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29.03.2022 | Research Article

Portfolio selection in quantile decision models

verfasst von: Luciano de Castro, Antonio F. Galvao, Gabriel Montes-Rojas, Jose Olmo

Erschienen in: Annals of Finance | Ausgabe 2/2022

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Abstract

This paper develops a model for optimal portfolio allocation for an investor with quantile preferences, i.e., who maximizes the \(\tau \)-quantile of the portfolio return, for \(\tau \in (0,1)\). Quantile preferences allow to study heterogeneity in individuals’ portfolio choice by varying the quantiles, and have a solid axiomatic foundation. Their associated risk attitude is captured entirely by a single dimensional parameter (the quantile \(\tau \)), instead of the utility function. We formally establish the properties of the quantile model. The presence of a risk-free asset in the portfolio produces an all-or-nothing optimal response to the risk-free asset that depends on investors’ quantile preference. In addition, when both assets are risky, we derive conditions under which the optimal portfolio decision has an interior solution that guarantees diversification vis-à-vis fully investing in a single risky asset. We also derive conditions under which the optimal portfolio decision is characterized by two regions: full diversification for quantiles below the median and no diversification for upper quantiles. These results are illustrated in an exhaustive simulation study and an empirical application using a tactical portfolio of stocks, bonds and a risk-free asset. The results show heterogeneity in portfolio diversification across risk attitudes.

Nächster Artikel Options on bonds: implied volatilities from affine short-rate dynamics

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Some studies suggesting that individuals did not always employ objective probabilities resulted in, among others, Prospect Theory (Kahneman and Tversky 1979), Rank-Dependent Expected Utility Theory (Quiggin 1982), Cumulative Prospect Theory (Tversky and Kahneman 1992), Regret (Bell 1982), and Ambiguity Aversion (Gilboa and Schmeidler 1989). Rabin (2000) criticized EU theory arguing that EU would require unreasonably large levels of risk aversion to explain the data from some small-stakes laboratory experiments. See also Simon (1979), Tversky and Kahneman (1981), Payne et al. (1992) and Baltussen and Post (2011) as examples providing experimental evidence on the failure of the EU paradigm.

See, e.g., Duffie and Pan (1997) and Jorion (2007). The VaR measure is one of the main practical tools for reporting the exposure to risk by financial institutions.

See, e.g., Apiwatcharoenkul et al. (2016) and Fanchi and Christiansen (2017).

Rostek (2010) discusses other advantages of the QP, such as robustness, ability to deal with categorical (instead of continuous) variables, and the flexibility of offering a family of preferences indexed by quantiles.

Intuitively, the monotonicity of quantiles allows one to avoid modeling individuals’ utility function. This is because the maximization problem is invariant to monotonic transformations of the distribution of portfolio returns.

Our model can deal with short sale, but we leave this to future work.

For instance, if the preference is given by EU, the belief is captured by the probability while the tastes by the utility function over outcomes or consequences (such as monetary payoffs). Beliefs and tastes are not completely separated, however, because if we take a monotonic transformation of the utility function, which maintains the same tastes over consequences, we may end up with a different preference. That is, the pair beliefs and tastes come together and are stable only under affine transformations of the utility function. In other words, the EU preferences, as many other preferences, do not allow a separation of tastes and beliefs.

See, e.g., Damodaran (2010) for an argument that every asset carries some risk.

In the context of risk management, Theorem 2 provides theoretical support to the lack of subadditivity of VaR measures in general settings, see Artzner et al. (1999).

The inferior limit m may be taken over a limited region, not over the whole support. That is, we can accommodate cases in which the support is infinite so that \(f(x,y) \rightarrow 0\) when \(y \rightarrow \infty \).

Of course, \(w^{*}=0\), when \({\underline{y}}>{\underline{x}}>-\infty \) and \({\underline{y}}- {\underline{x}} \geqslant \frac{M}{2m}\).

The two illustrations (a) and (b) correspond, respectively, to \(\tau \leqslant \Pr \left( \left[ \frac{X+Y}{2} \leqslant q \right] \right) \) and \(\tau > \Pr \left( \left[ \frac{X+Y}{2} \leqslant q \right] \right) \). In the first case (a), the line \(wX+(1-w)Y=q\) is below the dashed blue line joining (1, 0) to (0, 1). In case (b) \(\tau >\Pr \left( \left[ \frac{X+Y}{2} \leqslant q \right] \right) \), the line \(wX+(1-w)Y=q\) is above the dashed blue line.

The simulation algorithm reflects this issue by selecting either \(w=0\) or \(w=1\) for \(\tau _0=0.5\).

When \({\mathscr {I}}_{X}\) or \({\mathscr {I}}_{Y}\) are not \({\mathbb {R}}\), then we have to consider limits that make the derivatives more complex.

The case of two risky assets with different means is available from the authors upon request.

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Titel: Portfolio selection in quantile decision models
verfasst von: Luciano de Castro
Antonio F. Galvao
Gabriel Montes-Rojas
Jose Olmo
Publikationsdatum: 29.03.2022
Verlag: Springer Berlin Heidelberg
Erschienen in: Annals of Finance / Ausgabe 2/2022
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-021-00405-4

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