Market-Based Variance of Market Portfolio and of Entire Market

ArXiv ID: 2510.13790 “View on arXiv”

Authors: Victor Olkhov

Abstract

We present the unified market-based description of returns and variances of the trades with shares of a particular security, of the trades with shares of all securities in the market, and of the trades with the market portfolio. We consider the investor who doesn’t trade the shares of his portfolio he collected at time t0 in the past. The investor observes the time series of the current trades with all securities made in the market during the averaging interval. The investor may convert these time series into the time series that model the trades with all securities as the trades with a single security and into the time series that model the trades with the market portfolio as the trades with a single security. That establishes the same description of the returns and variances of the trades with a single security, the trades with all securities in the market, and the market portfolio. We show that the market-based variance, which accounts for the impact of random change of the volumes of consecutive trades with securities, takes the form of Markowitz’s (1952) portfolio variance if the volumes of consecutive trades with all market securities are assumed constant. That highlights that Markowitz’s (1952) variance ignores the effects of random volumes of consecutive trades. We compare the market-based variances of the market portfolio and of the trades with all market securities, consider the importance of the duration of the averaging interval, and explain the economic obstacles that limit the accuracy of the predictions of the returns and variances at best by Gaussian distributions. The same methods describe the returns and variances of any portfolio and the trades with its securities.

Keywords: Market-based variance, Markowitz portfolio variance, Trade volumes, Averaging interval, Gaussian distributions, Equities/Market Portfolio

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with advanced mathematical notation, derivatives, and theoretical framework (e.g., series expansions, variance decompositions), but lacks empirical data, backtests, or implementation details, focusing instead on theoretical extensions of Markowitz’s model.

  flowchart TD
    A["Research Goal: Unify market-based<br>descriptions of returns & variances<br>for single securities vs. market portfolio"] --> B{"Methodology"};
    B --> C["Model trades with all securities<br>as trades with a single security<br>via time-series conversion"];
    B --> D["Derive market-based variance formula<br>incorporating random trade volumes"];
    C --> E{"Comparison & Analysis"};
    D --> E;
    E --> F["Outcome 1: Markowitz (1952) variance is a<br>special case where trade volumes are constant"];
    E --> G["Outcome 2: Market-based variance captures<br>random volume effects ignored by Markowitz"];
    E --> H["Outcome 3: Unified framework applies<br>to any portfolio and its securities"];
    F --> I["Key Limitation: Gaussian distributions<br>represent best-case prediction accuracy"];
    G --> I;
    H --> I;