AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING 9-83 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING BY THOMAS O. CARLSON Probably the most lasrevinu hallmark fo the cifitneics mind si a tnetsisrep penchant rof pigeonholing. This ,tiart which ot the layman places the -itlu mate laes fo elituf noitacissed upon research and which has been the butt of a multitude fo jokes down the ,seirutnec si nevertheless one fo the most potent instruments of ;noitagitsevni rof ni the broad ti leads ot a proper evitcepsrep ni viewing a given ,dleif and ni eht narrow ti leads ot the -aeniled noit of distinctive relationships between the constituents of the field. The course often seems wayward and the ports of arrival startling to the even investigator, so that to maintain a balanced keel it is helpful for him to bear in mind Hosea Biglow's observation that "Facs are contrary 'z mules." This paper is, in epitome, the pigeonholing technique applied to the gamut of retrospective rating plans, both so-called and not so-called, that fall within the domain of practicability. And in no field that I know of is the Yankee apothegm just quoted better illustrated. The phrase "within the domain of practicability" affords a wide latitude to the author in the establishment of boundaries for his discussion. The term "Analysis" is used in a spirit which is the antithesis of pretentiousness, and has been resorted to only because the ostensibly humbler term "Note" has been so often applied to papers more unassailably definitive than I dare claim this one to be. Any presentation of such an analysis is simplified in proportion to the succinctness of the symbolism used, but there is a degree of succinctness beyond which one's fellow-workers will not bother to follow. It is hoped that the Table of Symbols set forth in Appendix A falls short of that degree. Properly speaking, an appendix should be reserved for notes of elaboration not essential to the continuity of the paper, but a glance at the length of Appen- dix A should produce immediate forgiveness for any breach of etiquette in this instance. The symbols conform to the accepted standard notation for common concepts, and have all been selected so as to permit typing as easily as possible. A few comments upon the loss and loss ratio symbols bearing subscripts may be helpful, and have been included in Appendix B. CLASSIFICATION OF PLANS Retrospective rating includes within sti scope any rating procedure which determines the premium for a risk after the expiration of the policy period for which the premium is being calculated, and in such a manner as to reflect 284 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING the loss experience incurred during that period. Since the loss experience of the risk is to be reflected in the rating, it follows that certain items in the final premium will be functions of the risk's losses. Other items will be func- tions of the standard premium or of the final premium in the general case. Consequently, the formula R=B+CL is implicit in the definition of retrospective rating; in this formula R is the final premium, B is a function of the standard premium, L represents the risk's losses, and the multiplier C in addition to including a credibility factor reflects those elements in the final rate which vary with the losses. Any item which is a function of the final premium, such as taxes, is included as a factor common to both of the terms B and CL. The restriction of R by the impo- sition of a minimum premium limitation H and a maximum premium limi- tation G will be expressed by writing the formula in the condensed form, H ~R=B+CL~ G By way of illustration, it may be noted that this is the exact formula for the application of the standard plan for workmen's compensation risks, ~1( B being termed the basic premium and C being termed the loss conversion factor. The factor C is constant for a given state, but the values of B, H and G vary by premium size and are set forth in a table of rating values. This plan is referred to hereafter as the standard plan. This fundamental formula has a most deceptively innocent appearance, for upon resolution into its elements an intricate mutability is discovered which is productive of widely varying particular formulas. It may be well to distinguish at once between what I shall call a particular formula and a particular plan: one formula may embrace a great variety of particular plans, each with its own definite schedule of rating values. A formula com- pletely defined specifies as respects expenses no more than the mode of allo- cating to the two terms the provisions for the respective items ; the aggregate provision for each such item is a characteristic of a particular plan, not of a particular formula. A formula may or may not specify that a limit per claim or per accident is imposed upon the risk's losses reflected in the rating, but a particular plan would have to define such a limit. Therefore, in the determination of a formula, the mode of distributing one expense item may constitute one condition to be imposed arbitrarily, or what we may call one degree of freedom, but the amount of the aggregate provision for that item is not pertinent to the analysis of the formula. In view of the fact that there may be more than a dozen degrees of freedom in a retrospective rating formula, a brief discussion of each