SOME STATISTICAL PROBLEMS ARISING FROM THE TRANSACTION OF MOTOR INSURANCE BUSINESS by R. E. BEARD

SOME STATISTICAL PROBLEMS ARISING FROM THE TRANSACTION OF MOTOR INSURANCE BUSINESS by R. E. BEARD (Substance of a talk given to a joint meeting of the Royal Statistical Society General Applications Section and the Students' Society on 12 November 1963) W H E N the word statistics is used in connexion with non-life insurance the first reaction of most of those concerned with the day-today requirements of the business is to think in terms of the relationship between premiums and claims. In my talk tonight I shall not be concerned with this aspect but will be concerned with the structure of the risk processes underlying the business. In order to try to bring out some of the essential characteristics I have used the minimum of mathematics, using numerical examples to illustrate the points. Furthermore, in order to limit my talk to a reasonable time and to make it' live' I have deliberately omitted many of the finer points which arise in practice to which regard must be had in using mathematical or statistical models to describe the actual situation. At rock bottom the insurance industry rests on the operation of laws of chance. A hazard is recognized as being liable to give rise to a potential pecuniary liability, and in no time at all someone is prepared to provide a policy against such liability. In devising the policy, and in assessing the rate of premium to be charged, it is necessary to define the event which must happen before a claim can be admitted under the policy and to be satisfied that the happening of the event is fortuitous as far as the proposer is concerned. In life insurance difficulty seldom arises under either count but in non-life insurance the position is frequently very troublesome. The attitude of people towards physical assets, whether belonging to them or to someone else, differs between quite wide limits. Some people are careful and a motor car will be treated with 18 ASS 17

28 R. E. BEARD immense care and affection; others will regard it as expendable and with it the cars and property of others. Some persons almost live in their cars and cover an immense mileage, while others hardly use them at all. In the U.K. third-party insurance is compulsory; what then should be the approach to this heterogeneous population so that the premiums are not only fair, but seen to be fair? The present form of cover granted in the U.K. for motor insurance has evolved during this century against the social background, the legal framework and the experience of the insurers who were prepared to underwrite the risks before compulsory insurance came into being. If there appear to be illogicalities when viewed against current conditions it is well to pause and consider the circumstances when the business was evolving. It is also worth while spending a moment on the alternatives. For example, it has often been argued that a more logical approach than the insurance of the car would be to issue two policies, one covering the vehicle and the other the driver. I do not propose to detail the arguments, but one of the consequences of such a change would be that the way would be clear for a much closer link to be established between the claims experience of the driver and the issue of licences to drive. At present any argument is between the policyholder and his insurer if he is dissatisfied he can seek another insurer; if his driving licence was conditional on his experience this would imply some arbitrary standard operated by the licensing authority, appeal against which would either be a lengthy and frustrating experience or an expensive matter, not only to him but to the country as a whole. As with this point, so with many others; competition among insurers has secured a reasonable, if sometimes slow, evolution of the types and conditions of cover. Nevertheless, the motor car is no longer the status symbol of a few but is now within reach of most of us on this small island with its limited road surface and inadequate parking spaces. With the motor becoming so universal, insurance will also be of concern to all; it is not a simple business and the need for the rating system to be seen to be fair will increase. I propose now to move from the realms of general comment to

MOTOR INSURANCE consider the problem of rating, not in the spirit of criticizing what has grown to be the pattern, but with the object of subjecting it to analysis in the light of recent statistical studies and so as to provide the self analysis which is so essential if proper progress is to be achieved. Technical progress in industry is achieved by a continued and active study of fundamentals. This is equally true in the insurance industry and some part of the resources should be devoted to basic research into the measurement of the risk processes underlying the business, not with the object of destructive criticism of what exists but in the search for fuller understanding and improvement. An insurance premium may be considered as built up from three fundamental components. The first is the probability that the event concerned arises, the second is the expected amount to be paid on the happening of the event and the third is the loading to be applied to a calculated net premium to reflect the commercial nature of the contract. In symbols P = qxsx(i+k). The loading k has to cover a great many items. It has to provide the necessary costs of running the business; it has to provide a profit margin and it has also to provide for contingencies such as fluctuations in the experience about the expected value. I do not however propose to discuss these commercial aspects and will limit the discussion to the net risk premium, i.e. q x S. Now q measures the claim frequency, i.e. the number of claims arising in a given period of time from a certain group of risks. I would like to diverge at this point and talk at some length about definitions which are clearly of critical importance, but this would take me too far from my main theme and I therefore assume that both the claims and the risks exposed have been properly defined. For insurance to be a practical commercial proposition there must be a reasonable expectation that a defined class of risk will experience a claim frequency which is reasonably stable. In statistical terms I think it necessary that frequency converges in probability to a limit. Without such convergence, no value can be attached to the expected number of claims and no fair premium could be 18-2 28l

