Practical Applications of Credibility Theory

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Practical Applications of Credibility Theory. Tom Rhodes, FSA, MAAA, FCA AVP &amp; Actuarial Director MIB Solutions February 16, 2007. Practical Applications of Credibility Theory. Importance to Practicing Actuaries Credibility Methods Formula and Variance Real Life Mortality Example

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Practical Applications of Credibility Theory

Tom Rhodes, FSA, MAAA, FCA

AVP & Actuarial Director MIB Solutions

February 16, 2007

Practical Applications of Credibility Theory
• Importance to Practicing Actuaries
• Credibility Methods
• Formula and Variance
• Real Life Mortality Example
• Selection of Method
• Elements of Credibility Theory
• Prior A/E Ratios
• Company A/E Ratios
• Estimate A/E Ratios
• Lapse Application
• Mortality Applications (VM-20, pricing, risk mgmt)
Importance of Credibility Theory
• Company results vary substantially from the intercompany averages
• Credibility Theory provides a statistically sound method of measuring variability of company results from intercompany results.
• Credibility Theory provides company specific estimates of mortality and lapse based on company results, intercompany results and the measured variability.
Importance of Credibility Theory

Prospect of Principles-Based Reserving raises its importance among life and annuity actuaries

• NAIC’s Life and Health Actuarial Task Force
• New Valuation Law and Valuation Manual (VM)
• VM-20 includes credibility practices in valuation of life insurance products
• Companies set reserves based on own experience
• Academy’s Credibility Practice Note highlights
• Determine the level of reliance that can be placed on company experience
• Determine assumptions for modeling company cash flows
Credibility Theory Practices Report
• The SOA’s Committee on Life Insurance Research, Financial Reporting and Product Development Sections sponsored a report on application and adoption of credibility theory for life insurance and annuities.
• MIB Solutions team of Stuart Klugman, Tom Rhodes, Marianne Purushotham and Stacy Gill produced Credibility Theory Practices report: http://www.soa.org/research/life/research-credibility-theory-pract.aspx
Credibility Theory Practices ReportTheory, Practical Methodology and More!

Theory: Formulas developed for two established statistical credibility methods, Limited Fluctuation and Bühlmann Empirical Bayesian

Practical Methodology: For Limited Fluctuation and Bühlmann Empirical Bayesian, formulas for ‘Z’ applied to sample of small, medium and large companies:

• Mortality and lapse results
• Excel files implementing formulas included
• Appendices document this Excel file implementation
• Companies can perform Limited Fluctuation method in-house

Survey and Bibliography

Survey

Bibliography

Credibility MethodsFormula
• Credibility formulas use a credibility factor ‘Z’ varies from 0 to 1.
• The standard form of credibility formula:

Estimate A/E Ratio =

Z × (Company A/E Ratio) + (1-Z) x (Prior A/E Ratio)

• Determine the level of reliance that can be placed on company experience (Z)
• Determine modeling assumptions for company cash flows (Estimate A/E Ratio)
Credibility Methods Variance
• Total Variance = Process Variance + Variance of Hypothetical Means
• Limited Fluctuation uses Process Variance
• Bühlmann Empirical Bayesian uses both Process Variance + Variance of Hypothetical Means
Real Life Mortality Example
• Experience is from ten companies that participated in the SOA 2004-05 experience study
• Companies were selected in order to give a mixture of large, medium and small size companies.
• To maintain strict confidentiality of individual company information, only a portion of each company’s mortality and lapse data was used.
Real Life Mortality Example
• The variability of the Company A/E Ratios around the Prior A/E Ratio of the inter-company study is evident
• In general, Credibility Factor ‘Z’ increases with increasing number of deaths.
• Difference between estimate A/E ratios and company A/E ratios varies substantially among different companies
Real Life Mortality ExampleNS Preferred Class Structure of 2 (NS PCS 2) Bühlmann Empirical Bayesian Method
Elements of Credibility Theory
• Prior A/E Ratios
• Company A/E Ratios
• Estimate A/E Ratios
Prior A/E Ratio in Credibility Theory Methods

Credibility method for PCS 2 NS

• Sample company experience was compared to expected basis 100% of 2001 VBT
• Admittedly NOT a preferred class table
• From 100% of 2001 VBT expected basis goes down to 65.6% of 2001 VBT
• 65.6% of 2001 VBT is the ‘Prior A/E Ratio’ used in credibility formula
Prior A/E Ratio in Credibility Theory Methods
• Fixed Table as Prior A/E Ratio
• Limited Fluctuation Method
• Could use fixed table as Prior A/E Ratio
• For example 100% 2001 VBT
• Intercompany experience does not modify fixed table
• Choice of fixed table can drive results, actuarial judgment is necessary
• For example 65% of 2001 VBT for preferred
Prior A/E RatioActuarial Judgment Needed

The selection of the Prior A/E Ratio affects estimate resulting from credibility method

Prior 65.6% , Company 112.4% , Estimate 70.4%

If one varies the Prior A/E assumption, one varies the Estimate A/E

The selection of the Prior A/E Ratio assumption requires actuarial judgment

Practical Sources of Prior A/E Ratios
• Fixed table (RR table, pricing assumption as % of standard table)
• Pivot tables from SOA Experience Reports
• Important to select appropriate mortality segment to correspond with assumption
• Subsets of industry studies for product/underwriting class/peer group
Company A/E Ratios
• In individual company experience study
• Select mortality segment consistent with mortality segment of Prior A/E Ratio
• Use policy by policy experience results
• Mandatory company studies under both NAIC Model Regulation 815 and proposed Valuation Law/Valuation Manual
• Statistical agent for Life needs to be named, comparable to statistical agent for P&C

