Selecting A Priori Loss Ratios Across Accident Years

Why A Priori Loss Ratios May Differ by Accident Year

In Bornhuetter-Ferguson (BF), the a priori loss ratio converts premium into an expected ultimate loss before considering how much loss has emerged. Selecting that ratio is therefore an assumption about the expected loss cost of the business in a particular accident year—not simply a calculation from that year’s reported losses. The selection should be supported by documented evidence and applied consistently with the premium and loss measures used in the BF calculation.

A common ratio across accident years can be reasonable when the years represent materially similar risks on a comparable basis. Separate ratios can be more appropriate when there is credible evidence of a difference, such as a documented change in business mix, coverage, underwriting, or expected loss cost. A difference in immature reported loss ratios, by itself, is not sufficient evidence: those ratios reflect both underlying losses and how much has emerged.

Definition: For accident year $i$, the a priori loss ratio $e_i$ is the selected expected ultimate loss per unit of the corresponding premium $P_i$. The a priori expected ultimate loss is $P_i e_i$. A common selection uses $e_i=e$ for all years in scope; differentiated selections allow $e_i$ to vary when documented assumptions support year-specific expected loss costs.

Selection and Application in Bornhuetter-Ferguson

Let $C_i$ be the losses reported to date for accident year $i$, and let $q_i$ be the selected proportion of ultimate losses expected to have emerged by the valuation date. The unreported share is $1-q_i$. The emergence measure must match the loss basis being used—for example, paid emergence with paid losses, or incurred emergence with incurred losses.

The BF method adds expected unreported losses to reported losses. Given the selected $e_i$ and $q_i$, it estimates the unreported portion as $P_i e_i(1-q_i)$. Therefore:

\widehat{U}^{BF}_i=C_i+P_i e_i(1-q_i)

Here, $\widehat{U}^{BF}_i$ is the estimated ultimate loss. The indicated unpaid amount on the same loss basis is $\widehat{U}^{BF}_i-C_i=P_i e_i(1-q_i)$. This form assumes the selected expected ultimate loss is $P_i e_i$ and that the selected emergence proportion applies to that expected ultimate. It does not require the reported losses to be a fixed proportion of the estimated ultimate.

Justification: If expected ultimate loss is $P_i e_i$, and the selected emergence pattern says the expected emerged share is $q_i$, then the expected unreported share is $1-q_i$. Multiplying gives expected unreported loss $P_i e_i(1-q_i)$. BF retains the actual reported amount $C_i$ and adds that expected unreported amount, giving the formula above. Dividing reported losses by $q_i$ would instead scale the observed amount and would not be this BF calculation.

The formula separates two judgments: $e_i$ reflects expected loss cost, while $q_i$ reflects expected emergence. When comparing accident years, document both selections. This tutorial focuses on selecting $e_i$; the emergence pattern still needs to be appropriate to the loss basis and valuation date.

When to Use a Common or Differentiated Selection

A common a priori ratio is most defensible when accident years are sufficiently comparable in expected loss cost and the underlying premium basis is consistent. It can avoid spurious year-to-year variation when the available evidence does not support separate estimates. “Common” is a modeling choice, not a rule that all years must share a ratio.

Differentiated ratios are more defensible when there is a meaningful, documented change in expected loss cost. Examples include a shift in the types of insureds or risks written, a coverage change, or a material change in underwriting criteria. The analyst should explain how the evidence supports both the direction and approximate size of each difference. If a change is uncertain or only weakly supported, a common or partially differentiated selection may be more stable than precise-looking year-specific ratios.

Useful documentation identifies the accident years in scope, the loss and premium measures, the common or year-specific selection, the evidence considered, and the reasons for any differences. It should also note limitations, such as sparse experience or a change whose effect is difficult to quantify. Historical experience can inform a selection, but an immature current-year reported loss ratio should not automatically be treated as that year’s a priori ratio.

Define the scope.
Specify the accident years and confirm that the premium and loss measures are intended to be comparable.

Assess expected loss-cost differences.
Review documented changes in risk mix, coverage, underwriting, or other relevant assumptions rather than treating immature reported ratios as direct measures of ultimate loss cost.

