Longevity risk · RiskAtoms Academy

By the end you will be able to

1Project mortality improvements and tell period from cohort life expectancy.
2Value an annuity as a discounted sum of survival probabilities, and see the geometric series inside it.
3Apply the Solvency II longevity stress (−20% on mortality) and explain why longevity is the largest life risk on the book.
4Compare longevity swaps, buy-ins and reinsurance, and price the capital relief a swap buys.

Why it matters

RiskAtoms Insurance SE pays 200,000 pensioners €1,696.9m this year, and will keep paying each of them until they die. On its table a 65-year-old lives another 22.9 years. If mortality turns out 20% lighter, that becomes 25.0 years, and the annuities cost €1,887.2m more. That one assumption is the largest risk in the insurer's life business: 9.1 times its mortality risk.

Core explanation

Living longer, priced today

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 annuity book: 200,000 annuitants in six age bands from 65 to 90, level pensions of €7,200 to €9,800 a year paid at each year end, and €40 a year of expenses each. The simplifications that matter: one unisex table, a constant 1.5% a year improvement to age 100 (none above), no indexation, and a longevity swap in the book that passes the stress on in full with no counterparty risk.

L3.8.06: Mortality risk built the table. For an annuity the question turns round: not how many die next year, but how long the survivors keep being paid. Longevity risk is the risk that they are paid for longer than assumed.

Improvements: the table moves

Mortality rates at each age have fallen for decades. The book assumes a mortality improvement of 1.5% a year to age 100: the rate for age x in year 2025 + t is q_x(1 − 0.015)^t, compounding like a negative interest rate (M-A03: Exponents). At 65 it is 1.044% in 2025, 0.897% in 2035 and 0.771% in 2045. At this pace a rate halves in 46 years.

That gives two life expectancies. The period figure uses 2025's rates at every age. The cohort figure follows today's 65-year-olds: at 66 they meet 2026's rate (1.128% rather than 1.145%), at 67 2027's, and so on.

At 65	Life expectancy	Reach 90	Annuity factor a₆₅
Period (2025 rates throughout)	20.85 years	36.2%	14.895
Cohort (1.5% improvement, the book)	22.92 years	45.4%	15.890
Cohort, mortality −20% (the stress)	25.04 years	53.3%	16.969

PeriodCohort (book)Cohort, −20% mortality

The curves differ most in the 80s and 90s, which is exactly where an annuity keeps paying. A 20% cut in every rate looks modest in a table, but it adds 2.12 years at 65. The small kink in the cohort curves just after 100 is by design, because the book's improvements stop at age 100.

What an annuity is worth

The annuity factor a_x adds up each future year's payment of 1, times the chance of being alive to receive it, times the discount factor. A 65-year-old's €9,800 pension is worth €155,718 on the book basis (a₆₅ = 15.890), and €166,293 after the stress.

Age	Lives	Pension	a_x	Best estimate	a_x stressed	Stressed	Change
65	30,000	€9,800	15.890	€4,696.6m	16.969	€5,015.8m	+€319.3m
70	45,000	€9,300	13.526	€5,691.7m	14.642	€6,161.7m	+€470.1m
75	50,000	€8,800	11.158	€4,937.3m	12.271	€5,430.5m	+€493.2m
80	40,000	€8,200	8.879	€2,930.0m	9.947	€3,282.5m	+€352.5m
85	25,000	€7,700	6.783	€1,314.1m	7.763	€1,504.1m	+€189.9m
90	10,000	€7,200	4.946	€358.6m	5.805	€420.9m	+€62.3m
Book	200,000			€19,928.3m		€21,815.5m	+€1,887.2m

The stress costs €1,887.2m, 9.5% of the best estimate, and the Solvency II longevity charge is that rise. Most of it sits with the 70- and 75-year-olds: young enough to have long to run, and the largest groups.

