Mortality risk · RiskAtoms Academy

By the end you will be able to

1Build a life table from the Gompertz–Makeham law and read q_x, p_x and survival over several years.
2Model deaths in a pool of lives as a binomial count and tell random fluctuation apart from a wrong table.
3Apply the Solvency II mortality (+15%) and life catastrophe (+0.15 points) stresses to a term book and explain the result.
4Work out what a pandemic year does to an insurer that sells both term assurance and annuities.

Why it matters

RiskAtoms Insurance SE has promised €58.4bn to the families of 400,000 people if they die before their cover ends. Next year it expects 1,028 of them to die, and €114.2m of claims. Chance alone moves that by about €3.8m either way. If the mortality table is 15% too light, claims run €17.1m higher next year, and again in every later year the cover runs. Solvency II puts that at €206.5m of capital.

Core explanation

Who dies is luck; how many die is a table

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 the term assurance book (four model points, ages 30 to 60, premiums set at 1.5 times the best-estimate risk cost, 6% lapses a year) and, for the pandemic, the annuities. The book's simplifications that matter here: one unisex mortality law, not a published table; annual cash flows; and the mortality and catastrophe stresses applied to term assurance only, the one book where more deaths cost money.

L2.2.06: Insurance risk named mortality risk: more deaths than assumed, hurting death cover. This lesson measures it. It starts from the assumption every life insurer rests on, the life table.

A life table from a law

The chance that a life aged x dies within a year is q_x, a probability (M-C01: Probability basics). Insurers build tables of q_x from their own and national data. The book uses a smooth law instead, the Gompertz–Makeham law: the force of mortality is μ_x = A + B·c^x with A = 0.0002, B = 0.00002 and c = 1.1. The B·c^x part grows by 10% with each year of age, so it doubles every 7.3 years. The constant A matters only for the young: it is 36% of μ at 30 and 2% at 65.

Age	μ_x	q_x (2025)	1 in …	Makeham share
30	0.00055	0.057%	1,767	36.4%
40	0.00111	0.115%	870	18.1%
50	0.00255	0.266%	376	7.8%
60	0.00629	0.657%	152	3.2%
65	0.01001	1.044%	96	2.0%
70	0.01599	1.663%	60	1.3%
80	0.04117	4.226%	24	0.5%
90	0.10646	10.568%	9	0.2%
100	0.27581	25.127%	4	0.1%

q_x, 2025 table+15% (mortality stress)+0.15 points (catastrophe)

On a log scale the law is almost a straight line from middle age. The two stresses look very different: +15% is the same small step at every age, while +0.15 points lifts the young most, where q is tiny. From the 2025 rates (improving by 1.5% a year) the life table for the 50-year-olds on the term book reads:

Age	q_x	p_x	l_x	d_x
50	0.266%	0.99734	100,000	266.0
51	0.286%	0.99714	99,734	285.4
52	0.308%	0.99692	99,449	306.4
53	0.332%	0.99668	99,142	329.0
54	0.358%	0.99642	98,813	353.4
55	0.386%	0.99614	98,460	379.7

Survival over several years multiplies: of 100,000 lives aged 50, 93,108 (93.1%) are expected to reach the end of a 15-year term.

Deaths in a pool are binomial

Take the 120,000 policyholders aged 50. If each dies independently with probability q₅₀ = 0.266%, the number of deaths next year is binomial (M-C04: The binomial distribution): mean 319.2, standard deviation 17.8, and a 99.5% count of 366. Now suppose the table is 15% too light for these lives. The true mean is 367.1, and the chance of more than 366 deaths, a 1-in-200 year on the old table, becomes 51%: a coin toss.

On the tableTable 15% too light

That is the core of mortality risk management. Random fluctuation, the volatility risk, shrinks relative to the mean as the pool grows: one sd is 5.6% of the mean here. A wrong table, the level risk, does not shrink at all. On the whole term book next year:

Age	Policies	q	Expected deaths	sd	Sum assured	Expected claims	sd of claims
30	80,000	0.057%	45.3	6.7	€200,000	€9.1m	€1.35m
40	120,000	0.115%	137.9	11.7	€180,000	€24.8m	€2.11m
50	120,000	0.266%	319.2	17.8	€120,000	€38.3m	€2.14m
60	80,000	0.657%	525.4	22.8	€80,000	€42.0m	€1.83m
Total	400,000		1,027.8	32.0		€114.2m	€3.77m

The four pools are independent, so their variances add. A 1-in-200 run of bad luck costs about €9.7m above the expected €114.2m (mean + 2.576 sd). A table 15% too light costs €17.1m next year and then again every year the policies run. A year with the catastrophe add-on costs €87.6m.

