Capital Modelling Under Solvency II

Written by Jonas Osman Abdelghafour — actuary and financial risk manager (FRM).

Solvency II turned eleven this year. For every European insurer above a certain size, the Solvency Capital Requirement is now the number that determines dividend capacity, reinsurance appetite, product strategy, and the tone of the annual regulator conversation. Yet the modelling choices that produce the SCR remain, in most firms, poorly understood outside a small circle in the actuarial and risk functions.

This guide is written for the CFO, the risk committee, and the incoming Head of Capital who need to know what a defensible capital model looks like — the standard formula, the internal model, and the space in between — without a semester of directives on their desk.

The standard formula: not just a fallback

Two-thirds of European insurers use the standard formula. It is often described as the conservative default, and for many small mono-line insurers that description is fair. For large multi-line groups, the standard formula is a set of specific, sometimes strong, and occasionally wrong assumptions about correlation, calibration, and diversification.

The most important thing to know about the standard formula is where it materially diverges from the firm's own risk profile. The typical suspects are the non-life premium and reserve module (whose factors are calibrated to a European industry-wide dataset and can misprice specialty lines), the market risk equity charge (whose symmetric adjustment mechanism does not always behave sensibly through a cycle), and the catastrophe module (whose scenarios are simplifications of what a modern cat model would output).

A standard-formula insurer should still run an internal challenge — a simplified economic capital calculation on the same risk drivers — every reporting cycle. If the internal number and the standard-formula SCR diverge materially, that is either a Pillar 2 conversation with the regulator or the start of the business case for partial internal model approval.

Internal models: what approval actually asks

The IMAP process is often described as bureaucratic, and parts of it are. But the substantive tests behind it — use, statistical quality, calibration, profit and loss attribution, and validation — encode good modelling discipline. A firm that treats them as compliance items will produce a model that passes on paper and misbehaves in practice.

Use test. The single most challenging test for most applicants. The internal model must be embedded in decisions: pricing, capital allocation, reinsurance, ORSA. If the model is a reporting engine that runs quarterly and informs nothing, no amount of statistical documentation will rescue the application. Evidence trail — board minutes, pricing committee papers, reinsurance decisions — is what the college of supervisors will want to see.

Statistical quality. The model has to produce the 1-in-200 loss distribution on total own funds, defensibly, with all material risks captured. In practice this means Monte Carlo with tens of thousands of paths, joint modelling of the assets and liabilities using proxy functions, and enough granularity that risk drivers do not aggregate away.

Calibration. Every distribution and every dependency in the model needs a documented calibration source. Where market data does not exist, expert judgement is permitted — provided it is documented, challenged, and reviewed. The proportion of the SCR driven by expert judgement is a metric supervisors will ask for.

P&L attribution. Every reporting cycle, the model must reconcile the movement in own funds to movements in modelled risk drivers, with a residual that is small and diagnosed. Persistent large residuals are a sign the model has drifted from the underlying business and need investigation before the next SCR run.

Validation. Independent, cyclical, risk-based. See the model validation service and the three-lines governance note for how I structure this in practice.

Proxy functions: the quiet centre of the model

For a life insurer, the SCR calculation would be computationally infeasible if the cash-flow projection had to be re-run under every Monte Carlo scenario. Proxy functions — polynomial or basis-function approximations of the balance sheet valuation as a function of risk drivers — are the compromise that makes internal models tractable.

The quality of the proxy fit determines how much of the reported SCR is real signal and how much is approximation error. Two diagnostics are essential. Out-of-sample fitting error, tested on a representative scenario set that was not used to calibrate the proxy, gives a scale for the residual noise. Sensitivity fits, which check that the proxy reproduces the balance sheet response to stresses of the underlying drivers, are what catches the failure mode where a proxy fits well in aggregate but wrong in the tail.

Proxy functions are also where the reviewer conversation gets sharp. Fitting errors that are acceptable at the mean can be material at the 99.5 percentile. Independent validation should include a targeted attack on the tail behaviour of the proxy — refitting under stressed calibrations, comparing to full nested stochastic runs on a small number of scenarios, and documenting the results.

Risk margin: no longer an afterthought

The 2020 review reduced the cost-of-capital rate and introduced the lambda factor, materially lowering the risk margin for long-tail life business. That has re-opened the market for retrospective reinsurance transactions and made the risk margin a live commercial number rather than a technical residual.

The methodological subtleties are worth attention. The projection of future SCRs uses simplifications that vary widely across firms; the choice of simplification can move the margin by ten percent or more on long-duration books. Any firm re-quoting reinsurance or writing new bulk annuity business should re-derive its projection method and document why it is representative.

Aggregation and diversification: where the SCR is really made

Ask a room of actuaries what the largest source of uncertainty in a life-and-non-life group SCR is, and the answer is not the individual module calibrations. It is the aggregation matrix. The reduction from the sum of module capitals to the diversified SCR is often thirty to fifty percent, and it is driven by correlations that are difficult to estimate and easy to under-challenge.

Two practical recommendations. First, run the SCR at multiple correlation assumptions — the point estimate, a plausible stress, and full comonotonicity — every cycle. The delta is the number the risk committee actually needs to see. Second, decompose the diversified SCR into stand-alone contributions using Euler allocation and reconcile the allocation year-on-year. Persistent drift in a contribution is a sign of a business-mix change, or of a calibration that has not kept up with it.

See the related Financial Risk & ALM services and the insurance and Solvency II practice for how these run in engagements I lead. The ORSA scenario design note covers how internal model output feeds board-level scenarios.

Model change: a governance problem, not a documentation one

Every internal model firm has a model change policy classifying changes as major or minor. The compliance layer is well-understood. The substantive question, which many firms handle less well, is whether the aggregate of minor changes over an annual cycle has drifted the model far enough that a re-approval is warranted. Supervisors have started to ask this question directly. Firms that can produce, on demand, a reconciled year-on-year model change summary — quantifying the SCR impact of each change and the aggregate — are in a materially better place.

The next cycle: 2027 review and beyond

The 2027 Solvency II review will re-open volatility adjustment, the ultimate forward rate, and the treatment of long-term equity investments. Firms should be modelling the sensitivities today. The models that are easiest to adapt are the ones that already separate the economic scenario generator, the balance sheet valuation, and the capital aggregation into modular components rather than a single monolithic pipeline. If your model does not have that separation, the 2027 review is a good time to invest in it.

Closing

Capital modelling under Solvency II is not primarily a technical problem. The mathematics is well-defined; the software is mature; the calibration data is available. What determines whether a firm's model runs the business, or merely reports on it, is the discipline around use, calibration, validation, and change control. The techniques above are the ones I return to on every engagement — from a standard-formula insurer investigating the case for a partial internal model, to a full internal model firm preparing for the next round of supervisory review.

If your capital model is due a refresh, or you are preparing for the 2027 review, see insurance and Solvency II services or get in touch.