Non-Traditional Reserving Methods: Back to Basics

MAF 2006-03-17 1March 17, 2006

Actuarial & Analytical Consulting

Non-Traditional Reserving Methods: Back to BasicsRajesh SahasrabuddheAon Risk Consultants

Midwestern Actuarial ForumFriday March 17, 2006

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Agenda

• Motivation• Just One Example• Questions and Answers

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Motivation

• Non-Traditional = Back to Basics: How can this be?– Exams 1-4: Actuarial Modeling: ∫, ∑, ∏, f(x), F(x),

Simulation– Exams 5-9: Arithmetic Models: +, / , *, -, deterministic– Why do we have this disconnect– How do we define “traditional”?

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Motivation

• The value proposition of traditional models: benefit v.s. cost– Zehnwirth / Barnett:

• “The standard link ratio models carry assumptions not usually satisfied by the data”– That same paper was later published in the Proceedings with softer

language: “Most loss arrays don’t satisfy the assumptions of standard link ratio techniques.”

– Why:• What does the nature of claims-made and occurrence

development patterns tell us? What is a model?• Consideration of Trend, Limits and Deductibles.• Information aggregated is information lost.• Maturity and predictive ability.

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Motivation

• The value proposition of traditional models: benefit v.s. cost– The output of traditional (deterministic) models – is it

good enough?– How do (will) stakeholders view the cost of actuarial

models?– Management is becoming increasing quantitative.

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Timeline for Case Study

Mar 31, 2001 Oct 1, 2001 Sep 30, 2002

Reserve Analysis – All events occurring

SIR Loss Forecast – All events occurring

Excess Insurance – All events reported

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True IBNR

• True IBNR is estimated using a frequency x severity approach

• Why? - This model is the most consistent with the real world!

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True IBNR Frequency

• IBNR Frequency is a direct function of exposure, initial expected ultimate frequency and report lag - i.e. IBNR frequency should be estimated using a B-F approach

• Critical Assumption – How long between accident occurrence and claim reporting – Use approach contained in Weissner – “Estimation of the Distribution of Report Lags by the Method of Maximum Likelihood” - Proceedings of the Casualty Actuarial Society (1978)

• Frequency is simulated as a Poisson distribution (or Negative Binomial)

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Report Lag

• The lag experience is truncated from above• Similar to a deductible problem in reverse (Hogg &

Klugman; Klugman, Panjer,& Wilmot)

(1) (2) (3) (4) (5) (6) (7)per loss database (2) - (1) (3) - (1) f (x ) / F (M ) ln (6)

Incident Date Report Date

Valuation Date

Report Lag (Days) (x )

Maximum Report Lag for Suit (M )

Conditional Likelihood

(L )

Log - Likelihood

ln (L )

10/17/90 10/07/92 03/31/01 721 3818 8.5E-04 -7.0677718209/28/91 05/10/93 03/31/01 590 3472 1.2E-03 -6.6979981510/19/91 10/17/92 03/31/01 364 3451 1.6E-03 -6.4650232611/22/91 11/09/92 03/31/01 353 3417 1.5E-03 -6.4695191701/15/92 01/21/93 03/31/01 372 3363 1.6E-03 -6.4628352104/20/92 05/12/93 03/31/01 387 3267 1.6E-03 -6.46110987

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Report Lag

• Use Maximum Likelihood Techniques (“Loss Models” – KPW)

• Use a B-F model

Report Lag Distribution

F (

x)

Report Lag (days)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

20

0

40

0

60

0

80

0

10

00

12

00

14

00

>5.0% 5.0%90.0%124.2852 949.8183

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Report Lag

• Use pattern to allocate to claims-made periods• May also apply model to estimate lag between report and

closing

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True IBNR Severity and Settlement Model

• Severity Model

Closed w/ Indemnity?

YesNo

Exp. Only ModelIndemnity Model

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Severity and Settlements Models

• Fit severity models using individual claim data• Myriad of references for estimating claim severity distributions.

My personal suggestions are:• Klugman, Panjer, & Wilmot - Loss Models• Keatinge – Modeling Losses with the Mixed Exponential

Distribution• Severity and Settlement models can be (should be?)

conditional on report and / or closing lag• Model is typically multi-modal

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IBNER

• IBNER may be estimated using:– Severity Models and Bayesian theory– Transition Matrices (Mahon paper)– Last Resort: Case reserve adequacy statistics for the

insurance industry – claims made coverage triangles from A.M. Best. (not a good option)

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Loss Forecast and Excess Insurance Analysis• Through our True IBNR reserve analysis, we have already

developed the parameters necessary for: the loss forecast and the excess insurance analysis!

• So we simply extend to the prospective year; but separately capture the results

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Simulation (Part 1)

• Model the entire claims process

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Simulated OccurrencePolicy Period

Suit No. Inception Expiry

Simulated Accident

DateSimulated

Report DateSimulated

Closed DatePV Factor

at 5%PV Factor

at 6%PV Factor

at 7%

1 10/1/2000 3/31/2001 10/5/2000 9/6/2002 6/27/2003 0.919 0.904 0.8892 10/1/2001 9/30/2002 3/8/2002 8/24/2002 4/30/2002 0.972 0.967 0.9613 10/1/1999 9/30/2000 12/11/1999 10/28/2001 3/31/2003 0.929 0.916 0.9034 10/1/2000 3/31/2001 11/1/2000 7/4/2002 5/30/2002 0.968 0.962 0.9565 10/1/1998 9/30/1999 9/15/1999 8/28/2000 6/12/2004 0.877 0.855 0.8336 10/1/2001 9/30/2002 6/29/2002 2/20/2003 5/6/2005 0.839 0.811 0.7847 10/1/2001 9/30/2002 9/25/2002 6/17/2004 8/29/2006 0.787 0.751 0.7178 10/1/1999 9/30/2000 3/29/2000 11/17/2001 12/21/2002 0.942 0.931 0.9219 4/1/2001 9/30/2001 6/17/2001 6/16/2002 10/27/2002 0.949 0.939 0.930

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Simulation (Part 2)

• Model the entire claims process

(10) (11) (12) (13) (14)

Simulated ParametersWith Indemnity (1); Expense Only (2); No Suit Simulated

(0) Parameter 1 Parameter 2

Provision for Expense on

Suits w/ Indemnity

Simulated Indemnity

and Expense

1 2.000E+00 1.621E-06 0.035 676,2592 9.412E+00 7.531E-01 0.000 8,6151 1.000E+00 2.894E-05 0.852 4,6072 9.353E+00 7.531E-01 0.000 21,6671 2.000E+00 1.528E-06 0.097 475,5122 9.412E+00 7.531E-01 0.000 8,1791 1.000E+00 3.162E-05 0.734 86,1921 2.000E+00 1.574E-06 0.058 204,3192 9.382E+00 7.531E-01 0.000 3,550

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Result

• A model that is both flexible and robust• A model that makes sense – ties with the real world• A model that provides results of interest to stakeholders

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Questions and Answers

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Non-Traditional Reserving Methods: Back to Basics