UEAInsurance Stats Overview

Steve Cant

Senior Statistics Manager, Aviva

steve.cant @ aviva.co.uk01603 686857

Modelling Opportunities at AvivaDoes modelling risk have any appeal ?Are you interested in an actuarial career but don’t fancy years more study in your free time ?Do you have graduate level maths skills ?Do you have any idea what financial statisticians do ? Have you ever wondered how insurance premiums are calculated ?

Key Aspects of RoleBuilding risk cost models – to predict who will claim on their motor or household insuranceBehavioural modelling – to predict how customers will react to pricing decisionsSpatial analysis of postcode area in order to produce world leading maps of insurance riskExtraction of deeper knowledge from large, already well understood data setsR & D into new modelling and analytical techniquesEducated guessworkWorking with colleagues across the business including those in pricing, marketing, finance, actuarial, claims and underwriting

You’ll apply your analytical enthusiasm to a range of business problems and produce mathematical and statistical models that drive real results.

The Elusive Advert - extracts

Products (Personal)

Motor

Household

Bike Van

Pet Travel

CreditorBreakdown

PRICING PROCESS

DATA RISK MODEL

MASS CUSTOMISEDPREMIUM MODEL

LIVE PREMIUMS

Cleanse Data

Recalibration

Competitive positioningProfitability ReviewsPrice Optimisation

DATA / STATS DATA / STATS

STATSSTATS

PRICING TEAMSPRICING TEAMS

Model Data

CHANNELS / FINANCE / CHANNELS / FINANCE / UNDERWRITINGUNDERWRITING

ACTUARIALACTUARIAL

Behavioural Models

STATSSTATS

BURNING COST

+ expenses + commission + profit

STREET RATES Maintenance EDDEDD

CORE RATES

AM80 and AF80 2 years select q[x-t]+t

0.000000

0.000500

0.001000

0.001500

0.002000

0.002500

0.003000

0.003500

0.004000

0.004500

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Death

Age

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Main Driver Age

Rela

tive F

req

uen

cy

FE

MA

Large Bodily Injury Claims – Major crashesTHIRD PARTY ONLY

10 years of data > £250K frequencyStill overwhelmingly random and rare – but can produce an index

About 1 in 5000 vehicle years

About 1 in 40000 vehicle years

Anything that influences risk is a rating factor

MOTOR RISK COST MODEL – Ranked by Information Gain

1 Bodily Injury Freq District 14 Property Damage Freq District

2 Young Additional Driver Age 15 Own Damage District

3 NCD 16 Vehicle Age

4 Main Driver Age 17 Transmission

5 Young Additional Driver Sex 18 Theft Freq District

6 Car Group 19 Fuel Type

7 Ritz 20 Duration

8 Driving Restriction 21 Convictions

9 Payment Frequency 22 Licence Length

10 Make Model 23 YAD Owns Car

11 Ownership Length 24 PNCD

12 Mileage 25 Voluntary Excess

13 At Fault Claims etc 30 other factors

Motor Insurance Rating Factors

Information gain is a weighted combination of factor range and exposure. E.g. age has high loadings for low exposure, payment method has lower loadings on high exposure.

Postcode

Vehicle

Insurance Premiums

Start with a base (average) premium E.g. £400 (40 year old, 3 year old Ford Focus in Norwich, with full No Claims)Then add various loadings and discounts

18 year old driver 200% loadingLives in Liverpool 100% loadingDrives a small car 40% discountDrives an old car 30% discountNo Claims Discount is zero 233% loading ! (5 years No Claims is a 70% discount)

£400 x 3 = £1200 x 2 = £2400x 0.6 = £1440x 0.7 = £1008x 3.33 = £3360 !

Harsh ?

How do we calculate these loadings ?

CHAOS

The Claims Universe

Undiscovered OrderNew factorsImproved modelling

ORDER

Modelling Process (Motor)

5 Perils: Accidental Damage, Bodily Injury, Theft, Glass, Property Damage 2 Models per peril:

Risk Modelling

Frequency = No. of Claims Exposure

Severity = Cost of Claim No. of Claims

Exposure is the time on risk

E.g. for 1000 cars, one year each this is 1000 ‘vehicle years’

120 claims from these 1000 vehicle years => 120/1000 = 12% frequency

But why bother risk modelling at all ?

Attempt to remove random effects (noise)

Avoid the illusions of variable association (Simpson’s paradox)

Consider all rating factors ‘together’ in order to discover ‘true effect’

Examine consistency over time

Ensure best possible prediction of future risk

Multivariate Modelling

Why bother ?

Simpson’s ParadoxBerkley Sex bias case (Source : Wikipedia)

Bias against women ?

1973 Admission figures

Breakdown by department

Tables are OK for two factors, no use for 50

Linear Modelling

LM expresses the relationship between an observed response (Y) and a set of predictors (X)

In its simplest form (first order) it can be conceptualised asE(Y) = β0 + β1X Y = β0 + β1X + ε

Where ε is an error term with expected value of 0

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

District

Cos

t

Simple Linear Modelling

Linear Modelling

Method of Least Squares

In order to calculate estimates of the parameters β0 and β1 we use the method of least squares.

This can be thought of as minimising the distance of each observed response y i is away from the predicted value ŷi.

yi – ŷi

x

Remove Outliers

We then extend this idea to n dimensions using matrices and Emblem software

Linear Modelling

Method of Least Squares

Minimize the Sum of Squared Errors;

By differentiating it can be shown that to minimise the SSE we must solve the following;

Linear Modelling

Multiple Linear Modelling

What happens when we believe a number of variables affect the distribution of our random variable Y ?

We still have the response variable Y but now instead of having a single predictor we have k predictors which we denote as X1, X2,.., Xk

Now we want to fit the model

So the same basic idea (least squares) but now we’re using matrix notation rather than simple algebra

Matrix notation

Generalized Linear Model (GLM)

Normal Distribution

• assumes each observation has the same fixed variance (no tail)

Poisson Distribution

• assumes the variance increases with the expected value of each observation (longer tail)

Gamma Distribution

• assumes variance increases with the square of the expected value (even longer tail !)

Basically An extension of Linear modelling that allows

Multiplicative models (using a ‘link function’) - more appropriate for insurance

A wider selection of errors (‘loss distributions’) from the exponential family

Emblem Software

• Raw data alone can lead to the wrong conclusion

Base 42% in Ritz 7-10

Age 50 +

Annual payers

Better Wealth Postcode

Policy Duration 3+

76% in Ritz 7-10Exposure 9%

73% in Ritz 7-10Exposure 17%

Ritz 7-10 SegmentationNUD (NB & Renewals)

67% in Ritz 7-10Exposure 27%

Proportion Ritz 1- 6

Proportion Ritz 7- 10

Data Mining

Decision Trees – Visual carve up of account

District – a quantum change in quality

2005 2010 X 10 Perils

© Aviva plc

THE END

Any questions ?