Market-consistent valuation and the risk-free curve · RiskAtoms Academy

By the end you will be able to

1Apply Art 75's exit-value principle and the Delegated Regulation's valuation hierarchy, and name where Solvency II departs from IFRS.
2Build the euro risk-free curve: swaps less the credit risk adjustment up to the last liquid point, then Smith–Wilson to the UFR with α set by the convergence test.
3Explain the 2027 extrapolation (first smoothing point, α from 20% to 11%, UFR weight of at least 77.5%) and compare it on an illustrative curve.
4Measure the curve's effect on RiskAtoms Insurance SE: +€12.3m of own funds on Smith–Wilson, +€14.7m on the 2027 method at α 11%.

Why it matters

An annuitant aged 65 today may be paid for 35 years or more. No euro swap is traded deep and liquid that far out, yet Solvency II needs a rate for every year of the payments. The answer is a curve that is market-based where the market is, and a rule where it is not. Change the rule and an insurer's liabilities move, without a single cash flow changing.

Core explanation

Exit values, and a curve that runs past the market

Illustrative dataRiskAtoms Insurance SE under Solvency II

The same made-up composite insurer as Modules 1.8, 2.2 and 3.8, reporting in € millions at 31 December 2025: €70bn of IFRS assets, €9.0bn of non-life premiums, annuities, savings with guarantees, term assurance and income protection. Level 3 booked own funds of €9,300m against an SCR of €5,000m, a ratio of 186%; those stay the insurer's figures. Module 4.6 reads them against the rules article by article: Directive 2009/138/EC and Delegated Regulation (EU) 2015/35 as in force at that date, and the 2025 review where it applies from 30 January 2027. Every policy, bond and figure is invented. Where a page adds an assumption of its own, it is labelled illustrative.

This page adds illustrative curves: a Smith–Wilson extrapolation and the 2027 method, both fitted to the book's own euro zero rates (L3.8.11: Insurer asset-liability management). None is EIOPA's published curve illustrative.

What a market-consistent balance sheet is

Art 75 sets one principle for the whole balance sheet: the exit value. Assets go in at the amount they could be exchanged for, liabilities at the amount they could be transferred or settled for, between knowledgeable willing parties at arm's length, and liabilities with no adjustment for the insurer's own credit standing. The Delegated Regulation turns that into rules:

IFRS where it agrees. Firms recognise assets and liabilities under IFRS and value them with IFRS methods, but only methods consistent with Art 75 (Art 9).
A hierarchy. Quoted prices in active markets for the same item by default; then quoted prices for similar items, adjusted; then alternative methods that use market inputs as much as possible (Art 10(2) to (7)).
Zeros. Goodwill is valued at zero, and other intangibles too unless they could be sold separately and quoted prices in active markets show a value for the same or similar assets (Art 12).
Technical provisions have rules of their own (Directive Arts 76–86): a best estimate discounted on the risk-free curve, plus a risk margin, in L4.6.03: Technical provisions: the best estimate and L4.6.04: The risk margin. There is no market price for them, so Solvency II builds one.

What goes on the Solvency II balance sheet?

The risk-free curve, piece by piece

The best estimate is discounted with “the relevant risk-free interest rate term structure” (Directive Art 77(2)), which EIOPA calculates and publishes every month for each currency. For the euro it is built in three steps:

Market rates where the market is reliable. Interest rate swap rates (Art 44), at the maturities where the market is deep, liquid and transparent (Art 43), up to the last liquid point: 20 years for the euro.
Less a credit risk adjustment. A swap's floating leg pays an interbank rate, which carries some bank credit risk. The credit risk adjustment takes off half the one-year average spread between those swaps and overnight index swaps, kept between 10bp and 35bp (Art 45):

One-year average IBOR–OIS spread	Half of it	CRA (Art 45)	What binds
12bp	6bp	10bp	floor
30bp	15bp	15bp	50% share
50bp	25bp	25bp	50% share
90bp	45bp	35bp	cap

Extrapolation to the UFR. Beyond the last liquid point, forward rates converge smoothly towards the ultimate forward rate (Directive Art 77a; Delegated Regulation Arts 46–47) using Smith–Wilson. For the euro the forward must be within 1bp of the UFR at the convergence point, the larger of LLP + 40 and 60 years, so 60; the speed α is the smallest that achieves it, and at least 0.05 (EIOPA-BoS-25-47, 9.4).

