Sensitivity and Reasonableness Checks for Claim Liabilities

Why Test a Claim Liability Estimate?

A selected claim liability is an estimate, not a directly observable amount. Its reasonableness therefore depends on more than whether a calculation has been completed correctly. An actuary should also ask how the estimate changes when important assumptions change, whether the implied paid and unpaid amounts fit together, and whether the aggregate result is compatible with the claim inventory.

A sensitivity test changes one selection while holding the other inputs fixed. This shows the estimate’s response to that selection; it does not establish that the alternative is more likely or create a statistical confidence interval. Reasonableness checks look for arithmetic inconsistencies, unexpected directions of movement, or material differences that warrant investigation. They do not prove that a reserve is adequate.

Definition: At a fixed valuation date and on a consistent loss basis, let $U$ be selected ultimate loss and $P$ be cumulative paid loss. The implied unpaid liability is $R=U-P$, and the paid-to-ultimate ratio is $q=P/U$, provided $U>0$. These quantities should use the same coverage, claim population, currency, and gross or net basis. Unless stated otherwise, amounts in this tutorial are undiscounted and exclude expenses and recoverables.

For a fixed paid amount, selecting a different ultimate changes the unpaid amount dollar for dollar: an increase of $10{,}000$ in $U$ increases $R$ by $10{,}000$. The paid-to-ultimate ratio changes at the same time. It is useful as a diagnostic, but a ratio by itself does not determine whether a liability is reasonable; maturity and the underlying claim experience matter.

A Basic Sensitivity Framework

A sound test starts with a documented base selection, then varies one key assumption over a defensible range. For each scenario, recalculate the implied unpaid amount and any relevant ratios. The comparison should show both dollar movement and percentage movement, because a seemingly small percentage change can still be material in dollars.

Fix the measurement basis.
Specify the valuation date, covered claims, paid-loss basis, and whether the amounts are gross or net. Do not compare figures built on different bases.

Record the base selection.
Set out $P$, $U$, $R=U-P$, and $q=P/U$. Retain enough precision in calculations to avoid rounding differences.

Change a specified assumption.
For example, test alternative ultimate selections or paid-to-ultimate ratios. Hold unrelated assumptions fixed for a one-at-a-time test.

Interpret and investigate.
Explain the direction and size of the movement, then compare the implied liability with relevant claim-level or operational information.

Worked Example: Changing the Selected Ultimate

A hypothetical portfolio has $P=\$420{,}000$ paid at the valuation date. The selected ultimate is $\$600{,}000$. Test the effect of selecting an ultimate 5% higher, leaving paid loss unchanged. Report the base and revised paid-to-ultimate ratios and the change in unpaid liability.

Method. Recalculate $R=U-P$ for each ultimate selection. This is a direct ultimate-selection sensitivity, not a new estimate of paid loss.

Base scenario. The unpaid amount is

R_{\text{base}}=\$600{,}000-\$420{,}000=\$180{,}000.

The base paid-to-ultimate ratio is

q_{\text{base}}=\frac{\$420{,}000}{\$600{,}000}=0.7000=70.00\%.

Revised scenario. A 5% increase gives an ultimate of

U_{\text{revised}}=\$600{,}000(1.05)=\$630{,}000.

The revised unpaid amount is $\$630{,}000-\$420{,}000=\$210{,}000$, and the revised paid-to-ultimate ratio is $\$420{,}000/\$630{,}000=0.6667$, or 66.67%. Thus unpaid liability increases by $\$210{,}000-\$180{,}000=\$30{,}000$, which is $\$30{,}000/\$180{,}000=16.67\%$ of the base liability.

Check and common mistake. The revised ultimate is $\$30{,}000$ higher, so with paid loss fixed the unpaid amount must also be $\$30{,}000$ higher. The arithmetic satisfies this identity. A common mistake is to report the 5% increase in ultimate as a 5% increase in unpaid liability. Here it is not: $\$30{,}000/\$180{,}000=16.67\%$.

Paid-to-Ultimate Ratios and Implied Unpaid Amounts

When the selected quantity is a paid-to-ultimate ratio, the ratio must be applied to paid loss in the correct direction. If $q=P/U$, then $U=P/q$, so the implied unpaid amount is $P/q-P$. A higher paid-to-ultimate ratio means a larger share of ultimate has already been paid and, for fixed $P$, a lower implied unpaid amount.

