Parametric Fitting Method

Two major approaches for fitting the term structures of interest rates can be differenti- ated.6 McCulloch (1971a, 1971b) introduced spline methods to approximate the discount function using a continuous piecewise polynomial or exponential function.7 Spline meth- ods provide sufficient fitting quality and require acceptable computational effort, however, they are omitted below due to their sensitivity to outliers, especially when data is scarce as in this case.

Alternatively, parametric models define the term structure of spot or forward rates as a functional form specified by a set of parameters. Parametric models are stable with respect to outliers and provide sufficient variation in the shape of fitted term structures. For its parsimonious parameterization with only four free parameters and its ability to fit normal, inverse and humped-back term structures, the exponential form provided by Nelson and Siegel (1987) is used to fit the term structures of riskless rates and for the term structures of risk classes of defaultable bonds. Nelson and Siegel propose the functional

6 _{A comprehensive overview of procedures for the fitting of term structures of interest rates is provided}

by Anderson et al. (1996), p. 57-64.

form Rc_t(τ;β_tc) =β₀c(t) + (β₁c(t) +β₂c(t))1−e −τ βc₃(t) − τ βc 3(t) −β₂c(t)e −τ βc 3(t) (4.5)

for the time t yield-to-maturityRc

t(τ, βtc) of a zero-coupon bond with time-to-maturity τ, termed as spot rate or zero rate, and the parameter vectorβc

t = (β0c(t), β1c(t), β2c(t), β3c(t)).

The short rate and the zero rate for an infinite time-to-maturity are directly related to parameters by the convergence characteristics:

lim τ−>0R c t(τ;β c t) =β c 0(t) +β c 1(t) (4.6) lim τ−>∞R c t(τ;β c t) =β c 0(t), (4.7)

so that β₀c(t) represents the long rate and β₁c(t) indicates the term spread of the term structure. Furthermore, β₂c(t)>0 (β₂c(t)<0) controls for the upward (downward) hump of the term structure in the short to medium term of the curve, whileβ₃c(t) characterizes the curvature.

An extension of the Nelson-Siegel form proposed by Svensson (1994) enables an even more flexible fit of term structures using an additional exponential term with two more parameters to set. However, limitations of data on defaultable bonds, especially at the beginning of the sample period, turn the fitting process unstable and result in term structures that are not robust. Even if a robust fitting of riskless term structures is possible, the Svensson model is not used, because the increase in fitting quality is negligible compared to the Nelson-Siegel model, so that the use of different parametric forms for the fitting of riskless curves and risk class curves is avoided.

The Nelson-Siegel term structures of spot rates at timetis specified by the parameter set

βc∗

t which minimizes the sum of the squared differences

X i∈RC b Di_t(τi, CFi(t);βtc)−D i t 2 (4.8)

of the present value Dbi_t(τ_i, CFi(t);β_tc) = Pt+τi

tj=etde

−Rc

t(tj−t;βct) ·CFi

tj of the coupon bond

i∈Ic _{in classc∈ {RC, rl}with deterministic cash flowsCF}i_{(t) = (CF}i

etd, ..., CFti+τi) and

time-to-maturity τi to the observed dirty price Dit of the bond.

Alternatively to (4.8), the Nelson-Siegel term structure can be fitted to minimize differ- ences in bond yields. Since the sensitivity of bond prices to a change in the term structure increases in the time-to-maturity of bonds, the fitting of bond prices implicitly accentu- ates observations of distant maturities, while yield fitting does not involve an implicit maturity-dependent weighting of observations. However, due to the illiquidity of bond

markets, yields show an increased spreading close to maturity which negatively affects the stability of term structures, so that price fitting is preferred.

Before riskless yield curves are fitted, mis-specified prices of government bonds are cor- rected according to the method proposed by Schwartz (1998). An outlier correction for credit curves is not feasible, because considerable changes in credit yields cannot easily be separated into pricing errors and actual changes in credit risk. In the fitting of risk class curves, the short rate of a risk class is fixed to the short rate of the riskless yield curve to stabilize credit curves in the short term and to ensure that short spreads converge to zero if the remaining time to maturity converges to zero, in line with the structural credit valuation model.

Fitted term structures of risk classes do not incorporate information on the evolution of residuals of obligor-specific bond yields from term-structure-implied yields. Since the fitting of obligor-specific yield curves is precluded due to data limitations, the yield curves of risk classes only qualify to estimate systematic factor processes.

In Section 4.4, the dynamics of risk class factor processes and the trajectories of systematic factors will be estimated based on yield series of coupon bonds which are derived from synthetic term structures of credit spot rates. The synthetic nature of the risk class curves is caused by the constant riskless short rate which is mandatory in the credit valuation model. The term structure of synthetic spot rates of a risk class is calculated by adding the riskless short rate Rrl

t(0;βtrl) to the credit spot spreads Strc(τ;βtrc, βtrl) =

Rrc_(τ;βrc

t ) −Rrl(τ;βtrl), defined as difference between the spot rate Rrct (τ;βtrc) of risk class rc and the riskless spot rate Rrl

t (τ;βtrl). In order to regain the lost information about specific yield variations in the estimation of specific factor weights of the risk class factor model, synthetic credit spot rates are used in Section 4.4.6 for bootstrapping obligor-specific changes in credit spreads.