ADJUSTING INDICATED INSURANCE RATES: FUZZY RULES THAT CONSIDER BOTH EXPERIENCE AND AUXILIARY DATA VIRGINIA R. YOUNG Abstract This paper describes how an actuary can use fuzzy logic to adjust insurance rates by considering both claim experience data and supplementary information. This supplementary data may be financial or marketing data or statements that reflect the philosophy of the actuary's company or client. The paper shows how to build and fine-tune a rate-making model by using workers com- pensation insurance data from an insurance company. ACKNOWLEDGEMENT I thank the Actuarial Education and Research Fund for support- ing me financially in this project. I thank my Project Oversight Chair, Charles Barry Watson, for encouraging me. I thank the ac- tuaries at WCI for providing data and insights into their rating process. I also thank Richard Derrig and anonymous referees for giving me valuable comments. 1. INTRODUCTION Through the education programs of the Society of Actuar- ies and the Casualty Actuarial Society, actuaries are equipped with statistical tools to analyze experience data and to determine necessary rate changes for their insurance products. Students are often surprised to learn that those rate changes are frequently not accepted "as is" by company management. Actuaries work with sales, marketing, and underwriting personnel to develop rates that will be competitive and adequate. 734
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`ADJUSTING INDICATED INSURANCE RATES 735 Actuaries frequently consider statistical data specific to rates, such as the results of experience studies. In setting premiums, actuaries also consider constraints that supplement experience data. These constraints may reflect company philosophy, such as "We wish to increase our market share moderately from year to year." They may also include financial data, such as "Raise the rates if we experience high loss ratios or low profit margins." The theory of fuzzy sets provides a natural setting in which to handle such statements. Through fuzzy sets, one can account for vague notions whose boundaries are not clearly defined, such as "large amount of business." Fuzzy logic provides a uniform way to handle such factors that influence the indicated rate change (Zadeh [20]). A fuzzy logic system is a type of expert system. An advantage of using a fuzzy logic system is that it provides a systematic way to develop mathematical rules from linguistic ones. This paper describes step-by-step how an actuary can ad- just rates by beginning with linguistic rules that consider both experience data and supplementary information. Fuzzy sets have only recently been applied to problems in ac- tuarial science. DeWit [5] and Lemaire [13] show how to apply fuzzy sets in individual underwriting, and Young [16] indicates how to use fuzzy sets in group health underwriting. Ostaszewski [15] suggests several areas in actuarial science in which fuzzy sets may prove useful. Cummins and Derrig [2] apply a form of fuzzy logic to calculate fuzzy trends in property-liability in- surance. Derrig and Ostaszewski [4] employ fuzzy clustering in risk classification and provide an example in automobile insur- ance. Cummins and Derrig [3] use fuzzy arithmetic in pricing property-liability insurance. In an earlier paper [18], I show how to develop a fuzzy logic model with which actuaries can adjust insurance rates by considering only constraints or information that are ancillary to experience data. Section 2 introduces fuzzy sets and defines operators corre- sponding to the linguistic connectors and and or and the modifier
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`736 ADJUSTING INDICATED INSURANCE RATES not. It also describes a simple fuzzy inference system. Refer- ences for fuzzy sets include Dubois and Prade [7], Kosko [12], and Zadeh [19]. Some references for fuzzy logic and fuzzy in- ference are Bellman and Zadeh [1], Driankov et al. [6], Kandel and Langholz [9], Klir and Folger [11], Mamdani [14], Zadeh [20], and Zimmermann [21]. Section 3 describes how to construct and fine-tune a pricing model using fuzzy inference. Section 4 shows how to build a pricing model using workers compensation insurance data from an insurance company. Finally, Section 5 summarizes the paper's key points. 2. FUZZY INFERENCE Fuzzy sets describe concepts that are vague (Zadeh [19]). The fuzziness of a set arises from the lack of well-defined bound- aries. This lack is due to the imprecise nature of language; that is, objects can possess an attribute to various degrees. A fuzzy set corresponding to a given characteristic assigns a value to an object, the degree to which the object possesses the attribute. Examples of fuzzy sets encountered in insurance pricing are stable rates, large profits, and small amounts of business renewed or written. Indeed, rates can be stable to different degrees de- pending on the relative or absolute changes in the premium rate. Also, profits can be large to different degrees depending on the relative or absolute amount of profits. Fuzzy sets generalize nonfuzzy, or crisp, sets. A crisp set, C, is given by a characteristic function: nc : X ~ {0,1}, in which Xc(X)= 1 if x is in C; otherwise, Xc(X)= 0. Fuzzy sets recognize that objects can belong to a given set to different degrees. They essentially expand the notion of set to allow partial membership in a set.
