UC-NRLF \s:.: -a-- ' -f > ,^,/ :»j-VA:;:' $B b3b 711 GIFT OF ENGINEERING LlBfiARY REPORT OF THE CO-INSURANCE COMMITTEE TO THE BOARD OF FIRE UNDERWRITERS OF THE PACIFIC ON PERCENTAGE CO-INSURANCE AND THE REI.ATIVE RATES CHARGEABLE THEREFOR ALSO ON THE COST OP CONFLAGRATION HAZARD OF LARGE CITIES San Francisco, California September, igos. REPORT OF THE CO-INSURANCE COMMITTEE TO THE BOARD OF FIRE UNDERWRITERS OF THE PACIFIC ON PERCENTAGE CO-INSURANCE AND THE REI.ATIVE RATES CHARGEABLE THEREFOR ALSO ON THE COST OF CONFLAGRATION HAZARD OF LARGE CITIES San Francisco, California September, 1905. S3B4 GIFT or ENGINEERING LIBRARY 3JC. ) J ) ) , J • 1 3 • , ' 5 ' . ) ''>,»'>!)' , ■ . »•> ^. »»", •> *» •^ "o' To ^Tie Executive Committee, Board of Fire Underwriters of the Pacific : Gentlemen : Your sub-Committee on the subject of Co-insurance being adopted for San Francisco beg to report : After careful consideration of what has been written on the subject and all the data obtainable, they are of the opinion that up to the present time all the different rate reductions for co-insurance are arbitrary to a greater or less extent, and that until other statistics are utilized than heretofore, they will so continue. As to cities and towns where sufficiently reliable data are obtainable as to losses and values, there is no reason why the value of the percentage co-insurance clause should not be ascertained within a small degree of exact- ness. Those who have given or who will give careful con- sideration to the question will find that nothing approach- ing facts regarding value of co-insurance can be learned without loss to value of property — that is the foundation stone; it has been claimed by some that loss to insurance is good enough, especially as any error on such a basis of calculation would result in favor of the insurance companies, but it is not wise to let non-co-operating com- panies have so wide a margin in their favor, and as the amount of insurance carried is a movable quantity at the will of the assured, it will readily be seen that facts can- not be learned therefrom; advocates of this plan argue that values of property as learned from proofs of loss are not altogether reliable; that as to individual losses is occasionally likely to be the case, but in a large number of losses under- and over-statements of values will adjust themselves very closely, and after everything that may be said against the reliability of such values of prop- erty, no better method of obtaining such values has been suggested. Your Committee are ambitious enough to wish they could settle the value of co-insurance over the entire United States, but their opportunities of getting sta- tistical information are too limited, having regard to the area under their jurisdiction, but they are not with- out strong hope that this report may pave the way for Eastern organizations to co-operate on similar lines and compile the data gathered so as to make an average that could be applied over an important part of the United States. One table would not answer because in the opin- ion of your Committee the value of co-insurance fluc- tuates according to construction, climatic conditions, water supply, fire department, etc. It would, therefore, be necessary to divide cities and towns into three or more groups and have loss to value and other data secured as to a sufficient number of each group to form a fair average of the whole of such group. All towns of inferior construction, those with inadequate water supply or fire department, should be thrown out, co-insur- ance being of little value in such towns, and it is of no value in the isolated risk unless under superior fire protection or sprinkler equipment. The question of mandatory co-insurance, eighty per cent co-insurance only, or leaving the percentage open and fixing rate accordingly was considered and having regard to the objection of some people to compulsory percentage of co-insurance and the enactment of anti- co-insurance laws in some States, it was deemed expedi- ent to recommend the open percentage method in order that the assured can select for himself what he wishes to purchase and companies can charge accordingly. The only available statistics of loss to value obtain- able by this Committee are those relating to San Fran- cisco fires contained in proofs of loss in various offices. Of course, similar figures could be obtained as to a number of other cities on the Coast, but in the opinion of the Committee such other cities would belong to a different class to that in which San Francisco should be placed. It was decided to employ Mr. Albert W. Whit- ney, Professor of Mathematics of the University of California, and almost without exception all the leading companies gave him the opportunity of examining their San Francisco proofs of loss for the five years 1899 to 1903, from which the record of 5642 fires was tabulated. From these facts under the instructions of the Committee, Mr. Whitney made certain classifications and calcula- tions as shown in his report attached hereto. Brick special hazards were not taken into account, as the number of important fires is too limited to be of material value. Brick dwellings are not separately dealt with as there are not enough of them in San Fran- cisco to form a class by themselves. We are aware that the Universal Mercantile Schedule adopts fifty per cent co-insurance as the basis from which to increase or reduce rates, but as there does not appear to be any evidence that tabulation of facts demonstrated that fifty per cent was the average insur- ance carried on which companies have been making their experience without co-insurance, the Committee can see no reason for recommending any figure as a base line for percentage co-insurance rates, other than the actual experience as to average insurance to value heretofore carried, and we are not experimenting in doing so be- cause the same results as to profit and loss heretofore obtained, or a little better, can be looked for in the future, provided we adhere to past experience as to per cent of insurance carried. The relative or percentage co-insurance rate for San Francisco based on the five years' experience shown in Table 17 of Mr. Whitney's report should produce the same underwriting results as in past years, with a prob- able improvement in years of general business depression (for reasons hereafter stated) by using present rates as a basis to calculate the percentage co-insurance rates. An important point to be considered is that the five years' losses in San Francisco on which results are fig- ured do not contain any conflagration losses, so that the question of a loading being necessary should be con- sidered. As stated heretofore, our opinion is that the value of co-insurance has to be grouped as to three or more classes of cities and towns ; therefore, conflagration hazards must be similarly dealt with, and taking (for the purpose of illustrating) as Class 1, cities showing over 200,000 population, as per 1900 census, we find nine- teen such cities, and the census returns of those cities for the last six decades are 1,591,588 for 1850, 2,926,732 for 1860, 4,159,425 for 1870, 5,586,268 for 1880, 7,858,595 for 1890, and 11,795,809 for 1900. This would make a mean average annual population of 5,444,943. The premium income (for 1900) of eleven of these cities (having a population in 1900 of 8,086,649) sufficiently distinctive and remote from one another to form a good average of the whole, shows the average annual premium per capita to be $3.77. Of course, the average rate has varied during the different periods, but not sufficiently to materially affect final results. An arbitrary period had to be taken for figuring con- flagration losses and it was decided to take