Credibility Theory

Lapse Application

Many lapses, ‘Z’ is high

Company Experience Rules

Application of Credibility Methods to Lapse
• Used term policies in the NS PCS 2 for the 7 companies studied for mortality sample
• Recent LIMRA study is basis for expected lapse
• Actual experience of 7 companies produces mean used for Prior A/E Ratio
Credibility Theory and VM-20
• NAIC’s Life and Health Actuarial Task Force
• New Valuation Law and Valuation Manual (VM)
• VM-20 covers valuation of life insurance products
• Credibility theory not used for less than 30 deaths
• Credibility theory used for 30 or greater deaths in credibility set of mortality segments
VM-20: For 30 or greater deaths in credibility set of mortality segments

Determine prudent estimate mortality by:

• Selecting credibility method
• Selecting industry basic table (UCS method)
• Determine mortality from experience studies and credibility method
• Determine margin
• Use credibility mortality plus margin to select industry basic table with higher mortality
Selecting credibility method
• Either the Limited Fluctuation method or the Bühlmann Empirical Bayesian method are options.
• Limited Fluctuation method
• Requires only your company’s data
• Can use Excel tables from MIB Solutions report
• Can be done in-house
• Bühlmann Empirical Bayesian method
• May give higher ‘Z’ than Limited Fluctuation method
• Requires data from multiple companies
• Can only be done by statistical agent
• For this example, use Limited Fluctuation Method
VM-20: For 30 or greater deaths in credibility set of mortality segments

Determine prudent estimate mortality by:

• Selecting credibility method
• Selecting industry basic table (UCS method)
• Determine mortality from experience studies and credibility method
• Determine margin
• Use credibility mortality plus margin to select industry basic table with higher mortality
Selecting industry basic table (UCS method)
• For each mortality segment
• Apply the UCS method for mortality segment
• Select the appropriate RR table
VM-20: For 30 or greater deaths in credibility set of mortality segments

Determine prudent estimate mortality by:

• Selecting credibility method
• Selecting industry basic table (UCS method)
• Determine mortality from experience studies and credibility method
• Determine margin
• Use credibility mortality plus margin to select industry basic table with higher mortality
• Determine mortality segments
• RR table for Prior A/E Ratio
• Company A/E Ratio calculated using policy by policy results from experience study
• Use MIB Solutions report
• Needed formulas, Appendices serve as guide
• Excel files provide template to do calculations
• Apply actuarial judgment

Apply Actuarial Judgment

• Credibility model produces an estimate of flat percentage of base mortality table
• Flat percentages may be modified for higher older age Qx’s
• Mortality curve varied by product characteristics ,e.g., term mortality after level period
• Other actuarial judgment changes possible
VM-20: For 30 or greater deaths in credibility set of mortality segments

Determine prudent estimate mortality by:

• Selecting credibility method
• Selecting industry basic table (UCS method)
• Determine mortality from experience studies and credibility method
• Determine margin
• Use credibility mortality plus margin to select industry basic table with higher mortality
Margins

For credibility theory methods, margin considerations:

• Reliability of Company Experience Studies
• Reliability of Prior A/E Ratio (If a fixed table and not adjusted by intercompany study, future mortality improvement eventually makes it out of date)
• Difference of Company and Estimate A/E Ratios:
• Company E has 46.2% Company and 63.6% Estimate
• Company F has 84.9% Company and 68.2% Estimate
• Future mortality trends
• Refer to SOA ‘s ‘Analysis of Methods for Determining Margins for Uncertainty under a Principles-Based Framework for Life Insurance and Annuity Products’
VM-20: For 30 or greater deaths in credibility set of mortality segments

Determine prudent estimate mortality by:

• Selecting credibility method
• Selecting industry basic table (UCS method)
• Determine mortality from experience studies and credibility method
• Determine margin
• Use credibility mortality plus margin to select industry basic table with higher mortality
Credibility TheoryOther Mortality Applications
• For evaluating pricing assumptions
• Use your company’s pricing assumptions as percentage of expected table as Prior A/E Ratio
• Use company experience study for Co A/E Ratio
• Limited Fluctuation method and Excel files from MIB Solution’s report to produce Estimate A/E Ratio
• Modify results using actuarial judgment
Credibility TheoryOther Mortality Applications
• For risk management application
• Use your best guess as Prior A/E Ratio
• Use company experience study for Co A/E Ratio
• Limited Fluctuation method and Excel files from MIB Solution’s report to produce Estimate A/E Ratio
• Modify results using actuarial judgment
Practical Applications of Credibility Theory
• Important – Co A/E varies from Intercompany A/E
• Credibility Methods applied with actuarial judgment
• Prior A/E Ratios
• Company A/E Ratios
• Estimate A/E Ratios
• Selection of Credibility Method
• Limited Fluctuation can be done in-house by company using MIB Solutions’ Report
• Lapse Application – Company results rule, insist on high quality company lapse experience studies
• Mortality Application - VM-20, evaluating pricing assumptions, risk management studies
Invitations

I.MIB Solutions is seeking current SOA contributor to ILEC study to develop practical implementation techniques for Credibility Theory:

• Use that company’s already submitted data
• Co-author practical credibility techniques paper
• Involvement will
• Improve knowledge of credibility techniques