Four ways the table can be wrong: guess the remedy, then reveal

The two Solvency II stresses

The standard formula prices level and catastrophe risk with two stresses on the best estimate. The mortality stress raises every mortality rate by 15%, permanently, but only on policies where more deaths increase the technical provisions (Article 137). The life catastrophe stress adds 0.15 percentage points (1.5 per mille) to the mortality rates for the next 12 months (Article 143). Each charge is the rise in the best estimate, which is the fall in own funds with assets unchanged.

Model point	Best estimate	Mortality +15%	Mortality charge	Catastrophe	Cat charge	Cat: q ×
Age 30, 25 years	−€39.7m	−€16.1m	+€23.6m	−€16.1m	+€23.6m	3.65×
Age 40, 20 years	−€147.8m	−€87.4m	+€60.4m	−€115.9m	+€31.9m	2.31×
Age 50, 15 years	−€205.9m	−€130.4m	+€75.5m	−€184.5m	+€21.4m	1.56×
Age 60, 8 years	−€136.1m	−€89.1m	+€47.0m	−€126.5m	+€9.6m	1.23×
Term book	−€529.5m		+€206.5m		+€86.5m

The best estimate is negative: the premiums, loaded by 50%, are worth more than the claims and expenses. The mortality stress eats 39% of that future profit, most of it on the long, large middle-aged model points. The catastrophe stress is a one-year event, so it is smaller, and it falls hardest on the young: at 30 it multiplies q by 3.65, so the 30-year-olds cost almost as much under it as under the permanent +15%. The two charges are correlated 0.25 (Article 136), so together they need €243.0m, not €293.0m.

A pandemic year, on both books

The book's pandemic is 15% more deaths than the table, at every age, for one year. Term claims rise by €16.8m. But the same pandemic takes annuitants too, and each of them takes their remaining pensions with them: the annuity best estimate falls by €73.2m. Net, RiskAtoms Insurance SE is €56.4m better off. This natural hedge (L2.2.06: Insurance risk) is why the standard formula correlates mortality and longevity at −0.25 rather than ignoring the offset. It is fragile: it rests on the extra deaths falling on annuitants as much as on insured workers, and it does nothing for an insurer that writes only protection.

Sort itWhich kind of mortality risk?

An experience study shows the insurer's smokers die 40% more often than its pricing table assumed.
A small group scheme of 300 lives has three deaths in a year against an expected 0.6.
An influenza pandemic raises deaths for one winter.
Death rates of the insured stop falling for a decade after years of steady improvement.
One policyholder with a €20m sum assured dies in a car accident.

The formal version

The term best estimate. For a model point of n lives aged x, sum assured S, premium P in advance, expense E_t and lapse rate w, the expected number in force at the start of year t is $n_t$ , with claims at the year end:

\begin{aligned} n_1 &= n, \qquad n_{t+1} = n_t\,\big(1 - q_{x+t-1}\big)(1 - w) \\ \text{BE} &= \sum_{t=1}^{T} \Big[\, n_t\,q_{x+t-1}\,S\,v^{t} \;+\; n_t\,(E_t - P)\,v^{t-1} \Big] \end{aligned}

Here $v^t$ is the discount factor from the book's curve. The two stresses change only q:

\begin{aligned} \text{Mortality:}\quad & q_{x+t-1} \to 1.15\,q_{x+t-1} \text{ for every } t, & \text{SCR}_{\text{mort}} &= \text{BE}^{\text{stressed}} - \text{BE} = 206.5\text{m} \\ \text{Catastrophe:}\quad & q_{x} \to q_{x} + 0.0015 \text{ in year 1 only}, & \text{SCR}_{\text{cat}} &= 86.5\text{m} \end{aligned}

