Can Volatility Solve the Naive Portfolio Puzzle?

Can 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/N

strategy. The portfolio strategies rely solely upon second moments. Using a diverse

group of econometric and portfolio 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 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, Jesus Fernandez-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) 610-519-8867

Address: Economics Dept., Villanova School of Business, Villanova University, 800 E Lancaster Ave, PA 19085, USA.‡Email: [email protected]; Phone: (+1) 412-759-5032

Address: Economics Dept., University of Pennsylvania, 133 South 36th Street, Philadelphia, PA 19104, USA.

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CAN VOLATILITY SOLVE THE NAIVE PORTFOLIO PUZZLE? 1

1 Introduction

Ever since mean-variance strategies were introduced by Markowitz (1952), their out-of-

sample performance has been criticized. One reason for weak performance is estimation

error, particularly in the mean return (Merton, 1980; Chopra and Ziemba, 2013). Variance

strategies, which rely solely upon second moments, avoid the pitfall of expected returns es-

timation. In comparing variance strategies relative to the naive 1/N portfolio benchmark,

while the portfolio selection literature advocates robust estimation procedures to reduce

estimation error, previous work such as DeMiguel et al. (2009b) include strategies account-

ing for parameter errors, but employ sample covariance estimates. Few papers in the naive

portfolio literature pursued improved estimation of the volatility of returns.1,2 Our marginal

contribution 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, our

study suggests considerable performance gains from employing modern econometric models

for estimating volatility in portfolio construction.

We investigate whether sophisticated volatility estimation improves the performance of

mean-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 rely

solely on estimation of the second moment of returns. We apply three portfolio strategies

1Instead 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.

2We 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).

2 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.3 We

assess 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 performances

relative 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).4 Specifically, we find that all four portfolio strategies estimated

using all fourteen econometric models perform at least as well as the naive benchmark,

and only rarely underperform it. These underperformances are mainly concentrated in the

Fama-French 3-factor dataset. If we discard this dataset, then all four portfolio strategies

estimated using twelve of the fourteen econometric models would not just perform well,

they would weakly dominate the naive benchmark.5

In general, the minimum-variance strategy, with or without short-sale constraints, achieves

higher Sharpe ratios, lower turnover costs, and lower portfolio volatility across the majority

of datasets. Wherever these strategies do not perform significantly better than the naive

rule, we often fail to reject the null hypothesis that their performances are identical to

the naive rule. Likewise, we find evidence that volatility-timing strategies achieve higher

Sharpe ratios and lower turnover costs but exhibit portfolio volatility that is comparable

to the naive rule. The tangency portfolio does reasonably well in certain datasets. Rela-

3Section 3 defines our financial portfolio models.4Our econometric estimation strategies yield improvements beyond the period and frequency differences.5A 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.


tive to the other strategies, however, its performance is lackluster, which we attribute to

estimation 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 naive

benchmark. 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) model

weakly dominates the naive rule when applied to the volatility-timing strategy. In the

tangency portfolio, although the RCOV model achieves higher Sharpe ratios and lower

portfolio 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 weakly

dominates the naive rule, many other econometric models do, such as the multivariate

GARCH models. Nonetheless, even econometric models such as the regime-switching vec-

tor autoregression (RSVAR) and exponentially-weighted moving-average (EWMA), which

perform worst in each of the four porfolio models, still perform at least as well as the naive

benchmark across every dataset except the Fama-French 3-factor.

Our third contribution is to compare the performance of econometric models relative to

the naive benchmark using each assessment criterion in isolation. That is, across portfolios

models and datasets, which econometric models consistently achieve the highest Sharpe

ratios? Which econometric models achieve the lowest turnover costs? Which econometric

models achieve the lowest portfolio volatilities? First, the combined parameter (CP) and

realized covariance (RCOV) models achieve significantly higher Sharpe ratios than the naive

benchmark. Second, the RCOV and a variety of GARCH models produce significantly

higher Sharpe ratios than the naive rule after adjustment for turnover costs. Finally, the

4 CURRAN AND ZALLA

exponentially-weighted moving-average (EWMA), multivariate stochastic volatility (MSV),

and RCOV models exhibit significantly lower portfolio volatility than the naive rule. These

performances are economically and statistically significant: relative to the naive strategy,

we achieve 30% higher Sharpe ratios and 9% lower portfolio volatility.6

Our paper exploits recent computational developments to estimate the covariance ma-

trix using multivariate, nonlinear, non-Gaussian econometric models. We extend the set

of models in Wang et al. (2015) beyond GARCH and discrete regime-switching models

to 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).7

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 strategies

consistently 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 return

performance. Considering a variety of mean-variance strategies including ones accounting

for parameter errors, DeMiguel et al. (2009b) find that no portfolio model consistently

outperforms naive diversification. One cause for the weak performance is that while mean-

variance strategies based on the minimum-variance portfolio outperforms naive diversifica-

6For 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.

7To reduce run-time, we employ fast, low-level languages, e.g., C++, that we program in parallel withhyperthreading and execute on clusters.


tion, portfolio turnover costs negates these benefits (Kirby and Ostdiek, 2012). To address

the 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. We

employ their volatility-timing strategy. Combining naive diversification with other con-

ventional portfolio strategies can also significantly enhance portfolio performance (Tu and

Zhou, 2011). We investigate combining parameters from the variance-covariance matrix

estimates as well as from the weights suggested by portfolio models.

A number of studies have increased the sophistication of parameter estimation since

DeMiguel et al. (2009b), which was based on rolling window sample estimates. Considering

a variety of mostly GARCH-based estimators, no method is universally superior (Trucıos

et al., 2019).8 A non-linear shrinkage estimator (Ledoit and Wolf, 2017) and an estimation

strategy for large portfolios (Ao et al., 2019) can improve performance.9 The use of implied

volatility 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.10 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 Giorgi

and Mahmoud, 2018; Gathergood et al., 2019), and identify the importance of volatility

for portfolio performance (Moreira and Muir, 2017, 2019). Consistent with Moreira and

8Using 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.

9Preliminary 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.

10To 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.

6 CURRAN AND ZALLA

Muir (2017, 2019), our paper finds that volatility-timing strategies generate higher Sharpe

ratios in these same datasets yet exhibit equal volatility to the naive portfolio. Taking

seriously the naive strategy as an important benchmark, we explore improved econometric

estimation of volatility as a potential source of out-of-sample performance.

Our benchmark is the naive portfolio diversification rule, allocating a fraction 1/N of

wealth to each of N assets available for investment at each rebalancing date.11 Three

reasons justify the naive rule as a benchmark. First, implementation is easy as it relies

on neither optimization nor estimation of the moments of asset returns. Second, investors

use 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 estimation

is zero, and thus the mean square error of the naive portfolio weights is simply the square

of the bias.12 The naive strategy therefore proxies as a challenging rival for mean-variance

strategies to outperform.

11We apply our methods to six different empirical datasets with N ≤ 11 portfolio choices. In preliminaryinvestigations, we experimented with datasets withN = 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.

