Why Audit Expected Losses and Plan Weights?
Expected losses and plan weights are inputs to experience rating calculations. An error in exposure, rate, component allocation, or weight can therefore affect a later calculation even when the arithmetic in that calculation is flawless. An audit checks whether the inputs agree with one another and with the plan’s stated assumptions. It does not, by itself, prove that an underlying rate or assumption is actuarially appropriate.
The central checks are straightforward: expected loss should reconcile to exposure multiplied by the applicable expected loss rate; component amounts should reconcile to the total; and weights should agree with the specified formula and stay within its allowable range. These checks are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
A Practical Audit Sequence
Start by identifying the basis of every input. “Exposure” might mean payroll, vehicle-years, earned units, or another measure; a rate is meaningful only when its denominator matches that exposure. Confirm the rate’s period and scope as well. An annual rate cannot be multiplied by a partial-year exposure as though the exposure represented a full year.
Record the exposure unit, time period, expected loss rate basis, and population or coverage to which each applies.
Multiply exposure by the rate, ensuring the units cancel to dollars of expected loss. Compare the result with the supplied total.
Check that the allocated primary and excess amounts sum to total expected loss, and that the allocation proportions agree with the stated D-ratio.
Apply the plan’s stated weight formula to the correct component expected loss volume and full-credibility threshold. Check bounds and component labels.
Separate arithmetic or data errors from plausible differences due to rounding, changed assumptions, or a different basis. Investigate rather than silently force a reconciliation.
Reconciliation Identities and Their Justification
For a single exposure group, total expected loss is the exposure multiplied by its rate. If the rate is stated per 100 exposure units, convert the exposure to groups of 100 (or divide the product by 100). If exposure is measured in partial years, first express it in the corresponding annual units when the rate is annual.
A D-ratio is the share of expected losses assigned to the primary component at the specified split point and under the stated loss basis. The excess share is the complement, not a second independently selected proportion.
The component identity follows by adding the allocated amounts:
Thus, if the components do not add to the stated total, possible causes include a calculation error, a mismatched D-ratio, inconsistent loss definitions, or rounding. The identity is a reconciliation check, not evidence that the D-ratio itself is valid for every risk or coverage.
Worked Audits
Worked Example: Reconcile a Rate, Total, and D-Ratio
A hypothetical plan lists 160 annual exposure units, an expected loss rate of $420 per unit, and total expected losses of $67,200. The plan’s D-ratio is 0.72. A worksheet reports $48,384 primary expected losses and $18,816 excess expected losses. Audit the figures.
Method. Recompute the total from its exposure basis, then independently apply the component proportions.
1. Recompute total expected loss. The rate is per annual exposure unit, and the supplied exposure is 160 annual units:
The recomputed total matches the worksheet total.
2. Recompute primary expected loss.
3. Recompute excess expected loss. The excess share is \(1-0.72=0.28\):
4. Check the allocation.
All three checks agree. The component amounts also imply the stated proportions: \(\$48{,}384/\$67{,}200=0.72\), and \(\$18{,}816/\$67{,}200=0.28\). Common mistake avoided: treating the excess share as 0.72 rather than as the complement of the primary share. That would assign $48,384 to each component and produce $96,768, contradicting the total.
Worked Example: Detect a Partial-Period Exposure Error
A hypothetical fleet has 240 delivery vans insured for nine months each. The expected loss rate is $510 per van-year. A summary reports expected losses of $122,400, calculated using 240 vans as though each were insured for a full year. The D-ratio is 0.65. Determine the correct expected loss and component amounts, and identify the summary error.
Method. Convert the exposure to van-years before applying the annual rate.
1. Convert the exposure period. Nine months is \(9/12=0.75\) of a year per van:
2. Recompute total expected loss.
The reported $122,400 equals \(240\times \$510\), so it used 240 full van-years rather than 180 van-years. It overstates the expected loss by \(\$122{,}400-\$91{,}800=\$30{,}600\).
3. Allocate the corrected total. The primary share is 0.65 and the excess share is \(1-0.65=0.35\):
4. Check. \(\$59{,}670+\$32{,}130=\$91{,}800\), the corrected total. Common mistake avoided: using the number of insured objects as annual exposure without adjusting for the insured period. Applying the D-ratio to the overstated total would not fix the underlying exposure error; its components would still sum to an incorrect total.
Worked Example: Audit Component-Specific Square-Root Weights
A hypothetical experience plan has total expected losses of $360,000 and a D-ratio of 0.75. Its stated square-root rule is \(Z_j=\min(1,\sqrt{L_j/L_{0,j}})\), where \(L_j\) is expected loss in component \(j\) and \(L_{0,j}\) is that component’s full-credibility expected loss threshold. The primary and excess thresholds are $400,000 and $100,000, respectively. A worksheet reports weights of 0.82 primary and 0.30 excess. Audit the allocation and weights.
