Chain Ladder And Bornhuetter-Ferguson Under Changing Emergence

Why Changing Emergence Matters

A development pattern describes how cumulative claims grow as an accident year matures. If recent claims are emerging faster than earlier claims, the factor from the current evaluation age to ultimate may be smaller than the historical factor. The choice matters: chain ladder projects from the observed cumulative amount, while Bornhuetter-Ferguson (BF) combines that amount with an expected ultimate and the proportion not yet emerged.

The comparison is not simply a choice between two formulas. It asks which pattern is suitable for the claims being projected, and whether the observed amount or an a priori expected ultimate should have more influence. The worked calculations below use hypothetical cumulative claim amounts. They are deterministic projections, not predictive distributions.

Definition: Let $C_j$ be cumulative claims at development age $j$, and let $f_k$ be the selected age-to-age development factor from age $k$ to age $k+1$. The cumulative development factor (CDF) from age $j$ to ultimate is the product of the selected future factors. Under the usual chain ladder emergence interpretation, the estimated proportion emerged at age $j$ is $1/\mathrm{CDF}_j$, and the proportion unpaid or unreported is $1-1/\mathrm{CDF}_j$.

Chain Ladder and Bornhuetter-Ferguson Under One Pattern

For a cumulative claims triangle, an age-to-age factor can be estimated from observations that are available at both ages. A common volume-weighted estimate is the sum of cumulative amounts at the later age divided by the sum at the earlier age. An actuary may select factors using broader experience or judgment instead; the selection and its rationale should be explicit.

Given selected factors, chain ladder estimates ultimate claims by multiplying the current cumulative amount by the CDF. BF estimates ultimate claims by adding the observed cumulative amount to an expected ultimate multiplied by the proportion expected to remain un-emerged. The expected ultimate must use a basis consistent with the claims being projected.

\widehat{U}_{\mathrm{CL}}=C_j\mathrm{CDF}_j, \qquad \widehat{U}_{\mathrm{BF}}=C_j+U_0\left(1-\frac{1}{\mathrm{CDF}_j}\right)

Here $U_0$ is the a priori expected ultimate, and the projected reserve is the estimated ultimate less the cumulative claims already observed. These methods use the same emergence assumption differently: chain ladder applies the CDF to the observed amount, whereas BF applies the estimated unpaid proportion to $U_0$.

Assumptions and limits: These formulas presume that the selected development pattern is appropriate for the accident year and that cumulative amounts, expected ultimate, and development factors use a consistent claims basis. The BF formula treats $U_0$ as an external expected ultimate rather than deriving it from the current observed claims. Chain ladder can be sensitive to immature observations and factor selection; BF can be sensitive to both the expected ultimate and the selected emergence pattern. Neither formula accounts automatically for a structural change in claims handling or mix.

Worked Example: One Accident Year Under Two Emergence Selections

A hypothetical accident year has cumulative claims of $600 at the current evaluation age. The historical selected CDF to ultimate is 1.50, while a recent pattern suggests a CDF of 1.25. The a priori expected ultimate is $900. Calculate chain ladder and BF ultimates and reserves under each pattern.

Model details. There is no discounting, so no interest rate is applied. No mortality assumption is relevant. No probability distribution is assumed: the inputs are deterministic cumulative claims, selected CDFs, and an a priori expected ultimate.

Historical-pattern chain ladder. The ultimate is $600(1.50)=\$900$. The reserve is $\$900-\$600=\$300$.

Recent-pattern chain ladder. The ultimate is $600(1.25)=\$750$. The reserve is $\$750-\$600=\$150$.

Historical-pattern BF. The emerged proportion is $1/1.50=2/3$, so the un-emerged proportion is $1-2/3=1/3$. The BF ultimate is $600+900(1/3)=\$900$, and the reserve is $\$900-\$600=\$300$.

Recent-pattern BF. The emerged proportion is $1/1.25=0.80$, so the un-emerged proportion is $0.20$. The BF ultimate is $600+900(0.20)=\$780$, and the reserve is $\$780-\$600=\$180$.

To the nearest cent, the historical-pattern reserve exceeds the recent-pattern reserve by $150.00 under chain ladder and by $120.00 under BF. The methods differ because BF retains an explicit expected-ultimate component; its recent-pattern projection is not simply the chain ladder result. The calculation is checked by subtracting $600 from each ultimate to recover the stated reserves.

Method and mistake avoided. Apply each CDF consistently, first converting it to an emerged or un-emerged proportion for BF. Do not treat a CDF of 1.25 as meaning that 25% remains unpaid.

Why the BF Emergence Formula Holds

The CDF links current cumulative claims to ultimate claims under the selected development pattern. If the pattern is applied proportionally, current cumulative claims represent $1/\mathrm{CDF}_j$ of ultimate claims. The complement is the fraction not yet emerged. BF applies that complement to the a priori expected ultimate to estimate future claims, then adds the claims already observed.

