Loss and Loss Adjustment Expense Exhibits

Purpose and Structure of Loss and LAE Exhibits

Loss and loss adjustment expense (LAE) exhibits show how claim experience moves from source records to the values used in a rate filing. They should let a reviewer trace the reported amounts, valuation date, development method, trend adjustment, and selected value without having to infer how one number became another. These exhibits complement earned premium and exposure exhibits: they document the claim-cost side of the experience.

A filing may show paid and incurred amounts separately. Paid amounts are cash payments made by the valuation date. Incurred amounts generally combine those payments with case reserves for reported claims. LAE should be identified by the basis used in the filing, such as allocated LAE (ALAE), unallocated LAE (ULAE), or a stated combination. The basis must remain consistent between the source data, development method, and selected result.

Definition: For a stated coverage, experience period, and valuation date, paid loss is the cumulative amount paid on claims; case incurred loss is paid loss plus the case reserves held for reported claims; and ultimate loss is the estimated total loss cost for the experience period after allowing for future development. Paid and incurred LAE are defined analogously. An exhibit should state whether its amounts include loss only, a particular LAE component, or total loss and LAE.

The valuation date matters because claims continue to develop after the end of an experience period. A triangle organizes cumulative claim amounts by experience period and development age. In an accident-year triangle, for example, the 12-month value for one accident year is compared with its 24-month value to estimate development from 12 to 24 months. The latest accident year may have only a 12-month observation, while an older one has more mature data.

Reconcile the Source Amounts First

A useful exhibit begins with values that can be reconciled to the underlying claim or accounting records. For a basic case-incurred basis, the identity is incurred equals paid plus case reserves. For LAE, show the corresponding paid and unpaid amounts under the stated LAE definition. Do not silently combine different bases—for example, loss-only incurred amounts with paid loss and LAE.

Worked Example: Reconciling Paid and Incurred Loss and LAE

A hypothetical accident-year source extract, in thousands of dollars, reports paid loss of $640, case loss reserves of $160, paid ALAE of $48, unpaid ALAE of $12, paid ULAE of $22, and unpaid ULAE of $8. Prepare the paid and incurred totals and check their reconciliation.

Method. Reconcile each component first, then sum the components. This avoids treating a reserve as a payment or omitting one LAE category.

Loss incurred is paid loss plus case reserves:

$$ 640+160=800 $$

ALAE incurred is paid plus unpaid ALAE: $48+12=60$. ULAE incurred is paid plus unpaid ULAE: $22+8=30$.

Total paid loss and LAE is $640+48+22=710$. Total unpaid case reserves and LAE are $160+12+8=180$. Thus total incurred loss and LAE is $710+180=890$. Summing the incurred components gives the same result: $800+60+30=890$.

Check. Total incurred less total paid is $890-710=180$, exactly the stated unpaid amount. The exhibit should preserve the component labels so a reviewer can tell that the $160 is a loss reserve, not paid loss.

Common mistake. Adding paid loss and paid LAE, then calling the result incurred, would give $710 and omit all unpaid amounts. Incurred amounts include the applicable reserves as well as payments.

Development and Ultimate Selections

Development exhibits show how observed amounts at a given age are adjusted for expected later emergence. A paid development method uses cumulative paid values; an incurred development method uses cumulative incurred values. Each method requires development factors appropriate to its own data basis. A paid factor should not be applied to incurred amounts merely because both are expressed in dollars.

Definition: For cumulative amounts $C_a$ and $C_b$ at development ages $a$ and $b$, where $b$ is later than $a$, the age-to-age factor is $C_b/C_a$. A selected age-to-ultimate factor at age $a$ estimates ultimate divided by the cumulative amount at age $a$. It is often formed from selected age-to-age factors, plus any selected tail factor beyond the last displayed age.

The factor relationship follows directly from ratios. If the amount at age $a$ is $C_a$, then applying the ratio $C_b/C_a$ gives the amount at age $b$. Applying the remaining ratio of ultimate to age $b$ gives:

C_a\left(\frac{C_b}{C_a}\right)\left(\frac{U}{C_b}\right)=U

Thus the product of sequential age-to-age factors and the tail factor is an age-to-ultimate factor. This multiplication is appropriate when the factors refer to successive cumulative ages on the same basis. It does not mean that every observed ratio must be selected without judgment; selections should be supported by the available experience and the actuary’s development analysis.

Worked Example: Developing Paid and Incurred Losses

A hypothetical insurer has the following cumulative loss triangles, in thousands of dollars. The rows are accident years; the columns are development ages in months.

Accident year	Paid at 12	Paid at 24	Paid at 36	Incurred at 12	Incurred at 24	Incurred at 36
2023	100	160	190	120	180	204
2024	110	176	—	132	198	—
2025	120	—	—	144	—	—

Select paid age-to-age factors of 1.60 from 12 to 24 months and 1.20 from 24 to 36 months. The observed 12-to-24 ratios are $160/100=1.60$ and $176/110=1.60$. The observed 24-to-36 ratio is $190/160=1.1875$, for which 1.20 is selected. Select incurred factors of 1.50 and 1.15. The observed 12-to-24 ratios are $180/120=1.50$ and $198/132=1.50$; the observed 24-to-36 ratio is $204/180=1.1333$ to four decimals, for which 1.15 is selected. Assume a tail factor of 1.02 from 36 months to ultimate for both bases.

