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Behavioral premium principles

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Abstract

We define a premium principle under the continuous cumulative prospect theory which extends the equivalent utility principle. In prospect theory, risk attitude and loss aversion are shaped via a value function, whereas a transformation of objective probabilities, which is commonly referred as probability weighting, models probabilistic risk perception. In cumulative prospect theory, probabilities of individual outcomes are replaced by decision weights, which are differences in transformed, through the weighting function, counter-cumulative probabilities of gains and cumulative probabilities of losses, with outcomes ordered from worst to best. Empirical evidence suggests a typical inverse-S shaped function: decision makers tend to overweight small probabilities, and underweight medium and high probabilities; moreover, the probability weighting function is initially concave and then convex. We study some properties of the behavioral premium principle. We also assume an alternative framing of the outcomes; then, we discuss several applications to the pricing of insurance contracts, considering different value functions and probability weighting functions proposed in the literature, and an alternative mental accounting. Finally, we focus on the shape of the probability weighting function.

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Notes

  1. The book of Wakker (2010) provides a thorough treatment on prospect theory.

  2. See Quiggin (1993), p. 56.

  3. See also Rieger and Wang (2008), Wakker (2010), and Kothiyal et al. (2011).

  4. When there is no ambiguity, we simply use the notation \(F(x) = {\mathbb {P}}(X\le x)\) and \(S(x) = 1- F(x) = {\mathbb {P}}(X > x)\), and f for the probability density function of the random loss X.

  5. The premium principle defined in Wang (1996) assumes an increasing and concave distortion function and maintains the second-order stochastic dominance.

  6. See Thaler (1985), p. 202.

  7. Time-value of money is normally disregarded when dealing with non-life insurance contracts, but may become important on a multi-year horizon.

  8. Note that \({\mathbb {E}}_{w^+w^-} (cX) = c {\mathbb {E}}_{w^+w^-} (X)\), for \(c\ge 0\).

  9. In general, linearity does not hold for the generalized Choquet integral. In Sect. 3.1, we discussed the case with \(w^+=w^-\). When \(w^+\ne w^-\), and \(c\in {\mathbb {R}}\), we apply the following result

    $$\begin{aligned} {\mathbb {E}}_{w^+w^-}(X+c) = {\mathbb {E}}_{w^+w^-}(X) + c + \int _0^c [w^-({\mathbb {P}}(-X>s))-\overline{w}^+({\mathbb {P}}(-X>s))]\,\mathrm{{d}}s, \end{aligned}$$

    where \(\overline{w}\) is the dual probability weighting function. See Kaluszka and Krzeszowiec (2012) for the proof and discussion of further properties of the generalized Choquet integral.

  10. We have \(u(0)=0\), \(u^\prime >0\), \(u^\prime (0)=b/a\), \(u^{\prime \prime }<0\). Heilpern (2003) considers the normalized case \(a=b\).

  11. The same result arises also when \(a=b\) and with \(W=0\).

  12. We have \(u(0)=0\), \(u^\prime >0\), \(u^\prime (0)=b/a\), and \(u^{\prime \prime }>0\), which may be useful to model the value function in the domain of losses.

  13. Observe that \(\int _0^c \psi (F(x))f(x)\mathrm{{d}}x = w(F(c))\).

  14. In the literature discontinuous (neo-additive) weighting functions are also considered.

  15. See e.g., Prelec (1998), Abdellaoui (2000), Bleichrodt and Pinto (2000), Bleichrodt et al. (2001), and Abdellaoui et al. (2007).

  16. In the same paper, Prelec derives two other probability weighting functions: the conditionally-invariantexponential-power and the projection-invarianthyperbolic-logarithm function.

  17. This is not the case for weighting function (25); when \(a \ne b\), both parameters controls for curvature and all parameters may influence elevation.

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Acknowledgements

The authors thank two anonymous referees for useful comments and suggestions of references.

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  1. Department of Economics, Ca’ Foscari University of Venice, San Giobbe - Cannaregio 873, 30121, Venezia, Italy

    Martina Nardon & Paolo Pianca

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  1. Martina Nardon
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Correspondence to Martina Nardon.

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Nardon, M., Pianca, P. Behavioral premium principles. Decisions Econ Finan 42, 229–257 (2019). https://doi.org/10.1007/s10203-019-00246-x

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