**Week 6: Default Probabilities – Part 3**

*“Introduction … CreditMetrics … C-VaR and F-IRB Capital Requirements … Credit Risk Plus”*

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**Summaries**

- Week 6: Default Probabilities - Part 3 > Lesson 1: CreditMetrics > Video Lesson
- Week 6: Default Probabilities - Part 3 > Lesson 2: C-VaR and F-IRB Capital Requirements > Video Lesson
- Week 6: Default Probabilities - Part 3 > Lesson 3: Credit Risk Plus > Video Lesson
- Week 6: Default Probabilities - Part 3 > Summary > Video

__Week 6: Default Probabilities – Part 3 > Lesson 1: CreditMetrics > Video Lesson__

__Week 6: Default Probabilities – Part 3 > Lesson 1: CreditMetrics > Video Lesson__

- In this week, we will continue our survey of models for the estimation, for the assessment of the probability of default of a counterparty.
- We consider a company that has been assigned to some credit rating class at the beginning of period [0,T]. This means that, for this company, we also have information about the transition probabilities to other rating classes and, naturally, about the probability of default by the end of the period.
- These transition probabilities can be computed using historical data, or other techniques.
- With p bar j, we indicate the probability that our company will be in rating class j at time T. To simplify notation, we order the j’s, so that j equal to 0 means “default” and j equal to n means that our company is in the best rating class, e.g. AAA.
- With p bar 0, we clearly denote the probability of default of the company at time T. Using the same notation we used in Merton’s model, with VT we indicate the value of the assets of the company at time T. If you remember, under Merton’s model, we have said that VT is linked to a normal distribution; and we have used the CDF of a Normal to compute the PD under Merton.
- For all the other d’s, we choose them so that the probability that VT is between dj tilde and dj 1 tilde is equal to p bar j. In this way, we translate the transition probabilities into a series of thresholds for the asset value of the company.
- All the other thresholds simply define the other rating classes, so that our company belongs to class j if VT is between dj tilde and d(j 1) tilde.
- The consequence of this transformation is that now our company belongs to class j if XT is between dj star and d(j 1) star.
- Let’s consider an example to better understand the mechanism thanks to which we transform transition probabilities into thresholds for XT. Consider a B-rated company over a 1-year period.
- The probabilities we are interested in are those in the green row.
- Using the table, the probability that our B-rated company will improve its rating to Aaa by the end of the year is 0.01% or 0.0001.
- If we consider the probability of default, this is 0.0473, and so on of all the other probabilities and classes.
- This probability, under the standard Normal quantile function, corresponds to a value of -1.6716.
- Starting from the lowest class, that is default, we also add the probability of moving to class Ca-C. Class Caa is then defined by thresholds -1.6009 and -1.1790.
- This last value is again obtained by inverting the cumulative probability.
- By ordering the rating classes, cumulating their transition probabilities and applying the quantile function, we can compute all the thresholds we are interested in.
- As you can imagine, such a topic requires a knowledge of probability which I cannot assume in this course.

