Deriving When Pure Premium and Loss Ratio Indications Agree

Why Compare the Two Indication Methods?

Pure premium and loss ratio methods arrange the same ratemaking information in different ways. The pure premium method starts with projected losses per exposure and converts that loss cost into a rate. The loss ratio method compares projected losses with premium and measures the result against a permissible loss ratio. If the methods use the same projected losses, exposure base, premium level, and provisions, their indicated rate changes should agree. The algebra makes both the equivalence and its conditions explicit.

The comparison is useful as a consistency check, not as a claim that the methods always produce the same answer. A disagreement can arise when, for example, premium is not adjusted to the rate level used as the comparison base, losses and exposures are projected on different bases, or the expense assumptions do not match. This tutorial isolates the assumptions needed for an exact agreement.

Definition: Let $L$ be the selected losses for a common experience basis, projected to the same cost and coverage level used for the indication. Let $E$ be the corresponding exposure volume, and let $P$ be the premium for those exposures adjusted to the current rate level. The observed pure premium is $L/E$, the current average premium per exposure is $P/E$, and the observed loss ratio is $L/P$. Let $q$ be the permissible loss ratio: the share of premium available for losses under the expense and profit assumptions used in both methods.

For the basic equivalence, assume the same loss and exposure totals are used in both methods; premium is on-level to the current rates; and $q$ is constant for the comparison. Also assume that non-loss provisions are expressed as a consistent percentage of premium, with no separate fixed dollar expense per exposure. These conditions make the target rate per exposure equal to the projected loss cost divided by $q$. If a method includes fixed expenses or uses a different provision structure, the simple formulas below do not necessarily apply without modifying the model.

The Two Indication Factors

The pure premium method first calculates projected losses per exposure. Because $q$ of each premium dollar is available for losses, the indicated premium per exposure is the pure premium divided by $q$. To obtain a rate level factor, compare that indicated premium with the current average premium per exposure.

\text{Pure premium indication factor} = \frac{(L/E)/q}{P/E}

The loss ratio method calculates the loss ratio $L/P$, then divides it by the permissible loss ratio. This gives the factor by which the current rate level would need to change under those assumptions.

\text{Loss ratio indication factor} = \frac{L/P}{q}

An indication factor of 1.00 means no rate level change; a factor above 1.00 indicates an increase, and a factor below 1.00 indicates a decrease. The percentage change is the factor minus 1, expressed as a percentage.

Deriving the Agreement

The equivalence follows by simplifying the pure premium factor. The exposure denominator appears in both the observed loss cost and the current average premium per exposure, so it cancels. What remains is precisely the loss ratio divided by the permissible loss ratio.

Write the pure premium factor.
Use projected losses per exposure for the indicated rate and on-level premium per exposure for the current rate.

Divide by the current average premium.
The pure premium indication factor is $\frac{(L/E)/q}{P/E}$.

Cancel the common exposure base.
Since $E$ is the same positive exposure total in both terms, $\frac{(L/E)/q}{P/E}=\frac{L}{Eq}\times\frac{E}{P}=\frac{L}{Pq}$.

Recognize the loss ratio factor.
Rearranging gives $\frac{L}{Pq}=\frac{L/P}{q}$, the loss ratio indication factor.

The proof depends on a shared positive exposure total, a shared loss amount $L$, and a shared premium amount $P$ at the current rate level. It also requires the same $q$ in both calculations. If any of these inputs differs, the cancellation or the final identification may no longer hold.

Theorem: Under the stated assumptions, the pure premium and loss ratio indication factors are equal: $$ \frac{(L/E)/q}{P/E} = \frac{L/P}{q} $$ The equivalence is an algebraic identity for consistent inputs; it is not a guarantee that differently constructed indications will agree.

Worked Example: Comparing Both Factors on One Exposure Basis

A hypothetical equipment policy portfolio has 15,000 earned exposure units. Projected losses on those units are $1,350,000. Premium for the same units, adjusted to the current rate level, is $1,650,000. The permissible loss ratio is 0.75. Calculate the pure premium and loss ratio indications, and verify that they agree. Assume there are no separate fixed dollar expenses per exposure.

Method. Calculate both indication factors using the same projected loss total, exposure total, on-level premium, and permissible loss ratio.