282 R. E. BEARD devised. Although this point may seem a little theoretical, it is important that it be recognized, if only to provide an answer to critics of the application of statistical methods to the business. It is, however, also a reminder that the central limit theorem is not just a piece of theory for students. If we make the assumptions (a) that the risks are homogeneous, (A) that the probability of multiple claims is o{dt), and (c) that there is no association between claims, then it can be shown that the expected distribution of claims will for all practical purposes follow a Poisson law. The qualification 'for all practical purposes' is inserted because there is a small proportion of cases which are totally destroyed as a result of a claim. This involves a small degree of binomial variation, but small enough to be treated as Poisson, particularly against the much greater number of partial claims. Thus the basic model of the distribution of number of claims starts from where P n is the probability that n claims arise in t years and the parameter A is of the order 2 for a year's exposure, i.e. the number of claims in a year is of the order of \ of the number of cases exposed. For a portfolio of 1, cases and using A = 2 the expected number of claims arising in one year would be as shown in Table 1 : Table 1 No. of claims Frequency 1 2 3 4 8,187 1.637 164 11 1 1, This table shows, of course, solely the variation arising from chance, but it means that even for a small portfolio on the average 12 cases out of 1, will experience 3 or more claims in a year. For 1,, cases the same model would throw up an average of 1 case with 6 or more claims.

MOTOR INSURANCE 283 In fact, of course, the assumptions of this model are not fulfilled. First, the risks are not homogeneous, secondly, multiple claims from a given event are possible, and thirdly, the claims are not independent. The difficulty with assumption (b) can be dealt with by suitable care in definition and, again provided the definitions are watched, the effect of assumption (c) can be eliminated from the mean, although it will still be of importance for the higher moments. Accordingly, as I do not wish to make my remarks too complicated it is sufficient if I deal only with assumption (a). We are concerned to build a rating structure which is seen to be fair; this seems to be a necessary commercial condition because if many policyholders think they are being unfairly treated it becomes a commercial embarrassment, either because of frequent letters to the press or because there will be selection against the company. This last point seems natural because if an opinion is formed that a particular rate is high, and the overall charge is not unreasonable, then some rates must appear low. The scale of premiums may also be fair, but if not seen to be fair there will still be trouble. In fact this could quite well mean that a scale of premiums which is not strictly fair may be more appropriate than one which is fair but which does not appear so. In the previous paragraph I have talked freely about a scale of premiums; this implies that some standard exists by which it can be measured and it is here that we meet the first difficulties. As I mentioned earlier, attitudes to chattels vary immensely and it is proper to ask against a commercial background whether allowance should be made for this type of variation. At one extreme a system could be operated in which the same premium was charged for each risk, the risk premium being I/nth of the total claims. This might be appropriate for a community in which the insurance of cars was compulsory, and the damages awarded for injuries were on a scale prescribed by the state, but it may well not be a reasonable system in other conditions. At the other extreme an attempt might be made to devise a rating system in which all the factors giving rise to risk variation were measured; this would become, in the limit, equivalent to charging each person for his own claims and would no longer be insurance. The practical answer must lie between the two.

284 R. E. BEARD Therefore, to get a rating system, we must be able to determine the factors which influence the risk. The situation can be visualized in terms of a number of risk factors, graded in order of importance. The classification then becomes a balance between the complications arising from a very detailed and graduated premium structure and the 'unfairness' which is implicit in treating some of the less important variations as random. What then are the factors which affect the risk? Much has been, and is, said but reliable quantitative data are sadly lacking. Evidence gained from studies by bodies such as the Road Research Laboratory in this country, and similar organizations overseas, and from statistics collected from insurance companies operating in overseas territories, do however throw some light on the inner statistical structure of the risk. As I have said earlier, there must be a stability in a statistical sense for the risk, otherwise insurance could not be practicable. For this reason alone the search for information of the right kind is justified. One important source of variation inherent to many rating schemes including the U.K., but to which I shall do no more than refer in this paper, arises from the fact that the insurance is effected on the car, yet the car can be driven by unspecified persons other than the named insured. This variation is not, and orobablv cannot be, allowed for. Let us now consider one of the other obvious variables, which is not expressly allowed for in the U.K. rating structure, namely mileage run per annum. Some information on this subject has recently been collected by the Ministry of Transport. Table 2 sets out the distribution determined from a sample investigation (the survey was made in October 1961 and published in Economic Trends, no. 116). The average mileage (excluding the first and last groups) is about 143, but the important feature about the table is the immense spread. The moment functions are σ ~ 11, β 1 ~ 2.1, β 2 ~ 5. These are near enough for a Pearson Type III curve to represent the data but an arithmetical approach rather than a mathematical one will best serve my present purpose. Suppose we make the assumption (which I shall discuss later) that the risk of a claim is proportional to the mileage and calculate the distribution of

MOTOR INSURANCE 285 individuals experiencing, 1, 2,... claims in a year. If it is assumed that the exposure for the 1-5 group is 25 miles and so on (51- assumed to be 6) it will be found that the 'A' p.a. for 25 miles per week is 3751 etc., corresponding to an overall frequency of 2. For 1, cars the distribution is as shown in Table 3. Table 2 Recorded mileage per week 1-5 51-1 11-15 151-4 21-25 251-3 31-35 351-4 41-45 451-5 51 or more Not obtained Percentage of cars 7 2 14 5 25 16 5 9 7 8 6 5 4 1 6 1 6 1 2 1 1 1 6 6 IOO O Mileage per week I- 5 51-1 11-15 151-2 21-25 251-3 31-35 351-4 41-45 451-5 51 or more Total no. of cases Table 3 Expected number of claims per annum o 1 2 3 4 No. of cases 766 1486 2376 1455 794 653 381 14 97 68 57 69 836 56 267 273 zo8 22 157 51 55 43 41 62 1433 1 15 26 27 37 32 12 15 14 15 28 222 1 2 2 4 5 2 3 3 4 8 34 1 1 1 2 5 Total claims 58 299 329 271 39 237 83 96 82 84 152 2 Average miles per week (incl. group) = 133 295.

