# Can Machine Learning-Based Portfolios Outperform Traditional Risk-Based Portfolios? The Need to Account for Covariance Misspecification

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## Abstract

**:**

## 1. Introduction

## 2. Risk-Based Portfolios

#### 2.1. Minimum Variance Portfolio (MVP)

#### 2.2. Inverse Volatility Weighted Portfolio (IVWP)

#### 2.3. Equal Risk Contribution Portfolio (ERC)

#### 2.4. Maximum Diversification Portfolio (MDP)

#### 2.5. Market-Capitalization-Weighted Portfolio (MCWP)

## 3. Covariance Matrix Forecasting Methods

#### 3.1. Sample-Based Covariance (SMPL)

#### 3.2. Exponentially Weighted Moving Average (EWMA)

#### 3.3. Dynamic Conditional Correlation GARCH (DCC-GARCH)

## 4. Hierarchical Risk Parity (HRP)

#### 4.1. Clustering

- Single Linkage: The single linkage (SL) clustering method keeps the distance between two clusters as the minimum of the distance between any two points in the clusters such that:$${d}_{{C}_{i},{C}_{j}}=\underset{x,y}{min}\{\tilde{D}(x,y)|x\in {C}_{i},y\in {C}_{j}\}$$
- Average Linkage: In the average linkage (AL) technique the distance is defined by the average of the distance between any two points in the clusters. For clusters ${C}_{i},{C}_{j}$:$${d}_{{C}_{i},{C}_{j}}={\mathrm{mean}}_{x,y}\{\tilde{D}(x,y)|x\in {C}_{i},y\in {C}_{j}\}$$
- Ward’s Method: The most popularly used method is Ward’s method (Ward 1963). It says that the distance between two clusters is how much the sum of squared errors will increase when they are merged:$${d}_{{C}_{i},{C}_{j}}=\frac{{m}_{i}{m}_{j}}{{m}_{i}+{m}_{j}}\u2225{c}_{i}-{c}_{j}\u2225.$$

#### 4.2. Quasi Diagonalisation

#### 4.3. Recursive Bisection

- Initialize a list of assets in the portfolio with $L=\left\{{L}_{0}\right\},$ with ${L}_{0}={\left\{n\right\}}_{n=1,\dots ,N}$.
- Initialize a vector of weights as ${w}_{i}=1,i\phantom{\rule{3.33333pt}{0ex}}=1,\dots ,N.$
- Stop if $|{L}_{i}|=1,\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}{L}_{i}\in L$
- For each ${L}_{i}\in L$ such that $|{L}_{i}|>1$
- Bisect ${L}_{i}$ into two subsets, ${L}_{i}^{1}\cup {L}_{i}^{2}={L}_{i},$ where $|{L}_{i}^{1}|=int[\frac{1}{2}|{L}_{i}|]$
- Calculate the variance of ${L}_{i}^{j},\phantom{\rule{4pt}{0ex}}j=1,2$ as ${\tilde{V}}_{i}^{j}=\tilde{{w}_{i}^{j}}{V}_{i}^{j}{\tilde{{w}_{i}^{j}}}^{\prime},$ where ${V}_{i}^{j}$ is the covariance matrix of the elements within cluster $j,$ and$${\tilde{w}}_{i}^{j}=\frac{tr{[{V}_{i}^{j}]}^{-1}}{{\sum}_{i}tr{[{V}_{i}^{j}]}^{-1}},$$
- Compute the split factor ${\alpha}_{i}=1-\frac{{\tilde{V}}_{i}^{1}}{{\tilde{V}}_{i}^{1}+{\tilde{V}}_{i}^{2}}$
- Rescale allocations ${w}_{n}$ by a factor of ${\alpha}_{i}\forall \phantom{\rule{4pt}{0ex}}n\in {L}_{i}^{1}$
- Rescale allocations ${w}_{n}$ by a factor $(1-{\alpha}_{i})\forall \phantom{\rule{4pt}{0ex}}n\in {L}_{i}^{2}$

- Loop to Step 2.