__Week 6: Default Probabilities – Part 3 > Lesson 2: C-VaR and F-IRB Capital Requirements > Video Lesson__

__Week 6: Default Probabilities – Part 3 > Lesson 2: C-VaR and F-IRB Capital Requirements > Video Lesson__

- Are you ready to see how we can use the PD of a counterparty to actually compute the capital requirements for credit risk for that counterparty? And are you ready to see how we can compute capital requirements for an entire portfolio of obligors, with which we have some sort of credit relation? Ok, the topic of today is the computation of capital requirements starting from the probability of default.
- Because, if you remember, this is the approach in which we start from the PD, and we plug this information into some specific formulas, the risk-weight functions, that will provide us with the desired results in terms of capital requirements.
- Let’s see how we can compute the capital requirements for credit risk, starting from the PD. Let’s go.
- A key quantity for the computation of capital requirements under the IRB approaches is the so-called worst case default rate.
- The point is that, in order to derive this formula completely and formally, we would need – all – a big probabilistic apparatus that we do not have, such as for example the copula model.
- That formula includes quantities we know, such as the exposure at default and the loss given default, for each obligor i. Those quantities are usually defined using historical data, empirical studies or, for certain types of instruments, they are provided by the regulator.
- Do you remember what we have said about the IRB approaches in Week 2? Do you remember risk factors and risk-weight functions? We are now ready to understand a little more about that! If you cannot remember something, go back and check.
- Here we are: on your screen you see how we are supposed to compute the correlation parameter for corporate, sovereign and bank exposures.
- Have you noticed that in this formula, based on some empirical studies, the PD and the correlation move in two opposite directions? If the PD increases, correlation decreases.
- Why? The idea is that if a company becomes less creditworthy, its PD increases and its probability of default becomes more idiosyncratic, company-specific, and less affected by the overall market conditions.
- For corporate, sovereign and bank exposures, the formula used for computing the capital requirements for each counterparty is the one you see on your screen.
- The total capital requirements for credit risk in a portfolio are simply given by summing the capital requirements of all the counterparties that belong to the portfolio.
- Those capital requirements need to be covered with Tier 1, additional Tier 1 and Tier 2 capital, according to the rules we have seen in Week 1.
- Since capital requirements are 8% of RWA, we have that RWA are 12.5 times the amount of capital requirements.
- Using the formulas we have seen so far, we compute rho, b, the maturity adjustment and the WCDR. For all these computations we can obviously use R. RWA are then easily obtained by substituting all values in the RWA formula.
- Notice that we can also do the opposite: first compute capital requirements, and then multiply them by 12.5, in order to get RWA.
- It is interesting to notice that, under the STA approach, RWA and capital requirements would be higher.
- For what concerns the other types of exposures, such as for example retail exposures, the mechanism, the philosophy to obtain RWA and capital requirements is always the same.
- We will use different formulas, such as for example those you can see on your screen, but the mechanism is always the same: we have the PD that we have computed using the way we prefer, and then we plug in this information into our formulas, into our risk-weight functions, and we get the desired results in terms of RWA and capital requirements.
- Now you are experts in the computation of capital requirements under the IRB approach.

__Week 6: Default Probabilities – Part 3 > Lesson 3: Credit Risk Plus > Video Lesson__

__Week 6: Default Probabilities – Part 3 > Lesson 3: Credit Risk Plus > Video Lesson__

- It’s a rather complex model from a mathematical point of view, because it relies on some quite difficult results in actuarial mathematics.
- Do you recognize the Binomial distribution? A further step in the Credit Risk Plus model is to assume p, the probability of default of each counterparty, to be small, while the total number of counterparties, that is n, is large.
- The probability of observing m defaults can then be combined with historical information about the probability distribution for the losses experienced when a certain type of counterparty defaults.
- The everyday business life and empirical studies show that the distribution of losses is typically skewed with a long fat right tail, indicating that the probability of observing large losses, extreme losses, is not that small.
- The simple approach we have just seen is just an unrealistic version of CR. The model actually used by banks is much more complex from a mathematical point of view, because it introduces more realistic components.
- In such a model, we do no longer assume defaults to be independent.
- We do not consider just risk- homogeneous counterparties, but rather a portfolio made up of loans and securities with different levels of risk, belonging to different rating classes.
- As you can imagine, from a mathematical point of view, the complexity of the model rapidly increases.
- A nice feature of a model like CR is the possibility of getting closed-form analytical results, even for rather complex versions of the model; obviously under specific configurations of the parameters.
- If we prefer to have a very realistic model, in which we cannot impose certain conditions on the parameters, we lose closed-form results, but nevertheless we can approach the model through simulations and computational techniques; obtaining, in any case, very useful results in terms of credit risk assessment.

__Week 6: Default Probabilities – Part 3 > Summary > Video__

__Week 6: Default Probabilities – Part 3 > Summary > Video__

- CreditMetrics is a Merton-like model introduced by JP Morgan.
- As Merton’s model it is a model that takes into account the value of the assets of a company.
- Differently from Merton’s model, in CreditMetrics thresholds are not provided by liabilities, but obtained from credit ratings.
- These ratings can be external or internal, depending on the sophistication of the model.
- First, we solve the problem of the unrealistic liability structure which affects Merton’s model, as well as the problem of correctly assessing the value of debt, as seen under Moody’s KMV. Second, the fact that thresholds are derived from credit ratings, allows for the definition of several thresholds, for the different rating classes, so that we can also take into account credit deterioration.
- Could you say why? Hence, all in all, CreditMetrics combines the strengths of both credit ratings and equity-based models.
- We have then seen how, under the F-IRB approach, the PD of a counterparty can be used to compute RWA and capital requirements, just using the formulas provided by the regulator.
- We have said that CR is not a Merton-like model, that is: it is not a structural model of default.
- Under Credit Risk Plus, we can derive not only the PD, but also all the other quantities we are interested in, for the computation of RWA and capital requirements.

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