Why the Sequence Matters
An experience rating plan may start with a raw modification and then require one or more adjustments, minimums, or maximums. The final modification is not found by collecting all the numbers and applying them in any convenient order. Each operation acts on the result of the preceding operation, so the plan’s stated sequence is part of the calculation.
This tutorial concerns adjustments and bounds applied to the modification factor. It does not change the underlying experience calculation that produced the raw modification. All examples are hypothetical and deterministic: the plan rules and raw modification are given, and no loss distribution, interest rate, or mortality assumption is needed.
A percentage increase of \(r\) is represented by multiplying by \(1+r\); a percentage decrease of \(r\) is represented by multiplying by \(1-r\). For example, a 10% increase uses factor \(1.10\), while a 5% credit uses factor \(0.95\). The percentage is applied to the current modification unless the plan explicitly defines a different base.
A floor and a cap can be combined into one clipping operation when they apply at the same stage and \(L\le U\):
Applying the floor and then the cap gives the same result as applying the cap and then the floor under this condition. The important order effects usually arise when an adjustment occurs between a bound and another operation, or when a bound is applied before rather than after an adjustment.
A Reliable Calculation Method
Treat the plan as an ordered list of operations. Keep the intermediate result after every step; do not jump directly from the raw modification to an intuitively plausible final value.
Use the value calculated from the experience-rating formula before the additional plan adjustments in the question.
Write a floor as \(\max(L,x)\), a cap as \(\min(U,x)\), and a percentage adjustment as multiplication by its factor.
After each operation, record the new value. A bound applies to the value at its position in the sequence, not automatically to the raw or final value.
Confirm whether each bound was binding when applied, retain full precision during the calculation, and round only as instructed.
Why Rearranging Operations Can Change the Result
A multiplicative adjustment changes the value to which a floor or cap applies. For a positive adjustment factor \(a\), a cap at \(U\) applied after the adjustment gives \(\min(U,ax)\). If the cap is applied first and the adjustment second, the result is \(a\min(U,x)\). These expressions need not be equal.
For instance, if \(x=1.12\), \(a=1.10\), and \(U=1.15\), capping after the increase gives \(\min(1.15,1.232)=1.15\). Capping first and then increasing gives \(1.10\min(1.15,1.12)=1.232\). The plan’s sequence determines which calculation is correct.
The same reasoning applies to a floor. A floor set before a decrease can raise the value that is then discounted; a floor set after the decrease instead guarantees a minimum on the reduced value. The operations are not interchangeable simply because they involve the same numerical floor, cap, or percentage.
Worked Example: Floor, Increase, Then Cap
A hypothetical plan gives a raw modification of \(R=0.74\). It specifies, in order: (1) a floor of \(0.80\), (2) an 8% increase, and (3) a cap of \(1.00\). Calculate the final modification. Then show the result if the increase is incorrectly applied before the floor.
Method. Translate the increase into multiplication by \(1.08\), then apply each operation in the plan’s stated order.
Step 1: Apply the floor.
Step 2: Apply the 8% increase.
Step 3: Apply the cap.
The final modification is 0.864. The floor binds, while the cap does not. To check the order effect, increase the raw value first: \(0.74(1.08)=0.7992\). Applying the floor then gives \(\max(0.80,0.7992)=0.80\), and the cap leaves it at \(0.80\). This differs from \(0.864\). The common mistake is to apply the percentage adjustment before the floor even though the plan places the floor first.
Worked Example: A Cap Between a Debit and a Credit
A hypothetical commercial policy has a raw modification of \(1.12\). Its plan specifies a 10% debit, then a cap of \(1.15\), then a 5% credit, and finally a floor of \(0.90\). Find the final modification and compare it with the result of moving the credit ahead of the cap.
Method. Follow the sequence literally. The debit factor is \(1.10\), and the credit factor is \(0.95\).
Step 1: Apply the debit.
Step 2: Apply the cap.
Step 3: Apply the credit.
Step 4: Apply the floor.