a period of fifty years. In these nineteen cities during the past fifty years we have had serious conflagrations in Chicago, Boston and Baltimore, causing an insurance loss of $180,116,620. The fifty years' premium income based on $3.77, the 1900 per capita, being $1,026,371,755, and the fifty years' conflagration loss to insurance com- panies on such premium income being $180,116,620, we find 17.55 per cent of premium is the cost of conflagration losses in cities at the present time having a population of over 200,000, and dividing the average annual con- flagration loss by the average annual population, we find the conflagration loss to be $0.66 (sixty-six cents) annually per capita. The Committee realize that the question of conflagra- tion hazard was not referred to them, but it is so closely allied to the co-insurance question that attention had to be called to it in this report. Before deciding, however, that any loading is needed to present rates for conflagration hazard, it would be necessary to ascertain the insurance losses in the nine- teen cities referred to. The losses, less the conflagra- tions named, for the past five years or any other period, should be secured and the average annual per capita loss ascertained, add the annual conflagration loss of $0.66 per capita, then compare the result with the average annual per capita premium paid for the corresponding years, and it would then be shown whether rates needed grading up or down, or whether present rates are fair to all interests involved. This method of calculation could be applied to any one city or group of cities. Another point to be considered is that with reduced rates on account of co-insurance, the public would be apt, especially in the better class of risks, to carry a larger amount of insurance, and that under such conditions a larger percentage of insurance to value would be in- volved in conflagration losses. The figures as to conflagration losses and premiums in large cities were obtained through the courtesy of General Agent Miller of the National Board of Fire Underwriters. Conflagrations at Seattle, Spokane, Jacksonville, Waterbury, Paterson, Eochester, etc., are not taken into account, as they would figure with the premium income and population in such group that they might be placed. Your Committee are of the opinion that the public having the option of purchasing any percentage of in- 8 surance to value tliey wished would be disarmed from making any valid objection to co-insurance, and the companies would be protected from adverse selection being made against them by having to carry a high per- centage of value on poor risks and a small percentage on good risks. With the percentage co-insurance clause and rates adjusted accordingly, the business of fire insurance will take care of itself much more evenly than at present. It is well knoAvn that in years of general depres- sion in business the loss ratio to premium income increases as a rule; is not that accounted for by the fact that property owners studying economy often reduce the amount of insurance carried, and the adverse expe- rience of insurance companies in such years can with little doubt (if any), in the Committee's opinion, be attributed to the falling off in contributing insurance to partial losses. In other words, the adverse selection by the insured reducing the average amount of insurance carried to value increases loss ratio and turns profit into loss. This is a much more reasonable solution of the unprofitable results to insurance companies during years of business depression than the easier and oft- reiterated cry of increased moral hazard. Eepeating our hope that this report may pave the way for Eastern associations to take this subject up on a similar or better basis (if ascertainable) and reduce to a science the subject of percentage co-insurance and rela- tive rates to be charged therefor. Eespectfully submitted, C. F. MULLINS, Chairman. Aethur M. Brown, r. w. osborn, B. J. Smith, V. Carus Driffield, Committee. Mr. C. F, Mullins, Chairman of the Co-insurance Committee of the Board of Fire Underwriters of the Pacific, Dear Sir:— I have the honor of presenting herewith, as requested by your committee, a report based on fire statistics of the city of San Francisco for the years 1899- 1903, inclusive. The primary object of this investigation has been the determination for a number of classes of the relative rates for co-insurance, the secondary object has been the putting on record of the elements involved in the scien- tific determination of fire insurance rates and of a plan for the calculations necessary thereto. I am. Yours very truly, Albert W. Whitney. Berkeley, California, August, 1905. 11 I. — Introduction. A determination of the rates for co-insurance involves, first, a thorough analysis of the structure of a rate, sec- ond, the gathering of the necessary statistics and, third, the application of the theory to the facts. I say, first, a thorough analysis of the rate. As a mat- ter of fact, co-insurance rates are nothing but the real rates guarded by the co-insurance clause against adverse selection, namely, the selection by the insured of less in- surance than the rate was designed for. In order to understand what, for lack of a better name, I have called the real rates, it is necessary to have clearly in mind the elements that make the fire insurance prob- lem; but in order to understand the fire insurance prob- lem, I propose, for the sake of the added clearness that comes from comparison, to measure it against the life in- surance problem whose elements are more easily grasped. If a man wished to buy an insurance of $1,000 for his whole life and if the element of interest were to be neg- lected the price of the insurance would be $1,000, for he is sure, sooner or later, to die. The time of his death is immaterial when the element of. interest is left out of account, but when this is admitted the time of his death is a matter of considerable importance. If it were known positively that he would live exactly twenty years and if an interest rate of five per cent prevailed the insurance could be sold to him for the present value of $1,000 twenty years hence, or about $377. But there would be no reason for calling this insurance ; it would be nothing more than an ordinary banking transaction. But as a matter of fact the time of his death is uncer- tain, and it is this element of uncertainty that brings the transaction into the field of insurance. Yet all must not be uncertainty or there will be no basis for an agreement. In reality we may assume that we know four things ; first, 12 a safe rate of interest to count upon ; second, the age of the applicant ; third, that he is sound physically and has a good environment ; fourth, that a large number of men of his age and of an average physical condition and en- vironment who have in the past been under observation have experienced a certain recorded mortality. We may treat this man then as though he were one of such a group. We may thereupon compute, taking account of interest, the present value of the sum needed to meet the death claims among this group as year by year they ma- ture ; this sum assessed equally among the insured is the net single premium. As a matter of fact life insurance is usually paid for in yearly installments, but as there is no analogue to this in fire insurance practice, we need not follow it out. I have supposed the insurance to be for the whole of life; this eliminates the question of whether or not the claim will mature, and makes it a question only of when it will mature. This fact that death is sure to occur, but that a loss by fire is not sure to occur, has been asserted to mark a vital distinction between these two types of insurance. This