Modern methods in personal lines pricing CAE/DAV Meeting

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

Modern methods in personal lines pricing CAE/DAV Meeting

Berlin

29 April 2005

Duncan Anderson

Copyright © Watson Wyatt Worldwide. All rights reserved.

Agenda

l The case for effective pricing

l Analysing claims

l Comparing claims models with current rates

l Competitor analyses

l Retention / new business analyses

l Price optimisation


0 100

200

300

400

500

600

700

1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

£ million

Direct Line (UK) – written premium 1985 to 1998

Source: DTI / HMT / FSA returns


UK motor market loss ratio

65%

70%

75%

80%

85%

90%

1986 1987 1988 1989 1990 1991 1992 1993 1994

Market Direct line Source: DTI / HMT / FSA returns


Progressive (US) – written premium 1994 to 2003

Source: The Progressive Corporation & Subsidiaries Ten Year Summary

0

2

4

6

8

10

12

14

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Written prem

ium (U

S$ billions)


US motor market combined ratio

80%

85%

90%

95%

100%

105%

110%

115%

120%

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Com

bine

d ratio

Progressive US motor market

Source: Insurance Information Institute, A.M. Best, The Progressive Corporation & Subsidiaries Ten Year Summary


Case study – company in Eastern Europe

@18 months

@ 6 months

@18 months

@ 6 months

Loss ratio Frequency

49% 6.4% 2004 (new rates)

86% 64% 7.2% 7.1% 2003 (old rates)


Objective

Excess

Vehicle

Age

etc.

Sex

Area Premium Tariff

"Tariff" is this context means "rating structure" it does not imply regulated fixed tariff, nor anything related to tax!


Rate level adjustment

Expense loadings

Lapse/takeup Model

Current Tariff

New Tariff

Competitor Model

Risk Model

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 office

The premium rating process


Modelling the cost of claims

Expected cost of claims

Model

Excess

Vehicle

Age

etc.

Sex

Area


Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

PI x = Cost 1

PD x = Cost 2

FT x = Cost 4

AD x = Cost 3

WS x = Cost 5




l Data & rating factors

l Statistical techniques


Example motor rating factors

l Standard factors: – Age of main driver – Sex – Marital status – Age of licence – Occupation – Residency – Convictions – Homeowner? – Postcode – Vehicle group – Age of vehicle – Value of vehicle – Alarm/immobiliser – Modifications? – Garaged? – Use of vehicle – Mileage – Cover – Age / number of additional driver(s) – Previous claims – Excess – Payment frequency – NCD – Protected NCD?

l External data: – individual data – vehicle data – geodemographic data – geophysical data

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


Generalised linear models

E[Y] = µ = g 1 (X.β + ξ)

Var[Y] = φ.V(µ) / ω

l Consider all factors simultaneously

l Allow for nature of random process

l Robust and transparent

l EU and increasingly global industry standard


Why GLMs over other methods

l Oneway and twoway analyses – distorted by correlations, no diagnostics

l Iteratively standardised oneways – no diagnostics, computationally inferior to GLMs (no faster), less flexibility for allowance of random process, not always tractable solution

l Neural networks – not transparent, hard to interpret, can be unstable with new types of policy, easy to over/under fit

l Cluster analyses / "segmenting" – suitable for marketing but less appropriate for assessing continuous risk; does not fit with rating structures


"A Practitioner's Guide to Generalized Linear Models"

l CAS 2004 Discussion Paper Program

l Copies available at www.watsonwyatt.com/glm

l Section 1 to be added to CAS Exam 9 syllabus in 2006


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

Log of m

ultiplier

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7

Exposure (policy years)

Exposure Onew ay relativities Approx 2 SE from estimate Smoothed GLM estimate

Example of GLM output (real UK data)


Model iteration

Standard errors of parameter estimates

Common sense

Consistency over time

Ftests / χ 2 tests on deviances (with ranks)


GLM output (significant factor)

154%

138%

105%

93%

72% 73% 84%

58%

45% 39%

31%

5% 0%

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Vehicle symbol

Log of m

ultiplier

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

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

Exposure (years)

Onew ay relativities Approx 95% conf idence interval Parameter estimate P value = 0.0%


GLM output (insignificant factor)

1%

4%

1%

3%

1%

5%

0%

5%

3%

7%

1%

4%

0%

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

Vehicle symbol

Log of multiplier

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

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

Exposure (years)

Onew ay relativities Approx 95% confidence interval Parameter estimate P value = 52.5%



