An Overview of Insurance Uses of Fuzzy Logic
`
`Arnold F. Shapiro
`
`Smeal College of Business, Penn State University, University Park, PA 16802
`afs1@psu.edu
`
`It has been twenty-five years since DeWit(1982) first applied fuzzy logic (FL) to in-
`surance. That article sought to quantify the fuzziness in underwriting. Since then, the
`universe of discourse has expanded considerably and now also includes FL applica-
`tions involving classification, projected liabilities, future and present values, pricing,
`asset allocations and cash flows, and investments. This article presents an overview
`of these studies. The two specific purposes of the article are to document the FL
`technologies have been employed in insurance-related areas and to review the FL
`applications so as to document the unique characteristics of insurance as an applica-
`tion area.
`
`Key words: Actuarial, Fuzzy Logic, Fuzzy Sets, Fuzzy Arithmetic, Fuzzy Inference
`Systems, Fuzzy Clustering, Insurance
`
`1 Introduction
`
`The first article to use fuzzy logic (FL) in insurance was [29] 1, which sought to quan-
`tify the fuzziness in underwriting. Since then, the universe of discourse has expanded
`considerably and now includes FL applications involving classification, underwrit-
`ing, projected liabilities, future and present values, pricing, asset allocations and cash
`flows, and investments.
`This article presents an overview of these FL applications in insurance. The spe-
`cific purposes of the article are twofold: first, to document the FL technologies have
`been employed in insurance-related areas; and, second, to review the FL applications
`so as to document the unique characteristics of insurance as an application area.
`
`1 While DeWit was the first to write an article that gave an explicit example of the use of
`FL in insurance, FL, as it related to insurance, was a topic of discussion at the time. Ref-
`erence [43], for example, remarked that “... not all expert knowledge is a set of “black and
`white” logic facts - much expert knowledge is codifiable only as alternatives, possibles,
`guesses and opinions (i.e., as fuzzy heuristics).”
`
`Liberty Mutual Exhibit 1024
`Liberty Mutual v. Progressive
`CBM2012-00002
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`26
`
`Arnold F. Shapiro
`
`Before continuing, the term FL needs to be clarified. In this article, we generally
`follow the lead of Zadeh and use the term FL in its wide sense. According to [85],
`
`Fuzzy logic (FL), in its wide sense, has four principal facets. First, the log-
`ical facet, FL/L, [fuzzy logic in its narrow sense], is a logical system which
`underlies approximate reasoning and inference from imprecisely defined
`premises. Second, the set-theoretic facet, FL/S, is focused on the theory
`of sets which have unsharp boundaries, rather than on issues which relate
`to logical inference, [examples of which are fuzzy sets and fuzzy mathe-
`matics]. Third is the relational facet, FL/R, which is concerned in the main
`with representation and analysis of imprecise dependencies. Of central im-
`portance in FL/R are the concepts of a linguistic variable and the calculus
`of fuzzy if-then rules. Most of the applications of fuzzy logic in control and
`systems analysis relate to this facet of fuzzy logic. Fourth is the epistemic
`facet of fuzzy logic, FL/E, which is focused on knowledge, meaning and
`imprecise information. Possibility theory is a part of this facet.
`
`The methodologies of the studies reviewed in this article cover all of these FL
`facets. The term “fuzzy systems” also is used to denote these concepts, as indicated
`by some of the titles in the reference section of this paper, and will be used inter-
`changeably with the term FL.
`The next section of this article contains a brief overview of insurance application
`areas. Thereafter, the article is subdivided by the fuzzy techniques 2 shown in Fig. 1.
`
`Fig. 1. Fuzzy Logic
`
`2 This article could have been structured by fuzzy technique, as was done by [75] or by
`insurance topic, as was done by [28] and [66]. Given the anticipated audience, the former
`structure was adopted.