element in the standard premium dollar ~1~ debircseD ni detail ni "The evitcepsorteR Rating Plan rof Workmen's -asnepmoC tion Risks," yb Sydney .D Pinney, ,.S.A.C.P Vol. XXIV, .p .192 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING 285 is in order, and it will be found that for practical consideration the problem may be greatly simplified in advance. Taxes must be paid upon the final premium. The provision for taxes may 1 therefore be included through the factor 1--f~- T- in each term of the formula, as in the standard plan, or it may be obtained by applying the same factor as a multiplier to the final premium, as in the Comprehensive Rating Plan for National Defense Projects. In the latter event, the basic premium and the loss conversion factor will include no provision for taxes; the minimum and maximum premiums may exclude or may include the provision for taxes according as they are imposed as limitations before or after the application of the tax multiplier to the formula result. It is not the purpose of this paper to discuss the advantages or disadvantages of these different methods. It is clear that the analysis of the respective formulas will not be essentially affected by the inclusion or the exclusion of the tax element. The production cost allowance and the provisions for general administra- tion, exposure audit and inspection expenses may be variously included in the formula. In the standard plan these four items are fundamentally in- cluded in the basic premium and thus vary with the standard premium: the administration, audit and inspection items are determined directly as func- tions of the standard premium, and the production cost allowance is for- mally determined as a function of the minimum premium which in turn depends upon the standard premium. Each of the four items could be inde- pendently apportioned to the two terms. To my knowledge such a differen- tiation in their treatment has never been proposed in principle, and since an interpretably simple analysis is dependent upon a reduction in the available number of conditions to be imposed, the four items have been handled herein as one, with the symbol V to designate them jointly. A reduction in the expense provisions from the amounts contemplated by the full manual rates is not inherent in the concept of retrospective rating but is customarily reflected therein. Any such reduction will involve one or more of these four items. The production cost allowance in the standard plan is equivalent to the full percentage provision in the manual rates applied to the minimum premium, thus effecting a reduction from the allowance under the guaranteed cost basis. It is almost universally accepted that actual administration and audit expenses decrease percentagewise in terms of the standard premium as the premium size increases: the introduction of expense constants recognized this situation to some extent, and a further gradation has been reflected in some of the approved retrospective rating plans. Inspection expense is not in general considered susceptible to reduc- tion from the provisions in the standard rates. In some instances such costs on the larger risks may even amount to a greater per cent of the standard premium than on the smaller risks; as an example may be cited the auto- 286 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING mobile liability line in which the small risks represent insurance on single automobiles and the larger risks represent the fleet business. The symbol Vr is used in this paper to designate the aggregate provision for these ex- penses in any retrospective rating formula, so that (V ~ Vr) represents the total reduction from the standard provisions for these four items. The expense of investigating and adjusting claims is more closely related to the losses than any of the other expense items, and is commonly reflected in the loss conversion factor. It is primarily because of this fact that claim expense is given consideration apart from other company expenses. It would not appear logical to reflect any of the other expense items, except taxes, in the loss multiplier unless claim expense were also reflected therein. The full provision for claim expense is represented in the following discussion by F. This provision may, like the other company expense items, be variously dis- tributed between the two terms in the fundamental formula. If the entire amount of F is provided through the loss conversion factor C, the com- ponent of C reflecting such provision has the form F/E, where E is the per- missible loss ratio or the expected losses. If there is any limitation upon the extent to which the risk's losses may affect the rating, as for example through the use of a credibility factor less than 100~ or through the imposition of a minimum or a maximum limit upon the final premium, the "rated" losses included in the final premium through the term CL will be less than the total expected losses in the aggre- gate, and in order to produce a technical balance within the plan the basic premium must be increased by an amount equal to the "non-rated" losses thus eliminated from the contribution made by the second term to the final premium. Any limitation of the losses entering the second term of the formula will result in a corresponding curtailment of those expense provi- sions which are functions of the losses; and