MOTOR PREMIUM RATI~G

Stewart Coutts

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&1_r71 This peper outlines a verr aiIIple structure tor the praotioal

analysis ot .,tor presiUlis trca a tJI( point ot view. It indioates

that it is possible to !IOdel dirterent types ot o1&illS oost. These

results are OOIIbined together with sOlIe st.ple e~nOllio assumptions,

to arrive at proeld.u... Then the peper develops a -points system­

"'ioh is st.llar to a nwaber ot presiwa systess operated by UJ(

Insurance Ccapanies. This -points syst.- is used to ocapare

ditterent sets ot assumptions. Finally, an analysis ot surplus is

described vith lID ex_pie.

1. INTRODUCTI<If

The purpose ot this paper is to outline sOlIe ot the statistical and

tec~.ni~al aspects ot a premium basis tor motor insurance. At present,

only broad outlines are availab.le in the tJI( Institute ot Actuaries'

literature and 10 particular, they do not highlight the pragmatic aspects

ot the problem. There have been sOlIe papers written in the ASTIN

bulletin, e.g. Pitkanen (1974), however, they are senerally theoretical

based. The tirst tull analysis tound in the literature was written by

Kahane and Levy (1975). However, the emphasis was statistical modelling

rather than discussing the practical p~oblems involved in premium

rating. Recently, two groups have published stUdies concerning the

problems ot.rating. In The Netherlands, a group ot actuaries were asked

to revise the motor rating structure and this is written up in Uethgrland

Gr~~p Report (1982). At the tJI( General Insurance Statistical Group

meeting in Strattord-upon-Avon (1982), there were tour case studies trom

different countries on premium rating. Hence, in seneral, there is

little written assistance to help either the actuarr new to motor

insurance or the actuarial student tryinS to appreciate the practical

problems ot premium ratins.

The theme ot this paper will be to em",asise a detailed within-porttolio

analysis, taking into account simultaneously, inter al1a, some ot the

major UK motor underwriting tactors and separate st4tistical analyses of

claims frequency and costs, together with expenses, to ar'rive at a

breakeven ploemiUIII. By pertorminS this detailed analysis, it is believed

that \','e pel"son responsible tor the premium decision l-rill be able to

1'(;:; ;;ri ct his at \;ention to the seroiti va are3S uhere jlld<3·-:I~ont h3.:l to be exercilled.

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This paper will describe a procedure which ill applicable in a competitive

.,tor premium urket situation, suohas in the U1C. However, it is

contended that in CO\mtries where fiud preadUIII rating set by a tariff is

in operation, an in-depth analysis is also necessary. This will enable

.an~nt to judge where in their ~tor portfolio the business is

profitable. In addition, the analysis will bring out warnings that where

all CCDpany pooled data is uaed, it oan be harmful, in particular to

.. ller companies.

The structure for this paper is lUI fo11<*s:-

Section 2

Sectio~ 3

Section 4

Section 5

Section 6

Section 7

Section 8

SecUc.n 9

Section 10

Section 11

Section 12

Section 13

Diacusses the reason for a detailed data breakdown.

Gi ves the general background to premium rating by asking,

wbJ project? ~

Introduces the Group which is involved in the decision

process.

Gives the fonnula to be used.

Defines the database.

Outlines the general principles for claim frequency and

cost statistical analyses.

Discusses bodily injury analysis.

Outlines the inclusion of expense allocation.

Discusses economic factors.

Performs the actual premium calculation.

Discusses the presentation of the premium rates.

Discusses a points table analysis.

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Section "

Section 15

Section 16

Compares different seta of assumptions used in the premium

bases.

Marketing aspects of preaiu. rating.

Anslysis of surplus.

Finally, to keep this paper to manageable length, it has been necessary

to restrict some of the discussion o~ statistical aspecta to a minimum

and only give references to papers Which will give details of these

analyses.

2. DATA BREAKDOWN

The first question to consider is Whether the premium rates are to be

reviewed in an overall fashion, e.g. adding 10J over the whole portfolio,

or whether to apply selective increases to different sections of the

portfolio. It is argued that with the use of computers, a selective

breakdown of underwriting factors should be undertaken, which by.

aggregating the results can automatically give the overall level of

premium adjustment needed. If, however, the system of analysis is only

gcared to the overall review, it is very much harder to obtain

information about the selective parts of the portfolio.

If the data are sub-divided by underwriting rating factors, we are left

with figures which are small in exposure and claim numbers, therefore,

simple averages will be.suspect. Hence, it is suggested that simple

statistical modelling be preferred. This will enable:-

1. Extension of actuarial judgement to small databases which implies

that the portfolios of small Insurance Companies can and should be

analysed. This statement is strongly worded, but it is believed that

it can be accomplished since the theory of statistics in the· past 10

years has made 'space age' progress in the analysis. of small

databases, in far more critical and sensitive. areas than motor

insurance, namely, medical statistics. An urgent plea is made to the

actuarial profession to step up research on small databases, in order

lhat small co:npanies ceon employ actuaries with the relevant expertise

to offset uninformed comments such as 'the data is too scanty to

support any meaningful analyses'.

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2. When the data is analysed in sufficient detail, then the effects of

portfolio changes are reduced and judgements on these effects can be

.. de with confidence.

3. The establishment of statistical structures, however simple, provides

'bench marks' that can be used as a basis to monitor actual results

as they emerge. This will be discussed in the final section of the

paper.

_. As will be indicated later, analysis reveals that the process of

premium rating involves many different assumptions. Changes in some

of these assumptions can affect the premium rates significantly, e.g.

different bodily injury assumptions, or in varying mathematical

models. The effects can only be assessed if the data is suffiCiently

detailed and the structure sufficiently defined.

In spite of these pOints, it has been argued by non-actuaries that any

actuarial input which might alter rates within the motor portfolio is

practically irrelevant, when compared with overall marketing

considerations or in countries where the motor rates are fixed. Hence,

the stati~tical process is considered a mere theoretical exercise, the

cost of which, gh'en the personnel involved, is hard to justify. In

addition, actuaries within the general insurance field have tended to

depreciate any detailed analyses, as recently as the UK General Insurance

Statistical Group conference at the one-day seminar on Premiums at

Stratrord-upon-Avon. One argument was that an experienced actuary did

not need to perform any detailed premium analysis, since he should ba

aware of the premium situation and adjust the rates accordingly. It ~~s

also al'gued that past analyses should be a sufficient basis for changes

in premiuM, if selected data were collected in order to monitor the

process.

It is sur.Le~ted tr.at all these remarks are half-truths. True, actuarial

rat.;:'! rc..y differ ct';1'llder3.bly frol!! meorket rates. Howev.cr, it is the

management's decision and they should be aware that the rates eharged may

in fa~t generate potential losses, the size·of which should be

quantified. If decisions are mC'.de without having all facts available,

t.hc'.n t.h1 r' l'::~.>:t. L~ (_I):.~: fl!::!".",!I~ r.oor mon,:l~C:·I~(~nt. A topl~~l example i~ that

tt>e undeJ'\ll'itir\f', r"tjnr factor 'car age' is generally ignored or given

insufficient weir:ht in market structure; in part:icular, from s\".atist~c~

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anilable, newer oars are being undercharged. On the opposite side, a

detailed analysis II&y reYeal unsuspected _rketins opportunities within

the present stNoture. Herein lies the area ot greatest potential tor statistical analysis. In addition, in countries where there is fixed

pr_ium ratins, set by a taritt, this _y eventually break down, e.g. in

the Netherlands. Hence, it is advisable to be able to pertol'll a detailed

analysis to cover this eventuality.

As tor the argument that the experienced actuary haa litUe need ot

analysis, it is aocepted that jud ..... nts could otten be _de without any

in-depth investigation, how is this aohie'ftld? The existins body ot knowledge has been accumulated largely by trial and error. In tact,

however, the actuarial protession ought to stri'ftl to establish detailed

procedures. It is contended that the present international actuarial

lit8~ature does not give sutticient intonaation tor this purpose.

The final argument, appeal1ns to past analyses, tacitly assumes that

premium stNctures are stable over time. It is agreed that frequent

changes to the system may be undesirable, but regular analyses are

necessary to 'ftIrity assumptions _de in the original caloulations. In

particular, if the database is small, this will necessitate regular

checks to judge whether the statistical interences were reasonable. For

the UK, in the late 1970's, _ny companies reviewed their premium leVels

quarterly because of the rising rate ot inflation. Furthermore, the

\.D'1derlying insurance risk pattern may also vary over time.

Finally, cost and time are put torward as reasons for not performins

regular analyses. However, with the advent ot microprocessor technology,

it is believed that the cost has been cut to a minimum and time reduced

to an irrelevant factor.

In sunmary, a breakdown ot the data to take into accoun"t rating factors,

claim frequency and cost is tundamental to establishing premium bases.

To analyse data in this way, modern statistical techniq\.les must be

understood and employed regularly since the systems are not necessarily

stable. To monitor the results, some torm ot analysis of surplus has to be carried out.

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3. WHY PROJECT?

Before discussing what background information is required in practice, to

make judgements about premium levels, it would be useful to visualise the

time span involved in the premium analysis. This is best explained by

means of a simple example.

Consider a company that reviews its premium rates on 1st October 1980 and

let us assume that these rates are expected to ~e in force for one year.