Why longevity is the largest life risk here

Three things make it large. It is long: an error in the table compounds over 25 or 30 years of payments. It is systematic: every annuitant benefits from the same medical progress, so 200,000 lives diversify the random part (L3.8.06: Mortality risk) but not the trend. And it is one-way for this book: nothing in the annuities gains if people live longer. On the book it is €1,887.2m of the €2,722.8m of life charges before diversification (69%).

The improvement assumption alone moves the best estimate by more than the mortality charge many times over:

Improvement a year	e₆₅ (cohort)	Annuity best estimate	Against 1.5%
0.0%	20.85 years	€18,775.5m	−€1,152.7m
1.0%	22.20 years	€19,535.6m	−€392.7m
1.5%	22.92 years	€19,928.3m	book
2.0%	23.64 years	€20,326.1m	+€397.8m
3.0%	25.10 years	€21,126.9m	+€1,198.6m

Sort itDoes the annuity best estimate rise or fall?

Each change is made to the book's assumptions or the world. Sort it.

A new study shows mortality at older ages improving at 2% a year, not 1.5%.
A severe flu season raises deaths among the over-75s for one winter.
Interest rates fall by 50 basis points.
The insurer finds that its annuitants, who are wealthier than average, die 10% less often than the table says.
The insurer writes 10,000 new annuities at a price above their best estimate.

Passing longevity on

Because the risk is systematic and long, insurers and pension schemes pay others to take it. On the book, a longevity swap on 50% of the annuities, at a fee of 2% a year of the hedged payments, halves the longevity charge (to €943.6m) and cuts the SCR by €323.4m, taking the solvency ratio from 186% to 199%. It costs €17.0m a year: 5.2% a year for each euro of capital released.

Four ways to transfer longevity: what goes, what stays?

The formal version

Improvement. The cohort survival from age x uses the rate for age x + s in calendar year 2025 + s:

\begin{aligned} q_{x+s}(s) &= q_{x+s}(2025)\,(1 - i)^{s}, \qquad i = 0.015 \\ {}_tp_x &= \prod_{s=0}^{t-1}\big(1 - q_{x+s}(s)\big), \qquad e_x \approx \sum_{t\ge1} {}_tp_x + \tfrac12 \end{aligned}

The period figures set i = 0 inside the projection; the half year allows for deaths during the year. The longevity stress multiplies every q by 0.8, improvement still applied.

The geometric series. If a 65-year-old never aged, surviving each year with p = 1 − q₆₅ = 0.98956, and the rate were a flat 2.5% (v = 0.97561), each term would be the last times r = pv = 0.96543 (M-A07: Geometric series):

\begin{aligned} a &= \sum_{t=1}^{n} r^{t} = \frac{r\,(1 - r^{n})}{1 - r} \\ n = 25:\quad a &= \frac{0.96543\,(1 - 0.96543^{25})}{1 - 0.96543} = 16.34 \\ n \to \infty:\quad a &= \frac{r}{1 - r} = 27.93 \end{aligned}

The book's a₆₅ is 15.89: real mortality doubles every seven years or so, which cuts the long tail of the series. The closed form is still useful as a check and as the backbone of a pension's duration.

Aggregation. Mortality, longevity and catastrophe enter the life module together, with Article 136's correlations. In full, in € millions:

\begin{aligned} s &= \begin{pmatrix} 206.5 \\ 1{,}887.2 \\ 86.5 \end{pmatrix}, \qquad C = \begin{pmatrix} 1.00 & -0.25 & 0.25 \\ -0.25 & 1.00 & 0.00 \\ 0.25 & 0.00 & 1.00 \end{pmatrix} \\ \sqrt{s^{\top} C\, s} &= 1{,}850.9\text{m} \quad \text{against} \quad \textstyle\sum s = 2{,}180.2\text{m} \end{aligned}

The three together need less capital than longevity alone (€1,887.2m). The −0.25 between mortality and longevity says that the scenarios which hurt the term book tend to help the annuities.

Try it

The sliders start at the Solvency II stress: mortality 0.8 times the table with the book's 1.5% improvement. Set the multiplier to 1 and change the improvement instead: which assumption matters more? Then move rates to see how far interest rates and longevity compound.

