Markowitz Portfolio Optimization Problem

• Markowitz Portfolio Optimization Problem is a framework that employs quadratic programming to balance expected returns against risk.
• It examines trade-offs under constraints by incorporating parameters like covariance and estimation uncertainty in asset returns.
• Relaxing constant trade volume assumptions reveals that real market risks can deviate significantly from classical model predictions.

The Markowitz Portfolio Optimization Problem is the foundational framework in quantitative finance for constructing an investment portfolio that optimally balances expected return against risk, with risk measured by the variance (or standard deviation) of portfolio returns. Originally introduced by Harry Markowitz in 1952, this problem formalizes portfolio selection as a quadratic program subject to budget and, often, return constraints. Subsequent decades of research have scrutinized, extended, and critiqued the model—examining issues arising from parameter uncertainty, the nonlinearity of risk, dynamic market features, and the complex interactions of estimation error, as well as integrating robust and Bayesian statistics, practical trading frictions, and computational innovations.

1. Mathematical Formulation and Fundamental Assumptions

In its classical form, the Markowitz optimization problem seeks portfolio weights $w \in \mathbb{R}^n$ that minimize risk at a fixed expected return: $$\begin{align*} &\min_{w} \quad w^\top \Sigma w \qquad \text{(portfolio variance)} \ &\text{subject to} \quad w^\top \mu = \mu_0, \quad \mathbf{1}^\top w = 1 \end{align*}$$ with $\mu$ the vector of expected asset returns, $\Sigma$ the covariance matrix of returns, and $\mu_0$ the target return. Equivalently, one may formulate the problem as: $$\max_{w} \quad w^\top \mu - \lambda w^\top \Sigma w$$ where $\lambda$ tunes the risk aversion.

Markowitz's foundational derivation hinges on viewing the portfolio random return $R$ as a weighted linear combination of random asset returns: $R = \sum_{j=1}^J R_j X_j$ where $X_j$ are fixed portfolio weights and $R_j$ are the random returns of each asset. The assumption is that the proportion of the portfolio invested in each asset remains constant over the averaging interval, which implicitly assumes constant trade volumes throughout that period (Olkhov, 11 Aug 2025).

2. The Hidden Assumption: Constant Trade Volumes

The classical Markowitz model describes the mean and variance of portfolio returns using time averages and/or ensemble expectations, effectively assuming that the volume of each security traded (within the portfolio) is constant during the observation period: $U_j(t_i) = \text{constant}, \quad \forall i, j$ Under this assumption, the time series of portfolio returns simplifies and the variance calculation aligns with the quadratic form: $$\sigma_M^2 = \sum_{j=1}^J \sum_{k=1}^J \mathrm{Cov}(R_j, R_k) X_j X_k$$ This simplification neglects potential stochasticity in the volume of trades, i.e., the fact that in real markets the actual amount of each asset bought or sold fluctuates over time.

This foundational approximation is unwitting and pervasive in the literature. When trade volumes are not constant, the effective realized (market-based) variance of the portfolio can depart meaningfully from that predicted by the classical formula.

3. Market-Based Variance: Incorporating Random Trade Volumes

When the assumption of constant trade volumes is relaxed, the portfolio return becomes: $$R(t_i, t_0) = \sum_{j=1}^J R_j(t_i, t_0) \; x_j(t_i)$$ with $x_j(t_i) = \frac{u_j(t_i)}{W(t_i)}$ the random relative volumes, $u_j(t_i)$ being normalized trade volumes of asset $j$ at time $t_i$ and $W(t_i)$ the portfolio's total volume at $t_i$. Portfolio weights thus become random variables.

In this augmented view, the variance of the portfolio's return acquires additional terms due to the variation and covariance of the external trade flows: $$\sigma^2(t, t_0) = \sigma_M^2(t, t_0) - 2 a_0 R(t, t_0) x(t) + [R^2(t, t_0) - \sigma_M^2(t, t_0)] x^2(t)$$ where $x(t)$ is the coefficient of variation of trade volumes, $a_0$ a parameter involving covariance between trade values and volumes, and $R(t, t_0)$ the portfolio's mean return (Olkhov, 11 Aug 2025). This series expansion demonstrates that depending on the sign and magnitude of the trade volume variation, the realized portfolio risk can be either higher or lower than that predicted by the Markowitz variance.

In the limiting case where all $u_j(t_i)$ are constant, this market-based variance reduces to $\sigma_M^2$ as in Markowitz's derivation.

4. Implications for Portfolio Risk Assessment and Optimization

The practical consequence of the constant volume assumption is that the classical Markowitz formula may systematically understate or overstate portfolio risk when actual market trade volumes are volatile.

• Underestimation of Risk: If trade volume stochasticity is positively correlated with large moves in asset returns, portfolio risk can be higher than indicated by $\sigma_M^2$.
• Overestimation of Risk: Conversely, if the trade volume randomization works to diversify or dilute exposure, realized risk could be lower.
• Estimation Error Magnification: The additional randomness from trade volumes may propagate nonlinearly and interact with estimation error in $\mu$ and $\Sigma$, amplifying the "Markowitz optimization enigma"—the empirical finding that plug-in estimators often yield portfolios with poor out-of-sample performance and erratic weights (Lai et al., 2011).

For advanced practical implementations, especially in environments with high-frequency trading or low liquidity, incorporating explicit models for trade volume stochasticity and its statistical dependencies with asset returns provides a more accurate risk assessment.

5. Comparison Table: Risk Estimation under Different Assumptions

Approach	Trade Volume Assumption	Portfolio Variance Formula	Limiting Cases
Classical Markowitz (Original)	Constant within averaging	$$\sigma_M^2 = \sum_{j,k} \mathrm{Cov}(R_j, R_k) X_j X_k$$	Accurate if $U_j(t_i)$ constant
Market-Based (with random volumes)	Fluctuates randomly	$$\sigma^2 = \sigma_M^2 -2a_0 R x + (R^2-\sigma_M^2)x^2$$	Reduces to Markowitz if $x=0$

The presence and estimation of $a_0$ and $x$ require empirical modeling of trade flows, not just price data.

6. Limitations, Extensions, and Best Practices

While the classical Markowitz approach offers unique analytic tractability and is widely used due to its computational convenience, its reliance on deterministic or static weights may be restrictive for markets with substantial activity-driven microstructure noise. The extended market-based framework can better accommodate dynamic trading environments, at the cost of model complexity and increased parameterization:

• Limitations of the Markowitz Framework: Best suited for environments where portfolio rebalancing is infrequent, and trade volumes are stable relative to the averaging interval of the model.
• Contexts Demanding Extensions: Particularly relevant for high-frequency portfolios, algorithmic trading systems, illiquid markets, or when rebalancing introduces stochastically varying exposure.
• Caveats: Incorporating trade volume randomness necessitates new data requirements (trade-level data), additional statistical modeling (possibly leveraging realized weights and their conditional distributions), and a clear understanding of the potential for "hidden" sources of estimation risk.

7. Concluding Synthesis

Markowitz's mean-variance optimization paradigm established a tractable method to balance return and risk, codifying the relationship between asset return covariances and portfolio risk. However, this framework is a simplified representation of market reality, critically predicated on the constancy of portfolio weights over time—which, in variable trading environments, is tantamount to assuming constant trade volumes. When this assumption is relaxed, realized portfolio risk includes extra terms reflecting random volume fluctuations, which can significantly alter both risk estimates and the optimality of portfolio allocations (Olkhov, 11 Aug 2025). For researchers and practitioners, recognizing and, where appropriate, modeling this additional source of randomness is necessary for accurate, robust portfolio risk and performance evaluation.