Select and explain the ratios.
Use a common $e$ if the evidence supports a shared expected loss cost; use $e_i$ by year if it supports specific differences. Record the rationale and uncertainty.

Apply each selection consistently.
Calculate expected ultimate loss $P_i e_i$, then expected unreported loss $P_i e_i(1-q_i)$, and add the latter to reported losses.

Worked Applications

Worked Example: Differentiating Ratios After a Documented Risk Shift

A hypothetical insurer’s commercial equipment portfolio changes between two accident years. Its documented underwriting review supports a 62% a priori loss ratio for Year 1 and a 70% ratio for Year 2, reflecting a shift toward risks with higher expected loss cost. Premium and reported losses are in thousands of dollars. Year 1 premium is $4,000, reported losses are $1,900, and 70% of ultimate losses are expected to have emerged. Year 2 premium is $5,000, reported losses are $1,650, and 45% are expected to have emerged. Apply BF separately by year. This is a deterministic estimate: no interest discounting, mortality assumption, or probability distribution is used.

Method and assumptions. Use the documented year-specific selections $e_1=0.62$ and $e_2=0.70$. The reported loss basis matches the stated emergence proportions $q_1=0.70$ and $q_2=0.45$. The premiums are stated on the basis intended for this calculation.

Year 1. The unreported share is $1-0.70=0.30$. Expected ultimate loss is $4{,}000(0.62)=2{,}480$. Expected unreported loss is $2{,}480(0.30)=744$. Thus the BF ultimate is $1{,}900+744=2{,}644$, and the indicated unpaid amount is $744$.

Year 2. The unreported share is $1-0.45=0.55$. Expected ultimate loss is $5{,}000(0.70)=3{,}500$. Expected unreported loss is $3{,}500(0.55)=1{,}925$. Thus the BF ultimate is $1{,}650+1{,}925=3{,}575$, and the indicated unpaid amount is $1{,}925$.

Check and common mistake avoided. For each year, ultimate less reported equals the expected unreported amount: $2{,}644-1{,}900=744$ and $3{,}575-1{,}650=1{,}925$. Applying the Year 1 ratio to both years would ignore the documented change in expected loss cost; using separate ratios is justified here by that evidence, not merely by differing reported ratios.

Worked Example: Retaining a Common Ratio for Comparable Years

Suppose two hypothetical accident years cover a stable group of small storage facilities. A documented review supports a common expected loss ratio of 68% for both years, with no identified change that warrants a different expected loss-cost selection. Year 1 has premium of $3,000 thousand, reported losses of $1,224 thousand, and 60% emergence. Year 2 has premium of $4,000 thousand, reported losses of $2,176 thousand, and 80% emergence. Calculate both BF ultimates and unpaid amounts. This is a deterministic estimate with no interest discounting, mortality assumption, or probability distribution.

Method and assumptions. Use $e_1=e_2=0.68$. The stated emergence proportions apply to the reported-loss basis: $q_1=0.60$ and $q_2=0.80$.

Year 1. Expected ultimate loss is $3{,}000(0.68)=2{,}040$. The unreported share is $1-0.60=0.40$, so expected unreported loss is $2{,}040(0.40)=816$. BF ultimate is $1{,}224+816=2{,}040$. The unpaid amount is $816$.

Year 2. Expected ultimate loss is $4{,}000(0.68)=2{,}720$. The unreported share is $1-0.80=0.20$, so expected unreported loss is $2{,}720(0.20)=544$. BF ultimate is $2{,}176+544=2{,}720$. The unpaid amount is $544$.

Check and common mistake avoided. The reported losses equal 60% of $2,040 thousand and 80% of $2,720 thousand, respectively, consistent with the given emergence assumptions. The fact that the reported ratios differ does not require different a priori ratios: here they differ in line with the different emergence proportions. The common selection follows the evidence about comparable expected loss cost.

Worked Example: Adjusting for a Change in Business Mix

A hypothetical liability portfolio has two risk groups. A documented analysis supports a 50% a priori loss ratio for Group A and 90% for Group B. In Year 1, the premium mix is $1,500 thousand for A and $500 thousand for B. In Year 2, it is $500 thousand for A and $1,500 thousand for B. Each year has reported losses of $800 thousand and $1,100 thousand, respectively. The expected emerged proportions are 40% in Year 1 and 70% in Year 2. Calculate an appropriate year-specific a priori ratio and BF ultimate for each year. All amounts are thousands of dollars. No interest discounting, mortality assumption, or probability distribution is used.

