Reconciling Loss Ratio and Pure Premium Results

Why Two Indications May Differ

Loss ratio and pure premium methods can express the same rate need in different forms. A loss ratio indication compares projected losses with premium and a permissible loss ratio. A pure premium indication compares projected losses and expenses per exposure with the rate needed to fund them. If both calculations use the same projected losses, exposures, premium level, and expense and profit assumptions, their indicated rate changes should reconcile. A difference is a prompt to check the bases before treating it as a substantive result.

This distinction matters when reviewing a rate level indication aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. A ratio can look internally correct while using a different loss projection or expense basis from the pure premium calculation. The goal here is to put both results on a common basis and determine what accounts for any gap.

Definition: A loss ratio indication expresses projected losses relative to premium and compares that ratio with a permissible loss ratio. A pure premium indication expresses projected losses, and any expenses stated per exposure, as a required rate per exposure. An indicated rate factor is the indicated rate divided by the current rate; the indicated percentage change is that factor minus one.

For a simple comparison, let $P_0$ be the current average premium per exposure, $L$ the projected losses per exposure, $v$ the variable expense ratio, and $q$ the target profit and contingency provision as a proportion of premium. Assume there are no fixed expenses. The permissible loss ratio is $1-v-q$. The assumptions must match: $L$ must use the same projection in both methods, and the premium and loss amounts must be on compatible exposure and rate-level bases.

\text{Loss ratio}=\frac{L}{P_0}, \qquad \text{Indicated rate factor}_{LR} =\frac{L/P_0}{1-v-q}

For the pure premium method, the required rate per exposure is $L/(1-v-q)$. Dividing that rate by the current average premium gives its indicated rate factor:

\text{Indicated rate factor}_{PP} =\frac{L/(1-v-q)}{P_0}

The expressions are algebraically identical. This simple result is a useful reconciliation benchmark, not a guarantee that two real indications agree: fixed expenses, differing projections, or incompatible premium and exposure bases can change the comparison.

Why the Simple Formulas Reconcile

The derivation is direct. The loss ratio indication divides the projected loss ratio, $L/P_0$, by the permissible loss ratio, $1-v-q$. Rearranging that expression gives the projected loss per exposure divided by the product of the current premium per exposure and the permissible loss ratio. The pure premium method first divides projected losses per exposure by the permissible loss ratio to obtain the indicated rate, then divides by the current rate $P_0$. Those are the same operations:

\frac{L/P_0}{1-v-q} = \frac{L}{P_0(1-v-q)} = \frac{L/(1-v-q)}{P_0}

The equality depends on using the same $L$, $P_0$, and $1-v-q$ in both calculations. It also assumes that expenses and profit are represented as percentages of premium. If an expense is a fixed dollar amount per exposure, simply inserting it as though it were a constant percentage can break the comparison.

Worked Example: Checking Agreement With Percentage-Based Provisions

A hypothetical personal property program has a current average premium of $600 per exposure. Projected losses are $420 per exposure. Variable expenses are 20% of premium, and the target profit and contingency provision is 5% of premium. There are no fixed expenses. Calculate and compare both indications.

Method. Use the loss ratio method and pure premium method with identical projected losses and percentage-based provisions.

Loss ratio method. The projected loss ratio is $420/600=0.70$, or 70%. The permissible loss ratio is $1-0.20-0.05=0.75$, or 75%. Therefore, the indicated rate factor is $0.70/0.75=0.9333$. The indicated rate change is $0.9333-1=-0.0667$, or $-6.67\%$.

Pure premium method. The required rate per exposure is $420/0.75=\$560$. Relative to the current premium of $600, the indicated factor is $560/600=0.9333$, giving an indicated change of $-6.67\%$.

Check and trap. Both methods give the same factor because they use the same loss amount and permissible loss ratio. A common mistake is to subtract 25% from the current premium and call the result the indication: that would give $450 and a $-25\%$ change. The provision percentage is applied to the indicated premium structure through the denominator; it is not the indicated rate change.

