Bornhuetter-Ferguson Across Accident Periods

Why Calculate Bornhuetter-Ferguson by Accident Period?

Accident periods in the same portfolio are rarely at the same stage of development. A recent period may have little paid or incurred loss to date, while an older period may be close to fully developed. Applying a single maturity percentage to the whole portfolio can therefore misstate expected unpaid losses. The Bornhuetter-Ferguson method handles this by calculating the expected unreported amount for each period using that period’s expected ultimate and maturity.

This tutorial uses hypothetical data. All monetary amounts in the worked examples are in thousands of dollars, denoted $$000$. The method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: For accident period $j$, let $E_j$ be its selected expected ultimate loss, $p_j$ its cumulative percent reported on the chosen basis, and $R_j$ its cumulative reported amount on that same basis. The Bornhuetter-Ferguson expected unreported amount is $E_j(1-p_j)$, and the Bornhuetter-Ferguson ultimate is $R_j+E_j(1-p_j)$. On a paid basis, the expected unreported amount is the paid-loss reserve. On an incurred basis, it is the IBNR estimate; case reserves are already included in incurred losses.

Set Up Each Accident Period Consistently

For each accident period, establish the expected ultimate, observed cumulative amount, and maturity percentage. The expected ultimate may be supplied directly or constructed from an exposure measure and a selected expected loss ratio. For example, if $P_j$ is earned premium and $L_j$ is the selected expected loss ratio, then $E_j=P_jL_j$. The exposure and loss ratio must use compatible bases, as must the observed amount and percent reported.

The reported percentage is cumulative and must match the loss measure. A paid amount requires a cumulative percent paid; an incurred amount requires a cumulative percent incurred. Do not use one period’s maturity for another merely because they are in the same portfolio. The selected percentages should reflect the development pattern appropriate to the data and should be between zero and one.

\begin{aligned} \text{Expected unreported}_j &= E_j(1-p_j),\\ \text{BF ultimate}_j &= R_j+E_j(1-p_j). \end{aligned}

For a portfolio of $n$ accident periods, add the period estimates:

\text{Portfolio BF reserve}=\sum_{j=1}^{n}E_j(1-p_j), \qquad \text{Portfolio BF ultimate}=\sum_{j=1}^{n}\left[R_j+E_j(1-p_j)\right].

Because summation is linear, the portfolio ultimate also equals total reported amount plus total BF reserve. This reconciliation is a useful arithmetic check, but it does not justify replacing the period-specific calculations with a single portfolio maturity assumption.

Worked Example: Incurred BF Across Three Accident Periods

A hypothetical insurer has three accident periods. Earned premium is $$1{,}000$, $$800$, and $$600$ thousand, and the selected expected loss ratio is 65% for each. Cumulative incurred amounts are $$500$, $$300$, and $$100$ thousand. The respective cumulative incurred percentages are 80%, 50%, and 20%. Calculate each BF IBNR and ultimate, then reconcile the portfolio.

Method. This is an incurred BF calculation. First calculate each expected ultimate as premium times expected loss ratio. Then apply the period-specific unreported proportion to that expected ultimate. The incurred amount is added to IBNR to obtain the BF ultimate.

Accident period 1. Expected ultimate is $1{,}000(0.65)=650$. The unreported proportion is $1-0.80=0.20$, so IBNR is $650(0.20)=130$. The BF ultimate is $500+130=630$.

Accident period 2. Expected ultimate is $800(0.65)=520$. The unreported proportion is $1-0.50=0.50$, so IBNR is $520(0.50)=260$. The BF ultimate is $300+260=560$.

Accident period 3. Expected ultimate is $600(0.65)=390$. The unreported proportion is $1-0.20=0.80$, so IBNR is $390(0.80)=312$. The BF ultimate is $100+312=412$.

Portfolio reconciliation. Total incurred is $500+300+100=900$. Total BF IBNR is $130+260+312=702$. Thus the portfolio BF ultimate is $900+702=1{,}602$. Adding the period ultimates gives $630+560+412=1{,}602$, which checks.