Method and assumptions. Calculate each year’s expected loss from its group premiums and group ratios, then divide by total premium to obtain the year-specific ratio. Assume the group ratios apply consistently in both years and that each stated emergence proportion matches the reported-loss basis.

Year 1 selection. Expected loss for A is $1{,}500(0.50)=750$. Expected loss for B is $500(0.90)=450$. Total expected loss is $750+450=1{,}200$, and total premium is $1{,}500+500=2{,}000$. Therefore $e_1=1{,}200/2{,}000=0.60$. The unreported share is $1-0.40=0.60$, so expected unreported loss is $1{,}200(0.60)=720$. BF ultimate is $800+720=1{,}520$.

Year 2 selection. Expected loss for A is $500(0.50)=250$. Expected loss for B is $1{,}500(0.90)=1{,}350$. Total expected loss is $250+1{,}350=1{,}600$, and total premium is $2{,}000$. Therefore $e_2=1{,}600/2{,}000=0.80$. The unreported share is $1-0.70=0.30$, so expected unreported loss is $1{,}600(0.30)=480$. BF ultimate is $1{,}100+480=1{,}580$.

Check and common mistake avoided. The premium-weighted group ratios give $0.75(0.50)+0.25(0.90)=0.60$ in Year 1 and $0.25(0.50)+0.75(0.90)=0.80$ in Year 2. Applying the simple, unweighted average of 50% and 90% to both years would ignore the different premium mixes and incorrectly assign 70% to each.

Common Selection Traps

Letting immature results dictate the prior. A low or high reported ratio can result from the selected emergence pattern, as well as from underlying loss cost. Assess the reasons for a difference before changing $e_i$.
Differentiating without evidence. A separate ratio for every year is not automatically more accurate. If expected loss costs are comparable and distinctions are unsupported, unnecessary variation can make the method less stable.
Using one ratio despite a material mix change. When groups have different expected loss ratios, a change in their premium weights can change the aggregate expected loss ratio even if group-level assumptions are unchanged.
Mixing the ratio and emergence assumptions. The a priori ratio estimates expected loss cost; $q_i$ specifies expected emergence. Keep the two assumptions distinct and use an emergence measure consistent with the losses being projected.
Applying the wrong unreported share. If 70% has emerged, the unreported share is 30%, not 70%. Check that $q_i+(1-q_i)=1$.

Key takeaway: Select a common a priori loss ratio when the evidence supports comparable expected loss costs; differentiate by accident year when documented assumptions support the differences. State the rationale, calculate expected unreported losses using the corresponding emergence pattern, and verify that each BF ultimate equals reported losses plus expected unreported losses.

Check Your Understanding

For each problem, identify the relevant selection and calculate the requested result. Amounts are in thousands of dollars unless stated otherwise.

A hypothetical portfolio has two comparable accident years. A documented analysis supports a common a priori ratio of 65%. What is the expected ultimate loss for a year with premium of $2,400? (Answer: $1,560 thousand.)
A year has premium of $1,800, a selected a priori ratio of 60%, reported losses of $700, and 50% emergence. Calculate its BF unpaid amount and ultimate. (Answer: unpaid $540 thousand; ultimate $1,240 thousand.)
Two hypothetical risk groups have selected ratios of 40% and 80%. A year has premium of $600 in the first group and $400 in the second. Find its aggregate a priori ratio. (Answer: 56%.)
Two years have the same documented risk profile and a common selected ratio of 70%. Their reported ratios differ, but their emergence proportions also differ. Should the reported-ratio difference alone require separate a priori ratios? (Answer: No; the difference alone does not justify separate selections.)
A documented portfolio mix supports expected loss ratios of 55% for Group A and 85% for Group B. A year has premiums of $800 in A and $1,200 in B, reported losses of $900, and 60% emergence. Calculate the mix-weighted a priori ratio, BF unpaid amount, and BF ultimate. (Answer: a priori ratio 73%; unpaid $584 thousand; ultimate $1,484 thousand.)

What You'll Learn