Integrated Bornhuetter-Ferguson and Cape Cod Selection Workshop

Why an Integrated Selection Requires More Than a Formula

Bornhuetter-Ferguson (BF) and Cape Cod (CC) both combine observed losses with an expectation of losses not yet observed, but they obtain that expectation differently. BF uses an externally selected expected loss ratio; CC estimates a loss ratio from the portfolio's observed experience and development. Either method can be applied to paid or incurred losses. The resulting estimates are not interchangeable unless their inputs, development assumptions, and reserve definitions are kept consistent.

This workshop uses a common set of accident-year data to calculate all four estimates, compare them, and select an unpaid liability for the most recent year. Amounts are in thousands of dollars unless stated otherwise. All projections are undiscounted: no interest rate, mortality assumption, or probability distribution is used. The calculations are deterministic estimates based on the stated premiums, observed losses, loss ratios, and development proportions.

Definition: Let $P$ be premium, $e$ a selected expected loss ratio, $E=Pe$ expected ultimate loss, $p$ the proportion of ultimate loss paid by the valuation date, and $q$ the proportion of ultimate loss incurred (reported) by that date. Paid-to-date loss is $C$, and incurred loss is $I$, where incurred loss includes paid loss and case reserves. Paid BF estimates unpaid loss as $E(1-p)$. Incurred BF estimates IBNR as $E(1-q)$; its total unpaid liability also includes case reserves, or equivalently is the incurred BF ultimate less paid loss.

BF Estimates and the Meaning of Unpaid Liability

The BF method does not apply a development factor directly to the observed amount to estimate the unreported portion. It uses the observed amount for the reported part and the expected loss for the part not yet reported. If $E$ is expected ultimate loss, the paid BF ultimate and unpaid liability are:

U_{\mathrm{BF,paid}}=C+E(1-p), \qquad R_{\mathrm{BF,paid}}=E(1-p).

For incurred BF, $I$ already includes both paid losses and case reserves. The expected addition is IBNR, not total unpaid liability:

\mathrm{IBNR}_{\mathrm{BF}}=E(1-q), \qquad U_{\mathrm{BF,inc}}=I+E(1-q), \qquad R_{\mathrm{BF,inc}}=I-C+E(1-q).

Here $I-C$ is the case reserve amount. This distinction matters whenever a question asks for a reserve or total unpaid liability rather than IBNR alone.

Worked Example: Reconciling Paid and Incurred BF

A hypothetical liability portfolio has premium of $500,000, a selected expected loss ratio of 0.70, paid proportion $p=0.50$, and incurred proportion $q=0.75$. At the valuation date, paid loss is $200,000 and incurred loss is $270,000. Calculate paid BF unpaid loss and incurred BF IBNR and total unpaid liability. Amounts in this example are in thousands of dollars; no discounting or probability distribution is used.

Step 1: Expected ultimate loss. Premium is 500 in the stated units, so $E=500(0.70)=350$.

Step 2: Paid BF. The unpaid proportion is $1-0.50=0.50$. Thus paid BF unpaid loss is $350(0.50)=175$, and its ultimate estimate is $200+175=375$.

Step 3: Incurred BF. The unreported proportion is $1-0.75=0.25$, so IBNR is $350(0.25)=87.5$. The incurred BF ultimate is $270+87.5=357.5$.

Step 4: Total unpaid liability. Case reserves are $270-200=70$. Total unpaid liability is $70+87.5=157.5$, which checks against $357.5-200=157.5$.

Result and trap avoided. Paid BF unpaid loss is 175; incurred BF IBNR is 87.5; incurred BF total unpaid liability is 157.5. Reporting 87.5 as the total incurred-based reserve would omit the 70 of case reserves.

Cape Cod Estimates from Paid and Incurred Experience

CC estimates a common loss ratio by comparing observed losses with the portion of premium expected to have emerged by the valuation date. For accident years $j=1,\ldots,n$, let $P_j$ be premium, $C_j$ paid loss, $I_j$ incurred loss, and $p_j$ and $q_j$ the corresponding paid and incurred proportions. The paid and incurred CC loss ratios are:

\widehat e_{\mathrm{CC,paid}} = \frac{\sum_j C_j}{\sum_j P_jp_j}, \qquad \widehat e_{\mathrm{CC,inc}} = \frac{\sum_j I_j}{\sum_j P_jq_j}.

Each ratio is exposure-weighted: the denominator is the premium expected to be represented in observed losses, not the sum of development proportions by themselves. Under the common-loss-ratio assumption, the estimated ultimate for year $j$ is $P_j\widehat e_{\mathrm{CC}}$. Paid CC unpaid loss is ultimate less paid loss, equivalently $P_j\widehat e_{\mathrm{CC,paid}}(1-p_j)$. Incurred CC total unpaid liability is incurred-based ultimate less paid loss; it includes case reserves as well as IBNR.

