), where a, b > 0 The exponential distribution with paraeter a then iplies the only eoryless (continuous) rando distribution in this class, which eerges as the special case b = 1. Thus, the null hypothesis of the likelihood-ratio Weibull duration test is H 0 : b = 1 and a = q, (3.2) where b = 1 is iplied under the null hypothesis of independence. Let {C j : j = 1,..., n} be the hit sequence of {0, 1} rando variables that indicate if a no-hit duration D j is censored (C j = 0) or if it is not (C j = 1) 9. For a given hit sequence and a given sequence of no-hit durations D = {D j : j = 1,.., n} the log-likelihood is given by log L(D; θ) = (1 C 1 ) log ( S(D 1 ) ) + (1 C n ) log ( S(D n ) ) + n j=1 ( C j log ( f W (D j ) )), (3.3) 8 See also Kiefer, 1988 or Gourieroux, 2000 for a general introduction to duration odelling. 9 If the hit sequence {C j j = 1,..., n} starts (ends) with 0 then D 1 (D n ) is the nuber of days until we get the first violation (nuber of days after the last violation) and C 1 = 0 (C n = 0). If instead the hit sequence starts (ends) with a 1, then C 1 = 1 and D 1 is siply the nuber of days until the second violation (then C n = 1 and D n = t n t n 1 ). 19

21 where in the case of a censored observation we erely know that no hit has been observed between tie 0 and D 1 or between tie n 1 j=1 D j and D n, respectively. In this case, the contribution to the likelihood is given by the survival function S(D j ) = exp ( (ad j ) b). The standard likelihood-ratio test statistic for testing (3.2) is then given by LR = 2 ( log L(D; â, ˆb) log L(D; q, 1) ), (3.4) where â, ˆb are the axiu likelihood estiators of the paraeters a, b. This statistic is asyptotically chi-square distributed with two degrees of freedo 10. Results of the above likelihood-ratio Weibull duration tests for 1-day ahead yield curve confidence bounds are reported in Table 3 below for our FGD-based historical siulation procedure (CCC-FGD), for the RiskMetrics TM one (RM) and for a ultivariate AR-CCC-GARCH odel based approach (CCC). TABLE 3 ABOUT HERE. As for the overall frequency tests an FGD-based historical siulation procedure is the one that clearly produces the lowest nuber of rejections of the relevant null hypothesis. Indeed, the only rejections are observed at the 95% confidence level for the one onth and the six onths aturities. The RiskMetrics TM approach yields too tight confidence bounds especially at the 99% confidence level while the AR-CCC-GARCH odel based approach produces 8 rejections at the different confidence levels, especially for tie to aturities up to one year. These findings confir that the joint non-linear dependence of the yield curve dynaics estiated by FGD iproves the accuracy of VaR confidence intervals coputed by historical siulation. 3.3 Back-testing confidence bounds for longer forecasting horizons Accuracy of the above confidence bound prediction ethodologies at forecasting horizons longer than one day is investigated next. In this context, we found that for ties to aturity up to about one year all historical siulation approaches under scrutiny produced a poor predictive power and inaccurate confidence interval estiates, with confidence bounds that were often violated several ties in a row. A 10 It is also possible to copute finite saple critical values for the above statistics by eans of Monte Carlo siulation. Our results do not change in an essential way when doing that. We therefore further use standard asyptotic critical values. 20

22 ore detailed data inspection showed that this is due principally to a sequence of ultiple big interest rate shocks on the Euro arket (often with changes larger than 0.3%-0.4%) caused by several adjustents in the Fed s target rate during the second part of our back-testing period. In the sequel we therefore focus on interest rate predictions for longer ters to aturity between two years and thirty years. Results of overall frequency tests on the total nuber of violations at prediction horizons of 3,5 and 10 days are suarized in Table 4 for the FGD-based approach (CCC-FGD), the RiskMetrics TM approach (RM) and the approach based on a ultivariate AR-CCC-GARCH odel (CCC). TABLE 4 ABOUT HERE. To correct for the autocorrelation in the series of violations in the presence of overlapping easureent intervals, we estiated the relevant standard errors using a Newey and West (1987) covariance atrix estiator with truncation paraeter x 1, where x is the forecasting horizon. Fro Table 4 we see that also for longer forecasting horizons the FGD-based approach produces clearly better back-testing results, with only one null hypothesis rejection at the ten days forecasting horizon for the two years interest rate. At the sae tie, both the Risketrics TM and the AR-CCC-GARCH ethodologies do provide a very bad back-testing perforance, with 17 and 20 null hypothesis rejections, respectively, across the different forecasting horizons and confidence levels. These findings suggest that the joint non-linear dependence of the yield curve dynaics estiated by FGD iproves even ore crucially the VaR confidence intervals coputed by historical siulation for longer forecasting horizons. Indeed, in ters of the pure nuber of null hypothesis rejections a standard AR-CCC-GARCH-based approach without FGD does not perfor better in our study than a very siple Risketrics TM approach. 4 Conclusions We proposed a ultivariate nonparaetric technique based on FGD and historical siulation to generate ore reliable scenarios and confidence intervals for the ter structure of interest rates fro historical data. The ethodology is coputationally feasible in large diensions and can account for a non-linear tie series dependence of interest rate at all available aturities. 21

23 We back-tested our ethodology on daily USD bond data and found that its out-of-saple accuracy is higher than the one of further scenario generating technologies based on principal coponents, a ultivariate AR-CCC-GARCH odel, or the exponential soothing volatility forecasting technique used by the RiskMetrics TM approach. At forecasting horizons of one day, FGD provided accurate ultivariate VaR coputations for tie to aturities between one onth and thirty years. For longer horizons (i.e. ten days) accurate VaR predictions are obtained for tie to aturities between roughly one and thirty years. 22

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25 Gourieroux, C. (2000). Econoetrics of qualitative dependent variables. Cabridge University Press. Jashidian, F. and Zhu, Y. (1997). Scenario siulation odel: theory and ethodology. Finance and Stochastics 1, Kiefer, N. (1988). Econoic duration data and hazard functions. Journal of Econoic Literature 26, Lekkos, I. (2000). A critique of factor analysis of interest rates. The Journal of Derivatives 8, Loretan, M. (1997). Generating arket risk scenarios using principal coponent analysis: ethodological and practical consideraions. Manuscript, Federal Reserve Board. Mardia, K. V. (1979). Multivariate Analysis. Acadeic Press, London. Reiers, M. and Zerbs, M. (1999). A ulti-factor statistical odel for interest rates. Algo Research Quarterly, Vol. 2, No. 3, Rodrigues, A. P. (1997). Ter structure and volatility shocks. Manuscript, Federal Reserve Board. 24

26 Yield (in %) Maturity index i Tie in days 1500 Figure 1: Ter structure data: the saple consists of 1760 daily observations between January 1, 1996 and Septeber 30, 2002 for twelve ties ro aturity T i = overnight, 1 week, 2 weeks, 1 onth, 2 onths, 3 onths, 6 onths, 1 year, 2 years, 5 years, 10 years, 30 years. 25