Paid ultimate estimates. Accident year 2023 is observed at 36 months, so its estimate is $190(1.02)=193.800$. Accident year 2024 is at 24 months: $176(1.20)=211.200$ at 36 months, then $211.200(1.02)=215.424$ ultimate. Accident year 2025 is at 12 months: $120(1.60)=192.000$ at 24 months; $192.000(1.20)=230.400$ at 36 months; and $230.400(1.02)=235.008$ ultimate.

Incurred ultimate estimates. Accident year 2023 is $204(1.02)=208.080$. Accident year 2024 is $198(1.15)=227.700$ at 36 months, then $227.700(1.02)=232.254$ ultimate. Accident year 2025 is $144(1.50)=216.000$ at 24 months; $216.000(1.15)=248.400$ at 36 months; and $248.400(1.02)=253.368$ ultimate.

Check and interpretation. Every factor is applied from the row’s observed age through the later ages, then through the tail. The paid and incurred estimates differ because their development patterns and starting values differ. The difference is a diagnostic to understand, not an instruction to average the estimates automatically. A selected filing value should reflect the chosen method and the support for its assumptions.

Common mistake. Applying only the 12-to-24 factor to the 2025 incurred amount would produce $144(1.50)=216.000$, which is an estimate at 24 months, not ultimate. The remaining 1.15 factor and the 1.02 tail are also required.

Show Trend After Development

Development estimates ultimate amounts for the experience period; trend adjusts a selected historical cost level to a prospective period. The exhibit should make these steps visible and state the trend rate and time interval. Where loss and LAE have different selected trend assumptions, show them separately before combining them. Trend should be applied to a clearly stated measure, such as ultimate cost per exposure, rather than to an aggregate amount whose exposure volume has not been accounted for.

Result: If a selected annual trend rate is $r$ and the interval from the historical cost level to the prospective cost level is $t$ years, compound trend is $(1+r)^t$. The trended value is the historical value multiplied by this factor. This assumes the selected annual rate applies consistently over the stated interval; a different trend pattern requires a different factor.

Worked Example: Trending Selected Ultimate Loss and LAE

For a hypothetical book, the selected ultimate values are $1,200 of loss and $120 of LAE per exposure. The midpoint of the experience period is two years before the prospective period. The selected annual loss trend is 5%, and the selected annual LAE trend is 3%. Calculate the trended amounts and combined cost per exposure.

Loss. The two-year loss trend factor is $1.05^2=1.1025$. Trended loss is $$1,200(1.1025)=\$1,323.00$.

LAE. The two-year LAE trend factor is $1.03^2=1.0609$. Trended LAE is $$120(1.0609)=\$127.308$, or $127.31 to the nearest cent.

Combined cost. Adding the separately trended components gives $$1,323.00+\$127.31=\$1,450.31$ per exposure. Before rounding, the sum is $1,450.308.

Check. Both factors cover two years, not one: $1.05^2$ and $1.03^2$. The combined result is slightly greater than the untrended $1,200+120=\$1,320$, as expected for positive trend assumptions.

Common mistake. Applying the 5% loss trend to LAE as well would give $$120(1.05^2)=\$132.30$. That is not the stated LAE selection; the given 3% assumption must be used for LAE.

What a Reviewable Exhibit Should Make Clear

A useful filing exhibit presents a traceable sequence, not just a final selected number. The display may be organized in separate schedules or in one linked table, but it should identify the scope and basis of every amount. In particular, a reviewer should be able to identify:

the coverage, experience period, valuation date, and whether the experience is organized by accident year or another stated period;
whether each amount is paid, incurred, or ultimate, and whether it includes loss, ALAE, ULAE, or a defined total;
the source totals and reserve reconciliation, along with any stated exclusions or adjustments;
the development ages, observed ratios, selected factors, and tail assumption used to reach ultimate;
the selected trend rates, time interval, and resulting trended values, with loss and LAE treatment shown consistently.

A selected factor may differ from a single observed link ratio, particularly when there is limited mature experience or volatility. The exhibit should distinguish observed data from the actuarial selection rather than presenting the selection as though it were directly reported. Likewise, show any rounding convention and retain enough precision in intermediate calculations to reproduce the final values.

The central control is reconciliation: paid plus applicable reserves should agree to incurred; factor products should agree to age-to-ultimate selections; and the selected ultimate values should reproduce the trended values when the stated rates and periods are applied. These checks help catch mismatched development ages, inconsistent LAE bases, and omitted trend periods. The approach is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

Work each problem from the stated basis and show the intermediate steps before checking the answer.

A hypothetical source record shows paid loss of $420, case reserves of $80, paid ALAE of $30, and unpaid ALAE of $10, all in thousands. What are incurred loss and total incurred loss plus ALAE? (Answer: incurred loss $500 thousand; total incurred loss plus ALAE $540 thousand.)
A cumulative incurred amount is $250 thousand at 12 months. Apply a selected 12-to-24 factor of 1.40, a selected 24-to-ultimate factor of 1.10, and find ultimate. (Answer: $385 thousand.)
A paid loss amount is $300 thousand at 12 months. The selected 12-to-24 factor is 1.40 and the selected 24-to-ultimate factor is 1.15. What is the estimated ultimate? (Answer: $483 thousand.)
A selected ultimate loss cost is $900 per exposure. Apply an annual trend rate of 4% for three years. Give the trended cost to the nearest cent. (Answer: $1,012.38 per exposure.)
A hypothetical selected ultimate is $500 of loss and $50 of LAE per exposure. The prospective period is one year later; selected annual trends are 6% for loss and 2% for LAE. What is the combined trended cost per exposure? (Answer: $581.00 per exposure.)

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