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Special Classification

Note Section 3.4 • Reading time: ~5 mins

This section covers special classification topics: territorial ratemaking, increased limits ratemaking, deductible pricing, workers’ compensation size of risk pricing, and insurance-to-value (ITV) modeling.

Territorial Ratemaking

Territorial ratemaking involves grouping geographic areas into rating zones based on expected loss potential.

Challenges

  • Exposure Correlation: Geography is highly correlated with other rating variables (e.g., driver age, vehicle type). Multivariate techniques (like GLMs) are required to avoid double-counting.
  • Credibility: Small basic geographic units (e.g., zip codes) often suffer from highly volatile data.

Implementation Steps

  1. Define Basic Geographic Units: Select granular units (e.g., zip codes, census tracts).
  2. Estimate Indicated Rates: Fit a GLM to isolate the systematic geographic risk from random noise and non-geographic systematic risk factors.
  3. Apply Spatial Smoothing: Smooth out remaining geographic residuals across neighboring areas.

Spatial Smoothing Techniques

To smooth geographic residuals, actuaries use two primary models:

1. Distance-Based Smoothing

Calculates a credibility-weighted average of the subject geographic unit’s data and all other units, with weights decreasing as distance increases.

  • Pros: Intuitively easy to explain and implement.
  • Cons: Assumes distance has the same impact in urban and rural areas; ignores physical boundaries (e.g., rivers, state borders).
  • Best Suited For: Weather perils (e.g., wind, hail) which are not constrained by physical or political boundaries.

2. Adjacency-Based Smoothing

Averages the subject unit’s data with neighboring units in concentric rings, with weights decreasing for wider rings.

  • Pros: Better reflects differences between urban and rural environments; respects physical/geographical boundaries.
  • Best Suited For: Socio-demographic perils (e.g., theft, vandalism).

[!TIP] Over- vs. Under-Smoothing

  • Over-smoothing: Incorporates too much data from distant, unrelated areas, diluting local systematic risk.
  • Under-smoothing: Gives too much weight to sparse local data, leading to volatile rate changes.

Increased Limits Ratemaking

In liability lines, basic limits rates cover losses up to a standard limit (e.g., $100,000). Increased Limits Factors (ILFs) scale this base rate to cover higher policy limits.

Mathematical Formulation

The Increased Limits Factor (ILF) for limit LL is:

ILFL=Expected Losses Capped at L+ALAELExpected Losses Capped at Basic Limit B+ALAEB\text{ILF}_L = \frac{\text{Expected Losses Capped at } L + \text{ALAE}_L}{\text{Expected Losses Capped at Basic Limit } B + \text{ALAE}_B}

Using the Limited Average Severity (LAS\text{LAS}):

LAS(L)=∫0Lx⋅f(x) dx+L⋅[1−F(L)]\text{LAS}(L) = \int_0^L x \cdot f(x) \, dx + L \cdot [1 - F(L)]

If estimating a layer from AA to A+LA+L:

LAS(A+L)=LAS(A)+P(X>A)×LAS(L xs A)\text{LAS}(A+L) = \text{LAS}(A) + P(X > A) \times \text{LAS}(L \text{ xs } A)

ILF vs. GLM Approaches

  • ILF Assumptions: Standard ILF methods assume that claim frequency does not vary by policy limit.
  • GLM Performance: GLMs can model limit-specific frequency. However, GLM indications for high limits (e.g., $250,000) may occasionally be lower than lower limits (e.g., $100,000) due to data sparsity. This counterintuitive result is typically overridden in practice.

Deductible Pricing

Deductibles shift the responsibility for the initial portion of a loss from the insurer to the insured, applying to Loss & ALAE.

