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The Theory of Risk Classification

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Handbook of Insurance

Abstract

Risk Classification is the avenue through which insurance companies compete in order to reduce the cost of providing insurance contracts. While the underwriting incentives leading insurers to categorize customers according to risk status are straightforward, the social value of such activities is less clear. This chapter reviews the theoretical and empirical literature on risk classification, which demonstrates that the efficiency of permitting categorical discrimination in insurance contracting depends on the informational structure of the environment, and on whether insurance applicants become informed by the classification signal.

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Notes

  1. 1.

    See Crocker and Snow (1986) for references to U.S. Supreme Court rulings disallowing gender-based categorization in pensions, and to discussions of the laws and public policies related to categorization practices. Tabarrok (1994) provides further references to the policy and popular debate on categorical discrimination.

  2. 2.

    Even though the shape of the locus FA is ambiguous, concavity is guaranteed around F. Indeed, the slope of this locus (see Crocker and Snow 1986, page 448) is the right-hand side of condition (c) evaluated at δ = 0: \(\frac{\lambda (1-{p}^{H}){U}^{{\prime}}(W_{1}^{L})+(1-\lambda )(1-{p}^{L}){U}^{{\prime}}(W_{2}^{H})} {\lambda {p}^{H}{U}^{{\prime}}(W_{2}^{L})+(1-\lambda ){p}^{L}{U}^{{\prime}}(W_{2}^{H})}\). Since we have \(W_{1}^{H} = W_{2}^{H} = W_{1}^{L} = W_{2}^{L}\) at F, the slope can be rewritten as follows: \(\frac{\lambda (1-{p}^{H})+(1-\lambda )(1-{p}^{L})} {\lambda {p}^{H}+(1-\lambda ){p}^{L}}\). This reduces to \(\frac{1-\overline{p}} {\overline{p}}\), which is the slope of the aggregate zero-profit line. So the AF locus is tangent to the aggregate zero-profit line (see Dionne and Fombaron 1996).

  3. 3.

    This nomenclature arises because this is the particular allocation supported as an equilibrium in the analyses of Miyazaki (1977), Wilson (1977), and Spence (1978).

  4. 4.

    Figure 11.4 depicts the portion of the utilities possibilities frontier that is better for L-types than the pooling contract F. As discussed in Crocker and Snow (1985a), there is a symmetric portion of the frontier above the 45∘ line that is better for H-types.

  5. 5.

    Both Harris and Townsend (1981) and Myerson (1979) have demonstrated that no alternative organization of the economy’s allocation process can dominate the allocations attainable by a social planner.

  6. 6.

    So, for example, in the efficiency problem just considered, the goal of the social planner is to maximize the expected utility of one arbitrarily selected agent (V L) subject to the constraints of (1) not making the other agent worse off than a specified level of expected utility \(\bar{{V }}^{H}({V }^{H} \geq {\overline{V }}^{H})\); (2) the economy’s resource constraint (11.5); and (3) the informational constraints of the market participants (11.6). By varying \({\overline{V }}^{H}\), the entire set of (second-best) efficient allocations may be determined.

  7. 7.

    Since Hoy was concerned with comparing equilibrium allocations in the pre- and post-categorization regimes, the pertinent efficiency issue—can be the winners from categorization compensate, in principle, the losers—was not considered. As Crocker and Snow (1986) demonstrate, the answer to this question, at least in the case of the Miyazaki equilibrium, is that they can.

  8. 8.

    An actual Pareto improvement requires that at least one type of agent be made better off while no agents are made worse off. A potential Pareto improvement requires only that the winners from the regime change be able, in principle, to compensate the losers, so that the latter would be made no worse off from the move. As Crocker and Snow (1985b) have demonstrated, there exists a balanced-budget system of taxes and subsidies that can be applied by a government constrained by the same informational asymmetries as the market participants, and which can transform any potential Pareto improvement into an actual improvement. In the discussion that follows, we will use the term “Pareto improvement” to mean “potential Pareto improvement,” recognizing throughout that any potential improvements can be implemented as actual improvements.

  9. 9.

    Since the expected utility of an uninformed agent is \(\lambda {V }^{H} + (1{ \textendash }\lambda ){V }^{L}\) where V i represents the agent’s utility in the informational state i, the slope of the associated indifference curve is \(\mathrm{d}{V }^{H}/\mathrm{d}{V }^{L} = { \textendash }(1{ \textendash }\lambda )/\lambda\).