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`An Overview of Insurance Uses of Fuzzy Logic
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`27
`
`As indicated, the topics covered include fuzzy set theory, fuzzy numbers, fuzzy
`arithmetic, fuzzy inference systems, fuzzy clustering, fuzzy programming, fuzzy re-
`gression, and soft computing. Each section begins with a brief description of the
`technique3 and is followed by a chronological review of the insurance applications
`of that technique. When an application involves more than one technique, it is only
`discussed in one section. The article ends with a comment regarding future insurance
`applications of FL.
`
`2 Insurance Application Areas
`
`The major application areas of insurance include classification, underwriting, pro-
`jected liabilities, ratemaking and pricing, and asset allocations and investments. In
`this section, we briefly describe each of these areas so that readers who are unfamiliar
`with the insurance field will have a context for the rest of the paper.
`
`2.1 Classification
`
`Classification is fundamental to insurance. On the one hand, classification is the pre-
`lude to the underwriting of potential coverage, while on the other hand, risks need to
`be properly classified and segregated for pricing purposes. Operationally, risk may be
`viewed from the perspective of the four classes of assets (physical, financial, human,
`intangible) and their size, type, and location.
`
`2.2 Underwriting
`
`Underwriting is the process of selection through which an insurer determines which
`of the risks offered to it should be accepted, and the conditions and amounts of the
`accepted risks. The goal of underwriting is to obtain a safe, yet profitable, distribution
`of risks. Operationally, underwriting determines the risk associated with an applicant
`and either assigns the appropriate rating class for an insurance policy or declines to
`offer a policy.
`
`2.3 Projected Liabilities
`
`In the context of this article, projected liabilities are future financial obligations that
`arise either because of a claim against and insurance company or a contractual ben-
`efit agreement between employers and their employees. The evaluation of projected
`liabilities is fundamental to the insurance and employee benefit industry, so it is not
`surprising that we are beginning to see SC technologies applied in this area.
`
`3 Only a cursory review of the FL methodologies is discussed in this paper. Readers who
`prefer a more extensive introduction to the topic, with an insurance perspective, are referred
`to [56]. Those who are interested in a comprehensive introduction to the topic are referred
`to [90] and [32]. Readers interested in a grand tour of the first 30 years of fuzzy logic are
`urged to read the collection of Zadeh’s papers contained in [74] and [45].
`
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`28
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`Arnold F. Shapiro
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`2.4 Ratemaking and Pricing
`
`Ratemaking and pricing refer to the process of establishing rates used in insurance
`or other risk transfer mechanisms. This process involves a number of considerations
`including marketing goals, competition and legal restrictions to the extent they affect
`the estimation of future costs associated with the transfer of risk. Such future costs
`include claims, claim settlement expenses, operational and administrative expenses,
`and the cost of capital.
`
`2.5 Asset Allocation and Investments
`
`The analysis of assets and investments is a major component in the management
`of an insurance enterprise. Of course, this is true of any financial intermediary, and
`many of the functions performed are uniform across financial companies. Thus, in-
`surers are involved with market and individual stock price forecasting, the forecast-
`ing of currency futures, credit decision-making, forecasting direction and magnitude
`of changes in stock indexes, and so on.
`
`3 Linguistic Variables and Fuzzy Set Theory
`
`Linguistic variables are the building blocks of FL. They may be defined ([82], [83])
`as variables whose values are expressed as words or sentences. Risk capacity, for
`example, a common concern in insurance, may be viewed both as a numerical value
`ranging over the interval [0,100%], and a linguistic variable that can take on values
`like high, not very high, and so on. Each of these linguistic values may be interpreted
`as a label of a fuzzy subset of the universe of discourse X = [0,100%], whose base
`variable, x, is the generic numerical value risk capacity. Such a set, an example of
`which is shown in Fig. 2, is characterized by a membership function (MF), µ high(x)
`here, which assigns to each object a grade of membership ranging between zero and
`one.
