Why the Reported Proportion Matters
Bornhuetter-Ferguson and Cape Cod estimates use an expected pattern of loss emergence. At a given valuation age, that pattern tells us what share of ultimate losses is expected to have emerged already and what share remains unreported. The selected age-to-ultimate factor provides a direct way to express those shares as proportions.
The key is to distinguish the reported proportion from the unreported proportion. The former is the share of ultimate losses represented by the current cumulative amount; the latter is its complement. A factor greater than one is not itself a proportion, so it must be converted before it is used as one.
An age-to-ultimate factor compares estimated ultimate losses with the cumulative losses at the current development age. It may be selected directly, or it may be obtained by multiplying selected age-to-age factors, including any applicable tail factor. For example, factors of 1.20 from 12 to 24 months and 1.25 from 24 months to ultimate combine to an age-to-ultimate factor of \(1.20(1.25)=1.50\) at 12 months.
The factor and the amount must use a consistent basis. A paid age-to-ultimate factor corresponds to cumulative paid losses; an incurred factor corresponds to the incurred measure used to construct that development pattern. The proportion is an expected share of ultimate losses on that basis. It is not necessarily the fraction of individual claims that have been reported.
From a Factor to Two Complementary Proportions
Suppose the current cumulative amount is \(C\), and the selected age-to-ultimate factor is \(L\). The factor projects ultimate losses as \(C L\). Thus the current amount as a fraction of that projected ultimate is \(C/(CL)\), which simplifies to \(1/L\). The remainder of ultimate is the unreported share.
This also shows why the complement, rather than the reciprocal itself, is used to represent unreported losses. If an expected ultimate amount is \(U\), the portion expected to remain unreported is \(U(1-1/L)\), not \(U/L\). The reciprocal \(1/L\) is the share already reported.
Use the factor appropriate to the accident period’s valuation age and to its cumulative paid or incurred measure.
If necessary, multiply the selected age-to-age factors from the current age through ultimate, including a selected tail factor when appropriate.
Calculate \(1/L\) for the reported proportion and \(1-1/L\) for the unreported proportion.
For an expected ultimate \(U\), the expected unreported amount is \(U(1-1/L)\). Check that both proportions sum to one.
Worked Applications
Worked Example: Converting a Factor to Expected Unreported Losses
A hypothetical insurer has selected age-to-age factors of 1.20 from 12 to 24 months and 1.25 from 24 months to ultimate for an accident period. Its expected ultimate losses are $360,000. Calculate the age-to-ultimate factor, both proportions, and the expected unreported amount. No interest-rate assumption is needed because this calculation uses loss-development proportions.
Method. First combine the development factors. The 12-month age-to-ultimate factor is \(L=1.20(1.25)=1.50\).
The reported proportion is the reciprocal: \(1/1.50=2/3=0.6667\), or 66.67% to two decimal places.
The unreported proportion is the complement: \(1-2/3=1/3=0.3333\), or 33.33% to two decimal places.
Apply the unreported proportion to the expected ultimate: \($360{,}000(1/3)=$120{,}000\). For a check, the complementary reported share of the expected ultimate is \($360{,}000(2/3)=$240{,}000\), and \($120{,}000+$240{,}000=$360{,}000\).
Answer. The selected age-to-ultimate factor is 1.50, the reported proportion is 66.67%, the unreported proportion is 33.33%, and expected unreported losses are $120,000. The mistake to avoid is multiplying expected ultimate by the reported share and calling that result unreported. Here that incorrect calculation would be \($360{,}000(2/3)=$240{,}000\), which is the reported share, not the unreported share.
Worked Example: Including a Tail Factor
For a hypothetical liability portfolio, selected age-to-age factors are 1.50 from 12 to 24 months and 1.20 from 24 to 36 months. A selected tail factor from 36 months to ultimate is 1.10. At 12 months, expected ultimate losses are $500,000. Find the reported and unreported proportions at 12 months and the expected unreported amount. No interest-rate assumption is needed.
Method. The age-to-ultimate factor at 12 months includes all three pieces: \(L=1.50(1.20)(1.10)=1.80(1.10)=1.98\).
The reported proportion is \(1/1.98=100/198=50/99\), or 50.5051% to four decimal places. The unreported proportion is \(1-50/99=49/99\), or 49.4949% to four decimal places.
The expected unreported amount is \($500{,}000(49/99)=$24{,}500{,}000/99=$247{,}474.747\ldots\). Rounded to the nearest cent, it is $247,474.75. The reported share of expected ultimate is \($500{,}000(50/99)=$252{,}525.25\), rounded to the nearest cent.
Check. The proportions sum to \(49/99+50/99=1\), and the rounded amounts sum to $500,000.00. The tail factor must be included because the requested factor runs from 12 months all the way to ultimate. Using only the observed-age factors would omit expected development after 36 months.
