#### 3.1. Data and Methods

In this section, we implement the procedures described in

Section 2 and use the predicted one-step-ahead conditional covariance matrix to construct the minimum variance portfolio (MVP) of the stocks used in the composition of the S&P 500 index, traded from 2 January 2000 to 30 November 2017. Because not all stocks of the index were traded during the whole period, we ended up with

$N=174$ stocks.

To evaluate the out-of-sample portfolio performance, we consider a rolling window scheme. The out-of-sample portfolio performance is evaluated in four different periods, namely: pre-crisis period (January 2004 to December 2007, 1008 days), subprime crisis period (January 2008 to June 2009, 378 days), post-crisis period (July 2009 to November 2017, 2218 days), and full period (January 2004 to November 2017, 3503 days). In each window, the one-step-ahead covariance matrix is estimated and the MVP values with and without short-sale constraints are obtained. The weights in the MVP portfolio are rebalanced with both daily and monthly frequencies. In the latter case, we follow

Engle et al. (

2017), that is, we obtain the portfolio returns daily but update the weights monthly (following the common convention we use 21 consecutive trading days as a month). Monthly updating is common in practice to reduce transaction costs.

The procedures described in

Section 2 are combined with the linear and non-linear shrinkage estimator described in Sub

Section 2.8. The linear and non-linear shrinkage are applied at the beginning and/or at the end of the estimation procedure. A detailed description of each combination of the estimation procedures is given in the

Appendix A. In addition, for the sake of comparison, we also implement the naive equal-weighted portfolio. In the line of

Engle et al. (

2017),

Gambacciani and Paolella (

2017),

Trucíos et al. (

2018) among others, we consider the following annualised out-of-sample performance measures. Denote by

${R}_{p}=\{{r}_{p,1},\dots ,{r}_{p,k}\}$ the observed out-of-sample returns from a given method where

k in the length of the out-of-sample period. The measures considered in this paper: the annualised average portfolio return (AV), standard deviation portfolio return (SD), information ratio (IR), Sortino’s ratio (SR) and average turnover (TO) are computed as follows:

AV: equal to $252\times {\overline{R}}_{p}$, where ${\overline{R}}_{p}$ is the average of the elements of ${R}_{p}$.

SD: equal to $\sqrt{252}\times {S}_{p}$, where ${S}_{p}$ is the standard deviation of the elements of ${R}_{p}$.

IR: AV/SD.

SR: AV/$\sqrt{252\times {S}^{\ast 2}}$, where ${S}^{\ast 2}$ is the mean of ${r}_{p,i}^{\ast},i=1,\dots ,k$, with ${r}_{p,i}^{\ast}={r}_{p,i}^{2}$ if ${r}_{p,i}$ less than the minimal acceptable return, which is taken to zero, and zero otherwise.

TO: ${k}^{-1}\sum _{t=2}^{k}\sum _{j=1}^{N}|{\omega}_{j,t}-{\omega}_{j,t-1}|$ where ${\omega}_{j,t}$ is the portfolio weight at time t for the j-th asset, and k is the number of the out-of-sample portfolio returns.

As pointed out by

Kirby and Ostdiek (

2012),

Santos and Ferreira (

2017),

Olivares-Nadal and DeMiguel (

2018), among others, transaction costs (

c) can have an impact on the portfolio’s performance. In order to take into account those costs, we also compute the portfolio returns net of transaction cost. For a given

c, the portfolio return net of transaction costs at time

t is given by

${r}_{p,t}^{net}=(1-c\times turnove{r}_{t})(1+{r}_{p,t})-1$ and then the annualised average portfolio return net of transaction costs is AV

${}^{net}=252\times {\overline{R}}_{p}^{net}$ where

${\overline{R}}_{p}^{net}$ is the average of the portfolio return net of transaction costs

${r}_{p,1}^{net},\dots ,{r}_{p,k}^{net}.$ We consider

$c=20bp$ (intermediate) and

$c=50bp$ (high level) transaction costs where a basis point (bp) is a unit of measure commonly used in finance and is equivalent to

$0.01\%.$ The annualised average portfolio return net of transation costs considering

$c=20bp$ and

$c=50bp$ are denoted by AV

${}_{20bp}^{net}$ and AV

${}_{50bp}^{net}$, respectively.