`
`Fig. 2. (Fuzzy) Set of Clients with High Risk Capacity
`
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`An Overview of Insurance Uses of Fuzzy Logic
`
`29
`
`In this case, which represents the set of clients with a high risk capacity, individ-
`uals with a risk capacity of 50 percent, or less, are assigned a membership grade of
`zero and those with a risk capacity of 80 percent, or more, are assigned a grade of
`one. Between those risk capacities, (50%, 80%), the grade of membership is fuzzy.
`In addition to the S-shaped MF depicted in Fig. 2, insurance applications also
`employ the triangular, trapezoidal, Gaussian, and generalized bell classes of MFs.
`As with other areas of application, fuzzy sets are implemented by extending many of
`the basic identities that hold for ordinary sets.
`
`3.1 Applications
`
`This subsection presents an overview of some insurance applications of linguistic
`variables and fuzzy set theory. The topics addressed include: earthquake insurance,
`optimal excess of loss retention in a reinsurance program, the selection of a “good”
`forecast, where goodness is defined using multiple criteria that may be vague or
`fuzzy, resolve statistical problems involving sparse, high dimensional data with cat-
`egorical responses, the definition and measurement of risk from the perspective of a
`risk manager, and deriving an overall disability Index.
`An early study was by [7], who used pattern recognition and FL in the evalua-
`tion of seismic intensity and damage forecasting, and for the development of models
`to estimate earthquake insurance premium rates and insurance strategies. The influ-
`ences on the performance of structures include quantifiable factors, which can be
`captured by probability models, and nonquantifiable factors, such as construction
`quality and architectural details, which are best formulated using fuzzy set models.
`For example, he defined the percentage of a building damaged by an earthquake
`by fuzzy terms such as medium, severe and total, and represented the membership
`functions of these terms as shown in Fig. 3.4
`
`Fig. 3. MFs of Building Damage
`
`Two methods of identifying earthquake intensity were presented and compared.
`The first method was based on the theory of pattern recognition where a discrimina-
`
`4 Adapted from [7, Figure 6.3].
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`30
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`Arnold F. Shapiro
`
`tive function was developed using Bayes’ criterion and the second method applied
`FL.
`
`Reference [49] envisioned the decision-making procedure in the selection of an
`optimal excess of loss retention in a reinsurance program as essentially a maximin
`technique, similar to the selection of an optimum strategy in noncooperative game
`theory. As an example, he considered four decision variables (two goals and two
`constraints) and their membership functions: probability of ruin, coefficient of varia-
`tion, reinsurance premium as a percentage of cedent’s premium income (Rel. Reins.
`Prem.) and deductible (retention) as a percentage of cedent’s premium income (Rel.
`Deductible). The grades of membership for the decision variables (where the vertical
`lines cut the MFs) and their degree of applicability (DOA), or rule strength, may be
`represented as shown Fig. 4.5
`
`Fig. 4. Retention Given Fuzzy Goals and Constraints
`
`In the choice represented in the figure, the relative reinsurance premium has the
`minimum membership value and defines the degree of applicability for this particular
`excess of loss reinsurance program. The optimal program is the one with the highest
`degree of applicability.
`Reference [22, p. 434] studied fuzzy trends in property-liability insurance claim
`costs as a follow-up to their assertion that “the actuarial approach to forecasting is
`rudimentary.” The essence of the study was that they emphasized the selection of
`a “good” forecast, where goodness was defined using multiple criteria that may be
`vague or fuzzy, rather than a forecasting model. They began by calculating several
`possible trends using accepted statistical procedures 6 and for each trend they deter-
`mined the degree to which the estimate was good by intersecting the fuzzy goals of
`historical accuracy, unbiasedness and reasonableness.
`
`5 Adapted from [49, Figure 2].