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`ADJUSTING INDICATED INSURANCE RATES 737 DEFINITION 2.1 A fuZZy set, A, in a universe of discourse, X, is a function m a on X that takes values in the unit interval [0, 1]: m A : X ~ [0, 1]. The function m a is called the membership function of A, and for any x in X, mA(x ) in [0, 1] represents the grade of membership of xinA. EXAMPLE 2.1 One may define stable rates by the following hy- pothetical fuzzy set: 0, if r <-0.10, r+0.10 if -0.10<r<-0.05, 0.05 ' mstable(r ) = 1, if - 0.05 < r < 0.05, 0.10-r__ if 0.05<r<0.10, 0.05 ' 0, if r_> 0.10, in which r is the relative rate change. For example, the degree to which a rate increase of 8% is stable is 0.40. It does not mean, however, that one will view an 8% rate increase as stable 40% of the time and unstable the rest of the time. See Figure 1 for the graph of this fuzzy set. The points + 0.05 and 4-0.10 depend on the line of business. Also, one may want to use a fuzzy set that is not necessarily piecewise linear. We now define three basic operations on fuzzy sets. DEFINITION 2.2 given by mauB(X) - max[ma(x),ms(x)], x E X, and the intersection, A M B, is given by manB(x ) =-- min[ma(x),mB(x)], x E X. The union, A U B, of two fuzzy sets, A and B, is
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`738 ADJUSTING INDICATED INSURANCE RATES FIGURE 1 GRAPH OF FUZZY SET OF STABLE RATES, EXAMPLE 2.1
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`I I I I I I -9.1 0 0.1 The complement, -A, of fuzzy set A is given by m_A(X ) ~ 1 --mA(X ), X E X. The union operation acts as an or operator, the intersection operation acts as and, and the complement operation acts as not. Thus, for example, mAnB(X) represents the degree to which x is a member of both A and B. The given definitions are not the only acceptable ones for these operations. Klir and Folger [11] specify axioms that union, intersection, and complement satisfy. Also, Dubois and Prade [7] and Young [16] discuss alternative operators. One in particular is the intersection operator called the algebraic product. The algebraic product of two fuzzy sets A and B is given by mAB(X ) = mA(X ) • roB(X). The algebraic product allows the fuzzy sets to interact in the in- tersection. That is, both fuzzy sets contribute to the value of the intersection, as opposed to the min operator in which the mini- mum of the two values determines the value of the intersection.
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`ADJUSTING INDICATED INSURANCE RATES 739 We will consider this intersection operator in some of the ex- amples below. Unless otherwise stated, however, the intersection operator is the min operator. EXAMPLE 2.2 Suppose we want to intersect the fuzzy set of stable rates from Example 2.1 with the fuzzy set of low actual- to-expected ratios given by 1, if x<0.90, 1.10 -x mt°w(x) = ] ---~-~-0---' if 0.90 < r < 1.10, 1.0, if 1.10 <x, in which x is the ratio of actual claims to expected claims (A/E ratio).l We first imbed these fuzzy sets in the product space of pairs {(r,x) : r >_ -1.00, x > 0}, as follows mstable(r, X) = mstable (r) mtow(r,x ) = mlow(X ). See Figure 2 for the graph of the intersection of these two fuzzy sets using the min operator and Figure 3 for the graph of the intersection of these two fuzzy sets using the algebraic product. Note that the algebraic product operator allows the two fuzzy sets to interact more than does the min operator. For exam- ple, suppose the rate decrease is 6% and the A/E ratio is 0.95. Then, the degree to which the rate change is stable is 0.80, and the degree to which the A/E ratio is low is 0.75. The degree to which the rate change is stable and the A/E ratio is low is min(0.80,0.75) = 0.75 if we use the min operator to intersect the lOne type of experience study is called an actual-to-expected study. In this study, one compares the actual (incurred) claims relative to the expected claims built into the pre- mium. If this study is performed before the claims have run out, then one develops the actual claims to an ultimate basis to estimate actual incurred claims. One result of this study is the ratio of actual claims to expected claims, called the actual-to-expected ra- tio, or briefly A/E ratio. A high A/E ratio indicates that the allowance for claims in the premium is too low.