#### 3.2. Results

Table 1,

Table 2,

Table 3,

Table 4,

Table 5,

Table 6,

Table 7 and

Table 8 report annualised out-of-sample performance measures for MVP with performance for the pre-crisis, crisis, post-crisis and full periods.

Table 1,

Table 2,

Table 3 and

Table 4 report the results for daily rebalanced portfolios whereas

Table 5,

Table 6,

Table 7 and

Table 8 report the results for monthly rebalanced portfolios. We also have results for MVP with no short-sale constraints. However, in this paper we focus on the results for MVP with short-sale constraints and give a short summary of the main findings for the case without short-sale constraints. A detailed analysis of the case without short-sale constraints is given in the

Supplementary Material.

In

Table 1,

Table 2,

Table 3,

Table 4,

Table 5,

Table 6,

Table 7 and

Table 8 we report (in parentheses) the rank of the methods according to the SD criterion in the second column. Moreover, for each criterion, the best five methods are highlighted in shadowed cells. The equal-weighted portfolio strategy is represented by

$1/N$.

Taking into account the fact that portfolios are chosen in order to have the minimum variance, the analysis is first done according to the SD criterion. For portfolios rebalanced daily or monthly, the largest SD is reported by the equal-weight portfolio strategy. For portfolios rebalanced daily (

Table 1,

Table 2,

Table 3 and

Table 4), the five smallest SDs are obtained by the DCC based-methods, except in the crisis period, in which case the five smallest SDs are spread among the DCC, OGARCH and GPVC based-methods. In the crisis-period, the smallest SD is obtained by the GPVC procedure with the non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. For portfolios rebalanced monthly (

Table 5,

Table 6,

Table 7 and

Table 8), the smallest SDs are obtained by the RM2006-LS

4, NLS-DCC, NLS-GPVC and RM2006-LS procedures for the full, pre-crisis, crisis and post-crisis periods, respectively.

The best performance in terms of the AV criterion differs depending on the period and rebalance strategy. For instance, for daily rebalancing the best performance in the full period is achieved by the RPVC followed by the RPVC with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. However, for the pre-crisis, crises and post-crisis periods, the best performance is achieved by the OGARCH with non-linear shrinkage applied to the unconditional covariance matrix (NLS-OGARCH), RPVC with linear shrinkage applied to the one-step-ahead conditional covariance matrix (RPVC-LS) and RiskMetrics method with linear shrinkage applied to the one-step-ahead conditional covariance matrix (RM1994-LS), respectively. For monthly rebalancing, the best performances in the full, pre-crisis, crisis and post-crisis periods are achieved by the RPVC, OGARCH-NLS, GPVC-LS and equal-weight portfolio strategy, respectively.

In terms of average turnover, the five smallest average turnovers are in the OGARCH and GPVC groups, with the best performance being achieved by the OGARCH with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix in almost all cases. The only two exceptions are observed in the crisis period, in which case the best performance is achieved by the GPVC procedure with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. Additionally, note that regardless of whether portfolio is rebalanced daily or monthly, the average turnover reported by all dimension reduction techniques is smaller than reported by the non-dimension reduction procedures.

As for the annualised average portfolio returns taking into account transaction costs, the procedures with the five largest values of AV${}_{20bp}^{net}$ and AV${}_{50bp}^{net}$ are the same procedures with the largest AV, except in some cases in the pre-crisis period, where one of five largest AV${}_{50bp}^{net}$ is obtained by the NLS-OGARCH-NLS procedure.

For each period, the five best methods in terms of information criteria are the same (except in

Table 8, where four methods are the same). We omit the analysis in the crisis period because these criteria values are negative. Overall, for daily rebalancing, RiskMetrics based methods are among the best in the full and post-crisis periods, RPVC and RPVC-NLS are among the best in the full and pre-crisis periods, and NLS-OGARCH and LS-OGARCH are among the best in the pre-crisis period. For monthly rebalancing, some OGARCH-based methods are among the best in the pre-crisis and full periods, some CCC-based methods are among the best in the post-crisis and full periods, RM1994-LS is among the best for the post-crisis period, and RPVC is among the best for the full period.

