Implied risk aversion: an alternative rating system for retail structured products

Abstract

This article proposes implied risk aversion as a rating methodology for retail structured products. Implied risk aversion is based on optimal expected utility risk measures as introduced by Geissel et al. (Stat Risk Model 35(1–2):73–87, 2017) and, in contrast to standard V@R-based ratings, takes into account both the upside potential and the downside risks of such products. In addition, implied risk aversion is easily interpreted in terms of an individual investor’s risk aversion: a product is attractive for an investor if his individual relative risk aversion is smaller than the product’s implied risk aversion. We illustrate our approach in a case study with more than 15,000 short-term warrants on DAX that highlights some differences between our rating system and the traditional V@R-based approach.

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Notes

  1. 1.

    For instance, the ratings of Deutscher Derivate Verband (DDV), which is the leading provider of rating information in the German RSP market, are based on V@R.

  2. 2.

    If r is the risk-free interest rate and the time period is T, we have \(\beta =\frac{1}{1+rT}\).

  3. 3.

    If u is a utility function, we define \(u(0)\,{:}{=}\,\lim _{x\downarrow 0}u(x)\in [-\infty ,\infty )\), \(u(\infty )\,{:}{=}\,\lim _{x\uparrow 0}u(x)\in (-\infty ,\infty ]\). Moreover, we denote the inverse of u by \(u^{-1}:[u(0),u(\infty ))\rightarrow [0,\infty )\). \(u^{-1}\) is well-defined and continuous, and \(u^{-1}(u(x))=x\) for all \(x\in [0,\infty )\). Finally, we set \(u^{-1}({\mathbb {E}}[u(Y)])\,{:}{=}\,-\infty \) if \(P(Y<0)>0\).

  4. 4.

    OEU remains well-defined in the limiting case \(\alpha =\beta \), but can imply unbounded leveraging (\(\eta =\infty \)). We refer to (Geissel et al. 2017, Remark 2.3, Proposition 2.12, Proposition 2.16) for an in-depth analysis of this boundary case.

  5. 5.

    As usual, we set \(u(x)=\ln (x)\) for \(\gamma =1\), so

    $$\begin{aligned} \rho ^u(X,1) = -\max _{\eta >-X_{\min }}\left\{ -\beta \eta +\alpha \exp \left( {\mathbb {E}}\left[ \ln (X+\eta )\right] \right) \right\} ,\quad X\in {\mathcal {X}}. \end{aligned}$$
  6. 6.

    We present results for alternative specifications of \(\alpha \) in Sect. 5.1 as well.

  7. 7.

    This and all upcoming estimations are based on daily price histories.

  8. 8.

    For ARMA-GARCH, Hansen’s skewed-tdistribution (with skew parameter 7.02721/10.51961 and tail parameter − 0.09461/− 0.11547) performed best based on an evaluation of AIC, BIC, log-likelihood and QQ-plots.

  9. 9.

    This implies in particular that, in DDV’s approach, risk factors other than the underlying’s market risk are not relevant.

  10. 10.

    For numerical stability, \(\gamma _0\)-values below 0.0001 are set to zero.

  11. 11.

    Here, volatility is fixed at the implied volatility of XM2HAG.

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Acknowledgements

All authors are grateful for the comments and suggestions of three anonymous referees.

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Fink, H., Geissel, S., Sass, J. et al. Implied risk aversion: an alternative rating system for retail structured products. Rev Deriv Res 22, 357–387 (2019). https://doi.org/10.1007/s11147-018-9151-0

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Keywords

  • Structured products
  • Risk measures
  • Optimal expected utility
  • Implied risk aversion

JEL Classification

  • G11
  • D81