PROCEEDINGS NOVEMBER 24, ~933 POLICY LIMITS IN CASUALTY INSURANCE PRESIDENTIAL ADDRESS~ PAUL DORW~-ILER

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1 VOL. XX, PART I No. 4 x PROCEEDINGS NOVEMBER 24, ~933 POLICY LIMITS IN CASUALTY INSURANCE PRESIDENTIAL ADDRESS~ PAUL DORW~-ILER Insurance, in theory, should either cover the whole risk or that portion of the risk involving the larger potential losses, even though their occurrence is less probable. The current practice in most lines of casualty insurance is somewhat in conflict with this theory. Generally, the standard coverage for third party insurance involving injury to persons is based on a fixed set of low policy limits per person and per accident with additional coverage for higher limits considered as secondary. It is proposed in this paper to discuss the place which policy limits, particularly for third party insurance involving injury to persons, occupy in the structure of casualty insurance and to consider some of the properties of policy limit factors. HISTORICAL DEVELOPMENT In the beginning of casualty insurance, there was no experience available and the rates were necessarily based on judgment. The underwriters proceeded with caution, under these conditions, and placed restrictions on the maximum liability assumed from the occurrence of a single event. The underwriters, too, had before them the practice in fire, marine, and life insurance, where definite amount limits on a single los4 were in general use. In these circumstances it was quite natural to restrict the losses in casualty insurance. As the business grew, experience was devel- I

2 9. POLICY LIMITS IN CASUALTY INSURANCE oped in which the losses were restricted through the presence of limits in the policies. The use of this experience for rate making purposes tended to fix and standardize these limits into basic coverages which came to be more and more widely accepted. There were other important reasons why caution should have been used and limits placed on the losses due to single accidents in lines involving injuries to persons. Injuries to persons differ essentially from injury or damage to property with respect to the possibility of replacement. Injured organs in persons cannot be replaced, so that the economic law of supply and demand does not operate in fixing the cost. Injuries to persons in third party insurance are the more difficult to evaluate, for, in addition to the severity of the injury, there are such factors as the negligence and fault of the injured or of others, and the degree of responsibility or the liability of the insured for the injury involved. Inherent in these difficulties is the further difficulty of expressing injuries to human beings in terms of dollars or of expressing the magnitude of something in terms of units for measuring magnitude in an entirely different category. These difficulties in evaluating losses do not appear in the same degree in the forms of insurance involving property, or in life, and accident and health insurance, which involve injuries to persons. In property insurance, replacement of damaged portions may ordinarily be secured in the open market, which permits the law of supply and demand to enter into the adjustment of losses whether by mutual agreement or by court verdict. While there can be no replacement in life insurance and accident and health insurance, there is a difference of major significance with respect to the time and plan of evaluating specific losses. In these types of insurance the values of the injuries in terms of money are left to the choice of the insured. Before he enters into the contract, the insured, with broad restrictions, can make his own estimates of the values of his injuries and have these estimates written into the contract, providing he pays the corresponding premiums. In third party insurance, the potential injured does not enter into the making of the contract; in fact, he is unknown at the time. The problem of evaluation does not arise until after the accident has occurred, when the economic interests of the injured and the assured or his carrier are at variance, which makes it difficult to bring about satisfactory adjustments.

3 POLICY LIMITS IN CASUALTY INSURANCE 3 The use of policy limits for lines of insurance which do not have natural limitations on losses from single events has become universal. State laws generally prohibit the writing of policies with limits so large that they make the carrier liable for a potential loss in excess of a definite percentage (usually 107'o) of its capital and surplus, after allowing for reinsurance provisions. With the growth of the carriers to their present size, these state laws have little effect on policy limits. In the beginning they probably placed definite limitations on many policies. The presence of these state laws would indicate that, from the public viewpoint, it is against sound policy to permit a carrier to risk an undue portion of its resources on any single event. These observations regarding the historical development of policy limits are based on fragmentary material and are given as plausible inferences and deductions rather than as an authentic history based on adequate research. POLICY LIMITS AND HAZARD There is a fundamental difference in the relation between the hazard and policy limits in insurance generally, and in casualty insurance, particularly third party insurance. In the other forms of insurance, and also in some casualty lines involving damage to property, the hazard varies about directly with the policy limits. It follows that the premium should vary accordingly. In third party insurance, losses do not vary even approximately with the size of the policy limits. The relative increase in losses is small compared with the increase in the policy limits even for increases just above standard limits. As the policy limits used increase, the relative change in hazard decreases, approaching zero for indefinitely large limits. In the other forms of insurance there is an increase in hazard which is approximately proportional to the increase in limits until an amount is reached representing the full value of the object insured. There can be no real loss to the assured in excess of such a limit and there is a moral hazard involved in issuing policies with such excess limits. In third party insurance there is no natural limitation of the amount that may be involved in a single accident. There is, however, a rapidly decreasing probability that very high amounts will be involved, which makes the corresponding increases in premium very small.

