Some Basic Credit Risk Modelling Concepts

As the world's banking and financial systems become increasingly stressed - and distressed, a common reaction is to blame the models that are used by banks to quantify risk.

If used properly, such models can contribute significantly to a sound risk management framework. However they are often used naively and misunderstood.

Economic Capital

This term is often used for quantitative measures which attempt to 'put a number' on the level of financial capital that would be required to 'cover' the riskiness of a portfolio.

There are several variations on this theme, which range from what are essentially empirical 'rules of thumb', to others that are based on statistical models of varying complexity.

Risk-Appetite

This term should be self-explanatory. By quantifying 'risk-appetite' it is possible to put flesh on a business strategy which involves the taking of financial risk, by identifying how much risk a business is prepared to take.

To be able to make clear statements about risk appetite also implies that there should also be a clear view of what 'risk' is in a particular context, and ways in which it might be represented or quantified.

What Causes A Simple Portfolio to Suffer Large Credit Losses ?

In the following we will restrict ourselves to considering some of the risk modelling issues that arise for a simple portfolio of conventional credit exposures - which facilitates a more transparent discussion of some of the key issues.

These arguments also have a significant bearing on more complex credit structures, including 'credit derivatives', simple and complex. This is because, being able to price and manage such instruments properly, must ultimately depend on understanding the 'riskiness' and dynamics of the underlying assets - which, for credit derivatives, ultimately reduces to understanding the behaviour of a credit risk portfolio.

In the author's view many of these factors are not sufficiently well understood.

Before considering some of the relevant modelling issues, it is perhaps worth 'getting back to basics', and considering a very simple, but important, question - which is why can a 'simple' portfolio of credit exposures suffer large losses ? - At the most basic level, there are two possible causes:

• There are large individual losses.
• Many smaller losses occur together.

This means that two of the critical 'inputs' to any portfolio credit risk model are: (i) the probabilities of default, and (ii) the factors which drive the likelihood of multiple defaults occurring together - 'joint distribution' or 'correlation' effects.

Some Problems Often Inherent In Economic Capital Credit Risk Portfolio Models

In the author's view, the most frequent problems that arise when using this type of model are as follows:

1. What The Model Results Actually Mean Is Frequently Misunderstood.

   The first and most important problem is that these models, and their results, are frequently misunderstood, even by those that use them.

   These misunderstandings do not just relate to their complex inner-workings, but much more fundamentally to what the results actually mean, what they can be used for, and perhaps more importantly, what they can not.

   The role of these models is primarily to produce a measure of risk - which is not to say that they are necessarily meant to be 'predictive' in the normal sense. Ideally they would, in statistics-speak, produce a true and fair 'unbiased' estimator or risk.

   However, this would only be an accurate measure - in the usual sense - if both the modelling assumptions, and all the input parameters were correct - which is not at all likely - in which case the result produced is a still a metric for risk, but one which should be interpreted with care and understanding.

   Another important point in this regard is that large complex risk models do not necessarily produce 'good' results. Some financiers can be fooled into thinking that, because they have very complex and expensive risk models, that the results that they produce will therefore be completely accurate and reliable - this is not the case.

   In the author's view, it is certainly not sensible to attach too much credibility to the results produced by these models - even if they are properly calibrated (which is rare) - for estimates for risk events that occur with a frequency of less than about once in every ten or twenty years.

2. Models Often Under-Estimate Extreme Risks Because Of A Simplistic Treatment of Multi-Dimensional Distribution Functions, Correlations And Risk-Diversification Effects

   One of the key features of a risk model 'under the bonnet', is the way that the 'joint distribution' for multiple risk events is treated. This consideration is an important one because, if a model does not correctly capture the shape of the multi-dimensional distribution of the factors which drive individual risk events, then it cannot be expected to properly quantify the likelihood of combinations of risk events occurring simultaneously - which is one of the key drivers of 'large' portfolio losses.

