TY - JOUR
T1 - Optimal investment problem under behavioral setting
T2 - A Lagrange duality perspective
AU - Bi, Xiuchun
AU - Cui, Zhenyu
AU - Fan, Jiacheng
AU - Yuan, Lvning
AU - Zhang, Shuguang
N1 - Publisher Copyright:
© 2023 Elsevier B.V.
PY - 2023/11
Y1 - 2023/11
N2 - In this paper, we consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points, and apply a Lagrange duality based relaxation approach for solving this problem. Existing literature has shown the equivalent relationships (strong duality) between the relaxed problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, combining both factors, we prove that the absence of concavity in the utilities may result in strictly positive gaps, thus the strong duality may not hold unconditionally. The necessary and sufficient conditions on eliminating such gaps have been provided under this circumstance. We have applied the solution method to obtain closed-form solutions for the optimal terminal wealth and corresponding investment strategies in a special cumulative prospect theory (CPT) example with an appropriately selected inverse S-shaped probability distortion function. Based on the solutions, the joint effects of non-concave utilities combined with the probability weighting on the trading behaviors can be explicitly characterized. We show that, under this particular example, the co-existence of both factors may water down the loss aversion effect induced by only S-shaped utility or probability distortion when the agents are more cash-strapped in the initial budgets. In addition, we find that the optimal strategies derived from distorted beliefs shall converge to a constant that can be expressed by the standard Merton ratio multiplied by an inflation factor which we name as “distorted” Merton ratio. More importantly, the inflation factor is solely dependent on the probability distortion rather than the features of the utility function.
AB - In this paper, we consider the optimal investment problem with both probability distortion/weighting and general non-concave utility functions with possibly finite number of inflection points, and apply a Lagrange duality based relaxation approach for solving this problem. Existing literature has shown the equivalent relationships (strong duality) between the relaxed problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, combining both factors, we prove that the absence of concavity in the utilities may result in strictly positive gaps, thus the strong duality may not hold unconditionally. The necessary and sufficient conditions on eliminating such gaps have been provided under this circumstance. We have applied the solution method to obtain closed-form solutions for the optimal terminal wealth and corresponding investment strategies in a special cumulative prospect theory (CPT) example with an appropriately selected inverse S-shaped probability distortion function. Based on the solutions, the joint effects of non-concave utilities combined with the probability weighting on the trading behaviors can be explicitly characterized. We show that, under this particular example, the co-existence of both factors may water down the loss aversion effect induced by only S-shaped utility or probability distortion when the agents are more cash-strapped in the initial budgets. In addition, we find that the optimal strategies derived from distorted beliefs shall converge to a constant that can be expressed by the standard Merton ratio multiplied by an inflation factor which we name as “distorted” Merton ratio. More importantly, the inflation factor is solely dependent on the probability distortion rather than the features of the utility function.
KW - Behavioral finance
KW - Lagrange duality
KW - Non-concave utility
KW - Portfolio optimization
KW - Probability distortion
KW - Relaxation method
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U2 - 10.1016/j.jedc.2023.104751
DO - 10.1016/j.jedc.2023.104751
M3 - Article
AN - SCOPUS:85173245020
SN - 0165-1889
VL - 156
JO - Journal of Economic Dynamics and Control
JF - Journal of Economic Dynamics and Control
M1 - 104751
ER -