Aggregation of Incidence and Intensity Risk Variables to Achieve Reconciliation

Clive Hunt,Ross Taplin

doi:10.3390/risks7040107

Abstract

The aggregation of individual risks into total risk using a weighting variable multiplied by two ratio variables representing incidence and intensity is an important task for risk professionals. For example, expected loss (EL) of a loan is the product of exposure at default (EAD), probability of default (PD), and loss given default (LGD) of the loan. Simple weighted (by EAD) means of PD and LGD are intuitive summaries however they do not satisfy a reconciliation property whereby their product with the total EAD equals the sum of the individual expected losses. This makes their interpretation problematic, especially when trying to ascertain whether changes in EAD, PD, or LGD are responsible for a change in EL. We propose means for PD and LGD that have the property of reconciling at the aggregate level. Properties of the new means are explored, including how changes in EL can be attributed to changes in EAD, PD, and LGD. Other applications such as insurance where the incidence ratio is utilization rate (UR) and the intensity ratio is an average benefit (AB) are discussed and the generalization to products of more than two ratio variables provided.

Highlights

Credit risk modelling for capital and provisioning in banking is an example of the modelling context where this paper is relevant
Whilst the joint-ratio means that we introduce in Section 4 appear to be a new type of mean, it is more helpful to understand them as a new setting for means—one that takes into account the entire system in which all variables exist and interact and where the means satisfy the reconciliation property, all flowing from Equation (1)
The argument in this paper for the joint ratio aggregates is self-contained and will be of immediate use to institutions managing credit risk, there are several avenues for future research. These include incorporating these ideas into the development of probability of default (PD) and loss given default (LGD) credit risk models, extending the ideas from expected loss to unexpected loss

Summary

Introduction

Table 1) for period 1 is the sum of the EL values for each exposure and is of great importance to the business, investors, auditors and regulators (for example, in the context of provisioning it forms an assessment of liabilities on the balance sheet and feeds directly into the profit and loss statement of the company). The aggregate for PD in period 1 of 2.21% is the weighted average of 0.5%, 5%, and 3% (with weights of 115,000, 25,000, and 160,000) These provide useful summaries of the PD and LGD risk components of the exposures within a portfolio but have the following undesirable properties. For period 1, the product of $300,000 × 2.21% × 46.5 = $3081 is far removed from the true value for EL of $1998 This brings into question the value of using these EAD weighted means as summary aggregates for PD and LGD. The discussion so far has exactly two ratio variables, PD and LGD, but Appendix A shows how the joint-ratio idea, and its reconciliation property, generalizes naturally to m ratio variables

Literature Review

Joint-Ratio Means

Attributing Change in EL to Changes in the Individual Components

Lack of Aggregation Path Invariance

Findings

Discussion

Conclusions