10 Data Commandments for Overcoming

the Law of Small Numbers Lessons learned during Seven Years of

Actuarial Work

By Stephen L. Kolk, ACAS

Vice President of Actuarial Services

& Chief Actuary of Grange Insurance Group

Three “R’s”

wRiting

aRithmetic

Reading

Three “R’s” of Actuarial

Reserves

Reports

Rates

Always advance, or else

• He who moves not forward, goes backward. – Johann Wolfgang von Goethe

• If everything's under control, you're going too slow. –Mario Andretti

• Even if you're on the right track, you'll get run over if you just sit there.

– Will Rodgers

TEAMWORK

• Coming together is a beginning

• Keeping together is progress

• Working together is success

- Henry Ford

The LAW of LARGE NUMBERS

• The Law of Large Numbers states that regardless of the shape of the original distribution, the taking of averages results in a normal distribution.

• Insurance is an uncertain business, so we have to use averages• The bigger the number of items in our average, the smaller the

error.• You need to average lots of items to be sure your average is

indicative.• This is why we looked and will continue to look at industry results…• This is also why we watch larger states more closely

GRANGE INSURANCE GROUPby Company

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

140,000,000

160,000,000

1997 1998 1999 2000 2001 2002

GIA

RM

GRANGE INSURANCE GROUPDWP by Major Product

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

140,000,000

160,000,000

1997 1998 1999 2000 2001 2002

PP Auto

Home

Farm

Comm Auto

Other

GRANGE INSURANCE GROUP DWP by State

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

140,000,000

160,000,000

1997 1998 1999 2000 2001 2002

CA

CO

ID

MT

OR

WA

WY

The LAW of SMALL NUMBERS

• The Law of Small Numbers states that there are not enough small numbers to satisfy all the demands placed on them.

- Richard Guy, statistician• You can't tell by looking [at a few examples]. • Superficial similarities spawn spurious statements. • Capricious coincidences cause careless conjectures. • Early exceptions eclipse eventual essentials. • Initial irregularities inhibit incisive intuition

TEN ACTUARIAL COMMANDMENTS

#10 – Ignore the law of small numbers and truth will bite you.

#9 – Find credible data. How? Graph thy data

#8 Skew is the rule

#7 Prune thy data

#6 Weed thy data

TEN ACTUARIAL COMMANDMENTS

#5 Explore thy data…

#4 Show thy data!

#3 Sort, shuffle and juggle thy data!

#2 Grow thy data!

#1 Know thy data!

COMMANDMENT #1

KNOW THY DATA

• To understand what has happened, find out what was counted and how.

Homeowners Pure Premium

PURE PREMIUMISO vs GIG companies

0

50

100

150

200

250

300

350

400

Calendar Quarter Ending

GIA

RM

ISO

All GIG StatesHomeowners (excluding catastrophes) All Policy Forms Combined

COMMANDMENT #2

GROW THY DATA

• Master actuarial measurements by studying and using knowledge

• Master business results by refining databases and reports from them

• Keep basics a part of the many ways, old & new, to capture and/or picture truth.

COMMANDMENT #3

SORT, SHUFFLE AND JUGGLE THY DATA

• What numbers do best is forget things. As soon as you write them down, they forget both where they have been and where they are going.

• So it takes actuarial work to make numbers remember the right things.

COMMANDMENT #4

• SHOW THY DATA!

• Depict stories whenever possible with graphs and pictures. The best way to say this is to paraphrase St. Francis of Assissi: “Use numbers only when absolutely necessary.”

• In actuarial work there are a lot of absolutely necessary times for numbers.

What rate change is neededto cover a 15% trend change?

Losses

Adjustment Costs

Expenses

Total Costs

Premium

Combined Ratio

Cost $60 10 30$100$100100%

Trend

Change

+15%

+15%

New Cost

$69.00

11.50

30.00

$110.50

$100

110.5%Required Rate Change = $110.50 / $100 = +10.5%

COMMANDMENT #5

• EXPLORE THY DATA

• To know, grow, and show data in a true manner, you have to understand the data.

• How? Spend time with thy data.

• DIG THY DATA! (pun intended)

COMMANDMENT #6

• WEED THY DATA!

• Flowers and fruits in thy data garden are grand.

• But weeds creep in.

• To keep data flowers and data fruit healthy… WEED THY DATA!

• Use the “Yellow Book”… make sure your figures match official Annual Statements

COMMANDMENT #7

PRUNE THY DATA

• Give non-number people appropriate focus. Match displayed precision to the accuracy of the facts.

• Don’t tell more than facts reveal.

• Pruning corollary: DO SOME GRAFTING.

• Get and use more credible data when you run out of truth.

What is happening to Industry Homeowners’ Rates?

Conning Average Homeowner Premium Changeversus Homeowner Loss Costs

-2%

0%

2%

4%

6%

8%

10%

1996

1997

1998

1999

2000

2001

Mar

'02

Jun

e '0

2

An

nu

al C

han

ge

in P

erce

nt

CPI - Total Housing Costs

CPI - HouseholdFurnishings

CPI - Medical Costs

CPI - Hospital and relatedservices

Conning AvgHomeowners' Ins. Prem.

COMMANDMENT #8

• SKEW IS THE RULE

• My college statistical training was incomplete. There I only learned about the Normal Distribution.

• On the actuarial job, there is no normal mountain of insurance data.

• Therefore, know that skew is the rule or…

be skewered!

GIA Farmowners vs Industry

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

1997 1998 1999 2000 2001 08/03

Year

Adj

uste

d Lo

ss R

atio

RM Farmowners vs Industry

0%

10%

20%30%

40%

50%

60%70%

80%

90%

100%110%

120%

130%

1997 1998 1999 2000 2001 08/03

Year

Adj

uste

d Lo

ss R

atio

COMMANDMENT #9

• GRAPH THY DATA

• This helps find the source of skewness

• Graphs also show clouds of truth

• Watch the clouds of truth to see which way the winds are blowing.

• Are thy clouds of truth cumulonimbus, or cirrus, or ??? …With gratitude to Luke Howard, the Inventor of Clouds

NEW TOOLS Improve Forecasting

Frequency Chart

Policies

.000

.006

.012

.017

.023

0

289.7

579.5

869.2

1159

158,000 160,250 162,500 164,750 167,000

50,000 Tr ials 49,826 Displayed

Forecast: GIG 12/31/04 PIF

COMMANDMENT #10

IGNORE THE LAW OF SMALL NUMBERS AND TRUTH WILL BITE YOU.

• In other words, get out of thine own niche.

Explore the insurance universe out yonder.

• Use your actuarial eyes to bring truth into focus.

COMMANDMENT #11

BONUS COMMANDMENT: USE HUMOR

•Look for real-life surprises in the data

•Share the surprise in story form to stir up laughter

•Tell funny data stories to stay connected with people you work with

THE GREATEST COMMANDMENT?

WHAT STORIES DO YOU LIKE TO TELL? FIND CREDIBLE DATA TO LOVE

• Use it to carve a path of truth that strikes a balance between stability and responsiveness.

KEEP THE STORIES IN THE NUMBERS• RESULT: Putting actuarial stories back

into the numbers. Do your part to keep insurance companies healthy.