introduction to pricing - actuaries.org.il · introduction to pricing part 2 presented by ... •...

Introduction to PricingPart 2

Presented byThomas Guidon & Andrzej Smolinski

21 September 2011

AgendaSession 2 – Personal lines pricing

Different LoBs need different pricing

Estimation of total risk cost volume

Pricing for personal lines

Two extreme approaches to setting profit margin

Example of motor renewals pricing project

Final remarks, suggested readings

Questions

Pricing for different LoBsOverview

• Detailed predictive modeling for risk & demand/policyholder behaviourMotor TPL & casco

• Medical care, income protection and workers compensation• Pricing to consider life and non-life techniquesAccident & Health

• Covers property and business interruption• Pricing based on

• Attritional & large loss models/GLMs• NatCat vendor models or proprietary models

Fire & natural catastrophe

• Aviation• Closed finite market, special model based on total market data is

recommended• Marine

• World-wide market, market know-how and special clauses need to be considered

• Financial insurance• High correlation with asset risk and across underwriting years, dependency

on economic scenarios • Engineering

• Multi-year exposures, exposure rating, expert driven

Specialty lines

Pricing for different LoBsOverview

• General liability generally on occurrence basis whereas subclasses like professional or product liability are often on claims made basis• Market knowledge, inflation sensitive, legal environment very important

General liability

• Mostly exposure pricing (due to limited loss history)• Claims behavior and exposure periods need to be considered (e.g.

travel vs. legal expenses coverage). Others

• Excess of loss reinsurance pricing is its own disciplineReinsurance

• Complex international setup for issuing of insurance polices• Risk often ceded to captives and reinsured on an aggregated basis• Captive reinsurance mainly based on exposure pricing

Industrial risks

Further we will focus on personal lines insurance pricing

Pricing for different LoBsPricing methods

Personal lines

Insurance pricing

Commercial lines

Reinsurance pricing

Personal lines

Commercial lines

Pricing for different LoBs Insurance premium breakdown, total European market 2010

Source: CEA








Questions

Estimation of risk cost volumePricing structure & data

• Risk exposure type and period • Occurrence or claims made policies• Exposed period (start & end date)• Portfolio composition (stability of history and expectation for future)

• Pricing data• Data accuracy, completeness and appropriateness

• Does data fit policy/exposure structure• Written/earned premium and occurrence/accident year loss & frequency data• claims made or multi year policies

• Indices• CPI & claims inflation Index, building cost index

• In Israel: Claims inflation exceeding CPI only (losses are indexed with CPI)

• Rate change index (company and/or market)

Estimation of risk cost volumeTwo principle methods to determine risk premium volume

1. Ultimate exposure method‒ Determine ultimate loss by period (chain ladder, BF, etc.)

‒ Normalize ultimate loss by period based on exposure metric

‒ Determine exposure ratio index for each selected method

‒ Select exposure ratio index

‒ Determine exposure ratio and expected ultimate loss for new period based on given exposure

2. Frequency-severity method‒ Determine ultimate frequency of claims-with-cost by period

‒ Determine ultimate loss cost by period (chain ladder etc.)

‒ Normalize ultimate frequency & loss cost by period based on exposure metric

‒ Select frequency and severity index

‒ Determine expected frequency and severity for new period based on given exposure

Method 1) is simpler but method 2) allows to monitor and compare realty with assumptions, hence trends might be noticed sooner.

Estimation of risk cost volumeAttritional and large loss model

• Split data into attritional and large losses as tail of loss severity distribution is badly fitted otherwise (extreme value theory (EVT))‒ Determine threshold index for large losses (iterative process)

• Index should reflect claims-inflation• Measure claims-inflation on several methods like

• Average ultimate claims series based on several development methods• Average of closed claims by accident/underwriting/reporting year• Average of closed claims by closing year

‒ Large loss threshold can be determined with EVT or rule-of-thumb: More than five claims per period should exceed indexed threshold• A Poisson-Pareto large loss model remains a Poisson-Pareto large loss

model for larger thresholds• Stability from one year to the next

‒ Exclude capped large losses form underlying attritional loss model

Estimation of risk cost volumeAttritional loss model

• Attritional loss model for new exposure period:1. Ultimate exposure method for attritional losses2. Frequency-severity method for attritional losses

• Method 1) is often applied for attritional losses inter alia in internal models

• The latest underwriting years may contain losses which in future years will develop to large losses and hence be excluded from the attritional data: Needs to be considered when developing triangles.

• Method 2) allows to measure frequency and severity trends on attritional losses

Estimation of risk cost volumeLarge loss model

• Large loss model for new exposure period:1. Ultimate exposure method is suboptimal for large losses2. Frequency-severity model is generally used for large losses

• Frequency-severity model for large loss model :‒ Are on-leveled losses “reasonable” (cross check on claims-inflation

index) ‒ Are on-leveled losses spread across the policy limits (impacts fitting)‒ Back-testing of frequency and severity selection (stability)‒ Past exposure or policy limit changes impact the result (e.g. increase

of young drivers over time)

What we have just learned

• Reserving, pricing and reinsurance division do similar calculations‒ Alignment between the three models. Cross checking results/assumptions‒ Avoid circular reference “Reserving uses pricing loss ratio and pricing uses

reserving loss ratio” for the last period.‒ Cross check loss ratio of last period with Loss-Ratio-Walk method: Use

rate change and claims inflation index to estimate latest loss-ratio.

