Converting Emergence Percentages to Unreported Shares

Why the Unreported Share Matters

In a Bornhuetter-Ferguson calculation, expected ultimate losses are allocated between the portion expected to have emerged by the valuation date and the portion expected to emerge afterward. The percentage that has emerged is not the percentage still unreported: the two shares are complements. Converting between them correctly is essential whenever an expected ultimate loss is used to estimate an expected unreported amount.

This tutorial focuses on that conversion and its consistent application. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. The calculation is simple, but a wrong evaluation age, an incremental percentage mistaken for a cumulative one, or a development factor used in the wrong direction can materially change the result.

Definition: Let $e_a$ denote the cumulative emergence percentage at development age $a$, expressed as a decimal. It is the selected proportion of ultimate losses expected to have emerged by age $a$, on a specified basis such as paid or incurred. The corresponding unreported share is $u_a=1-e_a$. For expected ultimate losses $L^{\mathrm{exp}}$, the expected unreported component implied by this pattern is $L^{\mathrm{exp}}u_a$.

From Cumulative Emergence to Its Complement

A cumulative emergence percentage is measured from the start of the loss period through the stated development age. For example, a cumulative emergence percentage of 64% at 24 months means that, under the selected pattern, 64% of ultimate losses is expected to have emerged by 24 months. The remaining 36% is the unreported share at that age.

The percentages must be on the same ultimate-loss basis. If the cumulative emergence percentage is 64%, multiplying expected ultimate losses by 64% gives the expected emerged amount, not the expected unreported amount. The unreported amount uses the complement, 36%.

$$ u_a=1-e_a $$

The relationship follows directly from partitioning the expected ultimate amount. Under the selected pattern, the expected ultimate is split into an emerged part and a part not yet emerged. If those parts are $e_aL^{\mathrm{exp}}$ and $u_aL^{\mathrm{exp}}$, respectively, they must sum to the expected ultimate:

e_aL^{\mathrm{exp}}+u_aL^{\mathrm{exp}}=L^{\mathrm{exp}}.

For positive $L^{\mathrm{exp}}$, divide by $L^{\mathrm{exp}}$ to obtain $e_a+u_a=1$, and therefore $u_a=1-e_a$. This is a share of expected ultimate losses, not an assertion about exactly which individual claims will be reported. It is the expected allocation implied by the selected emergence pattern.

Worked Example: Converting a Cumulative Percentage at the Correct Age

A hypothetical commercial equipment portfolio has expected ultimate losses of $1,350,000 for an accident year. The selected cumulative emergence percentages are 28% at 12 months, 57% at 24 months, and 81% at 36 months. The accident year is evaluated at 24 months. Calculate its expected unreported amount using the selected pattern.

Method. Use the cumulative emergence percentage at the accident year's evaluation age, then take its complement.

At 24 months, $e_{24}=0.57$. Therefore,

u_{24}=1-0.57=0.43.

The expected unreported amount is:

\$1{,}350{,}000(0.43) =\$1{,}350{,}000(0.40+0.03) =\$540{,}000+\$40{,}500 =\$580{,}500.

Check. The expected emerged amount is $0.57(\$1{,}350{,}000)=\$769{,}500$, and $\$769{,}500+\$580{,}500=\$1{,}350{,}000$. The split reconciles to expected ultimate losses.

Common mistake avoided: Using the 36-month emergence percentage because it is the most mature percentage in the table. The accident year is evaluated at 24 months, so the applicable share is 57%, not 81%.

Converting a Cumulative Development Factor

Sometimes the selected information is presented as a cumulative development factor to ultimate rather than as an emergence percentage. A cumulative development factor at age $a$ compares ultimate losses with the losses emerged at age $a$. If that factor and the emergence percentage use the same loss basis, valuation definition, and ultimate measure, they are reciprocals.

\mathrm{CDF}_a =\frac{\text{ultimate losses}}{\text{losses emerged at age }a} =\frac{1}{e_a}, \qquad e_a=\frac{1}{\mathrm{CDF}_a}, \qquad u_a=1-\frac{1}{\mathrm{CDF}_a}.