140% 149%

105%

86%

65%

76% 73%

56% 51%

34% 33%

5% 0%

140% 150%

108% 104%

81%

64%

81%

56%

40% 45%

33%

2% 4%

185%

118%

104%

89%

69% 78%

97%

63%

44% 38%

27%

10%

4%

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Vehicle symbol.Year of exposure

Log of multiplier

0

20000

40000

60000

80000

100000

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

Exposure (years)

Approx 95% confidence interval, Year of exposure: 2000 Approx 95% confidence interval, Year of exposure: 2001 Approx 95% confidence interval, Year of exposure: 2002

Parameter estimate, Year of exposure: 2000 Parameter estimate, Year of exposure: 2001 Parameter estimate, Year of exposure: 2002



30%

16% 14%

0%

15%

20% 21%

95%

49%

26%

3%

25%

13%

2%

22%

15%

9%

4%

29%

2%

23%

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Territory.Year of exposure

Log of multiplier

0

20000

40000

60000

80000

100000

1 2 3 4 5 6 7

Exposure (years)

Approx 95% confidence interval, Year of exposure: 2000 Approx 95% conf idence interval, Year of exposure: 2001 Approx 95% confidence interval, Year of exposure: 2002

Smoothed estimate, Year of exposure: 2000 Smoothed estimate, Year of exposure: 2001 Smoothed estimate, Year of exposure: 2002


R esidual

4

3

2

1

0

1

2

3

4

Log of fitted value 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5

Residuals

Pretium 04/05/2004 11:03


Example job Run 5 Model 3 Small interaction Third party material damage, Numbers

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 multiplier

0

50000

100000

150000

200000

250000

300000

1721 2224 2529 3034 3539 4049 5059 6069 70+

Exposure

Approx 2 SEs f rom 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

P level = 0.0% Rank 6/6

Interactions


BoxCox link function investigation

l GLM structure is E[Y] = µ = g 1 (X.β + ξ) Var[Y] = φ.V(µ) / ω

l Box Cox transforms defines g(x) = ( x λ 1 ) / λ for λ≠0, ln(x) for λ=0

l λ = 1 ⇒ g(x) = x 1 ⇒ additive (with base level shift) l λ → 0 ⇒ g(x) → ln(x) ⇒ multiplicative (via maths) l λ = 1 ⇒ g(x) = 1 1/x ⇒ inverse (with base level shift) l Try different values of λ and measure goodness of fit to see which fits experience best


BoxCox link function investigation Motor third party property frequencies

‐137910

‐137870

‐137830

‐137790

‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6

λ

Likelihood

Multiplicative Inverse Additive


BoxCox link function investigation Motor third party property average amounts

268569

268569

268568

268568

268567

268567

268566

0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3

λ

Likelihood

Inverse Multiplicative Additive


Tweedie distributions

l Incurred losses have a point mass at zero and then a continuous distribution

l Poisson and gamma not suited to this

l Tweedie distribution has point mass and parameters which can alter the shape to be like Poisson and gamma above zero

∑ ∞

=

−

> − − Γ

− =

1 0 0

1

0 for )] ( [ exp . ! ) (

) / 1 ( ) ( ) , , ; ( n

n

Y y y y n n y y f θ κ θ λω

α κ λω α λ θ α

α α

0

0.2

) ( exp ) 0 ( 0 θ λωκ α − = = Y p


60%

80%

100%

120%

140%

160%

180%

GY1 1

IV23 2

PH41 4

LL74 8

BA16 0

PL31 1

TQ12 4

PE12 8

BS24 9

LN6 0

AB10 6

HU11 4

OX17 2

ST1

6 1

BN20 7

BH15 1

LA4 4

MK44

3

SO53 3

FY7 6

GU16 8

KT2

4 6

LS21 2

ME16

9

FY4 3

B77 1

SS11 7

HU1 4

MK12

5

BN2 9

L64 2

B76 7

DY3 2

WD19

4

BS3 2

WD17

1

NE34 6

CF1

4

SR2 0

OL1

0 1

HA9 8

BD5 9

B66 4

L30 2

EC4V

4

SE17 2

N15

6

L13 0

Postcode Sector

Relative Premium

The problem with area

UK auto loadings


244

274

307

343

385

432

484

543

609

685

770

299 316 334 352 369 387 404 422 440 457 475 495 518 542 632

0

20

40

60

80

100

120

140

Num

ber of districts

Company F

Company I

Example of market disparity

UK motor


244

274

307

343

385

432

484

543

609

685

770

299 316 334 352 369 387 404 422 440 457 475 495 518 542 632

0

20

40

60

80

100

120

140

Num

ber of districts

Company F

Company I

Example of market disparity

UK motor


General approach

1

2

3

Assess true area risk as well as possible

Define "zones" containing areas of similar risk (may or may not be contiguous)

