The term structure of euro area sovereign bond yields

Transcription

1 The term structure of euro area sovereign bond yields Peter Hördahl y Bank for International Settlements Oreste Tristani z European Central Bank 3 May Preliminary and incomplete Abstract We model the dynamics of bond yields across a number of euro area sovereigns as functions of a default-free part and a component that is due to credit risk. The rst part re ects investors perceptions about expected ECB monetary policy (and associated term premia), while the second is largely associated with the perceived risk of sovereign default and other credit events in creditrisky countries. We let this latter credit spread component be a function of the perceived scal situation of each country, which in turn is assumed to be ltered out from de cit/gdp forecast data, and of country-speci c GDP growth. We allow the scal variable to a ect sovereign yields and spreads in a non-linear way using an a ne-quadratic model. We also include a latent common credit factor which can a ect spreads across all credit-risky sovereigns, in order to capture common systematic or contagion e ects. We nd that our model can t observed data well, including the extreme yield levels seen in some countries in recent months. We decompose the credit spreads of sovereigns bonds into a credit risk premium and a default risk component and examine their relative importance. JEL classi cation numbers: F34, G, G5 Keywords: Sovereign bond yields, a ne quadratic term structure, scal policy, credit risk, reduced form credit models. We would like to thank... for helpful comments and suggestions. The opinions expressed are personal and should not be attributed to the Bank for International Settlements or the European Central Bank. y Bank for International Settlements, Monetary and Economic Department, Centralbahnplatz, CH-4, Basel, Switzerland. Phone: ; Fax: ; peter.hoerdahl@bis.org. z European Central Bank, DG Research, Kaiserstrasse 9, D - 63, Frankfurt am Main, Germany. Phone: ; Fax: ; oreste.tristani@ecb.int.

2 Introduction From late 9 onwards, yields on bonds issued by peripheral euro area sovereigns rose sharply above comparable yields on German government bonds. Credit spreads for Greece, Ireland, Portugal and Spain, which had averaged only a few tens of basis points for most of the period since the introduction of the euro, surged to several hundred basis points. The underlying reason was rapidly eroding con dence in the ability of these governments to be able to meet commitments to their lenders as budget de cits and debt-to-gdp ratios jumped. Figure : -year spreads relative to German government bonds In annual basis points. Source: ECB. Although it seems clear that there were fundamental scal concerns for a number of peripheral euro area countries, the high degree of co-movement in the recent spread widening, visible in Figure, suggests that perhaps not only country-speci c scal factors were important in pushing spreads up. In this paper, we investigate the underlying factors driving the recent spread widening by modeling the dynamics of the term structure of sovereign credit spreads in a way that allows both countryspeci c and common factors to systematically a ect credit spreads. To capture country-speci c e ects, we include a de cit/gdp variable and a GDP growth variable that we lter out from available forecasts of these variables for each of the countries under consideration. To allow for the possibility that there may be spillover

3 e ects or global fundamental drivers of spreads, we include a latent common factor that can a ect spreads for all credit-risky countries. We explicitly model the German term structure of interest rates, which we take to be free of default (or other sovereign credit) risk, or at least perceived by investors to be free of such risk. German yields are assumed to re ect average expected interest rates set by the European Central Bank (ECB), plus term premia to compensate for the associated uncertainty. Speci cally, we model the perceived interest rate setting decisions of the ECB as a Taylor rule that depends on euro area in ation and output gap, as well as on monetary policy shocks. This setup makes it possible for us to examine the e ects of policy shocks on sovereign spreads, through their impact on the output gap and hence on scal sustainability. Our modeling approach is related to a number of papers that price credit risky securities using a reduced-form on-arbitrage approach, e.g. Du e, Pedersen and Singleton (3). We di er from their setting, as well as that in most of the litterature, in a number of ways. First, we include observable macro variables, including scal information, and we consider several sovereigns jointly. Amato and Luisi (6) also include macro factors within a reduced-form model of credit spreads, although they consider US corporates rather than sovereign entities. Dai and Philippon (6) examine the US term structure of interest rates using a model that includes scal information in addition to other macro variables. While they nd signi cant e ects of scal variables on bond yields, these e ects are, in their modelling framework, not at all related to credit risk, given that they explicitly assume that US bonds are risk-free. Here, instead, we view changes in spreads as re ecting variations in the perceived sovereign credit risk, and in the compensation required by investors to bear such risk. Our work is also related to Pan and Singleton (8) and Longsta, Pan, Pedersen and Singleton (), who examine sovereign credit risk using data on sovereign credit default swaps (CDS). They estimate a single-factor model for each of the sovereigns they consider and then investigate how credit risk covaries across countries. By contrast, we jointly estimate the credit spreads for four euro area peripheral countries, where we explicitly allow for a common factor. Our paper is closest to that of Borgy et al. (), who also study euro area credit spreads using observable scal information. Compared to their paper, we di er by including a euro-wide common credit factor that potentially can capture contagion e ects on spreads, and by extending the bond pricing framework from a linear to a non-linear setting. Speci cally, we allow the default intensity of a credit-risky country to de- 3