`6 Each forecast method was characterized by an estimation period, an estimation technique,
`and a frequency model. These were combined with severity estimates to obtain pure pre-
`mium trend factors. ([22, Table 1])
`
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`An Overview of Insurance Uses of Fuzzy Logic
`
`31
`
`The flavor of the article can be obtained by comparing the graphs in Fig. 5, which
`show the fuzzy membership values for 30 forecasts 7 according to historical accuracy
`(goal 1), ordered from best to worst, and unbiasedness (goal 2), before intersection,
`graph (a) and after intersection, graph (b).
`
`Fig. 5. The Intersection of Historical Accuracy and Unbiasedness
`
`They suggested that one may choose the trend that has the highest degree of
`goodness and proposed that a trend that accounts for all the trends can be calculated
`by forming a weighted average using the membership degrees as weights. They con-
`cluded that FL provides an effective method for combining statistical and judgmental
`criteria in insurance decision-making.
`Another interesting aspect of the [22] study was their α-cut for trend factors,
`which they conceptualized in terms of a multiple of the standard deviation of the
`trend factors beyond their grand mean. In their analysis, an α-cut corresponded to
`only including those trend factors within 2(1 − α) standard deviations.
`A novel classification issue was addressed by [51], who used FST to resolve sta-
`tistical problems involving sparse, high dimensional data with categorical responses.
`They began with a concept of extreme profile, which, for the health of the elderly,
`two examples might be “active, age 50” and “frail, age 100.”” From there, their focus
`was on gik, a grade of membership (GoM) score that represents the degree to which
`the i-th individual belongs to the k-th extreme profile in a fuzzy partition, and they
`presented statistical procedures that directly reflect fuzzy set principles in the estima-
`tion of the parameters. In addition to describing how the parameters estimated from
`7 Adapted from [22, Figures 2 and 3], which compared the membership values for 72 fore-
`casts.
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`32
`
`Arnold F. Shapiro
`
`the model may be used to make various types of health forecasts, they discussed how
`GoM may be used to combine data from multiple sources and they analyzed multiple
`versions of fuzzy set models under a wide range of empirical conditions.
`Reference [41] investigated the use of FST to represent uncertainty in both the
`definition and measurement of risk, from the perspective of a risk manager. His con-
`ceptualization of exposure analysis is captured in Fig. 6 8, which is composed of a
`fuzzy representation of (a) the perceived risk, as a contoured function of frequency
`and severity, (b) the probability of loss, and (c) the risk profile.
`
`Fig. 6. Fuzzy Risk Profile Development
`
`The grades of membership vary from 0 (white) to 1 (black); in the case of the
`probability distribution, the black squares represent point estimates of the probabili-
`ties. The risk profile is the intersection of the first two, using only the min operator.
`He concluded that FST provides a realistic approach to the formal analysis of risk.
`Reference [42] examined the problems for risk managers associated with knowl-
`edge imperfections, under which model parameters and measurements can only be
`specified as a range of possibilities, and described how FL can be used to deal with
`such situations. However, unlike [41], not much detail was provided.
`The last example of this section is from the life and health area. Reference [19]
`presented a methodology for deriving an Overall Disability Index (ODI) for mea-
`suring an individual’s disability. Their approach involved the transformation of the
`ODI derivation problem into a multiple-criteria decision-making problem. Essen-
`tially, they used the analytic hierarchy process, a multicriteria decision making tech-
`nique that uses pairwise comparisons to estimate the relative importance of each risk
`factor ([60]), along with entropy theory and FST, to elicit the weights among the
`attributes and to aggregate the multiple attributes into a single ODI measurement.
`
`4 Fuzzy Numbers and Fuzzy Arithmetic
`
`Fuzzy numbers are numbers that have fuzzy properties, examples of which are the
`notions of “around six percent” and “relatively high”. The general characteristic of a
`
`8 Adapted from [41, Figures 7, 8 and 9]
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`An Overview of Insurance Uses of Fuzzy Logic
`
`33
`
`fuzzy number ([82] and [31]) often is represented as shown in Fig. 7, although any of
`the MF classes, such as Gaussian and generalized bell, can serve as a fuzzy number,
`depending on the situation.
