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R. E. BEARD

(A paper discussed before the Society on 6 March 1959)

INTRODUCTION

JUST over four years ago I had the pleasure of presenting a paperto the Students' Society on some aspects of non-life insurance{J.S.S. 13, 139) and I would first like to thank the Society for againgiving me an opportunity of putting on paper some further ideaswhich have largely flowed from my interests in the non-life field.Frankly this paper is of an exploratory nature and suffers from allthe defects of reporting continuing experimental work. I hopetherefore that the many obvious loose ends will be viewed in thislight and that members will be able to sense the excitement of thisrelatively undeveloped field.

The purpose of this paper is not to provide a guide through thevarious papers that have appeared relating to the three relatedtopics of retention, reinsurance and risk but rather to provide anelementary approach to these problems with the object of stimu-lating interest in some aspects of actuarial practice which arelargely regarded as lying in the realm of 'experience judgment' butwhich are capable of precise formulation against a background ofmathematical statistics.

At the outset I would mention that the problems discussed inthis paper are of far greater significance in non-life branches ofinsurance than they are in the life field, but the differences are ofdegree and not of principle. I have, however, approached thesubject from the life side so that the nature of the problems willbe more readily appreciated. Furthermore, the study of basicprinciples will, it is hoped, be of some value in directing attentionto some of the more subtle aspects of life insurance finance referred

25-2

THREE R'S OF INSURANCE—RISK,RETENTION AND REINSURANCE

by

400 R. E. BEARD

to by the President in his address at the commencement of thissession (J.I.A. 85, 1).

In broad terms it can be asserted that the technical processes ofactuarial calculations, e.g. premium calculations and valuations,are based on mean values derived from a deterministic model. Theactuary approaches his problem from the basis that he is dealingwith large aggregates whose future behaviour follows the patternof a mortality table of his own selection. He recognizes that therewill be fluctuations, both random and systematic, of the actualexperience from that implied by his adoption of a particular set ofassumptions and he introduces margins, largely determined byexperience, to cover such fluctuations. He also relies on periodicaltests to ensure that his margins are adequate.

It would not be inappropriate at this point to digress into a dis-cussion on expenses, but although the expense question is critical,as will be apparent later, it would mean moving a long way fromthe main issues of this paper. I would however draw attention tothe lack of precision in much of this area and the almost completeabsence of scientific procedures as compared with some other com-mercial enterprises. Thus a cost analysis of life business by year ofentry would provide some shocks and bring out some featureswhich have been obscured by the relatively slow decline in thevalue of money in this country but which have given rise to seriousproblems in those countries where the decline has been muchmore rapid.

It is not necessary to discuss expenses in detail and I shall ingeneral restrict myself solely to the risk components of the business,in other words it will be assumed that actual expenses are just equalto the expense loadings provided.

There is however one aspect of expenses which is pertinent.It can be reasonably assumed that there is an element of costs whichis independent of the size of case, whether it is measured againstthe sum insured or against the premium. There is therefore a goodreason for writing as large a sum assured as possible on the groundsthat the relative expense component is thereby minimized. Theupper limit is provided by the fluctuation in surplus likely to becaused in the event of claim and this at once leads into the region

THREE R'S OF INSURANCE 4 0 1

of tonight's discussion. A failure to set objective standards formaximum net retentions will result in an unnecessarily highexpense ratio.

For most established life insurance companies random fluctua-tions in claims experience are likely to be small relative to the otherfactors involved so that the problem, although studied in the past,has not attracted much recent attention in this country. For non-life companies, however, the random fluctuations are frequently ofmuch greater relative significance and thus the problems assumea much greater importance. Unfortunately, suitable statisticalmaterial for study is not available in the U.K. and is relativelyscarce elsewhere. It is in this field that ASTIN is concerned in thehope of assisting in the scientific analysis of the problems involved.

NATURE OF THE PROBLEMS

To bring the nature of the problems into clearer relief it is con-venient to study a very simple insurance process, namely, businessconsisting wholly of one year temporary insurances of similaramounts on lives of the same age. Actual expenses are assumed toequal the loadings and interest will be ignored. It will be assumedthat a policy of £100 sum assured is issued on 1st January to eachof 1000 lives aged 50. It will also be assumed that the expectedmortality rate of this group is q50 by the Hypothetical Select Table,i.e. .00700.

The risk premium to be charged for each case is £.7 giving riseto a total premium income of £7oo, exactly sufficient to meet the7 expected claims of £100 each. However, due to random fluctua-tion it is unlikely that exactly 7 lives will die in the year. If lessthan 7 die a mortality profit will be made and if more than 7, a losswill emerge.