286 R. E. BEARD In Table 4 the column totals have been fitted by a negative binomial distribution. Perhaps, not unexpectedly, the claim distribution is well represented by the negative binomial distribution, but some other features may be noted. Thus in the pure Poisson case 365 claims arise from 176 cases which have 2 or more in the year and in the negative binomial case there are 561 claims from 259 such cases. Table 4 No. of Expected Neg. claims no. binomial Poisson 8,36 8,34 8,187 1 1.433 1.437 1,637 2 222 221 164 3 34 33 11 4 5 5 1 Neg. binomial frequencies 1, 1, 1, p = 1 286, c = 6 431. I would now like to turn this question the other way round, because it is the inverse problem which has over the years given rise to a great deal of loose talk under the general heading of accident proneness. Fortunately a very useful survey of this field has recently appeared in book form and I need not elaborate here, but I will illustrate the essential point by a personal experience. During the war some of my duties involved the recording and analysis of accidents to naval aircraft; included in the records was information relating to the pilot and we were able to develop figures showing the distribution of pilots with, 1, 2,... accidents in a given time. These showed a negative binomial distribution and the first reaction was to invert the reasoning and argue that this reflected differences in accident liability in the population of pilots. However, there was some opinion that the liability to accident might vary according to the previous accident history and it was not long before we developed an alternative model which also led to the negative binomial. We were then faced with two models:

MOTOR INSURANCE 287 (a) that the population was heterogeneous in regard to chance of accident, although within strata the distribution was Poisson, or (b) that the population was homogeneous but the chance of accident increased linearly according to the number of accidents already experienced. Our data would not allow us to discriminate immediately between the models, but remedial action depended on which model applied since the question arising in the former case was the suitability of the 'repeater' for flight duties and in the latter the desirability of a rest period. This lesson brought home very forcibly the dangers of inference when the negative binomial appeared, but unfortunately other workers have not always been so lucky in being reminded that different models can lead to the negative binomial. If the circumstances of the process being studied are such that the Poisson law can reasonably be expected to underlie the events, and that successive events are unrelated, then the arising of the negative binomial can be interpreted as showing heterogeneity in the underlying data. However, without further information it is not possible to be more precise about the nature of the heterogeneity. We can now return to our example. The appearance of the negative binomial is common in accident statistics and an excess of 'repeaters' as compared with a Poisson distribution has all too frequently been used as an argument that such drivers are accident prone. From an insurance company point of view the precise term used is not significant (apart from the public relations aspect which is not discussed here) because the company issues a policy covering certain risks, and it is the frequency with which the claims arise under the policy which is significant. Whether a high frequency arises from a high exposure or a high proneness the problem presented is the same, i.e. is a high claim rate covered by the averaging process used in fixing the premiums or does it lie outside? From the point of view of road accidents, and the social consequences thereof, the position is rather different. A high exposure will usually give rise to a high frequency; a 'repeater' may be a better than average driver and it is unjust to call such drivers ' bad' or ' dangerous' unless there are other factors which patently label them as such. In analysing the anatomy of road accidents a

288 R. E. BEARD suggestive line of thought has been started by Aimer, who conceives a 'risk situation', i.e. a sort of probability field which fluctuates in time according to the risk of accident. Following this idea the good driver would be one who recognizes that his 'field' has become denser and takes immediate steps to reduce its density; the bad driver is insensitive to the situation and thereby runs into trouble more often. This brings into relief the fundamental but often overlooked point that most accidents arise from a combination of an individual and his environment and underlines the danger of talking about accidents as an index of driving ability without regard to the exposure. It also suggests that with increasing traffic the relative contribution of the driver to accidents becomes less significant and accidents become more random; but I know of no statistical investigations which would throw any light on the suggestion. This line of thought also leads to some interesting speculations. Many and repeated efforts have been made to try to find physical and psychological characteristics which correlate with observed accident experience. Tests, and batteries of tests, have been devised to simulate industrial accidents. Instruments have also been constructed to test reaction times, depth perception and other factors which are thought to be of influence in driving. In general the results have been inconclusive unless the tests approach the complexity of actual driving, which, of course, is what should happen! Most, if not all, accidents can be regarded as preventable in the sense that a sufficiently acute observer could have anticipated them; some part must therefore lie in experience and some in a vague sort of intelligent anticipation. The former is measurable by statistical methods but the latter is much more difficult to quantify. At this point it is convenient to comment very briefly on an investigation made in drivers' records in California in 1958. I hope I will be forgiven if I do not give all the provisos, caveats and the rest appropriate to the analysis, but briefly a 1 in 1 sample was made of drivers licensed to drive in California and the experience extracted of these cases over the 3 years ending in 1958. Much very interesting information was extracted, including a tabulation of reported accidents and of convictions for traffic violations. Table 5

