## Rating Through-the-Cycle: What does the Concept Imply for Rating Stability and Accuracy? — _wp1364

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### Introduction and research objective
- CRAs face a tradeoff between producing accurate default-risk estimates and delivering stable ratings that do not change frequently.
- PIT approach: uses current information; provides the most accurate estimate of future default probabilities and expected losses.
- TTC approach: smooths ratings and incorporates a stress scenario for cyclical components to balance accuracy and stability.
- Paper objective: compare PIT and TTC within a stochastic framework by simulating asset values with permanent and cyclical shocks and mapping risk metrics into discrete ratings.

### Model structure and key formulas
- Asset decomposition:
  - x_t = x*_t + y_t
  - x_t: logarithm of observed asset value
  - x*_t: permanent (fundamental) value
  - y_t: cyclical component
- Dynamics:
  - Permanent component: dx* = μ dt + σ dW, where dW = ε √dt, ε vN(0,1).
  - Cyclical component: y_t = ρ y_{t−1} + u_t, with 0 < ρ < 1, u_t vN(0,σ_u^2), and Cov(ε_t, u_t) = 0.
- Forecast expectations for horizon T (t < T):
  - E(x*_T) = x*_t
  - E(y_T) = ρ^T y_t
  - E(x_T) = x*_t + ρ^T y_t
- Distance-to-default formulas:
  - PIT: DD_PIT = [E(x_{t+s}) − d] / σ(x)
  - TTC stress worst-case cyclical lower bound for v-period forecast:
    - ȳ_{t+v} = E(y_{t+v}) + Φ^{-1}[p(S)] σ_u √(1 + ρ^2 + ρ^4 + ... + ρ^{2(v−1)})
  - Stressed forecast: S(x_{t+v}) = E(x*_{t+v}) + Φ^{-1}[p(S)] σ_u √(1 + ρ^2 + ρ^4 + ... + ρ^{2(v−1)})
  - TTC: DD_TTC = [S(x_{t+v}) − d] / σ(x)
- Default probability mapping:
  - PD_i = Φ[−DD_i], i ∈ {PIT, TTC}
  - Continuous PDs mapped into discrete rating grades using Moody’s idealized default probability table (5-year table used).

### Numerical simulation setup and calibration
- Simulation grid:
  - Initial permanent component x* varied between 1.2 and 5.4 in increments of 0.2 (i.e., [1.2, 1.4, ... , 5.4]).
  - Volatility σ(ε) varied between 15 percent and 85 percent in increments of 5 percent (i.e., [15, 20, ... , 85 percent]).
  - Total of 330 value-volatility combinations.
- Other parameter values:
  - Cyclical component starting value y_0 = 0.
  - Volatility of autoregressive process σ_u = 20 percent.
  - ρ = 0.96.
  - μ = 0.
  - TTC stress-scenario probability p(S) = 20 percent.
  - Default when net asset value drops below zero at maturity of liability.
  - Monthly time steps: dt = 1/12.
  - Forecasting period: 5 years for TTC.
  - Number of replications: 10,000.
  - TTC mapped using Moody’s 5-year idealized default probabilities.
- PIT setup:
  - Forecast period equals 1 year; Moody’s 1-year idealized default probabilities used.
  - Using Moody’s 1-year table with PIT yields approximately 45 percent of assets AAA and around 75 percent at least AA initially.
- Calibration outcome:
  - Model-implied rating distribution under TTC resembles empirical S&P sovereign rating distribution as of March 2012 (majority investment-grade with the largest single fraction rated B).

