Can Volatility Solve the Naive Portfolio Puzzle?
CCan Volatility Solve the Naive Portfolio Puzzle? ∗ Michael Curran † Villanova University Ryan Zalla ‡ University of Pennsylvania
August 6, 2020
Abstract
We investigate whether sophisticated volatility estimation improves the out-of-sample performance of mean-variance portfolio strategies relative to the naive 1/Nstrategy. The portfolio strategies rely solely upon second moments. Using a diversegroup of econometric and portfolio models across multiple datasets, most modelsachieve higher Sharpe ratios and lower portfolio volatility that are statistically andeconomically significant relative to the naive rule, even after controlling for turnovercosts. Our results suggest benefits to employing more sophisticated econometric mod-els than the sample covariance matrix, and that mean-variance strategies often out-perform the naive portfolio across multiple datasets and assessment criteria.
Keywords: volatility, naive portfolio, mean-variance
JEL:
G11, G17 ∗ We thank Caitlin Dannhauser, Jes´us Fern´andez-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. † Corresponding author. Email: [email protected]; Phone: (+1) 610-519-8867Address: Economics Dept., Villanova School of Business, Villanova University, 800 E Lancaster Ave, PA 19085, USA. ‡ Email: [email protected]; Phone: (+1) 412-759-5032Address: Economics Dept., University of Pennsylvania, 133 South 36th Street, Philadelphia, PA 19104, USA. a r X i v : . [ q -f i n . GN ] A ug AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Ever since mean-variance strategies were introduced by Markowitz (1952), their out-of-sample performance has been criticized. One reason for weak performance is estimationerror, particularly in the mean return (Merton, 1980; Chopra and Ziemba, 2013). Variancestrategies, which rely solely upon second moments, avoid the pitfall of expected returns es-timation. In comparing variance strategies relative to the naive et al. (2009b) include strategies account-ing for parameter errors, but employ sample covariance estimates. Few papers in the naiveportfolio literature pursued improved estimation of the volatility of returns.
Our marginalcontribution is that a variety of econometric models of volatility improve performance rel-ative to the naive portfolio strategy. That is, relative to sample covariance estimation, ourstudy suggests considerable performance gains from employing modern econometric modelsfor estimating volatility in portfolio construction.We investigate whether sophisticated volatility estimation improves the performance ofmean-variance strategies depending only on conditional variance-covariance matrices (vari-ance strategies) relative to the naive 1/N strategy. Using a diverse set of fourteen econo-metric models, we evaluate the out-of-sample performance of portfolio strategies that relysolely on estimation of the second moment of returns. We apply three portfolio strategies 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 otherportfolio strategies, significantly beats the 1/N strategy for only 1 in 7 of their datasets. Jagannathanand Ma (2003) and Kirby and Ostdiek (2012) innovate on the portfolio model, illustrating that short-saleconstrained minimum-variance strategies and volatility timing strategies enhance performance. We consider a wide range of mostly parametric econometric models. Non-parametric models usinghigher-frequency data (DeMiguel et al. , 2013) and shrinkage approaches (Ledoit and Wolf, 2017) alsoimprove 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). Initial investigations reveal our results to be at least as strong as Ledoit and Wolf (2017).
CURRAN AND ZALLA – minimum-variance, constrained minimum-variance, and volatility-timing – to six empir-ical datasets of weekly and monthly returns; we also include the tangency portfolio. Weassess performance using the following three criteria: (i) the Sharpe ratio, (ii) the turnover(trading) costs, and (iii) the standard deviation of returns (portfolio volatility). Overall,we show that variance strategies perform consistently and significantly well out-of-sample.Our first contribution is to show that all four portfolio strategies, regardless of the econo-metric model used for estimating covariance, achieve improved out-of-sample performancesrelative to the naive benchmark. This assertion challenges the literature that has rarely re-ported superior minimum-variance performances relative to the equally-weighted portfolio(DeMiguel et al. , 2009b). Specifically, we find that all four portfolio strategies estimatedusing all fourteen econometric models perform at least as well as the naive benchmark,and only rarely underperform it. These underperformances are mainly concentrated in theFama-French 3-factor dataset. If we discard this dataset, then all four portfolio strategiesestimated using twelve of the fourteen econometric models would not just perform well,they would weakly dominate the naive benchmark. In general, the minimum-variance strategy, with or without short-sale constraints, achieveshigher Sharpe ratios, lower turnover costs, and lower portfolio volatility across the majorityof datasets. Wherever these strategies do not perform significantly better than the naiverule, we often fail to reject the null hypothesis that their performances are identical tothe naive rule. Likewise, we find evidence that volatility-timing strategies achieve higherSharpe ratios and lower turnover costs but exhibit portfolio volatility that is comparableto the naive rule. The tangency portfolio does reasonably well in certain datasets. Rela- Section 3 defines our financial portfolio models. Our econometric estimation strategies yield improvements beyond the period and frequency differences. 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 asthe naive benchmark across all datasets and performs significantly better in at least one dataset.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? tive to the other strategies, however, its performance is lackluster, which we attribute toestimation error in expected returns.Our second contribution is to identify pairings of volatility models and portfolio strate-gies that perform consistently and significantly well across datasets relative to the naivebenchmark. Multivariate GARCH models, particularly the constant conditional correla-tion (CCC), weakly dominate the naive rule when applied to minimum-variance and con-strained minimum-variance strategies. Similarly, the realized covariance (RCOV) modelweakly dominates the naive rule when applied to the volatility-timing strategy. In thetangency portfolio, although the RCOV model achieves higher Sharpe ratios and lowerportfolio volatility relative to the naive strategy than other econometric models, it ex-hibits higher portfolio volatility relative to the naive rule in data on international equities.Thus, although we cannot say the RCOV model applied to the tangency portfolio weaklydominates the naive rule, many other econometric models do, such as the multivariateGARCH models. Nonetheless, even econometric models such as the regime-switching vec-tor autoregression (RSVAR) and exponentially-weighted moving-average (EWMA), whichperform worst in each of the four porfolio models, still perform at least as well as the naivebenchmark across every dataset except the Fama-French 3-factor.Our third contribution is to compare the performance of econometric models relative tothe naive benchmark using each assessment criterion in isolation . That is, across portfoliosmodels and datasets, which econometric models consistently achieve the highest Sharperatios? Which econometric models achieve the lowest turnover costs? Which econometricmodels achieve the lowest portfolio volatilities? First, the combined parameter (CP) andrealized covariance (RCOV) models achieve significantly higher Sharpe ratios than the naivebenchmark. Second, the RCOV and a variety of GARCH models produce significantlyhigher Sharpe ratios than the naive rule after adjustment for turnover costs. Finally, the CURRAN AND ZALLA exponentially-weighted moving-average (EWMA), multivariate stochastic volatility (MSV),and RCOV models exhibit significantly lower portfolio volatility than the naive rule. Theseperformances are economically and statistically significant: relative to the naive strategy,we achieve 30% higher Sharpe ratios and 9% lower portfolio volatility. Our paper exploits recent computational developments to estimate the covariance ma-trix using multivariate, nonlinear, non-Gaussian econometric models. We extend the setof models in Wang et al. (2015) beyond GARCH and discrete regime-switching modelsto smooth multivariate stochastic volatility and non-parametric realized volatility models.Unlike Wang et al. (2015), our models employ multivariate ( N ≈
10) rather than bivariate( N =
2) sets of assets. Our study benefits from incorporating recent advances in the compu-tation of several models as in Vogiatzoglou (2017), Chan and Eisenstat (2018), and Kastner(2019b). We draw two conclusions from our results. First, relative to sample covariance estima-tion, our study suggests considerable performance gains from employing modern economet-ric models for estimating volatility in portfolio construction. Second, variance strategiesconsistently perform well relative to the naive strategy.To place our paper in context, our study builds on the literature of naive diversification.Mean-variance models struggle to compete with naive diversification in out-of-sample returnperformance. Considering a variety of mean-variance strategies including ones accountingfor parameter errors, DeMiguel et al. (2009b) find that no portfolio model consistentlyoutperforms naive diversification. One cause for the weak performance is that while mean-variance strategies based on the minimum-variance portfolio outperforms naive diversifica- For each portfolio strategy, we average the Sharpe ratios and portfolio volatility resulting from allfourteen econometric models across all six datasets. Then we average Sharpe ratio and portfolio volatilityacross all four portfolio models. To reduce run-time, we employ fast, low-level languages, e.g., C++, that we program in parallel withhyperthreading and execute on clusters.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? tion, portfolio turnover costs negates these benefits (Kirby and Ostdiek, 2012). To addressthe turnover issue, the authors propose two new approaches that reduce portfolio turnover,finding that the mean-variance strategies outperform the naive strategy out-of-sample. Weemploy their volatility-timing strategy. Combining naive diversification with other con-ventional portfolio strategies can also significantly enhance portfolio performance (Tu andZhou, 2011). We investigate combining parameters from the variance-covariance matrixestimates as well as from the weights suggested by portfolio models.A number of studies have increased the sophistication of parameter estimation sinceDeMiguel et al. (2009b), which was based on rolling window sample estimates. Consideringa variety of mostly GARCH-based estimators, no method is universally superior (Truc´ıos et al. , 2019). A non-linear shrinkage estimator (Ledoit and Wolf, 2017) and an estimationstrategy for large portfolios (Ao et al. , 2019) can improve performance. The use of impliedvolatility and skewness (DeMiguel et al. , 2013), vector autoregression (DeMiguel et al. ,2014), and portfolio constraints (DeMiguel et al. , 2009a; Kourtis et al. , 2012; Behr et al. ,2013) also enhances portfolio performance. Recent papers find comparably poor perfor-mance using sophisticated hedging strategies to attempt beating naive one-to-one hedg-ing (Wang et al. , 2015), provide behavioral evidence of naive choice strategies (De Giorgiand Mahmoud, 2018; Gathergood et al. , 2019), and identify the importance of volatilityfor portfolio performance (Moreira and Muir, 2017, 2019). Consistent with Moreira and Using a shorter time-sample across one dataset with a larger portfolio, they do not consider vectorautoregression, vector error correction for non-stationarity, or either regime-switching or stochastic volatilitymodels, which are computationally challenging and account for observed nuances of time-varying volatility. Preliminary evidence suggests that our results are at least as strong relative to the naive portfolio aswhat Ledoit and Wolf (2017) find. Direct comparisons are more complicated in Ao et al. (2019). Relativeto the naive portfolio, initial experiments suggest that their MAXSER estimator performs better than oureconometric models do in some comparisons, but that our models do better in most empirical comparisons. To isolate one study by DeMiguel et al. (2009a), our paper employs improved econometric methodsrather than more sophisticated portfolio constraints. Although not directly comparable, preliminary in-vestigations reveal that our models improve performance relative to the naive portfolio by a greater ratiothan DeMiguel et al. (2009a) in terms of Sharpe ratios, portfolio volatility, and turnover costs.