New recommendations would have to be made in practice 3 months in

advance. This time lag would be needed to alter computer output and

prepare documentation for the rate book to be passed to the broker.

The average policy will be effected halfway through the period over which

the'- premium series is expected to be in force, i.e. on 1st April 1981.

This policy will be on risk for one year and, should it have a claim, the

claim date will on average be 6 months after date on risk, i.e. 1st

October 1981.

The material damage costs will be expected to be settled within 3 months

from the date of accident. However, the third party bodily injury costs

will take on average 2 years to settle. Prospectively, the average

future time span is 3~ years from the date which the premium decision has

to be made, 1.e. by the 1st July 1980. Hence, data concerning claj.ms and

expenses have to be projected to the 1st October 1981 and beyond.

Expen~e data will he based on information which is reasonably up to dat.e

at the 1st July 1980 and is then projected to 1981. The claiM data will

be built up from claim numbers, material damage costs and third party

bodily injury costs. All but the latter costs will be based on recent

data, say 1980. However, the most reljable third party bodily injury

data is likely to be 5 years old, i.e. clai~s occurring in 1975. The

reason why the average settlement figure of 2 years is not appropriat.e i~

becalJ~c, :In I'rllctiC'e, the largp.r end proportionately more importa'"1t

claims take in excess of 5 years from date of accident to settlement.

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Hence, the total time span to be projected is on average 8l years. This

large tt8e span necessitates a nuaber of subjective decisions, the main

ones being:-

(i) Are third party claims occurring in 1975 relevant in respect of

liability olaillS expected to occur in 1981 and to be settled on

average in 1983 or later? The .. in probl.. lies in the

settlement figures, as court judg .. ents change with changing

social conditions. This has been commonly referred to as

judgement drift. If the Company has an individual liability

claim eati .. tion process (referred to as a manual basis) then it

is possible, though not necessarily reliable, to use later

liability information based on recent manual estimates as a

substitute for settled data.

(ii) Whatever method is employed as the base for projecting claim

costs and expenses, a view of past and future inflation has to

be taken. In particular, for liability claims, a view of the

rate of inflation to bring 1975 values up to 1980 is needed and

thereafter a future rate to project these costs into 1983 or

later.

(111)

4. THE GROUP

Finally, a view of future levels of claim frequency has to be

decided. Factors including future weather conditions, petrol

prices and speed restrictions have to be considered. It will be

shown later that these aspects may have a relatively large

effect on the profitability of results.

In a Company, it is desirable to make premium decisions within a Group.

The decision-maker is the person who ultimately says what the premlum

rates are going to be; he would usually be a General Manager or the Motor Underwriter.

The rest of the Group would act as advisers to the decision-maker,

supplying information on various aspects of the business. The size of

the Group might ranee from one person for each main function, to, in a

small co~pany, as few as two, the decision-maker and the underwriter.

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The following are the main aspects of the business which have to be

considered:-

1. General Underwriting principles, which vary from company to company

and reflects, i) prior views on occupations which the marketing

should attempt to attract, e.g. teachers, civil servants; ii) wording

in the policy conditions to take into account, the introduption of a

new rating factor such as protected No Claim Discount and so on.

2. The overall market position of the Company compared to its

competitors, with regard to growth and pricing. Within the Company,

production summaries will be available showing lapse and new business

figures. Harketing will also be pushing for sales increases over

selected parts of the portfolio, by trying to keep any rate increases

.' to a minimum

3. Analyses from the claim negotiator, who will report the latest manual

estimates on present third party liability claims.

~. Statistical analyses will produce information for various members of

the Group; concerning in particular, past claims frequency, claims

cost and production results,

5. General economic factors which will be used by the Group to project

into the future, past claim costs and expenses. This will involve,

inter alia, a view as to the effect of the Government's curren~

economic policy on inflation rates, salaries and prices.

5. FORKULAE

It might have seemed ideal to relate the time period basis for

calculation of premium rates to a cohort of policies a~l effected during

a particular calendar year. However, this would require an

overr-;op!list.ic::ted d"tahase, whiC'h is not available. Hene.", a pragmatic

approach has been taken, namely, the averaging process outlined in

Section 3, which uses basically a calendar rate year.

It would be ll.~.?f")] at t.hi.s po:i.nt. to eive the premium fO!'Tohlla to be

appHed by corr.liination of riSK factol's where appropriate.

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The risk prellliUII (RP), i.e. prea1_ excludins expensea.

RP .. to

where t .. clat. trequency .. number ot claims

exposure

o .. projected olaims cost

where AD .. average accident damage cost per claim it settled

1aIediately.

.'

and

,TPPD = average third party property damage cost per claim it settled

immediately •.

TPBI = average third party bodily injury cost per claim it settled

immediately.

M = average llliscellaneous cost per claim it settled immediately.

e,t,gth, are inflation rates tor respective types or claim cost.

n1,n2,n

3,n4 are average settlement periods tor respective

type~ ot claim cost.

++ An alternative suggested by Gunner Benktander is to replace (1+e)n1 by

Where the inflation rate is eCt, i.e. (e~ -1) p.a. and t is the average

time to settlement.

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The calculation of the brealceven praaium (or office praail.ll) P, depends

on how expenses are introduced. There are two main variants:-

(a)

or

(b)

P=~

1-8

where S is the total expenses

total premium income

1.e. a f1xed percentage of premium

(nb + L) P = RP + (cc.f + --t- + r + ed)

1-w

wh.ere cc = claims cost expenses and f is claim frequency

nb = new business expense

L = lapse expense

t = time period till lapse

r = renewal expense

ed = endorsement expense

w = commission plus another expense related to premium

and the costs per unit are inflattd to 'the relevant date of pr~~ium.

This method entails variable plus fixed expenses.

I. short dis:!ussion about these fonnulae appears in Sect!.on 9, but

brie:ly, the total expected expense will be the Saffie for cach method and

only thc division of expense alters.

Two omissions will be noticed:-

(a) No contingency loading (or solvency loading)

It is not certain l:hether co:np!lnies explicitly take this factor into

account. t.lt'1ouCh it has been o:nitted, ::.lr.!!bralcall~' its Inclu:oioi1

wwld be very easy. The p;"obler.JS are of estimating a value and its

effect on the final rate.

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(b) Investment Income

It is acknowledsed that the market does not explicitly take this

factor into account. ImpliCitly, income is taken into consideration

when the whole motor acoount is scrutinised, in that the ultimate

trading results can be caapared with the underwriting results. This

subject deserves a paper on its own and, hence, is left with the

simple comment that research should be carried out to investigate the

effect the factor haa on pralliums.

6. DATA (Sub-titled: Now the Story Begins)

It is now generally accepted that past clalll information must be matched

with the relevant underwriting factors at the t1llle of the cla1ll. In

addition, for IIIOtor insurance, the number of cars exposed to risk will

also be related to the claim period analysed.

There has been some work performed on the relevance of rating factors, in

particular traa the Belgian school, e.g. Hallin and Ingenbleek (1981),

but it is still in its early stages of development.

It will be assumed that all the existing market underwriting factors will

continue in use, since it is unlikely that any Underwriter would consider

altering theil, without other CClllpanies following suit. In order to

illustrate the principles under discussion, the rest of the paper will be

concerned with a worked example. The following rating factors will be

used:-

(1) Type of CO'ter - Caaprehensiw or Non-CClllprehenaive (2) Policyholder's Age - 17-20, 21-24, 25-29, 30-34, 35+ (3) Car Age - 0-3, 4-7, 8+ (II) Vehicle Group - A, B, C, D

Two ooaaents have to be made:-

(a) .sCllle sisnificant rating factors have been omitted from this list

(e.g. district, no claim discount and use of car), but the analysis

can easUy be extended to take th_ into account.

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(b) Following discussion with the Underwriter, it may be possible to

a~gregate some of the detailed data into relevant groupings, in order

to make the analysis more manageable. In our example, as an

illustration, Policyholder Age 35+ was grouped instead of

sub-dividing into 35-50, 50-60 and 60+.

7. THE GENERAL PRINCIPLES OF CLAIMS ANALYSIS

The principal objective of the analysis is to project past claims data

for the relevant period. If the example in Section 3 is taken, then on

1st July 1980 a premium for the 1st October 1980 has to be decided. The

premium rates will be in force for one year and the average date of

claims arising will be 1st October 1981. Hence, all claim information

has to be projected to 1st October 1981 and onwards. There are two

levels of decision to make, namely,

(a) t~e overall levels of frequency and claims cost,

and thereafter

(b) the within-portfolio levels, i.e. the relationship between rating

facto~s.

The first stage is dealt with by the Group with very little statistical

analYSiS, but the ultimate decision requires a great deal of judgement.

The second stage is basically where the statistical modelling takes over

and actuarial judgement comes into play.

The following sections give a brief resume of the process for different

parts of the analysis. Details of statistical modelling were contained

in a talk given to Post Oualification Course for UK Actuaries on General

Linear Models (GLM) by Professor A.P. Dawid and Messrs. Baxter, Coutts

and Ross in 1980. Further details were given in a paper entitled

"General Linear Models in Insurance" at the 1980 Congress of Actuaries in

Switzerland. More recently in the ASTIN 1982 a paper by Albrecht

discussed all the literature on this type of analysis. Even more

accurate models may be achieved by ascertaining the claims amounts

distribution by Ziai's methods (1919) and inputting into the powerful GL~

algorithm.