The reciprocal relationship comes from the definitions. If the losses emerged at age $a$ are a fraction $e_a$ of ultimate losses, then ultimate divided by emerged losses is $L/(e_aL)=1/e_a$. The unreported share is the complement of that emergence fraction. This conversion applies to a cumulative age-to-ultimate factor, not automatically to an age-to-age factor.

Worked Example: Converting a Cumulative Factor to an Unreported Share

A hypothetical specialty insurer selects a cumulative paid development factor to ultimate of 1.60 at 24 months. Its expected ultimate paid losses for an accident year are $960,000. A separate 12-to-24-month age-to-age factor is 1.40. Determine the expected unreported amount at 24 months.

Method. Convert the cumulative age-to-ultimate factor at 24 months to emergence, then take the complement and apply it to expected ultimate paid losses.

First, calculate the cumulative emergence percentage:

e_{24}=\frac{1}{1.60}=0.625.

Thus, 62.5% is expected to have emerged and the unreported share is:

u_{24}=1-0.625=0.375.

Apply this share to the expected ultimate:

\$960{,}000(0.375) =\$960{,}000\left(\frac{3}{8}\right) =\$120{,}000(3) =\$360{,}000.

Check. The implied expected emerged amount is $\$960{,}000(0.625)=\$600{,}000$; $\$600{,}000+\$360{,}000=\$960{,}000$.

Common mistake avoided: Taking the reciprocal of 1.40. That is the 12-to-24-month factor, not the cumulative factor from 24 months to ultimate. The age-24 emergence conversion uses 1.60, as specified.

Apply Shares by Age and Keep the Basis Consistent

When several accident years are evaluated together, each year may have a different development age and therefore a different unreported share. Apply the share for each year separately to that year's expected ultimate losses; only then add the resulting expected unreported amounts. Applying one average percentage to the combined expected ultimate can be wrong if the years have different ages or expected ultimate amounts.

The loss basis must also match. A paid emergence pattern describes the expected emergence of paid losses; an incurred pattern describes incurred losses under its own definition. Do not pair an expected ultimate amount on one basis with an emergence percentage or development factor selected on another. The same principle applies to the valuation date and the definition of ultimate used to produce the pattern.

Identify the evaluation age.
For each accident year, determine its development age as of the valuation date.

Select the matching pattern value.
Use cumulative emergence at that age, or the cumulative age-to-ultimate factor at that age if factors are provided.

Convert to an unreported share.
For emergence $e_a$, use $1-e_a$. For a compatible cumulative factor $\mathrm{CDF}_a$, use $1-1/\mathrm{CDF}_a$.

Apply to expected ultimate losses.
Multiply each year’s expected ultimate by its own unreported share, preserving the same loss basis.

Reconcile the split.
Check that emerged and unreported shares sum to 100%, and that their amounts sum to the expected ultimate for each year.

Worked Example: Combining Accident Years at Different Maturities

A hypothetical regional insurer has three accident years with expected ultimate losses on an incurred basis. At the current valuation date, the years are at 12, 24, and 36 months, respectively. The selected cumulative incurred emergence percentages are 40% at 12 months, 65% at 24 months, and 82% at 36 months. Expected ultimate losses are $700,000, $1,050,000, and $900,000, respectively. Calculate the total expected unreported component across the three years.

Method. Convert each age-specific cumulative emergence percentage to its complement, apply that share to the corresponding expected ultimate, and then sum.

First accident year, 12 months. The unreported share is $1-0.40=0.60$. Its expected unreported amount is:

\$700{,}000(0.60)=\$420{,}000.

Second accident year, 24 months. The unreported share is $1-0.65=0.35$. Its expected unreported amount is:

\$1{,}050{,}000(0.35)=\$367{,}500.

Third accident year, 36 months. The unreported share is $1-0.82=0.18$. Its expected unreported amount is:

\$900{,}000(0.18)=\$162{,}000.