4 4 POLICY LIMITS IN CASUALTY INSURANCE PoLIcY LIMITS AND EXCESS INSURANCE The presence of limits written into a policy places a restriction on the losses of the risk so that any losses in excess of the designated limit or limits are excluded from coverage under the policy. Excess insurance consists in coverage for losses in excess of specific limits which are designated in the contract. Excess insurance and the usual coverage under a policy containing the same limits may be considered complements of one another. The policy with its coverage restricted through policy limits and excess insurance for coverage beyond those limits constitute full coverage for all losses that may occur. CLASSIFICATION OF POLICY LIMITS Policy limits may be classified in two broad divisions: in one, restrictions are placed on an individual event in the risk, and in the other, restrictions are placed on a combination of all events or on the risk as a whole. In the first, limitations are placed upon the losses for an individual person which shall be referred to as a per person policy limit or upon a loss for an individual accident which shall be called a per accident policy limit. In the second, a limitation is placed upon the total amount of losses for the risk. This limitation may be a definite fixed amount for the risk, which shall be called per amoun~ policy limit, or the losses may be limited to a specific designated loss ratio, which shall be called per loss ratio policy limit. The per loss ratio limit is seldom, if ever, used with the assured as an upper policy limit, but it has been used as a basis for excess coverage and for supplementary reinsurance and coinsurance agreements. It is being included here primarily because it serves as the means for studying the relations involved between the per amount limit and the size of the risk premium. PER PERSON AND PER ACCIDENT POLICY LIMITS The per person and per accident policy limits, the two limitations placed on single events in casualty policies, will be considered together because they have many properties in common and also because they are chiefly used together. Policies for liability

5 POLICY LIMITS IN CASUALTY INSURANCE 5 for injuries to persons contain a maximum limit for injuries to one person in an accident and also a maximum limit for injuries to all persons in a single accident. A few standard tables showing the cost of policies containing various combinations of higher limits in terms of the cost of policies written on standard limits (usually $5,000 per person and $10,000 per accident) are now in general use by a large number of casualty companies for third party lines involving injuries to persons. The losses per person, i.e., the evaluations of injuries in dollars, are subject to the influence of human sympathy and public opinion. They are subject to variations due to changes in the public viewpoint which ultimately are reflected in direct settlements and in court verdicts. The trend has been toward placing higher monetary values on both fatal and non-fatal injuries to human beings. This may he a reflection of an increasing appreciation of the place and importance of human life. It is also in line with the course of the general price level for the last few decades. Apparently there will not result from this depression a corresponding recession of cost of injuries as of price level from the maximum of An increase in the cost per injury must necessarily be reflected in the cost per accident, for this is a sum of a number of individual injury costs. There is, however, another element in the cost per accident consisting of the number of injuries per accident. This second element is affected by such agencies as the mechanization of industry, the general development of industries involving 'a concentration of persons, or a change in our social institutions and custom~ bringing about larger and more frequent gatherings of people. The development of transportation facilities and the resulting social change undoubtedly increased public accidents and resulted in an increased cost of coverage for such accidents both per person and per accident. There are certain tacit assumptions underlying the present tables of policy limit factors for higher policy limits. It is assumed that there is no variation for per person limits by risk size for risks of the same kind or within a given classification. There may be some question as to whether big and prosperous concerns are not more likely to have higher verdicts rendered against them than smaller ones in the same classification. This same doubt may exist as to losses coming under standard limits.