The analysis of

Table 1,

Table 2,

Table 3,

Table 4,

Table 5,

Table 6,

Table 7 and

Table 8 reveals that none of the methods is the best in all scenarios and the performance depends on the criterion, the period and the rebalancing strategy. In this sense, the analysis will focus on the full period (

Table 1 and

Table 5) in order to account for periods with different volatility levels. When portfolios are rebalanced on a daily basis, we find that DCC-based methods are the best in terms of SD; RM2006-LS, RM2006-NL, RPVC and RPVC-NLS are the best in terms of {AV, AV

${}_{20bp}^{net}$, AV

${}_{50bp}^{net}$} and {IR, SR}, and some OGARCH-based are the best regarding TO. For monthly rebalanced portfolios, the best methods in terms of SD are DCC, LS-DCC, NLS-DCC, RM2006 and RM2006-LS, whereas the best performances in terms of {AV, AV

${}_{20bp}^{net}$, AV

${}_{50bp}^{net}$} and {IR, SR} are given by (RPVC, RPVC-NLS), (OGARCH-NLS, NLS-OGARCH-NLS) and CCC. In addition, the equal-weighted strategy is the second best in terms of AV, but the worst regarding SD, IR and SR criteria.

To show when the shrinkage method improves performance in terms of SD, the analysis is again focused on the full period (

Table 1 and

Table 5). For daily and monthly portfolio rebalancing: shrinkage always improves the performance of the RM2004 and GPVC methods (except LS-GPVC for monthly rebalancing) whereas it always worsens the DCC method; linear shrinkage at the end improves RM2006; just linear/non-linear shrinkage at the beginning improves DECO; OGARCH-NLS and NLS-OGARCH-NLS improves OGARCH; LS-CCC improves CCC (as well as NLS-DCC for daily rebalancing). Additionally, for daily rebalancing, shrinkage always improves the performance of RPVC (except LS-GPVC), whereas for monthly rebalancing, linear shrinkage applied at the beginning and/or end improves RPVC.

Nakagawa et al. (

2018) also reports that in some cases the use of non-linear shrinkage on the unconditional covariance matrix of the devolatilised returns in the DCC model increases the standard deviation of the out-of-sample portfolio returns.

We now discuss the effect of shrinkage in terms of AV${}_{50bp}^{net}$. For daily rebalancing, shrinkage improves the performance of the RM2006 and DECO methods, and worsens the performance of the DCC and RPVC methods. In addition, CCC-NLS is better than CCC, RM1994-NLS is better than RM1994, and LS-GPVC is better than GPVC. For monthly rebalancing, shrinkage does not improve the performance of the CCC, DCC, GPVC and RPVC methods. In addition, RM2006-LS is better than RM2006, RM1994-NLS is better than RM1994, DECO-NLS and NLS-DECO-NLS are better than DECO, and OGARCH-NLS and NLS-OGARCH-NLS are better than OGARCH.

Finally, we list next the main findings when short-selling is allowed for optimisation of the portfolio variance. A detailed analysis of these cases is given in the

Supplementary Material. First, none of the methods is the best in all scenarios and the performance depends on the criterion, the sample period and the portfolio rebalancing scheme. Second, the analysis of the full period reveals that for daily rebalancing, DCC methods are the best regarding SD and are among the best in terms of IR and SR. RM1994-LS and RM2006-LS are the best according to AV, AV

${}_{20bp}^{net}$, AV

${}_{50bp}^{net}$, IR and SR. For monthly rebalancing, DCC-LS and LS-DCC-LS are among the best in terms of SD, RM2006-NLS is the best in terms of SD and is among the best regarding IR and SR. RM 1994 and RM1994-LS are the first and second best in terms of AV, AV

${}_{20bp}^{net}$, AV

${}_{50bp}^{net}$ but are among the worst in terms of SD. Third, the analysis of the turnover and average net returns in the no short-sale constraints case must be carefully done. This is because since no limits are imposed on the weights of the portfolio, large turnover values can be obtained and consequently we can have a large loss (average return) but huge net gain (average net portfolio return taking into account transaction costs). Fourth, in many cases shrinkage improves the performance of the methods in terms of SD, and this improvement can be substantial. Fifth, the top-five models in terms of SD are the same in both restricted and unrestricted minimum variance portfolios for daily rebalancing, except in the crisis period.