the balance of such curtailed expense items must be included in the basic premium. The value of the basic premium is unaffected by the manner of making this adjustment and it is most feasibly accomplished by applying the expense component of the loss conversion factor to the loss provisions in the basic premium. The symbol will be used to designate the loss multiplier which reflects any expense items other than taxes. Then, as already noted, if all claim expense and no other expense is provided through such a factor, F --__ E :f we assume as a norm a procedure under which claim expense is a func- tion of losses and under which production cost, administration, audit and inspection expenses are functions of the standard premium, any variation from such a distribution of expenses may be handled by the introduction of a single symbol W to designate the amount by which any expenses pro- AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING 287 vided rof by the ssol reilpitlum l exceed the lluf claim expense provision F. Thus ni the general case =F+W E and a flat amount equal to W must be deducted from the basic premium, for W on the average the expense provision produced by the term ~ in J will be W equal to ~- • E, or W. The symbol W as thus used is unique among the symbols so far introduced in not being essentially positive. If W is nega- tive, then ! produces less than the equivalent of the full provision for claim expense, as in the case, for example, wherein an amount equal to W is in- cluded flat (i.e., as a direct percentage of the standard premium) in the basic premium for claim expense. If an amount equal to M is included flat in the basic premium for claim expense and an amount equal to N is provided for other expenses through the medium of J, then by setting W- N--M this interchange of functions is reflected in a simple manner. Thus W operates as a clearing house for any departures from what we have assumed to be the normal procedure in treating expense items other than taxes, a sort of factotum capable of handling two-way traffic if need be. If any expenses other than taxes are provided through the loss conversion factor, that is, if J is greater than zero, then J may be applied to the losses either modified or unmodified by credibility: the more logical of these two procedures would seem to be that which includes such expense provisions in direct proportion to the "rated" losses, i.e., which applies the multiplier to the credibility-modified losses. If we represent the credibility by the familiar Z, then in the case wherein J is applied to the credibility-modified losses, dis- regarding the tax multiplier, C=?Z+Z= (1 +J) Z, and in the case wherein J is applied to the losses without modification by credibility, C=J+Z The two classes of formulas thus produced do not comprehend the entire field, for it is evident at a glance that each of the formulas given for C is a special case of a more general formula, C=la + Jb • Z + Z, wherein Ja + lb = J, Fa -k- Wa Ja "-" E Fb + Wb and b = E 288 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING The first value given for C represents the case where a = O, the second where lb --~ .O This general formula, which does not appear to be of much if any practical importance, is treated briefly in Appendix C. Other modes of providing expenses may suggest themselves, such for ex- ample as having the provision V included by means of a multiplier applied to B rather than a term contained within B. These in my opinion are more of academic than practical interest. The formulas developed later in this paper can be readily adapted to reflect such variations, however, and in fact many variations reduce to the forms discussed in full. In the example just cited, for instance, since B is a function of the standard premium, a multi- plier of B is similarly a function of the standard premium, and the variation is reducible for analytic purposes to what I have chosen as a standard form. No differentiation between allocated and unallocated claim expense has been made in this discussion. Such a division is not important for analytical purposes. The natural procedure would be to treat allocated claim expense as it is treated in the determination of manual rates. For workmen's com- pensation, all claim expense is treated as a single item in calculating manual rates, and consequently in the standard plan no differentiation is made between the two types of claim expense. For liability lines, on the other hand, allocated claim expense is reported with Iosses and is treated as a loss in the determination of manual rates : the most natural procedure in develop- ing a retrospective rating plan for such lines would therefore be to include allocated claim expense with the losses; in this case F would represent un- allocated claim expense only, and E would represent expected losses plus allocated claim expense. In the Comprehensive Rating Plan for National Defense Projects the allocated claim expense is added to the losses after the latter have been increased by the unallocated claim expense multiplier, the multiplier in this case being applicable to the losses only. The provision for profit and contingencies in the manual rates will be designated by D, and in the retrospective premium by Dr. These symbols correspond to the symbols V and Vr introduced to designate expenses other than claim or