\text{emerged proportion}=\frac{C_j}{\widehat{U}_{\mathrm{CL}}} =\frac{C_j}{C_j\mathrm{CDF}_j} =\frac{1}{\mathrm{CDF}_j}

Therefore the BF estimate of future claims is $U_0(1-1/\mathrm{CDF}_j)$, and adding $C_j$ gives the BF ultimate. This justification depends on interpreting the CDF as a proportional emergence pattern. If that interpretation is inappropriate, the implied BF emergence percentage should not be used without adjustment.

Select a pattern.
State whether the CDF represents historical experience, recent experience, or another justified selection.

Convert to emergence.
Calculate the emerged proportion as $1/\mathrm{CDF}_j$ and the un-emerged proportion as $1-1/\mathrm{CDF}_j$.

Project both methods.
Multiply $C_j$ by the CDF for chain ladder; add the BF expected unpaid amount to $C_j$.

Compare reserves and assumptions.
Subtract $C_j$ from each ultimate, then identify whether differences arise from emergence, the a priori ultimate, or both.

Worked Example: Estimating a Historical Factor and Comparing a Recent Pattern

A hypothetical triangle has two historical 12-to-24-month observations: $700 develops to $1,050, and $800 develops to $1,120. A more recent year has $900 at 12 months and $1,080 at 24 months. Select a historical volume-weighted 12-to-24 factor, compare it with the recent factor, and project a new year with $1,000 at 12 months and an a priori ultimate of $1,500. Assume the 24-month-to-ultimate factor is 1.10 under either selection.

Model details. There is no discounting and no interest rate is applied. Mortality is not relevant. No probability distribution is assumed; all stated amounts and factors are deterministic inputs.

Historical factor. The later-age total is $1{,}050+1{,}120=2{,}170$, and the earlier-age total is $700+800=1{,}500$. Thus the volume-weighted factor is $2{,}170/1{,}500=1.4467$ to four decimal places. For the projection, select 1.45.

Recent factor. The recent 12-to-24 factor is $1{,}080/900=1.20$. This is lower than 1.45, indicating faster early emergence under this pattern. The historical CDF to ultimate is $1.45(1.10)=1.595$; the recent CDF is $1.20(1.10)=1.32$.

Historical-pattern chain ladder. The ultimate is $1{,}000(1.595)=\$1{,}595.00$, and the reserve is $\$1{,}595.00-\$1{,}000.00=\$595.00$.

Recent-pattern chain ladder. The ultimate is $1{,}000(1.32)=\$1{,}320.00$, and the reserve is $\$1{,}320.00-\$1{,}000.00=\$320.00$.

Historical-pattern BF. The un-emerged proportion is $1-1/1.595=0.3730408$. The expected unpaid amount is $1{,}500(0.3730408)=\$559.56$ to the nearest cent. The ultimate is $\$1{,}000.00+\$559.56=\$1{,}559.56$, giving a reserve of $559.56.

Recent-pattern BF. The un-emerged proportion is $1-1/1.32=0.2424242$. The expected unpaid amount is $1{,}500(0.2424242)=\$363.64$ to the nearest cent. The ultimate is $\$1{,}000.00+\$363.64=\$1{,}363.64$, giving a reserve of $363.64.

The recent-pattern selection reduces the chain ladder reserve by $275.00 and the BF reserve by $195.92, to the nearest cent. Each reserve equals its method's estimated future amount, and each ultimate equals the current $1,000 plus that reserve.

Method and mistake avoided. The historical factor is calculated from summed amounts, not by giving the two historical ratios equal weight. The recent factor is then kept distinct from the historical selection when forming the CDF.

Portfolio Reserve Impact When Patterns Change

A pattern change can affect accident years differently depending on their current cumulative amounts and BF expected ultimates. The following comparison shows why a portfolio-level difference should be calculated from the individual projections, rather than inferred from a single factor difference.

Worked Example: Aggregating the Reserve Impact Across Two Recent Years

Two hypothetical accident years are each evaluated at the same development age. Their observed cumulative claims are $900 and $1,100; their respective a priori expected ultimates are $1,400 and $1,600. Compare a historical CDF of 1.50 with a recent CDF of 1.25, for both chain ladder and BF.

Model details. There is no discounting and no interest rate is applied. Mortality is not relevant. No probability distribution is assumed; this is a deterministic aggregation of two projections.

Historical-pattern chain ladder. The two ultimates are $900(1.50)=\$1{,}350$ and $1{,}100(1.50)=\$1{,}650$. Total ultimate is $3,000, total observed claims are $2,000, and the aggregate reserve is $\$3{,}000-\$2{,}000=\$1{,}000$.