Determine relativities applicable to "zone"


Proximity

l Key assumption is that "close" areas are similar

l May not be a perfect assumption

l Nevertheless it seems consistently to yield good results in practice


Example residual risk

High residual

Low (negative) residual


r i * = Z(e i ).r i + (1 Z(e i )) Σ e j .r j .f(d ij ) / Σ e j .f(d ij ) where

r i * = smoothed residual r i = unsmoothed residual

Z(e i ) = e i / (e i + a) m e i = exposure in region i

d ij = (x i x j ) 2 + (y i y j ) 2 ½

f(d ij ) = 1/d ij n or 1/(d ij n + b n ) or exp(n.d ij ) etc

Watson Wyatt model

j j


Finding the parameters

Seek parameters which minimize

error

a, m, n, b

Calculate residuals Save for determining zoning relativities

Example job Run 2 Model 3 All claim types, all factors, N&A Third party material damage, Numbers

45%

15% 13%

10%

5%

0%

4% 6%

0.2

0.1

0

0.1

0.2

0.3

0.4

Area of garage

Log of m

ultiplier

0

50000

100000

150000

200000

250000

A B C D E F G H

Exposure (years)

Approx 2 s.e. from estimate Unsmoothed estimate Smoothed estimate P level = 0.0% Rank 11/13

1 1

3 3



Seek parameters which minimize

error

a, m, n, b

n e for Z=20%

UK 1.9 146 UK 2.2 152 UK 1.8 78

France 2.0 104 France 1.9 146

Netherlands 1.8 61 South Africa 2.2 106

USA 2.5 127 USA 1.9 106

Spain 2.1 17

Italy 1.4 87


Example results

Unsmoothed residuals Smoothed residuals

1 1


Contiguity clustering Exposure/risk based "manual" grouping Contiguity clustering algorithm

2 2



Save for determining zoning relativities

Example job Run 2 Model 3 All claim types, all factors, N&A Third party material damage, Numbers

45%

15% 13%

10%

5%

0%

4% 6%

0.2

0.1

0

0.1

0.2

0.3

0.4

Area of garage

Log of m

ultiplier

0

50000

100000

150000

200000

250000

A B C D E F G H

Exposure (years)

Approx 2 s.e. from estimate Unsmoothed estimate Smoothed estimate P level = 0.0% Rank 11/13

l Fit new zone definition in GLM to assess true predictive power

l Fresh data required to avoid selffulfilling prophesies

l Compare against existing territory definition

3 3


Finding the parameters Effect of smoothed residual zone on fresh data

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 multiplier

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

Exposure (th

ousand policy years)

2 S.E from GLM estimate GLM estimate

3 3


Amt Freq

Amt Freq

Amt Freq

Amt Freq

Amt Freq

PI x = Cost 1

PD x = Cost 2

FT x = Cost 4

AD x = Cost 3

WS x = Cost 5





Expense loadings

Risk Model

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




Expense loadings

Current Tariff

Risk Model 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


Demonstration job Run 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 multiplier

0

50000

100000

150000

200000

250000

1721 2224 2529 3034 3539 4049 5059 6069 70+

Exposure

Approx 2 SEs f rom 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 multiplier

0

50000

100000

150000

200000

2 to 7 8 9 10 11 12 13 to 17

Exposure





46%

15% 13% 10%

5%

0%

4% 6%

49%

16%

11% 7%

4%

0% 2%

5%

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

MAREA Area of garage

Log of multiplier

0

50000

100000

150000

200000

250000

A B C D E F G H

Exposure





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

Log of multiplier

0

100000

200000

300000

400000

500000

Yearly Halfyearly Quarterly

Exposure

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



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

Count of records

Currently unprofitable business

Currently profitable business


Impact analysis Example 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


Count of records

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss ratio

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


Count of records

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss ra

tio

1721 2224 2529 3034 3539 4049 5059 6069 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


Count of records

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss ra

tio

A B C D E F G H Claims / Earnedprem



Class of vehicle

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


Count of records

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss ra

tio

A & Y & Z B C D & E F & G & H & J 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


Count of records

30%

40%

50%

60%

70%

80%

90%

100%

110%

120%

130%

140%

150%

160%

170%

180%

Loss ra

tio

Yearly Halfyearly Quaterly Claims / Earnedprem


BonusMalus

Comparison of current and theoretical Bonus Malus Run 6 Model 2 Unrestricted standard risk premium run Unsmoothed standard risk premium model