4 pend on our scal forecast variable in a linear-quadratic way. As a result, we end up with an a ne-quadratic pricing framework, which we nd to work well in capturing some of the extreme sovereign credit spread widening witnessed in recent months. [more references to be added] This paper is organised as follows. Section presents our modelling approach and discusses the estimation method. Section 3 describes the data. Section 4 presents the results for the benchmark default-free term structure and for the creditrisky term structures. Section 5 concludes. Model and estimation Our empirical speci cation is based on the class of reduced-form credit pricing models in which assumptions are made about the process for default intensity, as in Lando (998) and Du e and Singleton (999). In this framework, default is assumed to be doubly stochastic, meaning that default arrives randomly according to a Poisson process with time-varying intensity and, in addition, this intensity process varies randomly over time. The advantage of this approach is that it gives rise to tractable pricing formulas. Speci cally, in continuous time, for a given risk-neutral default intensity process Q and a given risk-free interest rate process r; the price at t of a zero-coupon defaultable bond with maturity T and zero recovery is (see e.g. Du e and Singleton, 3): d (t; T ) = E Q t exp Z T t r (u) + Q (u) du : In case of non-zero (possibly stochastic) recovery Z; Lando (998) shows that the price is d (t; T ) = E Q t E Q t exp Z T t Z T Z (s) Q (s) exp t r (u) + Q (u) du + Z s t r (u) + Q (u) du ds ; where Z = L Q and L Q is the risk-neutral loss rate in the case of default. In some cases, notably under the assumption of fractional recovery of the market value (RMV) of the bond, it is possible to obtain closed-form solutions for defaultable bonds (Du e and Singleton, 999). We assume RMV and proceed by setting up our empirical speci cation in discrete time. 4

5 . Model: Basic setup We specify our model as a Gaussian macro- nance term structure model, in the spirit of Ang and Piazzesi (3). The state vector, denoted X t ; which will contain both observable and unobservable factors, follows an AR(): X t = X t + " t ; () where it is assumed that the state variables are expressed in deviations from their mean. In addition, we assume that the risk-free short-term interest rate r t is a ne in (some of) the state variables, r t = + X t : The pricing kernel m t;t+ is assumed to depend on X t : Speci cally, m t;t+ = exp ( r t ) t+ = t, where t+ is assumed to follow the log-normal process t+ = t exp t t t" t+ ; which results in m t;t+ = exp r t t t t" t+ : The vector t represents the market prices of risk associated with the systematic underlying sources of uncertainty in the economy. assume that the market prices of risk are a ne in X t : We follow Du ee () and t = + X t ; so that the market s required compensation for bearing risk can vary with the state of the economy. The default intensities of risky bonds, t, are assumed to depend on the state variables (or a subset of them) as well. Note that we allow credit-speci c state variables to be priced by the market, which means that we cannot in general assume that default intensities (or credit spreads) are independent of the risk-free interesty rate process. The macro- nance literature has grown tremendously over the past few years. Recent papers include Ang, Dong and Piazzesi (6), Ang, Piazzesi and Wei (6), Dewachter and Lyrio, (6), Hördahl, Tristani and Vestin (6), and Rudebusch and Wu (4). 5

