The BCBS published a paper in 2005 which offers an explanation of the Basel II IRB risk weight formula:
- describing the economic foundations
- as well as the underlying mathematical model and its input parameters.
While a lot has changed as a result of Basel III, the models underlying the calculation of Internal Rating Based Capital (IRB) requirements are still based on the core principles agreed under Basel II that are explained in this BCBS paper.
The notes below mostly summarise the July 2005 paper with some emphasis (bolded text) and comments (in italics) that I have added. The paper is a bit technical but worth reading if you want to understand not only the original thinking behind the Basel II risk weights for credit risk but also the distinction between expected and unexpected loss.
I initially found it useful for revisiting the foundation assumptions of the IRB framework as background to considering the regulatory treatment of Expected Loss as banks transition to IFRS9. The background on how the RW was initially intended to cover both Expected and Unexpected Loss, but was revised such that capital was only required to cover Unexpected Loss, is especially useful when considering the interaction of loan loss provisioning with capital requirements.
Reading the paper has also been useful for thinking through a range of related issues including:
- The rationale for and impact of prudential conservatism in setting the risk parameters used in the IRB formula
- The cyclicality of a risk sensitive capital requirement (and potential for pro cyclicality) and what might be done to mitigate the risk of pro-cyclical impacts on the economy
I am not a credit risk model expert, so the summary of the paper and my comments must be read with that in mind. I did this to help me think through some of the issues with bank capital adequacy. Hopefully others will find the notes useful. If you see something wrong or something you disagree with then let me know.
2. Economic foundations of the risk weight formulas
The paper sets out some of the core principles behind the BCBS analysis starting with Expected Loss:
- The IRB framework assumes that, while it is never possible to know in advance the losses a bank will suffer in a particular year, a bank can forecast the average level of credit losses it can reasonably expect to experience; i.e. Expected Losses (EL)
- Financial institutions view Expected Losses as a cost component of doing business, and manage them by a number of means, including through the pricing of credit exposures and through provisioning.
Comment: The concept of Expected Loss is useful when seeking to understand a bank’s capital requirement but the explanation set out in the paper is pretty simplistic. In practice provisions do not fully capture EL so the simple model presented here is not a description of what happens. Consequently, capital buffers do not just absorb Unexpected Loss (UL), they are also required to absorb some of what can be viewed to be EL. It is also a lot harder to forecast average level of credit losses than is assumed due to ambiguities regarding whether the average is for normal business conditions or a “through the cycle” average.
Next comes Unexpected Loss:
- The paper notes that one of the functions of bank capital is to provide a buffer to protect a bank’s debt holders against peak losses that exceed expected levels; i.e. Unexpected Losses (UL) –
- Institutions know these UL will occur now and then, but they cannot know in advance their timing or severity.
- Capital is needed to cover the risks of such peak losses, and therefore it has a loss-absorbing function.
The paper notes that there are a number of approaches to determining how much capital a bank should hold, but the IRB approach adopted for Basel II (and carried over into Basel III) focuses on the frequency of bank insolvencies arising from credit losses that supervisors are willing to accept.
- The paper notes that a stochastic credit portfolio model can be used to estimate the maximum amount of loss with a small, pre-defined probability.
- This probability can be considered the probability of bank insolvency.
- If capital is set according to the gap between EL and VaR, and if EL is covered by provisions or revenues, then the model adopted under Basel II assumes the likelihood that the bank will remain solvent over a one-year horizon is equal to the confidence level.
- Under Basel II, capital is set to maintain a supervisory fixed confidence level.
Comment: This is not an entirely accurate description of what actually happens in the real world mostly because banks can fail a long time before they become insolvent. That is not a criticism of regulators, most (if not all) economic capital models make the same assumption, probably because it seems to offer a very elegant means of calibrating a capital requirement to an objective target debt rating.
In the case of banks, the point of non viability (i.e. the point at which the market loses confidence in the bank’s viability as a going concern) is more important than insolvency and happens much sooner than technical insolvency. This was always true if you thought about it but became impossible to ignore under the GFC.
Also note that there is no consideration of the implications of the cyclicality of a risk sensitive capital measure. Also useful to remember that provisioning does not really attempt to measure EL under incurred loss accounting. This is why understanding the dynamic of the relationship between provisioning and Regulatory EL is so important. IFRS9 in theory moves provisioning to an EL basis but it will likely still be a much more cyclical measure than is commonly assumed. It could be argued that “downturn losses” calibrated to some level of severity and frequency are more important measures to the extent that they give a constant reminder of the inherent cyclicality of banking and where a bank is in the business cycle at any given point in time.