Why the CC ratio has this form: If all years share an underlying loss ratio $e$, the expected observed paid amount in year $j$ is $P_j e p_j$. Summing across years gives expected total paid loss $e\sum_j P_jp_j$. Equating this to the observed total $\sum_j C_j$ and solving for $e$ gives $\widehat e_{\mathrm{CC,paid}}=\sum_jC_j/\sum_jP_jp_j$. Replacing paid quantities with incurred quantities gives the incurred version. This justification depends on a common loss ratio and suitable development proportions; it does not establish that those assumptions fit every portfolio.

Worked Example: Four Estimates on One Accident-Year Data Set

A hypothetical portfolio has the following premium, current loss, and selected development proportions. Losses and premiums are in $000. The expected loss ratio for BF is 0.65. The development proportions are assumed appropriate for the respective paid and incurred bases. There is no discounting or stochastic distribution in this deterministic calculation.

Accident year	Premium	Paid $C_j$	Paid proportion $p_j$	Incurred $I_j$	Incurred proportion $q_j$
1	1,000	360	0.40	450	0.55
2	1,200	660	0.60	840	0.75
3	800	600	0.80	720	0.90

Step 1: Calculate the BF expected ultimate by year. Multiplying premium by 0.65 gives $E_1=1{,}000(0.65)=650$, $E_2=1{,}200(0.65)=780$, and $E_3=800(0.65)=520$.

Step 2: Calculate paid BF unpaid loss. By year, the estimates are $650(1-0.40)=390$, $780(1-0.60)=312$, and $520(1-0.80)=104$. The total paid BF unpaid estimate is $390+312+104=806$.

Step 3: Calculate incurred BF IBNR and total unpaid. IBNR by year is $650(0.45)=292.5$, $780(0.25)=195$, and $520(0.10)=52$. Case reserves are $450-360=90$, $840-660=180$, and $720-600=120$. Total incurred BF unpaid liability by year is therefore $292.5+90=382.5$, $195+180=375$, and $52+120=172$; the portfolio total is $929.5$.

Step 4: Calculate the paid CC loss ratio. Total paid loss is $360+660+600=1{,}620$. The premium-weighted paid denominator is $1{,}000(0.40)+1{,}200(0.60)+800(0.80)=400+720+640=1{,}760$. Thus $\widehat e_{\mathrm{CC,paid}}=1{,}620/1{,}760=0.9204545$.

The paid CC unpaid estimates are $1{,}000(0.9204545)(0.60)=552.27$, $1{,}200(0.9204545)(0.40)=441.82$, and $800(0.9204545)(0.20)=147.27$. The total is $1{,}141.36$.

Step 5: Calculate the incurred CC loss ratio and total unpaid. Total incurred loss is $450+840+720=2{,}010$. The incurred denominator is $1{,}000(0.55)+1{,}200(0.75)+800(0.90)=550+900+720=2{,}170$. Therefore $\widehat e_{\mathrm{CC,inc}}=2{,}010/2{,}170=0.9262673$.

Incurred CC ultimate estimates are $1{,}000(0.9262673)=926.27$, $1{,}200(0.9262673)=1{,}111.52$, and $800(0.9262673)=741.01$. Subtract paid losses of 360, 660, and 600: total unpaid by year is 566.27, 451.52, and 141.01, respectively. Their sum is 1,158.80. As a portfolio check, total incurred CC ultimate is $3{,}000(0.9262673)=2{,}778.80$; subtracting total paid of 1,620 also gives 1,158.80.

Result and trap avoided. For accident year 3, paid BF unpaid is 104, incurred BF total unpaid is 172, paid CC unpaid is 147.27, and incurred CC total unpaid is 141.01. The paid CC denominator is 1,760, not total premium of 3,000: using total premium would ignore the different paid proportions and produce an invalid ratio.

Comparing the Estimates and Selecting a Liability

The four methods answer related but distinct questions. BF holds the expected loss ratio fixed and lets the observed data influence the part considered emerged. CC estimates a portfolio loss ratio from observed experience and then applies it across accident years. Paid methods depend on the paid emergence pattern; incurred methods depend on the incurred emergence pattern and, for total unpaid liability, retain case reserves through the incurred ultimate less paid amount.

Method	AY 3 unpaid estimate ($000)	Key assumption or feature
Paid BF	104.00	Uses the selected BF loss ratio and the unpaid paid proportion
Incurred BF, total unpaid	172.00	Uses the selected BF ratio; includes case reserves and expected IBNR
Paid CC	147.27	Uses the portfolio paid CC ratio and paid emergence
Incurred CC, total unpaid	141.01	Uses the portfolio incurred CC ratio; subtracts paid from estimated ultimate

The BF expected loss ratio of 0.65 is materially below the CC ratios of approximately 0.92. That difference is not a rounding issue: BF uses a prior expectation, while CC reflects the observed losses relative to emerged premium in this data set. It calls for investigation of whether the expected ratio is stale, whether the CC experience is unusual, and whether the selected development proportions and premiums are comparable across years. The 31 difference between the AY 3 incurred BF and incurred CC estimates also illustrates that method choice can matter even when both are incurred-based.