   If we were dealing with simple 'normal' distributions, then such 'joint behaviour' is completely captured by the correlation or covariance matrix for the underlying Gaussian processes - in actual fact, it is not generally possible to estimate this matrix particularly well - but if we could, then the relevant multi-dimensional distribution would be completely described.

   A fundamental question in the modelling process is therefore, assuming that we can estimate the 'marginal' probability distributions for individual risk events accurately, how can we arrive at a useful estimate of the combined joint distribution ? - which is what we really need to properly estimate the overall risk.

   Mathematical tools, such as Copulas, can be used to synthesise a multi-dimensional distribution, given a set of individual 'marginal' distributions for the separate 'risk axes'. Unfortunately, in the author's view, assumptions of joint behaviour based on the use of 'normal' distributions (and therefore Gaussian Copulas) make unrealistic assumptions about joint risk behaviour.

   Another way to look at this is to think in terms of 'Portfolio Risk Diversification' - or from the opposite perspective - 'Concentration' effects. By having a diversified portfolio, the overall riskiness (in relative terms) is generally reduced - this is because there is less 'correlation' between separate risk events, and so it is less likely that many bad events will occur together.

   However, in the author's view, a crucial effect of this type of flaw in these models is that, because of the likely subtleties in the shape of the 'real' multi-dimensional joint distribution functions - in risky situations these models will tend to over-estimate the sanitising impact of risk-diversification, and therefore under-estimate the size of the tails of the overall loss distribution; and consequently underestimate the size and likelihood of significant losses occurring due to a combination of multiple simultaneous loss events.

3. The Statistical Measures That Are Used are Often Not Ideal

   Risk models that are based on a statistical risk analysis often compute probability distributions for loss, and then calculate various statistical 'risk' figures based on these loss distributions.

   Unfortunately, the 'measures' that are often used have serious practical flaws. One such measure is the 'Confidence Level', which is often used to quantify 'Value-At-Risk'. Unfortunately, in statistical-speak this is 'non-coherent', which means that it can be quite unstable, and the aggregation and disaggregation of these numbers is questionable, and can behave very strangely.

4. Calibration Issues And Questionable Estimates of Input Parameters

   Estimating Input Parameters

   The most obvious examples of this are the probabilities of default, or 'PD's that are used by such models. This is particularly true for better credit grades (lower PDs), where there are usually too few recent historical default events - if any - to get a proper estimate using simple statistics. Better results can be obtained using a more sophisticated statistical analysis - but then the 'error bars' inherent in any such process must be properly accounted for.

   This problem is particularly relevant because, in relative terms, there is often a disproportionately high sensitivity of the overall risk figures to changes to these particular estimated input values.

   In addition, the influence of 'joint distribution' and 'correlation effects', as discussed previously, usually depends on a range of input parameters, which are hard to estimate, in a rather complex way - and for which the estimation process is not especially robust.

   Model Calibration Issues

   Unfortunately, the process of arriving at suitable values for these model input parameters is not always as objective as it might be.

   Issues can arise, for example, if someone is unwilling to accept the risk figures that are produced by a model - because they consider them to be too high. This can result in the credibility of the model being questioned and the inputs being 'adjusted' - so that the outputs are more consistent with an individual's pre-conceived values.

   In the worst case, 'model calibration' can become a euphemism for adjusting the inputs to achieve a desired output. Which not only means that the results are misleading, but also that the figures produced do not respond as they should to changes in other inputs.

5. 'Cascade' Effects are Often Not Accounted For

   These risk models often only include 'first-order' cross-correlation effects between the various factors that drive outcomes. They rarely properly account for 'cascade' effects, which are common in the real world, particularly under extreme circumstances. Attempts are sometimes made to use multi-factor 'scenario analysis' to capture these effects - but, in the author's view it is questionable to what extent they succeed.

All of these issues do not mean that such models cannot make a significant contribution to quantifying and managing risk; they simply mean that they need to be properly understood by those that make use of them.

The worst outcome is when such models are not used properly, but their use gives the appearance of sound quantitative risk-management, which is then used as a justification for not properly using other complementary risk management techniques, resulting in even more risk being taken.