is wrong, however, for the certainty of death does not set any characteristic mark upon life insurance, and, as a matter of fact, term insurance in one form or another is a large part of the business of a life insurance company. Term insurance, however, yields a problem even if in- terest is neglected. If a man of 35 wishes to insure his life for $1,000 during only the next five years, we may go to the mortality table and consider the 81,822 persons alive at the age of 35 ; we shall find that of this number 3,716 die during the next five years. If each of this group of persons were insured, the death claims would amount to $3,716,000. Neglecting interest and assessing this equally among the 81,822 persons insured would give a net single premium of about $45. If interest is taken 13 into account the problem is essentially the same as the problem of whole-life insurance that we have already discussed. We have assumed that we are dealing with persons of a fairly definite type and standard of civilization, for instance, white persons living in the northern part of the United States. If we wish to transact an insurance busi- ness among, for instance, the natives of India, the prob- lem will be the same except that we shall find a mortality experience peculiar to the class. This is the form of a mortality table:— Table 1.- -The American Experience Table X dx 10 749 11 74G 12 743 93 58 94 18 95 3 Total, 100,000 The column headed x refers to the number of completed years, the column headed d^ refers to the number dying during the following year. The sum of the numbers in the second column is 100,000; that is, 100,000 persons began the 11th year together ; of these, the table says, 749 died during that year, 746 during the 12th year, and so on. Among the natives of India we should obtain a different set of numbers. We may now see clearly the elements that go to make up a life insurance rate ; they are two, the law of mortal- 14 ity and the rate of interest. The rate then will depend, first, upon the law of mortality, which will vary with the class, second, upon the age of the insured at the time at which the insurance is effected, third, upon the rate of interest, and, fourth, of course, upon the particular form of insurance desired. Now let us make a corresponding analysis of fire insur- ance conditions. In the first place fire insurance is written for such short terms that the element of interest enters in such a simple way as to be negligible in making the rate. We therefore sweep away at once what is the main factor in the structure of a life insurance rate. The mortality among lives changes with the age. This is not true, however, among fire risks to any great extent, that is, the age of a building is not a very important ele- ment in fixing the rate, at least it need not and indeed cannot be taken account of in any such systematic way as in life insurance. This, together with the fact that fire insurance is written for such short terms, eliminates this element from the problem. We see, therefore, that the two factors of life insur- ance rating, the rate of interest and the mortality in terms of age, are almost lacking in the fire insurance problem, at least they do not require a systematic treatment. What then are the elements of the fire insurance problem? In the first place and most important of all, the element of class. Just as the rate for a man with an hereditary tendency to an organic disease or for a stoker or for an inhabitant of a tropical country should be higher than the rate for a healthy life in a healthful environment, so the rate for a saw-mill or a frame build- ing without fire protection should be higher than for a fire-proof office building. In fire insurance the class is more important than any other element in making the rate, while in life insurance it has been, until recently, 15 of very little importance, for almost all life insurance has been effected in a single class, that of standard lives of white persons in non-tropical latitudes. We now come, however, to the element that really dif- ferentiates fire insurance from life insurance. In life insurance there is no such thing as partial loss (the analogy, if any, may be sought in accident insurance) ; when a man dies the full face-value of the policy is drawn upon. But when a building or a stock of goods burns, it seldom burns completely; the amount of dam- age is an exceedingly important factor in the problem, for it determines what part of the face of the policy will have to be paid. This is the analogue to the element of mortality among lives, but instead of the time ele- ment we have the quantity element. In life insurance the mortality question is when, in fire insurance it is how much. The mortality table for life insurance gives how many in terms of when, the mortality table for fire insurance gives how many in terms of how much. This for instance is the mortality table for frame business buildings in San Francisco: * Table 2.— Table of Paktial Loss fob the Class of Frame Business Buildings. X m^ 8293 1 576 2 326 3 215 4 139 5 97 6 69 7 49 8 42 9 194 Total 10,000 * It is not to be inferred that this table gives the actual number of fires that have occurred in some particular time ; the numbers are only relative, 10,000 having been chosen for convenience. 16 The column headed x refers to the number of tenths of sound value next lower than the amount destroyed, and the column headed m^, refers to the number of risks, among 10,000 losses altogether, that experience this particular range of loss. That is, out of 10,000 build- ings damaged by fire 8,293 may be expected to suffer a loss of less than 1/10 of the value, 576 a loss of more than 1/10 and less than 2/10 of the value, and so on. The analogy of this to a mortality table in life insurance is, I think, obvious. Now just as in life insurance there are different mor- tality tables for different classes, so in fire insurance there are different mortality tables for different classes (see Table 12). Inside of a single class then the ele- ments of the life insurance problem are the rate of inter- est and the law of mortality and the problem to be solved is: with a given rate of interest and at a given age, what is the rate for an insurance? Inside of a single class the element of the fire insur- ance problem is the law of mortality, or as I shall hence- forth call it, the law of partial loss, and the problem to be solved is: with a given ratio of insurance to value, what is the rate for an insurance t As a matter of fact when I say that this is the prob- lem of fire insurance I am not stating the problem of finding the ordinary rate, but the co-insurance rate. The ordinary rate takes no account of the ratio of insurance to value, that is, a man pays the same rate for $5,000 of insurance whether it is written on a building worth $5,500 or on one worth $10,000. But manifestly the ex- pected loss to the company is much greater in the second case than in the first; in the second case it will take only a 50 per cent loss to exhaust the insurance, in the first case it will take more than a 90 per cent loss to exhaust it. Now in the case of frame business buildings, according to our table, there are 451 chances of a 50 per cent loss to 194 chances of a 90 per cent loss. 17 The co-insurance rates are special averages, the ordinary rate is a general average in which the man who insures for a small amount, that is, a small ratio of insurance to value, gets his insurance too cheaply, lie who insures for a large amount pays for his insurance too dearly. The analogous rate in life insurance would be obtained by neglecting the element of age; then a young man would pay too much for his insurance in order that an old man might pay too little. Such a rate would be