If there were no other resources than the risk premiums availablethen in the latter case the 'company' would be insolvent—ruinedin the language of risk theory. Hence there is a need to havereserves available to meet adverse fluctuations in experience andwe may call such reserves' capital'. Clearly the capital required canbe accurately expressed in a probability form, i.e. given the prob-ability distribution of the expected claims, the probability that the

402 R. E. BEARD

actual claims exceed the risk premiums plus capital can be stated.In other words, to a given amount of capital can be attached a ' ruinprobability'.

Now since capital has to be serviced it is clear that the riskpremium must be loaded with an amount sufficient to meet theservice charge. It also follows that the amount of this loading isrelated to the underlying risk distribution and to the level of ruinprobability adopted. The lower the ruin probability adopted, thegreater will be the loading needed to service the larger capital.

We are now comfortably in the region of the first of the three' R's', namely, risk theory. The other two R's, retention and re-insurance, are different facets of the same problem. If it be con-sidered that the risk distribution is such that the ruin probabilityis too high there are two courses open. One is to increase thecapital (assuming the premium loadings are adequate) to such alevel that the ruin probability is reduced to an acceptable level. Theother is to modify the risk distribution so that the ruin probability(measured against the capital) is appropriate. This modification ofthe risk distribution is the essential function of reinsurance.

Some interesting points arise from considering the net retentionfrom a risk standpoint and whilst it may be noted that the'reduction' of the risk distribution is achieved in surplus linereinsurance by cutting off the tail of the distribution by settingmaximum retentions for individual cases, the quota share formachieves it by a proportionate overall reduction of all cases. Thenon-proportional forms, e.g. excess loss and stop loss coversachieve the same object, but have considerable advantagesexpense-wise.

EXAMPLE

The simple case set out at the beginning of this section will beused as the basis of numerical work. In Table I is set out theprobability distribution of claims assuming the lives to beindependent.

From Table I we see that the probability that 7 or less claimsarise in the year is .59871. The 'ruin' probability is thus .40129,i.e. the chance is about -4 that the claims will exceed the premiums.If it was decided that a ruin probability of .00092 was appropriate,

THREE R'S OF INSURANCE 403

the table shows that it would be necessary to be able to meet up to16 claims; since premiums of £700 are available, capital reserves of£900 would be needed. If it be assumed that the ' service charge'for the capital is 6% p.a. (adopted in this example purely forillustration purposes) then an amount of £54 is required forservice, so that the risk premiums require a loading of 7.7 %. Thecorresponding loadings for other values of the ruin probabilityhave been interpolated from Table 1 and are given in Table 2.

Ruin proby.Loading %

(w=1ooo) •21.4

•126

Table

•053.4

2

•01

5.4

•0056.1

•0017-6

•00058.2

•00019-6

The situation arising when reinsurance is introduced is nowconsidered. Assume that 1000 risks are written at a premium of£•742 each, i.e. a loading of 6 %, and that capital reserves of £700are available. From Table 1 we see the ruin probability is "00555;the loadings (£42) are sufficient to service the capital. The insurernow decides that the ruin probability is too high and decides toseek reinsurance. If he seeks a 50 % quota share arrangement onoriginal terms he will retain £350 risk premium plus loadings ofj£2i. Reference to Table 1 shows that his ruin probability would bereduced to -ooooi (since his total resources of £1050 will suffice tomeet up to 21 claims) but he cannot properly service his capital

Table I. Probability of n claims = (™°) •oo7" (.993)1000-n

No. ofclaims

0

I2

3456789

IOII

Proby.

•00089•00627•02209•05179•09099

'12778

•14937'14953'13083•10167•07101•04506

Cum. proby.

•00089•00716•02925•08104•17203•29981•44918

•5987I•72954•83121•90222•94728

No. ofclaims

I2

13141516

1718

1920

2122

Proby.

•02618•01402•00697•00323•00140•00058

'00022•00008

•00002•00001•00001

Cum. proby.

'97346•98748

•99445•99768•99908•99966•99988

•99996•99998•99999

1'ooooo

404 R. E. BEARD

since he has only £21 loadings available against the £42 assumedto be needed. To service his capital properly it would be necessaryfor him to retain all the original loading; this could only be doneat the expense of the reinsurer.

Of course, it may happen that the reinsurer by virtue of agreater spread of business would be prepared to accept a lowermargin in the premium he receives. But unless the reinsurer wasprepared to forego all loading, merely offloading a quota share willnot provide enough margin to service the capital in this example.If, on the other hand, the reinsurer agreed to pass to the companya further 1000 cases of £50 each in reciprocation for the businessceded then the company would have enough loading to service thecapital and his ruin probability would be reduced as a calculationon the basis of 2000 cases shows. The relevant figures are set outin Tables 3 and 4.

