GLMs in Personal Lines Pricing

W W W . W A T S O N W Y A T T . C O M

GLMs in Personal Lines PricingClaudine Modlin, FCAS

Watson Wyatt Insurance & Financial Services Inc.

www.watsonwyatt.com/pretium

MAF Fall MeetingSeptember 26, 2002

Agenda

Overview of GLMs in the rating process

GLMs in practice

– data– diagnostics– interactions

Territory analysis

How to get started

Objective

Claim

Vehicle

Age

Limit

Sex

AreaPremiumRate Scheme

Modeling the cost of claims

Claim

Vehicle

Age

Limit

Sex

Area

Expected cost of claims

Model

Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

BI x = Cost 1

PD x = Cost 2

COL x = Cost 4

MED x = Cost 3

OTC x = Cost 5



Rating factors

Statistical techniques

Example auto rating factors

Standard factors:– Age– Sex– Marital status– Number years licensed– Claim experience– Territory– Usage– Mileage– Limits– Deductibles– Make/Model of vehicle– Violations– Credit– Multi-line– Multi-car– Safety devices– Theft devices

External data:– geodemographic data– geophysical data

Data from other products:– banking data– other insurance data

T C

M 40% 20%

F 20% 10%

True risk ClaimsT C Total

M 80 20 100

F 20 20 40

Total 100 40 140

ExposureT C Total

M 200 100 300

F 100 200 300

Total 300 300 600

One-wayExp Claims Ratio

M 300 100 33.3%F 300 40 13.3%

T 300 100 33.3%C 300 40 13.3%

The failings of one way analysis

* 2

* 2.5

New

Old

Low

High

5

10

15

20

25

30

35

Nu

mb

er o

f p

oli

cie

s

Vehicle Value Vehicle Age

Example correlation

Generalized linear models

E[Y] = = g-1(X. + )

Var[Y] = .V() /

Consider all factors simultaneously

Allow for nature of random process

Robust and transparent

EU industry standard

Why GLMs over other methods

One-way and two-way analyses– Distorted by correlations, no diagnostics

Iteratively standardized one-ways– No diagnostics, no faster than GLMs, less flexibility for allowance of

random process, not always tractable solution

Neural networks– Not transparent, hard to interpret, can be unstable with new types of

policy, easy to over/under fit

Cluster analyses / "segmenting"– Suitable for marketing but less appropriate for assessing continuous

risk; does not fit with rating structures

Data mining– General term for all of the above but can often be merely one-way or

two-way analyses on subsets of data

-17%-19%

-15%

-20%

-4%-5%

0%

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Factor

Lo

g o

f m

ulti

plie

r

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7

Exp

osu

re (

po

licy

yea

rs)

Exposure Approx 2 SE from estimate GLM estimate

Example of GLM output (real UK data)

22%

7%6%

10%

-16%

-19%

0%

-17%-19%

-15%

-20%

-4%-5%

0%

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Factor

Lo

g o

f m

ulti

plie

r

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7

Exp

osu

re (

po

licy

yea

rs)

Exposure Oneway relativities Approx 2 SE from estimate GLM estimate

Example of GLM output (real UK data)

Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

BI x = Cost 1

PD x = Cost 2

COL x = Cost 4

MED x = Cost 3

OTC x = Cost 5


The premium rating process

Rate level adjustments

Risk model

Profit loadings

Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

BI x = Cost 1

PD x = Cost 2

COL x = Cost 4

MED x = Cost 3

OTC x = Cost 5


Rate level adjustments

Profit loadings

Current Rates

RiskModel

Compare

Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

BI x = Cost 1

PD x = Cost 2

COL x = Cost 4

MED x = Cost 3

OTC x = Cost 5

Demonstration jobRun 10 Model 2 - Third party material, standard risk premium run - Unsmoothed standard risk premium model

1%0%

-22%

0%

22%

30%

67%68%

89%

-18%

-10%

0%0%0%0%

11%

22%22%

-0.4

-0.2

0

0.2

0.4

0.6

0.8

MAGE - Age of driver

Log

of m

ultip

lier

0

50000

100000

150000

200000

250000

17-21 22-24 25-29 30-34 35-39 40-49 50-59 60-69 70+

Exp

osur

e

Approx 2 SEs from unsmoothed estimate Unsmoothed unrestricted estimate Unsmoothed restricted estimate Current rating structure

