Why Piecewise Modification Schedules Need Care
An experience modification schedule can respond differently to favorable and unfavorable experience. For example, a hypothetical plan might pass through only part of a raw decrease below the neutral factor, pass through a different proportion of an increase above neutral, and limit the result at stated endpoints. The calculation is then piecewise: the applicable formula depends on which range contains the raw indicated factor.
The central task is to keep two quantities separate. The raw indicated factor is the factor produced from the experience data before applying the schedule. The adjusted modification is the result after applying the plan’s specified piecewise rule. A raw factor is not itself the final modification whenever the schedule changes it.
A Hypothetical Schedule and Its Boundaries
Use the following hypothetical schedule throughout this tutorial. Its lower and upper limits are 0.80 and 1.20. Between those limits, the schedule applies a 50% adjustment to a favorable raw deviation below 1.00 and an 80% adjustment to an unfavorable raw deviation above 1.00.
The ranges are part of the definition, not decoration. First compare \(R\) with the thresholds 0.60, 1.00, and 1.25. Then use exactly one applicable line. At \(R=0.60\), the first line gives 0.80; the next line would also give \(0.80+0.50(0)=0.80\). At \(R=1.00\), the second line gives 1.00, matching the value approached from above. At \(R=1.25\), the third line approaches \(1.00+0.80(0.25)=1.20\), matching the upper-limit line.
| Raw factor range | Schedule rule | Interpretation |
|---|---|---|
| \(R\le 0.60\) | \(M=0.80\) | Lower flat portion |
| \(0.60<R\le 1.00\) | \(M=0.80+0.50(R-0.60)\) | Partial adjustment below neutral |
| \(1.00<R<1.25\) | \(M=1.00+0.80(R-1.00)\) | Partial adjustment above neutral |
| \(R\ge 1.25\) | \(M=1.20\) | Upper flat portion |
The schedule is continuous at each threshold: the value from one side agrees with the value from the other. That is a useful check, but continuity does not mean the slopes are the same. Within the lower sloped range, a 0.01 increase in \(R\) changes \(M\) by 0.005. Within the upper sloped range, a 0.01 increase in \(R\) changes \(M\) by 0.008. Outside those ranges, a further change in \(R\) does not change \(M\).
Calculating the Raw Indicated Factor
In some problems, the raw factor is supplied. In others, the experience data are supplied and the raw factor must be calculated first. For the examples below, assume the hypothetical plan defines the raw factor by
where \(A\) is actual loss on the plan’s specified basis, \(E\) is expected loss on that same basis, and \(Z\) is the credibility weight, with \(0\le Z\le 1\) and \(E>0\). This is a stipulated raw-factor formula for these examples; use it only when the problem specifies it. The piecewise schedule is then applied to \(R\), not directly to the loss difference.
Worked Calculations
Worked Example: A Raw Factor Below Neutral
A hypothetical risk has actual loss \(A=\$168{,}000\), expected loss \(E=\$200{,}000\), and credibility weight \(Z=0.75\). The plan uses \(R=1+Z(A-E)/E\) and the piecewise schedule above. Calculate the adjusted modification.
Step 1: Find the actual-to-expected difference.
Step 2: Calculate the credibility-weighted relative deviation.
Step 3: Calculate the raw factor.
Since \(0.60<0.88\le 1.00\), use the lower sloped branch, not the upper branch or a flat limit.
Check: The result is between 0.80 and 1.00, as expected for this range. The calculation avoids the common mistake of treating \(R=0.88\) as the final modification.
Worked Example: An Increase Above Neutral
A hypothetical account has \(A=\$208{,}000\), \(E=\$160{,}000\), and \(Z=0.75\). Use the same raw-factor formula and schedule. Find the adjusted modification.
Step 1: Calculate the experience difference and relative deviation.
Step 2: Apply credibility and calculate \(R\).
Because \(1.00<1.225<1.25\), use the upper sloped branch.
Check: The raw factor is below the upper threshold of 1.25, so the upper flat limit does not apply. The adjusted factor of 1.18 is lower than \(R=1.225\), consistent with the schedule passing through only 80% of the deviation above neutral. The common mistake is to apply the 50% lower-range adjustment simply because the schedule also has a range below 1.00; the raw factor determines the branch.
