Impact of A Priori Loss Ratio Selections

Why the A Priori Loss Ratio Matters

In Bornhuetter-Ferguson (BF), the a priori loss ratio translates premium into an expected ultimate loss amount. The method combines that expected amount with observed losses: observed losses are retained, while the expected loss estimate is used for the portion not yet emerged. Consequently, if observed losses and the emergence assumption are fixed, a different selected a priori loss ratio changes the estimate only through the unreported portion.

This distinction is useful when evaluating the effect of a selection without confusing it with changes in the data or development assumptions. The comparison in this tutorial holds earned premium, observed incurred losses, and the selected emergence percentage constant. It changes only the a priori loss ratio. All loss and premium amounts are assumed to be on a consistent basis, for the same coverage and valuation context, and at the same price level.

Definition: Let $EP$ be earned premium, $r$ the selected a priori loss ratio, $I$ observed incurred losses, and $q$ the expected percentage of ultimate losses emerged at the valuation date. The a priori expected ultimate loss is $EP r$. The unreported share is $1-q$, and the BF ultimate estimate is $$ U_{\text{BF}}=I+(1-q)EP r. $$ For an incurred-based estimate, the BF IBNR amount is $U_{\text{BF}}-I=(1-q)EP r$. This is not necessarily total unpaid loss: total unpaid loss is ultimate less paid loss, and incurred losses already include paid losses plus case reserves.

Model, Sensitivity, and Interpretation

The formula assumes that the selected emergence percentage $q$ is applicable to the losses being estimated, so the expected unreported fraction is $1-q$. It also assumes the a priori expected loss amount $EP r$ is appropriate for the same premium and loss basis as the observed incurred amount. BF does not replace the observed amount with a development-scaled value; it adds an estimate for the portion expected to remain unreported.

For this comparison, all inputs are treated as deterministic selections. No interest discounting or mortality assumption is used, and no stochastic loss distribution is assumed. These calculations estimate losses at the valuation date; they do not discount future payments.

\begin{aligned} E_{\text{prior}} &= EP r,\\ R_{\text{IBNR}} &= (1-q)E_{\text{prior}}=(1-q)EP r,\\ U_{\text{BF}} &= I+R_{\text{IBNR}}. \end{aligned}

The sensitivity to $r$ follows directly. Suppose two selections $r_1$ and $r_2$ are applied to the same $EP$, $I$, and $q$. Subtracting their BF estimates gives

\begin{aligned} U_{\text{BF}}(r_2)-U_{\text{BF}}(r_1) &=[I+(1-q)EP r_2]-[I+(1-q)EP r_1]\\ &=(1-q)EP(r_2-r_1). \end{aligned}

The observed losses cancel because they are held fixed. The IBNR estimates differ by exactly the same amount as the ultimate estimates, since $I$ is unchanged. Thus a one-percentage-point change in the a priori loss ratio does not change BF ultimate by one percent of premium in general; its dollar effect is one percent of premium multiplied by the unreported share.

Fix the comparison basis.
Confirm that premium, observed losses, and a priori loss ratios refer to the same coverage, valuation, and loss basis.

Calculate the unreported share.
For an emergence percentage $q$, use $1-q$, not $q$, as the fraction of expected ultimate losses still unreported.

Recalculate the estimate for each selection.
For each $r$, calculate $(1-q)EP r$, then add the same observed incurred amount $I$.

Check the change.
The difference between the estimates must equal $(1-q)EP$ multiplied by the difference in selected loss ratios.

Worked Comparisons

Worked Example: Comparing Two Selections for One Accident Year

A hypothetical commercial equipment portfolio has earned premium of $2,400,000 and observed incurred losses of $1,080,000. The selected expected emergence is 60%. Compare a priori loss ratios of 55% and 65%, and calculate BF ultimate losses and incurred-based IBNR for each selection.

Model and assumptions. Use the premium-based incurred BF formula. Premium, incurred losses, and both loss ratio selections are on the same basis. Inputs are deterministic; there is no interest discounting, mortality assumption, or assumed stochastic loss distribution.

Step 1: Find the unreported share. The emerged percentage is $q=0.60$, so the unreported share is $1-q=0.40$.

