Why Compare Indication Scenarios?
A rate indication depends on assumptions about projected losses, expenses, profit provisions, and the premium base used in the calculation. If reasonable alternatives exist for one or more of those inputs, they can produce different indications. Comparing the alternatives makes the consequences of those assumptions visible and helps identify which complete set is most appropriate for the prospective rating period.
A scenario is not simply a number substituted into a formula. It is a coherent package of assumptions. For example, a scenario might combine a particular loss trend with a projection period, a treatment of fixed expenses, and a profit provision. The selection question is whether that package is relevant and consistent, not merely whether its result is the largest, smallest, or closest to another result.
The calculations in this tutorial use the following aggregate premium formulation. Let \(L\) be projected losses and loss adjustment expenses, \(F\) be fixed expenses in dollars, \(v\) be the variable expense provision as a proportion of required premium, and \(p\) be the profit and contingency provision as a proportion of required premium. The current premium at the same rate level as the indication is \(P_0\).
This formulation assumes that the dollar amounts \(L\) and \(F\) are projected for the same exposure base, while the variable expense and profit provisions are percentages of required premium. The denominator must be positive. If expenses or profit are specified on a different basis, they must be converted appropriately before using this expression. The scenarios being compared must also use the same \(P_0\), exposure base, and effective date; otherwise, their rate changes are not directly comparable.
Why the Required-Premium Formula Works
The equation follows from balancing the required premium against its uses. Required premium must fund projected losses, fixed expenses, variable expenses, and the profit and contingency provision. Variable expenses equal \(vP_{\text{req}}\), and the provision for profit and contingencies equals \(pP_{\text{req}}\), under the stated percentage assumptions. Thus:
Move the two premium-based provisions to the left and factor out required premium:
When \(1-v-p>0\), division gives \(P_{\text{req}}=(L+F)/(1-v-p)\). The indicated rate change then compares that required premium with the current premium for the same business basis. This derivation also explains a frequent error: applying the variable expense and profit percentages to \(L+F\) rather than to required premium does not satisfy the stated balance equation.
A Consistent Comparison Process
Before interpreting a difference between indications, identify what differs between the scenarios and what must remain common. A large change can arise from one assumption, several interacting assumptions, or an accidental change in the comparison basis. The following sequence keeps the comparison interpretable.
Record each scenario’s projected losses, expenses, profit provision, and the assumptions that produced those inputs. Avoid changing an input without noting which scenario it belongs to.
Confirm the same current premium, exposure base, coverage, effective date, and treatment of premium and expenses. Normalize any input that is expressed on a different basis before comparing rate changes.
For each complete scenario, calculate required premium and then divide by the common current premium. Keep intermediate values so a calculation or basis error can be detected.
Identify the assumptions driving the spread. Select the scenario whose assumptions best represent the period being priced and are mutually consistent; do not treat a numerical average as automatically justified.
Worked Scenario Comparisons
Worked Example: Selecting Between Alternative Loss Projections
A hypothetical commercial equipment program has current premium of $2,000,000 on the exposure base being reviewed. The expense and profit assumptions are the same in all scenarios: fixed expenses of $200,000, variable expenses of 18% of required premium, and a profit and contingency provision of 4% of required premium. Three loss projections are under review: Scenario A projects $1,350,000, Scenario B projects $1,400,000, and Scenario C projects $1,460,000. The analysis indicates that B uses the current, documented trend period; A relies on a trend period that ends before the latest available observations, while C extends an unusual recent quarter. Calculate all three indications and select the scenario best supported by these facts.
Method. Apply the aggregate required-premium formula to each scenario. The common denominator is \(1-0.18-0.04=0.78\), and the common current premium is $2,000,000.
Scenario A. The numerator is $1,350,000 + $200,000 = $1,550,000. Required premium is $1,550,000 / 0.78 = $1,987,179.49. The indicated change is $1,987,179.49 / $2,000,000 - 1 = -0.0064103, or -0.64% rounded to two decimal places.
Scenario B. The numerator is $1,400,000 + $200,000 = $1,600,000. Required premium is $1,600,000 / 0.78 = $2,051,282.05. The indicated change is $2,051,282.05 / $2,000,000 - 1 = 0.0256410, or +2.56% rounded to two decimal places.
Scenario C. The numerator is $1,460,000 + $200,000 = $1,660,000. Required premium is $1,660,000 / 0.78 = $2,128,205.13. The indicated change is $2,128,205.13 / $2,000,000 - 1 = 0.0641026, or +6.41% rounded to two decimal places.
Selection and check. Scenario B is the most supportable selection from the information given because its trend period is documented as current, while the stated weaknesses in A and C concern the relevance of their projection periods. The results are ordered A, B, C from lowest to highest, as expected from the increasing loss projections and common positive denominator. The common mistake this avoids is selecting the middle result just because it is between the other two: here B is selected because its assumptions are relevant, not because it is numerically intermediate.
Worked Example: Checking That Scenarios Share a Premium Basis
A hypothetical regional service insurer is considering two indications for the same prospective period. The current premium for the common exposure base is $1,200,000. Both scenarios use fixed expenses of $90,000, variable expenses of 22% of required premium, and a profit provision of 4% of required premium. Scenario A projects losses of $780,000; Scenario B projects losses of $810,000. Scenario B uses an updated trend period that includes the latest complete experience year. Calculate both changes and explain why the two required premiums must be compared with the same current premium.
Method. The common denominator is \(1-0.22-0.04=0.74\). Both scenarios use the same current premium because they refer to the same exposure base and prospective rate level.
Scenario A. The numerator is $780,000 + $90,000 = $870,000. Required premium is $870,000 / 0.74 = $1,175,675.68. The change is $1,175,675.68 / $1,200,000 - 1 = -0.0202703, or -2.03% rounded to two decimal places.
