Reduced-form Models with Regime Switching: An Empirical Analysis for Corporate Bonds Hoi Ying Wong∗ and Tsz Lim Wong Department of Statistics Chinese University of Hong Kong, Hong Kong, China First version: March 4, 2007 March 28, 2007
Abstract Empirical evidence shows that there is a close link between regime shifts and business cycle fluctuations. A standard term structure of interest rates, such as the Cox, Ingersoll, and Ross (1985; CIR) model, is sharply rejected in the Treasury bond data. Only Markov regime-switching models on the entire yield curve of the Treasury bond data can account for the observed behavior of the yield curve. In this paper, we examine the impact of regime shifts on AAA-rated and BBB-rated corporate bonds through the use of a reduced-form model. The model is estimated by the Efficient Method of Moments. Our empirical results suggest that regime-switching risk has significant implications for corporate bond prices and hence has a material impact on the entire corporate bond yield curve, providing evidence for the approach of rating through the cycle employed by rating agencies. JEL classification: To be determined Keywords: Credit Risk, Regime Switching, Efficient Method of Moments.
∗
Corresponding author; fax: (852) 2603-5188; email:
[email protected].
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Electronic Electronic copy of copy this paper available is available at: http://ssrn.com/abstract=976826 at: http://ssrn.com/abstract=976826
1 Introduction The term structure of defaultable bonds is one of the most important entities in credit risk modeling because it describes the relationship between the yields on a defaultable discount bond and its maturity. It is one of the fundamentals for pricing credit risk derivatives. Many models of the term structure are based on the assumption that all information about the economy is contained in a finitedimensional vector of state variables, the dynamics of which are governed by stochastic processes. The dynamics may be derived by using absence of arbitrage arguments, obtained endogenously in a general equilibrium framework, or identified from market data using econometric methods. The exact expression for the price of defaultable bonds depends on the specification of the stochastic processes for the state variables and the associated market price of risk. Cox, Ingersoll, and Ross (1985; CIR) proposed the univariate square-root process for the instantaneous interest rate (spot rate) in a general equilibrium framework so that heteroscedasticity is introduced into the spot rate dynamics. Duffie and Singleton (1999) extended the framework to reduced-form models of credit risk. They introduced an adjusted spot rate to account for both the probability and timing of default, and for the effect of losses on default. These models are known as exponential affine models. The diffusion processes of the interest rate that are specified in these models provide closed-from expressions for transition and marginal densities of the interest rate, and also bond prices. As a result, these models are analytically tractable and easy to implement. However, Ghysels and Ng (1997) rejected current affine models in a semiparametric test. One potential problem of affine models is the assumption that all the model parameters are constant over time. Generally, business cycles and monetary policies can affect real rates and expected inflation, and can cause interest rates to behave differently in different time periods. An alternative to affine models in capturing these effects is the regime-switching model that is proposed by Hamilton (1988). This model allows the parameters of the interest rate process to be dependent on a discrete regime variable. Hence, the conditional density depends on the current regime, and a Markov transition matrix governs the evolution of the regime variable. Many papers demonstrate that the short interest rate process can be reasonably well modeled in time series as a regime-switching process (see Garcia and Perron (1996), Gray (1996), and Ang and Bekaert (1998)). The results have motivated recent empirical studies to estimate an entire term structure of interest rates that is based on regime-switching models (see Naik and Lee (1997), Evans (1998), 2
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Bansal and Zhou (2002), and Wu and Zeng (2005)). The regime dependence that is introduced by these empirical studies implies richer dynamic behavior of the market price of diffusion risk and regime-switching risk. Hence, they offer greater econometric flexibility for the term structure models to account for both the time series and cross-sectional properties of interest rates. In this paper, we extend this stream of literature to a reduced-form model of credit risk. We take seriously the idea that changes in regimes can potentially have sizable effects on the term structure of defaultable bonds; therefore, incorporating them can better account for the observed behavior of the term structure of defaultable interest rates. Motivated by this possibility, we develop a continuous time model with a general equilibrium framework of the term structure of defaultable interest rates that incorporates regime shifts. The actual yield curve of a given credit rating class fluctuates around the mean curve for the current regime. Sometimes, discrete changes in the economy lead to a jump in the term structure of yield volatilities and in the mean yield curve. At any time, the yield curve reflects the current regime and the expectation that the current regime may change. Our work contributes to both the theoretical and empirical literature on the term structure of defaultable interest rates. We obtain a closed-form solution of the term structure of defaultable interest rates using an affine-type model similar to that in Bansal and Zhou (2002) and Wu and Zeng (2005). We then show how the regime shifts affect the entire yield curve and dynamic behavior of bond yields for two credit rating classes - AAA-rated and BBB-rated. We use the Efficient Method of Moments (EMM), developed in Bansal (1995) and Gallant and Tauchen (1996), to estimate the models that are under consideration. The empirical exercise relies on the AAA-rated and BBB-rated defaultable bonds data from 1973 to 1997. We find that there is an improvement of model specification when the regime-switching risk component is incorporated in both AAA-rated and BBB-rated defaultable bonds. There are two major motivations for modeling defaultable interest rates with regime switches. First, there exists a large body of statistical empirical evidence in the finance literature (see Garcia and Perron (1996), Gray (1996), and Ang and Bekaert (1998)). This evidence motivated the studies of the impact of regime shifts on the entire yield curve of defaultable bonds with reduced-form models of credit risk. Second, the evidence in the macroeconomics literature suggests that models with regime shifts can help explain the movements in a number of real and nominal macroeconomic variables, which are intimately related to interest rates (see Evans and Lewis (1995), and Boudoukh, Richardson, Smith, and Whitelaw (1999)). For example, the transitions between economic expansion and 3
recession have effects on monetary policy, inflationary expectations, and nominal interest rates. However, term structure models such as the Cox, Ingersoll, and Ross (1985) and affine models do not incorporate them. The absence of these important components in the models may explain why the models have poor empirical performance. The rest of this paper is organized as follows. Section 2 presents a general equilibrium model of credit risk with regime-switching. Section 3 conducts an empirical analysis of the model in which the details of the empirical formulation will be given. Section 4 concludes the paper.
2 General Equilibrium Model This section develops a general equilibrium approach for a reduced-form model of defaultable bonds with systematic risk of regime shifts. A closed-form solution for the term structure of interest rates is obtained under an affine model using loglinear approximation. We start by stating the similarity of the short-rate process between defaultable and default-free bonds, and then describe the state variables, investment opportunities for the economy, and the lifetime utility function.
2.1 Recovery Models Three main specifications of modeling the recovery of defaultable claims have been adopted in the literature: recovery of face value (RFV), recovery of treasury (RT) and recovery of market value (RMV). Jarrow, Lando and Turnbull (1997) consider the RT specification in which the recovery rate is an exogenous fraction of the value of an equivalent default-free bond. Duffie and Singleton (1999) consider the RMV specification in which the recovery rate is equal to an exogenous fraction of the market value of the bond just before default. Houweling and Vorst (2001) consider the RFV specification in which the recovery rate is an exogenous fraction of the face value of the defaultable bond. Under the RMV specification, Duffie and Singleton (1999) show that this claim can be priced as if it were default-free by replacing the usual short-term interest rate process rt with the default-adjusted short-rate process yt = rt + λt Lt . Lt is the expected loss rate in the market value if default were to occur at time t,
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conditional on the information available up to time t Rτ = (1 − Lτ )Q(τ − , T ), Q(τ − , T ) = lim− Q(s, T ), s→τ
(1) (2)
where τ is the default time, Q(τ − , T ) is the market price of the bond just before default and Rτ is the market value of the defaulted bond. That is, under the technical conditions, the market value of the defaultable claim to (2.11) shown by Duffie and Singleton (1999) is Z T Q(t, T ) = E exp − ys ds M|Ft . (3) t
This is natural, in that λt Lt is the “risk-neutral mean-loss rate” of the instrument due to default. Discounting at the adjusted short rate yt therefore accounts for both the probability and timing of default, and for the effect of losses on default. A key feature of the valuation equation (3) is that if the mean-loss rate process λt Lt is exogenously given, standard term-structure models for default-free debt are directly applicable to defaultable debt by parameterizing yt instead of rt .