Try itValue the annuities on your assumptions

Model point (age of the annuitants)

Improvement a year1.50%

Mortality ÷ the table0.80×

Rates moved0bp

Book basis (1.5% improvement)Your basis

Life expectancy at 65, cohort25.04 years

Annuity factor a6516.969

Same, period basis (no improvement)22.81 years

Book basis22.92 years; a = 15.890

One €9,800 pension, valued€166,293

Model point (30,000 lives)€5,015.8m

Annuity book best estimate€21,815.5m

Change against the book+€1,887.2m

The multiplier applies to every future mortality rate; 0.80× with 1.5% improvement is the Solvency II longevity stress. Rates move the illustrative curve in parallel. Pensions are level and paid at each year end, expenses €40 a life a year. Illustrative data.

Worked example

The longevity charge, from one pensioner to the book

All figures from the book; € millions unless stated.

The 65-year-olds, then the annuity book

Step 1 of 6

1
Rates and improvement
$\begin{aligned} q_{65}(2025) &= 1.0435\% \\ q_{66}(2026) &= 1.1453\% \times 0.985 = 1.1281\% \\ q_{65}(2035) &= 1.0435\% \times 0.985^{10} = 0.8971\% \end{aligned}$

Is the swap worth it?

It costs 5.2% a year for each euro of SCR released. If shareholders want more than that on the capital the swap frees, it pays; the risk margin falls too, which the book holds fixed. But the comparison leaves out what the swap adds: a counterparty who must still be there in 2050.

Regulatory lens

Article 138 of Delegated Regulation (EU) 2015/35 sets the longevity charge as the loss from an instantaneous, permanent 20% fall in mortality rates, applied to the policies where lower mortality increases technical provisions. Because longevity cannot be hedged in markets, it also drives the risk margin, the cost of holding capital for it until the last annuity is paid. The 2025 review cuts the cost-of-capital rate from 6% to 4.75% from 30 January 2027 (Delegated Regulation (EU) 2026/269), with a new factor that shrinks the capital projected for later years. On the book's own cost-of-capital check, the rate change alone takes the risk margin from €1,848m to €1,463m, before the new factor. In the UK, where bulk annuity writers pass annuity books on through funded reinsurance, the PRA's SS5/24 (26 July 2024) expects limits on correlated counterparties and plans for recapturing the assets if a reinsurer fails. The full treatment is in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomTen years out of date

The UK's Office for National Statistics compares the two measures by birth year. For people born in England and Wales between 1950 and 2000, cohort life expectancy at birth is about 8 to 10 years above the period figure (ONS, “Period and cohort life expectancy explained”, revised January 2023). At 65 the gap is smaller, about two years on the book, but it lasts for every year an annuity is paid.

Spark AtomA £16bn swap

In 2014 the BT Pension Scheme passed £16bn of longevity risk, a quarter of its exposure, to The Prudential Insurance Company of America. It set up its own insurance company to take the risk and reinsure it, avoiding an intermediary. The deal was more than three times the previous record (IPE, 2014). In 2024 UK pension schemes completed £47.6bn of bulk annuities in 293 deals and £5.8bn of longevity swaps (Aon, reported by Reinsurance News, 1 April 2025).

Key takeaways

Improvements compound: q_x(2025 + t) = q_x(2025)·(1 − i)^t. At 1.5% a year a rate halves in 46 years.
Period life expectancy uses one year's rates; cohort follows the group. At 65 on the book: 20.85 against 22.92 years, and 25.04 after the −20% stress.
a_x = Σ ₜpₓ vᵗ, a geometric series when p and v are constant. a₆₅ = 15.89; the annuity best estimate is €19,928.3m.
The longevity charge is €1,887.2m (9.5% of the best estimate), 69% of the undiversified life charges: long, systematic and one-way.
A 50% longevity swap cuts the SCR by €323.4m for €17.0m a year, but keeps the assets and adds counterparty risk. Buy-ins move everything.