MOTOR INSURANCE 289 sets out the statistics relating to accidents with a representation of the data by the negative binomial. It will be seen that the negative binomial fits very well, indicating that the data are heterogeneous in regard to accident liability. A tabulation is also available of the number of convictions for No. of accidents 1 2 3 4 5 and over Total Table 5 No. of drivers 81,714 11,36 1,618 25 4 7, 94.935 Neg. binomial (p = 89265, c = 5 472) 81,726 11,271 1,648 246 37 7 94.935 No. of convictions o 1 2 3 4 5 6 7 8 9 + Total No. of accidents 1 2 3 4 5 + Total No. of drivers 55.757 2,613 8.753 4.32 2,297 1,242 725 45 266 512 94.935 No. of drivers 81,714 11,36 1,618 25 4 7 94.935 Table 6 Table 7 Total accidents 4,829 3.989 2.396 1,527 979 692 368 226 145 336 15,487 No. of convictions 58,38 17,841 3.986 718 156 23 81,14 Accidents/ drivers 87 193 274 353 426 557 57 52 545 656 163 Convictions/ drivers 71 1-57 246 2 87 3 9 3 29 86

29O R. E. BEARD traffic violations and from the double entry table the figures in Tables 6 and 7 can be derived. The feature found in Table 6, i.e. that the data appear to confirm a relationship between convictions and accidents, was regarded as very significant and an insurance rating scheme was developed in which the convictions (suitably weighted) formed the basis of premium calculation. It is instructive to look at the figures on the assumption that the underlying population can be stratified (in Type III form) according to ' exposure' and that the probabilities of accidents and convictions are basically Poisson distributions for unit exposure. It is assumed that an accident and a conviction for traffic violation arise independently of each other. If the population distribution is the distribution of accidents is e -λ λ j j! and the distribution of convictions exp ( kλ)(kλ) j /j! then the probability of j accidents and s convictions is For j accidents the mean value of convictions is [k(j +p)] l(c+i) and for s convictions the mean value of accidents is (s+p)/(c + k). Using the values of c and p derived from Table 5 (the data of Table 6 do not give a very good negative binomial fit) k from Table 6 is approx. 5 3 and from Table 7 approx. 5 18. The ratio of convictions to accidents is 5-25. The consistency of these figures shows that the figures are compatible with the hypothesis that the incidence of convictions and accidents is dependent on exposure, but the data cannot throw any light on whether the 'repeaters' are any more liable to accidents per unit exposure than others. Clearly the number of convictions recorded is a guide to the expected accidents, but this may be only because the drivers concerned have a common exposure. For some purposes, e.g. securing equity in premium calculations, it may be reasonable to use convictions as a more refined indicator of risk than accidents, but without more details there is no justifica-

MOTOR INSURANCE 291 tion for adjectives such as 'safe' or 'unsafe' drivers without qualification. I hope this example will be sufficient to highlight the problem of inverse reasoning from the negative binomial, but there are so many examples of loose argument about that the point cannot be overstressed. But the question may well be asked whether any data have been collected in which exposure in some sense has been used as a control. This is obviously a very difficult question but one published set in which mileage has been used may be of interest. This was a survey made in Sweden in 1961 (see ASTIN Bulletin, 2, pt. I, 96-1) and the figures in Table 8 are taken from this paper. Table 8. Claim frequencies in 1959 according to annual distance run Distance per annum Km. - 7.49 7,5-12,49 12,5-17,49 17,5-22,49 22,5-27,49 27.5-34,99 35,-44,99 45,-54,99 55.- Miles - 4,66-7,767-1,874-13,981-17,88-21,748-27,962-34.176 Claim frequency, P.a. 93 13 142 18 22 24 289 288 323 These figures suggest that frequency per thousand may be of some such form as (8 + 4 5 m ) (m in 1 km.). Table 9 Males Females Frequency, 161 161 Km. 18,35 14,67 Three other tabulations were made of the data; Table 9 shows that frequency was identical for males and females and suggests that the oft-quoted lower accident rate of females may be solely due to the different mileage run, and that in fact they may have a worse experience.

292 R. E. BEARD The tabulation by age at 31 December 1959 (males) is set out in Table 1. A rapid decrease in claim frequency is shown with age, but the average mileage also falls. A comparison of the 'expected' frequency on the basis of the average mileage using the approximate formula fr = (8+4 5m) is given in Table 11 and shows that there is very little variation after the mid-2's. Table 1 Age 18-19 2-24 25-29 3-34 35-39 4-49 5O-59 6-69 7- Age 18-19 2-24 25-29 3O-34 35-39 4O-49 5O-59 6-69 7- Frequency, 527 38 28 16 144 14 142 129 1 Table 11 Expected frequency,, 187 183 175 17 165 161 152 14 124 Km. 23,61 22,95 21,14 19,92 18,92 18,4 15,92 13,27 9,83 Actual frequency,, 527 38 28 16 144 14 142 129 1 The third tabulation relates to experience and is set out in Table 12. Rather unexpectedly the average mileage seems independent of experience but the figures show a steady improvement with experience during the first few years. This table read in conjunction with the previous one, where the young drivers must be inexperienced, leads to the suggestion that the frequency is of the form of a basic level plus a component dependent on mileage plus a

