# Aggregation of Incidence and Intensity Risk Variables to Achieve Reconciliation

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

The search for a mean has the purpose of simplifying a given question, by substituting to many values a single summary value, and leaving the overall picture of the problem under consideration unchanged […] One should not be thinking about the mean of two or more values, but only about the mean of those values with reference to the evaluation of a quantity that depends on them.

## 3. Aggregates (Means) for PD and LGD

## 4. Joint-Ratio Means

## 5. Attributing Change in EL to Changes in the Individual Components

## 6. Lack of Aggregation Path Invariance

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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1 | For now we consider these values to have come from models designed to estimate the respective quantities (prospective analysis) but they could also come from observed data (historical analysis) when monitoring observed exposures at default, default rates and losses given default. |

2 | We do not consider robust versions such as trimmed means and medians further because these can be misleading by hiding extreme observations (such as high LGD values close to 1 amongst the majority of values close to 0). Furthermore, means are useful because when multiplied by size they give a total (for example, the average PD multiplied by the size of the portfolio gives a total of defaults). Note that means exist and are useful irrespective of the distribution of PD, EAD and LGD. |

Exposure | Period 1 | Period 2 | ||||||
---|---|---|---|---|---|---|---|---|

EAD | PD | LGD | EL | EAD | PD | LGD | EL | |

1 | $115,000 | 0.50% | 90% | $518 | $115,000 | 1.00% | 90% | $1035 |

2 | $25,000 | 5.00% | 80% | $1000 | $25,000 | 5.00% | 80% | $1000 |

3 | $160,000 | 3.00% | 10% | $480 | $160,000 | 2.50% | 10% | $400 |

Aggregate ^{1} | $300,000 | 2.21% | 46.5% | $1998 | $300,000 | 2.13% | 46.5% | $2435 |

Product ^{2} | $300,000 × 2.21% × 46.5% = $3081 | $300,000 × 2.13% × 46.5% = $2976 |

^{1}Aggregates are: sum for exposure at default (EAD) and expected loss (EL), EAD-weighted arithmetic means for probability of default (PD) and loss given default (LGD).

^{2}Products of Aggregates are EAD × PD × LGD.

Context | $\mathbf{Weight}\text{}\left({\mathit{W}}_{\mathit{i}}\right)$ | $\mathbf{Ratio}\text{}\#1\text{}\left({\mathit{r}}_{1\mathit{i}}\right)$ | $\mathbf{Ratio}\text{}\#2\text{}\left({\mathit{r}}_{2\mathit{i}}\right)$ |
---|---|---|---|

Credit risk—historic | EAD | Default rate | LGD |

Credit risk—prospective | EAD | PD | LGD |

Insurance—generic | Exposure value | Incidence | Intensity |

Insurance—Private Health | Number of members | Utilisation rate (UR) | Average benefit (AB) |

**Table 3.**Weighted Means of Table 1 exposures and their reconciliation to the actual EL.

Aggregation Type | Period 1 | Period 2 | ||||
---|---|---|---|---|---|---|

PD | LGD | EL ^{1} | PD | LGD | EL ^{1} | |

Mean (weighted by EAD) | 2.21% | 46.5% | $3081 | 2.13% | 46.5% | $2976 |

Mean (weighted by EAD × other ratio) | 1.43% | 30.2% | $1295 | 1.75% | 38.0% | $1992 |

Joint-ratio mean | 1.78% | 37.4% | $1998 | 1.93% | 42.1% | $2435 |

^{1}EL values equal the product of the aggregate EAD, aggregate (mean) PD and aggregate (mean) LGD. The actual EL is $1998 (period 1) and $2435 (period 2).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hunt, C.; Taplin, R.
Aggregation of Incidence and Intensity Risk Variables to Achieve Reconciliation. *Risks* **2019**, *7*, 107.
https://doi.org/10.3390/risks7040107

**AMA Style**

Hunt C, Taplin R.
Aggregation of Incidence and Intensity Risk Variables to Achieve Reconciliation. *Risks*. 2019; 7(4):107.
https://doi.org/10.3390/risks7040107

**Chicago/Turabian Style**

Hunt, Clive, and Ross Taplin.
2019. "Aggregation of Incidence and Intensity Risk Variables to Achieve Reconciliation" *Risks* 7, no. 4: 107.
https://doi.org/10.3390/risks7040107