## 5. Data and Methodology

#### 5.1. Intra-Day Realized Covariance Estimator

#### 5.2. Portfolio Risk Measures

- (1)
- Portfolio variance: We use the total daily variance of the portfolio as the first measure of performance. The realized variance of the portfolio is given by,$${\sigma}_{t}^{2}({\overrightarrow{\widehat{w}}}_{t})={\overrightarrow{\widehat{w}}}_{t}{\mathsf{\Sigma}}_{\mathit{t}}{\overrightarrow{\widehat{w}}}_{t}^{\prime},$$
- (2)
- Conditional Value-at-Risk (CVaR), also known as the expected shortfall, is a measure of risk, which is defined as (see Acerbi and Tasche 2002). Let X be the profit-loss of a portfolio on a specified time horizon and let $\alpha \in (0,1)$ be some specified probability level. The expected $\alpha $ shortfall of the portfolio is defined as$${\mathrm{ES}}^{(\alpha )}(X)=-\frac{1}{\alpha}\left(\mathbb{E}\left[X{\overrightarrow{1}}_{X\le {x}^{(\alpha )}}\right]-{x}^{(\alpha )}(\mathbb{P}\left[X\le {x}^{(\alpha )}\right]-\alpha )\right),$$$${x}^{(\alpha )}(X)=sup\{x|\mathbb{P}[X\le x]\le \alpha \}.$$$${\mathrm{VaR}}^{(\alpha )}(X)=-{x}^{(\alpha )}(X).$$$${\mathrm{ES}}_{m}^{(\alpha )}(X)=-\frac{{\sum}_{i=1}^{s}{X}_{i:m}}{s}.$$
- (3)
- Herfindahl Index (${H}^{*}$) of percentage risk contribution: The normalized Herfindahl index is an indicator of concentration risk. It takes the value between 0 and 1, where 0 signifies a perfectly diversified portfolio. It is calculated as:$${H}^{*}(\%RC({\overrightarrow{\widehat{w}}}_{t}))=\frac{{\sum}_{i=1}^{N}{(\%R{C}_{i})}^{2}-\frac{1}{N}}{1-\frac{1}{N}}$$
- (4)
- Diversification Ratio (DR): It is computed as defined in Equation (4). In order to compute the realized DR. we use the portfolio weights computed using the forecasted covariance matrix, and the covariance matrix in the equation is substituted with the realized covariance matrix. It gives the measure of diversification in the portfolio and takes values $\ge 1$. As we know, a higher diversification ratio is a better performance indicator; we use $-DR$ as our loss function.
- (5)
- Sharpe Ratio: The Sharpe ratio, also called reward-to-variability ratio, is a measure of excess return per unit of deviation. It is defined as$$SR=\frac{\mathbb{E}({\tilde{\mathit{r}}}_{\mathit{t}}^{\mathit{p}}-{r}_{f})}{{\sigma}_{t}^{p}}$$

#### 5.3. Test for Superior Predictive Ability

#### 5.4. Stationary Bootstrap Based Implementation

## 6. Results

#### 6.1. Superior Method for Forecasting Covariance Matrix

- Forecast the covariance matrix ${\widehat{\mathsf{\Sigma}}}_{\mathit{t}}$ using the three approaches for the appropriate forecast horizon.
- Compute the portfolio weights, ${\overrightarrow{\widehat{w}}}_{t},$ using the above covariance matrix for the portfolio allocation method being considered.
- Compute using the intra day data, realized returns and realized covariance matrix ${\mathsf{\Sigma}}_{t}.$
- Use (2) and (3) to compute the time series of realized losses ${L}_{kt},$ using the appropriate loss function for the allocation method (as provided in Table 2), for each covariance forecast methodology.
- For different choices of benchmark covariance forecast models, compute the p-values for the null hypothesis, which is that a chosen model is as good as any other model.

#### 6.2. Benchmark Allocation Methods for Different Portfolio Performance Objectives

#### 6.2.1. Out-of-Sample Portfolio Variance

#### 6.2.2. Out-of-Sample Weekly CVaR

#### 6.2.3. Out-of-Sample Herfindahl Index and Diversification Ratio

#### 6.2.4. Out-of-Sample Sharpe Ratios

## 7. Conclusions

#### 7.1. Choice of Covariance Estimator

- The performance of all the portfolios, with respect to their corresponding objectives, is superior when DCC GARCH is used for forecasting the covariance in majority of the universes.
- MDP appears most sensitive to the choice of better covariance forecast methodology.

#### 7.2. Portfolio Variance

- HRP variants are superior in the majority of universes when the objective is to minimize out-of-sample variance.
- With a poor covariance estimator, IVWP and HRP (AL) are superior methods in majority of the universes.
- With longer rebalancing horizons, HRP variants have superior performance in majority of the universes with a few exceptions where IVWP and ERC have in expectation lower portfolio variance.

#### 7.3. Expected Shortfall

- HRP variants are superior when it comes to minimizing out-of-sample CVaR. Amongst them, HRP (SL) performs relatively better in majority of the universes.
- With inferior covariance estimates, IVWP and HRP (Ward) result in superior performance in majority of the universes.
- With longer rebalancing horizons ERC consistently results in superior performance when it comes to minimizing out-of-sample expected shortfall.