ob- tained by dividing the total amount of death claims by the total amount of insurance in force, just as in fire insurance the rate is actually obtained by dividing the total insurance loss by the insurance in force. The weakness of a system of this kind in life insur- ance is this, that it leads to adverse selection; young men will not insure, therefore the mortality increases, the rate increases, the svstem is unstable. Hence life companies are forced to take account of the element of age ; witness the experience of friendly societies. We might expect adverse selection in fire insurance; it would consist in a refusal to carry much insurance, that is, a large ratio of insurance to value. That it does not, as a matter of fact, operate to a greater extent is due to several causes, one of which is that the insurance is often needed at any reasonable price, especially as collateral security for loans. However, just as it is manifestly fairer and better that a man of age 25 should be rated according to the hazard at his age rather than be forced to help make up the deficit caused by a too small rate for a man of 55, so it is manifestly fairer and better that a man who wishes to insure for 90 per cent of the value of his prop- erty should be given a rate to meet the hazard rather than be forced to help make up the deficit caused by the under-rating of a man who carries 30 per cent of insur- ance to value. 18 I hardly need discuss the practical difficulties in the way of this. One of them arises from the expense of ascertaining sound values. For the company to deter- mine sound values accurately at the time of effecting the insurance would be practically out of the question, and in the case of stocks of merchandise which are con- tinually changing would be ineffective. The co-insurance clause is an agreement on the part of the insured to maintain a specified ratio of insurance to value. He will maintain this in insurance companies presumably, but in case he fails so to do he shall by the agreement be regarded as himself a co-insurer for the balance. He thereby becomes jointly responsible with the other insurers, each for his share of the loss. This agreement places upon the insured the responsi- bility for the ascertaining of sound values and with entire fairness for he should have a sufficiently accurate knowledge of his own affairs to obtain this information easily and to order his business with this agreement in mind. To fix the proper rate for a fire insurance it is as necessary to have this information as to ratio of insur- ance to value as to know the age of an applicant for life insurance. The responsibility for stating his age correctly is placed upon the insured with a penalty if he fails to do so; the co-insurance clause puts upon the insured the responsibility for keeping the ratio of insurance to value at a specified figure with a penalty if he fails to do so of having to act himself as insurer for the balance. Or again: when a man buys $8,000 worth .of insur- ance and pays for it at an 80 per cent co-insurance rate it is equivalent to the admission that his insurance protects just $10,000 worth of property (of course only partially protects, namely, up to 80 per cent of its value). The fact that he buys protection for just this 19 amount of property is an integral part of the transac- tion. If when a loss occurs the sound value of the property is greater than $10,000 it is quite obvious that the excess value is in this sense unprotected, or, if you like, is insured by himself for 80 per cent of its value. In case the damage is less than 80 per cent the insured bears of the loss only his part as a co-insurer; if the loss is greater than 80 per cent he loses not only as a co-insurer, but also because he has bought only incom- plete protection. The rates for co-insurance then are nothing but rates that take into account ratio of insurance to value. The necessary and sufficient data for their determination for a particular class are contained in what I have called the table of partial loss. . The mathematical statement of the method actually used in computing these rates will be reserved till later, for it is not necessary to a general understanding of the subject. I shall defer also the explanation of the method of obtaining this table of partial loss from the office statistics. II.— The Co-Insurance Problem Treated Arith- metically. As an example, let us consider the 60 per cent co- insurance* rate on frame business buildings. The table of partial loss has already been given (Table 2). As a rough approximation we might call the average loss under 10 per cent, 5 per cent, the average loss between 10 and 20 per cent, 15 per cent, and so on, but as a matter of fact it is worth while to examine our statistics closely enough to determine more accurate averages. The results are as follows: * Whenever rale is referred to in this report, net rate or fire cost is meant. The ofl&ce rate is obtained from this simply by loading. 20 Table 3.— Average Percentage of Property Loss to Sound Value. 1.8 per cent, among- losses between per cent. and 10 per c 14.2 '' " '' '' 10 '^ 20 " 24.5 '' '' " '' 20 CC 3Q .. 34.7 '' '' '' " 30 '' 40 " 44.8 '' '' '' '' 40 '' 50 '' 54.9 ^' " " " 50 '' 60 '' 65 " " '' " 60 '' 70 '' 75 '' '' '^ '' 70 '' 80 '' 85 '' '' " " 80 '' 90 '' 99.5 " '' '' '' 90 '' 100 '' cent. Let us now find the insurance loss on 10,000 claims, supposing that the sound value of each risk is $100 and that on each risk an insurance of 60 per cent of the value, that is, $60, is carried. But before we do this let us find the property loss. This will evidently be made up as fol- lows: Table 4. — The Property Loss; Sound Value of Each EiSK $100; 10,000 Claims. 8293 losses of $ 1.80 each .$14,927 40 8,179 20 7,987 00 7,460 50 6,227 20 5,325 30 4,485 00 3,675 00 3,570 00 19,303 00 576 14.20 '' 326 24.50 '' 215 34.70 ^' 139 44.80 '' 97 54.90 '' 69 65.00 '' 49 75.00 '' 42 85.00 '' 194 99.50 '' Entire property loss $81,139 60 Now the insurance loss ; since there is an insurance of $60 on each risk, any loss under $60 will be paid in full, but for losses over $60 only $60 on each. The in- surance loss will, therefore, be: 21 Table 5.— The Insurance Loss; Sound Value of Each Risk $100 ; Insurance $60; 10,000 Claims. 8293 losses of $ 1.80 each $14,927 40 8,179 20 7,987 00 7,460 50 6,227 20 5,325 30 4,140 00 2,940 00 2,520 00 11,640 00 576 14.20 '' 326 24.50 '' 215 34.70 '^ 139 44.80 '' 97 54.90 '' 69 60.00 '' 49 60.00 '' 42 60.00 " 194 60.00 " Insurance loss $71,346 60 Now if we divide this insurance loss by the number of risks of this kind among which these 10,000 losses have occurred, we shall obtain the average insurance loss per risk. This will be the average insurance loss per risk for $60 of insurance ; the average insurance loss per risk per dollar of insurance will be had by dividing this by 60. This will be then the net rate or fire cost per dollar of insurance when the insurance carried is 60 per cent of the sound value. But as a matter of fact the num- ber of risks upon which these 10,000 losses have occurred was not obtainable from the statistics at hand and therefore it has been impossible to determine the actual rate. But relative rates are easily enough determined; for just as the insurance loss has been determined for a ratio of insurance to value of 60 per cent, so the insur- ance loss may be determined for any percentage of insurance to value. The results are as follows: 22 Table 6.— Insurance Losses for Various Eatios of Insurance to Value; Sound Value of Each EiSK $100; 10,000 Claims. ratio op insurance to value. 