12Letting w denote our estimate of the optimal vector of portfolio weights w, the MSE bias-variancedecomposition from econometrics is MSE(w) = V ar(w) +Bias2(w,w), where Bias(w,w) = w −w.


2 Econometric Models

Table 1 lists the econometric models, some of which are included in Wang et al. (2015), who

consider out-of-sample performance of hedging strategies.13 Their analysis uses econometric

modeling of bivariate random variables, whereas we extend the analysis to multivariate

random 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 investment

periods. 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 = 1,

which corresponds to one week or month ahead, reduces compute time, and allows us to

compare our results with the literature, e.g., DeMiguel et al. (2009b). Let {Σjt}Tt=1 denote

the 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 expected

return over period t given model j by {µj}Tt=1. The portfolio models we focus on are based

solely on the variance-covariance matrix. We also experiment with the tangency portfolio

and use the sample mean, which is independent of econometric model j, as our estimate of

the mean. Mean-variance strategies are defined by the first two conditional moments of the

return for period t so that {(µjt , Σ

jt)}Tt=1 defines the sequence of mean-variance strategies

over the T out-of-sample investment periods with respect to model j. Section 3 details the

mean-variance strategies.

2.1 Sample Covariance

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

13While 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.

8 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.

2.2 Exponentially Weighted Moving Average

The recent past might be more informative for estimating the variance-covariance matrix

to use in selecting portfolios, motivating our first refinement, the exponentially weighted

moving average (EWMA) model. The EWMA model suggested by RiskMetrics places

decaying weight on the past: Σt = αΣt−1+(1−α)(rt− rt)′(rt− rt) where our decay parameter

suggested is 0.96 (weekly) or 0.97 (monthly).

2.3 Vector Autoregression

To exploit dependence along the cross-section and time-dimension (serial), we estimate

a vector autoregression (VAR): Y = XΦ + U using Bayesian methods. We experimented

with a VAR with a Normal-Wishart natural conjugate prior and another with a Minnesota

prior. To reduce the impact of our prior on our results, we choose the posterior based on a

non-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 two

lags to be optimal at weekly and monthly frequencies, especially as we look at financial

variables and given computational considerations. We therefore include a constant and two

lags of the variables, i.e., we estimate a VAR(2).

2.4 Vector Error Correction

Accounting for potential cointegration between the variables, we move beyond VAR to

estimate a parsimonious vector error correction (VEC) model: ∆yt = Πyt−1+∑p−1i=1 Φ∗

i ∆yt−i+

εt. We compute the number of cointegrating relations in the system following the Johansen


trace test (Johansen, 1988, 1991) and employ the variance-covariance matrix estimated from

the 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).

2.5 BEKK-GARCH and Asymmetric BEKK-GARCH

The volatility of financial return data varies over time. General Autoregressive Condi-

tional Heteroscedasticity (GARCH) provides a simple way of modeling the evolution of

volatility as a deterministic function of past volatility and innovations. Univariate GARCH

expresses the error term of a time series εt = H1/2t ξt where ξt is an i.i.d. innovation and

Ht = f({εt−i,Ht−j}q,pi=1,j=1) 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+1)/2 dimensional vector: vech(Ht) = ω+∑qi=1Aiηt−i+∑

pj=1Bjht−j. With this no-

tation, ht = vech(Ht), ω is the constant component of the covariances, and ηt = vech(εtεTt ).

To prepare the data for GARCH estimation, we fit an ARMA model to the data for each

series from which to obtain demeaned residuals εt. Diagnostics such as Ljung Box tests of

serial 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 lagged

values 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 of

equal magnitude to have different effects on conditional volatility by employing Asymmetric

BEKK (ABEKK). With ABEKK, vech(Ht) = ω +∑qi=1 [Aiεt−i +Ciηt−i] +∑

pj=1Bjht−j. We

allow (q = 1) one symmetric innovation when estimating BEKK. When estimating ABEKK,

we allow one symmetric innovation and one asymmetric innovation.

10 CURRAN AND ZALLA

2.6 Conditional Correlation: Constant, Dynamic, & Asymmetric

BEKK and its sibling ABEKK suffer from the curse of dimensionality that might render

them computationally infeasible for investors allocating capital across a large set of assets.

We therefore also estimate a constant conditional correlation (CCC) model. The CCC

model is a multivariate GARCH model, where all conditional correlations are constant and

conditional variances are modeled by univariate GARCH processes (Bollerslev et al., 1990).

Let us decompose the conditional covariance matrix into conditional standard deviations

and a correlation matrix Ht =DrRtDt where Dt = diag (h1/21t , . . . , h

1/2Nt ) is a diagonal matrix

of the standard deviations for the N assets. The conditional correlation matrix for the CCC

model is constant over time: Rt = R. The CCC model benefits from almost unrestricted

applicability for large systems of time series, but fails to account for the observation that

correlation 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 of

past 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 one

symmetric innovation and one asymmetric innovation.

2.7 Copula-GARCH

The assumption of multivariate normality is often called into question in practical applica-

tions. As motivation, consider Apple and Microsoft, who produce similar products. Shocks

that 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


assumes multivariate normality will underestimate the frequency and magnitude of rare

events. Such underestimation may be detrimental to the performance of the portfolio.

Modeling multivariate dependence among stock returns without assuming multivariate

normality has become popular in the 21st century. Copulas are functions that may be used

to 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 stock

returns 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 = 0.25 (proportion to

trim), δ = 0.01 (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 multivariate

Student copula distribution model, where the DCC copula is static, and we estimate the

correlation parameter in the static Student copula by maximum likelihood. We apply an

empirical (pseudo maximum likelihood) transformation to the marginal innovations of the

GARCH fitted models. In estimating the above specification for return data, we calculate

standard errors, require stationarity when optimizing the univariate GARCH, and do not

use any scale option during this first stage.14 We take the average robust estimate of the

14We 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 solnpsolver. 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 gosolnpfunction 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.

12 CURRAN AND ZALLA

covariance matrix.

2.8 Regime-Switching Vector Autoregression

To account for bull and bear phases of the market, we estimate a discrete time-varying

parameter model in the form of a regime-switching VAR (RSVAR) as in Chan and Eisenstat

(2018): B0,Styt = µSt+B1,Styt−1 + ⋯ +Bp,Styt−p + εt where εt ∼ N (0,ΣSt). We choose two

regimes. For each rolling window, we use a pre-sample of one year and estimate over the

remaining sample of nine years in the window. Our Bayesian estimation includes 20,000

simulations 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 using

highest probability, get part of the parameter set Θ corresponding to the state at each time,

and calculate the variance-covariance matrix as Σ = (Y −XΘ)′(Y −XΘ).

2.9 Multivariate Stochastic Volatility

Another nonlinear state-space model that allows for heteroscedasticity is the computation-

ally challenging multivariate stochastic volatility model (MSV). The curse of dimensionality

for 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 curse

of dimensionality by decomposing the variance-covariance matrix and using the pivoted

Cholesky algorithm of Higham (1990). The decomposition transforms the estimation prob-

lem from being quadratic in assets to becoming linear in assets.15

We demean the data and use r = 2 factors, where the factor loadings are unidentified.

