Why Input Selection Comes Before the Calculation
An overall rate indication can be arithmetically correct and still answer the wrong question. That happens when, for example, losses are projected to one future period but premiums represent another, or when an adjustment is applied twice. The essential work is to define the target and assemble compatible inputs before dividing one amount by another.
This workshop uses a hypothetical insurance program. It focuses on selecting and reconciling inputs, rather than on developing loss development or on-leveling methods. Any selected values described as ultimate or on-level are treated as already adjusted in that stated way.
For each candidate experience period, ask what the data represent: earned or written premium, the rate level embedded in the premium, reported or ultimate losses, and the dates or cost levels at which losses are measured. Then specify which adjustments are still needed. A selected ultimate loss is not necessarily trended, and an on-level premium is not necessarily projected for exposure or mix changes.
A Practical Input-Selection Sequence
The sequence below separates choices that are often mistakenly treated as one calculation. The ledger records the basis of each input and whether an adjustment has already been made.
State the future period and the cost level to which losses are to be projected. Use this target to determine the trend interval.
Consider data maturity, relevance, and consistency. Use the same selected period or a clearly documented weighting of periods for the paired loss and premium inputs.
Identify whether premium is earned or written, whether it is at historical or current rates, and whether it represents the coverage being indicated. Apply an on-level adjustment only if the base is not already at the selected rate level.
Identify whether losses are paid, incurred, or selected ultimate losses, and whether they include the intended coverage and loss components. Apply development or trend only when the selected loss basis does not already include that adjustment.
Check the adjustment ledger for mismatched periods or duplicate adjustments. Only then apply the overall indication relationship to the selected inputs.
The same named period does not guarantee matching bases. For example, earned premium from a calendar year can be paired with accident-year losses only if the analysis deliberately defines and supports that pairing. A candidate should be able to state what each value measures, not merely label both values with a year.
Worked Example: Selecting and Weighting Three Experience Years
Worked Example: Selecting and Weighting Three Experience Years
A hypothetical program uses three experience years, with losses selected ultimate but not trended. For each year, the loss and premium amounts below are expressed per the same standardized block of 10,000 exposures. Premium has already been adjusted to the current rate level. The selected annual loss trend is 4%. The indicated rates are intended for the midpoint of 2025. Use weights of 20%, 30%, and 50% for 2022, 2023, and 2024, respectively. Select the weighted loss and premium inputs.
| Experience year | Selected ultimate loss | Current-level premium | Weight | Years to target midpoint |
|---|---|---|---|---|
| 2022 | $1.20 million | $1.80 million | 20% | 3 |
| 2023 | $1.35 million | $2.00 million | 30% | 2 |
| 2024 | $1.50 million | $2.20 million | 50% | 1 |
Step 1: Trend each loss to the target. With a constant annual trend rate \(t=0.04\), the trend factor over \(n\) years is \((1+t)^n\). The trend periods are measured from each experience-year midpoint to the target midpoint.
Step 2: Apply the stated weights to the projected losses.
Step 3: Apply the same weights to the paired premium inputs. Because all premiums are already at the current rate level, no additional on-level factor is applied.
The selected inputs are therefore $1.48801536 million of losses projected to the target and $2.06 million of current-level premium, each on the stated standardized exposure basis. The weighting is valid for this setup because the paired amounts share that basis. Weighting raw totals from periods with substantially different exposure volumes without first addressing those differences could produce a misleading selection.
Worked Example: Auditing the Adjustment Ledger
Worked Example: Auditing the Adjustment Ledger
A hypothetical insurer is reviewing one experience year. Its earned premium is $2.40 million, and the average rate level in that premium is 96% of the selected current rate level. Selected ultimate losses are $1.32 million at the experience-period cost level; they are not trended. The target cost level is two years after the experience-period midpoint, with an annual trend rate of 5%. Determine the adjusted premium and projected loss inputs. State which adjustment applies to each.
Premium adjustment. The premium is at 0.96 of the selected rate level, so the on-level factor is the ratio of selected level to embedded level:
Thus, the adjusted premium is \(2.40(1.0416667\ldots)=2.50\) million dollars. Equivalently, \(2.40/0.96=2.50\). The premium is now at the selected current rate level.
Loss adjustment. The loss is already selected ultimate, so no further development adjustment is indicated by the facts. It is not yet at the target cost level, so apply two years of trend:
The reconciled inputs are $2.50 million of on-level premium and $1.4553 million of ultimate losses trended to the target. The premium and loss amounts now have the stated rate-level and cost-level adjustments, respectively; they should not be treated as though the same factor applies to both.
Common mistake and correction. A second trend application would give \(1.4553(1.05)=1.528065\) million dollars. That is wrong because the first calculation has already trended the selected ultimate loss through the full two-year interval. The ledger should mark the loss as “ultimate: yes; trended to target: yes,” preventing a duplicate adjustment.