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7. 1 CLAIM FREOUEICY

(a) OVerall Levels Data for past years would be available by, say, s~b-groups

Comprehensive and Non-Comprehensive, in the.following format in

Table 11-

Table 1

Frequency per 1,000 Comprehensive

vehicles Quarter of Accident Year

Year of Accident 2 3 II

1977 142 125 115 152 138

1978 152 137 138 154 148

1979 1811 134 141 164 156

1980 159 134-

- includes an IBNR estimate

Discussion on projecting the 1980 results into 1981 would be centred

around such items aSI-

(1)

(i1)

(U1)

(iv)

weather conditions,

petrol prices,

road repairs,

general economic conditions, since these ~isht affect'the

frequency with which policyholders have their cars serviced.

In our example, it was decided to use the 1919 overall level (i.e.

156) as the projection for 1981.

(b) Within-Portfolio Analysis

All the rating factors mentioned above are considered, with separate

analyses for Comprehensive and Non-Comprehensive. The method of

smoothing applied was basically the Johnson and Hey (1971) model,

slightly adjusted by using logit of proportions as the dependent

variable (see Baxter et al (1980) for details) to obtain the smoothed

claim frequency for all combinations of rating factors. The smoothed

values were adjusted to give a prOjected overall frequency equal to

the 1979 values, such as those shown for Comprehensive in Table 1.

By breaking down the data according to all the rating factors, the

effect of portfolio changes is minimised.

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7.2 PROPERTY DAMAGE AND MISCELLANEOUS COSTS

(a) Overall

For Accidental ~amage (or Fire or Theft for Non-Comprehensive), Third

Party Property Damage and Miscellaneous Costs, the object is to

obtain the average cost or a cla1m in October 1981 if settled

immediately. This value is then projected forward after taking into

account the average settlement date. In practice, inflation is

assumed to be the main projectin~ factor for overall levels. Hence,

for each respective claims cost, actual values are projected by

agreed specific inflation rates up to the projected settlement date.

(Inflation rates will be discussed in Section 10).

From detailed analyses, the average settlement periods used were 3 months, 6 months and 3 months respectively.

(b) Within-Portfolio Analysis

To date, statistical modelling using all relevant rating factors has

been seriously attempted only in accidental damage claims (see Baxter

et al (1980». A great deal more work i~ required to get reasonable

models to represent the data. Models similar to the Johnson-Rey

(1971) ~o not give reasonable results, since they do not take into

account the skew dist.ribution inherent in the accidental damage

claims (see Ziai (1979».

For Th.1rd Party Property Damage, separate averages were used based on

car grouping. For Miscellaneous, two overall values were usee,

Comprehensive and Non-Comprehensive. The reason for not modelling

these types of claims was simply lack of time.

8. BODILY INJUPY

This is by far the hardest part of the statistical analysis, since, on

average, only about 5J of all claims per year involve bodily injury

costs, yielding for the example portfolio about 1,000 claims. This

section deserves a paper to itself, so it is only possible to highlight

some of the main problems and present the prag~atic solutions used in

this example. However, before discussing details, it is necessary to

look at the arguments against pooling data from different companies to

arrive at an input to premium calculations.

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Since the n~ber of claims for each company is, in practice, small, it

has been s~gested that all companies should pool their data. That would

not, however, solve the problem, for a reason best illustrated by way of

an example. Table 2 shows some typical average bodily injury costs per

claim for an individual company. They have been adjusted for earnings

inflation plus 'judgement drift' (as previously defined) to bring t~em

all up to the value of 1982, the projected date.

Table 2

Year or Average Bodily Injury Costs per Claim

Accident Inflated uE to 1982

£

1912 119

1913 80

1914 115

1915 58

1916 104

1911 90

1918 19

1979 61

If inflation were the only factor operatinR on these averages, relatively

constant values would be expected. P.owever, it is clear that there is

wide variation. This is due to the extreme skewness of the underlying

distribution of bodily injury claims; there is no reason to suppose that

the distribution shifts from year to year.

Corresponding pooled ~ata for all companies is available in the UK from

the British Insurance Association motor statistics but is not available

for publication. However, if it ,,'ere, then the sample size would be far

greater and the variability smaller than for any single company. Suppose

an average claim of £90 (in 1982 values) resulted from pooled data for

years 12 to 19. If the Company based rates on this, large profits would

be shown if the 1915 experience were repeated, or large losses if the

1972 figures occurred again. The real problem is that the Company

experiences a small sa~ple of the total market experience and therefore

its premium rates should make an allowance for its own variability.

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(a) Overall

The Group has to decide the basis of projecting bodily injury

claims. Assumin~ that they have decided to use their company's data,

they have to ~elect the base period. The possibilities are:-

(i) Using the latest results (1978 and 1979), which, however,

entail predo~inantly manual estimates.

(ii) Groupin~ earlier years, which contain claims fully settled.

In our example, y~ars 72-76 and 72-77 were grouped and then inflated

(see Section 10) to the 1982 projections. The average settlement

period was established to be 2 years.

(b) Within-Portfolio Analysis

There are two basic decision areas:-

(i) The selection of rati~ factors is very difficult. In our

exa~ple, single factors based on Policyholder Age were

considered appropriate. This is a good example where GLM would

be very effective, as it could take the shape of the

distribution into account. There is another area of research

where the profession should lead. Some work by Chan~ and

Fairley (1980) shows that modelling is useful.

(ii) As the data are broken dcwn into even smaller groups, t~e effect

of large claims becomes very important, since a large claim in a

small cell will disproportionately affect the results. Hence

some method of smoothing has to be applied. In the example, all

claims over £10,000 were cut off at that value and the excess

was respread over the whole portfolio. This.method is open to

criticism, but it was felt appropriate at the time.

Another factor to be considered is whether the large claims are

in fact correlated with particular rating factors or randomly

spread.

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This completes the statistical analysis of costs and it would be

useful to summarise the analysis so far.

Table 3 - Summarises Analyses

Whether Modelling

TVEe of Anal vsis "atin~ Factors Technig:ue used Overall Level

Claim Frequency All YES 1979

Accidental Damage All YES Inflation

Rroperty Damage Compo & Non-Comp. NO Inflation Car Group

~iscellaneous Compo & Non-Comp. NO Inflation

Bodily Injury Comp., Non-romp. NO 1972-76 Policyholder age 1972-77 )

+ Inflation)

9. EYPENSES

It is not proposed to go into detail about the collection of data to

obtain a breakdown of expenses. A paper by Ruston (1977) has described

the process very well. As far as the Group is concerned, it will cbtain

this information directly from the accounting department within the .

Ccmpany, Which projects future expenses. The only factor it would

consider is the future rate of inflation related to expenses (see Section

10) •

As far as the premium formula is concerned, two approaches are available.

The first is to consider all expenses as fixed in the short term - see

(a) in Section 5. The total value of expenses is obtained from the

accountants and then expressed as a percentage of premium income.

CommisSion and other premium related expenses are added to this

percentage. In practice, the value should be between 25J to 45J,

depending on type of ccmpany. The rationale behind this is that in the

short term (say 2 years) staff levels (which make up 80S of expenses) are

Virtually fixed. Thus, it is argued, further sophistication is needless.

The second method (b) is to divide up costs into those which are fixed

and expressed as a percentage of premium, and those which are

identifiable as separate totals such as claim cost expense, new business

expense, lapse expense and endorsement expenses.

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For the example, the following were used and "then inflated to 1981

levels:-

Fixed as a percentage of premium (incl. commission)

Claim Cost

New Busi"ess

Lapse

Renewal

Endorsement

17J

£14 per claim

£6 per Pol~cy

£1.50 per Policy

£ 1.50 per Policy

£2.25 per Policy

Both methods will ~ive the same overall expense allocation, if the

portfolio is similar to that of the base period. However, if the

portfolio changes, the second method will give more variable results.

10. ECONO~IC ASS~TIONS

Tre Group will have to form some overall views on inflation. As

mentioned earlier, general economic factors and government policies will

dominate. Usually, several scenarios will be followed. Different rates

of inflation will be applied to different parts of the analysis. In this

respect, in the ur, the British Insurance Association Economic Advisory

Group is very helpful in reporting and projecting inflation separately

for each type of claim. Another problem is to establish which index will

be an appropriate indicator for each type of inflation risk.

The following is a summary of the rates and assumptions used by the Group

in this example.

Property Claims

Bodily Damage

Expenses

Table 4 - Inflation S~~ry

INFLATION INDEX

E8rni~s + ~Aterial C~ds

Farnings + Judgement Drift

Internal, based on Salary Increase

SCENARIOS

!!!2!i ~ 11J 7J

17J 13~

10J 10J

Since inflation has a major effect on the results, the decision maker

will have to dominate this Group decision.

-23-

11. 'DIE CALCDIATrON CF PREMIUM RATES

The stage has now been reaohed when the preJB1U11 oaloulation oan be

pertonaed. This was oarried out on aioroprooessor by applying both

formulae in Seotion 5 to the data generated by Seotions 7, 8, 9 and 10.

An exaaple of the final praailB is giwn in Appendix 1 ColUllll (4) (break

even preJB1U11) for all sets of ratins faotors. The underlyins assumptions

vere low inflation, bodily injury 1912-16 and expenses all r1xed.

