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Understanding the Long-Only Minimum Variance Portfolio

Nick L. Gunther, Alec N. Kercheval, Ololade Sowunmi

Abstract

For a covariance matrix coming from a factor model of returns, we investigate the relationship between the long-only global minimum variance portfolio and the asset exposures to the factors. In the case of a 1-factor model, we provide a rigorous and explicit description of the long-only solution in terms of the parameters of the covariance matrix. For $q>1$ factors, we provide a description of the long-only portfolio in geometric terms. The results are illustrated with empirical daily returns of US stocks.

Understanding the Long-Only Minimum Variance Portfolio

Abstract

For a covariance matrix coming from a factor model of returns, we investigate the relationship between the long-only global minimum variance portfolio and the asset exposures to the factors. In the case of a 1-factor model, we provide a rigorous and explicit description of the long-only solution in terms of the parameters of the covariance matrix. For factors, we provide a description of the long-only portfolio in geometric terms. The results are illustrated with empirical daily returns of US stocks.
Paper Structure (15 sections, 11 theorems, 114 equations, 5 figures, 2 tables)

This paper contains 15 sections, 11 theorems, 114 equations, 5 figures, 2 tables.

Table of Contents

  1. Long-only minimum variance
  2. Introduction
  3. Main Results
  4. Numerical Examples
  5. Single factor exposure estimation
  6. Empirical example for a single index model
  7. Empirical example for a two-factor model
  8. Proofs
  9. Proof of Theorem \ref{['thm:wL']}
  10. Proof of Theorem \ref{['thm:1']}
  11. Proof of Theorem \ref{['thm:multi']}
  12. Single factor beta estimation
  13. The JSE estimator
  14. Consistency of single index estimators
  15. Multifactor JSM estimation

Key Result

Theorem 1

Denote by $w^L$ the solution of problem eq: prob 1 and let $K$ denote the set of active assets in $w^L$: and $k = |K| \leq p$, the number of active assets. Let $\Sigma^{K,0}$ be the modified matrix obtained from $\Sigma$ by setting to zero the rows and columns not belonging to $K$. Then where $^+$ denotes the Moore-Penrose inverse.For a symmetric matrix $S$ with singular value decomposition $S =

Figures (5)

  • Figure 1: Comparison of portfolio asset weights for the three portfolios MJSE, JSE, and MS, plotted in percent.
  • Figure 2: MJSE portfolio weights against market beta for the top 1000 US stocks, estimated for daily returns in the first half of 2022.
  • Figure 3: MJSE portfolio weights against specific risk for the top 1000 US stocks, from daily returns in the first half of 2022.
  • Figure 4: MJSE market beta against specific risk for the top 1000 US stocks in the first half of 2022.
  • Figure 5: 2-vector exposures for each asset from a statistical 2-factor JSM return covariance matrix, plotted for each of the 1000 assets. On the right is a close-up showing portfolio weights.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3: Hyperplane separation for $q$-factor models
  • Corollary 2
  • Lemma 1
  • Corollary 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 1 more