Q310 End-10

Q3 2010Cut-off date for data to perform 2011 pricing

Year End 2010Start of new underwriting yearExposure data is as at this datePresent value of losses and premium are as of this dateUse risk free rate to discount

End 2011End of new underwriting yearMean time to exposure and mean time of premium income are need for present value

End 2012End of exposure, premium is now fully earned and all losses occurred.

End-11 End-12








Questions

Personal lines products marketImpact on products pricing

2. Customers• Very unloyal, especially in motor. However, usually companies make more profit on

renewals ! Hence retention critical.

• Large portfolios and hence huge volumes of data

• Many frauds (esp. in motor)

1. Products• Motor, home & other property, travel, general/private TPL, personal accident

• Commodity market, very little or no differences, much transparency

• Price and brand the only factors

3. Distribution channels• Traditional, direct (Internet, Call Centre), bancassurance, car dealers

• Each has a different business model and pricing challenges

Personal lines products marketImpact on products pricing

4. Legal environment• Usually MTPL is obligatory, in Israel it is MBI

• In some countries MTPL rates must be accepted by the regulator or are even

imposed by the regulator

6. Market-specific issues• Aggregators eroding profit margin in the UK

• Motor - policy attached to the vehicle (Poland) or to the owner (Switzerland)

• Multi-year policies in some countries (Switzerland)

5. Competition• Many different competitors, different distribution channels

• Direct operators aggressive with price, traditional companies have better client

relationship via agents

Personal lines products marketMain pricing challenges

• Large volumes of data, many different variablesØ Client, insured object, contract terms, external e.g. geographic coordinates,

population density

• Often poor data quality

• Market already uses sophisticated customer segmentation techniques, i.e. price

discrimination is a standard

• Many dimensions of considerationØ Risk cost, acquisition cost (high variable for traditional companies vs high fixed for

directs), demand (conversion, retention, cross-selling)

• Short-term vs long-term profitability horizonØ It is easier to get a client than to keep him

Ø Different pricing strategies for different horizons

Evolution of personal lines pricing

Balancing total cost

Price discrimination, computer data analysis

GLMs, risk modelling, rise in computational power

Demand modelling, price optimisation

More individualised price optimisation, real-time updates ?

1900

1950

1970

2000

2012+

19 19

Margin

Operational Cost

Risk Cost

Insurer View

Reliability

Brand

Competition

Convenience

Loyalty

Customer View

Customer’s view of the companyIndividual sensitivity to price determined by various subjective factors

20








Questions

How to set the profit margin ?

Technical premium Risk costs Acquisition

costsAdministration

costs Profit margin

Calculated Business decision

Here comes the real fun J

Two extremes (and a spectrum of choices between):

Flat margin (cost-plus)

Benchmarking against competition(e.g. new product)

Separating newbusiness and renewal tariffs

Modern price optimisation

Customer’s individual sensitivity to price








Questions

Example of a pricing projectAgenda

Assumptions

Product study

Data & KPIs

Modelling (GLM/GAM theory, risk & demand modelling,

competitors’ price benchmarking)

Combining the results – price optimisation vs flat margin strategy

Presenting the proposal

Implementation

Project assumptions

Four years have passed since the successful launch of your start-upcompany and there are already three cohorts of renewal policies inforce. The Pricing Committee decides that your personal motorproducts need repricing. As the most qualified pricing actuary in thecompany, you are to take the lead as the project manager. You haveto deliver the results to the Project Steering Committee before thedeadline, using the resources which are put at your disposal andcomplying with the limitations set by the PC.

Project assumptions• Direct operator with two distribution channels – Call Centre and Internet• Company was founded 4 years ago and since then has been growing

successfully, but still remains relatively small in the market (no major player)• Advanced pricing perceived as key strategic area

Company background

• Two types of motor policies: comprehensive (MTPL and casco) and MTPL only

• Each can be sold with two additional options (riders) – roadside assistance and personal accident

• The contracts are annual and can be cancelled prematurely only under specific circumstances (e.g. vehicle sale)

• The Pricing Committee has decided, that only privately used passenger cars (including SUVs and small vans/pickups) are in scope.

• Same tariffs for new business and renewals• Product is in the market for 4 years (3 cohorts of renewals)

Product info

• Separate renewal pricing from new business pricing and develop a coherent pricing strategy for them

• Create a pricing efficient frontier for motor renewals to analyse the potential profitability-volume trade-off

• Develop optimised pricing strategy for renewal book• Target combined ratio for renewals = 80%

Project scope

• Maintain the current overall retention rate of 80%Limitations

Project assumptionsOrganigram

Steering Committee

Peer review

Project manager

Project team

Pricing, reserving, reporting

Products, UW

Finance & Controlling Claims & fraud

Sales, renewals, servicing

IT

Reports to

Input

Feedback

Legend

Marketing

Project timeline

• Risk modelling• Demand

modelling• Price test in

progress• Price

benchmarking in progress

• Developing 2-3 alternative price strategies

• Preparing the presentation

• Presenting results to the Steering Committee

• Presenting results to the Pricing Committee

2 weeks 7 weeks 8 weeks 6 weeks 3 weeks

• Reengineering competitors tariffs

• Supplying your demand models with competitors rates

• Refitting demand models on price test data

• Combining the results

• Kick-off• Product study

• Data availability check

• Data extraction• Plausibility

checks• Price test

design and implementation

• Price benchmarking study design

6 months

• Preparing tariff document

• Preparing implementation document

• Implementation• Testing• Going live• Field testing


Assumptions

Product study

Data & KPIs





Implementation

Product study

The first step in a pricing project is understanding three things

Starting point

What are the boundary conditions

What are the delliverables

What do you price

Product study

1. What you price

• Product structure (e.g. covers)

