How the Number Is Built

The three companion essays argue about whether the matching adjustment is calibrated honestly. This one steps back from the argument to show the machine in motion: how, line by line, an insurer turns a portfolio of assets and a book of pension promises into a single discount rate and a day-one surplus. None of what follows is contentious as mechanics — it is the standard Solvency UK calculation, and any annuity actuary will recognise it. What is contentious is where the inputs come from, and those points are flagged as they arise rather than saved for a peroration. The aim is that by the end you could, in principle, build the number yourself — and therefore see exactly which of its joints bear the weight.

Throughout, a single illustrative deal runs as the worked example: a £1 billion bulk annuity, pensioners in payment, mean liability term twelve years. Figures are rounded and stylised; they are chosen to expose structure, not to reproduce any firm's return.

1. The two halves of the problem

The matching adjustment exists to answer one question: at what rate may an insurer discount its annuity liabilities? Solvency UK's answer is that the rate may exceed the risk-free curve by an amount reflecting the extra return the backing assets earn that is not compensation for risk the insurer bears. Formally the liability discount rate is

discount rate = risk-free rate + MA

and the matching adjustment itself is

MA = (portfolio yield − risk-free rate) − fundamental spread

which collapses, since the portfolio yield over risk-free is the asset spread, to

MA = asset spread − FS.

So the model has two halves. The first computes the asset spread — what the portfolio earns above risk-free, after stripping out the cashflows the liabilities don't need. The second computes the fundamental spread — the deduction representing retained credit risk. The MA is their difference, and everything downstream (liability value, own funds, capital, surplus) follows from it. We take each half in turn.

2. Projecting the cashflows

Before either yield can be struck, both sides must be reduced to dated cashflows.

Liabilities. The annuity book is projected as expected benefit payments by year, allowing for mortality (base tables plus a longevity-improvement projection), any attaching spouse's benefits, and inflation indexation where the pensions are linked. For our £1bn book, assume the projection produces benefit outgo declining from roughly £85m in year 1 as the population dies off, tailing to near zero beyond year 35. The present value of these cashflows at risk-free would be, say, £950m; the premium charged is £1,000m, of which the difference and more is what the asset strategy must justify.

Assets. The matching portfolio's contractual cashflows are projected similarly — coupons and redemptions for bonds, scheduled and modelled cashflows for the illiquids. The portfolio is constructed so that asset and liability cashflows are closely matched in timing and amount; the closeness of that match is what earns MA eligibility in the first place.

Judgement point one. For traded bonds, asset cashflows are contractual and unambiguous. For the illiquids — equity release, restructured property debt — the "cashflows" are themselves model output. An equity-release tranche has no fixed coupon; its cashflows depend on mortality, morbidity, voluntary redemption and the no-negative-equity guarantee, which in turn depends on a thirty-year house-price model. So before any spread or FS is calculated, the cashflows being matched are partly assumed. This is upstream of everything else, and every later figure inherits it.

3. The asset spread and the equivalent yield

With asset cashflows in hand, the model strikes the portfolio's internal rate of return — the single rate that discounts the asset cashflows back to their market (or model) value. Subtract the risk-free curve and you have the gross spread.

For our portfolio, assume the blend is:

The portfolio's weighted gross spread is 0.20×0 + 0.45×110 + 0.35×220 = 126.5 bps. Call it ~125bps. This is the asset spread before any deduction — the raw material from which both the MA and the FS are carved.

Judgement point two. The 220bps on the illiquid sleeve is, for self-originated assets, a number the insurer's own valuation produces — there is no screen price to read it from. Lift that sleeve's assumed spread by 30bps and the portfolio spread rises by ~10bps, flowing pound-for-pound (less FS) into the MA. The asset spread is thus partly an input the firm sets, not purely a market observable.

4. The fundamental spread, component by component

Now the deduction. The prescribed FS is built per asset, by mapping its rating, sector and term to the regulator's published tables, and summing two components.

Component A — expected loss (cost of default). For each asset, take the annual probability of default (PD) for its rating from long-run agency data, multiply by loss-given-default (LGD, i.e. 1 − recovery rate), and express as an annual basis-point charge:

EL ≈ PD × LGD (per annum, in spread terms)

For a BBB asset with PD ≈ 0.25% and recovery ≈ 40% (LGD 60%), EL ≈ 0.25% × 60% = 0.15% = 15bps.

Component B — cost of downgrade (CoD). This reserves for the loss on being forced to sell and replace an asset that migrates to a lower rating while still held. It is computed from the rating transition matrix: for each possible one-year migration, estimate the value loss from de-and-re-investing to maintain the match, weight by the migration's historical probability, and annualise over the asset's term. For our BBB asset, suppose this yields ≈ 20bps.

So the bottom-up FS for the BBB asset ≈ 15 + 20 = 35bps.

The floor. The total FS is then floored at a prescribed fraction of the long-term average spread (LTAS) for the asset's rating and sector — broadly 30% for non-financials, 35% for financials. If the LTAS for BBB is, say, 150bps, the floor is 0.30 × 150 = 45bps. Since 45 > 35, the floor binds: the asset's FS is 45bps, not the 35bps the bottom-up calculation produced. In benign markets this is the common case — the floor, not the risk calculation, sets the deduction.

Applying the same logic across the portfolio, assume the blended FS comes to ~45bps.

