Setting A Priori Loss Levels In Credible Reserving

Why the A Priori Loss Level Matters

A reserving method can be applied consistently and still produce a poor estimate if its expected-loss input is on the wrong basis. An a priori loss level should represent the expected ultimate losses for the same exposure, coverage, rate level, time period, portfolio mix, and gross-or-net basis as the reserve being estimated. It is an expected value selected from assumptions and external information, not a substitute name for the losses already observed.

This distinction matters especially for immature accident periods. The observed portion of losses may be small, while the expected ultimate remains sensitive to the selected loss level. Rate changes, claims inflation, and changes in the mix of insured risks can each affect that prior, but they do not affect it in the same way. A rate change alters premium; claims trends alter expected loss cost; and a mix change alters the weighted average loss cost.

Definition: An a priori expected ultimate loss estimate, denoted here by $B$, is the expected total ultimate loss for a specified portfolio and accident period, determined before giving weight to that period’s observed loss experience. It must be stated on a defined basis, including coverage, exposure, valuation or trend date, portfolio mix, and whether losses are gross or net of reinsurance.

A useful starting point is expected loss cost per exposure unit. If a portfolio has $e$ exposure units and the expected loss cost per unit is $m$, then the expected ultimate loss level is $B=em$. This calculation is only meaningful if $m$ and $e$ refer to compatible exposure definitions and the same portfolio composition. For example, a policy-year exposure count should not be multiplied by a loss cost stated per vehicle-year unless the conversion is justified.

Build the Prior on a Consistent Basis

A sound selection usually proceeds in a deliberate order: define the target portfolio and ultimate loss basis; establish a base expected loss cost or loss ratio; adjust the relevant data for rate level; project loss costs for claims trend; and apply the target exposures and mix. The exact order can vary when the data are segmented, but each adjustment should be applied once and only to the quantity it is intended to change.

Fix the target basis.
Specify accident period, coverage, exposure unit, gross-or-net treatment, and target portfolio mix. Identify the date to which loss costs are projected.

Establish the starting level.
Use a suitable expected loss cost per exposure or an expected loss ratio applied to premium expressed at the selected rate level. State the assumptions supporting the choice.

Adjust for rate level and trend.
On-level premium when the starting point is premium-based. Project expected losses for frequency and severity trend from the appropriate average accident date to the target period.

Apply target exposure and mix.
Multiply by the matching exposure base, or calculate class-level projected losses and sum them using target class exposures.

Use the prior in the reserving method.
Check that the prior and observed losses share a basis before combining them through a reserving or credibility formula.

For a premium-based selection, suppose historical earned premium $P_k$ was written at rate level $r_k$, where $r_k$ is the historical rate level as a fraction of the target level. On-level premium is $P_k/r_k$. If a selected expected loss ratio $q$ is applicable to that on-level premium, the corresponding expected loss level is $qP_k/r_k$. This assumes the loss ratio is appropriate for the target coverage and risk composition and that the rate change itself changes premium rather than expected claim cost.

For loss-cost trend, if the expected claim frequency trend per year is $g_f$ and expected severity trend per year is $g_s$, a standard multiplicative projection over $t$ years is $\big[(1+g_f)(1+g_s)\big]^t$. This assumes the stated annual trends apply over the full interval and that frequency and severity trends are modeled separately. Do not apply an additional aggregate loss trend on top of this factor unless it represents a distinct, justified adjustment.

Derivation: Let $N$ be claim count per exposure unit and let individual claim amounts have mean $\mu$. If claim amounts are independent of claim count, expected loss per exposure is $E[N]\mu$: conditional on $N=n$, expected total loss is $n\mu$, so averaging over $N$ gives $E[N]\mu$. If frequency changes by factor $1+g_f$ and severity changes by factor $1+g_s$, the new expected loss cost is therefore the old cost multiplied by $(1+g_f)(1+g_s)$. Over $t$ years of constant annual trends, the factor compounds to $\big[(1+g_f)(1+g_s)\big]^t$, rather than adding the two rates or applying either trend twice.

