Mapping Legacy Classes to Revised Classes

Why Mapping Requires Policy-Level Information

A revised classification scheme may divide an existing class into several new classes, combine old classes, or use different attributes altogether. To estimate experience under the revised scheme, the actuary needs to know how each policy would be classified under the proposed rules. A table of old class totals alone generally cannot answer that question: it does not show which policies, exposures, or losses belong in each new class.

The central task is therefore to apply the proposed classification rules to policy-level records, then aggregate the records under the revised classes. If an attribute needed by the new rules is missing, unclear, or governed by overlapping rules, the mapping is ambiguous. Such cases should be identified and measured, rather than silently assigned in a way that could distort the indicated differences among classes.

Definition: Let $p$ index policies, $e_p$ be policy $p$'s earned exposure on a consistent basis, and $L_p$ be its loss amount on a consistent basis. Let $g(p)$ denote its legacy class and $h(p)$ its revised class under the proposed mapping rules. For revised class $j$, mapped exposure and loss are $$ E_j=\sum_{p:h(p)=j} e_p,\qquad L_j=\sum_{p:h(p)=j} L_p. $$ When $E_j>0$, the class loss cost per exposure is $L_j/E_j$. The mapping must use the proposed rules and the same policy records and experience basis for both quantities.

The loss amounts used in a mapping analysis should already be on the basis intended for comparison, such as the applicable valuation and loss-cost basis. Mapping does not itself develop, trend, or otherwise adjust losses. Its job is to place the prepared exposure and loss records into the proposed classes.

Constructing and Checking a Crosswalk

A useful first step is to record how exposure moves from each legacy class to each revised class. Define $T_{ij}$ as the fraction of exposure in legacy class $i$ that maps to revised class $j$. If the mapping is complete and every policy has one revised assignment, then $T_{ij}\geq 0$ and $\sum_j T_{ij}=1$ for each legacy class $i$. This transition table is a summary of policy-level assignments; it is not a substitute for applying the rules to those policies.

Exposure shares and loss shares must not be confused. A policy-level crosswalk can produce a loss share $U_{ij}$ as well as an exposure share $T_{ij}$, where $U_{ij}$ is the fraction of legacy class $i$'s loss assigned to revised class $j$. The two shares need not match: policies with different loss experience may map to different new classes.

Important condition: If only legacy aggregate exposure $E_i$ and loss $L_i$ are available, allocating $E_i$ by $T_{ij}$ is possible when the exposure crosswalk is known. Allocating losses as $T_{ij}L_i$ additionally assumes that losses are distributed across the mapped portions in proportion to exposure within each legacy class. Without that assumption or more detailed loss information, revised class loss costs are not identified by the legacy totals alone.

Worked Example: Mapping Two Legacy Classes Into Three Revised Classes

A hypothetical insurer has policy-level records for two legacy classes. Applying the proposed rules produces the following exposure and loss subtotals. Losses are in dollars, and exposure is in earned exposure units.

Legacy class	Revised class	Exposure	Loss
A	R1	25	$25,000
A	R2	15	$23,000
B	R2	10	$16,000
B	R3	20	$29,000

Step 1: Aggregate each revised class. Revised class R1 receives 25 exposure units and $25,000 of loss. R2 receives $15+10=25$ exposure units and $$23{,}000+$16{,}000=$39{,}000$ of loss. R3 receives 20 exposure units and $29,000 of loss.

Step 2: Calculate loss cost per exposure. For R1, $$25{,}000/25=$1{,}000$. For R2, $$39{,}000/25=$1{,}560$. For R3, $$29{,}000/20=$1{,}450$.

Step 3: Check the totals against the legacy records. Legacy A has $25+15=40$ exposure units and $$25{,}000+$23{,}000=$48{,}000$ of loss. Legacy B has $10+20=30$ exposure units and $$16{,}000+$29{,}000=$45{,}000$. Across all classes, the mapped totals are $70$ exposure units and $$93{,}000$ of loss, matching the legacy totals. The overall loss cost is $$93{,}000/70=$1{,}328.57$ per exposure, rounded to the nearest cent.

Method and mistake avoided. This is direct policy-level aggregation under the proposed rules. A common mistake would be to split each legacy class's losses in the same proportions as its exposure without checking the policy-level loss records. The supplied subtotals show, for example, that class A's 25 R1 exposure units have $25,000 of loss; an exposure-proportional allocation of A's $48,000 would instead assign $30,000 to R1. That would not reproduce the observed mapped experience.

