Bornhuetter-Ferguson Versus Chain Ladder

Why Compare the Methods?

Bornhuetter-Ferguson (BF) and chain ladder use the same observed loss amount and selected emergence pattern, but they do not treat the observed amount in the same way. Chain ladder projects the observed losses to ultimate. BF adds an expected amount for the portion of losses not yet emerged. Comparing the methods shows exactly how much the BF estimate is pulled toward its a priori expected ultimate, and why that influence generally fades as experience matures.

The comparison is meaningful only when both estimates use the same loss basis, valuation date, and emergence pattern. For example, if the observed amount is incurred loss, the expected ultimate and selected emergence proportion must also be appropriate for incurred loss. The results below use a single cumulative observed loss amount and a selected cumulative proportion emerged.

Definition: Let $C$ be cumulative loss observed to date, $p$ the proportion of ultimate loss expected to have emerged by the valuation date, and $E$ the a priori expected ultimate loss. Assume $0<p\leq 1$, with $C$ and $E$ measured on a consistent basis. The chain ladder estimate is $U_{\mathrm{CL}}=C/p$. The Bornhuetter-Ferguson estimate is $U_{\mathrm{BF}}=C+(1-p)E$. The corresponding unpaid estimates are $R_{\mathrm{CL}}=U_{\mathrm{CL}}-C$ and $R_{\mathrm{BF}}=U_{\mathrm{BF}}-C$.

Here, $1-p$ is the expected unreported share of ultimate losses. In practice, $p$ is associated with a selected development pattern; it is not itself a loss development factor. The corresponding factor from the observed maturity to ultimate is $1/p$. Chain ladder applies that factor to $C$, while BF uses $E$ to estimate the unreported portion.

Algebraic Relationship

Subtract the BF estimate from the chain ladder estimate, keeping the sign convention explicit: the difference in this tutorial is $U_{\mathrm{CL}}-U_{\mathrm{BF}}$.

\begin{aligned} U_{\mathrm{CL}}-U_{\mathrm{BF}} &=\frac{C}{p}-\left[C+(1-p)E\right] \\ &=C\left(\frac{1-p}{p}\right)-(1-p)E \\ &=(1-p)\left(\frac{C}{p}-E\right) \\ &=(1-p)(U_{\mathrm{CL}}-E). \end{aligned}

This identity explains the comparison. The chain ladder estimate gives full weight to the loss experience observed so far, scaled to ultimate. BF adds only the expected unreported amount to the observed losses. Equivalently, because $C=pU_{\mathrm{CL}}$, BF can be written as:

U_{\mathrm{BF}}=pU_{\mathrm{CL}}+(1-p)E.

Thus, BF is a weighted average of the chain ladder indication and the a priori expected ultimate, with weights $p$ and $1-p$. This is an algebraic description of the estimate, not a statement that the selected prior is statistically independent of the experience.

Interpretation: If $U_{\mathrm{CL}}>E$, then $U_{\mathrm{CL}}-U_{\mathrm{BF}}>0$, so chain ladder is higher than BF. If $U_{\mathrm{CL}}<E$, then chain ladder is lower than BF. If the two indications agree, the estimates agree. For a fixed difference $U_{\mathrm{CL}}-E$, the gap between the methods is multiplied by the unreported share $1-p$, and therefore shrinks as $p$ approaches 1.

The same difference applies to unpaid estimates because both reserves subtract the same observed amount $C$:

R_{\mathrm{CL}}-R_{\mathrm{BF}} =(U_{\mathrm{CL}}-C)-(U_{\mathrm{BF}}-C) =U_{\mathrm{CL}}-U_{\mathrm{BF}}.

The “fixed difference” qualification matters. Across accident years, $U_{\mathrm{CL}}-E$ may also change with the observed losses, prior estimate, and selected emergence. It is therefore not universally true that every more mature year must have a smaller numerical difference. The algebra shows the effect of maturity holding the gap between the chain ladder indication and prior fixed.

Worked Comparisons

Worked Example: Comparing Ultimate Loss and Reserve for One Cohort

A hypothetical liability coverage cohort has cumulative incurred loss $C=\$360{,}000$. The selected incurred emergence proportion is $p=0.60$, and the a priori expected ultimate is $E=\$700{,}000$. Calculate both ultimate estimates, both unpaid estimates, and the signed difference $U_{\mathrm{CL}}-U_{\mathrm{BF}}$. All amounts are in dollars.

Method. Use the same incurred-loss emergence proportion for both methods. Chain ladder projects observed incurred loss by $1/p$; BF adds the expected unreported share of the prior ultimate.

Chain ladder. The factor to ultimate is $1/0.60=1.6667$. Therefore,

U_{\mathrm{CL}}=\frac{\$360{,}000}{0.60}=\$600{,}000.

