One-time changes

Workers’ Compensation Benefit Changes

Intuition for State Average Weekly Wage (SAWW) Problems

To analyze benefit changes linked to the State Average Weekly Wage (SAWW):

• Concept: Treat benefits as equivalent to claims.
• Relative Analysis: If the problem is stated in percentages of the SAWW, relate wages and workers to the average:
  • Percentage of Wages: $\dfrac{W}{TW}$ (where $W$ is class wage, $TW$ is total wage)
  • Percentage of Workers: $\dfrac{\#}{T\#}$ (where $\#$ is class workers, $T\#$ is total workers)
  • Ratio: $\dfrac{\%\text{ Wages}}{\%\text{ Workers}} = \dfrac{W / TW}{\# / T\#} = \dfrac{W / \#}{TW / T\#} = \dfrac{\text{Average Wage of Group}}{\text{SAWW}}$
• Average Benefit per Worker: Find the ratio of Average Benefit to SAWW for each wage interval, then calculate the sumproduct with the worker distribution to get the overall average benefit relative to the SAWW:
  • Below Minimum: $\dfrac{\text{Min Benefit}}{\text{SAWW}}$
  • In Range: $\dfrac{\text{Average Wage of Group} \times \text{Compensation Rate}}{\text{SAWW}}$
  • Above Maximum: $\dfrac{\text{Max Benefit}}{\text{SAWW}}$
• Critical Wage Points: Divide the benefit limits by the compensation rate to find the wage levels where limits begin to apply:
  • Minimum Limit Point: $\text{Wage} \times \text{Compensation Rate} \leq \text{Min Benefit} \implies \text{Wage} \leq \dfrac{\text{Min Benefit}}{\text{Compensation Rate}}$
  • Maximum Limit Point: $\text{Wage} \times \text{Compensation Rate} \geq \text{Max Benefit} \implies \text{Wage} \geq \dfrac{\text{Max Benefit}}{\text{Compensation Rate}}$

Benefit Limits Comparison

Type	Limit/Rate	Received	% of SAWW	Range Details
Current	Min Benefit	$550.00	100.0%	Or below
	Max Benefit	-	-	-
	Compensation Rate	-	50.0% (1/2)
Revised	Min Benefit	$550.00	75.0%	Or below
	Max Benefit	$1,375.00	187.5%	Or above
	Compensation Rate	-	66.7% (2/3)

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One-Time Premium Changes

Historical premiums must be adjusted to the current rate level (on-leveled) using one of two primary methods:

1. Extension of Exposures (EOE) Method

• Concept: Re-rates every historical policy using the current rate manual rules and rates.
• Pros/Cons:
  • ✅ Highly Accurate: Captures the exact distribution of rating variables.
  • ❌ Data Intensive: Requires detailed historical data for every single policy.
  • ❌ New Variables: Difficult or impossible to use if a new rating variable is introduced for which historical data was not captured.
  • ❌ Time-Consuming: Requires significant computational effort.

2. Parallelogram Method

• Concept: Adjusts historical earned premiums at an aggregate level using the average rate levels of the historical and current periods.
• Pros/Cons:
  • ✅ Simple: Does not require detailed policy-level data; only needs overall rate change history and earned premium aggregates.
  • ❌ Less Accurate: Relies on simplifying assumptions.
  • ❌ Uniform Writings Assumption: Assumes policies are written uniformly over time.
    • Mitigation 1: Analyze smaller time intervals (e.g., quarterly).
    • Mitigation 2: Adjust for actual writings distributions (non-uniform writings).
  • ❌ Classification Limitations: Less reliable for class-level ratemaking.
    • Mitigation: Run the parallelogram method at the class level rather than the aggregate level.

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Parallelogram Method Calculations

Portion of CY Earned Premium at a New Rate Level

Under the uniform writings assumption, the portion of a calendar year’s earned premium written at a particular rate level corresponds to the area of the geometric shape representing that rate level within the CY square:

• Annual Policies: The transition period is 24 months. The area of a triangle at a corner of the CY square (representing a rate change) is: $\text{Area} = \dfrac{1}{2} \times \text{Base} \times \text{Height}$
• Semi-Annual Policies: The transition period is 18 months.
• General Formula for Portion: $\text{Portion} = \dfrac{1}{2} \times \dfrac{\text{\# of months at new rate}}{\text{Policy Term (in months)}} \times \dfrac{\text{\# of months at new rate}}{\text{Calendar Year length (12 months)}}$

Handling Uneven Exposures

When the uniform writings assumption is violated, the geometric area does not represent the earned premium portion. We must weight the average rate levels by actual writings:

1. Divide the experience period into intervals between rate changes.
2. Determine the average rate level for each write-segment.
3. Apply weights based on the actual number of exposures written in each segment to find the overall average rate level for the policy year (PY) or calendar year (CY).

