Lapse and policyholder behaviour · RiskAtoms Academy

By the end you will be able to

1Explain why lapses can hurt in either direction, and compute the Solvency II lapse up, lapse down and mass lapse scenarios.
2Value a savings guarantee as intrinsic value plus time value, the second as an expectation over scenarios.
3Show how interest rates change which lapse scenario bites, and what dynamic lapses add.
4Draw the risk lessons of Equitable Life's guaranteed annuity rates.

Why it matters

750,000 savers hold €23,000m with RiskAtoms Insurance SE, and every one of them can ask for their money back tomorrow. Some of them hold a 3% guarantee that rates of 1.25% make valuable; they will not leave. Others hold 0.5% and will leave the moment a better offer appears. What policyholders choose to do is a risk in its own right, and on this book it needs €431.2m of capital: twice the mortality charge.

Core explanation

The policyholder holds the option

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 savings book (three generations of contracts with guaranteed minimum crediting rates of 3%, 1.75% and 0.5%) and the term book. The book's simplifications that matter here: lapse rates are fixed (3%, 5% and 7% on savings, 6% on term) and do not respond to rates; there is no profit sharing; the guarantee's time value is held fixed in every stress; and mass lapse uses 40%, because there is no group pension business.

L3.6.08: Optionality in loans showed the bank's version: a mortgage borrower may repay early, and does so when rates fall. A policyholder holds the mirror image. A lapse ends the policy; a surrender pays out its surrender value. Whether that hurts the insurer depends on what the policy was worth to it.

Term assurance is profitable: its best estimate is −€529.5m, future profit the insurer counts on. Every lapse gives some of that up, so more lapses hurt.
Savings with a guarantee in the money cost the insurer more than the account value. Every lapse takes a costly promise off its hands, so fewer lapses hurt.
Savings out of the money are worth less in the insurer's books than the surrender value their holders could take today. A rush for the exit hurts.

Three generations of guarantees

Each year the account is credited with max(g, forward rate − 0.75%): the guarantee g, or what the fund earns less the insurer's margin. On today's illustrative curve the fund earns 1.25% in year 1, rising to 2.37% by year 15. The 3% guarantee binds in all 8 years of its term, the 1.75% guarantee in the first 4, and the 0.5% guarantee never on this path.

Fund earns (forward − 0.75%)Guarantee 3.00%Guarantee 1.75%Guarantee 0.50%

Cohort	g	Account	Lapse	Charge	Surrender value	BE (det.)	Intrinsic	Time value	BE
2005–2012	3.00%	€6,500m	3%	0%	€6,500m	€6,851.8m	€600.8m	€149.5m	€7,001.3m
2013–2017	1.75%	€7,500m	5%	1%	€7,425m	€7,203.0m	€85.3m	€417.0m	€7,620.0m
2018–2025	0.50%	€9,000m	7%	3%	€8,730m	€8,403.6m	€0.0m	€234.6m	€8,638.2m
Savings		€23,000m			€22,655m	€22,458.4m	€686.1m	€801.1m	€23,259.5m

The guarantees cost €686.1m of intrinsic value, almost all on the 3% cohort, and €801.1m of time value. The time value is largest on the 1.75% cohort, whose guarantee sits close to what the fund earns: an at-the-money option is the most sensitive to uncertainty. It comes from 2,000 seeded fund scenarios (a yearly return of the forward rate plus 2% × a normal draw): the average cost across them less the cost on the average path (M-C02: Expectation).

Three lapse scenarios

Article 142 tests lapse rates 50% higher (capped at 100%), 50% lower (but by no more than 20 percentage points), and a mass lapse of 40% of policies at once. Each applies only where it raises the best estimate, and the charge is the largest of the three totals.

Model point	Lapse up	Lapse down	Mass lapse
Term, age 30	€0.0m	€10.6m	€15.9m
Term, age 40	€4.3m	€1.7m	€59.1m
Term, age 50	€13.0m	€0.0m	€82.4m
Term, age 60	€8.7m	€0.0m	€54.4m
Savings 2005–2012 (3.0%)	€0.0m	€13.6m	€0.0m
Savings 2013–2017 (1.75%)	€28.7m	€0.0m	€88.8m
Savings 2018–2025 (0.5%)	€38.4m	€0.0m	€130.6m
Total	€93.0m	€25.9m	€431.2m

Mass lapse binds at €431.2m. Of that, €211.8m is 40% of the term book's future profit, and the rest comes from the two savings cohorts whose surrender value is above their best estimate: €88.8m and €130.6m. The 3% cohort is the opposite: its best estimate is €351.8m above its surrender value, so mass lapse would release capital and only lapse down costs anything (€13.6m). Even the term book is not one-way: on the age-30 model point lapse down costs €10.6m and lapse up nothing, because level premiums paid early fund the expensive later years, and policies that stay longer reach them.