Worked Example: A Raw Factor Above the Upper Threshold
A hypothetical account has \(A=\$260{,}000\), \(E=\$200{,}000\), and \(Z=0.90\). The same raw-factor formula and schedule apply. Calculate the adjusted modification and explain whether the upper sloped formula applies.
Step 1: Calculate the relative deviation.
Step 2: Apply credibility to obtain \(R\).
Since \(R=1.27\ge 1.25\), use the upper flat portion:
Check: The value is exactly the schedule’s stated upper limit. Substituting \(R=1.27\) into the upper sloped formula would give \(1+0.80(0.27)=1.216\), but that formula is defined only for \(1<R<1.25\). The common mistake is to continue a sloped formula past its stated endpoint.
Why the Schedule Gives These Results
Each sloped branch adjusts a deviation from a reference point. Below neutral, the relevant reference is 0.60 and the adjusted factor starts at 0.80. Above neutral, the reference is 1.00 and the adjusted factor starts at 1.00. The slope specifies how much of a change in the raw factor is reflected in the adjusted factor.
For the lower sloped branch, the formula must start at \(M=0.80\) when \(R=0.60\), and each increase in \(R\) is multiplied by 0.50. Thus its form is \(M=0.80+0.50(R-0.60)\). For the upper sloped branch, the formula starts at \(M=1.00\) when \(R=1.00\), and each increase is multiplied by 0.80. Thus its form is \(M=1.00+0.80(R-1.00)\). Substituting the endpoints verifies that the branches meet their flat portions:
The flat portions follow from the schedule’s explicit limits: for \(R\le 0.60\), the result remains 0.80; for \(R\ge1.25\), it remains 1.20. They are not obtained by extending either slope indefinitely.
A Reliable Calculation Process
A disciplined order of operations keeps the raw-factor calculation separate from the schedule selection.
Use the formula specified by the problem, keeping actual and expected losses on a consistent basis.
Compare \(R\) with every stated threshold, paying attention to whether the inequalities include equality.
Use the formula or fixed value for that range; do not carry a sloped formula past its stated endpoint.
Confirm that the result fits the range, endpoints agree where the schedule is continuous, and rounding has not changed branch selection.
Common Mistakes and Exam Traps
- Using the wrong branch. Decide the range from the raw factor \(R\), not from the adjusted value being sought. Write the relevant inequality before substituting.
- Applying a slope to the wrong base. The lower branch adjusts \(R-0.60\), not \(R-1.00\). The upper branch adjusts \(R-1.00\).
- Confusing credibility with the schedule adjustment. In the stated raw-factor formula, \(Z\) is applied while calculating \(R\). The schedule’s 0.50 or 0.80 slope is then applied to the relevant deviation in \(R\).
- Extending a branch past its endpoint. A formula that is valid for \(1<R<1.25\) is not valid at or above \(1.25\); use the stated flat portion there.
- Rounding before selecting a range. Retain sufficient precision in \(R\) until the applicable branch is known. A rounded value near a threshold could appear to fall on the wrong side.
- Assuming the schedule is universal. Thresholds and slopes are plan-specific. Use the schedule given in the problem rather than importing values from another plan.
Check Your Understanding
Use the hypothetical schedule in this tutorial. Unless a question supplies \(R\) directly, use \(R=1+Z(A-E)/E\). Give the adjusted modification in each case.
- A hypothetical account has a raw factor \(R=0.74\). Which sloped branch applies, and what is \(M\)? (Answer: Lower sloped branch; \(M=0.87\).)
- A hypothetical account has \(R=1.10\). Calculate its adjusted modification and state how much of the raw deviation above 1.00 is passed through. (Answer: \(M=1.08\); 80% of the deviation is passed through.)
- For a hypothetical risk, \(A=\$145{,}000\), \(E=\$200{,}000\), and \(Z=0.80\). Calculate \(R\), identify the range, and find \(M\). (Answer: \(R=0.78\); lower sloped branch; \(M=0.89\).)
- A hypothetical account has \(R=1.30\). Calculate \(M\), and explain why the upper sloped formula is not used. (Answer: \(M=1.20\); \(R\ge1.25\), so the upper flat portion applies.)
- Evaluate the schedule at \(R=0.60\), \(R=1.00\), and \(R=1.25\). State whether the neighboring rules agree at each boundary. (Answer: \(M=0.80\), \(M=1.00\), and \(M=1.20\), respectively; the neighboring rules agree at all three boundaries.)