• Terms and conditions (e.g. exclusions)

• Customer behaviour

• Market (competitors & their offer)

• Business processes

Organise a series of meetings («interviews») with relevant functional departments, i.e. product management, UW, sales, servicing, renewals,

claims, marketing

Product study

2. What are the deliverables

• New pricing structure or just updated factors

• Expectations – lower combined ratio, bigger market share, etc.

• Other

Make sure you understand the expectations. Ask the Steering Committee – better too many questions than too few.

Product study

3. Boundary conditions

Limitations to what you can or are allowed to do:

• Arbitrary/soft: limitations set by the Pricing Committee, business/strategic

constraints

• Natural/hard: IT constraints, process constraints, legal and PR constrainst

Be aware: some boundary conditions may appear unexpectedly throughout the project

Product study

Examples of business constraints

• Maintain overall retention rate

• Do not touch customers with VIP status

• Keep the existing rating structure

• Or more extreme: 95% of the portfolio cannot change by more than +/-10%


Assumptions

Product study

Data & KPIs





Implementation

Data

We all have our data issues ... Be sure you understand yours J

Types and sources of data

Policy terms Customer Insured

object Claims External

policy data

Policy dataRisk modelling vs demand modelling

• Risk modelling and demand modelling require different data structures (way of organising your data) and different variables

• Risk modelling extract is the result of the risk process:

For any underwritten policy, we divide it into homogenous periods of earned exposure. To each earned exposure period, we merge the total number of claims and the total incurred claim amount separately for each claim type.

• Renewal demand modelling extract is about the dynamics of the renewal process - we take shapshots at different points in time throughout the renewal process to analyse how it develops:

For any renewing policy, observe what happens to it upon renewal offer issue, receipt by client, renewal moment and some time after renewal, to see how the customer behaves and how the new policy develops/lapses.

Policy dataRisk modelling vs demand modelling

• Risk process view of the policy data is the policy data extracted in such

a way that calculating earned exposure and earned premium for each

cover and each unique homogenous exposure period is possible

• Renewal process view of the policy data is the policy data extracted in

such a way that it reflects the development of renewal policy in time,

from issuing the offer to getting client’s final decision

• To prepare each modelling extract you have to combine the policy data

with the claims and the external data. For the purpose of merging these

databases, the policy data already has to be arranged in a proper way

before the merge.

Types and sources of dataRisk process view extract – examples of most important variables

1. Policyholder/driver characteristics• Age, place of residence, post code, bonus – malus class, etc.

2. Characteristics of insured object (i.e. motor vehicle)• Type and model of vehicle, age of vehicle, cubic capacity, car value, etc.

4. External data• Geographic and socio-economic data

3. Contract details• Payment frequency, distribution channel, tenure, other policies with the

company

• Cover start date, cover end date, exposure, written premium, earned

premium

Types and sources of dataRisk process view extract – volume of claims experience

What is the required volume of claims experience ? How many years of claims experience do you need ?

This answer is not simple. It depends mostly on:• Claims frequency• Your annual exposure volume• Number of years of available claims experience (you shouldn’ use too many)• Number and type of explanatory variables/effects you would like to fit• Computational power of your software (hardware is usually no bottleneck

today)

Indicative example of a small own damage collision portfolio:With claims frequency of 20%, annual exposure volume of 7k and 3 years of claims experience you can get a reasonable claims frequency model with 30 significant explanatory variables

Types and sources of dataRisk process view extract – example

2010-01-01 policy starts with cover A

and B

2010-03-01 client

changes name

2010-04-01 client moves from

Zurich to Geneva

2010-05-01 client drops

cover B

2010-07-01 policy is

cancelled

The above should be reflected in the risk extract as five records, representing separate

homogenous exposure periods:

PolicyNo

Cover type

Cover start

Cover end Age Gender ... City Other

covers Exposure Claim count

Totalclaim cost

P9999 A 1.1.2010 31.3.2010 34 Male ... Zurich B (31+28+31)/365 ? ?

P9999 A 1.4.2010 30.4.2010 34 Male ... Geneva B 30/365 ? ?

P9999 A 1.5.2010 30.6.2010 34 Male ... Geneva No(31+30)

/365 ? ?

P9999 B 1.1.2010 31.3.2010 34 Male ... Zurich A(31+28+31)

/365 ? ?

P9999 B 1.4.2010 30.4.2010 34 Male ... Geneva A 30/365 ? ?

