Why Audit a Credibility-Adjusted Classification Table?
A completed classification table can contain individually plausible numbers and still fail as a rating indication. A loss cost may use a different exposure denominator than the other classes, a credibility blend may use the wrong weight, or relativities may not reproduce the stated portfolio average. The audit is therefore not just a check of arithmetic: it tests whether the table’s quantities are comparable and whether its pieces agree with one another.
This tutorial focuses on four checks: consistent exposure bases, correct credibility blending, valid class relativities, and reconciliation to a stated portfolio average. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
These definitions assume that the experience losses, exposures, complements, and indicated costs have been put on comparable bases. In particular, all \(E_j\) must represent the same kind of exposure for the same coverage and relevant experience period, and the amounts blended in \(m_j\) must have the same units. A table that mixes vehicle-years and policies, or blends a loss cost per vehicle-year with a complement per policy, does not become valid merely because the arithmetic can be performed.
An audit should also distinguish a portfolio average computed from the class indications from an independently prescribed target average. If the former is intended, it must equal the exposure-weighted mean of the indicated costs. If the latter is fixed separately, the class relativities can be normalized to that target; this changes the overall level, not the relative pattern among classes.
A Reliable Audit Sequence
Work from the source quantities toward the final relativities. Do not begin by checking only whether the reported factors average to one: a normalized set of factors can still be based on an incorrect credibility blend or an inconsistent exposure basis.
Check the exposure definition, loss period, coverage, and units. Recalculate each experience loss cost as adjusted losses divided by the stated exposure.
Apply the class credibility to its own experience cost and the remaining weight to the stated complement. Confirm \(0\leq Z_j\leq1\).
When the table’s indications define the portfolio average, calculate the exposure-weighted mean of the blended costs.
Compare each relativity with the ratio of its indicated cost to the relevant average. Verify that exposure-weighted relativities average to one, subject to displayed rounding.
Why Exposure-Weighted Relativities Average to One
Suppose each class has a blended cost \(m_j\) on the same exposure basis, and the indicated portfolio average is calculated from those costs. The portfolio average is \(\bar m=\sum_j E_jm_j/\sum_jE_j\). Defining \(r_j=m_j/\bar m\) makes each relativity a ratio to that average. The weighted-average property follows directly:
The last equality holds because \(\bar m\) was defined as the exposure-weighted mean of the \(m_j\). It is a useful audit identity, not an independent guarantee that the table is sound: incorrect costs can still be normalized to weighted-average relativity one. If the stated portfolio average is an external target \(C_0\), the same relative pattern can be expressed as class rates \(C_0r_j\). Those rates then have exposure-weighted average \(C_0\), although the unscaled blended costs need not average to \(C_0\).
Worked Example: Finding a Misstated Credibility Blend
A hypothetical program reports the following three classes. All exposures are earned vehicle-years, and all losses and complements are on a common adjusted-dollar basis. The table states that the portfolio average is to be calculated from the class indications. Audit the reported blend for Class B and calculate the corrected relativities.
| Class | Exposure | Adjusted losses | Complement | Credibility | Reported blend |
|---|---|---|---|---|---|
| A | 100 | $80,000 | $1,000 | 0.25 | $950 |
| B | 200 | $240,000 | $1,000 | 0.50 | $1,050 |
| C | 100 | $90,000 | $1,000 | 0.75 | $925 |
Step 1: Recalculate the experience costs. Divide adjusted losses by vehicle-years: Class A has \(80{,}000/100=\$800\), Class B has \(240{,}000/200=\$1{,}200\), and Class C has \(90{,}000/100=\$900\).
Step 2: Recalculate the credibility blends. For A, \(0.25(800)+0.75(1{,}000)=200+750=\$950\). For B, \(0.50(1{,}200)+0.50(1{,}000)=600+500=\$1{,}100\). For C, \(0.75(900)+0.25(1{,}000)=675+250=\$925\). Thus the reported Class B blend of $1,050 is incorrect; it should be $1,100.
