The average cost per claim method: developing numbers and severity separately

CM2 · Financial Engineering & Loss Reserving · 10 min read

Two things can change, and the money shows only their product

Claims cost more this year than last. There are two possible reasons and they call for different responses:

• there were more claims — frequency has risen, perhaps because more policies were sold or the risk got worse;
• each claim cost more — severity has risen, perhaps because of inflation or a change in what the policy covers.

A triangle of claim amounts shows only the product of the two. The average cost per claim method separates them: develop a triangle of claim numbers, develop a triangle of average cost per claim, and multiply the two projections together.

In words: project how many, project how much each, multiply.

The extra arithmetic buys two things. The projection is often more stable, because frequency and severity usually develop in quite different patterns and averaging them together blurs both. And the output is diagnostic: it says which of the two is driving the answer, which an amounts-only projection cannot.

Building the three triangles

A question gives two of the three and you derive the third:

Compute that cell by cell, giving a triangle in money per claim. The numbers triangle and the average-cost triangle are then developed separately, and the amounts triangle is set aside — it is only used at the end, as a check.

One decision has to be made and stated. For the numbers triangle a volume-weighted development factor is natural, because the entries are counts and a year with more claims genuinely carries more information. For the average cost triangle it is not: the entries are already averages, so volume-weighting them weights an average by its own size, which is not obviously right. A simple average of the individual year-on-year ratios, or grossing-up factors, is the usual choice. Either is defensible. Saying which you used, and why, is the examinable part; the figures move very little either way on a well-behaved triangle.

What "number of claims" means, which matters more than it sounds

Three definitions are in use and they develop differently:

• Claims reported. Grows as late reports come in, then stops. This is the usual basis.
• Claims settled. Grows for much longer, because settlement lags reporting, and it will lag the amounts triangle.
• Claims outstanding. Rises and then falls, so it cannot be developed by this method at all.

The basis has to match the amounts. Dividing cumulative amounts paid by claims reported gives an average that is not the average cost of a claim at all — it is money paid so far divided by claims known so far, and it rises towards the true average as the claims settle. That ratio is still perfectly usable, because the development factors are measured on the same inconsistent basis throughout and the inconsistency cancels. What it is not is a quantity you can quote to anyone as "the average claim". A question that asks you to comment on a figure like that is testing whether you noticed.

Zero claims, and other traps in the data

Two things that look like data problems and are really definitional ones.

Nil claims. A claim reported and then settled for nothing still counts in the numbers triangle and contributes nothing to the amounts. A portfolio with many nil claims therefore has a low average cost that says more about the reporting convention than about the risk, and a change in how nil claims are recorded will move the whole average-cost triangle without anything real having happened.

Large claims. One very large claim moves an average far more than it moves a total made up of many claims. That is a real weakness of this method relative to the chain ladder: averages are less robust than aggregates. The standard practice is to strip out claims above a threshold, project the rest by this method, and reserve for the large ones separately.

When to prefer it, and when not

Use it when frequency and severity are known to be moving differently, when the numbers data is reliable and reasonably complete, or when the reserve has to be explained to somebody — because "we expect 1,397 claims at an average of ₹9,190" is a sentence a board can question, and "the development factor is 1.725" is not.

Prefer the chain ladder when the claim-number data is poor or inconsistently recorded, when the portfolio is dominated by a few large claims, or when nothing suggests frequency and severity are behaving differently — in which case the extra machinery buys nothing and the extra assumptions can only add error.

The check that should always be run

This method and a basic chain ladder on the amounts triangle are projecting the same claims from the same data. On coherent data they should land close together, and the worked example below has them 0.02% apart.

So run both, always. If they agree, the reserve is well supported and the split tells you something extra for free. If they disagree materially, that is the finding: something in the relationship between numbers and amounts has changed, and the disagreement points at where. A reserving answer that produces one number from one method and stops has left the most informative part of the exercise undone.

Where marks are lost

Developing the amounts triangle as well and then multiplying by something. The method develops numbers and average cost. The amounts triangle is derived, not projected.

Averaging the average costs across accident years to get a factor, when the question wanted grossing up. Both appear. State which.

Multiplying an ultimate number of claims by a current average cost. Both have to be at ultimate. Using the latest observed average understates the answer, because the average is still developing upwards.

Reporting the ultimate rather than the reserve. The reserve is ultimate minus paid to date, as always.

Mixing units between the two triangles. If amounts are in thousands then the average cost is in thousands per claim, and multiplying by a claim count returns thousands. Write the units at the top and keep them.

Questions students ask

Why not just develop the amounts, which is less work?

Because the extra work is what tells you whether the projection should be trusted. The example below has claim numbers rising 16% across three accident years while average cost rises 7%. A chain ladder on amounts gives almost the same reserve and cannot distinguish those two, so it also cannot tell you that the growth is mostly more claims rather than dearer ones — which is the difference between a book that is growing and a book that is deteriorating.