What the book holds fixed

The book adds the time value of the guarantees to the best estimate but holds it fixed in every stress. If the 40% who leave took their share of it with them, the savings cohorts' mass lapse would be measured against their full best estimate, and the charge would be €248.5m rather than €431.2m. A sensitivity, not the book's figure; a full model revalues the options in every scenario.

Rates decide which scenario bites

Move interest rates and the book's balance shifts. When rates rise, best estimates fall below surrender values, so leaving pays and mass lapse grows. When rates fall, guarantees move into the money and the danger becomes policyholders who stay.

Rates	Savings BE (det.)	Surrender value	Lapse up	Lapse down	Mass lapse
-100bp	€23,539.7m	€22,655m	€58.9m	€92.3m	€322.8m
-50bp	€22,908.3m	€22,655m	€76.1m	€43.2m	€347.5m
0bp	€22,458.4m	€22,655m	€93.0m	€25.9m	€431.2m
+50bp	€22,135.6m	€22,655m	€97.0m	€9.9m	€467.7m
+100bp	€21,915.8m	€22,655m	€108.4m	€7.2m	€510.1m
+200bp	€21,764.8m	€22,655m	€114.9m	€4.6m	€572.1m

Mass lapse binds throughout, helped by the term book, but it ranges from €322.8m at −100bp to €572.1m at +200bp, while lapse down rises from €25.9m to €92.3m as rates fall. On the savings book alone, a 200bp fall makes lapse down the only scenario that costs anything: €247.1m.

Illustrative dataLesson extraDynamic lapses

The book's lapse rates ignore rates. Real policyholders do not: they hold on to valuable guarantees. The lesson adds a dynamic lapse rule: in any year the guarantee is above what the fund earns, the base lapse rate is multiplied by 1 − 40 × the gap, with the gap as a decimal, so a gap of one percentage point cuts the rate by 40%. The rate never falls below 20% of the base rate. The 3% cohort's year-1 lapse falls from 3.0% to 0.90%.

Its best estimate rises by €16.9m today and by €66.5m if rates fall 100bp; its lapse-down charge falls from €13.6m to €5.4m. Dynamic lapses move the cost of rational policyholders out of the capital charge and into the best estimate, where it belongs. The rule and its parameters are made up for the lesson.

Sort itWhich lapse scenario hurts?

For each block of business, pick the Solvency II scenario that raises its best estimate most.

Savings policies with a 4% guarantee, when the fund earns 1%.
Savings policies with a 0.5% guarantee and a 3% surrender charge, after rates jump by 2 points.
A profitable term assurance book whose premiums are loaded 50% above expected claims.
Unit-linked policies with high initial expenses that the insurer recovers through annual charges.

The Equitable Life lesson

TimelineGuaranteed annuity rates at Equitable Life

1 / 6

1957–1988

A free option

The Equitable Life Assurance Society sells pension policies with a guaranteed annuity rate: typically 10%, so £100 of fund at retirement buys at least £10 a year for life. Market rates are high, the guarantee looks worthless, and no charge is made for it. It stops offering them in 1988.

Three lessons carry straight to RiskAtoms Insurance SE's 3% cohort. An option granted for free is still an option, and it will be valued by the people holding it. Policyholder behaviour follows value: when a guarantee is deep in the money, assume it will be used. And a guarantee on a long contract carries interest rate and longevity risk at once.

