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Necessary Criteria for Measures of Credibility
 1. Z must be greater than or equal to 0 and less than or equal to 1: No negative credibility and capped at fully credible
 2. Z should increase as the number of risks underlying the actuarial estimate increases (all else equal)
 3. Z should increase at a nonincreasing rate

Methods for Determining Credibillity of an Estimate
 Classical Credibility Approach
 Buhlmann Credibility
 Bayesian Analysis

Advantages of the classical credibility approach
 Most commonly used and therefore generally accepted
 Data required is readily available
 Computations are straightforward

Simplifying assumptions about observed experience using Classical Credibility Approach
 Exposures are homogeneous (i.e., same expected number of claims)
 Claim occurrence is assumed to follow a Poisson distribution
 No variation in the size of loss

Formula for credibility using Buhlmann Credibility
 Z = N / (N + K)
 K = EVPV / VHM

Assumptions using Buhlmann Credibility
 Complement of credibility is given to the prior mean
 Risk parameters and risk process do not shift over time
 Expected value of the process variance of the sum of N observations increases with N
 Variance of the hypothetical means of the sum of N observations increases with N

Desirable Qualities of a complement of Credibility
 1. Accurate
 2. Unbiased
 3. Statistically Independent from Base Statistic
 4. Available
 5. Easy to Compute
 6. Logical Relationship to Base Statistic

Complement of Credibility for First Dollar Ratemaking
 1. Loss Costs of a Larger Group that Include the Group being Rated
 2. Loss Costs of a Larger Related Group
 3. Rate Change for the Larger Group Applied to Present Rates
 4. Harwayne's Method
 5. Trended Present Rates
 6. Competitors' Rates

Evaluation of Loss Costs of a Larger Group that Include the Group being Rated
 Because data split into classes, believe that experience is different, so combining classes introduces bias and the true expected losses will differ
 Not independent because subject experience is included in group experience. However, may not be big issue if subject experience doesn't dominate the group
 Typically is available, easy to compute, and some logical connection

Evaluation of Loss Costs of a Larger Related Group
 Similar to large group including class in that it is biased and true expected losses differ: May make adjustment for bias to related experience to match exposure to loss
 Is independent  which may make it a better choice than large group including class
 Typically is available, easy to compute, and some logical connection if groups closely related: Note  if adjustment made for bias, may be more difficult to compute

Evaluation of Rate Change for the Larger Group Applied to Present Rates
 Current Loss Cost of Subject Experience (CLCSE)
 C = CLSCE x (LargerGrpIndLC / LargerGrpCurrLC)
 Largely unbiased and likely accurate over the long term assuming rate changes are small
 Independence depends on size of subject experience relative to the larger group
 Typically is available, easy to compute, and logical that rate change of bigger group is indicative of rate change of subject experience

Calculations in Harwayne's Method
 Compute the state overall means with the base state class distribution
 Compute individual state adjustment factors by dividing subject average PP by adjusted related state PP
 Multiply each related state's base class by state adjustment factor to get adjusted state class rates
 Complement equals the exposure weighted average of the adjusted related state rates

Evaluation of Harwayne's Method
 Unbiased as it adjusts for distributional differences
 Use of multistate data generally implies it is reasonably accurate: Need enough data to minimize process variance
 Mostly independent since subject and related experience from different states
 Data is available, but computations can be time consuming
 Logical relationship, but may be harder to explain due to calculation complexity

Trended Present Rates
 Current rates should be adjusted for the previously indicated rate, not what was implemented
 Changes in loss cost levels: May be due to inflation, distributional shifts, safety advances, etc.; Trend period (t) taken from original target eff date of current rates to planned eff date

Complement for the Pure Premium Approach
 Present Rate (PR)
 Loss Cost Implemented with Last Review (LCILR)
 C = PR x Trend ^ t x (PrevIndLC/LCILR)  1

Complement for an indicated rate change when using the Loss Ratio Approach
C = (LossTrndFact/PremTrndFact) x (1 + prior ind / 1 + prior rate chg)

Evaluation of Trended Present Rates
 Accuracy depends largely on process variance of historical loss costs: Primarily used for indications with large amounts of data
 Unbiased since pure trended loss costs are unbiased
 Independence depends on experience used: If complement comes from a review that used data from 2007  2009 and subject experience is from 2008  2010, then they are not independent
 Data is readily available, easy to compute, and is easily explainable

Evaluation of Competitors' Rates
 Must consider marketing practices and judgment of the competitor and effects of regulation: Can cause inaccuracy
 Competitors may have different underwriting and claims practices that creates bias
 Will be independent
 Calculations may be straightforward but getting the data may be difficult
 Generally accepted by regulators because of logical relationship: May be the only choice

Excess Ratemaking  products that cover claims that exceed some attachment point
 1. Issues:
 Excess ratemaking deals with volatile lines and low volumes of data
 Due to low volume, often use loss costs below attachment point to predict excess losses
 Slow development and trend in excess layers can also complicate projections
 2. Increased Limits Factors (ILF)
 3. Lower Limits Analysis
 4. Limits Analysis
 5. Fitted Curves

Evaluation of Increased Limits Factors (ILF)
 PA x (ILF @ A + L) / (ILF @ A  1)
 If subject experience has different size of loss distribution than used in developing the ILFs, procedure will be biased and inaccurate, but often best available estimate
 Error associated with estimate tends to be independent of error associated with base statistic
 Data needed incl ILFs and groundup losses that haven't been truncated below attachment
 Ease of computation  Easiest of the excess complements to compute
 Explainable relationship  Controversial; more logically related to losses below attach point

Evaluation of Lower Limits Analysis
 Pd x (ILF @ A + L  ILF @ A) / ILF @ d
 Even more prone to bias than first method because losses far below attachment point accentuates the impact of variations in loss severity distributions
 Losses capped at lower limit may increase stability and accuracy
 Error associated with estimate tends to be independent of error associated with base statistic
 Data a little more available since losses capped at lower limit
 Ease of computation  Just slightly more complex than 1st method
 Explainable relationship  Controversial for same reason as first method

Calculation of Limits Analysis
 LR x Sum(Pd) x (ILF @ min(d, A+L)  ILF @ A) / ILF @ d
 Analyze each limit of coverage separately
 Assume all limits will experience same loss ratio
 Calculate total loss cost (Prem x ELR) for each layer
 Use ILFs to calculate % loss in layer
 Multiply loss cost from layer by calculated %

Evalution of Limits Analysis
 Biased and inaccurate to same extent as prior two methods, plus assumes LR doesn't vary by limit
 Typically used by reinsurers that don't have access to the full loss distribution
 Calculations are straightforward but take more time than the first two methods
 Explainable relationship  Controversial for same reason as other methods

Evaluation of Fitted Curves
 Tends to be less biased and more stable, assuming curve replicates general shape of actual data, and signicantly more accurate when few claims in excess layer
 Less independent due to reliance on larger claims to fit curve
 Most complex procedure and requires data that may not be readily available
 Most logically related to losses in layer, but complexity may make it hard to communicate

