Internal models · RiskAtoms Academy

By the end you will be able to

1Explain how a full or partial internal model is approved, and what the supervisor can do afterwards.
2Name the six tests a model must pass (use, statistical quality, calibration, profit and loss attribution, validation, documentation) and recognise each in practice.
3Read a 99.5% one-year quantile from simulated years, and tell an occurrence figure from an aggregate one.
4Plug a partial model back into the standard formula with correlations, and say what it would change for RiskAtoms Insurance SE.

Why it matters

RiskAtoms Insurance SE already uses a model of its own. Level 3 took its cat charge, €703.2m, from its 17,000-event set rather than the formula's factors (L3.8.05: Catastrophe risk). L4.6.07: Counterparty and underwriting modules showed that the standard formula, in outline, gives €1,084.3m. Under Solvency II a firm may only replace part of the formula with its own model once its supervisor has approved it, and the approval comes with tests that most of the firm's day-to-day work has to pass. This lesson is what that would take, and what it would change.

Core explanation

Approval, six tests and a change policy

Illustrative dataRiskAtoms Insurance SE, under the rules

The same made-up composite insurer as M3.8: Insurance risk, in € millions at 31 December 2025. Its booked figures stand: own funds €9,300m, a Solvency Capital Requirement of €5,000m, a ratio of 186%. L4.6.05: The standard formula: structure rebuilt the standard formula article by article; where Level 3 stated a figure or used its own cat model, the “rules in outline” view computes the formula instead (SCR €5,155.9m, ratio 180.4%). The book applies no long-term guarantee measure, no transitional and no internal model. Every what-if on this page is labelled, holds the deferred-tax adjustment at the book's 19.4% of BSCR plus operational risk, and is before tax. Every policy, bond, treaty and counterparty is invented.

Added here. A partial internal model for cat, built on m3-8's own event set and its 50,000 simulated years, net of the cat XL. It is illustrative: the insurer has no approval, and nobody has validated the event set.

Full or partial, and how approval works

The Directive lets insurers calculate the SCR with a full or partial internal model approved by their supervisor (Art 112(1)). A partial model may cover one or more risk modules or sub-modules, operational risk or the deferred-tax adjustment, for the whole business or for major business units (Art 112(2)). Its limited scope must be justified, it must reflect the risk profile better than the formula, and it must be possible to integrate it into the formula (Art 113). The application carries documentary evidence that the model meets Arts 120–125, and the supervisor decides within six months of receiving a complete one (Art 112(3)–(4)).

ExploreFrom application to ongoing supervision

Click a box to see what happens there.

Board

The administrative, management or supervisory body approves the application and every later major change, and is responsible for the model working properly on a continuous basis.

The six tests

Approval rests on six sets of requirements. The use test (Art 120) is the one that changes how a firm works: a model built only for the supervisor fails it. Statistical quality (Art 121) asks for sound methods consistent with the technical provisions, and data that are accurate, complete and appropriate, updated at least yearly; no particular method is prescribed, and diversification counts only where the supervisor is satisfied with how it is measured (Art 121(5)). Calibration (Art 122) is the 99.5% one-year VaR of Art 101(3); another measure or period is allowed only where it protects policyholders as well. Profit and loss attribution (Art 123), validation (Art 124) and documentation (Art 125) complete the list. A vendor model does not exempt the firm from any of them (Art 126): the point L3.8.05: Catastrophe risk made about bought-in cat models.

Sort itWhich test is it?

Sort each piece of evidence into the article it belongs to.

The board sets the cat reinsurance budget using the model's 1-in-200 net loss.
Data on claims and exposures must be accurate, complete and appropriate, and updated at least once a year.
A firm wants to model the 99% expected shortfall instead of the 99.5% VaR, and must show policyholders are protected as well.
At the year end, the actual loss on the motor book is explained by the model's premium, reserve and cat risk categories.
Each year the model's 1-in-10 losses are tested against what actually happened, and its stability is tested.
The design, theory, assumptions and limitations of the model are written down, with each major change.

What a cat model would change here

Three figures for the same risk, all net of the cat XL and its reinstatement premiums. The standard formula in outline applies factors to sums insured by zone: €1,084.3m. The book took the 1-in-200 single event: €703.2m. But the SCR is a one-year measure, and a year can bring several events. The 99.5% quantile of each simulated year's total net cat cost is €1,125.1m. So a model calibrated as Art 101(3) asks would hold more than the book does, and slightly more than the formula. That is the difference Level 3 left open; it is shown here as a what-if, and the book's SCR stays €5,000.0m.

