Prevent Expense Double Counting In Retro Calculations

Why Expense Treatment Must Be Traced

A retrospective premium calculation can include adjustment expense in the loss charge, as a separate premium provision, or according to another formula specified by the plan. The arithmetic is not difficult once the treatment is clear. The risk is using an LCF that already includes an expense allowance and then adding that same allowance again as a separate line.

The correct calculation follows the plan’s definitions, not the label on a worksheet column. A line called “expense” does not establish whether it is already reflected in the LCF. Before calculating, identify the expense base, the factor’s components, and whether the plan says the provision is inside or outside the converted loss charge.

Definition: Let $R$ denote the loss amount to which the plan applies its conversion treatment, $A$ an adjustment-expense allowance, $C$ the converted loss charge, and $B$ the basic premium component. If the plan includes $A$ in the charge and also loads an expense equal to a fraction $q$ of $C$, its equation is $C=R+A+qC$. If the plan excludes $A$ from the charge, specifies $C=R+qC$, and provides for $A$ separately, the premium is $B+C+A$. Use only the treatment specified by the plan.

In these equations, $q$ is a proportion of the converted charge, not a proportion of the underlying losses. Thus $qC$ belongs on the right side of the equation for $C$, and solving for $C$ grosses up the other charge components. If the contract instead states an expense as a proportion of $R$, represent it as a proportion of $R$; do not substitute $qC$ without checking the stated base.

The examples use uncapped calculations so the expense treatment is visible. A plan’s minimum, maximum, or other premium term may affect the final amount; apply such terms as the plan specifies after identifying the correctly constructed components. A premium bound does not make a duplicated expense correct.

Deriving the Charge and the LCF

When adjustment expense $A$ is included in the charge and an additional expense is $q$ of that charge, start with the plan equation rather than adding a guessed percentage to the losses.

$$ C=R+A+qC $$

Subtract $qC$ from both sides. For $q<1$, divide by $1-q$ to obtain the charge. The LCF for this treatment is the resulting charge divided by $R$, provided $R$ is the base to which the plan applies the factor.

(1-q)C=R+A,\qquad C=\frac{R+A}{1-q},\qquad \mathrm{LCF}=\frac{C}{R} =\frac{1+A/R}{1-q}\quad (R>0).

This derivation explains why simply adding $A$ to a charge calculated as $R/(1-q)$ is not equivalent to including $A$ in the charge before the gross-up. Under the latter treatment, $A$ is also subject to the $q$ gross-up. Conversely, if the plan explicitly excludes $A$ from the LCF and provides it separately, the appropriate charge is $R/(1-q)$, followed by the separate addition of $A$. Neither arrangement is universally correct independent of the contract wording.

Locate the expense base.
Determine whether each stated expense is a percentage of losses, the converted charge, or another specified amount.

Read what the LCF includes.
List the expense components embedded in the factor. Do not infer this solely from the factor’s name or numerical size.

Write the plan equation.
Put included amounts in the charge equation and keep separately specified provisions outside it.

Reconcile the worksheet.
Check that each expense appears once in the premium formula, then substitute the calculated charge back into the equation.

Worked Applications

Worked Example: An Embedded Allowance Added Again

A hypothetical plan uses $R=\$190{,}000$. Its loss charge includes an adjustment-expense allowance equal to 9% of $R$, as well as an expense equal to 5% of the resulting charge $C$. The plan adds a basic premium of $\$24{,}000$. A worksheet also lists the 9% adjustment allowance as a separate premium provision. Find the correct premium and the worksheet overstatement.

Method. Follow the stated charge equation, then compare the correct premium with the worksheet that adds the embedded allowance again. The common mistake is to treat the same 9% allowance as both part of the charge and a new premium provision.

First calculate the adjustment allowance:

A=0.09(\$190{,}000)=\$17{,}100.

The plan’s equation is $C=R+A+0.05C$. Moving the charge-based expense to the left gives:

0.95C=\$190{,}000+\$17{,}100=\$207{,}100, \qquad C=\frac{\$207{,}100}{0.95}=\$218{,}000.

The correct premium is the charge plus the basic premium:

P=\$218{,}000+\$24{,}000=\$242{,}000.

The worksheet adds the already embedded $\$17{,}100$ again, so it reports $\$242{,}000+\$17{,}100=\$259{,}100$. Its overstatement is $\$17{,}100$. As a check, $\$190{,}000+\$17{,}100+0.05(\$218{,}000)=\$218{,}000$, which satisfies the plan equation exactly.

Worked Example: A Separate Provision That Belongs in the Premium

A hypothetical plan has $R=\$125{,}000$. It defines its LCF to cover a charge-based expense equal to 5% of $C$, but explicitly excludes adjustment expense from that factor. The adjustment-expense provision is $\$10{,}000$, payable separately, and the basic premium is $\$18{,}000$. Calculate the premium. Then quantify the overstatement if a worksheet mistakenly uses an LCF that embeds the $\$10{,}000$ allowance and still adds that allowance separately.

Method. First use the plan’s stated LCF treatment, which excludes the adjustment provision. The common mistake is to use a different, expense-inclusive LCF while retaining the plan’s separate provision.

For the stated plan, $C=R+0.05C$, so:

0.95C=\$125{,}000,\qquad C=\frac{\$125{,}000}{0.95} =\$131{,}578.947368\ldots

The adjustment provision and basic premium are separate additions. Before rounding, the total is:

P=\$131{,}578.947368\ldots+\$10{,}000+\$18{,}000 =\$159{,}578.947368\ldots

Rounded to the nearest cent, the correct premium is $\$159{,}578.95$.