Example of Uneven Writings

Suppose writing volumes differ:

• Jan 1, 2013 to Jul 1, 2013: 10 exposures/month (Total = 60)
• Jul 1, 2013 to Jan 1, 2014: 11 exposures/month (Total = 66)
• Total exposures written in PY 2013 = 126.
• Weights:
  • Segment A (first 6 months): $\dfrac{60}{126} = \dfrac{10}{21}$
  • Segment B (second 6 months): $\dfrac{66}{126} = \dfrac{11}{21}$

Average rate levels by area:

• $A_1 = 1.00$ (Portion earned in 2013 = $75\%$)
• $A_2 = 1.05$ (Portion earned in 2013 = $25\%$)
• $B_1 = 0.93$ (Portion earned in 2013 = $25\%$)
• $B_2 = 0.977$ (Portion earned in 2013 = $75\%$)

The average rate level for PY 2013 is: $\text{Avg Rate Level} = (1.00 \times 0.75 + 1.05 \times 0.25) \times \dfrac{10}{21} + (0.93 \times 0.25 + 0.977 \times 0.75) \times \dfrac{11}{21} \approx 0.9634$

Compare this to even exposures where the weight would be $50\%$ each: $\text{Avg Rate Level (Even)} = (1.00 \times 0.75 + 1.05 \times 0.25) \times \dfrac{1}{2} + (0.93 \times 0.25 + 0.977 \times 0.75) \times \dfrac{1}{2} \approx 0.9789$

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Calendar Year Written Premium (WP) On-Level Factors

Without Law Changes

For CY WP (without law changes), the average rate level is the weighted average of the rate levels in effect during the year, based on the portion of the year each rate level was active:

• Example: A $+10\%$ rate change occurs on July 1, 2025. $\text{Average Rate Level for CY 2025} = 1.0 \times 0.5 + 1.10 \times 0.5 = 1.05$ $\text{On-Level Factor for CY 2025 WP} = \dfrac{\text{Current Rate Level (1.10)}}{\text{Average Rate Level (1.05)}} \approx 1.0476$

With Law Changes (Adjusting In-Force Policies)

If a law change impacts in-force policies mid-term, the calculation must account for premium adjustments on policies written prior to the calendar year:

1. Partition the CY into segments between rate changes.
2. Find the written premium under base assumptions (e.g., 100 policies written uniformly per year, $1,000 base premium).
3. Add the premium adjustment from the law change for policies that were already in-force.
4. Calculate the average rate level and the On-Level Factor: $\text{OLF} = \dfrac{\text{CY Premium @ Current Rate Level}}{\text{Actual CY Written Premium}}$

Illustration of Law Change Adjustment

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• Average Rate Level Calculation:
  • Written in the CY: $(25\% \times 1.00) + (50\% \times 1.10) + (25\% \times 1.10 \times 1.05)$
  • Adjustment for Policies Written in Previous Year (In-Force during Law Change): $(50\% \times 1.00) \times \dfrac{3}{12} \times 5\% + (50\% \times 1.10) \times \dfrac{9}{12} \times 5\%$
  • Average Rate Level = $1.1156$
• Current Rate Level = $1.00 \times 1.10 \times 1.05 = 1.155$
• On-Level Factor = $\dfrac{1.155}{1.1156} = 1.0353$

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On-Leveling with Actual Writings Distributions

If exposures are not written uniformly, group data by rate level and calculate the earned exposure portion for the target calendar year:

Example Data Table

• Rate change: $+5\%$ on September 1, 2014.
• Historical writings distribution:

Interval	Written Exp.	Rate Level	Avg Written Date	% Earned 2014	Earned Exposures
2014 Q1	100	1.00	2/15/2014	0.878	87.78
2014 Q2	300	1.00	5/15/2014	0.628	188.33
Jul - Aug 2014	333	1.00	8/01/2014	0.417	138.89
Sept 2014	167	1.05	9/15/2014	0.294	49.07
2014 Q4	700	1.05	11/15/2014	0.128	89.44

Analysis Steps

1. Average Written Date: Determine the average date policies were written in each group.
2. % Earned in CY 2014: Calculate the portion earned in CY 2014. For annual policies, this is the time elapsed from the average written date to December 31, 2014.
3. Earned Exposures: $\text{Written Exposures} \times \text{\% Earned}$.
4. Average Earned Rate Level: $\text{Avg Earned Rate Level} = \dfrac{\sum (\text{CY Earned Exposures} \times \text{RL})}{\sum \text{CY Earned Exposures}}$
5. On-Level Factor: $\text{OLF} = \dfrac{\text{Latest Rate Level (1.05)}}{\text{Avg Earned Rate Level}}$

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Direct and Indirect Effects of One-Time Changes

When evaluating the impact of a one-time change, actuaries must distinguish between two types of effects:

• Direct Effects: The immediate, arithmetic impact on premiums, losses, or expenses under the assumption that human behavior remains unchanged.
  • Example: An increase in the statutory maximum benefit for workers’ compensation directly increases the amount paid for claims that exceed the old limit.
• Indirect Effects (Incentives): The secondary impact resulting from changes in human behavior triggered by the one-time change.
  • Frequency Effect: Higher benefit levels may incentivize workers to file more claims.
  • Duration Effect: Increased weekly benefit payments may reduce the financial incentive to return to work quickly, leading to longer claim durations.

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Miscellaneous Considerations

• Effective Through vs. Effective From:
  • “Effective through [Date]” means the rate level is active up to and including that date.
  • “Effective from [Date]” means the rate level starts on that date.
• Loss Ratio Method Denominator: When utilizing the Loss Ratio method, the premium in the denominator must be adjusted to the current rate level using the premium on-level factors (OLFs).
• On-Leveling for Loss Data (Benefit Changes):
  • If a benefit change applies to policies written on or after a certain date, the adjustment must be calculated using a policy-year/parallelogram approach.
  • If a benefit change applies to accidents occurring on or after a certain date, it behaves like a law change with a sharp cutoff. The losses should be adjusted using a step-function approach (no diagonal transitions).
• Equivalence of On-Leveling: On-leveling Calendar Year (CY) Earned Premium is methodologically equivalent to on-leveling Policy Year (PY) Written Premium (since for a given PY, ultimate Earned Premium equals Written Premium).