6 . Risk-free bond prices Given this setup, the price of an n-period default-free zero coupon bond, Pt n ; can be shown to be given by Pt n = exp A n + B nx t ; where A n and B n are de ned recursively as A n = A n B n + B n Bn ; () B n = B n ( ) ; (3) with initial conditions A = ; B = : The corresponding continuously compounded yield y n t is given by y n t = ~ A n + ~ B nx t ; (4) with ~A n = ~B n = A n n ; B n n :.3 Risky bond prices The price at t of a risky bond maturing at t + n can be written as with boundary condition B t+n t = E t mt;t+ B t+n t+ >t+ + Z t+ <t+ ; Bt+n t+n = E t+n mt+n ;t+n Bt+n t+n >t+n + Z t+n <t+n ; where Z t is the recovery payment, denotes the time of default and >t+ is an indicator variable that takes the value one if > t + : In general, the expectation 6

7 E t [ >t+k ] is the probability of survival until t + k :!# kx E t [ >t+k ] = E t "exp t+i : Under a RMV assumption, the expected recovery payment is a fraction of the bond price at t + ; conditional on no default, ie for an n-period bond i= E t [Z t+ ] = E t ( Lt+ ) B t+n t+ ; where L t+ is the fractional loss rate. Assuming that the loss rate is a constant L; we have, under RMV, which can be written B t+n t = E t mt;t+ B t+n t+ >t+ + ( L) B t+n t+ <t+ ; B t+n t = E t mt;t+ B t+n t+ exp ( t+) + B t+n t+ ( L) ( exp ( t+)) = E t mt;t+ (exp ( t+ ) + ( L) ( exp ( t+ ))) B t+n t+ = E t mt;t+ ( L ( exp ( t+ ))) B t+n t+ : Assume that we can make the following approximation L ( exp ( t+ )) exp ( t+ ) : This approximation holds exactly for L = : For L di erent from, we should view as re ecting adjusted default intensities, rather than actual intensities. This analogous to the use of recovery-adjusted default intensities in continuous time models with RMV (e.g. Du e and Singleton, 999). Given this assumption, we can write B t+n t = E t mt;t+ exp ( t+ ) B t+n t+ : We will assume that the (adjusted) default intensity of country j is a quadratic function of the states: j t = j + j X t + X t j X t : 7

8 The price of an n-period bond is therefore (supressing superscripts j) t = E t mt;t+ exp ( t+ ) Bt+ t+n = exp X t X t X t X t X t Xt X t E t exp + Xt + + Xt " t+ " t+ " t+ B t+n t+ B t+n We know, given the a ne-quadratic setup, that we can write the price of a bond as If we plug in B t+n t = exp A n + B n X t + X tc n X t : B t+n t+ = exp A n + B n X t+ + X t+c n X t+ = exp A n + B n (X t + " t+ ) + X t + " t+ C n (X t + " t+ ) = exp A n + B n X t + B n " t+ +X t C n X t + X t C n " t+ + " t+ C n " t+! into the bond price above we get B t+n t = exp E t "exp + A n + B n Xt +X t C n Xt B n + X t C n X t X t " t+ +" t+ (C n ) " t+!!# ; where the expectation can be written as E t exp awt+ + w t+ C n w t+, where w t+ " t+ a B n + X t C n X t X t ; C n C n : To evaluate the expectation we follow Realdon (6), who demonstrates that (if is of full rank) E t exp awt+ + w t+ C n w t+ = jj abs jj! NY exp (a i) i= 8

9 where ( ) C =, n i denotes the i-th column of, jj denotes the determinant of and absjj denotes the absolute value of the determinant of. We therefore get = E t exp awt+ + wt+ C n w t+ jj NY B n + X exp t C n Xt Xt! i ; abs jj i= so that ln Bt t+n = +X t + A n + B n Xt C n X t + ln jj abs jj + i= B n + X t C n X t X t i : Evaluating the squared term we get ln Bt t+n = +X t + ln + + i= + A n + B n Xt C n jj abs jj + i= B n i i X t B n i i Xt C n i i i= B n Cn X t Cn X t : We can therefore identify the recusive factor loadings of the bond price B t+n t = 9

