Actuarial and Quantitative Risk
The Global Risks Alliance (GRA)
GRA · Actuarial and Quantitative Risk
Spatial financeWhere capital meets hazard, exposure and transition
Capital and solvency models set how much capital insurers and other financial firms must hold to absorb severe losses. Regimes use standard formulas, internal models or both, calibrated to confidence levels over a time horizon. The IAIS Insurance Capital Standard adds a group-wide requirement for internationally active insurers. This page sets out how capital models work, where they mislead, the standards that govern them and who decides.
Internal models, standard formulas and their assumptions. The dossier below sets out how the risk develops, what is measured, where the data come from, how it is modelled and who decides.
Requirements are calibrated to a confidence level, such as the loss expected once in two hundred years. Calibration choices for each risk decide total capital.
Standard formulas apply common factors that may not fit a particular firm's risk, overstating some and understating others.
Internal models can reflect a firm's risk better but require supervisory approval, validation and governance, and can be optimised to reduce capital.
Solvency II, the ICS, US risk-based capital and other regimes measure capital differently, complicating group supervision and comparison.
Which places drive loss in a severe year, and how far it reaches into reinsurance. Illustrative design values.
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 |
|---|---|---|---|---|
| Solvency ratio | eligible own funds over capital requirement, percent | quarterly or annual | Supervisory disclosures | Published statistics |
| Capital requirement by risk module | currency | annual | Solvency and financial condition reports | Published statistics |
| Internal model users | count of firms | annual | Supervisor records | Published statistics |
| Diversification benefit in capital | currency | annual | Solvency reports | Published statistics |
| Internationally active insurance groups identified | count | periodic | IAIS register | Published statistics |
| Capital sensitivity to key assumptions | percent change | annual | Solvency reports | DECLARED |
Assembled from the Nexus Standards area profile Insurance and reinsurance: the shocks it names, the systems and sectors that carry the loss, the controls that dampen it and the data that observe it. Pick an initiating shock and step through the cascade to see which exposures it reaches and where losses could concentrate. The Map view places the run on an example geography with open data. Every weight is a stated assumption. An exercise, not a forecast, a valuation, a rating or investment advice.
Hazards
Systems and couplings
Controls
NS-0 Constitution and principlesNS-3 Risk record UNIs (grammar, legacy crosswalk, allocation authority)NS-9 Contextual UNIs (deployments, instances, programme instances)NS-10 Policies and clauses (machine-readable obligations: definition, binding, evaluation and enforcement)NS-11 Verification and monitoring (evidence, attestations, zero-trust conformance observation)NS-14 Conformance and crosswalk to external standardsNRS-F3 Insurance, catastrophe risk and protection gapsNRS-H113 Hazard family: Financial-system fragility (NHS-1 soc.financial-system)NRS-U17 Universe platform profile: Insurance NexusNRS-H71 Hazard family: Deep-time and cosmic catastrophes (NHS-1 planet.deep-time)NRS-H65 Hazard family: Planetary boundary transgressions (NHS-1 planet.boundary)NRS-F8 Parametric and index-based risk transferObservation
Example geography
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.
The exposure, loss, scenario and capital views a risk officer, supervisor or ministry of finance would ask for, with the frameworks they report against. Structures follow supervisory and actuarial practice. Values are illustrative: never prices, ratings or advice.
How losses from capital and solvency models grow with severity, gross and net of reinsurance, against capital.
| Return period | Annual probability | Gross loss | Net loss | Net, capital = 100 |
|---|---|---|---|---|
| 1 in 10 | 0.10 | 14 | 10 | 10 |
| 1 in 50 | 0.02 | 33 | 21 | 21 |
| 1 in 100 | 0.01 | 46 | 27 | 27 |
| 1 in 250 | 0.004 | 68 | 39 | 39 |
| Average annual loss | 3.1 | 2.2 | 2.2 |
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 |
|---|---|---|
| EIOPA Solvency II statistics and solvency and financial condition reports | Capital positions of EU insurers | Free, EU reuse terms |
| IAIS Insurance Capital Standard documents | ICS structure and specifications | Free, IAIS terms |
| NAIC risk-based capital resources | United States insurer capital framework | Free to read, NAIC terms |
| Bermuda Monetary Authority BSCR documents | Bermuda capital framework | Free, BMA terms |
| PRA and other supervisor internal model guidance | Expectations for internal models | 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 |
|---|---|---|
| Boards and chief risk officers | Whether to use a standard formula or internal model and what capital to hold | Capital results, sensitivities and validation |
| Supervisors | Whether to approve internal models and what capital to require | Model applications and reviews |
| The IAIS and jurisdictions | How the ICS is implemented and assessed | Implementation assessments |
| Rating agencies and investors | How to compare capital across regimes | Disclosures and reconciliation |
Most modern solvency frameworks define required capital as the amount needed to withstand losses at a chosen confidence level over a set horizon, using risk measures such as value at risk or expected shortfall. Expected shortfall, the average loss beyond the threshold, captures tail severity better than value at risk, which says nothing about losses beyond its threshold.
Standard formulas apply prescribed stresses and correlations, which makes them simple and comparable but may not fit unusual risk profiles. Internal models simulate a firm's risks and aggregate them, which can reflect its profile more closely but introduces model risk and requires extensive validation and supervisory approval. Key drivers include tail assumptions, dependence between risks and the treatment of management actions.
Capital models interact with behaviour. If many firms hold similar assets and respond to stress in similar ways, capital rules can amplify market moves. Countercyclical features and supervisory stress tests aim to address this.
Value at risk and expected shortfall; each has known strengths and weaknesses.
Combining risks through correlation matrices or simulation; dependence assumptions dominate.
Testing capital under severe scenarios as a complement to model results.
Limits: capital models estimate rare outcomes from limited data, so their precision is often overstated; one-year horizons may miss slow-building risks such as climate change.
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