Justification: By definition, $q=P/U$. Multiplying both sides by $U$ gives $qU=P$, and dividing by $q>0$ gives $U=P/q$. Subtracting paid loss gives $R=U-P=P/q-P=P(1/q-1)$. This relationship assumes the ratio and paid amount refer to the same claims, valuation date, and loss basis. It is not appropriate to apply a ratio derived for a different population or basis without adjustment.

This formula also explains why the relationship is nonlinear. For fixed $P$, the derivative of $R=P(1/q-1)$ with respect to $q$ is $-P/q^2$. Thus, locally, increasing $q$ reduces implied unpaid loss; the dollar response per unit change in $q$ depends on the current ratio. Ratios close to zero can imply very large ultimates, so a small absolute change in the ratio can have a substantial effect.

Worked Example: Stress-Testing a Paid-to-Ultimate Ratio

A hypothetical accident period has $\$360{,}000$ paid. The selected paid-to-ultimate ratio is 0.60. Test ratios of 0.56 and 0.64. Show the implied ultimate and unpaid amount in each scenario. Report dollar amounts in thousands to three decimal places.

Method. Use $U=P/q$, then calculate $R=U-P$. Keep paid loss fixed throughout.

Selected ratio, $q=0.60$. In thousands, $P=360$. Therefore $U=360/0.60=600.000$, and $R=600.000-360.000=240.000$.

Lower ratio, $q=0.56$. The implied ultimate is $360/0.56=642.857142\ldots$, or 642.857 thousand after rounding. The implied unpaid amount is $642.857142\ldots-360=282.857142\ldots$, or 282.857 thousand. Relative to the selected unpaid amount, this is an increase of $282.857142\ldots-240=42.857142\ldots$ thousand, or $42.857142\ldots/240=17.86\%$.

Higher ratio, $q=0.64$. The implied ultimate is $360/0.64=562.500$ thousand. The implied unpaid amount is $562.500-360=202.500$ thousand. Relative to the selected amount, this is a decrease of $240.000-202.500=37.500$ thousand, or $37.500/240=15.63\%$.

Check and common mistake. In every scenario, $P/U=q$: for example, $360/562.500=0.64$. Also, the higher ratio produces the lower unpaid amount, as required by $R=P(1/q-1)$. A common error is to calculate $U=Pq$, which would produce an ultimate below paid loss in this example. That contradicts the definition of a paid-to-ultimate ratio when unpaid loss is positive.

Comparing Aggregate Estimates With Case Reserves

Claim-level case reserves provide a useful, but limited, comparison. For a reported claim, the case reserve is the current estimate of its remaining payments recorded in the claim system. Summing case reserves gives a claim-inventory measure for the included reported claims. An aggregate unpaid estimate may also reflect amounts not yet reported and future development beyond the recorded case estimates. Therefore, the difference between aggregate unpaid loss and summed case reserves is a diagnostic residual, not automatically an IBNR estimate or a finding of error.

A claim-level review can reveal whether an aggregate result is driven by one claim, whether a case estimate is unusually high or low relative to the selected claim ultimate, or whether the claim inventory and triangle-based population appear inconsistent. A negative residual for an individual claim means that its case reserve exceeds the selected unpaid amount for that claim; it does not mean the liability is negative. Case adequacy judgments require claim facts and consistent definitions.

Worked Example: Checking Claim-Level Residuals

A hypothetical insurer reviews three reported claims. Amounts are in thousands of dollars. For each claim, the selected ultimate and paid amount are on the same valuation basis. Calculate selected unpaid amounts, compare them with case reserves, and reconcile the aggregate residual. Then increase every selected ultimate by 5% and assess the effect.

Claim	Paid	Case reserve	Selected ultimate
A	40	20	75
B	90	10	130
C	20	50	65

Method. For each claim, calculate selected unpaid as ultimate less paid. Define the claim-level residual as selected unpaid less case reserve. A positive residual means selected unpaid exceeds the recorded case reserve; a negative residual means the reverse.

Claim A. Selected unpaid is $75-40=35$. The residual is $35-20=15$.

Claim B. Selected unpaid is $130-90=40$. The residual is $40-10=30$.

Claim C. Selected unpaid is $65-20=45$. The residual is $45-50=-5$.

Aggregate check. Total paid is $40+90+20=150$, total case reserves are $20+10+50=80$, and total ultimate is $75+130+65=270$. Aggregate selected unpaid is $270-150=120$. The aggregate residual is $120-80=40$. The claim-level residuals also sum to $15+30-5=40$, so the comparison reconciles.

