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Univariate Classification

  • Whenever you are applying indicated rate changes to the existing base rates, YOU MUST off-balance the effect of the factor changes so that the total premium remains the same.

Remember, #Loss Ratio Approach and #Adjusted Pure Premium Approach just approximately correct for exposure correlation, they are not accurate like the Multivariate approaches like GLMs. So, don't use it as an answer for situations what exposure correlation is a problem. Band-aid not a solution.

General Procedure

We have certain adjustments to make just like we did for overall ratemaking. Following which we can obtain the rate relativities.

Adjustments

Large Events & Anomalies Are some individual classes more prone to large losses or catastrophes than others?
One-time changes Adjust all class data for one-time changes.
Continuous Changes Assumption: All classes are trending at the same rate (so ignore trending individual classes). Trend factors cancel out when dividing with the base level to get relativilities
Development Assumption: All classes are developing at the same rate. Dev Factors cancel out in a similar fashion.
- CY data for short-tailed lines
- AY ultimate data for long-tailed lines
Expenses & Profit Assumption: UW expenses, ULAE & profit doesn't vary by class.
Gets cancelled when div by base class.1 ALAE is included in losses
Credibility Individual classes have less data \(\implies\) Less credible. Thus it becomes more important. Complement class data: competitor, current rates

Steps

Indicated Class 1 Relativity = \(\dfrac{\text{Ind. Rate}_{1}}{\text{Ind. Rate}_{\text{Base}}}= \dfrac{\dfrac{PP_{1} + FE_{1}}{1 - V - Q}}{\dfrac{PP_{B} +FE_{B}}{1 - V - Q}}\) \(= \dfrac{PP_{1}+ FE_{1}}{PP_{B} + FE_{B}}\)

Fixed expenses are dealt with separately in the rating algo. So we ignore it here.

Indicated Class 1 Relativity = \(\dfrac{PP_{1}}{PP_{B}}\)

  • This is why and trends or development factors cancel out.

Pure Premium Approach

  • Calculate Pure Premium for each level of RL

Method (w/o Credibility)

  1. Earned exposures for each level of RV = \(E\)
  2. (Rept/Paid) Loss & ALAE = \(L\)
  3. Pure Premium = \(\dfrac{L}{E}\)
  4. Ind. Rel to Base = \(\dfrac{PP_{X}}{PP_{B}}\)

Method (w/ Credibility)

Distortion

Arises because we directly use Exposures (Pure Premium approach)

  • Distributional Bias
  • If exposures of levels of one rating variable are correlated with exposures of levels of another rating variable, this approach will "double count" the experience of those levels.
  • e.g. Terr A has more young drivers. Two variables (territory and age) are correlated. Thus the attribute of the reckless youth is captured by the experience of Territory A drivers. And any driver (whether young or old) in territory A will be penalized for that.

Method (w/o Credibility)

  1. Earned Premium @CRL
  2. (Rept) Loss & ALAE
  3. LR
  4. Ind. Rel chg. factor = \(\dfrac{LR_{X}}{LR_{total}}\) (divide by total if without credibility)
  5. (Credibility factor can be applied here)
  6. Current Rel
  7. Ind Relativity = (4) x (6)
  8. Ind. Rel to base

Loss Ratio Approach

  • Calculate LR for each level of RL
  • Divide by Overall total LR

Method (w/o Credibility)

Intuition

Improves over #Pure Premium Approach since it uses Premiums which reflect the higher (or lower) levels of premiums.

If the historical premiums are smaller than what the total indicated loss ratio suggests then the premiums have to be increased. \(LR_{X} > LR_{total} \implies \dfrac{LR_{X}}{LR_{total}}> 1\) has to be multiplied to the premiums, to equate the individual LR to the overall.

Mathematically,

  • Loss Ratio \(= \dfrac{\text{Losses}}{\text{Premiums}}\)
  • Initially, \(LR_{A} \neq LR_{B} \neq LR_{C} \neq LR_{total}\)
  • We don't want this to happen. We want the \(\text{Prem} \propto \text{Losses}\)
  • For that we need to multiply the premiums by a factor…
  • That factor is \(F_{X} = \dfrac{LR_{X}}{LR_{total}}\) with \(X\) rate level
  • Why? Because, the new premium would be \(\text{Premium}_{X}F_{X}\) so the new loss ratio would be \(\dfrac{\text{Losses}}{\text{Premiums}_{X}F_{X}} = \dfrac{\text{Losses}}{\text{Premium}_{X}}\left(\dfrac{LR_{total}}{LR_{X}}\right) = LR_{X}\left(\dfrac{LR_{total}}{LR_{X}}\right) = LR_{total}\)

Distortion

  • If the other rating variables are not priced to their indicated rate relativities, LR attempts to correct for that difference in the variable being analyzed

Adjusted Pure Premium Approach

  • Adjust exposures by weighted average current relativity (WACR)
    • = \(\dfrac{\text{SUMPRODUCT(...)}}{\text{Exposures}}\)
  • The WACR is a measure of the rate relativity inherent to a particular class based on the rate relativities of its exposures from the perspective of other rating variables
  • Then use #Pure Premium Approach
  • Adjusted pure premium method corrects for the variation in exposure distribution of other rating variables across the levels of the rating variables being analyzed.

WACR

Weighted Average current relativity. Just weight the relativities within a group by its exposures.

  • It can be done for a rating variable (if the exposure distribution is available)
  • It can be done for a total (which essentially tells you how much above/below the base class on average do you charge… based on your market share)

Method (w/o Credibility)

  1. Find the Weighted Average Territory/Class Relativity
  2. Adjust the exposures by multiplying with the WACR
  3. Apply the #Method (w/o Credibility)

Distortion

  • #Pure Premium Approach assumes there is no correlation between the exposures for different rating variables. If exposures of the levels of one RV is correlation to the levels of another RV then this approach will double count the experience of those levels. #Adjusted Pure Premium Approach attempts to correct this distributional bias.

Why use adjusted exposures?

Use of adjusted exposures corrects for varying exposure levels in other rating variables.
For example, when reviewing rating variable A, using adjusted exposures corrects for the possibility that Rating Variable B exposure distribution may vary by rating variable A.

Credibility consideration

Simple general rules that will help in credibility weighting. Always have the "total" as the denominator.

  • Rebase the indicated rates to total
  • Rebase the current rates to total
    • #Loss Ratio Approach:
      • If 'no-change' is the complement, use 1 as the complement to the change factors
      • If some other (e.g. competitor) current rates, normalize them
    • Pure premium approach
      • Divide pure premium by the total Pure premium
  • Credibility weight

Normalization

  • Find the WACR (Weight the current relativities with the (adjusted) exposures)
  • The relativities should be divided by the WACR
  • Done!

When using #Adjusted Pure Premium Approach, ensure you do this weighting with the same exposures, you used for finding the indicated pure premium in the first place, so use adjusted exposures.

  1. If a fixed expenses is material (significant) and a separate expense fee provision for that is not created then the rate relativities have to be adjusted in which case they all will be closer than 1 to than if provided. ↩