How insurers take risk · RiskAtoms Academy

By the end you will be able to

1Describe how an insurer takes risk: premiums in advance, claims later, and the float in between.
2Compute the combined ratio, the investment return on the float and the break-even combined ratio for a line of business.
3Map an insurer's risks to the Solvency II standard-formula modules and read their sizes off a real-looking book.
4Find your way around RiskAtoms Insurance SE's book, which every lesson in this module uses.

Why it matters

In 2025 RiskAtoms Insurance SE wrote €9,000m of non-life premiums. It had the money before a single claim was made. At the end of the year it still held €7,497m for claims from accidents that had already happened, some of them in 2016, and it expects to be paying them until 2034. An insurer sells a promise whose cost it learns later, and it lives on the money in between. This module is about everything that can make that cost larger than the price.

Core explanation

Paid first, tested later

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 2025 income statement (Module 1.8's figures, unchanged), its claims reserves by line and its standard-formula outline. The counterparty default charge is stated rather than computed, and the deferred-tax adjustment is set so the SCR equals Module 1.8's €5,000m.

A manufacturer buys materials, makes the product, then sells it: it knows its costs before it sets its price. An insurer works the other way round. This inverted production cycle is the source of every risk in this module (L1.2.03: Insurers: life and non-life introduced it). Follow one year of motor policies through the business.

ExploreA year of policies, from premium to the last claim

Click a box to see what happens there.

Write

The insurer sets a price from past claims (L3.8.02: Non-life underwriting and pricing) and collects it up front. In 2025 RiskAtoms Insurance SE wrote €9,000m gross and ceded €900m to reinsurers. If the price is too low, nothing can be done about it until the policies renew: premium risk.

The combined ratio, line by line

The combined ratio (L1.8.07: Key ratios for banks and insurers) measures the insurance on its own. In 2025 it was 95% net of reinsurance: claims of €5,200m and expenses of €2,400m on €8,000m of premium, an underwriting profit of €400m. Before reinsurance it was 90.3%. The book splits the year by line:

2025, €m. Prior years: the change in the cost of earlier accident years, inside the year's claims (+ is a strengthening).
Line	GWP	Net earned	Net claims	Expenses	Net combined ratio	Prior years
Motor liability	2,400	2,281.5	1,768.0	560	102.0%	+116.3
Motor own damage	1,600	1,580.5	1,058.1	380	91.0%	−7.4
Property (home and commercial)	3,400	2,646.5	1,411.8	1,020	91.9%	−17.2
General liability	1,600	1,491.5	962.1	440	94.0%	−147.1
Total	9,000	8,000	5,200	2,400	95.0%	−55.4

Motor liability lost money on its insurance: a net combined ratio of 102.0%, because claims inflation pushed up the cost of earlier years' injury claims by €116.3m. General liability released reserves; the total came out at a release of €55.4m. A good-looking total can hide a line in trouble.

The float

The float is mostly the claims reserves: money owed to claimants but not yet paid. Its size depends on the tail of each line.

End-2025, €m, gross and undiscounted. Mean term: the payment-weighted average time until the reserve is paid, counting each year's payments at its end.
Line	Tail	Claims reserve	Net earned premium	Reserve ÷ premium	Mean term (years)
Motor liability	medium	3,019.1	2,281.5	1.32	2.6
Motor own damage	short	244.5	1,580.5	0.15	1.2
Property (home and commercial)	short	1,204.9	2,646.5	0.46	1.7
General liability	long	3,028.6	1,491.5	2.03	3.1
Total		7,497.1	8,000	0.94	2.6

Own damage holds 0.15 years of premium for about 1.2 years; general liability holds 2.03 years for about 3.1. Invested at the book's illustrative 2.08% (the zero rate at the whole reserve's mean term), the float earns about €156m a year. That is why a liability insurer can run a combined ratio above 100% and still make money, and why the same insurer is hurt twice when rates fall: lower income, and reserves that are worth more once discounted.

The risks, measured in capital

Solvency II asks how much own funds the insurer could lose in a 1-in-200 year, risk by risk (L2.7.05: Solvency I to Solvency II). The standard formula breaks that into modules. On RiskAtoms Insurance SE's book the three insurance modules (non-life, life and health) are 59% of the total before diversification; the rest is market and counterparty risk.

ExploreThe risks on RiskAtoms Insurance SE's balance sheet

Each bar is a Solvency II standard-formula module, in €m of capital. Click one to open it.

Market risk

Equity€1,582m
Spread€1,429m
Property€750m
Interest rate (up binds)€393m

The largest module. The float and the life reserves are invested in €28bn of government bonds, €26bn of corporate and covered bonds, shares and property, and those can fall. Interest rate risk is small because assets and liabilities are roughly matched. Covered in L3.8.11: Insurer asset-liability management and L3.8.12: Insurer investment risk and the ORSA.