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`740 ADJUSTING INDICATED INSURANCE RATES FIGURE 2 INTERSECTION OF STABLE RATES AND LOW ACTUAL-TO-EXPECTED RATIO (USING THE MIN OPERATOR) 1.0 MIN o.,. 0 RateCl"
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`two fuzzy sets, and is (0.80)(0.75) = 0.60 if we use the algebraic product to intersect them. A few years after fuzzy sets were introduced, Bellman and Zadeh [1] developed the first fuzzy logic model in which goals and constraints were defined as fuzzy sets and their intersection was the fuzzy set of the decision. Cummins and Derrig [2] use the method of Bellman and Zadeh to calculate a trend factor in property-liability insurance. They calculate several possible trends using accepted statistical procedures. For each trend, they determine the degree to which the estimate is good by intersecting several fuzzy goals. They suggest that one may choose the trend
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`ADJUSTING INDICATED INSURANCE RATES 741 FIGURE 3 INTERSECTION OF STABLE RATES AND LOW ACTUAL-TO-EXPECTED RATIO (USING THE ALGEBRAIC PRODUCT) AIg_Prod RateChng J 1.0 I 0.5" 0.0 -0.1 0.0~~ 0.9 o.1 A/ERatio that has the highest degree of goodness. Cummins and Derrig also propose that one may calculate a trend that accounts for all the trends by forming a weighted average of these trends using the membership degrees as weights. It is this latter method that more closely relates to the technique proposed below. This paper shows how actuaries may incorporate supplemen- tary information in their pricing models, for example, amount of business written or profit earned. Instead of using the method designed by Bellman and Zadeh [1], we follow Zadeh [20] by applying fuzzy inference. In particular, we use a simple form of fuzzy inference proposed by Mamdani [14], who has been a pi-
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`742 ADJUSTING INDICATED INSURANCE RATES oneer in applying fuzzy logic in industry. We describe this fuzzy inference after the following example. 2 EXAMPLE 2.3 (a) If the A/E ratio is high and the amount of business is large, then raise the rates. (b) If the A/E ratio is moderate and the amount of business is moderate, then do not change the rates. (c) If the A/E ratio is low and the amount of business is small, then lower the rates. An actuary can only apply a crisp rate change, not a fuzzy expression such as "raise the rates." We therefore set the phrase "raise the rates" equal to the largest rate increase we are willing to administer; similarly, "lower the rates" is replaced by the largest rate decrease we are willing to administer. The reason for doing so will become evident as we proceed below. In general, our fuzzy system is a collection of n fuzzy rules: If x is A l, then y is Yl. If x is A2, then y is Y2. If x is An, then y is Yn" If we are given specific input, or explanatory, data J (possibly multi-dimensional if the A i are compound hypotheses, as in Ex- ample 2.3), then measure the degree to which J satisfies the hypothesis A i in rule i, i = 1 ..... n, namely, mA,(Yc). To calculate 2Throughout this paper, by default, assume that if none of the hypotheses is satisfied to a positive degree, then do nothing. In the following example, this would mean "do not change the rates." This convention is consistent with the weighting scheme defined below in Equation 2.1 if one sets 0/0 equal to 0.