MOTOR INSURANCE 293 loading for inexperience. It is worth noting that an investigation into accidents among motor cyclists in the U.K. showed that experience was the key factor after controlling on horse-power, mileage, etc. (Social Survey Report, No. 277A, H.M.S.O.: for review see ASTIN Bulletin, I, pt. v, 299). Years since licence - 1 1-2 2-3 3-4 4-5 5-1 1-2 2-3 3-4 Table 12 Frequency, 382 299 215 195 183 156 146 137 134 Km. 18,4 19,28 18,17 18,32 18,75 18,53 19,24 17,57 15,62 There are, of course, many other factors which have been used to classify the risks for premium rating purposes horse-power, value, geographical district and so on but none of these to my knowledge has been considered against mileage and it is quite reasonable to think that much of the variation according to these other factors arises from a high correlation with the mileage. An urgent necessity is a full-scale study of the various factors on a multi-factor basis. Some attempts are now being made in some countries to include mileage (in broad groups) as a rating factor, but there are considerable practical problems involved. Leaving the exposure problem let us now turn to some aspects of no claim bonus. This has been the subject of some fairly detailed study in recent years among the ASTIN group of actuaries and also the members of the Casualty Actuarial Society. The principle of no claim bonus in motor business in the U.K. arose many years ago probably as a competitive factor in the business but in the following I am concerned only with the use of the device as a means to achieve mathematical or statistical fairness in rating. There are obviously psychological factors involved which are far from unimportant but these cannot be considered here. An equivalent device is used in other classes of insurance such as workmen's.

294 R. E. BEARD compensation in America or the more recent idea of experience rating adjustments in group life schemes, but I cannot cover all the possible variations here. As mentioned before, from an analytical point of view we can approach the rating problem from two extremes; either we average the claims over the whole population of risks, charging a level premium to each, or we measure the variation factors of each risk and charge the proper premium to each. In the limit the second case is no longer insurance as each case gets charged for its own experience. The practical answer lies between these two extremes. Ideally the rating structure should take care of known factors, with the unknown factors in each of the classified groups random in their incidence. If for some reason the risk cannot be classified according to a factor which is known to be non-random in its incidence, then it may be practicable to adjust the premium retrospectively if the bias in the claims incidence can be measured. In motor business the adjustment is made to subsequent premiums; this presupposes that policies will continue in force. In so far as changes of insurer can be made so a selective option exists against the insurer. The consequences are not discussed, nor am I here concerned with arguments about whether particular types of accident should be ignored in determining whether a bonus is earned or not. This question is being approached on the basis that a bonus scale is to be used to make a retrospective adjustment for a non-random variation in risk rate which was not specifically allowed for in the premium. Clearly the variation in experience which can be catered for by this device must be contained between the limits (a) the standard premium, and (A) the premium obtained from maximum bonus. If, for example, the bonus scale rises to 4% then the premiums range between 6% and 1% of the standard. In point of fact, if the bonus 'period' is short the majority of policyholders will be charged the minimum rate and the variation catered for will be from +% to +66f % over the premium paid by the majority. This is clearly not a very wide margin and, having regard to the random fluctuations involved, it cannot be a very efficient adjustment device at this level (4 % bonus).

MOTOR INSURANCE 295 Various suggestions have been made for finding an index to measure the efficiency of a bonus scale and in order to give an idea of the sort of problem involved I have made some calculations on the assumption that the true risk is proportional to mileage, using the figures in Table 3 for this purpose. This is clearly an extreme case, since the variation in risk is large, ranging from an annual rate of 38 for the 1-5 mile per week group to 9 for the last group. To simplify I have ignored the question of' hunger for bonus' (see page 297). If the chance of at least one claim in a year is q then we have the following scheme for N cases at risk: Nq have had a claim in current year; Nq(I q) had a claim in previous year and none in current year; Nq(1 q) 2 had a claim in last year but one and none in the last two years; N(1 q) r have had at least r clear years. If it is assumed that the premiums are adjusted retrospectively, i.e. that all cases continue, we can find the expected reduction for any bonus scale and for any value of q which can then be compared with the actual premium charged. If consideration is limited to frequencies, i.e. the average claim cost is assumed to be unity, an easy method of studying the scale is obtained. Using the figures of Table 3, assuming a linear bonus scale, e.g. x% reduction for 1 clear year, 2X% for 2 years and so on, with a reversion- to a full premium following a claim, the figures in Table 13 are found in the ultimate situation for a 6-year scale. The column headed % gives the level premium charged to each risk to provide the expected claims of 2. Clearly the 5 % bonus scale (maximum = 3 %) only makes a small improvement in equating the premiums to the expected claims. The 15% scale (maximum = 9%) is making an impression, but a higher scale still is needed. The 2 % scale implies a refund of premium and is included solely for illustrative purposes. Other periods of bonuses show very much the same feature, the period of bonus playing a very small part compared with the maximum bonus allowed. The figures relating to standard premium are included to bring 19 ASS 17