Worked Examples

Worked Example: Projecting a Per-Exposure Prior

A hypothetical insurer is setting a gross expected ultimate for 1,000 property-year exposures. Its selected base expected loss cost is $400 per property-year at the base date, for the same coverage and portfolio mix as the target period. Assume annual claim frequency trend of $-2\%$, annual severity trend of $+7\%$, and a two-year interval from the base date to the target period. There are no rate-level or exposure-definition changes. Calculate the prior ultimate.

Method. Project the loss cost multiplicatively by frequency and severity trend, then multiply by the target exposure. The assumed frequency and severity trends are constant for both years.

The combined one-year factor is $0.98(1.07)=1.0486$. For two years, the factor is $1.0486^2=1.09956196$. Thus, the projected expected loss cost per property-year is $400(1.09956196)=\$439.824784$.

The prior ultimate is $1{,}000(\$439.824784)=\$439{,}824.784$, or $439,824.78 to the nearest cent. As a check, the result exceeds the base total of $400,000 because the severity trend more than offsets the frequency decline.

Common mistake avoided. Adding the annual trends gives $ -2\%+7\%=5\%$, which is not the specified multiplicative model. It would produce $400(1.05)^2(1{,}000)=\$441{,}000$, not the correct $439,824.78. The trend factors must be multiplied, then compounded for two years.

Worked Example: On-Level Premium Before Loss Trend

A hypothetical liability portfolio has $600,000 of earned premium at a historical rate level equal to 90% of the target rate level, and $400,000 at a rate level equal to 105% of the target. The selected expected loss ratio is 62% on the target rate basis. Assume the target losses are three years beyond the base date, with annual severity trend of $+4\%$ and annual frequency trend of $-1\%$. The portfolio mix and coverage are otherwise unchanged. Calculate the expected ultimate loss prior.

Method. First on-level each premium segment to the target rate level. Apply the expected loss ratio to the combined on-level premium, then trend the resulting loss estimate. This assumes the selected loss ratio is valid for each segment after on-leveling.

The first segment’s on-level premium is $\$600{,}000/0.90=\$666{,}666.67$. The second segment’s on-level premium is $\$400{,}000/1.05=\$380{,}952.38$. Combined on-level premium is $\$666{,}666.67+\$380{,}952.38=\$1{,}047{,}619.05$.

Expected losses at the base date are $\$1{,}047{,}619.05(0.62)=\$649{,}523.81$. The combined annual trend factor is $1.04(0.99)=1.0296$. Over three years, this becomes $1.0296^3=1.091454414336$. The projected prior is $\$649{,}523.81(1.091454414336)=\$708{,}925.63$, rounded to the nearest cent.

As a reasonableness check, the prior exceeds the base expected loss because the combined annual trend is positive and the initial premium has been adjusted to the target rate level. Common mistake avoided: multiplying the historical premium by the rate-level factor instead of dividing. A historical rate level of 90% of target means the same exposure would have generated more premium at target rates, so the conversion is $\$600{,}000/0.90$, not $\$600{,}000(0.90)$.

Worked Example: Reweighting Portfolio Mix and Using the Prior

A hypothetical insurer has 800 exposures in a target accident period: 70% are in class A and 30% in class B. At the base date, expected loss costs are $300 per exposure for A and $900 for B. Assume annual severity trend of $+4\%$ for A and $+6\%$ for B, no frequency trend, and a two-year projection. A chain-ladder ultimate estimate is $500,000. Assume 40% of ultimate losses are expected to have emerged by the valuation date, paid losses to date are $160,000, and the credibility weight on chain ladder in a chain-ladder/Bornhuetter-Ferguson blend is $Z=0.35$. Calculate the adjusted prior and the blended ultimate.

Method. Project each class loss cost separately, weight by the target class proportions, and multiply by total exposure. Use that prior in the Bornhuetter-Ferguson estimate, then apply the stated credibility blend. Assume all amounts are on the same gross-or-net basis.

