How it works
Illustrative dataCalverin Insurance SE, a new insurer for the Lab Test
The lessons use RiskAtoms Insurance SE's book. These problems apply the same methods to a new, much smaller composite insurer: Calverin Insurance SE, invented, with about €4.2bn of assets. Its accounts, motor data, triangle, cat events, treaties, life tables, policies, bonds and capital modules are all made up and kept hand-sized; amounts in € millions unless stated. The Solvency II stresses and correlations are the standard formula's (Delegated Regulation (EU) 2015/35, as the lessons apply them); the annuity uses a constant mortality rate, the duration problem a parallel shift and the first-order effect only, and the cat risk is the net 1-in-200 event, as on RiskAtoms Insurance SE's book, not the standard formula's factors.
Problem 1Paid first, tested later
L3.8.01Finance analyst, Calverin Insurance SE
Calverin Insurance SE (invented) is a small composite insurer in the euro area with about €4.2bn of assets: motor and property cover, liability, annuities, savings with guarantees, term assurance and income protection. Its 2025 non-life account, net of reinsurance: net earned premium €600m, net claims €414m and expenses €174m. Its net claims reserves of €900m are invested at a yield of 2.8% a year.
Lessons: L3.8.01: How insurers take risk
(a)What is the net combined ratio? Answer in %.
%(b)What did the non-life year earn: the underwriting result plus a year's income on the float? Answer in €m.
€m(c)Counting the float's income, up to what combined ratio would the year still break even? Answer in %.
%
Problem 2Pricing a motor book
L3.8.02Pricing actuary
Calverin's private-car book over three policy years, split by area (amounts in euros). Rural is the base level.
| Area | Vehicle-years | Claims | Claim amounts, € |
|---|---|---|---|
| Urban | 15,000 | 1,350 | 5,535,000 |
| Rural | 25,000 | 1,250 | 4,345,000 |
The technical premium loads the burning cost for expenses of 22% and a profit margin of 6%, both as shares of the premium. A third of urban vehicle-years have a driver aged 17–24, against a tenth of rural ones.
(a)What is the burning cost of the whole book (frequency × severity)? Answer in € per vehicle-year.
€ per vehicle-year(b)What is the one-way relativity of urban against rural, on burning cost?
×(c)Priced at that one-way burning cost, what technical premium does an urban car get? Answer in € per vehicle-year.
€ per vehicle-year(d)A multiplicative fit on driver age and area puts urban at 1.55 × rural. Why is the one-way relativity so much higher?
Problem 3A liability triangle
L3.8.03Reserving actuary
Cumulative incurred claims (paid plus case reserves) on Calverin's general liability book, € millions, with paid to date on the latest diagonal:
| Accident year | Dev 1 | Dev 2 | Dev 3 | Dev 4 | Paid to date |
|---|---|---|---|---|---|
| 2022 | 100 | 150 | 165 | 168 | 160 |
| 2023 | 110 | 150 | 168 | 150 | |
| 2024 | 160 | 200 | 130 | ||
| 2025 | 140 | 55 |
Use the volume-weighted chain ladder with no tail beyond development year 4. Separately, the paid chain ladder on the same claims gives accident year 2025 a cumulative factor of 4.6.
Lessons: L3.8.03: Reserve risk
(a)What is the development factor from year 1 to year 2?
(b)What is the chain-ladder ultimate for accident year 2025? Answer in €m.
€m(c)What is the total IBNR across the four accident years? Answer in €m.
€m(d)The paid projection puts accident year 2025 at 55 × 4.6 = €253m, well above the incurred one. What should the actuary do?
Problem 4A storm, a hailstorm and the treaties
L3.8.05 · L3.8.10Reinsurance buyer
Calverin's cat model for its property account reduces to five independent events (gross, € millions):
| Event | Peril | Loss, €m | Rate a year |
|---|---|---|---|
| E1 | Windstorm | 60 | 0.100 |
| E2 | Flood | 120 | 0.040 |
| E3 | Windstorm | 250 | 0.015 |
| E4 | Hail | 400 | 0.006 |
| E5 | Windstorm | 700 | 0.003 |
Its cat XL is 300 xs 200 (€m), at a rate on line of 8%, with one reinstatement at 100%, pro rata to the amount recovered.
A reinsurer offers a 30% quota share on the property account: next year's earned premium €400m, expected loss ratio 65%, ceding commission 28% of the ceded premium. Commercial risks also go through a surplus treaty with a €5m line and 6 lines of cover.
Lessons: L3.8.05: Catastrophe risk · L3.8.10: Reinsurance in depth
(a)What is the average annual loss (AAL)? Answer in €m.
€m(b)What is the return period of a loss of €250m or more in a year? Answer in years.
years(c)The €400m hailstorm happens. How much does the cat XL recover? Answer in €m.
€m(d)What reinstatement premium does it trigger? Answer in €m.
€m(e)How much expected margin would the quota share cost Calverin a year (ceded premium less commission less expected ceded claims)? Answer in €m.
€m(f)A fire causes an €8m loss at a factory insured for €20m. How much of it does Calverin keep after the surplus treaty? Answer in €m.