Factor effect analysis


31%

11%8%

0%

-7%

-15%-16%

82%

49%

22%

0%

-10%

-18%

-33%

-0.4

-0.2

0

0.2

0.4

0.6

0.8

MGROUP - Group of vehicle

Log

of m

ultip

lier

0

50000

100000

150000

200000

2 to 7 8 9 10 11 12 13 to 17

Exp

osur

e




28%

12%

0%5%5%

0%

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

MPFREQ - Payment frequency

Lo

g o

f m

ulti

plie

r

0

100000

200000

300000

400000

500000

Yearly Half-yearly Quarterly

Exp

osu

re



Impact analysis

Example job

0

1000

2000

3000

4000

5000

6000

7000

0.450 -

0.500

0.550 -

0.600

0.650 -

0.700

0.750 -

0.800

0.850 -

0.900

0.950 -

1.000

1.050 -

1.100

1.150 -

1.200

1.250 -

1.300

1.350 -

1.400

1.450 -

1.500

1.550 -

1.600

1.650 -

1.700

1.750 -

1.800

1.850 -

1.900

1.950 -

2.000

2.050 -

2.100

2.150 -

2.200

2.250 -

2.300

2.350 -

2.400

2.450 -

2.500

Ratio: Risk Premium / Current tariff

Cou

nt

of r

eco

rds

Currently unprofitable

business

Currently profitable

business

Impact analysisExample job

0

1000

2000

3000

4000

5000

6000

7000

0.450 -0.500

0.600 -0.650

0.750 -0.800

0.900 -0.950

1.050 -1.100

1.200 -1.250

1.350 -1.400

1.500 -1.550

1.650 -1.700

1.800 -1.850

1.950 -2.000

2.100 -2.150

2.250 -2.300

2.400 -2.450


Cou

nt

of r

eco

rds

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Lo

ss r

atio

Yearly Claims / Earnedprem


Age of driver

0

1000

2000

3000

4000

5000

6000

7000

0.450 -0.500

0.600 -0.650

0.750 -0.800

0.900 -0.950

1.050 -1.100

1.200 -1.250

1.350 -1.400

1.500 -1.550

1.650 -1.700

1.800 -1.850

1.950 -2.000

2.100 -2.150

2.250 -2.300

2.400 -2.450


Cou

nt o

f re

cord

s

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss

rat

io

17-21 22-24 25-29 30-34 35-39 40-49 50-59 60-69 70+ Claims / Earnedprem


Area of garage

0

1000

2000

3000

4000

5000

6000

7000

0.450 -0.500

0.600 -0.650

0.750 -0.800

0.900 -0.950

1.050 -1.100

1.200 -1.250

1.350 -1.400

1.500 -1.550

1.650 -1.700

1.800 -1.850

1.950 -2.000

2.100 -2.150

2.250 -2.300

2.400 -2.450


Cou

nt o

f re

cord

s

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss

rat

io

A B C D E F G H Claims / Earnedprem


Payment frequency

0

1000

2000

3000

4000

5000

6000

7000

0.450 -0.500

0.600 -0.650

0.750 -0.800

0.900 -0.950

1.050 -1.100

1.200 -1.250

1.350 -1.400

1.500 -1.550

1.650 -1.700

1.800 -1.850

1.950 -2.000

2.100 -2.150

2.250 -2.300

2.400 -2.450


Cou

nt o

f re

cord

s

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss

rat

io

Yearly Half-yearly Quaterly Claims / Earnedprem

Expense loadings

Profit loadings

Current Rates

NewRates

CompetitorModel

RiskModel

Compare

Amt

Amt

Amt

Amt

Amt

Freq

Freq

Freq

Freq

Freq

TPBI x = Cost 1

TPPD x = Cost 2

FT x = Cost 4

AD x = Cost 3

WS x = Cost 5


Competitive position

Survey market

– rate filings– quotation systems– question policyholder– mystery shopping

Investigate competitors' structures

Apply "cheapest" tariff to own portfolio

Use in retention / new business model

Age of main driver

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

2.10

2.20

2.30

2.40

2.50

25 30 35 40 45 50 55 60 65 70 75 80

Age of main driver

0

2

4

6

8

10

12

14

16

18

20

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Percentage change in premiumP

erce

nta

ge

of

con

trac

ts

Zone A

Zone B

Zone C

Expense loadings

Profit loadings

Lapse/take-upModel

Current Rates

NewRates

CompetitorModel

RiskModel

Compare

Amt

Amt

Amt

Amt

Amt

Freq

Freq

Freq

Freq

Freq

TPBI x = Cost 1

TPPD x = Cost 2

FT x = Cost 4

AD x = Cost 3

WS x = Cost 5


Model- rating factors - other products held- payment method - change in coverage- discount expectation plus…- source - change in premium- claims history - competitiveness

Claims

Vehicle age

Age

Premium / Competitors' premium

Sex

Premium

Probability of lapsing

Model

Modeling retention

Retention model - Policyholder age

Age of policyholder

Log

of m

ultip

lier

20 25 30 35 40 45 50 55 60 65 70

Approx 2 SEs from estimate Unsmoothed estimate

Retention model - Change in premium

-0.5

-0.2

0.1

0.4

0.7

1

Change in premium on renewal

Lo

g o

f m

ulti

plie

r

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

Approx 2 SEs from estimate Unsmoothed estimate

-47

Quote/Average of the three cheapest quotes on the market

Log

of m

ultip

lier

of p

/(1-

p)

