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Prompted by a few recent debates I feel some clarity on why banks actually exist would be interesting. The glib answer is “to make profits by borrowing and lending” – although many would put it in terms much more offensive than that.

Why are these profits available, though? What actual function does a bank perform? In short, what economic use is a bank? Read the rest of this entry »

Something that might be appreciated by the ozrisk community is a series of book reviews. Amazon reveals a couple of dozen books particularly relevant to credit risk analytics, and Basel. Would any of you readers out there like to offer a review or at least an opinion on books you have used?

I don’t currently have any books to hand, but recall favourable impressions of Lyn Thomas’s book and Naeem Siddiqi’s work in the shape of SAS training materials.

Credit Scoring and Its Applications 
by Lyn C. Thomas , David B. Edelman , Jonathan N. Crook

Credit Risk Scorecards: Developing and Implementing Intelligent Credit Scoring 
by Naeem Siddiqi

 

It would be convenient if one could assume independence of the two main agencies: default and churn.

Although this is likely to be assumed in the interests of keeping things simple, it is unfortunately a doubtful assumption. There may well be a correlation against the bank’s interests in the form of better credit risks finding it easier (then poor credit risks) to re-finance elsewhere on favourable terms. Then, higher churn (earlier closure) may be correlated with lower PD. Full modelling of such a situation would require the joint modelling of default and churn.

Churn is not a ‘risk’ in the Basel meaning(s) but is referred to as such in this post in the sense that it is an uncertain event with unfavourable financial consequence for the bank: opportunity loss of revenue. 

So far the event we’ve been considering as the subject of analysis has been default, with occasional mention of churn.

Default, being progressive, lends itself to analysis of its stages, such as the events of going 30DPD or 60DPD. In addition to default hazard, one can analyse 30DPD hazard and 60DPD hazard. One advantage, especially for monitoring, is that these events occur slightly sooner. A statistical advantage is that these events are more numerous than default events. Given an intuition, or perhaps a model, of how 30DPD and 60DPD profiles relate to default profiles, they could be a useful analytical tool.

That segues into the roll rates discussion AWML.

The relationship however need not be straightforward. For example, there may be a spike of 30DPD or 60DPD at MOB=2 or 3, due to bugs or carelessness with the administration of new re-payment schedules. Most of those would not roll through to default.

One of the uses of a hazard curve is as a sanity check on your data and the technicalities of the default definition.

If you regularly find yourself analysing millions of records, you will know that every conceivable weird and wobbly data bug will happen, as well as a few that could never have been conceived of. Recalling a typical example from a loan portfolio: there were 10,000 accounts that opened and closed on the same day. This not surprisingly was some artefact of how the data systems coped with proposals or quotes (or something), but in reality these accounts were NTU and there was never any exposure to risk in their respect. But, in amongst a quarter of a million accounts, it would be possible to miss their presence and to do some default analytics – and even some model building – including these accounts as “closed good” accounts.

<digress for a war story> One of those accounts even managed to span two months! It appeared in two consecutive calendar month snapshot datasets – somehow allowed by time zone differences and the exact timing of month-end processing. A casual analysis might have assumed that this represented two months of exposure to risk – see also the comments about time grains <end digression>

But coming to the point of this post, I have found that estimating the default and churn hazard is an excellent “sanity check” on the data that will quickly show up most issues that you would want to know about. The issue mentioned above showed up as a massive spike in the churn hazard at MOB=1.

Other features that might be noticeable in churn hazard curves are peaks of churn around key account ages, such as at MOB=6 if the product has a teaser rate for the first 6 months. Multiples of 12 MOB may also occur in certain pay-annual-interest-in-advance type of products. These examples would be features that one might be on the lookout for, so finding them would be “reassuring” feedback rather than “alerting” feedback.

Sanity checking is not only noticing what you didn’t expect, but also confirming what you did expect.