`
`Fig. 7. Flat Fuzzy Number
`
`This shape of a fuzzy number is referred to as trapezoidal or “flat” and its MF
`often is denoted as (a1,a2,a3,a4) or (a1/a2, a3/a4); when a2 is equal to a3, we get
`the triangular fuzzy number. A fuzzy number is positive if a 1 ≥ 0 and negative if
`a4 ≤ 0, and, as indicated, it is taken to be a convex fuzzy subset of the real line.
`
`4.1 Fuzzy Arithmetic
`
`As one would anticipate, fuzzy arithmetic can be applied to the fuzzy numbers. Using
`the extension principle ([82]), the nonfuzzy arithmetic operations can be extended to
`incorporate fuzzy sets and fuzzy numbers 9. Briefly, if ∗ is a binary operation such as
`addition (+), min (∧), or max (∨), the fuzzy number z, defined by z = x∗ y, is given
`as a fuzzy set by
`
`µz (w) = ∨u,v µx (u) ∧ µy (v) , u, v, w ∈ (cid:11)
`(1)
`subject to the constraint that w = u∗v, where µx , µy, and µz denote the membership
`functions of x, y, and z, respectively, and ∨ u,v denotes the supremum over u, v.10
`A simple application of the extension principle is the sum of the fuzzy numbers
`A and B, denoted by A ⊕ B = C, which has the membership function:
`
`9 Fuzzy arithmetic is related to interval arithmetic or categorical calculus, where the opera-
`tions use intervals, consisting of the range of numbers bounded by the interval endpoints,
`as the basic data objects. The primary difference between the two is that interval arith-
`metic involves crisp (rather than overlapping) boundaries at the extremes of each interval
`and it provides no intrinsic measure (like membership functions) of the degree to which a
`value belongs to a given interval. Reference [2] discussed the use interval arithmetic in an
`insurance context.
`10 See [90, Chap. 5], for a discussion of the extension principle.
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`34
`
`Arnold F. Shapiro
`
`µc(z) = max{min [µA (x) , µB (y)] : x + y = z}
`The general nature of the fuzzy arithmetic operations is depicted in Fig. 8 for
`A = (−1, 1, 3) and B = (1, 3, 5)11.
`
`(2)
`
`Fig. 8. Fuzzy Arithmetic Operations
`
`The first row shows the two membership functions A and B and their sum; the
`second row shows their difference and their ratio; and the third row shows their
`product.
`
`4.2 Applications
`
`This subsection presents an overview of insurance applications involving fuzzy arith-
`metic. The topics addressed include: the fuzzy future and present values of fuzzy cash
`amounts, using fuzzy interest rates, and both crisp and fuzzy periods; the computa-
`tion of the fuzzy premium for a pure endowment policy; fuzzy interest rate whose
`fuzziness was a function of duration; net single premium for a term insurance; the
`effective tax rate and after-tax rate of return on the asset and liability portfolio of a
`property-liability insurance company; cash-flow matching when the occurrence dates
`are uncertain; and the financial pricing of property-liability insurance contracts.
`Reference [10] appears to have been the first author to address the fuzzy time-
`value-of-money aspects of actuarial pricing, when he investigated the fuzzy future
`and present values of fuzzy cash amounts, using fuzzy interest rates, and both crisp
`
`11 This figure is similar to [55, Figure 18, p. 157], after correcting for an apparent discrepancy
`in their multiplication and division representations.