Types and sources of dataRisk modelling extract – IBNR claims issue & exposure censoring

• The IBNR claims and lenght of claim settlement process issueObserved earned exposure period has to be censored some time before the date the

extract is made (depends on risk type, e.g. notification lag and claim settlement time for

casco collision claims is very short and 3 months are enough). Then, claims are merged to

the observed earned exposure period and the correct claim evaluation is merged to each

claim• In this way we

1. Avoid most of the IBNR claims – we give them chance to get notified

2. Minimise the number of notified claims which have only initial reserve estimates and

no paid damages

• However, we cannot avoid IBNR claims completely (TALK TO YOUR RESERVING

ACTUARY)We need some IBNR claims loading in the final risk cost estimate

Types and sources of dataRenewal process view extract

1. Policyholder characteristics AT DIFFERENT POINTS IN TIME• Age, place of residence, bonus – malus class, etc.

2. Characteristics of insured object AT DIFFERENT POINTS IN TIME• Type and model of vehicle, age of vehicle, cubic capacity, etc.

3. Contract details AT DIFFERENT POINTS IN TIME• Payment frequency, distribution channel, tenure, other policies with the

company, etc.• Cover start date, cover end date, exposure, written premium, earned

premium

4. External data• Geographic and socio-economic data

5. Info is policy renewed

Types and sources of dataRenewal process view extract – volume of renewal experience

What is the required volume of renewal experience ? How many renewingpolicies do you need ?

This answer is not simple as well. It depends mostly on:• Retention rate• Monthly volume of policies maturing to renewal• Market stability• Price test (we will talk about it later)

Indicative example of a small MTPL portfolio:With retention rate of 70%, monthly volume of 6k renewing policies andrelatively dynamic market, 6 months of historical renewals were used to fitinitial renewal demand model. Then, a final demand model was fitted after 2months of price testing with proportionate allocation to 4 groups. It was enoughto get a reasonable demand model with 20 variables.

45

New business acquired last year

Policies not renewed due to

bad debt

Vehicle sales & immediate

declineIn force & paidRenewals

which came in force

Retention up to 2009-02-15

Retention ratio 60%-80%

Paid retention ratio <RR

Bad risks deliberately outpriced & customer

lapsed

Lapse rate 15%-30%

«Regular» lapses

Renewal processExample of new business cohort maturing to renewal

46

Renewal processRenewal process and its cascade of financial implications

MTPL only

Renewed

DiscountReduction in price, no influence on risk

cost

Add riders Increase in price and in risk cost

Drop riders Reduction in price and in risk cost

Add casco Increase in price and in risk pcost

Cancelled

Motor comprehensive

Renewed

DiscountReduction in price, no influence on risk

cost

Add riders Increase in price and in risk cost

Drop riders Reduction in price and in risk cost

Drop casco Reduction in price and in risk pcost

Cancelled

Renewal demand models

Additional customer behaviour

models

Types and sources of dataRenewal demand extract

1.1.2011 renewal offer issued to the

client

08.1.2011 client calls

and bargains,

new renewal

offer

20.1.2011 policy

renewed according to second

offer

26.01.2011 client drops

casco and keeps MTPL

PolicyNo

Cover type PY

Cover type O

Cover type R

Cover type F

Price PY Price O Price R Price F Claims countlast year

Claims amountlast year

Tenure

P9999 MTPL + casco

MTPL + casco

MTPL + casco + roadside assistance

MTPL + roadsideassistance

1000 1100 1100 400 0 0 1

Moment O Moment R Moment F

20.1.2010 new

business policy goes

in force

Moment PY

City lastyear

City O City R City F Cover type F

Price PY Price O Price R Other covers

Price F Claim count

Totalclaim cost

... MTPL + casco

MTPL + casco

31.3.2010 34 Male ... Zurich B (31+28+31)/365

1 10000 CHF

Types and sources of dataClaims data extract

1. Merging keys & identifiers• Policy no, cover type, claim date, claim code

2. Claim characteristics• Type of claim (casco theft, casco collision, casco fire, MTPL bodily injury,

MTPL collision, etc.), claim status (notified, open, closed, reopened), CAT

claim indicator

3. Value of claim• Paid damages, case reserve, paid ALAE, ALAE reserve

• Check the ultimate cost estimates with your reserving actuaries

Claims data extractImportant issues

• Separate record for each claim

• Accident <> claim in the risk modelling sense

If MTPL customer collides with another vehicle, damages it and injures twopassengers in that car, for the sake of risk modelling we have TWO claims – oneproperty claim and one bodily injury claim

• Attritional, large and CAT claims

Each claim should be classified into attritional, large or CAT category (see nextslide)

• Claims inflation (claims inflation <> CPI inflation !)

You price future policies based on past claims experience. Claim costs have tobe extrapolated using proper claims inflation indices to reflect expected futurecosts.

Claims data extractLarge claims analysis

• Large claims are a different type of claim and therefore should bemodelled separately from the attritional claims

• Typically, we have much less large claims than attritional claims and theirvalue is much more volatile. If we modelled all claims together, thepresence of large claims would distort the empirical severity distributionas they are clearly outliers.

• In personal lines motor portfolios, large claims exist mostly in the case ofMTPL bodily injury risk

• Large claims have to be separated from attritional claims. To do that, youhave to decide which claims are large and which are not. You need someclassification rule.

Claims data extractLarge claims analysis

Sufficient volume of data Limited volume of data

? ???

Where is the correct cut threshold ?

