Paper: arXiv 2312.04045
Authors: Yu-Jui Huang, Li-Hsien Sun
Abstract
This paper considers finitely many investors who perform mean-variance portfolio selection under relative performance criteria. That is, each investor is concerned about not only her terminal wealth, but how it compares to the average terminal wealth of all investors. At the inter-personal level, each investor selects a trading strategy in response to others’ strategies. This selected strategy additionally needs to yield an equilibrium intra-personally, so as to resolve time inconsistency among the investor’s current and future selves (triggered by the mean-variance objective). A Nash equilibrium we look for is thus a tuple of trading strategies under which every investor achieves her intra-personal equilibrium simultaneously. We derive such a Nash equilibrium explicitly in the idealized case of full information (i.e., the dynamics of the underlying stock is perfectly known) and semi-explicitly in the realistic case of partial information (i.e., the stock evolution is observed, but the expected return of the stock is not precisely known). The formula under partial information consists of the myopic trading and intertemporal hedging terms, both of which depend on an additional state process that serves to filter the true expected return and whose influence on trading is captured by a degenerate Cauchy problem. Our results identify that relative performance criteria can induce downward self-reinforcement of investors’ wealth–if every investor suffers a wealth decline simultaneously, then everyone’s wealth tends to decline further. This phenomenon, as numerical examples show, is negligible under full information but pronounced under partial information.
Complexity vs Empirical Score
- Math Complexity: 8.5/10
- Empirical Rigor: 2.5/10
- Quadrant: Lab Rats — theoretically deep, empirically untested
Why this score: The paper provides a rigorous mathematical treatment of mean-variance portfolio games with partial information, deriving explicit and semi-explicit Nash equilibria via filtering theory and degenerate PDEs. However, it is purely theoretical, relying on stylized numerical examples rather than empirical backtests or real market datasets.
Research Flowchart
flowchart TD
A[Goal: Solve Mean-Variance Portfolio Game<br/>with Relative Performance & Partial Info] --> B[Inputs: Stock Price Process,<br/>Wealth Dynamics & Unobserved Drift]
B --> C[Methodology: Combine Filtering Theory<br/>& Time-Consistent Intra-Personal Equilibrium]
C --> D[Solve Degenerate Cauchy Problem<br/>for Intertemporal Hedging]
D --> E[Explicit Full Info & Semi-Explicit<br/>Partial Info Nash Equilibria]
E --> F[Findings: Partial Info Drives Pronounced<br/>Downward Self-Reinforcement of Wealth]
Cited References (5)
- Asset fire sales (and purchases) in equity markets (2007)
- Fire-Sale Spillovers and Systemic Risk (2021)
- Dynamic Mean-Variance Asset Allocation (2010)
- Missing R & D (2015)
- What percentage of innovations are patented? empirical estimates for European firms (1998)
0 of 5 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.