Avoiding Reported-Proportion Formula Errors

Why Proportion Errors Matter

Bornhuetter-Ferguson and Cape Cod estimates use a development factor to describe how much of an ultimate loss is expected to have emerged by the valuation date. The factor itself is not that proportion. Confusing a factor with its reciprocal, or confusing the reported share with the unreported share, can materially distort a reserve even when every other input is correct.

The key discipline is to translate the factor into proportions before calculating. Use the reported proportion when asking how much of ultimate losses has emerged; use the unreported proportion when estimating the expected portion still to emerge. Keep that distinction visible throughout the calculation.

Definition: Let $L$ be a cumulative loss development factor from the current valuation age to ultimate, on the same paid or incurred basis as the observed losses. If reported losses at that age are $R$, then the factor relates ultimate losses $U$ to reported losses by $L=U/R$. The implied reported proportion is $r=1/L$, and the implied unreported proportion is $q=1-r=1-1/L$. These proportions apply when $L$ is a positive factor to ultimate and the selected development pattern is appropriate for the loss measure being used.

For example, if $L=1.5$, then the reported proportion is $1/1.5=2/3$, and the unreported proportion is $1-2/3=1/3$. It is not correct to call 1.5 the reported proportion or to use $1.5-1$ as the unreported proportion. If a factor is already expressed as a reported proportion, rather than as a loss development factor, it should not be inverted again.

The Proportion Logic Behind the Formulas

The proportion formulas follow directly from the definition of the development factor. If $L=U/R$, then dividing both sides by $L$ gives $R=U/L$. Thus reported losses are $1/L$ of ultimate losses. The part not yet reported is the complement of that share.

\text{Reported proportion}=\frac{R}{U}=\frac{1}{L}, \qquad \text{Unreported proportion}=1-\frac{R}{U}=1-\frac{1}{L}.

In Bornhuetter-Ferguson (BF), let $E[U]$ denote expected ultimate losses based on the selected expected loss ratio and exposure measure. The expected unreported amount is $E[U]q$. BF adds that expected unreported amount to observed reported losses. The method does not apply the development factor to the observed losses to calculate the BF reserve.

\text{BF reserve}=E[U]\left(1-\frac{1}{L}\right), \qquad \text{BF ultimate}=R+E[U]\left(1-\frac{1}{L}\right).

The expected ultimate loss is the starting point for the BF reserve, so it is multiplied by the unreported share. The observed reported loss is added once, without another development multiplier. This structure means the BF ultimate can differ from the chain-ladder estimate $RL$: it combines observed losses to date with an expected estimate for the portion not yet reported.

In Cape Cod, the fitted loss ratio is chosen so that expected reported losses across the included periods match their observed reported losses. If period $j$ has exposure $e_j$, reported losses $R_j$, and factor $L_j$, the expected reported amount under a fitted loss ratio $\widehat{\lambda}$ is $\widehat{\lambda}e_j/L_j$. Consequently, the denominator of the fitted loss ratio uses exposure multiplied by the reported proportion, not the unreported proportion.

\widehat{\lambda} = \frac{\sum_j R_j}{\sum_j e_j/L_j}, \qquad \text{Cape Cod reserve}_j = \widehat{\lambda}e_j\left(1-\frac{1}{L_j}\right).

These expressions assume that exposure and losses are on compatible bases, the development factors match the observed loss measure, and the selected development pattern is used consistently. A factor for paid losses should not be paired with incurred losses unless a valid basis conversion has been made.

Worked Calculations

Worked Example: Separating Reported and Unreported Shares in Cape Cod

A hypothetical Cape Cod portfolio has two accident periods. Their exposures are $600,000 and $400,000; their reported losses are $240,000 and $300,000; and their cumulative factors to ultimate are 1.5 and 2.0. Calculate the fitted loss ratio and each period’s reserve. Use the same exposure and loss bases throughout.

Method. First find each period’s reported share $1/L_j$. Use those shares to form the Cape Cod denominator, then use each complementary unreported share to calculate the reserve.

For the first period, $L_1=1.5$, so its reported proportion is $1/1.5=2/3$, and its unreported proportion is $1-2/3=1/3$. Its exposure contribution to the denominator is

\$600{,}000\left(\frac{2}{3}\right)=\$400{,}000.