#### 7.4. Herfindhal Index and Diversification Ratio

- ERC followed by IVWP are superior in majority of universes when the objective is to maximize out-of-sample diversification ratio.
- IVWP followed by ERC are superior in majority of universes when the objective is to minimize the Herfindhal index of $\%RC.$
- These results are consistent for different rebalancing horizons.

#### 7.5. Sharpe Ratio

- With daily rebalancing, and covariance estimated using DCC GARCH, many allocation methods including MVP, IVWP, MWP, and HRP results in not inferior performance when it comes to maximizing out-of-sample weekly Sharpe ratios.
- With an inferior covariance estimate Market Weighted portfolio has superior performance in majority of the universes, followed by IVWP.
- With increasing rebalancing horizons MVP clearly has superior performance in majority of the universes.

#### 7.6. Strengths and Weaknesses

- MVP: We see that out-of-sample performance of MVP is poor when it comes to minimizing the portfolio variance or expected shortfall. Its performance is good when it comes to maximizing Sharpe ratio, especially when the portfolio is supposed to be rebalanced less frequently. However, its performance with respect to Sharpe ratio is highly sensitive to covariance misspecification.
- IVWP: IVWP has superior performance when it comes to maximizing the out-of-sample Herfindhal index. It also has lower out-of-sample portfolio variance and expected short fall amongst the risk-based portfolios, especially when an inferior covariance estimator is used. However, with a superior covariance estimator, it often is not the best choice for most risk objectives.
- ERC: ERC has superior performance when it comes to maximizing the out-of-sample diversification ratio. It also appears to be the best choice for minimizing expected shortfall when the portfolio is not rebalanced often. It seems to have inferior performance when it comes to maximizing Sharpe ratio with longer rebalancing horizons.
- MDP: For our dataset, MDP had inferior performance for most objectives in the majority of the universes. It seems most sensitive to misspecification in a covariance matrix.
- MWP: Market weighted portfolios did not perform well with objectives of portfolio variance and expected shortfall minimization. They, however, showed up as superior models for maximizing weekly Sharpe ratios, especially when an inferior covariance matrix estimator is available. With longer rebalancing horizons and good covariance forecasts models, they result in inferior Sharpe ratios when compared to the other methods considered.
- HRP: HRP variants have superior performance when it comes to realized portfolio variance and expected shortfall when DCC GARCH is used to forecast the covariance matrix. They also are not inferior when the objective is to maximize weekly Sharpe ratio, when the portfolio is rebalanced daily and DCC GARCH is used to forecast the covariance matrix. They do seem to be sensitive to the choice of the covariance forecast model. The performance of different variants of HRP seems similar, although it might appear that HRP (SL) has a slight edge for our dataset. It is important to note that the key strength of HRP lies when the portfolio is constructed with many underlying assets, where inversion of the covariance matrix and good estimation of correlations becomes challenging. We have considered portfolios with only ten constituents; however, it would be interesting to see the outcomes of further studies carried out for portfolios with larger numbers of underlying assets.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Portfolio Composition

Universe | Ticker Name | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | AXISBANK | BANKBARODA | HDFCBANK | ICICIBANK | INDUSINDBK | KOTAKBANK | SBIN | TCS | YESBANK | HDFC |

2 | AXISBANK | HDFCBANK | HDFC | ICICIBANK | INFY | ITC | KOTAKBANK | LT | RELIANCE | TCS |

3 | BHARTIARTL | CIPLA | GAIL | ITC | MARUTI | NTPC | POWERGRID | TATAMOTORS | TATASTEEL | SUNPHARMA |

4 | BHEL | BPCL | GAIL | NTPC | ONGC | POWERGRID | RELIANCE | TATAPOWER | COALINDIA | HINDALCO |

5 | YESBANK | RELIANCE | ICICIBANK | IDEA | INFY | NTPC | CIPLA | HDFCBANK | WIPRO | ZEEL |

## Appendix B. Converting Covariance Matrix of Log Returns to Linear Returns

## Appendix C. Tables Containing p-Values for Weekly and Monthly Rebalancing

**Table A2.**${p}^{c}$-value of different benchmark portfolios based on the out-of-sample portfolio variance for different rebalancing horizons. The covariance forecast model is taken as DCC-GARCH.

Realized Portfolio Variance with Weekly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | 0.1080 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.0130 | 1.0000 | <0.0001 | 1.0000 |

ERC | 0.5710 | 0.4700 | <0.0001 | 1.0000 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.0990 | < 0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 1.0000 | 1.0000 | 0.0010 | 0.1230 | 0.2730 |

HRP (AL) | 0.0240 | 0.4650 | <0.0001 | 0.5420 | 0.0070 |

HRP (Ward) | 0.0110 | <0.0001 | <0.0001 | 0.3430 | 0.5160 |

Realized Portfolio Variance with Monthly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.0020 | 1.0000 | <0.0001 | 0.3380 |

ERC | <0.0001 | 0.0130 | <0.0001 | 1.0000 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.1720 | < 0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | <0.0001 | 1.0000 | <0.0001 | 0.0160 | 0.8380 |

HRP (AL) | 1.0000 | <0.0001 | <0.0001 | 0.0320 | 1.0000 |

HRP (Ward) | <0.0001 | 0.0560 | <0.0001 | 0.0730 | 0.3610 |

**Table A3.**The out-of-sample realized annual volatilities for different rebalancing horizons when DCC GARCH is used for covariance estimation.