Scenario B. The numerator is $810,000 + $90,000 = $900,000. Required premium is $900,000 / 0.74 = $1,216,216.22. The change is $1,216,216.22 / $1,200,000 - 1 = 0.0135135, or +1.35% rounded to two decimal places.
Selection and check. Given the information supplied, B is more relevant because it uses the latest complete experience year; the higher indicated change follows directly from its higher projected losses. Both calculations use the same $1,200,000 denominator, so the difference is attributable to the loss assumption. The common mistake is to divide one scenario’s required premium by a different current premium. That would mix a change in the indication with a change in the comparison base and would not isolate the effect of the scenario assumptions.
Worked Example: Selecting a Coherent Mix and Expense Scenario
A hypothetical home-repair coverage book has current premium of $2,400,000. Three complete projections are available. Scenario A projects losses of $1,560,000 and fixed expenses of $240,000. Scenario B projects losses of $1,680,000 and fixed expenses of $240,000. Scenario C projects losses of $1,620,000 and fixed expenses of $270,000. Each scenario uses variable expenses of 18% of required premium and a profit provision of 4%. The exposure review indicates that A retains an outdated customer mix, B assumes an unusually high-severity mix will persist, and C reflects the expected prospective mix and the associated servicing workload. Calculate the indications and select a scenario.
Method. Use a common denominator of \(1-0.18-0.04=0.78\), but preserve each scenario’s own loss and fixed-expense assumptions. Comparing only projected losses would miss the fixed-expense difference in C.
Scenario A. The numerator is $1,560,000 + $240,000 = $1,800,000. Required premium is $1,800,000 / 0.78 = $2,307,692.31. The indicated change is $2,307,692.31 / $2,400,000 - 1 = -0.0384615, or -3.85% rounded to two decimal places.
Scenario B. The numerator is $1,680,000 + $240,000 = $1,920,000. Required premium is $1,920,000 / 0.78 = $2,461,538.46. The indicated change is $2,461,538.46 / $2,400,000 - 1 = 0.0256410, or +2.56% rounded to two decimal places.
Scenario C. The numerator is $1,620,000 + $270,000 = $1,890,000. Required premium is $1,890,000 / 0.78 = $2,423,076.92. The indicated change is $2,423,076.92 / $2,400,000 - 1 = 0.0096154, or +0.96% rounded to two decimal places.
Selection and check. Select C on the stated facts: its assumptions jointly reflect the expected mix and servicing workload. Its indication lies between A and B, but that position is not the reason for selection. As a check, C has higher losses and fixed expenses than A, so its required premium is higher; it has lower combined losses and fixed expenses than B, so its required premium is lower. The common mistake avoided is choosing a loss projection from one scenario and fixed expenses from another without establishing that the combined assumptions describe a coherent prospective situation.
Interpreting the Spread and Avoiding Traps
The spread among scenario indications is useful because it shows how strongly the result depends on the assumptions under review. It does not, by itself, establish which scenario is correct. A scenario with a low indication may be based on stale information; a high indication may extend a short-lived observation. The selection should be tied to the exposure, experience period, and expense treatment relevant to the prospective period.
A reliable comparison keeps the arithmetic and the reasoning distinct. First calculate each scenario on the same basis. Then explain which assumption changes account for the differences. Finally, select the scenario whose assumptions are most applicable and internally compatible. If scenarios differ in several ways, state those differences rather than attributing the entire indication spread to only one input.
- Do not compare different bases. Required premium amounts are not comparable rate changes unless divided by the same current premium for the same exposure and rate level.
- Do not silently combine assumptions. A loss estimate from one scenario and expense provision from another create a new scenario; they do not automatically inherit either original scenario’s rationale.
- Do not mistake precision for support. Extra decimal places cannot make an irrelevant projection period more appropriate.
- Do not average simply to split the difference. An average is a separate selection method and needs its own justification. A central result is not automatically the most relevant one.
Check Your Understanding
For each problem, calculate or assess the scenarios before checking the final answer.
- A hypothetical program has current premium of $800,000, projected losses of $510,000, fixed expenses of $70,000, variable expenses of 20% of required premium, and a profit provision of 5%. Find the indicated rate change, rounded to two decimal places. (Answer: Required premium is $773,333.33; indicated change is -3.33%.)
- A hypothetical insurer compares two projections on the same $1,500,000 current premium base. Both use fixed expenses of $150,000, variable expenses of 20%, and profit of 5%. Scenario A projects losses of $1,050,000; Scenario B projects $1,110,000. Calculate both indicated changes, rounded to two decimal places. (Answer: A is +6.67%; B is +12.00%.)
- Two internally supported scenarios produce required premiums of $1,040,000 and $1,080,000 against the same current premium of $1,000,000. Calculate each indication. Which is larger, and by how many percentage points? (Answer: The indications are +4.00% and +8.00%; the second is larger by 4.00 percentage points.)
- A scenario requires $930,000 of premium on a $1,000,000 current premium base. An analyst instead divides by a $950,000 base from a different exposure segment. Calculate the analyst’s result and the comparable indication using the correct base. (Answer: The analyst obtains -2.11%; the comparable indication is -7.00%. The analyst’s result is not comparable because the denominator changes the exposure basis.)
- Scenario X indicates +1.00% using an experience period that excludes the latest complete year. Scenario Y indicates +4.00% using the latest complete year, and the review states that this period better represents prospective experience. Which scenario is supported by the stated facts, and why should the indications not simply be averaged? (Answer: Select Y because its experience period is stated to be more representative; averaging to +2.50% is not justified without a separate rationale for combining the scenarios.)