Sum of the five modules	€8,595m
Diversification	−€2,829m
Basic SCR (BSCR)	€5,766m
Operational risk	€439m
Loss-absorbing capacity of deferred taxes	−€1,204m
SCR	€5,000m

Sort itWhich module takes the hit?

Each event hurts RiskAtoms Insurance SE. Sort it into the standard-formula module that holds capital for it.

Hailstorms in Southern Germany damage 25,000 insured cars and roofs in an afternoon.
New mortality data show pensioners living a year longer than the annuity reserves assume.
Listed shares held against the reserves fall 30%.
A reinsurer that owes €80m on last year's storm becomes insolvent.
A claims system migration double-pays thousands of motor claims before anyone notices.
Inflation in car parts and labour makes last year's open claims cost 8% more than reserved.

The formal version

The underwriting year. With net earned premium $P$ , net claims $C$ and expenses $E$ , and reserves $R$ earning a yield $y$ :

\begin{aligned} \text{COR} &= \frac{C + E}{P} \\ \text{result} &= \underbrace{P - C - E}_{\text{underwriting}} + \underbrace{y\,R}_{\text{float}} = P\Big[(1 - \text{COR}) + y\,\frac{R}{P}\Big] \\ \text{break-even:}\quad \text{result} = 0 &\iff \text{COR}^{*} = 1 + y\,\frac{R}{P} \end{aligned}

On the book, with $R/P = 0.937$ and $y = 2.08\%$ :

\begin{aligned} (1 - 0.95) + 0.0208 \times 0.937 &= 0.05 + 0.0195 = 0.0695 \\ \text{COR}^{*} &= 1 + 0.0208 \times 0.937 = 102.0\% \end{aligned}

The float is worth about 2.0 points of combined ratio. This is a simplification: the true yield depends on what the reserves are invested in, and the reserves are themselves uncertain.

The capital. Each module $i$ gives a 1-in-200 loss $S_i$ . The standard formula combines them with a fixed correlation matrix $C$ , the same square-root rule as portfolio volatility (L3.1.06: Many assets and the covariance matrix):

\begin{aligned} \text{BSCR} &= \sqrt{\textstyle\sum_{i}\sum_{j} C_{ij}\,S_i\,S_j} = \sqrt{s^{\top} C\, s} \\ \text{SCR} &= \text{BSCR} + \text{Op} + \text{Adj} \\ &= 5{,}765.9 + 438.6 -1{,}204.4 = \text{€}5{,}000\text{m} \end{aligned}

$Adj$ here is the loss-absorbing capacity of deferred taxes (the book has no technical-provision adjustment). The matrix and the module calculations are the subject of L3.8.12: Insurer investment risk and the ORSA and M4.6: Solvency II and insurance regulation.

Try it

Start from the book's 2025 figures. Push the combined ratio above 100% and see how much yield it takes to stay profitable; then switch the float from the whole book to short-tail own damage and to long-tail liability.

Try itUnderwriting plus the float

Combined ratio95.0%

Yield on the float2.08%

Reserves ÷ premium (years)0.94

Underwriting margin5.00%

Investment return on the float1.96%

Total, per € of premium6.96%

Break-even combined ratio102.0%

Result on €8,000m of premium€556m

Of which underwriting€400m

Of which the float€156m

Before tax and before the cost of capital. The float earns the yield on reserves of the chosen size; the book's 2025 starting point is a 95% combined ratio, reserves of 0.94 years of premium, and the illustrative curve's 2.08% at the reserve's mean term. Illustrative data.

Worked example

Where the €1,050m comes from, and what it needs

RiskAtoms Insurance SE, 2025, € millions. Module 1.8's figures, read as a risk manager would.

From premium to profit to capital

Step 1 of 5

1
Underwriting
$\begin{aligned} P &= 9{,}000 - 900 - 100 = 8{,}000 \\ P - C - E &= 8{,}000 - 5{,}200 - 2{,}400 = \text{€}400\text{m} \\ \text{COR} &= (5{,}200 + 2{,}400) / 8{,}000 = 95\% \end{aligned}$

The book

Every later lesson opens one part of this book: motor pricing data (L3.8.02: Non-life underwriting and pricing), claims triangles (L3.8.03: Reserve risk), ten years of the market cycle (L3.8.04: Premium risk and the underwriting cycle), the catastrophe event set (L3.8.05: Catastrophe risk), the life books (L3.8.06: Mortality risk to L3.8.09: Expense, health and morbidity risk), the reinsurance programme (L3.8.10: Reinsurance in depth), and the assets, ALM and capital (L3.8.11: Insurer asset-liability management, L3.8.12: Insurer investment risk and the ORSA). The figures agree from one lesson to the next.

Regulatory lens

Solvency II (Directive 2009/138/EC) sets the SCR as the 99.5% value at risk of basic own funds over one year (Article 101(3)), and requires it to cover at least non-life, life and health underwriting risk, market risk, credit risk and operational risk (Article 101(4)). The standard formula splits those into the modules above; the detailed factors are in Delegated Regulation (EU) 2015/35. A review adopted in 2024 (Directive (EU) 2025/2) applies from 30 January 2027. How the modules are calculated, the risk margin, internal models and supervision are in M4.6: Solvency II and insurance regulation; how the same insurance contracts are reported in the accounts under IFRS 17 is in M4.8: Insurance accounting: IFRS 17 (overview).

