Expense, health and morbidity risk · RiskAtoms Academy

By the end you will be able to

1Apply the Solvency II expense stress to each book and explain why the inflation half grows with duration.
2Model income protection as a healthy, disabled and dead Markov chain and project it one year at a time.
3Compute the disability-morbidity stress and explain why recovery matters as much as incidence.
4Describe how the health module is built and where the book's charges sit in it.

Why it matters

RiskAtoms Insurance SE spends about €69m a year administering its life policies. Ten percent more, rising one point faster each year, would cost €111.4m today. Its income protection book collects €36m of premiums a year, and the disability stress on it needs €191.3m of capital: more than five years of premiums. Neither risk is about people dying.

Core explanation

Costs, claims and recoveries

Illustrative dataRiskAtoms Insurance SE's life and health book

The composite European insurer of Modules 1.8 and 2.2, reporting in € millions at 31 December 2025: 200,000 annuitants, 400,000 term assurance policies, three generations of savings contracts with guaranteed rates, and a small income protection book. One unisex Gompertz–Makeham mortality table with 1.5% a year improvement and an illustrative euro curve (not EIOPA's) value them all. Every policy, assumption and figure is made up for teaching and fixed in code; none is any insurer's data. Every lesson in Module 3.8 uses the same book, so the figures agree.

This lesson uses every life book and the small income protection book: 120,000 healthy lives aged 45, insured to 65 for €12,000 a year while disabled, plus 3,000 claimants aged 50 already in payment. The simplifications that matter here: expenses are a flat amount per policy rising with 2% inflation, with no fixed costs; incidence rises with age and recovery is a flat 18% a year whatever the claim's length; disabled lives die at twice the healthy rate; a claimant in payment who recovers leaves the model; and there is no health lapse, medical expense cover or health catastrophe charge.

Expense risk

A best estimate includes every future cost of running the policies: administration, systems, claims handling, investment management. Expense risk is the risk that those costs, or their inflation, come in higher. Article 140 stresses both at once: expenses 10% higher and expense inflation one percentage point higher, permanently.

Book	€ a policy	Policies	Year 1	PV of expenses	+10%	+1pt inflation	Both
Annuities	€40	200,000	€8.0m	€112.8m	€11.3m	€11.5m	€24.0m
Term	€40	400,000	€16.0m	€159.9m	€16.0m	€10.8m	€27.9m
Savings	€60	750,000	€45.0m	€387.6m	€38.8m	€18.9m	€59.5m
Income protection	€30	123,000	€3.7m	€63.9m	€6.4m	€6.0m	€13.0m
Life books			€69.0m				€111.4m

Three things stand out. The +10% half is exactly a tenth of the present value of expenses, because expenses enter the best estimate in proportion. The inflation half depends on how long the book runs: on the annuities it is larger than the +10% half (€11.5m against €11.3m), because pensions are paid for decades and a point of inflation compounds; on the shorter term book it is smaller. And the two halves interact: both together cost a little more than their sum (€1.2m more on the annuities). The life module's expense charge is €111.4m; the income protection row, €13.0m, goes to the health module.

Illustrative dataLesson extraFixed costs and the closed book

The book treats every euro of expense as a per-policy cost, so expenses shrink with the book. Real costs are partly fixed: the systems, the actuaries and the office stay when policies leave. Suppose half of the savings book's €60 a policy is fixed (€22.5m a year) and the book is closed to new business, with 7% of policies leaving each year.

Year	Policies	Cost a policy (today's money)
0	750,000	€60.00
5	521,766	€73.12
10	362,987	€91.99
15	252,526	€119.10

By year 15 the cost a policy has doubled before any inflation. This is why closed books are sold or merged onto shared platforms, and why expense risk is largest for an insurer that stops writing new business. The figures are made up for the lesson.

Income protection as a chain

Income protection pays an income while the insured cannot work. Each year a life is in one of three states: healthy (paying premiums), disabled (receiving €12,000 a year) or dead. Two rates drive the book: the incidence rate, how often healthy lives fall ill, and the recovery rate, how often claimants get better. Together with mortality they fill a transition matrix, and the book is a Markov chain (M-C13: Markov chains and transition matrices). At 45, in year 1:

From \ to	Healthy	Disabled	Dead
Healthy	99.53%	0.30%	0.17%
Disabled	17.94%	81.72%	0.35%
Dead	0.00%	0.00%	100.00%

Incidence at 45 is 0.30% (0.2% plus 0.02% for each year over 40) and q₄₅ is 0.173%. A claimant recovers with probability 17.94%, which is 18% of the 99.65% who survive. Multiply the 120,000 actives by this matrix and a year later 119,433.3 are healthy, 359.4 disabled and 207.4 dead. Repeat with each year's matrix and the book runs to 65:

On claim (best estimate)On claim (Art 156 stress)The 3,000 earlier claimants still on claim

Claims build up as the actives age and incidence rises: 3,447 on claim in the last year, against 359 after the first. The 3,000 claimants already in payment run off quickly through recovery. Benefits are paid to whoever is disabled; premiums less expenses come from whoever is healthy. Discounted, the book's best estimate is a liability of €49.9m: claims in payment and claims to come, plus costs, less future premiums.

