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Large Events

Note Section 1.2 • Reading time: ~5 mins

Anomalies in Loss Data

Ratemaking data often contains anomalies—such as shock losses or catastrophe losses—that can distort rate indications if left unadjusted.

Shock Losses vs. Catastrophe Losses

  • Shock Loss: A single, large claim that is fortuitous (e.g., a total fire loss on a high-value home, or a severe commercial liability claim). The definition of a shock loss varies by insurer and line of business (LOB) based on the size of the book:
    • A $1M loss has a massive impact on a book with $10M in annual premium, but is negligible for a book with $1B in annual premium.
  • Catastrophe Loss: A single event that causes multiple claims simultaneously (e.g., hurricanes, tornadoes, earthquakes).

Impact of Leaving Anomalies Unadjusted

If shock losses or catastrophes are not adjusted in historical loss data:

  • Overestimation: Future expected losses will be overestimated if these rare events occurred during the experience period.
  • Underestimation: Future expected losses will be underestimated if these events did not occur during the experience period (resulting in “lucky” years).

[!IMPORTANT] The goal of adjusting for large events is to produce rates that cover these costs over a long period, balancing stability (not overreacting to unlucky years) and responsiveness.

Adjustments to Shock Losses

Actuaries typically use one of three methods to adjust for shock losses:

1. Cap Losses at Basic Limits (Liability)

  • Method: Cap all losses at a basic coverage limit (e.g., $100k or $500k) and calculate the rate indication using only basic limits data.
  • Excess Pricing: Separately price the excess limits using increased limit factors (ILFs).
  • Limitation: Cannot be used for lines without policy limits, such as Workers’ Compensation.

2. Cap Losses and Apply an Excess Load (Property)

  • Method: Cap all losses at a specified threshold (non-excess losses) and apply a flat excess loading factor to the capped losses.
  • Excess Load Formula: Excess Load Factor=1+Excess LossesNon-Excess Losses\text{Excess Load Factor} = 1 + \dfrac{\text{Excess Losses}}{\text{Non-Excess Losses}}
  • Note: See Notes for definitions of how excess is calculated.

3. Remove Ground-Up Losses and Load for Excess (Less Common)

  • Method: Entirely remove the shock claims from the primary dataset and add a separate, long-term average load for excess losses.

Key Selections in Shock Loss Adjustments

Selecting Cap Levels

Actuaries choose capping levels based on:

  1. AJ: Actuarial Judgment.
  2. Percentile of Size-of-Loss Distribution: E.g., capping at the 95th or 99th percentile of historical losses.
  3. Loss as a Percentage of Insured Value: E.g., capping claims that exceed a certain percentage of the policy’s limit or insured value.

Selecting the Number of Years for the Load

[!TIP] Choosing the historical period for calculating the excess or catastrophe load requires balancing stability and responsiveness:

  • Too few years: Results in an unstable load that is overly sensitive to recent events.
  • Too many years: Fails to reflect recent changes in risk patterns, building materials, or climate.

Capping and Inflation

When trending capped and excess losses, the excess loss factor should be calculated using trended values to reflect changes in severity: Excess Loss Factor=Trended Excess LossesTrended Non-Excess Losses\text{Excess Loss Factor} = \dfrac{\text{Trended Excess Losses}}{\text{Trended Non-Excess Losses}}

Adjustments to Catastrophe Losses

For non-modeled catastrophes (where simulation models are not used), the indication includes a catastrophe provision based on historical experience:

  1. Project Exposures: E.g., Amount of Insurance Years (AIY) or earned exposures.
  2. Calculate the Non-Modeled Catastrophe Provision per Exposure: Catastrophe Provision per AIY=Arithmetic Average of Historical Catastrophe Losses per AIY×ULAE Factor\text{Catastrophe Provision per AIY} = \text{Arithmetic Average of Historical Catastrophe Losses per AIY} \times \text{ULAE Factor}
  3. Incorporate into Indication: Add this catastrophe pure premium to the projected non-catastrophe pure premium in the rate indication.

Notes

Definitions of “Excess” Loss

Excess loss can be calculated in two ways, which affects the capped (non-excess) and excess portions:

  1. Entire Claim Method (Claims Above Threshold): If a claim exceeds the threshold, the entire claim amount is classified as excess.

    • Excess Amount = Sum of all claims where claim size > threshold.
    • Non-Excess Amount = Sum of all claims where claim size ≤\leq threshold.

  2. Excess Portion Method (Amount Above Threshold): Only the portion of any claim that exceeds the threshold is classified as excess; the portion up to the threshold remains in the non-excess base.

    • Excess Amount = ∑max⁡(0,Claimi−Threshold)\sum \max(0, \text{Claim}_i - \text{Threshold})
    • Non-Excess Amount = ∑min⁡(Claimi,Threshold)\sum \min(\text{Claim}_i, \text{Threshold})

Example

Suppose we have a threshold of $500k and 5 claims:

  • N_1 = \100\text{k}, N_2 = $200\text{k}, N_3 = $300\text{k}$ (Non-excess claims)

  • E_1 = \600\text{k}, E_2 = $700\text{k}$ (Excess claims)

  • Under Method 1 (Entire Claim):

    • Excess Amount = E_1 + E_2 = \600\text{k} + $700\text{k} = $1.3\text{M}$
    • Non-Excess Amount = N_1 + N_2 + N_3 = \100\text{k} + $200\text{k} + $300\text{k} = $600\text{k}$

  • Under Method 2 (Excess Portion):

    • Excess Amount = (E_1 - 500\text{k}) + (E_2 - 500\text{k}) = \100\text{k} + $200\text{k} = $300\text{k}$
    • Non-Excess Amount = N_1 + N_2 + N_3 + 2 \times 500\text{k} = \600\text{k} + $1.0\text{M} = $1.6\text{M}$