Realized Annual Portfolio Volatility with Weekly Rebalancing | ||||||||

HRP | HRP | HRP | ||||||

Universe | MVP | IVWP | ERC | MDP | MWP | (SL) | (AL) | (Ward) |

1 | 11.16% | 11.76% | 11.15% | 11.03% | 16.65% | 10.81% | 11.47% | 11.54% |

2 | 10.38% | 9.48% | 9.36% | 9.88% | 10.56% | 9.48% | 9.45% | 9.77% |

3 | 11.35% | 11.51% | 11.48% | 11.60% | 16.31% | 11.45% | 11.41% | 11.56% |

4 | 11.93% | 11.89% | 11.86% | 11.89% | 13.01% | 11.73% | 11.66% | 11.53% |

5 | 10.00% | 9.92% | 10.03% | 10.80% | 11.52% | 9.72% | 9.82% | 9.77% |

Realized Annual Portfolio Volatility with Monthly Rebalancing | ||||||||

1 | 11.65% | 11.71% | 11.10% | 10.98% | 16.74% | 10.69% | 10.53% | 11.59% |

2 | 10.57% | 9.52% | 9.40% | 9.95% | 10.65% | 9.57% | 9.56% | 9.48% |

3 | 11.40% | 11.63% | 11.60% | 11.73% | 16.44% | 11.64% | 11.59% | 11.66% |

4 | 11.88% | 12.00% | 11.96% | 12.01% | 13.02% | 12.19% | 11.97% | 11.91% |

5 | 10.35% | 10.11% | 10.25% | 11.11% | 11.63% | 9.87% | 9.87% | 9.89% |

**Table A4.**${p}^{c}$-value of different benchmark portfolios based on the out-of-sample CVaR for different rebalancing horizons. The covariance forecast model is taken as DCC-GARCH.

Realized 5-day CVaR with Weekly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | 0.1040 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.3420 | 1.0000 | <0.0001 | 0.4200 |

ERC | 0.3040 | 1.0000 | 0.4330 | 0.4850 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.3170 | < 0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 1.0000 | 0.6970 | 0.0320 | 0.0650 | 0.0600 |

HRP (AL) | 0.0330 | 0.5570 | 0.0450 | 1.0000 | <0.0001 |

HRP (Ward) | 0.0190 | 0.0260 | <0.0001 | 0.4130 | 1.0000 |

Realized 5-Day CVaR with Monthly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.5400 | 1.0000 | <0.0001 | 0.0800 |

ERC | <0.0001 | 0.4900 | 0.3000 | 1.0000 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.5200 | < 0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | <0.0001 | 1.0000 | 0.0200 | 0.0500 | 0.0200 |

HRP (AL) | 1.0000 | <0.0001 | 0.0200 | 0.4700 | 0.0500 |

HRP (Ward) | <0.0001 | <0.0001 | 0.0100 | 0.5200 | 1.0000 |

**Table A5.**${p}^{c}$-value for different choices of benchmark model based on the out-of-sample ${H}^{*}(\%RC)$ and diversification ratios, when different rebalancing horizons are considered. The covariance forecast model is taken as DCC-GARCH.

Realized Diversification Ratio with Weekly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

IVWP | <0.0001 | <0.0001 | 1.0000 | 0.0940 | 1.0000 |

ERC | 1.0000 | 1.0000 | <0.0001 | 1.0000 | 0.1960 |

Realized Diversification Ratio with Monthly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

IVWP | <0.0001 | <0.0001 | 1.0000 | 0.0100 | 1.0000 |

ERC | 0.1810 | 1.0000 | <0.0001 | 1.0000 | 0.0270 |

Realized${\mathit{H}}^{*}(\%\mathit{RC})$with Weekly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

IVWP | 1.0000 | 0.1060 | 1.0000 | 0.2240 | 1.0000 |

ERC | 0.0260 | 1.0000 | <0.0001 | 1.0000 | <0.0001 |

Realized${\mathit{H}}^{*}(\%\mathit{RC})$with Monthly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