2.2 State Variables The economy is assumed to be driven by two types of state variables, x(t) = (x1 (t), ..., xM (t))′ and s(t). Following Duffie and Singleton (1999), the first type of variable x(t) has a continuous path, and is determined by the stochastic differential equation ˜t , dxt = µ(xt )dt + σ(xt )dB
(4)
˜ is the standard Brownian motion in RM , and the drift term µ(xt ) in where B RM and the diffusion term σ(xt ) in RM×M are regime dependent. The second type of variable s(t) is a continuous-time Markov chain that represents N distinct regimes, taking on values of 1, 2, ..., N. Following Landen (2000) and Wu and Zeng (2005), the marked point process is used to obtain a convenient representation of s(t). Let the mark space E be: E = {(i, j) : i ∈ {1, ..., N}, j ∈ {1, ..., N}, i 6= j}, 5
(5)
with σ-algebra E = 2E . Let z = (i, j) be a generic point in E, which represents a regime shift from state i to j. A marked point process, m(t, ·), is uniquely characterized by its stochastic intensity kernel and can be defined as γm (dt, dz) = h(z, x(t−))I{s(t−) = i}εz (dz)dt,
(6)
where h(z, x(t−)) is the regime-shift intensity from state i to state j at z = (i, j) conditional on x(t−), I{s(t−) = i} is an indicator function of the regime at time t−, and εz (A) is the Dirac measure for A, a subset of E, at point z = (i, j). It is defined by εz (A) = 1 if z ∈ A, and εz (A) = 0 if z 6∈ A. Hence, γm (dt, dz) is the conditional probability of shifting from state i to state j during [t, t + dt] given x(t−) and s(t−) = i. Moreover, γm (dt, dz) is in general state dependent. Let m(t, A) denote the cumulative number of regime shifts that belong to A, a subset of E, during (0, t]. Then m(t, A) has its compensator, γm (t, A), which is given by Z tZ γm (t, A) = h(z, x(τ −))I{s(τ −) = i}εz (dz)dτ, (7) 0
A
which implies that m(t, A) − γm (t, A) is a martingale. With the notation, it is now clear that the regime s(t) can be represented as Z ds = Ψ(z)m(dt, dz), (8) E
with the compensator given by γs (t)dt =
Z
Ψ(z)γm (dt, dz),
(9)
E
where Ψ(z) = Ψ((i, j)) = j − i.
2.3 Investment Opportunities Following Merton (1990) and Wu and Zeng (2005), it is assumed that expectations about the dynamics of the price per share in the future are the same for all investors. Thus, the prices depend on both state variables x(t) and s(t) as described by the stochastic differential equation Z dPk = µk dt + σk dBk + δk (z)m(dt, dz), k = 1, ..., n, (10) Pk E 6
where n ≤ M, both the instantaneous expected rate of return, µk , and the instantaneous standard deviation of return, σk , are functions of x(t−) and s(t−), and δk (z) k (t−,s(t−)) is the discrete percentage change in k due to a regime shift, i.e., Pk (t,s(t))−P . Pk (t−,s(t−)) Hence, we assume that regime shifts not only affect the drift µk and the volatility σk , but also directly result in discontinuous changes in the prices as the economy shifts from one regime to another. We further assume that one of the n assets is a defaultable pure discount bond, the price of which is given by Z dQ = µQ dt + σQ dBQ + δQ (z)m(dt, dz), (11) Q E where the drift µQ , the volatility σQ , and the discrete percentage change δQ (z) are to be determined by the equilibrium conditions, and δQ (z) is the discrete per. centage change in the bond prices due to a regime shift, i.e., Q(t,s(t))−Q(t−,s(t−)) Q(t−,s(t−)) In other words, we allow that regime shifts not only affect the drift µQ and the volatility σQ , but also directly result in discontinuous jumps in the prices. By manipulating (11), one can obtain the following: Z dQ = µQ dt + δQ (z)γm (dt, dz) Q E Z + σQ dBQ + δQ (z)[m(dt, dz) − γm (dt, dz)]. (12) E
As the last two terms in this equation are martingales, the instantaneous expected defaultable bond return should be Z dQ Et− = µQ dt + δQ (z)γm (dt, dz), (13) Q E where the first term is the regime-dependent expected defaultable bond return due to diffusion, and the second term is an additional component in the defaultable bond return due to discrete regime shifts.
2.4 Preferences Given the initial wealth w0 , we move to the problem of maximizing the expected lifetime utility of a representative agent, which is given by Z ∞ −ρt E0 e U(c(t))dt , (14) 0
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where c(t) is the flow of consumption and U(·) is the instantaneous utility function. U(·) is assumed to be strictly concave, increasing, and twice differentiable ′ with U(0) = 0 and U (0) = ∞. It is assumed that the representative agent is a mutual fund investing in a specified investment grade class. In other words, the agent only allocates wealth among the n − 1 assets in the same investment grade class, the defaultable borrowing and lending, and consumption. The dynamic of the wealth equation can be described by the stochastic differential equation: dw =
n X
wφk
k=1
dPk + (b − c)dt, Pk
(15)
where w(t) is the agent’s wealth atPtime t, φk is the fraction of wealth that is invested in the kth security (hence nk=1 φk ≡ 1), b(t) is her income and c(t) is her flow of consumption. Substituting for dPk /Pk from (3.7), and assuming that the nth asset is the defaultable borrowing and lending, we can rewrite (15) as Z dw = wµw dt + wσw dBt + wδw (z)m(dt, dz), (16) E
where wµw = w
" m X k=1
wσw
#
φk (µk − y) + y + (b − c),
v" # u m m u XX φi φj σij , = wt
(17)
(18)
i=1 j=1
wδw (z) = w
" m X
#
φk δk (z) ,
k=1
(19)
where m ≡ n − 1 and the φ1 , ..., Pφnm are unconstrained because φn can always be chosen to satisfy the constraint k=1 φk ≡ 1.
2.5 The Term Structure of Defaultable Bonds
In this section, we go on to obtain a closed-form solution for the term structure of defaultable bonds. Assuming that U(c) = log(c), the prices of the defaultable pure discount bonds are given by the following theorem. 8
Proposition 2.1. The price at time t of a defaultable discount bond Q(t, y(t), s(t), T ) in a specified investment grade class which matures at time T satisfies the following system of partial differential equations. 1 Qt + (µy − ηw σy )Qy + σy2 Qyy 2 Z + (1 − λs (z))∆s Qh(z, x)I{s = i}εz (dz) = yQ. E
with the boundary condition Q(T, y, s, T ) = 1 for all y and s.