Cat figure (net)	Cat	Non-life	SCR	Ratio
Standard formula in outline (nat cat factors)	1,084.3	3,002.0	5,155.9	180.4%
m3-8 as booked: 1-in-200 single net event	703.2	2,801.2	5,049.3	184.2%
Partial internal model (illustrative): 99.5% of the year's net cat cost	1,125.1	3,025.5	5,168.6	179.9%

All three on the rules in outline (counterparty computed at €267.3m), integrated with Art 239(1)'s default correlations. With the book's stated €150m for counterparty, the single-event row is the booked SCR of €5,000.0m.

The formal version

The calibration is a quantile (M-C09: Quantiles), and integration is aggregation with a correlation matrix (M-C10: Covariance and correlation). Directive Art 101(3) and 122; Delegated Regulation 2015/35 Art 239(1).

\begin{aligned} \text{SCR} &= \text{VaR}_{99.5\%}\big(-\Delta\text{BOF}_{0\to1}\big), \qquad \widehat{\text{VaR}}_p = L_{(\lceil p n\rceil)} \\ \text{Cat}_{\text{agg}} &= \widehat{\text{VaR}}_{99.5\%}\Big(\textstyle\sum_{k=1}^{N} X_k^{\text{net}} + \text{RIP}\Big) \;\ge\; \text{Cat}_{\text{occ}} = \widehat{\text{VaR}}_{99.5\%}\big(\max_k X_k^{\text{net}}\big) \\ \text{SCR}_{\text{nl}} &= \sqrt{\text{PR}^2 + 2\rho\,\text{PR}\cdot\text{Cat} + \text{Cat}^2}, \qquad \rho = 0.25 \ (\text{Del. Reg. Art 114}) \\ \text{BSCR} &= \sqrt{s^{\top} C\, s}, \quad s = (\text{Mkt}, \text{Def}, \text{Life}, \text{Health}, \text{SCR}_{\text{nl}}), \quad C = \text{Annex IV} \end{aligned}

$\Delta\text{BOF}_{0\to1}$ is the change in basic own funds over one year; $L_{(i)}$ the $i$ -th smallest of $n$ simulated losses.
$N$ is the number of events in a year, $X_k^{\text{net}}$ each one net of the cat XL, RIP the reinstatement premiums. The year total is at least its largest event, so the aggregate quantile is never below the occurrence one.
PR is premium and reserve risk (€2,541.3m); $C$ is the BSCR matrix of Directive Annex IV. Where the default does not fit, Art 239(2)–(3) allows an Annex XVIII technique or a justified alternative.

Try it

Choose where the cat figure comes from and the return period it is read at. Then change the correlation that plugs it back in. Watch how fast the single-event figure jumps once the cat XL is exhausted, and how little of any change survives into the SCR.

Try itA partial internal model for cat

Where the cat figure comes from

Return period (1-in-T years)

Quantile 99.50%. The rules ask for 99.5% over one year (1-in-200).

Correlation of cat with premium and reserve risk25% (Del Reg Art 114)

Cat charge

€1,125.1m

Premium and reserve

€2,541.3m

Non-life module

€3,025.5m

BSCR€5,975.1m

Operational risk€438.6m

SCR (LAC DT at 19.4%)€5,168.6m

Against the standard formula at 25%+€12.7m

Solvency ratio (own funds €9,300m)179.9%

Illustrative: the insurer holds no internal model approval. Other modules are the rules in outline (counterparty computed at €267.3m); the deferred-tax adjustment is held at 19.4% of BSCR plus operational risk. “Against the standard formula” always compares with the formula's own cat charge at its 25% correlation, wherever the slider is.

Worked example

A partial model for cat, on the rules in outline

€ millions, 31 December 2025, net of the cat XL with reinstatement premiums. illustrative

From simulated years to the solvency ratio

Step 1 of 5

1
Simulate a year's net cat cost, 50,000 times
Each simulated year draws its events from m3-8's set, runs them through the cat XL with its annual limits and adds the reinstatement premiums. The average year costs €143.6m.

Regulatory lens

Directive 2009/138/EC Arts 112–127, 37 and 45(3); Delegated Regulation (EU) 2015/35 Art 239. Status as at 4 November 2026 as at 4 Nov 2026.