Why the Mapping Preserves the Overall Total

When every policy is assigned to exactly one revised class, the mapped revised classes form a partition of the policy records: no record is omitted or counted twice. This gives a useful reconciliation test and explains why the overall loss cost is unchanged by reclassification, even though class-level loss costs can change.

\sum_j E_j =\sum_j\sum_{p:h(p)=j}e_p =\sum_p e_p, \qquad \sum_j L_j =\sum_j\sum_{p:h(p)=j}L_p =\sum_p L_p.

The equalities follow because each policy appears in one and only one revised class sum. Dividing the second total by the first gives the same overall loss cost before and after remapping, provided the policy set and experience basis are unchanged and total exposure is positive. This is a data reconciliation result, not a claim that every revised class has the same loss cost as its legacy class.

If a policy is excluded, duplicated, or left unassigned, the partition condition fails. Differences in the overall totals then signal a mapping or data-handling problem, unless a deliberate change in the included population has been made and documented.

Handling Ambiguous Assignments

An assignment can be ambiguous because a required policy attribute is missing, because the proposed rule does not specify how to handle a borderline value, or because two rules lead to different classes. A defensible analysis first identifies the affected records and the source of uncertainty. It then applies a documented resolution, or compares plausible resolutions when the available information does not support a unique choice.

Where possible, resolve uncertainty using the underlying policy record or a clearly specified rule hierarchy. If that cannot be done, calculate the impact of alternatives. A useful comparison includes revised-class exposure and loss, loss cost per exposure, and any relevant premium measure. The overall totals may remain fixed while the allocation among classes changes materially.

Worked Example: Measuring the Effect of Two Plausible Rule Interpretations

A hypothetical book has 20 definite R1 exposure units with $22,000 of loss and 10 definite R2 exposure units with $14,000 of loss. Four additional one-unit policies have incomplete usage records. Their losses are: policy A, $0; policy B, $1,000; policy C, $2,000; and policy D, $5,000. The proposed classification rules could reasonably be interpreted in either of two ways.

Scenario 1. Under interpretation 1, policies A, B, and C map to R1; policy D maps to R2. R1 has $20+3=23$ exposure units and losses of $$22{,}000+$0+$1{,}000+$2{,}000=$25{,}000$. Its loss cost is $$25{,}000/23=$1{,}086.96$ per exposure, rounded to the nearest cent. R2 has $10+1=11$ exposure units and losses of $$14{,}000+$5{,}000=$19{,}000$. Its loss cost is $$19{,}000/11=$1{,}727.27$ per exposure, rounded to the nearest cent.

Scenario 2. Under interpretation 2, policies A and B map to R1; policies C and D map to R2. R1 has $20+2=22$ exposure units and losses of $$22{,}000+$0+$1{,}000=$23{,}000$. Its loss cost is $$23{,}000/22=$1{,}045.45$ per exposure, rounded to the nearest cent. R2 has $10+2=12$ exposure units and losses of $$14{,}000+$2{,}000+$5{,}000=$21{,}000$. Its loss cost is $$21{,}000/12=$1{,}750.00$ per exposure.

Step 3: Compare and check. R1's indicated loss cost differs by $$1{,}086.96-$1{,}045.45=$41.51$ per exposure between the two interpretations; R2's differs by $$1{,}750.00-$1{,}727.27=$22.73$ per exposure. In both scenarios, total exposure is $34$ and total loss is $$22{,}000+$14{,}000+$0+$1{,}000+$2{,}000+$5{,}000=$44{,}000$. Thus the overall loss cost is $$44{,}000/34=$1{,}294.12$ per exposure, rounded to the nearest cent, in either scenario.

Method and mistake avoided. The analysis assigns each ambiguous policy's exposure and loss together under each rule interpretation. A mistake would be to move exposure but leave the policy's loss behind, or to split the ambiguous losses evenly without a stated basis. Either action breaks the policy-level correspondence or introduces an unsupported allocation.

When Only Legacy Totals Are Available

Sometimes a crosswalk can be developed for exposure, but losses are available only as old-class aggregates. In that case, the analyst should state which assumptions are needed to produce a new-class estimate. A proportional loss allocation is a possible approximation, not a consequence of knowing the exposure transition.

Worked Example: Estimating a Split From Legacy Aggregates Only

A hypothetical legacy class has 50 exposure units and $60,000 of total loss. A policy-level exposure crosswalk shows that 30 units map to proposed class N and 20 units map to proposed class M. No loss detail is available below the legacy class.

Step 1: State the allocation assumption. Assume, solely for this estimate, that loss is distributed in proportion to exposure within the legacy class. The exposure shares are $30/50=0.60$ for N and $20/50=0.40$ for M.