What if claim numbers are still developing when amounts have stopped?

Then the two triangles have different development lengths, which is normal and is handled by using each triangle's own factors over its own span. What you must not do is assume one is fully developed because the other is. If reported numbers are complete after two years but payments run for five, the numbers triangle has factors of 1 in its later columns while the average-cost triangle does not, and the method handles that without modification.

Does inflation need adjusting for here too?

It sits entirely in the average-cost triangle, which is a cleaner place for it than in the chain ladder, because severity is where inflation acts. The same restatement as the inflation-adjusted chain ladder can be applied to the average-cost triangle alone, leaving the numbers triangle untouched — and that combination is the most defensible of the four methods when both frequency and inflation are moving.

How is this different from Bornhuetter-Ferguson?

They fix different problems. This method addresses what is changing by splitting the claims in two, and still takes its level entirely from the triangle. Bornhuetter-Ferguson addresses too little data by bringing in premiums and a loss ratio, and does not split anything. They can be used together: develop numbers and average cost by this method for the older years, and apply Bornhuetter-Ferguson to the newest where neither triangle has much to say.

A worked example

A portfolio of general insurance policies, with claims fully run off by the end of development year 2. Amounts are in ₹ thousand, cumulative.

Both triangles are invented for this example. They are not a real insurer's experience.

Cumulative claims paid

Cumulative number of claims reported

Claims paid to date total 10,350 + 10,080 + 7,440 = ₹27,870 thousand.

Step 1: the average cost per claim triangle

Divide cell by cell. The result is in ₹ thousand per claim.

Check one: 9,000 / 1,150 = 7.826087. Note that these averages rise along each row, which is expected on a paid basis — money keeps arriving for claims already counted.

Step 2: develop the claim numbers (volume-weighted)

Step 3: develop the average cost (simple average of the ratios)

The individual year-on-year ratios for dev 0 to dev 1 are

and for dev 1 to dev 2 there is only one observation:

Step 4: multiply, and take off what has been paid

The reserve is ₹6,907.72 thousand. The 2023 row is a free check: fully developed, so its projected ultimate must come back to the ₹10,350 already paid, and 1,200 × 8.625 = 10,350.00 exactly.

The check: a basic chain ladder on the amounts

This uses the amounts triangle only and never touches the claim numbers, so it is a genuinely independent route to the same reserve.

Two methods, two different sets of development factors, one answer to within a fortieth of a per cent. The small residual is explained: the numbers and amounts factors were volume-weighted while the average-cost factors were a simple average, so the three sets of weights are not quite consistent. Had the gap been 5% rather than 0.02%, that would be the finding of the question rather than a footnote.

What the split tells you that the chain ladder cannot

Across the three accident years, claim numbers are up 16.4% and the average claim is up 6.6%. The total cost rose by 24%, and the split says that about two-thirds of that is more claims and one-third is dearer ones.

That is the sentence the exercise exists to produce, and the chain ladder cannot produce it. It also leads somewhere: 6.6% over two years is about 3.3% a year, which is a plausible claims inflation figure and needs no further explanation. The 16.4% growth in claim numbers does need one. If the policy count grew by a similar amount the book is simply bigger and the loss ratio is intact; if it did not, claim frequency has risen and the pricing basis is wrong. The reserve is the same either way, and the business consequence is not.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

• Utility theory and risk aversion: non-satiation, A(w) and R(w)
• Measures of investment risk: variance, semi-variance, VaR and Tail VaR
• CAPM and the security market line: beta and the market price of risk
• Ito's lemma explained term by term, and what it does to a share price
• The chain ladder method: estimating outstanding claims from a run-off triangle
• The Bornhuetter-Ferguson method: the chain ladder with a second opinion
• Binomial option pricing: where the risk-neutral probability comes from
• The Black-Scholes formula and put-call parity, read term by term
• Ruin theory: the adjustment coefficient and Lundberg bound explained
• Mean-variance portfolio theory: the efficient frontier and why diversification works
• The term structure: spot rates, forward rates and par yields
• Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared
• The Merton model: company equity as an option, and where a credit spread comes from
• The Greeks: delta, gamma, vega, theta and rho, and how a hedge decays
• The inflation-adjusted chain ladder: taking past inflation out and putting a future assumption in
• Valuing a maturity guarantee by simulation, and checking it against the formula
• The Efficient Markets Hypothesis: the three forms, and how each one is tested
• Single-index and multifactor models: getting a covariance matrix out of a few betas
• Excess-of-loss reinsurance: the retained claim, and choosing the retention
• State price deflators: pricing with real-world probabilities
• What makes a risk insurable, and what to do when it almost is
• Two-state and intensity credit models: backing a default rate out of a bond price
• The martingale representation theorem, and why hedging is possible at all

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.