As the sampler places no constraints on the loading matrix, we benefit from significant

15We adapt our method and exposition from Kastner (2019a,b). Observations of returns yt =(yt1, . . . , ytN)′ follow

yt∣Λ, ft, Σt ∼ NN(Λft, Σt)ft∣Σt ∼ N (0, Σt),


reductions in run time.16 Placing no constraints on the loading matrix possibly results

in unstable posteriors or multiple local optima. The instability of posterior estimates or

multiplicity of local optima cause no issues for our study, however, as inference is on the

covariance matrix rather than on the factor loadings (Kastner, 2019a). We find 10,000

draws with a burn in of 1,000 sufficient for convergence of the estimates.

2.10 Realized Volatility

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′iri over each

day in a given week for weekly frequency analysis or over each day in a given month for

monthly frequency analysis. Realized volatility is the Cholesky decomposition of the re-

alized variance. We focus on annualized realized volatility by pre-multiplying the realized

variance by the number of trading days in a year divided by the number of trading days

in a week for weekly frequency analysis or in a month for monthly frequency analysis and

obtaining the Cholesky decomposition of this product.

where ft = (ft1, . . . , ftr)′ is the vector of factors and Λ ∈ RN×r is the matrix of factor loadings. Thecovariance matrices Σt and Σt are diagonal and represent stochastic volatility processes

Σt = diag(exp ht1, . . . , exp htN)Σt = diag(exp ht1, . . . , exp htr)hti ∼ N (µi + ψi(ht−1,i − µi), σ2

i ) i = 1, . . . ,N

htj ∼ N (µj + ψj(ht−1,j − µj), σ2j ) j = 1, . . . , r.

With latent factor models, few shocks drive the system and we can reduce the number of unknowns throughthe decomposition Σt = Σt + Σt where rank(Σ) = r < N and Σt is a diagonal matrix where the diagonalentries are the idiosyncratic errors. Using the pivoted Cholesky algorithm of Higham (1990), Σt = ΨΨ′

where Ψ ∈ RN×r has Nr − r(r − 1)/2 free elements; therefore, N(r + 1) − r(r − 1)/2 free elements are left inΣt, which is now linear in N . Thus, Σt = ΛΣtΛt + Σt.

16We 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.

14 CURRAN AND ZALLA

3 Portfolio Models

Table 2 details the portfolios we compare. In order to avoid the significant issues with

estimating mean returns (Merton, 1980) and the relatively large impact of errors in the

mean vector on out-of-sample portfolio performance (Chopra and Ziemba, 2013), we choose

to concentrate on portfolio models that depend only on the covariance matrix.

3.1 Naive Diversification

With N assets, the portfolio held over investment period t, wNVt , is given by

wNVt = (1/N, . . . ,1/N) ∀t. (1)

3.2 Minimum-Variance Portfolio

The minimum-variance portfolio for investment period t, wMV Pt , minimizes conditional

portfolio variance. For each econometric model j discussed in Section 2, we calculate the

minimum-variance portfolio by

wMV Pt,j = argmin

w∈RN ∣w′1=1wΣj

tw′. (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.

3.3 Constrained Minimum-Variance Portfolio

The constrained minimum-variance portfolio for investment period t, wCon−MV Pt , minimizes

conditional portfolio variance subject to no short selling, which has been shown to improve

performance (Jagannathan and Ma (2003); DeMiguel et al. (2009b)). For each econometric


model j discussed in Section 2, we calculate the minimum-variance portfolio by

wCon−MV Pt,j = argmin

w∈RN ∣w′1=1,w≥0

wΣjtw

′. (3)

For each econometric model, we use the conditional estimate of the covariance matrix over

period t as an input to find the constrained minimum-variance portfolio.

3.4 Volatility-Timing Strategies

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 wV Ti = 1/Σii

∑Ni=1 1/Σii

. We similarly define the

volatility-timing (VT) strategy given conditional estimate of the covariance matrix Σjt by

(wV Tt,j )

i=

1/ (Σjt)i,i

∑Ni=1 1/ (Σj

t)i,ii = 1, . . . ,N. (4)

3.5 Tangency Portfolio

While our main focus is on minimum-variance portfolio strategies, we also include the

tangency portfolio (TP) for illustrative purposes. The TP with respect to econometric

model j, wTPt,j , is given by

wTPt,j = argmax

w∈RN ∣w′1=1

wµjt

wΣjtw

′. (5)

3.6 Combined Parameter Model

We combine portfolios by inputting the arithmetic average over the econometric estimates

of the covariance matrix into the portfolio optimization strategies. We form a combined

parameter estimate of the covariance matrix Σ by equally weighting over estimates of

the covariance matrix taken from each of the thirteen econometric models. We use the

16 CURRAN AND ZALLA

combined parameter estimate for Σ in each of (2)–(5) to get four combined parameter

portfolios. Taking the example of the minimum-variance portfolio, using Σcomv as the

arithmetic average of the covariance matrices across the thirteen econometric models, our

combined portfolio strategy is

wMV P,comvt,j = argmin

w∈RN ∣w′1=1wΣcomv

t w′. (6)

Our combined parameter model is motivated by the finding that combining different hedging

forecasts leads to more consistent hedging performance across datasets (Wang et al., 2015).

The result echoes the forecasting literature finding that combined models tend to perform

more consistently over time than individual models (Stock and Watson, 2003, 2004).

In preliminary investigations, we explored two other approaches: (i) naively weighting

across 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 weighting

across the four weights suggested by the four financial portfolio models for a given econo-

metric model.17 These alternative combined parameter models are less relevant for our

17First, 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, wMV P,com

t , which is equally invested across the thirteenestimates of the true minimum-variance portfolio, i.e.,

wMV P,comt = 1

13

13

∑j=1

wMV 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 VAR

econometric model. We form a fifth portfolio strategy, wV AR,compt that is equally invested across the

four vectors of portfolio weights suggested by inputting the volatility estimates from the VAR model intofinancial portfolio strategies (2)–(5), i.e.,

wV AR,compt = 1

4

4

∑k=1

wkt,V AR.


study. With the first variation, averaging over thirteen weights suggested by the portfolio

model is less direct than averaging over the variance-covariance matrix estimates. With

the second variation, averaging over four portfolio models for a given econometric model is

similar to that of Wang et al. (2015). Consider instead the realistic situation that we are

unsure of the data generating process underlying the return series. Rather than choosing

one econometric model, we benefit from using all the information by hedging equally across

the various nuances captured by each of the thirteen econometric models. Results are

broadly similar across the three versions of combined parameter models. Thus, we report

the results from our combined parameter model (6), which we denote CP.

