Why the Bases Must Stay Separate
A rate indication may include expenses that behave differently: a fixed amount allocated to each exposure, a percentage of premium, and a percentage of losses. The components can be added only after each has been expressed on a consistent basis, such as dollars per exposure. Applying every percentage to premium, or adding a dollar amount directly to a percentage, mixes unlike quantities and produces an invalid total.
The earlier material on expense allocation distinguishes fixed expenses from premium- and loss-related expenses. Here the task is to combine them transparently. The central discipline is to identify, for each component, both its amount or rate and its application base.
The formula assumes the bases are measured for the same coverage, period, and exposure unit, and that the expense provisions are additive as specified. For example, an annual per-policy fixed amount should not be combined with a six-month premium per policy unless the amounts have first been put on a consistent time basis. The formula does not decide whether a particular expense should be allocated by premium or loss; that choice must be established from the expense’s definition and the ratemaking assumptions.
Combining the Components per Exposure
The most direct calculation converts each component to dollars per exposure. The fixed amount is already on that basis. Multiply the premium provision by premium per exposure, and multiply the loss provision by expected losses per exposure. Then add the dollar amounts.
If a single expense ratio is useful for comparison, divide the combined expense dollars per exposure by premium dollars per exposure. The fixed component’s ratio is therefore not a constant: it depends on the premium per exposure.
This ratio form is valid when \(R>0\) and the amounts in the numerator and denominator use the same exposure and period basis. It is not appropriate to add \(F\), which is in dollars, directly to percentage rates. Nor should \(e_l\) be treated as a percentage of premium: its defined base is \(L\).
Worked Example: Combining Three Provisions for One Exposure
A hypothetical annual policy has premium of $560 per exposure and expected losses of $320 per exposure. The expense plan allocates fixed expenses of $18 per exposure, premium-related expenses of 7.5% of premium, and loss-related expenses of 10% of losses. Find the combined expense provision per exposure and its ratio to premium.
Method. Convert each provision to dollars per exposure before adding. This avoids treating a dollar amount as if it were a percentage.
Fixed provision: $18 per exposure.
Premium-related provision: \(0.075\times \$560=\$42\) per exposure.
Loss-related provision: \(0.10\times \$320=\$32\) per exposure.
Combined provision: \(\$18+\$42+\$32=\$92\) per exposure.
Expense ratio: \(\$92/\$560=0.1642857\ldots\), or 16.43% when rounded to two decimal places. As a component check, the fixed component is \(18/560=3.2143\%\), the premium component is 7.5%, and the loss component is \(32/560=5.7143\%\); their sum is 16.4286%, subject to rounding.
The result is $92 per exposure and 16.43% of premium. The common mistake avoided is applying the 10% loss provision to the $560 premium: that would incorrectly produce $56 instead of $32.
Why the Ratio Form Holds
The ratio expression follows by dividing every expense dollar by the same premium amount. This is useful because it shows exactly how a fixed amount behaves when expressed as a percentage: it becomes \(F/R\), rather than a constant percentage.
For one exposure, the total is \(F+e_pR+e_lL\), with each component calculated on its own base.
When \(R>0\), the expense ratio is \((F+e_pR+e_lL)/R\).
Division distributes across the sum: \(F/R+(e_pR)/R+(e_lL)/R\).
Since \((e_pR)/R=e_p\), the result is \(F/R+e_p+e_l(L/R)\). Thus the fixed expense ratio changes with premium per exposure, while the two percentage provisions retain their stated bases.
This is an identity, not an assumption that the expense ratio is constant across classes. If classes have different premium per exposure or different expected loss-to-premium relationships, their combined expense ratios can differ even when they use the same fixed amount and percentage provisions.
Combining Portfolio Amounts and Class Ratios
For a portfolio, add expense dollars across classes and divide by the portfolio’s matching base to obtain an aggregate ratio. An unweighted average of class ratios generally does not equal that ratio. A class with a small base must not receive the same portfolio weight as a class with a large base unless equal weighting is specifically intended.
Worked Example: Aggregate Loads Without Averaging Ratios Incorrectly
A hypothetical portfolio has two groups. Group A has loss-related expense of $1,400,000 on a loss base of $800,000. Group B has loss-related expense of $236,000 on a loss base of $1,200,000. Across both groups, fixed expenses are $60,000 for 3,000 exposures, and premium-related expenses are 5% of $2,000,000 of premium. Find the loss-related expense ratios by group, their simple average, the aggregate loss-related ratio, and the total combined expense per exposure.
Group ratios. For Group A, \(\$1{,}400{,}000/\$800{,}000=1.75=175.00\%\). For Group B, \(\$236{,}000/\$1{,}200{,}000=0.1966667\ldots=19.67\%\), rounded to two decimal places.
Simple average of the group ratios. \((175.00\%+19.6667\%)/2=97.33335\%\), or 97.33% to two decimal places. This is the arithmetic average of the two group ratios; it is not the portfolio ratio because the loss bases differ.
Aggregate loss-related ratio. Add the expense amounts and the loss bases: \(\$1{,}400{,}000+\$236{,}000=\$1{,}636{,}000\), and \(\$800{,}000+\$1{,}200{,}000=\$2{,}000{,}000\). The aggregate ratio is \(\$1{,}636{,}000/\$2{,}000{,}000=81.80\%\).
Convert the combined provisions to dollars per exposure. Fixed expenses are \(\$60{,}000/3{,}000=\$20\) per exposure. Premium-related expenses are \(0.05\times \$2{,}000{,}000=\$100{,}000\) in total, or \(\$100{,}000/3{,}000=\$33.3333\ldots\) per exposure. Loss-related expenses total $1,636,000, or \(\$1{,}636{,}000/3{,}000=\$545.3333\ldots\) per exposure.
