Can volatility solve the naive portfolio puzzle?

Abstract

We investigate whether sophisticated volatility estimation improves the out-of-sample performance of mean-variance portfolio strategies relative to the naive 1/N strategy. The portfolio strategies rely solely upon second moments. Using a diverse group of portfolios and econometric models across multiple datasets, most models achieve higher Sharpe ratios and lower portfolio volatility that are statistically and economically significant relative to the naive rule, even after controlling for turnover costs. Our results suggest benefits to employing more sophisticated econometric models than the sample covariance matrix, and that mean-variance strategies often outperform the naive portfolio across multiple datasets and assessment criteria.

Keywords:

• Mean-variance
• Naive portfolio
• volatility

JEL:

• G11
• G17

Acknowledgments

We thank Caitlin Dannhauser, Jesús Fernández-Villaverde, Alejandro Lopez-Lira, Rabih Moussawi, Michael Pagano, Nikolai Roussanov, Paul Scanlon, Frank Schorfheide, John Sedunov, Raman Uppal, and Raisa Velthuis for helpful comments. Christopher Antonello provided diligent research assistance.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/14697688.2023.2249996.

Notes

1 Instead of the portfolio strategy, our innovation explores a wide variety of econometric models. DeMiguel et al. (2009b) find that the minimum-variance portfolio, though performing well relative to other portfolio strategies, significantly beats the 1/N strategy for only 1 in 7 of their datasets. Jagannathan and Ma (2003) and Kirby and Ostdiek (2012) innovate on the portfolio strategy, illustrating that short-sale constrained minimum-variance strategies and volatility-timing strategies enhance performance.

2 We consider a wide range of mostly parametric econometric models. Non-parametric models using higher-frequency data (DeMiguel et al. 2013) and shrinkage approaches (Ledoit and Wolf 2017) also improve the accuracy of estimation. Daily frequency option-implied volatility reduces portfolio volatility, but never statistically significantly improves the Sharpe ratio relative to the 1/N strategy (DeMiguel et al. 2013). Although Johannes et al. (2014) account for both estimation risk and time-varying volatility through eight variations of a similar class of constant and stochastic volatility models, we expand to more varied classes of volatility types with 14 econometric models. Initial investigations reveal our results to be at least as strong as Ledoit and Wolf (2017).

3 Our econometric estimation strategies yield improvements beyond the period and frequency differences.

4 A portfolio strategy, whose covariance is estimated using a given econometric model, weakly dominates the naive benchmark if, for each performance criterion, the portfolio strategy performs at least as well as the naive benchmark across all datasets and performs significantly better in at least one dataset.

5 For each portfolio strategy, we average the Sharpe ratios and portfolio volatility resulting from all 14 econometric models across all six datasets. Then we average Sharpe ratio and portfolio volatility across all three portfolio strategies.

6 Our study benefits from incorporating recent advances in the computation of several models as in Vogiatzoglou (2017), Chan and Eisenstat (2018), and Kastner (2019b). To reduce run-time, we employ fast, low-level languages, e.g. C++, that we program in parallel with hyperthreading and execute on clusters.

7 Using a shorter time-sample across one dataset with a larger portfolio, they do not consider vector autoregression, vector error correction for non-stationarity, or either regime-switching or stochastic volatility models, which are computationally challenging and account for observed nuances of time-varying volatility.

8 Preliminary evidence suggests that our results are at least as strong relative to the naive portfolio as what Ledoit and Wolf (2017) find. Direct comparisons are more complicated in Ao et al. (2019). Relative to the naive portfolio, initial experiments suggest that their MAXSER estimator performs better than our econometric models do in some comparisons, but that our models do better in most empirical comparisons.

9 To isolate one study by DeMiguel et al. (2009a), our paper employs improved econometric methods rather than more sophisticated portfolio constraints. Although not directly comparable, preliminary investigations reveal that our models improve performance relative to the naive portfolio by a greater ratio than DeMiguel et al. (2009a) in terms of Sharpe ratios, portfolio volatility, and turnover costs.

10 The use of shrinkage covariance matrices following Ledoit and Wolf (2004, 2017) is less relevant with smaller portfolio choices as in our study.

11 Letting $\hat{\mathbf{w}}$ denote our estimate of the optimal vector of portfolio weights $\mathbf{w}$, the MSE bias-variance decomposition from econometrics is $MSE\left(\hat{\mathbf{w}}\right)=Var\left(\hat{\mathbf{w}}\right)+Bia{s}^2(\hat{\mathbf{w}},\mathbf{w})$, where $Bias(\hat{\mathbf{w}},\mathbf{w})=\hat{\mathbf{w}}-\mathbf{w}$.

12 While we attempt to cover the broad classes of econometric models, our set of econometric models is not exhaustive. For instance, we omit the shrinkage estimators of Hafner and Reznikova (2012) and Ledoit and Wolf (2003, 2017). Although these and other econometric models are interesting, our study is the most expansive in its coverage of econometric models.

13 Detailed model, implementation, and robustness descriptions are relegated to the online appendix.

14 Inference from finite samples is also much more informative regarding the expected return volatility rather than the expected mean return (Andersen and Teräsvirta 2009).