Claims across pools. With independent pools k, each Bin(n_k, q_k) and sum assured S_k, the year's claims C have

\begin{aligned} E[C] &= \textstyle\sum_k n_k q_k S_k = 114.2\text{m} \\ \operatorname{Var}[C] &= \textstyle\sum_k n_k q_k (1 - q_k) S_k^2, \qquad \operatorname{sd}[C] = 3.77\text{m} \end{aligned}

Double every pool and the mean doubles but the sd grows by only √2. A 15% error in every q_k moves the mean by 15% whatever the size. That asymmetry is why capital for mortality is mostly capital for the table, not for luck.

Try it

Choose an age and a pool size, then change the true mortality against the table. With 1,000 lives a 15% error disappears in the noise. With 120,000 it moves the whole distribution. Switch on the catastrophe add-on at age 30 and at age 60 to see why it hurts the young.

Try itDeaths next year in a pool of lives

Age of the lives (a term model point)

Lives in the pool

True mortality ÷ the table1.15×

Catastrophe add-on (+0.15 points)

Deaths on the tableDeaths on your assumption

Expected deaths (table)319.2

Expected deaths (yours)367.1

sd of deaths (table)17.8 (5.6% of the mean)

99.5% death count (table)366

Shift in the mean, in sds2.68

P(deaths > the table's 99.5% count)50.8%

q on the table0.266%

Extra claims at €120,000 each€5.7m

Lives die independently with the same rate q in a binomial pool. The table is the book's 2025 Gompertz–Makeham rate for the age; the sums assured are the term model point's. Illustrative data.

Worked example

From one model point to the mortality charge

All figures from the book, € millions unless stated.

The 50-year-olds, then the term book

Step 1 of 6

1
The mortality rate
$q_{50} = 1 - \exp\!\Big(-0.0002 - \frac{0.00002\times 1.1^{50}\,(1.1-1)}{\ln 1.1}\Big) = 0.2660\%$

How big is that?

€206.5m is 39% of the term book's future profit, but the longevity charge on the same insurer is 9.1 times larger (€1,887.2m). RiskAtoms Insurance SE is an annuity writer with some protection on the side, and its capital follows that.

Regulatory lens

Under the Solvency II standard formula, mortality and life catastrophe are two of the seven life underwriting sub-modules in Delegated Regulation (EU) 2015/35: Article 137 (+15% on mortality rates, permanent, where it increases technical provisions) and Article 143 (+0.15 percentage points for the next 12 months). Article 136 aggregates the seven with a fixed correlation matrix. Firms with an internal model split the risk the way this lesson does, into level, trend, volatility and catastrophe, and calibrate each to their own data. The 2025 review (Directive (EU) 2025/2 and Delegated Regulation (EU) 2026/269, applying from 30 January 2027) cuts the risk margin's cost-of-capital rate from 6% to 4.75%, which lowers what non-hedgeable risks such as mortality cost in technical provisions. The detail of the standard formula is in M4.6: Solvency II and insurance regulation; IFRS 17's risk adjustment for the same risk is in M4.8: Insurance accounting: IFRS 17 (overview).

Spark Atoms

Spark AtomA law from 1825

Benjamin Gompertz, an actuary, proposed in 1825 that a person's power to resist death falls by equal proportions in equal intervals of time, so that mortality rises geometrically with age. William Makeham added the constant term in 1860. Adult mortality is often said to double about every eight years; the book's c = 1.1 doubles it every 7.3.

Spark AtomWhat a pandemic looked like

Eurostat measures excess mortality against the average deaths of 2016–2019. Across the EU it peaked at +25% in April 2020 and +40% in November 2020, and at +21% and +27% in April and November 2021 (Eurostat, 17 February 2023). Those are single months. The book's pandemic, +15% for a whole year at every age, is a severe year, and the standard formula's catastrophe add-on is larger still for the young.