10 per 20 30 40 50 60 70 80 90 100 cent, THE insurance LOSls. ,$31,997 40 , 45,726 60 , 55,243 60 62,154 10 , 67,331 30 , 71,346 60 , 74,541 60 , 77,146 60 79,296 60 , 81,139 60 With full insurance the insurance loss is equal to the property loss as it should be. Now just as we proposed to obtain the 60 per cent rate by dividing first by the number of risks and then by 60 so we might propose to obtain the 10 per cent rate by dividing by the number of risks and then by 10, and so on. Now since the unknown element, the number of risks, enters into all the rates in the same way, in forming the relative rates it may be neglected and we may divide the insurance losses in Table 6 simply by the corresponding amounts of insurance carried. This gives: 23 Table 7.— Relative Rates for Various Ratios of Insur- ance TO Value. ratio of insurance to value. 10 per 20 30 40 50 60 70 80 90 100 cent, relative rates. ,3200 ,2286 .1841 .1554 .1347 .1189 .1065 . 964 . 881 . 811 From Table 6 we have insurance losses : by subtraction of successive Table 8.— Cost to the Company of Carrying Successive Tenths of Insurance to Value ; Sound Value OF Each Risk $100; 10,000 Claims. The 1st tenth $31,997 40 2d 3d 4th 5th 6th 7th 8th 9th 10th 13,729 20 9,517 00 6,910 50 5,177 20 4,015 30 3,195 00 2,605 00 2,150 00 1,843 00 It is hardly necesary to point out how severely the first few tenths of insurance carried tax the company in comparison with the later tenths. The cost to the company of carrying an insurance of 20 per cent of the value is more than half the cost of carrying full insur- ance (see Table 6). The rate for 10 per cent insurance is four times the rate for full insurance (see Table 7). 24 The question that now comes up to be answered is this: what relation has the ordinary rate to the co- insurance rates'? This cannot be answered without additional statistical information. The information that we need is this: how much insurance do people buy in general? If on the average they insure their buildings for 20 to 30 per cent of the value the rate will be high, if on the other hand they insure well up, on the aver- age say for 80 per cent of the value, the rate will be low. Roughly we might say that the ordinary rate will be the same as the co-insurance rate for the average ratio of insurance to value. This, however, in general will not be exactly true and we have at hand data that will allow us to make a closer determination. The same statistics that yield the table of partial loss yield also data regard- ing amount of insurance carried. Out of 127 risks under observation in the class of frame business buildings under discussion, two carried between 20 and 30 per cent of insurance, five carried between 30 and 40 per cent, or in tabular form : * Table 9. — The Distribution of Risks as Regards Ratio OF Insurance to Value. ratio of insurance to value. Between per cent and 10 per cent 10 '* '' 20 '' 20 30 40 50 60 70 80 Over 90 per cent 20 30 40 50 60 70 80 90 Total 127 number of risks. 2 5 14 14 29 21 22 20 * I shall refer to this as the table or law of insurance to value. 25 The average ratio of insurance to value taken from the actual figures was 70.77 per cent. If we were to divide each of the insurance losses in Table 6 by the number of risks we should obtain the average insurance losses per risk, but as I have said we have no information as to the number of risks. How- ever, if we divide each of these losses by the number of risks upon which loss occurred, that is 10,000, we should obtain the average insurance loss per claim. For instance the average insurance loss per claim when 60 per cent of insurance to value is carried is $7.13466; when 70 per cent is carried is $7.45416, and for risks that range in ratio of insurance to value from 60 to 70 per cent as do the 29 in Table 9 we may with sufficient accuracy take for the average insurance loss per claim the average of the 60 and the 70 per cent values or $7,294 and call this the average insurance loss per claim when 65 per cent of insurance to value is carried, and so for the other intervals. These intermediate values of the average insurance loss per claim are : Table 10.— The Average Insurance Loss per Claim for Various Intermediate Values of the Eatio op Insurance to Sound Value. ratio of insurance to VALUE. 5 per cent, 15 25 35 45 55 65 75 85 95 a a i i i i n AVERAGE INSURANCE LOSS PER CLAIM. $1,600 3.886 5.048 5.869 6.474 6.934 7.295 7.584 7.822 8.022 26 Now then, assuming that the experience in Table 9 may be taken to be typical, 2/127 of 10,000 claims, on which the insurance averages 25 per cent of the value, will have an average insurance loss per claim of $5,048, or altogether $795.00 ; 5/127 of 10,000 claims, on which the insurance averages 35 per cent of the value, will have an average insurance loss per claim of $5,869, or altogether $2311.00 ; and so on, or in tabular form : Table 11. — The Actual Insurance Loss on 10,000 Claims; Sound Value of Each Risk $100; Amount OF Insurance Carried as in Table 9. AVERAGE ratio OF INSURANCE TO VALUE. 5 15 25 35 45 55 65 75 85 95 per cent, actual insurance loss. ,$ 795 00 . 2,311 00 . 7,137 00 . 7,644 00 . 16,656 00 12,540 00 13,550 00 . 12,633 00 Actual insurance loss on 10,000 claims $73,266 00 Dividing this by 10,000 we obtain $7.3266 as the actual average insurance loss per claim; I say actual since this is based upon figures as to insurance actually sold. $7.3266 then is the actual average insurance loss per claim, but there is an average insurance per claim of 70.77 per cent of the value, or 70.77 dollars. The rate then per dollar of insurance actually in force will be $7.3266 divided by 70.77, or .1035. This is the rate per dollar of insurance actually in force on risks that have become claims. If this were multiplied by the ratio of the number of claims to the number of risks it would 27 be the burning ratio or net rate. The corresponding rate (see Table 7) with 70 per cent co-insurrance is .1065, with 80 per cent co-insurance is .0964; this actual rate of .1035 lies between these two and by interpolation it is found to agree with the rate for 73 per cent co-insurance. The corresponding problem in life insurance would be to determine the age for which the rate would be the same as the rate irrespective of age got by dividing the total amount of death claims by the total amount of insurance in force. If then the rate in use has been properly obtained, that is, if it is the true rate, then if it is taken for the 73 per cent co-insurance rate and the other co-insurance rates are taken in proper ratios to this as determined by Table 7, the business will produce with these rates an amount sufficient to meet the expected loss just as well as with the old single rate; in fact even if the old rate is not the true rate but if it is, for instance, too large, the business conducted with the co-insurance rates will continue to produce the same income as with the old single rate, provided the law of insurance to value remains the same. A very important thing to observe is that the co-insur- ance rates are entirely independent of the law of insur- ance to value; that was introduced only when we came to a consideration of the ordinary rate. As a matter of fact the tendency would be with a change to co-insur- ance rates for the average amount of insurance carried to increase. The co-insurance rates, however, would still continue perfectly to produce the income requisite to meet the expected loss. The expected loss would now be larger but it would not increase as fast as the amount of insurance in force so that the ratio of the two or the ordinary rate would fall. As the profit is proportional to the expected loss the ratio of profit to insurance in force will be proportional to the ordinary rate; a 28 change then to co-insurance rates would be likely to be followed by an increase in the profit, but not as great an increase as that in the insurance in force. While then the