Bornhuetter-Ferguson. The unreported share is $1-0.60=0.40$. The expected unreported amount is $0.40(\$700{,}000)=\$280{,}000$, so

U_{\mathrm{BF}}=\$360{,}000+\$280{,}000=\$640{,}000.

Unpaid estimates. Subtract the observed incurred amount from each ultimate:

R_{\mathrm{CL}}=\$600{,}000-\$360{,}000=\$240{,}000,\qquad R_{\mathrm{BF}}=\$640{,}000-\$360{,}000=\$280{,}000.

Signed comparison and check. The difference is $\$600{,}000-\$640{,}000=-\$40{,}000$. The algebraic identity gives the same answer:

(1-p)(U_{\mathrm{CL}}-E) =0.40(\$600{,}000-\$700{,}000) =0.40(-\$100{,}000) =-\$40{,}000.

The negative sign means chain ladder is below BF. The reserve difference is also $\$240{,}000-\$280{,}000=-\$40{,}000$, as it must be because the same observed amount was subtracted from both ultimates. The common mistake avoided here is to assume that chain ladder must exceed BF; the direction depends on the comparison of the chain ladder indication with $E$.

Worked Example: How the Difference Changes with Maturity

Consider three hypothetical valuation points for a cohort. For each point, suppose the chain ladder indicated ultimate remains $\$800{,}000$, while the a priori expected ultimate is $\$700{,}000$. The selected emergence proportions are $0.20$, $0.50$, and $0.80$. Calculate the observed amount implied by each maturity and the BF estimate, then compare the results.

Method. Since $C=pU_{\mathrm{CL}}$, calculate $C$ at each valuation point. Then calculate $U_{\mathrm{BF}}=C+(1-p)E$. Amounts are rounded to the nearest dollar only at the end; the arithmetic here is exact in dollars.

At $p=0.20$, the implied observed amount is $0.20(\$800{,}000)=\$160{,}000$. BF adds $0.80(\$700{,}000)=\$560{,}000$:

U_{\mathrm{BF}}=\$160{,}000+\$560{,}000=\$720{,}000,\qquad U_{\mathrm{CL}}-U_{\mathrm{BF}}=\$800{,}000-\$720{,}000=\$80{,}000.

At $p=0.50$, $C=0.50(\$800{,}000)=\$400{,}000$, and the BF unreported amount is $0.50(\$700{,}000)=\$350{,}000$:

U_{\mathrm{BF}}=\$400{,}000+\$350{,}000=\$750{,}000,\qquad U_{\mathrm{CL}}-U_{\mathrm{BF}}=\$50{,}000.

At $p=0.80$, $C=0.80(\$800{,}000)=\$640{,}000$, and the BF unreported amount is $0.20(\$700{,}000)=\$140{,}000$:

U_{\mathrm{BF}}=\$640{,}000+\$140{,}000=\$780{,}000,\qquad U_{\mathrm{CL}}-U_{\mathrm{BF}}=\$20{,}000.

Check. The identity gives the three differences directly: $0.80(\$100{,}000)=\$80{,}000$, $0.50(\$100{,}000)=\$50{,}000$, and $0.20(\$100{,}000)=\$20{,}000$. With the gap between the chain ladder indication and prior fixed at $\$100{,}000$, the difference narrows as the unreported share falls.

Worked Example: Aggregating Two Cohorts

A hypothetical insurer evaluates two distinct cohorts on a consistent loss basis. For Year A, $C_A=\$450{,}000$, $p_A=0.75$, and $E_A=\$600{,}000$. For Year B, $C_B=\$240{,}000$, $p_B=0.40$, and $E_B=\$700{,}000$. Calculate aggregate chain ladder and BF ultimates, and the signed aggregate difference.

Method. Calculate each cohort separately using its own emergence proportion and prior expected ultimate, then sum the cohort estimates. The aggregate difference is the sum of the cohort-level differences.

For Year A, the chain ladder ultimate is $\$450{,}000/0.75=\$600{,}000$. The BF unreported amount is $0.25(\$600{,}000)=\$150{,}000$, giving

U_{\mathrm{BF},A}=\$450{,}000+\$150{,}000=\$600{,}000.

For Year B, the chain ladder ultimate is $\$240{,}000/0.40=\$600{,}000$. The BF unreported amount is $0.60(\$700{,}000)=\$420{,}000$, giving

U_{\mathrm{BF},B}=\$240{,}000+\$420{,}000=\$660{,}000.

Sum the two chain ladder estimates and the two BF estimates:

U_{\mathrm{CL,agg}}=\$600{,}000+\$600{,}000=\$1{,}200{,}000,

U_{\mathrm{BF,agg}}=\$600{,}000+\$660{,}000=\$1{,}260{,}000.