CURRAN AND ZALLA
Muir (2017, 2019), our paper finds that volatility-timing strategies generate higher Sharperatios in these same datasets yet exhibit equal volatility to the naive portfolio. Takingseriously the naive strategy as an important benchmark, we explore improved econometricestimation of volatility as a potential source of out-of-sample performance.Our benchmark is the naive portfolio diversification rule, allocating a fraction 1 / N ofwealth to each of N assets available for investment at each rebalancing date. Threereasons justify the naive rule as a benchmark. First, implementation is easy as it relieson neither optimization nor estimation of the moments of asset returns. Second, investorsuse such simple rules for allocating their wealth across assets (Benartzi and Thaler, 2001;Baltussen and Post, 2011; De Giorgi and Mahmoud, 2018; Gathergood et al. , 2019). Third,the naive rule consistently outperforms mean-variance strategies (Jobson and Korkie, 1980;Michaud and Michaud, 2008; DeMiguel et al. , 2009b; Duchin and Levy, 2009; Pflug et al. ,2012). Without needing to estimate the 1/N weights, the variance of parameter estimationis zero, and thus the mean square error of the naive portfolio weights is simply the squareof the bias. The naive strategy therefore proxies as a challenging rival for mean-variancestrategies to outperform. We apply our methods to six different empirical datasets with N ≤
11 portfolio choices. In preliminaryinvestigations, we experimented with datasets with N =
25 portfolio choices. As indicated by DeMiguel et al. (2009b), however, the critical minimal value for the estimation period (rolling window sample) grows in N ,requiring windows much larger than 10 years. Larger N reduces performance when the estimation windowis fixed. The reason is that there are more parameters in the covariance matrix to estimate and thereforethere is more room for estimation error. Large portfolio choice sets preclude using most empirical datasetsbecause we typically need over 100 years of data. Employing larger datasets also significantly increases thecomputing time. Fortunately, our multiple datasets are broad and varied. For instance, we cover datasetsallowing investors to diversify across international equities in the form of national stock market aggregateindices, across entire sector and industry indices, across expansive Fama-French portfolios, and across otherencompassing indices based on size/book-to-market and momentum portfolios. The best application forour methods from a practical standpoint, therefore, is to an investor holding multiple mutual funds inequities. Letting ˆ w denote our estimate of the optimal vector of portfolio weights w , the MSE bias-variancedecomposition from econometrics is M SE ( ˆ w ) = V ar ( ˆ w ) + Bias ( ˆ w , w ) , where Bias ( ˆ w , w ) = ˆ w − w . AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 1 lists the econometric models, some of which are included in Wang et al. (2015), whoconsider out-of-sample performance of hedging strategies. Their analysis uses econometricmodeling of bivariate random variables, whereas we extend the analysis to multivariaterandom variables, where N >
2. We use a rolling window approach with T + M returns for N assets. M is the estimation window length so that there are T out-of-sample investmentperiods. Picking ten-year rolling windows, M will be set by the frequency of the data.For instance, M =
520 for weekly data and M =
120 for monthly data. We choose T = et al. (2009b). Let { ˆΣ jt } Tt = denotethe conditional estimate of the variance-covariance matrix of returns for investment period t based on econometric model j . Similarly, we define the conditional estimate of the expectedreturn over period t given model j by { ˆ µ j } Tt = . The portfolio models we focus on are basedsolely on the variance-covariance matrix. We also experiment with the tangency portfolioand use the sample mean, which is independent of econometric model j , as our estimate ofthe mean. Mean-variance strategies are defined by the first two conditional moments of thereturn for period t so that {( ˆ µ jt , ˆΣ jt )} Tt = defines the sequence of mean-variance strategiesover the T out-of-sample investment periods with respect to model j . Section 3 details themean-variance strategies. Our first econometric model to estimate the variance-covariance matrix is the sample covari-ance matrix (Cov). Sample-based in-sample estimation of the variance-covariance matrix While we attempt to cover the broad classes of econometric models, our set of econometric models isnot exhaustive. For instance, we omit the shrinkage estimators of Ledoit and Wolf (2003, 2017). Althoughthese and other econometric models are interesting, we exclude them in the interest of space.
CURRAN AND ZALLA is standard in the literature on the naive diversification puzzle (DeMiguel et al. , 2009b;Fletcher, 2011; Tu and Zhou, 2011; Kirby and Ostdiek, 2012). Some studies examine im-proved estimation, but they typically focus on a small set of models (DeMiguel et al. , 2013).The sample covariance matrix places equal weight on past observations.
The recent past might be more informative for estimating the variance-covariance matrixto use in selecting portfolios, motivating our first refinement, the exponentially weightedmoving average (EWMA) model. The EWMA model suggested by RiskMetrics placesdecaying weight on the past: Σ t = α Σ t − + ( − α )( r t − ¯ r t ) ′ ( r t − ¯ r t ) where our decay parametersuggested is 0.96 (weekly) or 0.97 (monthly). To exploit dependence along the cross-section and time-dimension (serial), we estimatea vector autoregression (VAR): Y = X Φ + U using Bayesian methods. We experimentedwith a VAR with a Normal-Wishart natural conjugate prior and another with a Minnesotaprior. To reduce the impact of our prior on our results, we choose the posterior based on anon-informative prior, and we report the posterior mode of the variance-covariance matrix.Long lags are helpful to approximate the Wold representation, but we typically find twolags to be optimal at weekly and monthly frequencies, especially as we look at financialvariables and given computational considerations. We therefore include a constant and twolags of the variables, i.e., we estimate a VAR(2). Accounting for potential cointegration between the variables, we move beyond VAR toestimate a parsimonious vector error correction (VEC) model: ∆ y t = Π y t − +∑ p − i = Φ ∗ i ∆ y t − i + (cid:15) t . We compute the number of cointegrating relations in the system following the Johansen AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? trace test (Johansen, 1988, 1991) and employ the variance-covariance matrix estimated fromthe VEC model. If the Johansen test fails to reject the null of no cointegrating relations,then our VEC reduces to a VAR in first differences (error-correction coefficient is zero). The volatility of financial return data varies over time. General Autoregressive Condi-tional Heteroscedasticity (GARCH) provides a simple way of modeling the evolution ofvolatility as a deterministic function of past volatility and innovations. Univariate GARCHexpresses the error term of a time series (cid:15) t = H / t ξ t where ξ t is an i.i.d. innovation and H t = f ({ (cid:15) t − i , H t − j } q,pi = ,j = ) and can be extended to the multivariate setting through the vech (⋅) operator that stacks the lower triangular part of a symmetric N × N matrix into a N ∗ = N ( N + )/ vech ( H t ) = ω +∑ qi = A i η t − i +∑ pj = B j h t − j . With this no-tation, h t = vech ( H t ) , ω is the constant component of the covariances, and η t = vech ( (cid:15) t (cid:15) Tt ) .To prepare the data for GARCH estimation, we fit an ARMA model to the data for eachseries from which to obtain demeaned residuals (cid:15) t . Diagnostics such as Ljung Box tests ofserial correlation suggest ARMA(1,1) fits the data well.Our first multivariate GARCH specification is BEKK-GARCH (Engle and Kroner,1995), which allows for the dependence of conditional variances of one variable on laggedvalues of another so that causalities in variances can be modeled. Empirically, BEKK is gen-eral, but easy to estimate. Relaxing symmetry, we also allow positive and negative shocks ofequal magnitude to have different effects on conditional volatility by employing AsymmetricBEKK (ABEKK). With ABEKK, vech ( H t ) = ω + ∑ qi = [ A i (cid:15) t − i + C i η t − i ] + ∑ pj = B j h t − j . Weallow ( q =
1) one symmetric innovation when estimating BEKK. When estimating ABEKK,we allow one symmetric innovation and one asymmetric innovation. CURRAN AND ZALLA
BEKK and its sibling ABEKK suffer from the curse of dimensionality that might renderthem computationally infeasible for investors allocating capital across a large set of assets.We therefore also estimate a constant conditional correlation (CCC) model. The CCCmodel is a multivariate GARCH model, where all conditional correlations are constant andconditional variances are modeled by univariate GARCH processes (Bollerslev et al. , 1990).Let us decompose the conditional covariance matrix into conditional standard deviationsand a correlation matrix H t = D r R t D t where D t = diag ( h / t , . . . , h / Nt ) is a diagonal matrixof the standard deviations for the N assets. The conditional correlation matrix for the CCCmodel is constant over time: R t = R . The CCC model benefits from almost unrestrictedapplicability for large systems of time series, but fails to account for the observation thatcorrelation increases during financial crises.Dynamic conditional correlation (DCC) permits correlation to vary over time (Engle,2002). In addition to DCC, we also estimate its asymmetric version (ADCC). Without ac-counting for dynamics of asymmetric effects, DCC cannot distinguish between the effect ofpast positive and negative shocks on the future conditional volatility and levels (Cappiello et al. , 2006). As when we estimate BEKK and ABEKK, we allow ( q =
1) one symmet-ric innovation when estimating CCC and DCC. When estimating ADCC, we allow onesymmetric innovation and one asymmetric innovation.