0.6 0.7 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Approx 2 SD from estimate Smoothed estimate

New business modelCompetitiveness of premium

Customer lifetime value

RiskModel

Profitability

Current Rates -

Re

ten

tion

La

pse

mo

de

l

High

Hig

hL

ow

Low

Actively target atrenewal (discountvouchers / phone

calls)

Target marketing atthese

Increase premiums

Price elasticity

-5

0

5

10

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90 100 110 120

Premium change ($)

Number of policies Profit

Expense loadings

Profit loadings

Lapse/take-upModel

Current Rates

NewRates

CompetitorModel

RiskModel

Compare

Amt

Amt

Amt

Amt

Amt

Freq

Freq

Freq

Freq

Freq

TPBI x = Cost 1

TPPD x = Cost 2

FT x = Cost 4

AD x = Cost 3

WS x = Cost 5

Modeloffice


Agenda


GLMs in practice


Territory analysis

How to get started

Data required

Linked policy + claims data

Record: one insured risk (eg car) for one policy period or portion of policy period for which risk has not changed

Fields:– explanatory variables - rating, underwriting,

marketing, external– stats - earned exposure, incurred claim count,

incurred loss, earned premium (optional)

Minimum of 100,000 earned exposures

Data considerations

Reflect cancellation/endorsement

Include time lag to reduce effect of IBNR

Include dummy variables to standardize for geography (if countrywide study) and time

Display rating factors applicable at time of exposure, categorized on current basis

Model iteration diagnostics

Standard errors of parameter estimates

F-tests / 2 tests on deviances (with ranks)

Consistency over time

Common sense

Factor 1

Factor 3

Factor 6

Factor 5

Factor 7

Factor 3

Factor 2

Factor 4

Factor 6

Factor 5

101%

82%

65%

49%

35%

22%

11%

0%

-10%

-18%

-26%

-33%

-39%

-45%

-50%

-1

-0.7

-0.4

-0.1

0.2

0.5

0.8

Log

of m

ultip

lier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-18%-5%-10%

28%

57%

16%11%0%

-39%

-18%

22%

-26%

28%

-10%-18%

-2

-1.7

-1.4

-1.1

-0.8

-0.5

-0.2

0.1

0.4

0.7

1

1.3

1.6

1.9

Log

of m

ultip

lier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Standard errors ofparameter estimates

Fittedvalue

Multi-car

Vehicle

Age

Claims

Sex

ZoneModel A

Fittedvalue

Multi-car

Vehicle

Age

Claims

Sex

Model B

Deviance = 9585df = 109954

Deviance = 9604df = 109965

?

Deviances

Consistency over time

A B C D

1996 1997 1998

A B C D

1996 1997 1998

Common sense

Does it make sense given correlations?

Are ordered categorical variables well behaved?

Can you believe it?

Can underwriters believe it?

Consider results for frequency and amounts at the same time

Consider results for each claim type at the same time

Example jobRun 12 Model 3 - Small interaction - Third party material damage, Numbers

-11%-6%

-20%

0%

24%26%

56%

105%

122%

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Age of driver

Log

of m

ultip

lier

0

50000

100000

150000

200000

250000

17-21 22-24 25-29 30-34 35-39 40-49 50-59 60-69 70+

Exp

osur

e

Approx 2 SEs from estimate Unsmoothed estimate Smoothed estimate

P level = 0.0%Rank 7/7

Interactions


0%

-13%

-0.18

-0.15

-0.12

-0.09

-0.06

-0.03

0

0.03

Sex of driver

Log

of m

ultip

lier

0

100000

200000

300000

400000

500000

600000

Female Male

Exp

osur

e

Approx 2 SEs from estimate Unsmoothed estimate Smoothed estimate


Interactions

Interactions


13%6%

-18%

-2%

20%19%

40%46%

63%

-11%-6%

-19%

0%

24%28%

63%

138%

155%

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Age of driver.Sex of driver

Log

of m

ultip

lier

0

50000

100000

150000

200000

250000

300000

17-21 22-24 25-29 30-34 35-39 40-49 50-59 60-69 70+

Exp

osur

e

Approx 2 SEs from estimate, Sex of driver: Female Approx 2 SEs from estimate, Sex of driver: Male Unsmoothed estimate, Sex of driver: Female

Unsmoothed estimate, Sex of driver: Male Smoothed estimate, Sex of driver: Female Smoothed estimate, Sex of driver: Male


CompanyInteraction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Age / Sex Age / vehicle group

