Why the Loss Basis Must Match the Permissible Ratio
A loss ratio indication compares an adjusted loss ratio with a permissible loss ratio. That comparison is meaningful only when both ratios refer to the same costs. If the actual loss ratio includes loss adjustment expenses (LAE), the permissible ratio must allow for those expenses too. If the actual loss ratio excludes LAE, the permissible ratio must account for LAE elsewhere, rather than leaving its premium provision available for losses.
LAE are expenses associated with investigating, evaluating, and settling claims. For a ratemaking indication, an actuary may consider losses alone or losses and LAE together. Either basis can be appropriate for a stated purpose. The key is not to mix the bases. This tutorial assumes that premium and loss amounts have already been adjusted to a common experience and rate level as appropriate; it focuses on the treatment of LAE in the indication.
For example, if the permissible share of premium for losses and LAE together is 70%, that 70% is not automatically the permissible ratio for losses alone. If a separate 8% of premium is provided for LAE, the permissible ratio for losses alone is 62%. The 8% provision must not be counted once as an LAE allowance and again as part of the loss allowance.
The Indication on a Consistent Basis
Let \(P\) be adjusted on-level premium and \(C_B\) be adjusted costs on a specified basis \(B\). The actual ratio on that basis is \(C_B/P\). Let \(q_B\) be the permissible loss ratio for the same basis. Under the loss ratio indication method, the indicated rate factor is the actual ratio divided by the permissible ratio.
The indicated percentage rate change is the factor minus 1. A factor above 1 indicates an increase; a factor below 1 indicates a decrease. This calculation assumes that the adjusted premium and costs represent the intended prospective level and that the supplied permissible ratio is appropriate for the selected basis. If the permissible ratio is not appropriate, the arithmetic may be correct while the indication is not.
The relationship follows directly from the premium needed to support the costs. If the relevant costs are \(C_B\), and those costs may use a fraction \(q_B\) of premium, then the premium required is \(C_B/q_B\). Dividing that required premium by the current adjusted premium gives the rate factor:
This derivation also explains the matching requirement: the numerator \(C_B\) and the permitted share \(q_B\) must describe the same cost set. Otherwise the calculated premium does not support the costs actually placed in the numerator.
How to Check the Basis Before Calculating
Check whether the adjusted amount includes losses only or losses plus LAE. Do not infer the basis from a label such as “loss ratio” without checking what is included.
Determine which share of premium is available for the costs in the numerator and which provisions are allocated to other costs, including LAE if excluded from the numerator.
Divide the adjusted costs by adjusted premium, then divide that ratio by the matching permissible ratio.
Convert the factor to a percentage change and verify that no component was omitted or counted twice.
The permissible ratios in the examples are supplied or derived from stated provisions. In practice, the actuary must confirm that the provisions match the intended indication. The examples are hypothetical and deterministic; no interest rate, mortality assumption, or probability distribution is used.
Worked Example: Indication Including Losses and LAE
A hypothetical book has $2,400,000 of adjusted on-level premium, $1,320,000 of adjusted losses, and $240,000 of adjusted LAE. The permissible share of premium for losses and LAE together is 70%. Calculate the indicated rate change on a losses-and-LAE basis.
Method. Use the combined cost amount in the actual ratio and the combined permissible loss ratio. This avoids excluding LAE from only one side of the comparison.
Step 1: Combine losses and LAE. The adjusted cost amount is $1,320,000 + $240,000 = $1,560,000.
Step 2: Calculate the actual combined ratio. Divide the combined amount by premium: $1,560,000 / $2,400,000 = 0.65, or 65%.
Step 3: Calculate the indicated factor. Divide the actual ratio by the permissible ratio: 0.65 / 0.70 = 0.928571.
Step 4: Convert to a rate change. The percentage change is 0.928571 − 1 = −0.071429, or a 7.1429% decrease. Rounded to two decimal places, the indicated change is a 7.14% decrease.
Check. The premium needed to support $1,560,000 at a 70% permissible ratio is $1,560,000 / 0.70 = $2,228,571. Dividing this by $2,400,000 gives the same factor, approximately 0.928571. The common mistake avoided here is comparing the combined loss ratio with a permissible ratio that does not include an LAE allowance.
Losses-Only Indications and LAE Provisions
A losses-only indication is consistent when the permissible ratio is the share of premium available for losses after all other provisions have been allowed for. In particular, if LAE are excluded from the numerator, their provision must be excluded from the loss-only permissible ratio. That adjustment may be supplied directly or calculated from the total permissible ratio and the stated LAE provision.
This expression applies when the stated combined permissible ratio includes a distinct LAE provision expressed as a percentage of premium. It is not a universal shortcut for every expense structure. If LAE are treated differently, use the permissible ratio specifically developed for the losses-only basis.
Worked Example: Losses-Only Basis With an LAE Provision
A hypothetical segment has $1,000,000 of adjusted on-level premium, $680,000 of adjusted losses, and $120,000 of adjusted LAE. The combined permissible ratio for losses and LAE is 70%, and the indicated LAE provision is 8% of premium. Calculate (a) the indication on a losses-only basis and (b) the indication on a combined basis. Explain why the results differ.
Method. For (a), remove the LAE provision from the combined permissible ratio and use losses alone in the actual ratio. For (b), include losses and LAE in the actual ratio and use the combined permissible ratio.
(a) Losses-only calculation. The losses-only permissible ratio is 0.70 − 0.08 = 0.62, or 62%. The actual losses-only ratio is $680,000 / $1,000,000 = 0.68, or 68%. The indicated factor is 0.68 / 0.62 = 1.096774. The indicated change is 1.096774 − 1 = 0.096774, or a 9.68% increase rounded to two decimal places.