0%

32% 29% 35% 33%

41% 43% 53%

69% 78% 75%

97% 113%

93% 109% 113% 116% 121%

133% 145%

164%

0% 0% 0% 0% 0% 0%

20%

40%

60%

80%

100% 120%

140% 160%

180% 200%

220% 240%

260% 280%

300%

0.3

0

0.3

0.6

0.9

1.2

1.5

Bonus Malus

Log of multiplier

0

20000

40000

60000

80000

100000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Exposure (years)

Approx 95% conf idence interval R6M2 Smoothed estimate R6M2 Current tarif f Third party tarif f


Restricted models

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1.5

1

0.5

0

0.5

1

1.5

2

BonusMalus

Log of multiplier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1.5

1

0.5

0

0.5

1

1.5

Vehicle group

Log of multiplier

Geographical zone A B C D E F G H I J K L M N O

2

1.5

1

0.5

0

0.5

1

1.5

Log of multiplier

18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 1

0.5

0

0.5

1

1.5

Age of driver

Log of multiplier


Restricted models

Restriction

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1.5

1

0.5

0

0.5

1

1.5

2

BonusMalus

Log of multiplier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1.5

1

0.5

0

0.5

1

1.5

Vehicle group

Log of multiplier


2

1.5

1

0.5

0

0.5

1

1.5

Log of multiplier

18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 1

0.5

0

0.5

1

1.5

Age of driver

Log of multiplier


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1.5

1

0.5

0

0.5

1

1.5

2

BonusMalus

Log of multiplier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1.5

1

0.5

0

0.5

1

1.5

Vehicle group

Log of multiplier


2

1.5

1

0.5

0

0.5

1

1.5

Log of multiplier

18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 1

0.5

0

0.5

1

1.5

Age of driver

Log of multiplier

Restricted models

Restriction

Model compensates (as best it can)

to allow for restriction



Expense loadings

Current Tariff

New Tariff

Risk Model

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

l Survey market – broker quotation systems

– question policyholder – mystery shopping

l Investigate competitors' structures

l Apply "cheapest" tariff to own portfolio

l 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 premium Percentage of con

tracts

Zone A

Zone B

Zone C


Considering the competitive position

Example of competitor analysis Third party cover

50% 45% 40% 35% 30%

25% 20%

15% 10%

5% 0%

5% 10%

15% 20%

25% 30%

35% 40%

45%

125% 107%

91% 80%

70% 62%

48% 35%

28%

11%

0% 5% 4%

17% 19% 23%

28%

40% 43%

50%

82%

65%

44% 41% 48%

30% 31%

18% 15% 7%

0%

9% 3%

10%

18% 20% 24%

28% 31%

39%

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Vehicle group

Log of multiplier

0

5000

10000

15000

20000

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

Exposure (years)

Current tarif f Approx 95% confidence interval Third cheapest market quote Smoothed estimate P value = 0.0% Rank 9/11


Considering the competitive position

Competitiveness (market premium/current premium)

Profitability (theoretical premium/current premium)

Competitive

Profitable



Expense loadings

Current Tariff

New Tariff

Competitor Model

Risk Model

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



l Model normal factors other products held payment method change in cover BonusMalus expectation plus… source change in premium claims history competitiveness

Claims

Vehicle age

Age

Premium / Competitors' premium

Sex

∆ Premium Probability of lapsing

Model

Modelling retention


l If details of individual quotes known, can be modelled in similar way

l Otherwise approximations required

Vehicle age

Age

Quotation / competitors' premium

Sex

Probability of accepting

Model Excess

Modelling new business rates


Effect of premium change on lapses

0.5

0.2

0.1

0.4

0.7

1

Change in premium on renewal (£)

Parameter estimate: high = high probability of lapsing

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

(Splines often better to use in practice)


47

Quote/Average of the three cheapest quotes on the market

Param

eter estimate: high = high probability of quote being accepted

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

Effect of competitiveness on new business


Risk Model

Profitability

Current Tariff

Retention

Lapse model

High

High

Low

Low

renewal (discount vouchers / phone

calls)

Treat well after claims

Target marketing at these

Increase premiums

Tougher claims handling

Customer value

Custome

r value

Treat well after claims

Actively target at


Lifetime loadings l Expenses per policy

– acquisition 100 – renewal 30

l Expected lifetime – young 2 years – old 5 years

l Lifetime expense loadings – young ( 100 + 1 * 30 ) / 2 = 65 – old ( 100 + 4 * 30 ) / 5 = 44



