How IBNR Works
On the day an insurer closes its books, some of the claims it owes have already happened and nobody has told it yet. A hospital bill still in the post; an injury that will not be reported for a year. Incurred but not reported: the insurer has to put a number on claims it cannot see. Press Next and the demo builds that number one step at a time, for a health book and for a motor book, or skip the story and reserve it yourself.
What you are looking at
Claims arrive after the event
A claim is owed from the day the accident or the treatment happens, not the day the insurer hears about it. On the day the accounts close there is always a layer of claims that have happened and not been reported. The insurer cannot see them, but it owes them, so it has to hold a reserve for them: the IBNR.
The past says how the young periods will grow
Nobody tries to guess the missing claims one by one. Actuaries look at how older periods filled in as they aged, in the development triangle, and assume the recent periods will fill in the same way. The chain ladder applies that pattern to what has been reported; Bornhuetter-Ferguson applies it to what the business was expected to cost.
A book where the answer is known
Every claim is simulated on this page, with a day it happened, a day it was reported and an amount, so the future the insurer cannot see is sitting in memory. Reveal lifts the curtain and scores the estimate. New data draws a fresh book from the same business, and the twist breaks the method the way it breaks in practice.
The maths, briefly
Three lines carry the whole method. The demo shows each of them: the factors along the bottom of the triangle are the first, the teal cells rolling forward are the second, and the Bornhuetter-Ferguson switch is the third.
The age-to-age factor
f(j) = Σᵢ C(i, j+1) / Σᵢ C(i, j) over every period i that has reached age j+1
C(i, j) is the cumulative amount on the books for accident period i after j periods of development: one cell of the triangle. The factor is how much the total grew in one step of age, pooled over every period old enough to have made that step. Pooling by volume lets the big periods count for more than the small ones.
The chain ladder
ultimate(i) = latest(i) × f(a) × f(a+1) × … IBNR(i) = ultimate(i) − latest(i)
Each period starts from its latest actual, at age a on the diagonal, and is multiplied along by every factor still ahead of it. The product of those factors is the cumulative development factor; its inverse is the share already reported, the percentage line under the demo's triangle. What is left after taking away the latest actual is the reserve.
Bornhuetter-Ferguson
IBNR(i) = premium(i) × ELR × (1 − 1 / CDF(a))
The same development pattern, applied to a different starting point: the expected cost of the business, premium times an expected loss ratio set when it was priced. What has been reported is kept as it is and never multiplied, so a slow month of reporting or a single huge claim in the youngest period does not ripple through the reserve. The price is that the answer leans on the expected loss ratio being right.
Where this gets used
Every insurer, every quarter, on every line of business. The method is the same; what changes is how long the tail is and what bends the pattern.
Health schemes, every month
Medical and health books reserve monthly with exactly this triangle, usually called a lag or completion-factor triangle in that world. The tail is short, so the reserve is mostly the last two or three months, and the thing that moves it is the speed of claims processing. A backlog, a new claims system or a change in how providers bill all bend the pattern, which is why the latest months are often reserved on an expected cost basis instead.
Motor, and anything with injuries
Where injuries are involved, claims are reported late and valued in stages as treatment, rehabilitation and legal costs become known, so the triangle keeps developing for years. The reserve covers both the claims not yet reported and the growth still to come on the ones that have been (IBNER). Large claims are usually taken out of the triangle and reserved on their own merits, which is the lesson of the demo's twist.
The balance sheet
Under IFRS 17 the reserve for claims that have already happened is part of the liability for incurred claims, and under Solvency II it is the claims provision in the best estimate. Either way it is usually one of the largest numbers on an insurer's balance sheet, it is audited, and a few percent of error on it can be the whole year's profit.
Pricing and planning
The ultimates from the triangle, not the amounts reported so far, are what an actuary compares with premium to see whether a product is priced right. Recent periods always look cheap before they have finished developing; reading them at face value is the classic way a book looks profitable for two years and then isn't.
A demonstration of the method, not a reserving tool: a real book needs its large claims, its changes in claims handling and its tail looked at by someone who knows the business. That judgement, backed by the same triangles and methods, is the reserving and actuarial work we do for clients.
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