taxes. The rates for workmen's compensation insurance do not include a profit factor except in one state; the contingency factor in the manual rates varies according to the accumulated past experience and is de- signed to produce neither an underwriting profit nor an underwriting loss over a period of years. On lines other than workmen's compensation the manual rates include a definite provision for profit and contingencies. There are many ways in which this item may be included in a retrospec- tive rating formula. Any of the modes which have already been discussed for reflecting the respective expense items could be applied to this item as well. In the workmen's compensation line in order to avoid a sharp break in the provision at the eligibility point for application of the rating plan, the introduction could be graded in a variety of ways ; for example, the provision AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING 289 for any particular size of risk could be included as some per cent of that portion of the standard premium in excess of the premium required to qualify for rating; or it could be included as D per cent modified either by the credibility or by the ratio of rated losses to expected losses. Although these and other methods produce a slight variation in the individual formulas, we may in the interest of simplicity consider that the provision Dr is included as a fiat item in the basic premium, with any reflection of a split in the aggregate provision between the two terms of the fundamental formula taken care of through the "happy" medium of the symbol W, corresponding to the treatment of the expense items. Under any practicable means for introduc- ing this provision, it can ultimately be reduced to the sum thus indicated or it can be reflected by extending the tax multipliers applicable either to the first or to both terms of the formula to reflect both T and D, i.e., by making 1 the multiplier equal to1_ T- D" Any changes in the formulas to reflect the latter mode of including the item will be apparent and should require no elaboration. The actual losses L reflected in the final premium are limited in the aggre- gate to those losses lying between the loss allowances in the minimum and in the maximum premiums. These losses may be limited by the application of a credibility factor, and they may be further constrained by a limit per claim or per accident; as examples may be cited the $10,000 limitation per claim in the New York plan for workmen's ,noitasnepmoc and the limitation to the experience rating normal loss amount per case which was a feature of a proposal given extensive consideration some years ago. ~2~ The credibility factor, represented by Z, if explicitly expressed is a com- ponent of the loss conversion factor C; in such a case some function of Z is also involved in the basic premium. As an independent variable in theory, Z could follow any specified law or no law. In the light of practical con- siderations, however, we may again considerably reduce the scope of our investigations, since we are interested in only two cases, (1) that in which Z increases between the limits of 0 and 1.00 as the premium size increases, and (2) that in which Z is constant. The standard plan for workmen's compen- sation is a special case of the latter category, with Z equal to 1.00 for every size of risk. The minimum and maximum premiums H and G may be subjected to par- ticular conditions. The loss provision in the basic premium, and conse- quently each term in the fundamental formula, is affected by the variation of H and G. The basic premium may be adapted to a particular progres- sion of values, with corresponding adjustments in H or G or both. The loss conversion factor is determined fundamentally when the credibility and the distribution of the expense items in the formula are known; as will be seen, >2( ,.S.A.C.P Vol. XXIV, .p .033 290 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING however, a rounding of the loss conversion factor may produce an effect upon the basic premium. Of the foregoing elements of variability, all but one in general represent separate degrees of freedom. It should be noted, however, that the determi- nation of C is equivalent to the determination of Z and vice versa. What started out as a field of several dimensions for classification has by lopping and squeezing now been reduced to an order sufficiently low to permit a reasonably simple treatment, and a design can be developed which is somewhat less like a crazy quilt than an ordered pattern. The order of handling the separate elements of variability in the develop- ment of a formula is immaterial for our purposes. To make a start, how- ever, let us review the expense elements, the limitation of losses, the credi- bility and the rating factors in succession. Expense Elements In the general case the provisions for expenses included flat in the basic premium are equal to Vr--W and the remaining expense provisions other than taxes are produced by a loss multiplier j= w_____+v E If W--0, we may consider that claim expense is provided wholly through a loss multiplier and that other expenses except taxes are pro- vided wholly as a direct function of the standard premium. In case any other distribution is contemplated, it can be indicated by using the symbols N and M already explained, setting N -- M -- W. For