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`ADJUSTING INDICATED INSURANCE RATES 743 the output ~, form the weighted average n ~-'~yimA, (~C) _ i=1 (2.1) n i=1 A fuzzy hypothesis A may be a compound statement, such as "our company has been writing a great deal of business and earning a small amount of profit." In this case, we intersect the fuzzy sets corresponding to a great deal of business and a small amount of profit with the min operator, as in Definition 2.2. Alternatively, one may use the algebraic product operator to in- tersect the fuzzy sets, as in Example 2.2. Also, if a compound hypothesis involves the connector or and modifier not, then use the max and negative operators, respectively, to combine the in- dividual fuzzy sets. In Section 3, we describe how to obtain a specific output Yi, i = 1 ..... n, if the conclusion is expressed as a fuzzy statement, such as "raise the rates a great deal." EXAMPLE 2.4 To continue with Example 2.3, suppose that we have determined the following values of Yi that correspond to the conclusions in the fuzzy rules that we state in that example: (a) If the A/E ratio is high and the amount of business is large, then raise the rates 15%. (b) If the A/E ratio is moderate and the amount of business is moderate, then do not change the rates. (c) If the A/E ratio is low and the amount of business is small, then lower the rates 10%. Again, if none of the hypotheses is satisfied, then do not change the rates. We are given that the actual-to-expected (A/E) ratio is 1.05, and the amount of business is 3.0 (on some appro-
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`744 ADJUSTING INDICATED INSURANCE RATES priate scale). Given fuzzy sets for the components of the hypothe- ses, the next step is to calculate the degree to which the input satisfies each hypothesis. Evaluate the degree of membership of the A/E ratio, 1.05, in the fuzzy sets for high, moderate, and low. Hypothetically, suppose that the A/E ratio is high to degree 0.75, moderate to degree 0.25, and low to degree 0.0. Similarly, eval- uate the degree of membership of the amount of business, 3.0, in the fuzzy sets for large, moderate, and small. Suppose that the amount of business is large to degree 0.50, moderate to degree 0.50, and small to degree 0.0. The hypothesis of the first rule is, thus, satisfied to degree rnin(0.75,0.50) = 0.50; the second rule, min(0.25,0.50) = 0.25; and the third rule, min(0.0,0.0)= 0.0. Our rate change is, therefore, = 0.50(0.15) + 0.25(0.00) + 0.0(-0.10) = 0.10, 0.50 + 0.25 + 0.0 or increase the rates 10%. Compare the expression for ~ with Equation 2.1. If, instead of the min operator, we were to use the algebraic product operator for intersection, the rate change would be = 0.375(0.15) + 0.125(0.00) + 0.0(-0.10) = 0.1125. 0.375 + 0.125 + 0.0 In Examples 2.3 and 2.4, we incorporate experience data, the actual-to-expected ratio, in the hypotheses of our fuzzy rules. One may also include experience data in the conclusion, as in the following example. EXAMPLE 2.5 The following fuzzy rules may more accurately reflect the philosophy of the company: (a) If the amount of business is increasing greatly and the profit margin is decreasing greatly, then raise the rates more than indicated by the A/E ratio.
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`ADJUSTING INDICATED INSURANCE RATES 745 (b) If the amount of business is stable and the profit margin is stable, then change the rates as indicated by the A/E ratio. (c) If the amount of business is decreasing greatly and the profit margin is increasing greatly, then lower the rates more than indicated by the A/E ratio. 3. BUILDING A FUZZY INFERENCE MODEL The previous section describes how to obtain a crisp output given a fuzzy inference model and crisp input ~. This section ex- plains how to construct and fine-tune a fuzzy logic model. Young [18] presents steps that may be followed to build a fuzzy logic model. They are repeated here so that this work is self-contained. Section 4 shows how to follow these steps in creating and fine- tuning a fuzzy logic model. Because the following procedure formalizes the discussion in Section 2, the casual or first-time reader may wish to skip to Section 4. . Verbally state linguistic rules. These rules may reflect current or desired company philosophy. They may arise from the business sense of actuaries. They may result from the combined input of several functions in the insurance company. . Create the fuzzy sets corresponding to the hypotheses. As- sume that the linguistic variables used are naturally or- dered. For example, the linguistic variable of amount of business is naturally ordered because large amounts of business correspond with large numbers that measure the amount of business, and similarly for small amounts of business. (a) To create the fuzzy sets for the jth dimension of the input, partition the input space Xj = [Xj, I,Xj.n(j) ] into n(j) -- 1 disjoint subintervals, one fewer than the num-
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`746 ADJUSTING INDICATED INSURANCE RATES ber, n(j), of fuzzy sets defined on Xj. Write the bound- ary points of the subintervals: xj, 1 < xj, z <... < xj,,~/). Even though the input space Xj may be infinitely long, the example below describes how to determine xj. l and xj,,,(j) so that we can effectively limit Xj to the finite interval [Xj,l,Xj,n(j) ]. (b) The graph of the leftmost fuzzy set ALl is defined to be the line segment joining the points (xj, 1, 1) and (x j2,0) and 0 elsewhere. The graph of each of the middle n(j)- 2 sets Ajj¢O. ) is the triangular fuzzy set that connects the points (xj,k(j)_ 1,0), (xj,k(j~, 1), and (xj,k(j)+l,O) and 0 elsewhere, k(j) = 2 ..... n(j)- 1. Fi- nally, the n(j)-th fuzzy set Aj,n(j) is the line segment joining the points (xj,n(j)_ l, 0) and (xj,n(j), 1) and 0 else- where. Note that for any input value of x j, the sum (over k(j)) of its membership values in the sets Aj,k(j) is 1; thus, we say that the Aj.k(j) form a fuzzy partition
`See Figure 4 for an illustration of a partition of the variable of amount of business into four fuzzy sets. Other forms of fuzzy sets may be used to partition a variable, but triangular fuzzy sets are easy to compute and are completely determined by the points in the partition of Xj. (c) Combine the fuzzy sets that comprise each hypothe- sis into one fuzzy set using the operators min, max, and negative, corresponding to the linguistic connec- tors and and or and modifier not, respectively. 3. Determine the output values {Yi} for the conclusions. Set the output value Yi to the desired output if the hypothesis of rule i is met to degree 1.0.