In a later aeotion, the erfeots of different sets of assumptions on the

praai. rates are di80usaed.

12. PRESENTATI~ CF RESULTS

The traditional way of presentins rates was via a rate book, where every

oaabination of ratlns faotors was defined with its respeotive rates. In

the early 1970's, the Co-operative Insuranoe Caapany introduoed a points

system and several oaapao1es are nov usins similar systems. In the

follOWing seotions, we look at the problem of derlving a premium for a

points table. Firstly, we must define a points table. Then we will

examine why a Caapany would want to detenaine praaiums in this way. We

will then oonsider the mathematioal background and develop an algorithm

for deriving a points table. Finally, we will seek to interpret the

results. However, it is emphasised that the "points system" is only an

alternative to the rate book and is used as a selling point.

WHAT IS A POIl-1TS SYSTEM?

The explanation of a points systea 1s probably best explained by way of

an example. Consider the followins:-

Table 5 - Points Table

CCItP NCIf-caU' Cover 19 0

Policyholder's 17-20 21-2~ 25-29 30-34 35+ Age 30 12 1) 2 0

O-J II-I 8+ Car Age 8 ij 0

Vehicle A B C D Group 0 5 10 20

-24-

It can be 'seen that we have four rating factors and with each is

associated a scale. For each point on the scale, there is a value,

expressed in pOints. The differences in the points on each scale,

reduced to percentages, are called relativities.

The procedure used is to look up the points score for each rating factor

and then a~regate them. We then use a pOints conversion table to arrive

at the premium to be charged. The oonversion table is simply a list of

points scores with associated monetary values -'see Appendix 2. For

example, oonsider Comprehensive, Polioyholder Age 35+, Car Age 4-7 and

Vehiole Group A -

19 + 0 + 4 + 0 pts. = 23 pts. £12.35

13. WHY USE A POINTS SYSTEM?

The advantages of a pOints system are mainly praotical. They inolude,

not necessarily in order of importance:-

(a) It. is cheaper to produce than the rate book.

(b) It is easy to revise rates.

(c) The calculation is relatively straightforward.

(d) The method is easy to understand.

There are however problems associated with such a system. Examples ~re:-

(a) It may be regarded by outsiders as over-simplified.

(b) Fine tuning of the rate structure is no longer possible for marketjng

purposes.

(c) The pOil'ts valt'~z (a~ opposed to tr.e n;(1nctar:: valued ch&.n('.e only

rarely, and changes over time in the underlying variabl~s tend to be

ncslt'cted.

(rl) B(·c?u~c U'e po.i.pt.~ syster.> 11' eazy t.o jr>tc'rpj-"l, an:! ad.1ul't~·~nts to

the points whjch reflect statistical experience, arc th~ur.ht. by

oulsidere to be errors.

(e) The re~ult.inp: premium slrllcture is not exact, because the underlyjne

llroalysj s is ulUmatcly a simpU fication.

-25-

Returning to equation (1), take lop to the baH 1.0325, whence

log 1.0325 Pijkl = 9

We then fit this relationship by weighted least squares:-

.:;- A2 min,- Wijkl (log Pijkl - e)

"~,

" where denotes the least squares estimator

Wijkl Is some set of weights to be selected

A ",.. A Ci PHj C'k and VGl etc. are the estimated points for the

table.

We now require to know whether we can dispense with any of the terms in

(2), particularly the interaction terms (about which more will be said

later). We must then estimate values for the factors retained.

The method of Johnson and Hey (1971) can be used to arrive at a points

systell' rolloHing equation (2). To ascertain whether any of the terms can

be discard~d use General Linear Models. However, for simple assumptions,

the methods of Orthogonal \Oleighted Least Squares (OWLS) and of Johnson

and Fey and the General Line?r '.!odel (Baxter et al (1980» all give

similar 'ref;ults.

OELS l"eHef; on tl"e fact that. it is possible to factorif;e the ",eie.hts in

equaUon (1), that is

This is demonstrated in Appendix 3.

This ap~",'oxi:1'l'1tj on h~s one hj,p:hly pract.ic!'l advantage. It does not

require the invercion of matrices. Hence, it'is possible to analyse a

large nu~ber of factors without computational capacity becoming a

conct;r'nint, ~nd this method is c:c'opted here.

-26-

13.1 METHOD OF CALCULATION

We wish to convert the break even pre.ium into a points system. Then,

the problem is to compare our fitted points premium (est1lllated from our

~odel) with the break even pre.ium. That is

fOl' each combination of rating factors, where

~ P = fitted premium

P = break even premium

Proceed as follows. Let

• Pijkl = (1.0325)

Pijkl = premium associated with a given rating

where i = level of cover (C)

j = level of Policyholder's age (PH)

k = level of car age (CA)

1 = level of vehicle group (VG)

Then define

where M is a constant

In = all interactions between terms

factor

(2)

The choice of the figure 1.0325 is quite arbitrary. This is an example

of a multiplicative sy~tem.

This complicated analysis is used instead of, say, single tables, in

order to explain the relationships between the factors. The interested

reader i.s rE'ferred to Professor A.P. Dawid (Post Qualification Seminar

for Vi: 1:(Jtu"des in 19[.0).

-27-

13.2 AN ALGORITHM

An algorithm follows for the computation of a points table.

(1) The first step is to choose a set of suitable weights. A suitable

choice would be standing business, since, as will later be

demonstrated, the method is reasonably robust to the choice ot weights selected.

(2) A standard analysis of variance is performed, using

log Wijkl (log of standing business).

A" " A (3) T,h~-factors ci (ph)j (ca)k (vg)l are selected, after checking

for interactions from the Analysis of Variance.

A '" '" (4) The Wijkl is replaced by the product of the estimates of c, ph, ca,

~ from the previous step, i.e. we find 0ijkl from the expression

min 2. ~i (~) j (~)k (~)l (log p-8}2 -J"'l

(5) The Weighted Analysis of Variance of log P is then checked to ensure

that there are no significant interactions.

(6) Assuming that no si~nificant interactions appear, then the estimates

of the main effects serve as the points wi thin the table.

'(1) The fitted values for premiums computed through the model are

co~pared with the break even premiums, in order to assess the

goodness of tit.

(8) Then an overall adjustment is made to make sure that the sum of tl1e

breakeven prE'mium 1s equal to the sum of fitted premiums.

A!l ey'p",p1 n I-Jill now t-p prf'<:'?nted. Appendix 1 ShOHS the bnsic data. The

ratin,!!; fael co,'1! can toe seen 0:1 the left. (Column (1) sl10ws the Standard

BUSiness, which is being used for the weip.hts, i.e. Wijkl •

-28-

Analysis of Variance on log Wijkl 8ives the following results:-

Table 6 - ANOVA of loS W

Degrees of ~ Sum of ~uares Freedom Mean Sguare Ratio

Cover 1.5 1 258 Car Age 5.9 2 487 Vehicle Group 4.6 3 253 Policyholder's Age 33.0 4 1360 Cover x Car Age 12.4 2 1028 Cover x Vehicle Group 0.0 3 1 Cover x PH Age 3.9 4 161 Car Age x Vehicle Group 2.3 6 62 Car Ae;e x PH Age 0.2 8 3 Vehicle Group x PH Age 0.5 12 7 Residual 0.5 74 TOTAL 64.8 119

From this, it can be seen that, in addition to the main effects, there is

a significant association between cover and car age. This point will be

returned to later.

Th~ standing business distribution yields weights for the factors:-

Table 7 - "'eillht.s for Stl'nding Flu~j ne:o.s

COII!P.

11.8

Car Aile

0.3 1:1i

Non-Comp. ~8--

4.7 5.3

Vehicle Group

A B 2.3 6:1

Po1ic~'h(lld(!r lire

17-20 21_;>ij Q.9 --;-:g-

C D 5.5 2.ii

25-20 30-34 35+ 2:8 11:3- 32.3

Tt"., valu"s reprc:"'.ent tl'e cli;et.ri bution of standin!': bu~ine:;s wi thin the portfolio. For f)x~r.>plet there are 32 times as Jn~ny policie~ in Age 35+ a3 in Al~e 17 -20.

-29-

Using these estimates Wijkl , 0 can be estimated from equation (3).

The Analysis of Variance on log PiJkl yields:-

Table 8 - ANOVA log P

Degrees of Sum of Squares Freedom Hean Squares Hean Square Ratio

Cover Car Age Vehicle Gr6up Policyholder's ARe Cover x Car Age Cover x Vehicle Gp. Cover x PH Ap;e Car Age x Vehicle Gp. Car Age x PH Age Vehicle Group x PH Age Residual TOTAL

(1) 4 545 659 3 2511 634 4 0011 061 2 758 813

231 3111 811 683 110 913 5 869 5 576

21 566 "i 191

14 99 306

(2) 1 2 3 4 2 3 II 6 8

12 711

119

(3)=(1).(2) (II)=(3)/Residual(3 4 5115 659 7 788 1 627 317 2 788 1 334 687 2 287

689 703 1 102 115 670 198 28 228 118 10 228 18

978 2 697 1 797 3 584

It is important to note that this analysis does not allow us to make any

statement about the goodness of fit of the model, since we do not have

available an independent estimate of the residual variance. All that the

analysis shows is that, relatively, the main effects are far more

important than the interaction terms, which can therefore be neglected.

