Why Construct an Expected Ultimate?
Bornhuetter-Ferguson (BF) begins with an estimate of the losses an accident period is expected to produce in total. That expected ultimate is a prior estimate: it is built from a measure of the period’s exposure and an assumption about the losses generated by that exposure. The method then uses the proportion of losses expected to remain unreported to estimate the unreported amount.
This tutorial focuses on constructing that expected ultimate. The calculation can use earned premium multiplied by an expected loss ratio, or an exposure count multiplied by an expected loss cost per exposure unit. These are alternative ways to express a prior; they are not interchangeable unless their units and underlying assumptions are consistent.
From Exposure to an Expected Ultimate
An exposure measure describes the amount of insurance business being considered. Examples include earned premium in dollars, policy-years, vehicle-years, or another count of insured units. An expected loss ratio is expected losses divided by the premium basis to which it applies. An expected loss cost per exposure unit is expected loss dollars per unit, such as dollars per vehicle-year.
State the units before multiplying. If \(E_i\) is earned premium in dollars and \(r_i\) is a dimensionless ratio, \(E_i r_i\) is dollars of expected loss. If \(E_i\) is vehicle-years and \(c_i\) is dollars per vehicle-year, \(E_i c_i\) is again dollars of expected loss. A numerical product can be arithmetically valid yet actuarially meaningless if its units do not match.
Why the BF Prior Is Multiplied by the Unreported Proportion
Suppose the expected ultimate is \(U_i^{\mathrm{prior}}\), and the expected reported proportion is \(p_i\). Under the assumed development pattern, the expected reported portion of that prior is \(p_iU_i^{\mathrm{prior}}\), leaving \((1-p_i)U_i^{\mathrm{prior}}\) expected to emerge after the valuation date. BF uses this expected emergence for the unreported portion, rather than applying a development factor directly to the observed amount.
The construction can also be checked by imagining that the actual reported amount equals its prior expectation. In that case, \(I_i=p_iU_i^{\mathrm{prior}}\), so $$ U_i^{\mathrm{BF}} =p_iU_i^{\mathrm{prior}}+(1-p_i)U_i^{\mathrm{prior}} =U_i^{\mathrm{prior}}. $$ Thus, when observed experience matches the prior’s expected reported amount, the BF ultimate reproduces the prior ultimate. When observed experience differs, the observed amount replaces the prior expectation for the reported portion, while the prior still supplies the expected unreported portion.
Worked Applications
Worked Example: Premium-Based Prior and Incurred BF
A hypothetical accident period has earned premium of $2,400,000. The assumed expected loss ratio is 62.5%. At the valuation date, 40% of ultimate incurred losses are expected to be reported, and observed incurred losses are $630,000. Calculate the prior expected ultimate, expected IBNR, and BF ultimate. Assume the premium and loss ratio use the same dollar and loss basis.
Method. Construct the prior as earned premium times the assumed expected loss ratio, then multiply that prior by the expected unreported proportion. This avoids treating the prior ultimate itself as the reserve.
Step 1: Calculate the prior expected ultimate. $$ U^{\mathrm{prior}} =\$2{,}400{,}000(0.625) =\$1{,}500{,}000. $$
Step 2: Calculate the expected unreported proportion. The reported proportion is \(p=0.40\), so $$ 1-p=1-0.40=0.60. $$
Step 3: Calculate expected IBNR. $$ \operatorname{IBNR} =0.60(\$1{,}500{,}000) =\$900{,}000. $$
Step 4: Calculate the BF ultimate. $$ U^{\mathrm{BF}} =\$630{,}000+\$900{,}000 =\$1{,}530{,}000. $$ The prior’s expected reported amount would have been \(0.40(\$1{,}500{,}000)=\$600{,}000\). Observed incurred losses of $630,000 are $30,000 above that amount, so the BF ultimate is $30,000 above the prior, as the result shows. The common mistake avoided is calling $1,500,000 the BF reserve: it is the prior ultimate, while the expected IBNR is $900,000.
Worked Example: Prior from Exposure Units
A hypothetical portfolio has 8,000 earned vehicle-years. The assumed expected loss cost is $175 per vehicle-year. At the valuation date, 65% of ultimate losses are expected to be reported, and incurred losses are $980,000. Calculate the prior expected ultimate, expected IBNR, and BF ultimate.
Method. Use exposure units multiplied by expected loss cost per unit. Check that vehicle-years cancel and the product is in dollars.
Step 1: Calculate the prior. $$ U^{\mathrm{prior}} =8{,}000\text{ vehicle-years} \times \$175\text{ per vehicle-year} =\$1{,}400{,}000. $$
Step 2: Find the unreported proportion. $$ 1-p=1-0.65=0.35. $$
Step 3: Calculate expected IBNR and the BF ultimate. $$ \operatorname{IBNR} =0.35(\$1{,}400{,}000) =\$490{,}000, $$ $$ U^{\mathrm{BF}} =\$980{,}000+\$490{,}000 =\$1{,}470{,}000. $$ The prior expected reported amount is \(0.65(\$1{,}400{,}000)=\$910{,}000\); actual incurred losses exceed it by $70,000. The BF ultimate is consequently $70,000 above the prior. The mistake avoided is multiplying a count of vehicle-years by a loss ratio: a count needs a loss cost per vehicle-year, not a dimensionless ratio.