Pure premium calculation. First calculate the projected loss cost:

\frac{L}{E} = \frac{\$1{,}350{,}000}{15{,}000} = \$90.00\text{ per exposure}

The indicated premium per exposure is the loss cost divided by the permissible loss ratio:

\frac{\$90.00}{0.75} = \$120.00\text{ per exposure}

The current average premium per exposure is:

\frac{P}{E} = \frac{\$1{,}650{,}000}{15{,}000} = \$110.00\text{ per exposure}

Therefore, the pure premium indication factor is $120/110=1.0909$, or a 9.09% indicated increase.

Loss ratio calculation. The observed loss ratio is:

\frac{L}{P} = \frac{\$1{,}350{,}000}{\$1{,}650{,}000} = 0.8181818

Dividing by the permissible loss ratio gives $0.8181818/0.75=1.0909$, also a 9.09% indicated increase. As a check, increasing the current average premium of $110.00 by a factor of 1.0909 gives $120.00 per exposure, matching the pure premium indicated amount. The common mistake avoided here is comparing the $90.00 loss cost directly with the $110.00 current premium; that comparison omits the permissible loss ratio.

Consistency Across Multiple Periods

The identity does not require a single experience period. Multiple periods can be combined, provided that the totals used by both methods refer to the same exposure basis and that losses and premium have been brought to the intended comparison basis. The pure premium uses total losses divided by total exposures. The loss ratio uses total losses divided by total on-level premium. Neither method should replace those aggregate ratios with an unweighted average of period-by-period ratios unless that averaging is demonstrably equivalent.

Worked Example: Aggregating Two Experience Periods

For a hypothetical specialty property portfolio, two periods have the following selected projected losses, exposures, and on-level premiums. Assume the selected losses and premiums are already on a common basis, and the permissible loss ratio is 0.80. Find each indication factor from the aggregate experience.

Period 1: losses are $420,000, exposures are 5,000, and on-level premium is $500,000.
Period 2: losses are $660,000, exposures are 7,000, and on-level premium is $770,000.

Method. Add the amounts first, then calculate the aggregate ratios. This ensures that both methods use the same combined experience.

Total losses are $\$420{,}000+\$660{,}000=\$1{,}080{,}000$. Total exposures are $5{,}000+7{,}000=12{,}000$. Total on-level premium is $\$500{,}000+\$770{,}000=\$1{,}270{,}000$.

The aggregate pure premium is:

\frac{\$1{,}080{,}000}{12{,}000} = \$90.00\text{ per exposure}

The indicated premium per exposure is $\$90.00/0.80=\$112.50$. Current average premium per exposure is $\$1{,}270{,}000/12{,}000=\$105.8333$. Thus, the pure premium factor is $112.50/105.8333=1.06299$, a 6.30% increase.

The aggregate loss ratio is $\$1{,}080{,}000/\$1{,}270{,}000=0.850394$. The loss ratio factor is $0.850394/0.80=1.06299$, again a 6.30% increase. As a check, $12{,}000$ exposures at $112.50 per exposure produce $1,350,000 of indicated premium, and $1,350,000 divided by the current on-level premium of $1,270,000 is 1.06299. The common mistake avoided is taking an unweighted average of the two periods’ loss ratios, which would not generally equal total losses divided by total premium.

Rate-Level and Loss-Projection Consistency

The premium in the loss ratio denominator must be at the current rate level used as the comparison base for the indicated change. Likewise, the losses in the numerator must be projected to the cost level used for the indication. Rate-level adjustment of premium and trend of losses serve different purposes: one adjusts premium for changes in rates, while the other projects losses for changes in claim costs. If one method uses adjusted premium but the other uses recorded premium, the calculations are not using the same $P$, and the derivation no longer applies.

Worked Example: Reconcile an On-Level Premium and a Loss Trend

A hypothetical small commercial portfolio has $1,200,000 of selected losses before trend, 14,000 exposure units, and $1,440,000 of recorded earned premium. For the indication, apply a 5% loss trend for the selected projection period and a 1.05 premium on-level factor. Use a permissible loss ratio of 0.80. Assume the factors are supplied for this hypothetical calculation and there are no separate fixed dollar expenses per exposure. Calculate the two factors using consistent projected losses and on-level premium.