-30-

From these results, we estimate the main effects as fo11ows:-

~

Caup. Non-Caup. 6.5 -10.9

Car Ase

0.3 11.7 '1 .. 0 li:1

Vehicle Group

A B -9.0 -w

Po1iclho1der ABe

17-20 21-24 27.8 20.6

Table 9 - Estimates of Points

Weighted Grand Mean 132.9

8+ -7.8

C D 0.8 17.2

25-20 30-34 6:9 2.4

35+ -2.3

It ~ou1d be easier to make comparisons if there were no negative va1ups.

This is simply accomplished, by transforming the smallest value in each

1i~e to zero, making the same addition to the other values in the line,

and adjusting the overall mean subsequently.

For e1.:am;>10, with car cg~, we can alter the value for '8+' to zero by

adding 7.8, whence 18.8 for '0-3' and 11.9 for '4-7'. The overall mean

is 1ate~ reduced by 7.8, as in the following table.

-31-

Table 10 - Altering the Base to Facilitate Comparison

Unadjusted Adjusted

Overall 132.9 102.8

~

Caap. 6.5 ·17.11 Non-Comp. -10.9 0

Policlholder A!e

17-20 27.8 30.2 21-211 20.6 23.0 25-29 6.9 9.3 30-311 -2.11 0 35+ -2.3 .1

Car Ae;e

'0-3 11.0 18.8 11-7 11.1 11.9 8+ -7.8 0

Vehicle Group

A "9.0 0 B -11.3 11.7 C 0.8 9.8 D 17 .2 26.2

We now wish to examine the goodness of fit of the new premium structure.

There are three tests:-

(1) The fitted premiums can be compared with the break even premiums for

each rating factor (i.e. E/A).

(2) The premium income we would expect, taking account of our standing

business, from our fitted premiums can be compared' with the break

even premi ums •

(3) The expected premium income from the fit.ted premiums can also be

compared with that of the present premium structure.

-32- I 1

ColUllll (6) ot Appendix 1 shows the relationship between the tit ted

preJlliuJIIS and the break even prellliuJIIS, expressed as a percentage. It can

be seen that, generally, the divergence is only tour percentage points,

though in a tew oases, e.g. CCJllprehensive oover tor an eight-year-old

whicle in Group D driven by a teenager, it is very large. Note also

that tor CCJllprehensive whiole age 0-7, the ratio is less than 100,

141llst tor the oorresponding Non-CCJllprehensive values, it exceeds 100. This suggests that CCJllprehensi ve and Non-CCllprehensi ve require separate.

tables, as discussed below.

Column (7) ot Appendix 1 shows the ditterence between the tit ted premiums

and break even prelllium, weighted tor the standing business. Column 7

reveals that the actual ditterence by cell in required premium income is

in s,OIIie cases considerable. It would seem that Canprehensive is being

mdercharged and Non-Canprehensive overcharged. We will return to this.

However, overall the total prelllium income is reasonable, as shown by the

tollawing:-

(A) Total Standing Business 90,362

(B) Point~ Premiums (Fitted using main effects) £7,898,372

(C) B~eak Even Pre=i~s £8,268,308

(D) Difterence (B)-(e) -£369,936

(E) Ratio (B)/(e) (~) 96

Hence t:1e fitted fJre:nium would have to be increased by IIJ to match the

breakeven premium.

A s1t:!:i.lat' co",parison r..ay be made with our present premium structure, for

example:-

(A) Total Standing Business 90,362

(B) Pcir~tn Pl'''eiirl.ur~ (Fitted usine main effects) £g,012~1I51l

(C) Actual Premiums (charged in the existing £8,812,028 rate book)

(D) Differcnce (B)-(C) £200,1126

(E) r:~~' .L'7'J (C)I (i.;) (,. , ,to' 91

This jndicates that a prcmil~ increase of about 3J 1s requir~d to break

even.

-33-

13.3 PROBLEMS

There are a number of problems which such a method might encounter in

practice, such as:-

(1) What are the appropriate weights?

(2) \ihat happens if we later change. the assumptions, e.g. as to expenses

or the impact of inflation on claim costs?

(3) What if we discover interactions between the main effects at the

first stage?

To investigate the effect of changing the weights, an experiment was

und'ertaken. The weights used in the previous example - Column (1) of

Appendix 1 - form a series, starting 32, ~2, 103 ••• and ending ••• 151,

167, 858. This series was reversed, i.e. the weights used on the second

occasion began 858, 167, 151 ••• and ended 103, 62, 32.

The calculations were repeated with the following results:-

Table 11 - Comparison of Different Portfolios

Ori,.inal Revised

Overall 103 103

Cover

Compo 17 19 Non-Comp. 0 0

Policyho] der Aee

17-20 30 32 21-24 23 23 25-29 9 9 30-34 0 0 35+ 0 2

Car Ase

0-3 19 13 4-7 12 10 8+ 0 0

Vehicle GI'~

A 0 0 B 5 5 .r. 10 9 D 26 21

-34-

Although some differences are apparent, these are perhaps not so great as

mjght have been expected, given the drastic nature of the change. We

conclude, therefore, that the method is fairly robust with respect to

changes in weights.

111. COMPARISON OF DIFFERENT SETS OF ASSIJtofPTIONS

One of the great advantages of this method is the relative ease with

which calculations may be performed. Hence the.points system can help

examine the effect various input assumptions have on the premium

structure. We will investigate the following changes:-

(a) Inflation: We will have two scenarios - High, with inflation at 11%

for property damage and 17% for liability settlements, and Low, with

"inflation at 7% for property damage and 13% for liability settlements.

(b) \"Po will compare data for Third Party Bodily Injury (TPBI) claims

based on the period 1972-1976 against data based on the period

1972-1977.

(c) Expenses: We will compare the effect of treating all expenses as

fixed, against the effect of distinguishing bettleen fixed and

variable expenses.

-35-

The results are shown in the following table:-

Table 12 - SUBDarl of Results

AssumE!tions (1) (2) (3) (4) (5) (6) Inflation Low High Low Low Low Present TPBI 72-76 72-76 72-77 72-76 72-76 Points Expenses Fixed Fixed Fixed Fixed & Var. Fixed System Weil5!!ts Present Present Present Present Reversed

Overall 103 106 100 109 103 104

Cover

Coa!p. 17 16 19 15 19 18 Non-Comp. 0 0 0 0 0 0

Pol1c~holder Afie

17-20 30 30 33 27 32 29 21-24 23 23 23 20 23 23 25-29 9 9 10 8 9 8 30-34 0 0 0 0 0 0 35+ 0 0 1 0 2 0

Car A!e

0-3 19 19 19 16 13 7 4-7 12 12 12 10 10 4 8+ 0 0 0 0 0 0

Vehicle GrouE!

A O· 0 0 0 0 0 B 5 5 5 4 5 5 C 10 10 10 8 9 12 D 26 26 26 23 21 23

The effect of varying the inflation assumption can he seen by comparing

Columns (1) and (2). As we might expect, the overall level has risen,

but the relativities are virtually unchanged.

Changes in the base year assumptions for TPBI are reflected in Columns

(1) and (3). The lower overall level suggests that the 72-77 basis

yields a lower overall average. This time there has been a change in the

Poiicyholder Age relativities. The factors affected are Cover and

Policyholder Age, which again agrees with our intuitive expectations.

Altering the expense assulI'ptions from Fixed (Column (1» to Fixed plus

Variable (Column (4», gives results as expected, na~ely, an incrense in

overall level combined with a narrowing of the relativities.

-36-

Column (5) repeats the results obtained when the weights were reversed.

It can be seen that t~e overall level does not change but the

relativities do.

We now turn to the question of what is to be done if associations between

main effects are observed at the first stage. It will be recalled that

in our worked example, an association was noticed between cover and age

of car. One possible solution is to have separate rating structures for

the two different types of cover. The results are summarised below:-

Table 13 - Separate Tables for Cover

Overall

Policyholder AFe

17-20 21-24 25-29 30-34 35+

0-3 4-7 8+

Vehicle GrouD

A B C D

These results suggest that

Comprehensive

118

28 23 10 2 0

23 13 0

0 5

11 28

there may be valid

Non-Comprehensive

105

36 25 11 0 3

11 9 0

0 4 7

21

theoretical grounds fo:-

having separate premium structures. However, we must never forget that

these decisions are taken on pr"ctical as well as theoretical r;rounds.

It is highly probl'ble that despite these results, ~:e will opt fOI' a

single premium structure, becau~e of marketing considerations.

A further advantage flows fr~~ th~ ease of computation. We can take

another c0mpany's rates and perform an analysis upon them. In this way,

their rating structure can be compared with our own.

-37-

15. fo4ARKETING ASPECTS

The Group will now be in a position to discuss the premium recommendation.

Factors they will have to consider will include inter alia:-

(i) What other companies have done since the last meeting.

(ii) Whether the competitive position will allow all or some ot the

recommendatIons to be implemented. For example, in.Table 12,

compare the actuarial premium recommendations Column (1) with

the existing structure Column (6). The outstanding differences

occur when comparing the points tor Car Age. Hence the

actuarial recommendation might indicate an increase in rates for

new cars. However, this is unlikely to be acceptable because of

the market conditions, which give little weight to this rating

tactor.

