Short Communication
Consistent modeling of risk averse behavior with spectral risk measures: Wächter/Mazzoni revisited

https://doi.org/10.1016/j.ejor.2016.12.027Get rights and content

Highlights

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    We revisit Wächter/Mazzoni (2013)’s linking procedure.

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    The general procedure does not yield a spectral risk measure.

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    The specific procedure violates expected utility and spectral risk measure-axioms.

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    A general consistency between the two decision rules is impossible.

Abstract

Wächter and Mazzoni (2013) (W/M) have proposed a procedure to consistently link traditional expected utility (EU)-theory with modern spectral risk measures (SRMs). They construct a corresponding W/M-risk measure which induces the same preference ordering as the decision maker’s initial utility function does. In this note, we revisit W/M’s procedure and show that it violates the axiomatic foundation of the underlying decision rules: Within the general part of the procedure that builds on an auxiliary equivalent probability measure, the emerging W/M-risk measure does not satisfy the axiomatic properties of SRMs. The specific part of the procedure that singles out a preferred equivalent probability measure links two decision rules of which neither does the initial one respect the axioms of EU-theory, nor is the emerging one in line with the SRM-axioms.

Introduction

Wächter and Mazzoni (2013) (W/M) recently have introduced a novel procedure to consistently link modern spectral risk measures (SRMs) as proposed by Acerbi (2002) with traditional expected utility (EU-)theory as founded by von Neumann and Morgenstern (1947). Given an initial utility function u and associated expected utility under the physical probability measure P, EP(u(·)), W/M by means of an auxiliary equivalent probability measure Q construct a corresponding risk spectrum ϕ(u;p)=φ(u;F−1(p))=dQ/dP and associated risk measure ρϕ(u), resp. The authors show that, “inside P&L distribution families” (p. 493), these two decision rules yield identical preference orderings. They also make a proposal of how to single out a preferred equivalent probability measure. W/M’s linking procedure, so it proved to be valid, could be of theoretical relevance as SRMs have gained increasing attention as a counterpart to classical EU-theory for modeling rational decision behaviour under risk (e.g., Adam, Houkari, Laurent, 2008, Brandtner, Kürsten, 2015).

In this note, we revisit W/M’s procedure and argue that, even under the restriction of “inside P&L distribution families”, it violates the axiomatic foundation of the underlying decision rules: Within the general part of the procedure based on the auxiliary equivalent probability measure Q (see W/M’s Section 5), the emerging risk measure does not satisfy the axiomatic properties of SRMs. The specific part of the procedure (see W/M’s Section 6) links two decision rules of which neither does the initial one respect the axioms of EU-theory, nor is the emerging one in line with the SRM-axioms.

We proceed as follows. Section 2 briefly recalls the fundamentals of SRMs and EU-theory, and discusses W/M’s notion of consistency. Section 3 addresses the general part of the procedure and shows that the emerging risk measure violates relevant properties of SRMs, with even popular subadditivity among them. Sections 4 is devoted to the specific part of the procedure and proves that neither is the input an EU-functional, nor is the outcome an SRM. Section 5 concludes.

Section snippets

Preliminaries: decision making with SRMs and EU-theory

Let L0(Ω,F,P) be the space of all measurable, real valued random variables on some probability space (Ω,F,P). Under the theories of choice considered in W/M’s paper, the random variables – or risky prospects – X∈L0(Ω,F,P) are interpreted as the profit and loss (P&L) of financial positions. They are completely described by their cumulative distribution functions (cdfs) and quantile functions, which are given by FX(x)=P(X≤x) and FX−1(p)=inf{x∈R:F(x)≥p},p∈(0,1],FX−1(0)=essinf{X}, resp.

SRMs

W/M’s linking procedure: the general part

The general part and genuine idea of W/M’s linking procedure is introduced in their Section 5, Eqs. (24)–(26): for some given probability measure P, random variable X, and utility function u, assume that there exists an equivalent (with respect to P) auxiliary probability measure Q on (Ω,F), including cdf GX induced by Q, which fulfills EQ(X)=∫−∞∞xdGX(x)=∫−∞∞u(x)dFX(x)=E(u(X)).Then, according to the theorem of Radon–Nikodym, there exists a normalized Radon–Nikodym derivative φ(x)=dQdP=gX(x)fX(x)

Preliminary remarks

Subsequently, W/M specify the general part of their procedure by singling out a preferred equivalent probability measure (see W/M’s Section 6). This specific part, however, does not represent a mere sub-case of the general part in (8) (see (21) below), so the issue of whether the specific part may provide consistency between SRMs and EU-theory needs to be addressed further. The essentials of W/M’s specific part can be summarized as follows: Let the utility function u:R→R satisfy the

Conclusions

In this note, we have shown that in both the general and the specific part of the procedure of linking SRMs with EU-theory as proposed by Wächter and Mazzoni (2013), the axiomatic foundation of the underlying decision rules is violated: The general part of the procedure does not link SRMs and EU-theory, but merely rewrites the EU-functional in a different functional form that resembles SRM. Accordingly, the emerging W/M-risk measure still satisfies the axioms of EU-theory, but violates a number

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