Worked Example: Applying Different Proportions by Accident Period
A hypothetical book has three accident periods with expected ultimate losses of $240,000, $360,000, and $400,000. Their selected age-to-ultimate factors, on a consistent paid-loss basis, are respectively 2.00, 1.50, and 1.25. Calculate the reported and unreported proportions and expected unreported amount for each period, then find the total expected unreported amount. No interest-rate assumption is needed.
Period 1. The reported proportion is \(1/2.00=0.50\), and the unreported proportion is \(1-0.50=0.50\). Expected unreported losses are \($240{,}000(0.50)=$120{,}000\).
Period 2. The reported proportion is \(1/1.50=2/3\), and the unreported proportion is \(1-2/3=1/3\). Expected unreported losses are \($360{,}000(1/3)=$120{,}000\).
Period 3. The reported proportion is \(1/1.25=0.80\), and the unreported proportion is \(1-0.80=0.20\). Expected unreported losses are \($400{,}000(0.20)=$80{,}000\).
The total expected unreported amount is \($120{,}000+$120{,}000+$80{,}000=$320{,}000\). For a check, the corresponding reported shares are $120,000, $240,000, and $320,000. Their total is $680,000, and $680,000 plus $320,000 equals the total expected ultimate of $1,000,000.
Answer. Expected unreported amounts are $120,000, $120,000, and $80,000, totaling $320,000. Each accident period uses its own selected factor; using one period’s proportion for another would ignore their different development ages and patterns.
Interpretation and Conditions
The reciprocal relationship is an algebraic consequence of the age-to-ultimate factor’s definition, but interpreting it as an expected emergence proportion requires a suitable development pattern. The selected factor should represent the expected relationship between the current cumulative amount and ultimate losses for the relevant accident period. The method assumes that the selected pattern is appropriate for the data and valuation age in question.
Factors are commonly at least one when cumulative losses are expected to develop upward to ultimate. In that case, the reported proportion is between zero and one, as is its complement. If a selected factor is below one, the formula still gives a reciprocal algebraically, but it gives a reported proportion above one and a negative unreported proportion. That result is a warning to investigate the selection, data basis, or development behavior rather than to treat the negative amount as an ordinary unreported share.
Also check whether the stated factor is truly age-to-ultimate. An age-to-age factor, such as a factor from 12 to 24 months alone, does not generally describe the full remaining development from 12 months to ultimate. The relevant age-to-ultimate factor is the product of the remaining age-to-age factors and tail factor, if selected. The factor must also match the cumulative measure: applying a paid development factor to incurred losses, or vice versa, can produce a misleading proportion.
Common Mistakes and Checks
- Using the reported share for unreported losses. The reported share is \(1/L\); the unreported share is \(1-1/L\). Multiply expected ultimate by the complement when calculating expected unreported losses.
- Subtracting one from the factor and stopping. \(L-1\) is not the unreported proportion. The proportion is \((L-1)/L\), since it must be measured relative to ultimate.
- Using a single age-to-age factor as though it reached ultimate. Multiply all remaining selected factors, including any applicable tail factor, to obtain the age-to-ultimate factor.
- Mixing bases or valuation ages. Match the factor to the accident period’s age and to its paid or incurred cumulative amount.
- Failing the complement check. The reported and unreported proportions must sum to one, and their applications to the same expected ultimate must sum to that ultimate.
Check Your Understanding
For each question, use the selected age-to-ultimate factor on the stated basis. Round proportions to four decimal places where needed and dollar amounts to the nearest cent.
- A selected age-to-ultimate factor is 1.25. What are the reported and unreported proportions? (Answer: Reported \(0.8000\); unreported \(0.2000\).)
- Selected factors are 1.40 from 12 to 24 months and 1.25 from 24 months to ultimate. Find the 12-month reported and unreported proportions. (Answer: Age-to-ultimate factor \(1.75\); reported \(4/7=0.5714\); unreported \(3/7=0.4286\).)
- Expected ultimate losses are $280,000 and the selected age-to-ultimate factor is 1.40. Calculate expected unreported losses. (Answer: Unreported proportion \(1-1/1.40=2/7\); expected unreported losses \(=$80,000.00\).)
- Three accident periods have expected ultimate losses of $150,000, $240,000, and $300,000 and factors of 1.50, 1.20, and 2.50, respectively. Calculate the total expected unreported amount. (Answer: Unreported amounts are $50,000, $40,000, and $180,000; total \(=$270,000\).)
- A hypothetical portfolio has expected ultimate losses of $800,000 and an age-to-ultimate factor of 1.60. A calculation uses \(1/1.60\) as the unreported proportion. Identify the error and calculate the correct expected unreported amount. (Answer: \(1/1.60=0.6250\) is the reported proportion; unreported proportion \(=0.3750\), so expected unreported losses \(=$300,000.00\).)