Key takeaways

Gompertz–Makeham: μ_x = A + B·c^x, q_x = 1 − exp(−∫μ). With c = 1.1 mortality doubles every 7.3 years; q₅₀ = 0.266%.
Deaths in a pool are Bin(n, q): mean nq, sd √(nq(1 − q)). For 120,000 lives aged 50 that is 319.2 ± 17.8.
Random fluctuation shrinks relative to the mean as the pool grows; a wrong table does not. Mortality capital is mostly capital for the table.
Solvency II: +15% on mortality rates for term business (€206.5m) and +0.15 percentage points for 12 months (€86.5m), correlated 0.25: €243.0m together.
A pandemic year costs the term book €16.8m but releases €73.2m of annuity reserves: a natural hedge for an annuity-heavy insurer.

Reaction

Check what stuck

Question 1 · objective 2

A pool of 10,000 lives each has a 0.2% chance of dying this year, independently. What is the standard deviation of the number of deaths? Give it to two decimal places.

Question 2 · objective 3

A 40-year-old's mortality rate for next year is 0.11%. What rate does the Solvency II life catastrophe stress use for the next 12 months, in %?

%

Question 3 · objective 3

Why does the book apply the +15% mortality stress to the term book but not to the annuities?

AArticle 137 applies it only to policies where higher mortality increases the technical provisions; more deaths reduce what the annuities costBAnnuities are covered by the catastrophe stress insteadCAnnuitants are too old for mortality riskDBecause the stress is applied to non-life business only

Question 4 · objective 1

The force of mortality is a constant 0.01 a year over the coming year. What is the one-year mortality rate q, in %, to three decimal places?

%

Question 5 · objective 4

In a pandemic year, deaths in the book run 15% above the table. Term claims rise by €16.8m. What happens to RiskAtoms Insurance SE overall?

AIt gains about €56.4m: annuitants who die take €73.2m of future pensions with themBIt loses the term claims and nothing else changesCIt loses on both books, because all life business suffers in a pandemicDIt gains on term assurance and loses on annuities

Glossary

Mortality risk: Mortality risk: the risk that more insured people die, or die sooner, than the assumptions used to price and reserve the business. It hurts death cover such as term assurance.
q_x and p_x: q_x: the probability that a life aged exactly x dies within the next year. p_x = 1 − q_x is the probability of surviving it.
Force of mortality: Force of mortality μ_x: the instantaneous rate of dying at exact age x, per year. Over a short time h, the chance of dying is about μ_x·h.
Gompertz–Makeham law: Gompertz–Makeham law: μ_x = A + B·c^x. A constant part (accidents) plus a part that grows by the factor c with every year of age.
Life table: Life table: q_x by age, often shown with l_x (the number alive at age x out of a starting group) and d_x (the number who die between x and x + 1).
Term assurance: Term assurance: pays a fixed sum if the insured person dies before the policy ends, in return for regular premiums. Nothing is paid if they survive.
Level risk: Level risk: the risk that the mortality table is simply wrong for this group of lives, every year, by a similar proportion. More lives do not reduce it.
Volatility (process) risk: Volatility (process) risk: random fluctuation in the number of deaths around the true rate. It shrinks, relative to the mean, as the pool grows.
Catastrophe risk: Catastrophe risk: a one-off spike in deaths in a single year, from a pandemic, a war or a natural disaster.
Trend risk: Trend risk: future improvements in mortality turning out different from those assumed.
Best estimate: Best estimate: the probability-weighted present value of future cash flows out (claims, expenses) less those in (premiums). For a profitable term book it is negative: future profits.

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: chapters 2 and 3, survival models, life tables and the Gompertz and Makeham laws. Advanced.
B. Gompertz, “On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies”, Philosophical Transactions of the Royal Society of London 115, 1825, pp. 513–583. Historical.

Origins

Commission Delegated Regulation (EU) 2015/35, Articles 136 (life correlations), 137 (mortality risk) and 143 (life catastrophe risk), as adopted, via legislation.gov.uk.
B. Gompertz, Philosophical Transactions of the Royal Society of London 115, 1825; W. M. Makeham, “On the law of mortality and the construction of annuity tables”, Journal of the Institute of Actuaries 8, 1860, pp. 301–310.
Eurostat, “Excess mortality rose sharply to 19% in December 2022”, Eurostat news, 17 February 2023.
Commission Delegated Regulation (EU) 2026/269, Article 39 as amended (cost-of-capital rate 4.75%), EUR-Lex.

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