Additional and Return Premium from Audits

Why Audited Exposure Changes Premium

For some hypothetical insurance policies, an initial premium is based on an estimate of an exposure that will not be known precisely until coverage ends. A final audit determines the exposure used for premium calculation. If the audited exposure differs from the estimate, the premium calculated from the audited amount may also differ from the initial premium.

The central calculation is a comparison of two premiums for the same coverage and rating terms: the premium based on the audited exposure and the comparable initial premium. The difference is the audit adjustment. A positive difference means additional premium; a negative difference means return premium. This tutorial focuses on that calculation, aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $P_{\text{initial}}$ be the premium calculated using the exposure estimate used for the initial premium, and let $P_{\text{audited}}$ be the premium recalculated using audited exposure and the applicable rating terms. Define the audit adjustment as $A=P_{\text{audited}}-P_{\text{initial}}$. If $A>0$, additional premium is due relative to the initial premium. If $A<0$, the insured is entitled to a return relative to the initial premium. The adjustment is not necessarily the amount still payable on the account: deposits or other payments must be considered separately.

The Basic Calculation

First identify the exposure basis and the unit in which the rate is stated. For example, a rate of $18 per $1,000 of payroll requires payroll to be expressed in thousands of dollars. Next apply the rate and any specified rating factors. The initial and audited calculations must use comparable terms: otherwise, their difference can combine an exposure change with an unrelated change in fees, coverage, or rating rules.

For a single class with a rate $r$ per exposure unit and a multiplicative rating factor $m$, suppose the policy premium being compared contains only the exposure-based component. The initial and audited premiums are calculated from their respective exposures. Their difference follows directly:

P_{\text{initial}}=E_{\text{initial}}rm,\qquad P_{\text{audited}}=E_{\text{audited}}rm,\qquad A=(E_{\text{audited}}-E_{\text{initial}})rm.

This identity is a useful check, not a replacement for calculating both premiums. It holds when the same rate and factor apply to both exposure amounts and there are no other premium components that change differently. If the audited classification or rate changes, calculate each class separately. If fixed charges, taxes, or other components apply, compare the complete premiums under the stated rules.

Unit check: If exposure is measured in payroll dollars but the rate is per $1,000 of payroll, divide payroll by $1,000 before multiplying by the rate. The exposure unit and rate denominator must match; otherwise, the premium will be wrong by a conversion factor.

Worked Example: Lower Audited Exposure and a Return

A hypothetical policy has an initial estimate of 1,200 exposure units. Its variable rate is $15 per unit, and the contract specifies a 0.10 participation factor applied to this variable premium. The audit finds 800 exposure units. There are no fixed charges or taxes. Find the audit adjustment and state whether it is an additional or return premium.

Method. Recalculate the variable premium at the audited exposure using the same rate and participation factor, then subtract the initial premium. The factor applies to the variable premium in both calculations.

Initial base premium: $1{,}200\times \$15=\$18{,}000$. After the factor, the initial premium is $\$18{,}000\times0.10=\$1{,}800$.

Audited base premium: $800\times \$15=\$12{,}000$. After the factor, the audited premium is $\$12{,}000\times0.10=\$1{,}200$.

Therefore, $A=\$1{,}200-\$1{,}800=-\$600$. The audit adjustment is $-\$600$, so it is a $600 return relative to the initial premium. As a check, the exposure fell by $1{,}200-800=400$ units; $400\times \$15\times0.10=\$600$, the absolute amount of the adjustment.

Common mistake avoided. Do not report the audited premium of $1,200 as the amount of the return. The return is the difference between the audited and initial premiums: $\$1{,}200-\$1{,}800=-\$600$.

Multiple Classes and Revised Classification

If an audit changes both the amount of exposure and its classification, a single exposure difference multiplied by one rate is not generally valid. Let $E_j$ be audited exposure in class $j$, and let $r_j$ be that class's rate per exposure unit. When a common multiplicative factor $m$ applies across classes, the audited variable premium is the sum of the class premiums:

P_{\text{audited}}=m\sum_j E_jr_j.

The initial premium must likewise be calculated from the initial class exposures and their applicable rates. Subtract the initial total from the audited total. This sum-by-class approach follows because each class contributes its own exposure multiplied by its own rate; combining exposures before applying rates would ignore the different prices for exposure in each class.

Worked Example: Audited Exposure Split Between Two Classes

A hypothetical contractor initially reports 300 payroll units in Class A at $8 per unit and 200 units in Class B at $12 per unit. The initial premium is based on those estimates. The audit finds 260 units in Class A and 240 units in Class B. No factor, fixed charge, or tax applies. Calculate the adjustment.

Method. Calculate the premium class by class for both the initial estimate and the audit, then compare totals.

Initial Class A premium: $300\times \$8=\$2{,}400$. Initial Class B premium: $200\times \$12=\$2{,}400$. Thus $P_{\text{initial}}=\$2{,}400+\$2{,}400=\$4{,}800$.

Audited Class A premium: $260\times \$8=\$2{,}080$. Audited Class B premium: $240\times \$12=\$2{,}880$. Thus $P_{\text{audited}}=\$2{,}080+\$2{,}880=\$4{,}960$.

The adjustment is $A=\$4{,}960-\$4{,}800=+\$160$. The audit produces $160 of additional premium relative to the initial premium.

As a check, Class A exposure decreased by 40 units, reducing premium by $40\times \$8=\$320$. Class B exposure increased by 40 units, increasing premium by $40\times \$12=\$480$. The net change is $-\$320+\$480=+\$160$, which agrees with the total-premium comparison.