co-insurance rates have the great advantage of being independent of the law of insurance to value so that in any case the business will take care of itself so long as the law of partial loss does not change, the ordinary rate on the other hand depends very decid- edly on the law of insurance to value so that with this rate the business would not take care of itself if there were a change in this law; for instance a drop in the average ratio of insurance to value from say 70 to 50 per cent would probably convert a profitable business into a losing business. Before proceeding to give tabular results for the sev- eral classes examined, I propose to state once more the elements involved in a rate and to suggest some terms. The basis on which the computation is made is what I have called the table of partial loss. This will vary from one class to another. The table lacks, however, one element of being competent to give us absolute rates; it tells us how the claims are distributed as to size, but it does not tell us what proportion of risks become claims. The table, however, is competent to give us relative rates and to give us the relation between the co-insur- ance rates and the ordinary rate. Let us suppose for a moment that we had this addi- tional information. Then we should be able to compute the insurance loss for any given percentage of insurance to value, not merely per claim, but per risk, and from this per dollar of insurance. This would be the net co-insur- ance rate, the fire cost or, I should like to call it, the meas- ure of the hazard. We should find on examination that this consists of two factors, one being the ratio of the number of claims to the number of risks ; this can be in- terpreted as the probability that a fire will occur; this I 29 propose to call the ignition hazard. The second factor is the probability that, a fire having started, the amount at stake, namely, the amount of insurance on the risk, will be lost to the company ; this I propose to call the damage hazard. The product of these two, the ignition hazard and the damage hazard, is the measure of the hazard or the fire cost for this particular ratio of insurance to value ; it is the probability first that a fire will occur, and, sec- ond, that having occurred, there will be a loss of the insurance.* The ignition hazard might be the same under some con- ditions on a stock of millinery in a brick building as on a stock of groceries, but the damage hazard in the case of the millinery would certainly be much larger than in the case of the groceries. The ignition hazard is independent of the ratio of in- surance to value, the damage hazard on the other hand depends upon this as well as upon the susceptibility to damage of the risk since it is the probability of a loss of the insurance. Just as we have analyzed the co-insurance rates so we may analyze the ordinary rate. It will be found to con- sist in the same way of two factors, the ignition hazard and the damage hazard. The ignition hazard, since it de- pends only upon the class, will have exactly the same value as in the co-insurance rates; the damage hazard, which is the probability that the amount at stake, namely, the average amount of insurance per risk, will be a loss to the company, will, however, differ from the damage hazards with co-insurance. * It should be noted that in reality the insurance loss per risk from which the fire cost is got is a very complex thing made up as it is of the sum of a number of partial losses, some of which do and some of which do not ex- haust the insurance It however reduces to the form of a single simple ex- pectation of losing the whole amount at stake, so that for instance in the case of 60 per cent co-insurance already discussed, the insurance loss per claim made up of separate items as in Table 5 (dividing by 10000) reduces to a single quantity 17.1346 which is the amount at stake, |60, multiplied by ,1189, the damage-hazard. The fire-cost comes by multiplying this by the ignition-hazard. 30 The ignition hazard involves the very element that our statistics fail to give and is therefore treated as unknown in this report. The damage hazards, on the other hand, we can compute completely. As the practical problem of relative rates for co-insurance involves only the damage hazards it is therefore entirely soluble. III.— Eesults foe Eight Classes in Tabular Form. It was originally intended to embrace in the investiga- tion ten classes, first, frame business buildings; second, contents of the same; third, brick business buildings; fourth, contents of the same; fifth, dwellings, (frame); sixth, contents of the same; seventh, frame special haz- ards; eighth, contents of the same; ninth, brick special hazards ; tenth, contents of the same. It was found, how- ever, upon examination of the statistics that they were in- sufficient in number in the last two classes to give reliable results. The results for the other classes are given here- with in tabular form:' 31 w. OS O P O M O m to o 1-^ PL, -c!t o oooooooooo i-ic^eO'^iocDC^oooJO 1—1 ggaaaaaaaa OOOOOOOOOO O ,-1 (M CO -^ iC CD t^ 00 Oi a u ^ : of Ivosses Total.. 1 32 CO w M ii's o « 5 ■a o u N * 5 "2K&. o sj n (Ll N rt a« ^ «.K£. « M a c -won a u P S '^ Q tn bo e »-< V s Q 5 SS= a .i^ i* a r.iM o a-- Cont Bri Busi Build , iJ so M ij s ■C "''0 "^ tn « if;, ■fj u a) 00 a c 1' s I'm-, s a-z: a ° t- 0) — o t=^^'3 u «m a,Sa J; D a a "^ -^ M — 'O fe s-s «K 05t>.»-iCD'-|i;0«Di-ICDC^ csoooocoost^t-cocoa: (MCO(MCDO-^I>-0>T-IC<1CC i-ItH(M(NC0O«DCDt-I — CO Tj^C^OCC^ICOiO'^COOl'— 1 »0'-iCC^OCDvOCO'<4^CD io" 00 CO (n" t-^ O CO lO" t^ oo" CO-^iOCDCDt^t^l^t^t' €©• ^ T-H "^ i>^ oi" i-T eO-^ii5CDCDt^t^l>l^00 =w= 1 a tn c a) 1 " ^ ■M Ml o oooooooooo tH^C^ CO "* »0 CO t- 00 05 O a o M* u a 2 s a M - 33 O O Pi • w H (in <) W w a) O H h-^ M r:» w !zi o ^ w O Q '^ N w W © O C5 ^ -* CO »o (M © 00 (.4 Ih C CO (M O o lO CO CO Hi J-Q CO -^ O a; N CG cs tH CO (M 00 lO (M © CO CO o :o iq iq '^ Hi Hi • • CO CO CO —1 <" ~ ^ ■H tH tH '^ Hi CO Hi tH iO y cd ^ 05 00 CO 00 t- tH go OO © Hi S S g 05 o lO r^ 00 CO CO tH © 00 »o iO ^. '^ CO • CO CO • • CO CO cq 5 1 ^ CO o tH tH 00 tH GO rH CO 5'"'-^ lO o 00 CO 'tH t- lO Hi CO 00 c°-Z o '^ 00 >o CO r-i © © 00 t- R ^ CO C.« «^ N^ ^^ S^ N^ ^^ *^ N^ C8 s^ y^ S^ S^ W U M cd e CS Q V J3 H 34 w P o H H O I— I o o H lO pq -si EH tH i5 ^ <«•—• ot^o:)THO CO OiOs^fM o o t^ t- • O rH tH ^ rH tH (M ^ >— ' T— ( r-l tH 1-H r-i Cont Spec Haz! (Fra rtV SI OC<^COCOTt^COOGOOiC Tf ^ 00*^. • ^ ^ F^ 5 rt S rH .— . >— 1 t^ 1—1 00 OS ^S2 rH rH CO t&K£l OtH^tHO»OC0Q01>.IC .TH00t:^<:005Q0 CC »o cq t^ '^. T-H rH CO ^ »0 (M Oq CO CO 05 at) "3 (M ^ CO tH 1-1 r-i <« a oc O t^ (M rH iH (M Q > > '~'~^ .^ (U • »H o ■ W) > be 0000000000°: 2 "So rir-tiHT— IrHrHTH-r-', tHt—I-^- rH(MC<0'^»i^Ot--00aso| 1^ -s »— » > ui"*"^"'''* ■"*^o- > ,^ 'O OOOOOOOOOoS: THrHrHTHiHTHrHrHiHTHS; OrHCqcO'^lOCOI^QO^'^; P l> to "S tn (u p (u a ■? S .a 1^ Vj «- .S <*-! to "+H ;2 o S o g 2 nj Ml- P i ^ &. i a ^ E —1 "^ ^ rt S « H iz; h ! c8 n t o > V c^ t^ O ^ CM (M CO iH lO tH !> • r— I be 02 > o 02 s «. ^ OC 0) g 02 02 ^ _ C^ r^ !^ — , "^ '^ —J ■^^-^ § s § S •p^ .y -^ g M ^ M .9 ^ SL 03 ^ ^ "^ ^r-" 02 o o c-i Qi O O rj 13 03 03 o3 o O O OJ 02 a; 35 CD PQ Contents of Special Hazards (Frame) CO <£> o (M CO CO • 00 CO Special Hazards (Frame) 05 O tH Oi • T-l CO 00 CO • tH Oi CO Contents of Dwellings CO tH 00 co' CO Oi CO O • CO oo' CO a) be a % Q o -^ CO CO ai o 00 o CO 00 Contents of Brick Business Buildings 00 tH CD CO Oi CO 00 tH Brick Business Buildings CO CO CO CO 1— 1 CO CO O Oi* CO Contents of Frame Business Buildings o (M 00 O zi W H H 1— 1 < Q w ^ H O Ph H C/J <1 S m < M O O K <1 ^ o 1— 1 ^ M W P3 o <1 U H P M 1— 1 t^ Ph tH O w l-J w «*J H i -S^E ' O ?^ S S o to o > be a C U u s 1-. 