(b) Combined calculation. The combined adjusted cost is $680,000 + $120,000 = $800,000. Its actual ratio is $800,000 / $1,000,000 = 0.80, or 80%. The combined factor is 0.80 / 0.70 = 1.142857, giving a 14.29% increase rounded to two decimal places.
Interpretation and check. The observed LAE ratio is $120,000 / $1,000,000 = 12%, while the LAE provision used in the losses-only calculation is 8%. The combined indication responds to that higher observed LAE amount; the losses-only indication does not. The two answers differ because they answer different questions about which costs the indication is intended to reflect. They should not be substituted for one another without a reasoned choice of basis.
A common mistake is to use the 68% losses-only actual ratio with the 70% combined permissible ratio. That produces 0.68 / 0.70 = 0.971429, or a 2.86% decrease. It is inconsistent: the numerator excludes LAE, but the denominator still leaves the LAE provision available to losses. The correctly matched losses-only calculation is 0.68 / 0.62 = 1.096774.
Reconciliation and the Effect of Inconsistency
When the observed LAE ratio agrees with the LAE provision used to derive the losses-only permissible ratio, the losses-only and combined approaches can reconcile. The next example shows this check. It also illustrates how excluding LAE from the numerator while retaining a combined permissible ratio can materially change the result.
Worked Example: Reconciling Two Consistent Bases
For two hypothetical experience years, adjusted on-level premiums are $2,000,000 and $3,000,000. Adjusted losses are $1,100,000 and $1,500,000, and adjusted LAE are $200,000 and $450,000. The combined permissible ratio is 65%, including a 13% LAE provision. Calculate the indications using both the combined and losses-only bases.
Method. Aggregate the adjusted amounts on each basis. Use the 65% permissible ratio for combined costs and subtract the 13% LAE provision to obtain the losses-only permissible ratio.
Combined basis. Total premium is $2,000,000 + $3,000,000 = $5,000,000. Total losses and LAE are $1,100,000 + $1,500,000 + $200,000 + $450,000 = $3,250,000. The combined actual ratio is $3,250,000 / $5,000,000 = 0.65. The factor is 0.65 / 0.65 = 1.0000, so the indicated change is 0%.
Losses-only basis. Total losses are $1,100,000 + $1,500,000 = $2,600,000. The losses-only permissible ratio is 0.65 − 0.13 = 0.52. The actual losses-only ratio is $2,600,000 / $5,000,000 = 0.52. The factor is 0.52 / 0.52 = 1.0000, again giving a 0% indicated change.
Check and trap. The actual LAE ratio is ($200,000 + $450,000) / $5,000,000 = 0.13, exactly matching the 13% provision. If losses alone were divided by the combined 65% permissible ratio, the factor would be 0.52 / 0.65 = 0.80, falsely indicating a 20% decrease. The matched calculations reconcile because the observed LAE ratio equals its provision; the mismatched calculation incorrectly treats the premium reserved for LAE as available to pay losses.
Common Errors and Practical Checks
- Omitting LAE from only one side. Including LAE in actual costs but using a losses-only permissible ratio, or excluding LAE from costs but using a combined permissible ratio, breaks the comparison.
- Counting an LAE provision twice. If the combined permissible ratio already includes the LAE allowance, do not subtract that allowance again when the numerator also includes LAE.
- Assuming alternative bases must give the same result. They reconcile when their provisions and observed cost proportions are aligned. If actual LAE differs from the provision, an indication that includes LAE may differ from one that excludes it.
- Using inconsistent adjusted amounts. Premium, losses, and LAE must be on the intended common basis. For example, including developed losses but undeveloped LAE can undermine an otherwise correctly structured ratio.
Before finalizing an indication, write down the components included in the numerator and state what the permissible ratio allows for. Then verify that the same costs appear on both sides of the comparison, and that each other provision is included exactly once. This basis check is essential whether the indication is losses-only or includes losses and LAE.
Check Your Understanding
For each hypothetical situation, calculate the requested indication and check that the loss basis matches the permissible ratio.
- A book has $800,000 of adjusted premium, $420,000 of adjusted losses, and $60,000 of adjusted LAE. The permissible ratio for losses and LAE is 65%. Calculate the combined indicated rate change. (Answer: \(480{,}000/800{,}000=60\%\); factor \(=0.60/0.65=0.9231\), a 7.69% decrease.)
- A segment has $2,000,000 of adjusted premium and $1,180,000 of adjusted losses. Its combined permissible ratio is 68%, including a 9% LAE provision. Calculate the losses-only indicated rate change. (Answer: Losses-only permissible ratio \(=68\%-9\%=59\%\); actual ratio \(=59\%\); factor \(=1.0000\), so 0%.)
- Two experience years have adjusted premiums of $1,200,000 and $1,800,000; losses of $780,000 and $1,080,000; and LAE of $120,000 and $270,000, respectively. The combined permissible ratio is 68%. Calculate the aggregate combined indication. (Answer: Total premium \(=\$3{,}000{,}000\); total losses and LAE \(=\$2{,}250{,}000\); actual ratio \(=75\%\); factor \(=0.75/0.68=1.1029\), a 10.29% increase.)
- Adjusted premium is $1,000,000, adjusted losses are $620,000, and adjusted LAE are $100,000. The combined permissible ratio is 72%. Compare the correct combined factor with the factor from incorrectly using losses alone and the combined permissible ratio. (Answer: Correct factor \(=(620{,}000+100{,}000)/1{,}000{,}000/0.72=1.0000\); the mismatched factor is \(0.62/0.72=0.8611\), falsely suggesting a 13.89% decrease.)