Rating Dynamics of Fallen Angels and Their Speculative Grade-Rated Peers: Static vs. Dynamic Approach

Abstract

This study adopts the survival analysis framework (Allison, P. D. (1984). Event history analysis. Beverly Hills: Sage) to examine issuer-heterogeneity and time-heterogeneity in the rating migrations of fallen angels (FAs) and their speculative grade-rated peers (FA peers). Cox’s hazard model is considered the preeminent method to estimate the probability that an issuer survives in its current rating grade at any point in time t over the time horizon T. In this study, estimation is based on two Cox’s hazard models, including a proportional hazard model (Cox, Journal of Royal Statistical Society Series B (Methodological) 34:187–220, 1972) and a dynamic hazard model. The first model employs a static estimation approach and time-independent covariates, whereas the second uses a dynamic estimation approach and time-dependent covariates. To allow for any dependence among rating states of the same issuer, the marginal event-specific method (Wei et al., Journal of The American Statistical Association 84:1065–1073, 1989) was used to obtain robust variance estimates. For validation purpose, the Brier score (Brier, Monthly Weather Review 78(1):1–3, 1950) and its covariance decomposition (Yates, Organizational Behaviour and Human Performance 30:132–156, 1982) were applied to assess the forecast performance of estimated models in forming time-varying survival probability estimates for issuers out of sample.

It was found that FAs and their peers exhibit strong but markedly different dependences on rating history, industry sectors, and macroeconomic conditions. These factors jointly, and in several cases separately, are more important than the current rating in determining future rating changes. A key finding is that past rating behaviors persist even after controlling for the industry sector and the evolution of the macroeconomic environment over the time for which the current rating persists. Switching from a static to a dynamic estimation framework markedly improves the forecast performance of the upgrade model for FAs. The results suggest that rating history provides important diagnostic information and different rating paths require different dynamic migration models.

Appendices

Appendix 1: Maximum Partial Likelihood Estimation

The expression in Eq. 72.1 for the Cox’s proportional hazard model and the expression in Eq. 72.3 for the full partial likelihood are repeated here as Eqs. 72.17 and 72.18 for convenience:

$$ {h}_m\left[t,Z\right]=h\left(0,t\right) \exp \left[{Z}^m\beta \right] $$

(72.17)

$$ PL={\displaystyle \prod_{m=1}^{{}_N}{L}_{t_m}^m}={\displaystyle \prod_{m=1}^{{}_N}\left[\frac{ \exp \left(\beta \kern0.5em {Z}^m\right)}{{\displaystyle \sum_{i\in R\left({t}_m\right)} \exp \left(\beta \kern0.5em {Z}^i\right)}}\right]} $$

(72.18)

The log partial likelihood function can be written as

$$ PL={\displaystyle \sum_{m=1}^N\left[\left(\beta \kern0.5em {Z}^m\right)-\operatorname{l}\mathrm{n}\left[{\displaystyle \sum_{i\in R\left({t}_m\right)} \exp \left(\beta \kern0.5em {Z}^i\right)}\right.\right]} $$

(72.19)

The derivative of Eq. (72.19) with respect to β is

$$ \frac{\partial PL}{\partial \beta }={\displaystyle \sum_{m=1}^N\left[{Z}^m-\frac{{\displaystyle \sum_{i\in R\left({t}_m\right)}{Z}^i \exp \left(\beta \kern0.5em {Z}^i\right)}}{{\displaystyle \sum_{i\in R\left({t}_m\right)} \exp \left(\beta \kern0.5em {Z}^i\right)}}\right]} $$

(72.20)

$$ \frac{\partial PL}{\partial \beta }={\displaystyle \sum_{m=1}^N\left[{Z}^m-{\displaystyle \sum_{i\in R\left({t}_m\right)}{w}_{im}\left(\beta \right){Z}^i}\right]} $$

$$ \frac{\partial PL}{\partial \beta }={\displaystyle \sum_{m=1}^N\left[{Z}^m-{\overline{Z}}^{w_{im}}\right]} $$

where ${w}_{im}\left(\beta \right)=\frac{ \exp \left(\beta \kern0.5em {Z}^i\right)}{{\displaystyle \sum_{i\in R\left({t}_m\right)} \exp \left(\beta \kern0.5em {Z}^i\right)}}$ and ${\overline{Z}}^{w_{im}}={\displaystyle \sum_{i\in R\left({t}_m\right)}{w}_{im}\left(\beta \right){Z}^i}$