0)r-c Bl U «) to p a o H O p o tH (>1 kO ^DT^^fOC^^HOOaiCOt^ ^ iH tH tH tH iH tH o «o o t^OCOCMTHOOOiODOO OOOiOiGOCOCMOO CO Ol tH tH T— I tH tH CM Oi 00 00 Oi u^ 1— I 'rti »0 O CTi O CM »0 T— icMcx)»ococ\iooaioo COCvItHi— IrHrHrHiH lO CO CO (M lO Ti^ 0:1 C£) ^ (M C^ tH tH rH tH O O iH O Oi 00 O . CTi t>^ t>^ oi 00 L-^ iX> 05 O t^ T— I CM CM C^l -^ tH -^ o ct: O 01 »o t— 10 CO 0-1 O O Oi GO O VO CO ?0 CO CM O tH t^ -rfH CM T— I CO CM tH tH iH 1— i 10 t>; CM O O cm' CO O Oi 00 t^ -(-3 o u ft 0000000000 1— iCMCO^»OCOt^OOOiO 37 IV.— The Statistical Pkoblem. In the foregoing pages I have tried to show that the co- insurance problem may be solved when the laws of partial loss and of ratio of insurance to value are known, but as a matter of fact by far the most laborious part of this investigation has been the determination of these laws, even taking no account therein of the difficulty of getting the statistics. No data in the least adequate for such an investigation as this were immediately at hand and there was no possi- bility of obtaining them except from proofs of loss in the separate offices. A letter addressed to the Board offices by the Chairman of the Co-insurance Committee request- ing that access to proofs of loss for the years 1899-1903, inclusive, be given to myself or a representative, brought a favorable response from about ninety companies. The work of examining this material and collecting from it the necessary data was excellently done by Mr. A. H. Mowbray, now in the Actuarial Department of the New York Life Insurance Company; Mr. Mowbray was assisted for a time by Mr. Hart Greensfelder. The card system was used; each card represented a risk upon which a fire loss had occurred; on each card were places for the date and location of the fire and marks for avoiding and detecting the multiple recording of a risk which was insured in more than one company. The information really desired was the class to which the risk belonged, the sound value, the amount of insur- ance carried and the value of the property destroyed; there were places for these data on each card. If it had been possible to obtain information on every risk with regard to each of these four items the problem and its treatment would have been comparatively simple. Three of the items, the class, the property loss and the amount of insurance carried were indeed obtainable on practically every risk, but the fourth item, the sound value, in only about twenty per cent of the cases. 38 We have seen that the co-insurance problem arises en- tirely from the circumstance of varying ratios of loss to sound value. To have information as to the sound value is then absolutely essential for the solution of the problem and yet the situation was this, that in eighty per cent of the cases the sound value was not obtainable. This was so discouraging as to make it seem almost im- possible to succeed with the investigation. The ray of hope that came to one for an instant of being able to throw aside the eighty per cent of cases where sound >alue was not given and treat only the remaining cases disappeared completely as soon as one realized that the cases in which sound value is given are highly selective, they are, namely, in general, or at least largely, just those cases in which the loss has been relatively large, and they would therefore yield entirely misleading results. The question then is : how can we introduce the element of sound value into this eighty per cent of cases in which we have only property loss and amount of insurance given 1 Fortunately this is not quite so hopeless as it seems at first. If every risk were insured for just three-quarters of its value we should be able to infer the value from the amount of the insurance and therefore the ratio of loss to value, a fifty per cent ratio of loss to insurance, for ex- ample, would mean a 37i/^ per cent ratio of loss to value. Now it can indeed be shown that the average ratio of insurance to value is somewhere near 75 per cent and it is evident that a fairly good approximate result would be obtained if we assumed that 75 per cent of insurance (or whatever the exact value of the average might be) were carried on each risk, relying upon the risks carrying over 75 per cent to offset those carrying less than 75 per cent. But as a matter of fact this offsetting would by no means be perfectly effective. Suppose, for instance, the ratio of insurance to value in one case to be 120 per cent (as might 39 easily happen on stocks of goods) and in another case to be 30 per cent, an average of 75 per cent, and suppose that the ratio of loss to insurance has been in each case 5/6 ; then, as a matter of fact, on one the ratio of loss to value would be 120 per cent of 5/6, that is a total loss, in the other case a loss of only 30 per cent of 5/6, or one- fourth of the value. Now if the average ratio of insur- ance to value, 75 per cent, were used we should obtain two losses of 75 per cent of 5/6, or 62i/^ per cent of the value. But from the point of view of co-insurance two losses of 621/2 per cent of the value are very different from one total loss and another loss of 25 per cent. This is a case where, in forming an average, we lose the very facts that we need, that is this is a problem that needs something finer than an ordinary average. It is to be said, however, that 75 per cent (or whatever it may more exactly be) is not only the average ratio of insurance to value but, as a matter of fact, the greater part of all risks are written for nearly this ratio and for these the use of the average ratio instead of the actual ratios would lead to sufficiently accurate conclusions. The cases in which we should be led into error are, how- ever, of considerable importance. We should lose for one thing nearly all cases of total loss, a fully insured total loss, for instance, would count only as a 75 per cent loss. There is, however, again this to be said that in nearly all cases where there have been relatively large losses the sound values are given and we should, therefore, not be driven to the expedients that we are discussing. However, the point is simply this, that as a matter of fact there is a better method of procedure available than that involved in assuming a uniform 75 per cent ratio of insurance to value and that the importance of the prob- lem demands its use. I will explain the method actually used by applying it to the class of frame business buildings. In this class our 40 statistics furnished records, during the five years, of 567 losses; for 127 of these, or about 22 per cent, the sound value was given. For these 127 risks for which sound value was given the table of partial loss and the table of ratio of insurance to value were as follows : Table 18, Partial Loss. Table 19, Insurance to Value. X IDx 71 1 17 2 12 3 6 4 4 5 4 6 2 7 1 8 2 9 8 Total, 127 X n. 1 2 2 3 5 4 14 5 14 6 29 7 21 8 22 9 13 10 4 11 1 12 1 13 14 1 Total, 127 There is, furthermore, a table of ratios of loss to in- surance starting out with 64 risks for which this ratio is less than 10 per cent, 14 for which the ratio is greater tlian 10 and less than 20 pev cent, and so on. Such a table as this contains all the information to be had on such risks as the 440 upon which sound value is not given. However, even these three tables do not exhibit all the information obtainable from our statistics; we exhibit, for instance, 17 losses of between 10 and 20 per cent of the value, but we do not exhibit just what per cent of insur- ance is carried on each of these particular losses. To exhibit our information completely we must use a double entry table such as the following: 41 « H O n m flcS <* H g « •< % > O Q H H b. O !5 :3 t3 o 00 iz; CO 1—4 ::4 fA (i< tf o o t^ 1— 1 «M < I— 1 fa (« o o o Szi •< »-4 H 9 Sz; ea PCS H p CO (0 !zi ►-4 H o » H H O M (Z! Q hJ § § OQ 3 o ^ H ^ M o a »H ^H 1-1 T-H vH tH o y^ —4 r-l - o I— t <— < »-< 1—1 o o 1— « 1-1 (N o tH T-t f»H eo «-^ »-t »-< eo »-4 r-t 1— ( e» c» r- t-H 1^ G^ 1— t t-l - f^ •* T— < ^-i 1— ( •-H ■^ 1-^ .H t— 1 •V 1-* «^ <3i eo (5) 1^ = JVX, = VioVPvx. (7) Ix is the expectation of insurance loss per risk ; it is made up by (3) of two factors, J and VX^. J is the probability that a given risk will become a claim ; it may be called the ignition hazard. VA-x is the expectation of insurance loss per claim ; it is made up by (2) of ten ele- mentary expectations. The product of VX^, the expectation per claim, by J, the probability that a given risk will become a claim, is the expectation per risk or Ix- But Ix, instead of being looked at as the sum of a number of elementary expectations, may be thrown into the form (7) in which it is an equivalent single expectation ; the amount at stake is the insurance, VioV; the probability of its being called out on a given risk is Rx> the measure of the hazard. 