296 S. E. BEARD out the fact that as the bonus scale is increased it becomes necessary to increase the standard premium to offset the higher reductions so that the total premiums are sufficient to meet the expected claims. For further study the 15% scale will be used (i.e. maximum bonus = 9%). Table 14 sets out the numbers of policyholders in each stratum who will be paying different scales of premiums for a total population of 1,. Mileage 1-5 -1 -IS 151-2 21-35 251-3 31-35 351-4 41-45 451-5O 5 or more Total Expected claims 58 299 329 271 3O9 237 83 96 82 84 152 2OOO Standard premium, Table 13 Net premiums received with annual bonus increment of» 5% 1% 15% % 153 39 532 351 26 183 115 34 34 26 23 34 2 2 131 277 515 358 218 199 128 38 39 3 27 4 2 244 96 228 488 369 237 224 147 45 47 36 33 5 2 313 Table 14 Number paying 34 14 442 388 271 269 182 57 6 46 44 67 2 435 2% -19-59 334 433 347 37O 262 86 91 71 68 16 2 714. Mileage 1-5 51-1 11-15 151-2 21-25 251-3 31-35 351-4 41-45 451-5 51- Total Nos. 766 1.543 2.659 1.755 1,32 915 575 17 17 128 117 17 1, 1% 57 283 3 238 262 194 66 73 6 6 11 1.694 85% 55 253 249 183 187 128 41 42 32 29 41 1,24 7% 53 226 26 141 133 85 25 24 17 14 17 941 55% 5» 22 171 18 95 56 15 14 9 7 7 735 4O% 49 181 142 83 68 37 9 8 5 3 3 588 25% 47 162 118 64 48 24 6 5 3 1 1 479 1% 766 1,231 1.352 569 215 122 51 8 4 2 3 4,323

MOTOR INSURANCE 297 Overall 43 % of the policyholders will be paying premiums at the rate of 1% of the standard. This immediately throws up one of the disadvantages of the idea of 'bonus' since for these persons the standard scale is 1 times their payment. Of course it would be possible to quote the reduced rate as the base and deal with claims as a malus, i.e. to charge a loading according to the claims experience. Psychologically this would probably be worse, and commercially less desirable since the transfer option would probably be of greater impact. Between the limits there is, of course, a whole range of alternatives, depending where the standard is set and various alternatives can be found in use commercially. When a bonus scale covers a wide range other problems begin to emerge. The obvious one is that the jump of 1 times for one claim implied by the above, for example, would be very unpopular as a commercial proposition. There would be a marked 'hunger for bonus', a term devised by Philipson to describe the comparison made by the claimant between the value of the bonus lost and the cost to repair the damage, and which incidentally is reflected in quite a marked shift in the numbers in the lowest premium class as compared with the theoretical figures. The effect of this shift can in practice be estimated from the frequency distribution of claims by amounts and is not to be regarded as a defect in the theory but rather an important refinement in the model which cannot be dealt with here. This has led to the introduction of bonus scales which may be described as 'one up, two down', e.g. for a claim-free year the premium is reduced by one step in the scale but on a claim arising it is increased by two steps. This type of model will obviously take a long time to reach a stationary condition, and with the current rate of growth in business a 'good' calculation would be lengthy. I have not thought it necessary to illustrate the effect of such a scale as its general behaviour can be inferred as a respreading of cases between the extreme premium limits, it being noted that the proportion of cases entitled to full bonus remains unchanged. There are some important logical consequences which follow from the above. If, for instance, the bonus scheme is assumed to take care of systematic variation not allowed for specifically in the 19-2

298 R. E. BEARD premium calculation, and the bonus scale is inefficient in the sense that it only takes care of part of the variation, then a proper question to ask is whether the accepted inequity is greater than that attempted in bringing the defined factors into the rating, i.e. if, for example, it is known that a range of 5 to 1 in claim frequency arises from mileage, but the rating schedule does not allow for mileage and the bonus scheme can only deal with a range of 2 to 1, is there then a justification for charging a finely graduated scale according to horse-power? Table 15. Discordance index Max. 1 2 3 4 5 6 7 8 9 1 bonu8(%) Index 98 95 92 88 82 76 69 59 47 34 Table 16 No. of claims 1 2 3 4 Total Expected no. 8,218 1,584 181 16 1 1, Neg. binomial 8,219 1,582 182 16 1 1, In order to provide some kind of index for measuring the effect of the bonus scheme in achieving equity among the groups a ' discordance index' has been calculated by taking the total of the absolute differences between the expected claims and the premiums received, using the o % scale as standard. It was found that this index was almost solely dependent on the maximum rate of bonus, the period of bonus being of very little effect (see Appendix II). The figures are given in Table 15. These figures suggest that the linear bonus scale at lower bonus levels is not a very efficient instrument for dealing with non-random variations, but no studies have been made to see whether other systems could be devised which are more efficient. A set of calculations was also made to investigate the effect if, instead of assuming that the accident liability was directly proportional to mileage, it was assumed that it was of the form of a