4 Data

We employ six datasets at weekly and monthly frequencies. Lower frequencies smooth

out too much volatility, and thus, are inappropriate for our study. Higher frequencies are

computationally infeasible in light of estimating some of our econometric models using

rolling windows with the length and frequency of our samples. Comparisons could thus be

made only for a subset of our econometric models.18 We use value-weighted returns and

assess 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 ΠNDj=1 (1 + rj)1/ND − 1 where rj is the

daily return and ND denotes the number of trading days in the week. We adopt similar

procedures to scale to monthly frequency. For the realized covariance (RCOV) model, we

use daily data to calculate RCOV over each week for weekly frequency analysis and over

each month for monthly frequency analysis. We omit data prior to July 1963.19

18Higher 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.

19Standard 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

18 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 window

that allow these strategies to beat the naive portfolio. We choose datasets with a small

number of assets (N ≈ 10) because the minimum critical value grows with the number of

assets. Large estimation windows reduce the number of out-of-sample periods and become

unrealistic with most weekly and monthly frequency empirical datasets for N > 25. We also

motivate this choice because of computational considerations.20 Rather than examining

simulated datasets, such as randomized selections of stocks, we restrict our attention to

empirical datasets because our focus is on the econometric model as the source of improve-

ment in performance.

4.1 Dataset 1: Fama-French Portfolios

Our first dataset consists of returns obtained from Wharton Research Data Services.21 We

focus on the three-factor Fama-French portfolio: Small Minus Big, High Minus Low, and

Market portfolios. Small Minus Big (SMB) is the average return on three small portfolios

minus the average return on three big portfolios. High Minus Low (HML) is the average

return on two value portfolios minus the average return on two growth portfolios. The

Market (MKT) return is the weighted return on all NYSE, AMEX, and NASDAQ stocks

from CRSP and is obtained by adding the risk-free return to the excess market return.22 The

benchmark 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.20DeMiguel 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 Ngrows beyond about 15 or the usual number of assets most individual investors hold.

21Kenneth French provides full description at https://mba.tuck.dartmouth.edu/pages/faculty/

ken.french/22The 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.

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/


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 all

US trading days from July 1st, 1963 to December 31st, 2018.

4.2 Dataset 2: Industry Portfolios

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 on

weekly and 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 all US trading days from July 1st, 1963 to December 31st, 2018.

4.3 Dataset 3: Sector Portfolios

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 from

Bloomberg. 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. Data

are end-of-period returns for weekly and monthly frequencies and span all US trading days

from January 2nd, 1995 to December 31st, 2018.

4.4 Dataset 4: International equity indices

This dataset includes returns on eight MSCI countries, Canada, France, Germany, Italy,

Japan, Switzerland, UK, and USA, along with a developed countries index (MXWO). The

returns are total gross returns with dividends reinvested. For robustness, we also use the

20 CURRAN AND ZALLA

world index (MXWD) and look at the regular return index. We source data from Bloomberg

and the MSCI. Data are end-of-period returns for weekly and monthly frequencies and span

all US trading days from January 4th, 1999 to December 31st, 2018.

4.5 Dataset 5: Size/Book-to-Market

We employ returns on the 6 (2×3) portfolios sorted by size and book-to-market. We source

this data from Kenneth French’s website. The benchmark analysis uses value-weighted

returns. We also employ equal-weighting in robustness checks. We focus on weekly and

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 all

US trading days from July 1st, 1963 to December 31st, 2018.

4.6 Dataset 6: Momentum Portfolios

This dataset consists of returns on the 10 portfolios sorted by momentum. We obtain data

on momentum portfolios from Kenneth French’s website. The benchmark analysis uses

value-weighted returns. We also employ equal-weighting in robustness checks. We focus on

weekly and 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 all US trading days from July 1st, 1963 to December 31st, 2018.

5 Assessment Criteria

We assess portfolio performance through three metrics, following convention in the litera-

ture: Sharpe ratio, portfolio volatility, and turnover costs. For each measure, we estimate

the statistical significance of the difference in the estimated measure from that of the naive

portfolio strategy.


5.1 Sharpe Ratio

The Sharpe ratio measures reward to risk from a portfolio strategy, i.e., expected return

per standard deviation. To test for differences between the Sharpe ratio from investing

according to the naive strategy and the Sharpe ratio from investing according to the strategy

in question, we employ the robust inference methods of Ledoit and Wolf (2008).

5.2 Turnover Cost

We assume a proportional turnover cost of 5% and calculate the expected returns net of

the cost of rebalancing similar to DeMiguel et al. (2014)

rt+1 = (1 − κN

∑i=1

∣wi,t −wi,(t−1)∗∣) (wt)′rt+1,

where wki,(t−1)∗ is the portfolio weight in asset i and time t prior to rebalancing, wi,t is

the portfolio weight suggested by the strategy at time t, i.e., after rebalancing, κ is the

proportional transaction cost, wt is the vector of portfolio weights, and rt+1 is the return

vector.23 Rebalancing may occur each period. We compare the difference in Sharpe ratios

between the expected net returns following the specific strategy and the expected net returns

following the naive strategy.

5.3 Portfolio Volatility

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 also

apply the Diebold-Mariano test in comparing forecast errors from a naive strategy with

23Several 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.

22 CURRAN AND ZALLA

forecast errors from the strategy under consideration (Diebold and Mariano, 1995).24 The

procedures allow testing whether the strategy is significantly more volatile or less volatile

relative to the naive strategy.

6 Empirical Results

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 tangency

portfolios have equivalent or superior Sharpe ratios, portfolio volatility, and turnover costs

relative to the naive portfolio in datasets 2, 3, 5, and 6, regardless of the econometric model

used 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 strongly

relative to the naive when applied to country stock-market indices. The lessons from our

results 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 better

performance 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 attempt

to 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 perform

in an absolute sense across datasets. Often, however, one model has the highest Sharpe

ratio yet the highest portfolio volatility. In the third subsection, we therefore undertake a

holistic analysis that incorporates all three performance metrics to rank econometric models

24The 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.


and portfolio strategies that consistently outperform the naive strategy.

6.1 Sharpe Ratio

We first evaluate the Sharpe ratio performance metric. In Tables 3–4, we provide the

Sharpe ratios associated with each of our thirteen econometric models when used as the

input in each of the four portfolio strategies. Sharpe ratios are assessed across all six

datasets with value-weighting at weekly frequency. We empirically test the difference in

Sharpe ratios between each of the econometric models relative to the naive strategy and

report significance levels. The row ordering reflects our attempt to rank the econometric

models according to consistency of performance relative to the naive benchmark.

In the minimum-variance and constrained minimum-variance portfolios, the combined

parameter (CP) model yields Sharpe ratios that are consistently and significantly higher

than 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 3.25 Consequentially, most economet-

ric models, when combined with either the constrained or unconstrained minimum-variance

portfolio, weakly dominate the naive benchmark. The few exceptions, which occur only in

dataset 1, are the vector autoregression (VAR) and vector error-correction (VEC) models in

the minimum-variance portfolio, and the regime-switching vector autoregression (RSVAR)

model in both the constrained and unconstrained minimum-variance portfolios. Even our

lowest 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) model

delivers Sharpe ratios that are consistently and significantly higher than those of the naive

25To 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.