Step 3: Calculate the portfolio average. The exposure-weighted indication is \((100(950)+200(1{,}100)+100(925))/400=(95{,}000+220{,}000+92{,}500)/400=\$1{,}018.75\).
Step 4: Calculate the corrected relativities. Divide each corrected blend by $1,018.75. The resulting factors are A: \(950/1{,}018.75=0.9325\), B: \(1{,}100/1{,}018.75=1.0798\), and C: \(925/1{,}018.75=0.9080\), rounded to four decimal places.
Check. Using the unrounded factors, the exposure-weighted mean is exactly one by the derivation above. Using the displayed factors, it is \((100(0.9325)+200(1.0798)+100(0.9080))/400=1.0000\) to four decimals. The mistake to avoid is accepting the table’s reported blend simply because its formula looks plausible: recalculate the underlying experience cost and each weighted component.
When the Stated Portfolio Average Is a Target
A table may instead provide a target average that is not calculated from the class indications. In that case, first find the indicated relative pattern, then normalize it to the target. This distinction prevents a common audit error: calling any difference between a raw indicated mean and the target a failed reconciliation when the target was explicitly set independently.
Worked Example: Reconcile Indications to a Prescribed Average
A hypothetical coverage has two classes measured in the same exposure unit. Class A has 80 exposures, an experience cost of $900, a complement of $1,000, and credibility 0.40. Class B has 120 exposures, an experience cost of $1,300, the same complement, and credibility 0.60. The prescribed portfolio average is $1,100. Find the blended costs, their indicated average, and the class rates consistent with the prescribed average.
Step 1: Blend each class. For A, \(m_A=0.40(900)+0.60(1{,}000)=360+600=\$960\). For B, \(m_B=0.60(1{,}300)+0.40(1{,}000)=780+400=\$1{,}180\).
Step 2: Compute the indication-weighted average. The total exposure is \(80+120=200\), and the weighted cost total is \(80(960)+120(1{,}180)=76{,}800+141{,}600=\$218{,}400\). Thus \(\bar m=218{,}400/200=\$1{,}092\). This does not equal the independently prescribed $1,100, and that difference should be reported rather than hidden.
Step 3: Calculate relativities to the indication-weighted average. \(r_A=960/1{,}092=0.8791209\) and \(r_B=1{,}180/1{,}092=1.0805861\), retaining extra digits for the check.
Step 4: Apply the prescribed level. The consistent class rates are \(1{,}100(0.8791209)=\$967.03\) for A and \(1{,}100(1.0805861)=\$1{,}188.64\), rounded to cents.
Check. The exposure-weighted rate is approximately \((80(967.03297)+120(1{,}188.64469))/200=(77{,}362.64+142{,}637.36)/200=\$1{,}100.00\), using unrounded rates. The relativities preserve the indicated class pattern while the prescribed average sets the overall level.
Validity, Normalization, and Rounding
For positive indicated costs and a positive portfolio average, ratios of indicated costs to that average are positive. A reported negative relativity is therefore invalid in this setting. A zero factor would imply a zero class rate; unless the underlying indication is zero and the rating plan explicitly permits that outcome, it is a warning that the table or its transformation is wrong. Also check that the factors preserve the ordering of the class indications: if \(m_A<m_B\), then \(r_A<r_B\) when both use the same positive denominator.
A reported factor can fail for two different reasons. It may not equal the class indication divided by the stated denominator, or the set of factors may have the wrong weighted average. If the class indications and their ratios are correct but displayed factors have been rounded, a small departure from one may be harmless. If the departures are material, calculate with unrounded values before deciding whether the problem is only rounding.
Worked Example: Detecting Invalid and Unreconciled Relativities
A hypothetical three-class table has exposures of 50, 100, and 150 units and credibility-adjusted costs of $800, $1,000, and $1,200. Its portfolio average is stated to be calculated from these indications. The reported relativities are 0.75, 0.95, and 1.15. Determine whether they are valid and provide corrected values and rates.