• There are formal statistical approaches to the problem

• We believe, however, that a «common sense» approach yields results whichsuffice. We treat large BI claims as typical outliers and set the theshold to the 95thor 90th quantile of the empirical severity distribution.

Claims data extractExample

Claim no Policy no

Cover type

Claim date

Claim type

Claimstatus

CAT indicator

Paid Casereserve

Paid ALAE

ALAE reserve

...

C111111 P111111 Casco 31.3.2010 Collision Notified 0 0 4500 0 100 ...

C222222 P222222 MTPL 30.4.2010 Property Open 0 1478 600 210 50 ...

C333333 P333333 Personalaccident

30.6.2010 n/a Closed 0 1579 0 150 0 ...

C444444 P444444 Casco 31.3.2010 Theft -partial

Closed –denied

0 0 0 0 0 ...

C555555 P555555 MTPL 30.4.2010 Bodily injury

Open 0 0 2000000 2000 5000 ...

Five different claims:• One notifiedà No paid amounts, no good reserve estimates (initial reserves only)

• One potential large claimà BI with high case reserve, probably annuity, to be checked

• One «denied» nil claimà nil claims should be removed (they affect the estimates)

Risk modelling extractMerging policies with claims

• Merge by merging keysPolicy number, risk code, exposure period start and end date, date of claim

• One-to-many relationship - «full outer join» in SQL terminologyAll claims should have a policy but not all policies have a claim (hopefully J) à

check that !

• Seems OK ? Go to plausability checks (see next slides).

• Is not OK?Do not panic, data extracting is a tiring, «trial & error» – type iterative process.

Types and sources of dataRisk modelling extract – example

2010-01-01 policy starts with cover A

and B

2010-03-01 one claim

from cover A

2010-04-01 client moves from

Zurich to Geneva

2010-05-01 client drops

cover B

2010-07-01 policy is

cancelled

The above should be reflected in the risk extract as five records:

PolicyNo

Cover type

Cover start

Cover end

Age Gender ... City Other covers

Exposure

Claim count

Totalclaim cost

P9999 A 1.1.2010 31.3.2010 34 Male ... Zurich B (31+28+31)/365

1 10000 CHF

P9999 A 1.4.2010 30.4.2010 34 Male ... Geneva B 30/365 0 0

P9999 A 1.5.2010 30.6.2010 34 Male ... Geneva No (31+30)/365

0 0

P9999 B 1.1.2010 31.3.2010 34 Male ... Zurich A (31+28+31)/365

0 0

P9999 B 1.4.2010 30.4.2010 34 Male ... Geneva A 30/365 0 0

Competitors benchmarking data

• Competitive market analysis is a vital part of pricing project

• Even if your company uses typical cost-plus approach, you cannotignore the existence of competition and have to take their pricingstrategies into account

• Especially for price optimisation, very detailed competitive marketanalysis is necessary, including detailed price benchmarking

• The goal is to get competitors’ rates or at least approximate them basedon a statistical sample

• Approximated rates are then used to create a market price index

• The ratio of your company’s renewal price to the market index will beone of the most important explanatory variables in the renewal demandmodel

Price elasticity testDesign & test data

• Historical rate changes are not random – they are systematic and

correlated with risk

• To estimate customer’s real sensitivity to price – his price elasticity of

demand - you need to have some random variation in prices

• The best way to get proper data is a price test

• Renewing customers are randomly allocated to one of 3 – 4 test groups

• Each group gets a random price modifieE.g. +10%, +5%, no price change (control group), -5%, -10%

• After 2-3 months you stop the test and extract the data

Data cleaning

Costs a lot of time and is absolutely crucial. Checks include:

• Number of records

Correct merge should not lose records

• Empty fields

• Impossible values

1. Univariate outliers, e.g. negative age or very young/old, driving licence issued in 1915, passenger cars with 20 seats, etcà histograms, first and last quintiles analysis, manual inspection

2. Multivariate outliers, e.g. 18 years old and driving licence age since 10 years, Toyota Yaris shouldn’t have an engine with 4000 cc, etc) à well, not that easy ...

• Feedback to IT and risk management

Plausibility (sanity) checks

Compare main KPIs with some INDEPENDENT sources (i.e. not the same

data source – e.g. accounting data bank vs claims data bank)• loss ratio

• avg premium

• total exposure

• claims frequency

• average claim

• number of active policies

Explain the differences, perhaps your data extract is wrong (data extraction

is an iterative process, quite offen being trial & error)

Product portfolio KPIs

• (Exposure) Period = part of policy year

• Homogenous period = period for which no risk change took place (see exampleon slide 41)

• Exposure of a policy in a period = number of days the policy is active in thatperiod / 365

• Written premium (WP) = 1. In case of real data, written premium is equal to itscommon accounting definition. 2. In case of projecting based on hypotheticalportfolio (pricing sample), we define written premium as expected premium(insurance price) due for insurance cover over assumed period.

• Number of policies = underwritten policies in given period

We analyse policy years, unless stated otherwise. All the quantities, unless stated otherwise, are defined on single risk (cover) basis (e.g. casco theft for one policy, mtpt, etc.).


