Options, Futures & Other Derivatives, - John C. Hull
Credit-Metrics Model
We discussed the Vasicek model that address to question: “how many borrowers may default in an adverse economic scenario?”
This model is particularly useful when we have a very large and homogeneous portfolio — many similar credit instruments, the same probability of default, and a common correlation parameter.
Under these conditions, borrower-specific risk is largely diversified away, and the portfolio’s credit loss can be estimated using a direct algebraic formula.
However, the Vasicek model has one important limitation: a company may not default, but its credit quality may still deteriorate significantly and cause losses.
For example, a company’s rating may change as follows:
A → Baa
Baa → Ba
Ba → B
The company may continue meeting its obligations, but investors now perceive it as riskier.
As a result, the required credit spread increases, the market value of the bond or loan declines, and the bond portfolio may incur a loss even before default occurs.
CreditMetrics addresses this broader problem.
What Does CreditMetrics Measure?
CreditMetrics asks:
How much can a credit portfolio lose because of changes in credit ratings?
For example, if a company is rated Baa today, one year later it may be rated:
Aaa, Aa, A, Baa, Ba, B, Caa, Ca-C, or Default.
Each possible outcome has a different probability and a different financial impact.
Rating Transition Matrix
The first important component of CreditMetrics is the credit rating transition matrix.
The table answers the following question:
If a company has a particular credit rating today, what is the probability that it will have each possible rating one year later?
In the table:
- the first column shows the company’s current rating;
- the following columns show its possible rating one year later;
- each cell shows the probability of a particular rating transition.
For example, if the probability of moving from A to A is 90.91%, this means:
Based on historical data, approximately 90.91% of similar A-rated companies were still rated A one year later.
This is the probability we later use in the simulation.
During the Monte Carlo simulation, Excel generates a credit outcome for each company. However, the outcome is not selected arbitrarily. It must be consistent with the probabilities shown in the transition matrix.
The rating transition matrix is therefore the model’s initial probability map.
From Probabilities to Rating Thresholds
The transition matrix gives us the probability of each possible rating.
However, in a Monte Carlo simulation, Excel does not directly generate a rating.
It first generates a random standard normal value:
The model must then determine which rating corresponds to that value.
For this purpose, the area under the standard normal distribution is divided into several zones:
Aaa | Aa | A | Baa | Ba | B | Caa | Ca-C | Default
Each rating is assigned a specific area under the normal distribution. The size of each area is determined by the probabilities in the rating transition matrix.
The horizontal axis of the diagram shows the thresholds, or boundaries, that determine how much cumulative probability lies to their left.
The threshold values are calculated from cumulative probabilities by adding the migration probabilities from left to right.
For example:
This means that 0.04% of the standard normal distribution lies to the left of −3.3528.
This 0.04% area represents the probability that a Baa-rated company will migrate to Aaa.
Therefore, in the simulation, x < -3.3528, means that the company becomes Aaa.
If the simulated value does not fall in the Aaa zone, but is still below −2.8202, the company becomes Aa.
In other words, each rating has its own zone between two thresholds.
Improvement on the Left, Deterioration on the Right
In this version of CreditMetrics, the normal variable (x) can be interpreted as an indicator of credit deterioration.
A low value of (x) represents an improvement, while a high value represents deterioration.
A value close to zero usually indicates that the company remains near its current rating.
Under this sign convention:
- Baa → Aaa is a rare positive outcome and lies in the left tail;
- Baa → Caa is a rare negative outcome and lies in the right tail;
- Baa → Default is located in the extreme right tail.
Instrument Value Versus Credit Rating
Using the thresholds, we can now determine what rating a company will have one year later.
However, knowing the new rating is not enough.
The final objective of CreditMetrics is not only to simulate a rating transition, but also to determine the financial gain or loss that the portfolio experiences because of that transition.
For this purpose, we must know:
What will the value of the company’s credit instrument be under each possible rating?
In the learning model, we use a revaluation table.
The table contains two companies:
- Company A, whose instrument initially has a value of $100;
- Company B, whose instrument also initially has a value of $100.
However, the companies begin with different credit ratings.
Company A starts with an Aaa rating, while Company B starts with a Baa rating.
The yellow columns show the estimated value of each instrument depending on the rating assigned to the company one year later.
In this learning model, these valuation figures are provided as assumptions.
In a full CreditMetrics model, the values would normally be calculated using rating-specific credit spreads, interest-rate curves, contractual cash flows, remaining maturity, and recovery assumptions.
Simulation
We then simulate the rating migrations.
The model assumes a 20% correlation between the companies’ credit variables.
This correlation is important because companies’ credit conditions are not completely independent. A common economic shock may cause several companies to deteriorate at the same time.
For each simulation trial, the model:
- generates correlated normal credit variables;
- compares them with the relevant rating thresholds;
- determines each company’s rating after one year;
- selects the corresponding instrument value from the revaluation table;
- calculates the new portfolio value;
- calculates the portfolio credit loss.
The final column of the simulation table shows the portfolio loss for each simulated scenario.
After repeating the process many times, we obtain a distribution of possible credit losses.
A 99% confidence level is then applied to this loss distribution to calculate Credit Value at Risk.
The chart below shows the simulated Value at Risk based on the selected confidence level.
Excel Model — CreditMetrics
Adapted from:
Options, Futures, and Other Derivatives — John C. Hull