Why a Premium-Level Mismatch Can Mislead
A pure premium indication and a loss ratio indication can appear to disagree even when they use the same projected losses. One frequent reason is that the loss ratio calculation uses earned premium at an older rate level while the pure premium calculation uses exposures and a current rate-level benchmark. The loss ratio then reflects a change in premium rates as well as the adequacy of the rates being evaluated.
The diagnostic question is not simply whether the two methods produce different numbers. It is whether they compare the same projected losses against the same rate level, exposure basis, and permissible loss provision. This comparison isolates a premium-level mismatch; other differences in assumptions may also cause indications to diverge.
For the calculations below, let \(L^*\) denote projected losses and loss adjustment expenses on a consistent basis, \(E\) denote the corresponding exposures, and \(q\) denote the permissible loss ratio. The permissible loss ratio is the share of premium available for losses and loss adjustment expenses under the assumptions being compared. It is treated as an input here; differences in expenses or profit provisions are outside this diagnostic.
These formulas require a common basis: the same projected losses, exposures, coverage, and permissible loss ratio. In addition, the premium in the loss ratio method must represent the current rate level, and the current average rate \(r_0\) must correspond to that premium and exposure base. If these conditions are not met, the factors need not agree—and that difference alone does not establish that either method is wrong.
Why the Indications Agree When Bases Match
The relationship follows from expressing the current-level premium as exposures multiplied by the average current rate. If \(P_{\mathrm{cur}}=E r_0\), then dividing projected losses by the premium loss provision gives the same factor as dividing projected loss cost per exposure by the current loss provision per exposure.
The second line uses \(P_{\mathrm{cur}}=E r_0\); the third divides numerator and denominator by \(E\). Thus, under the stated assumptions, the methods are algebraically equivalent. A mismatch between historical premium and current-level premium breaks the substitution: \(P_{\mathrm{hist}}\) is not generally equal to \(E r_0\).
Confirm that projected losses and exposures refer to the same coverage, experience period, and projection basis.
Apply the appropriate on-level adjustment to historical earned premium before using it in the loss ratio indication.
Check that the loss provision allowed in each method is based on the same expense and profit assumptions.
Verify that current-level premium divided by the matching exposures equals the average rate used in the pure premium comparison.
Worked Comparisons
Worked Example: Historical Premium Makes the Loss Ratio Indication Look Too High
A hypothetical insurer has 1,000 earned exposures. Historical earned premium is $400,000, and the on-level factor to current rates is 1.25. Projected losses and loss adjustment expenses are $300,000, and the permissible loss ratio is 0.60. Compare the correctly calculated loss ratio and pure premium indication factors, then show the effect of using historical premium without adjustment.
Method and assumptions. This is a deterministic comparison using the same projected losses, exposures, and permissible loss ratio in both methods. No probability distribution or interest assumption is needed.
Step 1: On-level the earned premium. The current-level premium is \(400{,}000 \times 1.25=\$500{,}000\). The implied current average rate is \(\$500{,}000/1{,}000=\$500\) per exposure.
Step 2: Calculate the pure premium factor. Projected pure premium is \(\$300{,}000/1{,}000=\$300\) per exposure. The current rate’s loss provision per exposure is \(0.60 \times \$500=\$300\). Therefore, the pure premium indication factor is \(\$300/\$300=1.0000\).
Step 3: Calculate the loss ratio factor on current-level premium. The available loss provision is \(0.60 \times \$500{,}000=\$300{,}000\). The loss ratio indication factor is \(\$300{,}000/\$300{,}000=1.0000\). The two methods agree.
Common mistake and correction. Using historical premium gives a raw loss ratio of \(\$300{,}000/\$400{,}000=0.75\), and \(0.75/0.60=1.25\), which appears to indicate a 25% increase. That calculation compares projected losses with premium at the old rate level. Correct the premium first: \(\$400{,}000 \times 1.25=\$500{,}000\), giving a current-level loss ratio of \(0.60\) and an indication factor of 1.0000. The apparent difference was caused by the premium basis, not by the projected loss cost.
Worked Example: Checking a Weighted Current-Level Premium
For a hypothetical book, historical earned premium is $480,000. A documented on-level factor of 1.10 applies to that earned premium. The matching experience has 900 exposures, projected losses and loss adjustment expenses of $330,000, and a permissible loss ratio of 0.625. Calculate both indication factors and quantify the false result from using historical premium.
Method and assumptions. Treat the stated on-level factor as applicable to the entire earned premium in this example. Use the same 900 exposures and projected losses in both methods.
Step 1: Establish the current premium base. On-level premium is \(\$480{,}000 \times 1.10=\$528{,}000\). Its implied current average rate is \(\$528{,}000/900=\$586.666\ldots\) per exposure.
Step 2: Calculate the pure premium indication. Projected pure premium is \(\$330{,}000/900=\$366.666\ldots\) per exposure. The loss provision at the current average rate is \(0.625 \times \$586.666\ldots=\$366.666\ldots\) per exposure. The pure premium factor is therefore 1.0000. To the nearest cent, both per-exposure amounts are $366.67.
Step 3: Calculate the loss ratio indication. The current-level premium loss provision is \(0.625 \times \$528{,}000=\$330{,}000\). Thus the loss ratio factor is \(\$330{,}000/\$330{,}000=1.0000\), matching the pure premium result.
Step 4: Identify the mismatch effect. The unadjusted historical loss ratio is \(\$330{,}000/\$480{,}000=0.6875\). Dividing by 0.625 gives \(0.6875/0.625=1.1000\), a false 10% indication. The 1.1000 factor simply repeats the 1.10 premium-level change in the denominator. The correction is to use $528,000, not $480,000, in the loss ratio indication.