For the mistaken worksheet, embedding the adjustment allowance produces $C_{\text{wrong}}=(\$125{,}000+\$10{,}000)/0.95=\$142{,}105.263157\ldots$. Adding the same separate provision and basic premium gives an erroneous total of $\$170{,}105.263157\ldots$, or $\$170{,}105.26$ to the nearest cent. The unrounded overstatement is:

\frac{\$10{,}000}{0.95} =\$10{,}526.315789\ldots \quad\Longrightarrow\quad \text{overstatement}=\$10{,}526.32.

The overstatement is the increase in the converted charge caused by wrongly embedding the allowance; the separate $\$10{,}000$ is present in both totals. Use the unrounded totals to calculate the difference, then round the result to cents.

Worked Example: Auditing Several Worksheet Lines

A hypothetical plan has $R=\$240{,}000$. Its stated conversion treatment includes an adjustment allowance of 6% of $R$ and an expense equal to 4% of the converted charge $C$. The plan also adds a basic premium of $\$32{,}000$ and a separate, non-adjustment provision of $\$7{,}500$. A worksheet adds the 6% allowance as another line. Calculate the correct total and diagnose the error.

Method. Separate the embedded adjustment allowance from the distinct $\$7{,}500$ provision. The common mistake is to remove all expense lines when correcting a duplicate, rather than removing only the expense already included in the charge.

The adjustment allowance included in the charge is:

A=0.06(\$240{,}000)=\$14{,}400.

The charge equation is $C=R+A+0.04C$, giving:

0.96C=\$240{,}000+\$14{,}400=\$254{,}400, \qquad C=\frac{\$254{,}400}{0.96}=\$265{,}000.

The basic premium and the distinct provision still apply:

P=\$265{,}000+\$32{,}000+\$7{,}500=\$304{,}500.

If the worksheet adds the embedded $\$14{,}400$ again, it reports $\$318{,}900$, an overstatement of $\$14{,}400$. The separate $\$7{,}500$ remains because the plan specifies it as a different provision. Substitution checks the charge: $\$240{,}000+\$14{,}400+0.04(\$265{,}000)=\$265{,}000$.

Common Traps and a Reliable Audit

A useful audit is to write the premium formula in words and symbols before accepting any worksheet total. For example: “converted charge, plus basic premium, plus adjustment expense only if not already included, plus other separately specified provisions.” Then mark each expense component as embedded or separate. If one component has both marks, investigate; if it has neither, check whether the plan intentionally excludes it.

Using a factor without its definition. An LCF is not self-explanatory. Confirm which cost components it converts or includes.
Mixing percentage bases. An expense stated as 5% of the converted charge is not automatically 5% of reported losses. Write the stated base in the equation.
Adding a displayed allowance twice. A worksheet may show an expense component for transparency even though it is already included in the factor. A displayed line is not necessarily an additional charge.
Removing a distinct provision along with the duplicate. Correct the duplicated component only. Keep other amounts the plan specifies separately.
Rounding too early. Keep full precision through the charge, premium, and overstatement calculations; round money to cents at the final step.

The governing check is whether the calculated $C$ satisfies the plan’s charge equation and whether the final premium adds only the provisions that remain outside $C$. This approach is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus and is also useful when reviewing an actual rating worksheet.

Check Your Understanding

For each problem, use the plan’s stated treatment rather than assuming every expense is separate. Round dollar answers to the nearest cent.

A plan has $R=\$60{,}000$, includes an adjustment allowance of 10% of $R$, and includes an expense equal to 5% of $C$. Its basic premium is $\$8{,}000$. A worksheet adds the $\$6{,}000$ allowance again. Find the correct premium and overstatement. (Answer: Correct premium $\$77{,}473.68$; overstatement $\$6{,}000.00$.)
A plan has $R=\$90{,}000$, an LCF that excludes adjustment expense and grosses up for an expense equal to 4% of $C$, a separate adjustment provision of $\$7{,}200$, and a basic premium of $\$12{,}000$. Find the correct premium. (Answer: $\$112{,}950.00$.)
A plan’s LCF is 1.125. It is defined by $\mathrm{LCF}=(1+a)/0.96$, where $a$ is the adjustment allowance as a proportion of $R$. For $R=\$160{,}000$, the plan includes that allowance in the charge and adds a basic premium of $\$15{,}000$. Find $a$, the charge, and the correct premium; state the overstatement if the allowance is added separately too. (Answer: $a=8\%$; charge $\$180{,}000.00$; correct premium $\$195{,}000.00$; overstatement $\$12{,}800.00$.)
A plan includes an adjustment allowance of 7% of $R=\$210{,}000$ and an expense equal to 6% of $C$. It adds a basic premium of $\$25{,}000$. Find the correct premium and the erroneous premium if the adjustment allowance is also added separately. (Answer: Correct premium $\$264{,}042.55$; erroneous premium $\$278{,}742.55$.)
Compare two hypothetical treatments for $R=\$100{,}000$, adjustment expense $A=\$5{,}000$, charge-based expense rate $q=5\%$, and basic premium $B=\$14{,}000$. Under Plan X, $A$ is included before the gross-up and $P=B+(R+A)/(1-q)$. Under Plan Y, $A$ is excluded from the charge and $P=B+R/(1-q)+A$. Calculate both premiums and their difference; explain whether the separate $A$ is a duplicate under either stated treatment. (Answer: Plan X $\$124{,}526.32$; Plan Y $\$124{,}263.16$; Plan X is $\$263.16$ higher; neither treatment duplicates $A$ when applied as stated.)

What You'll Learn