6 6 POLICY LIMITS IN CASUALTY INSURANCE The general spread of third party insurance in recent decades would tend to reduce such variations on the ground that the losses are covered by an insurance carrier. A policy limit factor table attempts to express the cost of policies with higher limits in terms of the cost with standard limits. Any influence that would affect the pure cost for injuries equally under standard limits and higher limits should have no effect on the limit table. It is also generally assumed in the present tables that there are no variations in the cost per accident by size of risk within a classification. This may be questioned for there are certain causes of accidents, such as explosion and fire, which are quite likely to embrace the whole or a very large part of a risk. Large risks are subject to greater hazards from these sources than small risks where the total number exposed is more limited. For this same reason, however, the smaller risk would be less likely to select high limits, which tends to obviate a need for distinction. Recognition of variation within a classification for both per person and per accident limits is a refinement hardly warranted at present with our limited information of these hazards. Variations by lines of insurance and by classifications within some lines are recognized to a degree at present through selection of the particular policy limit factor table which is to be applied. The use of the same table in all territories implies that there is no variation in the policy limit factors by place. The variations over periods of time are probably greater for per person limits than for per accident limits. Such variations by time periods can be recognized by changes in the tables or the adoption of new tables. There are possibly unrecognized conditions which cause greater variations than some of those that have received recognition. The functions representing the policy limit factors in terms of standard limits have certain properties which may be stated as a general law. For equal consecutive increases in the policy limits there will be correlated increases in the policy limit factors which decrease as the policy limits become larger. In other words, in the general arrangement of per person and per accident limit factor tables (see Appendix I), the difference between adjacent factors for equally distant limits should decrease as one passes to the right along the same line or as one

7 POLICY LII~fITS IN CASUALTY INSURANCE 7 passes downward along the same column. The properties of the limit factors are treated in detail in Appendix I. PER A~OUNT AND PER LOSS RATIO POLICY LIMITS Under certain conditions it is difficult to restrict losses by limiting the amount per accident. It is almost impossible to establish a causal relation to a specific event when there are a series of events, of which one or more may be causally involved. Damage to buildings from continued blasting nearby, or the occurrence of many occupational disease cases arising from long exposure are examples. In such cases the use of the policy limit per accident is impracticable. It is found more practicable to use as a limit a fixed amount per policy period, usually a year. In place of the fixed amount, an amount that is somewhat less definite and related to the loss ratio of the risk may be used for fairly stable risks. It is obvious that the same amount for a policy limit on a small and a large risk would produce very different results in the limiting effect on the total losses developed with each size of risk. An amount that is many times the expected losses on a small risk might actually be less than the expected losses Oil a large one. Such an amount limit used in the policies would have very little effect on the losses of the small risk and a very limiting effect on the losses of the large risk whic_h it might even restrict to below the normally expected. What should be the relation between the amount limit of a risk and the size of the risk? What are the per risk amount limits for risks of different size that may be said to be equally effective? This is the particular problem which has been treated in detail in Appendix II. Consider two groups of homogeneous risks, designated by Group I and Group II, which differ only in size, all risks within a group being of equal size but those in Group I being smaller than those in Group II, and let it be assumed these groups are adequate in volume. First consider these risks under coverage without policy limit restrictions of any kind, then let the losses under Group I be evaluated on the basis of each risk having a policy limit by which the losses covered are limited to a definite amount AI per policy per year. Let it be found that, if in every policy the losses had been restricted to a maximum amount At, then there

8 8 POLICY LIMITS IN" CASUALTY II~SURANCE would have been r~ of the total losses not covered. In this case let the r~o when expressed decimally be r and be called the pure premium ratio* corresponding to a per risk amount limit AI on a risk of Group I size. If the experience of the risks of Group II could be studied so as to ascertain what amount limit A,x per risk per year would produce the same pure premium ratio r, or what amount limitation per policy is necessary on risks of this size to exclude r% of the total losses from coverage, the amount AlI, corresponding to the pure premium ratio r, would be found to be larger. In Appendix II, one carrier's compensation experience for several states and a few policy years was used in studying the effect of amount limitations on policies. This experience, consisting of some 80,000 risks, was divided into eleven groups according to size of annual premium. Pure premium ratios were then determined for each group for the losses in excess of various loss ratios according to the method explained in a previous paperjsee Proceedings, Volume XIII, pp The results are shown in Table I and are represented graphically in Chart I. From Chart I, per amount policy limits corresponding to various pure premium ratios were determined for each size group of risks. These per amount limits with the corresponding average sized risks were plotted for the selected pure premium ratios as shown in Chart III-A. The carrier's experience was supplemented by using the classification experience of the National Council on Compensation Insurance for policy years , the classification experience for one policy year being considered a unit. These units were separated into eight size groups according to the volume of expected losses. The actual losses of the units were expressed as indices of the expected losses by dividing the policy year pure premiums of each classification by the average pure premium for the five policy year period. For each of the eight size groups, pure premium ratios were calculated which represent the losses of the classifications in excess of selected indices per unit when expressed in terms of the total losses. These ratios are recorded in Table II and are presented graphically in Chart II. Then, by a method similar to that used in treating the carrier's experience in Chart I, per * Pure premium ratio is here used as in a previous paper on Excess Insurance. Proceedings, Vol. XlII, page 165. It refers here to the excluded losses.