Proposition (2.1) defines a system of N partial differential equations if there are N distinct regimes. As pointed out by Wu and Zeng (2005), the system generally does not admit a closed-form solution to the bond price. Following Duffie and Kan (1996), Dai and Singleton (2000), Duffie, Pan, and Singleton (2000), and Wu and Zeng (2005), we have the following affine specification, which is known to offer a tractable model of the term structure of interest rate. We assume µy = a0 (s) + a1 (s)y, p σ(s)y. σy =
(20) (21)
Under (20) and (21), the default-adjusted short rate y(t) follows the squareroot process with regime-dependent drift and diffusion terms p dy = (a0 (s) + a1 (s)y)dt + σ(s)ydBy . (22)
By manipulating (22), we have the standard CIR model that follows the squareroot process p dy = κ(s)(¯ y (s) − y)dt + σ(s)ydBy , (23) where κ(s) = −a1 (s) and y¯(s) = Furthermore, we assume that
a0 (s) . −a1 (s)
h(z, x) = eνs (z) , p ηw = θ(s) σ(s)y, λs (z) = 1 − eφs (z) .
(24) (25) (26)
Equation (24) assumes that the Markov chain s(t) has constant transition probabilities that are given by eνs (z) for simplicity. Equation (25) implies the market 9
price of the diffusion risk in equilibrium and θ(s) is the coefficient that determines the market price of diffusion risk. Equation (26) parameterizes the market price of the regime-switching risk, and φs (z) is the coefficient that determines the market price of regime-switching risk. Assuming that U(c) = log(c), the term structure of defaultable bonds can be solved and given by the following theorem. Proposition 2.2. Under the assumptions (20)-(21) and (24)-(26), the price at time t of a defaultable pure discount bond in a specified investment grade class with time to maturity τ is given by Q(t, τ ) = eA(τ,s(t))+B(τ,s(t))y(t) , and the observed τ period defaultable interest rate is given by Y (t, τ ) = − A(τ,s(t)) − B(τ,s(t))y(t) , where τ τ A(τ, s(t)) and B(τ, s(t)) are determined by the following system of differential equations: −
∂B(τ, s) 1 + [a1 (s) − θ(s)σ(s)]B(τ, s) + σ(s)B 2 (τ, s) ∂τ 2 Z + (eφs (z)+νs (z) )e∆s (A) ∆s (B)I{s = i}εz (dz) = 1, E
and
∂A(τ, s) + a0 (s)B(τ, s) + − ∂τ
Z
E
(eφs (z)+νs (z) )(e∆s (A) − 1)I{s = i}εz (dz) = 0,
with the boundary conditions A(0, s) = 0 and B(0, s) = 0, where ∆s A = A(τ, s + Ψ(z)) − A(τ, s) and ∆s B = B(τ, s + Ψ(z)) − B(τ, s).
3 Empirical Results 3.1 Methodologies In the estimation, our focus is to evaluate whether different term structure models of defaultable bonds can justify the observed behavior of two defaultable interest rates — the six-month and five-year AAA-rated and BBB-rated defaultable bond rates. We explore the ability of two types of models to justify the observed conditional distribution of the two interest rates under consideration. Model 1 is the standard one-factor CIR model that is proposed by Cox, Ingersoll, and Ross (1985). Model 2 is the one-factor CIR model that is developed in Proposition (2.2), in which the risk of regime shifts is priced. Because the main purpose of 10
this paper is to highlight the potential impact of the systematic risk of regime shifts on the term structure of defaultable bonds, we do not consider multi-factor term structure models. We assume that there are two distinct regimes (N = 2) for s(t). Therefore, Proposition (2.2) defines a system of four differential equations that must be solved simultaneously. There are 12 parameters in the model. We fit the model to the data for the six-month and five-year rates. However, the data cannot be fitted to the model directly because Y (t, τ ) in Proposition (2.2) is a kind of spot rate and we only have corporate bond data. We need a model to extract zero-coupon rates from current fixed coupon-bearing bond prices. There is a direct method to fit a yield curve, known as the bootstrapping method, but it somewhat lacks robustness. Indirect methods are therefore usually preferred. The common character of all indirect models is that they involve fitting data to a pre-specified form of the zero-coupon yield curve. The general approach is to first select a reference set of bonds with market prices and cash flows that are taken as given. Then, one postulates a specific form of the discount function or zero-coupon rates.1 Finally, a set of parameters is estimated that best approximates given market prices. Among all indirect methods, the Extended Nelson and Siegel (ENS) model is chosen to fit the yield curve. The curve-fitting technique first described by Nelson and Siegel (1987) has been applied and modified in a number of ways,2 so that it is sometimes described as a family of curves. The ENS model offers a conceptually simple and parsimonious description of the term structure of interest rates. It avoids over-parametrization while allowing for monotonically increasing or decreasing yield curves and hump-shaped yield curves. It also avoids the problem in spline-based models of choosing knot points subjectively. After obtaining Y (t, τ ) in Proposition (2.2) using the ENS model, the model is fit to Y (t, τ ) on the six-month and five-year AAA-rated and BBB-rated defaultable bond rates. To utilize a consistent approach for evaluation and estimation across the different models, we apply the simulation-based efficient method of moments (EMM) estimator, developed by Gallant and Tauchen (1996). The EMM estimator consists of two steps. First, the empirical conditional density of the observed defaultable interest rates is estimated by an auxiliary model that is a close approximation of the true data generating process. Gallant and 1
Examples of indirect methods include exponential spline, polynomial spline, Nelson and Siegel model and Vasicek model. 2 Examples of Extended Nelson and Siegel models include those of Svensson (1994), Bliss (1997), and Landschoot (2003).
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Tauchen (1996) suggest a semi-nonparametric (SNP) series expansion as a convenient general purpose auxiliary model. As pointed out by Bansal and Zhou (2002), the advantage of using the semi-nonparametric specification for the auxiliary model is that it can asymptotically converge to any smooth distributions, including the density of Markov regime-switching models. Second, the score functions from the log-likelihood of the SNP density are used as moments to construct a GMM-type criterion function. The scores are evaluated using the simulation output from a given term structure model, and the criterion function is minimized with respect to the parameters on the term structure model under consideration. A nonlinear optimizer is used to find the parameter setting that minimizes the criterion function.3 Further details regarding SNP density and EMM estimation are provided in Appendix B. Finally, note that the short-rate factor y(t) in (23) is a standard one-factor CIR model. It defines interest rate movements in terms of the dynamics of the short rate. The variance of the short rate is related to the level of interest rates, and this feature has the effect of not allowing negative interest rates. It also reflects a higher interest rate volatility in periods of relatively high interest rates, and correspondingly lower volatility when interest rates are lower. However, the short-rate factor in the standard one-factor CIR model cannot be directly simulated using a discrete time counterpart. Hence, the step-wisely moment-matched log-normal scheme is applied to simulate the short-rate factor under the CIR model. Further details of the step-wisely moment-matched log-normal scheme are provided in Appendix C.
3.2 Data Description The data in this empirical study were obtained from the University of Houston’s Fixed Income Database. This database consists of monthly information on most publicly traded bonds since 1973. Each issue is identified by a CUSIP number and includes information on the issue date, maturity date, flat price, coupon, accrued interest, bond rating, industry sector, and call and put features. As the latest update we have is March 1997, we use monthly data that run from January 1973 through March 1997, 291 months in all. To study the term structure of defaultable bonds, we focus on two credit rating classes - AAA-rated and BBB-rated. Several filters are imposed to construct the sample of defaultable corporate bonds. First, we choose non-callable and non-putable bonds that are issued by 3
Gallant and Tauchen (2006) provide user guides for the SNP and EMM programs.
