Bornhuetter-Ferguson Sensitivity to Development and Expected Losses

Why Sensitivity Matters

A Bornhuetter-Ferguson (BF) estimate depends on both an expected ultimate loss amount and an expected loss emergence pattern. Those selections can be uncertain. A sensitivity table makes their effects visible: it shows how the estimated unpaid amount and ultimate loss change when one or both selections change, while holding the observed loss amount fixed.

The key distinction is between the emerged and unreported portions of expected ultimate losses. If the selected cumulative emergence percentage is $p$, then the selected unreported percentage is $1-p$. The BF method uses the observed cumulative losses for the emerged portion and the a priori expected ultimate for the unreported portion. Sensitivity analysis changes these inputs systematically; it does not, by itself, establish which selection is most appropriate.

Definition: For an accident period at a particular evaluation age, let $C$ be observed cumulative losses, $E$ be the selected expected ultimate losses on the same basis, and $p$ be the selected cumulative expected emergence percentage. The unreported percentage is $q=1-p$. The BF estimate of unpaid losses is $\hat R^{BF}=E(1-p)$, and the BF estimate of ultimate losses is $\hat U^{BF}=C+E(1-p)$.

Here, $p$ is a proportion from 0 to 1, not a development factor. The inputs must be consistent: $C$ and $E$ should refer to the same loss definition, coverage, accident period, and evaluation basis, while $p$ must describe the expected emergence of that same loss measure. The formula assumes that $p$ is an appropriate selected emergence percentage for the evaluation age and that $E$ is the expected ultimate for the period.

Why the BF Sensitivity Formula Works

The formula follows by dividing the expected ultimate into two parts. Under the selected emergence pattern, the expected emerged amount is $Ep$, and the expected unreported amount is $E(1-p)$. BF replaces the expected emerged amount with the observed cumulative amount $C$, while retaining the expected unreported amount. Therefore:

\hat U^{BF}=C+E(1-p)

This construction explains the different roles of the inputs. A change in $C$ changes the ultimate estimate dollar for dollar. A change in $E$ changes only the expected unreported component, scaled by $1-p$. A change in $p$ also acts only through that component. Holding the other inputs fixed, the sensitivities are:

\frac{\partial \hat R^{BF}}{\partial E}=1-p,\qquad \frac{\partial \hat R^{BF}}{\partial p}=-E,\qquad \frac{\partial \hat U^{BF}}{\partial C}=1

Thus, when $p$ is high, relatively little of a change in $E$ flows into the BF estimate because little is expected to remain unreported. Conversely, a one-percentage-point increase in $p$ reduces the estimate by $0.01E$, if $E$ is held fixed. These are local, one-input-at-a-time effects; changing $E$ and $p$ together can produce a different combined result.

Building and Reading a Sensitivity Table

A useful table states what is held fixed, identifies the alternative assumptions, and reports a clearly defined output. A table of unpaid amounts isolates the effect of $E$ and $p$. To obtain ultimate losses, add the applicable observed cumulative losses $C$ to each unpaid amount. If comparing multiple maturity levels, use the observed cumulative loss for each maturity when calculating the corresponding ultimate estimate.

Fix the observed data.
Record cumulative losses at each evaluation age. Do not change these values merely because an expected-loss selection changes.

Choose alternative assumptions.
List plausible expected ultimate amounts and emergence percentages. Convert each emergence percentage to an unreported share using $1-p$.

Calculate each table cell.
For every combination, calculate $E(1-p)$. Add the relevant observed cumulative losses if the table is to show ultimate estimates.

Interpret the pattern.
Check that unpaid amounts rise with $E$ and fall with $p$, when the other input is fixed. Describe the size of the changes in context.

Worked Example: Two Selections at One Maturity

A hypothetical commercial equipment portfolio has $C=\$4.20$ million in cumulative losses at the evaluation date. The selected expected ultimate is either $\$6.00$ million or $\$6.60$ million, and the selected emergence percentage is either 60% or 70%. Construct a table showing both unpaid and ultimate estimates.

Method. Apply the BF formula separately to each combination. The observed amount remains $\$4.20$ million in all four cells.

For $E=\$6.00$ million and $p=60\%$, the unreported share is $1-0.60=0.40$. The unpaid estimate is $6.00(0.40)=\$2.40$ million, so the ultimate estimate is $4.20+2.40=\$6.60$ million.

For $E=\$6.00$ million and $p=70\%$, the unreported share is $0.30$. The unpaid estimate is $6.00(0.30)=\$1.80$ million, and the ultimate estimate is $4.20+1.80=\$6.00$ million.

For $E=\$6.60$ million and $p=60\%$, the unpaid estimate is $6.60(0.40)=\$2.64$ million, and the ultimate estimate is $4.20+2.64=\$6.84$ million.