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`An Overview of Insurance Uses of Fuzzy Logic
`
`35
`
`and fuzzy periods. His approach, generally speaking, was based on the premise that
`“the arithmetic of fuzzy numbers is easily handled when x is a function of y.” ([10,
`p. 258]) For a flat fuzzy number and straight line segments for µ A(x) on [a1, a2] and
`[a3, a4], this can be conceptualized as shown in Fig. 9
`
`Fig. 9. MF and Inverse MF
`
`where f1(y|A) = a1 + y(a2 − a1) and f2(y|A) = a4 − y(a4 − a3). The points aj,
`j = 1, 2, 3, 4, and the functions fj(y|A), j = 1, 2, “A” a fuzzy number, which are
`inverse functions mapping the membership function onto the real line, characterize
`the fuzzy number.
`If the investment is A and the interest rate per period is i, where both values are
`fuzzy numbers, he showed that the accumulated value (S n), a fuzzy number, after n
`periods, a crisp number, is
`
`Sn = A ⊗ (1 ⊕ i)n
`because, for positive fuzzy numbers, multiplication distributes over addition and is
`associative. It follows that the membership function for S n takes the form
`µ(x|Sn) = (sn1, fn1(y|Sn)/sn2, sn3/f n2(y|Sn), sn4)
`where, for j = 1, 2,
`
`(4)
`
`(3)
`
`fnj(y|Sn) = fj(y|A) · (1 + fj(y|i))n
`and can be represented in a manner similar to Fig. 9, except that a j is replaced with
`Snj.
`Then, using the extension principle ([31]), he showed how to extend the analysis
`to include a fuzzy duration.
`Buckley then went on to extend the literature to fuzzy discounted values and
`fuzzy annuities. In the case of positive discounted values, he showed ([10, pp. 263-
`4]) that:
`
`(5)
`
`If S > 0 then P V2(S, n) exists; otherwise it may not, where :
`P V2(S, n) = A iif A is a fuzzy number and A = S ⊗ (1 ⊕ i)−n
`
`(6)
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`36
`
`Arnold F. Shapiro
`
`The essence of his argument was that this function does not exist when using it
`leads to contradictions such as a2 < a1 or a4 < a3.
`The inverse membership function of PV 2(S, n) is:
`fj(y|A) = fj(y|S) · (1 + f3−j(y|i))
`−n
`Both the accumulated value and the present value of fuzzy annuities were dis-
`cussed.12
`Reference [49], using [10] as a model, discussed the computation of the fuzzy
`premium for a pure endowment policy using fuzzy arithmetic. Figure 10 is an adap-
`tation of his representation of the computation.
`
`, j = 1, 2
`
`(7)
`
`Fig. 10. Fuzzy Present Value of a Pure Endowment
`
`As indicated, the top left figure represents the MF of the discounted value after
`ten years at the fuzzy effective interest rate per annum of (.03, .05, .07, .09), while
`the top right figure represents the MF of 10p55, the probability that a life aged 55 will
`survive to age 65. The figure on the bottom represents the MF for the present value
`of the pure endowment.
`Reference [56, pp. 29-38] extended the pure endowment analysis of [49]. First,
`he incorporated a fuzzy interest rate whose fuzziness was a function of duration.
`This involved a current crisp rate of 6 percent, a 10-year Treasury Note yield of 8
`percent, and a linearly increasing fuzzy rate between the two. Figure 11 shows a
`conceptualization of his idea.
`Then he investigated the more challenging situation of a net single premium for
`a term insurance, where the progressive fuzzification of rates plays a major role.
`Along the same lines, [72] explored the membership functions associated with
`the net single premium of some basic life insurance products assuming a crisp moral-
`ity rate and a fuzzy interest rate. Their focus was on α-cuts, and, starting with a
`
`12 While not pursued here, the use of fuzzy arithmetic in more general finance applications
`can be found in [12] and [68].
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`An Overview of Insurance Uses of Fuzzy Logic
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`37
`
`Fig. 11. Fuzzy Interest Rate
`
`fuzzy interest rate, they gave fuzzy numbers for such products as term insurance
`and deferred annuities, and used the extension principle to develop the associated
`membership functions.