A selection should be supported by the quality and relevance of the assumptions, not by averaging methods automatically. For this hypothetical example, suppose review supports using the recent portfolio experience in CC, while the BF expectation remains useful as a counterweight; also suppose the incurred information is given somewhat more weight because it includes case estimates. The following explicit judgmental weights illustrate a documented selection, not a universal formula.

Worked Example: Selecting AY 3 Total Unpaid Liability

Use the four AY 3 estimates above. Assign weights of 20% to paid BF, 30% to incurred BF total unpaid, 20% to paid CC, and 30% to incurred CC total unpaid. These weights sum to $0.20+0.30+0.20+0.30=1.00$. All amounts are in $000, undiscounted, with no probability distribution assumed.

Step 1: Apply each weight. The contributions are $104(0.20)=20.80$, $172(0.30)=51.60$, $147.27(0.20)=29.454$, and $141.01(0.30)=42.303$.

Step 2: Sum the contributions. The selected liability is $20.80+51.60+29.454+42.303=144.157$, or $144.16 thousand to the nearest cent in thousands (equivalently, approximately $144,157).

Step 3: Check the selection. The result lies between the smallest estimate, 104, and the largest, 172, as it must for a weighted average with nonnegative weights summing to one. It is also close to the two CC estimates, which receive half the total weight. The selection is a reasoned judgment conditional on the stated support for the weights; the calculations alone do not prove that those weights are appropriate.

Common mistake avoided. Do not add the four estimates and call the sum the reserve, and do not treat the methods as four independent observations. The estimates share loss data and may share development assumptions. The weighted result is a documented actuarial selection, not a statistical credibility formula.

Diagnostics Before Finalizing

A useful comparison asks why estimates differ, rather than choosing the method with the most appealing result. In this workshop, an actuary would confirm the selected paid and incurred proportions, verify that premium is on a consistent basis across accident years, and investigate the gap between the BF and CC loss ratios. The incurred estimate should also be reviewed for case-reserve adequacy; a change in case practices can affect incurred methods without an equivalent change in ultimate loss. Assessment of case-reserve changes is a separate adjustment problem, so the estimate here assumes the incurred data are suitable as provided.

Keep these checks explicit:

Use the matching emergence proportion. Paid losses pair with paid proportions, and incurred losses pair with incurred proportions.
Use premium-weighted emergence in CC. The denominator is $\sum_j P_jp_j$ or $\sum_j P_jq_j$, not an unweighted average of proportions.
State the reserve basis. Incurred BF's $E(1-q)$ is IBNR; total unpaid liability adds case reserves, or subtracts paid from ultimate.
Explain the expected-loss assumption. BF and CC differ in where the loss ratio comes from; a selection should address the relevance of each source.
Avoid false precision. Intermediate ratios can be carried at full precision; round only the reported results, here to two decimal places in $000.

Key takeaway: A defensible integrated selection preserves the distinction between paid and incurred development, between IBNR and total unpaid liability, and between BF's selected expectation and CC's experience-derived ratio. Compare assumptions and diagnostics before assigning weights. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical problem, calculate the requested estimate before checking the answer. Amounts are undiscounted; no mortality or probability distribution is involved.

A property portfolio has premium of $900,000, a BF expected loss ratio of 0.60, paid proportion 0.70, and paid loss of $350,000. Calculate paid BF unpaid loss and ultimate loss. (Answer: unpaid loss = $162,000; ultimate loss = $512,000.)
A liability year has premium of $600,000, BF expected loss ratio 0.75, incurred proportion 0.80, incurred loss $390,000, and paid loss $300,000. Calculate incurred BF IBNR and total unpaid liability. (Answer: IBNR = $90,000; total unpaid liability = $180,000.)
For a hypothetical paid CC calculation, year A has premium $4 million, paid proportion 0.50, and paid loss $1.1 million; year B has premium $6 million, paid proportion 0.25, and paid loss $1.0 million. Calculate the paid CC loss ratio and the total paid CC unpaid estimate for both years. (Answer: loss ratio = 0.6000; total unpaid = $3.90 million.)
Incurred CC data, in $000, are: year A premium 500, incurred proportion 0.60, incurred loss 240; year B premium 700, incurred proportion 0.80, incurred loss 560. Year B paid loss is 430. Calculate the incurred CC ratio and year B total unpaid liability. (Answer: incurred CC ratio = 0.9302; year B total unpaid liability = $221.16 thousand.)
Four estimates of a recent year's total unpaid liability are paid BF 120, incurred BF 160, paid CC 150, and incurred CC 140, all in $000. Weights are 10%, 30%, 25%, and 35%, respectively. Calculate the weighted selection and check that it lies within the range of estimates. (Answer: selected liability = $146.50 thousand, within the range 120 to 160.)

What You'll Learn