Hope, fear, and aspirations (Q2788689)

The paper under review is to propose a rank-dependent portfolio choice model in continuous time that captures three emotions (hope, fear and aspiration) in decision making. The main contribution of this paper is to introduce a fear index, a hope index and a lottery-likeness index to quantify hope, fear, and aspiration.NEWLINENEWLINESection 2 recalls the SP/A theory [\textit{L. L. Lopes}, ``Between hope and fear: the psychology of risk'', Adv. Exp. Soc. Psychol. 255--295 (1987; \url{doi:10.1016/s0065-2601(08)60416-5})] security, potential and aspiration theory) and the RDUT [\textit{J. Quiggin}, ``A theory of anticipated utility'', J. Econ. Behav. Organ. 3, No. 4, 323--343 (1982; \url{doi:10.1016/0167-2681(82)90008-7})]. AP/A theory is a two-factor (dispositional factor and a situation factor) theory to explain risky perferences and choices with the decumulative wieghting function \(w(z) = \nu z^{q_s+1} + (1-\nu) [1- (1-z)^{q_p+1}]\) for \(\nu \in [0, 1]\) and \(q_s, q_p \geq 0\) (where \(z^{q_s+1}\) is for risk-aversion and \(1- (1-z)^{q_p+1}\) for risk-loving) between hope and fear. The situational factor is modeled by the probabilistic constrain \(P(X\geq A)\geq \alpha\) for the aspiration level \(A\) and the confidence level \(\alpha \in [0,1]\). The RDUT perference measures a prospect \(X\) by NEWLINE\[NEWLINEV(X) = \int_0^{\infty} u(x) d[-w(1-F_X(x))],NEWLINE\]NEWLINE where \(w(\cdot )\) is a probability weighting/distortion function and \(u(\cdot )\) is the utility function, and SP/A theory measures a prospect \(V(X) = \int_0^{\infty} x d[-w(1-F_X(x))]\) as a special case of RDUT. The typical weighting function is inverse S-shaped which has small probabilities of both very good and very bad events are overweighted. Authors construct another weighting functions from pasting smoothly two 1-parameter weighting functions of \textit{S. S. Wang} [``A class of distortion operators for pricing financial and insurance risks'', J. Risk Insur. 67, No. 1, 15--36 (2000; \url{doi:10.2307/253675})] as following: NEWLINE\[NEWLINEw(z) = \begin{cases} ke^{(a+b)\Phi^{-1}(\overline{z}) +\frac{\sigma^2}{2}}\Phi (\Phi^{-1}(z) +a) & z\leq \overline{z} \\ A + ke^{b^2/2}\Phi (\Phi^{-1}(z) -b) & z \geq \overline{z} \end{cases}NEWLINE\]NEWLINE where \(\Phi (\cdot )\) is the CDF of a standard normal random variable, \(a, b\geq 0\), and \(k\) and A are determined. This class of weighting functions is concave down on \((0, \overline{z})\) and concave up on \((\overline{z}, 1)\) to be inverse-S shaped. Various inverse-S shaped functions are compared in Figure 1.NEWLINENEWLINESection 3 formulates the HF/A portfolio choice model under the market arbitrage-free and complete assumption to maximize a portfolio \(\pi (t) = (\pi_i (t))\) that invested in stock i at time t subject to the situational factor \(P(X(T)\geq A)\geq \alpha\) and \(E[\rho X]\leq x_0\) in (3.3). The authors further assume that the CDF \(F_{\rho}(\cdot)\) is continuous and \(E\rho <\infty\), and \(w(\cdot): [0, 1]\to [0, 1]\) is continuous and strictly increasing with \(w(0)=0\) and \(w(1)=1\).NEWLINENEWLINEAn optimization problem is feasible if the set of decision variables satisfying all the constraints is nonempty. Section 4 starts with Proposition 4.1 that HF/A problem is feasible if and only if \(AE[{\rho 1_{\rho \leq F_{\rho}^{-1}(\alpha)}]}\leq x_0\) and feasible solution is unique whenever \(AE[{\rho 1_{\rho \leq F_{\rho}^{-1}(\alpha)}}] =x_0\) and \(X=A1_{\rho \leq F_{\rho}^{-1}(\alpha)}.\) The proof mainly follows from the authors' earlier work [Manage. Sci. 57, No. 2, 315--331 (2011; Zbl 1214.91099)]. Theorem 4.2 shows that the SP/A theory can lead to an ill-posed portfolio choice problem in the continuous time setting. The fear of possible catastrophic situation is insufficient to prevent the agent from taking excessive risky exposures.NEWLINENEWLINEUnder the integrability assumption and the feasible situation, authors prove Theorem 5.2 in subsection 5.1 to show th unique optimal solution to the HF/A portfolio choice model. This follows from the Lagrange dual method and the pointwise maximization procedure leads to a nondecreasing function \(G^*_{\lambda}(\cdot)\) that is optimal to the problem. Theorem 5.4 shows that an optimal strategy needs to set a strictly positive deterministic floor in wealth at the terminal time in sharp contrast to the strategy with monotonicity conditions. The fear index is defined from the weighting function \(w(\cdot)\): NEWLINE\[NEWLINEI_{w}(z) = \frac{w^{''}(z)}{w^{'}(z)}, \;\;\;0< z < 1.NEWLINE\]NEWLINE The higher this index, the more convex the weighting function is, and the mode fear the agent has. Theorem 5.9 shows that the HF/A portfolio has a unique optimal solution without monotonicity condition with \(A=0\) and further assumption 5.5 on differentiability and behaviors of the function \(M(\cdot)\). The unique root \(c^*\) is used to distinguish the good states (\(\rho \leq c^*\)) and bad states (\(\rho >c^*\)). Theorem 5.10 provides a complete solution to the HF/A portfolio choice problem to characterize the exactly the unique optimal solution under different cases. The hope index NEWLINE\[NEWLINEH_w(z) = \frac{w^{'}}{1-w(z)}, \;\;0< z < 1,NEWLINE\]NEWLINE tells in Theorem 5.11 that the critical point \(c^*\) is increasing with respect to the hope index as \(z \to 0\). Proposition 5.12 shows that the higher hope index leads to a higher payoff in sufficiently goog scenarios than that with a lower hope index. The lottery-likeness index NEWLINE\[NEWLINEL(A) = \frac{ ess-inf \{X^*|\rho \leq F_{\rho}^{-1}(\alpha)\}}{ ess-sup \{X^*| \rho > F_{\rho}^{-1}(\alpha)\}} ,NEWLINE\]NEWLINE gives the ratio between the worst winning payoff and the best losing payoff.NEWLINENEWLINESection 6 illustrates some numerical experiments from the data set in [\textit{R. Mehra} and \textit{E. C. Prescott}, ``The equity premium: a puzzle'', J. Monetary Econ. 15, No. 2, 145--161 (1985; \url{doi:10.1016/0304-3932(85)90061-3})], and vary the hope index and fear index under the changes of a and b with the aspiration level at \(A=0\) in Figure 6.4, Figure 6.5 and Figure 6.6. Authors conclude in Section 7 with remarks. Appendix A gives the proof of Theorem 5.10. It would be interesting to see if the HF/A theory, the hope index, the fear index and the lottery-likeness index can be used in practice to guide the investment or regulations.