Common mistake avoided. The total exposure is 500 units both before and after audit, but that does not mean the premium is unchanged. The exposure moved between classes with different rates. Multiplying the unchanged total by one rate would discard the classification information.

Fixed Charges, Taxes, and the Account Balance

Not every premium component changes with audited exposure. A fixed policy charge that is unchanged by the audit appears in both premium calculations and cancels when their difference is taken. A tax or other percentage charge may change if it is applied to a premium amount that changes. Follow the stated calculation rule and apply it consistently to both the initial and audited premiums.

Also keep the audit adjustment separate from payments already made. If $D$ is the deposit or other amount already paid toward the initial premium, the amount still due after audit is $P_{\text{audited}}-D$, if positive. If this amount is negative, its absolute value is a credit or amount to be returned, subject to the applicable contract terms. This account balance is not necessarily equal to the audit adjustment: the adjustment compares premium amounts, while the balance also reflects payments.

Worked Example: A Return Adjustment and the Amount Paid

A hypothetical policy has estimated exposure of 500 units at $6 per unit, plus a fixed charge of $250. A 5% tax applies to the sum of the variable premium and fixed charge. The insured has paid $3,400. The audit finds 420 exposure units; the rate, charge, and tax rule remain unchanged. Find the audit adjustment and the balance after crediting the payment.

Method. Calculate the two pre-tax subtotals, apply the 5% tax to each, compare the complete premiums, and then subtract the amount paid from the audited premium.

Initial variable premium: $500\times \$6=\$3{,}000$. Initial pre-tax subtotal: $\$3{,}000+\$250=\$3{,}250$. Initial tax: $\$3{,}250\times0.05=\$162.50$. Therefore, $P_{\text{initial}}=\$3{,}250+\$162.50=\$3{,}412.50$.

Audited variable premium: $420\times \$6=\$2{,}520$. Audited pre-tax subtotal: $\$2{,}520+\$250=\$2{,}770$. Audited tax: $\$2{,}770\times0.05=\$138.50$. Therefore, $P_{\text{audited}}=\$2{,}770+\$138.50=\$2{,}908.50$.

The adjustment is $\$2{,}908.50-\$3{,}412.50=-\$504.00$, a $504 return relative to the initial premium. After crediting the $3,400 already paid, the account balance is $\$2{,}908.50-\$3{,}400=-\$491.50$. Thus the payment exceeds the audited premium by $491.50, subject to the applicable terms.

The fixed charge cancels in the difference, but the tax changes with the pre-tax subtotal. The premium decreases by $480 before tax, and the tax decreases by $0.05\times \$480=\$24$, for a total decrease of $\$480+\$24=\$504$. This independently checks the adjustment.

Common mistake avoided. The audit adjustment is $-\$504$, not the $-\$491.50$ account balance. The latter also reflects the amount already paid.

Calculation Sequence and Common Traps

A reliable audit calculation keeps units, class rates, and premium components visible. Use the following sequence when the policy terms provide the required information:

Identify the exposure basis and units.
Convert the estimate and audit amounts to the rate's stated unit before multiplying.

Apply the applicable class rates and factors.
If classification changed, calculate each class separately for both the initial and audited premiums.

Include other applicable components.
Apply fixed charges and taxes according to the stated terms; do not assume every component varies with exposure.

Subtract in the defined order.
Calculate audited premium minus initial premium. The sign determines additional versus return premium.

Reconcile payments separately.
Subtract deposits or payments from the audited premium only when finding the amount due or credit on the account.

A frequent exam trap is to use the sign backwards, subtract audited premium from initial premium, or call the entire audited premium the adjustment. Another is to confuse a return relative to the initial premium with a refund of every dollar paid; only after accounting for deposits can the account credit be determined. The calculation also does not by itself establish whether a minimum premium, fee, or other contractual provision changes the final amount. If such a term is specified, include it as directed rather than silently assuming it is absent.

Key takeaway: Compute the initial and audited premiums on a comparable basis, then use $A=P_{\text{audited}}-P_{\text{initial}}$. A positive result is additional premium; a negative result is return premium. Payments affect the account balance, not the definition of the audit adjustment.

Check Your Understanding

For each hypothetical policy, determine the requested audit adjustment or account balance. Treat all exposure units, rates, charges, and taxes as stated.

A policy has estimated exposure of 900 units and audited exposure of 960 units. The rate is $4.50 per unit, with no other premium components. What is the audit adjustment and its direction? (Answer: $+\$270$, additional premium.)
Estimated payroll is $1,250,000 and audited payroll is $1,180,000. The rate is $18 per $1,000 of payroll. Find the adjustment. (Answer: $-\$1,260$, a return premium.)
Initial exposures are 400 units in Class A at $7 per unit and 150 units in Class B at $10 per unit. Audit exposures are 360 units in Class A and 180 units in Class B. Find the adjustment. (Answer: $+\$20$, additional premium.)
Estimated exposure is 700 units; audited exposure is 640 units. The rate is $5 per unit, and a 0.08 factor applies to the variable premium both initially and after audit. There are no other components. Find the adjustment. (Answer: $-\$24$, a return premium.)
A policy's initial exposure is 240 units at $11 per unit, plus a fixed charge of $180. A 4% tax applies to that subtotal. The audit finds 210 units, with the same rate, charge, and tax rule. The insured has paid $2,700. Find both the audit adjustment and the amount due or credit after payment. (Answer: audited premium $2,589.60; adjustment $-\$343.20$, a return; $110.40 credit after payment.)

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