• Earned premium (EP)

For a real portfolio and for a hypothetical portfolio earned premium is definedin the same, following sense:

Example: Given a customer who buys a policy and after 3 months moves from Zurich toGeneva (apart from this, the risk remains constant for entire policy year). Annualisedpremium for Zurich is APZ, for Geneva it is APG. Written premium for given policy in themiddle of policy year would then be 3/12*APZ+9/12*APG, whereas its earned premium3/12*APZ+3/12*APG.

Product portfolio KPIs• Claim count

Expected number of claims in given period per analysed risk. Example: if twopeople got injured in one incident it is one claim, if a car was damaged and aperson injured, it is considered as two claims (BI and property). Nil claims areexcluded.

• Paid claims costs = per period: paid damages + paid ALAE costs – receivedrecoveries

• Ultimate claims costs (ultimates) = per period: paid claims costs + casereserves + ALAE reserves + reserves for recoveries + IBNR reserves earned inthe period (allocated to the period according to the exposure earning principle)

• Severity = sum of ultimates per risk and period / claim count in period

• Loss ratio = ultimates / earned premium

• Expected combined ratio = «all the earned costs» / earned premium =(ultimates + earned administrative costs + earned acquisition costs) / earnedpremium


• Conversion rate = number of accepted quotes / number of issued quotes ingiven period

• Retention rate = number of renewed polices / number of issued renewal offersin given period (mid-term cancellations should not be taken into account)

• Lapse rate = 1 – retention rate


Assumptions

Product study

Data & KPIs





Implementation

A small introduction to GLMs/GAMs

What do we use GLMs/GAMs for in pricing• Estimating risk costs (risk premium)

• Estimating demand (probability of purchase)

GLM = generalised linear model

GAM = generalised additive model, extension of GLMs

What software tools can we use• Open source, e.g. R (inefficient and cannot process large volumes of data)

• Statistical packages – SAS, Statistica, etc. (inefficient)

• Dedicated actuarial pricing software – Earnix Optimizer, Emblem, Pretium (best

tools for the task, much more efficient)

Risk modellingIntroduction

• If possible, model attritional claims for each peril separately (collision

partial claims, collision total claims, partial thefts, total thefts, etc.). If not,

at least model each risk separately.

• Large claims – common loading vs propensity models vs complete freq

& sev modelling, each peril or each risk or in total ?

• CAT claims – Are there any ? How to include them ?

• risk_cost = claims_frequency1 * claim_severity1 + …• Frequency and severity are often influenced by different effects

• They have different distributions

• Statistically better to separate them

Risk modellingRisk level or peril level

How deep can we go ? Depends on available data and the required time-

quality trade-off.

• MTPL property

• MTPL bodily injury

• Casco collision (total vs partial)

• Casco theft (total vs partial)

• Casco fire & natural disasters

• Other market-specific

GLMs vs classical linear regression

Linear regression – OLS model

• Response variable Y is normally distributed

• Its expectation is a linear combination of explanatory variables

• Its variance is independent of mean and constant for each observation (homoscedasticity)

Model Reality

Generalised linear model

• Response variable Y has a distribution which belongs to the exponential family of distributions

• Its expectation depends on the linear combination of explanatory variables via the link function g(.)

• Variance can be mean-dependent, i.e. homoscedasticity is not required

Insurance data• Continuous variables (e.g. claim severity) are generally

unsymmetrically distributed. Even more, many modelled variables are discreet (in particular binary).

• We often assume some multiplicative structure, rather than additive (in particular multiplicative rating structure)

• Often clear heteroscedasticity of empirical distributions (e.g. claim severity)

Insurance data

• The exponential family of distribution contains many distributions widely adopted in non-life insurance mathematics – gamma, inverse gaussian, Poisson, negative binomial, binomial. They are well-suited to model insurance phenomena.

• The choice of log as the link function allows us to reflect the multiplicative character of the modelled relationship

• We have both discreet variables and heteroscedastic distributions

( ) ( )

−=

φθθ

φayycyf exp,)(

( ) ( ) ∑=

+==p

jjjxgEYg

10 ββµ

∑=

+==p

jjjxEY

10 ββµ

GLMs most important in pricing

Poisson regression• Claim counts (frequency)• Anything occuring according to Poisson process

Gamma regression• Claim amount (severity)• Other continuous variables without upper value limit

(loss ratio, premium amount, etc.)

Logistic regression• Probability of sale (conversion rate)• Probability of renewing (retention/lapse rate)• Any indicators (probabilities, proportions)

Poisson regression for claim countsBasics

GLM with log link function and Poisson error structure. Exposure used either

as offset or as weights to rescale observed claim counts (results are the

same).

Typically most important variables:• Bonus-malus class (NCD class)

• Age of driving licence

• Age * gender

• Deductible

Can be enhances by using GAM components (smoothing splines) for• Age * gender

• Geographical coordinates (spatial smoothing)

• Vehicle value, etc.

Modelling claims freq – MTPL propertyExample of modelling output – parameter estimates

Source: Earnix

Modelling claims freq – MTPL propertyExample of modelling output – model diagnostics, goodness-of-fit

Source: Earnix

Gamma regression for claim amountBasics

• Claim severity Y (total claim amount of a single claim) is assumed to

be gamma distributed with mean varying from risk to risk.