Judgement point three. Every input above — PD, LGD, the transition matrix, the LTAS — is calibrated on traded-bond history and applied to the illiquids by way of their assigned rating. For self-originated assets the rating is frequently the firm's own internal rating (PRA-approved as a methodology, not validated outcome by outcome). The FS is therefore only as meaningful as the rating that selects it, and for a third of the portfolio the rating is produced in-house. Nothing in components A or B prices the fact that the asset has no market price and no default history of its own — that was the job of the proposed valuation uncertainty component, which was not enacted.

5. Striking the MA and revaluing the liability

Now assemble. Portfolio spread ~125bps, blended FS ~45bps:

MA = 125 − 45 = 80 bps.

The liabilities are rediscounted at risk-free + 80bps. Against a book whose risk-free PV was £950m and mean term ~12 years, an 80bps uplift reduces the value by approximately 80bps × 12 ≈ 0.96%, i.e. ~£90m, though the exact figure comes from rediscounting the actual cashflows rather than the duration approximation. Take the modelled reduction as ~£90m, giving a best-estimate liability (BEL) of ~£860m.

Add a small risk margin (post-reform, ~£15m) and the technical provisions are ~£875m. Against £1,000m of assets received:

Own funds at inception ≈ £1,000m − £875m ≈ £125m

— created not by any cash received but by the act of discounting the liabilities at the asset-derived rate. This is the "day-one surplus" the companion essays discuss; here you can see precisely which line produces it. Had the MA been zero (discounting at risk-free), technical provisions would be ~£965m and own funds ~£35m — the difference, ~£90m, is the MA benefit.

6. The capital requirement, and the circularity

The deal also generates a Solvency Capital Requirement — the 1-in-200 stress. Its largest annuity components are:

• Spread risk: asset values fall under a prescribed widening of credit spreads. Critically, under the rules the MA recalculates in the stress — wider spreads raise the MA, lowering stressed liabilities and offsetting much of the asset hit.
• Longevity risk: liabilities rise under a permanent mortality-improvement stress.
• Counterparty / concentration / property components for the specific assets.

Suppose the diversified SCR is ~£90m. The own funds covering it are the ~£125m the MA just created. Coverage ≈ 125 / 90 ≈ 139%, before the firm adds its target buffer.

Pause on what just happened. The capital held against the assets failing is substantially the capitalised expectation that they will not fail; and within the spread-risk module, the MA's rise under stress is itself assumed to absorb the blow. The same favourable spread assumption appears three times — in the asset yield that creates the surplus, in the MA that lowers the liability, and in the stressed MA that shrinks the capital requirement the surplus must cover. This is not an accounting trick; it is the prescribed methodology. But it means the entire structure's resilience reduces to one question — is the FS partition correct? — asked at three points at once.

7. What flows out

The ~£125m inception surplus, less the capital that must be retained against the new SCR and the firm's buffer, becomes distributable over time as the SCR amortises with the runoff. Under IFRS 17 the same economics are reported differently — the excess of premium over fulfilment cashflows is parked in the Contractual Service Margin and released over the ~40-year coverage period, so the income statement shows the profit slowly while Solvency capital generation shows it fast. Dividends follow the Solvency measure, not the IFRS one. (The two-measure divergence is developed in the companion essays.)

8. The four joints that bear the weight

Strip the model to its load-bearing judgements and there are four, each identified above:

1. Illiquid cashflows (§2) — for self-originated assets, the cashflows being matched are themselves modelled, upstream of everything.
2. Illiquid spreads (§3) — the yield on the self-originated sleeve is a valuation the firm produces, not a market price it reads.
3. The rating bridge (§4) — the FS is calibrated on traded bonds and applied to illiquids via an often-internal rating; nothing prices non-tradability itself.
4. The circularity (§6) — the same spread assumption creates the capital, lowers the liability, and shrinks the requirement, so an error compounds rather than cancels.

Notice that all four converge on the same place: the treatment of assets that are not traded bonds but the insurer's own investments. The mechanics are robust where the assets are public and market-priced; they become assumption-driven exactly where the portfolio has migrated. The model is not wrong. It is precise about quantities that, for a growing share of the book, are estimates produced by the party that benefits from them — and the one prescribed, regulator-owned component designed to charge for that very fact was proposed by the supervisor and left out of the final rules.

That is the whole of it. The number is built from a projection, an equivalent yield, a prescribed deduction and a stress — four clean steps, three of which silently inherit the same set of in-house assumptions about assets that have never been tested by a market or a cycle. Whether the number is right is, in the end, whether those assumptions are. The model cannot tell you; it can only propagate them, elegantly, into a surplus that is paid out long before the assumptions are ever put to the test.

C.J. Marsden writes on political economy at ir35andmore.com. Companion essays — "The Overruled Objection," "The Profession That Welcomed It," and "The Missing Invoice" — cover the 2022 reform decision, the actuarial profession's response, and the distribution of risk across pensioners, shareholders and the state.

Note on figures. The worked example is illustrative and stylised to expose structure; spreads, FS components, durations and capital figures are plausible round numbers, not any firm's disclosure. The mechanics (equivalent-yield approach, FS = expected loss + cost of downgrade subject to an LTAS floor, MA recalculation within the spread-risk stress, CSM release under IFRS 17) follow the published Solvency UK and IFRS 17 frameworks. Practitioners should consult the PRA's matching adjustment rules and technical information, and the relevant TASs, for calibration detail.