## Abstract

The aim of this manuscript is to analyze the monotonicity and limit properties of the optimal investment strategy in a behavioral portfolio choice model under cumulative prospect theory over risk aversion coefficient, loss aversion coefficient, and the market opportunity. We show that the optimal investment strategy is nonincreasing of the loss aversion coefficient, and strictly increasing of the Sharpe ratio for normal distributions. The monotonicity properties over risk aversion coefficient depend on the position of the investor and the goodness of the actual and perceived market. The piecewise-linear utility is also discussed. An interesting finding is that when the excess return follows an elliptical distribution, the optimal investment strategy over small mean for piecewise-power and piecewise-linear utility exhibits different limit behavior.

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## Notes

In Sect. 4 we will consider the piecewise-linear utility, i.e., \(\alpha =1\).

For two functions

*g*and*h*, \(g(t)=O(h(t))\) represents that \(\limsup _{t\rightarrow \infty }\frac{g(t)}{h(t)}\le M\) for some constant \(M>0\), while \(g(t)=o(h(t))\) represents that \(\lim _{t\rightarrow \infty }\frac{g(t)}{h(t)}=0\).Corollary 2 in [5] shows that the CPT preference value function \(V(\cdot )\) is nonconcave on either \({\mathbb {R}}^+\) or \({\mathbb {R}}^-\). The nonconcavity of CPT preference value function brings greater difficulty in solving the optimization problem (4) analytically (see the statements in Section 5 in [5]).

Specifically, it shows that if the excess return

*R*has a positive mean and is bounded from above, i.e., \(\mathrm {E}[R]>0\) and \(R\le M\) with probability one for some \(M>0\), then \(|\gamma _-|M>1\).

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## Acknowledgements

The author would like to thank the anonymous referees for improving the quality of the paper. This research was supported by the National Natural Science Foundation of China under Grant 71971208.

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Lou, Y. Some properties of the optimal investment strategy in a behavioral portfolio choice model.
*Optim Lett* **14**, 1731–1746 (2020). https://doi.org/10.1007/s11590-019-01467-0

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DOI: https://doi.org/10.1007/s11590-019-01467-0