The assumption of multivariate normality is often called into question in practical applica-tions. As motivation, consider Apple and Microsoft, who produce similar products. Shocksthat affect Apple may be expected to affect Microsoft. Each company may experience sim-ilar nonlinear extreme events, hence exhibiting tail dependence. A portfolio manager who
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? assumes multivariate normality will underestimate the frequency and magnitude of rareevents. Such underestimation may be detrimental to the performance of the portfolio.Modeling multivariate dependence among stock returns without assuming multivariatenormality has become popular in the 21st century. Copulas are functions that may be usedto bind univariate marginal distributions to produce a multivariate distribution (Sklar,1959). Parameters can vary over time as an autoregression in a copula-GARCH model.Copulas have become the standard tools for modeling multivariate dependence among stockreturns without assuming multivariate normality with many general applications in finance(Stric and Granger, 2005; Zimmer, 2012; Christoffersen et al. , 2012; Aloui et al. , 2013;Christoffersen and Langlois, 2013; Creal et al. , 2013; Xiao, 2014; Adrian and Brunnermeier,2016; Bodnar and Hautsch, 2016; Solnik and Watewai, 2016; Bekiros et al. , 2017).We specify a copula-GARCH process without fitting a VAR for the conditional mean.We set the following tuning parameters to the robust regression: γ = .
25 (proportion totrim), δ = .
01 (critical value for the re-weighted estimator), 500 subsets, and 10 steps.We allow for a symmetric DCC autoregressive order of (1,1) and choose the multivariateStudent copula distribution model, where the DCC copula is static, and we estimate thecorrelation parameter in the static Student copula by maximum likelihood. We apply anempirical (pseudo maximum likelihood) transformation to the marginal innovations of theGARCH fitted models. In estimating the above specification for return data, we calculatestandard errors, require stationarity when optimizing the univariate GARCH, and do notuse any scale option during this first stage. We take the average robust estimate of the We alternate between using R’s solnp solver, which is a nonlinear optimization using the augmentedLagrange method, and gosolnp , which randomly initializes and conducts multiple restarts of the solnp solver. When the objective function is non-smooth or has many local minima, it is hard to judge theoptimality of the solution, and this usually depends critically on the starting parameters. The gosolnp function enables the generation of a set of randomly chosen parameters from which to initialize multiplerestarts of the solver. We chose solnp , as it is faster, but when our solver encounters difficulties, we switchto gosolnp .2 CURRAN AND ZALLA covariance matrix.
To account for bull and bear phases of the market, we estimate a discrete time-varyingparameter model in the form of a regime-switching VAR (RSVAR) as in Chan and Eisenstat(2018): B ,S t y t = µ S t + B ,S t y t − + ⋯ + B p,S t y t − p + (cid:15) t where (cid:15) t ∼ N ( , Σ S t ) . We choose tworegimes. For each rolling window, we use a pre-sample of one year and estimate over theremaining sample of nine years in the window. Our Bayesian estimation includes 20,000simulations with a burn-in of 5,000 periods. We set our lag length at 2 for parsimony.To derive the variance-covariance matrix from the RSVAR, we back out the states usinghighest probability, get part of the parameter set Θ corresponding to the state at each time,and calculate the variance-covariance matrix as Σ = ( Y − X Θ ) ′ ( Y − X Θ ) . Another nonlinear state-space model that allows for heteroscedasticity is the computation-ally challenging multivariate stochastic volatility model (MSV). The curse of dimensionalityfor the MSV is that the degrees of freedom in the variance-covariance matrix scales quadrat-ically with the number of assets. Multivariate factor stochastic volatility breaks the curseof dimensionality by decomposing the variance-covariance matrix and using the pivotedCholesky algorithm of Higham (1990). The decomposition transforms the estimation prob-lem from being quadratic in assets to becoming linear in assets. We demean the data and use r = We adapt our method and exposition from Kastner (2019a,b). Observations of returns y t =( y t , . . . , y tN ) ′ follow y t ∣ Λ , f t , ¯Σ t ∼ N N ( Λ f t , ¯Σ t ) f t ∣ ˜Σ t ∼ N ( , ˜Σ t ) , AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? reductions in run time. Placing no constraints on the loading matrix possibly resultsin unstable posteriors or multiple local optima. The instability of posterior estimates ormultiplicity of local optima cause no issues for our study, however, as inference is on thecovariance matrix rather than on the factor loadings (Kastner, 2019a). We find 10,000draws with a burn in of 1,000 sufficient for convergence of the estimates.
Our final econometric model is a non-parametric model: realized volatility (RCOV). Re-alized variance is the summation matrix of the return vector outerproduct r ′ i r i over eachday in a given week for weekly frequency analysis or over each day in a given month formonthly frequency analysis. Realized volatility is the Cholesky decomposition of the re-alized variance. We focus on annualized realized volatility by pre-multiplying the realizedvariance by the number of trading days in a year divided by the number of trading daysin a week for weekly frequency analysis or in a month for monthly frequency analysis andobtaining the Cholesky decomposition of this product. where f t = ( f t , . . . , f tr ) ′ is the vector of factors and Λ ∈ R N × r is the matrix of factor loadings. Thecovariance matrices ¯Σ t and ˜Σ t are diagonal and represent stochastic volatility processes¯Σ t = diag ( exp ¯ h t , . . . , exp ¯ h tN ) ˜Σ t = diag ( exp ˜ h t , . . . , exp ˜ h tr ) ¯ h ti ∼ N ( ¯ µ i + ¯ ψ i ( ¯ h t − ,i − ¯ µ i ) , ¯ σ i ) i = , . . . , N ˜ h tj ∼ N ( ˜ µ j + ˜ ψ j ( ˜ h t − ,j − ˜ µ j ) , ˜ σ j ) j = , . . . , r. With latent factor models, few shocks drive the system and we can reduce the number of unknowns throughthe decomposition Σ t = ˜Σ t + ¯Σ t where rank ( ˜Σ ) = r < N and ¯Σ t is a diagonal matrix where the diagonalentries are the idiosyncratic errors. Using the pivoted Cholesky algorithm of Higham (1990), ˜Σ t = ΨΨ ′ where Ψ ∈ R N × r has N r − r ( r − )/ N ( r + ) − r ( r − )/ t , which is now linear in N . Thus, Σ t = Λ ˜Σ t Λ t + ¯Σ t . We also speed up computation by storing only the conditional covariance matrix and the square rootsof its diagonal elements and parallelizing the factorstochvol function of Kastner (2019a) in R and C++with hyperthreading.4
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Table 2 details the portfolios we compare. In order to avoid the significant issues withestimating mean returns (Merton, 1980) and the relatively large impact of errors in themean vector on out-of-sample portfolio performance (Chopra and Ziemba, 2013), we chooseto concentrate on portfolio models that depend only on the covariance matrix.
