Univariate Classification

Univariate classification methods analyze a single rating variable at a time, calculating the indicated rate relativity for each level (or class) of that variable.

[!WARNING] Off-Balancing Required Whenever you apply indicated rate changes to existing base rates, you must off-balance the effect of the factor changes so that the total premium remains the same.

[!IMPORTANT] The Loss Ratio and Adjusted Pure Premium approaches only approximately correct for exposure correlation. They are not as accurate as multivariate techniques like Generalized Linear Models (GLMs). Do not suggest them as solutions when severe exposure correlation is present.

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General Procedure & Adjustments

Before calculating rate relativities, historical data must be adjusted to reflect the conditions expected during the future period when the rates will be in effect.

Adjustments Table

Adjustment Type	Description & Assumptions
Large Events & Anomalies	Identify and smooth out large losses or catastrophes if certain classes are disproportionately affected.
One-time Changes	Adjust historical class data for known one-time changes (e.g., benefit level changes).
Continuous Changes	Assumption: All classes trend at the same rate. Under this assumption, trend factors cancel out when dividing by the base class level.
Development	Assumption: All classes develop at the same rate. Development factors cancel out during division. • Use Calendar Year (CY) data for short-tailed lines. • Use Accident Year (AY) ultimate data for long-tailed lines.
Expenses & Profit	Assumption: Underwriting expenses, ULAE, and profit provisions do not vary by class. They cancel out during relativity derivation.1 ALAE is typically included directly in losses.
Credibility	Individual classes have smaller volumes of data and lower credibility. Apply credibility weighting using a suitable complement (e.g., current rates, competitor rates, or larger group data).

Mathematical Relativities

The indicated relativity for Class 1 compared to the Base Class is:

$$\text{Indicated Relativity}_{1} = \frac{\text{Indicated Rate}_{1}}{\text{Indicated Rate}_{\text{Base}}} = \frac{\frac{PP_{1} + FE_{1}}{1 - V - Q}}{\frac{PP_{B} + FE_{B}}{1 - V - Q}} = \frac{PP_{1} + FE_{1}}{PP_{B} + FE_{B}}$$

If fixed expenses ($FE$) are handled separately in the rating algorithm (e.g., through a flat expense fee), they are ignored in the relativity calculation:

$$\text{Indicated Relativity}_{1} = \frac{PP_{1}}{PP_{B}}$$

Because of this division, any uniform trend or development factor applied to both the numerator and denominator cancels out.

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Pure Premium Approach

The Pure Premium approach calculates the loss cost per unit of exposure for each rating class.

Methodology (Without Credibility)

1. Earned Exposures ($E$): Summarize for each level.
2. Losses ($L$): Summarize reported or paid losses + ALAE.
3. Pure Premium ($PP_X$): Calculate $PP_X = \frac{L_X}{E_X}$.
4. Indicated Relativity: $\text{Rel}_X = \frac{PP_X}{PP_B}$, where $B$ is the base class.

Distortions & Limitations

• Distributional Bias: Assumes there is no correlation between the rating variable being analyzed and other rating variables.
• Double-Counting: If exposures of levels of one variable are correlated with levels of another (e.g., Territory A has a high concentration of young drivers), the pure premium approach will double-count the risk. Territory A drivers will be penalized for the age correlation, and young drivers will also be penalized.

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Loss Ratio Approach

The Loss Ratio approach calculates the loss ratio for each level of a rating variable and compares it to the overall (average) loss ratio.

Methodology (Without Credibility)

1. Determine Earned Premium at Current Rate Level (CRL) for each class level.
2. Summarize Losses & ALAE for each level.
3. Calculate the Loss Ratio ($LR_X$): $LR_X = \frac{\text{Losses}_X}{\text{Premiums}_X}$.
4. Calculate the Indicated Relativity Change Factor: $F_X = \frac{LR_X}{LR_{\text{total}}}$.
5. Calculate the Indicated Relativity: $\text{Indicated Relativity}_X = F_X \times \text{Current Relativity}_X$.
6. Rebase the indicated relativities to the selected base class.

Actuarial Intuition

The Loss Ratio approach improves upon the Pure Premium approach by utilizing premiums, which reflect the existing rate structure (already accounting for some differences in risk).

If the historical premiums for a class are too low relative to its losses, the loss ratio will exceed the average:

$$LR_X > LR_{\text{total}} \implies F_X = \frac{LR_X}{LR_{\text{total}}} > 1$$

Multiplying the current rate by $F_X$ adjusts the premium to target the overall loss ratio:

$$\text{New Loss Ratio}_X = \frac{\text{Losses}_X}{\text{Premium}_X \times F_X} = LR_X \times \frac{LR_{\text{total}}}{LR_X} = LR_{\text{total}}$$

Distortions & Limitations

If other rating variables are not priced to their true indicated levels, the Loss Ratio approach will attempt to correct for those pricing errors in the variable currently being analyzed.

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Adjusted Pure Premium Approach

The Adjusted Pure Premium approach attempts to remove the distributional bias (exposure correlation) of other variables by adjusting the exposure base.

Actuarial Concept: Weighted Average Current Relativity (WACR)

The WACR measures the average rate relativity inherent to a class level based on its exposure distribution across other rating variables.

[!INFO] WACR Formula

$$\text{WACR}_X = \frac{\sum (\text{Exposures}_{X, i} \times \text{Relativity}_i)}{\sum \text{Exposures}_{X, i}}$$

Where $i$ represents the levels of the other rating variables.

Methodology (Without Credibility)

1. Calculate the WACR for each level of the rating variable being reviewed.
2. Calculate Adjusted Exposures: $\text{Adjusted Exposures}_X = \text{Exposures}_X \times \text{WACR}_X$.
3. Calculate the Adjusted Pure Premium: $\text{Adj } PP_X = \frac{\text{Losses}_X}{\text{Adjusted Exposures}_X}$.
4. Calculate the Indicated Relativity: $\text{Rel}_X = \frac{\text{Adj } PP_X}{\text{Adj } PP_B}$.

Distortions & Limitations

The use of adjusted exposures corrects for varying exposure distributions of other rating variables across the levels being analyzed, reducing the double-counting issue present in the standard Pure Premium approach.

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Credibility & Normalization in Relativity Calculations

When data volumes are low, the indications must be credibility-weighted with a complement (typically the current relativity).

Rebasing and Normalizing Steps

1. Rebase Indicated Rates to the Total:
   • Loss Ratio Approach: Dividing by the overall total loss ratio automatically yields a change factor.
   • Pure Premium Approach: Divide individual pure premiums by the overall average pure premium (rather than the base class pure premium).
2. Rebase Current Rates to the Total:
   • If “no-change” is the complement, use $1.0$ as the complement to the change factors.
   • If utilizing competitor rates or present rates, normalize them.
3. Normalization:
   • Calculate the WACR using the same exposures (or adjusted exposures if using the Adjusted Pure Premium approach) that were used to find the indications.
   • Divide the relativities by this WACR to ensure the average relativity is $1.0$ (revenue neutrality).
4. Credibility Weight: $$\text{Selected Relativity Factor} = Z \times \text{Indicated Factor} + (1 - Z) \times \text{Complement}$$

Footnotes

1. If fixed expenses are material and a separate flat fee is not utilized, the rate relativities must be compressed toward 1.0. ↩