24 CURRAN AND ZALLA

benchmark. Let us first examine the volatility-timing portfolio. Most econometric models

achieve 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 dataset

1, where all models except RCOV underperform the naive rule. Turning our attention to

the tangency portfolio, we observe significantly higher Sharpe ratios relative to the naive

benchmark 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 Sharpe

ratio. As with the minimum-variance portfolios, the exponentially-weighted moving-average

(EWMA) achieves the worst Sharpe ratios relative to the other econometric models yet still

performs well in comparison to the naive strategy.

Tables 5–6 show that the results are robust to adjusting the Sharpe ratios for turnover

costs. The only difference is that the BEKK- and ABEKK-GARCH models achieve the

highest and most consistent turnover-cost-adjusted Sharpe ratio with the minimum-variance

portfolio relative to the naive benchmark. Moreover, we determine results are robust to

both equal-weighting and monthly frequency.

The Sharpe ratios of our portfolio models relative to the naive strategy are not just

statistically 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 obtain

a 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 performs

better than the naive portfolio. Their samples end at the turn of the millenium, whereas our

datasets extend almost two decades to 2018. In line with their findings, applying strategies

to longer datasets improves the performance relative to the naive strategy. Moving from


Figure 1: Sharpe Ratio Percentage Difference Relative to Naive

−20

00

20

40

60

80

100

120

140

160

1 2 3 4 5 6

Minimum−Variance Portfolio

−20

00

20

40

60

80

1 2 3 4 5 6

Constrained Minimum−Variance Portfolio

−40

−30

−20

−10

00

10

20

30

40

50

60

1 2 3 4 5 6

Volatility−Timing Portfolio

−40

00

40

80

120

160

200

1 2 3 4 5 6

Tangency Portfolio

EconometricModel

NAIVE

ABEKK

ADCC

BEKK

CCC

COPULA

COV

CP

DCC

EWMA

MSV

RCOV

RSVAR

VAR

VEC

Dataset

Sh

arp

e R

atio

(P

erc

en

tag

e D

iffe

ren

ce

fro

m N

aiv

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.

26 CURRAN AND ZALLA

Figure 2: Sharpe Ratio Net of Turnover Costs Percentage Difference Relative to Naive

−40

00

40

80

120

160

200

1 2 3 4 5 6


−20

00

20

40

60

80

100

1 2 3 4 5 6


−40

−20

00

20

40

60

1 2 3 4 5 6


−40

00

40

80

120

160

200

1 2 3 4 5 6

Tangency Portfolio

EconometricModel

NAIVE

ABEKK

ADCC

BEKK

CCC

COPULA

COV

CP

DCC

EWMA

MSV

RCOV

RSVAR

VAR

VEC

Dataset

Sh

arp

e R

atio

(P

erc

en

tag

e D

iffe

ren

ce

fro

m N

aiv

e,

Ad

juste

d fo

r Tu

rnove

r C

ost)

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.


monthly to weekly frequency is less important to this result than the period length of the

samples. However, we find that there are econometric models for each portfolio model that

weakly dominate COV in terms of (turnover-cost-adjusted) Sharpe ratios. We also find

in subsection 6.2 that multivariate stochastic volatility (MSV) weakly dominates COV in

terms of portfolio volatility. We thus show that performance improves by employing more

sophisticated econometric models than COV.

6.2 Portfolio Volatility

We next evaluate portfolio volatility. In Tables 7–8, we provide the standard deviations of

the returns associated with each of our thirteen econometric models when used as the input

in each of the four portfolio strategies. We assess portfolio volatility across all six datasets

with value-weighting at weekly frequency. We empirically test the difference in portfolio

volatility between each of the econometric models relative to the naive strategy and report

significance levels.26 The row ordering reflects our attempt to rank the econometric models

according 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 the

naive 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 to

be the worst ranking models, still weakly dominate the naive benchmark.

For the volatility-timing portfolio, the EWMA model delivers the best results in terms of

portfolio volatility across datasets. Moreover, every econometric model weakly dominates

26Significance 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.

28 CURRAN AND ZALLA

the naive benchmark. For the tangency portfolio, the COPULA achieves the lowest portfolio

volatility across datasets. All econometric models, except RCOV and VEC in dataset

4, weakly dominate the naive benchmark. In addition, the MSV and RCOV models are

consistent runner-ups in both the volatility-timing and tangency portfolio models. Although

the RSVAR and CP models yield the highest volatility, both still weakly dominate the naive

portfolio.

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 at

10% lower volatility.

6.3 Portfolio Models

We undertake a holistic evaluation of the econometric models. In Tables 9–12, for each

econometric model, we compare the Sharpe ratio, turnover cost, and portfolio volatility of

each portfolio model across all six datasets. A “✓” indicates that the strategy outperforms

at 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 respect

to the given performance metric. All of our econometric models are broadly successful at

outperforming the naive portfolio; that is, they achieve higher Sharpe ratios, lower turnover

costs, lower portfolio volatility, or some combination of these superior performance indica-

tors. The few exceptions are concentrated where the volatility-timing portfolio is applied

to dataset 1. In order to aid our identification of the best performers, we develop a simple

heuristic to score individual econometric models: we add up the instances where the model

outperforms the naive benchmark and subtract the instances where the model underper-


Figure 3: Portfolio Volatility Percentage Difference Relative to Naive

−30

−25

−20

−15

−10

−5

00

5

1 2 3 4 5 6


−30

−25

−20

−15

−10

−5

00

5

1 2 3 4 5 6


−25

−20

−15

−10

−5

00

5

1 2 3 4 5 6


−25

−20

−15

−10

−5

00

5

1 2 3 4 5 6

Tangency Portfolio

EconometricModel

NAIVE

ABEKK

ADCC

BEKK

CCC

COPULA

COV

CP

DCC

EWMA

MSV

RCOV

RSVAR

VAR

VEC

Dataset

Po

rtfo

lio V

ola

tilit

y (

Pe

rce

nta

ge

Diffe

ren

ce

fro

m N

aiv

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.

30 CURRAN AND ZALLA

forms.27

The multivariate GARCH models achieve the highest scores relative to other econo-

metric models when applied to the minimum-variance and constrained minimum-variance

strategies. 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 the

volatility-timing and tangency portfolios, the realized covariance (RCOV) model exhibits

the 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. The

sample covariance matrix (COV) performs at the median relative to the other economet-

ric models assessed by our ranking. While the convenience of implementing COV makes

it an attractive option to researchers, our analysis shows there are returns to using more

sophisticated methods to forecast volatility. The worst-ranking econometric models are

the regime-switching vector autoregression (RSVAR) and exponentially-weighted moving-

average (EWMA) models. Nonetheless, both of these models still perform at least as well

as the naive benchmark in every dataset except the Fama-French 3-factor.

As a final assessment, we naively average the estimated conditional volatilities of all

thirteen econometric models to form a combined parameter (CP) model; see Table 12. The

main takeaway from this exercise is that controlling for volatility in a portfolio, no matter

how volatility is estimated, delivers performance metrics that are generally at least as strong

27To clarify, “✓” = 1, “✓*” = 2/3, “ ” (blanks) = 0, and “×” = −1. We discount results that are significantat the 10% level by assigning a value of only 2/3 instead of 1.


as the naive strategy.

