Why Permissible Ratios Can Differ
In the loss ratio indication method, a permissible loss ratio expresses the portion of premium available for losses after allowing for expenses and the target underwriting profit or contingency provision. When those provisions are percentages of premium, their percentages directly determine the permissible ratio.
The calculation is straightforward for one segment, but a portfolio may contain segments with different commission rates, premium taxes, other premium-based expenses, or profit provisions. A permissible ratio calculated for one segment should not automatically be applied to another. The ratio depends on that segment’s assumptions.
Here, a provision stated as 5% of premium means 5% of the premium denominator, not 5% of losses. Premium-based expense provisions might include commissions, premium taxes, and other expenses that are modeled as percentages of premium. For this calculation, the expense and profit provisions must use the same premium basis as the loss ratio. Losses should also be on the prospective basis being indicated, rather than on an inconsistent historical basis.
This topic concerns provisions that vary in proportion to premium. If a provision is not premium-based, the simple permissible-ratio formula may not represent it correctly; it should not be silently treated as a percentage of premium.
Deriving the Permissible Loss Ratio
Suppose a segment needs premium \(P\) to fund its losses \(L\), its premium-based expenses, and its target profit or contingency provision. Assume those expenses equal \(eP\), and the profit or contingency provision equals \(qP\). Assume there are no other required amounts omitted from this allocation. The premium requirement is then:
Subtract the expense and profit provisions from both sides and divide by \(P\). The resulting share of premium available for losses is:
This is the permissible loss ratio. The derivation also explains why the denominator matters: both provisions are defined as percentages of premium, so they are subtracted from 100% of premium. If, for example, a profit target were instead defined as a percentage of losses, it could not be inserted as \(q\) in this formula without first translating it to a premium basis.
The formula follows by finding the premium required to fund the selected losses: \(P_{\text{required}}=L/(1-e-q)\). Dividing this required premium by the current on-level premium gives the indicated premium factor. Subtracting 1 converts that factor to a percentage rate change. A positive result indicates an increase; a negative result indicates a decrease.
Calculating Ratios by Segment
The following example isolates the permissible-ratio calculation. All figures are hypothetical. The percentages are provisions as a share of premium, and the loss amounts below are allowable loss amounts at a stated premium—not historical loss selections.
Worked Example: Two Segments With Different Expense Provisions
A hypothetical insurer is reviewing two segments, each at $1,000,000 of premium. Segment A has a 14% commission provision, 2% premium tax provision, 5% other premium-based expense provision, and 3% target profit provision. Segment B has a 20% commission provision, 2% premium tax provision, 5% other premium-based expense provision, and the same 3% target profit provision. Calculate each permissible loss ratio and the allowable loss amount at $1,000,000 of premium.
Method. Add the premium-based expense provisions within each segment, then add the profit provision. Subtract the total provision from 100% of premium.
For Segment A, the total expense provision is \(14\%+2\%+5\%=21\%\). Including the 3% profit provision gives a total provision of \(21\%+3\%=24\%\). Therefore its permissible loss ratio is \(100\%-24\%=76\%\), and its allowable loss amount is \(0.76\times \$1{,}000{,}000=\$760{,}000\).
For Segment B, the total expense provision is \(20\%+2\%+5\%=27\%\). Including profit gives \(27\%+3\%=30\%\). Its permissible loss ratio is \(100\%-30\%=70\%\), and its allowable loss amount is \(0.70\times \$1{,}000{,}000=\$700{,}000\).
Check. For Segment A, \$760,000 of losses plus \$210,000 of expenses and \$30,000 of profit totals \$1,000,000. For Segment B, \$700,000 plus \$270,000 and \$30,000 also totals \$1,000,000. The lower permissible ratio for Segment B follows from its higher commission provision. This calculation avoids applying Segment A’s ratio uniformly simply because the two segments have the same premium.
A difference in expense provisions is enough to change the permissible ratio, even when the loss measure and profit provision are identical. The same principle applies when profit or contingency provisions differ: each segment’s own expense and profit percentages determine its ratio.
From a Segment Loss Ratio to a Rate Indication
A permissible ratio by itself is not a rate indication. The indication also depends on the segment’s adjusted loss ratio. For a segment with on-level premium \(P\) and selected losses \(L\), the adjusted loss ratio is \(L/P\). The required premium for those losses is \(L/(1-e-q)\); comparing that amount with \(P\) gives the indicated change.
Worked Example: Indications With Different Segment Provisions
A hypothetical book has two segments. For Fleet, on-level premium is $1,200,000 and selected losses are $780,000. Its provisions are 16% commission, 2% premium tax, 6% other premium-based expenses, and 6% profit. For Artisan, on-level premium is $800,000 and selected losses are $600,000. Its provisions are 12% commission, 2% premium tax, 4% other premium-based expenses, and 5% profit. Calculate each segment’s permissible loss ratio and indicated rate change. Round rate changes to two decimal places.
Method. Calculate the provision total and permissible ratio separately for each segment. Then divide the adjusted loss ratio by that segment’s permissible ratio and subtract 1.
For Fleet, total expenses are \(16\%+2\%+6\%=24\%\). Adding 6% profit gives total provisions of 30%, so the permissible ratio is \(100\%-30\%=70\%\). Its adjusted loss ratio is \(\$780{,}000/\$1{,}200{,}000=65\%\). Thus the indicated change is \(65\%/70\%-1=-0.0714286\), or a 7.14% decrease.