Recent-pattern chain ladder. The ultimates are $900(1.25)=\$1{,}125$ and $1{,}100(1.25)=\$1{,}375$. Total ultimate is $2,500, so the aggregate reserve is $\$2{,}500-\$2{,}000=\$500$.

Historical-pattern BF. The un-emerged proportion is $1-1/1.50=1/3$. Expected unpaid amounts are $1{,}400/3=\$466.67$ and $1{,}600/3=\$533.33$, to the nearest cent. Their total is $1,000.00, so total BF ultimate is $3,000.00 and the reserve is $1,000.00.

Recent-pattern BF. The un-emerged proportion is $1-1/1.25=0.20$. Expected unpaid amounts are $1{,}400(0.20)=\$280$ and $1{,}600(0.20)=\$320$. Their total is $600, so total BF ultimate is $2,600 and the reserve is $600.

The recent pattern reduces the aggregate chain ladder reserve by $500 and the aggregate BF reserve by $400. The BF ultimate remains above the chain ladder ultimate under the recent selection because BF's expected unpaid amount is based on the $3,000 aggregate a priori ultimate. The check is that each aggregate reserve plus $2,000 of observed claims equals its aggregate ultimate.

Method and mistake avoided. Calculate expected unpaid amounts separately from each year's $U_0$, then aggregate. Applying one aggregate expected ultimate is equivalent only when the same emergence proportion and a consistent expected-ultimate basis are used.

Interpreting the Difference and Avoiding Traps

A lower CDF implies a higher proportion emerged at the current age and a smaller proportion remains unpaid. Thus, if recent claims are genuinely emerging faster, a recent lower CDF will generally reduce both chain ladder and BF future amounts. The size of the reduction need not match: chain ladder scales observed claims, whereas BF scales the a priori expected ultimate.

The result alone does not establish that the recent pattern is superior. A factor may change because of claim handling, settlement speed, changes in claim mix, data limitations, or random variation. The actuary should assess whether the recent observations are representative of the years being projected and whether the expected ultimate is on the right basis. If the pattern differs by development age, use the selected age-specific factors to form the CDF rather than applying one factor indiscriminately.

Factor versus unpaid percentage: A CDF of 1.25 implies 80% emerged and 20% un-emerged, not 25% un-emerged.
BF uses the expected ultimate for future claims: Calculate $U_0(1-1/\mathrm{CDF})$, then add observed claims. Do not multiply $U_0$ by the CDF.
Keep bases consistent: Do not combine paid claims, incurred claims, or expected ultimates from incompatible bases without a justified adjustment.
Report the assumption behind the result: A projection under a recent pattern is conditional on that pattern being suitable, not proof that the emergence change will continue.

Exam takeaway: For each proposed emergence pattern, calculate its CDF, convert the CDF to the un-emerged proportion, and then calculate chain ladder and BF separately. Compare reserves only after subtracting observed cumulative claims from each ultimate, and state what assumption drives the difference.

Check Your Understanding

For each problem, use the stated pattern and expected ultimate as given. Amounts are hypothetical; round money to the nearest cent where needed.

A recent accident year has cumulative claims of $500, CDF 1.25, and a priori expected ultimate $700. Calculate the chain ladder and BF ultimates and reserves. (Answer: Chain ladder ultimate $625 and reserve $125; BF ultimate $640 and reserve $140.)
A year has $750 cumulative claims and a priori expected ultimate of $1,050. Compare historical CDF 1.50 with recent CDF 1.20 under both methods, giving reserves. (Answer: Historical chain ladder ultimate $1,125 and reserve $375; recent chain ladder ultimate $900 and reserve $150; historical BF ultimate $1,100 and reserve $350; recent BF ultimate $925 and reserve $175.)
Two historical 12-to-24 observations are $600 to $900 and $800 to $1,040. Select the volume-weighted factor. If the 24-to-ultimate factor is 1.10, calculate the chain ladder ultimate for a new year with $900 at 12 months. (Answer: The selected 12-to-24 factor is $1{,}940/1{,}400=1.3857$; the CDF is approximately 1.5243; the chain ladder ultimate is approximately $1,371.86.)
A year has $1,200 cumulative claims and a priori expected ultimate $1,800. Its historical CDF is 1.50 and the recent CDF is 1.25. How much does the recent pattern reduce the BF reserve compared with the historical pattern? (Answer: Historical reserve is $1{,}800(1-1/1.50)=\$600$; recent reserve is $1{,}800(1-1/1.25)=\$360$; reduction is $240.)
Two years have observed claims of $700 and $900, and a priori ultimates of $1,000 and $1,500. Using recent CDF 1.25, calculate the aggregate BF reserve and aggregate ultimate. (Answer: The un-emerged proportion is 0.20; the reserve is $0.20(1{,}000+1{,}500)=\$500$; aggregate ultimate is $\$700+\$900+\$500=\$2{,}100$.)

What You'll Learn