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industrial, utility, and transportation firms. Firms in broad industries such as finance, real estate finance, insurance, and banking are excluded from our sample. Second, notes and bonds under one year to maturity, and bills under one month to maturity are eliminated from the sample due to liquidity problems.4 At this stage, our sample, on average, consists of 52 AAA-rated bonds and 74 BBB-rated bonds from January 1973 through March 1997. We then apply the Extended Nelson and Siegel (ENS) model to fit the yield curve. Two sets of data, August 1975 and December 1984, are reported missing in the database. Hence, we use the mean of the preceding and the following month as proxies for these two yields. The summary statistics of the term structures of AAA-rated and BBB-rated defaultable bonds data are given in Table 1. It is clear that, on average, the yield curve is upward sloping. Moreover, the positive skewness and kurtosis suggest departure from Gaussian distribution. To incorporate important time-series and cross-sectional aspects of term structure data, we focus on a short-term yield on six-month bills and a long-term yield on five-year notes. The two models that are mentioned in Chapter 4 are forced to match the conditional bivariate joint dynamics of the two yields. As pointed out by Bansal and Zhou (2002), one-month or three-month bill is not used to represent short end, because it is more likely to be affected by liquidity needs.5 The time-series plots of the yields are given in Figure 1.
3.3 Estimation Results Tables 2-5 contain the results of the chosen SNP specifications. The EMM estimator consists of two steps. In the first step of the EMM procedure, we fit an SNP density of the bivariate joint dynamics of the short-term and long-term yields. The SNP density employs an expansion in Hermite functions as a convenient general purpose auxiliary model that is a close approximation of the true data generating process. The dimension of this auxiliary model can be selected by the Bayesian Information Criterion (BIC) proposed by Schwarz (1978). The technical notations of the SNP specifications are given in Appendix B. Table 2 reports different choices of SNP density and their corresponding BIC values of the term structures of AAA-rated defaultable bonds. We find that the overall best fit based on BIC is the SNP specification with two lags (Lµ = 2) in the VAR-based conditional mean, five lags (Lr = 5) in the ARCH specification, 4 5
See Bliss (1997). See Bansal and Coleman (1996) and Bansal and Zhou (2002).
13
Table 1: Summary Statistics of Term Structures of AAA-rated and BBB-rated Defaultable Bonds Data, ranging from January 1973 to March 1997 Panel A: Term Structures of AAA-rated Defaultable Bonds Data Maturity Mean Std. Dev. Skewness Kurtosis Maximum Minimum Range
3 Months
6 Months
1 Year
2 Years
3 Years
5 Years
7 Years
10 Years
0.0790 0.0277 0.9001 0.9860 0.1696 0.0292 0.1404
0.0806 0.0265 0.8112 0.7523 0.1679 0.0315 0.1364
0.0821 0.0256 0.7976 0.5887 0.1651 0.0357 0.1294
0.0837 0.0245 0.8730 0.5184 0.1617 0.0420 0.1196
0.0851 0.0237 0.9508 0.5167 0.1605 0.0460 0.1145
0.0876 0.0231 1.0783 0.5992 0.1617 0.0520 0.1096
0.0897 0.0231 1.1697 0.7203 0.1648 0.0564 0.1083
0.0922 0.0236 1.2543 0.8759 0.1695 0.0605 0.1089
Panel B: Term Structures of BBB-rated Defaultable Bonds Data Maturity Mean Std. Dev. Skewness Kurtosis Maximum Minimum Range
3 Months
6 Months
1 Year
2 Years
3 Years
5 Years
7 Years
10 Years
0.1003 0.0284 0.8359 0.5674 0.1869 0.0476 0.1393
0.1012 0.0272 0.7489 0.3565 0.1824 0.0499 0.1325
0.1024 0.0264 0.7134 0.2138 0.1768 0.0539 0.1229
0.1041 0.0254 0.7083 0.0359 0.1724 0.0583 0.1141
0.1055 0.0248 0.7463 0.0002 0.1713 0.0619 0.1094
0.1075 0.0239 0.8721 0.1722 0.1738 0.0683 0.1056
0.1089 0.0233 0.9775 0.3991 0.1766 0.0731 0.1034
0.1107 0.0229 1.1111 0.8066 0.1821 0.0780 0.1042
and a polynomial of order four (Kz = 4) in the standardized residual z. The total number of parameters for this SNP specification is 31 (lθ = 31). Table 3 reports different choices of SNP density and their corresponding BIC values of the term structures of BBB-rated defaultable bonds. We find that the overall best fit based on BIC is the SNP specification with one lag (Lµ = 1) in the VAR-based conditional mean, four lags (Lr = 4) in the ARCH specification, and a polynomial of order four (Kz = 4) in the standardized residual z. The total number of parameters for this SNP specification is 26 (lθ = 26). The information criterion for choosing the preferred SNP specification is the minimum value of BIC given in (47). Parameter estimates of the preferred SNP density are reported in Table 4 and Table 5. After obtaining the estimated nonparametric SNP density, we can estimate the 14
Six−Month Yield Level 0.2 BBB AAA
0.15 0.1 0.05 0
1975
1980
1985
1990
1995
Five−Year Yield Level 0.2 BBB AAA
0.15 0.1 0.05 0
1975
1980
1985
1990
1995
Figure 1: Observed short-term yield and long-term yield parameters of different term structure models using EMM estimation. Table 6 shows the main EMM estimation results for two models. Panels A and B give the results for AAA-rated defaultable bonds and BBB-rated defaultable bonds respectively. Model 1 is the standard one-factor CIR model that is proposed by Cox, Ingersoll, and Ross (1985). Model 2 is the one-factor CIR model that is developed in Theorem 3.2, in which the risk of regime shifts is priced. As seen from Panel A of Table 6, the chi-square statistic of model 1 is 216.39 and the chi-square statistic of model 2 drops to 79.46. This indicates an improvement when the regime-switching risk component is incorporated in AAA-rated defaultable bonds. In model 1, the standard one-factor CIR model, the shortterm AAA-rated defaultable interest rate has an average long-run mean level of 7.13% (¯ y = a0 / − a1 = 0.0063/0.0884) and an average conditional standard dep √ viation of 1.31% ( σ¯ y = (0.0024)(0.0713) ). The mean reversion parameter is 0.0884 (κ = −a1 = 0.0884), therefore the short-term AAA-rated defaultable 15
Table 2: SNP Score Generator for AAA-rated Defaultable Interest Rates Lµ
Lr
Lp
Kz
Iz
Kx
Ix
lθ
˜ sn (θ)