The formal version

A one-year example. A guarantee g = 2%; the fund earns 4% or 0% with equal probability, so E[R] = 2%:

\begin{aligned} \text{On the expected path:}\quad & \max\big(g,\ E[R]\big) = \max(2\%,\ 2\%) = 2\% \;\Rightarrow\; \text{IV} = 0\% \\ \text{Across scenarios:}\quad & E\big[\max(g,\ R)\big] = \tfrac12 \max(2\%,\ 4\%) + \tfrac12 \max(2\%,\ 0\%) = 3\% \\ & \text{TV} = 3\% - 2\% = 1\% \text{ of the account} \end{aligned}

The lapse scenarios. With a best-estimate lapse rate w, Article 142 sets

\begin{aligned} w^{\text{up}} &= \min\big(1.5\,w,\ 1\big), \qquad w^{\text{down}} = \max\big(0.5\,w,\ w - 0.20\big) \\ \Delta\text{BE}^{\text{mass}} &= 0.4\,\big(\text{SV} - \text{BE}\big) \quad \text{(policies paid their surrender value today)} \\ \text{SCR}_{\text{lapse}} &= \max\Big(\textstyle\sum \Delta\text{BE}^{\text{up}}_+,\ \sum \Delta\text{BE}^{\text{down}}_+,\ \sum \Delta\text{BE}^{\text{mass}}_+\Big) \\ &= \max(93.0,\ 25.9,\ 431.2) = 431.2\text{m} \end{aligned}

The mass-lapse line is exact in the book's model because cash flows scale with the number in force: the 60% who stay keep their best estimate, the 40% who leave cost SV. The subscript + keeps each model point's change only where positive, the way the book applies Article 142 policy by policy.

Try it

Move rates from −200bp to +200bp and watch the bars swap places: lapse down on the guarantees as rates fall, mass lapse on the low-guarantee cohort as they rise. Then switch dynamic lapses on with rates down: the lapse-down bars shrink, because rational policyholders are already in the best estimate.

Try itWhich lapse scenario hurts, and when?

Interest rates moved0bp

Lapses

Lapse up (×1.5)Lapse down (×0.5)Mass lapse (40%)

Read out a cohort

Guarantee3.00%

Fund earns in year 1 (forward − 0.75%)1.25%

Intrinsic value of the guarantee€600.8m

Lapse rate in year 13.00%

Best estimate (deterministic)€6,851.8m

Surrender value today€6,500.0m

Savings book: worst scenarioMass lapse

Its cost (savings only)€219.4m

Each bar is the rise in that cohort's best estimate, kept only where positive (Article 142). Mass lapse pays 40% of policyholders their surrender value today. Dynamic lapses fall when the guarantee is in the money (the base rate × (1 − 40 × the gap), but never below 20% of the base rate). The time value is held fixed, as in the book. Term assurance (always hurt by mass lapse) is left out. Illustrative data.

Worked example

Two cohorts, three scenarios

All figures from the book, € millions.

The 0.5% and 3% cohorts, then the lapse charge

Step 1 of 5

1
The 0.5% cohort: out of the money
The fund earns 1.25% in year 1, above the 0.5% guarantee all the way: intrinsic value zero. Its deterministic best estimate is €8,403.6m, below the €8,730m its policyholders could take today (€9,000m less a 3% charge). The margin and expenses make the policy worth less to the insurer than the cash it would pay out.

Regulatory lens

Article 142 of Delegated Regulation (EU) 2015/35 defines the lapse sub-module as the largest of a permanent 50% rise in option exercise rates (capped at 100%), a permanent 50% fall (by no more than 20 percentage points) and a mass lapse: 40% of policies, or 70% for group pension business where the policyholder is not a natural person or acts for the scheme's members. It covers every right to end, surrender, reduce or extend cover, and each scenario applies only to the options whose exercise moves the technical provisions the wrong way. The time value of the guarantees is part of the best estimate under Solvency II, which is why firms with guaranteed business run stochastic models of the kind the book uses. The detail of the standard formula is in M4.6: Solvency II and insurance regulation; how IFRS 17 measures the same guarantees, in M4.8: Insurance accounting: IFRS 17 (overview).

Spark Atoms

Spark AtomWhy time value is never negative

A guarantee pays max(g, R). Averaging the payoff over scenarios always gives at least the payoff of the average scenario, because the max function bends upwards (Jensen's inequality). So uncertainty can only add to a guarantee's cost. On the book it adds €801.1m, more than the €686.1m of intrinsic value.

Spark Atom£10 for every £100

Equitable Life's typical guarantee converted £100 of pension fund into £10 a year for life, and the Society charged nothing for it. It sold such policies from 1957 to 1988. The House of Lords' ruling in 2000 added about £1.5bn to its liabilities (Geneva Association, 2016).

