Premium risk and the underwriting cycle · RiskAtoms Academy

By the end you will be able to

1Measure rate adequacy from rate changes and claims inflation, and read it against accident-year combined ratios.
2Describe the underwriting cycle, soft and hard markets and their causes, using the book's 2016–2025 history.
3Compute premium and reserve risk in the Solvency II standard formula in outline: volumes, segment standard deviations, correlations and 3σV.
4Explain where the factor 3 comes from and what the standard formula leaves out.

Why it matters

In 2025 RiskAtoms Insurance SE's markets raised prices by 1% while the cost of a claim rose 3%. Nothing went wrong, and the accident year still ran at a 91.0% combined ratio. But the margin the hard market had rebuilt started to leak away, and every insurer that has lived through a soft market knows how that story goes. Premium risk is the chance that next year's price is wrong, and the cycle is how a whole market gets it wrong together.

Core explanation

Prices, costs and the cycle between them

Illustrative dataRiskAtoms Insurance SE's insurance book

The same composite European insurer as Modules 1.8 and 2.2, in € millions at 31 December 2025: four non-life lines with ten years of claims triangles, a 1.2 million vehicle-year motor pricing data set, a 17,000-event catastrophe set, its reinsurance, its life books, its assets and an outline of its Solvency II standard formula. Every policy, claim, event and figure is made up for teaching and seeded so it never changes; none is any insurer's data. Every lesson in Module 3.8 uses this book, so the figures agree.

This lesson uses the book's market cycle (one rate index and one claims-inflation index for all lines, 2016–2025), its accident-year results by line (incurred chain-ladder ultimates, 2025's expense ratios in every year) and its standard-formula outline, which simplifies the premium volume to the larger of next year's and this year's net earned premium and sets the geographical diversification factor to its default.

Rate adequacy

Premium risk starts with two numbers each year: the rate change on renewals and claims inflation. Compound each into an index and divide one by the other, and you have rate adequacy: how much claims cost the premium now buys, relative to a reference year. Below 1, the same policy is underpriced relative to 2025.

Premium-rate indexClaims-cost indexRate adequacy

The cycle on the book

Step through the ten years. The book's cycle is illustrative, but its shape follows what economists describe: prices drift down while capital is plentiful, then jump after losses or a shock to costs.

TimelineRiskAtoms Insurance SE's markets, 2016 to 2025

1 / 10

2016

Soft: Ample capital; rates drift down

Rates −2.0%, claims inflation +1.5%. Rate adequacy 1.091 (2025 = 1). Accident-year gross combined ratio 84.9%, on €6,351m of written premium.

Three phases show. In the soft market of 2016–2019 rates fell 7.3% while claims costs rose 6.6%: adequacy dropped 13%. In the hard market of 2020–2024 prices rose, but the 2022 inflation shock (9%) briefly outran them: adequacy fell to 0.959 before rates of 9% and 8% caught up. From 2021 to 2024 rates rose 22.4% against claims costs of 20.7%. In 2025 the market softened: adequacy fell 1.9% in one year.

Accident-year combined ratios

The accident-year combined ratio puts each year's claims, at their current estimate, against that year's premium. It tracks adequacy, with noise from catastrophes:

All linesExcluding property

Accident-year gross combined ratio by line, %. Above 100% in red.
Line	2016	2017	2018	2019	2020	2021	2022	2023	2024	2025
Motor liab.	89	92	94	94	92	96	97	94	98	95
Own damage	86	84	85	93	94	87	98	91	86	91
Property	83	78	98	86	83	104	83	91	81	83
Gen. liab.	82	90	92	91	91	88	99	94	85	101

Across the ten years the correlation between rate adequacy and the combined ratio is −0.60, and −0.78 without property, whose catastrophe years (the 2021 flood took property to 104.2%) are noise as far as pricing is concerned. Note the years on the right are the least developed: their ultimates are chain-ladder estimates, and motor liability's recent years have just been strengthened (L3.8.03: Reserve risk).

Why markets cycle

Economists offer several reasons, and they combine. Capital is costly to raise, so after large losses capacity shrinks and prices rise until profits rebuild it; then capital flows back in and competition pushes prices down. Prices are set on data that is a year or two old, so they lag claims costs, as in 2022. Interest rates matter too: when investment returns are high, insurers accept thinner underwriting margins. And because loss estimates adjust with a lag, prices tend to overshoot in both directions.

Claims inflation is premium risk and reserve risk at once

A rise in the cost of claims hits next year's business, which was priced before it, and every claim still open. On the book it shows up twice: as 2022's accident-year combined ratio of 92.2% with adequacy at 0.959, and as the 2025 strengthening of motor liability's most recent accident years, whose claims cost more than was assumed when they were reserved (L3.8.03: Reserve risk).