Thus a 6 % loading on 2000 cases of £50 each will amount to£42, sufficient to service a capital of £700; premiums will amount

Table 3. Probability of n claims = (2™°) '007" (•993)2000-n

No. ofclaims

I2

3456789

10

11

I21314I51617

Proby.

•00001•00008

•00037•00130•00366•00856•01721•03023

•04715•06618•08440•09862•10631•10636•09927•08682•07142

Cum. proby.•00001•00009•00046•00176'00542'01398

•03119•06142•10857•I7475•25915•35777•46408•57O44•66971•75653•82795

No. ofclaims

181920

21

22

23242526272829303I3233

Proby.

•O5547•04079•02848•01893•01201•00727•00423•00235

•00126•00066•00032•00015•00007•00004•00001•00001

Cum. proby.

•88342•92421•95269•97162•98363•99090•99513•99748•99874•99940•99972•99987•99994•99998•99999

I'ooooo

Ruin proby. (nLoading %

= 2000)

I•2

•I' I

19•°52'5

•0I3'8

•005

4'3•001

5'3•0005

5'7•0001

6'6

Table 4

THREE R'S OF INSURANCE 4°5to £700 so that a total of 28 claims could be met. The probability ofthis or a lower number is .99972 so that the ruin probability is•00028, only one-twentieth of the value when 1000 cases wereinvolved.

The position can be looked at from another angle. Assume thatthe direct writing company has capital of £700 on which theservice charge is £42; assume that the expected writings are 1000cases. It is decided that a ruin probability of .00012 is acceptable.Table 1 shows that this probability is reached at 18 claims and thatcapital reserves of £1100 are needed, calling for a loading of 9.4 %or £66. If business is obtained at this rate and the full amountretained the ruin probability would be .00555, rnuch higher thandesired, but the loading of £66 would be more than adequate tomeet the service charge of £42 on the actual capital of £700. If4/11 of the business was reinsured on a quota share basis therequired conditions would just be met.

Instead, however, of reinsuring on a quota share basis, the directinsurer could decide to reinsure on some kind of excess basis. Tomaintain his desired ruin probability while retaining the fullamount of each case, would mean obtaining cover for the excessclaims over his resources up to 18 claims. The theoretical net riskpremium may be obtained by noting that if P is the premium, thetotal risk premiums + capital resources are £1400 — P, so that thereinsurer has to provide £P if there are 14 claims, £ 100 + P for 15claims, £200 + P for 16, £300 + P for 17 and £400 + P for 18claims, £500 for 19 claims, £600 for 20 claims and so on. (Toensure consistency in the calculations it is necessary to adopt asituation in which the cedant has just enough resources to meet thenet claims when the total number of claims lies between 14 and18 inclusive.) Using the figures in Table 1 we find P = £-944.The remaining question is how much of the loading of £66 shouldbe passed on to the reinsurer? The probability that the totalnumber of claims is 19 or more is .00012, when he has to pay atleast £500. If the total claims are 18 or less he will have sufficientto meet his commitments if his capital is £400, hence to meet thesame conditions as in the quota share example, the loading passedon should be £24.

406 R. E. BEARD

This example is, of course, highly theoretical, but the importantthing to notice from it is that whilst in the quota share case the'service charge' amounted to only 9.4% of the risk premiumceded, in the excess case the 'service charge' amounted to over25 times the risk premium ceded.

By way of a finishing comment on this section it is of interest tonote the different structure of the ruin position in the quota shareand excess cases. In Table 5 the balance available in the event ofvarious numbers of claims is set out for the two cases. In the excesscase there is a sequence of years in which the company is ' nearlyruined', a reminder that the spread of the components of the ruinprobability may on occasion be of significance.

No. of

claims

o

i

2

3

4

S6

78

9

Proby.

•00089

•00627•02209•05179•09099

•12778

•14937

•14953

•13083•10167

Table 5. Balance of resources

Balance

, " , No. of

Q.S. X.S. claims Proby.

1145 1399

1082 1299

1018 1199

955 i°99891 999

827 899

764 799700 699

636 599

573 499

10

11

12

13

14

1516

1718

•07101

•04506

•02618

•01402•00697

•00323

•00140•00058

•00022

Balance

Q.S. X.S.

5°9 399445 299382 199

318 99

255 0191 0

127 0

64 00 0

FURTHER DEVELOPMENT

The previous part of this paper has dealt with the analysis of asimple portfolio in essentially theoretical conditions. It is hopedthat the critical underlying principles have emerged and that thebasis for further development is secure. The position may berecapitulated in that we have shown that a relationship existsbetween the net retention, the reserves and premium loadings, thisrelationship being expressed in probability terms. Thus, if Z is thetotal amount of claims, u the 'free' reserves, P the net risk pre-mium and a the ruin probability we have

Pr{Z > u + P} = α.