Features found in the default hazard curves may give important feedback about the way the default definition works. For example, with a 90DPD definition one may be expecting zero hazard for MOB=1,2,3 but there may in fact be genuine defaults in that zone triggered by supplementary business rules. However, what can happen is that the totality of rules in the default definition don’t quite produce the desired effect in practice. One example I recall caused the year-in-advance loans to reflect as default after only 30DPD. This showed up as a spike at 12,24,36 MOB and caused a review of the default definition as applied to this (relatively small) portion of the loan book. 

The data cleaning and sanity checking stage is helped by having some experience in similar analyses on similar products. But even in a completely new context, some data wobblies will produce such an unnatural effect on the hazard curve that you will be immediately alerted to follow up.

Hazard curves, being longitudinal, only help you examine default tendencies that relate to MOB. Cross-sectional effects, such as a sudden worsening in credit conditions in the economy, would be monitored in other ways.   

 

What shape does a typical default hazard curve have?

Note that this post is about default hazard – the churn hazard curve is a completely different matter.

Recall that the hazard at any particular MOB is indicating the instantaneous chance that a good account of that MOB age might go bad. So, where the curve is highest is showing the most dangerous age for accounts.

For most products, the hazard will be very close to zero for the first 3 or 4 months. This depends on the details of your default definition, but for example a simple 90DPD type of definition can’t produce a default in MOB 1,2 or 3. Some default definitions can be triggered even in those first MOBs via business rules about bankruptcy etc.

For some situations – like a new product to market – there can be an issue of “application fraud” or “soft fraud” whereby new accounts come on book that perhaps never had an intention to make any repayments. Such a situation would show up as a spike in hazard around the 4-5 MOB.

Aside from application fraud, typical CC hazard curves tend to rise rapidly to a maximum by 9-12 MOB and then to decline slowly to stable plateau at maybe half the peak hazard level. Hazard doesn’t decline to zero because no matter how old an account is, there remains a residual chance that it can go into default.

In practice, one gets relatively little chance to study the hazard behaviour at long MOB – say, over 36 months – because that calls for data going back more than 3 years – rather a long time in credit markets.

On a technical point, a constant hazard corresponds to an exponential distribution for the waiting time until first default.  

It would be fairly easy to confuse the notions of hazard curve and probability density function, since (for default on a typical credit product) both start at zero and climb to a peak and then decline.

The more data used in the analysis, the smoother the curves will be, but whatever the case the cumulative density function (“emergence curve”) will always be much smoother than the hazard and pdf.

For the reasons in the above two paragraphs, I recommend presenting default analytical work via the cdf graph using a non-technical name like “emergence curve” or “default profile”. Please send in your preferred nomenclatures in case there is some consensus we could publicise. My slight preference is for “default profile” which is neutral and non-technical and easily accommodates “churn profile” or “cross-sell profile” when one analyses some other waiting time quantity such as these.

The above paragraph is about presenting and communicating the results; but for analytical insight, I recommend that the analyst should be looking at the hazard curves as well – for discussion next time.

Continuing part 1, perhaps we should note that the subject ‘hazard’ here is in its very specific statistical sense, and is not the moral hazard issue which has been a serious subject of debate in the context of taxpayer rescues of financial institutions.

Mathematically, the hazard function is defined by h(x) = f(x) / (1 – F(x) )

Although its definition has a continuous context, the time granularity of our data imposes a discreteness on the hazard: for example, if our data is monthly then the hazards we calculate will be “1-month” hazards.

The meaning of the formula is that the hazard is a conditional probability : the probability of default in the time grain immediately following time=x , given that the account hasn’t defaulted yet (i.e. anywhere in the time interval from 0 up to x). Thus, h(12) would be the probability that an account that has been good for its first 12 MOB, might go bad in MOB=13.

1 – F(x) is also called S(x) and given the name survival function, i.e. the probability of not defaulting before time x.

