# Default Model

Default models are a category of models that assess the likelihood of default by an obligor. They differ from credit scoring models in two ways:

• Credit scoring is usually (but not always) applied to smaller credits—individuals or small businesses. Default models are applied more to larger credits—corporations or sovereigns.
• Credit scoring models are largely statistical, regressing instances of default against various risk indicators, such as a obligor’s income, home renter/owner status, etc. Default models directly model the default process, and are typically calibrated to market variables, such as the obligor’s stock price or the credit spread on its bonds.

Default models find many applications within financial institutions. They are used

Default models may be integrated with some sort of correlation model to facilitate modeling the credit risk of portfolios with exposures to multiple obligors. Such extensions of default models—called portfolio credit risk models—can be used

Consider a time horizon staring at the current time 0 and ending at some future time t. A one year horizon is typical, but financial institutions usually consider credit risk over several horizons. Let L represent the financial loss, if any, due to default on a particular obligation—a bond, loan, derivative instrument, etc.—over the horizon. L is a random variable. Its expected value E(L) is a metric of the credit risk of the obligation. It can be calculated as the product

[1]

where

• Pr(default) is the probability of default on the obligation during the horizon—what is called the default probability.
• EAD is exposure at default—the credit exposure on the obligation at the time of default. In [1], this is treated as a known constant.
• LGD is loss given default—the fraction of EAD that will not be recovered following default. LGD is simply 1 minus the recovery rate. In [1], it too is treated as a known constant.

The essential purpose of a default model is to calculate the default probability. However, sophisticated models may do more than this. For example, models might treat EAD and LGD as random and substitute their expectations into [1]. Treating both in this manner requires an assumption that they are independent. Such an assumption is difficult to justify, but it may be made to simplify models.

A simple default model can be constructed by calibrating credit ratings to historical frequencies of migrations between ratings. Exhibit 1 indicates a ratings transition matrix constructed by Standard & Poor’s. It indicates one-year ratings migration probabilities based upon bond rating data from the period 1981-2000.

Exhibit 1: One-year ratings migration probabilities based upon bond rating data from 1981-2000. Data is adjusted for rating withdrawals. Numbers in each row should sum to 100%. Due to round-off error, they may not do so exactly. Source: Standard & Poor’s.

For example, based upon the matrix, a BBB-rated bond has a 4.44% probability of being downgraded to a BB-rating by the end of one year. The matrix is based upon raw data, so it exhibits statistical anomalies. A CCC-rated bond is given a 0.16% probability of being upgraded to AAA, but a B-rated bond has a 0.00% probability of such an upgrade. If it were used to model defaults, the numbers in the matrix might be smoothed.

To use a ratings transition matrix as a default model, we simply take the default probabilities indicated in the last column and ascribe them to bonds of the corresponding credit ratings. For example, with this approach, we would ascribe an A-rated bond a 0.04% probability of default within one year.

If we want two-year default probabilities, we simply multiply the matrix by itself once to obtain a two-year ratings transition matrix. The last column of that matrix will provide the desired default probabilities. For three-year default probabilities, we multiply the matrix by itself three times, etc. Exhibit 2 indicates a five-year ratings transition matrix obtained by multiplying the one-year matrix of Exhibit 1 by itself five times.

Exhibit 2: Five-year ratings migration probabilities obtained by multiplying the matrix of Exhibit 1 by itself five times.

Default models that base default probabilities on empirical ratings transition matrices are called ratings migration models. CreditMetrics is an example of a commercial portfolio credit risk model that calculates default probabilities with a ratings migration model. CreditMetrics also uses its ratings migration matrices to model the evolution of bonds’ credit spreads based upon migrations in their ratings. This allows for the modeling of bond portfolios’ market (or mark-to-model) values over time.

Ratings migration models have a number of shortcomings. First, credit ratings reflect overall credit quality, which depends on both probabilities of default as well as likely recovery rates. If two bonds have the same credit rating, but one bond is senior and the other is subordinated, the senior bond is likely to have a higher default probability offsetting its likely higher recovery rate.

Second, ratings migration models are not dynamic. Because they are based upon long-term empirical probabilities of ratings transitions, they are not sensitive to business cycles or other fluctuations in the business environment.

Ratings migration models are just one type of default model. Many different default models have been proposed in the literature or implemented by financial institutions. With few exceptions, those that are not ratings migration models are implementations of:

Both types of models are sophisticated, flexible approaches to credit risk modeling that support a variety of analyses. They can be calibrated to current business conditions (typically using a firm’s stock price or bond spreads for this purpose). They can be implemented with “real probabilities” to support credit risk measurement or with risk neutral probabilities to support financial engineering applications.

Sophistication is not the same as accuracy. Default models were widely employed during the decade preceding the 2008 financial crisis. In tasks such as pricing collateralized debt obligations or assessing bank capital adequacy, they failed to identify looming risks.

### 4 Responses to Default Model

1. Syarif Surbakti at #

Dear Sir,

I pretty like this article. I’ve tried to develop a spreadsheet (Excel based) to prove the methodology providing transition matrix for 5 years horizon (Tabel 2) by multiplying the one-year matrix of Exhibit 1 by itself five times. For example, probability AAA-AAA in Tabel 1 is 93.66. In Tabel 2 (the 5 years horizon), according the guideline in the article, probability of AAA-AAA should be 93.66^5 = 72.07. This is not same as Tabel 2 in the article which shows 72.39.

The other AA-BBB in Tabel 1 = 0.49. Probability of AA-BBB in Tabel 2 should be 0.49^5 = 2.82. While Tabel 2 shows 4.45.

Maybe I was wrong in this case.Whould you please advice me how to fix this problems?

Best regard and thank you for your kind support
Syarif Surbakti

• Based on what you say, you may not be multiplying the matrices correctly. Do a web search for “how to multiply matrices”. I am on the road, but let me know if that doesn’t resolve the problem, and I will look at this on my return.

• Hi Syarif,

Glyn is right: your calculation of the probability of an AAA-bond having an AAA rating after 5 years is only based on the assumption that no migration to a different rating class takes place in between (i.e. you only assume AAA-AAA-AAA-AAA-AAA). But there are quite a number of other possibilities for an AAA bond to end up with an AAA-rating after 5 years, e.g. from AAA to AA and back to AAA again: AAA-AAA-AA-AAA-AAA.

All these different paths leading from AAA to AAA after 5 years will be taken into account by using the correct matrix multiplication.

By the way, you are also right: I like the article as well!

Cheers,

Marcel de Lange, PhD, MBA, FRM

2. Sizwe Ngubane at #

How do you compute the one year ratings migration? Do you take tow points as in today and twelve months later, then create a migration matrix? how do you use such long history to get one year migrations?