• Its mean EY is estimated as a function of customer, vehicle and

policy characteristics: = , , …• GLM regression is used, with log link function and Gamma error

structure

• Modelling is basically similar to Poisson regression

Logistic regressionLogistic regression as a GLM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Response variable Y is binary distributed (binomially with one trial):

Our link function is the logistic function:

Hence the model:

( ) ( ) ( ) { } ( )1,0,1,0,1 1 ∈∈−=== − pyppyYPyf yy

( )

−

=µ

µµη

1ln

Graph of logistic function – real line mapped into the [0,1] interval

( )

−

=µ

µµη

1ln

Why logistic transformation

There are two other possible choices – probit and complementary log-log. They yield, however, similar results for different link functions.

Additionally, within the (0.1 , 0.9) interval the logit and probit functions are appox. linearly dependent.

There are at least two arguments in favour of the logistic one:

1. More intuitive «mathematically», less «exotic»

2. (Relative) Ease of explaining the results

j

p

jj xp

p ∑=

+=

−

=

− 1

01ln

1ln ββ

µµ

Individual demand for insurance

Demand for new business

Number of quotes which are accepted by the customers at the given price level

Demand for policy renewals

Number of renewal offers which are accepted by the customers at the given price level

Two types of demand

Conversion rate

CR = P / Q

Retention rate

RR = R / RO

Assumption

Number of quotes and renewal offers is independent of the price

CR and RR are(statistical) probabilities that the offer is accepted

Individual demand for insurance

• We do not model the value of demand function at the given point (at the given price), but rather the

relative PROBABILITY of buying (in the sense of CR or RR)

• Different consumers (buyers of insurance services) have different price elasticity of demandà CR and

RR vary not only according to price, but also according to market segment (portfolio segment)

• Customers generally compare available market offers (some more, others less actively) à we need to

take competitors’ prices into account

• We therefore need some multivariate regression model for the probability of purchase

• Logistic regression!

Suggested approach

Price elasticity of demand

Price elasticity – isolated reaction to price

while controlling all other factors

Modelling the probability of purchaseHow to measure customer’s individual sensitivity to price

Ω∆ ∆

Price elasticity will vary among the customers affected by customer’s profile, policy characteristics and

competitors’ prices, and other variables.

Inelastic demand (|E| < 1)

Customers with inelastic demand are less sensitive to price changes. Raising their prices will decrease

their demand by a smaller percentage. Increasing price for an “inelastic customer” may increase the

expected revenue from him/her.

Elastic demand (|E| > 1)

Customers with elastic demand are more sensitive to price changes. Raising their prices will decrease

their demand by a larger percentage. For an “elastic customer” decreasing the price may result in

an increase of the expected revenue from him/her.

Elasticity changes along the demand curve

77

TPL – example of individual demand function

Variable Value

Age 49

Premium 443.91

Previous Premium 447.84

Tenure 2

CARWEIGHT W5

COMPREHENSIVE 1

GENDER M

INSTALLMENTS 3

MARKETPREMIUM 448.83

POSTALCODES 01067

PROFESSION Employed

TOTALCLAIMS 0

Source: Earnix


Assumptions

Product study

Data & KPIs





Implementation (tariff document, IT specification, price

optimisation issues)

Combining the resultsChoosing a recipe for technical premium

Two ways to go:• Either choosing some rating structure (introducing or not new factors), overall

price level and relativities

• Introducing some individual price optimisation setup

Office premium

Risk costs

Variable costs

Fixed costs

loadingProfit

margin

• Risk models

• Demand models

• Competitors benchmarking results/reengineered tariffs

• Qualitative input from UW, product managers, sales, claims, etc.

• Strategic input from Pricing Committee (including limitations/border conditions)

Rating structure vs. technical modelsDeriving tariff relativities

In a typical multiplicative rating structure for each risk we have:• Base premium

• A set of rating factors and corresponding relativities for each level of factor, e.g.

bonus-malus class, fuel type, policyholder age, deductible

• Some functional factors/effects, e.g. degressive rates for vehicle value, 1/x for

driving licence age, given by a specific mathematic formula

• Room for commercial discretionary discounts

Rating structure vs. technical modelsDeriving tariff relativities

These are usually derived from the original risk models through some

arbitrary combination of• ‘Flattening’ of risk models (exclusion of some factors / simplyfing)

• Reconciliating of portfolio profitability projections under the chosen tariff with

company’s strategy, business model, shareholders’ expectations

• Comparison with competitors

• Input from product managers, underwriters, claims department, sales,

marketing, etc.

There is no smart actuarial algorithm to do that. It is more of a trial & error

procedure than a mathematical algorithm.

Price optimisationBasic idea

Margin vs. Price

Retention probability vs. Price

Profit vs. Price

Optimal price

Risk cost models

Demand models

The purpose is to find the price to balance margin and volume such that profit is

maximized

Price optimisationFramework

Historical CR & RR data

Benchmarking

R test

NB & R behavioural models

Demand functions

Other models (cover

reduction, discounts, etc.)

