On the Combination of Naive and Mean-Variance Portfolio Strategies

Abstract

We study how to best combine the sample mean-variance portfolio with the naive equally weighted portfolio to optimize out-of-sample performance. We show that the seemingly natural convexity constraint—the two combination coefficients must sum to one—is undesirable because it severely constrains the allocation to the risk-free asset relative to the unconstrained portfolio combination. However, we demonstrate that relaxing the convexity constraint inflates estimation errors in combination coefficients, which we alleviate using a shrinkage estimator of the unconstrained combination scheme. Empirically, the constrained combination outperforms the unconstrained one in a range of generally small degrees of risk aversion, but severely deteriorates otherwise. In contrast, the shrinkage unconstrained combination enjoys the best of both strategies and performs consistently well for all levels of risk aversion.

Keywords:

• Equally weighted portfolio
• Estimation risk
• Parameter uncertainty
• Portfolio constraints
• Portfolio optimization

Supplementary Materials

The Supplementary Material is divided in four sections. In Section A, we provide a detailed literature review. In Section B, we give simulation and theoretical evidence regarding the impact of estimation errors on constrained and unconstrained combination coefficients. In Section C, we provide additional theoretical results and corresponding empirical results. In Section D, we detail the proofs of all theoretical results in the main body of the paper and in the Supplementary Material.

Acknowledgments

The authors thank Victor DeMiguel, Alberto Martìn-Utrera, Majeed Simaan, Valeri Sokolovski, Xiaolu Wang, Morten Wilke, Guofu Zhou, and participants at the Belgian Financial Research Forum, 16th International Conference on Computational and Financial Econometrics, IFABS 2022 conference, CORE brown bag seminar, HEC Montréal seminar, and Louvain Finance seminar for helpful comments.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 Although we focus on combining the SMV and EW portfolios, in Section C.3 of the supplementary material we also derive counterparts of our main results for the combination of the SMV and SGMV portfolios.

2 In contrast with EW that never invests in the risk-free asset, SMV does so whenever $\gamma \ne \mathit{1}\prime {\stackrel{\mathbf{⁁}}{\mathbf{\Sigma }}}^{\mathit{-1}}\widehat{\mathit{\mu }}$.

3 For the dataset of 25 size-and-book-to-market portfolios (25SBTM) spanning July 1926 to December 2022, a sample size T = 120 months, and a risk aversion γ = 10, the constrained and unconstrained strategies allocate 57% and 20% to SMV, respectively.

4 Suppose asset returns are t distributed with six degrees of freedom, that is, a realistic excess kurtosis of three for monthly returns. For the 25SBTM dataset, $N/T=0.2$, and a risk aversion γ = 5, the unconstrained combination coefficients are $({\kappa }_1,{\kappa }_2)=(0.19,0.29)$, while those under normality are $({\kappa }_1,{\kappa }_2)=(0.20,0.29)$. The expected out-of-sample utility loss by relying on the normal coefficients is a mere 0.007% per month.

5 A detailed literature review is available in Section A of the supplementary material.

6 In Section C.2 of the supplementary material, we derive theoretical results in the high-dimensional asymptotic regime, that is, $N,T\to \infty$ and $N/T\to \rho \in (0,1)$, which is often considered in recent literature; see, for example, Bodnar, Parolya, and Schmid (2018), Ao, Li, and Zheng (2019), Ledoit and Wolf (2020), and Bodnar, Okhrin, and Parolya (2023).

7 In Section C.7 of the supplementary material, we also study the expected out-of-sample Sharpe ratio.

8 We make this assumption to simplify the exposition of our theory. The results that need this assumption can also be extended to the case in which ${\mu }_{ew} < 0$.

9 The parameter γ_ew has the following interpretation: an investor with risk aversion γ investing in the EW portfolio and the risk-free asset fully allocates her wealth to the EW portfolio if and only if $\gamma ={\gamma }_{ew}$.

10 All the proofs are available in Section D of the supplementary material.

11 Kan and Wang (2023, sec. 2.3.2) use similar coefficients in a different context, that of combining a benchmark factor model with a set of test assets.

12 For the 25SBTM dataset and T = 120, $d>{\theta }^2$ holds as soon as N > 8, and ${\gamma }_{\mathit{neg}}=5.27$.

13 Provided $\mathbf{\Sigma }$ is a constant diagonal matrix, ${\psi }^2=0$ if and only if $\mathit{\mu }=c\mathbf{1}$ for some $c\in \mathbb{R}$.

14 ${\Delta }_{\mathit{smv}}\le 0$ if ${\psi }^2+d-{\theta }_{ew}^2\le 0$, and else ${\Delta }_{\mathit{smv}}\ge 0$ if $\gamma \ge {\gamma }_{ew}(1+\sqrt{\frac{2({\psi }^2+d)}{{\psi }^2+d-{\theta }_{ew}^2}})$, and ${\Delta }_{\mathit{smv}}<0$ otherwise.

15 If d is small enough such that $d^2({\psi }^2+d)-8{\theta }_{ew}^2{\psi }^2(2{\psi }^2+d)\le 0$, then ${\Delta }_{ew}\le 0$ if $\gamma \in [{\gamma }_{ew},({\theta }^2+d)/{\mu }_{ew}]$, and else ${\Delta }_{ew}>0$. Otherwise, ${\Delta }_{ew}\ge 0$ if either $\gamma \le {\gamma }_{ew},\gamma \ge ({\theta }^2+d)/{\mu }_{ew}$, or $\gamma \in \left[\frac{d^2+d({\theta }^2+{\theta }_{ew}^2)+4{\theta }_{ew}^2{\psi }^2\pm \sqrt{({\psi }^2+d)(d^2({\psi }^2+d)-8{\theta }_{ew}^2{\psi }^2(2{\psi }^2+d))}}{2{\mu }_{ew}(2{\psi }^2+d)}\right].$

16 (Assumption-BB) ensures that the distribution of the λ_t’s does not put too much mass near zero.

17 We recover the case of iid normal asset returns when $\nu \to \infty$, that is, ${\lambda }_t\to 1$. This situation leads to $\alpha =\beta =1$, and thus, ${\overline{\mathit{\kappa }}}_n^c={\overline{\mathit{\kappa }}}^c$ and ${\overline{\mathit{\kappa }}}_u^c={\overline{\mathit{\kappa }}}^u$.

18 We also consider the nonlinear shrinkage estimate of Ledoit and Wolf (2020) in Section C.9 of the supplementary material and find the conclusions are similar to those obtained with linear shrinkage.

19 The MCS test needs as an input a performance difference for each pair of models and for all time t. However, our performance measure, the net out-of-sample utility, can only be computed over several out-of-sample monthly returns. Therefore, to provide a performance difference for all time t, we collect daily returns for all our datasets and evaluate the net out-of-sample utility of a given portfolio at time t from the next-month daily returns, using daily rebalancing.

20 In Section C.5 of the supplementary material, we also derive a four-fund strategy that combines SMV, EW, and SGMV. However, we find that it is generally outperformed by UNC3F that does not invest in SGMV. This can be explained because the theoretical gain from adding SGMV is small and thus is offset by the estimation risk coming from the additional combination coefficient to estimate.

Additional information

Funding

This work was supported by the Fonds de la Recherche Scientifique (F.R.S.-FNRS) under Grant Number J.0115.22 and T.0221.22, and by the Belgian Federal Science Policy Office under Grant Number ARC 18-23/089.