As a check using required premium, Fleet requires \(\$780{,}000/0.70=\$1{,}114{,}285.71\). Relative to current on-level premium, the change is \(\$1{,}114{,}285.71/\$1{,}200{,}000-1=-0.0714286\), consistent with the ratio calculation.
For Artisan, total expenses are \(12\%+2\%+4\%=18\%\). Adding 5% profit gives 23%, so the permissible ratio is 77%. Its adjusted loss ratio is \(\$600{,}000/\$800{,}000=75\%\). The indicated change is \(75\%/77\%-1=-0.0259740\), or a 2.60% decrease.
The required premium check for Artisan is \(\$600{,}000/0.77=\$779{,}220.78\). Relative to \$800,000, this is a 2.60% decrease after rounding. Fleet and Artisan have different indicated changes because their adjusted loss ratios and permissible ratios differ. This method avoids using one segment’s provisions as the denominator for the other segment’s indication.
Combining Segments for One Uniform Change
Sometimes an analysis needs a single uniform rate change across segments even though the permissible ratios differ. In that case, calculate the premium required for each segment using its own permissible ratio, add the required premiums, and compare their total with current on-level premium. In general, do not combine different permissible ratios by taking a simple average.
Worked Example: One Overall Indication From Segment Requirements
A hypothetical insurer wants one uniform rate change for two segments. Segment C has $900,000 of current on-level premium, $630,000 of selected losses, 20% premium-based expenses, and a 5% profit provision. Segment D has $600,000 of current on-level premium, $480,000 of selected losses, 15% premium-based expenses, and a 5% profit provision. Calculate the uniform indicated change. Round to two decimal places.
Method. Determine each segment’s permissible ratio, calculate its required premium, and aggregate the required and current premiums.
For Segment C, the permissible ratio is \(100\%-20\%-5\%=75\%\). Its required premium is \(\$630{,}000/0.75=\$840{,}000\). The current premium is \$900,000.
For Segment D, the permissible ratio is \(100\%-15\%-5\%=80\%\). Its required premium is \(\$480{,}000/0.80=\$600{,}000\). The current premium is \$600,000.
The total required premium is \(\$840{,}000+\$600{,}000=\$1{,}440{,}000\), and total current premium is \(\$900{,}000+\$600{,}000=\$1{,}500{,}000\). The uniform indicated change is \(\$1{,}440{,}000/\$1{,}500{,}000-1=-0.04\), or a 4.00% decrease.
Check and trap. The adjusted aggregate loss ratio is \(\$1{,}110{,}000/\$1{,}500{,}000=74\%\). A tempting but incorrect shortcut is to average the segment permissible ratios using current premium weights: \((\$900{,}000\times75\%+\$600{,}000\times80\%)/\$1{,}500{,}000=77\%\). Dividing 74% by 77% and subtracting 1 gives a 3.90% decrease, not the correct 4.00%. That shortcut fails because each segment’s losses are converted to required premium using its own denominator. The correct total required premium is \$1,440,000, which verifies the 4.00% decrease.
Common Mistakes and Practical Checks
A permissible loss ratio is tied to a segment’s premium basis and its stated provisions. Before using one, check which expenses and profit provisions are included, whether each is expressed as a percentage of premium, and whether the loss ratio uses a consistent loss measure. A percentage cannot be moved between denominators without changing its meaning.
- Using one ratio for every segment. If expense or profit provisions vary by segment, calculate a separate permissible ratio for each one.
- Adding provisions to the loss ratio. The permissible ratio is what remains after provisions: \(1-e-q\), not \(e+q\).
- Using the wrong denominator. A provision defined as a percentage of premium is subtracted as a percentage of premium. Do not interpret it as a percentage of losses.
- Averaging ratios before calculating required premium. For a single overall indication with different segment ratios, calculate segment required premiums first, then aggregate.
- Using inconsistent loss and expense bases. If the adjusted loss measure includes LAE, the permissible ratio must be constructed for that same loss measure.
These checks matter because a small difference in the permissible ratio can change an indicated rate level. Reporting the components and calculation by segment makes the result auditable and helps explain why a ratio appropriate for one part of a book may not be appropriate for another. This method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
For each hypothetical problem, calculate the requested result using the stated premium-based provisions.
- A segment has a 17% premium-based expense provision and a 4% profit provision. What is its permissible loss ratio? (Answer: 79%)
- A segment has $500,000 of on-level premium and $360,000 of adjusted losses. Its premium-based expenses are 22% of premium and its profit provision is 3% of premium. Calculate its permissible ratio and indicated rate change, rounded to two decimal places. (Answer: Permissible loss ratio 75%; indicated change 4.00% decrease)
- Segment A has $400,000 of current premium, $300,000 of selected losses, 20% premium-based expenses, and a 5% profit provision. Segment B has $600,000 of current premium, $510,000 of selected losses, 15% premium-based expenses, and a 5% profit provision. Calculate one uniform indicated change, rounded to two decimal places. (Answer: 3.75% increase)
- Two segments have current premiums of $900,000 and $600,000, adjusted losses of $630,000 and $480,000, and permissible ratios of 75% and 80%, respectively. Calculate total required premium and the uniform indicated change, rounded to two decimal places. (Answer: Required premium $1,440,000; 4.00% decrease)
- Segment X has $750,000 of on-level premium, $570,000 of selected losses, a 25% premium-based expense provision, and a 4% profit provision. Segment Y has $250,000 of on-level premium, $175,000 of selected losses, a 19% expense provision, and a 4% profit provision. Calculate the overall uniform indicated change by summing segment required premiums, rounded to two decimal places. (Answer: 3.01% increase)