Building a Single-Coverage Expense Provision
A single-coverage rate must fund more than expected losses. It must also provide for the coverage’s fixed expenses, claim-related costs, commissions, taxes, and any other expenses included in the rate. The calculation depends on the base for each item: a fixed amount per exposure is not treated the same way as an expense charged as a percentage of premium.
The central task is to make those bases explicit before solving for the rate. Costs stated per claim must first be converted to an expected amount per exposure, and premium-based expenses must be recognized as a share of the premium being calculated. The resulting equation can then be solved and checked.
Sort Expenses by Their Base
For this calculation, use one consistent unit: dollars per exposure. A fixed expense stated per exposure enters the numerator directly. An expense stated per claim must be multiplied by expected claim frequency per exposure. An expense expressed as a percentage of expected losses is applied to expected losses, not to premium. These amounts are combined in the numerator as non-premium-based costs.
By contrast, commissions, premium taxes, and assessments stated as percentages of premium are collected from the premium itself. If their rates are \(q_1,q_2,\ldots,q_k\), and each applies to the same premium base, then \(q=q_1+q_2+\cdots+q_k\). This sum is the share of premium used for those expenses. The remaining share, \(1-q\), is available to fund losses and non-premium-based expenses.
This setup assumes the listed expenses are additive, apply to the same single coverage and exposure basis, and are measured for the same rating period. It also assumes a constant expected cost per exposure and constant premium-based percentages. If an expense uses a different base, depends on the premium in a nonlinear way, or is already included in another cost, the equation must be adjusted rather than applied mechanically.
The premium-based expenses equal \(qR\), because they are calculated on the unknown premium \(R\). Rearranging gives \((1-q)R=L+F+C\), which yields the rate formula in the definition. The division by \(1-q\) is a gross-up: it provides enough premium that, after the premium-based expense share is taken, the remainder still funds the numerator. For positive costs, a finite positive indicated rate requires \(q<1\).
Convert per-claim costs using expected claim frequency, and calculate any ledger allocation or loss-based expense before assembling the numerator.
Add only the rates that apply to the same premium base. The denominator is one minus this total rate.
Divide total non-premium-based cost per exposure by the denominator, then verify that the premium-based expenses plus the remaining premium equal the indicated rate.
Worked Calculations
Worked Example: Converting Claim Costs to a Per-Exposure Rate
A hypothetical coverage has expected losses of $540 per exposure and expected claim frequency of 0.80 claims per exposure. Claim handling costs $36 per claim, while fixed expenses are $43.20 per exposure. Commission is 8% of premium and premium tax is 2% of premium. Calculate and check the indicated rate.
Method. Convert the claim handling cost to a per-exposure amount, add it to losses and fixed expenses, and gross up for the premium-based expenses.
Step 1: Expected claim handling expense. The frequency is claims per exposure, so multiplying by dollars per claim gives dollars per exposure:
Step 2: Non-premium-based costs.
Step 3: Premium-based percentage and denominator. Commission and tax both apply to premium, so \(q=0.08+0.02=0.10\), and \(1-q=0.90\).
Check. Commission is \(0.08\times \$680=\$54.40\), and tax is \(0.02\times \$680=\$13.60\). Together they are $68.00. The remaining \(90\%\) of premium is \(0.90\times \$680=\$612.00\), exactly the losses and other costs in the numerator. The check reconciles: \(\$612.00+\$68.00=\$680.00\).
Common mistake avoided: Treating $36 per claim as $36 per exposure would overstate handling expense. The relevant expected amount is $28.80 per exposure because the frequency is less than one claim per exposure.
Worked Example: Combining an Allocated Fixed Cost and Loss-Based Expense
A hypothetical coverage has $18,000 of annual fixed expenses allocated over 1,200 exposures. Expected losses are $450 per exposure. Loss adjustment expense is 4% of expected losses, and a separate handling charge is $20 per claim at a frequency of 0.50 claims per exposure. Commission is 9% of premium and tax is 3% of premium. Calculate the indicated rate per exposure.
Method. First allocate the fixed expense, then calculate both claim-related items on their stated bases. Only commission and tax enter the premium percentage.
Step 1: Fixed expense per exposure.
Step 2: Non-premium-based claim costs. The 4% loss adjustment expense applies to expected losses, while the handling charge applies per claim:
Step 3: Numerator and denominator. Total non-premium-based cost is \(\$450+\$15+\$18+\$10=\$493\) per exposure. Commission and tax total \(q=0.09+0.03=0.12\), so the premium share left to fund that cost is \(1-q=0.88\).