THREE R'S OF INSURANCE 407

Clearly we can immediately extend the model to cover the casewhen the sums at risk are no longer constant by calculating theprobability distribution of expected claims. Following the analysisgiven in J.S.S. 13, 141, if there is an aggregate of N1 identical riskseach subject to a probability of p1 that a claim arises in a year andsuch that the proportion of claims falling between a and a + da isφ1(a)da then the moment generating function of the total amountof claims arising is

If we now consider a mixed portfolio consisting of N1 N2, ..., risks,subject to claim probabilities p1,p2, •••, and claim distributionsφ1(a), φ2(a),..., the m.g.f. of the whole portfolio will be given by

the product being taken over all the different groups. By takingthe logarithm we get the cumulant generating function, namely

whence

etc.

If a Poisson distribution be used instead of a binomial themoments of the distribution of total claims would reduce to

from which it will be seen that the moments are based on productsof expected numbers of claims (Nrpr) and moments (about zero) ofthe φ distributions. The expected claim rate (pr) enters only in-directly (unless it happens to be high), a feature which is of use insorting out questions such as the attitude to be adopted to, say,

408 R. E. BEARD

insurances on old lives. It is also useful to note that the usualmeasure of skewness and kurtosis may be written in the form

where the m's lie between the extreme values taken by thesefunctions.

The purpose of this very general treatment is to sort out theprinciples upon which a net retention policy should be based. Itsets out the theory behind the combination of claims observationsto build up the total claim distribution for the portfolio beingconsidered. The mean is given by the average of the observationsand the variance by the sum of the variances of the respectivestrata. The approach to normality, needed to decide whether anormal curve can be safely used as an approximation, follows fromthe and β1 and β2estimates. Once the moments of the distribution oftotal claims are known the calculation of the one year ruinprobability for the appropriate free reserves can be made. If thisis too high consideration can then be given to the effect of differentreinsurance methods until the ruin probability is reduced to thedesired level.

In calculating the ruin probability allowance will have beenmade for the risk premium receivable and it is assumed that thereare sufficient loadings to provide the necessary service on thereserves employed. Implicit in the assumption is that the aggre-gate of risk premiums is equal to the expected value of total claims.Suppose now that it is found that the maximum claim on any onecase should be limited to say £10,000 to produce the desired ruinprobability how should detailed underwriting instructions bebuilt up?

On the basis that the distribution and level of business will bethe same as that on which the experience is based, and that riskpremiums are fair, then clearly the instruction should be to retainthe maximum amount on each case, subject to an upper limit of£10,000. By retention we refer to 'probable maximum claim' andassume that a lower limit has been fixed on expense considerations,involving special premium treatment. However, for many reasons

the β1 and β2 estimates. Once the moments of the distribution oftotal claims are known the calculation of the one year ruinprobability for the appropriate free reserves can be made. If thisis too high consideration can then be given to the effect of differentreinsurance methods until the ruin probability is reduced to thedesired level.

THREE R'S OF INSURANCE 409

risk premiums may not be fair (competition, accommodation, etc.)and it is important to see what variations should be applied. First,it must be assumed that business is not accepted at 'under claimcost', i.e. that the expected claims exceed the risk premiums. Iffor special reasons this is done then (theoretically) the shortfallin risk premium should be charged to an expense account appro-priate to the special reason. On this basis, no change in retentionpolicy is called for.

Regard must however be given to the 'service loading'. If thisis too low then there will be insufficient money to service thecapital employed. If it is too high there is a danger that businessgrowth will be inhibited. It is, however, implicit that the serviceloadings are in the aggregate sufficient for capital needs and it maywell be that the loading in different cases may be considered out ofline. Thus suppose that for the conditions of the company theservice loading required overall is 5 % of the expected risk pre-miums and it is considered that for 20 % of the business the marginwill be zero, for another 20% the margin is 71/2% and for theremaining 60 % the margin is 5 %. On this distribution the overallmargin will be 41/2% only, but if one-half of the first group isreinsured the margin will rise to 5 %, but on a total of businessreduced by 10 %. If therefore the business budgeting is raised by10% then the total premiums after reinsurance will provide therequired amount for service.

It is assumed that the pattern of business is largely dictated byfactors beyond the control of the underwriter so he has to deviserules to balance the portfolio. In this very simple example he doesthis by fixing his retentions for the substandard class at 50 % of thestandard so that the adjustment is secured automatically. Theo-retically this gives rise to a change in his expected claim distribu-tion by a slight shift towards the lower end and thus the conditionswould permit a slightly higher standard retention. However, sincethe critical aspect, namely the judgment of margins in the pre-miums, cannot be precisely defined, there is little justification forfine adjustments.