With N assets, the portfolio held over investment period t , w NVt , is given by w NVt = ( / N, . . . , / N ) ∀ t. (1) The minimum-variance portfolio for investment period t , w MV Pt , minimizes conditionalportfolio variance. For each econometric model j discussed in Section 2, we calculate theminimum-variance portfolio by w MV Pt,j = argmin w ∈ R N ∣ w ′ = w ˆΣ jt w ′ . (2)For each econometric model, the conditional estimate of the covariance matrix over period t is used as an input to find the minimum-variance portfolio. The constrained minimum-variance portfolio for investment period t , w Con − MV Pt , minimizesconditional portfolio variance subject to no short selling, which has been shown to improveperformance (Jagannathan and Ma (2003); DeMiguel et al. (2009b)). For each econometric
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? model j discussed in Section 2, we calculate the minimum-variance portfolio by w Con − MV Pt,j = argmin w ∈ R N ∣ w ′ = , w ≥ w ˆΣ jt w ′ . (3)For each econometric model, we use the conditional estimate of the covariance matrix overperiod t as an input to find the constrained minimum-variance portfolio. The volatility-timing strategy ignores off-diagonal elements of the covariance matrix, i.e.,assumes all pair-wise correlations are zero (Kirby and Ostdiek, 2012). The minimum-variance portfolio given covariance matrix Σ is w V Ti = / Σ ii ∑ Ni = / Σ ii . We similarly define thevolatility-timing (VT) strategy given conditional estimate of the covariance matrix ˆΣ jt by ( w V Tt,j ) i = / ( ˆΣ jt ) i,i ∑ Ni = / ( ˆΣ jt ) i,i i = , . . . , N. (4) While our main focus is on minimum-variance portfolio strategies, we also include thetangency portfolio (TP) for illustrative purposes. The TP with respect to econometricmodel j , w T Pt,j , is given by w T Pt,j = argmax w ∈ R N ∣ w ′ = w ˆ µ jt w ˆΣ jt w ′ . (5) We combine portfolios by inputting the arithmetic average over the econometric estimatesof the covariance matrix into the portfolio optimization strategies. We form a combinedparameter estimate of the covariance matrix ˆΣ by equally weighting over estimates ofthe covariance matrix taken from each of the thirteen econometric models. We use the CURRAN AND ZALLA combined parameter estimate for ˆΣ in each of (2)–(5) to get four combined parameterportfolios. Taking the example of the minimum-variance portfolio, using ˆΣ comv as thearithmetic average of the covariance matrices across the thirteen econometric models, ourcombined portfolio strategy is w MV P,comvt,j = argmin w ∈ R N ∣ w ′ = w ˆΣ comvt w ′ . (6)Our combined parameter model is motivated by the finding that combining different hedgingforecasts leads to more consistent hedging performance across datasets (Wang et al. , 2015).The result echoes the forecasting literature finding that combined models tend to performmore consistently over time than individual models (Stock and Watson, 2003, 2004).In preliminary investigations, we explored two other approaches: (i) naively weightingacross the thirteen vectors of weights suggested by the portfolio model using each econo-metric model’s variance-covariance matrix estimate as an input; and (ii) naively weightingacross the four weights suggested by the four financial portfolio models for a given econo-metric model. These alternative combined parameter models are less relevant for our First, a variation of our benchmark combined parameter model (6), for each of the financial portfoliomodels (2)–(5), we examine the corresponding portfolio given by naive investments across the thirteenportfolios with respect to each of the econometric models. More precisely, consider the minimum-varianceportfolio. We form a fourteenth portfolio strategy, w MV P,comt , which is equally invested across the thirteenestimates of the true minimum-variance portfolio, i.e., w MV P,comt = ∑ j = w MV Pt,j . Second, with respect to each of the econometric models, we examine the corresponding portfolio givenby naive investments across the four financial portfolio models (2)–(5). More precisely, consider the VAReconometric model. We form a fifth portfolio strategy, w V AR,compt that is equally invested across thefour vectors of portfolio weights suggested by inputting the volatility estimates from the VAR model intofinancial portfolio strategies (2)–(5), i.e., w V AR,compt = ∑ k = w kt,V AR . AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? study. With the first variation, averaging over thirteen weights suggested by the portfoliomodel is less direct than averaging over the variance-covariance matrix estimates. Withthe second variation, averaging over four portfolio models for a given econometric model issimilar to that of Wang et al. (2015). Consider instead the realistic situation that we areunsure of the data generating process underlying the return series. Rather than choosingone econometric model, we benefit from using all the information by hedging equally acrossthe various nuances captured by each of the thirteen econometric models. Results arebroadly similar across the three versions of combined parameter models. Thus, we reportthe results from our combined parameter model (6), which we denote CP. We employ six datasets at weekly and monthly frequencies. Lower frequencies smoothout too much volatility, and thus, are inappropriate for our study. Higher frequencies arecomputationally infeasible in light of estimating some of our econometric models usingrolling windows with the length and frequency of our samples. Comparisons could thus bemade only for a subset of our econometric models. We use value-weighted returns andassess robustness to equally-weighted returns. We use end of period data where possible.When weekly or monthly frequency is unavailable, we scale data geometrically. For instance,to scale returns from daily to weekly frequency, we use Π
NDj = ( + r j ) / ND − r j is thedaily return and N D denotes the number of trading days in the week. We adopt similarprocedures to scale to monthly frequency. For the realized covariance (RCOV) model, weuse daily data to calculate RCOV over each week for weekly frequency analysis and overeach month for monthly frequency analysis. We omit data prior to July 1963. Higher frequency data generate more accurate estimates of the covariance matrix, relevant for ourstudy, but daily and higher frequency data are also troubled with the problem of asynchronous trading. 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. The8
CURRAN AND ZALLA
Our first four datasets closely correspond to datasets 4, 2, 1, and 3 of DeMiguel et al. (2009b). The authors suggest minimum critical values for the size of the estimation windowthat allow these strategies to beat the naive portfolio. We choose datasets with a smallnumber of assets ( N ≈
10) because the minimum critical value grows with the number ofassets. Large estimation windows reduce the number of out-of-sample periods and becomeunrealistic with most weekly and monthly frequency empirical datasets for N >
25. We alsomotivate this choice because of computational considerations. Rather than examiningsimulated datasets, such as randomized selections of stocks, we restrict our attention toempirical datasets because our focus is on the econometric model as the source of improve-ment in performance.
Our first dataset consists of returns obtained from Wharton Research Data Services. Wefocus on the three-factor Fama-French portfolio: Small Minus Big, High Minus Low, andMarket portfolios. Small Minus Big (SMB) is the average return on three small portfoliosminus the average return on three big portfolios. High Minus Low (HML) is the averagereturn on two value portfolios minus the average return on two growth portfolios. TheMarket (MKT) return is the weighted return on all NYSE, AMEX, and NASDAQ stocksfrom CRSP and is obtained by adding the risk-free return to the excess market return. Thebenchmark analysis uses value-weighted returns to generate MKT. We focus on weekly and result is significantly sparser coverage prior to 1963 for a selected sample of well performing firms. DeMiguel et al. (2009b) find that performance weakens for large N due to estimation error. In pre-liminary investigations with N =
25, we observed that performance weakens relative to the naive portfolio.The curse of dimensionality applies for some of the econometric model estimations, however, renderingcomparisons and investigating sensitivity checks across the full set of models infeasible time-wise as N grows beyond about 15 or the usual number of assets most individual investors hold. Kenneth French provides full description at https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ The risk-free (RF) asset is the one-month Treasury bill rate from Ibbotson Associates and proxies thereturn 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.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? monthly frequencies, where we use end-of-period returns for daily and monthly frequencies,and we scale daily data to weekly data as described above. We limit the data to span allUS trading days from July 1st, 1963 to December 31st, 2018. We take returns from Kenneth French’s website covering ten industries: Consumer-Discretionary, Consumer-Staples, Manufacturing, Energy, High-Tech, Telecommunications,Wholesale and Retail, Health, Utilities, and Others. The benchmark analysis uses value-weighted returns. We also employ equal-weighting in robustness checks. We focus onweekly and monthly frequencies, where we use end-of-period returns for daily and monthlyfrequencies, and we scale daily data to weekly data as described above. We limit the datato span all US trading days from July 1st, 1963 to December 31st, 2018.
This dataset includes returns for eleven value-weighted industry portfolios formed by us-ing the Global Industry Classification Standard (GICS) developed by Standard & Poor’s(S&P) and Morgan Stanley Capital International (MSCI). We obtained the returns fromBloomberg. The ten industries are Energy, Materials, Industrials, Consumer-Discretionary,Consumer-Staples, Healthcare, Financials, Information-Technology, Telecommunications,Real Estate, and Utilities. The expected returns are based on equity investments. Dataare end-of-period returns for weekly and monthly frequencies and span all US trading daysfrom January 2nd, 1995 to December 31st, 2018.
This dataset includes returns on eight MSCI countries, Canada, France, Germany, Italy,Japan, Switzerland, UK, and USA, along with a developed countries index (MXWO). Thereturns are total gross returns with dividends reinvested. For robustness, we also use the CURRAN AND ZALLA world index (MXWD) and look at the regular return index. We source data from Bloombergand the MSCI. Data are end-of-period returns for weekly and monthly frequencies and spanall US trading days from January 4th, 1999 to December 31st, 2018.
We employ returns on the 6 (2 ×
3) portfolios sorted by size and book-to-market. We sourcethis data from Kenneth French’s website. The benchmark analysis uses value-weightedreturns. We also employ equal-weighting in robustness checks. We focus on weekly andmonthly frequencies, where we use end-of-period returns for daily and monthly frequencies,and we scale daily data to weekly data as described above. We limit the data to span allUS trading days from July 1st, 1963 to December 31st, 2018.
This dataset consists of returns on the 10 portfolios sorted by momentum. We obtain dataon momentum portfolios from Kenneth French’s website. The benchmark analysis usesvalue-weighted returns. We also employ equal-weighting in robustness checks. We focus onweekly and monthly frequencies, where we use end-of-period returns for daily and monthlyfrequencies, and we scale daily data to weekly data as described above. We limit the datato span all US trading days from July 1st, 1963 to December 31st, 2018.
We assess portfolio performance through three metrics, following convention in the litera-ture: Sharpe ratio, portfolio volatility, and turnover costs. For each measure, we estimatethe statistical significance of the difference in the estimated measure from that of the naiveportfolio strategy.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? The Sharpe ratio measures reward to risk from a portfolio strategy, i.e., expected returnper standard deviation. To test for differences between the Sharpe ratio from investingaccording to the naive strategy and the Sharpe ratio from investing according to the strategyin question, we employ the robust inference methods of Ledoit and Wolf (2008).
We assume a proportional turnover cost of 5% and calculate the expected returns net ofthe cost of rebalancing similar to DeMiguel et al. (2014) r t + = ( − κ N ∑ i = ∣ w i,t − w i, ( t − )∗ ∣) ( w t ) ′ r t + , where w ki, ( t − )∗ is the portfolio weight in asset i and time t prior to rebalancing, w i,t isthe portfolio weight suggested by the strategy at time t , i.e., after rebalancing, κ is theproportional transaction cost, w t is the vector of portfolio weights, and r t + is the returnvector. Rebalancing may occur each period. We compare the difference in Sharpe ratiosbetween the expected net returns following the specific strategy and the expected net returnsfollowing the naive strategy.
Assuming that the investor’s goal is to minimize portfolio volatility, by analogy with Wang et al. (2015), we examine ranking portfolio strategies by out-of-sample volatility of returns.To be precise, we conduct the Brown-Forsythe F* test of unequal group variances. We alsoapply the Diebold-Mariano test in comparing forecast errors from a naive strategy with 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 throughtime (Brandt et al. , 2009). With high turnover, assuming 5% transactions costs conservatively biases ourmodels away from beating the 1/N strategy.2
CURRAN AND ZALLA forecast errors from the strategy under consideration (Diebold and Mariano, 1995). Theprocedures allow testing whether the strategy is significantly more volatile or less volatilerelative to the naive strategy.