`Reference [26] and [27] illustrated how FL can be used to estimate the effective
`tax rate and after-tax rate of return on the asset and liability portfolio of a property-
`liability insurance company. They began with the observation that the effective tax
`rate and the risk-free rate fully determine the present value of the expected investment
`tax liability. This leads to differential tax treatment for stocks and bonds, which,
`together with the tax shield of underwriting losses, determine the overall effective
`tax rate for the firm. They then argued that the estimation of the effective tax rate
`is an important tool of asset-liability management and that FL is the appropriate
`technology for this estimation.
`The essence of their paper is illustrated in Fig. 12 13, which shows the mem-
`bership functions for the fuzzy investment tax rates of a beta one company 14, with
`assumed investments, liabilities and underwriting profit, before and after the effect
`of the liability tax shield.
`
`Fig. 12. Fuzzy Interest Rate
`
`13 Adapted from [27, Figure 1]
`14 A beta one company has a completely diversified stock holding, and thus has the same
`amount of risk (β = 1) as the entire market.
`
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`38
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`Arnold F. Shapiro
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`As suggested by the figure, in the assets-only case, the non-fuzzy tax rate is
`32.4 percent, but when the expected returns of stocks, bonds, dividends and capital
`gains are fuzzified, the tax rate becomes the fuzzy number (31%, 32.4%, 32.4%,
`33.6%). A similar result occurs when both the assets and liabilities are considered.
`The authors conclude that, while the outcomes generally follow intuition, the benefit
`is the quantification, and graphic display, of the uncertainty involved.
`Reference [8] investigates the use of Zadeh’s extension principle for transform-
`ing crisp financial concepts into fuzzy ones and the application of the methodology
`to cash-flow matching. They observer that the extension principle allows them to
`rigorously define the fuzzy equivalent of financial and economical concepts such as
`duration and utility, and to interpret them. A primary contribution of their study was
`the investigation of the matching of cash flows whose occurrence dates are uncertain.
`The final study of this section is [23], who used FL to address the financial pric-
`ing of property-liability insurance contracts. Observing that much of the information
`about cash flows, future economic conditions, risk premiums, and other factors af-
`fecting the pricing decision is subjective and thus difficult to quantify using conven-
`tional methods, they incorporated both probabilistic and nonprobabilistic types of
`uncertainty in their model. The authors focused primarily on the FL aspects needed
`to solve the insurance-pricing problem, and in the process “fuzzified” a well-known
`insurance financial pricing model, provided numerical examples of fuzzy pricing,
`and proposed fuzzy rules for project decision-making. Their methodology was based
`on Buckley’s inverse membership function (See Fig. 9 and related discussion).
`Figure 13 shows their conceptualization of a fuzzy loss, the fuzzy present value
`of that loss, and the fuzzy premium, net of fuzzy taxes, using a one-period model. 15
`
`Fig. 13. Fuzzy Premium
`
`They concluded that FL can lead to significantly different pricing decisions than
`the conventional approach.
`
`15 Adapted from [23], Fig. 5.
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`An Overview of Insurance Uses of Fuzzy Logic
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`39
`
`5 Fuzzy Inference Systems
`
`The fuzzy inference system (FIS) is a popular methodology for implementing FL.
`FISs are also known as fuzzy rule based systems, fuzzy expert systems (FES), fuzzy
`models, fuzzy associative memories (FAM), or fuzzy logic controllers when used
`as controllers ([40, p. 73]), although not everyone agrees that all these terms are
`synonymous. Reference [5, p.77], for example, observes that a FIS based on IF-
`THEN rules is practically an expert system if the rules are developed from expert
`knowledge, but if the rules are based on common sense reasoning then the term
`expert system does not apply. The essence of a FIS can be represented as shown in
`Fig. 14.16
`
`Fig. 14. Fuzzy Inference System (FIS)
`
`As indicated in the figure, the FIS can be envisioned as involving a knowledge
`base and a processing stage. The knowledge base provides MFs and fuzzy rules
`needed for the process. In the processing stage, numerical crisp variables are the
`input of the system.17 These variables are passed through a fuzzification stage where
`they are transformed to linguistic variables, which become the fuzzy input for the
`inference engine. This fuzzy input is transformed by the rules of the inference engine
`to fuzzy output. These linguistic results are then changed by a defuzzification stage
`into numerical values that become the output of the system.