So far Expected Loss has been regarded from a top-down perspective, i.e. from a portfolio view. It can also be viewed bottom-up, namely from its components.
- The Expected Loss of a portfolio is assumed to equal the proportion of obligors that might default within a given time frame (1 year in the Basel context), multiplied by the outstanding exposure at default, and once more multiplied by the loss given default rate (i.e. the percentage of exposure that will not be recovered by sale of collateral etc.).
- Of course, banks will not know in advance the exact number of defaults in a given year, nor the exact amount outstanding nor the actual loss rate; these factors are random variables.
- But banks can estimate average or expected figures.
Comment: The model assumes these are random exogenous factors but it is important to not lose sight of how they can become endogenous systematic factors if the bank engages in poor lending. It can be argued that scenarios that threaten the solvency or viability of the bank are more likely to be the result of an endogenous systematic failure of credit risk management than the result of random adverse events impacting a well managed bank. Also note that average and expected are not necessarily the same thing. Averages in bank modelling tend to be historical averages which may or may not be a good measure of expected outcomes.
As such, the three factors mentioned above correspond to the risk parameters upon which the Basel II IRB approach is built:
- probability of default (PD) per rating grade, – which gives the average percentage of obligors that default in this rating grade in the course of one year
- exposure at default (EAD), – which gives an estimate of the amount outstanding (drawn amounts plus likely future drawdowns of yet undrawn lines) in case the borrower defaults
- loss given default (LGD), – which gives the percentage of exposure the bank might lose in case the borrower defaults. These losses are usually shown as a percentage of EAD, and depend, amongst others, on the type and amount of collateral as well as the type of borrower and the expected proceeds from the work-out of the assets.
3. Regulatory requirements to the Basel credit risk model
The Basel risk weight functions used for the derivation of supervisory capital charges for Unexpected Losses (UL) are based on a specific model developed by the Basel Committee on Banking Supervision (cf. Gordy, 2003). The model specification adopted by the BCBS was subject to an important restriction in order to fit supervisory needs: The model should be portfolio invariant, i.e. the capital required for any given loan should only depend on the risk of that loan and must not depend on the portfolio it is added to.
- This characteristic has been deemed vital in order to make the new IRB framework applicable to a wider range of countries and institutions.
- Taking into account the actual portfolio composition when determining capital for each loan – as is done in more advanced credit portfolio models – would have been too complex a task for most banks and supervisors alike.
- The desire for portfolio invariance, however, makes recognition of institution-specific diversification effects within the framework difficult: diversification effects would depend on how well a new loan fits into an existing portfolio.
- As a result the Revised Framework was calibrated to well diversified banks.
- Where a bank deviates from this ideal it is expected to address this under Pillar 2 of the framework. If a bank failed at this, supervisors would have to take action under the supervisory review process (pillar 2).
4. Model specification
4.1. The Asymptotic Single Risk Factor(ASRF) framework
In the specification process of the Basel II model, it turned out that portfolio invariance of the capital requirements is a property with a strong influence on the structure of the portfolio model. It can be shown that essentially only so-called Asymptotic Single Risk Factor (ASRF) models are portfolio invariant (Gordy, 2003). ASRF models are derived from “ordinary” credit portfolio models by the law of large numbers.
When a portfolio consists of a large number of relatively small exposures,
- idiosyncratic risks associated with individual exposures tend to cancel out one-another and
- only systematic risks that affect many exposures have a material effect on portfolio losses.
In the ASRF model, all systematic (or system-wide) risks, that affect all borrowers to a certain degree, like industry or regional risks, are modelled with only one (the “single”) systematic risk factor.
Given the ASRF framework, it is possible to estimate the sum of the expected and unexpected losses associated with each credit exposure. This is accomplished by calculating the conditional expected loss for an exposure given an appropriately conservative value of the single systematic risk factor.
Under the particular implementation of the ASRF model adopted for Basel II, the conditional expected loss for an exposure is expressed as a product of
- a probability of default (PD), which describes the likelihood that an obligor will default,
- and a loss-given-default (LGD) parameter, which describes the loss rate on the exposure in the event of default.
Probability of Default
The implementation of the ASRF model developed for Basel II makes use of average PDs estimated by the banks that reflect expected default rates under normal business conditions.