Common mistake avoided. The same expected loss ratio does not imply the same maturity. Using 80% reported for all three periods would incorrectly calculate IBNR as $0.20(650+520+390)=312$, rather than the correct $$702$ thousand.

Why the Formula Uses Expected Unreported Loss

The BF estimate combines observed experience for the portion expected to have emerged with an a priori expected amount for the portion not yet expected to emerge. The selected cumulative percent reported $p_j$ implies an unreported fraction $1-p_j$. Applying that fraction to the expected ultimate gives the expected amount still to emerge.

Partition expected loss.
For period $j$, the expected amount associated with the reported fraction is $E_jp_j$, and the expected amount associated with the unreported fraction is $E_j(1-p_j)$. These sum to $E_j$.

Use the actual reported amount where available.
The BF method replaces the expected reported portion with observed cumulative losses $R_j$, rather than projecting all losses from the observed amount.

Add the expected remainder.
The resulting estimate is $R_j+E_j(1-p_j)$. It uses actual experience for what has emerged and the expected-loss basis for what has not.

E_jp_j+E_j(1-p_j)=E_j \quad\Longrightarrow\quad \text{BF ultimate}_j=R_j+E_j(1-p_j).

This argument explains the formula’s structure; it does not mean that the observed amount must equal $E_jp_j$. In fact, the difference between observed and expected reported losses is not multiplied by a development factor in the BF formula. That distinction is why the method generally gives less weight to unusually high or low early experience than a straight chain-ladder projection would.

Worked Example: Paid BF With Different Expected Loss Ratios

For two hypothetical accident periods, earned premiums are $$500$ and $$600$ thousand, with selected expected loss ratios of 60% and 60%. Paid losses to date are $$210$ and $$100$ thousand. The cumulative percent paid is 75% for the first period and 40% for the second. Calculate the paid BF reserves and ultimates, and check the portfolio totals.

Method. This is a paid BF calculation, so use cumulative percent paid and paid losses. Expected ultimates are calculated separately, even though the selected loss ratios happen to be equal.

Accident period 1. Expected ultimate is $500(0.60)=300$. The unpaid proportion is $1-0.75=0.25$. The paid BF reserve is $300(0.25)=75$, and the BF ultimate is $210+75=285$.

Accident period 2. Expected ultimate is $600(0.60)=360$. The unpaid proportion is $1-0.40=0.60$. The paid BF reserve is $360(0.60)=216$, and the BF ultimate is $100+216=316$.

Portfolio check. Total paid is $210+100=310$; total reserve is $75+216=291$; total ultimate is $310+291=601$. Directly adding period ultimates gives $285+316=601$. The expected ultimates total $300+360=660$; the BF ultimate need not equal that total because actual paid losses replace the expected reported portions.

Common mistake avoided. Applying the 75% paid percentage to both periods would produce a reserve of $0.25(300+360)=165$, which is too low. The second period is only 40% paid, so its own unreported proportion must be used.

Reconcile the Portfolio Without Averaging Maturity Incorrectly

A portfolio reserve can be written as total expected ultimate multiplied by a weighted average unreported proportion, where the weights are the accident periods’ expected ultimates. This is an algebraic restatement of summing the period reserves. It is not generally valid to use the simple average of the periods’ percentages because that treats a small expected period and a large expected period as equally important.

\sum_{j=1}^{n}E_j(1-p_j) = \left(\sum_{j=1}^{n}E_j\right) \left( \frac{\sum_{j=1}^{n}E_j(1-p_j)}{\sum_{j=1}^{n}E_j} \right).

The term in parentheses is the expected-ultimate-weighted average unreported proportion. Equivalently, the expected-ultimate-weighted average reported percentage is $\sum_j E_jp_j/\sum_jE_j$. Neither is usually the unweighted arithmetic average of the period percentages.

