practical applications of credibility theory

© 2007 MIB Solutions, Inc. All Rights Reserved

Practical Applications of Credibility Theory

Tom Rhodes, FSA, MAAA, FCA

AVP & Actuarial Director MIB Solutions

February 16, 2007



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

http://www.soa.org/research/life/research-credibility-theory-pract.aspx

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)

Addresses Academy Practice Note Highlights

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


Mean of Company

A

Mean ofCompany

B

Mean of Company

C

Mean of Company

F

Mean of Company

E

Mean of Company

D

Mean ofAll

Companies

For Company A’s Credibility FactorLimited Fluctuation Method

Uses the variation of a Co A’s observations about Co A’s mean & the variation between

Co A’s mean and the overall mean


Mean of Company

A

Mean ofCompany

B

Mean of Company

C

Mean of Company

F

Mean of Company

E

Mean of Company

D

Mean ofAll

Companies

For Company A’s Credibility FactorBühlmann empirical Bayesian Method

Uses the variation of a Co A’s observations about Co A’s mean & the variation

between each Co’s mean and overall mean



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.



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)Limited Fluctuation Method

2001 VBT Prior

Company 2001 VBT Credibility

Limited Fluctuation

A/E Ratio A/E Ratio Factor Number A/E Ratioby Amount Company by Amount Z of Deaths by Amount

65.6% B 66.4% 0.060 9 65.7%65.6% D 109.3% 0.075 17 68.9%65.6% E 46.2% 0.108 36 63.6%65.6% F 84.9% 0.134 65 68.2%65.6% H 65.3% 0.152 554 65.6%65.6% I 57.8% 0.205 153 64.0%65.6% J 112.4% 0.102 63 70.4%


Real Life Mortality ExampleNS Preferred Class Structure of 2 (NS PCS 2) Bühlmann Empirical Bayesian Method

2001 VBT Prior

Company 2001 VBT Credibility

Buhlmann empirical Bayesian

A/E Ratio A/E Ratio Factor Number A/E Ratioby Amount Company by Amount Z Of Deaths by Amount

65.6% B 66.4% 0.072 9 65.7%

65.6% D 109.3% 0.068 17 68.6%

65.6% E 46.2% 0.266 36 60.5%

65.6% F 84.9% 0.231 65 70.1%

65.6% H 65.3% 0.336 554 65.5%

65.6% I 57.8% 0.511 153 61.7%

65.6% J 112.4% 0.118 63 71.1%


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

Intercompany study adjusts

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


Term Lapse for NS PCS 2Limited Fluctuation Method

Seven Co Prior Company Credibility

Limited Fluctuation

A/E Ratio A/E Ratio Factor Number A/E Ratioby Amount Company by Amount Z of Lapses by Amount

150.1% B 130.9% 0.822 1,069 134.3%150.1% D 123.8% 0.091 30 147.7%150.1% E 97.3% 1.000 1,221 97.3%150.1% F 112.1% 0.263 95 140.1%150.1% H 174.8% 1.000 15,540 174.8%150.1% I 142.8% 1.000 7,966 142.8%150.1% J 73.7% 1.000 994 73.7%


Term Lapse for NS PCS 2 Bühlmann Empirical Bayesian Method

Seven Co Prior Company Credibility

Buhlmann empirical Bayesian

A/E Ratio A/E Ratio Factor Number A/E Ratioby Amount Company by Amount Z Of Lapses by Amount

150.1% B 130.9% 0.982 1,069 131.3%150.1% D 123.8% 0.390 30 139.8%150.1% E 97.3% 0.985 1,221 98.1%150.1% F 112.1% 0.826 95 118.7%150.1% H 174.8% 0.998 15,540 174.8%150.1% I 142.8% 0.997 7,966 142.8%150.1% J 73.7% 0.988 994 74.7%


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:

1. Selecting credibility method

2. Selecting industry basic table (UCS method)

3. Determine mortality from experience studies and credibility method

4. Determine margin

5. 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







4. Determine margin



Selecting industry basic table (UCS method)

For each mortality segment

Apply the UCS method for mortality segment

Select the appropriate RR table







4. Determine margin



Determine Mortality from Experience Studies and Credibility Method

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


Determine Mortality from Experience Studies and Credibility MethodApply Actuarial Judgment

Start with credibility model’s Estimate A/E Ratio

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







4. Determine margin



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’







4. Determine margin



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



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

Provide roadmap for applying in your company

II. If you have questions on implementing credibility

theory, call me for free advice!


Questions?

[email protected]

917-400-0213

mailto:[email protected]

practical applications of credibility theory

Documents

credibility practices

adoption of credibility

credibility factor z

z company ae ratio

intercompany results

modeling company cash

life insurance research

limited fluctuation