Credibility

Credibility theory provides a mathematical framework for combining a subject group’s historical experience with an alternative source of data (the complement) to estimate future expected costs.

$$\text{Estimate} = Z \times \text{Observed Experience} + (1 - Z) \times \text{Complement}$$

Where $Z \in [0, 1]$ represents the credibility factor.

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Actuarial Criteria for Credibility Formulas

A mathematically sound credibility factor $Z$ must satisfy three criteria:

1. Boundedness: $0 \leq Z \leq 1$.
2. Monotonicity with volume: $\frac{dZ}{dN} > 0$ (credibility should increase as the exposure volume $N$ increases).
3. Diminishing marginal credibility: $\frac{d}{dN}\left(\frac{Z}{N}\right) < 0$ (each additional unit of exposure adds less marginal credibility than the previous unit).

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Credibility Models

1. Classical (Partial) Credibility

The classical model (or limited fluctuation credibility) defines the full credibility standard $N_0$ based on a probability $P$ of being within $k\%$ of the true mean. For volumes below $N_0$:

$$Z = \min\left(\sqrt{\frac{n}{N_0}}, \, 1.0\right)$$

2. Bühlmann Credibility (Greatest Tractability)

Bühlmann credibility minimizes the expected squared error and defines the credibility parameter $K$:

$$K = \frac{\text{Expected Value of Process Variance (EPVP)}}{\text{Variance of Hypothetical Means (VHM)}}$$

• EPVP (Process Variance): Measures the average variance within each individual risk group over time. A smaller EPVP indicates higher risk homogeneity and increases credibility.
• VHM (Between-Group Variance): Measures the variance between the means of different risk groups. A larger VHM indicates distinct risk clusters and increases credibility.
• Credibility Factor: $$Z = \frac{N}{N + K}$$ Where $N$ is the number of years or exposure units. A smaller $K$ results in a higher $Z$.

3. Bayesian Analysis

Computes the exact posterior probability distribution of losses by combining a prior distribution with observed data. Due to its computational complexity, Bühlmann credibility is often used as a linear approximation of the Bayesian posterior mean.

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Complements of Credibility

When the subject experience is not fully credible ($Z < 1.0$), the remaining weight $(1-Z)$ must be assigned to a complement of credibility.

Desirable Qualities of a Complement

An ideal credibility complement should possess six characteristics:

1. Accurate: Close to the true expected value of the subject group.
2. Unbiased: Expected value equals the target expectation over time.
3. Independent: Statistically independent from the subject statistic (to prevent compounding errors).
4. Available: Practical to obtain and verify.
5. Easy to Compute: Straightforward implementation.
6. Logical Relationship: Clear business or risk relationship to the subject group.

Comparison of Credibility Complement Methods

Method	Accurate	Unbiased	Indep.	Available	Easy Comp.	Logical Rel.
Competitor’s Rates	❓ Variable	❌ Biased (assumptions)	✅ Yes	❌ Difficult to obtain	✅ Yes	✅ Yes
Larger Related Group (excluding subject)	❓ Possibly	❌ Usually biased	✅ Yes	✅ Yes	✅ Yes	❓ If reasonable
Larger Group (including subject)	✅ Yes	❌ Biased	❌ No	✅ Yes	✅ Yes	✅ Yes
Rate Changes from Larger Group Applied to Present Rates	✅ Yes	❓ Less biased	❓ Yes (if excl.)	✅ Yes	❓ Slightly harder	✅ Yes
Harwayne’s Method	✅ Yes	✅ Yes	✅ Mostly	✅ Yes	❌ Harder	✅ Yes
Trended Present Rates	❓ Dep. on stability	✅ Yes	❓ Variable	✅ Yes	✅ Yes	✅ Yes

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First-Dollar Ratemaking Complements

1. Loss Costs of a Larger Group (Including Subject)

• Example: Using countrywide or statewide data to complement a specific territory.
• Pros: Highly stable, readily available, and logically related.
• Cons: Biased, as the subject territory was separated precisely because its risk profile differs from the larger group.