The UFR is set once a year as an expected real rate plus expected inflation (Art 47). For 2025 EIOPA calculated 3.20% (1.20% real plus 2% inflation), but the applicable rate moves only when the calculated one is more than 15bp away, and then by 15bp. So it stayed at 3.30% for 2025, and again for 2026 as at 31 Mar 2025.

The book's curve against Smith–Wilson

Level 3 valued the book on a simple curve: linear between pillars, 3.05% from 60 years and flat beyond. Smith–Wilson through the same rates at 1–10, 12, 15 and 20 years, with a UFR of 3.30%, needs α = 0.0914 to pass the 1bp test: the forward at 60 years is 3.237% against ω = 3.247%. Its zero rates sit above the book's from 20 years to just before 50, a fraction of a basis point below them from 50 to 61, and above again after that, rising towards the UFR where the book's curve stays flat. The one-year forwards in the chart cross more often: the book's curve is straight between pillars, so its forwards jump at each one:

Book's curveSmith–Wilson (today)2027 method, α 11%

Zero rate	20y	30y	50y	60y	100y
Book's curve	2.75%	2.80%	3.00%	3.05%	3.05%
Smith–Wilson	2.75%	2.84%	3.00%	3.04%	3.15%
2027 method, α 20%	2.75%	2.88%	3.04%	3.08%	3.17%
2027 method, α 11%	2.75%	2.85%	3.01%	3.06%	3.15%

From 2027: a first smoothing point and a slower convergence from 2027

The 2025 review replaces the method from 30 January 2027. The amended Art 77a calls the end of the market part the first smoothing point, 20 years for the euro, and requires the UFR's weight to be at least 77.5% 40 years past it. Beyond it, rates start from the last liquid forward rate and move towards the UFR at speed α: 11%, phased in from 20% (Delegated Regulation 2026/269, Arts 46 and 46a):

Year from 1 January	2027	2028	2029	2030	2031	2032
α	20.0%	18.2%	16.4%	14.6%	12.8%	11.0%

At α = 11% the UFR has 77.55% of the weight 40 years past the FSP, just above the minimum; at 20%, 87.5%. On the book the LLFR is 2.859% (continuous), below ω, so long rates rise towards the UFR from a lower start but still sit a little above Smith–Wilson's. With a curve that slopes more steeply at 20 years, or a UFR further from the market, the two methods differ far more.

The formal version

In full, with the 2027 method beside it (rates continuous; zero rate R(t), annual r(t)):

\begin{aligned} P(t) &= e^{-\omega t} + \sum_{j=1}^{N} \zeta_j\, W(t, u_j), \qquad \omega = \ln(1 + \text{UFR}), \qquad R(t) = \ln\big(1 + r(t)\big) = -\tfrac{1}{t}\ln P(t) \\ W(t, u) &= e^{-\omega (t+u)} \Big[\alpha \min(t,u) - e^{-\alpha \max(t,u)} \sinh\big(\alpha \min(t,u)\big)\Big] \\ \alpha &= \min\big\{\alpha \ge 0.05 : |f(60) - \omega| \le 0.0001\big\}, \qquad f(t) = -\tfrac{d}{dt} \ln P(t) \\[6pt] \text{2027:}\quad \bar f_h &= \omega + (\text{LLFR} - \omega)\, B(\alpha, h), \qquad B(\alpha, h) = \frac{1 - e^{-\alpha h}}{\alpha h}, \qquad h = t - \text{FSP} \\ R(\text{FSP} + h) &= \frac{\text{FSP}\cdot R(\text{FSP}) + h\, \bar f_h}{\text{FSP} + h}, \qquad 1 - B(0.11, 40) = 77.55\% \ \ge\ 77.5\% \end{aligned}

$u_j$ are the input maturities and $\zeta_j$ the weights that make the curve fit every input exactly (EIOPA-BoS-25-47, 9.7 and 9.10).
$\bar f_h$ is the average continuous forward from the FSP to $\text{FSP} + h$ . Delegated Regulation 2026/269 prints its formula as an image; the shape here is the one in EIOPA's 2020 Opinion, and its parameters reproduce the Directive's 77.5% test illustrative. The LLFR here is the 15-to-20-year forward on the book's curve.

Try it

Move the UFR and watch Smith–Wilson re-fit α to the 1bp test. Then set α by hand: too low and the test fails. Switch to the 2027 method and slide α from 20% to 11%. The annuities, about 14% of whose payments fall after 20 years, show what it is worth.