Loss Elimination Ratio (LER)

The LER represents the percentage of ground-up losses eliminated by the deductible DD:

LER(D)=Expected Losses & ALAE below DExpected Ground-Up Losses & ALAE\text{LER}(D) = \frac{\text{Expected Losses \& ALAE below } D}{\text{Expected Ground-Up Losses \& ALAE}}

The indicated relativity factor for deductible DD is the Excess Ratio:

Indicated RelativityD=1−LER(D)\text{Indicated Relativity}_D = 1 - \text{LER}(D)

LER Under Non-Zero Base Deductible

If the base deductible is B>0B > 0, the LER for a higher deductible DD relative to BB is:

LER(D,base B)=∑min⁡(Loss,D)−∑min⁡(Loss,B)∑max⁡(0,Loss−B)\text{LER}(D, \text{base } B) = \frac{\sum \min(\text{Loss}, D) - \sum \min(\text{Loss}, B)}{\sum \max(0, \text{Loss} - B)}

Deductible Interaction Example

When summarizing reported losses across different deductible levels, use a structured grid to re-evaluate losses under uniform deductible scenarios:

DeductibleNet LossNet Loss ($250 Ded)Net Loss ($500 Ded)
Full Coverage$680,000$590,000$525,000
$250 Deductible$2,900,000$2,900,000$2,600,000
$500 Deductible$5,200,000N/A$5,200,000

Workers’ Compensation: Size of Risk Pricing

Standard workers’ compensation rating utilizes expense constants and premium discounts to ensure equitable pricing across policy sizes.

1. Expense Constant

A flat fee added to small policies to cover fixed administrative costs that do not vary with premium size (preventing small risks from being undercharged relative to expense).

2. Premium Discount

Reflects the economies of scale on larger policies where fixed expenses represent a lower percentage of the total premium.

  • Discount Calculation: Discount %=Expense % Reduction1−Truly Variable Expense %\text{Discount \%} = \frac{\text{Expense \% Reduction}}{1 - \text{Truly Variable Expense \%}} Truly variable expenses include taxes, licenses, fees, and profit provisions, which remain constant as a percentage of premium.

3. Loss Constant

Historically, smaller risks show higher loss ratios due to:

  • Less formal safety training and risk mitigation.
  • Lack of eligibility for experience rating, reducing incentives to control losses.

To equalize loss ratios between small and large risks:

Small Risk LossesSmall Risk Premium+(Number of Small Policies×Loss Constant)=Large Risk Loss Ratio\frac{\text{Small Risk Losses}}{\text{Small Risk Premium} + (\text{Number of Small Policies} \times \text{Loss Constant})} = \text{Large Risk Loss Ratio}

Insurance to Value (ITV)

In property insurance, rates are typically quoted per $100 of coverage. Insurance to Value (ITV) is the ratio of selected coverage to the property’s replacement cost:

ITV=Coverage Amount SelectedReplacement Cost\text{ITV} = \frac{\text{Coverage Amount Selected}}{\text{Replacement Cost}}

Impact of Underinsurance

Because the majority of property losses are partial rather than total, the expected loss per $100 of coverage is higher for underinsured properties.

  • Example: If two houses have different replacement costs ($200k vs $250k) but both purchase $200k of coverage:
    • Fully Insured ($200k replacement, $200k coverage): Expected Claim Severity is lower.
    • Underinsured ($250k replacement, $200k coverage): Expected Claim Severity is higher because partial losses (e.g., kitchen fire costing $120k) are fully covered up to the policy limit.
    • Therefore, the indicated rate per $100 of coverage must be higher for underinsured properties.

Coinsurance Clause

To encourage policyholders to insure to full value, a coinsurance clause imposes a penalty if the ITV falls below a specified limit (typically 80%80\%).

Penalty Factor (a)=min⁡(1.0, Coverage AmountRequired Coinsurance %×Replacement Cost)\text{Penalty Factor } (a) = \min\left(1.0, \, \frac{\text{Coverage Amount}}{\text{Required Coinsurance \%} \times \text{Replacement Cost}}\right)

The indemnity payment (before deductible) is:

Indemnity Payment=min⁡(a×Loss Amount, Coverage Amount)‘‘‘\text{Indemnity Payment} = \min(a \times \text{Loss Amount}, \, \text{Coverage Amount}) ```