MATH 3640 Watt Lecture 16 Credibility Methods PDF

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MATH 3640

Credibility Methods

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Credibility

Introduction • Law of Large Numbers – as the volume of similar, independent exposure units increases, the observed experience will approach the “true” experience. • When the volume of data for overall ratemaking or classification ratemaking is not sufficient to produce accurate and stable results, other information should be supplemented (i.e. complement of credibility). • Science of credibility in ratemaking deals with blending an actuarial estimate based on the observed experience with one (or more) related experience (e.g. industry information) in order to improve the estimate of the expected value. • ASOP #25 defines credibility as “a measure of the predictive value in a given application that the actuary attaches to a particular body of data.”

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Credibility

Necessary Criteria for Measures of Credibility • Amount of credibility given to the observed experience is denoted as Z • Z should meet three criteria – 0≤Z ≤1 – Z should increase as the number of risks underlying the actual experience increases – Z should increase at a non-increasing rate

Estimate = Z x Observed Experience + (1 – Z) x Related Experience 3

Credibility

Common Credibility Methods • Classical Credibility (i.e. Limited Fluctuation Credibility) • Buhlmann Credibility (i.e. Least Squares Credibility) • Bayesian Credibility

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Credibility - Classical Credibility

Classical Credibility Approach • Also referred to as ‘Limited Fluctuation Credibility’ – Most frequently used in insurance ratemaking • Goal: Limit the effect that random fluctuations in the observations have on the risk estimate • Value of credibility (Z) is used to assign weights to the observed experience and to some related experience Estimate = Z x Observed Experience + (1 – Z) x Related Experience After assumptions and simplification… • Y = total number of claims

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Credibility - Classical Credibility

Example

80%  90th  1.28 90%  95th  1.645 95%  97.5th  1.96 99%  99.5th  2.58

• You may determine that the data is fully credible when there is a 90% probability (p) that the observed experience is within 5% of the expected value (k). – 90% probability  Standard normal table, 95th percentile  1.645 standard deviations above the mean. – Expected number of claims for fully credibility given these assumptions:

• Observed pure premium of $200 is based on 100 claims • Pure premium of the related experience is $300 Credibility weighted pure premium estimate: 0.30 x $200 + 0.70 x $300 = $270 6

Credibility - Classical Credibility

Practice Using ‘Classical Credibility’, calculate the credibility-weighted indication, given the following information: • It is determined that the data is fully credible when there is a 95% probability that the observed experience is within 10% of the expected value. • The indication based on internal data is +10% based on 200 claims • The indicated change based on related experience is +5%

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Credibility - Classical Credibility

Practice - Answer Using ‘Classical Credibility’, calculate the credibility-weighted indication, given the following information: • It is determined that the data is fully credible when there is a 95% probability that the observed experience is within 10% of the expected value. • The indication based on internal data is +10% based on 200 claims • The indicated change based on related experience is +5%

Credibility-Weighted Indication 8

Credibility - Classical Credibility

Practice Using ‘Classical Credibility’, calculate the credibility-weighted indication, given the following information: • It is determined that the data is fully credible when there is a 99% probability that the observed experience is within 4% of the expected value. • Pure premium based on internal data is $400 based on 3,000 claims • Pure premium based on related experience is $500 • LAE ratio is 10% • Average fixed expense is $50 • Variable expenses are 12% • The target profit is 15%

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Credibility - Classical Credibility

Practice - Answer Using ‘Classical Credibility’, calculate the credibility-weighted indicated premium, given the following information: • It is determined that the data is fully credible when there is a 99% probability that the observed experience is within 4% of the expected value. • Pure premium based on internal data is $400 based on 3,000 claims • Pure premium based on related experience is $500 ଶ • LAE ratio is 10% • Average fixed expense is $50 • Variable expenses are 12% • The target profit is 15%

Credibility-Weighted Indicated Premium

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Credibility - Classical Credibility

Practice Using ‘Classical Credibility’ fill in the following table using the given information: Remember that frequency = # claims / # exposures

k 7.5% 7.5% 15.0% 15.0%

p 90.0% 99.0% 90.0% 99.0%

Projected Frequency 5.0% 5.0% 5.0% 5.0%

# Claims Full # Exposures Credibility Full Credibility

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Credibility - Classical Credibility