7 Conclusion

We evaluate the out-of-sample performance of mean-variance strategies relying solely upon

the second moment relative to the naive benchmark. Using fourteen econometric models

across six datasets at weekly frequency, we show that the minimum-variance, constrained

minimum-variance, and volatility-timing strategies generally achieve higher Sharpe ratios,

lower turnover costs, and lower portfolio volatility that are economically significant relative

to naive diversification. Whenever mean-variance strategies do not significantly outperform

the naive rule, they usually match and only rarely lose to it.

We identify the econometric models that most consistently and significantly outperform

the 1/N benchmark. First, we show that the multivariate GARCH models weakly dominate

the naive rule when applied to the minimum-variance and constrained minimum-variance

strategies. Next, we demonstrate that the realized covariance model achieves impressive

results 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 in

all but one dataset. Third, we illustrate that if one wishes to prioritize the Sharpe ratio, then

the combined parameter and realized covariance models are excellent choices, even after

controlling for turnover costs. Finally, we show the exponentially-weighted moving-average

and 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 empirically

demonstrate that one important source of the naive portfolio puzzle is the quality of the

econometric 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

32 CURRAN AND ZALLA

that while considerable energy has been devoted to optimizing the mathematical design of

portfolio theory models, more progress may be warranted in improving the estimation of

the moments of asset returns.

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Table 1: Econometric models

1 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

1 Naive portfolio model based on equal investment across assets (1/N)2 Minimum-variance portfolio (MVP)3 Constrained minimum-variance portfolio (Con-MVP)4 Volatility-timing strategies (VT)5 Tangency portfolio (TP)

6Naive portfolio inputting average cov-matrix

in Table 1 into each of M2-M5 (CP)

7Naive portfolio across each of the cov-matrices

in Table 1 based on each of M2-M5

8Naive portfolio across M2-M5 based on each

of the cov-matrices in Table 1

Notes: We describe combined parameter (CP) model 6 in equation (6) and theother combined parameter models 7 and 8 in footnote 17.

40 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 0.112 0.101 0.035 0.058 0.100 0.087

Minimum-VarianceCP 0.089 0.106** 0.042 0.095** 0.117*** 0.099***ABEKK 0.098 0.131** 0.056 0.149** 0.181*** 0.113**BEKK 0.098 0.131** 0.057 0.148** 0.181*** 0.113**DCC 0.100 0.131** 0.055 0.137** 0.172*** 0.110**ADCC 0.100 0.131** 0.055 0.137** 0.172*** 0.110**CCC 0.099 0.131** 0.057 0.130** 0.176*** 0.108**VAR (0.093*) 0.134** 0.057 0.145** 0.191*** 0.110**VEC (0.090*) 0.132** 0.056 0.149** 0.185*** 0.113**COV 0.095 0.132** 0.056 0.153** 0.182*** 0.110*RSVAR (0.087*) 0.101 0.041 0.089** 0.113*** 0.098***RCOV 0.109 0.134* 0.074 0.128** 0.196*** 0.105COPULA 0.100 0.129** 0.056 0.111* 0.118*** 0.086EWMA 0.102 0.124 0.043 0.123* 0.166*** 0.093MSV 0.101 0.125 0.070 0.121* 0.156*** 0.112

ConstrainedMinimum-VarianceCCC 0.102 0.119** 0.047 0.102** 0.118*** 0.098***CP 0.090 0.107** 0.042 0.094** 0.115*** 0.098***COV 0.101 0.118* 0.044 0.103*** 0.117*** 0.099***ABEKK 0.106 0.118* 0.046 0.102*** 0.119*** 0.098***ADCC 0.102 0.119* 0.043 0.102*** 0.116*** 0.098***VAR 0.099 0.118* 0.046 0.099*** 0.120*** 0.098**VEC 0.098 0.119* 0.048 0.096** 0.116*** 0.099***DCC 0.102 0.119* 0.043 0.102** 0.116*** 0.098**MSV 0.104 0.118* 0.059 0.090** 0.121*** 0.098*RCOV 0.111 0.121 0.055 0.098** 0.125*** 0.099**BEKK 0.106 0.118 0.046 0.102** 0.118*** 0.098**COPULA 0.102 0.120** 0.044 0.094** 0.112*** 0.090RSVAR (0.087*) 0.101 0.041 0.089** 0.113*** 0.098***EWMA 0.111 0.121** 0.056 0.078 0.122*** 0.094

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.


Table 4: Sharpe Ratio: Volatility-Timing and Tangency


Naive 0.112 0.101 0.035 0.058 0.100 0.087

VolatilityTimingRCOV 0.093 0.110*** 0.043* 0.066*** 0.107*** 0.092***CCC (0.088*) 0.108*** 0.040 0.068*** 0.106*** 0.093***BEKK (0.089*) 0.109*** 0.040 0.067*** 0.105*** 0.093***ABEKK (0.089*) 0.109*** 0.040 0.067*** 0.105*** 0.093***COPULA (0.088*) 0.108*** 0.040 0.068*** 0.105*** 0.093***ADCC (0.088*) 0.108*** 0.040 0.068*** 0.105*** 0.093***DCC (0.088*) 0.108** 0.040 0.068*** 0.105*** 0.093***COV (0.088**) 0.108*** 0.040 0.067*** 0.105*** 0.093***VAR (0.081***) 0.109*** 0.040 0.067** 0.106*** 0.093***VEC (0.078***) 0.108** 0.040 0.067*** 0.105*** 0.094***MSV (0.074**) 0.109** 0.044* 0.065* 0.107*** 0.094**RSVAR (0.084*) 0.101 0.041 0.090** 0.113*** 0.098***CP (0.085*) 0.103 0.041 0.081** 0.113*** 0.098***EWMA (0.085*) 0.110** 0.042 0.065 0.108*** 0.093**

TangencyRCOV 0.124 0.161*** 0.104** 0.032 0.215*** 0.160***RSVAR 0.102 0.099 0.027 0.091** 0.114*** 0.100***CP 0.104 0.103 0.040 0.097** 0.118*** 0.103**DCC 0.113 0.114 0.038 0.137 0.182*** 0.130***VAR 0.109 0.115 0.045 0.135 0.199*** 0.133**VEC 0.108 0.113 0.046 0.132 0.195*** 0.135**BEKK 0.113 0.113 0.041 0.142 0.192*** 0.134**ABEKK 0.113 0.113 0.041 0.142 0.192*** 0.134**COV 0.113 0.113 0.043 0.142 0.193*** 0.133**CCC 0.113 0.114 0.040 0.134 0.190*** 0.130**ADCC 0.113 0.114 0.038 0.137 0.182*** 0.130**MSV 0.110 0.109 0.043 0.144* 0.158*** 0.128*EWMA 0.112 0.101 0.042 0.138 0.178*** 0.117COPULA 0.112 0.112 0.038 0.126 0.131*** 0.105