BIC
1 2 3
0 0 0
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
9 13 17
0.02777 -0.03253 -0.03987
0.11550 0.09419 0.12585
2 2 2 2 2 2 2
1 2 3 4 5 6 7
1 1 1 1 1 1 1
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
15 17 19 21 23 25 27
-0.09418 -0.10796 -0.15274 -0.17222 -0.19230 -0.19312 -0.19571
0.05204 0.05776 0.03247 0.03249 0.03190 0.05058 0.06749
2 2 2 2 2 2 2 2 2
5 5 5 5 5 5 5 5 5
1 1 1 1 1 1 1 1 1
4 4 4 4 5 5 5 5 5
3 2 1 0 4 3 2 1 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
31 32 34 37 33 34 36 39 43
-0.30793 -0.31045 -0.31187 -0.31545 -0.31297 -0.31448 -0.31563 -0.32173 -0.33176
-0.00574 0.00149 0.01956 0.04523 0.00871 0.01695 0.03530 0.05844 0.08740
2 2 2
5 5 5
1 1 1
4 4 4
3 3 3
1 2 2
0 1 0
49 67 76
-0.36162 -0.41532 -0.44405
0.11603 0.23779 0.29680
interest rate is very persistent. In model 2, the one-factor CIR model in which the risk of regime shifts is priced is developed in Theorem 3.2. In Regime 0, the short-term AAA-rated defaultable interest rate has an average long-run mean level of 3.95% (¯ y = a0 / − a1 = 0.0031/0.0784) and an average conditional p √ y = (0.0029)(0.0395) ). The mean reverstandard deviation of 1.07% ( σ¯ sion parameter is only 0.0784 (κ = −a1 = 0.0784), and thus the short rate is very persistent in this regime. In Regime 1, the short-term AAA-rated defaultable interest rate has a higher average long-run mean level of 11.78% (¯ y = a0 / − a1 = 0.0103/0.0874) and a higher average conditional standard deviap √ tion of 2.40% ( σ¯ y = (0.0049)(0.1178) ). The mean reversion parameter is 16
Table 3: SNP Score Generator for BBB-rated Defaultable Interest Rates Lµ
Lr
Lp
Kz
Iz
Kx
Ix
lθ
˜ sn (θ)
BIC
1 2 3
0 0 0
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
9 13 17
0.42634 0.39259 0.37350
0.51407 0.51931 0.53922
1 1 1 1 1 1 1
1 2 3 4 5 6 7
1 1 1 1 1 1 1
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
11 13 15 17 19 21 23
0.35436 0.32504 0.30713 0.28526 0.26921 0.25263 0.25093
0.46159 0.45176 0.45335 0.45098 0.45442 0.45734 0.47513
1 1 1 1 1 1 1 1 1
4 4 4 4 4 4 4 4 4
1 1 1 1 1 1 1 1 1
4 4 4 4 5 5 5 5 5
3 2 1 0 4 3 2 1 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
25 26 28 31 27 28 30 33 37
0.21360 0.19567 0.18959 0.18600 0.20346 0.18934 0.18829 0.18367 0.16685
0.45730 0.44912 0.46253 0.48819 0.46666 0.46228 0.48073 0.50535 0.52753
1 1 1
4 4 4
1 1 1
4 4 4
2 2 2
1 2 2
0 1 0
46 66 76
0.12105 0.08479 0.03974
0.56946 0.72816 0.78059
0.0874 (κ = −a1 = 0.0874), and thus the short rate is less persistent in this regime. Also, note that the coefficients on the market price of diffusion risk and regime-switching risk are different among regimes. This shows that the yield curve has regime-dependent properties. The transition probability from Regime 0 to Regime 1 is 0.0438 (h(z, x) = eνs (z) = e−3.1279 ). The transition probability from Regime 1 to Regime 0 is 0.0414 (h(z, x) = eνs (z) = e−3.1846 ). As seen from Panel B of Table 6, the chi-square statistic of model 1 is 238.77 and the chi-square statistic of model 2 drops to 84.10. This indicates an improvement when the regime-switching risk component is incorporated in BBB-rated defaultable bonds. In model 1, the standard one-factor CIR model, the short17
term BBB-rated defaultable interest rate has an average long-run mean level of 10.38% (¯ y = a0 / − a1 = 0.0112/0.1079) and an average conditional standard dep √ y = (0.0063)(0.1038) ). The mean reversion parameter viation of 2.56% ( σ¯ is 0.1079 (κ = −a1 = 0.1079), therefore the short-term BBB-rated defaultable interest rate is very persistent. In model 2, the one-factor CIR model in which the risk of regime shifts is priced is developed in Theorem 3.2. In Regime 0, the short-term AAA-rated defaultable interest rate has an average long-run mean level of 4.43% (¯ y = a0 / − a1 = 0.0029/0.0654) and an average conditional p √ y = (0.0049)(0.0443) ). The mean reverstandard deviation of 1.47% ( σ¯ sion parameter is only 0.0654 (κ = −a1 = 0.0654), and thus the short rate is very persistent in this regime. In Regime 1, the short-term AAA-rated defaultable interest rate has a higher average long-run mean level of 13.31% (¯ y = a0 / − a1 = 0.0282/0.2119) and a higher average conditional standard deviap √ y = (0.0063)(0.1331) ). The mean reversion parameter is tion of 2.90% ( σ¯ 0.2119 (κ = −a1 = 0.2119), and thus the short rate is less persistent in this regime. Also, note that the coefficients on the market price of diffusion risk and regime-switching risk are different among regimes. This shows that the yield curve has regime-dependent properties. The transition probability from Regime 0 to Regime 1 is 0.0425 (h(z, x) = eνs (z) = e−3.1592 ). The transition probability from Regime 1 to Regime 0 is 0.0394 (h(z, x) = eνs (z) = e−3.2334 ).
4 Conclusion Many papers demonstrate that the short interest rate process can be reasonably well modeled in time series as a regime-switching process (see Garcia and Perron (1996), Gray (1996), and Ang and Bekaert (1998)). In addition to statistical evidence, there are also economic reasons to believe that regime shifts are important to understanding the behavior of the entire yield curve. However, term structure models such as the Cox, Ingersoll, and Ross (1985) and affine models do not incorporate them. Their absence in these models may explain why the models have poor empirical performance. The main contribution of this paper is to show that there is an improvement of model specification when the regime-switching risk component is incorporated in both AAA-rated and BBB-rated defaultable bonds. To be more specific, we develop and estimate a model for a term structure that incorporates regime-switching risk. The empirical results have important implications for both AAA-rated and BBB-rated defaultable bonds term structure models. First, the regime shifts af18
fect the forecasting power in the term structure of defaultable bonds. Second, the regime shifts also affect the price of credit derivatives. The non-linear regime switching specification seems to add both statistical and economic values in pricing credit derivatives. It would be an interesting and valuable extension of this paper.
