Overall Indication

Key Ratemaking Fundamentals

Loss Ratio (LR) vs. Pure Premium (PP) Method

Actuaries choose between the Loss Ratio and Pure Premium methods based on data availability and characteristics:

• Use the Loss Ratio Method when:
  • Exposures are not clearly defined or are non-homogeneous (e.g., commercial lines with complex, varying underlying risk exposures).
  • Exposure data is unavailable or unreliable.
• Use the Pure Premium Method when:
  • Pricing a new line of business (LOB) where historical premiums do not exist.
  • On-leveling historical premiums is difficult or inaccurate (e.g., highly uneven written exposures that make the parallelogram method unreliable).

Stability vs. Responsiveness

When selecting rates, trend factors, or development factors, actuaries must balance stability and responsiveness:

• Responsive Approach: Select the latest year or a short average (e.g., 2-year average) to reflect recent trends quickly.
• Conservative/Stable Approach: Select a longer average (e.g., 5-year average) to smooth out random fluctuations, particularly when future trends are highly uncertain.
• Balanced Approach: Select an intermediate average (e.g., 3-year average) to achieve a compromise:

  “To balance stability and responsiveness, we select the $n$-year average.”

Accident Year (AY) Weighting

If weighting is applied to different Accident Years, weight the resulting loss ratios or pure premiums from each year individually rather than weighting the raw inputs.

Premium Development and Audits

Premium development occurs due to audits and retrospective adjustments. If audits cause an increase in premium:

• Audited Premium Factor = $\dfrac{\text{Ultimate Premium Level}}{\text{Latest Reported Premium Level}}$
• Example Calculation: If first-half policies are audited (resulting in a $2.5\%$ increase) but second-half policies are not:
  • $\text{Latest Premium Level} = (0.5 \times 1.0 \times 1.025) + (0.5 \times 1.1)$
  • $\text{Ultimate Premium Level} = 1.025 \times (0.5 \times 1.0 + 0.5 \times 1.1)$

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Selection Rules

Pure Premium (PP) Selection

• If there is no clear trend or pattern, select the weighted average of the experience years.

Loss Ratio (LR) Selection

• If there is no clear trend or outliers, select the weighted average of the experience years.

Average Premium Selection

• Analyze historical average premiums to identify structural shifts or mix-of-business trends before selecting a trend rate.

Age-to-Age Development Factors

• If a downward trend in factors occurs, evaluate if it is due to a systematic change (e.g., changes in reserving philosophy) or random fluctuation.
• Justification: Clearly document the reasoning for the selection (e.g., “Medical inflation slowdown justifies selecting the latest 3-year average”). Mention alternative reasonable selections if appropriate (e.g., medial average).

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Trend Periods for Expenses

• Fixed Expenses: Trended from the average written date of the historical period to the average written date of the prospective period.
• Reason: Most fixed expenses are incurred at policy inception.
• Exception: General expenses incurred throughout the policy term are trended using average earned dates.

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Big Picture: Catastrophe and Shock Loss Loadings

To calculate an indication adjusted for shock losses or catastrophes:

1. Remove Shock Losses: Exclude the catastrophe/shock loss events from the historical data.
2. Calculate Non-Catastrophe LR: Compute the Loss and ALAE ratio excluding these events.
3. Calculate Catastrophe Load:
   • Determine the historical weighted average of catastrophe losses relative to Amount of Insurance Years (AIY) or exposures over a long-term period.
   • Project exposures to the prospective period.
   • Apply a ULAE factor to the catastrophe pure premium.
4. Combine: Add the catastrophe load to the projected non-catastrophe pure premium to get the final rate indication.

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Practical Problem Strategies

Example: Agency Commission Reductions (2013 Q5)

Suppose an insurer reduces agency commissions by $3\%$ starting mid-year, affecting the prospective policy period:

• If uniform writings are assumed, and $2/3$ of the policies in the future period will be written under the new commission structure: $\text{Reduction in Variable Expense Rate} = \dfrac{2}{3} \times 3\% = 2\%$
• Reduce the projected variable expense ratio by $2\%$ in the indication formula.

Identifying Extrapolated Values

Always look for directly available information before starting calculations:

• Example: If asked to find the projected non-modeled catastrophe pure premium, check if projected exposures or average values are already provided or can be averaged directly from the given data.

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