Interpreting Conflicting Pure Premium and Loss Ratio Indications

Why Two Indications Can Disagree

A pure premium indication and a loss ratio indication can appear to recommend different rate changes even when both use the same experience period. That difference is a signal to reconcile the inputs before interpreting the result. The methods express expected losses against different denominators: exposures in a pure premium calculation and premium in a loss ratio calculation. If those denominators do not describe the same business at the same rate level, the resulting indications need not agree.

This tutorial focuses on diagnosing that conflict. The calculations below isolate losses and use a stated target loss ratio; they do not add separate expense or profit provisions. The methods and checks are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For a selected experience set, let $E$ be the number of exposures, $L_p$ the projected losses for the future rating period, $P_{OL}$ the premium from that experience set adjusted to the current rate level, and $q$ the target loss ratio. The pure premium is $L_p/E$. The loss-only rate per exposure implied by that pure premium is $(L_p/E)/q$. A loss ratio indication factor is $L_p/(qP_{OL})$, the factor applied to the current rate level under these assumptions.

A valid comparison requires more than putting two calculated percentages side by side. The selected exposures and on-level premium must represent the same policies and coverage; $P_{OL}$ must be at the current rate level; the average current rate must be computed from that same premium and exposure basis; and $L_p$ must be the same projected loss amount in both methods. The loss projection must also use consistent assumptions about such matters as loss development and trend.

A rate-level indication factor is not itself the indicated dollar rate. To compare the methods, first express the pure premium as a required rate per exposure using $q$, then compare that rate with the current average rate per exposure. This makes the denominator and rate-level assumptions visible.

Why Consistent Inputs Produce the Same Indication

Suppose $P_{OL}$ is the on-level premium for exactly the exposures counted in $E$. The current average rate per exposure is then $P_{OL}/E$. Under the stated loss-only setup, the pure premium method’s indicated rate per exposure is $(L_p/E)/q$. Dividing that indicated rate by the current average rate gives its rate-level factor:

\frac{(L_p/E)/q}{P_{OL}/E} = \frac{L_p}{qP_{OL}}.

The expression on the right is the loss ratio indication factor. The exposures cancel because both the indicated rate and current rate use the same exposure denominator. Thus, with a common $L_p$, $E$, $P_{OL}$, and $q$, the methods agree by algebra—not by coincidence.

This equivalence depends on the stated matching conditions. It does not establish that two selections are comparable merely because both are labelled “pure premium” or “loss ratio.” In particular, if the exposure count describes one book of business while the premium describes another, cancellation does not correct the mismatch.

Check the experience population.
Confirm that the exposures, premium, and losses refer to the same policies, coverage, and experience period.

Check the exposure basis.
Confirm that the pure premium denominator is the exposure volume appropriate to the selected losses, and that it matches the premium basis used for comparison.

Check the premium level.
Verify that historical premium has been adjusted to the current rate level before it is used in a current-level loss ratio indication.

Check the current average rate.
Calculate it as on-level premium divided by the matching exposures. Use that same rate as the comparison point for the pure premium result.

Check projected losses.
Use the same selected losses in both methods, including consistent development, trend, coverage, and other projection assumptions.

Diagnosing Common Sources of Conflict

Each input check addresses a different potential mismatch. An exposure selection may omit policies included in the premium, or include policies whose losses are not in the selected loss amount. A premium may be earned across several historical rate levels and therefore need on-level adjustment before it represents the current level. A current average rate calculated from the wrong exposure base then compounds the problem.

Losses can also differ between calculations. For example, a pure premium calculation may use projected ultimate losses while the loss ratio calculation uses an earlier, less-developed selection, or one may apply a trend adjustment that the other omits. A different loss amount can generate a different indication even when the exposure and premium denominators are reconciled. First make the selections consistent; only then interpret any remaining difference as a consequence of genuinely different assumptions.

Worked Example: An Exposure-Basis Mismatch

A hypothetical commercial equipment program has projected losses of $1,560,000 and a target loss ratio of 0.65. The on-level premium is $2,400,000, reported for 10,000 exposures. A preliminary pure premium calculation instead divides the same projected losses by 8,000 exposures. Diagnose the disagreement, assuming the current average rate is to be derived from the on-level premium and its matching exposure count.