9 POLICY LIMITS IN CASUALTY INSURANCE 9 amount limits corresponding to a specific index were determined from Chart II for each loss group. These per amount limits were plotted against the average expected classification losses of the groups and are shown in Chart IILB. It will be noted that, allowing for some fluctuations in Chart III-A, which is based on a rather limited volume of experience, particularly for the smaller risks at the left and for the low pure premium ratios at the top, these amount limits, when plotted, appear to lie along a straight line. The per amount limits in Chart III-B, which is based on larger units of experience, apparently are even more nearly on a straight line. As indicated in Appendix II, these ratios in reality lie on long drawn out curves whose curvature decreases as the size of the unit experience (risk or classification) increases. These curves represent per amount policy limits having the same effectiveness with regard to their associated risks or classifications. That is, they exclude the same proportion of the total losses from coverage by their inclusion in the policy. The curves are convex downward and for the larger units approach radial lines through the origin as shown in Chart IV. The curves labeled R in Chart IV indicate the relation of the per amount limit to the size of the risk for specific ratios of excluded losses to total losses. The per amount limit on a risk corresponding to a definite per amount policy limit factor should vary with the size of the risk not exactly at the same rate but at a somewhat lower rate. Per amount policy limit factors may be shown in a two-way table showing various per amount policy limits at the head of the columns and size of risk groups at the left of the lines, with the corresponding per amount policy limit factors in the body of the table. CO~IBINATION OF POLICY LIMITS Of the four types of policy limits discussed, the per person and per accident are used constantly in certain lines of insurance, the per amount limit is used at times, and the per loss ratio is used rarely. These limits may be used singly or in combination. The question arises as to the relative effectiveness of each type in limiting losses when used in combination.

10 i0 POLICY LIMITS IN CASUALTY INSURANCE No general statement can be made covering all these relations, for they vary by line of insurance and they vary within a line depending upon the size of the limits which are used in the comparison. No data are available for lines of insurance written with policy limits to show what effect the standard per person and per accident limits have on losses. It would seem safe to state, however, that for nearly all' lines the standard limit per person is more effective than the standard limit per accident. When other than the standard limits are compared, the results depend entirely on the particular limits considered. The effect of a combination of per person and per accident policy limits on the experience of an individual risk is at most equal to the sum of the effects of each policy limit separately and at least equal to the more effective of the individual policy limits. The maximum effect of a combination of two effective limits is attained when no elemental part of the experience is affected by both limits. The minimum is obtained when the less effective limit applies only to elements already so restricted through the operation of the other limit that the second becomes non-effective. In general it may be stated that the effect on an aggregate of experience of a combination of limits is, at a maximum, equal to the sum of the individual effects and, at a minimum, equal to the larger of the individual effects. There are certain properties of the types of policy limits which may aid in a general comparison. Specific policy limits per person are equally effective for the small and the large risk. Specific policy limits per accident are not quite equally effective for the small and the large risk but this variation may be disregarded. A specific per amount policy limit is least effective on small risks and most effective on large risks and a specific per loss ratio policy limit is most effective on small risks and least effective on large risks. EFFECT OF POLICY LIMITS ON THE DEVELOPMENT OF CASUALTY INSURANCE It is interesting to speculate on the effect that the presence of policy limits may have had on the development of casualty insurance. With the introduction of insurance there develops a claim consciousness which results in an increase in the loss cost of accidents. It is probable that the presence of policy limits has