Worked Example: Weighted Maturity and Portfolio Reconciliation

Three hypothetical accident periods have expected ultimates of $$400$, $$500$, and $$300$ thousand. Their cumulative incurred percentages are 90%, 60%, and 25%, and incurred amounts are $$350$, $$250$, and $$80$ thousand. Calculate period IBNR and ultimates, then show why the simple average maturity gives the wrong portfolio reserve.

Period calculations. Period 1 has unreported proportion $0.10$, so IBNR is $400(0.10)=40$ and ultimate is $350+40=390$. Period 2 has unreported proportion $0.40$, so IBNR is $500(0.40)=200$ and ultimate is $250+200=450$. Period 3 has unreported proportion $0.75$, so IBNR is $300(0.75)=225$ and ultimate is $80+225=305$.

Sum the periods. Total IBNR is $40+200+225=465$. Total incurred is $350+250+80=680$. Therefore total BF ultimate is $680+465=1{,}145$, which also equals $390+450+305=1{,}145$.

Weighted check. Expected ultimates total $400+500+300=1{,}200$. The weighted unreported proportion is $465/1{,}200=0.3875$, so $1{,}200(0.3875)=465$, matching the summed IBNR.

Trap check. The simple average unreported proportion is $(0.10+0.40+0.75)/3=0.4167$ to four decimals. Multiplying by total expected ultimate gives approximately $1{,}200(0.4167)=500.04$, not the correct $465$. The discrepancy occurs because the simple average ignores the different expected-ultimate weights.

Common Checks and Scope

Before finalizing a multi-period estimate, verify that each input belongs to the intended accident period and that the expected ultimate, observed amount, and development percentage share a consistent basis. A paid estimate needs paid emergence; an incurred estimate needs incurred emergence. If reported percentages are supplied as percentages, convert them to proportions before calculating the unreported share.

Check the definition of the reserve. Paid BF reserve is expected unpaid loss relative to paid amounts. Incurred BF IBNR is expected unreported loss beyond incurred amounts, not the total unpaid amount including case reserves.
Check the portfolio arithmetic. Sum period reserves, sum reported amounts, then verify that their sum equals the sum of period ultimates.
Do not average period percentages casually. Use period-specific maturity, or derive a weighted portfolio maturity using expected ultimates as weights.
Do not confuse an expected ultimate with a BF ultimate. The BF ultimate includes actual reported experience plus expected unreported loss, so it can differ from the sum of expected ultimates.

The multi-period calculation is therefore a set of consistent accident-period calculations followed by addition. The central discipline is to preserve each period’s own expected ultimate and maturity until its estimate has been calculated; only then should the results be reconciled to portfolio totals.

Check Your Understanding

For each problem, use the specified basis and keep monetary amounts in thousands of dollars.

A period has expected ultimate loss of $$240$, incurred loss of $$130$, and 65% cumulative incurred. Calculate its BF IBNR and ultimate. (Answer: IBNR $=$84$; ultimate $=$214$.)
A paid BF period has expected ultimate $$500$, paid loss $$120$, and 30% cumulative paid. Calculate the reserve and ultimate. (Answer: reserve $=$350$; ultimate $=$470$.)
Period A has expected ultimate $$200$, incurred loss $$170$, and 80% reported. Period B has expected ultimate $$400$, incurred loss $$190$, and 50% reported. Calculate each IBNR and the portfolio ultimate. (Answer: IBNR $=$40$ and $$200$; portfolio ultimate $=$600$.)
Two paid periods have expected ultimates $$100$ and $$200$, cumulative paid percentages of 90% and 25%, and paid losses of $$85$ and $$70$. Find the total paid BF reserve and ultimate. (Answer: reserve $=$160$; ultimate $=$315$.)
Two periods have expected ultimates $$300$ and $$500$, with cumulative reported percentages of 80% and 40%. Calculate the total expected unreported amount and its expected-ultimate-weighted proportion. (Answer: expected unreported amount $=$360$; weighted unreported proportion $=0.45$.)

What You'll Learn