Reinsurance Pricing Diagnostics and Common Errors

Why Pricing Diagnostics Matter

A reinsurance indication can be arithmetically correct and still be unreliable. A loss total divided by the wrong exposure base, a per-occurrence layer applied separately to individual claims, or historical losses left at old cost levels can each produce a plausible-looking but inconsistent result. Diagnostics are the checks that reveal whether the data, assumptions, treaty terms, and calculation are aligned.

The objective is not to force every indication to agree. Different methods may use different assumptions and therefore produce different estimates. The objective is to identify which differences are explainable, quantify their effect, and avoid presenting unsupported precision. The checks below are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A pricing reconciliation is a structured comparison of indications after putting their losses, exposure measures, time basis, coverage, and treaty terms on a consistent basis. A sensitivity test changes one uncertain assumption at a time, such as a tail probability or trend factor, to show how much the indicated recovery or premium depends on that assumption.

Reconcile the Exposure Basis Before Comparing Rates

An aggregate loss total is not comparable across portfolios or periods unless its exposure denominator is also understood. The exposure unit might be insured vehicle-years, payroll, insured values, policies, or another measure. Even when two indications use the same label, their coverage, limits, mix of business, and time period may differ.

For a simple indication with losses $L$ and exposure $E$, the observed loss cost per exposure unit is $L/E$. If losses are adjusted by a factor $T$ to a common cost level and the current exposure is $E_{\text{current}}$, a basic projected loss estimate is:

\text{Projected losses} = \frac{L T}{E}E_{\text{current}}.

This calculation assumes that the historical exposure unit is comparable with the current one and that the adjustment factor addresses the relevant change in loss costs. It does not automatically account for changes in coverage, deductibles, limits, business mix, or claim handling. Those differences require separate adjustments or a clear qualification of the comparison.

Worked Example: Reconciling Two Loss Indications

A hypothetical liability portfolio incurred $3,600,000 of losses over 24,000 insured vehicle-years. For a simple comparison, assume those losses require a 10% trend adjustment to the current cost level, with no other changes in coverage or business mix. The current portfolio has 30,000 insured vehicle-years. A second indication reports $5,400,000 for the current portfolio. Reconcile the indications.

Method. First calculate the historical loss cost per insured vehicle-year, then trend it and apply it to current exposure.

Historical loss cost is $ \$3,600,000/24,000=\$150$ per insured vehicle-year. Applying the 10% trend factor gives $ \$150(1.10)=\$165$ per current-level insured vehicle-year.

The projected losses at current exposure are $ \$165(30,000)=\$4,950,000$. The second indication is higher by $ \$5,400,000-\$4,950,000=\$450,000$, or $ \$450,000/\$4,950,000=0.090909\ldots$, which is 9.09% relative to the first indication.

Check. Multiplying the historical total directly by the trend factor gives $ \$3,600,000(1.10)=\$3,960,000$, but that amount still reflects only the historical exposure. Scaling it by $30,000/24,000=1.25$ gives $ \$3,960,000(1.25)=\$4,950,000$, matching the loss-cost method.

The remaining $450,000 is a reconciliation item, not proof that either indication is wrong. Check whether the second indication uses the same exposure definition, loss coverage, trend basis, and business mix. The common error avoided here is comparing aggregate losses without scaling for the exposure change.

Apply Treaty Terms to the Correct Loss Unit

A layer’s attachment and limit apply to the unit defined by the contract. That unit may be an individual loss, an occurrence, or another specified aggregation. Let $X$ be the loss amount at that treaty-defined unit, $A$ the attachment, and $L$ the layer limit. For a layer of $L$ excess of $A$, the recovery before any other contractual terms is:

Y=\min\bigl(\max(X-A,0),L\bigr).

The formula assumes that the stated loss amount is already measured on the contractual basis and that no aggregate limit, reinstatement, exclusion, or other term changes the payment. It cannot be applied claim by claim when the treaty attaches per occurrence and claims must first be combined.

Worked Example: Aggregating Claims Before Applying an Occurrence Layer

A hypothetical treaty provides $800,000 of cover excess of $400,000 per occurrence. Occurrence A has three covered claims of $300,000, $250,000, and $200,000. Occurrence B has two covered claims of $500,000 and $700,000. Assume the claims are allocated to occurrences as stated, all amounts are on the same coverage basis, and there are no other terms affecting recovery. Calculate total recovery.

Method. Aggregate covered claims within each occurrence, then apply the layer formula to each occurrence total.

Occurrence A totals $ \$300,000+\$250,000+\$200,000=\$750,000$. Its amount above attachment is $ \$750,000-\$400,000=\$350,000$. Since this is below the $800,000 limit, recovery is $350,000.