Hazard is not the same thing as the probability distribution. I tend to illustrate this point with a familiar example from human mortality. What is the probability that a person would die in their 100th year? The likely interpretation of this is to visualise a distribution of all the ages 0-125 and a distribution curve with a peak somewhere in grandparent zone and tailing off sharply such that the chance a person dies during their 100th year would be very low – less than 1%. This is the chance that a newly born person might die in their 100th year. By contrast, the one-year hazard at age 99 is rather high – over 30%. This is a chance that someone who has survived to age 99 dies during the next year (their 100th year).

Upcoming posts will discuss uses and interpretations of all these items in the context of default (and churn) analytics.

Introducing some technical terms that arise in default analytics.

This nomenclature comes from the field of statistics called survival analysis, which is well established and readily found in text books or wiki entries etc. If you don’t mind reading maths you will find better guidance there than in this post. The name ‘survival’ arose because the subject matter was/is mostly mortality or onset/re-occurrence of disease in populations or test cohorts. This is not too far from the study of the onset of default, so happily (?if this is an appropriate word) a great deal of well established statistical theory and practice is available for the study of default. This applies mainly to PD rather than LGD modelling.

Some of these terms and their equivalents in banking terminology are covered below.

Survival analysis is an essentially longitudinal activity, although the data it is based on will often be cross-sectional in structure.

The key variable x is the waiting time until default. This means the MOB of the first default. This variable x will have a distribution (probability density function) f(x), from which can be derived (by integration) the cumulative density function F(x). The pdf is not intuitive for non-technical audiences and I recommend only showing the cdf which is a monotonic rising curve that is easy to interpret: F(24), for example, would show the probability of going bad on or before MOB=24. This can also be interpreted as “what proportion of this population will have gone bad within 2 years”.

Note that PD alwyas needs to be related to some time window, so “PD” alone is a vague concept and one needs to be specifying something like “probability of going bad within the first 24 MOB” (for a longitudinal PD) or “probability of going bad within the next 12 calendar months” (for a cross-sectional PD).

I avoid using the stats terminology of pdf or cdf because they don’t sound intuitive, and particularly the word “cumulative” can mean so many different things in various contexts. Some more business-intuitive term is preferable. Some colleagues have called the cdf the “emergence curve” which is quite descriptive as it makes one think of the bads “emerging” with the passing of time, as the curve climbs up. An emergence curve is visually comfortable to absorb (being an integral, it will be quite smoothe) and shows at a glance the values of the 12-month PD or 24-month PD or any other x-month PD. Another business-friendly term is “default profile”, which sits comfortably with “churn profile” for the cdf of waiting time until closed-good.

But none of these is the hazard curve > continued next time…

As I discussed a while back the big four are unlikely to be allowed to consolidate between themselves, so, with St. George being the largest of the second tier this is as big as they are likely to get within the Oz banking community. Banking deals in Australia are unlikely to ever be much bigger than this, but, to be frank, I can’t see the point.

The dangers for Westpac I would have thought are large. From a business / strategic sense this means that they are increasing their bet on New South Wales, with both of the entities being heavily concentrated there. NSW has been looking unhealthy for a while, So I would not be making this call. The real advantage is cost cutting – only one headquarters would be needed even if all of the branches are to be kept*.

That said, few people could claim to be as knowledgable about St. George as Gail Kelly, so perhaps she has spotted some real hidden value there. She is also likely to know which senior executive she wants, so the integration hit list should be a fairly easy thing to sort out – and my guess would be the hit list has quite a few Westpac names on it.

Having one of the other banks come in and trump the deal cannot be ruled out – someone offering a mix of cash and shares at the election of the holder would be good. The question is, who? The NAB I would have thought unlikely – they have their own issues to sort out. CBA or ANZ? Possible, with CBA as the more likely.

If HBOS had not so recently looked at selling BankWest I would have thought them a strong possibility – the price tag would not be large for them and it would fit with their strategy. They may choose to do it anyway to gain the sort of scale they need – it would mean that there were 5 large banks in Australia as a result. This would probably be the best outcome from a consumer’s point of view as it would reduce the stranglehold the big 4 have on the industry.