The problem of net retentions then reduces to (1) determinationof the maximum net line, and (2) a pattern of reductions from this

410 R. E. BEARD

maximum to secure the necessary 'service' loading. Clearly fora given portfolio there are an unlimited number of ways of adjustingfor (2) depending on the relative amounts of business in the variousloading groups. It is therefore necessary to set up some criteria sothat a specific solution may be obtained.

Let it be assumed that the reserves are u, risk premiums nP andthat the risk distribution may be assumed to be normal (i.e. n large).Then we may set up the following approximate equation

u = α (nP),

where a is a factor which includes the multiple of the standarddeviation corresponding to the desired ruin probability and thefactor {m2/m1)

1/2. If A is the loading and r the service rate on thereserve items we then have ru — λnP. If the expected loading(λ1 say) is less than A then it will be necessary to write more casesto provide enough loading to service u. However, the effect ofwriting more cases is to increase the premiums and the standarddeviation so that the basic risk equation will no longer be satisfied.If, however, the amount per case is reduced, then a counteractingfactor can be introduced into the S.D. and a solution found whichbalances the equation. Thus if k be the reducing factor, the pre-mium required being λnP/λ1 the number of cases must be increasedfrom n to nλ|Kλ1. The S.D. then becomes {nλPk2/kλ1

)1/2; since thishas to remain constant at (nP) we find that k = λ1/λ..

This simple and, of course, approximate result applies to a singletype of risk and leads to a rule that the net retention should beproportional to the relative loading in the premium. To extend thecalculation to the case where a mixed portfolio is concerned, someconditions must be imposed for a solution to be obtained. Weassume that the proportions of gross business available in thedifferent loading groups are fixed and have to find a system ofreduction factors for the lower loading groups to provide an overallaverage loading factor equal to that required having regard to theruin probability and the reserves available. We must then increasethe total net premiums to the level required to provide the amountof loading required. Finally we must increase the number of cases,and reduce the sum insured per case to bring the standard deviation

THREE R'S OF INSURANCE 4 1 1

to the required level. To get a unique solution we can then imposesome kind of maximum or minimum condition and a convenientone is to minimize the total loading with respect to the reductionfactors.

Thus, consider the position of a mixed portfolio in which thereare nrPr premiums with a loading of A,.. We make the assumptionthat enough business is written (from a specified population) toprovide sufficient loadings to meet the service charge on reservesof u. We then have for the basic equation u = α ( nrPr) andru = λ nrPr. The total loadings on net premiums are

and the number of cases must then be increased by a factor

The revised S.D. is then

which has to be equal to ( nr,Pr). We then find

or

Clearly one solution of this is kr = λr/λ for all r. By the use ofLagrange multipliers and the condition that λrnrPrkr is a maxi-mum the same solution will be found.

One other aspect of the problem needs consideration before theresults can be properly summarized, namely the position of anexpanding business. The foregoing analysis is built up on theassumption that the premium loadings are just sufficient (on theaverage) to provide service for the capital reserves, which are them-selves sufficient to produce a ruin probability of specified amount.If the business expands there will be an increase in the standarddeviation and thus a rise in the ruin probability and there will alsobe a rise in the available loadings. Various theoretical ways ofmeeting the position are available: (a) the capital can be increased,(b) the reinsurance can be adjusted, and (c) the premium can be

L = T,\nrPrkr

412 R. E. BEARD

loaded to provide the necessary increase in reserves, (a) will notbe discussed in the present context as being a method required inexceptional circumstances and (i) will be discarded as implying nogrowth in the retained account; we are then left with (c).

If there is an increase of h% in the premiums the standarddeviation will rise by a factor of approximately (1+h) = I +1/2hand the reserves should therefore be increased by 1/2hu to maintainthe ruin probability level. This will require an increase in the servicecharge of 1/2hur and as the increase in the loading is hλnP = hurthere is a balance of 1/2hur available towards the required reserveincrease. This leaves a shortage of 1/2[hu(1 — r)] which has to be madeup by a premium loading. The rate of this loading is thus

We thus see that to finance an increase of h % in premium incomethere must be a loading in the premium of 1/2h multiplied by afactor depending on the relative levels of the capital reserves andthe premiums.