What moves a disability book: guess the cause, then reveal

The disability stress

Article 156 stresses income protection three ways at once: incidence 35% higher for the next 12 months and 25% higher after that, and recovery 20% lower where it is below 50%. The stressed year-1 matrix at 45:

From \ to	Healthy	Disabled	Dead
Healthy	99.42%	0.40%	0.17%
Disabled	14.35%	85.30%	0.35%
Dead	0.00%	0.00%	100.00%

Piece of the stress	Change in BE
Incidence +35% in year 1	€7.2m
Incidence +25% after year 1	€91.0m
Recovery −20%	€80.5m
All three together	€191.3m

Year-1 incidence barely matters: it adds a year's worth of extra claims. Incidence after year 1 and recovery cost similar amounts, and all three together they cost €12.6m more than the pieces add to, because the extra claimants from higher incidence also recover more slowly. Slower recovery works on every claim, old and new: a claimant aged 50 expects 4.18 years of benefit at 18% recovery and 5.17 years at 14.4%, a rise of 24%.

The health module

Solvency II gives health its own module with three parts: SLT health, for business written like life insurance; NSLT health, for business written like non-life; and health catastrophe. Income protection that runs to 65 on a life basis is SLT health. Its sub-modules mirror the life module's: mortality, longevity, disability-morbidity, lapse, expense and revision. The book charges two of them and aggregates with their correlation of 0.5:

\text{SCR}_{\text{health}} = \sqrt{191.3^2 + 13.0^2 + 2 \times 0.5 \times 191.3 \times 13.0} = 198.1\text{m}

That is small beside the life module's €2,054.0m, and it enters the basic SCR with a correlation of 0.25 to life. A specialist health insurer would see it the other way round. The life module's own disability sub-module is zero here, because no life policy carries disability cover.

Sort itWhich sub-module?

For each event, pick the charge that captures it.

A five-year IT replacement runs over budget and wage inflation rises.
After a recession, income protection claimants take longer to return to work.
Healthy income protection policyholders cancel their cover after a premium rise.
A stadium collapse injures hundreds of insured lives at once.
A new diagnosis becomes common and doubles claims for one condition.

The formal version

The expense stress. With n_t policies in force in year t, expense E a policy today and inflation i, the present value of expenses and its stressed version are

\begin{aligned} \text{PV}_E &= \sum_t v^t\, n_t\, E\,(1+i)^{t-1} \\ \text{PV}_E^{\text{stressed}} &= \sum_t v^t\, n_t\, 1.1\,E\,(1+i+0.01)^{t-1} \end{aligned}

The first factor scales everything by 1.1; the second grows with t, so long books feel it most. For one €40 expense paid for 30 years, discounted at a flat 2.5% with nobody dying, the present value goes from €1,087 to €1,376: up 27%, well over the 10% headline.

Try it

Slow recovery to 0.8× and then raise incidence to 1.25× instead: the two land close together. Then pick the Art 156 preset and look at the matrix: four small cells move (incidence, recovery and the two diagonal cells that keep each row summing to one), and the best estimate jumps.

Try itAn income protection book as a Markov chain

Assumptions

Incidence ÷ best estimate1.00×

Recovery ÷ best estimate1.00×

Expenses ÷ best estimate1.00×

On claim (best estimate)On claim (yours)

Best estimate€49.9m

Change on the book's best estimate€0.0m

On claim at 64 (actives)3,447

Benefits in the last year€41.4m

Year-1 transition matrix at 45 (row: from; column: to)
From \ to	Healthy	Disabled	Dead
Healthy	99.53%	0.30%	0.17%
Disabled	17.94%	81.72%	0.35%
Dead	0.00%	0.00%	100.00%

120,000 healthy lives aged 45 insured to 65 for €12,000 a year while disabled, plus 3,000 claimants aged 50 in payment. The sliders scale incidence in every year and recovery; the Art 156 preset uses +35% incidence in year 1, +25% after and recovery −20%. Illustrative data.

Worked example

One year of the chain, then the charge

All figures from the book; € millions unless stated.

Income protection: the first year and the health charge

Step 1 of 5

1
Start: everyone healthy
π₀ = (120,000, 0, 0) at age 45, plus 3,000 claimants aged 50 already in payment. The year-1 matrix at 45 is in the table above.