(11i) Whether there are to be rate changes, their timing, and the

likely reaction of the market

(iv) Any special marketing campaign proposed, e.g. introducing new

factors, offering discounts for older drivers, or intro~ucing no

claim discount protected.

As a final thought, the market place for selling motor insurance is

changing, owing to the advent of direct telecommunication on televiSi.on.

The time will come (when the next generation of televisions are made)

whe" quotations will be available to the public via the television in

their own homes. Insurance companies and brokers will have to consider

the implications of this new dimension. This just highlights the dynamic

world of marketing.

The issues raised by marketing considerations are very important, and in

fact dominate any decision process. However, the subject is too complex

to discuss formally in this paper.

16. Al:ALYSIS OF SURPLUS

Premium rates are simply an attempt to forecast claim and expense

experience. For proper control an analysiS of surplus should be

rei.uliJl'ly pel'formed to Ifleasure ,,-here .the forecasts arc failing and the

effects on the prof! tability of business.

-38-

The following analysis of surplus for motor is based on a note published

by Andrew Grant in the UK General Insurance research bulletin (1981).·

Additional papers by Brennan (1968) and Taylor (1974) are also relevant.

In 1977, a projection based on data then available was made for 1978; the

main categories of data are listed below. The results actually

experienced differ from those projected because the assumption$ made in

the projection do not wholly agree with the actual experience. We need a

method of analysing these differences into their component contributions,

which can later be addp.d back to yield the differences between actual and

projected results. This problem is similar to an analysis of surplus and

the information gained will be useful in fine-tuning the assumptions in

the model, as the monetary effect of the differences between the

assumptions made in the model and the actual experience is discovered.

It Would also provide a useful format for analysing these differences for

management.

Companies use models of which the simplest use only premiums and loss

ratio~, while the more complex ones involve assumptions about exposure,

average premium rate, average claim cost, claim frequency and expenses.

In the model consi~ered, assumptions on the following factors are made

for each period and risk group:-

SB Standing Business (number of vehicles at the end of the year)

A\:F Average Written Premium

ACF Average Claim Frequency

Ace Average Claim Cost

VER Variable Expen~e Rates (this would include the co~is~ion rate)

FE Fixed Expensee

The Earned Premiull' EP and the Exposure EX can be easily calculated usinp:

the 1/8 method for ouarterly projections or the 1/24 method for

monthly projections. The Average Earned Premium AEP can be calcu16t:ed hy

dividilY th~ EP 1:Iy tt,c FY. (At p:-esent., tax and ir-vcstr.:E'nt inccmE' m'e

ignored) •

The result from the year's Undel'w!"iting Profit UP (i.e. excludlne-: any

adjtl:>tr.'(';1!; to Out :'.:' ,,:.,~j.ne Clairl.:) for ,prim" years) eN' be E'Xp)':::,:~e,l in the forlll

UP = Earned Premhim - Claims - Expenses (inclusive of commission)

-39."

The Underwriting Profit UP projected by the model is

UP = EX x AEP - EX x ACF x Ace - SB x VER x AWP - FE

We will denote by , the figures derived from the actual results. It is

possible to derive the AEP' fro. the Earned Premium EP' and the Exposure

EX'. The claillll can easily be expressed in the form EX' x ACFI X ACC' as

the Exposure EX' and the Claim Frequency ACF' are known. The

Underwriting Profit oan, therefore, be expressed as

UP' = EX' X AEP' - EX' X ACF' X Ace' - SB' X VER' X AWP' - FE'

The differences between the actual experience and that projected are the

result of differences in the exposure, claim frequency, level of

expenses, inflation rate, 'etc. We need a method of assessing the

numerical contributions from each of the factors. The formulae for a

possible analysis are given below.

Effects of

Exposure

(EX' - EX) x (AEP - ACF x Ace) - (58' - 58) x VER x AWP

Average Prell'ium

EX' x (AEP' - AEP) - SB' x VER X (A~~' - AWP)

Claims Frequency

- (EX' x (ACF' - ACF) x ACC)

Claims Inflation

- (EX' x ACF' x (ACC. - ACC»

Claims Cost

- (F.X' X ACF' x (ACC' - ACC·»

Expenses

- (FE' - FE + 58' X VER' X AWP' - 58' x VER x AWP')

(where ACC. is the forecast of the average claims cost using either the

known or the latest estimates of claims cost inflation rates).

The fonnulae above can easily be checked from the expression for

IJndcrwr~t.jnp: Profit. -40-

The effect of Claim Frequency and Claim Cost and Claims Inflation can be

expressed in several different ways. Two of these are given belowl-

Orisinal Alternative Claim Frequency - El' x (ACF' - ACF) xACC - EX' x (ACF' - ACF) x ACC'

Claim Cost - EX' x ACF' x (ACC' - ACC·) - EX' x ACF x (ACC' - ACC·) Claims Inflation - El' x ACF' x (ACC· - ACC) - EX' x ACF x (ACC· - ACC)

The Original is preferred and is used in in the analysiS because the effect

of the claim frequency is independent of the actual claim cost. The actual

claim cost wil be partly based on the Outstanding Claims estimated and will

change over time until the ultimate settlement. Any adjustment to these

estimates in the analysis will affect only the result attributed to Claim·

Cost ••.

Various other alternative breakdowns of tbe analysis exist and the choice

largely depends on the projection procedure, the data and the factors to be

hl@hll~ted. As an example, the effect of average claim cost can be divided

into its component parts, namely, Accidental Damage, Third Party.Bodily

Injury, Third Party Property Damage and Miscellaneous. Agaj~ the effect of

inflation can be highlighted and the model would be similar to that used for

making the all()\olance ror inflation in the original projection. If it is

required to highlight both factors - that is inflation and types of claim -

and adequa.te data a~e available, the year's claim experience using the year

of account bases can be expressed as

- EX' Y. (ACF1' x AD' + ACF2' x TPBI' + ACF3' x TPPD' + ACF4' x M')

- (ACF1 x AD + ACF2 x TPBI + ACF3 x TPPD + ACFII x H)

The analy~is would pive the following results:-

Claim Frequency

- EJ' x (AD x (ACF1' - ACF1) + TPBI x (ACF2' - ACF2) + TPPD

x (ACF3' - ACF3) + H x (ACFII' - ACFII»

Claims Inflation

- EJ' x (ACF1' x (AD. - AD) + ArF2' x (TPBI. - TPBI) + ACF3'

x (TPPDI - TPPj) + ACFlp X (W' - M»

-41-

Accidental Damage

- £1' x ACF1' x (AD' - AD-)

Third Party Bodily Injury

- EX' x ACF2' x (TPBI' - TPBI-)

Third Party Property Damage

- EX' x ACF3' x (TPPD' - TPPo-)

Miscellaneous

- EX' x ACF_' x (M' - ~)

In the analysis above, AD, TPBI, TPPD and M represent the average

estimated cost per claim, for the Accidental Damage, Third Party Bodily

InjUry, Third Party Propepty Damage and Miscellaneous types of claim and

their respective claim frequencies are denoted by ACF1, ACF2, ACF3 and

ACF_. The - denotes estimates using the updated estimates of the claims

inflation rates.

Further, since ACC' will vary from year to year until utllmate

settlement, it may be useful to extend the analysis to measure what

effect changes in ACC' have on the overall surplus. The extension is

straightforwad but is not carried out.

Anal vais' of Surplus - Example

The following example is taken from the forecasts of part of the Private

Car portfolio of a medium sized UK insurance company. The forecasts are

made on a year of accident basis, this example being 1978. We give four

sets of forecasts for 1918, corresponding to a prOjection at the end of

each quarter in 1977.

-42-

The actual results for claim costs included the latest case estimates and

therefore this analysis would be subject to some adjustments as the claim

p8)'111ents run off.

1st 2nd 3rd 4th

(Figures in 'OOOs) Forecast Forecast Forecast Forecast Results

Standing Business 101.4 88.'4 87.4 85.4 78.3

Exposure 97.5 87.8 87.8 85.8 80.8

Written Premium 6244. 5702. 6114. 6339. 5794.

Earned Premium 6061. 5684. 5902. 6035. 5600.

No. of Claims 13.1 12.6 12.4 12.2 11.7

Claims Costs 3993. 3938. 3821. 3712. 3348.

Expen~es + 2077. 1987. 2184. 2295. 2342.

Profit (Loss) • (9) (241) (103) 28 (90)

• Profit : Earned Premium - Claims Cost - Expenses + including Variable Expenses (15J of Written Premium)

To calculate the effects for the analysis of surplus we need, in

addition, to reculculate the forecast~ of claims cost using the latest

estimates of claims inflation. These averages are given below:-

1st 2nd 3rd 4th

Forecast Forecast Forecast Forecast Results

Fo~ecasted Avera~e

Adjusted Forecast

304.81

289.85

The effect of each factor is then:

312.54

300.13

308.15

304.13

304.26

302.84

286.15

(Figures in 'OOOs) 1st Forecast 2nd Forecast 3rd Forecast 4th Forecast

Exposure -141 -41 -71 -56 Premiums 431 258 121 -81 Claim Frequency -257 -33 -89 -64 r.lai~s Jnrlatjon 175 145 47 17 ('laj",:". ro . .,t s 43 1611 210 195 Expen5Cs -333 -341 -206 -129

-43-

The initial underestimation of the average premium is partly due to the

premium increases during 1977 and 1978 which were not allowed for in the

earlier forecasts. It would appear to be relatively straightforward to

extend the ideas presented in this paper to separate.off the effects of

premium increases and therefore examine more closely the actual method of

projecting premiums.