42 CURRAN AND ZALLA

Table 5: Turnover Costs: Minimum-Variance and Constrained Minimum-Variance


Naıve 0.113 0.101 0.035 0.054 0.100 0.087

Minimum-VarianceBEKK 0.098 0.132** 0.057 0.146** 0.181*** 0.113**ABEKK 0.098 0.131** 0.056 0.147** 0.181*** 0.113**DCC 0.100 0.132** 0.055 0.134** 0.172*** 0.111**ADCC 0.100 0.132** 0.055 0.134** 0.172*** 0.111**CCC 0.100 0.131** 0.057 0.126** 0.176*** 0.108**VEC (0.090*) 0.133** 0.056 0.146** 0.185*** 0.113**VAR 0.093 0.134** 0.057 0.142** 0.191*** 0.111*COV 0.095 0.132** 0.056 0.151** 0.182*** 0.110*CP 0.089 0.107* 0.041 0.090** 0.117*** 0.099***RSVAR (0.087*) 0.101 0.041 0.084** 0.114*** 0.098***RCOV 0.108 0.137* 0.074 0.132** 0.198*** 0.106COPULA 0.100 0.129** 0.056 0.107* 0.118*** 0.087MSV 0.102 0.127 0.071 0.121* 0.158*** 0.113EWMA 0.103 0.125 0.045 0.119 0.170*** 0.094

ConstrainedMinimum-VarianceCCC 0.102 0.120** 0.047 0.098*** 0.119*** 0.099***CP 0.090 0.107** 0.041 0.090** 0.115*** 0.098***COV 0.101 0.118* 0.044 0.099*** 0.118*** 0.099***ADCC 0.102 0.119* 0.043 0.098** 0.117*** 0.098***DCC 0.102 0.119* 0.043 0.098*** 0.117*** 0.098**VEC 0.099 0.119* 0.048 0.092** 0.117*** 0.099***VAR 0.099 0.119* 0.046 0.095** 0.120*** 0.098**BEKK 0.106 0.118* 0.046 0.098** 0.119*** 0.098**ABEKK 0.106 0.118* 0.046 0.098** 0.119*** 0.098**COPULA 0.103 0.120** 0.044 0.089** 0.113*** 0.090RSVAR 0.087 0.102 0.041 0.084** 0.113*** 0.098***RCOV 0.111 0.123 0.054 0.100** 0.124*** 0.098**MSV 0.104 0.118* 0.060 0.086** 0.121*** 0.099*EWMA 0.112 0.121** 0.056 0.074 0.123*** 0.094



Table 6: Turnover Costs: Volatility-Timing and Tangency


Naıve 0.113 0.101 0.035 0.054 0.100 0.087

VolatilityTimingRCOV 0.092 0.110*** 0.043* 0.065*** 0.105*** 0.091***BEKK (0.089*) 0.109*** 0.040 0.063*** 0.105*** 0.093***ABEKK (0.089*) 0.109*** 0.040 0.063*** 0.105*** 0.093***DCC (0.088**) 0.109*** 0.040 0.063*** 0.105*** 0.093***COPULA (0.088**) 0.109*** 0.039 0.063*** 0.105*** 0.094***VAR (0.081**) 0.109*** 0.040 0.063** 0.106*** 0.093***CCC (0.088**) 0.109*** 0.040 0.063** 0.106*** 0.094***ADCC (0.088**) 0.109*** 0.040 0.063** 0.105*** 0.093***COV (0.088**) 0.109*** 0.040 0.063** 0.105*** 0.094***VEC (0.079**) 0.109*** 0.040 0.062** 0.106*** 0.094***MSV (0.074**) 0.110** 0.044* 0.061* 0.108*** 0.094**RSVAR (0.084*) 0.102 0.041 0.085** 0.113*** 0.099***CP (0.085*) 0.103 0.040 0.076** 0.113*** 0.098***EWMA (0.085*) 0.111** 0.042 0.061 0.109*** 0.093**

TangencyRCOV 0.123 0.166*** 0.107** 0.057 0.217*** 0.164***CP 0.104 0.104 0.038 0.093** 0.118*** 0.103**RSVAR 0.102 0.099 0.027 0.086** 0.115*** 0.100***VAR 0.109 0.116 0.046 0.130 0.200*** 0.133**VEC 0.108 0.114 0.047 0.127 0.196*** 0.134**ABEKK 0.113 0.114 0.041 0.137 0.193*** 0.135**BEKK 0.114 0.114 0.042 0.136 0.193*** 0.134**COV 0.113 0.114 0.043 0.137 0.193*** 0.133**CCC 0.113 0.115 0.040 0.129 0.190*** 0.130**DCC 0.113 0.114 0.039 0.131 0.183*** 0.130**ADCC 0.113 0.114 0.039 0.131 0.183*** 0.130**MSV 0.110 0.111 0.046 0.139 0.161*** 0.130*EWMA 0.112 0.102 0.044 0.134 0.182*** 0.118COPULA 0.112 0.113 0.039 0.119 0.131*** 0.105


44 CURRAN AND ZALLA

Table 7: Portfolio Volatility: Minimum-Variance and Constrained Minimum-Variance


Naive 0.002 0.005 0.024 0.024 0.005 0.005

Minimum-VarianceEWMA 0.001*** 0.004*** 0.019*** 0.017*** 0.004*** 0.005***MSV 0.001*** 0.004*** 0.019*** 0.019*** 0.004*** 0.004***RCOV 0.002*** 0.004*** 0.018*** 0.018*** 0.004*** 0.005***DCC 0.002*** 0.004*** 0.019*** 0.018*** 0.004*** 0.005***ADCC 0.002*** 0.004*** 0.019*** 0.018*** 0.004*** 0.005***VAR 0.002*** 0.004*** 0.019*** 0.018*** 0.004*** 0.005***VEC 0.002*** 0.004*** 0.019*** 0.019*** 0.004*** 0.005***BEKK 0.002*** 0.004*** 0.019*** 0.019*** 0.004*** 0.005***ABEKK 0.002*** 0.004*** 0.019*** 0.019*** 0.004*** 0.005***CCC 0.002*** 0.004*** 0.019*** 0.019*** 0.004*** 0.005***COV 0.002*** 0.004*** 0.019*** 0.019*** 0.004*** 0.005***COPULA 0.002*** 0.004*** 0.019*** 0.019*** 0.005*** 0.005***CP 0.002*** 0.005 0.022 0.022** 0.005* 0.005**RSVAR 0.002*** 0.005 0.023 0.022* 0.005 0.005

ConstrainedMinimum-VarianceEWMA 0.001*** 0.004*** 0.017*** 0.020*** 0.004*** 0.005***MSV 0.001*** 0.004*** 0.018*** 0.020*** 0.005*** 0.005***RCOV 0.002*** 0.004*** 0.019*** 0.020*** 0.005*** 0.005***VEC 0.002*** 0.004*** 0.019*** 0.020*** 0.005*** 0.005**CCC 0.002*** 0.004*** 0.019*** 0.020*** 0.005*** 0.005**VAR 0.002*** 0.004*** 0.019** 0.020*** 0.005*** 0.005**BEKK 0.002*** 0.004*** 0.020** 0.020*** 0.005*** 0.005**ABEKK 0.002*** 0.004*** 0.020** 0.020*** 0.005*** 0.005**DCC 0.002*** 0.004*** 0.020** 0.020*** 0.005*** 0.005**ADCC 0.002*** 0.004*** 0.020** 0.020*** 0.005*** 0.005**COV 0.002*** 0.004*** 0.020** 0.020*** 0.005*** 0.005**COPULA 0.002*** 0.004*** 0.020** 0.020*** 0.005** 0.005*CP 0.002*** 0.005 0.022 0.022** 0.005* 0.005*RSVAR 0.002*** 0.005 0.023 0.022* 0.005 0.005

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.


