The Value of Information in Insurance Pricing: Comment

Abstract

Introduction In estimating the expected losses underlying an insurance premium, the insurer must consider both actuarial issues, such as the accuracy of prior information and historical data, and market effects, such as the impact of adverse selection. To analyze these types of issues, Gogol (1993) proposes a sequential model of the estimation of expected losses. Gogol's analysis is based largely upon two mathematical propositions, which he uses to provide insights into various aspects of insurance pricing. In this note it is shown that Gogol's Proposition 2 is incorrect. After providing a corrected version of this result, it is further demonstrated that two general statements made by the author are also not true. The Basic Model and Results To summarize Gogol's basic model of the estimation process, let [Lambda] = expected losses (the unknown quantity to be estimated),(1) m = the prior estimate of [Lambda], X = a subsequent estimator of [Lambda], [X.sub.1] = ln([Lambda]/m), [X.sub.2] = ln([Lambda]/X), [X.sub.1], [X.sub.2] [similar to] Bivariate Normal with E[[X.sub.1]] = E[[X.sub.2]] = 0, Var[[X.sub.1]] = [[[Sigma].sub.1].sup.2], Var[[X.sub.2]] = [[[Sigma].sub.3].sup.2] and Corr[[X.sub.1], [X.sub.2]] = [Rho]. Using this framework, and the notation [v.sup.2] = [[[Sigma].sub.1].sup.2] + [[[Sigma].sub.2].sup.2] - 2[Rho][[Sigma].sub.1][[Sigma].sub.2], the author's two propositions may be stated as follows. Proposition 1 Given that X = x, the posterior probability distribution of [Lambda] has mean E[[Lambda][where]X = x] = m exp([Mu] + [[Sigma].sub.2]/2) and variance Var[[Lambda][where]X = x] = [m.sup.2] exp(2[Mu] + [[Sigma].sup.2]) [exp([[Sigma].sup.2]) - 1], where [Mu] = [([[[Sigma].sub.1].sup.2] - [Rho][[Sigma].sub.1][[Sigma].sub.2])ln(x/m)]/[v.sup.2] and [[Sigma].sub.2] and [[Sigma].sup.2] = (1-[[Rho].sup.2])[[[Sigma].sub.1].sup.2][[[Sigma].sub.2].sup.2]/[v.sub.2] Proposition 2 Given that X [is less than or equal to] E, where E is a positive constant, the posterior distribution of [Lambda] has mean E[[Lambda][where]X [is less than or equal to] E] = m exp[([[Sigma].sup.2]+[a.sup.2])/2] [Phi][(1/v)ln(E/m) - a]/[Phi][(1/v)ln(E/m)], where a = [([[Sigma].sub.1] - [Rho][[Sigma].sub.2]).sup.2]/v. Also, Pr{X [is less than or equal to] E} = [Phi]{(1/v)ln(E/m)]. As noted above, Proposition 2 is incorrect. The following is a corrected version of this result. Proposition 2' Given that X [is less than or equal to] E, where E is a positive constant, the posterior distribution of [Lambda] has mean E[[Lambda][where]X [is less than or equal to] E] = m exp([[[Sigma].sub.1].sup.2]/2) [Phi][(1/v)ln(E/m) - [Alpha]]/[Phi][(1/v)ln(E/m)], where [Alpha] = ([[[Sigma].sub.1].sup.2] - [Rho][[Sigma].sub.1][[Sigma].sub.2]) Also, Pr{X [is less than or equal to] E} = [Phi][(1/v)ln(E/m)]. The source of error in Proposition 2 is found in its proof, offered in the Appendix to Gogol's article. In this proof, he makes the substitution t = (1/v)ln(x/m) in the expression for E[[Lambda][where]X = x] given by Proposition 1 and finds that [Mu] = vt [([[Sigma].sub.1] - [Rho][[Sigma].sub.2]).sup.2]/[v.sup.2], when in fact [Mu] = vt ([[[Sigma].sub.1].sup.2] - [Phi][[Sigma].sub.1][[Sigma].sub.2])/[v.sup.2]. By correcting this error, and following all of the remaining steps of the author's proof, one easily obtains the result stated in Proposition 2'.(2) New Insights Following his presentation of Propositions 1 and 2, Gogol provides examples in which he applies these mathematical results to the analysis of pricing problems. In the course of these examples, he makes two general statements that may be disproved by counterexamples. In Example 1, Gogol considers the effect of the accuracy of the estimator X on an insurer's pricing. …