In the previous paragraphs, and in fact throughout the paper,it has been assumed that the loading λr can be assessed in someway. Formally, for non-life insurance it is a question of knowingthe risk premium rate for a particular class of business and theproperly costed expense for this class. The margin then gives an

estimate of the loading available. To determine the risk premiumrate is not an easy statistical problem when regard is had to thenature of the variation of the claims, but this is a question in itself.The expense element should be easier, but in fact there appearsto have been very little in the way of true cost analysis of insurancebusiness and a whole area of research is available.

The problems of life business are complicated by the long-termnature of the contracts and the fact that other factors are of morecommercial significance than the pure risk factor. However, fornew companies the problem is not without significance and it wouldnot be without value to develop the ideas for a life portfolio.

It will be noted that in the foregoing analysis no allowance hasbeen made for interest which may be earned on the 'reserves' and

THREE R'S OF INSURANCE 4I3which will be available towards the service charges. Theoreticallythe reserves and premiums should be held in cash if the conditionsimposed are to be met, so that interest earnings will be smallrelative to the other factors involved. If there is any departurefrom this then there might be a capital depreciation at the timewhen the risk reserve was required to meet an exceptional fluctua-tion in claims. To meet the originally specified ruin probability itwould then seem necessary to start with a larger reserve, based onthe expected maximum depreciation. This would call for a largerservice charge which could be regarded as made up of the premiumloadings, plus the interest earnings on the invested reserves.This approach leads to the idea that if reserves are not invested'dead short' it would be reasonable to use an appropriatelylower figure for u in the risk equation and ignore the interestearnings.

All the foregoing has been based on the assumption that we areconcerned only with a I-year business and it is desirable to examinethe position where a continuing business is concerned. Havinggiven the distribution of total net claims, the initial reserve and thepremium income (it being assumed that the loading is appropriate)the probability of not being ruined by the end of the tth year canbe written down on the basis of a constant premium income. Ifthis probability was too low, the underwriting policy would needadjustment (e.g. by reducing the maximum probable loss) untilit was considered that a desirable level had been reached. The netretentions would then be reviewed in the light of this revisedpolicy, and its effect on the claim distribution. It will be notedthat basing the underwriting policy on a t-year ruin probabilityinstead of a I-year affects only the maximum acceptance and notthe relative levels of the various subclasses, which depend on therelative loadings in the premiums.

Provided some idea of the relative values of the t-year andI-year ruin probabilities are available there is no need to calculatethe t-year values, it being sufficient to take a sufficiently low valuefor the I-year ruin probability. Unfortunately the relationship iscomplicated and does not seem to lend itself to elementary methods.Thus if Q(x) be the probability that claims in a year do not exceed x,

26 ASS I5

414 R. E. BEARD

i.e. p(z)dz where p{z)dz is probability that claims between z and

z + dz arise we have:Probability not ruined in year I = Q(u+P).

Probability not ruined in year 2

Probability not ruined by end of year tand so on until

This gives the form of the required probability, but it is difficult tomake any precise statements about the value of w, ... z. Numericalcalculation would be feasible, particularly if an electronic computeris used.

If p(z) = αe-αz an explicit solution can be found for Qt i.e.

but other forms of p(z) do not lead to simple expressions.An approximate value of Qt may be found by repeated use of

mean values, the method lending itself very conveniently to agraphical method of calculation. Thus

where

where

etc.

THREE R'S OF INSURANCE 415

If values of Q(z) and m(z) wp(w)dw p{w)dw are calcu-

lated the various Q factors can be obtained successively.Before concluding, it is proper to add a few words about the

reinsurance angle. Implicit in the principle of reinsuring thebusiness showing a low loading is the assumption that a reinsurermay be found who is prepared to accept this business. This can bejustified on the basis that this is possible because the reinsurer hasa greater spread of business and so can operate with a lower loadingfactor. I do not propose to analyse this aspect further but merelymake the comment that some nice practical problems can arise.Discussion of the reinsurance position is a subject in itself, parti-cularly the question of excess loss and stop loss where the loadingfactor is frequently of far greater significance than the net riskpremium.

I have not provided a bibliography but there are two referenceswhich are important background, namely Dubourdieu's TheorieMathematiques des Assurances (1952) and Thepaut, Bulletin de I'lnst.Actuaires Francais(Dec. 1953). A fuller list of reference was providedin papers read to ASTIN in New York by Depoid and de Finetti,published in the ASTIN Bulletin, vol. 1, part 11 (December 1959).

CONCLUSION

As I stated at the outset, this paper is largely of an experimentalnature, and is an attempt to use the principles of classical risktheory in the study of the problem of net retentions. It is originalin the sense that it has not been derived from other published workon the subject, but having regard to the many continental paperson risk theory it may well be that the ideas have been recordedbefore. It is of course possible to start from Collective RiskTheory and develop a theory of retention. To my mind, however,there are certain features about the basic assumptions of CollectiveRisk Theory that do not seem to fit in nicely with commercialrealities, and accordingly I have preferred to tread the classicalrather than the elegant path. It will be obvious that I have merelyscratched at the surface of the problem and I hope that others betterequipped than I may find the problems worth solving.