Table 8: Portfolio Volatility: Volatility-Timing and Tangency


Naive 0.002 0.005 0.024 0.024 0.005 0.005

VolatilityTimingEWMA 0.001*** 0.004*** 0.021* 0.022 0.005 0.005**MSV 0.002*** 0.004** 0.021 0.023 0.005 0.005**RCOV 0.002*** 0.004** 0.022 0.023 0.005 0.005COPULA 0.002*** 0.005* 0.022 0.023 0.005 0.005BEKK 0.002*** 0.005* 0.022 0.023 0.005 0.005ABEKK 0.002*** 0.005* 0.022 0.023 0.005 0.005CCC 0.002*** 0.005* 0.022 0.023 0.005 0.005DCC 0.002*** 0.005* 0.022 0.023 0.005 0.005ADCC 0.002*** 0.005* 0.022 0.023 0.005 0.005COV 0.002*** 0.005* 0.022 0.023 0.005 0.005RSVAR 0.002** 0.005 0.023 0.022* 0.005 0.005*VAR 0.002** 0.005* 0.022 0.023 0.005 0.005VEC 0.002** 0.005* 0.022 0.023 0.005 0.005CP 0.002*** 0.005 0.023 0.023 0.005 0.005

TangencyCOPULA 0.002*** 0.004*** 0.022 0.020*** 0.005*** 0.005**MSV 0.002*** 0.004*** 0.022 0.023 0.005*** 0.005**RCOV 0.002*** 0.004*** 0.022 (0.025*) 0.004*** 0.005**VAR 0.002*** 0.004*** 0.022 0.024 0.004*** 0.005BEKK 0.002*** 0.004*** 0.022 0.024 0.004*** 0.005ABEKK 0.002*** 0.004*** 0.022 0.024 0.004*** 0.005CCC 0.002*** 0.004*** 0.021 0.022 0.004*** 0.005DCC 0.002*** 0.004*** 0.022 0.022 0.004*** 0.005ADCC 0.002*** 0.004*** 0.022 0.022 0.004*** 0.005COV 0.002*** 0.004*** 0.022 0.025 0.004*** 0.005EWMA 0.001*** 0.004*** 0.024 0.023 0.004*** 0.005VEC 0.002** 0.004*** 0.023 (0.026*) 0.004*** 0.005CP 0.002*** 0.005 0.021 0.022** 0.005 0.005**RSVAR 0.002*** 0.005 0.021 0.022* 0.005 0.005

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.

46 CURRAN AND ZALLA

Table 9: COV, EWMA, and VAR

COV

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓ ✓*Sharpe Ratio ✓ ✓ ✓ ✓*Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓

(a) Minimum Variance

Dataset 1 2 3 4 5 6

Turnover Cost ✓* ✓ ✓ ✓Sharpe Ratio ✓* ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓

(b) Constrained Minimum–Variance

Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓*

(c) Volatility-Timing

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓Sharpe Ratio ✓ ✓Portfolio Volatility ✓ ✓ ✓

(d) Tangency

EWMA

Dataset 1 2 3 4 5 6

Turnover Cost ✓Sharpe Ratio ✓* ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓

(e) Minimum Variance

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓Sharpe Ratio ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓

(f) Constrained Minimum–Variance

Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓* ✓

(g) Volatility-Timing

Dataset 1 2 3 4 5 6

Turnover Cost ✓Sharpe Ratio ✓Portfolio Volatility ✓ ✓ ✓

(h) Tangency

VAR

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓ ✓*Sharpe Ratio × ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓

(i) Minimum Variance

Dataset 1 2 3 4 5 6


(j) Constrained Minimum–Variance

Dataset 1 2 3 4 5 6


(k) Volatility-Timing

Dataset 1 2 3 4 5 6


(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.


Table 10: VEC, BEKK, ABEKK, and CCC

VEC

Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓ ✓*Sharpe Ratio × ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓Sharpe Ratio ✓ ✓Portfolio Volatility ✓ ✓ × ✓

(d) Tangency

BEKK and ABEKK

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6


(h) Tangency

CCC

Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6


(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; ×: strategy underperforms naive; else, insignificant difference.

48 CURRAN AND ZALLA

Table 11: DCC, ADCC, COPULA, and RSVAR

DCC and ADCC

Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6


(d) Tangency

COPULA

Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓* ✓Sharpe Ratio ✓ ✓* ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓*


Dataset 1 2 3 4 5 6



Dataset 1 2 3 4 5 6

Turnover Cost ✓Sharpe Ratio ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓

(h) Tangency

RSVAR

Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓Portfolio Volatility ✓ ✓*


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓Portfolio Volatility ✓ ✓*


Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓Portfolio Volatility ✓ ✓* ✓*


Dataset 1 2 3 4 5 6

Turnover 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.


Table 12: MSV, RCOV, and CP

MSV

Dataset 1 2 3 4 5 6

Turnover Cost ✓* ✓Sharpe Ratio ✓* ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓* ✓ ✓ ✓*Sharpe Ratio ✓* ✓ ✓ ✓*Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓* ✓* ✓ ✓Sharpe Ratio × ✓ ✓* ✓* ✓ ✓Portfolio Volatility ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓*Sharpe Ratio ✓* ✓ ✓*Portfolio Volatility ✓ ✓ ✓ ✓

(d) Tangency

RCOV

Dataset 1 2 3 4 5 6

Turnover Cost ✓* ✓ ✓Sharpe Ratio ✓* ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓ ✓ ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓* ✓ ✓ ✓Sharpe Ratio ✓ ✓* ✓ ✓ ✓Portfolio Volatility ✓ ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ × ✓ ✓

(h) Tangency

CP

Dataset 1 2 3 4 5 6

Turnover Cost ✓* ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓* ✓


Dataset 1 2 3 4 5 6

Turnover Cost ✓ ✓ ✓ ✓Sharpe Ratio ✓ ✓ ✓ ✓Portfolio Volatility ✓ ✓ ✓* ✓*


Dataset 1 2 3 4 5 6

Turnover Cost × ✓ ✓ ✓Sharpe Ratio × ✓ ✓ ✓Portfolio Volatility ✓ ✓


Dataset 1 2 3 4 5 6

Turnover 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; ×: strategy underperforms naive; else, insignificant difference.

Can Volatility Solve the Naive Portfolio Puzzle?

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