Every answer and its explanation appears here once you have finished the path. Each one then links to the matching glossary entry, where the concept is set out in full with its worked example.
1. In March 2017 the United Kingdom cut the Ogden discount rate from 2.5% to minus 0.75%, raising at a stroke the value of severe bodily injury claims settled as lump sums. In British motor triangles, development factors break along a single diagonal, several accident years hit at once. What should an actuary read in that shape?
A calendar year effect, hitting every open accident year together, which chaining cannot handle as it stands
A development factor is the ratio of cumulative losses at one age to cumulative losses at the prior age, within a single accident year. Chaining rests on a strong and rarely stated assumption: each accident year develops the way earlier ones did. So a triangle reads in two directions, and telling them apart is most of the craft. An anomaly along a row is an accident year effect, particular to one year of claims, a single large file for instance. An anomaly along a diagonal is a calendar year effect: something happened on a date, and that thing reaches every still open accident year on the same day. The Ogden change is the textbook case, because it does not alter the claims of any one year but the value of every unsettled bodily injury file, whatever year it arose in. Averaging factors without seeing the diagonal then spreads a one off shock across the whole future projection, as though it were due to recur annually. Treatments exist, restating the diagonal or methods that separate calendar inflation explicitly, but all of them start from having recognised the shape.
Glossary entry · ldf2. Two accident years are projected on the same day: last year's, seen at twelve months of development, and 2021's, seen at forty-eight months. Both receive a cumulative factor drawn from the same triangle. Which carries the greater estimation risk, and why?
The more recent one, because its ultimate rests on a long chain of factors in which the first error multiplies
The cumulative factor is the product of every development factor chained from a given age to ultimate, and the estimated ultimate equals the observed cumulative multiplied by that product. The reserve for claims not yet reported then follows as the difference. The consequence is arithmetic and worth seeing plainly: a year seen at twelve months has observed almost nothing and owes everything to its chain, while a year seen at forty-eight months has already run off most of its development and owes only a tail factor close to one. Each factor in the chain is itself an estimate, computed on a limited number of historical years, and the uncertainties do not cancel: they compound. A five point error on the first factor of a long chain moves the ultimate far more than the same error on a tail factor. This is why the most recent accident year is at once the one most discussed at closing and the one least known, and why methods that anchor it on something other than its own data exist at all.
Glossary entry · cdf3. Across financial years 2022 to 2024, several insurers and reinsurers strengthened reserves on their 2015 to 2019 United States casualty accident years, under the effect of drifting court awards. What is the name for the gap thus revealed between the ultimate first estimated and the ultimate that emerges?
The run off result, adverse in this case
Ultimate loss is the expected final cost of an accident year's claims once all development is done. It gathers what has been paid, what is reserved on open files, and what has not yet been reported. Its peculiarity is that it becomes certain only on the day the year is fully run off, which takes years in a long tailed line and sometimes decades in casualty. Until then it is not a fact but an estimate revised at every closing. The gap between the ultimate you announced and the one that emerges has a name, the run off result, and its sign is readable in the accounts: favourable when old years settle for less than expected, adverse when they cost more. The United States sequence on the late 2010s accident years is instructive because it shows a slow mechanism: severe bodily injury files settle late, so drifting court awards do not show in the early development years, and the correction arrives in a bunch when it arrives. A run off result that is adverse year after year does not say that one year went badly, it says the estimation method itself was running behind.
Glossary entry · charge-ultime4. An accident year carries an a priori loss ratio of 68%, taken from the pricing plan. At twelve months the observed loss ratio comes out at 52%, and the year is only about three tenths developed. The Bornhuetter-Ferguson method still weights the a priori more heavily than the observation. Why?
Because the weighting follows the developed share, and at this stage the observation is not yet credible
The a priori is a prediction made before claims have had time to speak, drawn from the pricing plan, from market analysis, or from a credibility weighted average of earlier years. It anchors the Bornhuetter-Ferguson method, whose idea fits in a sentence: on the share of the year already developed, believe what you observe, and on the share still to come, believe what you forecast. At twelve months in a long tailed line the developed share is small, so the a priori dominates mechanically, which is the intended behaviour, since pure chaining on a tiny cumulative would amplify every fluctuation. The trade off is the real subject, and it is formidable. The method is only as good as its anchor, and a systematically optimistic a priori produces under reserving that does not show at once, precisely because the method declines to let observation contradict it early. Chaining reacts faster, and harder. Revising the a priori every year, testing past anchors against the ultimates that have since emerged, is therefore less a formality than an audit of the tool by its own results.
Glossary entry · loss-ratio-a-priori5. Two notes present the same liability book. One reports an average severity of 18,000 EUR, the other 47,000 EUR. Neither is wrong. What is missing for the two figures to become comparable?
