Large Loss Effects on the Loss Ratio Indication

Why Large Losses Matter in a Loss Ratio Indication

A single severe claim can make an experience period’s losses unusually high relative to its premium. In a loss ratio indication, that claim can therefore have a visible effect on the indicated rate change. The effect is not automatically a reason to remove the claim: a rate must fund expected losses, including large losses. The relevant actuarial question is whether the observed large-loss experience is a suitable estimate of the future large-loss cost.

A large-loss adjustment is a documented way to address that question. For example, an analyst may cap individual claims at a selected amount and replace the removed excess with a provision for expected excess losses. The cap, dollar level, treatment of the excess, and premium basis must all be stated. Without those details, the adjusted loss ratio is not interpretable.

Definition: For this tutorial, a large-loss adjustment is a specified change to experience-period losses intended to reduce the influence of unusually large individual claims while preserving an appropriate estimate of expected losses. If each claim is capped at $C$, a claim $X_j$ contributes $\min(X_j,C)$ to capped losses. A documented expected excess-loss provision may then be added to represent the expected cost above the cap. The adjustment is not a universal rule that every claim above $C$ should be discarded.

Place the Adjustment in the Indication

Use losses developed and trended to the indication level, and premium on-level to the same rate level. The loss ratio method compares the experience loss ratio with the permissible loss ratio. In the simple variable-expense form, the permissible loss ratio is one minus the variable expense, profit, and contingency provisions expressed as percentages of premium. The exact provision structure depends on the indicated rate plan.

\text{Experience loss ratio}=\frac{\text{selected losses}}{\text{on-level premium}}, \qquad \text{Indicated change}= \frac{\text{experience loss ratio}}{\text{permissible loss ratio}}-1.

For a large-loss-adjusted indication, “selected losses” means the selected developed and trended losses after the stated adjustment. If the adjustment caps claims and adds an expected excess-loss provision, the numerator is capped losses plus that provision. The premium and permissible loss ratio remain on the same basis as in the unadjusted comparison unless the rating plan itself requires a change.

L_{\text{adj}}=L_{\text{all}}-E_{\text{actual}}+E_{\text{expected}}, \qquad \text{Adjusted indicated change} = \frac{L_{\text{adj}}/P_{\text{on-level}}}{\text{permissible loss ratio}}-1.

Here $L_{\text{all}}$ is the selected loss total including the full observed claims, $E_{\text{actual}}$ is the actual amount removed by the specified cap, $E_{\text{expected}}$ is the provision added for expected excess losses, and $P_{\text{on-level}}$ is on-level premium. The symbols describe an adjustment at a consistent dollar level; the actual loss excess and expected provision need not be equal.

Why the Indication Formula Works

The indication formula follows from the premium requirement. If the selected losses are $L$ and the on-level premium is $P$, the experience loss ratio is $L/P$. If the permissible loss ratio is $q$, then the premium needed to support those losses and the allowed non-loss provisions is $L/q$. Relative to the existing premium $P$, the indicated rate factor is $(L/q)/P$. Rearranging gives:

\frac{L/q}{P}=\frac{L/P}{q}, \qquad \text{indicated percentage change}=\frac{L/P}{q}-1.

Thus, an adjustment to losses changes the indicated change through the experience loss ratio. It does not by itself change the permissible loss ratio. The formula assumes the selected losses, premium, and permissible ratio are measured on compatible bases and that the chosen loss ratio method is appropriate for the indication. If the adjustment also changes the provision structure or premium basis, those changes must be reflected explicitly rather than hidden in the loss numerator.

Worked Comparisons

Worked Example: One Large Claim and an Expected Excess Provision

A hypothetical regional equipment insurer has $10,000,000 of on-level premium. Selected developed and trended losses total $8,200,000, including one $1,400,000 claim. The analyst’s documented procedure caps that claim at $1,000,000 and adds an expected excess-loss provision of $250,000. The permissible loss ratio is 65%. Calculate the indication before and after adjustment.

Method. First calculate the unadjusted aggregate loss ratio and indication. Then remove only the observed amount above the cap and add the stated expected provision.

Unadjusted calculation. The full-loss ratio is $8,200,000/10,000,000=0.82$, or 82.00%. Its indicated change is $0.82/0.65-1=1.26153846-1=0.26153846$, which is a 26.15% increase when rounded to two decimal places.