11 POLICY LIMITS IN CASUALTY INSURANCE 11 tended to stem the rising cost of severe cases. While no data are available as to the degree in which the upward trend has been retarded, individual cases are known to those familiar with the adjustment of losses where the presence of policy limits has had a marked effect in securing the adjustment of losses within these limits. Also many instances are known where the absence of insurance combined with the low credit standard of the individuals liable for the accidents has prevented the claimants from instituting suits or pressing claim settlements. It is reasonable to conclude from these individual cases that the presence of the policy limits has tended to induce claimants to make settlements which would not have been made under higher limits. In addition to including losses above standard limits in severe cases, there is another respect in which the presence of policy limits higher than standard may influence the cost from the carrier's viewpoint. It is contended by some that the wide variations obtained in jury awards for similar accidents prompt some carriers to settle cases of little merit, but of a nature adapted to appeal to human sympathy, for amounts far beyond a reasonable appraisal for fear that the juries may be moved to give extraordinary amounts. It is claimed that the cost of the presence of higher limits is shown to a large degree, not in the losses extending beyond the standard limits, but in the increased cost of the losses below standard limits due to such settlements. There arises also the question as to the difference, if any, in the type of policyholders who insure for higher limits and for standard limits. Is the assured who is unwilling to take a chance with low limits more conservative generally and is this quality reflected in better environment and methods which produce a better loss ratio? Might not a test which shows a better standard limit loss ratio for higher policy limits be a reflection of a better type of policyholder on the whole under the higher limit policies rather than a lack of evidence of forced settlements for fear of excessive verdicts?

12 , ~.+i 12 POLICY LIMITS IN CASUALTY INSURANCE APPENDIX I. PER PERSON AND PER ACCIDENT LIMIT FACTORS. The policy limit factor is the index which expresses the rate for coverage with specific policy limits in terms of the rate for coverage with standard limits. Since the rates for various policy limits have the same pro rata loading, the policy limit factor may be considered the index number which represents the losses for an aggregate of adequate experience when evaluated under specific policy limits in terms of the losses for the same experience when evaluated under standard policy limits. In considering the properties of policy limit factors this latter view will be taken. The policy limit factor tables may be represented generally as follows :-- L! l l.-t Per Person Policy Limits I l. I i L,~-t I L,, FF 1 im+l... [ Fm~-~l F,~-I F~+~ Where 1, denotes the policy limit per Person, the maximum amount for any one person involved in an accident. L,n denotes the policy limit per accident, the maximum amount for all persons involved in an accident. F~ denotes the policy limit factor corresponding to combined policy limits of 1, per person and L.+ per accident. It and Lt denote standard limits, and the corresponding limit factor F = 1.00.

13 POLICY LIMITS IN CASUALTY INSURANCE 18 It will be assumed in the following that I. and L~ each represent a set of equidistant limits in the table. In the actual tables the limits are taken farther apart as they increase in size. It will also be assumed that in a representative frequency distribution of losses by size for either losses per person or losses per accident some cases will fall within each policy limit interval represented in the table. Under these assumptions it may be shown that these general relations hold :-- F~ +I - F~, < F~- F~ -I F~+I- F~ < F,~- F~ The differences between the members of the inequalities will become too small to express by three decimal places as the limits increase. This accounts in part for the larger intervals between the higher policy limits in the tables in use. If further assumptions are made regarding the frequency of high cost individual cases within a distribution of losses by cost per accident then these general relations follow: 1. If there are relatively more high cost individual cases in accidents as the cost per accident increases then F~ +~ - F,~ > F~, - F,~_~ F~ +1 - F,~ > F,~- F~-I 2. If there is no change in the frequency of high cost individual cases with an increase in cost per accident then F~ +~ - F,~ = F,~ - F~ F~ +~ - F~ ~ F,~ ~ - F~_l It would be expected that there is some positive correlation of high cost individual cases with high cost accidents, particularly at the lower per accident policy limit intervals beyond the standard limit interval. For the cost of accidents in one of these lower cost intervals is in large part due to the presence of a high cost case. If there were a decrease, as seems improbable, in the frequency of high cost individual cases with an increase in high cost accidents, the inequality in the above would be reversed.

14 14 POLICY LII~ITS IN CASUALTY INSURANCE APPENDIX II. PER AMOUNT LIMIT FACTOR The per amount limit factor is the index number which expresses the rate for coverage with a specific per amount limit for the risk in terms of the rate for coverage with standard per amount limits. Assuming the same pro rata expense loading for all per amount policy limits, this is equivalent to an index which expresses the losses for an aggregate of adequate experience when evaluated with a maximum limit of a given amount per policy period in terms of the losses for the same experience when evaluated with a standard amount limit per policy period. There are no data available directly showing the risk experience by policy amount limits for any line of insurance, from which the properties of per amount limit factors could be studied. It is proposed that this problem be approached by using unit experience developed under policies without limits. Two sources of such experiences are readily available. 1. Actual compensation experience for 80,259 policies for all industries from several states and a few policy years as developed by one carrier. 2. National Council on Compensation Insurance classification experience for policy years as compiled in This experience was used by considering each classification in each policy year as though it were an individual policy. Compensation insurance is more standardized than third party insurance and is less subject to extreme fluctuation for that reason. Results obtained from compensation data are not applicable to other lines without modification, nor can the results obtained from classification data be applied directly to risks. The object here is to study the per amount limit factor in relation to small and large units by using the available compensation experience. The changes observed in the effect of the per amount limit factor between the small and the large risk will apply in all probability to other lines of insurance though in different degrees. COMPENSATION EXPERIENCE OF ONE CARRIER A large volume of individual risk experience developed under compensation laws was divided into eleven groups according to size of premium. For each of these groups, pure premium ratios