Reaction

Check what stuck

Question 1 · objective 1

The mortality rate at age 75 is 2.0% today and improves by 1.5% a year. What will the rate at age 75 be in ten years' time, in %, to two decimal places?

%

Question 2 · objective 2

Suppose an annuitant survived each year with the same probability 0.98 and the rate were a flat 3%. What would 1 a year for life be worth? Use the geometric series, to one decimal place.

Question 3 · objective 1

The book gives a 65-year-old a period life expectancy of 20.85 years and a cohort life expectancy of 22.92. Which should price an annuity sold today?

ACohort: the annuitant will meet each future year's (improved) mortality rates, not today'sBPeriod: it is based only on observed dataCEither: they always agreeDPeriod, because improvements are not allowed in Solvency II

Question 4 · objective 3

The annuity best estimate is €19,928.3m and the −20% longevity stress raises it by €1,887.2m. By what percentage does the stress raise the best estimate?

%

Question 5 · objective 4

A 50% longevity swap cuts RiskAtoms Insurance SE's SCR by €323.4m for a fee of €17.0m a year. What does the insurer keep after the swap?

AThe assets and their investment and interest rate risk, the other half of the longevity risk, and a new counterparty risk on the swapBNothing: all its annuity risks have goneCOnly the expense riskDThe longevity risk; the swap passes on the interest rate risk

Glossary

Longevity risk: Longevity risk: the risk that annuitants and pensioners live longer than assumed, so pensions are paid for more years. It hurts annuities and defined benefit pension schemes.
Mortality improvement: Mortality improvement: the yearly fall in the mortality rate at a given age. At 1.5% a year, q at age 65 in ten years' time is 0.985¹⁰ times today's.
Period life expectancy: Period life expectancy: computed from one year's mortality rates, as if they never changed. It ignores future improvements.
Cohort life expectancy: Cohort life expectancy: follows one group born in the same year through time, using each future year's (projected) rates. Usually higher than period life expectancy.
Annuity factor: Annuity factor a_x: the value today of 1 a year paid at each year end for as long as a life aged x survives. A pension's value is the pension times a_x.
Longevity swap: Longevity swap: the insurer pays a fixed schedule of expected pensions plus a fee; the counterparty pays the pensions actually due. The insurer keeps its assets and passes on the longevity risk.
Longevity reinsurance: Longevity reinsurance: an insurer passes part of its annuitants' longevity risk to a reinsurer, often in the form of a swap.
Buy-in: Buy-in: a pension scheme pays an insurer a premium for a bulk annuity policy that pays the scheme its pensions. The scheme keeps the members; the insurer takes the longevity, investment and interest rate risk.
Buy-out: Buy-out: as a buy-in, but the members become the insurer's policyholders and the scheme can wind up.
Basis risk: Basis risk: the gap between a hedge and what it hedges. An index-based longevity hedge pays on national mortality, not on the insurer's own annuitants.

Deep Dives & Origins

Deep Dives

Office for National Statistics, “Period and cohort life expectancy explained”, methodology, revised 19 January 2023. Intermediate.
D. C. M. Dickson, M. R. Hardy and H. R. Waters, Actuarial Mathematics for Life Contingent Risks, 3rd edition, Cambridge University Press, 2020: chapter 5, annuities and their valuation. Advanced.

Origins

Commission Delegated Regulation (EU) 2015/35, Articles 136 and 138, as adopted, via legislation.gov.uk.
Commission Delegated Regulation (EU) 2026/269, Article 39 as amended (cost-of-capital rate 4.75%), EUR-Lex.
Office for National Statistics, “Period and cohort life expectancy explained”, 19 January 2023.
IPE, “BT Pension Scheme transfers £16bn of longevity risk in mammoth deal”, 2014.
Reinsurance News, “UK bulk annuity market hits record £47.6bn in 2024: Aon”, 1 April 2025.
Prudential Regulation Authority, letter “SS5/24 Funded reinsurance: implementation approach”, G. Truran, 26 July 2024.

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