Adjusted losses. The amount above the cap is $1,400,000-1,000,000=400,000$. Adjusted losses are $8,200,000-400,000+250,000=8,050,000$. The adjusted loss ratio is $8,050,000/10,000,000=0.805$, or 80.50%.

Adjusted indication. The indicated change is $0.805/0.65-1=1.23846154-1=0.23846154$, or a 23.85% increase rounded to two decimal places. The adjustment reduces the indicated change by about 2.31 percentage points compared with the unadjusted result.

Check and trap avoided. The adjusted loss total is $150,000 below the original total, equal to the $400,000 removed less the $250,000 provision added. Removing the full $1,400,000 claim, rather than just its $400,000 excess, would incorrectly discard the claim amount up to the cap.

Worked Example: Pooling Two Experience Years

A hypothetical commercial property program has two years of on-level premium: $8,000,000 and $12,000,000. The respective selected developed and trended losses are $8,000,000 and $8,100,000. Those totals include claims of $1,300,000, $2,100,000, and $1,400,000. Under a documented $1,000,000 per-claim cap, the analyst replaces actual excess losses with a $1,500,000 expected excess-loss provision for the combined period. The permissible loss ratio is 70%. Find the aggregate indication with and without the adjustment.

Unadjusted calculation. Aggregate premium is $8,000,000+12,000,000=20,000,000$. Aggregate losses are $8,000,000+8,100,000=16,100,000$. The aggregate loss ratio is $16,100,000/20,000,000=0.805$. The indication is $0.805/0.70-1=1.15-1=0.15$, or a 15.00% increase.

Calculate actual excess. The three claims exceed the cap by $1,300,000-1,000,000=300,000$, $2,100,000-1,000,000=1,100,000$, and $1,400,000-1,000,000=400,000$. Total actual excess is $300,000+1,100,000+400,000=1,800,000$.

Adjusted calculation. Adjusted losses are $16,100,000-1,800,000+1,500,000=15,800,000$. The adjusted aggregate loss ratio is $15,800,000/20,000,000=0.79$. The adjusted indicated change is $0.79/0.70-1=1.12857143-1=0.12857143$, or a 12.86% increase rounded to two decimal places.

Check and trap avoided. The adjusted losses are $300,000 below unadjusted losses because actual excess of $1,800,000 is replaced by an expected provision of $1,500,000. Capping claims and adding no provision would produce a loss ratio of $14,300,000/20,000,000=0.715$ and an indication of $0.715/0.70-1=2.14\%$. That is a different assumption about expected losses, not the documented adjustment in this problem. Pooling ratios by dividing total losses by total premium also correctly weights the years by premium.

Worked Example: Applying the Cap at the Indication-Level Dollar Amount

A hypothetical specialty insurer has $5,000,000 of on-level premium and $3,600,000 of selected losses at the historical dollar level. A claim within those losses is $900,000 at that historical level. All losses are trended upward by 10% to the indication level. The large-loss procedure caps the claim at $950,000 at the indication level and replaces actual excess with an expected $80,000 provision at that same level. The permissible loss ratio is 64%. Calculate both indications.

Unadjusted calculation. The trended losses are $3,600,000(1.10)=3,960,000$. The unadjusted loss ratio is $3,960,000/5,000,000=0.792$. The indicated change is $0.792/0.64-1=1.2375-1=0.2375$, or a 23.75% increase.

Apply the cap consistently. The identified claim after trend is $900,000(1.10)=990,000$. Its excess over the $950,000 indication-level cap is $990,000-950,000=40,000$. Adjusted losses are $3,960,000-40,000+80,000=4,000,000$.

Adjusted indication. The adjusted loss ratio is $4,000,000/5,000,000=0.80$. The indicated change is $0.80/0.64-1=1.25-1=0.25$, or a 25.00% increase. The adjusted result is 1.25 percentage points higher than the unadjusted result because the expected provision added exceeds the actual excess removed.

Check and trap avoided. The $950,000 cap is applied at the same dollar level as the trended claim and expected provision. Equivalently, the historical-level cap would be $950,000/1.10=863,636.36$ dollars. Capping at $950,000 before trend would leave the historical $900,000 claim uncapped, then trend it to $990,000; that would fail to apply the stated indication-level cap.