IVWP | 1.0000 | 1.0000 | 1.0000 | 0.1740 | 1.0000 |

ERC | 0.0260 | 0.4400 | <0.0001 | 1.0000 | <0.0001 |

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1 | In practice, typically, a rolling window smaller than the number of observations in the estimation period is used. We here, however, use the longest possible rolling window. |

2 | The dataset are available from the author upon request. |

3 | It ensures that ${lim}_{n\to \infty}P({\widehat{\mu}}_{k}^{c}=0\mid {\mu}_{k}=0)=1$ and ${lim}_{n\to \infty}P({\overline{Z}}_{b,k,n}^{*}\le 0\mid {\mu}_{k}<0)=1$ which is important for the consistency, as the models with ${\mu}_{k}<0$ do not influence the asymptotic distribution of ${T}_{n}^{SPA}$, see Hansen (2005). |

**Figure 1.**A Sequence of clusters formed in hierarchical clustering represented as a dendrogram with y-axis representing the distance between the two merging leaves.

**Table 1.**List of universes constructed: N denotes the number of assets considered in each universe. The Min, Med, Max reports the minimum, median and maximum values of the unconditional volatilities, respectively, and the unconditional pairwise correlations in percentage for each universe.

# | Universe | N | Volatility | Correlation | ||||
---|---|---|---|---|---|---|---|---|

Min | Med | Max | Min | Med | Max | |||

1 | Financial Services Sector | 10 | 14.43 | 23.97 | 44.10 | −13.01 | 04.90 | 87.32 |

2 | Top 10 Market−Cap | 10 | 14.43 | 21.88 | 29.52 | −16.46 | 08.76 | 44.84 |

3 | Random 10 | 10 | −16.46 | 08.76 | 44.84 | −06.38 | 14.47 | 38.99 |

4 | Energy sector | 10 | −06.38 | 14.47 | 38.99 | 02.30 | 17.04 | 32.73 |

5 | Random 10 | 10 | 17.81 | 23.24 | 51.28 | −07.30 | 06.72 | 28.70 |

**Table 2.**The portfolio allocation method and corresponding loss function to determine benchmark covariance forecast models.

Category | Method Name | Loss Function Used for SPA |
---|---|---|

MVP | ${\sigma}_{t}({\overrightarrow{\widehat{w}}}_{t})$ | |

IVWP | ${H}^{*}(\%RC({\overrightarrow{\widehat{w}}}_{t}))$ | |

Traditional Risk-Based | ERC | ${H}^{*}(\%RC({\overrightarrow{\widehat{w}}}_{t}))$ |

MDP | $-DR({\overrightarrow{\widehat{w}}}_{t})$ | |

HRP (SL) | ${\sigma}_{t}({\overrightarrow{\widehat{w}}}_{t})$ | |

Machine Learning | HRP (AL) | ${\sigma}_{t}({\overrightarrow{\widehat{w}}}_{t})$ |

HRP (Ward) | ${\sigma}_{t}({\overrightarrow{\widehat{w}}}_{t})$ |

**Table 3.**p-values from the SPA test when DCC, SMPL, and EWMA are chosen as the benchmark model respectively for the different portfolio allocation techniques. The corresponding loss functions considered for each portfolio allocation technique are provided in Table 2.

MVP | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

DCC | 1.000 | 0.108 | 1.000 | 1.000 | 1.000 |

SMPL | 0.185 | 1.000 | 0.095 | 0.090 | 0.178 |

EWMA | 0.136 | 0.034 | 0.123 | 0.087 | 0.202 |

IVWP | |||||

DCC | 1.000 | 0.039 | 1.000 | 0.096 | 1.000 |

SMPL | 0.251 | 1.000 | 0.010 | 1.000 | 0.005 |

EWMA | 0.106 | 0.513 | 0.048 | 0.034 | 0.009 |

ERC | |||||

DCC | 1.000 | 1.000 | 1.000 | 0.221 | 1.000 |

SMPL | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

EWMA | <0.0001 | <0.0001 | 0.084 | 1.000 | <0.0001 |

MDP | |||||

DCC | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

SMPL | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

EWMA | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | |||||

DCC | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

SMPL | 0.004 | <0.0001 | 0.0001 | 0.002 | <0.0001 |

EWMA | 0.090 | <0.0001 | 0.001 | 0.321 | <0.0001 |

HRP (AL) | |||||

DCC | 1.000 | 1.000 | 1.000 | 0.093 | 1.000 |

SMPL | 0.083 | 0.001 | <0.0001 | 1.000 | <0.0001 |

EWMA | 0.122 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (Ward) | |||||

DCC | 1.000 | 1.000 | 1.000 | 0.467 | 1.000 |

SMPL | 0.061 | 0.055 | <0.0001 | 0.141 | <0.0001 |

EWMA | 0.085 | 0.121 | 0.003 | 1.000 | <0.0001 |

**Table 4.**${p}^{c}$-values for different portfolio allocation benchmark models considered when DCC-GARCH is used for estimating the covariance matrix and the portfolio is rebalanced daily. The highlighted cells are outcomes for variants of HRP.