References [1] Ang, A. and Bekaert, G. 1998. Regime Switches in Interest Rates, working paper, NBER. [2] Bansal, R. and Coleman, W. J. 1996. A Monetary Explanation of the Equity Premium, Term Premium, and Risk-Free Rate Puzzles, Journal of Political Economy, 104, 1135-1171. [3] Bansal, R., Gallant, R. and Tauchen, G. 1995. Nonparametric estimation of structural models for high-frequency currency market data, Journal of Econometrics, 66, 251-287. [4] Bansal, R. and Zhou, H. 2002. Term Structure of Intereset Rates with Regime Shifts, Journal of Finance, 57, 1997-2043. [5] Bliss, R. 1997. Testing Term Structure Estimation Methods, Advances in Futures and Options Research, 9, 197-231. [6] Boudoukh, J., Richardson, M., Smith, T. and Whitelaw, R. 1999. Regime Shifts and Bond Returns, working paper, New York University. [7] Cox, J., Ingersoll, J. and Ross, S. 1985. A Theory of the Term Structure of Interest Rates, Econometrica, 53, 385-407. [8] Dai, Q. and Singleton, K. 2000. Specification Analysis of Affine Term Structure Models, Journal of Finance, 55, 1943-1978. [9] Duffie, D. and Kan, R. 1996. A Yield-Factor Model of Interest Rates, Mathematical Finance, 6, 379-406. [10] Duffie, D., Pan, J. and Singleton, K. J. 2000. Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica, 68, 1343-1376. 19
[11] Duffie, D. and Singleton, K. J. 1999. Modeling Term Structures of Defaultable Bonds, The Review of Financial Studies, 12, 687-720. [12] Evans, M. 1998. Real Risk, Inflation Risk, and the Term Structure, working paper, Georgetown University. [13] Evans, M. and Lewis, K. 1995. Do Expected Shifts in Inflation Affect Estimates of the Long-Run Fisher Relation?, Journal of Finance, 50, 225-253. [14] Fons, J. 1994. Using Default Rates to Model the Term Structure of Credit Risk, Financial Analysts Journal, September-October, 25-32. [15] Gallant, R. and Tauchen, G. 1996. Which moment to match, Econometric Theory, 12, 657-681. [16] Gallant, R. and Tauchen, G. 2006a. SNP: A Program for Nonparametric Time Series Analysis, working paper, Duke University. [17] Gallant, R. and Tauchen, G. 2006b. EMM: A Program for Efficient Method of Moments Estimation, working paper, Duke University. [18] Garcia, R. and Perron, P. 1996. An Analysis of the Real Interest Rate under Regime Shifts, Review of Economics and Statistics, 78, 111-125. [19] Ghysels, E. and Ng, S. 1997. A Semiparametric Factor Model of Interest Rates and Tests of the Affine Term Structure, Review of Economics and Statistics, 80, 535-548. [20] Gray, S. 1996. Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process, Journal of Financial Economics, 42, 27-62. [21] Hamilton, J. 1988. Rational Expectations Econometric Analysis of Changes in Regimes: An Investigation of the Term Structure of Interest Rates, Journal of Economic Dynamics and Control, 12, 385-423. [22] Houweling, P. and Vorst, A. C. F. 2001. An Empirical Comparison of Default Swap Pricing Models, working paper, Erasmus University Rotterdam. [23] Hughston, L. P. and Turnbull, S. 2001. Credit Risk: Constructing the Basic Building Blocks, Economic Notes, 30, 281-292.
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[24] Jarrow, R., Lando, D. and Turnbull, S. 1997. A Markov Model for the Term Structure of Credit Spreads, Review of Financial Studies, 10, 481-523. [25] Kushner, H. J. and Dupuis, P. 2001. Numerical Methods for Stochastic Control Problems in Continuous Time, Springer. [26] Landen, C. 2000. Bond Pricing in a Hidden Markov Model of the Short Rate, Finance and Stochastics, 4, 371-389. [27] Landschoot, A. V. 2003. The Term Structure of Credit Spreads on Euro Corporate Bonds, working paper, Tilburg University. [28] Litterman, R. and Iben, T. 1991. Corporate Bond Valuation and the Term Structure of Credit Spreads, Journal of Porfolio Management, Spring, 5264. [29] Martin, M. 1997. Credit Risk in Derivative Products, Ph.D. dissertation, University of London. [30] Merton, R. C. 1990. Continuous-Time Finance, Cambridge, Mass. [31] Naik, V. and Lee, M. 1997. Yield Curve Dynamics with Discrete Shifts in Economic Regimes: Theory and Estimation, working paper, University of British Columbia. [32] Nelson, C. R. and Siegel, A. F. 1987. Parsimonious Modeling of Yield Curves, Journal of Business, 60, 473-489. [33] Nielsen, S. and Ronn, E. 1995. The Valuation of Default Risk in Corporate Bonds and Interest Rate Swaps, working paper, University of Texas at Austin. [34] Protter, P. 1990. Stochastic Intergration and Differential Equations, Springer Verlag, Berlin. [35] Pye, G. 1974. Gauging the Default Premium, Financial Analyst’s Journal, January-February, 49-50. [36] Ramaswamy, K. and Sundaresan, S. M. 1986. The Valuation of FloatingRate Instruments, Theory and Evidence, Journal of Financial Economics, 17, 251-272. 21
[37] Sch¨ onbucher, P. J. 1997. The Term Structure of Defaultable Bond Prices, working paper, University of Bonn. [38] Sch¨ onbucher, P. J. 2003. Credit Derivatives Pricing Models, Wiley Finance. [39] Schwarz, G. 1978. Estimating the Dimension of a Model, Annals of Statistics, 6, 461-464. [40] Svensson, L. E. 1994. Estimating and Interpreting Forward Interest Rates, working paper, NBER. [41] Wu, S. and Zeng, Y. 2005. A General Equilibrium Model of the Term Structure of Interest Rates under Regime-Switching Risk, International Journal of Theoretical and Applied Finance, 8, 839-869. [42] Wu, L. and Zhang, F. 2006. Libor Market Model with Stochastic Volatility, Journal of Industrial and Management Optimization, 2, 199 - 227.
22
Appendix A Proof A.1 Proof of Proposition 2.1 R ∞ Let J(w(t), s(t), y(t)) = sup(φ1 ,...φm,c) Et t e−ρ(τ −t) U(c(τ ))dτ be the indirect utility function. We assume that a solution to the agent’s problem exists. We also assume that indirect utility function J(w(t), s(t), y(t)), the optimal consumption, and portfolio choice satisfy the Bellman equation6 1 0 = sup U(c) − ρJ(w, s, y) + (wµw )Jw + µy Jy + (wσw )2 Jww 2 (φ1 ,...φm ,c) Z 1 ∆s Jγm (dz) , (27) +(wσwy )Jwy + σy2 Jyy + 2 E where " m X
#
(28)
∆s J = J(w(1 + δw (z)), s + Ψ(z), y) − J(w, s, y), γm (dz) = h(z, x(t−))I{s(t−) = i}εz (dz),
(29) (30)
wσwy = w
φk σky ,
k=1
and y(t) represents the defaultable interest rate that is associated in defaultable bonds of a given investment grade class defined in (2.14). Moreover, µw , σw and δw are given in (17), (18), and (19), respectively, and the subscripts on the J(w, s, y) function denote partial derivatives. The state variable y(t) is xn+1 (t) as defined in (4), while µy and σy are the drift and the volatility term of y(t), respectively. Substituting µw , σw , and δw from (17), (18), and (19), respectively into (27), the m + 1 first-order conditions are ′
0 = U (c) − Jw , ! m X φj σkj Jww + (wσky )Jwy 0 = w(µk − y)Jw + w 2
(31)
j=1
+
Z
wδk (z)Jw (w(1 + δw (z)), s + Ψ(z), y)γm(dz), k = 1, ..., m.(32)
E
6
See Kushner and Dupuis (2001) Ch. 3.
23
From (32), the first-order condition for a defaultable pure discount bond defined in (11) is given by ! m X 2 0 = w(µQ − y)Jw + w φj σQj Jww + (wσQy )Jwy j=1
+
Z
wδQ (z)Jw (w(1 + δw (z)), s + Ψ(z), y)γm (dz).
