P&C Reserve Basic

HUIYU ZHANG, Principal Actuary, Goouon

Summer 2008, China

Where are we

Two types of actuaries

Life Actuary: Society Of Actuary. Website: http://www.soa.org

Property & Casualty Actuary: Casualty Actuary Society.

Web site: http://casact.org One combined organization:

American Academy of Actuaries. Web site: http://www.actuary.org/

Actuaries are paid big $$$$, especially P&C Actuary. Here is the information collected by the biggest actuary recuiter DW Simpson http://www.dwsimpson.com/salary

P&C Actuary Salary Survey_DW Simpson 09/2007

  0-0.5 yrs 0.5- 2.5- 4.5- 6.5- 9.5- 14.5- 19.5+

 (excl.

sign-on) 2.5 yrs 4.5 yrs 6.5 yrs 9.5 yrs 14.5 yrs* 19.5 yrs*

              yrs*  

1 exam 46-61 49-63 52-68          

2 exams 47-63 54-67 56-73 61-78        

3 exams 50-65 57-72 60-79 64-85 70-95 76-106    

4 exams 53-68 60-78 64-87 70-92 73-104 88-129    

5 exams   63-85 69-97 74-102 80-117 94-140    

6 exams   70-88 77-108 81-117 88-124 106-161    

ACAS   78-101 82-115 88-130 101-149 114-183 128-251 141-268+

8 exams     85-122 95-143 106-166 127-198    

FCAS       111-170 123-196 143-283 150-405 158-439+

Key functions for insurers

Underwriting Claim Actuary

Pricing Actuary

Reserving Actuary Investment

Why reserving is important?

The financial condition of an insurance company can not be

adequately assessed without sound loss reserve estimates. A loss

reserve is a provision for an insurer’s liability for claims.

Reserving is the term used to describe the actuarial process of

estimating the amount of an insurance company’s liabilities for loss and

loss adjustment expenses.

It is a challenge for the casualty actuary because the estimation process

involves not only complex technical tasks but considerable judgment

as well. No formula provides the real answer.

Why reserve?

To ensure that the company has made “adequate provision” for liabilities associated with all losses, whether reported or not, that were incurred through a specific evaluation date (e.g. 12/31/XX).

The liabilities are related to …– The losses themselves, and– The cost of adjusting the losses, that is, loss

adjustment expense (LAE).

Claim Cycle

A. Event occurs

B. Insurer is notified of the event

C. Insurer gathers facts concerning the event

D. Insurer determines whether the event is covered

E. Insurer may established a case reserve consistent with the expected ultimate payment

F. Insurer may, or may not, issue payment

Major Volatility Drivers:

InternalChanges in contract languageChanges in claim handling processChanges in claim handling staffChanges in claim authorities

ExternalChanges in economic conditionsChanges in legal environmentChanges in inflationChanges in health care

Reserving Terms

A Case Payment is a payment made by the Claims Dept. on individual claims.

A Case Reserve is a liability established by the Claims Depart. on an individual claim.

A Case Incurred Loss is the sum of all case payments and case reserves.

Reserve Terms_Continue

Incurred But Not Reported (IBNR) is the additional reserves that a company must carry such that the sum of the case reserves and the IBNR makes “adequate provision” for liabilities associated with all losses that were incurred through a specific evaluation date. IBNR includes amounts for…

– Development on reported claims, and– Unreported claims that have been incurred through the

valuation date.

Standard Methods of Property – Casualty Loss Development Analysis:

Incurred Loss Development Paid Loss Development Numerous other methods of increasing

analytic sophistication

Paid Loss Development

Assumes past patterns of changes in paid losses from one period to the next are on average predictive of future changes in paid losses. Assumes case reserves are not meaningful.

Incurred Loss Development

Assumes past patterns of changes in the incurred losses from one period to the next are on average predictive of future changes in incurred losses.

Assumes paid dollars and case reserves are equally meaningful.

Incurred Loss Development

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

12 24 36 48 60

2001 2002 2003 2004 2005

Paid & Case Incurred Losses12 24 36 48 60

2001 1,000 1,200 1,320 1,390 1,3802002 1,100 1,320 1,450 1,5202003 1,210 1,450 1,6002004 1,330 1,600 2005 1,460

Age to Age Factors12-24 24-36 36-48 48-60

2001 120% 110% 105% 99%2002 120% 110% 105%2003 120% 110%2004 120%Ave. 120% 110% 105% 99%

0%

20%

40%

60%

80%

100%

120%

140%

2001 2002 2003 2004

12-24 24-36 36-48 48-60

Incurred Age to Age Factors

Paid Losses12 24 36 48 60

2001 200 600 990 1,251 1,3802002 220 660 1,088 1,3682003 242 725 1,2002004 266 8002005 292