The signed aggregate difference is $\$1{,}200{,}000-\$1{,}260{,}000=-\$60{,}000$. As a check, Year A contributes $0.25(\$600{,}000-\$600{,}000)=\$0$; Year B contributes $0.60(\$600{,}000-\$700{,}000)=-\$60{,}000$. The total is therefore $-\$60{,}000$. The common aggregation trap is to calculate each cohort correctly but make an arithmetic error when summing BF estimates: $\$600{,}000+\$660{,}000$ is $\$1{,}260{,}000$, not $\$1{,}270{,}000$.

Assumptions and Common Exam Traps

These formulas are comparisons of selected estimates, not guarantees that either estimate equals the eventual observed ultimate. Their use requires a consistent basis and a selected emergence proportion that is meaningful for the cohort and loss measure being estimated. The chain ladder calculation also requires $p>0$; at $p=0$, $C/p$ is undefined. BF can still be represented as $E$ when no losses have emerged, but that boundary case does not provide a chain ladder estimate.

A frequent error is to use the chain ladder ultimate and then add the BF expected unreported amount. In the first example, that incorrect calculation would be $\$600{,}000+\$280{,}000=\$880{,}000$. It counts the observed losses projected to ultimate and then adds a further expected unreported amount, even though the chain ladder ultimate already includes an estimate for all losses. The correct BF calculation uses the observed amount $C$, not $U_{\mathrm{CL}}$, as its first term: $\$360{,}000+\$280{,}000=\$640{,}000$.

Another trap is reversing the signed difference. Always state the order before interpreting the sign. Here the convention is chain ladder minus BF. A negative result means BF is higher; a positive result means chain ladder is higher. Finally, a higher maturity does not by itself determine which estimate is higher: the sign is set by $U_{\mathrm{CL}}-E$, while $1-p$ controls how much of that gap remains in the comparison.

Check the inputs.
Confirm that $C$, $E$, and $p$ use the same loss basis and valuation date, and that $0<p\leq 1$.

Calculate both estimates.
Use $C/p$ for chain ladder and $C+(1-p)E$ for BF. Do not add the BF unreported amount to the chain ladder ultimate.

Compare and interpret.
Calculate $U_{\mathrm{CL}}-U_{\mathrm{BF}}=(1-p)(U_{\mathrm{CL}}-E)$, then use the sign convention to identify which estimate is larger.

Verify the reserve comparison.
Subtract $C$ from each ultimate. The difference between unpaid estimates must equal the difference between ultimate estimates.

Key takeaway: The difference between chain ladder and BF is the unreported share, $1-p$, multiplied by the gap between the chain ladder indication and the a priori expected ultimate. This relationship explains both the direction of the comparison and why, for a fixed gap, the difference diminishes as losses emerge.

Check Your Understanding

For each problem, use chain ladder minus BF as the signed difference unless stated otherwise.

A hypothetical cohort has $C=\$315{,}000$, $p=0.70$, and $E=\$420{,}000$. Calculate both ultimate estimates and their signed difference. (Answer: $U_{\mathrm{CL}}=\$450{,}000$, $U_{\mathrm{BF}}=\$441{,}000$, difference $=\$9{,}000$.)
A cohort has $C=\$280{,}000$, $p=0.35$, and $E=\$900{,}000$. Calculate both ultimates and the signed difference. (Answer: $U_{\mathrm{CL}}=\$800{,}000$, $U_{\mathrm{BF}}=\$865{,}000$, difference $=-\$65{,}000$.)
For Cohort A, $C=\$250{,}000$, $p=0.50$, and $E=\$400{,}000$. For Cohort B, $C=\$480{,}000$, $p=0.80$, and $E=\$550{,}000$. Calculate aggregate chain ladder and BF ultimates and their signed difference. (Answer: aggregate $U_{\mathrm{CL}}=\$1{,}100{,}000$, aggregate $U_{\mathrm{BF}}=\$1{,}040{,}000$, difference $=\$60{,}000$.)
A cohort has chain ladder ultimate $\$1{,}200{,}000$, prior expected ultimate $\$1{,}050{,}000$, and BF ultimate $\$1{,}110{,}000$. Find the emergence proportion $p$ and observed amount $C$. (Answer: $p=0.40$, $C=\$480{,}000$.)
A cohort has $C=\$210{,}000$, $p=0.30$, and $E=\$500{,}000$. Calculate each unpaid estimate and their signed difference. (Answer: $R_{\mathrm{CL}}=\$490{,}000$, $R_{\mathrm{BF}}=\$350{,}000$, $R_{\mathrm{CL}}-R_{\mathrm{BF}}=\$140{,}000$.)

What You'll Learn