Occurrence B totals $ \$500,000+\$700,000=\$1,200,000$. Its amount above attachment is $ \$1,200,000-\$400,000=\$800,000$. Recovery is $\min(\$800,000,\$800,000)=\$800,000$.

Total recovery is $ \$350,000+\$800,000=\$1,150,000$. As a check, the recovery for each occurrence cannot exceed the $800,000 layer limit, and neither recovery exceeds that occurrence’s loss above attachment.

Common mistake. Applying the layer separately to each claim would produce $0 for all three claims in occurrence A. For occurrence B it would produce $ \$500,000-\$400,000=\$100,000$ and $ \$700,000-\$400,000=\$300,000$, or $400,000 in total. That is not the contractual calculation: the attachment is per occurrence, so claims must be aggregated first. The incorrect method understates recovery by $ \$1,150,000-\$400,000=\$750,000$.

Adjust Historical Losses Before Pricing a Layer

A historical loss amount is not necessarily at the cost level relevant to a current price. When the indicated method calls for trending, each observation should be adjusted from its own valuation or loss date to the common target date. For an annual trend rate $r$, a loss from $k$ years earlier is multiplied by $(1+r)^k$, assuming a constant annual compound trend over that period.

After adjustment, apply the treaty terms to each loss at the appropriate contractual unit. Do not trend a calculated recovery if the intended assumption is that the underlying loss amount changes before applying a fixed nominal attachment and limit: in general, changing the order changes the result.

Worked Example: Trending Losses Before Applying a Layer

A hypothetical portfolio has three equally weighted historical losses: $60,000 from two years ago, $100,000 from one year ago, and $140,000 at the current level. Trend older losses at 5% annually to the current level. The treaty is $100,000 excess of $50,000 per loss. Assume each observation is one separate treaty loss, the attachment and limit are fixed nominal amounts, and no other adjustments apply. Find the average recovery per loss.

Method. Compound the trend factor for each loss’s age, apply the per-loss layer, and average the three recoveries.

The two-year-old loss trends to $ \$60,000(1.05)^2=\$60,000(1.1025)=\$66,150$. The one-year-old loss trends to $ \$100,000(1.05)=\$105,000$. The current loss remains $140,000.

For the first loss, recovery is $\min(\$66,150-\$50,000,\$100,000)=\$16,150$. For the second, recovery is $\min(\$105,000-\$50,000,\$100,000)=\$55,000$. For the third, recovery is $\min(\$140,000-\$50,000,\$100,000)=\$90,000$.

The total recovery is $ \$16,150+\$55,000+\$90,000=\$161,150$. The average is $ \$161,150/3=\$53,716.666\ldots$, or $53,716.67 to the nearest cent.

Check. Each recovery is nonnegative, below the $100,000 layer limit, and no larger than the corresponding trended loss above $50,000. The common errors avoided are omitting the second year of compounding and reporting the total recovery as the average.

Test How Much the Tail Assumption Drives the Price

Expected layer recovery depends on the full loss distribution, including the probability assigned to severe losses. If the tail is supported by sparse or uncertain evidence, a single expected value can conceal substantial model sensitivity. A useful diagnostic is to change a tail probability or tail severity while holding the other assumptions fixed, then recompute the indicated recovery.

For a discrete distribution with possible losses $x_j$ and probabilities $p_j$, where $\sum_j p_j=1$, the expected layer recovery is the probability-weighted sum of the recoveries. This follows by partitioning outcomes into the mutually exclusive loss values: recovery equals the corresponding layer payment $Y_j$ in outcome $j$, so its mean is $\sum_j p_jY_j$.

E[Y]=\sum_j p_j\min\bigl(\max(x_j-A,0),L\bigr).

This formula is exact for the stated discrete model. It does not establish that the probabilities are credible; the sensitivity analysis is a diagnostic of the consequences of the model assumptions, not evidence that a particular tail probability is correct.

Worked Example: Sensitivity to a Rare Severe Loss

A hypothetical per-loss layer is $300,000 excess of $200,000. A pricing model assigns probability 0.90 to a $100,000 loss, 0.08 to a $300,000 loss, and 0.02 to a $1,000,000 loss. Assume these outcomes are exhaustive and mutually exclusive. A 25% load is applied to expected recovery. Calculate the indicated premium, then test a model that moves 0.01 probability from the $100,000 outcome to the $1,000,000 outcome.

Method. Calculate each layer payment, weight by its probability, and then apply the load. Repeat with the changed probabilities.