10 exp (A n + B n X t + X tc n X t ) as jj A n = A n + ln abs jj + B n i i Bn ; i= B n = B n + B n i i i= C n = C n + C n i i i= with initial conditions A = B = + C = + ln i= + i= abs jj + Cn ; i= i i Cn ; i i i i ; : The general setup of the model allows us to estimate the dynamics of the default-free term structure separately from the credit-risky dynamics. Before turning to estimation issues, we rst specify the speci cs of the model below, starting with the setup of the default-free term structure..4 Model: speci c setup Given our focus on the euro area, we need a default-free euro-denominated reference. We take the German term structure as our reference yield curve, and we assume that it is driven by four factors: in ation, the output gap, and two latent factors. We let the default-free (de-meaned) short rate process (which can be viewed as the process of the ECB policy rate) be determined by these same four variables: r t =! t +! x x t +! ;t + ;t : (5)

11 This formulation is consistent with typical speci cations of monetary policy rules, such as Taylor rules, in structural macro models. In such rules, the short-term policy rate is assumed to react to in ation (possibly in deviation from some implicit or explicit objective of the central bank) and to the output gap. With this type of set-up in mind, the latent factors ;t and ;t could, at least informally, be seen as capturing changes in preferences of the central bank. In the empirical implementation, we will allow ;t to be serially correlated while ;t is assumed to be uncorrelated. As such, these factors could potentially pick up perceived persistent changes to the central bank s in ation objective and transitory monetary policy shocks, respectively. The two macro factors and the policy shock, in turn, are assumed to follow a VAR process. We restrict this process in several ways. First, to keep the number of parameters manageable, we allow only one lag (we will be working with a monthly data frequency). The rst latent factor is assumed to depend only on its own lag, while the second latent factor is assumed to be a white noise shock. Moreover, both of these factors are allowed to impact on in ation and the output gap (after one lag). We assume that the credit-speci c state variables consist of three variables: a latent credit variable that is common across sovereigns (C t ), the GDP growth rate of country j g j t ; and the scal debt/gdp ratio of country j d j t : Given this, we specify our state vector as (suppressing superscripts j) X t = t ; x t ; ;t ; ;t ; C t ; g t ; d t : The state vector is assumed to follow an AR() process, where we impose various restrictions to keep the number of parameters manageable. Speci cally, we assume the following structure: 6 4 t x t ;t ;t C t g t d t 3 3 ; ; ;3 ;4 ; ; ;3 ;4 3;3 = 5; ; ;6 7;7 t x t t ;t C t g t d t 3 + " t ; 7 5

12 while is assumed to be diagonal with elements diag () = [ ; x ; ; ; ; g ; d ] : The risk-free short rate is given by r t = X t ; with and! ;! : =! ;! x ;! ;! ; ; (3) For the speci cation of the market prices of risk, we impose some restrictions to reduce the number of parameters. We restrict the market price of risk coe cient matrix to take the same form ar the AR coe cient matrix above (but also allwoing for non-zero risk price for ;t t = 6 4 x C g d ). We then get the following speci cation: ; ;x ; ; x; x;x x; x; ; ; C;C g;g d;g d;d 3 X t : 7 5 The risk parameters associated with the euro area macro and monetary policy variables are the same across all countries, while the parameters associated with C; g, and d are allowed to vary across countries. Finally, we need an assumption regarding the default intensities. In principle, we can allow these to depend directly on the "risk-free" factors, but initially we will assume that they depend only on the credit factors, in the following way: t = + C C t + g g t + d d t + dd d t : As a result, j in j t = j + j X t + X t j X t is a matrix of zeros with the lowest right-hand element equal to dd :

13 .5 Estimation of the reference term structure We estimate the risk free block in a rst step using maximum likelihood, based on the Kalman lter. To construct the likelihood function, we rst de ne a vector W t containing the observable contemporaneous variables, W t 6 4 where Y t denotes the vector of default-free zero-coupon yields and t and x t are assumed to be de-meaned. The dimension of W t is denoted n w. We can then write the measurement equation as W t = 6 4 A Y t t x t B ; (8) X t + w t ; K + H X t + w t ; E w t w t = R; where w t is a vector of serially uncorrelated measurement errors corresponding to the observable variables W t. We assume that, apart from the short-term (onemonth) rate, the n y yields are measured with error (assumed to be cross-sectionally uncorrelated), while the two macro variables have zero measurement error: 3 y. R=.. : y The transition equation is X r t = r X r t + v r t ; E [v r t v r t ] = Q; (6) where Q = r r : 3