The total is:

\$420{,}000+\$367{,}500+\$162{,}000=\$949{,}500.

Check. The respective emerged amounts are $280,000, $682,500, and $738,000. For each year, emerged plus unreported equals its expected ultimate; in total, $1,700,500+$949,500=$2,650,000, the sum of the three expected ultimates.

Common mistake avoided: Applying the 18% share from the oldest year to all three expected ultimates. The years have different maturities, so their unreported shares are 60%, 35%, and 18%.

Incremental Percentages, Checks, and Exam Traps

A table may report incremental emergence rather than cumulative emergence. An incremental percentage describes the portion of ultimate expected to emerge during one development interval. To use it as an age-specific emergence percentage, add the increments from the beginning through the evaluation age. Only then take the complement.

Worked Example: Accumulating Incremental Emergence Before Taking the Complement

A hypothetical liability program has expected ultimate losses of $2,400,000. Its selected incremental emergence percentages are 15% in the first development interval, 25% in the second, 30% in the third, and 20% in the fourth. The program is evaluated after the third interval. Calculate the expected unreported amount.

Method. Sum the increments through the third interval to obtain cumulative emergence, then take the complement.

Cumulative emergence after three intervals is:

$$ e_3=0.15+0.25+0.30=0.70. $$

The unreported share is $1-0.70=0.30$, so:

\$2{,}400{,}000(0.30)=\$720{,}000.

Check. The expected emerged amount is $\$2{,}400{,}000(0.70)=\$1{,}680{,}000$, and $1,680,000+$720,000=$2,400,000.

Common mistake avoided: Treating the third-interval increment, 30%, as the cumulative emergence percentage. In this example the resulting unreported share happens to equal 70%, but that is a coincidence; the correct cumulative emergence is 70%, and the correct unreported share is 30%.

The final example also illustrates a useful diagnostic: the total expected emerged and unreported amounts should reconcile to expected ultimate losses under the chosen pattern. For a standard emergence pattern, cumulative emergence is ordinarily between 0% and 100%, and the unreported share is also between 0% and 100%. A calculated negative unreported share or a share above 100% is a reason to check the data, basis, factor direction, and interpretation before proceeding. Do not silently replace such a result with a plausible-looking percentage.

The unreported component calculated here is the expected amount associated with the selected pattern and expected ultimate. It is an input to the Bornhuetter-Ferguson framework, not by itself a complete explanation of every reserve component or a substitute for checking the selected expected ultimate. The central discipline is to pair each expected ultimate with the correct age-specific unreported share on a consistent basis.

Key takeaway: If cumulative emergence at age $a$ is $e_a$, the unreported share is $1-e_a$. If a compatible cumulative age-to-ultimate factor is provided, first convert it to emergence with $e_a=1/\mathrm{CDF}_a$. Match the age and loss basis before applying the resulting share to expected ultimate losses.

Check Your Understanding

For each problem, identify the applicable age and basis before calculating the unreported share.

A hypothetical accident year has expected ultimate paid losses of $640,000 and cumulative paid emergence of 55% at its current age. Calculate its expected unreported amount. (Answer: $288,000)
A hypothetical portfolio has a cumulative incurred development factor to ultimate of 1.25 at 36 months. Expected ultimate incurred losses are $950,000. Calculate the unreported share and expected unreported amount. (Answer: 20%; $190,000)
A hypothetical program's incremental emergence is 22% in the first interval, 31% in the second, and 27% in the third. At the end of the second interval, expected ultimate losses are $1,100,000. Calculate the expected unreported amount. (Answer: $517,000)
Two hypothetical accident years have expected ultimate losses of $500,000 and $800,000. Their matching cumulative emergence percentages at their respective ages are 40% and 65%. Calculate the combined expected unreported amount. (Answer: $580,000)
For a hypothetical accident year, cumulative emergence at 24 months is 70% and expected ultimate losses are $600,000. An analyst calculates the unreported amount as 70% of expected ultimate. State the correct unreported share and amount. (Answer: 30%; $180,000)

What You'll Learn