At $100,000, recovery is $0 because the loss is below the $200,000 attachment. At $300,000, recovery is $ \$300,000-\$200,000=\$100,000$. At $1,000,000, recovery is capped at the $300,000 limit.

The baseline expected recovery is $0.90(\$0)+0.08(\$100,000)+0.02(\$300,000)=\$0+\$8,000+\$6,000=\$14,000$. Applying the 25% load gives $ \$14,000(1.25)=\$17,500$.

In the sensitivity model, the probabilities are 0.89, 0.08, and 0.03; they still sum to 1. The expected recovery becomes $0.89(\$0)+0.08(\$100,000)+0.03(\$300,000)=\$8,000+\$9,000=\$17,000$. The loaded premium is $ \$17,000(1.25)=\$21,250$, an increase of $3,750, or $ \$3,750/\$17,500=21.43\%$ over the baseline premium.

Check. The expected recovery increase is $3,000, equal to the 0.01 probability shift multiplied by the difference between the severe-loss recovery and the low-loss recovery: $0.01(\$300,000-\$0)=\$3,000$. Loading that increase by 25% gives $3,750. The common mistake avoided is treating a small change in tail probability as immaterial without calculating its effect on the layer.

Reconciliation Checks to Use in Practice

A discrepancy is most useful when it is decomposed into identifiable causes. The following sequence keeps the comparison focused on the inputs that can change a reinsurance indication:

Align the exposure denominator.
Confirm the exposure unit, period, and scale. Recalculate loss cost per unit before projecting to current exposure.

Align the loss basis.
Check that the compared amounts refer to consistent coverage, loss definitions, and valuation dates. Identify adjustments rather than silently assuming equivalence.

Rebuild treaty recoveries.
Aggregate losses at the treaty-defined unit, then apply attachment, limit, and any stated contract terms in the specified order.

Isolate uncertain assumptions.
Change one trend, exposure, or tail assumption at a time and record its effect on expected recovery or premium.

Explain the residual.
After known differences are reconciled, report what remains and which assumptions or data limitations could account for it.

These checks matter because errors can compound. For example, an exposure mismatch can distort the expected loss before a layer calculation, while an incorrect loss unit can distort the recovery after that. A final total alone will not reveal which step caused the problem. Retaining intermediate values makes the indication reproducible and makes disagreements easier to resolve.

Common Errors to Avoid

Comparing totals with different exposure bases. Divide by the relevant exposure first, then scale to the target exposure.
Applying a treaty attachment at the wrong unit. Determine whether the contract applies per loss, occurrence, or another defined unit before calculating recovery.
Leaving historical losses at an old cost level. If the method requires trend, adjust each loss by the factor appropriate to its age before applying fixed treaty terms.
Using an unsupported tail estimate as if it were certain. State the assumed probabilities and severities, and show how plausible alternatives affect the result.
Failing to reconcile intermediate arithmetic. Check that probabilities sum to one, recoveries respect the layer limit, and totals and averages are not confused.

Key takeaway: A defensible reinsurance indication requires consistency from data to price: exposure and loss bases must align, historical amounts must be adjusted as intended, treaty terms must be applied to their contractual unit, and uncertain tail assumptions must be tested for their effect.

Check Your Understanding

For each problem, identify the relevant basis or assumption, show the key calculation, and compare your result with the answer.

A hypothetical portfolio has $2,700,000 of historical losses over 18,000 exposure-years. Apply a 8% trend factor and project to 20,000 current exposure-years, assuming no other changes. What is the projected loss amount? (Answer: $3,240,000)
A hypothetical occurrence layer is $500,000 excess of $250,000. Two separate occurrences have aggregate losses of $600,000 and $1,100,000. What is total recovery? (Answer: $850,000)
Three equally weighted losses are $40,000 from two years ago, $80,000 from one year ago, and $120,000 at the current level. Trend at 10% annually. A per-loss layer is $75,000 excess of $25,000. What is average recovery? (Answer: $53,800.00)
A model assigns probabilities 0.95, 0.04, and 0.01 to losses of $0, $200,000, and $800,000, respectively. A layer is $300,000 excess of $100,000. What is expected recovery? Then shift 0.005 probability from the $0 outcome to the $800,000 outcome; what is the revised expected recovery? (Answer: $7,000 initially; $8,500 revised)
A hypothetical indication has $12,000,000 of losses over 80,000 exposure units. Apply a 5% trend factor and project to 100,000 current exposure units. A second indication is $16,000,000. What is the first indication and how much higher is the second relative to the first? (Answer: $15,750,000; the second is $250,000, or 1.59%, higher)

What You'll Learn