Worked Example: Constructing Priors by Exposure Group
A hypothetical insurer divides an accident period into two groups. Group A has 3,000 exposure units and an assumed expected loss cost of $240 per unit. Group B has 5,000 exposure units and an assumed expected loss cost of $160 per unit. At valuation, 50% of Group A’s ultimate and 25% of Group B’s ultimate are expected to be reported. Incurred losses are $400,000 for A and $700,000 for B. Calculate each group’s prior, the combined prior, expected IBNR, and combined BF ultimate.
Method. Calculate each prior within its own exposure group, then add the dollar amounts. This preserves differences in expected loss cost and development.
Step 1: Calculate each group’s prior. $$ U_A^{\mathrm{prior}}=3{,}000(\$240)=\$720{,}000, \qquad U_B^{\mathrm{prior}}=5{,}000(\$160)=\$800{,}000. $$ Therefore, $$ U_{\mathrm{total}}^{\mathrm{prior}} =\$720{,}000+\$800{,}000 =\$1{,}520{,}000. $$
Step 2: Apply each group’s unreported proportion. Group A’s unreported proportion is \(1-0.50=0.50\), and Group B’s is \(1-0.25=0.75\): $$ \operatorname{IBNR}_A=0.50(\$720{,}000)=\$360{,}000, \qquad \operatorname{IBNR}_B=0.75(\$800{,}000)=\$600{,}000. $$
Step 3: Add the group results. $$ \operatorname{IBNR}_{\mathrm{total}} =\$360{,}000+\$600{,}000 =\$960{,}000. $$ Total observed incurred losses are \(\$400{,}000+\$700{,}000=\$1{,}100{,}000\), so $$ U_{\mathrm{total}}^{\mathrm{BF}} =\$1{,}100{,}000+\$960{,}000 =\$2{,}060{,}000. $$ The check is that each group’s prior reported expectation is $360,000 for A and $200,000 for B. Actual incurred amounts exceed those expectations by $40,000 and $500,000, respectively; adding the combined $540,000 difference to the combined prior of $1,520,000 gives $2,060,000. The common mistake avoided is using one development proportion for both groups when their expected maturity differs.
Choosing and Combining the Basis
The exposure measure and the loss assumption jointly determine the prior. For example, earned premium paired with an expected loss ratio gives a dollar estimate. A count of insured units paired with expected loss cost per unit also gives a dollar estimate. Where two formulations are genuinely equivalent, they produce the same prior. If $2,800,000 of earned premium and a 50% expected loss ratio describe the same business and loss basis as 8,000 vehicle-years with a $175 expected loss cost, both give $1,400,000.
When the portfolio has materially different groups, calculate exposure-based expected losses at the appropriate group level and then sum them. A single portfolio average can be used only if it is weighted consistently with the exposure and loss basis. The following wrong calculation illustrates why:
A loss ratio must also be interpreted on the basis for which it is assumed. Before multiplying, identify the premium or exposure measure, the loss definition, and the period to which both refer. Do not combine a ratio with an unrelated exposure simply because the units appear convenient. Selecting or adjusting an expected loss ratio is a separate judgment; here, the ratio or loss cost is treated as a stated assumption.
Practical Checks
Before using an exposure-based prior in BF, check the calculation in a fixed order:
Record whether it is premium, policy-years, vehicle-years, or another measure, and confirm that it represents the accident period being estimated.
Use an expected loss ratio with its premium basis, or an expected loss cost with its exposure unit. Confirm that the product gives dollars of expected loss.
Calculate the prior at the level needed to preserve relevant group differences, then add group dollar amounts.
Use \(1-p_i\) on the prior expected ultimate to calculate expected IBNR, then add observed incurred losses if the requested result is a BF ultimate.
These checks separate three quantities that are easy to confuse: exposure, prior expected ultimate, and expected unreported losses. Keeping them distinct makes the calculation auditable and reduces errors in signs, proportions, and units. This exposure-based prior construction is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, calculate the requested prior or BF quantity and check that the units and basis are consistent.
- A hypothetical accident period has earned premium of $1,600,000 and an assumed expected loss ratio of 58%. Calculate its prior expected ultimate. (Answer: $928,000)
- A hypothetical portfolio has 12,000 earned exposure units and an expected loss cost of $90 per unit. The expected reported proportion is 25%, and observed incurred losses are $300,000. Calculate expected IBNR and the BF ultimate. (Answer: IBNR $810,000; BF ultimate $1,110,000)
- A hypothetical group has 2,000 exposure units at an expected loss cost of $300 per unit and 5,000 units at $120 per unit. Calculate the combined prior expected ultimate. (Answer: $1,200,000)
- A hypothetical period has earned premium of $4,500,000, an assumed expected loss ratio of 64%, an expected reported proportion of 70%, and observed incurred losses of $2,050,000. Calculate expected IBNR and the BF ultimate. (Answer: IBNR $864,000; BF ultimate $2,914,000)
- A hypothetical portfolio has 6,000 exposure units at an expected loss cost of $140 and 4,000 units at $210. A colleague uses the simple average loss cost for all 10,000 units. Calculate the correct prior and identify the colleague’s result. (Answer: Correct prior $1,680,000; colleague’s simple-average result $1,750,000)