For the second period, $L_2=2.0$, so its reported proportion is $1/2$, and its unreported proportion is $1-1/2=1/2$. Its exposure contribution is

\$400{,}000\left(\frac{1}{2}\right)=\$200{,}000.

The denominator is therefore $\$400{,}000+\$200{,}000=\$600{,}000$. Total reported losses are $\$240{,}000+\$300{,}000=\$540{,}000$, so the fitted loss ratio is

\widehat{\lambda} = \frac{\$540{,}000}{\$600{,}000} =0.90=90\%.

The first-period reserve is fitted loss ratio times exposure times unreported proportion:

0.90(\$600{,}000)\left(\frac{1}{3}\right) =\$540{,}000\left(\frac{1}{3}\right) =\$180{,}000.

The second-period reserve is

0.90(\$400{,}000)\left(\frac{1}{2}\right) =\$360{,}000\left(\frac{1}{2}\right) =\$180{,}000.

Check. The reserves are $180,000 and $180,000, and the fitted loss ratio is 90%. A common error is to use $1/L_1=2/3$ as the first period’s unreported proportion; that would give $360,000, not the correct $180,000. Another error is to use the unreported proportions in the fitted-ratio denominator, even though that denominator represents expected reported losses.

Worked Example: Bornhuetter-Ferguson Does Not Develop the Observed Losses

A hypothetical coverage portfolio has expected ultimate losses of $520,000. At the valuation date, reported losses are $390,000 and the cumulative development factor to ultimate is 1.25. Calculate the BF reserve and ultimate. Then compare the BF ultimate with the chain-ladder ultimate.

Method. Translate the factor into the reported and unreported shares, calculate the expected unreported amount, and add that amount to reported losses.

The reported proportion is

\frac{1}{1.25}=0.80.

Therefore, the unreported proportion is $1-0.80=0.20$. The BF reserve is

\$520{,}000(0.20)=\$104{,}000.

The BF ultimate is reported losses plus the BF reserve:

\$390{,}000+\$104{,}000=\$494{,}000.

For comparison, the chain-ladder ultimate is reported losses multiplied by the development factor:

\$390{,}000(1.25)=\$487{,}500.

Check. The BF reserve is $104,000, the BF ultimate is $494,000, and the chain-ladder ultimate is $487,500. The estimates differ because BF uses the expected ultimate for the unreported part, while chain ladder develops the observed reported losses. A common mistake is to calculate the BF reserve as $\$390{,}000(1.25-1)=\$97{,}500$. That is the chain-ladder increase over reported losses, not the BF reserve.

Worked Example: Correcting a Development-Factor Multiplier

A hypothetical accident period has exposure of $500,000, a selected expected loss ratio of 70%, reported losses of $300,000, and a cumulative factor to ultimate of 1.5. Calculate the BF reserve and ultimate. A reviewer proposes multiplying the reported losses by $1-1/1.5$ to obtain the reserve. Assess that proposal.

Method. Calculate expected ultimate losses from exposure and expected loss ratio. Apply the unreported proportion to expected ultimate losses, not to observed reported losses.

Expected ultimate losses are

\$500{,}000(0.70)=\$350{,}000.

The reported proportion is $1/1.5=2/3$; hence the unreported proportion is $1-2/3=1/3$. The BF reserve is

\$350{,}000\left(\frac{1}{3}\right) =\$116{,}666.666\ldots =\$116{,}666.67.

The BF ultimate is

\$300{,}000+\$116{,}666.67=\$416{,}666.67.

The reviewer’s proposed calculation is

\$300{,}000\left(1-\frac{1}{1.5}\right) =\$300{,}000\left(\frac{1}{3}\right) =\$100{,}000.

That $100,000 is not the BF reserve: it applies the unreported proportion to observed reported losses rather than to expected ultimate losses. The calculation $\$300{,}000(1.5-1)=\$150{,}000$ would also be wrong as a BF reserve; it is the chain-ladder increase over reported losses in this case.

Check. The BF reserve is $116,666.67 and the BF ultimate is $416,666.67, rounded to the nearest cent. The reserve plus reported losses equals the stated BF ultimate.