The capping threshold applied, and the separate treatment given to losses above it
Capping means limiting each loss to a threshold for the purposes of analysis, and handling the excess separately. This is not presentational convenience but statistical necessity: attritional claims, frequent and moderate, and large claims, rare and heavy, do not follow the same laws. Blending them into one average yields a number that describes neither, unstable from year to year according to whether a large file happens to be present. Kept apart, the lower layer yields to classical modelling and the upper layer to distribution tail tools. The practical consequence is the one in the question, and it is worth keeping as a rule of reading: an average severity without its threshold means nothing. The same book can show eighteen thousand euros capped at five hundred thousand and forty seven thousand uncapped, and the gap then measures no difference in risk but a difference in convention. Any comparison between two books, two years or two brokers begins with checking that the threshold is the same.
Glossary entry · ecretement6. A burning cost is computed on a book's 2015 to 2023 experience, adding up losses as they were paid at the time and relating them to the premiums of their own year. What structural defect affects the result?
Old years are expressed in the money and the structure of their time, which understates today's cost
As-if restating means recomputing historical losses at today's conditions: cost level, exchange rates, exposure, and cover structure. Without it, a long experience blends euros of unequal value and files that never met the same architecture of cover, and the resulting average describes no particular year. The bias is not symmetric, which makes it a defect rather than mere noise: the oldest years are those whose cost is most understated, so the longer the observation window, the further the raw burning cost is dragged down. In reinsurance the restatement covers a second dimension that inflation alone does not reach, and often the heavier one: simulating what past losses would have cost the treaty as it is structured today, at present retentions and limits. The same 2016 loss may have touched nothing at the time and cut through several layers under the current structure. The two restatements compound, they do not substitute for each other.
Glossary entry · as-if7. A construction claim paid 200,000 EUR in 2018 must enter an analysis run in 2024. The construction cost index used stands at 100 in 2018 and 126 in 2024, after the pressure on materials and labour through 2021 and 2022. What indexed value is used, and what caution goes with the calculation?
252,000 EUR, provided the chosen index matches the nature of the claim being indexed
Indexation applies to a past value the ratio of two indices, target over origin, and the arithmetic itself is trivial: two hundred thousand times one hundred and twenty six over one hundred gives two hundred and fifty two thousand. All the judgment sits in the choice of index, and that is where the reasoning is won or lost. A construction claim follows construction costs, a bodily injury claim follows medical and wage indices, a motor claim follows parts and repair labour. Using the consumer price index for everything looks neutral and is not: 2021 and 2022 showed it starkly, materials and freight having risen well beyond general inflation, so that a construction book indexed on consumer prices would have been restated several points short. Two cautions complete the gesture. The first is not to confuse the inflation bearing on the cost of each claim with drift in the number of claims, which calls for different treatment. The second is to check that the chosen index has not itself been rebuilt or rebased in the middle of the observation window.
Glossary entry · indexation8. A motor frequency series is used to project claims experience. The year 2020 sits in it, with frequency collapsed by lockdowns, then a return to earlier levels from 2021 and 2022. What should be done with it when estimating the trend?
Treat it as a level shock, distinct from a trend, on pain of biasing the projection
Trend is the systematic annual variation of a claims indicator, and the word systematic is what separates it from ordinary volatility. It usefully splits in two: a frequency trend, often slowly declining in motor under the effect of safety equipment, and a severity trend, generally rising with repair costs. Their product gives the pure premium trend, and projecting means applying the compounded factor across the years between observation and the target period. The year 2020 is the counterexample that illuminates the definition. Frequency did not follow a new trend that year, it took a level shock, caused by a dated external event, and it came back afterwards. Fitting a line through a series containing that trough produces one of two false readings depending on where the window is cut: an artificially negative slope if the series stops soon after, or an artificially positive one if it starts in 2020 and climbs. Sensitivity to the choice of window is the ordinary weakness of any trend, and an identified shock is handled by removing it or modelling it as such, never by letting it pass for a trend.
Glossary entry · trend9. Two insurers publish loss ratios that one wishes to compare. At the first, the claims department's overhead sits in general expenses; at the second, it is reserved with incurred losses. What does that single difference produce?
It moves the loss ratio and the expense ratio by several points in opposite directions, leaving the combined ratio unchanged
Claims handling expenses fall into two families worth telling apart. Those attaching to an identified file, expert fees, legal costs, investigation charges, are tracked file by file. The others cover the running of the claims department without being attributable to any single file: staff salaries and charges, rent, systems, management. This second family is not observed but estimated, usually as a percentage of gross reserves, and its order of magnitude is not trivial, a few points of incurred losses. Where the question turns practical is that the classification of these expenses is not uniform between insurers or between countries. Counting them in incurred losses inflates the loss ratio and lightens the expense ratio by as much; counting them in general expenses does the reverse. The combined ratio, which adds the two, is insensitive to the choice, and that is exactly what makes the error easy: one can compare two combined ratios safely and go badly wrong comparing two loss ratios. Before comparing any separated ratios, the convention used on each side is therefore the first thing to check.
Glossary entry · ulae