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`ADJUSTING INDICATED INSURANCE RATES 747 FIGURE 4 A FUZZY PARTITION OF AMOUNT OF BUSINESS 0.5
`~l /IIINx %% I I I • • / I I "X t f %,~ ^ / \ / \ / \ / ! :( / \ I
`(dash-dot) Large Amount of Business 4. Fine-tune the fuzzy rules, if applicable. If learning data is available, either historical data that is still relevant or hy- pothetical data from experts, then use that data to modify the values Xj,k(j) and the values Yi. This is done to opti- mize any one of a number of objectives. In this work, we minimize a squared-error loss function. Given data of the form {(x~,y;) • l = 1 ..... L}, pairs of input and output values, either from prior rating periods or from experts' opinions, the model may be fine-tuned using the following simple method: Perturb the parameters {Xj,k(j) } and {Yi} to minimize the squared error ~-~(y; - ~(x;)) 2, I=l
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`(solid) Very Small Amount of Business
`" (dot) Somewhat Small Amount of Busin
`(dash) Moderate Amount of Business
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`748 ADJUSTING INDICATED INSURANCE RATES in which ~(xT) is the output of the fuzzy logic model, given the input x~. These errors may also be weighted to reflect the relative importance of each ordered pair. In the next section, we minimize such a weighted sum of squared errors: L wt(y 7 - ~(xT)) z, (3.1) /=1 in which wt, l = 1 ..... L, is the weight for the pair (xT,yT). The data, {(x~,y~) • 1 = 1 ..... L}, is called learning data because one "trains" the fuzzy logic system to follow the data to the degree measured by Equation 3.1. The interested reader may wish to explore other meth- ods for fine-tuning a fuzzy logic model. Glorennec [8], Katayama et al. [10], and Driankov et al. [6] describe sev- eral methods for adjusting the parameters to fit learning data. Also, Young [17] proposes using a measure of impli- cation derived from fuzzy subsethood to fine-tune fuzzy logic models. This measure of implication measures the degree to which the input implies the output. To fine-tune a given model, therefore, perturb the parameters of the model to maximize this measure of implication. 4. WORKERS COMPENSATION EXAMPLE Here is an example of building and fine-tuning fuzzy logic models, using workers compensation insurance data from an in- surance company for four consecutive rating periods. Call the insurance company Workers Compensation Insurer (WCI). To protect the interests of this insurance company, the data has been masked by linearly transforming it and by relabeling the geo- graphic regions and the dates involved. There is a distinction between prescriptive and descriptive modeling. The first part of this section briefly explains the de- cision process that WCI works through every six months, and
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`ADJUSTING INDICATED INSURANCE RATES 749 proposes and builds fuzzy logic systems that model that process. That is, fuzzy models are built based on the expert opinions of the actuaries and other managers at WCI. This is prescriptive modeling, and it corresponds to Steps 1 through 3 in Section 3. The second part of this section fits three fuzzy logic models based on the data that WCI provides, using Step 4 in Section 3. That is, we seek to find fuzzy models that describe what WCI has actually done in the past. WCI files rates for its workers compensation insurance line in various states. Every six months, WCI determines the adequacy of those filed rates. WCI represents that adequacy by an indicated target. For example, an indicated target of +5% in a state means that WCI requires premiums equal to 105% of its filed rates to reach a specified return on surplus. Similarly, an indicated target of -7% means that WCI requires premium equal to 93% of its filed rates. In the fuzzy models, the indicated target is based on the ex- perience data. WCI calculates it by comparing the filed rates in a state with the sum of the experience loss ratio and expense ratio in that state, among other items. Based on the indicated target and supplementary (financial and marketing) data, WCI then chooses a selected target for each state. (See Section 4.1.1 for more about how WCI selects a target.) Financial data include competitively-driven rate departures with respect to previous se- lected targets. For example, a rate departure of -1% means that actual premium was 99% of (filed rates),(1 + selected target). Marketing data include retention ratios and actual versus planned initial premium. 4.1. Prescriptive Modeling 4.1.1 Verbally state linguistic rules. To develop linguistic rules for a prescriptive model, the pricing actuaries and prod- uct developers at WCI provide information about how an ideal "target selector" would use the data for choosing a target. As a