0

200

400

600

800

1,000

1,200

1,400

1,600

12 24 36 48 60

2001 2002 2003 2004 2005

Paid Loss Development

Age to Age Factors12-24 24-36 36-48 48-60

2001 300% 165% 126% 110%2002 300% 165% 126%2003 300% 166%2004 301%Ave. 300% 165% 126% 110%

0%

50%

100%

150%

200%

250%

300%

350%

2001 2002 2003 2004

12-24 24-36 36-48 48-60

Paid Age to Age Factors

Projection of Ultimate Paid Losses12 24 36 48 60

2001 200 600 990 1,251 1,3802002 220 660 1,088 1,368 1,5092003 242 725 1,200 1,513 1,6692004 266 800 1,321 1,665 1,8372005 292 876 1,447 1,824 2,012

0

500

1,000

1,500

2,000

2,500

12 24 36 48 60

2001 2002 2003 2004 2005

Ultimate Paid Losses

Case Reserved Losses12 24 36 48 60

2001 800 600 330 139 02002 880 660 362 1522003 968 725 4002004 1,064 8002005 1,168

0

200

400

600

800

1,000

1,200

1,400

12 24 36 48 60

2001 2002 2003 2004 2005

Case Reserve Development

Ratio of Current Case to Prior Case24 36 48 60

2001 75% 55% 42% 0%2002 75% 55% 42%2003 75% 55%2004 75%Ave. 75% 55% 42% 0%

0%

10%

20%

30%

40%

50%

60%

70%

80%

2001 2002 2003 2004

24 36 48 60

Case Age to Age Factors

Projection of Case Reserves12 24 36 48 60

2001 800 600 330 139 02002 880 660 362 152 02003 968 725 400 168 02004 1,064 800 440 185 02005 1,168 876 482 203 0

0

200

400

600

800

1,000

1,200

1,400

12 24 36 48 60

2001 2002 2003 2004 2005

Ultimate Case Reserves

Paid Losses12 24 36 48 60

2001 200 600 990 1,251 1,3802002 220 660 1,088 1,3682003 242 725 1,2002004 266 8002005 292

0

200

400

600

800

1,000

1,200

1,400

1,600

12 24 36 48 60

2001 2002 2003 2004 2005

Paid Loss Development

Change In Paid24 36 48 60

2001 400 390 261 1292002 440 428 2802003 483 4752004 534

0

100

200

300

400

500

600

24 36 48 60

2001 2002 2003 2004

Change In Paid

Ratio of Change In Paid to Prior Case24 36 48 60

2001 50% 65% 79% 93%2002 50% 65% 77%2003 50% 66%2004 50%Ave. 50% 65% 78% 93%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

24 36 48 60

2001 2002 2003 2004

Ratio of Change in Paid to Prior Case

Paid Projection12 24 36 48 60

2001 200 600 990 1,251 1,3802002 220 660 1,088 1,368 1,5092003 242 725 1,200 1,513 1,6692004 266 800 1,321 1,665 1,8372005 292 876 1,447 1,824 2,012

0

500

1,000

1,500

2,000

2,500

12 24 36 48 60

2001 2002 2003 2004 2005

Ultimate Paid Loss Projection

Why Age, why triangle

People Comparison AY Loss Comparison Past trend for same age group could be

used to analyze future unknown

_People

_Loss

How to age

Examples:

HowToAgeData1

HowToAgeData2 End of appetizer, next, main dish.

Paper Introduction

Blame received for reserving actuary

2003 S&P #1 CAS Story Caveat of link ratio

Simplest Liner Relationship

Go to paper Triangle 1 and Triangle 2

Reserve Variability Estimates

Parameter Risk, Process Risk, Model Risk and other risks.

Two approaches

Bootstrap_ Parameter Risks

Simulating LDF_ Process Risks

2 Stage Bootstrap_ Parameter and Process Risks

2 Stage Bootstrap

Bootstrap: Resample with replacement,

“new” generation of simulation. Resample residuals. Residuals are generally

iid. What is the residual?

Pearson residuals (good for Poisson GLMs): (A-E)/E^0.5

How to get residual

Triangle 3: residuals Triangle 4: A- Actual Payments (incremental) Triangle 5: E- Expected Payments

(incremental)

Bootstrap and Pseudo triangles

Triangle 7: Resampled Residuals

Compare with triangle 3 Triangle 8: Pseudo triangles Repeat End of First Stage Explain Results: Table 2

Stage 2: resolve process risk

Interrupt First Stage Mean, variance, distribution

Appendix: triangle 9, triangle 10 Explain Result: table 4. End of main dish. Next: Dessert