Realized Portfolio Variance with DCC-GARCH | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | <0.0001 | 1.0000 | <0.0001 | <0.0001 |

ERC | <0.0001 | 0.0030 | <0.0001 | 0.0450 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.0100 | <0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 1.0000 | 0.0570 | 0.2020 | 0.2970 | 0.1460 |

HRP (AL) | 0.0130 | 1.0000 | 0.2320 | 1.0000 | 0.0470 |

HRP (Ward) | 0.0440 | <0.0001 | 0.0670 | 0.4760 | 1.0000 |

Realized Portfolio Variance with SMPL | |||||

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.587 | 1.000 | <0.0001 | 0.2400 |

ERC | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 1.0000 | 0.0180 | <0.0001 | <0.0001 | 0.1680 |

HRP (AL) | 0.1890 | 1.000 | <0.0001 | 1.0000 | 0.0020 |

HRP (Ward) | 0.0560 | 0.1200 | <0.0001 | 0.006 | 1.0000 |

**Table 5.**The out-of-sample realized annual portfolio volatilities for different covariance forecast models when they were rebalanced daily.

Annual Portfolio Volatility with DCC GARCH | ||||||||

HRP | HRP | HRP | ||||||

Universe | MVP | IVWP | ERC | MDP | MWP | (SL) | (AL) | (Ward) |

1 | 11.08% | 11.31% | 10.77% | 10.84% | 17.52% | 10.36% | 10.58% | 10.49% |

2 | 9.70% | 9.39% | 9.26% | 9.78% | 10.60% | 9.26% | 9.21% | 9.45% |

3 | 11.05% | 11.45% | 11.40% | 11.47% | 16.21% | 11.13% | 11.03% | 11.22% |

4 | 11.78% | 11.86% | 11.82% | 11.86% | 13.00% | 11.76% | 11.65% | 11.54% |

5 | 9.54% | 9.83% | 9.93% | 10.62% | 11.56% | 9.36% | 9.47% | 9.46% |

Annual Portfolio Volatility with SMPL | ||||||||

1 | 11.34% | 11.64% | 11.09% | 11.58% | 17.52% | 10.31% | 10.36% | 10.38% |

2 | 10.26% | 9.42% | 9.41% | 10.44% | 10.60% | 9.48% | 9.52% | 9.59% |

3 | 11.26% | 11.47% | 11.46% | 11.60% | 16.21% | 11.25% | 11.19% | 11.40% |

4 | 11.56% | 11.80% | 11.78% | 11.89% | 13.00% | 11.59% | 11.54% | 11.44% |

5 | 9.87% | 9.95% | 10.17% | 11.43% | 11.56% | 9.77% | 9.88% | 9.74% |

**Table 6.**${p}^{c}$-values for different portfolio allocation benchmark models considered when weekly expected shortfall is taken as the loss function and the portfolio is rebalanced daily. The highlighted cells are outcomes for variants of HRP.

Realized 5-Day CVaR with DCC-GARCH | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | 0.0160 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.0870 | 1.0000 | <0.0001 | <0.0001 |

ERC | <0.0001 | 0.2170 | 0.8150 | <0.0001 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | 0.0310 | <0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 1.0000 | 0.0960 | 0.1510 | 0.1760 | <0.0001 |

HRP (AL) | <0.0001 | 1.0000 | 0.1260 | 1.0000 | <0.0001 |

HRP (Ward) | <0.0001 | 0.0290 | <0.0001 | 0.3270 | 1.0000 |

Realized 5-day CVaR with SMPL | |||||

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | 0.0860 | 1.000 | <0.0001 | 1.000 |

ERC | <0.0001 | <0.0001 | 0.1470 | <0.0001 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

MWP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP (SL) | 0.357 | <0.0001 | <0.0001 | 0.005 | 0.040 |

HRP (AL) | 1.000 | <0.0001 | <0.0001 | 1.0000 | <0.0001 |

HRP (Ward) | 0.325 | 1.000 | <0.0001 | 0.234 | 0.493 |

**Table 7.**${p}^{c}$-values for different choices of portfolio allocation benchmark models when DCC-GARCH is used for estimating the covariance matrix and the portfolio is rebalanced daily.