Try itExtrapolate the curve beyond 20 years

Method

Ultimate forward rate3.30%

Speed of convergence α

Book's curve (flat after 60)Smith–WilsonUFR

Annuity best estimate€19,915.8m

Change against the book's curve−€12.5m

α used0.0914 (calibrated)

Forward at 60 years less ω1.00bp (meets 1bp)

Zero rate at 100 years3.15%

Illustrative, € millions. Inputs are the book's zero rates at 1–10, 12, 15 and 20 years, with no credit risk adjustment (the book's curve already stands for the risk-free rate). The 2027 method starts from the Smith–Wilson curve to 20 years and takes the last liquid forward from 15 to 20 years. Only the annuities are revalued here; the lesson's table revalues the whole book.

Worked example

The curve's effect on RiskAtoms Insurance SE

€ millions at 31 December 2025. Each curve revalues every bond and every best estimate; the risk margin, the time value of the guarantees and the SCR are held illustrative.

From the book's curve to Smith–Wilson and the 2027 method

Step 1 of 4

1
How much sits on the extrapolated curve
Of the book's undiscounted liability cash flows, 6.0% fall after 20 years, almost all of them annuity payments. Every bond matures within 20 years, so the extrapolation does not touch the assets. They still move by −€0.28m: between its inputs Smith–Wilson interpolates slightly differently from the book's straight lines, so coupons at 11, 13, 14 and 16–19 years are discounted at slightly different rates. Almost all the change is in the best estimate.

Curve (illustrative)	Best estimate	Change	Own funds	Ratio
m3-8 curve (linear, flat after 60)	50,921.6	–	9,300.0	186.0%
Smith–Wilson, UFR 3.30% (illustrative, today)	50,909.0	−12.6	9,312.3	186.2%
2027 method, α 20% (illustrative)	50,900.8	−20.7	9,320.5	186.4%
2027 method, α 11% from 2032 (illustrative)	50,906.6	−15.0	9,314.7	186.3%

Regulatory lens

Valuation: Directive Art 75 and Delegated Regulation Arts 9, 10 and 12; the curve: Directive Arts 77(2) and 77a, Delegated Regulation Arts 43–47, and EIOPA's RFR technical documentation (EIOPA-BoS-25-47 from 30 June 2025; EIOPA-BoS-25-471 from 1 January 2026, with the same euro LLP, convergence point, tolerance and α floor). The 2027 method applies from 30 January 2027 as at 4 Nov 2026. The volatility and matching adjustments, which add to the curve, are in L4.6.10: Long-term guarantee measures and transitionals.

In the UK

Since 31 December 2020 UK firms have used the risk-free curves the PRA publishes each month, for sterling, the dollar, the euro and the Canadian dollar, with Smith–Wilson extrapolation parameters among its files. The EU's 2027 method does not apply there. The UK comparison is in L4.6.13: Solvency UK and global standards.

Art 45 The CRA's 10bp floor and 35bp cap, Art 47 the UFR's stability and no term premium, Art 77a the 77.5% test from 2027.

Spark Atoms

Spark AtomA rate that moves 15bp at a time

The UFR is meant to be stable: Art 47 says it changes only with long-term expectations. EIOPA's method moves the applicable rate by at most 15bp a year. For 2025 it calculated 3.20%, 10bp below the 3.30% in force, so nothing changed.

Spark AtomA curve with no market behind it

Past 20 years the euro curve is not a price anyone trades at. A UFR above market long rates makes the discount rate on long liabilities higher, and the liabilities smaller, than market swaps would. The 2025 review slows the convergence: with α at 11%, the market's last forward keeps more weight for longer.

Key takeaways

Art 75: assets at the price they could be exchanged for, liabilities at the price they could be transferred for, with no adjustment for own credit; IFRS only where consistent; goodwill at zero.
The euro curve is swap rates less a credit risk adjustment of 50% of the IBOR–OIS spread (10bp to 35bp), market-based to 20 years (Arts 43–45).
Beyond 20 years Smith–Wilson takes forwards to a UFR of 3.30%, within 1bp at 60 years, with the smallest α of at least 0.05.
From 30 January 2027: a first smoothing point of 20 years, then convergence from the last liquid forward at α = 11% (20% in 2027, falling to 11% by 2032), with a UFR weight of at least 77.5% at FSP + 40.
On RiskAtoms Insurance SE (illustrative): Smith–Wilson +€12.3m, 2027 method +€20.5m at α 20% and +€14.7m at 11%, against the book's curve.