Key takeaways

Lapses hurt whichever way makes the insurer worse off: more lapses lose future profit, fewer lapses keep costly guarantees, and a mass exit pays surrender values above best estimates.
Article 142: lapse rates × 1.5 (capped at 100%), × 0.5 (by at most 20 points), or 40% leaving at once; the largest total counts. On the book: €93.0m, €25.9m and €431.2m, so €431.2m.
Guarantees cost intrinsic value (€686.1m, the expected path) plus time value (€801.1m, the average over scenarios less the expected path).
Rates decide which scenario bites: mass lapse €322.8m at −100bp and €572.1m at +200bp; lapse down grows as guarantees move into the money. Dynamic lapses put rational behaviour into the best estimate.
Equitable Life: a free guarantee, set when rates were high, went deep into the money as rates fell and lives lengthened. Value options as if they will be used.

Reaction

Check what stuck

Question 1 · objective 1

A cohort of savings policies has a surrender value of €1,000m and a best estimate of €950m. What does a 40% mass lapse cost, in €m?

€m

Question 2 · objective 1

A cohort's 3% guarantee is well above what the fund earns. Which Solvency II lapse scenario raises its best estimate?

ALapse down: fewer policyholders give up a valuable guarantee, so the insurer pays it for longerBLapse up: more surrenders always cost moneyCMass lapse: losing 40% of the cohort is always the worst caseDNone: lapses never matter for guaranteed business

Question 3 · objective 2

A €1,000 account has a 1% guaranteed rate for one year. The fund earns +3% or −1% with equal probability. What is the time value of the guarantee, in €?

€

Question 4 · objective 4

What is the main risk lesson of Equitable Life's guaranteed annuity rates?

AAn option given to policyholders for free, deep in the money as rates fell and lives lengthened, will be used; it must be valued and reserved as if it will beBAnnuities should never be soldCCourts always side with insurers on bonus policyDInterest rates always rise after a long fall

Question 5 · objective 1

A policy's best-estimate lapse rate is 60% a year. What rate does the Solvency II lapse-down scenario use, in %?

%

Glossary

Lapse: Lapse: a policyholder ending a policy early, by stopping premiums or by surrendering it for cash. Lapse risk is the risk that they do so more or less often than assumed.
Surrender value: Surrender value: what a policyholder receives on leaving a savings policy early, here the account value less a surrender charge.
Guaranteed minimum crediting rate: Guaranteed minimum crediting rate: the lowest rate the insurer may add to the account each year, whatever its investments earn.
Intrinsic value: Intrinsic value: what an option is worth on the expected path alone. For a guarantee, the extra cost on today's forward rates.
Time value: Time value: what an option is worth on top of its intrinsic value because the future is uncertain. For a guarantee, the average extra cost across fund scenarios.
TVOG: TVOG: the time value of options and guarantees, part of the best estimate of a life book with guarantees.
Mass lapse: Mass lapse: many policyholders leaving at once, for example after bad news about the insurer or a jump in rates elsewhere. Solvency II tests 40% at once (70% for some group pension business).
Dynamic lapse: Dynamic lapse: a lapse rate that depends on market conditions, such as whether a guarantee is in the money or how the credited rate compares with rates elsewhere.
Guaranteed annuity rate: Guaranteed annuity rate: a promise to convert a pension fund into an annuity at a fixed rate, such as £10 a year for every £100 of fund.

Deep Dives & Origins

Deep Dives

D. Baranoff and C. O'Brien (ed. E. Baranoff), Equitable Life U.K.: a Decade of Regulations and Restructuring, The Geneva Association, 2016. Intermediate.
Lord Penrose, Report of the Equitable Life Inquiry, HC 290, March 2004: chapter 1, “Annuity guarantees and the Hyman case”. Advanced.

Origins

Commission Delegated Regulation (EU) 2015/35, Article 142, as adopted, via legislation.gov.uk.
D. Baranoff and C. O'Brien, Equitable Life U.K.: a Decade of Regulations and Restructuring, The Geneva Association, September 2016.
Lord Penrose, Report of the Equitable Life Inquiry, chapter 1, HC 290, 8 March 2004 (HM Treasury copy, National Archives).
Equitable Life Assurance Society v Hyman [2000] UKHL 39, House of Lords, 20 July 2000.

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