The effect of claims cost unexpectedly increases between the first and

last forecasts, highlighting either a need to examine the methods used to

project claims cost, or possibly an unexpected change in the claims

experience.

The relatively large effect of expenses is due to two problems. Firstly,'

there~is a slight difference between forecasts and results in the basis

for·' alloca ting fixed expenses between classes; also, the inflation ra tes

used to project expenses during 1978 coul~ be out of line with those

actually experienced. An obvious extension of these methods would be to

separate the effect of inflation from effect of expenses in a similar

manner to that employed for claims costs.

CONCLUSION

This paper has argued for a detailed breakdown of motor insurance data. This

allows for an indepth analysiS of claim experience which in turn require~

statistical modelling, which assists in understanding the underlying

structure. In addition, it is suggested that premium rates can be structured

to take into account explicitly the underlying assumptions involved in

rating. This will help actuaries concerned with motor rating to appreCiate

the problems and advise the Underwriter from a position of knowledge.

-44-

A(X~WLEIXlEHmTS :

I would like to giYe credit to an Insurance CClllpany. tor allolling the data to

be produced. Thank Michael Brockman tor his etforts in getting the cClllputer

to perform the various oalculations and Russell Devitt and Miohael Weitzman

tor their help in preparing the paper.

In addition, I would like to thank nUJDerous colleagues tor tbeir instructive

ori ticiSlll, which have helped generate IIOst of the ideas in this paper, in

partioular Grahaa Ross 3nd Graham Masters. Finally, a word ot congratulations

to Jan Stow tor reading my writing and typing the paper aoourately and quickly.

-45-

Albrecht, P. (1982)

Baxter, L.A./Coutts, S.M.

and Ross, G.A.P. (1980)

REFERENCE

Parametric Multiple Regression Risk Models

I Theory and Statistical Analysis 16th

ASTIN Colloquium Liege

Applications ot Linear Models in Motor

Insurance 21st ICA Vol.2

'Test Hypotheses by Statistical

Investigations'

Brennan, A.R. (1968) A note on the Theory ot Analysis ot Surplus

TlAANZ p. 1311

Chang, L. and Fairley, W. (1980) An estimation Model for Multivariate

Insurance Rate Classifications : Journal of

Risk and Insurance Vol.1I7

David, A.P. (1980)

Grant, A.(1981)

Hallin, M. and In~enbleek, J.F.

( 1981)

Statistical Models tor Smoothing and

Predictions. (Talk given at a Post

Qualification Seminar on the Application of

Linear Models to the Analysis ot Motor"

Insurance Data).

Available from Statistical Department

University College London, UK.

Analysis of Surplus in Motor Insurance GIRO

Bulletin December

Etude Statistique des facteurs influencant

Ie risque automobile

15th ASTIN Colloquium Loen

Johnson", P.D. and Hey, G.B. (1971) Statistical' Studies in Motor Insurance JIA

Vol.97 p.199

Kat'ane, Y. and Levy, H. (1775) Regulation in the Insurance Industry

Determination of Premiums in Automobile

Insurance. Journal of Risk and Insurance

Vol. XLII No.1

-46-

Netherlands Group Report (1982)

(Chairman Wit, G.W.)

Pitkanen, P. (197~)

Ruston, I.L. (1977)

Taylor, G.C. (197~)

Ziai, Y. (1979)

New Motor Rating Structure in the

Netherlands

(Publication of the ASTIN-GROEP Nederland)

Tariff Theory - ASTIN. Colloquium in Turku

Expenses in General Insurance. OK Actuarial

Student Society Paper

Symmetry Between Components of Analysis of

Surplus TIAANZ p.37~

Statistical Models of Claim Amount

Distributions in General Insurance

Phd Thesis at The City University, London

-47-

-----------~..;..:..:.-.:.-.:..:..:.

PRIV;'TE CAR ?REM ANALYSIS TPBI 72-76. MAIN EFFECTS Fitted C;OYk::R V5H. ADE vtH. ORP P/H AGi stAND1N() NHNT9 I PR[MIUM

BREAK [VEil om. RAllO SB • BUSINE~S PR[~IUM OIFF

11~ (2) (3) (4) (5) (6) '(7)

(3) - (4) (3)-(4) (2) • (5)

COI1r 0-3 01 17-20 :32. 16" 224.~ 2:$7.77 ··1~.44 94 04'14.13 21-24 62. 162 173.34 101:S.06 ,017.:S2 '11 -10"J(..25 25-29 103. 140 114.06 1~7.(.O 0'12.74 90 -1312.26 30-34 152. 1;59 0:;.34 90.41 -'1;.1; 07 \ 137 -1906.77 3~+ 1446. 1:$9 E;~.C:3 ?3.US o-O.~ 91 -11~0.70

02 17-20 52. 174 260.64 274.10 "13.46 9$ 06".SO 21-24 146. IG., 207.21 20!6.13 -10."2 92 -2"1(.2.21 25-29 .207. 153 t.~3.4~ 140.51 -15.06 90 -4321.96 30-34 472. 144 9~.16 11:>.17 -16.02 06 -7:;39.46 35+ ~63. 144 9?73 lOS. 17 ·'S.44 , 92 ·42741.0S

03+04 17-20 60. 179 306.72 31S.06 -11.34 96 '680.34 21-24 180. 1"12 243.S4 2<.3.'10 "20.14 92 -37~.S6 25-29 .44. 150 157.04 176.64 -19.60 S9 -El702.S2 30-34 820. 149 116.69 139.32 -22.6.3 04 -113:55'1.'12 35+ S142. 149 117.36 130.40 -13.12 90 -106860.34

~+ 17-20 21. 195 ~17.16 478.~7 38.59 0108 S10.32 21-24 15.3. las 411.13 409.69 1.44 100 220.23

I 25-29 291. 174 264.79 2El7.65 -22.El7 '12 -6654.55 ~ • 30-34 630. 165 196.74 234.lt -31 • .36 84 -23~3a.3~

CIII I 35+ 4498. 165 197.S7 21S.70 -20.El3 90 -93672.76 ~

4-7 01 17-20 35. 162 179.6~ 100.96 -"1.31 99 -45.9S ~ .... 21-24 93. 155 142.02 147.:.so '-4.413 97 "416.96 >< 2~-29 126. I'll 91.90 92.91 -.9:5 99 ·117.00 ... 30-34 i' 101. 132 G3.34 70.2'1 ·'1.?4 97 '-351.34 35+ : 2OS1. 132 . 60.7. 67.01 1.72 103 358S.71

02 17-20 94. 167 200.73 ::Zll.~ "2.60' 99 -244.19 21-24 229. 160 1(.:;."4 171.14 "5.20 97 -11?l.07 25-29 30S. 146 106.07 100.55 "1.60 90 0652.15 30-34 744. 0 137 79.41 02.42 -3.02 96 -2244.03 35+ 7911. 137 79.86 70.2.1 1.59 102 12571.22

03+04 17-20 65. 172 245.63 244.76 .07 100 154.69 21-24 199. 165 101:>.27 201.16 "5.09 97 -1171.37 25-29 437. 1~1 125.76 130.96 -5.20 96 -2270.47 30-34 001. 142 93.4:> 101.41 -7.9<' 92 -6376.06 35+ 6077. 14~ 93.90 94.?O -1.00 99 -68S1.97

05. o 17-20 21. lOS 414/15 375.40 30.60 llO S12.18 21-24 113. 1131 329.2:> '320.20 9.05 103 1022.22 25-29 215. 167 212.05 219.23 -7.18 97 -1544.60 30-34 427. 1:59 157.56 176.52 -18.96 89 -009'5.25 35+ 2S18. l~EI l~a.46 165.21 -6.74 96 -19005.80 ,

\

S+ 01 17-20 17. 150 122.89 114.91 7.90 107 135.67 21-24 39. 143 9".70 95.90 1.79 102 69.92

---- --------

F

PRIVATE C~~ ~EM ANA~Y~IS rrOl 72,:76

FITTED COli:.;1 VEH. f,OE VCH. en? P/H AOE: ZTAtll'l;.o) rolt"T~ rm:MlIJM lInrAK tvt:N Dtrr nAno SD •

nu3INE:;~ r-m:l1lL1M Dl1'"F (1) (21 C~) (41 (:)1 (6) (7)

(3)-(4) (;$11(4) (21·(S)

COl\? s+ f. 25-~9 76. 130 62.92 :;0..92 4 •. 00 i07 303.37 30-34 170. 120 46.7:5 43.27 ~.4:3 100 591.62 3:;+ 2<)26. 120 0.02 42.20 4.32 111 13613.49

B . 17-20 43. l~S 142.79 142.39 .40 100 17.16 . 21-2'- 103. 14~ Uo3.:a I1S.2;$ -1.71 99 -lC~.21

25-29 219. 134 73.11 71.05 2.()6 . 103 4~0.16 30-3~ 497. 1:25 54.32 51.0.s 2.49 1015 1212.97 35+ 6792. 125 54.63 50.12 4.52 109 31573.06