14 We start the lter from the unconditional mean E [X r t ] = ; and the unconditional MSE matrix, whose vectorised elements are vec P j = In u F F vec (Q) ; (see Hamilton, 994). The Kalman lter will produce forecasts of the states and the associated MSE according to ^X t+jt r = r ^Xr ;tjt + r P tjt H H P tjt H + R W t K H ^Xr tjt (7) P t+jt = r P tjt r r P tjt H H P tjt H + R H P tjt r + Q: (8) Given this, the likelihood can be expressed as where TX t= log f (W t j W t ; ) = T (n y ) ln () TX ln j t [ ]j t= TX (W t t [ ]) ( t [ ]) (W t t [ ]) (9) t= t [ ] H [ ] P tjt [ ] H [ ] + R [ ] ; t [ ] K [ ] + H [ ] ^Xr tjt [ ] : Given that the parameter space is quite large, we employ the method of simulated annealing, introduced to the econometric literature by Go e, Ferrier and Rogers (994). The method is developed with an aim towards applications where there may be a large number of local optima..6 Estimation of credit risky term structures In our setup, yields on credit risky bonds are non-linear functions of the state variables. As a result, we can no longer use the standard Kalman lter to estimate the model. [discussion of possible alternatives] Instead, we rely on the unscented Kalman lter of Julier and Uhlmann (997, 4). The unscented Kalman lter relies on 4

15 a deterministic sampling technique to pick points around the mean of some underlying random variable. These so-called sigma points are then propagated through the non-linear functions of interest, in order to recover the rst two moments of the non-linear system. These can subsequently be used in the updating step of the lter. In our application, the transition equation is X t = X t + " t ; () while the observation equation is z t = (X t ) + t ; () where z t is a vector of observables (e.g. yields on risky bonds), () is a non-linear function, and where the observation error vector t is assumed to have zero mean and a diagonal covariance matrix R: ~ Similar to the standard Kalman lter, the unscented lter relies on a linear updating rule according to ^X tjt = ^X tjt + ~ K t z t ^z tjt ; () where ^X tjt = ^X t jt ; ~K t = P xz(tjt ) P h i ^z tjt = E ^Xtjt ; zz(tjt ) ; and where P zz is the innovation covariance matrix and P xz is the cross covariance matrix. The updated state is associated with updated covariance P xx(tjt) = P xx(tjt ) ~ Kt P xx(tjt ) ~ K t ; (3) where P xx(tjt ) = P xx(t jt ) + : For an n x -dimensional state vector X; a set of n x + sigma points { ; { ; :::; { nx with associated weights $ ; $ ; :::; $ nx are chosen (see the Appendix [to come] 5

16 for details). For each sigma point i, the nonlinear transformation in () is applied Z i = ({ i ) : The covariance matrices P xx ; P zz and P xz are then approximated using { i and the transformed points Z i (see Appendix [to come]). In our case, the observation vector consists of n s risky zero-coupon bond yields for each country j; stacked in s j t ; and a vector f j t that contains data on the expected scal position and expected GDP growth rate of country j; based on forecasts of the de cit to GDP ratio and GDP growth (the exact nature of the data is discussed below). Given data for m countries, we can de ne the observation vector as z t 6 4 Note that we keep the systematic state variables of the risk-free block that we obtained in the rst estimation step xed at their ltered values when performing the second step. s t. s m t fṭ. f m t 3 : Data Our data is monthly and covers the period from the introduction of the euro, January 999 to December [soon to be updated to end-]. The default-free term structure data consists of German zero-coupon yields with maturities, 3, 6 months, and, 3, 5, 7, and years, which are estimated by the German Bundesbank. To model the perceived behaviour of the ECB, we use euro area aggregate values of in ation and the output gap. The measure of in ation is monthly y-o-y HICP log-di erences. For the output gap, we follow Clarida, Galí and Gertler (998) and measure the output gap as deviations of real GDP from a quadratic trend. To obtain a monthly series for the gap, we t an ARMA(,) model to the quarterly gap series, then forecast the gap one quarter ahead and compute one- and two-month ahead values by means of linear interpolation. This exercise is conducted in "real time", in 6