### Stability of ratings — quantified comparisons and cliff effects
- General pattern:
  - PIT: initially homogeneous but require substantial downgrades after the first period and continue to deteriorate.
  - TTC: initially more stable ex-ante but prone to large, abrupt downgrades (“rating cliff effect”) when cushions are eroded and adjustments delayed.
- Tail-risk experiment design:
  - Asset realizations set below the 5th percentile of each asset’s distribution; the average of realizations below the 5th percentile used to assess rating evolution.
- Aggregate outcomes (periods 1–5, tail realization):
  - End of experiment under PIT: only 45 percent of underlying assets receive an investment-grade rating.
  - Under PIT: a total of 38 percent would be rated below B at the end.
- Downgrade frequency under Cantor and Mann (2006) smoothing rule (ratings adjusted only if new rating is at least two notches below old one and persists > 1 period):
  - Average downgrades per period:
    - TTC methodology: 57 downgrades (on average) per period.
    - PIT methodology: 196 downgrades (on average) per period.
  - Detailed per-period counts:
    - Period 2: 225 downgrades under PIT.
    - Period 3: 62 downgrades under TTC; 202 downgrades under PIT.
    - Period 4: 44 downgrades under TTC; 186 downgrades under PIT.
    - Period 5: 66 downgrades under TTC; 172 downgrades under PIT.
- Rating cliff examples (period 3 and others):
  - Period 3 left-panel: x* = 2.8 and volatility σ(ϵ) = 0.8 → TTC: BB- to CCC- (5-notch downgrade).
  - Period 3 right-panel: x* = 2.0 and σ(ϵ) = 0.55 → TTC: 4-notch downgrade.
  - Period 4:
    - Upper-left: x* = 3.2 and σ(ϵ) = 0.75 → 5-notch downgrade under TTC.
    - Upper-right: x* = 1.6 and σ(ϵ) = 0.45 → 4-notch downgrade under TTC.
  - Period 5:
    - Lower-left: x* = 2.2 and σ(ϵ) = 0.45 → 4-notch downgrade under TTC.
    - Lower-right: x* = 3.0 and σ(ϵ) = 0.60 → 4-notch downgrade under TTC.
  - Edge case: if rating already CCC-, no room for a 2-notch downgrade; rating adjusted to D with a 1 period lag.
- Interpretation and mitigation:
  - TTC builds in pessimistic forecasts and downgrades less often ex-ante.
  - Delayed adjustments after PIT realizations breach TTC-implied levels produce abrupt large downgrades (up to 5 notches) or immediate jumps to default.
  - Immediate switching from TTC to PIT once the initial stress scenario is breached would mitigate cliff effects and improve the stability–accuracy trade-off.

### Predictive power and temporal CAP analysis
- Method: Cumulative Accuracy Profile (CAP) curves compare cumulative proportion of defaulters by rating grade to cumulative proportion of entities by rating grade.
- Period 1:
  - Initial CAP: TTC only slightly inferior to PIT in forecasting defaults for period 1.
- Period 2:
  - PIT performs better because TTC ratings remain unaltered from the previous period.
- Period 3:
  - Once TTC ratings are updated, both approaches predict defaults roughly the same; TTC can be slightly better in period 3 because TTC downgrades capture significant, persistent deterioration and create more granularity among low-quality entities.
- Periods 4 and 5:
  - PIT again dominates on average when predicting future defaults.
  - Reason: After TTC downgrades target the worst-performing assets, subsequent TTC ratings are not updated immediately, reducing ex-post forecasting performance relative to PIT.
- Overall CAP conclusion:
  - PIT ratings generally provide more accurate information about default probability over time because TTC’s lagged reactions reduce forecasting ability.

### Policy implications and recommendations
- Core problem: the second-stage smoothing and lagged reaction in TTC that delays updates to new market information, causing rating cliffs and potential market disruption including forced selling.
- Policy-relevant observations:
  - Debates on TTC usefulness should focus on CRAs’ reaction once asset values breach initial stress scenarios.
  - “Through the crisis” methodology (TTC with more severe ex-ante stress) reduces cliff effects partially but does not address slow adjustment after the TTC cushion is eroded.
- Recommended operational response:
  - Implement immediate switching to PIT-like updates after the TTC stress scenario is breached to mitigate cliff effects while preserving some ex-ante stability benefits.
- Trade-off summary:
  - TTC: more stable ex-ante, prone to cliff effects, inferior overall forecasting accuracy when adjustments are delayed.
  - PIT: more volatile, superior forecasting accuracy for defaults.