Method. Allocate expected losses first, then calculate each weight from the matching component threshold. The stated rule limits a weight to 1.
1. Allocate the expected losses.
The allocation checks: \(\$270{,}000+\$90{,}000=\$360{,}000\).
2. Calculate the primary weight.
3. Calculate the excess weight.
To four decimal places, the weights are 0.8216 and 0.9487. The reported primary weight of 0.82 is consistent with rounding to two decimals, but the reported excess weight of 0.30 is not. The excess expected loss is close to its threshold, so the stated formula gives a weight close to 1, not 0.30. Common mistake avoided: dividing excess expected losses by the primary threshold, or using a total-plan threshold for both components. Component weights must use the corresponding component expected loss and threshold.
How to Interpret an Unusual Result
An audit flag is a request to investigate, not necessarily a finding that the underlying assumption is wrong. A low weight may be correct if a component has little expected loss volume relative to its threshold. A high weight may be correct if volume meets or exceeds the full-credibility threshold, subject to the plan’s stated rule. Conversely, a weight outside the permitted range, a component total that does not reconcile, or a rate applied to the wrong exposure basis is a concrete inconsistency to resolve.
| Audit signal | Possible explanation | Next check |
|---|---|---|
| Exposure multiplied by rate does not match total | Wrong unit, period, rate basis, or arithmetic | Reconcile the exposure definition and rate denominator |
| Primary and excess do not sum to total | Wrong D-ratio, mismatched component basis, or rounding | Recompute both shares from the same total and D-ratio |
| Component weight appears implausible | Wrong component amount or threshold used | Check the plan formula and component labels |
| Small discrepancy in rounded amounts | Intermediate rounding or allocation precision | Recalculate at full precision, then round once at the end |
Do not alter a rate, D-ratio, or threshold merely to make totals reconcile. First establish whether all values refer to the same exposure period, coverage, loss definition, and split point. If the documentation does not resolve a material discrepancy, the appropriate audit result is an open question requiring clarification.
Common Audit Traps
- Mixing units. A rate per vehicle-year cannot be multiplied directly by a count of vehicles insured for a partial year without converting the count to vehicle-years.
- Using inconsistent periods. A rate updated to one period and exposure from another period may not produce a meaningful expected loss total.
- Allocating components from different totals. Primary and excess amounts must be shares of the same total on the same loss basis.
- Applying the wrong threshold. A component-specific weight must use the threshold for that component, if the plan specifies separate thresholds.
- Rounding too early. Keep adequate precision through the calculation. Round displayed money to cents and weights to the stated precision only at the end.
- Treating a plausible number as proof. A weight between zero and one can still be calculated from the wrong inputs; arithmetic plausibility is not a substitute for tracing the basis.
Check Your Understanding
For each problem, check the exposure basis before calculating and verify that component amounts reconcile.
- A hypothetical policy group has 75 annual exposure units, an expected loss rate of $840 per unit, and a D-ratio of 0.68. Calculate total, primary, and excess expected losses. (Answer: Total $63,000; primary $42,840; excess $20,160.)
- A group of 90 hypothetical service vans is covered for eight months each. The annual expected loss rate is $720 per van-year, and the D-ratio is 0.55. Calculate annualized exposure, total expected loss, and both components. (Answer: 60 van-years; total $43,200; primary $23,760; excess $19,440.)
- A plan uses \(Z=\min(1,\sqrt{L/L_0})\) separately for each component. Primary expected loss is $160,000 with a $250,000 threshold; excess expected loss is $40,000 with a $90,000 threshold. Calculate both weights to four decimals and state whether either is capped. (Answer: Primary 0.8000; excess 0.6667; neither is capped.)
- Two hypothetical classes have exposures of 120 and 80 units and expected loss rates of $500 and $650 per unit. Calculate their combined expected loss and the aggregate rate per exposure unit. If the aggregate D-ratio is 0.70, calculate primary and excess expected losses. (Answer: Combined expected loss $112,000; aggregate rate $560 per unit; primary $78,400; excess $33,600.)
- A worksheet allocates $200,000 of expected losses using a D-ratio of 0.60, but reports $125,000 primary and $75,000 excess. Identify the inconsistency and give the correct allocation. (Answer: The reported amounts sum to $200,000 but imply a 0.625 primary share, not 0.60; correct allocation is $120,000 primary and $80,000 excess.)