Step 2: Allocate exposure and loss. N receives 30 exposure units and $0.60\times$60{,}000=$36{,}000$ of loss. M receives 20 exposure units and $0.40\times$60{,}000=$24{,}000$ of loss.

Step 3: Calculate and check the estimates. N's estimated loss cost is $$36{,}000/30=$1{,}200$ per exposure. M's estimated loss cost is $$24{,}000/20=$1{,}200$ per exposure. The allocated totals return to 50 exposure units and $60,000 of loss. The equal estimated loss costs result from the proportional-allocation assumption; they are not evidence that the two proposed classes actually have equal experience.

For contrast, if later policy-level information showed that N had $48,000 of the loss and M had $12,000, the observed loss costs would be $$48{,}000/30=$1{,}600$ for N and $$12{,}000/20=$600$ for M. Those amounts have the same $60,000 total, but a very different class allocation. The legacy aggregates alone cannot distinguish this allocation from the proportional estimate.

Method and mistake avoided. The calculation is an exposure-proportional estimate, explicitly conditional on its assumption. The mistake is reporting the resulting $1,200 amounts as policy-level findings when no class-specific loss records support them.

Practical Controls and Exam Traps

Mapping results are most useful when the assignment rule, data limitations, and reconciliation checks are visible. A compact transition summary can show, for each legacy class, the exposure moving to each revised class. Where policy-level losses are available, a parallel loss summary helps reveal whether high- or low-loss records are concentrated in particular destinations.

Keep exposures and losses paired at policy level. Apply a policy's revised class to both its exposure and its corresponding loss record.
Do not infer loss shares from exposure shares without an assumption. If legacy aggregates are the only loss data, label any proportional allocation as an estimate and explain its limitation.
Count each record once. A mutually exclusive and exhaustive mapping should preserve total exposure and total loss.
Separate rule ambiguity from random variation. An unresolved classification rule is a data or definition issue; it is not resolved merely by having more experience.
Show class and portfolio effects separately. Reclassification can redistribute indicated class costs while leaving aggregate loss and exposure unchanged.

The value of this work is not just a set of revised class totals. A transparent crosswalk makes it possible to distinguish a genuine difference in mapped experience from a result driven by uncertain rules or an unsupported allocation. This approach is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Key takeaway: Apply the proposed classification rules to policy-level records whenever possible. Reconcile mapped totals to the legacy portfolio, and quantify alternative outcomes when an assignment cannot be resolved uniquely.

Check Your Understanding

For each problem, map both exposure and loss consistently, and check totals where possible.

A legacy class has 18 exposure units and $27,000 in loss. Policy-level records map 10 units with $12,000 of loss to revised class A and the remainder to B. Calculate each revised class's exposure and loss cost. (Answer: A: 10 units, $12,000, $1,200 per exposure; B: 8 units, $15,000, $1,875 per exposure.)
A legacy class has 40 exposure units and $52,000 in loss. A crosswalk assigns 25 units to X and 15 to Y. Under an explicitly proportional loss-allocation assumption, calculate each class's allocated loss and loss cost. (Answer: X: $32,500 and $1,300 per exposure; Y: $19,500 and $1,300 per exposure.)
A hypothetical book has definite class C experience of 12 units and $18,000 loss, and definite class D experience of 8 units and $8,000 loss. One ambiguous one-unit policy has $6,000 loss. Compare class loss costs if it maps to C versus D. (Answer: If assigned to C, C is 13 units and $24,000, or $1,846.15 per exposure; D is $1,000 per exposure. If assigned to D, C is $1,500 per exposure; D is 9 units and $14,000, or $1,555.56 per exposure.)
A legacy class has 30 exposure units and $45,000 loss. Policy-level exposure mapping sends 20 units to P and 10 to Q, but no class-specific loss data exist. What are the proportional estimates, and what limitation must be stated? (Answer: Each class is estimated at $1,500 per exposure, with allocated losses of $30,000 for P and $15,000 for Q; this depends on assuming losses follow exposure shares, and actual class costs are not identified by the legacy totals.)
Two legacy classes have combined totals of 60 exposure units and $84,000 loss. A complete policy-level mapping produces 22 units and $26,400 loss in revised class M, and 38 units and $57,600 loss in revised class N. Calculate both loss costs and verify the portfolio result. (Answer: M is $1,200 per exposure; N is approximately $1,515.79 per exposure; the portfolio is $1,400 per exposure because $84,000/60=$1,400.)

What You'll Learn