Re-Rating Historical Exposures

Why Re-Rate Historical Exposures?

A historical premium total reflects the rates that applied when the underlying policies were written or earned. To compare historical losses with a common target rate level, an actuary needs a premium measure at that target level. Re-rating provides one way to calculate it: retain the historical exposure quantities and rating characteristics, but price those exposures using target rates.

The central discipline is to change the rate basis and hold the rest of the comparison fixed. Do not quietly change the number of insured units, the coverage provided, or the classification mix while calculating the re-rated premium. This tutorial is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Re-rating historical exposures means applying a specified target rate schedule to the historical exposure quantities and rating characteristics, with the relevant coverage basis held constant. The resulting target-level premium is compared with the corresponding historical premium to form an extension-method factor.

Let $j$ index exposure records or groups. Let $E_j$ be the historical exposure quantity, $r_j$ the applicable historical rate per exposure unit, and $r_j^*$ the corresponding target rate. In a simple rating structure, the historical premium is $P_j=E_jr_j$, and the re-rated premium is $P_j^*=E_jr_j^*$. If a row-level target rate factor is available, define $f_j=r_j^*/r_j$; then $P_j^*=P_jf_j$.

P_{\mathrm{hist}}=\sum_j P_j,\qquad P_{\mathrm{target}}=\sum_j P_j^*,\qquad F_{\mathrm{ext}}=\frac{P_{\mathrm{target}}}{P_{\mathrm{hist}}}.

Here the extension-method factor $F_{\mathrm{ext}}$ converts the selected historical premium to its equivalent at the target rate level. This convention uses target-level premium divided by historical premium; a factor above 1.000 indicates an increase in the aggregate premium under the target rates. Apply the factor only to the premium basis represented in the numerator and denominator.

Why the Aggregate Factor Is Premium-Weighted

A portfolio usually contains records with different rate changes. Its overall factor is therefore not generally the simple average of the row-level factors. The appropriate weight for a row is its share of historical premium, because that is the amount of premium to which the row’s factor is applied.

F_{\mathrm{ext}} =\frac{\sum_j P_j f_j}{\sum_j P_j} =\sum_j \left(\frac{P_j}{\sum_k P_k}\right)f_j.

To justify the result, start with the row-level definition $f_j=P_j^*/P_j$, assuming $P_j>0$. Rearranging gives $P_j^*=P_jf_j$. Add the target premiums across all records and divide by the total historical premium:

\frac{\sum_j P_j^*}{\sum_j P_j} =\frac{\sum_j P_j f_j}{\sum_j P_j} =\sum_j w_j f_j,\qquad w_j=\frac{P_j}{\sum_k P_k}.

The weights sum to 1, since $\sum_j w_j=1$. Thus, the aggregate extension factor is a weighted average of the row factors, with historical premium shares as weights. This also explains why a row with a large premium has more influence on the portfolio result than a small row, even if their rate factors are the same.

Worked Example: Combining Different Rate Changes

A hypothetical insurer has three groups of historical exposure records. Their historical premiums are $10,000, $20,000, and $10,000. The respective target-to-historical rate factors are 1.10, 0.95, and 1.20. Calculate the target-level premium and extension-method factor. Name the method and identify the averaging error to avoid.

Method. Re-rate each historical premium by its applicable target factor, then divide the sum of target premiums by the sum of historical premiums. The historical exposure quantities and rating characteristics are represented as fixed within each group.

Group 1. Target premium $=\$10{,}000(1.10)=\$11{,}000$.

Group 2. Target premium $=\$20{,}000(0.95)=\$19{,}000$.

Group 3. Target premium $=\$10{,}000(1.20)=\$12{,}000$.

The historical premium is $\$10{,}000+\$20{,}000+\$10{,}000=\$40{,}000$. The target premium is $\$11{,}000+\$19{,}000+\$12{,}000=\$42{,}000$. Therefore,

F_{\mathrm{ext}}=\frac{\$42{,}000}{\$40{,}000}=1.05.

Check. The premium shares are 0.25, 0.50, and 0.25. Their weighted average factor is $0.25(1.10)+0.50(0.95)+0.25(1.20)=0.275+0.475+0.300=1.050$, which agrees with the ratio of aggregate premiums.

Common mistake avoided: The simple average, $(1.10+0.95+1.20)/3=1.0833$, gives each group equal weight even though the middle group has twice the historical premium of either other group. It is not the portfolio extension factor.

Re-Rate the Exposure Basis, Not Just the Total

When records provide exposure quantities and rates, calculate the target premium from those quantities rather than applying an unrelated portfolio-wide adjustment. This makes the calculation auditable: each target amount can be traced to an exposure basis and a target rate. If a row’s rate is expressed per 100 units, for example, convert the exposure to that same basis before multiplying.

The target rate must correspond to the same coverage and rating characteristics as the historical record. A difference in target rate should represent the intended rate-level change, not an unexamined change in deductible, limit, territory, or classification. If such a characteristic changes, the comparison no longer isolates the effect of rate level.

Worked Example: Re-Rating Exposure Quantities Directly

A hypothetical equipment insurer is re-rating two groups of machine-year exposures. Group A has 150 machine-years at a historical rate of $24 per machine-year; its target rate is $26.40. Group B has 80 machine-years at a historical rate of $35; its target rate is $33.25. The coverage and rating characteristics are unchanged. Find both premium totals and the extension-method factor.

Method. Multiply each fixed exposure quantity by its historical rate and by its target rate, then compare the aggregate premiums.

Group A historical premium. $150\times \$24=\$3{,}600$. Its target premium is $150\times \$26.40=\$3{,}960$, equivalent to applying the factor $\$26.40/\$24=1.10$.