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`750 ADJUSTING INDICATED INSURANCE RATES rule of thumb, if the indicated target increases over the previous six months, then the selected target increases, and vice versa. However, this rate change is tempered by how well the region met its previous targets and by how much business is written in the region. For example, if the region had a positive rate depar- ture recently, then WCI might consider increasing the selected target. Also, if the amount of business is low relative to planned, then WCI might consider decreasing the selected target in or- der to stimulate growth. On the other hand, a large amount of initial business (relative to planned initial business) may not be desirable because of the legal or competitive climate in a given state. In view of the opinions of the experts at WCI, the following linguistic rules are developed on which to base a prescriptive fuzzy logic model: (a) (b) (c) If the change in indicated target from time t - 1 to time t is positive, and if the recent rate departure is positive, and if the amount of business is good, then the change in selected target from time t - 1 to time t is positive. If the change in indicated target from time t- 1 to time t is zero, and if the recent rate departure is zero, and if the amount of business is moderate, then the change in selected target from time t- 1 to time t is zero. If the change in indicated target from time t - 1 to time t is negative, and if the recent rate departure is negative, and if the amount of business is bad, then the change in selected target from time t- 1 to time t is negative. Methods for measuring the amount of business include pre- mium, number of accounts, retention ratio, close ratio (percent- age of new business written to new business quoted), and pre- mium for new business. This paper measures amount of business by the sum of the retention ratio and the minimum of the ratio
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`ADJUSTING INDICATED INSURANCE RATES 751 of actual initial premium to planned initial premium and the in- verse of that ratio, that is, min(actual/planned, planned/actual). This minimum lies between 0.0 and 1.0, and it takes into account that writing a great deal of business (relative to the planned ini- tial premium) is not necessarily a profitable goal. The closer the minimum is to 1.0, the better the region has met its tar- get. If the ratio actual/planned is very small or very large, then min(actual/planned, planned/actual) is close to 0.0. Therefore, a good amount of business is measured relative to a maximum of 2.0, after expressing the retention ratio as a decimal. 4.1.2 Create the fuzzy sets corresponding to the hypotheses and determine the output values for the conclusions. In the above linguistic rules, each hypothesis is a compound statement that combines three fuzzy sets with the connector and. Denote the space of change in indicated target by X l , the space of rate depar- tures (RD) by X 2, and the space of amount of business by X 3. On each of these spaces, define three fuzzy sets---one for each fuzzy rule. To get the endpoints of each of these spaces and the interme- diate boundary points, work backwards as follows: Determine the maximum and minimum changes in the selected target from time t- 1 to time t. For example, suppose that the maximum allowable change in selected target is + 10%, and the minimum is -10%. Then, determine the changes in indicated target, the rate departures, and the sum of retention ratio and rain(A/P, P/A) that would lead to those maximum and minimum changes. Sup- pose that the selected target would be increased 10% if the in- dicated target increased by at least 15%, if the rate departure were at least +3%, and if the measure of the amount of business were greater than or equal to 1.8. Also, suppose that the selected tar- get would be decreased by 10% if the indicated target decreased by at least 10%, if the rate departure were at least -5%, and if the measure of the amount of business were less than or equal to 1.0.
