Calculate Premium Changes Across Parameter Alternatives

Why Compare Retrospective Parameters on Common Losses?

When evaluating alternative retrospective rating parameters, the loss experience should be held constant. Otherwise, a premium difference could reflect different claims rather than a different loss conversion factor, claim cap, or adjustment expense provision. A side-by-side comparison makes the source of the difference visible and supports a more reliable review of the indicated premium under each option.

The comparison begins with the contract terms, not with the largest or smallest parameter. In particular, identify what amount the cap applies to, whether the loss conversion factor is then applied, and how the adjustment expense provision is calculated. This tutorial uses hypothetical plans with no premium tax, minimum or maximum premium, aggregate cap, or discounting unless stated otherwise. Those features would require additional terms in the premium calculation.

Definition: For a plan option $j$, let $X_k$ be the reported loss for claim $k$, $u_j$ the per-claim cap on reported losses, $c_j$ the loss conversion factor (LCF), $B_j$ the basic premium, and $A_j$ the adjustment expense provision in dollars. Define limited losses as $T_j=\sum_k\min(X_k,u_j)$. If the contract applies the LCF to those limited losses, converted losses are $C_j=c_jT_j$, and the retrospective premium is $P_j=B_j+C_j+A_j$. The contract controls the order and basis of each calculation; this formula does not apply unchanged if, for example, the cap is defined on converted losses or an adjustment provision is calculated on a different base.

An adjustment expense provision may be a fixed dollar amount or a percentage applied to a specified base. If it is a proportion $e_j$ of converted losses, then $A_j=e_jC_j$, and the premium is $P_j=B_j+(1+e_j)c_jT_j$. A percentage of basic premium is instead $A_j=e_jB_j$. These expense bases are not interchangeable.

For a fair comparison, use the same claim experience in every option, but apply each option’s own cap, LCF, basic premium, and expense terms. Keep units consistent: if losses are in dollars, the premium and expense amounts must also be in dollars.

A Consistent Comparison Method

A compact calculation sequence reduces the chance of applying a term to the wrong amount. First transcribe each option’s provisions, especially the cap basis and expense basis. Then calculate the limited losses, converted losses, expenses, and premium separately for each option. Only after each premium is complete should the options be compared.

Hold the experience constant.
Use the same claim amounts in every option. Confirm whether the amounts are reported losses and whether any other adjustment to them is specified.

Apply each cap on its contractual basis.
If the cap is on reported loss, calculate $\min(X_k,u_j)$ for each claim before applying the LCF.

Calculate converted losses and expenses.
Multiply limited losses by the option’s LCF. Calculate each adjustment provision using its stated dollar amount or percentage base.

Calculate and reconcile premiums.
Add basic premium, converted losses, and adjustment expenses as the stated plan requires. Compare the completed premiums and identify which components explain the difference.

Worked Comparisons

Worked Example: Different Caps, LCFs, and Expense Bases

A hypothetical insured has four reported claims of $40,000, $75,000, $130,000, and $220,000. Both plan options have a basic premium of $90,000. Option A has a $100,000 cap on each reported claim, an LCF of 1.10, and a fixed adjustment expense provision of $20,000. Option B has a $150,000 cap on each reported claim, an LCF of 1.05, and an adjustment expense provision equal to 2% of converted losses. Calculate both premiums and the difference.

Method and cap step. Apply each cap to each reported claim before multiplying by the LCF. For Option A, the limited claims are $40,000, $75,000, $100,000, and $100,000. Their total is $T_A=\$315{,}000$.

Option A converted losses are $C_A=1.10\times \$315{,}000=\$346{,}500$. Its adjustment expense is fixed at $20,000, so its premium is $\$90{,}000+\$346{,}500+\$20{,}000=\$456{,}500$.

For Option B, the limited claims are $40,000, $75,000, $130,000, and $150,000. Their total is $T_B=\$395{,}000$. Converted losses are $C_B=1.05\times \$395{,}000=\$414{,}750$.