Rounding only the final rate to cents gives $560.23 per exposure.
Check. Using the unrounded rate, the premium-based expenses are \(0.12\times \$560.227272\ldots=\$67.227272\ldots\). The remaining \(88\%\) is \(\$493\), matching the numerator. Thus the unrounded premium reconciles as \(\$493+\$67.227272\ldots=\$560.227272\ldots\).
Common mistake avoided: The 4% loss adjustment expense is based on expected losses, not premium. It belongs in the numerator as $18 per exposure; it must not be added to the premium-percentage denominator.
Worked Example: Why Fixed Expenses Must Also Be Grossed Up
For a hypothetical coverage, expected losses are $540 per exposure, a claim-related expense is $36 per exposure, and another loss-related expense equals 8% of expected losses. Fixed expense is $28 per exposure. Commission is 9% of premium and tax is 3% of premium. Calculate the indicated rate and assess the tempting approach of grossing up the first three costs before adding fixed expense.
Method. All non-premium-based costs belong together in the numerator before applying the premium denominator.
Step 1: Calculate the loss-based expense.
Step 2: Add all non-premium-based costs.
Step 3: Calculate the premium denominator. The premium-based rates total \(q=0.09+0.03=0.12\), leaving \(1-q=0.88\) of premium for the numerator.
Rounding only the final rate to cents gives $735.45 per exposure.
Check. At the unrounded rate, the premium-based expenses are \(0.12\times \$735.454545\ldots=\$88.254545\ldots\). The remaining \(88\%\) is $647.20, so the premium funds all stated costs.
Common mistake and correction. A tempting but incorrect calculation grosses up the losses and claim-related costs first, then adds fixed expense:
That calculation rounds to $731.64, not $730.27. It is still incorrect because the added $28 of fixed expense does not generate the premium-based expenses associated with that amount. At the unrounded incorrect rate, the \(88\%\) remaining after commission and tax is only \(0.88\times \$731.636363\ldots=\$643.84\), which is $3.36 short of the required $647.20. Including fixed expense in the numerator before dividing gives the correct $735.45 rate.
Interpreting and Checking the Indication
An indicated rate is a cost-based result under the stated assumptions, not a guarantee that actual results will equal expected results. It also depends on the expense bases being classified correctly. The same expense amount can produce a different indication if it is stated per claim rather than per exposure, or as a percentage of premium rather than as a percentage of losses.
A practical check is to reconstruct the rate from its components. At an unrounded indicated rate, the amount remaining after premium-based expenses must equal the full numerator:
If this equality does not hold, check the arithmetic and whether an expense was omitted, counted twice, or assigned to the wrong base. Keep the calculation specific to the single coverage: the exposure count used to allocate fixed costs, the claim frequency used for per-claim costs, and the premium base for percentage expenses must all correspond to that coverage and period.
Check Your Understanding
For each hypothetical single-coverage problem, identify the expense bases, calculate the indicated rate, and check the premium denominator. Answers are rounded to cents.
- A coverage has expected losses of $360 per exposure and fixed expenses of $24 per exposure. Commission is 6% of premium and tax is 2% of premium. What is the indicated rate? (Answer: $417.39 per exposure)
- Expected losses are $420 per exposure, fixed expenses are $18 per exposure, and claim handling costs $40 per claim at a frequency of 0.30 claims per exposure. Commission and tax are 7% and 3% of premium. What is the indicated rate? (Answer: $500.00 per exposure)
- Annual fixed expenses of $14,400 are allocated over 900 exposures. Expected losses are $520 per exposure, loss adjustment expense is 5% of expected losses, commission is 10% of premium, and tax is 2% of premium. Calculate the indicated rate. (Answer: $638.64 per exposure)
- Expected losses are $780 per exposure. Fixed expense is $25 per exposure, claim handling costs $15 per claim at a frequency of 0.40, and another claim-related expense equals 2% of expected losses. Commission, tax, and an assessment are 7%, 2%, and 1% of premium, respectively. Calculate the indicated rate. (Answer: $918.44 per exposure)
- A coverage has expected losses of $700 per exposure, fixed expenses of $27 per exposure, claim handling costs $30 per claim at a frequency of 0.20, and loss adjustment expense equal to 2% of expected losses. Commission and tax are 8% and 3% of premium. Find the indicated rate, then determine whether a premium of $850 per exposure leaves a surplus or deficit after the premium-based expenses. (Answer: Indicated rate $839.33 per exposure; at $850, the amount remaining after premium-based expenses is $756.50, a $9.50 surplus over the $747.00 of non-premium-based costs.)