I would also expect the regulators to have a say, although I think Westpac would have already discussed this with APRA and they would have some very heavyweight legal advice on the likely ACCC response. That one will be fun for the economists to sort out. If HBOS step in, have a good look – the regulatory and reporting situation there is interesting with a foreign parent, so this one would cause the regulators, lawyers and everyone else a bit more work.

The offer on the table is not a killer one – so it looks like we will have interesting times ahead as this one plays out. Fun for all and big fees for the investment banks and brokers as we all buy and sell shares of the various banks.

*Not something I can believe. I do not know of any, but there would have to be a few Westpac branches really close to some St. George’s ones. Having two branches of the same bank on one block to me at least makes little sense.

[Update] I just thought – Gail could be buying it to hide some mistake she made earlier – but I would have thought this unlikely. Disclaimer – this is just a thought and I have no further information on this.[/Update]

Harking back to the issue of time granularity, and anticipating some default analytic calculations yet to come, let’s get back to small details and look inside the smallest data unit i.e. the time grain.

For typical retail products this granularity would be a month, which is the example carried forward in this post.

Monthly data warehoused for analysis purposes would typically be on a calendar month basis. An alternative (for CC?) might be data on a monthly payment cycle basis.

Even though a grain is ‘small’ there is still latitude for vagueness because data recorded against a month may relate in several ways to the time axis within that month:

  • point in time at the beginning of the month
  • point in time in the middle of the month
  • point in time at the end of the month
  • the whole time window comprising that month

For most cross-sectional studies, the time axis is calendar date and the ‘status’ variables like account balance would usually relate to the end of the month, as that would be their most up-to-date value. Other variables that summarise or count transactions (for example) would relate to the whole time window. Certain calculated values (like hazards AWML) may relate to the mid-point of the month.

In cross-sectional studies there is no difficulty in finding the point-in-time variables as at the beginning of a month, because these will be the (end-of-month) values from the previous month’s record – i.e. closing balance for Feb = opening balance for March etc.

If numeric date values are used as the key on a data table, they would most logically perhaps be set equal to the last day of each month, which is unfortunately a bit messy and harder (for a human) to remember than the obvious choice of the 1st of each month.

A non-numeric-date month key like “200805” avoids specifying any particular part of the month, and leaves it up to the user to figure the time relationships from the metadata. A slight disadvantage of such a key is that date arithmetic (figuring out the difference between two dates) becomes non-trivial.

Longitudinal studies would typically rely on performance data for each individual account that is stored cross-sectionally i.e. by calendar month. This introduces a slight wrinkle because the account opening date can be anywhere within a month, whereas the performance data is only available at month ends. So the first performance measurement point an account reaches may come up in only 1-2 days (if the account opened on the 29-30th of a month) or alternatively may represent up to 30 days of exposure-to-risk. Longitudinal studies have MOB rather than calendar date as their time axis, and this means that the MOB=1 analysis really represents on average about 0.5 months of exposure, and likewise all subsequent MOB points really represent on average half a month less. (This example assumes your MOB counting convention starts at 1 rather than from 0.) But in any case, it would be most representative to start at 0.5 and count upwards as 1.5, 2.5, etc.

The above may sound picky, but it can quite easily come about that one analyst’s 12-month OW is another analyst’s 13-month OW due to choices at this level, and this could make a significant change to risk measures.

Further intra-grain issues will be met when calculating hazards. This basically means dividing the number of defaults (at a certain MOB) by the number of accounts that were exposed-to-risk of default. In a longitudinal study the number of accounts exposed-to-risk will always be declining, as accounts close good or go into default. Good practice would therefore be to find the average number (month_start + month_end)/2 of exposed-to-risk accounts during that month for use in the denominator of the hazard.

Actuaries are good at these deliberations because of the care and expertise put into estimation of mortality statistics. If you can’t find a tame actuary, the recommended approach is large diagrams on whiteboards and a bottle of headache tablets.  

 

 

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