Risk Weight Functions

In one my first articles (here) in this series, I presented the concept of Expected Loss Amount (ELA) based on the concepts of PD (Probability of Default), LGD (Loss Given Default) and EAD (Exposure at Default):

ELA = PD x LGD x EAD

The EL/ELA measures the average level of credit losses that a bank can expect to be impacted by and considered as a normal part of a bank’s business. Typically, banks manage it by adjusting the pricing of credit exposures and through provisioning – see concept of IRB shortfall in this article.

The application of EL/PD/LGD/EAD concepts in the calculation of capital requirements /VaR/Unexpected Loss was nevertheless not yet presented.

Bank’s capital, mainly equity, has a function to work as a buffer to protect a bank’s debt holders and taxpayers against losses that exceed expected levels. Losses above expected levels (EL) are usually referred to as Unexpected Losses (UL).?

?In the figure above, the graph represents a probability density function of the losses of a bank. The likelihood that losses will exceed the EL and UL (VaR) equals the marked area under the right hand side of the curve.?

If capital is set according to the gap between EL and VaR, and if EL is covered by provisions or revenues, then the likelihood that the bank will remain solvent over a one-year horizon is equal to the confidence level.?

Under the Basel framework, capital is set to maintain a supervisory fixed confidence level.?

The model aimed to 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, so we consequentially have that the model was calibrated to reflect a well-diversified portfolio. Deviations from this should be reflected in the Pillar II of the framework (see Porfolio Credit Models article here).?

To reflect the portfolio invariance principle, an Asymptotic Single Risk Factor (ASRF) model was used. The basic formula is given by:

PD

Long story short, a mapping function is used to derive conditional PDs from average PDs. The derivation is based on Merton’s (1974) single asset model to credit portfolios where he also described the change in value of the borrower’s assets with a normally distributed random variable. From this framework we have:

Standard normal distribution (N) applied to threshold and conservative value of systematic factor:?

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Inverse of the standard normal distribution (G) applied to PD to derive default threshold:

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Inverse of the standard normal distribution (G) applied to confidence level to derive conservative value of systematic factor:

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LGD

Supervisors considered a function that transforms average LGDs into downturn LGDs but given that these could depend on many different factors and given the evolving nature of bank practices in LGD quantification, it would be inappropriate to apply a single supervisory LGD mapping function, so banks are required to estimate their own downturn LGDs.

There are two LGD in the formula, one as part of the ASRF model and the other as part of the Expected Loss:

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You can also observe here that the PD x LGD is the EL component deducted from the VaR in order to represent the UL.

Asset Correlations

The asset correlations determine the shape of the risk weight formulas and define the degree of the obligor’s exposure to the systematic risk factor, i.e., it shows how the asset value of one borrower depends on the asset value of another borrower.?

They are asset class dependent and will be presented in a separate article.

?Maturity Adjustments

Intuition and empirical evidence suggest that long-term credits are riskier than short-term credits and therefore capital requirement should increase with maturity.?

Additionally, a less intuitive logic makes maturity effect relate to potential for down-grades and loss of market value of loans. Low PD borrowers have more room for down-gradings than high PD borrowers.

The Basel maturity adjustment has therefore the following properties:

? the adjustments are linear and increasing in the maturity M

? the slope of the adjustment function with respect to M decreases as the PD increases

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To be continued…

Source:?An Explanatory Note on the Basel II IRB Risk Weight Functions: https://www.bis.org/bcbs/irbriskweight.pdf

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