Method. Calculate the loss ratio factor and the preliminary pure premium indication, then reconcile the exposure basis. The common mistake this avoids is dividing premium by one exposure count and losses by another, then treating the resulting difference as a rate need.

Loss ratio indication. Its denominator is the target loss ratio times on-level premium:

0.65 \times \$2{,}400{,}000 = \$1{,}560{,}000.

Therefore, the loss ratio factor is:

\frac{\$1{,}560{,}000}{\$1{,}560{,}000}=1.000.

Preliminary pure premium indication. Dividing by 8,000 gives:

\frac{\$1{,}560{,}000}{8{,}000}=\$195

per exposure. The implied rate per exposure at a 0.65 target loss ratio is $\$195/0.65=\$300$. But the current average rate derived from the premium and its matching 10,000 exposures is $\$2{,}400{,}000/10{,}000=\$240$. The preliminary pure premium factor is $\$300/\$240=1.250$, which conflicts with the loss ratio factor of 1.000.

Reconciliation. If the selected losses and premium both represent 10,000 exposures, the matching pure premium is $\$1{,}560{,}000/10{,}000=\$156$. Its indicated rate is $\$156/0.65=\$240$, exactly the current average rate. The corrected factor is 1.000. The original 1.250 result arose from using 8,000 exposures for losses against a premium and current rate based on 10,000; whether 8,000 is the appropriate count must be verified against the actual experience selection.

Check. The corrected indicated rate, $240 per exposure, equals the current average rate, $240 per exposure, and agrees with the loss ratio factor of 1.000.

Worked Example: Historical Premium Not On-Leveled

A hypothetical residential service policy book has 12,000 exposures and projected losses of $1,872,000. Its earned premium at historical rate levels is $2,400,000. Adjusting those premiums to the current rate level increases them by a factor of 1.20. The current average rate is $240 per exposure, and the target loss ratio is 0.65. Compare the indications with and without on-leveling.

Method. Calculate the pure premium indication and the loss ratio indication using on-level premium; then show the effect of omitting the adjustment. The common mistake this avoids is treating historical earned premium as though it had already been charged at current rates.

Pure premium result. The selected pure premium is:

\frac{\$1{,}872{,}000}{12{,}000}=\$156

per exposure. The required rate per exposure is $\$156/0.65=\$240$. Dividing by the current average rate of $240 gives a pure premium factor of 1.000.

On-level premium. Applying the stated factor to historical earned premium gives:

\$2{,}400{,}000 \times 1.20 = \$2{,}880{,}000.

The loss ratio indication using on-level premium is:

\frac{\$1{,}872{,}000}{0.65 \times \$2{,}880{,}000} = \frac{\$1{,}872{,}000}{\$1{,}872{,}000} =1.000.

If the historical premium is mistakenly used without on-leveling, the calculated factor is $\$1{,}872{,}000/(0.65\times \$2{,}400{,}000)=\$1{,}872{,}000/\$1{,}560{,}000=1.200$. That apparent 20% indication conflicts with the pure premium result because the premium denominator is at a lower rate level, not because the two methods imply different needs.

Check. The on-level premium per exposure is $\$2{,}880{,}000/12{,}000=\$240$. The target loss ratio applied to this current-level premium is $0.65\times \$2{,}880{,}000=\$1{,}872{,}000$, equal to projected losses. Both methods therefore indicate a factor of 1.000.

Worked Example: Different Projected Loss Selections

A hypothetical specialty repair program has 9,000 exposures, on-level premium of $2,250,000, current average rate of $250 per exposure, and a target loss ratio of 0.60. The pure premium calculation uses projected losses of $1,620,000. A separate loss ratio worksheet uses $1,350,000. Determine whether the disagreement can be attributed to the methods themselves.

Method. Calculate each result using its stated loss selection, then recompute the loss ratio indication with the pure premium loss selection. The common mistake this avoids is assuming that two indications use the same losses merely because they use the same experience period.

Pure premium indication. The pure premium is $\$1{,}620{,}000/9{,}000=\$180$ per exposure. At a 0.60 target loss ratio, the indicated rate is $\$180/0.60=\$300$ per exposure. The factor relative to the $250 current average rate is $\$300/\$250=1.200$.

