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Credit Rating Transition Matrix

The transition_matrix function calculates a credit rating transition matrix, which shows the probability of credit ratings migrating from one level to another over a specified time period. This is a fundamental tool in credit risk management for understanding rating stability and modeling future rating changes.

Usage in Pypulate

from pypulate.credit import transition_matrix

# Calculate transition matrix from historical rating data
result = transition_matrix(
    ratings_t0=['AAA', 'AA', 'A', 'BBB', 'BB', 'B', 'CCC'],  # Ratings at time 0
    ratings_t1=['AA', 'A', 'BBB', 'BB', 'B', 'CCC', 'D']     # Ratings at time 1
)

# Access the results
prob_matrix = result["probability_matrix"]
trans_matrix = result["transition_matrix"]
ratings = result["ratings"]

Parameters

Parameter Type Description Default
ratings_t0 array_like Array of credit ratings at the beginning of the period Required
ratings_t1 array_like Array of credit ratings at the end of the period Required

Return Value

The function returns a dictionary with the following keys:

Key Type Description
transition_matrix list of lists Count of transitions from each rating to each other rating
probability_matrix list of lists Probability of transitioning from each rating to each other rating
ratings list Unique ratings found in the input data

Risk Level Classification

Credit ratings are typically categorized into risk levels:

Rating Category Risk Level
AAA, AA Investment Grade - Very Low Risk
A, BBB Investment Grade - Low to Moderate Risk
BB, B Non-Investment Grade - Moderate to High Risk
CCC, CC, C Non-Investment Grade - Very High Risk
D Default

Comprehensive Example

Here's a complete example demonstrating how to calculate and analyze credit rating transitions:

from pypulate.credit import transition_matrix
import matplotlib.pyplot as plt
import numpy as np
from collections import Counter

# Sample historical rating data
# Let's create a dataset with 100 companies and their ratings over two periods
np.random.seed(42)  # For reproducibility

# Define possible ratings
ratings = ['AAA', 'AA', 'A', 'BBB', 'BB', 'B', 'CCC', 'D']

# Generate initial ratings with a distribution skewed toward investment grade
weights = [0.05, 0.15, 0.25, 0.25, 0.15, 0.10, 0.05, 0.00]  # No defaults at t0
initial_ratings = np.random.choice(ratings, size=100, p=weights)

# Create a simple model for rating transitions
# Higher ratings are more stable, lower ratings have higher probability of downgrade
def generate_next_rating(current_rating):
    current_idx = ratings.index(current_rating)

    if current_rating == 'D':
        return 'D'  # Default is an absorbing state

    # Probability of staying at the same rating
    stay_prob = 0.7 - 0.05 * current_idx  # Higher ratings are more stable

    # Probability of upgrading (less likely for higher ratings)
    if current_idx == 0:
        upgrade_prob = 0  # AAA can't be upgraded
    else:
        upgrade_prob = 0.05 * (8 - current_idx) / 7  # More room for upgrade at lower ratings

    # Probability of downgrading (more likely for lower ratings)
    downgrade_prob = 1 - stay_prob - upgrade_prob

    # Determine direction of movement
    r = np.random.random()
    if r < stay_prob:
        return current_rating
    elif r < stay_prob + upgrade_prob:
        return ratings[current_idx - 1]  # Upgrade
    else:
        # For downgrades, allow for multi-notch downgrades for lower ratings
        if current_idx >= 5:  # B or lower
            # Possible to skip directly to default
            possible_downgrades = ratings[current_idx+1:]
            return np.random.choice(possible_downgrades)
        else:
            return ratings[current_idx + 1]  # Single notch downgrade

# Generate ratings at time 1
final_ratings = [generate_next_rating(rating) for rating in initial_ratings]

# Calculate the transition matrix
result = transition_matrix(initial_ratings, final_ratings)

# Print the transition probability matrix
print("Credit Rating Transition Matrix (Probabilities)")
print("==============================================")
prob_matrix = result['probability_matrix']
unique_ratings = result['ratings']

# Print the matrix with proper formatting
print(f"{'':5}", end="")
for r in unique_ratings:
    print(f"{r:6}", end="")
print()

for i, row in enumerate(prob_matrix):
    print(f"{unique_ratings[i]:5}", end="")
    for val in row:
        print(f"{val:.2f}  ", end="")
    print()

# Calculate some statistics
downgrades = sum(1 for i, j in zip(initial_ratings, final_ratings) 
                if ratings.index(j) > ratings.index(i))
upgrades = sum(1 for i, j in zip(initial_ratings, final_ratings) 
              if ratings.index(j) < ratings.index(i))
same = sum(1 for i, j in zip(initial_ratings, final_ratings) if i == j)
defaults = sum(1 for rating in final_ratings if rating == 'D')

print("\nTransition Statistics:")
print(f"Total Entities: {len(initial_ratings)}")
print(f"Upgrades: {upgrades} ({upgrades/len(initial_ratings):.1%})")
print(f"Downgrades: {downgrades} ({downgrades/len(initial_ratings):.1%})")
print(f"Unchanged: {same} ({same/len(initial_ratings):.1%})")
print(f"Defaults: {defaults} ({defaults/len(initial_ratings):.1%})")

# Visualize the transition matrix as a heatmap using matplotlib
plt.figure(figsize=(10, 8))
plt.imshow(prob_matrix, cmap='YlGnBu', aspect='equal')
plt.colorbar(label='Transition Probability')