Total and check. The combined amount is \(\$20+\$33.3333\ldots+\$545.3333\ldots=\$598.6667\ldots\), or $598.67 per exposure to the nearest cent. Multiplying the unrounded amount by 3,000 gives $1,796,000, which equals $60,000 + $100,000 + $1,636,000. The correct portfolio loss-related ratio is 81.80%; 97.33% is only the unweighted average of the two group ratios. The common mistake avoided is reporting that simple average as the portfolio result.
Class-Specific Provisions and Portfolio Checks
When classes have different expected premium or losses per exposure, calculate the combined provision within each class and then aggregate dollars. A single portfolio average can hide those differences. The following example uses a common fixed allocation but shows why class provisions should be computed on each class’s own premium and loss amounts.
Worked Example: Applying Common Percentages to Different Class Bases
A hypothetical annual book has 1,000 exposures in each of two classes. Annual fixed expenses of $48,000 are allocated equally per exposure. Class X has premium of $500 and expected losses of $300 per exposure; Class Y has premium of $800 and expected losses of $420 per exposure. Both classes use premium-related expenses of 4% of premium and loss-related expenses of 8% of losses. Find each class’s expense provision per exposure and the book’s combined provision per exposure and expense ratio.
Fixed amount. The fixed allocation is \(\$48{,}000/(1{,}000+1{,}000)=\$24\) per exposure in each class.
Class X. Its premium-related provision is \(0.04\times \$500=\$20\). Its loss-related provision is \(0.08\times \$300=\$24\). Its combined provision is \(\$24+\$20+\$24=\$68\) per exposure.
Class Y. Its premium-related provision is \(0.04\times \$800=\$32\). Its loss-related provision is \(0.08\times \$420=\$33.60\). Its combined provision is \(\$24+\$32+\$33.60=\$89.60\) per exposure.
Aggregate expense dollars. Fixed expenses are $48,000. Premium-related expenses are \(1{,}000\times \$20+1{,}000\times \$32=\$52{,}000\). Loss-related expenses are \(1{,}000\times \$24+1{,}000\times \$33.60=\$57{,}600\). Total expense is \(\$48{,}000+\$52{,}000+\$57{,}600=\$157{,}600\), or \(\$157{,}600/2{,}000=\$78.80\) per exposure.
Aggregate expense ratio and check. Total premium is \(1{,}000\times \$500+1{,}000\times \$800=\$1{,}300{,}000\). The ratio is \(\$157{,}600/\$1{,}300{,}000=0.1212307\ldots\), or 12.12% to two decimal places. The class-level check gives \((\$68+\$89.60)/2=\$78.80\) per exposure because the class exposure counts are equal. The common mistake avoided is using Class X’s loss base or premium as the base for Class Y’s percentage provisions.
Common Errors to Avoid
- Adding dollars to percentages. A fixed $24 cannot be added directly to 7% and 10%. First calculate each percentage provision in dollars, or express every component as a ratio using a stated denominator.
- Applying every percentage to premium. A loss-related percentage uses its defined loss base. In the first worked example, 10% applies to $320 of expected losses, not $560 of premium.
- Averaging class ratios without weights. A simple average describes the arithmetic mean of class ratios, not the portfolio ratio when class bases differ. Aggregate the amounts and the matching bases.
- Using inconsistent periods or units. Do not combine an annual fixed expense with a partial-year premium or a per-policy amount with a per-vehicle exposure unless the quantities have been made consistent.
- Assuming a fixed expense is a fixed percentage. The ratio \(F/R\) changes when premium per exposure changes, even if the fixed dollar allocation remains the same.
Check Your Understanding
For each problem, keep the expense amount and its application base explicit. Round dollar amounts to the nearest cent and ratios to two decimal places where needed.
- A hypothetical policy has premium of $450 per exposure, expected losses of $270 per exposure, fixed expenses of $30 per exposure, a premium-related expense provision of 6%, and a loss-related provision of 9%. Find the combined expense dollars per exposure. (Answer: $81.30 per exposure)
- A coverage has premium of $640 per exposure and expected losses of $400 per exposure. Its fixed allocation is $16 per exposure, its premium-related provision is 2.5%, and its loss-related provision is 11%. Find the combined provision and its ratio to premium. (Answer: $76.00 per exposure; 11.88%)
- A hypothetical book has 2,400 exposures, fixed expenses of $72,000, total premium of $1,600,000, and total expected losses of $900,000. Premium-related expenses are 4.5% of premium and loss-related expenses are 7% of expected losses. Find total expense dollars and expense per exposure. (Answer: $207,000 total; $86.25 per exposure)
- Group A has loss-related expenses of $180,000 on a loss base of $900,000. Group B has loss-related expenses of $120,000 on a loss base of $600,000. Find the two class ratios and the aggregate ratio. (Answer: 20.00% for each class; 20.00% aggregate)
- Two equally sized hypothetical classes each have 1,000 exposures. Class A has loss-related expenses of $450,000 on losses of $900,000; Class B has $100,000 of loss-related expenses on losses of $500,000. Each exposure has premium of $700 and expected losses of $500. If the fixed allocation is $20 per exposure and the premium-related provision is 5%, use the aggregate loss-related ratio to find the combined expense provision per exposure. (Answer: Aggregate loss-related ratio \(=\$550{,}000/\$1{,}400{,}000=39.29\%\); combined provision \(=\$20+\$35+(0.3928571\ldots\times \$500)=\$251.43\) per exposure)