Option B’s expense is 2% of converted losses: $A_B=0.02\times \$414{,}750=\$8{,}295$. Its premium is $\$90{,}000+\$414{,}750+\$8{,}295=\$513{,}045$. The difference is $\$513{,}045-\$456{,}500=\$56{,}545$, so Option B’s premium is $56,545 higher.

Check. Both options use the same four reported claims and the same $90,000 basic premium. Option B has $68,250 more converted losses and $11,705 less adjustment expense than Option A. The net difference is $\$68{,}250-\$11{,}705=\$56{,}545$, matching the premium comparison. This method avoids comparing LCFs alone while ignoring the cap and expense basis.

Worked Example: A Fixed Expense Versus a Percentage of Losses

Consider three reported claims of $55,000, $95,000, and $175,000. The basic premium is $70,000 under either hypothetical option. Option A applies a $125,000 per-claim reported-loss cap, uses an LCF of 1.20, and has a fixed $15,000 adjustment expense. Option B applies a $160,000 per-claim reported-loss cap, uses an LCF of 1.12, and has an adjustment expense equal to 3% of converted losses. Calculate and compare the premiums.

Option A. After the $125,000 cap, the claims are $55,000, $95,000, and $125,000. Thus $T_A=\$55{,}000+\$95{,}000+\$125{,}000=\$275{,}000$. Converted losses are $C_A=1.20\times \$275{,}000=\$330{,}000$. The premium is $\$70{,}000+\$330{,}000+\$15{,}000=\$415{,}000$.

Option B. After the $160,000 cap, the claims are $55,000, $95,000, and $160,000. Thus $T_B=\$310{,}000$. Converted losses are $C_B=1.12\times \$310{,}000=\$347{,}200$. The expense is $0.03\times \$347{,}200=\$10{,}416$. The premium is $\$70{,}000+\$347{,}200+\$10{,}416=\$427{,}616$.

Option B is $\$427{,}616-\$415{,}000=\$12{,}616$ higher. Although its LCF is lower, its higher cap produces more limited losses. Its percentage expense is also based on converted losses, so it changes with that amount. The check is that the converted-loss difference is $17,200 and the expense difference is $-\$4,584$; together they give $\$17{,}200-\$4{,}584=\$12{,}616$. The common mistake avoided here is treating a percentage expense as a fixed dollar amount.

Worked Example: Reconciling a Premium Difference by Component

A hypothetical plan comparison uses reported claims of $30,000, $60,000, $90,000, and $140,000. Both options have a basic premium of $50,000. Option A has a $100,000 per-claim reported-loss cap, an LCF of 1.15, and a fixed $9,000 adjustment expense. Option B has the same cap, an LCF of 1.10, and an expense provision of $4,000 plus 2.5% of converted losses. Find each premium and reconcile the difference.

The capped claims under both options are $30,000, $60,000, $90,000, and $100,000. Therefore, $T_A=T_B=\$280{,}000$. Option A’s converted losses are $1.15\times \$280{,}000=\$322{,}000$, giving a premium of $\$50{,}000+\$322{,}000+\$9{,}000=\$381{,}000$.

Option B’s converted losses are $1.10\times \$280{,}000=\$308{,}000$. Its percentage expense is $0.025\times \$308{,}000=\$7{,}700$. Total adjustment expense is $\$4{,}000+\$7{,}700=\$11{,}700$, and its premium is $\$50{,}000+\$308{,}000+\$11{,}700=\$369{,}700$.

Option A is $\$381{,}000-\$369{,}700=\$11{,}300$ higher. To check the reconciliation, Option A has $14,000 more converted losses, while Option B has $2,700 more adjustment expense. Thus the net difference is $\$14{,}000-\$2{,}700=\$11{,}300$. This comparison shows why expense provisions should be stated in dollars after applying their specified bases, rather than described only as “higher” or “lower.”

