Ebook Evaluating Design Choices in Economic Capital Modeling: A Loss Function
The concept of economic capital (also referred to as "risk capital" or "risk-based capital") is increasingly being adopted by banks and other financial institutions as a standard by which to determine the amount of capital needed to protect against financial distress in the event of unexpectedly large losses.
When calculating portfolio economic capital requirements, most models estimate critical values corresponding to extreme tail percentiles of a portfolio or whole-bank loss distribution. Economic capital requirements are then set to cover a measure of "unexpected loss," defined as the difference between the estimated mean of the loss distribution and the estimated loss level corresponding to the chosen critical tail percentile.
By their design, economic capital models are complex, and usually take as input the output of several other modeling exercises, including but not limited to the estimation of asset-level default probabilities (PD), loss given default (LGD) rates, and cross-asset correlations of these same parameters. Because these models are inherently complex, financial institutions must assess the risks of using them, as well as their associated driver-models. Model risk is defined for this purpose as the potential for loss from incorrect predictions or incorrect decisions resulting from the misuse of models. Such misuse usually occurs when a model is misapplied or its results are misinterpreted. Model risk is assessed in the context of the intended use of models and best known practices used to build models. Credit risk decision models are evaluated with respect to sample design, modeling techniques, validation procedures, and re-validation procedures. This paper considers issues relating to the segmentation or grouping of credit exposures and the potential impact upon economic capital allocation and attribution. When discussing capital allocation, we refer to assessing total capital at the portfolio level, while our discussion of capital attribution refers to assigning capital appropriately at the bucket level. We discuss whether a model’s logical structure fits its application. As referenced in OCC Bulletin 2000-16, “Risk Modeling Model Validation,” this assessment is essential to the first stage of model validation.
In most quantitative approaches to assessing expected loss and reserves, or the appropriate amount of economic capital to support a portfolio of assets, the risk ratings of assets and their associated estimates of PD and LGD are key inputs. PD and LGD can be estimated using a variety of techniques including simple descriptive statistical analysis, statistical and econometric regression models, and structural finance models. Whatever the approach, these metrics are almost impossible to estimate uniquely for each asset there is simply not enough available information. Assets are therefore grouped, or segmented, into categories buckets and PDs are estimated by bucket. This results in PD estimates that are actually average PDs for assets within categories.
Since models that yield estimates of economic capital requirements are typically nonlinear in PD, how assets are grouped or bucketed has implications for economic capital. That is, estimation usually poses the following trade-off: As the size of each group increases, PD estimates of group averages, although more precise, are less relevant because more heterogeneous assets are grouped together. And as the size of each group decreases, PD estimates become less accurate.
This paper analyzes exactly this trade-off in the context of economic capital allocation and attribution. We employ the Basel II specification in our analysis since it is built upon a very simplified economic capital model, the Asymptotic Single Risk Factor (ASRF) model, which allows for marginal portfolio capital charges to be computed based upon exposure-level characteristics. (See Vasicek (1997) and Gordy (2000) for a detailed discussion of the ASRF.) The ASRF model enables a bank to calculate its minimum regulatory capital requirement for total portfolio credit risk as the sum of exposure-level capital charges, which in turn are strictly functions of PD, LGD, and a single portfolio-level asset correlation coefficient. However, this simplicity does not come without cost, since one can justify computing portfolio capital charges in this way only if there is a single systematic risk factor driving correlations across obligors and no exposure in a portfolio accounts for more than an arbitrarily small share of total exposure.
The Basel II implementation process is devoting considerable resources to defining standards and procedures by which to judge the readiness and ability of financial institutions to estimate loan characteristics including PD and LGD. Supervisory authorities are developing detailed specifications of the validation standards for these drivers. We therefore do not focus on issues relating to the validation of models used to estimate the drivers of, or inputs to, economic capital models. Our focus is instead on the application of the economic capital model, and we emphasize that a loss or value function must be specified so as to quantify the gains and losses from choosing a more or less granular scheme of asset segmentation. The numbers and types of alternate loss
functions that could be specified are great, and they vary with the ultimate business uses of the capital estimates. Nevertheless, a natural starting point is to consider the mean-square error implications (MSE) of alternate segmentations or groupings of assets for economic capital. We illustrate the implications with several numerical examples.
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