Spark Atoms

Spark AtomLosing money on every policy, on purpose?

With reserves of 2.03 years of premium and a 2.08% yield, general liability breaks even at a combined ratio of about 104.2%. Long-tail insurers can price above 100% and still profit, which works until rates fall or the reserves prove too low. Then both supports go at once.

Spark AtomA third of the capital is diversification

The five modules add up to €8,595m, but the BSCR is €5,766m: a saving of 33%. Most of it comes from the modest correlations of 0.25 between most pairs of modules. Some comes from two zeros: the standard formula treats life and health underwriting as uncorrelated with non-life. At 0.25 the BSCR would be €6,033m, €267m more. A composite needs less capital than a life insurer and a non-life insurer apart.

Key takeaways

An insurer is paid before it knows its costs; premium, reserve and catastrophe risk are the three ways the cost can exceed the price.
The combined ratio measures the insurance alone: 95% net and 90.3% gross on the book in 2025, with motor liability at 102.0% after a €116m reserve strengthening.
The float, here €7,497m of claims reserves (0.94 years of premium), earns an investment return; break-even is a combined ratio of 1 + yield × reserves ÷ premium.
Solvency II measures each risk as a 1-in-200 loss and combines the modules with a correlation matrix: €8,595m becomes a BSCR of €5,766m and an SCR of €5,000m.
Insurance risk is 59% of the stand-alone capital on the book; the largest modules are market (€3,391m), non-life (€2,801m) and life (€2,054m).

Reaction

Check what stuck

Question 1 · objective 2

A motor insurer earns €2,000m of net premium, pays net claims of €1,400m and has expenses of €560m. What is its combined ratio, in %?

%

Question 2 · objective 2

A liability insurer holds reserves of 1.5 times a year's premium and earns 3% on them. At what combined ratio does it break even before tax, in %?

%

Question 3 · objective 1

RiskAtoms Insurance SE's general liability reserve is 2.03 years of premium; own damage's is 0.15. Why the difference?

ALiability claims take years to be reported, argued and paid; car repairs are paid within monthsBLiability premiums are much largerCOwn damage is reinsured more heavilyDLiability is priced with a higher combined ratio

Question 4 · objective 3

The five Solvency II modules of RiskAtoms Insurance SE add up to €8,595m; combined with the correlation matrix they give a BSCR of €5,766m. What share of the sum is saved by diversification, in %?

%

Question 5 · objective 3

Which of these is not insurance risk in the Solvency II sense, even though it hits an insurer?

ABond spreads widen and the float's corporate bonds fall in valueBA windstorm damages 40,000 insured homesCAnnuitants live longer than the pricing assumedDCourt awards for injury claims rise, so old motor claims cost more

Glossary

Inverted production cycle: The insurer's cycle of production runs backwards: it is paid before it knows its costs, which arrive months or decades later.
Float: Money an insurer holds between collecting premiums and paying claims, mostly the claims reserves. It is invested meanwhile.
Combined ratio: Net claims plus expenses, divided by net earned premium. Below 100% the insurance itself makes a profit.
Break-even combined ratio: The combined ratio at which underwriting losses are exactly paid for by the investment return on the float: 1 + yield × reserves ÷ premium.
Short-tail and long-tail: How long claims take to be reported and paid. Own damage is short-tail (paid within a year or two); liability is long-tail.
Solvency Capital Requirement (SCR): Solvency Capital Requirement: under Solvency II, the capital that covers a 1-in-200 loss of own funds over one year.
Standard-formula module: One block of the Solvency II standard formula: market, counterparty default, life, health and non-life. Each has its own sub-modules.
Basic SCR (BSCR): Basic SCR: the five modules combined with the standard formula's correlation matrix, before operational risk and tax.
Diversification: The capital saved because the modules' 1-in-200 losses are not expected to happen together: the sum of the modules less the BSCR.

Deep Dives & Origins

Deep Dives

Swiss Re Institute (T. Holzheu and J. Finucane), “Understanding the current underwriting cycle: difficult times extend hard market”, Reinsurance News, Society of Actuaries, August 2021: how capital, losses and interest rates move insurance prices. Intermediate.
Directive 2009/138/EC (Solvency II), Articles 100 to 111: the SCR and its standard formula. Advanced.

Origins

Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 (Solvency II), Article 101(3) and (4), as adopted (legislation.gov.uk).
Commission Delegated Regulation (EU) 2015/35, Article 114 (non-life underwriting risk module), version at 31 December 2020 (legislation.gov.uk).
Module 1.8 and the Module 3.8 insurance book (RiskAtoms Academy), illustrative figures for RiskAtoms Insurance SE at 31 December 2025.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.