Reaction

Check what stuck

Question 1 · objective 1

An insurer's own subordinated bond falls in price after the insurer is downgraded. On the Solvency II balance sheet the liability…

AIs not reduced for the insurer's own worse credit standingBFalls to the new market priceCIs valued at zeroDMoves to IFRS amortised cost

Question 2 · objective 2

The one-year average spread between euro IBOR swaps and overnight index swaps is 60bp. What credit risk adjustment applies, in bp?

bp

Question 3 · objective 2

What continuous rate ω corresponds to a UFR of 3.30%, in % (three decimals)?

%

Question 4 · objective 3

Under the 2027 method, what weight does the UFR carry in the average forward 40 years past the first smoothing point when α = 11%, in %?

%

Question 5 · objective 4

Moving RiskAtoms Insurance SE from the book's curve to the illustrative Smith–Wilson curve changes own funds by how much, in €m?

€m

Glossary

Exit value: Exit value: Art 75 values assets at the amount they could be exchanged for, and liabilities at the amount they could be transferred or settled for, between knowledgeable willing parties at arm's length.
Deep, liquid and transparent: Deep, liquid and transparent (DLT): a market where large trades do not move the price much, instruments can readily be bought or sold, and current trade and price information is public (Delegated Regulation Art 1(32) to (34)). The risk-free curve uses market rates only where they are DLT (Art 43).
Credit risk adjustment: Credit risk adjustment (CRA): a cut to swap rates for the bank credit risk in the floating leg: 50% of the one-year average spread between IBOR swaps and overnight index swaps, at least 10bp and at most 35bp (Art 45).
Last liquid point: Last liquid point (LLP): the longest maturity at which market rates are still used. For the euro it is 20 years (EIOPA RFR technical documentation, 9.2.1).
Ultimate forward rate: Ultimate forward rate (UFR): the long-run forward rate the extrapolated curve converges to: an expected real rate plus expected inflation, with no term premium (Art 47). 3.30% for the euro in 2025 and 2026.
Smith–Wilson: Smith–Wilson: the method EIOPA uses to fit the market inputs exactly and extrapolate beyond the LLP, with forward rates converging smoothly to the UFR.
Convergence speed α: α (alpha): the speed of convergence. EIOPA picks the smallest α of at least 0.05 that brings the forward rate within 1bp of the UFR at the convergence point (60 years for the euro).
First smoothing point: First smoothing point (FSP): from 2027 the longest maturity at which market rates are used, under the amended Art 77a; 20 years for the euro.
Last liquid forward rate: Last liquid forward rate (LLFR): from 2027 the forward rate at the end of the liquid part of the curve, which the extrapolation starts from.

Deep Dives & Origins

Deep Dives

L1.3.08: The yield curve: spot and forward rates: spot and forward rates, the language the curve is built in. Foundation.
L4.6.10: Long-term guarantee measures and transitionals: the volatility and matching adjustments that sit on top of this curve. Intermediate.

Origins

Directive 2009/138/EC, Arts 75, 77 and 77a, legislation.gov.uk copy of the EU text.
Commission Delegated Regulation (EU) 2015/35, Arts 1(32) to (34), 9, 10, 12, 43, 44, 45 and 47 as adopted (legislation.gov.uk); Art 46 (EIOPA Solvency II Single Rulebook).
Directive (EU) 2025/2, amended Art 77a; Commission Delegated Regulation (EU) 2026/269 of 29 October 2025, recital 11 and Arts 46, 46a and 2 (EUR-Lex; LawPlayer copy of the OJ text).
EIOPA, Technical documentation of the methodology to derive EIOPA's risk-free interest rate term structures, EIOPA-BoS-25-47 (June 2025) and EIOPA-BoS-25-471 (October 2025), sections 7, 9.2, 9.4, 9.7 and 9.10.
EIOPA, EIOPA publishes the Ultimate Forward Rate (UFR) for 2026, 31 March 2025; Report on the calculation of the UFR for 2025, EIOPA-BoS-24-099, 26 March 2024; monthly RFR technical information release, May 2025.
Prudential Regulation Authority, Technical information for Solvency II firms (web page, read November 2026).

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