# Add text annotations to the heatmap
for i in range(len(unique_ratings)):
    for j in range(len(unique_ratings)):
        plt.text(j, i, f"{prob_matrix[i][j]:.2f}", 
                 ha="center", va="center", 
                 color="black" if prob_matrix[i][j] < 0.7 else "white")

# Set ticks and labels
plt.xticks(range(len(unique_ratings)), unique_ratings)
plt.yticks(range(len(unique_ratings)), unique_ratings)
plt.xlabel('Rating at Time 1')
plt.ylabel('Rating at Time 0')
plt.title('Credit Rating Transition Matrix')
plt.tight_layout()
plt.show()

# Visualize rating distribution before and after
plt.figure(figsize=(12, 6))

# Count ratings at each time period
t0_counts = {r: 0 for r in ratings}
t1_counts = {r: 0 for r in ratings}

# Update with actual counts
for r in initial_ratings:
    t0_counts[r] += 1
for r in final_ratings:
    t1_counts[r] += 1

# Create bar chart
x = np.arange(len(ratings))
width = 0.35

fig, ax = plt.subplots(figsize=(12, 6))
ax.bar(x - width/2, [t0_counts[r] for r in ratings], width, label='Time 0')
ax.bar(x + width/2, [t1_counts[r] for r in ratings], width, label='Time 1')

# Add labels and legend
ax.set_xlabel('Credit Rating')
ax.set_ylabel('Number of Entities')
ax.set_title('Rating Distribution: Before vs After')
ax.set_xticks(x)
ax.set_xticklabels(ratings)
ax.legend()
ax.grid(axis='y', linestyle='--', alpha=0.7)

plt.tight_layout()
plt.show()

# Calculate and visualize the default rates by initial rating
default_rates = []
for rating in ratings[:-1]:  # Exclude 'D' as initial rating
    entities_with_rating = sum(1 for r in initial_ratings if r == rating)
    if entities_with_rating > 0:
        defaults_from_rating = sum(1 for i, j in zip(initial_ratings, final_ratings) 
                                  if i == rating and j == 'D')
        default_rate = defaults_from_rating / entities_with_rating
    else:
        default_rate = 0
    default_rates.append(default_rate)

plt.figure(figsize=(10, 6))
plt.bar(ratings[:-1], [rate * 100 for rate in default_rates], color='darkred')
plt.title('Default Rate by Initial Credit Rating')
plt.xlabel('Initial Credit Rating')
plt.ylabel('Default Rate (%)')
plt.grid(axis='y', linestyle='--', alpha=0.7)
plt.tight_layout()
plt.show()

Example Output

Credit Rating Transition Matrix (Probabilities)
==============================================
     A     AA    AAA   B     BB    BBB   CCC   D     
A    0.52  0.00  0.00  0.00  0.00  0.48  0.00  0.00  
AA   0.23  0.68  0.09  0.00  0.00  0.00  0.00  0.00  
AAA  0.00  0.33  0.67  0.00  0.00  0.00  0.00  0.00  
B    0.00  0.00  0.00  0.50  0.00  0.00  0.00  0.50  
BB   0.00  0.00  0.00  0.53  0.35  0.12  0.00  0.00  
BBB  0.00  0.00  0.00  0.00  0.33  0.67  0.00  0.00  
CCC  0.00  0.00  0.00  0.00  0.00  0.00  0.40  0.60  
D    0.00  0.00  0.00  0.00  0.00  0.00  0.00  0.00  

Transition Statistics:
Total Entities: 100
Upgrades: 4 (4.0%)
Downgrades: 40 (40.0%)
Unchanged: 56 (56.0%)
Defaults: 7 (7.0%)

Visualizations

Transition Matrix Heatmap

This visualization shows the probability of transitioning from one rating to another as a color-coded heatmap, with darker colors indicating higher probabilities.

Transition Matrix Heatmap

Rating Distribution

This bar chart compares the distribution of ratings at the beginning and end of the period, showing how the overall credit quality of the portfolio has changed.

Transition Matrix Heatmap

Default Rates by Initial Rating

This visualization shows the relationship between initial credit rating and default rate, illustrating the higher default probabilities associated with lower credit ratings.

Practical Applications

Credit rating transition matrices can be used for:

  1. Credit Portfolio Management: Modeling the evolution of portfolio credit quality over time
  2. Economic Capital Calculation: Estimating potential credit losses under various scenarios
  3. Pricing Credit-Sensitive Instruments: Determining appropriate spreads for bonds and loans
  4. Stress Testing: Analyzing the impact of economic downturns on credit quality
  5. Regulatory Compliance: Meeting requirements for internal ratings-based approaches under Basel frameworks

Methodological Considerations

When calculating transition matrices, several methodological issues should be considered:

  1. Time Horizon: Transition probabilities depend on the length of the observation period (e.g., 1-year vs. 5-year)
  2. Rating Withdrawals: How to handle entities whose ratings are withdrawn during the period
  3. Point-in-Time vs. Through-the-Cycle: Whether to use ratings that reflect current conditions or long-term averages
  4. Cohort vs. Hazard Rate Method: Different approaches to calculating transition probabilities
  5. Economic Conditions: Transition probabilities vary across different phases of the economic cycle

Limitations

Credit rating transition matrices have several limitations:

  1. Rating Stability Bias: Rating agencies may be slow to change ratings, leading to underestimation of transition probabilities
  2. Limited History: For newer rating categories or markets, historical data may be insufficient
  3. Non-Markovian Behavior: Future rating changes may depend on rating history, not just current rating
  4. Heterogeneity Within Ratings: Entities with the same rating may have different default probabilities
  5. Time Variation: Transition probabilities change over time with economic conditions