Premium risk in the standard formula, in outline

Solvency II measures premium risk and reserve risk together. For each segment it takes a volume and a standard deviation from Annex II of the Delegated Regulation, combines premium and reserve within the segment, then combines segments with a correlation matrix (L3.1.06: Many assets and the covariance matrix), and holds three standard deviations. On the book:

€m. V prem: the larger of next year's and 2025's net earned premium. V res: the discounted best estimate of claims, net of reinsurance. Total row: σ_nl and the total volume V_nl.
Segment	V prem	σ prem (× NP)	V res	σ res	σ segment	σ × V
Motor liability	2,338.5	8.0%	2,712.1	9%	7.41%	374.5
Motor own damage	1,620.0	8.0%	238.7	8%	7.54%	140.1
Property (home and commercial)	2,712.7	6.4%	990.2	10%	6.46%	239.0
General liability	1,528.8	11.2%	2,606.0	11%	9.69%	400.7
Total					5.74%	14,746.9

The standard deviations for premium are not the history of this book: the observed ten-year standard deviation of motor liability's accident-year combined ratio is 2.7% against Annex II's 10%, and general liability's 5.7% against 14%. Annex II is calibrated on data from many insurers, so it carries more than one firm's ten years of noise; a ten-year history also cannot show whether the expected loss ratio itself is wrong. The adjustment factor for non-proportional reinsurance trims three of them.

The formal version

Rate adequacy and the expected loss ratio. With renewal rate changes $r_t$ and claims inflation $i_t$ , indexed to a reference year:

\begin{aligned} R_t &= \prod_{u \le t}(1 + r_u), \qquad C_t = \prod_{u \le t}(1 + i_u), \qquad A_t = \frac{R_t}{C_t} \\ \text{LR}_t &= \frac{\text{LR}_{\text{ref}}}{A_t}, \qquad \text{COR}_t = \text{LR}_t + \text{expense ratio} \end{aligned}

Why 3. If a segment's result is lognormal with mean 1 and coefficient of variation $v$ (M-C03: Variance and standard deviation), its 99.5% point lies $k(v)$ standard deviations above the mean:

\begin{aligned} s^2 &= \ln(1 + v^2), \qquad q_{99.5\%} = e^{-s^2/2 + z_{0.995}\,s}, \qquad k(v) = \frac{q_{99.5\%} - 1}{v} \\ k(5\%) &= 2.719 \\ k(8\%) &= 2.806 \\ k(10\%) &= 2.866 \\ k(14\%) &= 2.985 \\ k(20\%) &= 3.166 \end{aligned}

For the standard deviations the standard formula ends up with, a few to fifteen percent, a lognormal's 1-in-200 point is about three standard deviations out ( $z_{0.995} = 2.5758$ for a normal). The regulation simply writes 3. Where a segment is much more volatile, 3σ understates a lognormal tail.

Try it

Pick a line and price 2026. If claims inflation returns to 2022's 9%, what rate change does motor liability need just to break even, and what combined ratio does a rise of 1%, as in 2025, leave it with? How far above 100% does general liability's 1-in-200 reading go, and why is it higher than motor liability's?

Try itNext year's combined ratio, and a 1-in-200 year

Line

Rate change on renewal, 2026+1.0%

Claims inflation, 2026+3.0%

Expected combined ratio95.0%

1-in-200 reading: expected + 3σ125.0%

The thin line marks 100%.

Expected loss ratio71.4%

Expense ratio (2025)23.6%

σ of premium (Annex II, gross)10%

Rate change that breaks even−5.6%

Loss ratio = 2025's expected gross loss ratio × (1 + inflation) ÷ (1 + rate change); exposure and mix held. For motor liability, property and liability the standard formula then multiplies σ by an 80% adjustment factor for non-proportional reinsurance (Art 117(3)). Illustrative data.

Worked example

Premium and reserve risk on the book

The standard formula in outline, € millions, at 31 December 2025.

From four segments to €2,541m

Step 1 of 5

1
Motor liability's volumes
$\begin{aligned} V_p &= \max(P_{2026}, P_{2025}) = 2{,}338.5, \qquad \sigma_p = 10\% \times 80\% = 8\% \\ V_r &= 2{,}712.1, \qquad \sigma_r = 9\% \end{aligned}$

Regulatory lens

The formulas above are Articles 115 to 117 of Delegated Regulation (EU) 2015/35, with the standard deviations in Annex II and the segment correlations in Annex IV; Article 114 combines premium and reserve risk with catastrophe risk at 0.25. EIOPA's 2011 calibration report describes the target as a 99.5% level over one year. The full regulation also counts future premiums on multi-year contracts in the volume, allows a geographical diversification factor, and lets firms replace the market-wide σ with undertaking-specific parameters under conditions. All of that, and the 2025 review, is in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomThe hard market after the pandemic

Swiss Re Institute economists, writing in August 2021, cited Marsh's figures for global commercial insurance prices: up 22% in the fourth quarter of 2020 and 18% in the first quarter of 2021 against a year earlier. They described the episode as a hard market driven by several things at once: catastrophe losses, low interest rates, uncertainty over pandemic claims and rising liability costs.