Lapse/takeup Model

Current Tariff

New Tariff

Competitor Model

Risk Model 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


Expense loadings



Lapse/takeup Model

Current Tariff

New Tariff

Competitor Model

Risk Model

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 office


Expense loadings


Portfolio

Now Year 1

Portfolio Lapse

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

Tariff

Expenses € 1

Comp Tariff New Bsns

Scenario testing


Problems (1)

l What will the competition do?

l Things change – age of policyholder – age of vehicle – vehicle – address – NCD / BonusMalus

l Inputs to some models are outputs from others


Period of projection

0

500

1000

1500

2000

2500

3000

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6

Change in base rate

Profit / value

0

5,000

10,000

15,000

20,000

25,000

Volum

e

Yr 1 profit Volume


Portfolio

Now Year 1

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 1

Comp Tariff

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 2

Comp Tariff

Year 2

Multiple year projections


Multiple year projections

Portfolio

Now Year 1

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 1

Comp Tariff

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 2

Comp Tariff

Year 2

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 3

Comp Tariff

Year 3

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 4

Comp Tariff

Year 4

Lapse Portfolio

New Bsns

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

Tariff

Expenses £ 5

Comp Tariff

Year 5

/ (1+i) 1 / (1+i) 2 / (1+i) 3 / (1+i) 4 / (1+i) 5

l In theory project many years

l In practice assumptions become too uncertain and model becomes too complex


A pragmatic compromise

+ €X per policy in force, perhaps

modified to make some allowance for lifetime

value

Portfolio Portfolio

Now Year 1

Lapse Lapse Portfolio Portfolio

New Bsns New Bsns

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

Tariff Tariff

Expenses € 1

Comp Tariff Comp Comp Tariff Tariff

Lapse Portfolio

New Bsns

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

Tariff

Expenses € 2

Comp Tariff

Year 2

Lapse Lapse Portfolio Portfolio

New Bsns New Bsns

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

Tariff Tariff

Expenses 2

Comp Tariff Comp Comp Tariff Tariff

Year 2


Base rate change consider profit vs volume

0

500

1000

1500

2000

2500

3000

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6

Change in base rate

Profit / value

0

5,000

10,000

15,000

20,000

25,000

Volum

e

Yr 1 profit Volume


Base rate change single success criteria

0

500

1000

1500

2000

2500

3000

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6

Change in base rate

Success criteria

0

5,000

10,000

15,000

20,000

25,000

Volum

e

Success criteria Volume


0% 5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

55%

60%

65%

70%

1.075 1.1

1.125 1.15

1.175 1.2

1.225 1.25

1.275 1.3

1.325 1.35 3,000

3,050

3,100

3,150

3,200

3,250

3,300

3,350

3,400

3,450

3,500

3,550

Base rate change with simple relativity change

3570

Base rate change Movement to

theoretically correct relativities


Optimisation

l Methods exist which allow optimal rating structure (considering all parameters) to be derived


Optimised rating structure Optimised premium

Comparison with claims model and current premium

135%

95%

75%

55% 47%

25%

12%

0% 0% 5% 2% 3% 6%

15%

31%

188%

134%

109%

82% 73%

41%

20%

0% 1% 2%

2% 2% 1%

12%

40% 64%

48% 38%

26% 18%

12% 5%

0% 0% 2% 2% 2% 5% 10%

16%

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Policyholder age

Log of multiplier

0

5000

10000

15000

20000

< 20 2021 2223 2425 2627 2829 3034 3539 4044 4549 5054 5559 6064 6569 70+

Exposure (years)

Optimised premium model Claims cost +/ 2SE Unsmoothed claim cost model Smoothed claims cost model Current premium


Optimised rating structure Optimised premium

Comparison with claims model and current premium

0%

5% 6% 7% 4%

1% 3% 5%

8% 10%

12% 14%

16% 17% 17% 17% 17%

0% 0% 0% 0% 0%

10% 13%

16% 18%

20% 23%

27% 29% 30% 30% 31% 31%

0% 1% 2% 4%

6% 8%

10% 11% 13% 14%

16% 17% 18% 19% 20% 21% 21%

0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

Policy duration (years)

Log of multiplier

0

10000

20000

30000

40000

50000

60000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 +

Exposure (years)

Optimised premium model Claims cost +/ 2SE Unsmoothed claim cost model Smoothed claims cost model Current premium

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

Modern methods in personal lines pricing CAE/DAV Meeting

Berlin

29 April 2005

Duncan Anderson

Documents

Modern methods in personal lines pricing CAE/DAV Meeting