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Title
Abstract
Key Takeaways
Figures
Motivation and Objectives
Main Results
Unconstrained Case
Constrained Case
Discussion
Conclusion
References
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Mathematics
Applied Mathematics

A paradox in time-consistency in the mean–variance problem?

Profile image of Alain BensoussanAlain Bensoussan

2018, Finance and Stochastics

https://doi.org/10.1007/S00780-018-00381-0
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35 pages

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Abstract

We establish new conditions under which a constrained (no short-selling) time-consistent equilibrium strategy, starting at a certain time, will beat the unconstrained counterpart, as measured by the magnitude of their corresponding equilibrium mean-variance value functions. We further show that the pure strategy of solely investing in a risk-free bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, we also illustrate that the constrained strategy can dominate the unconstrained one for most of the commencement dates (even more than 90%) of a prescribed planning horizon. Under a precommitment approach, the value function of an investor increases with the size of the admissible sets of strategies. However, this may fail to be true under the game-theoretic paradigm, as the constraint of time-consistency itself affects the value function differently when short-selling is and is not prohibited.

Key takeaways
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  1. Constrained strategies can outperform unconstrained ones in over 90% of planning horizons.
  2. Investing solely in risk-free bonds can dominate both constrained and unconstrained strategies.
  3. Time-consistency constraints can limit investment flexibility, impacting equilibrium outcomes.
  4. Conditions 2.1 and 2.2 are essential for observing the paradoxical results described.
  5. The study explores time-consistency in mean-variance portfolio optimization, challenging conventional assumptions.
Figures (13)
Table 1 The construction of a non-cooperative game problem under the game-theoretic approach
Table 1 The construction of a non-cooperative game problem under the game-theoretic approach
Table 2 The counter-intuitive numerical observation of Equilibrium objective value for different investors in different T with
Table 2 The counter-intuitive numerical observation of Equilibrium objective value for different investors in different T with
Fig. 1 An illustration of admissible sets. The black and gray circles represent the admissible sets of unconstrained and constrained investors respectively. The solid lined ones contain the admissible sets of the basic optimization, while the dotted lined ones represent the admissible sets of optimization after imposing “‘time-consistency” constraints
Fig. 1 An illustration of admissible sets. The black and gray circles represent the admissible sets of unconstrained and constrained investors respectively. The solid lined ones contain the admissible sets of the basic optimization, while the dotted lined ones represent the admissible sets of optimization after imposing “‘time-consistency” constraints
Fig. 2. A graphical illustration to demonstrate how to find the commencement time such that both un- constrained and constrained investors have the same expected terminal payoff. The solid line and the dashed line represent the equilibrium investment-to-wealth ratio of the unconstrained and constrained in-
Fig. 2. A graphical illustration to demonstrate how to find the commencement time such that both un- constrained and constrained investors have the same expected terminal payoff. The solid line and the dashed line represent the equilibrium investment-to-wealth ratio of the unconstrained and constrained in-
A paradox in time-consistency in the mean-variance problem?
A paradox in time-consistency in the mean-variance problem?
allx > 0.
allx > 0.
Table 3. Times when the constrained investor dominates the unconstrained investor for different t* gets closer to the expiry. Hence the portion of time over which the unconstrained strategy remains more favorable than the riskless one reduces as yr decreases. In contrast, as yr decreases, t/ is moved further away from f*. Thus the short-selling constraints enlarge the portion of time on which the equilibrium strategy is more fa- vorable than the riskless one, which suggests that short-selling constraints are more useful for a time-consistent investor when he becomes less risk-averse. Perhaps we can interpret this observation as a less risk-averse attitude misleading the investor into over-investing in stock, while the short-selling constraints can protect the investor
Table 3. Times when the constrained investor dominates the unconstrained investor for different t* gets closer to the expiry. Hence the portion of time over which the unconstrained strategy remains more favorable than the riskless one reduces as yr decreases. In contrast, as yr decreases, t/ is moved further away from f*. Thus the short-selling constraints enlarge the portion of time on which the equilibrium strategy is more fa- vorable than the riskless one, which suggests that short-selling constraints are more useful for a time-consistent investor when he becomes less risk-averse. Perhaps we can interpret this observation as a less risk-averse attitude misleading the investor into over-investing in stock, while the short-selling constraints can protect the investor
Fig. 3. Fix yr = 1, k = 0.025. The investment-to-wealth ratio c; (left) and the equilibrium value function V(t, 1) (right) are plotted against the commencement time ¢, for unconstrained (solid line), constrained (dashed line) and riskless (dotted line) investors
Fig. 3. Fix yr = 1, k = 0.025. The investment-to-wealth ratio c; (left) and the equilibrium value function V(t, 1) (right) are plotted against the commencement time ¢, for unconstrained (solid line), constrained (dashed line) and riskless (dotted line) investors
Fig. 4 Fix yr =3, k = 0.025. The investment-to-wealth ratio c; (left) and the equilibrium value function V(t, 1) (ight) are plotted against the commencement time ¢, for unconstrained (solid line), constrained (dashed line) and riskless (dotted line) investors
Fig. 4 Fix yr =3, k = 0.025. The investment-to-wealth ratio c; (left) and the equilibrium value function V(t, 1) (ight) are plotted against the commencement time ¢, for unconstrained (solid line), constrained (dashed line) and riskless (dotted line) investors
Table 4 Times when the riskless strategy dominates the equilibrium strategies for different y;
Table 4 Times when the riskless strategy dominates the equilibrium strategies for different y;

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Related topics

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  • Applied Mathematics
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  • Mathematical Finance
  • Time Consistency

    • Cited by

    Time-Consistent Investment and Consumption Strategies under a General Discount Function
    Josep Vives

    Journal of Risk and Financial Management, 2021

    In the present paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that gives rise to time-inconsistency of the decision-maker. We consider equilibrium policies within the class of open-loop controls that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

    View PDFchevron_right
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