Area / vehicle group

Area / garaged Age / occupation

Use / mileage Occupation / use Age / marital status / sex

Age Factor

17 2.52

18 2.05

19 1.97

20 1.85

21-23 1.75

24-26 1.54

27-30 1.42

31-35 1.20

36-40 1.00

41-45 0.93

46-50 0.84

51-60 0.76

60+ 0.78

Group 1 2 3 4 5 6 7 8 9 10 11 12 13

Factor 0.54 0.65 0.73 0.85 0.92 0.96 1.00 1.08 1.19 1.26 1.36 1.43 1.56

1.00

1.00

1.17

1.00

Group > 1 2 3 4 5 6 7 8 9 10 11 12 13Age v

17 1.36 1.64 1.79 2.09 2.27 2.42 2.56 2.65 3.27 3.71 4.08 4.36 4.8418 1.12 1.31 1.47 1.76 1.84 2.00 2.11 2.19 2.43 2.97 3.29 3.55 3.9019 1.08 1.30 1.46 1.63 1.82 1.91 2.02 2.11 2.53 2.88 3.30 3.35 3.6320 0.98 1.18 1.36 1.54 1.68 1.79 1.83 1.97 2.19 2.66 3.02 3.20 3.38

21-23 0.96 1.13 1.24 1.51 1.65 1.64 1.80 1.85 2.04 2.26 2.55 2.53 2.8924-26 0.82 0.99 1.10 1.31 1.43 1.52 1.51 1.64 1.81 1.93 2.13 2.22 2.4727-30 0.78 0.90 1.07 1.19 1.32 1.39 1.41 1.51 1.65 1.77 1.91 2.01 2.2431-35 0.63 0.78 0.86 0.99 1.09 1.17 1.22 1.32 1.42 1.54 1.66 1.71 1.8836-40 0.55 0.64 0.71 0.85 0.91 0.93 0.99 1.07 1.18 1.29 1.40 1.41 1.5341-45 0.51 0.61 0.66 0.79 0.88 0.88 0.94 0.99 1.09 1.15 1.29 1.31 1.4246-50 0.46 0.55 0.61 0.70 0.76 0.81 0.84 0.92 1.02 1.07 1.12 1.18 1.3151-60 0.40 0.49 0.56 0.64 0.68 0.71 0.78 0.82 0.90 0.99 1.02 1.12 1.20

60+ 0.43 0.52 0.55 0.67 0.72 0.73 0.78 0.83 0.93 0.98 1.04 1.11 1.25

Interactions

Agenda


GLMs in practice


Territory analysis

How to get started

Geographic rating

Territory is one of the main drivers of cost

Considerable variety in how insurers rate for territory

One insurer will have limited exposure in any one area

Spatial smoothing

Fit GLM (excluding current territories)

Map "residual" risk by "region"

Make this residual risk more predictive

Categorize into territories to derive appropriate loadings

Residual risk

High residual

Low residual

A model form

ri* = Z.ri + ( 1 - Z ) . neighboring experience

where

ri*= smoothed residual risk

ri = unsmoothed residual risk

Definitions of "neighboring"

Example results

Unsmoothed residuals Smoothed residuals

Finding the parametersEffect of smoothed vs unsmoothed residual zone

Zone based on smoothed residuals

650%

464%418%

345%

287%

189%

139%

101%

74%

26%15%

0%

-22%-18%-16%

-34%

-18%

-37%

-22%

-35%

-1

-0.5

0

0.5

1

1.5

2

2.5

Zone

Log

of m

ultip

lier

0

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osur

e (t

hous

and

polic

y ye

ars)

2 S.E from GLM estimate GLM estimate

Zone based on unsmoothed residuals

-1%6%

-4%

63%49%

74%64%

79%99%91%

24%

0%

19%

2%

16%30%

49%41%29%

-3%

-1

-0.5

0

0.5

1

1.5

2

2.5

Zone

Log

of m

ultip

lier

0

50

100

150

200

250

300

350

400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osur

e (t

hous

and

polic

y ye

ars)

2 S.E from GLM estimate GLM estimate

Zone based on smoothed residuals

Zone based on unsmoothed residuals

Agenda


GLMs in practice


Territory analysis

How to get started

How can I start?

Programming from scratch

Software applications– tailored to personal lines– easy to navigate– fast, even on PC– clear output– cost is often less than the annual

compensation of one actuary

W W W . W A T S O N W Y A T T . C O M

GLMs in Personal Lines PricingClaudine Modlin, FCAS

Watson Wyatt Insurance & Financial Services Inc.

www.watsonwyatt.com/pretium

MAF Fall MeetingSeptember 26, 2002

GLMs in Personal Lines Pricing

Documents

Transcript of GLMs in Personal Lines Pricing