17 the sense that the model is reestimated at each quarter using data only up to that quarter. For the estimation of the credit risky term structures, we use, 5, and -year yields for Greece, Portugal, Spain, France and Italy. We estimate zero-coupon yields for these countries based on prices of all available government bonds at each point in time, as reported by Bloomberg, using the Nelson-Siegel model. We use data on zero-coupon bonds with maturities, 3, 4, 5, 7 and years when estimating the model. Finally, we include forecast data on the de cit/gdp ratios and GDP growth gures for these ve countries. Twice a year, the European Commission provides data on such forecasts for horizons roughly one and two years ahead. For the forecast released in the Spring, the forecasts cover the current and next year, i.e. until the end of the current year and until the end of the following year. For the Fall forecast, the horizons extend through the next and the following years. By including this data, we are implicitly assuming that investors make similar de cit forecasts as the European Commission, or that they take these forecasts into account in their pricing decisions when they are made public. By using forecasts rather than o cial data on de cits, we hope to be able to capture some of the forward-lookingness of nancial markets. 4 Estimates We rst discuss the estimates of the benchmark German term structure, which we take to be free of default risk. We then go on to look at the estimates for the four credit risky countries included in our sample. 4. Estimates for Germany The rst step of our estimation procedure gives us the dynamics of the German term structure, including the implied interest rate setting behaviour of the ECB. For the default-free short rate, we obtain the following estimates, r t = :97 t + :3x t + :6 ;t + :76 ;t ; 7

18 with state variable dynamics 6 4 t x t ;t ;t 3 = :84 : : :49 : :99 : : : t x t t ;t 3 + " t ; 7 5 with = 6 4 :59 : [standard errors to be added]. As displayed in Figure, the t of the model is quite good in terms of how well it can capture the dynamics of observed German yields. The standard deviations of the yield measurement errors range from 5 to basis points per year for the seven imperfectly observed yield series. This is in line with typical results reported in the empirical term structure literature (e.g. Dai and Singleton () and Du ee ()). 8

19 Figure : Actual and estimated German yields: 3-month maturity (left) and -year maturity (right) 4. Estimates for Greece, Portugal, Spain, France and Italy [text to come] Model t: 5 y fitted 5y fitted y fitted y actual 5y actual y actual Greece

20 8 7 6 y fitted 5y fitted y fitted y actual 5y actual y actual Portugal y fitted 5y fitted y fitted y actual 5y actual y actual Spain

21 3 y fitted 5y fitted y fitted y actual 5y actual y actual France y fitted 5y fitted y fitted y actual 5y actual y actual Italy Credit risk premium / default risk decompositions:

22 w ith premia zero premia Greece, maturity w ith premia zero premia Portugal, maturity

23 .5 w ith premia zero premia Spain, maturity w ith premia zero premia France, maturity

24 .5 w ith premia zero premia Italy, maturity Conclusions [to come] 4

25 A Appendix: A. Credit-risky bond prices The price at t of a risky bond maturing at t + n can be written as B t+n t = E t mt;t+ B t+n t+ >t+ + Z t+ <t+ ; with boundary condition Bt+n t+n = E t+n mt+n ;t+n Bt+n t+n >t+n + Z t+n <t+n ; where Z t is the recovery payment, denotes the time of default and >t+ is an indicator variable that takes the value one if > t + : In general, the expectation E t [ >t+k ] is the probability of survival until t + k : E t [ >t+k ] = E t "exp!# kx t+i : Under a RMV assumption, the expected recovery payment is a fraction of the bond price at t + ; conditional on no default, ie for an n-period bond i= E t [Z t+ ] = E t ( Lt+ ) B t+n t+ ; where L t+ is the fractional loss rate. Assuming that the loss rate is a constant L; we have, under RMV, which can be written B t+n t = E t mt;t+ B t+n t+ >t+ + ( L) B t+n t+ <t+ ; Bt t+n = E t mt;t+ Bt+ t+n exp ( t+) + Bt+ t+n ( L) ( exp ( t+)) = E t mt;t+ (exp ( t+ ) + ( L) ( exp ( t+ ))) Bt+ t+n = E t mt;t+ ( L ( exp ( t+ ))) Bt+ t+n : Assume that we can make the following approximation L ( exp ( t+ )) exp ( t+ ) : This approximation holds exactly for L = : For L di erent from, we should view as re ecting adjusted default intensities, rather than actual intensities. This analogous to the use of recovery-adjusted default intensities in continuous time models with RMV (e.g. Du e and Singleton, 999). Given this assumption, we can write Bt t+n = E t mt;t+ exp ( t+ ) Bt+ t+n : We will assume that the (adjusted) default intensity of country j is a quadratic function of the states: j t = j + j X t + X t j X t : 5