15 POLICY LIMITS IN CASUALTY INSURANCE 15 were calculated for losses in excess of various selected loss ratios. The procedure was similar to that described in a previous paper in the Proceedings, Volume XIII, pp The pure premium ratios thus calculated, the average premium, and the number of risks for each group are shown in Table I. For each group of risks in Table I, the pure premium ratios for losses in excess of selected loss ratios per risk were plotted against the selected loss ratios as shown in Chart I. It was found expedient to use semi-logarithmic 4-cycle cross section paper in order to place proper stress on the portion of the field desired (i.e., ,0003) within the limits of one sheet. This also has the advantage of larger angles of intersection between the curves and the horizontal lines toward the lower part of the Chart representing the smaller pure premium ratios. The object of Chart I is to show the pure premium ratio corresponding to each selected risk loss ratio for each risk group. The relation between pure premium ratios and selected loss ratios was used in Chart III-A to establish a relation between sizes of risks and per amount limits for risks. For various specific loss ratios corresponding per amount limits were determined from Chart I by applying the specific loss ratios to the average risk premiums; for the losses in excess of such amount limits for each risk would be the same as the losses in excess of the corresponding specific risk loss ratios. The per amount limits as determined were plotted against the corresponding risk premiums for the various pure premium ratios. It will be noted from Chart III-A that, allowing for fluctuations, particularly for the low pure premium ratios which are least reliable, the results appear to lie approximately on straight lines or on long curves. NATIONAL COUNCIL ON COMPENSATION EXPERIENCE The individual risk data of the carrier were supplemented by using the classification data of the National Council on Compensation Insurance for policy years as compiled in The procedure followed in preparing this experience was substantially as given in these items: 1. The experience of one classification for one year was selected as the unit of experience and the expected losses of the unit taken as the measure of size. The whole range of

16 16 POLICY LIMITS IN CASUALTY INSURANCE expected losses 0-$9,000,000 was divided into 56 size groups numbered consecutively. The interval included in each size group was made larger than the interval of the group next below in size according t6 a plan of maintaining roughly the same relative interval throughout, i.e., a constant ratio between the lower and upper limit in each size group. 2. For every classification each policy year payroll was multiplied by the average pure premium for the five year period to determine annual expected losses. Each policy year of the classification was then assigned a size group number from 1 to 56 so that the expected losses fell within the group interval. 3. Each policy year pure premium was divided by the average pure premium for the five year period. The quotient was taken as the index for the given policy year in terms of the average for the period. 4. A rectangular table or cell diagram was formed showing at the top of the columns indices (item 3 above) with.10 intervals and at the left of the lines the size groups (item 1 above). The unit experiences were then entered in the table by placing a dot in the proper rectangle or cell for each, according to the index and size. The result was a scatter diagram in which the dots appeared in a V-shaped form with the apex in the.90 index column and on the last line in the largest size group. The dots in each cell were counted and recorded. 5. The fifty-six consecutive size groups in this table were combined into eight groups except that the lowest size groups were discarded, largely because they contained many cases of exposure not extending over the five year period. 6. Frequency distributions were formed for each of the eight new groups showing the number of units of experience falling into each of the index intervals at the top of the table described in item The frequency distributions (item 6) were graduated by the method given in the Proceedings, Volume VI, page 52, Carver, and their number was extended to 10,000 cases for each group. In the ungraduated distributions there were so few cases that chance fluctuations controlled the extremes. It was the intention, in raising the graduated distributions to 10,000 cases, to extend the distributions so as to include some cases with higher indices, which is equivalent to predicating ratios beyond the actual experience. 8. Each of these new distributions of 10,000 cases was then used to calculate ratios of the losses of the group in excess

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