Our evidence suggests that the out-of-sample performance of portfolios whose only in-puts are volatility estimates often weakly dominate that of the naive diversification portfo-lio. The minimum-variance, constrained minimum-variance, volatility-timing, and tangencyportfolios have equivalent or superior Sharpe ratios, portfolio volatility, and turnover costsrelative to the naive portfolio in datasets 2, 3, 5, and 6, regardless of the econometric modelused to estimate volatility. If we average the volatility estimates of all thirteen economet-ric models, we continue to obtain similar results. Our portfolio models perform stronglyrelative to the naive when applied to country stock-market indices. The lessons from ourresults are robust to value- and equal-weighting and to weekly and monthly frequency es-timation. We thus show that controlling for volatility in portfolio strategies delivers betterperformance than the naive portfolio.In the next two subsections, we evaluate the performance of the econometric models.Specifically, we select an individual performance metric (e.g., the Sharpe ratio) and attemptto rank econometric models by performance consistency across datasets within a given port-folio strategy (e.g., minimum-variance). Ranking allows us to observe how models performin an absolute sense across datasets. Often, however, one model has the highest Sharperatio yet the highest portfolio volatility. In the third subsection, we therefore undertake aholistic analysis that incorporates all three performance metrics to rank econometric models The forecast error is defined as the difference between expected returns using estimated portfolioweights and mean returns. The loss differential underlying the test looks at the difference of the squaredforecast errors, and we calculate the the loss differential correcting for autocorrelation.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? and portfolio strategies that consistently outperform the naive strategy. We first evaluate the Sharpe ratio performance metric. In Tables 3–4, we provide theSharpe ratios associated with each of our thirteen econometric models when used as theinput in each of the four portfolio strategies. Sharpe ratios are assessed across all sixdatasets with value-weighting at weekly frequency. We empirically test the difference inSharpe ratios between each of the econometric models relative to the naive strategy andreport significance levels. The row ordering reflects our attempt to rank the econometricmodels according to consistency of performance relative to the naive benchmark.In the minimum-variance and constrained minimum-variance portfolios, the combinedparameter (CP) model yields Sharpe ratios that are consistently and significantly higherthan those of the naive benchmark. In fact, nearly all econometric models achieve signifi-cantly higher Sharpe ratios relative to the naive rule in datasets 2, 4, 5, and 6, while display-ing broadly equivalent Sharpe ratios in datasets 1 and 3. Consequentially, most economet-ric models, when combined with either the constrained or unconstrained minimum-varianceportfolio, weakly dominate the naive benchmark. The few exceptions, which occur only indataset 1, are the vector autoregression (VAR) and vector error-correction (VEC) models inthe minimum-variance portfolio, and the regime-switching vector autoregression (RSVAR)model in both the constrained and unconstrained minimum-variance portfolios. Even ourlowest ranked econometric model, the exponentially-weighted moving-average (EWMA)model, still manages to weakly dominate the naive portfolio.In the volatility-timing and tangency portfolios, the realized covariance (RCOV) modeldelivers Sharpe ratios that are consistently and significantly higher than those of the naive To explain the poorer performance of datasets 1 and 3, first, the literature consistently finds weakperformance with the Fama-French dataset (DeMiguel et al. , 2009b); second, a simple correlation matrix ofthe six datasets shows that dataset 3 is the only dataset to be negatively correlated with the other datasets.4
CURRAN AND ZALLA benchmark. Let us first examine the volatility-timing portfolio. Most econometric modelsachieve significantly higher Sharpe ratios relative to the naive rule in datasets 2, 4, 5,and 6, and similar Sharpe ratios in dataset 3. The only weakness again lies in dataset1, where all models except RCOV underperform the naive rule. Turning our attention tothe tangency portfolio, we observe significantly higher Sharpe ratios relative to the naivebenchmark across models in datasets 5 and 6, and similar Sharpe ratios in the rest. Thus,we conclude the tangency portfolio weakly dominates the naive portfolio in terms of Sharperatio. As with the minimum-variance portfolios, the exponentially-weighted moving-average(EWMA) achieves the worst Sharpe ratios relative to the other econometric models yet stillperforms well in comparison to the naive strategy.Tables 5–6 show that the results are robust to adjusting the Sharpe ratios for turnovercosts. The only difference is that the BEKK- and ABEKK-GARCH models achieve thehighest and most consistent turnover-cost-adjusted Sharpe ratio with the minimum-varianceportfolio relative to the naive benchmark. Moreover, we determine results are robust toboth equal-weighting and monthly frequency.The Sharpe ratios of our portfolio models relative to the naive strategy are not juststatistically significant but economically significant. Figure 1 helps to illustrate this point.On average, our portfolio models achieve Sharpe ratios that are 30% higher than the naive,across all six datasets, punctuated by the minimum-variance portfolio at 47%. We obtaina similar message in Figures 2 when we adjust the Sharpe ratios for turnover costs.In contrast to DeMiguel et al. (2009b), even when looking at the sample covariance-matrix (COV) for similar datasets, we find that the minimum-variance portfolio performsbetter than the naive portfolio. Their samples end at the turn of the millenium, whereas ourdatasets extend almost two decades to 2018. In line with their findings, applying strategiesto longer datasets improves the performance relative to the naive strategy. Moving from
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Figure 1: Sharpe Ratio Percentage Difference Relative to Naive −200020406080100120140160 1 2 3 4 5 6
Minimum−Variance Portfolio −200020406080 1 2 3 4 5 6
Constrained Minimum−Variance Portfolio −40−30−20−1000102030405060 1 2 3 4 5 6
Volatility−Timing Portfolio −40004080120160200 1 2 3 4 5 6
Tangency Portfolio
EconometricModel
NAIVEABEKKADCCBEKKCCCCOPULACOVCPDCCEWMAMSVRCOVRSVARVARVEC
Dataset S ha r pe R a t i o ( P e r c en t age D i ff e r en c e f r o m N a i v e ) Notes: Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset 3: sectorportfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted by size/book-to-market; Dataset 6: momentum portfolios. See Table 1 for explanations of econometric modelabbreviations. Results are for value-weighted data at weekly frequency. Note that standard errorsof the Sharpe ratio for the tangency portfolio are mostly large, rendering most Sharpe ratios forthe tangency portfolio statistically insignificantly different to Sharpe ratios for the naive portfolio. CURRAN AND ZALLA
Figure 2: Sharpe Ratio Net of Turnover Costs Percentage Difference Relative to Naive −40004080120160200 1 2 3 4 5 6
Minimum−Variance Portfolio −200020406080100 1 2 3 4 5 6
Constrained Minimum−Variance Portfolio −40−2000204060 1 2 3 4 5 6
Volatility−Timing Portfolio −40004080120160200 1 2 3 4 5 6
Tangency Portfolio
EconometricModel
NAIVEABEKKADCCBEKKCCCCOPULACOVCPDCCEWMAMSVRCOVRSVARVARVEC
Dataset S ha r pe R a t i o ( P e r c en t age D i ff e r en c e f r o m N a i v e , A d j u s t ed f o r T u r no v e r C o s t ) Notes: Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset 3: sectorportfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted by size/book-to-market; Dataset 6: momentum portfolios. See Table 1 for explanations of econometric modelabbreviations. Results are for value-weighted data at weekly frequency. Note that standard errorsof the Sharpe ratio for the tangency portfolio are mostly large, rendering most Sharpe ratios forthe tangency portfolio statistically insignificantly different to Sharpe ratios for the naive portfolio.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? monthly to weekly frequency is less important to this result than the period length of thesamples. However, we find that there are econometric models for each portfolio model thatweakly dominate COV in terms of (turnover-cost-adjusted) Sharpe ratios. We also findin subsection 6.2 that multivariate stochastic volatility (MSV) weakly dominates COV interms of portfolio volatility. We thus show that performance improves by employing moresophisticated econometric models than COV. We next evaluate portfolio volatility. In Tables 7–8, we provide the standard deviations ofthe returns associated with each of our thirteen econometric models when used as the inputin each of the four portfolio strategies. We assess portfolio volatility across all six datasetswith value-weighting at weekly frequency. We empirically test the difference in portfoliovolatility between each of the econometric models relative to the naive strategy and reportsignificance levels. The row ordering reflects our attempt to rank the econometric modelsaccording to consistency of performance relative to the naive benchmark.In the minimum-variance and constrained minimum-variance portfolios, the exponentially-weighted moving-average (EWMA), realized covariance (RCOV), and multivariate stochas-tic volatility (MSV) models exhibit significantly lower portfolio volatility relative to thenaive portfolio across all datasets. More importantly, most econometric models strictly dominate the naive portfolio in terms of volatility performance. The two exceptions, regime-switching vector autoregression (RSVAR) and combined parameter (CP), which happen tobe the worst ranking models, still weakly dominate the naive benchmark.For the volatility-timing portfolio, the EWMA model delivers the best results in terms ofportfolio volatility across datasets. Moreover, every econometric model weakly dominates Significance corresponds to the Brown-Forsythe F* test for unequal group variances. Results from theDiebold-Mariano (Diebold and Mariano, 1995) test for differences in variance relative to the naive portfolioare similar.8
CURRAN AND ZALLA the naive benchmark. For the tangency portfolio, the COPULA achieves the lowest portfoliovolatility across datasets. All econometric models, except RCOV and VEC in dataset4, weakly dominate the naive benchmark. In addition, the MSV and RCOV models areconsistent runner-ups in both the volatility-timing and tangency portfolio models. Althoughthe RSVAR and CP models yield the highest volatility, both still weakly dominate the naiveportfolio.The volatility of our portfolio models relative to the naive strategy is economically significant. Figure 3 helps to illustrate this point. On average, our portfolio models are 9%less volatile than the naive, across all six datasets, with the minimum-variance portfolio at10% lower volatility.