Why the Difference Reconciles

For two options with fixed-dollar adjustment provisions, the premium definition gives $P_j=B_j+c_jT_j+A_j$. Subtracting Option B’s premium from Option A’s premium yields the exact component reconciliation below:

$$ P_A-P_B=(B_A-B_B)+(c_AT_A-c_BT_B)+(A_A-A_B). $$

This equality follows by replacing each premium with its defined components and collecting like terms: $(B_A+c_AT_A+A_A)-(B_B+c_BT_B+A_B)$. It is useful as a check because the premium difference must equal the sum of the differences in basic premium, converted losses, and adjustment expenses.

If each adjustment provision is a percentage $e_j$ of converted losses, replace $A_j$ with $e_jc_jT_j$. The same subtraction then gives:

P_A-P_B=(B_A-B_B)+\bigl((1+e_A)c_AT_A-(1+e_B)c_BT_B\bigr).

The formula is an accounting identity under the stated premium structure, not a claim that one option is generally preferable. It does not incorporate any minimum or maximum premium, tax, aggregate limitation, or other contractual provision omitted from the stated structure. If such a term applies, calculate it as specified before drawing a conclusion.

Common Mistake: Applying the Cap to the Wrong Amount

Suppose a contract caps each reported claim at $100,000 before applying an LCF of 1.20, and one reported claim is $130,000. The correct calculation is to cap first: $\min(\$130{,}000,\$100{,}000)=\$100{,}000$, then convert: $1.20\times \$100{,}000=\$120{,}000$.

A wrong calculation converts first and then applies the $100,000 reported-loss cap: $\min(1.20\times \$130{,}000,\$100{,}000)=\min(\$156{,}000,\$100{,}000)=\$100{,}000$. That incorrectly treats the reported-loss cap as a cap on converted losses and understates the converted amount by $20,000. The contract’s cap basis determines the order; do not infer that basis from the numerical value of the cap.

Do not compare LCFs in isolation. A lower LCF can accompany a higher cap and therefore a larger converted-loss amount.
Do not add an expense percentage to the wrong base. A percentage of converted losses is not the same dollar amount as that percentage of basic premium.
Do not count an expense twice. Add only the expense provisions stated for the option, and check whether a quoted provision is already included in another component.
Do not change the experience between options. A side-by-side parameter comparison should use the same reported claims unless the question explicitly asks for a different scenario.

Careful side-by-side calculations are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. The essential discipline is to separate the loss basis, conversion, and expense calculations before adding them into a premium.

Check Your Understanding

For each problem, apply the stated cap basis and expense basis, then compare only the completed premiums.

A hypothetical option has reported claims of $20,000, $50,000, and $80,000, a $60,000 per-claim reported-loss cap, an LCF of 1.25, basic premium of $30,000, and fixed adjustment expense of $5,000. Calculate the premium. (Answer: $197,500.)
Reported claims are $40,000, $100,000, and $150,000. A plan has a $100,000 per-claim reported-loss cap, LCF 1.10, basic premium $25,000, and adjustment expense of 4% of converted losses. Calculate the premium. (Answer: $299,560.)
For claims of $60,000, $90,000, and $140,000, Option A has a $100,000 reported-loss cap, LCF 1.20, basic premium $40,000, and fixed expense $8,000. Option B has a $120,000 reported-loss cap, LCF 1.10, the same basic premium, and expense of $3,000 plus 1% of converted losses. Which option is higher, and by how much? (Answer: Option A is higher by $5,030; premiums are $348,000 and $342,970.)
Claims are $50,000, $90,000, and $130,000. Two options have basic premium $10,000 and fixed adjustment expense $2,000. Option A has a $75,000 per-claim reported-loss cap and LCF 1.30; Option B has a $100,000 cap and the same LCF. Calculate both premiums and the difference. (Answer: Option A is $272,000 and Option B is $324,000; Option B is higher by $52,000.)
A hypothetical plan has basic premium $80,000, claims of $25,000, $75,000, and $125,000, a $100,000 per-claim reported-loss cap, and LCF 1.20. Calculate the premium if the adjustment expense is (a) 5% of basic premium and (b) 5% of converted losses. (Answer: (a) $324,000; (b) $332,000; the second is $8,000 higher.)

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