*Source: Rating Through-the-Cycle: What does the Concept Imply for Rating Stability and Accuracy?, John Kiff, Michael Kisser and Liliana Schumacher, WP/13/64, March 2013.*

### Section 1

### Rating Through-the-Cycle: What does the Concept Imply for Rating Stability and Accuracy? — Section 1

### Introduction: trade-off between accuracy and stability
- Credit Rating Agencies (CRAs) face a tradeoff between delivering accurate estimates of default risk and producing stable ratings that do not change frequently.
- Point-in-Time (PIT) approach: uses current information to compute default risk metrics; ratings provide the most accurate estimate of future default probabilities and expected losses.
- Through-the-Cycle (TTC) approach: intended to balance accuracy and stability by smoothing ratings and incorporating a stress scenario for cyclical components.
- Paper objective: compare PIT and TTC approaches within a stochastic framework, assessing impacts on rating stability and accuracy by simulating asset values subject to transitory and cyclical shocks and mapping risk metrics into discrete ratings.

### Literature overview: definitions and prior findings on TTC
- No single, simple definition of TTC exists in the literature or CRA practice.
- Key prior results summarized:
  - Altman and Rijken (2006): CRAs focus on the permanent credit risk component; slow rating adjustments are a main source of stability.
  - Loeffler (2004, 2005): asset values have permanent and cyclical components; TTC imposes stress on the cyclical component when forecasting; slow reaction helps avoid rating reversals.
  - Topp and Perl (2010): actual ratings show cyclical patterns despite TTC claims.
  - Carey and Hrycay (2001): TTC estimates default risk over a long horizon and applies an explicit stress scenario.
  - Cantor and Mann (2006) and Standard and Poor’s commentaries: CRAs consider the likelihood of experiencing large adverse changes under moderate stress and may assign lower ratings accordingly.
- Operational definition used in this paper (following the literature):
  - TTC rating is a two step process: (i) ex-ante ratings are calculated conditional on a stress scenario for the cyclical component; (ii) ex-post rating changes are smoothed and not adjusted immediately.

### The model: structural credit risk with permanent and cyclical components
- Asset decomposition:
  - x_t = x*_t + y_t
  - x_t: logarithm of observed asset value
  - x*_t: permanent (fundamental) value
  - y_t: cyclical component
- Dynamics:
  - Permanent component: dx* = μ dt + σ dW, where dW = ε √dt, ε vN(0,1).
  - Cyclical component: y_t = ρ y_{t−1} + u_t, with 0 < ρ < 1, u_t vN(0,σ_u^2), and Cov(ε_t, u_t) = 0.
- Forecast expectations for horizon T (t < T):
  - E(x*_T) = x*_t
  - E(y_T) = ρ^T y_t
  - E(x_T) = x*_t + ρ^T y_t
- Distance-to-default (PIT) for forecasting horizon s:
  - DD_PIT = [E(x_{t+s}) − d] / σ(x)
  - d is the face value of liabilities; σ(x) is volatility of x_t.
- TTC stress-test construction:
  - Stress scenario probability p(S); worst-case cyclical lower bound for v-period forecast:
    - ȳ_{t+v} = E(y_{t+v}) + Φ^{-1}[p(S)] σ_u √(1 + ρ^2 + ρ^4 + ... + ρ^{2(v−1)})
  - Forecasted stressed asset value:
    - S(x_{t+v}) = E(x*_{t+v}) + Φ^{-1}[p(S)] σ_u √(1 + ρ^2 + ρ^4 + ... + ρ^{2(v−1)})
  - Distance-to-default (TTC):
    - DD_TTC = [S(x_{t+v}) − d] / σ(x)
- Default probability mapping:
  - PD_i = Φ[−DD_i], where i ∈ {PIT, TTC}.
  - Continuous PDs are mapped into discrete rating grades using Moody’s idealized default probability table (5-year table referenced and used in the Appendix).
- Ex-post smoothing (filtering) assumption to capture CRA slow updates:
  - A CRA will update a rating only if (i) the implied rating change is larger than one notch downgrade and (ii) the change is persistent.

