Feasible fraud and auditing probabilities for insurance companies and policyholders

Abstract

Insurance claims fraud is counted among the major concerns in the insurance industry, the reason being that excess payments due to fraudulent claims account for a large percentage of the total payments each year. We formulate optimization problems from the insurance company as well as the policyholder perspective based on a costly state verification approach. In this setting—while the policyholder observes his losses privately—the insurance company can decide to verify the truthfulness of incoming claims at some cost. We show simulation results illustrating the agreement range which is characterized by all valid fraud and auditing probability combinations both stakeholders are willing to accept. Furthermore, we present the impact of different valid probability combinations on the insurance company’s and the policyholder’s objective quantities and analyze the sensitivity of the agreement range with respect to a relevant input parameter. This contribution summarizes the major findings of a working paper written by Müller et al. (Working Papers on Risk Management and Insurance (IVW-HSG), No. 92, 2011).

Zusammenfassung

Die Bekämpfung von Versicherungsbetrug gehört zu den zentralen Herausforderungen in der Versicherungswirtschaft. Für den Versicherungsnehmer besteht regelmässig die Möglichkeit einer Falschangabe bezüglich der tatsächlichen Schadenhöhe. Das Versicherungsunternehmen behält sich vor, den Wahrheitsgehalt eingehender Forderungen zu überprüfen. Im vorliegenden Beitrag werden zulässige Betrugs- sowie Prüfwahrscheinlichkeiten aus der Sicht beider Vertragspartner hergeleitet und in Form eines Einigungsbereichs (in einem solchen sind Versicherungsnehmer und -unternehmen bereit, einen Vertrag abzuschliessen) illustriert. Auf dieser Grundlage wird im Anschluss das jeweilige optimale Verhalten ermittelt. Zusätzlich wird der Einfluss relevanter Parameter auf Form und Ausmass des Einigungsbereiches analysiert. Der vorliegende Beitrag stellt eine Zusammenfassung der zentralen Erkenntnisse aus einem Arbeitspapier von Müller et al. (Working Papers on Risk Management and Insurance (IVW-HSG), No. 92, 2011) dar.

Appendix: Proof of Proposition 1

• (i) Using (1), (2) and the assumptions B=θ, $\hat{\theta}= \alpha\theta$ with α≥1, we get

  

  (24)

  Deriving (24) with respect to q leads to

  $$ \frac{\partial}{\partial q} \mathit{NPV} = \alpha p \mathbb{E}(\theta) - k, $$

  (25)

  which can be distinguished into two cases with respect to its sign.

  • (a) If for the given fraud strategy $p > \frac{k}{\alpha\mathbb {E}(\theta)}$ holds, the NPV as defined in (2) has a positive slope with respect to the parameter q. Consequently, the optimal auditing strategy q ^opt has to be chosen as large as possible in order to maximize the value of NPV.

  • (b) If the given fraud strategy p is given such that $p \leq\frac {k}{\alpha\mathbb{E}(\theta)}$ holds, the NPV has a negative slope with respect to the parameter q. Hence, the optimal auditing strategy q ^opt has to be chosen as small as possible for the NPV to be maximized.

• (ii) Applying the assumptions a≠0, B=θ, $\hat{\theta}= \alpha\theta$ with α≥1 to (1) and (11), we obtain

  

  (26)

  Deriving (26) with respect to p results in

  $$ \frac{\partial}{\partial p} U\bigl(W_1^A \bigr) = -(1 - \alpha+ q \alpha) \mathbb {E}(\theta) - a p (1 - \alpha+ q \alpha)^2 \operatorname{Var}(\theta). $$

  (27)

  Based on (27), three cases can be identified:

  • (a) For $\frac{- \mathbb{E}(\theta)}{ap(1- \alpha(1-q) ) \operatorname{Var}(\theta)} \geq1$, the policyholder can choose any fraud strategy p∈[0,1], especially any p in the agreement range, such that $p \leq\frac{- \mathbb{E}(\theta)}{a(1- \alpha(1-q) ) \operatorname{Var}(\theta)}$. Applying this inequality to (27), we obtain $\frac{\partial}{\partial p} U(W_{1}^{A}) \geq0$. From this can be concluded that $U(W_{1}^{A})$ has a positive slope. Consequently, the optimal fraud strategy p ^opt has to be chosen as large as possible in order to maximize the value of $U(W_{1}^{A})$.

  • (b) Similarly, for $\frac{- \mathbb{E}(\theta)}{ap(1- \alpha(1-q) ) \operatorname{Var}(\theta)} \leq0$, the policyholder can choose any fraud strategy p∈[0,1], especially any p in the agreement range, such that $p \geq\frac{- \mathbb{E}(\theta)}{a(1- \alpha(1-q) ) \operatorname{Var}(\theta)}$. For (27) this implies that $\frac{\partial}{\partial p} U(W_{1}^{A}) \leq0$. This means that in this case $U(W_{1}^{A})$ has a negative slope and hence, the optimal fraud strategy p ^opt needs to be chosen as small as possible for $U(W_{1}^{A})$ to be maximized.

  • (c) For $0 < \frac{- \mathbb{E}(\theta)}{a(1- \alpha(1-q) ) \operatorname{Var}(\theta)} < 1$, no general statement about the corresponding optimal fraud strategy p ^opt can be made. □