49 Rx in turn is made up of two factors, J, the ignition hazard, and /3x which may be called the damage hazard. J is the probability that a given risk will become a claim ; Px is the probability that, having become a claim, the insurance VioV will be called out ; p^^ is the measure of the hazard among the claims, R^ is the measure of the hazard among the risks. Neither J nor p^ depend upon the sound value V. J does not depend upon the ratio of insurance to value ; /o^ however is a function both of the ratio of insurance to value, since it is the probability of insurance loss, and of the damageability of the risk as given by the /j-'s. This, so far, is all under the assumption that the insurance in every case is j ast VioV. Let us now suppose that the N risks are not all insured for the same amount. Let Ux be the number of risks insured for more than Vio V and less than ''+V10V, on the average (call it) yV ; then n^ + Di + + ng =2 nj = N. Leti'x = °^'^. Just as Ix is the insurance loss per risk when there is an in- surance of Vio V so let I'x be the insurance loss per risk when there is an insurance of V V. In general let primed sym- /lO bols refer to the case where the insurance is / V. The /lO expressions for this case are the same as those already given after a change of Vio to / for instance : I'x = / V R'x, /lO, /lO the product of the amount at stake, namely the insurance, / V, and the measure of the hazard, R'^, where R'x=J/>'x /lO and p'x = /u As a matter of fact X'x may with sufficient ' Vio. accuracy be taken to be Xx + X^ ^1 , although a closer determi- 2 nation might be made if it were thought desirable. 50 Let L be the actual insurance loss among N risks insured. Let us use actual to refer to the case where the distribution of risks as to amount of insurance carried is described by the n's, and in general let the barred symbols refer to this actual case. Then L = n„l'o + nj/ + + ng I'g = 2 nJi^}^J\ivp<.\. 9 _ Let 2 Vi \\ = X; , then L = N JVX = M Vx", (8) 1, the actual average insurance loss per risk, = /{j=J V X. (10) T=noV V + n^y V + iigVv^N V I i^.Y . /lO /lO /lO /lo 9 \)/ 2 V, y is the actual average ratio of insurance to value; call this /lO b/. 10 Then I = N y V. /lO ^ L/ N J V X /lO j/b/ Let yb/-7; then R = J /3 (12) (11) andT= J VX = y VK: (13) /lO 1 is in reality made up of 100 elementary expectations by (10), (9) and (2) but it reduces by (13) to an equivalent single expectation in which the amount at stake is the actual average insurance / V and the probability of this being called out is K, the burning- ratio or ordinary net rate; R again is the product of the ignition hazard J and the damage hazard p. 51 J is independent of the v's but not so p. The number N is not known and therefore J cannot be found directly. The damage hazard p iiowever can be com- puted and if the ordinary frate R has already been ascer- tained J may be computed from the relation K = J /a. Since R^ = J p^ and K, == J /a, Rx = R-' / This gives the coinsurance- rates R^i^ terms of the theoretically correct single rate R. Let us consider however instead of R the rate actually in use R which may or may not be correct. Let then R^ = R /— These rates R^ will produce exactly the same income as the single rate R, for : the income is 2 Uj /" V R' /lo .= in,y V ^py /lo /p NVll ». b,/ , p /lo p /lO If R" = R, this income is the expected loss L. This equivalence in the results of using R^ and R" is con- ditional however upon the i^'s remaining the same. The rates ^^ rV Rx will produce an income equal to /-^ times the expected /R loss whatever the value of the v's; the rate IT however will produce an income equal to =^times the expected loss where R* is determined from a new set of numbers, v *, just as R is determined from the i^'s. The second multiplier is a function of the v*'s while the first is not; the two will agree in general only if the v**s are the same as the v's. tSince R is independent of V it may be obtained in the ordinary way, without any assumption as to V by dividing the entire actual insurance- loss by the entire actual amount of insurance in force. 52 The quantities that it is important to determine are the damage-hazards /j^ and p. It is convenient to throw the work into tabular form by means of recursion formulas, and in this form some of the auxiliary quantities will be of interest. The formulas are obtained thus : I X-l 9 1 L'x = V/io -j 2 a, mi -1- X 2 m, V . Lx+i= V/io I 2 aj mi + (x + 1) 2mi I . Lx+i = Lx -f I (ax— x) m^ + 2 mi Iv/io. Let j (a^— x) m^-\- 2 mj I V/io = C^; then L^^i = L,, + C^. 9 9 Let 2 mi = M^, then 2 nij = Mx.i . X+l X M^.i = Mx + m^. Furthermore M9 = o and Lo = 0. By these formulas the values L^ may be computed. '^x = i^ and p, = y/_^, ■v '' ^x "T ^x + 1 ^ X 2~^' _^ 9 •\ = 2 I/, X'l, - V- p=/h/- Ao For actual computation we may conveniently take V to be one hundred. The formulas then are : Mg = 0. M^.i = Mx + m,. 53 a = l0|(a,-x)m, + mJ Lo = 0. Lx4-i = Lx + Cj 100 m; ^^" /Vio- ^ 2 9 Ao The work may be arranged in a table as follows, using for an example the figures for the class of frame mercantile buildings: 54 o • TtH r-t iH • CD ci CO I-H I-H 00 T-H I-H CO q I-H CO q 0:1 10 ifi ~J^ 05 01 CO 00 CD CO CO CO ci '^ ^ 't CO CD t^ 03 CO 00 "* Oi I— I i-H I-H CO I-H CO lO 10 't IC CO I-H CD 06 t^ 02 cq ■* 1— 1 10 CD t-^ ^ 05 1—1 IM ^ 10 I-H q t^ lO 'i^ 10 10 iO iO f~) CD 10 CD l>^ ^ C5 CD -t i-O I-H t^ I-H ^ CO (M cq CD q CD 10 10 S 10 lO 10 CO CO CO CD C2 LO ^ C5 16 CD iH l-H r-H CD CO CO rH Ci -* q CI CO I-H CO CO I-H iO C5 Ci CO CO CO T^ '^^ -+I 10 IC CO CO CO ■* >o r- ^ t^ rH lb I-H t^ CO I—H C5 >o -+ ?-H CO q ro 'f CO M^ CO ■* GO CO (M * 1- I-H i-H CO '*. CI Oi I-H CO H ksH <=; 1 >«; 1 a X 1 X 3^ ::^ J ^ cS a X S O O o Oj O -«^ CQ • rH a a o ri4 O a> o a • rH a a o o 0} a o O d 55 Table 22.— The Computatioit for the Class of Frame Business Buildings. X K ^x + ^x+l ^\ ^x ^x^'x 1 .03200 .07773 .01600 .03886 .03200 2 .04573- .10097 .05048 2 .000795 3 .05524 .11739 .05869 5 .002311 4 .06215 .12948 .06474 1 4 .007137 5 .06733 .13868 .06934 1 4 .007644 6 .07135 .14589 .07295 2 9 .016656 7 .07454 .15169 .07584 21 .012540 8 .07715 .15645 .07822 2 2 ITST .013550 9 .07930 .16044 .08022 2 .012633 10 .08114 X = .073266 / the actual average ratio of insurance to value, is found ./ lOj to be .7077 /lO By interpolation it is found that for x =7.3, p^ =~p that is in order for the coinsurance rates to produce the same income as the ordinary rate the 73 per cent coinsurance rate should equal the ordinary rate. •t-tAiujUs':.'?'" -ns BOOK «^^T^,— ^- AN INITIAL FINE OJf^f fr„' W,UU BE A==^=,^° ;°%"oUE ?HB PENALTY THIS BOOK °?'/"%°''J|nTS ON THE FOURTH ^I^V^rnrTO S^rorTHE seventh 0.V OVERDUE. LD21-20m-5,' 39 (9269s) 8Gfi423 THE UNIVERSITY OF CALIFORNIA LIBRARY