Realized Diversification Ratio with DCC-GARCH | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | <0.0001 | <0.0001 | 1.0000 | <0.0001 | 1.0000 |

ERC | 1.0000 | 1.0000 | <0.0001 | 1.0000 | 0.0210 |

MDP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

Realized${\mathit{H}}^{*}(\%\mathit{RC})$with DCC-GARCH | |||||

MVP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

IVWP | 1.0000 | 0.1120 | 1.0000 | 0.2220 | 1.0000 |

ERC | 0.0220 | 1.0000 | <0.0001 | 1.0000 | <0.0001 |

MDP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

HRP | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

**Table 8.**${p}^{c}$-value for different choices of benchmark models when the loss function considered is negative of weekly Sharpe ratio and the portfolio is rebalanced daily.

Realized Weekly Sharpe Ratio with DCC GARCH | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | 1.000 | 1.000 | 0.692 | 0.074 | 0.011 |

IVWP | 0.043 | 0.065 | 0.145 | 1.000 | 0.080 |

ERC | 0.013 | 0.004 | 0.166 | 0.854 | 0.105 |

MDP | 0.001 | <0.0001 | 1.000 | 0.771 | 0.095 |

MWP | 0.011 | 0.635 | 0.103 | 0.189 | 1.000 |

HRP (SL) | 0.122 | 0.226 | 0.192 | 0.467 | 0.060 |

HRP (AL) | 0.044 | 0.755 | 0.045 | 0.137 | 0.080 |

HRP (Ward) | 0.073 | 0.090 | 0.117 | 0.297 | 0.079 |

Realized weekly Sharpe Ratio with SMPL | |||||

MVP | 1.000 | 0.003 | 0.002 | 0.062 | 0.002 |

IVWP | 0.125 | 0.001 | 0.122 | 0.617 | 0.033 |

ERC | 0.005 | <0.0001 | 0.150 | 0.804 | 0.043 |

MDP | 0.003 | <0.0001 | 1.000 | 1.000 | 0.045 |

MWP | 0.156 | 1.000 | 0.158 | 0.192 | 1.000 |

HRP (SL) | 0.403 | <0.0001 | 0.031 | 0.076 | 0.009 |

HRP (AL) | 0.497 | <0.0001 | 0.018 | 0.003 | 0.008 |

HRP (Ward) | 0.356 | <0.0001 | 0.015 | 0.001 | <0.0001 |

**Table 9.**${p}^{c}$-value for different choices of benchmark models when the loss function considered is negative of weekly Sharpe ratio and different rebalancing horizons are considered. The covariance matrix forecast is made using DCC-GARCH.

Realized Weekly Sharpe Ratio with Weekly Rebalancing | |||||

Benchmark Method | Univ 1 | Univ 2 | Univ 3 | Univ 4 | Univ 5 |

MVP | 1.000 | 1.000 | 1.000 | 1.000 | 0.606 |

IVWP | 0.027 | 0.102 | 0.004 | 0.592 | 0.174 |

ERC | 0.007 | 0.056 | 0.003 | 0.743 | 0.206 |

MDP | <0.0001 | 0.004 | 0.022 | 0.551 | 0.093 |

MWP | 0.001 | 0.242 | 0.015 | 0.081 | 1.000 |

HRP (SL) | 0.046 | 0.054 | 0.002 | 0.697 | 0.017 |

HRP (AL) | 0.028 | 0.083 | <0.0001 | 0.798 | 0.023 |

HRP (Ward) | 0.026 | 0.063 | 0.001 | 0.507 | 0.096 |

Realized Weekly Sharpe Ratio with Monthly Rebalancing | |||||

MVP | 1.000 | 1.000 | 1.000 | 1.000 | 0.508 |

IVWP | 0.002 | 0.014 | 0.006 | 0.331 | 0.122 |

ERC | 0.001 | 0.004 | 0.001 | 0.356 | 0.065 |

MDP | 0.000 | 0.000 | 0.005 | 0.227 | 0.015 |

MWP | 0.002 | 0.071 | 0.010 | 0.047 | 1.000 |

HRP (SL) | 0.004 | 0.001 | 0.003 | 0.189 | 0.018 |

HRP (AL) | 0.002 | 0.002 | 0.002 | 0.174 | 0.024 |

HRP (Ward) | 0.002 | 0.000 | 0.018 | 0.119 | 0.251 |

**Table 10.**The out-of-sample realized annual Sharpe ratio for the portfolios considered when they were rebalanced daily.