Practice - Answers Using ‘Classical Credibility’ fill in the following table using the given information: Remember that frequency = # claims / # exposures

k 7.5% 7.5% 15.0% 15.0%

p 90.0% 99.0% 90.0% 99.0%

Projected Frequency 5.0% 5.0% 5.0% 5.0%

# Claims Full # Exposures Credibility Full Credibility

481 1,179 120 295

9,621 23,576 2,405 5,894

ଶ

ଶ

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Credibility – Buhlmann Credibility

Buhlmann Credibility • Referred to as “least squares” credibility • Goal: Minimize the square of the error between the estimate (i.e. sample mean) and the true expected value (i.e. true mean or population mean) of the quantity being estimated • Estimate = Z x Observed Experience + (1 – Z) x Prior Mean • Assumption: Observations are assumed to be independent and identically distributed (iid) • Classical credibility considers related experience, this method considers a prior mean o Prior Mean = Prior assumption of the risk estimate

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Credibility – Buhlmann Credibility • How much should the class rate be modified for the experience rating? How much credibility should be given to the actuarial experience of the individual risk? Two factors to answer these questions: – 1) How homogenous are the classes? • If all of the risks in the classes were identical there would be no need to do individual rating. If there is significant difference in the expected outcomes for the risks, more weight should be given to the individual loss experience. • Each risk in the class has its own individual risk mean called its hypothetical mean. VHM = “Between Variance” – Small VHM  More class homogeneity – Large VHM  More class heterogeneity – 2) How much variation is there in the individual loss experience? • If there is a large amount of variation in the individual risk experience, the actual observed experience may be far from the expected value. • EVPV = “Within Variance” is the avg value of the process variance over the entire class of risks

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Credibility – Buhlmann Credibility

• Each risk in the class has its own individual risk mean called its hypothetical mean. VHM = “Between Variance” – Small VHM  More class homogeneity – Large VHM  More class heterogeneity

• EVPV = “Within Variance” is the avg value of the process variance over the entire class of risks

let K =





n is the number of observations (it may be easier to think on n as the number of rows in your dataset) • Note that if VHM is larger than EVPV, Z increases closer to 1.0. This should make sense… The overall volatility of your data is relatively good (lower EVPV) and your data is good enough to be able to distinguish well between the individual (large VHM), so it is better at predicting. Thus, it should be more credible. • If EVPV is large, particularly in comparison to VHM, Z will decrease. Think through this… 15

Credibility – Buhlmann Credibility Example Calculate the Buhlmann credibility-weighted estimate assuming the following: • • • •

The observed pure premium is $200 based on 21 observations The expected value of the process variance (EVPV) is 2.00 The variance of the hypothetical means (VHM) is 0.50 The prior mean is $225

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Credibility – Buhlmann Credibility Example You are given: • Annual claim frequency for an individual policyholder has mean • The prior distribution for is uniform on the interval [0.5,1.5]. • The prior distribution for ଶ is exponential with mean 1.25.

and variance

ଶ

.

A policyholder is selected at random and observed to have no claims in Year 1. Using Buhlmann credibility, estimate the number of claims in Year 2 for the selected policyholder.

ଶ ଶ

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Credibility – Buhlmann Credibility

Practice You are given: • The number of claims made by an individual insured in a year has a Poisson distribution with mean . • The prior distribution for is gamma with parameters α = 1 and θ = 1.2. Three claims are observed in Year 1, and no claims are observed in Year 2. Using Buhlmann credibility, estimate the number of claims in Year 3. Note: Expected value of a Gamma is αθ and variance is αθ2

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Credibility – Buhlmann Credibility

Practice - Answer You are given: • The number of claims made by an individual insured in a year has a Poisson distribution with mean . • The prior distribution for is gamma with parameters α = 1 and θ = 1.2. Three claims are observed in Year 1, and no claims are observed in Year 2. Using Buhlmann credibility, estimate the number of claims in Year 3.

ଶ

ଶ

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Credibility – Bayesian Credibility

Bayesian Credibility • There is no Z parameter – distributional assumption is made about the data • Prior estimate is adjusted to reflect new information • The new information is incorporated into the prior estimate in a probabilistic manner via Bayes Theorem o This is different than classical and least squares credibility where new information is incorporated via credibility weighting • Bayesian credibility tends to be more complex than classical and least squares credibility o Not used in practice as often, but I believe that will change! • Buhlmann/Least-squares credibility is the weighted least squares line associated with the Bayesian estimates o In certain situations, the Bayesian estimate is equal to the least squares estimate • Bayes Theorem o P(A | B) =

[P (B | A) x P(A)] P(B)

o P (Risk Type | Observation) =

[P(Observation | Risk Type) x P(Risk Type)] P(Observation) 20...