2. Loss Costs of a Related Group (Excluding Subject)

• Example: Using data from neighboring territories.
• Pros: Statistically independent and available.
• Cons: Subject to bias if the neighboring territories have different risk characteristics.

3. Rate Changes from a Larger Group Applied to Present Rates

Adjusts the subject’s present rate using the indicated change factor of the larger group:

$$\text{Complement} = \text{Current Subject Loss Cost} \times \frac{\text{Larger Group Indicated Loss Cost}}{\text{Larger Group Current Average Loss Cost}}$$

This reduces the bias associated with simply using the larger group’s raw loss cost.

4. Harwayne’s Method

Used in class ratemaking (e.g., Workers’ Compensation) to adjust for differences in exposure distributions between states.

Implementation Procedure (e.g., state $A$, class $1$):

1. Calculate the weighted average pure premium for all classes in State $A$.
2. Use State $A$‘s exposure distribution to calculate the weighted average pure premium for States $B$ and $C$.
3. Derive adjustment factors for each external state: $$\text{Adjustment Factor}_S = \frac{\text{Weighted Avg PP}_A}{\text{Weighted Avg PP}_S}$$
4. Apply adjustment factors to the class $1$ pure premium of external states: $$\text{Adjusted PP}_{S, 1} = \text{PP}_{S, 1} \times \text{Adjustment Factor}_S$$
5. The complement is the exposure-weighted average of these adjusted pure premiums.

5. Trended Present Rates

Used when no external group data is available. Project historical indications to the future period.

Pure Premium Method Formulation

To adjust the present average rate for past filings that were not fully approved, use:

$$\text{Complement} = \text{Present Avg Rate} \times \frac{1 + \text{Requested Rate Change}}{1 + \text{Approved Rate Change}} \times (1 + \text{Annual Loss Trend})^{\text{Trend Period}}$$

• Trend Period: Evaluated from the date the previous actuary performed the requested calculation to the future period when rates will be effective.

Loss Ratio Method Formulation

$$\text{Complement} = \frac{1 + \text{Loss Trend Factor}}{1 + \text{Premium Trend Factor}} \times \frac{1 + \text{Prior Indicated Rate Change Factor}}{1 + \text{Prior Implemented Rate Change Factor}}$$

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Excess Ratemaking Complements

In excess ratemaking, limited data makes credibility weighting crucial for estimating losses in layers above an attachment point $A$.

1. Increased Limits Analysis

Utilizes ground-up losses capped at the attachment point $A$ to estimate losses in the layer $L \text{ xs } A$ using Increased Limits Factors ($\text{ILF}$):

$$\text{Complement} = \text{Losses Capped at } A \times \frac{\text{ILF}_{A+L} - \text{ILF}_A}{\text{ILF}_A}$$

• Assumptions: Assumes the ground-up loss distribution is appropriate for the subject group.
• Pros/Cons: Practical and independent, but can be biased if the external size-of-loss distribution does not match the subject group.

2. Lower Limits Analysis

Uses losses capped at a lower limit $d < A$ to improve data volume:

$$\text{Complement} = \text{Losses Capped at } d \times \frac{\text{ILF}_{A+L} - \text{ILF}_A}{\text{ILF}_d}$$

• Pros/Cons: Increases stability (lower variance) but introduces additional distributional bias.

3. Limits Analysis (Reinsurance Generalization)

Reinsurers without access to ground-up loss data utilize this approach, assuming the loss ratio is constant across different policy limits:

$$\text{Complement} = \text{Expected Loss Ratio} \times \sum_{d > A} \text{Premium}_d \times \frac{\text{ILF}_{\min(d, A+L)} - \text{ILF}_A}{\text{ILF}_d}$$

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Contribution to Complement by Limit $d$:

• If $d \leq A$: No contribution.
• If $d_2 = A$: $ILF_A - ILF_A = 0$ (no contribution).
• If $A < d_3 < A+L$: Contributes proportional to $\frac{\text{ILF}_{d_3} - \text{ILF}_A}{\text{ILF}_{d_3}}$.
• If $d_4 \geq A+L$: Contributes proportional to $\frac{\text{ILF}_{A+L} - \text{ILF}_A}{\text{ILF}_{d_4}}$.