Worksheet loss ratio indication. Its factor is:

\frac{\$1{,}350{,}000}{0.60 \times \$2{,}250{,}000} = \frac{\$1{,}350{,}000}{\$1{,}350{,}000} =1.000.

The factors conflict because the worksheet uses a different projected loss amount. Holding premium, target loss ratio, and all other assumptions fixed, the comparable loss ratio factor is:

\frac{\$1{,}620{,}000}{0.60 \times \$2{,}250{,}000} = \frac{\$1{,}620{,}000}{\$1{,}350{,}000} =1.200.

Check. The on-level premium per exposure is $\$2{,}250{,}000/9{,}000=\$250$. Applying the pure premium factor of 1.200 gives an indicated rate of $\$250\times1.200=\$300$, matching the pure premium result. The discrepancy is in loss selection, not in the algebra of the methods. The two loss selections would need to be reconciled—for example, by checking whether they use the same development and trend assumptions—before choosing between them.

Common Exam Traps and Practical Checks

The main trap is to interpret a conflicting percentage immediately as evidence that one method is preferable. A difference may instead be caused by a denominator or selection mismatch. Keep the calculation auditable: write down which exposures, premium, current rate, target loss ratio, and projected losses enter each method. Then substitute a common set of inputs and see whether the disagreement disappears.

Mixing exposure populations. Verify that exposure counts match the policies represented by the selected losses and premium.
Skipping on-leveling. Historical premium at earlier rate levels is not current-level premium. Use the applicable premium adjustment before calculating a current-level loss ratio indication.
Using an inconsistent current rate. If current average rate is needed for a factor comparison, calculate it as matching on-level premium divided by matching exposures.
Mixing loss selections. Check that both calculations use the same projected loss amount and compatible coverage, development, and trend assumptions.
Comparing unlike quantities. A pure premium is dollars per exposure; a rate-level indication is a factor. Convert the pure premium to an indicated rate per exposure and compare it with the current average rate.

When all inputs are consistent, the identity $(L_p/E)/q$ divided by $P_{OL}/E$ equals $L_p/(qP_{OL})$. When the indications differ, that identity provides a diagnostic framework: identify which quantity is not shared, confirm its basis, and correct the mismatch before drawing a rate conclusion.

Check Your Understanding

For each problem, reconcile the stated data before interpreting the indication. Dollar amounts are hypothetical.

A program has projected losses of $840,000, target loss ratio 0.70, and on-level premium of $1,200,000. The premium represents 6,000 exposures, but the pure premium worksheet uses 5,000. The current average rate based on the premium is $200. What is the loss ratio factor, and what factor results after putting the pure premium calculation on the matching 6,000-exposure basis? (Answer: Both factors are 1.000.)
A book has 7,000 exposures, projected losses of $975,000, historical earned premium of $1,250,000, and an on-level factor of 1.12. The current average rate is $200 per exposure and the target loss ratio is 0.65. What common factor do the two methods indicate when premium is on-leveled? (Answer: $1.0714$, or approximately a 7.14% increase.)
A hypothetical pool has 4,000 exposures, on-level premium of $1,000,000, current average rate of $250, and target loss ratio of 0.625. The pure premium worksheet uses projected losses of $700,000, while the loss ratio worksheet uses $625,000. Calculate the pure premium factor and the loss ratio factor using the $625,000 selection. What loss ratio factor results if the losses are reconciled to $700,000? (Answer: Pure premium factor $1.120$; worksheet loss ratio factor $1.000$; reconciled loss ratio factor $1.120$.)
A selected historical loss amount is $1,080,000 and is to be trended by a factor of 1.10. There are 6,000 exposures, on-level premium is $1,800,000, and the target loss ratio is 0.66. What projected losses and pure premium result, and what common indication factor follows? (Answer: Projected losses $1,188,000; pure premium $198 per exposure; factor $1.000$.)
A comparison uses the same projected losses of $1,440,000 and target loss ratio of 0.60. The pure premium basis has 8,000 exposures, while the on-level premium of $2,000,000 represents only 7,500 exposures. What pure premium rate and loss ratio factor are calculated as stated, and why is their comparison not yet valid? (Answer: Pure premium $180 per exposure and indicated rate $300; loss ratio factor $1.200$. The comparison is not valid until the exposures and premium are reconciled to a common business basis.)

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