[Paper Review] Asymptotic distribution of the Markowitz portfolio

Steven E. Pav|arXiv (Cornell University)|Dec 2, 2013

Financial Risk and Volatility Modeling60 references3 citations

TL;DR

This paper derives the asymptotic distribution of the Markowitz portfolio under general moment conditions and multivariate normality, enabling robust inference under heteroskedasticity and autocorrelation. It provides a likelihood ratio test for linear combinations of the precision matrix and Markowitz portfolio, with applications to error decomposition and portfolio shrinkage.

ABSTRACT

The asymptotic distribution of the Markowitz portfolio is derived, for the general case (assuming fourth moments of returns exist), and for the case of multivariate normal returns. The derivation allows for inference which is robust to heteroskedasticity and autocorrelation of moments up to order four. As a side effect, one can estimate the proportion of error in the Markowitz portfolio due to mis-estimation of the covariance matrix. A likelihood ratio test is given which generalizes Dempster's Covariance Selection test to allow inference on linear combinations of the precision matrix and the Markowitz portfolio. Extensions of the main method to deal with hedged portfolios, conditional heteroskedasticity, conditional expectation, and constrained estimation are given. It is shown that the Hotelling-Lawley statistic generalizes the (squared) Sharpe ratio under the conditional expectation model. Asymptotic distributions of all four of the common `MGLH' statistics are found, assuming random covariates. Examples are given demonstrating the possible uses of these results.

Motivation & Objective

• To derive the asymptotic distribution of the sample Markowitz portfolio under general moment conditions (fourth moments exist), enabling robust statistical inference.
• To extend inference to handle heteroskedasticity and autocorrelation in return moments up to order four.
• To estimate the proportion of portfolio error attributable to covariance matrix misestimation, challenging the common belief that return mean estimation dominates error.
• To generalize Dempster’s covariance selection test to allow inference on linear combinations of the precision matrix and Markowitz portfolio.
• To extend the framework to hedged portfolios, conditional heteroskedasticity, conditional expectations, and constrained estimation.

Proposed method

• Formulates the Markowitz portfolio as the solution to a linear regression problem, enabling asymptotic normality derivation under general moment conditions.
• Uses the augmented second moment matrix $\mathsf{\Theta} = \mathbb{E}[\tilde{\mathbf{x}} \tilde{\mathbf{x}}^\top]$, where $\tilde{\mathbf{x}} = [1, \mathbf{x}^\top]^\top$, to link the Markowitz portfolio to the inverse of $\mathsf{\Theta}$.
• Derives the asymptotic distribution of $\hat{\boldsymbol{\nu}}_*$ using the central limit theorem on the sample second moment of $\tilde{\mathbf{x}}$, assuming i.i.d. returns with finite fourth moments.
• Derives the asymptotic covariance of $\hat{\boldsymbol{\nu}}_*$ and $\hat{\mathsf{\Sigma}}^{-1}$, enabling error decomposition and shrinkage estimation.
• Constructs a likelihood ratio test under multivariate normality for linear combinations of $\boldsymbol{\nu}_*$ and $\mathsf{\Sigma}^{-1}$, generalizing Dempster’s test.
• Shows that the Hotelling-Lawley statistic generalizes the squared Sharpe ratio under the conditional expectation model, linking classical statistics to portfolio performance metrics.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the sample Markowitz portfolio when only fourth moments exist, and how can inference be made robust to heteroskedasticity and autocorrelation?
• RQ2 To what extent is estimation error in the Markowitz portfolio attributable to misestimation of the covariance matrix versus the mean vector?
• RQ3 Can a likelihood ratio test be developed to perform inference on linear combinations of the precision matrix and the Markowitz portfolio, generalizing existing tests?
• RQ4 How do the asymptotic distributions of the four common MGLH statistics (e.g., Hotelling-Lawley) behave under random covariates and general moment conditions?
• RQ5 Can the framework be extended to handle hedged portfolios, conditional heteroskedasticity, and constrained estimation in a unified way?

Key findings

• The sample Markowitz portfolio $\hat{\boldsymbol{\nu}}_*$ is asymptotically normal under the assumption that the returns have finite fourth moments.
• The asymptotic covariance of $\hat{\boldsymbol{\nu}}_*$ is feasible to estimate and robust to heteroskedasticity and autocorrelation in moments up to order four.
• The proportion of error in the Markowitz portfolio due to covariance matrix misestimation can be estimated directly from the derived asymptotic covariance structure.
• Under multivariate normal returns, a likelihood ratio test is derived that generalizes Dempster’s covariance selection test to include linear combinations of the precision matrix and the Markowitz portfolio.
• The Hotelling-Lawley statistic is shown to generalize the squared Sharpe ratio under the conditional expectation model, providing a unified statistical framework for performance evaluation.
• Asymptotic distributions of all four MGLH statistics (e.g., Pillai’s trace, Wilks’ lambda) are derived under random covariates, extending classical multivariate analysis to portfolio contexts.