A Reliable Calculation Sequence

A short sequence of checks helps prevent these errors, especially when multiple periods have different factors. Do not substitute a single “development percentage” for both proportions: calculate the reported and unreported shares separately.

Confirm what the factor means.
Check that $L$ is a cumulative factor from the valuation age to ultimate, and that it matches the loss basis being analyzed.

Write both proportions.
Calculate reported proportion $1/L$ and unreported proportion $1-1/L$. Verify that they sum to 1.

Choose the correct starting amount.
For a BF reserve, multiply the unreported proportion by expected ultimate losses. For Cape Cod, use the reported proportion in the fitted-ratio denominator and the unreported proportion in the reserve.

Reconcile the result.
For BF, check that ultimate equals reported losses plus reserve. For Cape Cod, check that the fitted loss ratio reproduces the portfolio’s reported losses when multiplied by exposure and reported proportion.

Common Exam Traps

The arithmetic often looks plausible even when the chosen multiplier is conceptually wrong. These checks identify the most frequent errors.

Calling $L$ the reported proportion. A factor of 1.5 means ultimate is 1.5 times reported losses; the reported proportion is $1/1.5$, not 1.5.
Using $L-1$ as the unreported proportion. $L-1$ is a multiplier used in the chain-ladder increase $R(L-1)$. The unreported proportion is $1-1/L$.
Swapping the two Cape Cod proportions. The fitted loss ratio denominator uses exposure times the reported proportion, $e_j/L_j$. The reserve uses exposure times the unreported proportion, $e_j(1-1/L_j)$.
Applying the unreported share to observed losses in BF. BF applies it to expected ultimate losses. Observed reported losses are added separately to form the ultimate.
Applying the development factor twice. If reported losses have already been developed to ultimate, do not then add a BF reserve to that developed amount. BF starts with reported losses, not a chain-ladder ultimate.
Forgetting that a reserve is not an ultimate. Under BF, the reserve is only the expected unreported component; the ultimate is reported losses plus that reserve.

Key takeaway: For a cumulative factor $L$, reported losses represent $1/L$ of ultimate and the unreported share is $1-1/L$. Use the reported share to fit Cape Cod to reported experience; use the unreported share with expected ultimate losses for a BF reserve. Do not interchange these proportions or substitute $L$ itself for either one.

Check Your Understanding

For each problem, identify the relevant reported and unreported proportions before calculating.

A hypothetical BF estimate has expected ultimate losses of $720,000, reported losses of $480,000, and a factor of 1.2. Find the BF reserve and ultimate. (Answer: Reported proportion $=0.8333$; unreported proportion $=0.1667$; reserve $=\$120{,}000$; ultimate $=\$600{,}000$.)
A hypothetical coverage has expected ultimate losses of $600,000, reported losses of $210,000, and a factor of 2.5. Calculate the BF reserve and ultimate. (Answer: Unreported proportion $=0.60$; reserve $=\$360{,}000$; ultimate $=\$570{,}000$.)
A two-period hypothetical Cape Cod estimate has exposures of $300,000 and $500,000, reported losses of $120,000 and $250,000, and factors of 1.5 and 2.0. Find the fitted loss ratio and both reserves. (Answer: Denominator $=\$450{,}000$; fitted loss ratio $=82.2222\%$; reserves $=\$82{,}222.22$ and $\$205{,}555.56$.)
A hypothetical BF period has exposure of $900,000, expected loss ratio 60%, reported losses of $430,000, and a factor of 1.25. Calculate the reserve and ultimate. (Answer: Expected ultimate $=\$540{,}000$; unreported proportion $=0.20$; reserve $=\$108{,}000$; ultimate $=\$538{,}000$.)
A hypothetical period has exposure of $400,000, expected loss ratio 75%, reported losses of $160,000, and a factor of 1.6. A proposed reserve is calculated as reported losses multiplied by $1-1/1.6$. Calculate the correct BF reserve and ultimate, and identify the proposed reserve. (Answer: Expected ultimate $=\$300{,}000$; unreported proportion $=0.375$; correct reserve $=\$112{,}500$; ultimate $=\$272{,}500$; proposed reserve $=\$60{,}000$, which incorrectly applies the unreported proportion to reported losses.)

What You'll Learn