€m
Problem 5Deaths in a term book, years in an annuity book
L3.8.06 · L3.8.07Life actuary
Calverin's life table has l50 = 98,000 and l51 = 97,706. Its term book has 5,000 independent lives aged 50, each with a sum assured of €100,000.
It also pays 1,500 pensions of €12,000 a year, at each year end while the pensioner is alive, for at most 20 years. For a quick reading, take a mortality rate of 2.5% every year (as if the pensioners did not age) and a flat discount rate of 3.5%.
Lessons: L3.8.06: Mortality risk · L3.8.07: Longevity risk
(a)What is the standard deviation of the number of deaths in the term book next year?
deaths(b)Solvency II's mortality stress raises mortality rates by 15%. How much does it add to next year's expected term claims? Answer in euros.
€(c)What is the annuity factor for one pension (the value of €1 a year)?
(d)The longevity stress cuts mortality rates by 20%. By how much does the value of the pensions rise? Answer in €m.
€m
Problem 6Policyholders' options, costs and claims
L3.8.08 · L3.8.09Life and health risk manager
Calverin's life model points, € millions. Surrender value = account value less the surrender charge. The changes in best estimate (ΔBE) under lapse up and lapse down come from the projection model:
| Model point | Account value | Surrender charge | Best estimate | ΔBE lapse up | ΔBE lapse down |
|---|---|---|---|---|---|
| Savings, 0.5% guarantee | 500 | 2% | 470 | 6.0 | −4.0 |
| Savings, 2.5% guarantee | 300 | 0% | 320 | −3.0 | 5.0 |
| Term assurance | – | – | −15 | 2.5 | −1.5 |
Term policies have no surrender value. Cohort B's account of €300m has 5 years to run with a guaranteed 2.5% a year; on the expected path the fund earns 1.5% a year after the insurer's margin, and the discount rate is a flat 2.25%.
The present value of all future life expenses is €80m at 2% expense inflation, and would be €86m at 3%.
Income protection: 10,000 healthy policyholders aged 50 and 200 claimants. Over the year a healthy life dies with probability 0.4%; survivors fall disabled with probability 1.5%. A claimant dies with probability 0.8%; survivors recover with probability 20%.
Lessons: L3.8.08: Lapse and policyholder behaviour · L3.8.09: Expense, health and morbidity risk
(a)In the year a surrender penalty ends, one cohort's best-estimate lapse rate is 50%. What rate does the lapse-down scenario use for that year? Answer in %.
%(b)What is the lapse risk charge on the three model points? Answer in €m.
€m(c)What is the intrinsic value of cohort B's guarantee? Answer in €m.
€m(d)What does the expense stress (+10% on expenses, +1 point of inflation) cost? Answer in €m.
€m(e)How many claimants will there be at the end of the year?
lives
Problem 7Matching the bonds to the promises
L3.8.11ALM manager
Calverin holds €3,000m of bonds with a modified duration of 6.0 years against best-estimate liabilities of €2,600m with a duration of 8.5 years. Rates move in parallel; take the first-order (duration) effect only, before tax.
(a)What is the duration gap, D_A − (L/A)·D_L? Answer in years.
years(b)How much do own funds change if rates fall by 100bp? Answer in €m, with its sign.
€m(c)How much would Calverin have to move from 2-year bonds (duration 1.9) into 20-year bonds (duration 16.4) to close the gap? Answer in €m.
€m
Problem 8From modules to a solvency ratio
L3.8.04 · L3.8.12 · L3.8.10Chief risk officer
Net premium (Vprem) and reserve (Vres) volumes and the standard formula's standard deviations for Calverin's two non-life segments, € millions (σprem already includes the 80% factor for non-proportional reinsurance):
| Segment | V prem | σ prem | V res | σ res |
|---|---|---|---|---|
| Motor vehicle liability | 300 | 8.0% | 500 | 9% |
| Fire and other damage to property | 400 | 6.4% | 250 | 10% |
Within a segment, premium and reserve risk are correlated 0.5; the two segments are correlated 0.25. Combined with the cat risk of Problem 4 (€216m, the 1-in-200 event net, with its reinstatement premium) at a correlation of 0.25, the non-life module comes to €368.8m.
The other modules: market €450m and life €300m. They are aggregated with the standard formula's correlations:
| Market | Life | Non-life | |
|---|---|---|---|
| Market | 1.00 | 0.25 | 0.25 |
| Life | 0.25 | 1.00 | 0.00 |
| Non-life | 0.25 | 0.00 | 1.00 |
Operational risk adds €40m and the deferred-tax adjustment is −€120m. Own funds are €1,100m. Taking up Problem 4's quota share would cut the non-life module to €340.1m; operational risk and the tax adjustment would not change.
Lessons: L3.8.04: Premium risk and the underwriting cycle · L3.8.12: Insurer investment risk and the ORSA · L3.8.10: Reinsurance in depth
(a)What is the capital charge for premium and reserve risk (3σV)? Answer in €m.
€m(b)What is the BSCR? Answer in €m.
€m(c)How much does diversification save, as a share of the sum of the market, life and non-life modules? Answer in %.
%(d)What is the solvency ratio? Answer in %.
%(e)What would the quota share cost a year per €1 of SCR it releases?
€ per €1 of SCR