Comment: Note that the average PD estimates specified by the IRB formula are meant to reflect expected average default rates under “normal” business conditions. The question of what is “normal” is not covered in this note and, to the best of knowledge, continues to be ambiguous in other BCBS guidance. There are requirements that PD reflect long run averages over a full cycle but the cycle frequency, duration and severity are open to interpretation and judgement. Stress testing does offer some guidance though even here it has taken many years for some kind of consensus to emerge on what kind of cycle downturn is the benchmark.
To calculate the conditional expected loss, bank-reported average PDs are transformed into conditional PDs using a supervisory mapping function (see section 4.2 below).
The conditional PDs reflect default rates given an appropriately conservative value of the systematic risk factor.
- The same value of the systematic risk factor is used for all instruments in the portfolio.
- Diversification or concentration aspects of an actual portfolio are not specifically treated within an ASRF model.
Loss Given Default
In contrast to the treatment of PDs, Basel II does not contain an explicit function that transforms average LGDs expected to occur under normal business conditions into conditional LGDs consistent with an appropriately conservative value of the systematic risk factor. Instead, banks are asked to report LGDs that reflect economic-downturn conditions in circumstances where loss severities are expected to be higher during cyclical downturns than during typical business conditions.
Conditional Expected Loss
The conditional expected loss for an exposure is estimated as the product of the
- conditional PD and
- the “downturn” LGD for that exposure.
Under the ASRF model the total economic resources (capital plus provisions and write-offs) that a bank must hold to cover the sum of UL and EL for an exposure is equal to that exposure’s conditional expected loss. Adding up these resources across all exposures yields sufficient resources to meet a portfolio-wide Value-at-Risk target.
4.2. Average and conditional PDs
The mapping function used to derive conditional PDs from average PDs is derived from an adaptation of Merton’s (1974) single asset model to credit portfolios. The appropriate default threshold for “average” conditions is determined by applying a reverse of the Merton model to the average PDs.
Since in Merton’s model the default threshold and the borrower’s PD are connected through the normal distribution function,the default threshold can be inferred from the PD by applying the inverse normal distribution function to the average PD in order to derive the model input from the already known model output.
Likewise, the required “appropriately conservative value” of the systematic risk factor can be derived by applying the inverse of the normal distribution function to the pre-determined supervisory confidence level. A correlation-weighted sum of the default threshold and the conservative value of the systematic factor yields a “conditional (or downturn) default threshold”.
In a second step, the conditional default threshold is used as an input into the original Merton model and is put forward in order to derive a PD again – but this time a conditional PD. The transformation is performed by the application of the normal distribution function of the original Merton model.
In addition, the Revised Framework requires banks to undertake credit risk stress tests to underpin these calculations. Stress testing must involve identifying possible events or future changes in economic conditions that could have unfavourable effects on a bank’s credit exposures and assessment of the bank’s ability to withstand such changes. As a result of the stress test, banks should ensure that they have sufficient capital to meet the Pillar 1 capital requirements. The results of the credit risk stress test form part of the IRB minimum standards. Since this paper is restricted to an explanation of the risk weight formulas, no more detail of the stress testing issue is presented here.
4.3. Loss Given Default
Under the implementation of the ASRF model used for Basel II, the sum of UL and EL for an exposure (i.e. its conditional expected loss) is equal to the product of a conditional PD and a “downturn” LGD
As discussed earlier, the conditional PD is derived by means of a supervisory mapping function that depends on the exposure’s average PD. The LGD parameter used to calculate an exposure’s conditional expected loss must also reflect adverse economic scenarios.
The Basel Committee considered two approaches for deriving economic-downturn LGDs.
- One approach would be to apply a mapping function similar to that used for PDs that would extrapolate downturn LGDs from bank-reported average LGDs.
- Alternatively, banks could be asked to provide downturn LGD figures based on their internal assessments of LGDs during adverse conditions (subject to supervisory standards).
In principle, a function that transforms average LGDs into downturn LGDs could depend on many different factors including the overall state of the economy, the magnitude of the average LGD itself, the exposure class and the type and amount of collateral assigned to the exposure.
The Basel Committee determined that given the evolving nature of bank practices in the area of LGD quantification, it would be inappropriate to apply a single supervisory LGD mapping function. Rather, Advanced IRB banks are required to estimate their own downturn LGDs that, where necessary, reflect the tendency for LGDs during economic downturn conditions to exceed those that arise during typical business conditions.