Regulatory lens

Delegated Regulation (EU) 2015/35 sets the expense stress for life business in Article 140 and, in the same terms, for SLT health in Article 157: a 10% rise in expenses and a 1 percentage point rise in expense inflation. Article 144 builds the health module from NSLT health, SLT health and health catastrophe; Article 151 lists the SLT health sub-modules and their correlations (0.5 between disability-morbidity and expense). Under Article 154 the SLT disability-morbidity charge is the sum of a medical expense part and an income protection part, and Article 156 sets the income protection scenario: disability and morbidity rates up 35% for 12 months and 25% after, recovery rates below 50% cut by 20%, and persistency rates of 50% or less raised by 20% (the book has no persistency assumption). The standard formula is in M4.6: Solvency II and insurance regulation; NSLT health uses premium and reserve risk of the kind in L3.8.04: Premium risk and the underwriting cycle.

Spark Atoms

Spark AtomOne point of inflation is not a stress any more

The book assumes expenses rise 2% a year, and the stress adds one point. Euro area annual inflation reached 10.6% in October 2022, against 4.1% a year earlier (Eurostat, 17 November 2022). A stress calibrated in calm years can be overtaken by events.

Spark AtomRecovery is nearly half the stress

On the book, slowing recovery from 18% to 14.4% adds €80.5m, nearly as much as 25% more claims every year after the first. A claims team that gets people back to work is a capital tool.

Key takeaways

The expense stress is +10% on expenses and +1 point on expense inflation. The first half is a tenth of the PV of expenses; the second grows with duration. On the life books: €111.4m.
Income protection is a Markov chain: healthy, disabled and dead, with a transition matrix by age. One year is a row vector times a matrix.
Article 156 raises incidence (35%, then 25%) and cuts recovery by 20%. On the book: €191.3m, with slower recovery costing nearly as much as higher incidence after year 1.
The health module has SLT, NSLT and catastrophe parts. The book's SLT health charge is €198.1m, beside €2,054.0m for life.
Fixed costs on a shrinking book are the expense risk the standard formula's stress does not see directly.

Reaction

Check what stuck

Question 1 · objective 1

A book's expenses have a present value of €200m. What does the +10% part of the Solvency II expense stress add to its best estimate, in €m?

€m

Question 2 · objective 2

1,000 healthy lives. Each dies within the year with probability 0.5%; a survivor becomes disabled with probability 1%. How many are expected to be disabled at the year end?

Question 3 · objective 3

On the book's income protection, which part of the Article 156 stress costs least?

AIncidence +35% in the first yearBIncidence +25% after the first yearCRecovery −20%DThey all cost the same

Question 4 · objective 3

The best-estimate recovery rate is 18% a year. What rate does the Article 156 stress use, in %?

%

Question 5 · objective 4

Where does long-term income protection sit in the Solvency II standard formula?

AIn the health module, under SLT health, because it is written on a similar technical basis to life insuranceBIn the life module's disability sub-moduleCIn non-life premium and reserve riskDIn operational risk

Glossary

Expense risk: Expense risk: the risk that the cost of administering policies, or its inflation, turns out higher than assumed in the technical provisions.
Closed book: Closed book: a block of policies no longer open to new business, which runs off as policies end.
Income protection: Income protection: insurance that pays a regular income while the insured is too ill or injured to work, usually until recovery, death or a set age.
Incidence rate: Incidence rate: the probability that a healthy insured life becomes disabled (makes a claim) within the year.
Recovery rate: Recovery rate: the probability that a life on claim recovers and stops receiving benefit within the year.
Markov chain: Markov chain: a model in which lives move between states each period with probabilities that depend only on the current state (and here, age).
Transition matrix: Transition matrix: the table of one-period probabilities of moving from each state (row) to each state (column). Each row adds to one.
SLT health: SLT health: health insurance written on a similar technical basis to life insurance, such as long-term income protection. It is stressed with life-style scenarios.
NSLT health: NSLT health: health insurance not written on a life basis, such as annual medical expense cover. It is measured with non-life style techniques.

Deep Dives & Origins

Deep Dives

D. C. M. Dickson, M. R. Hardy and H. R. Waters, Actuarial Mathematics for Life Contingent Risks, 3rd edition, Cambridge University Press, 2020: chapter 8, multiple state models. Advanced.

Origins

Commission Delegated Regulation (EU) 2015/35, Articles 140, 144, 151, 154, 156 and 157, as adopted, via legislation.gov.uk.
Eurostat, “Annual inflation up to 10.6% in the euro area”, euro indicators 130/2022, 17 November 2022.

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