Method. Project losses and adjust premium separately, then use those results in both methods.

The trended losses are $\$1{,}200{,}000(1.05)=\$1{,}260{,}000$. The on-level premium is $\$1{,}440{,}000(1.05)=\$1{,}512{,}000$. The common exposure total remains 14,000.

The projected pure premium is $\$1{,}260{,}000/14{,}000=\$90.00$ per exposure. The indicated premium per exposure is $\$90.00/0.80=\$112.50$. Current on-level premium per exposure is $\$1{,}512{,}000/14{,}000=\$108.00$. Therefore, the pure premium factor is $112.50/108.00=1.04167$, an indicated increase of 4.17%.

The loss ratio method gives $\$1{,}260{,}000/\$1{,}512{,}000=0.833333$. Dividing by 0.80 gives $1.04167$, also a 4.17% increase. As a check, multiplying $108.00 by 1.04167 gives $112.50 per exposure, the indicated premium from the pure premium method. The common mistake avoided is dividing projected losses by recorded premium: that would give a loss ratio of $1{,}260{,}000/1{,}440{,}000=0.875$ and a spurious factor of $0.875/0.80=1.09375$. The error is the use of a premium denominator at the wrong rate level.

When the Identity Does Not Apply

The derivation is deliberately specific. It compares indications built from one coherent set of inputs. If a pure premium calculation uses one exposure definition while the loss ratio calculation uses a different exposure population, cancelling $E$ is not a valid reconciliation. Similarly, if one method uses losses before trend and the other uses trended losses, the two methods do not share $L$. These are not algebraic contradictions; they are different calculations.

The assumed permissible loss ratio also matters. If one method uses $q=0.75$ and the other uses $q=0.80$, their resulting indicated factors can differ even when $L$, $P$, and $E$ match. And where fixed expenses are modeled as a dollar amount per exposure, the target premium per exposure is not generally just $(L/E)/q$ for a constant $q$. A consistent formulation must account for those expenses directly rather than assuming the basic proportional-provision formula.

Use total amounts consistently. For an aggregate comparison, form $L/E$ and $L/P$ from matching totals.
Match the premium level. Use on-level premium for the rate level being tested, not an unadjusted premium amount.
Match projected loss assumptions. Use the same selected loss amount in both methods.
Match provisions. The permissible loss ratio must represent the same expense and profit assumptions.

The practical value of the derivation is diagnostic: when the inputs and assumptions match, the factors should agree exactly apart from rounding. If they do not, check the denominators, projection bases, and permissible loss ratio before interpreting the difference as a substantive signal.

Check Your Understanding

For each hypothetical case, use the stated consistent inputs to find the indication factor. Round factors to four decimal places and percentage changes to two decimal places where applicable.

A portfolio has projected losses of $540,000, 6,000 exposures, on-level premium of $720,000, and a permissible loss ratio of 0.75. Find the pure premium factor and verify it with the loss ratio method. (Answer: Both factors are 1.0000, or a 0.00% indicated change.)
A hypothetical liability book has projected losses of $936,000, 12,000 exposures, on-level premium of $1,440,000, and a permissible loss ratio of 0.80. Find the indicated factor and percentage change using either method. (Answer: 0.8125, or a 18.75% indicated decrease.)
Selected losses of $800,000 are projected with a 10% trend factor. The matching exposure total is 8,000, on-level premium is $1,100,000, and the permissible loss ratio is 0.80. Find the shared indication factor after trending losses. (Answer: Trended losses are $880,000; both factors are 1.0000, or a 0.00% indicated change.)
A hypothetical portfolio has projected losses of $900,000, 10,000 exposures, and recorded earned premium of $1,200,000. Apply a 1.10 on-level factor to premium, and use a permissible loss ratio of 0.75. Find the indication factor. (Answer: On-level premium is $1,320,000; both methods give 0.9091, or a 9.09% indicated decrease.)
For a consistent indication, $L=\$1{,}050{,}000$, $E=15{,}000$, $P=\$1{,}500{,}000$, and $q=0.70$. Calculate the indicated premium per exposure, current average premium per exposure, and common indication factor. (Answer: $100.00 indicated premium per exposure, $100.00 current average premium per exposure, and a factor of 1.0000.)

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