Supervisors will continue to monitor and encourage the development of appropriate approaches to quantifying downturn LGDs.
Comment: Note the different treatment of PD and LGD. The IRB formula uses long run PD’s estimated under “normal” business conditions as an input whereas the LGD input variable is based on a “downturn” measure. Understanding the different approaches to these two key input parameters and their rationale seems to be quite important but possibly not widely understood. At a minimum, the BCBS model makes the regulatory measure of Expected Loss difficult to reconcile with an intuitive understanding of expected loss since it seems to be combining measures drawn from two separate parts of the loss distribution. PD is based on a long run average drawn from “normal” business conditions while LGD is a downturn measure with the duration/severity of the downturn open to interpretation and judgement.
The downturn LGD enters the Basel II capital function in two ways.
- The downturn LGD is multiplied by the conditional PD to produce an estimate of the conditional expected loss associated with an exposure.
- It is also multiplied by the average PD to produce an estimate of the EL associated with the exposure.
Capital requirement (K) =
[LGD * N [(1 – R)^-0.5 * G (PD)
• (R / (1 – R))^0.5 * G (0.999)]
• PD * LGD] * (1 – 1.5 x b(PD))^ -1 × (1 + (M – 2.5) * b (PD))
LGD (as part of the ASRF model)
LGD (as part of the Expected Loss)
4.4. Expected versus Unexpected Losses
As explained above, banks are expected in general to cover their Expected Losses on an ongoing basis, e.g. by provisions and write-offs, because it represents another cost component of the lending business.
Comment: This may not happen fully in practice, especially for the more cyclical component where the exact level of expected loss across the full credit cycle is a matter of interpretation and judgement regarding the frequency, duration and severity of cycle downturns.
The Unexpected Loss, on the contrary, relates to potentially large losses that occur rather seldomly.
According to this concept, capital would only be needed for absorbing Unexpected Losses.
Nevertheless, it has to be made sure that banks do indeed build enough provisions against EL.
Comment: The Committee was very concerned to address differences in loan loss provisioning. The requirement to deduct the shortfall between Provisions and Reg EL deduction mitigates the impact of these differences on capital adequacy. Note however that standardised banks are not subject to a Regulatory EL capital deduction so differences in provisioning are not mitigated by their regulatory capital measure.
Up to the Third Consultative Paper of the Basel Committee, banks had thus been required to include EL in the risk weighted assets as well. Provisions set aside for credit losses could be counted against the EL portion of the risk weighted assets – as such only reducing the risk weighted assets by the amount of provisions actually built. In Figure 2 above, this would have meant to hold capital for the entire distance between the VaR and the origin (less provisions).
In the end, it was decided to follow the UL concept and to require banks to hold capital against UL only. However, in order to preserve a prudent level of overall funds, banks have to demonstrate that they build adequate provisions against EL. In above Figure 2, the risk weights now relate to the distance between the VaR and the EL only. what the Reg EL shortfall effectively does
As the ASRF model delivers the entire capital amount from the origin to the VaR, the EL has to be taken out of the capital requirement. The Basel II Framework accomplishes this by defining EL as the product of the bank-reported “average” PD and the bank-reported “downturn” LGD for an exposure. Note that this definition leads to a higher EL than would be implied by a statistical expected loss concept because the “downturn” LGD will generally be higher than the average LGD. Subtracting EL from the conditional expected loss for an exposure yields a “UL-only” capital requirement.
For performing loans the Committee decided to use downturn LGDs in calculating EL. Applied for non-performing loans, this rule would result in zero capital requirements. For defaulted assets, in the risk-weight formula both the N term as well as the PD would equal one, and thus the difference in the brackets equals zero (and consequently, LGD equals the EL as calculated above).
However, a capital charge for defaulted assets would be desirable in order to cover systematic uncertainty in realised recovery rates for these exposures. Therefore, the Committee determined that separate estimates of EL and LGD are needed for defaulted assets. In particular, banks are required to use their best estimate of EL, which in many cases will be lower than the downturn LGD. The difference of the downturn LGD and the best estimate of EL represents the UL capital charge for defaulted assets.
4.5. Asset correlations
The single systematic risk factor needed in the ASRF model may be interpreted as reflecting the state of the global economy. The degree of the obligor’s exposure to the systematic risk factor is expressed by the asset correlation. The asset correlations, in short, show how the asset value (e.g. sum of all asset values of a firm) of one borrower depends on the asset value of another borrower. Likewise, the correlations could be described as the dependence of the asset value of a borrower on the general state of the economy – all borrowers are linked to each other by this single risk factor.