Step 2: Apply the 55% selection. The a priori expected ultimate is $EP r=\$2{,}400{,}000(0.55)=\$1{,}320{,}000$. The IBNR estimate is $0.40(\$1{,}320{,}000)=\$528{,}000$. Therefore, $U_{\text{BF}}=\$1{,}080{,}000+\$528{,}000=\$1{,}608{,}000$.

Step 3: Apply the 65% selection. The a priori expected ultimate is $\$2{,}400{,}000(0.65)=\$1{,}560{,}000$. The IBNR estimate is $0.40(\$1{,}560{,}000)=\$624{,}000$. Therefore, $U_{\text{BF}}=\$1{,}080{,}000+\$624{,}000=\$1{,}704{,}000$.

Step 4: Check the difference. The loss ratio increased by 0.10. The formula for the change gives $0.40(\$2{,}400{,}000)(0.10)=\$96{,}000$. This agrees with $\$1{,}704{,}000-\$1{,}608{,}000=\$96{,}000$. The method avoids attributing any change to observed losses or emergence, which were fixed.

Worked Example: Comparing Selections Across Two Accident Years

For a hypothetical specialty insurer, two accident years have earned premiums of $1,500,000 and $2,000,000. Their observed incurred losses are $750,000 and $1,300,000, and their expected emergence percentages are 75% and 50%, respectively. Compare a 60% a priori loss ratio with a 70% selection for the combined BF ultimate and incurred-based IBNR.

Model and assumptions. Apply the premium-based incurred BF formula to each year, then sum the year-level estimates. All figures are on a consistent basis. This is a deterministic, undiscounted calculation: no interest, mortality, or stochastic loss distribution is assumed.

Step 1: Determine each unreported share. The first year has unreported share $1-0.75=0.25$. The second has unreported share $1-0.50=0.50$.

Step 2: Calculate IBNR at 60%. For the first year, the a priori expected ultimate is $\$1{,}500{,}000(0.60)=\$900{,}000$; IBNR is $0.25(\$900{,}000)=\$225{,}000$. For the second year, expected ultimate is $\$2{,}000{,}000(0.60)=\$1{,}200{,}000$; IBNR is $0.50(\$1{,}200{,}000)=\$600{,}000$. Combined IBNR is $\$225{,}000+\$600{,}000=\$825{,}000$. Combined observed incurred losses are $\$750{,}000+\$1{,}300{,}000=\$2{,}050{,}000$, so combined BF ultimate is $\$2{,}050{,}000+\$825{,}000=\$2{,}875{,}000$.

Step 3: Calculate IBNR at 70%. For the first year, expected ultimate is $\$1{,}500{,}000(0.70)=\$1{,}050{,}000$; IBNR is $0.25(\$1{,}050{,}000)=\$262{,}500$. For the second year, expected ultimate is $\$2{,}000{,}000(0.70)=\$1{,}400{,}000$; IBNR is $0.50(\$1{,}400{,}000)=\$700{,}000$. Combined IBNR is $\$962{,}500$, and combined BF ultimate is $\$2{,}050{,}000+\$962{,}500=\$3{,}012{,}500$.

Step 4: Check the change. The increase is $\$3{,}012{,}500-\$2{,}875{,}000=\$137{,}500$. The sensitivity formula gives $[0.25(\$1{,}500{,}000)+0.50(\$2{,}000{,}000)](0.10)=(\$375{,}000+\$1{,}000{,}000)(0.10)=\$137{,}500$. The years must be weighted by their own unreported shares; a single average emergence percentage would not preserve this calculation automatically.

Worked Example: Solving for a Loss Ratio Under an Ultimate Target

A hypothetical liability portfolio has earned premium of $3,600,000, observed incurred losses of $1,440,000, and 80% expected emergence. Compare selections of 60% and 75%, then determine the largest a priori loss ratio consistent with a BF ultimate no greater than $1,950,000.

Model and assumptions. Use $U_{\text{BF}}=I+(1-q)EP r$, treating all inputs as fixed and consistently based. There is no interest discounting, mortality assumption, or stochastic loss distribution.