Step 1: Recompute the portfolio average. The total exposure is \(50+100+150=300\). The weighted cost total is \(50(800)+100(1{,}000)+150(1{,}200)=40{,}000+100{,}000+180{,}000=\$320{,}000\). Therefore, \(\bar m=320{,}000/300=\$1{,}066.6667\).
Step 2: Test the reported factors. Their weighted mean is \((50(0.75)+100(0.95)+150(1.15))/300=(37.5+95+172.5)/300=305/300=1.0167\). They do not reconcile to one. More importantly, the second and third factors are not the class costs divided by the average.
Step 3: Replace them with the indicated ratios. The correct factors are \(800/1{,}066.6667=0.7500\), \(1{,}000/1{,}066.6667=0.9375\), and \(1{,}200/1{,}066.6667=1.1250\). Their weighted mean is \((37.5+93.75+168.75)/300=300/300=1\).
Step 4: Check the corresponding class rates. Multiplying the factors by the average gives $800, $1,000, and $1,200. Their weighted mean is \((40{,}000+100{,}000+180{,}000)/300=\$1{,}066.6667\), confirming reconciliation. The trap is to “fix” an incorrect factor set only by dividing all reported factors by 1.0167. That would force a weighted mean of one but would retain the wrong relative pattern; the factors must first agree with the class indications.
Common Audit Traps
- Mixing denominators. A loss total divided by policy count cannot be compared directly with a loss total divided by earned vehicle-years. Recompute or restate the data on one common basis before blending.
- Using the credibility weight twice. The blend is \(Z_jc_j+(1-Z_j)C_j\). Do not multiply the complement by \(Z_j\) or give the experience cost the complement’s weight.
- Normalizing too early. First verify the class loss costs and blends. A final set of relativities averaging to one does not validate incorrect inputs.
- Using the wrong weights in the reconciliation. When the rating basis is exposure, the portfolio check uses the stated class exposures, not an unweighted average of class factors.
- Confusing level with pattern. Rescaling to an independently prescribed portfolio average changes the overall rate level. It does not change the relative indication among classes.
Check Your Understanding
For each problem, state the relevant audit conclusion and keep unrounded values until the final reported answer.
- A class has 60 earned units and adjusted losses of $54,000. Its complement is $900 and credibility is 0.40. A table reports an experience cost of $900 and a blend of $900. Is the experience cost correct, and what is the corrected blend? (Answer: The experience cost is $900; the blend is \(0.40(900)+0.60(900)=\$900\).)
- Two classes have exposures 40 and 60, blended costs $700 and $1,100, and a portfolio average derived from those indications. Find the average and the exposure-weighted relativities, to four decimals. (Answer: Average \(=(40(700)+60(1{,}100))/100=\$940\); relativities are 0.7447 and 1.1702.)
- A portfolio has 100 exposures in Class A and 300 in Class B. Their indicated costs are $800 and $1,000. A report uses relativities 0.80 and 1.00. Do they reconcile, and what are the correct factors if the average is derived from indications? (Answer: Reported weighted mean \(=(100(0.80)+300(1.00))/400=0.9500\), so they do not reconcile; the indicated average is \(\$950\), and the correct factors are 0.8421 and 1.0526.)
- Two classes have exposures 75 and 125 and blended costs $960 and $1,120. A separate target average is $1,050. Calculate the indication-weighted average and the class rates after anchoring the relativities to the target, rounded to cents. (Answer: Indicated average \(=(75(960)+125(1{,}120))/200=\$1{,}060\); target-consistent class rates are \(1{,}050(960/1{,}060)=\$950.94\) and \(1{,}050(1{,}120/1{,}060)=\$1{,}109.43\).)
- A completed table reports positive relativities with exposure-weighted mean one, but one class’s displayed factor is below another’s even though its recalculated blended cost is higher. What audit conclusion follows? (Answer: The factors pass the weighted-mean check but fail the class-indication check; investigate the factor calculation or class labeling, because a common positive denominator preserves the ordering of indicated costs.)