We undertake a holistic evaluation of the econometric models. In Tables 9–12, for eacheconometric model, we compare the Sharpe ratio, turnover cost, and portfolio volatility ofeach portfolio model across all six datasets. A “ ✓ ” indicates that the strategy outperformsat the 5% significance level or better, a “ ✓ *” indicates a 10% significance level, an “ × ”indicates that the strategy underperforms, and a blank space indicates there is no signifi-cant difference in the strategy’s performance relative to the naive benchmark with respectto the given performance metric. All of our econometric models are broadly successful atoutperforming the naive portfolio; that is, they achieve higher Sharpe ratios, lower turnovercosts, lower portfolio volatility, or some combination of these superior performance indica-tors. The few exceptions are concentrated where the volatility-timing portfolio is appliedto dataset 1. In order to aid our identification of the best performers, we develop a simpleheuristic to score individual econometric models: we add up the instances where the modeloutperforms the naive benchmark and subtract the instances where the model underper- AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Figure 3: Portfolio Volatility Percentage Difference Relative to Naive −30−25−20−15−10−5005 1 2 3 4 5 6
Minimum−Variance Portfolio −30−25−20−15−10−5005 1 2 3 4 5 6
Constrained Minimum−Variance Portfolio −25−20−15−10−5005 1 2 3 4 5 6
Volatility−Timing Portfolio −25−20−15−10−5005 1 2 3 4 5 6
Tangency Portfolio
EconometricModel
NAIVEABEKKADCCBEKKCCCCOPULACOVCPDCCEWMAMSVRCOVRSVARVARVEC
Dataset P o r tf o li o V o l a t ili t y ( P e r c en t age D i ff e r en c e f r o m N a i v e ) Notes: Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset 3: sectorportfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted by size/book-to-market; Dataset 6: momentum portfolios. See Table 1 for explanations of econometric modelabbreviations. Results are for value-weighted data at weekly frequency. CURRAN AND ZALLA forms. The multivariate GARCH models achieve the highest scores relative to other econo-metric models when applied to the minimum-variance and constrained minimum-variancestrategies. In particular, with GARCH estimates of the covariance matrix, these portfo-lio strategies weakly dominate the naive benchmark. The constant conditional correlation(CCC) performs especially well when the no-short-sale constraint is imposed. For thevolatility-timing and tangency portfolios, the realized covariance (RCOV) model exhibitsthe most impressive results relative to the naive rule. Specifically, RCOV weakly domi-nates the naive rule across every dataset when paired with the volatility-timing strategy,and across five out of six datasets when paired with the tangency portfolio. In general,our results suggest the multivariate GARCH and RCOV models represent better alterna-tives to the often-used sample covariance (COV) matrix for portfolio construction. Thesample covariance matrix (COV) performs at the median relative to the other economet-ric models assessed by our ranking. While the convenience of implementing COV makesit an attractive option to researchers, our analysis shows there are returns to using moresophisticated methods to forecast volatility. The worst-ranking econometric models arethe regime-switching vector autoregression (RSVAR) and exponentially-weighted moving-average (EWMA) models. Nonetheless, both of these models still perform at least as wellas the naive benchmark in every dataset except the Fama-French 3-factor.As a final assessment, we naively average the estimated conditional volatilities of allthirteen econometric models to form a combined parameter (CP) model; see Table 12. Themain takeaway from this exercise is that controlling for volatility in a portfolio, no matterhow volatility is estimated, delivers performance metrics that are generally at least as strong To clarify, “ ✓ ” =
1, “ ✓ *” = /
3, “ ” (blanks) =
0, and “ × ” = − . We discount results that are significantat the 10% level by assigning a value of only 2/3 instead of 1.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? as the naive strategy. We evaluate the out-of-sample performance of mean-variance strategies relying solely uponthe second moment relative to the naive benchmark. Using fourteen econometric modelsacross six datasets at weekly frequency, we show that the minimum-variance, constrainedminimum-variance, and volatility-timing strategies generally achieve higher Sharpe ratios,lower turnover costs, and lower portfolio volatility that are economically significant relativeto naive diversification. Whenever mean-variance strategies do not significantly outperformthe naive rule, they usually match and only rarely lose to it.We identify the econometric models that most consistently and significantly outperformthe 1/N benchmark. First, we show that the multivariate GARCH models weakly dominatethe naive rule when applied to the minimum-variance and constrained minimum-variancestrategies. Next, we demonstrate that the realized covariance model achieves impressiveresults when paired with the volatility-timing and tangency portfolios. Even our “worst-performing” econometric models still manage to perform at least as well as the naive rule inall but one dataset. Third, we illustrate that if one wishes to prioritize the Sharpe ratio, thenthe combined parameter and realized covariance models are excellent choices, even aftercontrolling for turnover costs. Finally, we show the exponentially-weighted moving-averageand multivariate stochastic volatility models consistently deliver low portfolio volatility.With the difficulty in consistently outperforming the strategy, the 1/N naive diver-sification should serve as a benchmark for practitioners and academics. We empiricallydemonstrate that one important source of the naive portfolio puzzle is the quality of theeconometric volatility inputs to the mean-variance portfolio strategies. With improved esti-mates, the mean-variance models can beat the naive portfolio strategy. Our findings imply CURRAN AND ZALLA that while considerable energy has been devoted to optimizing the mathematical design ofportfolio theory models, more progress may be warranted in improving the estimation ofthe moments of asset returns.
References
Adrian, T. and
Brunnermeier, M. K. (2016). CoVaR.
The American Economic Re-view , (7), 1705–1741. Aloui, R. , A¨ıssa, M. S. B. and
Nguyen, D. K. (2013). Conditional Dependence Struc-ture between Oil Prices and Exchange Rates: a Copula-GARCH Approach.
Journal ofInternational Money and Finance , (1), 719–738. Ao, M. , Yingying, L. and
Zheng, X. (2019). Approaching Mean-Variance Efficiency forLarge Portfolios.
The Review of Financial Studies , (7), 2890–2919. Baltussen, G. and
Post, G. T. (2011). Irrational Diversification: An Examinationof Individual Portfolio Choice.
Journal of Financial and Quantitative Analysis , (5),1463–1491. Behr, P. , Guettler, A. and
Miebs, F. (2013). On Portfolio Optimization: Imposingthe Right Constraints.
Journal of Banking & Finance , (4), 1232–1242. Bekiros, S. , Boubaker, S. , Nguyen, D. K. and
Uddin, G. S. (2017). Black SwanEvents and Safe Havens: The Role of Gold in Globally Integrated Emerging Markets.
Journal of International Money and Finance , , 317 – 334. Benartzi, S. and
Thaler, R. H. (2001). Naive Diversification Strategies in DefinedContribution Saving Plans.
American Economic Review , (1), 79–98. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Bodnar, T. and
Hautsch, N. (2016). Dynamic Conditional Correlation MultiplicativeError Processes.
Journal of Empirical Finance , , 41 – 67. Bollerslev, T. et al. (1990). Modelling the Coherence in Short-Run Nominal ExchangeRates: a Multivariate Generalized ARCH Model.
The Review of Economics and Statis-tics , (3), 498–505. Brandt, M. W. , Santa-Clara, P. and
Valkanov, R. (2009). Parametric PortfolioPolicies: Exploiting Characteristics in the Cross-Section of Equity Returns.
The Reviewof Financial Studies , (9), 3411–3447. Cappiello, L. , Engle, R. F. and
Sheppard, K. (2006). Asymmetric Dynamics in theCorrelations of Global Equity and Bond Returns.
Journal of Financial Econometrics , (4), 537–572. Chan, J. C. and
Eisenstat, E. (2018). Bayesian Model Comparison for Time-VaryingParameter VARs with Stochastic Volatility.
Journal of Applied Econometrics , (4),509–532. Chopra, V. K. and
Ziemba, W. T. (2013). The Effect of Errors in Means, Variances, andCovariances on Optimal Portfolio Choice. In
Handbook of the Fundamentals of FinancialDecision Making: Part I , World Scientific, pp. 365–373.
Christoffersen, P. , Errunza, V. , Jacobs, K. and
Langlois, H. (2012). Is thePotential for International Diversification Disappearing? A Dynamic Copula Approach.
The Review of Financial Studies , (12), 3711–3751. — and Langlois, H. (2013). The Joint Dynamics of Equity Market Factors.
The Journalof Financial and Quantitative Analysis , (5), 1371–1404. CURRAN AND ZALLA
Creal, D. , Koopman, S. J. and
Lucas, A. (2013). Generalized Autoregressive ScoreModels with Applications.
Journal of Applied Econometrics , (5), 777–795. De Giorgi, E. G. and
Mahmoud, O. (2018).
Naive Diversification Preferences andTheir Representation . Working paper.
DeMiguel, V. , Garlappi, L. , Nogales, F. J. and
Uppal, R. (2009a). A GeneralizedApproach to Portfolio Optimization: Improving Performance by Constraining PortfolioNorms.
Management Science , (5), 798–812. — , — and Uppal, R. (2009b). Optimal Versus Naive Diversification: How Inefficient isthe 1/N Portfolio Strategy?
The Review of Financial Studies , (5), 1915–1953. — , Nogales, F. J. and
Uppal, R. (2014). Stock Return Serial Dependence and Out-of-Sample Portfolio Performance.
The Review of Financial Studies , (4), 1031–1073. — , Plyakha, Y. , Uppal, R. and
Vilkov, G. (2013). Improving Portfolio Selection UsingOption-Implied Volatility and Skewness.
Journal of Financial and Quantitative Analysis , (6), 1813–1845. Diebold, F. X. and
Mariano, R. S. (1995). Comparing Predictive Accuracy.
Journalof Business & Economic Statistics , (3), 253–263. Duchin, R. and
Levy, H. (2009). Markowitz Versus the Talmudic Portfolio DiversificationStrategies.