MOTOR INSURANCE 299 constant plus a linear component, i.e. A = 1 + 1875m where m is in units of 25 miles per week. Table 16 sets out the negative binomial curve corresponding to Table 4. In Table 17 the pattern of premiums for accident liability of the form λ = 1 + 1875m is set out in the same form as Table 13. Whilst the ' discordance' index shows a similar pattern to the values obtained in Table 15 it should be noted that the absolute value of the discordance is only about one-half as compared with the form A = 3751m. Mileage 1-5 51-1 11-15 151-2 21-25 251-3 31-35 351-4 41-45 451-5 Expected claims 77 183 416 34 239 246 176 58 65 53 53 51 or more 94 Total 2 Discordance index % Net 153 39 532 351 26 183 115 34 34 26 23 34 2 I OO Table 17 premiums received with annual increment (6-year bonus) 5% 144 295 522 352 211 I9 121 36 37 28 26 38 2 9 1% 13 274 56 354 218 22 131 4 41 31 29 44 2 75 15% 17 239 481 357 23 22 147 45 48 37 35 54 2 51 bonus 2% 63 172 432 364 253 255 177 57 6 48 46 73 2 There is clearly considerable scope for original study into the question of finding bonus systems which are more effective in dealing with non-random variations. To make the exercise of value it would be necessary to have some reliable information of the true underlying risk structure, and, of course, the simple models would require elaboration to allow for the neglected factors such as 'hunger for bonus'. A suggestion for an alternative approach to a rating structure was recently put forward by P. Delaporte, based on risk-theoretical considerations (17th International Congress of Actuaries). In this scheme the premium for a particular risk in a given year depends 19

3 R. E. BEARD on the premium in the first year of experience multiplied by a factor depending on the total number of claims incurred since inception. This is clearly an interesting variation of a 'bonus system' but its successful application would also depend on the accident variation in the classification groups being so reduced as to fall within the practical range of adjusted premiums. I have made no reference to the incidence of costs of administration, but these cannot be ignored in a full study. Frequent changes of premium are expensive to administer, quite apart from the effect on the policyholders, but beyond mentioning the point the subject will not be further discussed. All the foregoing has been based on the frequency component of the risk premium and although there are many interesting aspects solely concerned with frequency, the remaining component, i.e. the average amount for claims, gives rise to many more, on some of which I would now like to spend some time. In the first place it is necessary to point out that motor insurance covers two main types of claims, property damage and personal injury. The distribution of claims in the former group tends to be relatively compact since most of the claims are in respect of damage to cars for which there is a fairly low upper limit of values. The range of variations in the latter group is much greater and gives rise to some troublesome practical points. An analysis of property damage claims by amount is generally J -shaped, a high proportion of small claims tailing off as the amount increases. There is a complication from zero claims which I will not discuss. Normally physical damage claims are settled fairly promptly and there is no great problem from estimating the expected value of such claims. No theory apart from the studies by Zipf has yet been advanced which leads to an analytical form for such claims, although experimental data show that a log-normal curve will give a very reasonable fit. Other transformations, such as a log- Type I will often deal with cases not well represented by the log-normal but introduce complications into the mathematical properties. It is not uncommon for the assumption to be made that the Pareto distribution applies; this has the theoretical objection that the higher moments may be infinite, although this feature can

MOTOR INSURANCE 31 be dealt with by truncation. Some authors have used exponential polynomials. Liability claims show the same sort of general feature, although in this case the range is much greater and the tail is very long. The difficult aspect is that it may take many years before the exact amount of the claim can be determined; first there may be a question of liability and secondly the amount of damages may be in question. Unfortunately it is the larger claims where settlements tend to be longest deferred and conditions can change during this period; the attitude of courts towards damages may change, legislation may alter and the value of money may change. For many years the effect of these factors has been in the sense of increasing the claim cost at settlement as compared with comparable cases at the time of arising. Efforts are made to allow for such factors but the problem is not easy. To give some idea of the nature of the variation Table 18 gives a distribution based on a sample of settled claims, both property damage and liability claims being included. Table 18. Approximate distribution of 1, claims log] amount of claims in No. of claims -5 6826 5-6 1373 6-7 924 7-8 514 8-9 236 9-1 9 1-11 28 11-12 7 12-2 The mean of the distribution is approximately 53 and the ratio of the second moment about the origin to the mean approximately 56. A log-normal curve with x = (.44 log 2 claim 1.725) gives reasonable representation for practical work as Table 19 shows. These figures show that something like half the amount of claims arise from about 4% of the number. As an indication of the effect of the 'time lag to settle', the figures in Table 2 relate to a small sample of claims, but much longer periods will in fact be found in practice. With this very sketchy view of the claim structure we can now revert to the combination with frequency. Whether we are concerned with premium calculations or with estimates for financial

32 R. E. BEARD purposes our current data are in the form of a number or claims arising, to each of which must be associated an expected settlement. If we wait until the claims are settled, the data may be 5 years or more out of date. To do a scientific job we should therefore find Table 19. Cumulative distributions No. of claims Amount of claim log, c Observed Calculated Observed Calculated I 2 3 4 5 6 7 8 9 1 11 12 13 15 3 83 247 48 72 841 917 965 99O 995 995 999 1..42 99 199 343.514 683 82.912 964 987 96 999 1. 1. 2 6.38 23 749 171 2923 4266.586 7447 8183 957 957 1. 5 21 78 239.618 1357.2546 4129.5871 7454 8643.9382 9761 9922 Table 2 Amount ( ) 2-3- 4-5- No. of claims 8 4 % 2 Av. time in months to settle 21 3 38 4 some means of estimating the expected average using all the information at our disposal. If there is no provision for central collation of statistics for a country we may be forced to rely on the records for a sample only and having regard to the nature of the underlying distribution this will make estimation difficult if the numbers are small. The facts that can be tabulated are the distributions of settled claims in certain periods of time but, remembering that almost universally motor business is growing fast, this distribution will require adjustment because of the time lag to settle large claims.