The asset correlations finally determine the shape of the risk weight formulas. They are asset class dependent, because different borrowers and/or asset classes show different degrees of dependency on the overall economy.
4.6. Maturity adjustments
Credit portfolios consist of instruments with different maturities. Both intuition and empirical evidence indicate that long-term credits are riskier than short-term credits. As a consequence, the capital requirement should increase with maturity. Alternatively, maturity adjustments can be interpreted as anticipations of additional capital requirements due to downgrades. Downgrades are more likely in case of long-term credits and hence the anticipated capital requirements will be higher than for short-term credits.
Economically, maturity adjustments may also be explained as a consequence of mark-to-market (MtM) valuation of credits.
- Loans with high PDs have a lower market value today than loans with low PDs with the same face value, as investors take into account the Expected Loss, as well as different risk-adjusted discount factors.
- The maturity effect would relate to potential down-grades and loss of market value of loans. Maturity effects are stronger with low PDs than high PDs: intuition tells that low PD borrowers have, so to speak, more “potential” and more room for down-gradings than high PD borrowers.
- Consistent with these considerations, the Basel maturity adjustments are a function of both maturity and PD, and they are higher (in relative terms) for low PD than for high PD borrowers.
The actual form of the Basel maturity adjustments has been derived by applying a specific MtM credit risk model, similar to the KMV Portfolio ManagerTM, in a Basel consistent way. This model has been fed with the same bank target solvency (confidence level) and the same asset correlations as used in the Basel ASRF model. Moreover, risk premia observed in capital market data have been used to derive the time structure of PDs (i.e. the likelihood and magnitude of PD changes). This time structure describes the probability of borrowers to migrate from one rating grade to another within a given time horizon. Thus, they are vital for modelling the potential for up- and downgrades, and consequently for deriving the maturity adjustments that result from up- and down-grades.
4.7. Exposure at Default and risk weighted assets
The capital requirement (K) as laid out in the Revised Framework is expressed as a percentage of the exposure. In order to derive risk weighted assets, it must be multiplied by EAD and the reciprocal of the minimum capital ratio of 8%, i.e. by a factor of 12.5:
Risk weighted assets = 12.5 * K * EAD
5. Calibration of the model
Within the above model, two key parameters have to be determined by supervisory authorities: the confidence level supervisors feel comfortable to live with, and the asset correlation that determines the degree of dependence of the borrowers on the overall economy.
5.1. Confidence level
The confidence level is fixed at 99.9%, i.e. an institution is expected to suffer losses that exceed its level of tier 1 and tier 2 capital on average once in a thousand years. This confidence level might seem rather high. However, Tier 2 does not have the loss absorbing capacity of Tier 1. The high confidence level was also chosen to protect against estimation errors, that might inevitably occur from banks’ internal PD, LGD and EAD estimation, as well as other model uncertainties.
Comment: Note that the original rationale for high confidence level was concerned to defend against arguments that the capital requirement was too high. It was argued that high confidence was necessary to protect against estimation errors but this conservatism was mitigated by the fact that it could be covered in part by Tier 2 capital.
It is also worth noting the use of fixed confidence level irrespective of the state of the credit cycle. I can see why this is mathematically convenient and also consistent with the rating agency convention that a debt rating could be represented as being consistent with an expected probability of default/insolvency. Intuitively, however, I would expect the degree of confidence in the solvency of the banking system to vary depending on the state of the credit cycle. During the downturn part of the cycle that confidence would decline. Conversely, immediately post a downturn, bank balance sheets tend to be much cleaner than they are on average as the value of marginal credits have been written down or off against capital. At that time, the level of confidence in the quality of the bank’s balance sheet and hence its solvency would tend to be higher, all other things being equal.
5.2. Supervisory estimates of asset correlations for corporate, bank and sovereign exposures
The supervisory asset correlations of the Basel risk weight formula for corporate, bank and sovereign exposures have been derived by analysis of data sets from G10 supervisors. Time series of these systems have been used to determine default rates as well as correlations between borrowers. The analysis of these time series has revealed two systematic dependencies:
- Asset correlations decrease with increasing PDs. This is based on both empirical evidence and intuition. Intuitively, for instance, the effect can be explained as follows: the higher the PD, the higher the idiosyncratic (individual) risk components of a borrower. The default risk depends less on the overall state of the economy and more on individual risk drivers.