Step 1: Calculate the unreported share and sensitivity coefficient. The unreported share is $1-0.80=0.20$. Thus $(1-q)EP=0.20(\$3{,}600{,}000)=\$720{,}000$ per unit of loss ratio.

Step 2: Calculate both selected estimates. At 60%, IBNR is $\$720{,}000(0.60)=\$432{,}000$, and ultimate is $\$1{,}440{,}000+\$432{,}000=\$1{,}872{,}000$. At 75%, IBNR is $\$720{,}000(0.75)=\$540{,}000$, and ultimate is $\$1{,}440{,}000+\$540{,}000=\$1{,}980{,}000$.

Step 3: Solve the target condition. Require $\$1{,}440{,}000+\$720{,}000r\leq\$1{,}950{,}000$. Subtracting observed losses gives $\$720{,}000r\leq\$510{,}000$, so $r\leq\$510{,}000/\$720{,}000=0.708333\ldots$. The largest ratio is 70.83% to two decimal places.

Step 4: Check. At the unrounded threshold, IBNR is $510,000 and ultimate is exactly $1,950,000. At 75%, the estimate is $30,000 above the target; this also equals $\$720{,}000(0.75-0.708333\ldots)=\$30{,}000$, subject to rounding.

Common Mistake and Practical Checks

A frequent error is to use the emerged percentage as the factor on the a priori expected ultimate when calculating IBNR. In the first worked example at a 65% loss ratio, that incorrect calculation would be $0.60(\$1{,}560{,}000)=\$936{,}000$, producing an incorrect ultimate of $\$1{,}080{,}000+\$936{,}000=\$2{,}016{,}000$. The 60% is the share expected to have emerged, not the share expected to remain unreported. The correct IBNR is $0.40(\$1{,}560{,}000)=\$624{,}000$, and the correct ultimate is $1,704,000.

Another check is that, with positive premium and an emergence percentage below 100%, increasing the selected a priori loss ratio must increase the BF ultimate. If $q=1$, the unreported share is zero and the BF estimate equals observed losses, so changing $r$ has no effect. If $q$ is close to 1, the effect is correspondingly small. If a comparison changes observed losses, premium, or emergence as well as $r$, it no longer isolates the impact of the a priori loss ratio.

Keep the reserve label precise. In an incurred BF calculation, $(1-q)EP r$ is the estimated IBNR amount added to observed incurred losses. It does not include case reserves already contained in incurred losses. A total unpaid estimate would require paid losses to calculate ultimate less paid; that is a different quantity.

Key takeaway: With premium, observed incurred losses, and emergence held fixed, a change in the selected a priori loss ratio changes BF ultimate and incurred-based IBNR by $(1-q)EP$ times the change in that ratio. The lower the unreported share, the smaller the impact of the selection.

Check Your Understanding

For each problem, use the stated incurred BF assumptions and keep amounts on the given basis.

A hypothetical portfolio has earned premium of $800,000, observed incurred losses of $310,000, a selected a priori loss ratio of 62.5%, and 70% expected emergence. Find incurred-based IBNR and BF ultimate. (Answer: IBNR $150,000; ultimate $460,000.)
Premium is $1,200,000, observed incurred losses are $540,000, and expected emergence is 60%. By how much does BF ultimate increase when the selected loss ratio changes from 55% to 65%? Give the ultimate under each selection. (Answer: Increase $48,000; ultimate $804,000 at 55% and $852,000 at 65%.)
For two hypothetical years, premiums are $900,000 and $1,400,000; observed incurred losses are $400,000 and $720,000; emergence is 80% and 50%. At a common 68% loss ratio, find total incurred-based IBNR and total BF ultimate. (Answer: IBNR $598,400; ultimate $1,718,400.)
Premium is $2,500,000, observed incurred losses are $1,050,000, and expected emergence is 65%. What is the largest a priori loss ratio that gives a BF ultimate of no more than $1,400,000? (Answer: 40.00%.)
A portfolio has premium of $4,000,000, observed incurred losses of $2,200,000, and 75% expected emergence. Calculate BF ultimate at a 58% loss ratio, then find the increase in ultimate if the selection rises by 3 percentage points. (Answer: Ultimate at 58% is $2,780,000; increase is $30,000.)

What You'll Learn