The final modification is 1.0925; the cap binds, and the floor does not. If the credit is incorrectly moved before the cap, the calculation becomes \(1.232(0.95)=1.1704\), followed by \(\min(1.15,1.1704)=1.15\). The floor leaves that value unchanged. The reordered result is \(1.15\), not \(1.0925\). The common mistake is to assume the cap can be applied after all percentage adjustments without changing the outcome.
Worked Example: A Floor Before an Increase
A hypothetical plan starts with a raw modification of \(0.68\). It requires a 25% increase, then a floor of \(0.80\), then a 8% credit, and finally a cap of \(0.90\). Calculate the final modification and assess what would happen if the floor were applied before the increase instead.
Method. The 25% increase uses factor \(1.25\), and the 8% credit uses factor \(0.92\).
Step 1: Apply the increase.
Step 2: Apply the floor.
Step 3: Apply the credit.
Step 4: Apply the cap.
The final modification is 0.782. Here the floor does not bind in the prescribed sequence, and neither does the cap. If the floor were incorrectly applied before the increase, it would raise \(0.68\) to \(0.80\); the increase would then produce \(0.80(1.25)=1.00\). After the credit, \(1.00(0.92)=0.92\), and the cap reduces this to \(0.90\). That result differs from the prescribed result. The common mistake is to treat a floor as a guarantee that must be imposed at the end, even though the plan places it before the increase.
Reading and Checking Plan Language
A calculation is only as accurate as its interpretation of the plan rule. In an exam-style problem, identify the operation’s trigger, the value it applies to, and its position in the sequence. Wording such as “after the debit, subject the modification to a maximum” makes the cap apply to the post-debit value. Wording such as “apply the minimum modification before the credit” places the floor before the credit, even if a different ordering might seem more intuitive.
It is useful to maintain a compact calculation table while working. The “operation” column records what the plan requires; the “current value” column prevents a later bound from being mistakenly applied to an earlier number.
| Stage | Operation | Value after operation |
|---|---|---|
| Starting point | Raw modification | \(R\) |
| 1 | First stated adjustment or bound | Intermediate result |
| 2 | Next stated adjustment or bound | Intermediate result |
| Final | Last stated adjustment or bound | Final modification |
Before finishing, test each bound at the moment it applies. A floor at \(L\) binds only if the current value is below \(L\); a cap at \(U\) binds only if the current value is above \(U\). A nonbinding bound still belongs in the sequence, but it leaves the value unchanged. Do not replace the plan’s ordered rules with one overall floor and cap unless the rules actually apply at the same stage.
Common Exam Traps
- Changing the order because it seems simpler. Keep the given order, even when another arrangement uses the same factors and bounds.
- Applying a final cap to an intermediate value. A cap applies where the plan places it, not wherever it is easiest to calculate.
- Interpreting a percentage as an addition. A 10% increase to \(x\) is \(1.10x\), not \(x+0.10\), unless a rule explicitly specifies an additive change of 0.10.
- Assuming every bound binds. Check the current value against the threshold; a floor or cap can leave it unchanged.
- Rounding intermediate values. Carry sufficient precision through every step and round only at the final reporting stage when the instructions specify a precision.
Check Your Understanding
For each hypothetical plan, apply the operations in the stated order and retain full precision until the final answer.
- A raw modification is \(0.76\). The plan specifies a floor of \(0.82\), a 5% increase, and a cap of \(1.00\), in that order. What is the final modification? (Answer: \(0.861\))
- A raw modification is \(1.09\). Apply a 12% debit, a cap of \(1.18\), a 4% credit, and a floor of \(1.00\), in that order. What is the final modification? (Answer: \(1.1328\))
- A raw modification is \(0.72\). The plan specifies a floor of \(0.80\), a 15% increase, and a cap of \(0.95\). What is the final modification? (Answer: \(0.92\))
- A raw modification is \(1.16\). Apply a 6% credit, a cap of \(1.10\), a 3% debit, and a floor of \(0.95\), in that order. Give the final modification to three decimals. (Answer: \(1.123\))
- A raw modification is \(0.64\). Apply a 30% increase, a floor of \(0.85\), a 5% credit, and a cap of \(0.90\), in that order. What is the final modification? (Answer: \(0.8075\))