- Asset correlations increase with firm size. Again, this is based on both empirical evidence and intuition. Although empirical evidence in this area is not completely conclusive, intuitively, the larger a firm, the higher its dependency upon the overall state of the economy, and vice versa. Smaller firms are more likely to default for idiosyncratic reasons.
The asset correlation functions eventually used in the Basel risk weight formulas exhibit both dependencies:
- The asset correlation function is built of two limit correlations of 12% and 24% for very high and very low PDs (100% and 0%, respectively).
- Correlations between these limits are modelled by an exponential weighting function that displays the dependency on PD.
- The exponential function decreases rather fast; its pace is determined by the so-called “k-factor”, which is set at 50 for corporate exposures.
- The upper and lower bounds for the correlations, as well as the shape of the exponentially decreasing functions, are in line with the findings of above mentioned supervisory studies.
- In addition to the exponentially decreasing function of PD, correlations are adjusted to firm size, which is measured by annual sales
- The asset correlation function for bank and sovereign exposures is the same as for corporate borrowers, only that the size adjustment factor does not apply.
5.3. Specification of the retail risk weight curves
The retail risk weights differ from the corporate risk weights in two respects:
- First, the asset correlation assumptions are different.
- Second, the retail risk weight functions do not include maturity adjustments – but they do have an implicit maturity adjustment via the asset correlation assumption
As for the other risk weight curves (see section 4.2), stress test requirements also apply to the retail portfolio. The differences relate to the actual calibration of the curves. The asset correlations that determine the shape of the retail curves have been “reverse engineered” from
- economic capital figures from large internationally active banks, and
- (historical loss data from supervisory databases of the G10 countries.
Both data sets contained matching PD and LGD values per economic capital or loss data point.
- The banks’ economic capital data have been regarded as if they were the results of the Basel risk weight formulas with their matching PD and LGD figures being inserted into the Basel risk weight formulas. Then, asset correlations that would approximately result in these capital figures within the Basel model framework, have been determined. Obviously, the asset correlation would not exactly match for each and every bank, nor for each and every PD-LGD-Economic Capital triple of a given bank, but on average the figures work.
- With the second data set (supervisory time series of loss data), Expected Loss (as the mean of the time series) and standard deviations of the annual losses were computed. Moreover, the Expected Loss has been split into a PD and a LGD component by using LGD estimates from supervisory charge-off data. Then again, these figures have been regarded as PD, LGD and standard deviations of the Basel risk weight model, and asset correlations that would produce approximately the same standard deviation within the Basel framework have been sought.
Both analyses showed significantly different asset correlations for different retail asset classes. They have led to the three retail risk weight curves for
- residential mortgage exposures,
- qualifying revolving retail exposures and
- other retail exposures, respectively.
The three curves differ with respect to the applied asset correlations:
- relatively high and constant in the residential mortgage case,
- relatively low and constant in the revolving retail case, and,
- similarly to corporate borrowers, PD-dependent in the other retail case
Implied maturity adjustment to Retail RW
The implicit maturity effect also explains the relatively high mortgage correlations: not only are mortgage losses strongly linked to the mortgage collateral value and the effects of the overall economy on that collateral, but they have usually long maturities that drive the asset correlations upwards as well. Both effects are less significant with qualifying revolving retail exposures and other retail exposures, and thus the asset correlations are significantly lower.
Comment: Useful to note that retail RW do not have an explicit maturity adjustment even though some (residential mortgages in particular) are long term exposures. The paper explains how the asset correlation assumption employed was calibrated in a way that implicitly takes account of the maturity adjustment. Also interesting background now that APRA applies higher correlation to selected residential mortgage portfolios.
In the above analysis, both the economic capital data from banks and the supervisory loss data time series implicitly contained maturity effects. Consequently, the reverse engineered asset correlations implicitly contain maturity effects as well, as the latter were not separately controlled for. In the absence of sufficient data for retail borrowers (similar to the risk premia used to deriving the time structure of PDs for corporate exposures), this control would have been difficult in any case.
Thus, the maturity effects have been left as an implicit driver in the asset correlations, and no separate maturity adjustment is necessary for the retail risk weight formulas.
This site page was updated on 23 May 2018. The only material changes were to add comments in S4.1 and 4.4. I also edited the comments in S4.3 hopefully for greater clarity. I also added some new comments in S5.1 on 25 May 2018.