For $E=\$6.60$ million and $p=70\%$, the unpaid estimate is $6.60(0.30)=\$1.98$ million, and the ultimate estimate is $4.20+1.98=\$6.18$ million.

Expected ultimate $E$ ($ millions)	Emergence $p$	Unpaid estimate ($ millions)	Ultimate estimate ($ millions)
6.00	60%	2.40	6.60
6.00	70%	1.80	6.00
6.60	60%	2.64	6.84
6.60	70%	1.98	6.18

The table passes two directional checks: increasing $E$ increases the unpaid estimate at either emergence level, while increasing $p$ decreases it at either expected ultimate. For example, increasing $E$ from $\$6.00$ million to $\$6.60$ million at $p=60\%$ raises unpaid losses by $\$0.24$ million, equal to $0.40(0.60)$ million. The method avoids treating the entire change in expected ultimate as a change in BF ultimate; only the unreported share of that change enters.

Sensitivity Across Maturity Levels

Maturity matters because the unreported share is generally smaller at a more mature evaluation age. Consequently, the same range of expected ultimate selections can have a larger effect on the BF unpaid estimate at an earlier maturity. This statement concerns the effect of changing $E$ with $p$ fixed or compared on a consistent basis; actual sensitivity comparisons must use the selected emergence percentages appropriate to the ages being analyzed.

Worked Example: Expected-Loss Sensitivity at Three Maturities

A hypothetical liability portfolio has selected expected ultimate losses of $\$7.20$, $\$8.00$, or $\$8.80$ million. At three evaluation ages, cumulative emergence is selected as 35%, 60%, and 80%, respectively. Observed cumulative losses at those ages are $\$2.50$, $\$4.60$, and $\$6.10$ million. Calculate the unpaid sensitivity table and the ultimate estimates for the central expected ultimate selection of $\$8.00$ million.

Method. At each maturity, calculate $E(1-p)$ for all three expected ultimates. To form central-selection ultimates, add the observed amount for that maturity to the unpaid amount.

At 35% emergence, $1-p=0.65$. The unpaid estimates are $7.20(0.65)=\$4.68$ million, $8.00(0.65)=\$5.20$ million, and $8.80(0.65)=\$5.72$ million.

At 60% emergence, $1-p=0.40$. The unpaid estimates are $7.20(0.40)=\$2.88$ million, $8.00(0.40)=\$3.20$ million, and $8.80(0.40)=\$3.52$ million.

At 80% emergence, $1-p=0.20$. The unpaid estimates are $7.20(0.20)=\$1.44$ million, $8.00(0.20)=\$1.60$ million, and $8.80(0.20)=\$1.76$ million.

Emergence	Unpaid at $E=7.20$ ($ millions)	Unpaid at $E=8.00$ ($ millions)	Unpaid at $E=8.80$ ($ millions)	Ultimate at $E=8.00$ ($ millions)
35%	4.68	5.20	5.72	2.50 + 5.20 = 7.70
60%	2.88	3.20	3.52	4.60 + 3.20 = 7.80
80%	1.44	1.60	1.76	6.10 + 1.60 = 7.70

The range in unpaid estimates across the three expected ultimate selections is $\$1.04$ million at 35% emergence ($5.72-4.68$), $\$0.64$ million at 60% emergence ($3.52-2.88$), and $\$0.32$ million at 80% emergence ($1.76-1.44$). The range narrows with maturity because the expected unreported share falls. The central-selection ultimate estimates are not required to move monotonically across ages: each combines an age-specific observed amount with an age-specific expected unpaid amount. This example avoids comparing unpaid amounts across ages as if they were ultimate estimates.

Joint Changes and Interpretation

A two-way sensitivity table can vary both assumptions together. Its corner values show the combined effect of the selected alternatives, not the isolated effect of either assumption. To explain a result, compare cells that differ in only one input first; then describe the combined range. This prevents attributing a joint change entirely to the expected ultimate or entirely to emergence.

Worked Example: A Joint Range for Expected Loss and Emergence

For a hypothetical specialty policy group, observed cumulative losses are $C=\$3.10$ million. The central expected ultimate is $\$5.00$ million, with alternatives 8% lower and 8% higher. The central emergence selection is 62%, with alternatives four percentage points lower and higher. Calculate the reserve sensitivity table and the ultimate range.

Method. The expected ultimate alternatives are $5.00(0.92)=\$4.60$, $\$5.00$, and $5.00(1.08)=\$5.40$ million. The emergence alternatives are 58%, 62%, and 66%; their unreported shares are 42%, 38%, and 34%.

For 58% emergence, the unpaid estimates are $4.60(0.42)=\$1.932$ million, $5.00(0.42)=\$2.100$ million, and $5.40(0.42)=\$2.268$ million.