Reconciling Fixed Expenses

A fixed expense stated as dollars per exposure does not remain a constant percentage of premium when the rate changes. If $F$ is the fixed expense per exposure, and variable expenses and the target profit provision remain proportions $v$ and $q$ of premium, the pure premium indicated rate is:

R_{PP}=\frac{L+F}{1-v-q}

This formula assumes the rate must cover projected losses $L$, fixed expense $F$, variable expenses $vR_{PP}$, and profit provision $qR_{PP}$. In other words, $R_{PP}=L+F+vR_{PP}+qR_{PP}$. Rearranging gives the stated expression. The loss ratio calculation must recognize the same fixed expense at the indicated premium level. If it does not, the methods are not using the same expense basis.

Worked Example: Reconciling a Fixed Expense Per Exposure

A hypothetical specialty program has a current average premium of $600 per exposure. Projected losses are $420 per exposure, fixed expenses are $30 per exposure, variable expenses are 20% of premium, and the target profit provision is 5% of premium. Calculate the indicated change using the pure premium method, then reconcile the loss ratio method.

Method. First include the fixed expense in the required dollars per exposure. Then express that same expense as a ratio of the indicated premium for the loss ratio calculation.

Pure premium method. The premium share available for losses and fixed expenses is $1-0.20-0.05=0.75$. The required rate per exposure is $(420+30)/0.75=450/0.75=\$600$. The indicated factor is $600/600=1.0000$, so the indicated change is 0%.

Loss ratio reconciliation. Let $r$ be the indicated rate factor, so the indicated premium is $600r$. The projected loss ratio is $420/(600r)$, and the fixed-expense ratio is $30/(600r)$. Together, those amounts must use the 75% of premium available after variable expenses and profit:

\frac{420}{600r}+\frac{30}{600r}=0.75

Thus $450/(600r)=0.75$, so $450=450r$ and $r=1.0000$. The reconciled indicated change is 0%.

Check and trap. Ignoring fixed expenses would produce a loss ratio indication of $(420/600)/0.75=0.9333$, or $-6.67\%$. That is not a competing answer under the stated assumptions; it omits the $30 fixed expense. The fixed expense ratio is $30/(600r)$, not a fixed percentage independent of the indicated rate.

When Different Loss Projections Explain the Gap

Even when each calculation is correct on its own, the methods will differ if their projected losses differ. For example, a pure premium calculation may include a loss trend that was not applied to the losses in the loss ratio calculation. The resulting gap is explainable by the different loss basis; it is not evidence that the formulas themselves disagree.

Worked Example: Isolating a Trend-Basis Difference

A hypothetical warranty portfolio has a current average premium of $1,000 per exposure. The loss ratio calculation uses projected losses of $700 per exposure. The pure premium calculation applies an additional 5% trend to that loss amount. Both calculations use a permissible loss ratio of 0.75, with no fixed expenses. Calculate the two indicated changes and quantify the source of their difference.

Method. Calculate each indication from its stated loss basis, then bridge the results by comparing the projected loss amounts.

Loss ratio indication. The loss ratio is $700/1{,}000=0.70$. The indicated factor is $0.70/0.75=0.9333$, so the indicated change is $-6.67\%$.

Pure premium indication. The trended losses are $700(1.05)=\$735$ per exposure. The indicated rate is $735/0.75=\$980$ per exposure. The factor relative to the current $1,000 premium is $980/1{,}000=0.9800$, so the indicated change is $-2.00\%$.

Reconciliation. The pure premium result uses losses 5% higher than the loss ratio result. Its factor, $0.9800$, is the loss ratio factor, $0.9333$, multiplied by 1.05: $0.9333(1.05)=0.9800$, allowing for rounding. The 4.67 percentage-point difference in indicated changes is therefore explained by the different loss projection.

Check and trap. If both methods had used $700 of projected losses, both would have indicated $-6.67\%$. Comparing the two answers without checking the trend basis could lead to the false conclusion that the methods inherently produce different indications.