26-2

416 R. E. BEARD

APPENDIX

In this paper I have referred to various types of reinsurance whichare in common usage in the non-life field. This appendix hastherefore been prepared to give a very brief outline of the variousreinsurance arrangements so that the significance of the argumentsof the paper may be appreciated.

Fire insurance is historically the oldest form of non-life businessand it will be convenient first to sketch the development of rein-surance facilities against the background of fire business. In thisbusiness the amount of a risk retained for the insurers net accountis referred to as a net line. In the early days amounts in excess ofthe company's net line were offered to other companies on afacultative basis, each company specifying its acceptance in termsof the ceding company's line.

As the business grew the facultative arrangements developedinto obligatory treaties under which the cedant passed his excesslines (according to a schedule) to a treaty; reinsurers would thentake a share of this treaty, expressed in terms of lines or on apercentage basis. This automatic arrangement clearly was a greatbenefit to direct writers, and apart from convenience was a sourceof some economies in operation. Facultative business was stillexchanged for cases in excess of the treaty capacity or for specialrisks.

Such treaties, being based on the surplus of each risk over thecompany's net retention, go by the general name of surplustreaties. The normal practice was to list the cessions to the treatyon bordereaux, copies of which were sent to the reinsurers with theaccounts and enabled the reinsurers to watch closely the risksbeing placed on the treaty. It was common to look for accumula-tions and make special arrangements to avoid excessive exposureson individual risks.

Terms were fixed on the results of these treaties and the marketsoon developed along the lines of looking for substantial profitmargins in these treaties. Between the wars the surplus treatyflourished and exchanges between offices grew on the basis ofproviding a wider spread of risks. However, the competition for

THREE R'S OF INSURANCE 417

better profit margins had its reaction and after the second worldwar much greater attention became directed to the question of'reciprocity'. Premiums ceded under surplus treaties amounted toperhaps one-third of the direct writings and with the generalincrease in competition gross profit margins became thinner andthe effect of such substantial cessions could not be ignored.Furthermore, even the clerical work involved in treaty bordereauxbecame expensive and various devices were developed to try andeconomize in operation.

A number or interesting variations on the surplus treaty weretried in France, these being essentially methods of cutting downclerical work whilst still retaining the benefits of the surplus treaty.A more significant development was however the movement toexcess covers of some kind. These covers had come into generalusage between the wars in connexion with catastrophe covers andfor the growing classes of liability insurance. Basically they providefor the reinsurer to pay the excess of such claims as exceed a certainfigure in return for a premium expressed as a percentage of thegross premium income.

Clearly such covers afford a considerable economy in operationsince it becomes necessary only to deal with the exceptional claims.They have, however, given rise to a number of practical difficultieswhich have to some extent inhibited their growth. In general termsthey are grouped under the heading of non-proportional reinsurance.

In excess of loss reinsurance the reinsurer agrees to pay theexcess of each and every claim in excess of an agreed amount. Thisform is commonly employed in Motor insurance. There is astatistical justification in that the frequency distribution of claimsby amount from a portfolio of business may be assumed to havea basic underlying form. When this form is known the proportionof claims falling in the excess area can be found and a rate deter-mined for the treaty. The difficulties present are (a) the persistentinflation of post-war years has meant that the scale of the claimdistribution has been changing so that rates have been behindexperience, (b) even apart from (a) the estimation problem isdifficult enough, (c) the loadings required to cover fluctuations aresubstantial and not always appreciated.

418 R. E. BEARD

Various devices have been tried to get over some of thesedifficulties. In many treaties an index-linked clause has been usedwith some limited success against inflationary tendencies. In oneother treaty an attempt was made to link the excess figure to aquantile of the claim-distribution curve; unfortunately otherfeatures of this treaty were unsatisfactory. Obviously treaties inwhich the excess varies are not completely satisfactory becauseneither party can be certain of the position at any point of time.Nevertheless, such treaties have a great attraction because of thecost saving and are slowly growing in application in spite of thehandicaps.

A later development has been to develop the so-called 'stoploss' treaties which are closely linked with' aggregate excess' covers.If it be assumed that the claim distribution is known then theprobability distribution of the total amount of claims over a totalfigure can be found. In the aggregate excess cover the reinsureragrees to pay claims over the agreed total. In the stop loss cover theagreement is to pay in excess of a given claim ratio. Either of thesemay be expressed in terms of a fixed amount (i.e. the cover placedin layers) or in terms of a ratio. The premium estimation problemswill be apparent.