26 The price of an n-period bond is therefore (supressing superscripts j) t = E t mt;t+ exp ( t+ ) Bt+ t+n = E t exp r t t t t" t+ exp X t+ Xt+X t+ B t+n t+ X = E t exp t + X t + Xt X t + Xt "t+ (X t + " t+ ) Xt + " t+ B t+n (X t + " t+ ) t+ = exp X t X t X t X t X t E t exp + Xt + " t+ Xt X t Xt " t+ " t+ " t+ B t+n t+ = exp X t X t X t X t X t Xt X t E t exp + Xt + + Xt " t+ " t+ " t+ B t+n t+ B t+n We know that we can write the price of a bond as We plug in B t+n t = exp A n + B n X t + X tc n X t : Bt+ t+n = exp A n + B n X t+ + Xt+C n X t+ = exp A n + B n (X t + " t+ ) + Xt + " t+ C n (X t + " t+ ) = exp A n + B n X t + B n " t+ +X t C n X t + X t C n " t+ + " t+ C n " t+ into the bond price above to get Bt t+n = exp X t X t X t X t X t Xt X t + Xt + + Xt " t+ " 3 t+ " t+ E t +A n + B n X t + B n " t+ A5 +Xt C n X t + Xt C n " t+ + " t+ C n " t+ = exp + A n + B n Xt +Xt C n Xt Bn E t exp + Xt C n Xt Xt " t+ +" : t+ (C n ) " t+ Rewrite the expectation as Bn E t exp + Xt C n Xt Xt " t+ +" t+ (C n ) " t+ Bn = E t exp + Xt C n Xt Xt " t+ +" t+ (C n ) " t+ = E t exp awt+ + w t+ C n w t+ ; 6

27 where w t+ " t+ a B n + X t C n X t X t ; C n C n : To evaluate the expectation we follow Realdon (6), who demonstrates that (if is of full rank) where E t exp awt+ + w t+ C n w t+ = jj abs jj! NY exp (a i) ( ) C n =, i denotes the i-th column of, jj denotes the determinant of and absjj denotes the absolute value of the determinant of. We therefore get E t exp awt+ + wt+ C n w t+ = so that jj abs jj NY exp i= ln Bt t+n = +X t i= B n + Xt C n Xt Xt! i ; + A n + B n Xt C n X t + ln jj abs jj + i= B n + X t C n X t X t i : Evaluating the squared term: B n + X t C n X t X t i = B n + Xt C n Xt Xt i B n + Cn X t X t X t i = B n i i B n + B n i i Cn X t +Xt C n i i Cn X t ; 7

28 we get ln Bt t+n = +X t + ln + + i= + A n + B n Xt C n jj abs jj + i= B n i i X t B n i i Xt C n i i i= B n Cn X t Cn X t : We can therefore identify the recusive factor loadings of the bond price B t+n t = exp (A n + B n X t + X tc n X t ) as jj A n = A n + ln abs jj + B n i i Bn ; i= B n = B n + B n i i i= C n = C n + C n i i i= Cn ; Cn : To get the initial conditions, consider the price of a -period bond: Bt t+ = exp X t X t X t X t X t Xt X t E t exp + X t + + X t " t+ " t+ " t+ : Rewrite the expectation as E t exp + X t + + X t " t+ " t+ " t+ = E t exp X t X t " t+ " t+ " t+ = E t exp a w t+ w t+w t+ ; where a X t X t ; 8

29 so that where Bt t+ = exp E t exp a w t+ w t+w t+ = ( ) + =. We get abs jj NY exp i= a i X t X t X t X t X t Xt X t E t exp + X t + + X t " t+ " t+ " t+ : ln Bt t+ = + ln The squared term is: so that abs jj + ln Bt t+ = + ln + + Xt Xt i=! + X t X t X t i : Xt Xt i = Xt Xt i i X t X t = i i + i i X t +Xt i i X t ; + + i= abs jj + i= + + Xt Xt i i i= i i X t i i X t X t ; + X t 9

30 which gives A = B = + C = + ln i= + i= abs jj + i= i i i i i i ; : 3

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