Actuarial and Quantitative Risk
The Global Risks Alliance (GRA)
Diversification lowers capital only if risks do not move together. Dependence between lines, perils, regions and assets, especially in the tail, decides how much diversification credit is justified. Linear correlation often fails to capture tail dependence. This page sets out how dependence is modelled, where models understate joint losses, the data used and who decides.
GRA · In development · service level R0
Dependence and correlation are measures of how risks move together, which decides how much diversification reduces capital. Linear correlation can miss joint extremes, so copulas and scenarios are used, with scarce data on tail behaviour. Capital model owners, risk officers, supervisors and validators hold the decisions.
Which places drive loss in a severe year, and how far it reaches into reinsurance. Illustrative design values.
The working screen behind this subject: its structure, terms and states. Names are placeholders and values illustrative.
Copulas, contagion and why correlations break in stress. The dossier below sets out how the risk develops, what is measured, where the data come from, how it is modelled and who decides.
Linear correlation measures average co-movement. Risks can look uncorrelated in normal times and move together in extremes.
Copulas model dependence separately from each risk's distribution. The choice of copula changes tail dependence and capital.
A pandemic, a financial crisis or a large catastrophe can hit several lines and assets at once. Scenario-based dependence captures such drivers.
Few observations exist of several risks at their extremes together, so tail dependence parameters rest heavily on judgement.
Each term as this page uses it. Where a standard-setting body defines the term, the wording follows that body.
| Term | Definition |
|---|---|
| Copula | A function that combines individual risk distributions into a joint distribution with a specified dependence. |
| Tail dependence | The tendency of risks to produce extreme outcomes at the same time. |
| Diversification benefit | The reduction in total capital from combining risks that do not all fail together. |
Each indicator carries its unit, its cadence, the kind of source behind it and the evidence label a value would carry. No values are shown: a programme supplies them under its own authority.
| Indicator | Unit | Cadence | Source type | Label |
|---|---|---|---|---|
| Correlation matrix entries used in capital | correlation coefficients | per model | Capital model documentation | DECLARED |
| Tail dependence coefficient | dimensionless | per analysis | Copula fits | MODELLED |
| Diversification benefit | percent reduction in capital | per model run | Capital models | MODELLED |
| Joint extreme events in data | count | per analysis | Historical data | OBSERVED |
| Sensitivity of capital to dependence assumptions | percent change | per analysis | Sensitivity testing | MODELLED |
| Copula family used | categorical | per model | Model documentation | DECLARED |
The topic as a computational graph: named sources feed the methods, an evidence review gates the outputs, and the decisions stay with the authorities that hold them.
Fig. 02Nodes carry their type, two metric rows and a state; connectors show what feeds what; dashed frames name the owners; the timeline is the cadence. Design of the pipeline, not a record of a run.
Named public datasets and institutional records, each with its licence class. Sources by agreement are used only under their terms and never republished.
| Source | What it gives | Licence class |
|---|---|---|
| Solvency II standard formula correlation parameters | Regulatory correlation matrices | Free, EU reuse terms |
| IAIS ICS technical specifications | Correlation and aggregation in the ICS | Free, IAIS terms |
| McNeil, Frey and Embrechts, Quantitative Risk Management | Reference text on dependence modelling | Proprietary, published book |
| R copula package | Open-source copula fitting tools | Open source, GPL |
| Supervisory reviews of internal model dependency calibration | Findings on dependence assumptions | Free, terms per supervisor |
Actuarial and Quantitative Risk prepares evidence. The decisions below are taken by the authorities named, under their own rules, and never by the platform.
| Who decides | The question | Evidence this page serves |
|---|---|---|
| Capital model owners | Which dependence structure to use | Data, expert judgement and sensitivity |
| Chief risk officers | How much diversification credit to accept | Diversification benefit and stress results |
| Supervisors | Whether dependence calibrations are prudent | Model reviews and peer comparison |
| Validators | Whether dependence assumptions are supported | Back-testing and scenario evidence |
Linear correlation measures average co-movement and is adequate for jointly normal variables, but financial and insurance risks are rarely normal. Two portfolios can have the same correlation yet very different behaviour in extremes. Tail dependence measures the tendency of variables to be extreme together, and it is what matters for capital and solvency.
Copulas separate the dependence structure from the marginal distributions, allowing modellers to choose dependence with or without tail dependence. Choice of copula matters: Gaussian copulas have no tail dependence, while t and Archimedean copulas can capture it. Misuse of Gaussian copulas in structured credit was widely discussed after the global financial crisis.
Mechanisms behind rising dependence include common shocks, such as a recession or a single large catastrophe affecting many lines; feedback, such as fire sales that push prices down for all holders; and contagion through counterparty and funding links. Common models and data across firms can also create correlated decisions.
Dependence also arises between assets and liabilities. A catastrophe can raise insurance claims while depressing the value of investments held to pay them, and a financial crisis can weaken reinsurers when cedants most need them; such wrong-way risk is easy to miss when risks are modelled separately.
Modelling joint distributions with chosen dependence structures; tail parameters are hard to estimate.
Using correlations observed in stress periods; stress periods are few.
Modelling the mechanisms that link risks; demanding in data.
Limits: dependence in the tail is estimated from very few joint extremes, and dependence structures can change with market conditions and policy.
The current service level is R0: information and enquiries. An enquiry receives a reference and names the next step; applications, payments, compute allocations and registrations open only when their team, terms and review capacity exist.