Spark AtomProfits can lag the price

The book's accident year 2019 still shows a 90.0% combined ratio at the bottom of the soft market. A market that judges itself by this year's reported profit sees the damage only once those years develop, which is one of the institutional lags that economists blame for the cycle.

Key takeaways

Rate adequacy is the rate index divided by the claims-cost index: price increases below claims inflation make the book worse, not better.
On the book the soft market cut adequacy by 13% from 2016 to 2019; the 2022 inflation shock outran the hard market's prices; 2025's +1% against +3% started the next softening.
Accident-year combined ratios move with adequacy (correlation −0.78 excluding property), with catastrophes and reserve development as noise.
The standard formula holds 3σV: per segment σ combines premium and reserve risk at a correlation of 0.5, segments combine with CorrS. On the book σ_nl = 5.74% on €14,747m, so €2,541m.
The 3 is close to a lognormal's 1-in-200 point for a coefficient of variation of 5% to 14% (2.72 to 2.98 standard deviations).

Reaction

Check what stuck

Question 1 · objective 1

Rates rise 4% on renewal while claims inflation is 9%. Starting from a rate adequacy of 1, what is it a year later (to three decimals)?

Question 2 · objective 2

On the book, accident year 2019 had a combined ratio of 90.0% and 2016 84.9%. Both are below 100%. Why call 2019 the bottom of a soft market?

ARate adequacy had fallen 13% since 2016: the same book was earning much less margin, and only good luck with claims or prior years could hide itBBecause the combined ratio was above 100%CBecause premium volumes grew fastest that yearDBecause the chain ladder overstates recent years

Question 3 · objective 3

A motor liability segment has 2,000 of net premium next year. With σ = 10% and the 80% adjustment factor, what is premium risk on its own (3σV, ignoring reserves), in €m?

€m

Question 4 · objective 3

In one segment σ_prem·V_prem = 160 and σ_res·V_res = 240. What is σ_s·V_s, the segment's combined standard deviation in €m (premium and reserve correlated 0.5)?

€m

Question 5 · objective 4

Why does a correlation of 0.5 appear as σ_p V_p σ_r V_r with no 2 in front in Article 117?

ABecause 2 × 0.5 = 1: the usual cross term 2ρab with ρ = 0.5BBecause premium and reserve risk are independentCBecause the regulation leaves out the cross termDBecause σ is already squared

Glossary

Premium risk: The risk that premiums for business written or renewed in the coming year do not cover its claims and costs.
Rate change: The average change in price on renewing policies, for the same cover and the same risk.
Claims inflation: The change in the cost of a like-for-like claim: parts, labour, rebuilding costs, court awards.
Rate adequacy: Premium rates relative to claims costs, as an index. Here 2025 = 1: below 1 the price buys less cover than in 2025.
Underwriting cycle: The tendency of insurance prices and profits to swing between soft markets (prices falling, cover easy to buy) and hard markets (prices rising, cover scarce).
Accident-year combined ratio: Claims and expenses of the policies covering accidents in one year, over that year's earned premium, with claims at their latest estimate of ultimate.
Volume measure: In the standard formula: for premium risk, the larger of next year's and last year's net earned premium (plus future premiums on longer contracts); for reserve risk, the net best estimate of the claims reserve.
Adjustment factor for non-proportional reinsurance: A factor of 80% on the premium standard deviation for motor vehicle liability, fire and other damage, and general liability (Art 117(3)).

Deep Dives & Origins

Deep Dives

M. A. Weiss, “Underwriting cycles: a synthesis and further directions”, Journal of Insurance Issues 30(1), 2007, pp. 31–45. Intermediate.
EIOPA, Calibration of the Premium and Reserve Risk Factors in the Standard Formula of Solvency II, EIOPA-11/163, December 2011. Advanced.

Origins

Commission Delegated Regulation (EU) 2015/35, Articles 114–117 and Annexes II and IV, version at 31 December 2020 (legislation.gov.uk; EIOPA Single Rulebook).
EIOPA, Calibration of the Premium and Reserve Risk Factors in the Standard Formula of Solvency II, EIOPA-11/163, 12 December 2011.
M. A. Weiss, “Underwriting cycles: a synthesis and further directions”, Journal of Insurance Issues 30(1), 2007.
T. Holzheu and J. Finucane (Swiss Re Institute), “Understanding the current underwriting cycle: difficult times extend hard market”, Reinsurance News, Society of Actuaries, August 2021.

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