MOTOR INSURANCE 33 The large claims among those settled will, in general, have arisen from the business exposed some years previously. If the business is increasing this means that the proportion of large claims among those currently settled is relatively lower than it would be amongst a group of claims currently incurred. Thus a first adjustment is to raise the average of the settled by an amount dependent on the rate of growth of business. Now the average amount is to be applied to a currently incurred group; the larger ones will have deferred settlements and will be subject to increases if the value of money falls, more so than the smaller claims. Equally they will be more exposed to other causes leading to increase. Philipson has recently shown how this model can be described as a random process and hence opened the way for the formal treatment to determine the usual statistical parameters for the various distributions involved. I have suggested a model to sort out these factors which simply consists in the very reasonable postulate that a population of claims can be classified according to a ' degree of damage' and that this distribution is stable. The various derived distributions, i.e. incurred claims, settled claims and outstanding claims, can then be written down with allowance for the time lags and changes in the value of money. Provided the basic distribution is known a simple method would become available for evaluating the claims (collectively) under various assumptions. Furthermore, statistical measures (e.g. σ) would be calculable for significance testing. I assume that the claim distribution has been determined and the premium (q x c) then follows. What can be learnt about the significance of total claim payments? I cannot do more at this stage than just indicate the method of approach. Let us assume that the probability that exactly r claims arise in a given interval t is p r (t). If the moment generating function for one claim is K(x), then for a sample of r claims the m.g.f. for the total amount of claims from the r is {K(x)} r ; hence the m.g.f. for the total claims arising ia p r (t){k(x)} r. Given p r (t) and K(x) the problem of the distribution of claims is formally solved although the practical problems are quite difficult. If p r (t) is a Poisson distribution the moments of z,. where z is the total amount of claims, can be easily expressed in terms of the moments of the distribution of one claim, but from

34 R. E. BEARD what has been shown of the form of K(x) this method may not be very satisfactory. Much has been written on this problem and considerable research is now going on, but the development of these ideas is beyond the scope of this paper. It will be appreciated that the current pattern of rating motor-car insurance has evolved over the years and it may well be asked if there is any alternative approach which would be a better method in practice. Those who have to administer insurance based on indemnities have suffered grievously in recent years from the recurring need for premium revision. It is thus pertinent to ask whether there is any alternative approach by which this continual revision could be avoided. In approaching the question one can be fortified by the fact that it seems that the basis for equitable premiums can be quite wide, judged by the present methods of rating. Recently in looking at certain statistics of claim frequency in terms of year of manufacture, it was noted that there was a pronounced declining tendency, the newer cars showing the higher frequencies. This seemed rather surprising at first sight since it might be expected that older cars would give more trouble than newer cars. A closer look at the figures suggested that both duration (i.e. age of car) and year of experience were complicating factors so that an appropriate 3-variate analysis was made on the figures to try to sort out the true relationship. (The technique used was similar to one that I recently used in the study of the relationship between smoking and lung cancer in England and Wales over the last 5 years and which was contributed at the Royal Society discussion on Demography in November 1962.) The interesting fact to emerge was that the run of figures could be argued as supporting the mathematical model that each year of manufacture had its weighting factor, as had the exposure years, and that when these were eliminated the resulting claim frequency showed an approximate exponential decline with age of car. A similar calculation for accidental damage claims showed that the figures could be explained on the basis that each year of manufacture had its appropriate weighting factor and that when this was allowed for the average claim varied approximately with the value of money.

MOTOR INSURANCE 35 Following these ideas further and assuming that the average claim increases in geometric progression with time, we can write down the following notional formula for rating, where T represents calendar time, r age of car assumed to be insured at that time P = K T - r e- λr e δr = K T - r e- ( λ - δ ) r. From the statistical calculations A was of the order of.8 and δ ~ 6 so that the formula becomes approximately P ~ K T ~ r e- 2r. In other words the risk premium for a given year of manufacture will decrease slowly with duration. Since we know that there are some loadings to be applied to the risk premium, and we know that current rating methods apply the concept of 'fairness' over a wide area it looks as though it would be possible to devise a basic rating structure determined for each new car and constant throughout the life of that car. If this could be shown to be generally true a basis exists for avoiding premium increases with changes in money values. The tariff could, of course, have a no claim bonus scheme and the K factor could vary according to the features of the risk. The small rate of decrease in the risk premium is on the safe side and would provide a buffer against unexpected movements in the underlying risk structure. It would not be necessary to guarantee these rates and they could be treated as annual rates as at present, subject to adjustment if the experience moves significantly in either direction. There would however be much less variation than under present methods, which average the experience over different risk groups and which must be adjusted for changes in the value of money. There is some justification for an approach on these lines, although it should be stressed that it depends on the market structure for new and second-hand cars in the U.K. The pattern seems to be that most new cars go to commercial users and in their early life they are exposed to a high mileage; as the car gets older it moves down the second-hand market, successive classes of users tending to drive fewer miles. (Table 21, taken from Economic Trends, illustrates the feature.) Thus, provided no real change occurred in this pattern and wage rates continued to rise a possible alternative rating scheme could be envisaged.