5% higher-ultimate sensitivity. The revised claim ultimates are $75(1.05)=78.75$, $130(1.05)=136.50$, and $65(1.05)=68.25$, totaling $283.50$. Paid remains 150, so revised unpaid is $283.50-150=133.50$. The increase is $133.50-120=13.50$, or $13.50/120=11.25\%$. Revised residuals are: A, $78.75-40-20=18.75$; B, $136.50-90-10=36.50$; and C, $68.25-20-50=-1.75$. They sum to $18.75+36.50-1.75=53.50$, equal to $133.50-80$.

Check and common mistake. The aggregate residual agrees with the sum of claim-level residuals in both scenarios. Claim C’s negative residual is not a negative reserve; it says only that the selected unpaid amount is below the recorded case reserve. A common mistake is to label the aggregate residual automatically as IBNR. The calculation alone cannot distinguish unreported claims from additional development on reported claims or differences in estimation basis.

What Makes a Sensitivity Test Informative?

The range of alternatives should be tied to uncertainty in a specific selection, not chosen merely to produce a desired result. A narrow range may be appropriate when the estimate is well supported; a wider range may be useful when the period is immature or key inputs are uncertain. If several assumptions are changed together, the result measures a combined scenario and does not identify which assumption caused the movement.

A practical set of checks includes:

Direction. Confirm that the result moves as the formula implies. For fixed paid loss, higher ultimate means higher unpaid; higher paid-to-ultimate ratio means lower implied unpaid.
Magnitude. Report changes in dollars and relative to the base liability. Do not compare percentages without also considering the dollar scale.
Internal reconciliation. Verify that ultimate equals paid plus unpaid and that aggregate totals agree with the sum of included claim-level amounts where the bases match.
Population and basis. Confirm that paid, ultimate, ratios, and case reserves cover compatible claims and use consistent gross or net definitions.
Concentration. Identify whether a change is driven by a small number of large claims or is distributed across the portfolio. A stable aggregate can conceal offsetting claim-level movements.

Reasonableness checks should lead to questions, not unsupported conclusions. A large difference between an aggregate unpaid estimate and case reserves may reflect expected development, unreported claims, timing, or mismatched definitions. The actuary should investigate the cause before changing a selection. Conversely, agreement between two measures is not proof that both are correct if they share the same weak assumptions.

Key takeaway: A sensitivity test is most useful when its basis, changed assumption, and resulting movement are explicit. Recalculate unpaid amounts consistently, use paid-to-ultimate ratios in the correct direction, and treat case-reserve comparisons as diagnostics rather than automatic classifications of the liability.

Check Your Understanding

For each problem, use a consistent valuation date and basis. Amounts are hypothetical; where amounts are stated in thousands, retain that unit.

A portfolio has paid loss of $\$275{,}000$ and selected ultimate of $\$500{,}000$. Increase ultimate by 6%, holding paid fixed. Find the base and revised unpaid amounts and the percentage change in unpaid. (Answer: base unpaid $\$225{,}000$; revised unpaid $\$255{,}000$; increase $\$30{,}000$, or 13.33%.)
Paid loss is $\$420{,}000$. Compare implied unpaid amounts at paid-to-ultimate ratios of 0.70 and 0.75. What is the change from the 0.70 scenario? (Answer: $\$180{,}000$ at 0.70 and $\$140{,}000$ at 0.75; decrease $\$40{,}000$, or 22.22%.)
In thousands, an aggregate estimate has paid of 265, case reserves of 135, and selected ultimate of 440. Calculate selected unpaid and the residual relative to case reserves. (Answer: unpaid 175; residual $175-135=40$ thousand.)
For three claims, the paid, case reserve, and selected ultimate amounts in thousands are respectively: Claim 1: 25, 15, 50; Claim 2: 40, 30, 80; Claim 3: 10, 20, 25. Find aggregate unpaid, aggregate case reserves, and aggregate residual; verify by summing claim-level residuals. (Answer: aggregate unpaid 80, case reserves 65, residual 15 thousand; claim residuals are 10, 10, and $-5$, summing to 15.)
Paid loss is 840 thousand and selected ultimate is 1,200 thousand. Test ultimates 5% below and above the selection, holding paid fixed. Find each implied unpaid amount and the difference between the high and low scenarios. (Answer: low ultimate 1,140 and unpaid 300 thousand; high ultimate 1,260 and unpaid 420 thousand; high-minus-low unpaid difference 120 thousand.)

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