C .~ 17-20 27. 160 163.03 164.31 3.72 102 100.31 21-~4 60. 1:;3 . 133.~3 1;:4.76 -1.17 . 99 -'O.4~ . ;:S-~7 1:!·3. 139 ~'S.03 96.151 -.47 99 -74.95 30-34 310. 1:S0 "_~.92 65.29 '1.37 9a -4203.41 35+ 3346. 1:::0 ~4.2? 62.34 1." 103 6529.111

I D 17-20 3. 177 ~3.51 144.26 139.05 . 196 '. tl7.1' .. 21-24 19. 16';> 22''';.23,' 207.159 1'1.64 100 33:;.17 '" , ::'5-29 61. 1:;6 145.06 1:;1.06 "6.00 • 96 -366.20 30-34 126. 146 107. '10 120.'13 -13.15 09 -165'.3~ 35" 966. 147 1 ()c·.40 114.90 -6.S0 94 -6357.6'

1"0)1 CO!1? 0-3 A- 17-20 6. 152 120.52 110.39 10.14 116 10S.Sa 21-24 6. 145 10;:.17 7J.6:5 ". 23.53 13'1 1;'1.1S :!~-2'9 6. 131 ~·5.(;0 20.63 37.17 230 223.0' 3')-34 12. 1'''~ 4;).C9 23.46 20.43 172 24S.1~ ,,-35+ 52. 122 4'}.17 44.9" 4.10 l09 217.~'

B ;' 17-20 . IS. 1~7 .. 149.33 132.62 16.71 113 250.6' 21-24 29. 14'';> 110. '11 100.67 10.04 110 523.2, 25-2~ 30. 136 76.46 62.24 14.21 123 426.41 30-34 23: 126 :;6.01 41.613 15.13 136 347.9~ 35+ lSI. 126 57.14 52.22 4.92 109 899.69

C· ." 17-20 17. 162 17:;.73 1:;1.57 24.16 116 410.67 21-24 34. 154 ' 137.70 111.07 27.03 125 946.14 '~-27 'Z7. 141 89.97 71.52 10.46 126 632.92 30-~'l '12. 1.51 66.C:; 50.27 16.53 133 696.::;' 35+ 291. 132 67.24 57.95 9.29 116 2702.0$

D',··, 17-20 12. 17g. 2-:>6.2? 214.S6 81.43 139 977.12 21-24 :25. 171 \ 23'5.~5 170.06 65.49 139 163-'.:!~ 2~'-29 23. 1:;7 1~1. 70 lOS. Sf.> '42.(;4 139 9S'5.37 30-34 2::!. 140 112.72 CO.36 32.36 140 711.9, :)5+ 121. 14~ 113.37 91.7? 21.50 . 124 2611.0~

':-7 I~ •• ' n-~'" $3. .If,:: 10::'.92 11?51 ~·16. 5? Sf.> -14'10.0, :' t-:~··. , -" .. •..... c:. t~:! CJ .. 7 .. t \, ... ~.11 9'1 -164.5Q , .. " ,~ ... "'1 ." -71,.:'4

"-PRIVAT;: CAR fRIZM ANALY~l$ TrCl r:t.-76

FITTED COV:.R VEH. AOE: VEH. GRP P/H AGE >;TANDINO POINTS F'REI1IUM CH<:AK EVEN DIFF RATIO SO •

DUSI NE:::S rR011UI1 DIFF (1) ( '~l .(31 (41 (SI H.I n'l

(31-(41 (;$'''41 (21.(~1

NON COI'J' 4-7 .1. 3~+ 482. 11~ '39.39' 42..2!; -2.87 93 -13e~.02 . \

'B ' 17-20 147. 1~0 119.59 1~:I.28 -1~.6? 83 -2307.08 21-~4 194. 142 9:5.07 94.90 .17 I 100 33.fJ8 25-29 211. 129 61.23 61.:12 -.29 100 -60.ea 30-34- 293. 119 4:1.49 'IJ.44 ~.O:l 105 601.12 35+ 1723. 120 45.76 47.20 ·'1.~2 97 -2621.39

C ... iO. 17-20 140. 1:1~ 140.73 147.00 -6.20 96. -878.93 21-24 220. 147', , 111.00 '103.?6 7.91 100 17"0.~1 2~-29 2:13. . 134 72.05 67.9~ 4.11 106 1038.70 30-34 300. 124 ' 5:1.54 ' 40.20 5. a4 ' 111 1(,01.84 , . 3~+ 163&. 12~ ~~.o .. ~2.10 1.6& 103 271S.n

",

D .... 17-20 6.'. 171 237.20 21~.99 23.29 111 . 1560.2S I 21-24 122. 164 100.(.a 157.69 30.'14 120 :577/).7' ....

2~-29 130. 1:10 121.4'1 106.12 15.36 114 • 1997.20 0 I 30-34 174. 141 90.2/,' 73.23 12.04 115 2014.84

35+ 69:1. 141 90.79 02.41 8.37 ,10 ~820.SP .'

s+ A. 17-20 346. 1:s3 70'.41 ", 93.87 -23.46 ' ri -8117.91 21-24 303. 126 5'5.'17 t..4.69 "0.72 87 -2642.16 25-29 392. '112 36.0~ 41.81 -~.76 86 -22'6.7~ 30-34 .": l504. .. 103 " 26.7'1 29.02 -2.2S· 92' -112S.0~ 3~+ 3217. 10~ 26.94 31.23 -4.29 86 -13815.31

B. 17-20 ~28. 133 81.80 104.49 -22.69 78 -11979.27 21-24 609. 131 65.03 72.03 .. '1.0~ 90 -4211.87' 25-29 774. ' 117 , u.sa 4S.9? -4.11 91 -3178. Ie) • 3"-34 90'1. 107 . 31.12 31.74 -.62 9G -613.66 35+ 6036. 108 ' 31.30 3:S.0~ -3.75 ' 87 -22608.4,

,C' ,_ 17-20 209. 143 ' .... 27 113.~ -17.~8 a -3674.', 21-24 204. U6 76.~;S 79.14 -2.61 97 -742.11 2~-29 346. 122 4?29 lSO.75 -1.46 97 -lSOS.7' 30-34 558. 113 36.62 35.~7 1.06 103 :5G7.0? 35+ 2035. 113 36.&3 30.43 -1.:'9 96 -4~17.9' '

11-20 ~6. 1~9' 1~2.31 .

166.29 -3.97 98 -222.4l1 '0' , 21-24 136. l!i2 129.04 120.32 0.72 ' 107 UO:S.91 2~-2.? 1~1. p~ 03.11' 79.74 3.37 . 104 l508.1' 30-3" 167. 129 61.7.5 58.30 3.37 106 1562.27 3~+ ~8. 129 62.10 62.24 -.13 100 . -'U3.96

.' TOTALS

90362. I iA" STA~DING 9US:NCSS (:)) POH.:iS F'R!ZMIUMS 7890372.00 • , (el DFl!;,;AK EVE~: PIlEI11UMS 826~307.50 " (01 DIFFER:::NC::: iBI-IC) -36990$5.5<>

Appendix 2

-51-

APPENDIX 3

Orthogonal Weighted Least Squares (OWLS)

It has been suggested that weights, which are the recriprocals of the

estimated binomial variance ViJk, where

Hence, with an interaction term included the expression to be minimised would

be:-

Choice of Weights

The minimisation can be simplied by assuming V1Jk factorises such thatl-

(2A)

where a1 is the' age effect di is the district effect and ck is the car effect. By substituting equation (2A) in expression (lA) it can be shown that

for a given set of constraints the model is 'orthogonal'. Hence, the

estimates of the parameters and an ANOVA on accident proportions are

relatively straightforward to compute.

The necessary constraints and estimators are as follows:-

[4\.' '(. : f. :olj\(j~ 2clC~'" ':. ~

. .. .. ~

\

L4;'~~~ L .... (~\()~ = 0 .a.h-;

. .l J

-52-

)

Estimators:

and similar

and similar

where

(I 2- 2 2 ~ . .l~ ~ ( ~~) f'""

~

• • ~ Q.. ;.. C.

"-L~

,J..' el( (~) 'ls"c:.

J ~. v. ~. 111':''''

.... equations for \(. and ""CI(

;)

"-

(Y\{l.j .. equations for

~q., • A. ...

" 2. £!l <. C!L><) - t-e. 1'\:,,, '" ~ ~)~ ... and l\(~) ,v.

}

/\~) ... ... t"-

... y. ..

z Co", :. c.. ...

" \(. ~

When we apply an orthogonal weighted least squares analysis (OWLS)

To justify the assumption of factorisation in equation (2A) the following

model is considered:-

Where H is the overall mean, A is the age effect,'Dj is the district, Ck is the car erfect, etc. Then I:. standal'd ANOVA on -10g10 Vijk is carried

out to see which effects are significant, by examining the mean squares. If

the mean squares of the interaction effects are small compared to the mean

squarcs of the main effects, then the interaction effect~ can be omitted

we can write " " " Ai " to" c.-" " ,J

A.' d.. ID ..l. t:< 10 ~.\ c..' I- lO ~ ~ J

/' ,.. :stimates of ai' OJ and ck can then be calculated, where ai' dj and

ck are only defined wil-ilin an arl>itrary multiplicative constant which for convenience \1111 be taker. as 10M!3

-53-

and