The standard formula: structure

Level 3 gave RiskAtoms Insurance SE an SCR of €5,000m and a ratio of 186%. Look at how that figure is made and three things stand out. Two of the five modules were not computed by the rules at all. Market and life together are 63% of the modules' sum. And €1,204.4m of the requirement is taken off for tax the insurer would no longer pay after a disaster: a credit worth 36 points of ratio, set in Level 3 so the answer came out round. This lesson opens each line.

The frame

The Directive sets the shape and the Delegated Regulation fills it. The SCR is the 99.5% value at risk of basic own funds over one year (Art 101(3), recalled from L4.6.01: The Solvency II architecture). Under the standard formula it is the sum of three items (Art 103): the basic SCRBasic Solvency Capital Requirement (BSCR): the module charges for market, counterparty default, life, health and non-life risk, combined with the correlation matrix of Annex IV of the Directive, plus the intangible asset charge (Directive Art 104; Delegated Regulation Art 87)., the operational risk chargeOperational risk charge: a factor formula on earned premiums and technical provisions, the larger of the two, capped at 30% of the BSCR, plus 25% of unit-linked expenses (Art 204)., and an adjustment for the loss-absorbing capacity of technical provisions and deferred taxes. The BSCR is built from at least five risk modulesRisk module: one block of the standard formula, such as market risk, made of sub-modules (interest rate, equity …). Each is calibrated to a 99.5% value at risk over one year (Directive Art 104(4)). (Art 104(1)), each calibrated to the same 99.5% one-year standard (Art 104(4)), and each module from sub-modules that apply a prescribed shock and measure the fall in own funds. Click through the tree.

Aggregation, twice

L3.8.12: Insurer investment risk and the ORSA combined the market sub-modules and then the five modules as √(s′Cs). The rule does this at two levels: sub-modules into a module with the module's own matrix, then modules into the BSCR with Annex IV's (point 1). The matrix is fixed for every insurer in the EU, set so that the square-root rule approximates the 1-in-200 of the sum when risks are not jointly normal; the correlations are meant for the tail, not for ordinary years (M-C10: Covariance and correlation).

The modules add to €8,594.8m; the BSCR is €5,765.9m. Had every correlation been 1, the BSCR would be the sum; at 0 it would be €4,861.0m. The diversified contributionsDiversified (Euler) contribution: a module's share of the BSCR, s_i (Cs)_i / √(s′Cs). The contributions add up to the BSCR. show where the capital sits: market is 48% of the BSCR, because it correlates 0.25 with every other module while life and non-life offset each other at 0.

Operational risk: a max inside a min

Operational risk is not modelled at all. Art 204 takes the larger of two volume measures, caps it at 30% of the BSCR, and adds a quarter of unit-linked expenses:

Provisions bind (€438.6m against €358.2m), and the cap of €1,729.8m is far away. Premium growth above 20% a year would add a growth term (Art 204(3)); the book's premiums grew less. Operational risk is added outside the matrix: the formula gives it no diversification with anything.

The two adjustments

The adjustment for technical provisionsLoss-absorbing capacity of technical provisions (LAC TP): the part of a shock that future discretionary benefits, such as bonuses on with-profits policies, would absorb; at most the value of those benefits (Art 206). recognises that a with-profits insurer can cut future bonuses after a loss. The firm reruns the modules allowing for that (the net BSCR, nBSCR), and the adjustment is the difference, never more than the future discretionary benefits (FDB) themselves (Art 206). RiskAtoms' savings book credits a guaranteed rate or a formula rate with no discretion, so its FDB and its adjustment are nil.

The adjustment for deferred taxesLoss-absorbing capacity of deferred taxes (LAC DT): the fall in deferred tax liabilities, or rise in recognisable deferred tax assets, if the firm suffered a loss equal to BSCR + LAC TP + operational risk at once (Art 207). is the large one. Suppose the firm lost BSCR + LAC TP + operational risk at once, €6,204.4m here (Art 207(1)). Its tax position would change: deferred tax liabilities on revaluation gains would fall, and a loss carried forward would be a deferred tax asset. That change reduces the SCR. At a 25% tax rate it can never exceed €1,551.1m, the tax on the whole loss; that ceiling is arithmetic, not a stated cap. Level 3 set the adjustment at €1,204.4m, 19.4% of the loss or 77.7% of the ceiling, as the number that made the SCR €5,000m. The rule asks the firm to earn it.

How firms justify it. A tax asset created by the loss counts only if the firm shows the supervisor that future taxable profit is probable (Art 207(2a)). The usual recoverability testRecoverability test: the evidence a firm gives that a deferred tax asset created by the loss could be used: existing deferred tax liabilities, carry-back where tax law allows, and future taxable profits, under the limits of Art 207(2a)–(2d). stacks three sources: first, existing net deferred tax liabilities that the loss would release; second, carry-back of the loss against tax paid in earlier years, where national tax law allows; third, future taxable profits. For those, the assumptions may not be more favourable than in the balance sheet's own deferred tax valuation (2b); new business may not exceed the business plan, nor run past both its horizon and five years; investment returns are the post-shock risk-free forwards unless there is credible evidence for more; profits from new business beyond the plan's horizon need growing haircuts (2c); management actions count only if they meet Art 23 (2d).

On these assumptions illustrative the test supports €1,225m: just above the €1,204.4m booked, with €20.6m to spare. Count only the three plan years and it supports €1,015m: the SCR would rise to €5,189.4m and the ratio fall to 179.2%. Module 1.8's bridge books no deferred tax on its €700m of revaluations; the €175m DTL here assumes the IFRS figures are the tax base. The recoverability test leans on the same business plan as the ORSA (L4.6.11: Governance, ORSA and reporting).