Annual Sharpe Ratio with DCC GARCH | ||||||||

HRP | HRP | HRP | ||||||

Universe | MVP | IVWP | ERC | MDP | MWP | (SL) | (AL) | (Ward) |

1 | 2.382 | 2.476 | 2.485 | 2.007 | 1.768 | 2.521 | 2.241 | 2.431 |

2 | 2.384 | 2.935 | 2.839 | 2.365 | 2.860 | 2.747 | 2.813 | 2.611 |

3 | 1.267 | 2.080 | 2.041 | 1.972 | 0.696 | 1.714 | 1.653 | 1.684 |

4 | 0.151 | 1.728 | 1.672 | 1.401 | 1.553 | 1.376 | 1.233 | 1.243 |

5 | 1.315 | 2.384 | 2.301 | 2.020 | 2.659 | 2.011 | 2.038 | 2.034 |

Annual Sharpe Ratio with SMPL | ||||||||

1 | 1.012 | 2.171 | 2.174 | 1.738 | 1.768 | 2.007 | 2.043 | 1.996 |

2 | 0.781 | 2.680 | 2.503 | 1.879 | 2.860 | 2.018 | 2.125 | 2.114 |

3 | 0.493 | 1.907 | 1.861 | 1.796 | 0.696 | 1.317 | 1.278 | 1.352 |

4 | 0.176 | 1.716 | 1.664 | 1.375 | 1.553 | 1.265 | 1.332 | 1.281 |

5 | 0.558 | 2.246 | 2.115 | 1.679 | 2.659 | 1.659 | 1.644 | 1.638 |

**Table 11.**The out-of-sample realized annual Sharpe ratio for the portfolios considered when they were rebalanced daily.

Annual Sharpe Ratio with Weekly Rebalancing | ||||||||

HRP | HRP | HRP | ||||||

Universe | MVP | IVWP | ERC | MDP | MWP | (SL) | (AL) | (Ward) |

1 | 3.156 | 2.720 | 2.752 | 2.401 | 1.662 | 2.658 | 2.587 | 2.591 |

2 | 3.591 | 3.188 | 3.076 | 2.557 | 2.659 | 2.925 | 3.047 | 2.906 |

3 | 2.333 | 2.343 | 2.309 | 2.247 | 0.503 | 2.068 | 2.039 | 2.079 |

4 | 1.789 | 2.106 | 2.046 | 1.739 | 1.444 | 2.144 | 2.153 | 2.015 |

5 | 2.741 | 2.737 | 2.633 | 2.258 | 2.511 | 2.663 | 2.730 | 2.892 |

Annual Sharpe Ratio with Monthly Rebalancing | ||||||||

1 | 3.054 | 2.754 | 2.767 | 2.266 | 1.711 | 2.782 | 2.878 | 2.614 |

2 | 3.494 | 3.194 | 3.069 | 2.489 | 2.570 | 3.045 | 2.914 | 3.099 |

3 | 2.384 | 2.319 | 2.277 | 2.197 | 0.274 | 2.087 | 2.083 | 2.177 |

4 | 2.055 | 2.089 | 2.034 | 1.753 | 1.266 | 2.198 | 1.995 | 1.979 |

5 | 2.783 | 2.684 | 2.545 | 2.053 | 2.389 | 2.752 | 2.752 | 2.750 |

**Table 12.**The annual Sharpe ratio for the different allocation methods when the portfolio is rebalanced monthly and SMPL is used to forecast the covariance matrix.

Annual Sharpe Ratio with Monthly Rebalancing Using SMPL | ||||||||
---|---|---|---|---|---|---|---|---|

Universe | MVP | IVWP | ERC | MDP | MWP | HRP (SL) | HRP (AL) | HRP (Ward) |

1 | 2.431 | 2.526 | 2.532 | 2.057 | 1.711 | 2.826 | 2.726 | 2.291 |

2 | 2.489 | 3.041 | 2.861 | 2.175 | 2.570 | 2.808 | 2.910 | 2.853 |

3 | 2.057 | 2.248 | 2.207 | 2.155 | 0.274 | 2.054 | 2.051 | 2.090 |

4 | 1.978 | 2.065 | 2.019 | 1.747 | 1.266 | 2.086 | 2.078 | 2.064 |

5 | 2.368 | 2.637 | 2.489 | 1.978 | 2.389 | 2.493 | 2.487 | 2.621 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jain, P.; Jain, S.
Can Machine Learning-Based Portfolios Outperform Traditional Risk-Based Portfolios? The Need to Account for Covariance Misspecification. *Risks* **2019**, *7*, 74.
https://doi.org/10.3390/risks7030074

**AMA Style**

Jain P, Jain S.
Can Machine Learning-Based Portfolios Outperform Traditional Risk-Based Portfolios? The Need to Account for Covariance Misspecification. *Risks*. 2019; 7(3):74.
https://doi.org/10.3390/risks7030074

**Chicago/Turabian Style**

Jain, Prayut, and Shashi Jain.
2019. "Can Machine Learning-Based Portfolios Outperform Traditional Risk-Based Portfolios? The Need to Account for Covariance Misspecification" *Risks* 7, no. 3: 74.
https://doi.org/10.3390/risks7030074