Worked Example: Diagnosing an Incorrect On-Level Factor
An analyst reports a 1.1111 loss ratio indication for a hypothetical book. The data are 1,500 exposures, $720,000 of historical earned premium, projected losses and loss adjustment expenses of $450,000, and a permissible loss ratio of 0.50. The current average rate used in a separate pure premium calculation is $600 per exposure. Determine whether the reported loss ratio factor is consistent with that rate, and find the premium-level factor required for consistency.
Method and assumptions. The $600 current average rate is assumed to apply to all 1,500 exposures. Projected losses and the permissible loss ratio match across the two calculations.
Step 1: Find the premium implied by the current rate. The current-level premium consistent with the stated rate is \(1{,}500 \times \$600=\$900{,}000\).
Step 2: Calculate the pure premium indication. Projected pure premium is \(\$450{,}000/1{,}500=\$300\) per exposure. The current rate’s loss provision is \(0.50 \times \$600=\$300\) per exposure, so the pure premium factor is 1.0000.
Step 3: Check the premium level implied by the reported factor. A loss ratio indication factor of 1.1111 implies a premium denominator of approximately \(\$450{,}000/(0.50 \times 1.1111)=\$810{,}000\). That is not the $900,000 consistent with the stated current rate and exposures.
Step 4: Calculate the required on-level factor. The required factor is \(\$900{,}000/\$720{,}000=1.25\). Using it gives a loss ratio factor of \(\$450{,}000/(0.50 \times \$900{,}000)=1.0000\). To the nearest cent, the projected pure premium and current loss provision are each $300.00 per exposure.
Common mistake and correction. Accepting the reported 1.1111 factor without checking its premium denominator would create a false conflict with the pure premium result. The reported factor uses an implied $810,000 premium base; the stated current rate requires $900,000. Reconcile the on-level premium before interpreting the difference as a rate adequacy signal.
Practical Interpretation and Exam Traps
A loss ratio is sensitive to its premium denominator. If rates have increased since the historical period, unadjusted historical premium is lower than premium at the current rate level. Dividing the same projected losses by that lower amount raises the loss ratio and can create an apparent indication increase. A rate decrease can create the opposite distortion. In either case, the pure premium calculation may seem to conflict with the loss ratio calculation even though the underlying loss cost is unchanged.
Premium on-leveling does not by itself make two indications comparable. Check that projected losses and loss adjustment expenses, exposures, coverage, and permissible loss ratio are also consistent. For example, changing the treatment of loss adjustment expenses or using a different permissible loss ratio changes the comparison for a reason other than premium level. This tutorial isolates the premium-level issue; it does not establish that two calculations with different loss or expense assumptions should agree.
| Check | What to reconcile | Diagnostic sign of a mismatch |
|---|---|---|
| Premium level | Historical earned premium versus earned premium at the current rate level | The loss ratio method uses a different rate level from the current rate benchmark |
| Exposure base | Exposures used for projected pure premium and current average rate | Current premium divided by exposures does not equal the rate used in the pure premium comparison |
| Loss basis | Projected losses and loss adjustment expenses in both methods | One calculation uses a different coverage or loss basis |
| Permissible loss ratio | The loss provision available under each method’s assumptions | Different allowable loss shares are used in the two factors |
- Do not divide projected losses by a rate-level-mismatched premium. First identify which rate level the premium represents.
- Do not apply the on-level factor to the wrong quantity. Here, it adjusts earned premium to the comparison rate level; it does not replace the projected loss calculation.
- Do not interpret every method difference as a true conflict. First test whether the inputs satisfy the conditions for equivalence.
Check Your Understanding
For each problem, use the same loss, exposure, and permissible-loss-ratio basis in both methods. Round indication factors to four decimal places where needed.
- A hypothetical portfolio has 800 exposures, historical earned premium of $320,000, an on-level factor of 1.25, projected losses and loss adjustment expenses of $240,000, and a permissible loss ratio of 0.60. Calculate the current-level premium and both indication factors. (Answer: Current-level premium = $400,000; both factors = 1.0000.)
- A hypothetical program has 1,200 exposures, historical premium of $600,000, an on-level factor of 1.08, projected losses of $405,000, and a permissible loss ratio of 0.625. Find the pure premium per exposure, the implied current average rate, and both indication factors. (Answer: Pure premium = $337.50; current average rate = $540.00; both factors = 1.0000.)
- A hypothetical book has 2,000 exposures and a current average rate of $450 per exposure. Historical premium is $750,000, projected losses are $495,000, and the permissible loss ratio is 0.55. Calculate the correctly matched indication factor and the factor obtained if historical premium is used without on-leveling. (Answer: Correct current-level premium = $900,000 and both matched factors = 1.0000; unadjusted historical-premium loss ratio factor = 1.2000.)
- For 1,000 hypothetical exposures, historical earned premium is $500,000 and projected losses are $315,000. The permissible loss ratio is 0.60, and the current average rate is $560 per exposure. Calculate the current-level premium implied by the rate, the on-level factor, and the matching indication factor. (Answer: Current-level premium = $560,000; on-level factor = 1.12; both matched factors = 0.9375.)
- A hypothetical portfolio has 1,000 exposures, current-level earned premium of $500,000, projected losses of $330,000, and a permissible loss ratio of 0.60. Calculate the matched loss ratio and pure premium indication factors. Then calculate the loss ratio factor if someone mistakenly uses historical premium of $400,000 instead. (Answer: Matched factors = 1.1000; factor using historical premium = 1.3750.)