Interpreting an Adjustment

A large-loss adjustment can increase or decrease an indication. It decreases the selected loss total when the actual excess removed is greater than the expected provision added; it increases the total when the provision is greater. In either case, the indicated change moves in the same direction as the adjusted loss ratio when the permissible loss ratio and premium are fixed.

Specify the basis.
State whether claim amounts are historical, developed, trended, or otherwise adjusted, and identify the level at which the cap applies.

Calculate actual excess.
For each capped claim, subtract the cap from the claim only when the claim exceeds that cap. Sum the excess amounts.

Apply the stated replacement.
Subtract actual excess and add the documented expected provision, if one is specified. Do not silently assume that the two amounts are equal.

Recalculate and compare.
Divide the resulting losses by the same on-level premium, then apply the same permissible loss ratio to obtain the adjusted indication.

The unadjusted indication answers what the selected experience implies when the full observed losses are retained. The adjusted indication answers what it implies under the stated large-loss method. Neither result is automatically superior. The choice depends on the credibility and representativeness of the observed large losses and the support for the expected provision. A cap selected merely because it produces a preferred rate change is not a sound adjustment.

Common Errors to Avoid

Removing an entire claim. Under a cap, only the amount above the cap is removed. A $1,400,000 claim capped at $1,000,000 contributes $1,000,000, not zero.
Ignoring the expected cost above the cap. Removing actual excess without adding an expected provision may understate the losses the future rate must fund. Do so only when that is the explicitly intended method.
Using inconsistent dollar levels. A cap, actual excess, and expected provision must refer to a compatible level, particularly when losses are trended.
Comparing unlike bases. Keep premium, selected losses, and the permissible loss ratio consistent between the unadjusted and adjusted calculations. If the adjustment changes the basis of another quantity, show that change.
Subtracting ratios instead of recalculating the indication. First calculate adjusted losses and the adjusted loss ratio; then divide by the permissible loss ratio and subtract one.
Taking a simple average of annual ratios. For a combined indication, aggregate selected losses and premium, or use an equivalent premium-weighted calculation.

Key takeaway: A large-loss adjustment is a stated loss-selection procedure, not an automatic deletion of severe claims. Calculate the actual excess at the selected dollar level, add any documented expected excess provision, and recompute the loss ratio indication on a consistent premium and expense basis. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, use the stated large-loss method and report the indicated change to two decimal places.

A hypothetical insurer has $4,800,000 of on-level premium and $3,700,000 of selected losses, including a $1,200,000 claim. Cap the claim at $900,000, add an expected excess-loss provision of $150,000, and use a 68% permissible loss ratio. What are the adjusted losses and indicated change? (Answer: Adjusted losses are $3,550,000; indicated change is 8.76%.)
A hypothetical program has $7,500,000 of on-level premium and $5,400,000 of losses, including two claims whose amounts above the stated cap total $800,000. Replace that excess with a $500,000 expected provision. If the permissible loss ratio is 64%, what is the adjusted loss ratio and indication? (Answer: Adjusted loss ratio is 68.00%; indicated change is 6.25%.)
A hypothetical insurer has $9,000,000 of on-level premium and $7,200,000 of selected losses. A documented cap removes $700,000 of actual excess, and the expected excess-loss provision is $400,000. With a 70% permissible loss ratio, calculate the unadjusted and adjusted indications. (Answer: Unadjusted indication is 14.29%; adjusted indication is 9.52%.)
A hypothetical book has $3,000,000 of on-level premium and $2,000,000 of historical-level losses. A claim within those losses is $600,000 at the historical level. Trend all losses by 20%; at the indication level cap that claim at $650,000 and add an expected excess-loss provision of $100,000. If the permissible loss ratio is 67.5%, what is the adjusted indication? (Answer: Adjusted losses are $2,430,000; adjusted loss ratio is 81.00%; indicated change is 20.00%.)
Two years of on-level premium are $6,000,000 and $14,000,000; their selected losses are $4,200,000 and $10,500,000. Across the combined period, a documented cap removes $1,200,000 of actual excess and an expected provision adds $800,000. With a 65% permissible loss ratio, calculate the aggregate adjusted indication. (Answer: Adjusted losses are $14,300,000; aggregate adjusted loss ratio is 71.50%; indicated change is 10.00%.)

What You'll Learn