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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 non-increasing rate
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Methods for Determining Credibillity of an Estimate
- Classical Credibility Approach
- Buhlmann Credibility
- Bayesian Analysis
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Advantages of the classical credibility approach
- Most commonly used and therefore generally accepted
- Data required is readily available
- Computations are straightforward
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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
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Formula for credibility using Buhlmann Credibility
- Z = N / (N + K)
- K = EVPV / VHM
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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
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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
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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
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Evaluation of Loss Costs of a Larger Group that Include the Group being Rated
- Because data split into classes, believe that experience is diff erent, so combining classes introduces bias and the true expected losses will diff er
- 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
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Evaluation of Loss Costs of a Larger Related Group
- Similar to large group including class in that it is biased and true expected losses diff er: 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
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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
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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
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Evaluation of Harwayne's Method
- Unbiased as it adjusts for distributional diff erences
- Use of multi-state data generally implies it is reasonably accurate: Need enough data to minimize process variance
- Mostly independent since subject and related experience from diff erent states
- Data is available, but computations can be time consuming
- Logical relationship, but may be harder to explain due to calculation complexity
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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
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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
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Complement for an indicated rate change when using the Loss Ratio Approach
C = (LossTrndFact/PremTrndFact) x (1 + prior ind / 1 + prior rate chg)
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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
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Evaluation of Competitors' Rates
- Must consider marketing practices and judgment of the competitor and effects of regulation: Can cause inaccuracy
- Competitors may have di fferent 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
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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
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Evaluation of Increased Limits Factors (ILF)
- PA x (ILF @ A + L) / (ILF @ A - 1)
- If subject experience has diff erent 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 ground-up 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
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Evaluation of Lower Limits Analysis
- Pd x (ILF @ A + L - ILF @ A) / ILF @ d
- Even more prone to bias than fi rst 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
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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 %
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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 fi rst two methods
- Explainable relationship - Controversial for same reason as other methods
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Evaluation of Fitted Curves
- Tends to be less biased and more stable, assuming curve replicates general shape of actual data, and signi cantly 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
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