  10. 10.

    The Rothschild and Stiglitz allocation is the Pareto dominant member of the class of informationally consistent allocations, which is defined as the set of contracts that satisfy self-selection, and that each make zero profit given the class of customers electing to purchase them.

    While the analysis of the previous sections indicates that these allocations are not always elements of the efficient set (for some parameter configurations), we will, in the interests of expositional ease, assume that they are in the arguments that follow. This is without loss of generality, for in cases where cross-subsidization between risk types is required for efficiency, the same arguments will apply, except with the zero-profit loci relabeled to effect the desired level of subsidy.

  11. 11.

    The problem arises because the H-types have no insurable risks when p H = 1. Whenever p H ≠ 1, the allocations B and L depicted in Fig. 11.6 are non-degenerate (in the sense that they do not correspond with the origin). This holds even when p L = 0, although in this particular case the allocation L would reside on the horizontal axis. In contrast, when p H = 1, B and L necessarily correspond with the origin, so there are no insurance opportunities for the uninformed agent (since B is degenerate). This argument holds for any p L ≥ 0.

  12. 12.

    For example, the expected utility of α L-types is given by P(\(\beta {}^{2}\vert {\alpha }^{L})V ({p}^{2},\,\hat{H}) + P{(\beta {}^{1}\vert \alpha }^{L})V ({p}^{1}\), A), where the allocation \(\hat{H}\) is depicted in Fig. 11.11 below. Using the self-selection condition V(\({p}^{2},\,\hat{H}) = V ({p}^{2}\), A), we can rewrite this expression as P(\(\beta {}^{2}\vert {\alpha }^{L})V ({p}^{2},\,A) + P{(\beta {}^{1}\vert \alpha }^{L})V ({p}^{1}\), A), which is equal to V(p L, A) since \(P{(\beta {}^{2}\vert \alpha }^{L}){p}^{2}\; +\; P{(\beta {}^{1}\vert \alpha }^{L}){p}^{1}\; =\; {p}^{L}\). Thus, the pair (\(\hat{H}\), A) provides α L-types the same expected utility that they enjoy at A.

  13. 13.

    These profits could then be rebated to the consumers through lower premiums, so that they would be made strictly better off in the post β-experiment regime.

  14. 14.

    By construction in Fig. 11.13, the α L-types are indifferent between A, and observing the β-experiment followed by a selection of H 2 or A 1.

  15. 15.

    Marang-van de Mheen et al. (2002).

  16. 16.

    See Joly et al. (2010) for details. See also Hoy and Ruse (2005) for a discussion of the broader issues.

  17. 17.

    The result of Crocker and Snow (1992, p. 334) showing that public information always has positive social value applies in a linear signaling environment with risk neutral consumers, so the classification risk has no social cost.

  18. 18.

    In environments where risk class is not known by consumers, as in Sect. 11.2.2, the veil of ignorance is an actual veil with respect to risk class, leading to the same measure of consumer welfare.

  19. 19.

    Differentiating the zero-profit condition (11.22) with respect to λ and evaluating the result with λ = 0, while recognizing that p(0) = p L and I L(0) = D < I H(0) yields \(\partial p(\lambda )/\partial \lambda \vert _{\lambda =0} = ({p}^{H} - {p}^{L})[{I}^{H}(0)/D]\). Hence, the premium increases by (p H − p L)I H(0).

  20. 20.

    Hoy and Polborn (2000) obtain a yet stronger result showing that when some consumers are uninformed demanders in the life insurance market, social welfare can increase when they become informed. From an ex ante perspective, uninformed consumers gain from the opportunity to purchase insurance knowing the risk class to which they belong in a manner similar to the analysis in Sect. 11.4.2. Further, in the linear-pricing equilibrium, newly informed demanders may be less risky than the average of those initially in the market, in which case the equilibrium price declines to the benefit of all demanders.

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Authors and Affiliations

  1. Smeal College of Business, The Pennsylvania State University, University Park, PA, 16802-3603, USA

    Keith J. Crocker

  2. Department of Economics, University of Georgia, Athens, GA, 30602-6254, USA

    Arthur Snow

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  1. Keith J. Crocker
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  2. Arthur Snow
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Correspondence to Keith J. Crocker .

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Editors and Affiliations

  1. Finance, HEC Montreal, Montreal, Québec, Canada

    Georges Dionne

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Crocker, K.J., Snow, A. (2013). The Theory of Risk Classification. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_11

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