The Journal of Portfolio Management , , 71–74. Engle, R. (2002). Dynamic Conditional Correlation: A Simple Class of MultivariateGeneralized Autoregressive Conditional Heteroskedasticity Models.
Journal of Business& Economic Statistics , (3), 339–350. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Engle, R. F. and
Kroner, K. F. (1995). Multivariate Simultaneous Generalized ARCH.
Econometric Theory , (1), 122–150. Fletcher, J. (2011). Do Optimal Diversification Strategies Outperform the 1/N Strategyin UK Stock Returns?
International Review of Financial Analysis , (5), 375–385. Gathergood, J. , Hirshleifer, D. , Leake, D. , Sakaguchi, H. and
Stewart, N. (2019).
Na¨ıve *Buying* Diversification and Narrow Framing by Individual Investors .Working Paper 25567, NBER.
Higham, N. J. (1990).
Analysis of the Cholesky Decomposition of a Semi-Definite Matrix .Oxford University Press.
Jagannathan, R. and
Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposingthe Wrong Constraints Helps.
The Journal of Finance , (4), 1651–1683. Jobson, J. D. and
Korkie, B. (1980). Estimation for Markowitz Efficient Portfolios.
Journal of the American Statistical Association , (371), 544–554. Johansen, S. (1988). Statistical Analysis of Cointegration Vectors.
Journal of EconomicDynamics and Control , (2-3), 231–254. — (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian VectorAutoregressive Models. Econometrica , pp. 1551–1580.
Kastner, G. (2019a). factorstochvol: Bayesian Estimation of (Sparse) Latent FactorStochastic Volatility Models.
R package version 0.9.2 . — (2019b). Sparse Bayesian Time-Varying Covariance Estimation in Many Dimensions. Journal of Econometrics , (1), 98–115. CURRAN AND ZALLA
Kirby, C. and
Ostdiek, B. (2012). It’s All in the Timing: Simple Active PortfolioStrategies that Outperform Naive Diversification.
Journal of Financial and QuantitativeAnalysis , (2), 437–467. Kourtis, A. , Dotsis, G. and
Markellos, R. N. (2012). Parameter Uncertainty inPortfolio Selection: Shrinking the Inverse Covariance Matrix.
Journal of Banking & Fi-nance , (9), 2522–2531. Ledoit, O. and
Wolf, M. (2003). Improved Estimation of the Covariance Matrix ofStock Returns with an Application to Portfolio Selection.
Journal of Empirical Finance , (5), 603–621. — and — (2008). Robust Performance Hypothesis Testing with the Sharpe Ratio. Journalof Empirical Finance , (5), 850–859. — and — (2017). Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection:Markowitz Meets Goldilocks. The Review of Financial Studies , (12), 4349–4388. Markowitz, H. (1952). Portfolio Selection.
The Journal of Finance , (1), 77–91. Merton, R. C. (1980). On Estimating the Expected Return on the Market: An Ex-ploratory Investigation.
Journal of Financial Economics , (4), 323–361. Michaud, R. O. and
Michaud, R. O. (2008).
Efficient Asset Management: A PracticalGuide to Stock Portfolio Optimization and Asset Allocation . Oxford University Press,2nd edn.
Moreira, A. and
Muir, T. (2017). Volatility-Managed Portfolios.
The Journal of Fi-nance , (4), 1611–1644. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? — and — (2019). Should Long-Term Investors Time Volatility? Journal of FinancialEconomics , (3), 507–527. Pflug, G. C. , Pichler, A. and
Wozabal, D. (2012). The 1/N Investment Strategy isOptimal Under High Model Ambiguity.
Journal of Banking & Finance , (2), 410–417. Sklar, M. (1959). Fonctions de Repartition an Dimensions et Leurs Marges.
Publ. Inst.Statist. Univ. Paris , , 229–231. Solnik, B. and
Watewai, T. (2016). International Correlation Asymmetries: Frequent-but-Small and Infrequent-but-Large Equity Returns.
Review of Asset Pricing Studies , (2), 221 – 260. Stock, J. H. and
Watson, M. W. (2003). Forecasting Output and Inflation: The Roleof Asset Prices.
Journal of Economic Literature , (3), 788–829. — and — (2004). Combination Forecasts of Output Growth in a Seven-Country Data Set. Journal of Forecasting , (6), 405–430. Stric, C. and
Granger, C. (2005). Nonstationarities in Stock Returns.
The Review ofEconomics and Statistics , (3), 503–522. Truc´ıos, C. , Zevallos, M. , Hotta, L. K. and
Santos, A. A. (2019). CovariancePrediction in Large Portfolio Allocation.
Econometrica , (2), 19. Tu, J. and
Zhou, G. (2011). Markowitz Meets Talmud: A Combination of Sophisticatedand Naive Diversification Strategies.
Journal of Financial Economics , (1), 204–215. Vogiatzoglou, M. (2017). Dynamic Copula Toolbox.
Available at SSRN 2956888 . CURRAN AND ZALLA
Wang, Y. , Wu, C. and
Yang, L. (2015). Hedging with Futures: Does Anything Beatthe Na¨ıve Hedging Strategy?
Management Science , (12), 2870–2889. Xiao, Z. (2014). Right-Tail Information in Financial Markets.
Econometric Theory , (1),94–126. Zimmer, D. M. (2012). The Role of Copulas in the Housing Crisis.
The Review of Eco-nomics and Statistics , (2), 607–620. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 1: Econometric models1 Sample covariance matrix (Cov)2 Exponentially weighted covariance matrix (EWMA)3 Vector autoregressions (VAR)4 Vector error corrections (VEC)5 Multivariate BEKK-GARCH (BEKK)6 Asymmetric multivariate BEKK-GARCH (ABEKK)7 Constant conditional correlation GARCH (CCC)8 Dynamic conditional correlation GARCH (DCC)9 Asymmetric dynamic conditional correlation GARCH (ADCC)10 T-copula with GARCH margins (COPULA)11 Regime-switching vector autoregressions (RSVAR)12 High dimensional multivariate stochastic volatility (MSV)13 Realized covariance (RCOV)
Table 2: Portfolio models
Notes:
We describe combined parameter (CP) model 6 in equation (6) and theother combined parameter models 7 and 8 in footnote 17. CURRAN AND ZALLA
Table 3: Sharpe Ratio: Minimum-Variance and Constrained Minimum-Variance
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Naive .
112 0 .
101 0 .
035 0 .
058 0 .
100 0 . Minimum - Variance
CP 0 .
089 0 . .
042 0 . . . .
098 0 . .
056 0 . . . .
098 0 . .
057 0 . . . .
100 0 . .
055 0 . . . .
100 0 . .
055 0 . . . .
099 0 . .
057 0 . . . ( . ) . .
057 0 . . . ( . ) . .
056 0 . . . .
095 0 . .
056 0 . . . ( . ) .
101 0 .
041 0 . . . .
109 0 . .
074 0 . . . .
100 0 . .
056 0 . . . .
102 0 .
124 0 .
043 0 . . . .
101 0 .
125 0 .
070 0 . . . ConstrainedMinimum - Variance
CCC 0 .
102 0 . .
047 0 . . . .
090 0 . .
042 0 . . . .
101 0 . .
044 0 . . . .
106 0 . .
046 0 . . . .
102 0 . .
043 0 . . . .
099 0 . .
046 0 . . . .
098 0 . .
048 0 . . . .
102 0 . .
043 0 . . . .
104 0 . .
059 0 . . . .
111 0 .
121 0 .
055 0 . . . .
106 0 .
118 0 .
046 0 . . . .
102 0 . .
044 0 . . . ( . ) .
101 0 .
041 0 . . . .
111 0 . .
056 0 .
078 0 . . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Ledoit and Wolf (2008) robust test for differences between the Sharpe ratioand that of the naive strategy. Numbers in parentheses are statistically significantly worse thanthose of the naive strategy.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 4: Sharpe Ratio: Volatility-Timing and Tangency
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Naive .
112 0 .
101 0 .
035 0 .
058 0 .
100 0 . VolatilityTiming
RCOV 0 .
093 0 . . . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . . . . . ( . ) .
101 0 .
041 0 . . . ( . ) .
103 0 .
041 0 . . . ( . ) . .
042 0 .
065 0 . . Tangency
RCOV 0 .
124 0 . . .
032 0 . . .
102 0 .
099 0 .
027 0 . . . .
104 0 .
103 0 .
040 0 . . . .
113 0 .
114 0 .
038 0 .
137 0 . . .
109 0 .
115 0 .
045 0 .
135 0 . . .
108 0 .
113 0 .
046 0 .
132 0 . . .
113 0 .
113 0 .
041 0 .
142 0 . . .
113 0 .
113 0 .
041 0 .
142 0 . . .
113 0 .
113 0 .
043 0 .
142 0 . . .
113 0 .
114 0 .
040 0 .
134 0 . . .
113 0 .
114 0 .
038 0 .
137 0 . . .
110 0 .
109 0 .
043 0 . . . .
112 0 .
101 0 .
042 0 .
138 0 . . .
112 0 .
112 0 .
038 0 .
126 0 . . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Ledoit and Wolf (2008) robust test for differences between the Sharpe ratioand that of the naive strategy. Numbers in parentheses are statistically significantly worse thanthose of the naive strategy. CURRAN AND ZALLA
Table 5: Turnover Costs: Minimum-Variance and Constrained Minimum-Variance
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Na¨ıve .
113 0 .
101 0 .
035 0 .
054 0 .
100 0 . Minimum - Variance
BEKK 0 .
098 0 . .
057 0 . . . .
098 0 . .
056 0 . . . .
100 0 . .
055 0 . . . .
100 0 . .
055 0 . . . .
100 0 . .
057 0 . . . ( . ) . .
056 0 . . . .
093 0 . .
057 0 . . . .
095 0 . .
056 0 . . . .