Group B historical premium. $80\times \$35=\$2{,}800$. Its target premium is $80\times \$33.25=\$2{,}660$, equivalent to applying the factor $\$33.25/\$35=0.95$.

The historical total is $\$3{,}600+\$2{,}800=\$6{,}400$. The target total is $\$3{,}960+\$2{,}660=\$6{,}620$. Thus,

F_{\mathrm{ext}}=\frac{\$6{,}620}{\$6{,}400}=1.034375\approx 1.0344.

The target rates produce a 3.44% aggregate increase, rounded to four decimal places for the factor. As a check, the weighted rate-factor calculation is $(\$3{,}600(1.10)+\$2{,}800(0.95))/\$6{,}400=\$6{,}620/\$6{,}400$, the same result.

Common mistake avoided: Applying the unweighted average $(1.10+0.95)/2=1.025$ would produce $\$6{,}560$, not $\$6{,}620$. Group A has the larger historical premium, so its higher factor receives more weight.

Separate Rate-Sensitive Premium From Fixed Charges

Not every amount included in a policy’s total charge necessarily changes with the rate schedule. A fixed policy fee, for example, should not be multiplied by a rate-level factor if it remains unchanged under the target basis. Identify the premium component being re-rated, calculate the target amount for that component, and then add unchanged charges only if they belong in the required total.

The same principle applies to taxes or other charges: include them according to their actual basis. A charge that is a fixed dollar amount is not re-rated as premium; a charge calculated as a percentage of premium may change when premium changes. The calculation should follow the stated rating and accounting rules, rather than treating every billed dollar as rate-sensitive.

Worked Example: Keeping a Fixed Fee Unchanged

A hypothetical policy portfolio has two rate-sensitive historical premium components: $8,000 with a target factor of 1.10, and $12,000 with a target factor of 0.95. It also includes $600 in fixed policy fees, which do not change at the target rate level. Calculate the rate-sensitive extension factor and the factor for the total including fees.

Rate-sensitive premium. The historical amount is $\$8{,}000+\$12{,}000=\$20{,}000$. The target amount is $\$8{,}000(1.10)+\$12{,}000(0.95)=\$8{,}800+\$11{,}400=\$20{,}200$. The extension factor for this premium basis is $\$20{,}200/\$20{,}000=1.0100$.

Total including fixed fees. Historical total is $\$20{,}000+\$600=\$20{,}600$. Target total is $\$20{,}200+\$600=\$20{,}800$. The total factor is $\$20{,}800/\$20{,}600=1.0097087$, or 1.0097 to four decimal places.

Check. The total increases by $200, which is the change in rate-sensitive premium; the fixed fee remains $600. The total factor is therefore slightly closer to 1.000 than the factor for rate-sensitive premium alone.

Common mistake avoided: Multiplying the full historical total by 1.0100 would give $\$20{,}600(1.0100)=\$20{,}806$. That incorrectly increases the fixed fee by $6.

Conditions for a Meaningful Re-Rating

A re-rating factor is useful only when its numerator and denominator refer to compatible premium and exposure bases. Before calculating, verify the records and the target schedule. In particular:

Match the scope. Use the same experience period, coverages, and included policy population in the historical and target calculations.
Hold exposures fixed. Do not substitute a different exposure count or quantity when pricing at target rates.
Hold rating characteristics fixed. Use the historical characteristics unless a specified adjustment is deliberately part of the target calculation.
Use consistent premium definitions. For example, do not compare a historical premium excluding a component with a target premium including it.
Isolate non-rate amounts. Treat fixed fees and other non-rate-sensitive amounts according to their actual basis.

If a required exposure or rating characteristic is missing, a direct re-rating may not be possible from the available aggregate premium alone. A single portfolio factor may still be supplied or estimated by another method, but it should not be presented as a record-level re-rating without the data needed to support it.

Key takeaway: Re-rate each historical exposure using its corresponding target rate while holding exposure quantities and the relevant coverage basis fixed. Sum the target premiums, then divide by the matching historical premium. Equivalently, weight each row’s target factor by its share of historical premium.

Check Your Understanding

For each problem, calculate the requested target premium or extension factor using only the stated premium basis.

A hypothetical portfolio has historical premiums of $4,000 and $6,000, with target factors of 1.08 and 0.97. Calculate the total target premium and extension factor. (Answer: target premium $10,140; factor 1.0140)
A book has 200 exposure units at a historical rate of $18 and a target rate of $19.80, plus 50 units at a historical rate of $40 and a target rate of $44. Calculate the aggregate historical and target premiums and the factor. (Answer: historical premium $5,600; target premium $6,160; factor 1.1000)
Rate-sensitive historical premiums are $15,000 with target factor 0.96 and $5,000 with target factor 1.15. An unchanged fixed charge of $300 is included in both totals. Calculate the factor for the rate-sensitive premium and the factor for the total including the fixed charge. (Answer: rate-sensitive factor 1.0075; total factor 1.0074)
Three exposure groups represent 50%, 30%, and 20% of historical premium. Their respective target factors are 1.04, 0.90, and 1.15. Find the aggregate extension factor and state why the simple average is inappropriate. (Answer: factor 1.0200; the simple average is inappropriate because the groups have unequal historical premium weights)
A hypothetical liability book has 100 exposure units at a historical rate of $32 and target rate of $33.60, and 40 units at a historical rate of $50 and target rate of $47.50. Coverage is unchanged. Calculate both premium totals and the extension factor to four decimal places. (Answer: historical premium $5,200; target premium $5,260; factor 1.0115)

What You'll Learn