(33)
E
This can be further rewritten as 0 = w(µQ − y)Jw + w 2 ηw σQ Jww + (wσQy )Jwy Z + wδQ (z)Jw (w(1 + δw (z)), s + Ψ(z), y)γm (dz),
(34)
E
Pm where ηw = j=1 φj ρQj σj and ρQj is the correlation between the defaultable bond and the jth asset. By manipulating (34), we have Jwy Jww (wηw σQ ) + − (σQy ) µQ − y = − Jw Jw Z ∆s Jw − 1+ δQ (z)γm (dz). (35) Jw E Under logarithm utility function U(c(t)) = log c(t), it can be proven that the indirect utility function is separable in w(t) and y(t). It can also be proven that the indirect utility function is separable in w(t) and s(t). Therefore, J(w, s, y) can be written as ρ1 log w + f (s, y), where f (s, y) solves the system of differential equation after substituting J(w, s, y) and the optimal choice of consumption (c∗ ) and portfolio (φ∗1 , ..., φ∗m ) into (27). This separability implies that Jwy = 0. Following Protter (1990), we can apply Ito’s formula to Q(t, w, s, y), and we have 1 1 2 2 dQ = Qt + (wµw )Qw + µy Qy + (wσw ) Qww + (wσwy )Qwy + σy Qyy dt 2 2 Z +[(wσw )Qw + σy Qy ]dBQ + ∆s Qm(dt, dz). (36) E
24
Matching the coefficients between (36) and (11), we have 1 [Qt + (wµw )Qw + µy Qy Q 1 1 + (wσw )2 Qww + (wσwy )Qwy + σy2 Qyy ], 2 2 1 [(wσw )Qw + σy Qy ] , σQ = Q ∆s Q δQ (z) = . Q µQ =
(37) (38) (39)
Substituting Jwx = 0 and (37), (38), and (39) into (35), we have 1 1 (wσw )2 Qww + (wσwy )Qwy + σy2 Qyy 2 2 Jww + µy − − wηw σy Qy Jw Jww 2 w ηw σw Qw + wµw − − Jw Z ∆s Jw + 1+ ∆s Qγm (dz) = yQ. Jw E
Qt +
(40)
Moreover, we assume that the change of wealth of the representative agent is independent of the change of price of a defaultable bond. In other words, the representative agent is not the issuer of the defaultable bond. It implies Qw = 0, 1 and Qww = 0, and Qwy = 0. Moreover, J = 1ρ log w + f (s, y) implies Jw = ρw 1 Jww = − ρw2 . Therefore, (40) can be further simplified as 1 Qt + σy2 Qyy + (µy − ηw σy )Qy + 2 In addition, because J = 1+
1 ρ
Z E
∆s Jw 1+ Jw
∆s Qγm (dz) = yQ.
(41)
log w + f (s, y), we have
∆s Jw 1 = = 1 − λs (z), Jw 1 + δw (z)
δw (z) . where λs (z) = 1+δ w (z) Substituting (42) into (41), the proof is completed.
25
(42)
2
A.2 Proof of Proposition 2.2 Substituting Q(t, τ ) = eA(τ,s(t))+B(τ,s(t))y(t) and τ = T − t into Proposition (2.1), we have y = −
∂A(τ, s) ∂B(τ, s) − y ∂τ ∂τ
1 +[a0 (s) + (a1 (s) − θ(s)σ(s))y]B(τ, s) + [σ(s)y]B 2 (τ, s) 2 Z + (eφs (z)+νs (z) )(e∆s (A)+∆s (B)y − 1)I{s = i}εz (dz),
(43)
E
where ∆s A = A(τ, s + Ψ(z)) − A(τ, s) and ∆s B = B(τ, s + Ψ(z)) − B(τ, s). Because y is small, by applying the log-linear approximation, we can obtain e∆s (B)y ≈ 1 + ∆s (B)y. By matching the coefficients of y on both sides of the equation, the proof is completed. 2
B SNP Density and EMM Estimation B.1 SNP Density The method is termed semi-nonparametric (SNP) to suggest that it lies halfway between parametric and nonparametric procedures. The method employs an expansion in Hermite functions as a general purpose nonparametric estimator to approximate the conditional density of a multivariate process. Following Gallant and Tauchen (1996), any smooth conditional density function can be approximated arbitrarily close by a Hermite polynomial expansion. Let y denote the vector of the interest rates under consideration and x be the vector of lagged y. The auxiliary f -model has a density function that is defined by a modified Hermite polynomial f (y|x, θ) = R
[P(z, x)]2 nM (y|µx, Σx ) , {[P(z, x)]2 nM (y|µx, Σx )}dz
(44)
where P(z, x) is a polynomial with degree Kz and z, which is a standardized ′ transformation z = Rx−1 (y − µx ) with Σx = Rx Rx . The square of P(z, x) makes the density positive, and the argument of the polynomial is z. The coefficients of the polynomial are allowed to be another polynomial of degree Kx in x. The constant in the polynomial of z is set to 1 for identification. In addition, nM (·) is a Gaussian density of dimension M with a mean vector µx and variance-covariance 26
matrix Σx , where µx is estimated by using a VAR specification, and Σx is estimated by using an ARCH specification, which parameterizes Rx . Note that both µx and Rx depend only on lags of y. The length of the auxiliary model parameter is determined by the number of lags of x used in constructing the coefficients of the polynomial Lp , the degree Kz of the polynomial in z, the degree Kx of the polynomial in x, lags in the VAR mean specification Lµ , and lags in the ARCH specification Lr . The polynomial P(z, x) take the form X a(λ1 , λ2 , x)z1λ1 z2λ2 , (45) P(z, x) = λ1 ,λ2
where a(λ1 , λ2 , x) are the coefficients of the polynomial in z, and the sum is over all pairs of nonnegative integers (λ1 , λ2 ) such that λ1 + λ2 ≤ Kz . In general, a positive Iz means all interactions of order exceeding Kz − Iz are suppressed to 0. Similarly, the interactions terms of Kx of order exceeding Kx − Ix are suppressed to 0. Putting certain of the tuning parameters to zero implies sharp restrictions on the process {yt }. Let {˜ yt }nt=1 be the observed data and x˜t−1 be the lagged observations. The parameters θ˜n are estimated by minimizing sn (θ) = (−1/n)
n X t=1
log[f (˜ yt |˜ xt−1 , θ)].
(46)
The dimension of the auxiliary f -model, the length of θ, is selected by the Schwarz Bayes information criterion (Schwarz, 1978), which is computed as BIC = sn (θ˜n ) + (lθ /2n) log(n),
(47)
where lθ is the length of the auxiliary model. The criterion rewards good fits as represented by small sn (θ˜n ) but uses the term (lθ /2n) log(n) to penalize good fits obtained by means of excessively rich parameterizations.
B.2 EMM Estimation Let {ˆ y t }N t=1 be a long simulation from a candidate value of ρ, the parameter vector of the maintained structural model. The moment criterion is m ˆ N (ρ, θ˜n ) =
N 1 X (∂/∂θ) log f [ˆ yt (ρ)|ˆ xt−1 (ρ), θ˜n ], N τ =1
27
(48)
and the GMM estimator of the structural parameter vector is ′ arg min m ˆ N (ρ, θ˜n ) (I˜n )−1 m ˆ N (ρ, θ˜n ),
ρ
(49)
where I˜n is the weighting matrix. It is estimated by the mean-outer-product of SNP scores ′ n 1X ∂ ∂ ˜ ˜ ˜ In = ( ) log f [˜ yt (ρ)|˜ xt−1 (ρ), θn ] ( ) log f [˜ yt (ρ)|˜ xt−1 (ρ), θn ] .(50) n t=1 ∂θ ∂θ The normalized criterion function value in the EMM estimation forms a specification test for the overidentifying restrictions ′ nm ˆ N (ρ, θ˜n ) I˜n−1 m ˆ N (ρ, θ˜n ) ∼ χ2 (lθ − lp ),
(51)
where the degree of freedom equals lθ − lp , that is, the number of scores moment conditions in the auxiliary model, lθ , less the number of structural parameters, lp . It is assumed that lp is smaller than lθ .