089 0 . .
041 0 . . . ( . ) .
101 0 .
041 0 . . . .
108 0 . .
074 0 . . . .
100 0 . .
056 0 . . . .
102 0 .
127 0 .
071 0 . . . .
103 0 .
125 0 .
045 0 .
119 0 . . ConstrainedMinimum - Variance
CCC 0 .
102 0 . .
047 0 . . . .
090 0 . .
041 0 . . . .
101 0 . .
044 0 . . . .
102 0 . .
043 0 . . . .
102 0 . .
043 0 . . . .
099 0 . .
048 0 . . . .
099 0 . .
046 0 . . . .
106 0 . .
046 0 . . . .
106 0 . .
046 0 . . . .
103 0 . .
044 0 . . . .
087 0 .
102 0 .
041 0 . . . .
111 0 .
123 0 .
054 0 . . . .
104 0 . .
060 0 . . . .
112 0 . .
056 0 .
074 0 . . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Ledoit and Wolf (2008) robust test for differences between the Sharpe ratioand that of the naive strategy. Numbers in parentheses are statistically significantly worse thanthose of the naive strategy.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 6: Turnover Costs: Volatility-Timing and Tangency
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Na¨ıve .
113 0 .
101 0 .
035 0 .
054 0 .
100 0 . VolatilityTiming
RCOV 0 .
092 0 . . . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
039 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . .
040 0 . . . ( . ) . . . . . ( . ) .
102 0 .
041 0 . . . ( . ) .
103 0 .
040 0 . . . ( . ) . .
042 0 .
061 0 . . Tangency
RCOV 0 .
123 0 . . .
057 0 . . .
104 0 .
104 0 .
038 0 . . . .
102 0 .
099 0 .
027 0 . . . .
109 0 .
116 0 .
046 0 .
130 0 . . .
108 0 .
114 0 .
047 0 .
127 0 . . .
113 0 .
114 0 .
041 0 .
137 0 . . .
114 0 .
114 0 .
042 0 .
136 0 . . .
113 0 .
114 0 .
043 0 .
137 0 . . .
113 0 .
115 0 .
040 0 .
129 0 . . .
113 0 .
114 0 .
039 0 .
131 0 . . .
113 0 .
114 0 .
039 0 .
131 0 . . .
110 0 .
111 0 .
046 0 .
139 0 . . .
112 0 .
102 0 .
044 0 .
134 0 . . .
112 0 .
113 0 .
039 0 .
119 0 . . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Ledoit and Wolf (2008) robust test for differences between the Sharpe ratioand that of the naive strategy. Numbers in parentheses are statistically significantly worse thanthose of the naive strategy. CURRAN AND ZALLA
Table 7: Portfolio Volatility: Minimum-Variance and Constrained Minimum-Variance
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Naive .
002 0 .
005 0 .
024 0 .
024 0 .
005 0 . Minimum - Variance
EWMA 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
005 0 .
022 0 . . . . .
005 0 .
023 0 . .
005 0 . ConstrainedMinimum - Variance
EWMA 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
005 0 .
022 0 . . . . .
005 0 .
023 0 . .
005 0 . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Brown-Forsythe F* test for unequal group variances. Results from the Diebold-Mariano (Diebold and Mariano, 1995) test for differences in variance relative to the naive portfolioare similar.
AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 8: Portfolio Volatility: Volatility-Timing and Tangency
Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6
Naive .
002 0 .
005 0 .
024 0 .
024 0 .
005 0 . VolatilityTiming
EWMA 0 . . . .
022 0 .
005 0 . . . .
021 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . .
005 0 .
023 0 . .
005 0 . . . .
022 0 .
023 0 .
005 0 . . . .
022 0 .
023 0 .
005 0 . . .
005 0 .
023 0 .
023 0 .
005 0 . Tangency
COPULA 0 . . .
022 0 . . . . . .
022 0 .
023 0 . . . . . ( . ) . . . . .
022 0 .
024 0 . . . . .
022 0 .
024 0 . . . . .
022 0 .
024 0 . . . . .
021 0 .
022 0 . . . . .
022 0 .
022 0 . . . . .
022 0 .
022 0 . . . . .
022 0 .
025 0 . . . . .
024 0 .
023 0 . . . . . ( . ) . . . .
005 0 .
021 0 . .
005 0 . . .
005 0 .
021 0 . .
005 0 . Notes:
See Table 1 for explanations of econometric model abbreviations; CP denotes combinedparameter model (6). Dataset 1: Fama-French portfolios; Dataset 2: industry portfolios; Dataset3: sector portfolios; Dataset 4: international equity indices; Dataset 5: portfolios sorted bysize/book-to-market; Dataset 6: momentum portfolios. Results are for value-weighted data atweekly frequency. * significant at 10%; ** significant at 5%; *** significant at 1%. Significancecorresponds to the Brown-Forsythe F* test for unequal group variances. Results from the Diebold-Mariano (Diebold and Mariano, 1995) test for differences in variance relative to the naive portfolioare similar. CURRAN AND ZALLA
Table 9: COV, EWMA, and VAR
COV
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓ *Sharpe Ratio ✓ ✓ ✓ ✓ *Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (a) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (b) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (c) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (d) Tangency
EWMA
Dataset 1 2 3 4 5 6Turnover Cost ✓ Sharpe Ratio ✓ * ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (e) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (f) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ * ✓ (g) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ Sharpe Ratio ✓ Portfolio Volatility ✓ ✓ ✓ (h) Tangency
VAR
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓ *Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (i) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (j) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (k) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (l) Tangency
Note : COV, EWMA, and VAR models are described in Sections 2.1– 2.3. See Table 1 forexplanations of econometric model abbreviations. See Table 3 and Section 4 for explanationsof datasets. Results are for value-weighted data at weekly frequency. ✓ : strategy outper-forms naive at 5% level or lower; ✓ *: strategy outperforms naive at 10% level; × : strategyunderperforms naive; else, insignificant difference. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 10: VEC, BEKK, ABEKK, and CCC
VEC
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓ *Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (a) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (b) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (c) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ × ✓ (d) Tangency
BEKK and ABEKK
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (e) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (f) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (g) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (h) Tangency
CCC
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (i) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (j) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (k) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (l) Tangency
Note : VEC, BEKK, ABEKK, and CCC models are described in Sections 2.4–2.6. SeeTable 1 for explanations of econometric model abbreviations. See Table 3 and Section 4for explanations of datasets. Results are for value-weighted data at weekly frequency. ✓ :strategy outperforms naive at 5% level or lower; ✓ *: strategy outperforms naive at 10%level; ××
Note : VEC, BEKK, ABEKK, and CCC models are described in Sections 2.4–2.6. SeeTable 1 for explanations of econometric model abbreviations. See Table 3 and Section 4for explanations of datasets. Results are for value-weighted data at weekly frequency. ✓ :strategy outperforms naive at 5% level or lower; ✓ *: strategy outperforms naive at 10%level; ×× : strategy underperforms naive; else, insignificant difference. CURRAN AND ZALLA
Table 11: DCC, ADCC, COPULA, and RSVAR
DCC and ADCC
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (a) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (b) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (c) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓
Sharpe Ratio ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (d) Tangency
COPULA
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ * ✓ Sharpe Ratio ✓ ✓ * ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (e) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ * (f) Constrained Minimum–Variance Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (g) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ Sharpe Ratio ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ (h) Tangency
RSVAR
Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (i) Minimum Variance Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (j) Constrained Minimum–Variance Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * ✓ * (k) Volatility-Timing Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓
Portfolio Volatility ✓ ✓ * (l) Tangency Note : DCC, ADCC, COPULA, and RSVAR models are described in Sections 2.6– 2.8. SeeTable 1 for explanations of econometric model abbreviations. See Table 3 and Section 4for explanations of datasets. Results are for value-weighted data at weekly frequency. ✓ :strategy outperforms naive at 5% level or lower; ✓ *: strategy outperforms naive at 10%level; × : strategy underperforms naive; else, insignificant difference. AN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? Table 12: MSV, RCOV, and CP
MSV
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ Sharpe Ratio ✓ * ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (a) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ *Sharpe Ratio ✓ * ✓ ✓ ✓ *Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (b) Constrained Minimum–Variance Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ * ✓ * ✓ ✓ Sharpe Ratio × ✓ ✓ * ✓ * ✓ ✓ Portfolio Volatility ✓ ✓ ✓ (c) Volatility-Timing
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ *Sharpe Ratio ✓ * ✓ ✓ *Portfolio Volatility ✓ ✓ ✓ ✓ (d) Tangency RCOV
Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ Sharpe Ratio ✓ * ✓ ✓ Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (e) Minimum Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓ (f) Constrained Minimum–Variance
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ ✓ * ✓ ✓ ✓ Portfolio Volatility ✓ ✓ (g) Volatility-Timing
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ × ✓ ✓ (h) Tangency CP Dataset 1 2 3 4 5 6Turnover Cost ✓ * ✓ ✓ ✓ Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ * ✓ (i) Minimum Variance Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ * ✓ * (j) Constrained Minimum–Variance Dataset 1 2 3 4 5 6Turnover Cost × ✓ ✓ ✓
Sharpe Ratio × ✓ ✓ ✓
Portfolio Volatility ✓ ✓ (k) Volatility-Timing
Dataset 1 2 3 4 5 6Turnover Cost ✓ ✓ ✓
Sharpe Ratio ✓ ✓ ✓
Portfolio Volatility ✓ ✓ ✓ (l) Tangency
Note : MSV, RCOV, and CP models are described in Section 2.9, 2.10, and 3.6. SeeTable 1 for explanations of econometric model abbreviations. See Table 3 and Section 4for explanations of datasets. Results are for value-weighted data at weekly frequency. ✓ :strategy outperforms naive at 5% level or lower; ✓ *: strategy outperforms naive at 10%level; ××