The estimated coefficient vector $\widehat{\beta}$ can be obtained by setting the derivative in Eq. 72.20 equal to zero and solving for the unknown parameter (Hosmer et al. 2008, pp. 75–76).

Appendix 2: Covariance Graph

Brier score B _t = 0.1578; outcome index variance ${\overline{d}}_t\left(1-{\overline{d}}_t\right)=0.0837$; bias ${\overline{f}}_t-{\overline{d}}_t=-0.2198$; slope ${\overline{f}}_1-{\overline{f}}_0=0.0677$; forecast variance (Scatter) ${S}_{f_t}^2=0.0372$

Yates (1982, pp. 143–148) and Arkes et al. (1995, pp. 121–123) provide detailed descriptions of a covariance graph. For illustration purpose, the above covariance graph depicts the characteristics of the 1-year survival forecasts generated by the proportional Cox’s hazard upgrade model for FAs.

The abscissa shows the survival outcome index. The two possible outcomes for a FA in the upgrade model are “upgrade,” which is denoted as 0 on the left, and “survival” (non-upgrade), which is denoted as 1 on the right. Of 141 holdout FAs available at 1-year lead time (forecast time t = 1 year), 128 FAs survived, and 13 FAs were upgraded. A vertical dotted line is located at the survival base rate, or the overall mean survival outcome index $\overline{d}=0.9078$, on the abscissa. On the ordinate are the probability survival forecasts, categorized in deciles. A horizontal dotted line is located at the overall mean survival forecasts $\overline{f}=0.688$ on the ordinate. The 45° solid line represents unbiased estimates. Bias can be measured as the vertical distance from the 45° line to the point where the vertical survival base rate line and the horizontal mean survival forecast line cross (marked as ◊). If a model produces unbiased forecasts, the vertical and horizontal dotted lines will cross on the 45° line, corresponding to a zero bias. If the two dotted lines meet below (above) the diagonal line, the model underestimates (overestimate) the survival outcome, corresponding to a negative (positive) bias. In the covariance graph, the static upgrade model for FAs is 21.98 % too pessimistic in making 1-year survival forecasts.

On the vertical lines above the survival outcome (1) and the upgrade outcome (0) indices are the histograms for survival forecasts of 128 FAs that actually survived and 13 FAs that were upgraded, respectively. Survived and non-survived holdout FAs are stratified into distinct decile categories in the order of estimated survival probabilities. In this setting, FAs with survival forecasts varying from 0 % to 10 % are put together, those with forecasts ranging from 11 % to 20 % in another decile category and so on. The bars on the histograms illustrate the percentage of survival forecasts made at the individual probability deciles. The number of survival forecasts observed within each decile was attached to the corresponding bar for an easy reference. The further the histogram bars spread along the vertical lines, the greater the scatter (variance) of the survival forecasts.

The outcome index line extending vertically from 1 (on the right edge) includes the mean survival forecasts given to FAs that actually survived, ${\overline{f}}_1=0.6943$. The outcome index line drawn vertically from 0 (on the left edge) contains the average survival forecasts given to FAs that were actually upgraded, ${\overline{f}}_0=0.6265$. The dotted line linking ${\overline{f}}_1$ and ${\overline{f}}_0$ is the regression line for survival forecast on outcome index. The slope of the regression line is the difference between ${\overline{f}}_1$ and ${\overline{f}}_0$, or $\left({\overline{f}}_1-{\overline{f}}_0\right)=6.77\kern0.5em \%$. The further the regression line diverges from the horizontal line, the more discriminative the forecasts of the survived and upgraded groups.