From Selected Inputs to an Overall Indication
A compact case can use the overall indication relationship already established in the rate-level indication framework. Suppose \(P\) is the premium base at the selected current rate level, \(L\) is projected loss, \(F\) is fixed expense for the target business, and \(v\) is the variable expense ratio. A variable expense ratio represents expenses that change proportionally with premium; fixed expenses are treated as a dollar amount in this calculation.
Why the relationship holds. Let \(I\) be the factor applied to the current-level premium \(P\). The resulting premium is \(IP\), of which \(vIP\) is used for variable expenses. The amount remaining for losses and fixed expenses is therefore \(IP-vIP=IP(1-v)\). At the indicated level, this must equal \(L+F\):
This derivation assumes the stated expenses can be represented by the simple fixed-plus-proportional model. It does not independently select losses, estimate trend, or decide whether a premium base is appropriate; those are input-selection tasks.
Worked Example: Completing a Consistent Indication
A hypothetical specialty program has selected ultimate losses of $1.62 million, which have been trended to the projection target. Its premium base is $2.50 million, already on-level to the selected current rate level and matched to the selected experience basis. Fixed expenses for the target business are $0.18 million, and the variable expense ratio is 22%. Calculate the indicated factor and percentage change. Use the exact factor in the check, and round the reported factor to four decimal places and the percentage change to two decimal places.
Step 1: Confirm the inputs. The loss input is ultimate and trended to the target; the premium input is already on-level. Therefore neither input needs another adjustment in this calculation.
Step 2: Calculate the denominator. The share of premium remaining after variable expenses is \(1-0.22=0.78\). Thus,
Step 3: Calculate the numerator and factor.
The indicated factor is \(0.9231\) to four decimal places. The percentage change is \((I-1)\times100\%=-7.6923077\ldots\%\), or a 7.69% decrease to two decimal places.
Step 4: Check by substitution. Using the unrounded factor,
The check agrees with the required $1.80 million for losses and fixed expenses. The result is conditional on the stated expense model and on the chosen inputs; it does not establish that those selections are appropriate for every hypothetical program.
Input Controls and Common Traps
A short reconciliation before calculation often catches more errors than redoing the division. In particular, distinguish the kind of adjustment from the value being adjusted: on-leveling applies to premium rate level, while loss development and loss trend apply to loss amounts for different reasons. An adjustment is not justified merely because it is common in an indication.
| Input question | Check before calculation | Typical error |
|---|---|---|
| Period and target | Are losses and premium paired on an explicitly defined experience basis, and are losses projected to the stated target? | Using a trend interval measured to a different target date. |
| Premium basis | Is the premium earned or written, and is its rate level known? | Applying an on-level factor to premium already at the selected level. |
| Loss basis | Are losses reported or ultimate, and have development and trend already been applied? | Developing an ultimate selection again, or trending twice. |
| Expense basis | Are fixed expenses dollar amounts and variable expenses a proportion of premium for the same target business? | Putting the variable expense share in the numerator instead of allowing for it in the premium share. |
The denominator trap is especially consequential. In the worked indication, the correct denominator is \(P(1-v)=2.50(0.78)=1.95\) million dollars. Using \(P\) alone would yield \(1.80/2.50=0.7200\), a different factor that fails to allow for variable expenses. Using \(P(1+v)\) would also be wrong: variable expenses reduce the premium available for losses and fixed expenses, so the relevant share is \(1-v\).
Check Your Understanding
For each original hypothetical case, determine the requested quantity. Treat any adjustment explicitly described as already completed as complete.
- A selected premium amount is $1.80 million at 90% of the current rate level. What is its on-level premium at the current level? (Answer: $2.00 million)
- Selected ultimate losses are $0.80 million at the experience-period cost level. Trend them for two years at 6% per year. (Answer: $0.89888 million)
- Two experience periods have projected losses of $1.10 million and $1.40 million and matching current-level premiums of $1.60 million and $2.00 million, all on the same standardized exposure basis. If their weights are 40% and 60%, what are the weighted loss and premium inputs? (Answer: $1.28 million of losses and $1.84 million of premium)
- A hypothetical indication has projected losses of $1.25 million, fixed expenses of $0.10 million, current-level premium of $1.70 million, and a variable expense ratio of 20%. Calculate the indicated factor to four decimal places. (Answer: \(1.35/[1.70(0.80)]=0.9926\))
- A selected loss of $1.05 million is ultimate but not trended; the target is one year later, with 4% annual trend. Premium of $1.50 million is already at the selected current rate level. Fixed expenses are $0.06 million and variable expenses are 20% of premium. Calculate the factor to four decimals, without duplicating any adjustment. (Answer: Projected loss \(=1.05(1.04)=1.092\); \(I=(1.092+0.06)/[1.50(0.80)]=0.9600\))