### Numerical analysis: simulation setup and parameterization
- Goal: illustrate differences between ratings under TTC and PIT and choose parameters so TTC-implied rating distribution resembles actual Standard and Poor’s sovereign ratings (as of March 2012).
- Simulation grid:
  - Initial permanent component x* varied between 1.2 and 5.4 in increments of 0.2 (i.e., [1.2, 1.4, ... , 5.4]).
  - Volatility σ(ε) varied between 15 percent and 85 percent in increments of 5 percent (i.e., [15, 20, ... , 85 percent]).
  - Total of 330 value-volatility combinations.
- Additional parameter choices:
  - Cyclical component starting value y_0 = 0 (unconditional mean zero).
  - Volatility of the autoregressive process σ_u = 20 percent.
  - ρ = 0.96.
  - μ = 0.
  - TTC stress-scenario probability p(S) = 20 percent.
  - Default occurs when net asset value drops below zero at maturity of liability.
  - Monthly time steps: dt = 1/12.
  - Forecasting period: 5 years.
  - Number of replications: 10,000.
  - For TTC, business cycle and forecasting period length set to 5 years; mapped into ratings using Moody’s 5-year idealized default probabilities.
- Calibration target and outcome:
  - The model-implied rating distribution under the TTC approach is similar to the empirical S&P sovereign rating distribution as of March 2012 (majority investment-grade with the largest single fraction rated B).

### Main conclusions reported (from the Abstract and synthesis)
- Through-the-cycle ratings:
  - Are initially more stable relative to PIT ratings because they smooth and delay rating changes.
  - Are prone to rating cliff effects (large, abrupt rating changes when smoothing breaks down).
  - Suffer from inferior performance in predicting future defaults compared with PIT, because smoothing delays rating changes and TTC imposes stressed cyclical assumptions that reduce predictive accuracy.
- Through-the-crisis methodology (TTC with a more stringent stress test):
  - Goes halfway toward mitigating cliff effects.
  - Remains prone to discretionary rating change delays despite stronger stress assumptions.

*Source: Rating Through-the-Cycle: What does the Concept Imply for Rating Stability and Accuracy?, John Kiff, Michael Kisser and Liliana Schumacher, WP/13/64, March 2013.*

### Section 2

### _wp1364 - Section 2

### Methodology: forecasting horizons and rating mapping
- The length of the forecasting period for the TTC approach is 5 years and model implied default probabilities are mapped into credit ratings using Moody’s 5-year idealized default probabilities.
- For the PIT approach, the forecast period equals 1 year while leaving all other parameter values unchanged.
- Using Moody’s 1-year idealized default probabilities with the PIT approach yields an initial distribution in which approximately 45 percent of the assets would receive a AAA rating and around 75 percent would be at least AA rated.
- Tail-risk experiment design:
  - To analyze rating dynamics under stress, asset realizations are set below the 5th percentile of each asset’s distribution (i.e., the ex-ante probability of observing this value or lower is 5 percent).
  - For each asset, the average value of all realizations below the 5th percentile is used to assess rating evolution.

### A. Stability of ratings — main findings and quantified comparisons
- General patterns:
  - Initial PIT ratings are homogeneous but require substantial downgrades after the first period and continue to deteriorate in subsequent periods.
  - TTC ratings are initially more stable ex-ante but are prone to large, abrupt downgrades (“rating cliff effect”) if the cushion is eroded and adjustments are delayed.
- Aggregate outcomes (periods 1–5, tail realization experiment):
  - At the end of the experiment, under the PIT approach, only 45 percent of the underlying assets receive an investment-grade rating.
  - Under the PIT approach, a total of 38 percent would be rated below B at the end.
- Downgrade frequency under smoothing rule (Cantor and Mann (2006) smoothing): ratings adjusted only if (1) the new rating is at least two notches below the old one and (2) the change persists for more than 1 period.
  - Average downgrades per period:
    - TTC methodology: 57 downgrades (on average) per period.
    - PIT methodology: 196 downgrades (on average) per period.
  - Detailed per-period counts (precision from the text):
    - Period 2: 225 downgrades under the PIT approach.
    - Period 3: 62 downgrades under TTC; 202 downgrades under PIT.
    - Period 4: 44 downgrades under TTC; 186 downgrades under PIT.
    - Period 5: 66 downgrades under TTC; 172 downgrades under PIT.
- Rating cliff examples (period 3 illustrations):
  - Left-panel example:
    - Initial fundamental asset value x* = 2.8 and volatility σ(ϵ) = 0.8.
    - Under the TTC policy, the credit rating would drop from BB- to CCC-: a 5-notch downgrade.
    - The downgrade is more smoothed under PIT or if the CRA immediately switched to PIT once the initial pessimistic forecast was breached.
  - Right-panel example:
    - x* = 2.0 and σ(ϵ) = 0.55.
    - TTC ratings are downgraded by 4 notches.
  - Note: If the rating is already CCC-, there is no room for a 2-notch downgrade; in that case the rating is adjusted to D with a 1 period lag.
- Additional period-specific cliff examples:
  - Period 4:
    - Upper-left: x* = 3.2 and σ(ϵ) = 0.75 → 5-notch downgrade under TTC.
    - Upper-right: x* = 1.6 and σ(ϵ) = 0.45 → 4-notch downgrade under TTC.
  - Period 5:
    - Lower-left: x* = 2.2 and σ(ϵ) = 0.45 → 4-notch downgrade under TTC.
    - Lower-right: x* = 3.0 and σ(ϵ) = 0.60 → 4-notch downgrade under TTC.
- Interpretation:
  - TTC ratings build in pessimistic forecasts, so they are downgraded less often ex-ante.
  - However, when PIT realized values fall below TTC-implied ratings and CRAs delay adjustments, TTC methodology can produce abrupt large downgrades (up to 5 notches) or immediate jump to default in extreme cases.
  - Immediate switching from TTC to PIT once the initial stress scenario is breached would mitigate the cliff effect and maximize the trade-off between rating stability and accuracy.