analysis, such a distinction is immaterial, and the symbols N and M have there- fore been omitted from the table in Appendix A. No further mention of such a differentiation will be made. If W ---- Vr, we may consider that all expenses are provided through a loss multiplier, a case which would be practicable only for a very large risk. If W = -- F, it follows that J = 0 and no expenses other than taxes vary with the losses. W, in its representation of expense provisions, may then vary between the limits -- F and Vr, or --F<.W<.Vr All these variations are reflected by including W in the formula, and consequently need not be studied further as producing formula types. The method of including the loss multiplier J in the loss conversion factor C is, however, productive of two types requiring individual consideration : 192 AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING Type i. c --1--T Type n. c + z --1--T where T is the percentage provision for taxes. Each of these two types is deducible from a more general formula wherein Ja +b " Z+Z C= 1--T Ja + b : J Type I is deduced by setting a -- ,O Type II by setting b -~ .O The general formula is discussed in Appendix C. The provision for profit and contingencies may be included in several ways, none of which would seem to affect our analysis fundamentally. This provision has throughout been represented by Dr as an element in the basic premium, with the understanding that the symbol W would absorb any portion of D reflected in ./. As already indicated, the mode of providing for taxes, although divid- ing all formulas into two groups according as the loading is applied to each term separately or to the final sum of the two terms, is not impor- tant for our present purposes. Limitation o Losses In the general case, the risk's losses reflected in a rating are restricted by a limit per claim or per accident in addition to the limitation im- posed by the specification of minimum and maximum premiums. In the discussion, unless otherwise noted, the insurance charge for losses in excess of the loss allowance in the maximum premium will be con- sidered as including losses above a specified limit per claim or per accident. Excess pure premium ratio tables which reflect such a limi- tation may be constructed and in fact have been constructed in at least one state for workmen's compensation risks. )3( One particular plan in which the limit per case is the normal loss amount under the experience rating plan will be accorded separate consideration. Credibility Two cases are of particular importance : Class A. Credibility increases with premium size, 0 <~ Z < 1.00 Class B. Credibility is constant. An important case under Class B is that for which the credibility equals 1.00 throughout, as illustrated by the standard plan. (a) "On Graduating Excess Pure Premium Ratios," by Paul Dorweiler, Vol. XXVIII, p. .231 29.P AN ACTUARIAL ANALYSIS OF RETROSPECTIVE RATING Rating srotcaF In the general formula no .particular conditions are imposed upon the minimum or maximum premmms. The only relationshit~s specified are that the sum of B and ,LC equal to R, shall be not less than H nor more than G. The predetermination of certain conditions to be satisfied by H or G or both gives rise to four cases of sufficient practical importance to be worthy of special attention. Case (a). Minimum premium is greater than basic premium, maxi- mum premium equals standard premium : jrt> B,G=P Case (b). Minimum premium equals basic premium, maximum pre- mium is greater than standard premium : H= B,G>P Case (c). Minimum premium equals basic premium, maximum pre- mium equals standard premium : H--B,G=P Case (d). Same as (c), but in addition the loss allowance in the maxi- mum (i.e., standard) premium equals the expected losses : H = B, G = P, G' = P' =E The imposition of these conditions cannot of course be made unless the requisite degrees of freedom are available. For example, if the mode of providing for all expense items and the limitation of losses per claim or per accident have been specified, three conditions remain to be deter- mined: the establishment of a constant credibility factor would remove one of these, leaving as possibilities among the foregoing special cases only (a), (b) and (c) ; the three conditions under (d) could not under such circumstances be satisfied except by coincidence. For our purposes, therefore, there are two important cases of the general formula, giving rise to two broad categories which will be referred to as Types I and II according as the loss conversion factor excluding taxes takes the form (JZ + Z) or (l + Z). Each of these categories has two important subdivisions designated as Classes A and B according as the credibility in- creases with the premium size or is constant. And within each subdivision consideration will be given to certain conditions which may be imposed on the rating values, denoted as Cases (a), (b), (c) and (d). The mode of providing for taxes does not affect our analysis of other variables because in one way or another the tax multiplier is common to all terms. The mode of distributing all other expense provisions is conveniently removed as an issue by the versatile symbol W. And any loss limitation per claim or per acci- dent is offset by an increase in the charge for excess losses included in the basic premium. We are then ready to proceed with our study.