In the EU

EU from 2027 Directive (EU) 2025/2 applies from 30 January 2027. It replaces Art 112(7): internal model users must give their supervisor an estimate of the SCR on the standard formula every two years, and supervisors may ask more often by a reasoned decision. A new Art 122(5) lets Member States allow a model to reflect how credit spread moves change the volatility adjustment, only under conditions (L4.6.10: Long-term guarantee measures and transitionals). Every user must still recalibrate its ORSA figures to the SCR's measure (Art 45(3)).

In the UK

UK From 31 December 2024 the PRA uses a streamlined set of internal model rules with fewer prescriptive requirements, and two safeguards: a residual model limitation capital add-on and model use requirements (PS2/24, 28 February 2024). More in L4.6.13: Solvency UK and global standards.

Spark Atoms

Spark AtomSame assets, different models, different answers

EIOPA compares the market and credit risk models of approved firms on common portfolios. Its year-end 2024 study, published on 16 April 2026, had 22 participants from 7 Member States, covering close to all the euro investments held by firms with approved models for those risks. It found “moderate to significant dispersion” in some outputs, partly explained by known model and business differences, and called for continued supervisory attention.

Spark AtomThe 1-in-250 event costs almost twice the 1-in-200

On the book, the net single-event loss is €703.2m at 1-in-200 and €1,210.2m at 1-in-250. Once a storm passes €2,000m gross, the cat XL is used up and every further euro is the insurer's own. A quantile just above the top of a reinsurance programme can jump: one reason supervisors look hard at where a model puts its 99.5%.

Key takeaways

A full or partial internal model replaces the standard formula only after approval; the supervisor decides within six months of a complete application (Art 112).
Six tests: use, statistical quality, calibration (99.5% one-year VaR or equivalent), profit and loss attribution, validation and documentation (Arts 120–125). Vendor models do not exempt the firm (Art 126).
A model change policy splits major changes (prior approval) from minor ones; reverting to the formula needs approval; a model that misses risks can bring a capital add-on.
On the book, a cat model calibrated to the year's total gives €1,125.1m against the formula's €1,084.3m and the book's single event of €703.2m: SCR €5,168.6m, ratio 179.9% (illustrative).
A partial model is plugged back in with the formula's correlations by default (Art 239); from 2027 every model user also reports a standard formula estimate every two years.

Reaction

Check what stuck

Glossary

Internal model: Internal model: the insurer's own model of the probability distribution of its basic own funds over one year, approved by its supervisor to calculate the SCR instead of the standard formula (Directive Arts 112–127).
Partial internal model: Partial internal model: an internal model for some risk modules or sub-modules, operational risk or the deferred-tax adjustment, or for some major business units, with the standard formula for the rest (Art 112(2)).
Use test: Use test: the model must be widely used in, and play an important role in, the system of governance, risk management and decisions, including the ORSA (Art 120).
Profit and loss attribution: Profit and loss attribution: at least once a year, explaining the actual profits and losses of each major business unit with the model's risk categories (Art 123).
Model change policy: Model change policy: the rules, approved with the model, that split changes into major ones (prior approval needed) and minor ones (no prior approval) (Art 115).
Integration technique: Integration technique: how a partial model's result is combined with the standard formula's modules. By default, with the standard formula's own correlation matrices (Delegated Regulation 2015/35 Art 239(1)).
Occurrence loss: Occurrence loss: the largest single event in a year. Its 1-in-200 value is read from the occurrence exceedance curve.
Aggregate loss: Aggregate loss: the total of all events in a year. Its 1-in-200 value is the one-year 99.5% quantile, the measure the SCR is calibrated to.
Capital add-on: Capital add-on: an extra requirement a supervisor may set, in exceptional circumstances, for example where an internal model captures some quantifiable risks insufficiently (Art 37(1)(b)).

Deep Dives & Origins

Deep Dives

L3.8.05: Catastrophe risk: the event set and exceedance curves this model is built on. Intermediate.
M6.7: Model risk, validation and machine learning: model validation in depth. Advanced.

Origins

European Parliament and Council, Directive 2009/138/EC (Solvency II), Arts 37, 45, 101 and 112–127, as adopted (legislation.gov.uk).
European Parliament and Council, Directive (EU) 2025/2 of 27 November 2024, Art 1 points replacing Art 112(7) and adding Art 122(5) (LawPlayer copy of the Official Journal text).
European Commission, Delegated Regulation (EU) 2015/35, Art 239 (integration of partial internal models), as adopted (legislation.gov.uk).
Prudential Regulation Authority, PS2/24 Review of Solvency II: Adapting to the UK insurance market, 28 February 2024.
EIOPA, EIOPA's year-end 2024 study on market and credit risk, press release, 16 April 2026.

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