### B. Predictive power of ratings — CAP analysis and temporal performance
- Method:
  - Cumulative Accuracy Profile (CAP) curves are computed to measure how accurately ratings rank default risk. CAP curves compare cumulative proportion of defaulters by rating grade to cumulative proportion of entities by rating grade.
- Period 1 (defaults at end of period 1):
  - Initial CAP shows the TTC approach is only slightly inferior in forecasting future defaults relative to PIT.
- Periods 2–5 (ratings at beginning of period vs. end-of-period defaults):
  - Period 2:
    - PIT performs better in period 2 because TTC ratings remain unaltered with respect to the previous period.
  - Period 3:
    - Once TTC ratings are updated (period 3), both approaches predict defaults roughly the same; TTC can even provide slightly better default forecasts in this period.
    - Explanation: TTC downgrades reflect significant and persistent deterioration, producing a more granular rating distribution for lower-quality entities and improving discriminatory power at that moment.
  - Periods 4 and 5:
    - PIT again dominates and on average performs better when predicting future defaults.
    - Reason: After TTC downgrades occur (targeting the worst-performing assets), subsequent TTC ratings are not updated immediately, so ex-post forecasting performance deteriorates relative to PIT.
- Overall CAP-based conclusion:
  - PIT ratings generally provide more accurate information regarding default probability over time, particularly because TTC ratings’ lagged reaction reduces forecasting ability.

### V. Summary — conclusions and policy implications
- Comparative summary:
  - TTC implied credit ratings are initially more stable ex-ante (as long as the underlying stress scenario has not been breached).
  - TTC is prone to rating cliff effects and suffers an inferior ability to predict future defaults once realized asset values drop below TTC-implied levels and ratings are not updated immediately.
  - PIT ratings are more volatile but provide superior forecasting accuracy for defaults overall.
- Key mechanism identified:
  - The core problem with the TTC approach is the second-stage smoothing and lagged reaction to new market information. This lag can cause abrupt several-notch downgrades (rating cliffs) and dangerous market disruptions, including forced selling.
- Policy-relevant observations:
  - Current discussions on the usefulness of the TTC approach should focus on CRAs’ reaction to new information once lower asset values related to the initial stress scenario are reached.
  - Implementing a “through the crisis” approach (as mentioned by CRAs) requires a more severe stress test ex-ante but does not address the slow adjustment once the TTC cushion is eroded nor the potential cliff effects due to inefficient smoothing policies.
- Trade-off:
  - The trade-off between rating stability and accuracy is largely determined by whether and how quickly rating agencies update ratings after stress scenarios are breached; immediate switching to PIT-like updates after a breach would mitigate cliff effects while preserving some stability benefits ex-ante.

*Source: _wp1364 - Section 2*

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