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`752 ADJUSTING INDICATED INSURANCE RATES Then, the space of change in indicated target is effec- tively X l = [-10%, 15%], the space of rate departures is X 2 = [-5%,3%], and the space of amount of business is X 3 = [ 1.0, 1.8]. In this representation, all rate departures less than -5% are identified with -5% because the change in selected target resulting from any rate departure less than -5% is the same as the change in selected target if the rate departure were identically -5%. Observed values outside the ranges selected for other vari- ables are treated similarly. To get the intermediate points at which no change in select- ed target occurs, decide what values of change in indicated tar- get, rate departure, and amount of business would lead to no change. Suppose that these values are 0%, 0%, and 1.6, respec- tively. The defining equations of the fuzzy sets for positive, zero, and negative changes in indicated target (chind) are, re- spectively, mp°sitive(chind)=max[ O'min(chind~O\ 15-0 ' 1)] [ (15-chind chind+!O~] mze~o(chind) = max 0,min -1-5-O ' 0 + 10 JJ mnesative(chind) = max [O'mJn ( l' O - +-l'O J ] " Similarly, the defining equations of the fuzzy sets for positive, zero, and negative rate departures are, respectively, [0, mJn f rd - 0 mposiave(rd) max [ (3-rd rd+5)l mze~o(rd) = max O,min 3-0' 0 + 5 = max l. (1' ()+5)1 ' [0, rain 0 - rd mnegaave(rd)
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`ADJUSTING INDICATED INSURANCE RATES 753 and the defining equations of the fuzzy sets for good, moderate, and bad amounts of business are, respectively, [0, min { bus--1.6 \ 1.8- i~'l)] mgood(bUs) max mm°derate(bUS) =max[O'min( l'8-busl.8 1.6'bus-1.6 1-~1"0)] mb~(bus)=max[O, min(1,1"6-bus~ ] 1.6- 1.0Jj " Finally, the change in selected target is given by [mal (chind, rd, bus). 10 + maz (chind, rd, bus). 0 + ma3 (chind, rd, bus). (- 10)] - [mal (chind, rd, bus) + maz(chind, rd, bus) + ma3 (chind, rd, bus)], (see Equation 2.1) in which mal (chind, rd, bus) = mJn[mpositive(chind), mpositive(rd), mgooa(bus)] m t2 ( chind, rd, bus) = min[mzero(chind), mzero(rd), mmoderate(bus)] ma3 (chind, rd, bus) = min [mnegative(chind), mnegative(rd), mbad(bus)]. (4.1) Figure 5 plots contours of the change in selected target against rate departure and amount of business while fixing the change in indicated target at +10%. Amount of business is along the vertical and rate departure lies along the horizontal. Note that the region for "no change" is relatively large. If the three variables--change in indicated target, rate depar- ture, and amount of business--may interact when connected by
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`754 ADJUSTING INDICATED INSURANCE RATES FIGURE 5 CONTOURS OF CHANGE IN SELECTED TARGET AGAINST AMOUNT OF BUSINESS AND RD (USING AND IN ALL THREE RULES) 1.8 1.6-- o 1.4-- o E < 1.2--
`--4 --'2 0 2 Rate Departure 1 ./.J and, then consider replacing the min operator with the algebraic product. Also other intersection operators may be used, including those that form weighted averages of the values of the member- ship functions. There are many ways to formulate the fuzzy rules, but an actuary should, at a minimum, check contour plots to see which formulation coincides with the philosophy or practices of the company. For example, in Figure 5, it should be verified that such a large area of no change is consistent with the com- pany's pricing philosophy when the change in the indicated tar- get is + 10%.
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`ADJUSTING INDICATED INSURANCE RATES 755 TABLE 1 Variable 1 Variable 2 Weighted Correlations Current Indicated Current Selected 0.952 Change in Indicated Change in Selected 0.812 Previous Selected Current Selected 0.909 Previous RD Change in Selected 0.281 Current RD Change in Selected 0.151 Previous Retention Current Selected -0.412 Previous Retention Change in Selected 0.236 Current Retention Current Selected -0.429 Current Retention Change in Selected -0.035 Actual/Planned Initial Current Selected -0.115 Actual/Planned Initial Change in Selected -0.149 min(Act/Plan,Plan/Act) Current Selected -0.270 min(Act/Plan,PlardAct) Change in Selected 0.005 4.2. Descriptive Modeling Turning to the descriptive portion of fuzzy modeling, fuzzy models are fit to the data that WCI provided. In selecting a tar- get, the actuaries consider the relative amount of business in each state. For this reason, the fuzzy models were fine-tuned by min- imizing a weighted sum of squared errors, as in Equation 3.1. The data for each period