Booked against the rules in outline

Level 3 built the formula in outline, and every parameter it uses matches the text. Two figures were not the rule's: counterparty default was stated, and the cat charge came from the insurer's own event model. Applied, the rules give:

The counterparty charge rises by €117.3m and the non-life module by €200.8m; after diversification the BSCR rises €193.4m, and with the deferred tax credit held at 19.4% the SCR rises €155.9m to €5,155.9m: a ratio of 180.4% rules. The bigger loss also asks more of the tax credit: €1,242.0m at 19.4%, more than the €1,225m the same recoverability test supports, which would leave 179.8%. The other places where Level 3 simplifies:

In full, for modules $i, j$ (Directive Art 103, Annex IV; Delegated Regulation Arts 87, 203, 204, 206, 207):

• $C$ is the Annex IV matrix above; $s^{\top}Cs$ is a quadratic form (M-E02: Matrix multiplication and quadratic forms). Each $\text{SCR}_i$ is itself a 99.5% quantile of a one-year loss (M-C09: Quantiles), so the square root is exact only if losses are jointly normal; otherwise it is an approximation the matrix is calibrated to.
• Earned premiums are gross of reinsurance; provisions exclude the risk margin and are gross of recoverables. Growth terms add 4% or 3% of premium growth above 20% a year.
• $\Delta\text{DT}$ is negative when liabilities fall or recognisable assets rise; a positive change gives nil (Art 207(3)–(4)). With tax rate $t$, $|\text{Adj}_{\text{DT}}| \le t \times (\text{BSCR} + \text{Adj}_{\text{TP}} + \text{SCR}_{\text{op}})$.

Start from the booked €5,000m. Switch counterparty and cat to the rules and watch diversification absorb part of the rise. Then change how the deferred tax credit is set: the test, the ceiling, nothing. With the test chosen, cut the profits or the years counted and see how quickly the credit, and the ratio, fall.

€ millions at 31 December 2025, Level 3's book with the counterparty and cat charges of L4.6.07: Counterparty and underwriting modules illustrative.

Directive 2009/138/EC Arts 101 and 103–108 and Annex IV; Delegated Regulation (EU) 2015/35 Arts 87, 203–207 as amended by Delegated Regulation (EU) 2019/981, which added the conditions of Art 207(2a)–(2d). Statements on what changes are as at 4 November 2026 as at 4 Nov 2026.

Insurers disclose the module split, operational risk and both adjustments in the SFCR every year (template S.25.01, L4.6.11: Governance, ORSA and reporting). Internal models replace some or all of the modules, but must still allow for tax and discretionary benefits (L4.6.09: Internal models).

• SCR = BSCR + operational risk + adjustments for technical provisions and deferred taxes (Directive Art 103). The BSCR is √(s′Cs) over five modules with Annex IV's fixed matrix, plus 80% of intangibles.
• Every module and sub-module is a 99.5% one-year shock to own funds; the matrix stands in for tail dependence and is the same for every EU insurer.
• Operational risk = min(30% × BSCR, max(Op_premiums, Op_provisions)) + 25% of unit-linked expenses, with no diversification: €438.6m on the book.
• The deferred tax adjustment is the change in deferred taxes after a loss of BSCR + LAC TP + Op, at most the tax rate times that loss, and must be backed by DTLs, carry-back or probable profits (Art 207(2a)–(2d)). RiskAtoms books €1,204.4m of a €1,551.1m ceiling.
• Rules in outline: counterparty €267.3m and non-life €3,002.0m give an SCR of €5,155.9m and 180.4%, against €5,000m and 186.0% booked.

Basic Solvency Capital Requirement (BSCR)

Basic Solvency Capital Requirement (BSCR): the module charges for market, counterparty default, life, health and non-life risk, combined with the correlation matrix of Annex IV of the Directive, plus the intangible asset charge (Directive Art 104; Delegated Regulation Art 87).

Risk module

Risk module: one block of the standard formula, such as market risk, made of sub-modules (interest rate, equity …). Each is calibrated to a 99.5% value at risk over one year (Directive Art 104(4)).

Diversified (Euler) contribution

Diversified (Euler) contribution: a module's share of the BSCR, s_i (Cs)_i / √(s′Cs). The contributions add up to the BSCR.

Operational risk charge

Operational risk charge: a factor formula on earned premiums and technical provisions, the larger of the two, capped at 30% of the BSCR, plus 25% of unit-linked expenses (Art 204).

Intangible asset risk

Intangible asset risk: 80% of intangible assets recognised on the Solvency II balance sheet (Art 203). Goodwill is valued at zero, and other intangibles only if they can be sold separately and a value for the same or similar assets can be shown from quoted prices in active markets (Arts 10(2), 12).

LAC of technical provisions

Loss-absorbing capacity of technical provisions (LAC TP): the part of a shock that future discretionary benefits, such as bonuses on with-profits policies, would absorb; at most the value of those benefits (Art 206).

LAC of deferred taxes

Loss-absorbing capacity of deferred taxes (LAC DT): the fall in deferred tax liabilities, or rise in recognisable deferred tax assets, if the firm suffered a loss equal to BSCR + LAC TP + operational risk at once (Art 207).

Recoverability test

Recoverability test: the evidence a firm gives that a deferred tax asset created by the loss could be used: existing deferred tax liabilities, carry-back where tax law allows, and future taxable profits, under the limits of Art 207(2a)–(2d).