For 62% emergence, they are $4.60(0.38)=\$1.748$ million, $5.00(0.38)=\$1.900$ million, and $5.40(0.38)=\$2.052$ million.

For 66% emergence, they are $4.60(0.34)=\$1.564$ million, $5.00(0.34)=\$1.700$ million, and $5.40(0.34)=\$1.836$ million.

Emergence	Unpaid at $E=4.60$ ($ millions)	Unpaid at $E=5.00$ ($ millions)	Unpaid at $E=5.40$ ($ millions)
58%	1.932	2.100	2.268
62%	1.748	1.900	2.052
66%	1.564	1.700	1.836

The smallest ultimate estimate is $3.100+1.564=\$4.664$ million, at $E=\$4.60$ million and 66% emergence. The largest is $3.100+2.268=\$5.368$ million, at $E=\$5.40$ million and 58% emergence. The resulting range is $\$0.704$ million. As a check, the central assumptions give an unpaid estimate of $5.00(0.38)=\$1.900$ million and an ultimate estimate of $3.100+1.900=\$5.000$ million. The equality to the central expected ultimate is coincidental to these selected values; it is not a general BF identity.

Common Mistake and Practical Checks

A frequent error is multiplying expected ultimate by the emergence percentage to estimate unpaid losses. For example, with $E=\$5.00$ million and $p=62\%$, that incorrect calculation gives $5.00(0.62)=\$3.10$ million as unpaid losses. It uses the expected emerged share, not the expected unreported share. The correction is:

\hat R^{BF}=5.00(1-0.62)=5.00(0.38)=\$1.90\text{ million}

The incorrect amount would overstate unpaid losses by $\$1.20$ million because $3.10-1.90=1.20$. When the observed cumulative amount is $\$3.10$ million, the corrected ultimate is $3.10+1.90=\$5.00$ million. A second trap is to change $E$ and $p$ in a table but leave the unreported share unchanged. Recalculate $1-p$ in every column or row where emergence changes.

Before relying on a sensitivity table, check that all percentages are expressed as proportions in the calculation, that every cell uses the correct maturity-specific $p$, and that observed losses are added only when calculating ultimate estimates. A sensitivity range is a measure of responsiveness to the chosen scenarios; it is not automatically a confidence interval, a probability-weighted estimate, or evidence that the endpoints are equally plausible. The analysis is most useful when assumptions and the reason for varying them are stated explicitly.

Key takeaway: Under BF, the sensitivity of unpaid losses to expected ultimate losses is scaled by the unreported share. At a fixed expected ultimate, a higher emergence percentage reduces unpaid losses. Separate those effects in the table, then interpret the combined scenarios without confusing a sensitivity range with a measure of statistical uncertainty.

Check Your Understanding

For each problem, calculate the requested BF quantities using the stated expected ultimate and emergence selections.

A hypothetical accident period has observed cumulative losses of $\$1.80$ million, expected ultimate losses of $\$3.00$ million, and 40% emergence. Find unpaid and ultimate estimates. (Answer: unpaid $\$1.80$ million; ultimate $\$3.60$ million.)
A hypothetical cargo portfolio has $C=\$4.25$ million, $E=\$5.50$ million, and emergence of 70%. Find the BF unpaid and ultimate estimates. (Answer: unpaid $\$1.65$ million; ultimate $\$5.90$ million.)
For a hypothetical claim group, $E=\$4.80$ million, $C=\$2.20$ million, and emergence is initially 55%. If emergence increases by five percentage points while $E$ and $C$ remain fixed, calculate the change in unpaid and ultimate estimates. (Answer: both estimates decrease by $\$0.24$ million; the new unpaid estimate is $\$1.92$ million and the new ultimate is $\$4.12$ million.)
A hypothetical program has $C=\$5.10$ million. Consider $E=\$6.40,\ \$7.00,\ \$7.60$ million and emergence of 75% or 80%. Find the smallest and largest ultimate estimates among all combinations. (Answer: smallest $\$6.38$ million at $E=\$6.40$ million and 80% emergence; largest $\$7.00$ million at $E=\$7.60$ million and 75% emergence.)
At an earlier maturity for a hypothetical portfolio, $C=\$2.60$ million and emergence is 30%. At a later maturity, $C=\$6.00$ million and emergence is 70%. Use $E=\$9.00$ million at both ages to calculate each unpaid and ultimate estimate, then compare the unpaid estimates. (Answer: earlier unpaid $\$6.30$ million and ultimate $\$8.90$ million; later unpaid $\$2.70$ million and ultimate $\$8.70$ million; unpaid decreases by $\$3.60$ million.)

What You'll Learn