C Moment-Matching of the CIR Model The short-rate factor in the standard CIR model is assumed to follow the squareroot process √ dr = κ(θ − r)dt + σ rdW, (52) where κ is the mean reversion parameter, θ is the long-run mean parameter, and σ is the local volatility parameter. The short-rate factor in the standard CIR model cannot be simulated using a discrete-time counterpart directly. Wu and Zhang (2006) show that the short-rate factor can be simulated according to the following step-wisely moment-matched log-normal scheme 1 2 Q r(t + ∆t) = Et [r(t + ∆t)] exp − Γt ∆t + Γt ∆Wt , (53) 2 where Γ2t
EQ [r 2 (t + ∆t)] 1 , log tQ = ∆t (Et [r(t + ∆t)])2 28
(54)
with −κ∆t EQ , (55) t [r(t + ∆t)] = θ + (r(t) − θ)e 2 2 σ −2κ∆t 2 σ 2 2 (EQ e r (t). (56) EQ 1+ t [r(t + ∆t)]) − t [r (t + ∆t)] = 2κθ 2κθ
29
Table 4: Parameter Estimates of SNP Density for AAA-rated Defaultable Bonds Parameter
Estimate
Standard Error
Hermite: a(0,0) a(0,1) a(1,0) a(0,2) a(2,0) a(0,3) a(3,0) a(0,4) a(4,0)
1.00000 0.00638 0.20973 -0.62401 -0.08536 -0.15094 0.23417 0.16303 -0.10477
(0.00000) (0.10447) (0.08471) (0.09267) (0.07443) (0.11192) (0.05429) (0.22990) (0.05595)
Mean: µ(2,0) µ(1,0) µ(2,4) µ(2,3) µ(2,2) µ(2,1) µ(1,4) µ(1,3) µ(1,2) µ(1,1)
-0.04549 -0.04989 -0.02216 -0.13827 0.92669 0.17952 -0.43903 0.11032 0.45267 0.83201
(0.02630) (0.02827) (0.05423) (0.04049) (0.05741) (0.03770) (0.06370) (0.05407) (0.05964) (0.05886)
ARCH: R(1,0) R(1,2) R(3) R(1,1) R(2,1) R(1,2) R(2,2) R(1,3) R(2,3) R(1,4) R(2,4) R(1,5) R(2,5)
0.08474 0.19452 0.11277 0.44506 0.62252 -0.27063 -0.20607 0.03205 0.44619 -0.31903 -0.33327 0.04459 0.34684
(0.02959) (0.02276) (0.04153) (0.08765) (0.12043) (0.10311) (0.16883) (0.13836) (0.10371) (0.06837) (0.12196) (0.11355) (0.09001)
30
Table 5: Parameter Estimates of SNP Density for BBB-rated Defaultable Bonds Parameter
Estimate
Standard Error
Hermite: a(0,0) a(0,1) a(1,0) a(0,2) a(1,1) a(2,0) a(0,3) a(3,0) a(0,4) a(4,0)
1.00000 -0.03099 -0.10675 -0.32663 0.15431 0.37065 0.01324 -0.08556 0.17069 0.18115
(0.00000) (0.05858) (0.10617) (0.09267) (0.04718) (0.05042) (0.10127) (0.04141) (0.07264) (0.05296)
Mean: µ(2,0) µ(1,0) µ(2,2) µ(2,1) µ(1,2) µ(1,1)
0.00172 0.06437 0.92679 0.04129 0.37820 0.55571
(0.02304) (0.02865) (0.03035) (0.02798) (0.05805) (0.05642)
ARCH: R(1,0) R(1,2) R(3) R(1,1) R(2,1) R(1,2) R(2,2) R(1,3) R(2,3) R(1,4) R(2,4)
0.09587 0.16905 0.11637 -0.34786 -0.63774 -0.17104 -0.74861 0.03381 0.71594 -0.29285 -0.19392
(0.02267) (0.01430) (0.02255) (0.06822) (0.11182) (0.05820) (0.12060) (0.06624) (0.14145) (0.05839) (0.16617)
Note: The parameter in the Hermite polynomial function a(i, j) stands for the term with ith power on the short yield and jth power on the long yield. The parameters in the VAR conditional mean function µ(1, 0), µ(2, 0), µ(1, 1), µ(2, 1), µ(1, 2), and µ(2, 2) are, respectively, constant in short yield, constant in long yield, lag one short yield in short yield equation, lag one long yield in short yield equation, lag one short yield in long yield equation, and lag one long yield in long yield equation. The parameter in the ARCH standard deviation function R(1, m) and R(2, n) is, respectively, short yield with lag equals m, and long yield with lag equals n. The parameter R(3) is the constant off-diagonal term.
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Table 6: Model Estimation by Efficient Method of Moments Panel A: Model 1 Regime 0: a0 a1 σ θ φ
0.0063 (0.0003) -0.0884 (0.0008) 0.0024 (0.0003) -8.3550 (0.0182)
Model 2 0.0031 (0.0005) -0.0784 (0.0016) 0.0029 (0.0006) -12.2900 (0.0328) 0.1279 (0.0382)
Panel B: Model 1 0.0112 (0.0003) -0.1079 (0.0004) 0.0063 (0.0003) -7.4468 (0.0046)
Model 2 0.0029 (0.0005) -0.0654 (0.0005) 0.0049 (0.0005) -10.8970 (0.0557) 0.1561 (0.0216)
Regime 1: a0 a1 σ θ φ
0.0103 (0.0006) -0.0874 (0.0005) 0.0049 (0.0006) -12.9640 (0.0285) -0.3154 (0.0498)
0.0282 (0.0005) -0.2119 (0.0028) 0.0063 (0.0006) -13.0460 (0.0403) -0.3205 (0.0227)
Transitional Probability: νs (0, 1) νs (1, 0)
-3.1279 (0.0204) -3.1846 (0.0069)
-3.1592 (0.0272) -3.2334 (0.0339)
Specification Test: χ2 d.o.f.
216.39 27
79.46 19
238.77 22
84.10 14
Note: The two term structure models are given in Chapter 4. Model 1 is the standard one-factor model that is proposed by Cox, Ingersoll, and Ross (1985), without regime shifts. Model 2 is the one-factor CIR model that is developed in Theorem 3.2, in which regime shift is priced. First, a0 (s), a1 (s) and σ(s) are the coefficients that are given in the diffusion process of defaultable p interest rate y(t): dy = (a0 (s) + a1 (s)y)dt + σ(s)ydBy . Second, θ(s) is the coefficient on the p market price of diffusion risk that is given in ηw = θ(s) σ(s)y, and φ(s) is the coefficient on the market price of regime-switching risk that is given in λs (z) = 1 − eφs (z) . Third, νs (z) is the parameter which determines the transitional probability that is given in h(z, x) = eνs (z) .
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