15 Regularization implied by no short selling ensures invertibility of the resulting covariance matrix.

16 First, a variation of our benchmark CP strategy (5(5) $\begin{array}{r}{\mathbf{w}}_{t,j}^{MVP,comv}=\underset{\mathbf{w}\in {\mathbb{R}}^N|{\mathbf{w}}^{\prime }\mathbf{1}=\mathbf{1}}{argmin}\mathbf{w}{\hat{\mathrm{\Sigma }}}_t^{comv}{\mathbf{w}}^{\prime }.\end{array}$(5) ), for each of the strategies (2(2) $\begin{array}{r}{\mathbf{w}}_{t,j}^{MVP}=\underset{\mathbf{w}\in {\mathbb{R}}^N|{\mathbf{w}}^{\prime }\mathbf{1}=\mathbf{1}}{argmin}\mathbf{w}\widehat{{\mathrm{\Sigma }}_t^j}{\mathbf{w}}^{\prime }.\end{array}$(2) )–(4(4) $\begin{array}{r}{\left(w_{t,j}^{VT}\right)}_i=\frac{1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}{\sum _{i=1}^{N}1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}i=1,\dots ,N.\end{array}$(4) ), we examine the corresponding portfolio given by naive investments across the 13 portfolios with respect to each of the econometric models. More precisely, consider the minimum-variance portfolio. We form a 14th portfolio strategy, $w_t^{MVP,com}$, which is equally invested across the 13 estimates of the true minimum-variance portfolio, i.e. ${\mathbf{w}}_t^{MVP,com}=\frac{1}{13}\sum _{j=1}^{13}{\mathbf{w}}_{t,j}^{MVP}.$ Second, with respect to each of the econometric models, we examine the corresponding portfolio given by naive investments across the three strategies (2(2) $\begin{array}{r}{\mathbf{w}}_{t,j}^{MVP}=\underset{\mathbf{w}\in {\mathbb{R}}^N|{\mathbf{w}}^{\prime }\mathbf{1}=\mathbf{1}}{argmin}\mathbf{w}\widehat{{\mathrm{\Sigma }}_t^j}{\mathbf{w}}^{\prime }.\end{array}$(2) )–(4(4) $\begin{array}{r}{\left(w_{t,j}^{VT}\right)}_i=\frac{1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}{\sum _{i=1}^{N}1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}i=1,\dots ,N.\end{array}$(4) ). More precisely, consider the VAR econometric model. We form a fourth portfolio strategy, $w_t^{VAR,comp}$ that is equally invested across the three vectors of portfolio weights suggested by inputting the volatility estimates from the VAR model into strategies (2(2) $\begin{array}{r}{\mathbf{w}}_{t,j}^{MVP}=\underset{\mathbf{w}\in {\mathbb{R}}^N|{\mathbf{w}}^{\prime }\mathbf{1}=\mathbf{1}}{argmin}\mathbf{w}\widehat{{\mathrm{\Sigma }}_t^j}{\mathbf{w}}^{\prime }.\end{array}$(2) )–(4(4) $\begin{array}{r}{\left(w_{t,j}^{VT}\right)}_i=\frac{1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}{\sum _{i=1}^{N}1/{\left({\hat{\mathrm{\Sigma }}}_t^j\right)}_{i,i}}i=1,\dots ,N.\end{array}$(4) ), i.e. ${\mathbf{w}}_t^{VAR,comp}=\frac{1}{3}\sum _{k=1}^3{\mathbf{w}}_{t,VAR}^k.$

17 Standard and Poor's established Compustat in 1962 to serve the needs of financial analysts and back-filed information only for the firms that were deemed to be of the greatest interest to the analysts. The result is significantly sparser coverage prior to 1963 for a selected sample of well performing firms.

18 Kenneth French provides full description at https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/

19 The risk-free (RF) asset is the one-month Treasury bill rate from Ibbotson Associates and proxies the return from investing in the money market. We exclude the risk-free rate from the investor's choice set; therefore, we exclude returns in excess of the risk-free rate.

20 We also employ equal-weighting in robustness checks.

21 Dataset 2's industry portfolios are popular among academics with the older Standard Industry Classification (SIC) scheme and longer data, while the broader dataset 3's sector portfolios with newer GICS codes are popular amongst practitioners.

22 Covariance-based methods such as the minimum-variance portfolio may lower variance relative to the 1/N portfolio and thus raise Sharpe ratios. We therefore also consider returns. While the naive strategy performs well on dataset 1, and despite with weaker dominance over the naive strategy on dataset 6, other strategies dominate the naive strategy on datasets 2 through 5. Results are available upon request.

23 Several papers in the literature consider transaction costs of 10 or 50 basis points (Kirby and Ostdiek 2012, DeMiguel et al. 2014) and others consider transactions costs that vary across stock size and through time (Brandt et al. 2009). With high turnover, assuming 50 basis points transactions costs conservatively biases our models away from beating the 1/N strategy.

24 The forecast error is defined as the difference between expected returns using estimated portfolio weights and mean returns. The loss differential underlying the test looks at the difference of the squared forecast errors, and we calculate the the loss differential correcting for autocorrelation.

25 We report only results for value-weighted data.

26 Tables S2.1–3 in the online appendix report pairwise comparisons between the three portfolio variance strategies. For each evaluation criterion across most econometric models and datasets, the minimum-variance strategy performs the best and the volatility-timing strategy performs the weakest.

27 To explain the poorer performance of datasets 1 and 3, first, the literature consistently finds weak performance with the Fama-French dataset (DeMiguel et al. 2009b); second, a simple correlation matrix of the six datasets shows that dataset 3 is the only dataset to be negatively correlated with the other datasets.

28 Allocations can shift, requiring rebalancing turnover even for the naive portfolio. With turnover, expected returns are no larger, but standard deviations may be smaller or larger.

29 To clarify, ‘$✓$’ $=1$, ‘$✓$*’ $=2/3$, ‘ ’ (blanks) $=0$, and ‘×’ $=-1.$ We discount results that are significant at the 10% level by assigning a value of only 2/3 instead of 1.