Claims data

Financial costs data Variable cost model

Frequency &

severity modelsBurning cost

Variable costs

NB & R margin

+

+

PREMIUMBusiness goals &

pricing constraintsCompany strategy

=

OPTIMISATION

84

Impact on results

Financial projectionsSimulating customer behaviour

Discounts

Drop rider A

Drop own damage

Effectsat renewal time

PremiumDiscountDiscountRenewal ⋅⋅⋅= %_)Pr()Pr(

( )riderAtburningriderApriceDropRiderARenewal _cos__)Pr()Pr( −⋅⋅=

( )damageowntburningdamageownpriceageDropOwnDamRenewal __cos___)Pr()Pr( −⋅⋅=

Renewals ( )VCtBurningCosPremiumRenewal −−⋅= )Pr(

Sum up results for all offered renewals

Drop rider B ( )riderBtburningriderBpriceDropRiderBRenewal _cos__)Pr()Pr( −⋅⋅=

Individual price optimisation• Commercially the most powerful use of demand models

• Idea – profit margin set individually for each customer (in practice – identifiable & separate segment of customers)

• How to calculate the profit margin – conditional maximisation of a selected target function

• Examples

Ø Annual underwriting profit maximisation for a portfolio of renewals, with the border conditions of keeping the retention rate unchanged

Ø Customer lifetime value maximisation while keeping CR and RR at a certain level

• Multi-period CLV optimisation (e.g. 5 years)

•

• Each year demand is a function of the current price and the previous priceà dynamic programming

( ) ( ) ( ) max1

→= ∑∞

=kkk

k jPjpvjCLV

( ) ( ) )( jPMFCLVCLjj kkkriskk

officek +++Π=Π

Price optimisationHow much we can gain – industry benchmark results

• Price elasticity for renewals is usually much lower than for new

business. Typically, it is around 1 for average profile. There is much to

gain here.

Individual optimisation2% - 4%

on combined ratio

General formula1% - 3%

on combined ratio

• Depending on how you implement your price optisimation setup for

renewals, the benchmark results are

87

MTPL renewals – efficient frontierPrice optimisation as a trade-off between two KPIs

Individually optimised

prices, a set of trade-offs between

retention rate and profit

Source: Earnix


Assumptions

Product study

Data & KPIs





Implementation


• Do not save time on the presentation - it is the MOST important part

of the project, not the least important one. If you fail to convince the

stakeholders to accept your proposal, the entire project is a failure.

• Remember – they are no actuaries. Imagine you are preparing a

presentation on something new to yourselves.

• Short messages, each supported by results. More in backup.

• Train your presentation. Show your presentation to non-actuarial

colleagues and ask for feedback.


Argumenting

• Pure business reasoning (plus some politics L)• No fancy PhD stuff !• Throw in some carrots for marketing, sales, claims

Language

• Speak common, business language• Be confident, address people directly and personally• Avoid scientific terms and reasoning

Prepare for questions

• Image you were the head of sales, etc.• Imagine you were to invest your own money in the

proposal based only on the presentation

Smoothing splines are

cool, but your CEO DOES NOT CARE

There is already

enough jokes about

actuaries

Eachexecutive has different KPIs in his contract

...


Assumptions

Product study

Data & KPIs





Implementation

Implementation

• OK, all the stakeholders have agreed to implement your proposal for

new pricing. Good news, but you are still far from being successful.

• Work on implementation document requires your constant cooperation

with the IT. Better ask too many questions than too few. Be VERY

friendly – these guys can really help you or make your task much more

difficult, your call.

• Test the implementation results thoroughly. Analyse business processes

which involve pricing, prepare test cases for typical activities (renewal

offer, mid-term adjustment, etc.), try to predict most probable errors.

Your CEO says «Good job» J








Questions

Suggested readings1. Anderson D., Feldblum S., Modlin C., Schirnmacher D., Schirnmacher E., Thandi M. ‘A Practitioner’s Guide

to Generalized Linear Models’, CAS Study Note, Casualty Actuarial Society 2005

2. Anderson D., Bolton C., Callan G., Cross M., Howard S., Mitchell G., Murphy K., Rakow J., Stirling P., Welsh G., Report of the General Insurance Premium Rating Issues Working Party (GRIP), Faculty & Institute of Actuaries 2007

3. Bland R., Carter T., Coughan D., Kelsey R., Anderson D., Cooper S., Jones S. ‘Customer Selection and Retention’, report presented at General Insurance Convention (GIRO), Faculty & Institute of Actuaries 1997

4. Denuit M., Marechal X., Pitrebois S., Walhin J.-F. ‘Actuarial Modelling of Claim Counts’, Wiley, Chippenham2007

5. de Jong P., Heller G. Z. ‘Generalized Linear Models for Insurance Data’, Cambridge University Press, Cambridge 2008

6. Kelsey R., Anderson D., Beauchamp R., Black S., Bland R., Klauke P., Senator I. ‘Price/Demand Elasticity’, General Insurance Convention (GIRO), Faculty & Institute of Actuaries 1998

7. Krikler S., Dolberger D., Eckel J. ‘Method and Tools for Insurance Price and Revenue Optimisation’, Journal of Financial Services Marketing Vol. 9/2004, p. 68-79

8. McCullagh P., Nelder J. A. ‘Generalized Linear Models’, Chapman & Hall, Washington, D.C. 1992

9. Murphy K. P., Brockman M. J., Lee P. K. W. ‘Using Generalized Linear Models to Build Dynamic Pricing Systems for Personal Lines Insurance’, CAS Forum, Winter 2000, p. 107 – 139

10. Tanser J., Light J., Mealy S., Morris O., report of the Demand Modelling Group, Faculty & Institute of Actuaries 2008








Questions

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