Statistically speaking the estimation of the tail of the claim curveis similar in the case of individual claims or of the claim ratios(J.S.S. 13, 139). However, there is a certain amount of 'built in'protection from inflation in the claim ratio since there will be atendency for sums insured to be revised and premiums adjustedaccordingly. In many cases the application of the average clausewill secure the same effect. If average were applied universally,the protections would be better. The main disadvantage is, ofcourse, the fact that until the results are fully known the truerelationship of the parties cannot be determined.

To complete this very brief outline it is necessary to mentionthose treaties which are based on 'burning cost'. The premium isbased on the actual claim experience on perhaps a 5-year averagewith a suitable loading. Clearly if the treaty continued indefinitelyit would amount to no more than a financing arrangement betweenthe parties. A measure of true reinsurance is however introduced

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by the fixing of maximum and minimum premium rates. Statisti-cally speaking the problem of rate fixing is formally determinedwhen the claim distribution is known.

Many variations are introduced into these basic forms of rein-surance and different combinations have been devised in attemptsto meet practical demands. Suffice to say here that one of the mainfeatures is the general tendency to treat the treaty result as a wholeand reduce to an absolute minimum any exchange of detailedinformation about the finer structure.

One of the problems which lies at the back of all this is, ofcourse, the incidence of taxation. Briefly, tax is assessed on theresults of a year's working in which the outgoing reserve is expressed(in the U.K.) as a percentage of the premiums written in theyear. Claims are estimated as closely as can be, subject to a safetymargin. Since, however, the results of the business fluctuate fromboth secular trends and random fluctuations, the method of asses-sing profit tends to encourage insurers to iron out fluctuations toas great a degree as possible. For example, if a catastrophe cover iswritten, and the event happens, the year in which it arises will showa substantial loss to be met by premiums over many years. Thesepremiums will fall into profits as they arise and be taxed at the fullrate so that unless other measures exist for spreading the incidenceof the large loss, the company is at a marked disadvantage.

This feature has been recognized in some countries where thefiscal authorities are now prepared to recognize a calculated riskequalization reserve. (In the U.K. some limited recognition hasbeen given in regard to Lloyds underwriters.) The mathematicalproblem is clear and is based on developing for non-life insurancea reserve position parallel to that developed by the actuary for lifebusiness. At present, however, the technical processes involvedare still in course of development and existing actuarial techniquesare unsuited for the particular problems involved.

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Subsequent to the discussion some members raised points in corre-spondence with the author, who has replied as follows

In answering a question from Mr W. J. Courcouf it would seemthat the position would be clarified if Table 5 were extended asfollows:

No. ofclaims

192 0

2 1

2 2

Proby.•00008•00002•00001•00001

Q.S.- 6 4

-127-191

- 2 5 5

X.S.-P-P-P-P

Balance

Mr Courcouf also raised some points on the general interpretationof the first part of the paper and his own comments adequatelycover the position:

'...it might give us a clearer understanding of the position andthe analysis, if we had regard more to the total market than tothe individual companies serving it. I think that the moralof this part is that if there is 'maximum' reciprocity, theruin probability scale and the consequent loadings should bebased on the total market. If there is no reciprocity the smallercompanies cannot afford to write at the same rates as large ones.If two companies separately decide on a ruin probability of•0012 for each, and the market consists of 2000 cases to bewritten, they may hope for 1000 cases each and therefore employa loading of 9.4 %. If in fact they get 1500 and 500, then theloading will have been too much in the one case and too littlein the other, and the one with too little cannot rectify the positionby insurance without reciprocity. If there is reinsurance withreciprocity, then the smaller office can just rectify the position—it would have 1000 cases at 50 and the larger company wouldhave 1000 at 100 and 1000 at 50. If reinsurance were effected onthe basis of obtaining an equal share in each case in the market,then the smaller company would give off 75 % of each it obtainedand receive 25 % of each of the larger company's cases, and thisis what I would define as "maximum reciprocity".'As a result of Mr S. Benjamin's inquiries on the appropriateness

THREE R'S OF INSURANCE 4 2 1

of the relation λ = ru a subtle but interesting point emerged inthat some care is needed in specifying the premium basis if con-sistent mathematical results are to be achieved. The risk premium

is defined as the expected value of claims, i.e. zp(z)dz. However,

if the business is reinsured in such a way that there is a risk of ruinby the reinsurer, then to the extent that the cedant cannot obtainpayment from the reinsurer (because the latter is ruined) thepremium is too large. The true risk premium should thus be

P = zp(z)dz + zαp{z)dz,

where zα is the limit beyond which the reinsurer cannot pay.