A Practical Reconciliation

Before interpreting a difference, restate each indication in common terms: projected losses per exposure, current premium per exposure, variable expenses, fixed expenses, and profit or contingency provision. Confirm the units and the period represented by each amount. A loss amount in dollars per exposure cannot be compared directly with an aggregate premium unless the exposure count and corresponding premium are handled consistently.

Check the loss projection.
Confirm that both methods use the same loss amount, including consistent treatment of trend and any other projection adjustments.

Check the premium basis.
Make sure the premium in the loss ratio is earned at a rate level comparable to the current rate used to form the pure premium factor.

Check the provision basis.
Match variable expense and profit percentages, and include fixed expenses in dollars per exposure or as a consistently calculated ratio.

Convert both outputs to rate factors.
Compare indicated rates with current rates on the same basis, then subtract one to express each result as a percentage change.

Explain any remaining gap.
Identify which input or basis differs and quantify its effect before deciding whether the difference is explainable.

Other differences can arise if the loss ratio uses aggregate amounts while the pure premium calculation uses a different exposure base, or if the current average premium does not correspond to the policies underlying the projected losses. These are comparability issues. The algebraic equality applies only after the inputs have been made comparable.

Common Reconciliation Errors

Comparing rate change with premium change. An indicated rate factor compares rates on a common basis. A change in total written premium can also reflect a change in exposure count or policy mix, so it is not automatically the same percentage.
Using inconsistent provisions. A loss ratio calculation with a 75% permissible loss ratio cannot be compared directly with a pure premium denominator of 0.80. Reconcile what expenses and profit amounts each denominator includes.
Dropping fixed expenses. A fixed dollar amount per exposure affects the required rate and generally changes as a percentage of premium when the rate changes.
Applying trend only once—or twice. Check whether the projected losses already include the trend used in the other method. Applying it in one calculation but not the other creates a basis difference; applying it twice distorts the result.
Mixing aggregate and per-exposure values. Divide aggregate losses and premium by compatible exposure counts, or keep both calculations in aggregate terms. Do not combine a per-exposure loss cost with an unrelated aggregate premium.

Key takeaway: With the same projected losses, current premium basis, and premium-proportional provisions, loss ratio and pure premium methods produce the same indicated rate factor. When they differ, first reconcile the inputs—especially fixed expenses, trends, and units—before interpreting the results.

Check Your Understanding

For each problem, calculate the requested indication and check that the stated bases are used consistently.

A hypothetical fleet program has a projected loss ratio of 0.72 and a permissible loss ratio of 0.80. Calculate the indicated rate change. (Answer: indicated factor $0.9000$, or $-10.00\%$)
A hypothetical product has projected losses of $510 per exposure, variable expenses of 15% of premium, and a target profit provision of 5%. With no fixed expenses and a current average premium of $600, calculate the pure premium indicated rate and percentage change. (Answer: indicated rate $637.50 per exposure; indicated change $+6.25\%$)
A hypothetical book has projected aggregate losses of $2,400,000 and premium of $3,200,000 on the same basis. If the permissible loss ratio is 0.70, calculate the loss ratio indication. (Answer: loss ratio $0.7500$, factor $1.0714$, or $+7.14\%$)
A hypothetical program has projected losses of $480 per exposure and fixed expenses of $20 per exposure. Variable expenses are 20% of premium, the target profit provision is 4%, and the current average premium is $650. Calculate the pure premium indication, rounding the rate to cents and the percentage change to two decimal places. (Answer: indicated rate $$500/0.76=\$657.89$ per exposure; indicated change $+1.21\%$)
A hypothetical policy group has current premium of $1,000 per exposure. A loss ratio calculation uses projected losses of $640 per exposure; a pure premium calculation trends those losses by 5%. Both use a permissible loss ratio of 0.80, with no fixed expenses. Calculate both indicated changes and state the source of the difference. (Answer: loss ratio change $-20.00\%$; pure premium change $-16.00\%$; the pure premium calculation uses losses 5% higher.)

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