`The Mamdani FIS has been the most commonly mentioned FIS in the insurance
`literature, and most often the t-norm and the t-conorm are the min-operator and max-
`
`16 Adapted from [58], Fig. 2.
`17 In practice, input and output scaling factors are often used to normalize the crisp inputs and
`outputs. Also, the numerical input can be crisp or fuzzy. In this latter event, the input does
`not have to be fuzzified.
`
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`
`

`
`40
`
`Arnold F. Shapiro
`
`operator, respectively.18 Commonly, the centre of gravity (COG) approach is used for
`defuzzification.
`
`5.1 Applications
`
`This subsection presents an overview of insurance applications of FISs. In most in-
`stances, as indicated, an FES was used. The application areas include: life and health
`underwriting; classification; modeling the selection process in group health insur-
`ance; evaluating risks, including occupational injury risk; pricing group health in-
`surance using fuzzy ancillary data; adjusting workers compensation insurance rates;
`financial forecasting; and budgeting for national health care.
`As mentioned above, the first recognition that fuzzy systems could be applied
`to the problem of individual insurance underwriting was due to [29]. He recognized
`that underwriting was subjective and used a basic form of the FES to analyze the
`underwriting practice of a life insurance company.
`Using what is now a common approach, he had underwriters evaluate 30 hypo-
`thetical life insurance applications and rank them on the basis of various attributes.
`He then used this information to create the five membership functions: technical as-
`pects (µt), health (µh), profession (µp), commercial (µc), and other (µo). Table 1
`shows DeWit’s conceptualization of the fuzzy set “technical aspects.”
`
`Table 1. Technical Aspects
`
`Description Example
`
`Fuzzy value
`
`remunerative, good policy
`good
`unattractive policy provisions
`moderate
`sum insured does not match wealth of insured
`bad
`impossible child inappropriately insured for large amount
`
`1.0
`0.5
`0.2
`0.0
`
`Next, by way of example, he combined these membership functions and an array
`of fuzzy set operations into a fuzzy expert underwriting system, using the formula:
`(cid:10)
`(cid:12)[1−max(0,µc−0.5)]
`√
`(cid:11)
`µpµ2
`2 min (0.5, µc)
`o
`
`(8)
`
`W =
`
`I(µt)µh
`
`where intensification (I(µt)) increases the grade of membership for membership
`√
`functions above some value (often 0.5) and decreases it otherwise, concentration
`(µ2
`µp) increases the grade of
`o) reduces the grade of membership, and dilation (
`membership. He then suggested hypothetical underwriting decision rules related to
`the values of W.19
`
`18 Reference [35] shows that many copulas can serve as t-norms.
`19 The hypothetical decision rules took the form:
`0.0 ≤ W < 0.1 refuse
`0.1 ≤ W < 0.3 try to improve the condition, if not possible: refuse
`
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`An Overview of Insurance Uses of Fuzzy Logic
`
`41
`
`Reference [49] used a FES to provide a flexible definition of a preferred pol-
`icyholder in life insurance. As a part of this effort, he extended the insurance un-
`derwriting literature in three ways: he used continuous membership functions; he
`extended the definition of intersection to include the bounded difference, Hamacher
`and Yager operators; and he showed how α-cuts could be implemented to refine the
`decision rule for the minimum operator, where the α-cuts is applied to each mem-
`bership function, and the algebraic product, where the minimum acceptable product
`is equal to the α-cut. Whereas [29] focused on technical and behavioral features,
`Lemaire focused on t