Class A’s projected cost is $\$300(1.04)^2=\$300(1.0816)=\$324.48$. Class B’s projected cost is $\$900(1.06)^2=\$900(1.1236)=\$1{,}011.24$. The target average cost is $0.70(\$324.48)+0.30(\$1{,}011.24)=\$227.136+\$303.372=\$530.508$ per exposure. The prior ultimate is $800(\$530.508)=\$424{,}406.40$.

The expected un-emerged proportion is $1-0.40=0.60$. Therefore the Bornhuetter-Ferguson estimate is $\$160{,}000+0.60(\$424{,}406.40)=\$160{,}000+\$254{,}643.84=\$414{,}643.84$. The blended estimate is $0.35(\$500{,}000)+0.65(\$414{,}643.84)=\$175{,}000+\$269{,}518.496=\$444{,}518.50$, rounded to the nearest cent.

The prior is below the chain-ladder estimate, and the blend is between the chain-ladder and Bornhuetter-Ferguson estimates, as expected for a weight of 0.35. Common mistake avoided: using the old 50%/50% class mix. That would give a projected average cost of $(\$324.48+\$1{,}011.24)/2=\$667.86$, overstating the appropriate prior by $(\$667.86-\$530.508)(800)=\$109{,}881.60$. The reserve prior must reflect the target mix, not an unadjusted historical mix.

Checks Before Selecting a Prior

A prior is not validated merely because its arithmetic is correct. Before using it, check the underlying assumptions and confirm that the quantity being adjusted is the right one. A premium rate change does not automatically imply an equal change in expected claim cost: the rate change affects the premium basis, while claim costs require their own trend or exposure adjustments. Conversely, when using a loss ratio to infer losses, failure to put premium on a consistent rate level can distort the inferred loss level.

Similarly, trend should begin at a date representative of the losses underlying the base estimate, often an average accident date, and end at a date appropriate for the target accident period. If class trends differ, project by class before weighting. If the target mix differs from the base mix, use target exposures or target proportions. Do not adjust both the class loss costs and the overall portfolio average for the same mix change.

In a Bornhuetter-Ferguson estimate, the prior ultimate $B$ determines the expected un-emerged losses. If the expected emerged proportion is $p$ and losses observed to date are $L$, the estimate is $L+(1-p)B$. The credibility blend with chain ladder $C$, when $Z$ is the weight on chain ladder, is $ZC+(1-Z)[L+(1-p)B]$. These formulas require the emergence proportion, observed losses, and prior to use compatible definitions and valuation bases. A more precise prior cannot repair inconsistent bases among those inputs.

Key takeaway: Select the prior for the target portfolio, not simply from the most convenient historical total. Separate premium rate-level adjustments from loss-cost trend, project class costs consistently, reweight to the target mix, and verify that the prior matches the observed losses used in the reserving method.

Check Your Understanding

For each problem, determine the requested prior or reserving estimate. Amounts are hypothetical.

A selected loss cost is $250 per exposure at the base date. Annual frequency trend is $-1\%$, annual severity trend is $+5\%$, and the target period is two years later with 600 exposures. Calculate the expected ultimate prior to the nearest cent. (Answer: $162,084.04)
Earned premium of $300,000 is at 80% of the target rate level and $500,000 is already at target. With an expected loss ratio of 60%, unchanged mix, and no loss trend, calculate expected losses. (Answer: $525,000.00)
A target portfolio has 1,000 exposures: 40% in a class with expected loss cost $200 and 60% in a class with expected loss cost $700. There is no trend. Calculate the prior ultimate. (Answer: $500,000.00)
For a period, the prior ultimate is $400,000, paid losses are $120,000, and 50% of ultimate losses are expected to have emerged. The chain-ladder estimate is $600,000 and its credibility weight is 0.25. Calculate the Bornhuetter-Ferguson estimate and the blended estimate. (Answer: Bornhuetter-Ferguson $320,000; blended estimate $390,000)
Earned premium of $420,000 is at 84% of the target rate level. The selected target-basis expected loss ratio is 65%, and there is no trend or mix change. Calculate the prior expected loss level. (Answer: $325,000.00)

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