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Portfolio optimization in fuzzy asset management with coherent risk measures derived from risk averse utility

  • S.I. : India Intl. Congress on Computational Intelligence 2017
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Abstract

A portfolio optimization problem with fuzzy random variables is discussed using coherent risk measures, which are characterized by weighted average value-at-risks with risk spectra. By perception-based approach, coherent risk measures and weighted average value-at-risks are extended for fuzzy random variables. Coherent risk measures derived from risk averse utility functions are introduced to discuss the portfolio optimization with randomness and fuzziness. The randomness is estimated by probability, and the fuzziness is evaluated by lambda-mean functions and evaluation weights. By mathematical programming approaches, a solution is derived for the risk-minimizing portfolio optimization problem. Numerical examples are given to compare coherent risk measures. It is made clear that coherent risk measures derived from risk averse utility functions have excellent properties as risk criteria for these optimization problems. Not only pessimistic and necessity case but also optimistic and possibility case are calculated numerically to deal with uncertain information.

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Abbreviations

\({\mathrm{VaR}}, {\mathrm{AVaR}}\) :

Value-at-risk and average value-at-risk

\({\mathrm{AVaR}}^{\nu }\,(\widetilde{{\mathrm{AVaR}}^{\nu }})\) :

(Extended) Weighted average value-at-risk with \(\nu\)

\(\rho \,({\tilde{\rho }})\) :

(Extended) Coherent risk measure

\(\nu , C\) :

Risk spectrum and its component function

\({{\mathcal {N}}}\) :

The set of all fuzzy numbers

\({\tilde{n}}, {\tilde{n}}_{\alpha } = {[}{\tilde{n}}_{\alpha }^{-}, {\tilde{n}}_{\alpha }^{+}{]}\) :

Fuzzy number and its \(\alpha\)-cut

\({\tilde{X}}, {{\tilde{X}}}_{\alpha }={[}{\tilde{X}}_{\alpha }^{-},{\tilde{X}}_{\alpha }^{+}{]}\) :

Fuzzy random variable and its \(\alpha\)-cut

\({{\mathcal {X}}}\,(\tilde{{\mathcal {X}}})\) :

The family of all integrable real-valued (fuzzy-valued) random variables

\(E, {\tilde{E}}\) :

Expectation and perception-based expectation

\(E^{\lambda }\) :

Mean of fuzzy numbers

\(\lambda\) :

Optimistic/pessimistic index

\(w(\alpha )\) :

Possibility/necessity evaluation weight

f :

Utility function

\(S^{i}_{t}\,({\tilde{S}}^{i}_{t})\) :

(Fuzzy-valued) Stock price for asset i at time t

\(R^{i}_{t}\,({\tilde{R}}^{i}_{t})\) :

(Fuzzy-valued) Rate of return for asset i at time t

\(w_t = (w_t^{1},\ldots , w_t^{n})\) :

Portfolio weight vector

\({{\mathcal {W}}}_t\) :

The set of all portfolio weight vectors

\(\mu _{t} = {[}\mu _{t}^{i} {]}\) :

Vector of expected rates of return

\(\varSigma _{t} = {[}\sigma _{t}^{ij}{]}\) :

Variance–covariance matrix for rates of return

\(\gamma _t^*\) :

The optimal expected rate of return

\(\rho _t^*\) :

The optimal risk value

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Acknowledgements

This research is supported from JSPS KAKENHI Grant Number JP 16K05282.

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Authors and Affiliations

  1. Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu, 802-8577, Japan

    Yuji Yoshida

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Correspondence to Yuji Yoshida.

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Yoshida, Y. Portfolio optimization in fuzzy asset management with coherent risk measures derived from risk averse utility. Neural Comput & Applic 32, 10847–10857 (2020). https://doi.org/10.1007/s00521-018-3683-y

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