# Smart Beta Allocation and Macroeconomic Variables: The Impact of COVID-19

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## Abstract

**:**

## 1. Introduction

- gaining exposure to factors that potentially provide excess returns (smart factor strategy); and
- diversifying exposure across factors to potentially reduce overall volatility (smart weighted strategy).

- (i)
- Size premium: small capitalization stocks tend to outperform large capitalization stocks. This evidence was first encountered by Banz (1981) and later confirmed by Fama and French (1992).
- (ii)
- Value premium: a security is considered valuable if it has a low market price when compared to some measure of the fundamental value of the underlying company. It was first considered by Basu (1977), and Fama and French (2012) also found the same occurrence in markets outside the United States.
- (iii)
- Momentum premium: stocks that have outperformed in recent months (1 to 12 mos.) tend to show higher returns even in the subsequent time interval (see Asness et al. 2018).
- (iv)
- Volatility premium: the reward for bearing an asset’s risk. Multi-factor models argue that higher exposure to factors with excess returns (higher β) implies a higher risk premium.
- (v)
- Investment and profitability/quality premium: Profitability measures are based on fundamental values directly tied to the profitability of the company, for example return on equity (Fama and French 2006) and gross profits compared to accounting activities (Novy-Marx 2013), or indicators that consider financial stability and debt ratios. These measures show a positive correlation with net expected return of the size, value, and momentum effects. We can refer to the profitability factor in terms of the quality factor; likewise, some measures that capture the effect of investments such as the growth of capital expenditure (Xing 2008) and total assets (Fama and French 2006; Hou et al. 2015) show the same relationship with net expected returns.
- (vi)
- Dividend premium: Historically, stocks with a high dividend have outperformed the market by about 1.5% per year (evidence from 1927 to 2015). This factor describes net excess returns of traditional factors, with even higher returns in emerging markets. However, this premium presents a series of risks tied, for example, to temporary high profits, high payout ratios, or lower future prices.
- (vii)
- Illiquidity premium: Less liquid stocks are traded at lower prices and offer higher expected returns than more liquid ones. This premium is tied to the greater risk of holding an asset that is more difficult to convert into liquidity and to the possibility of an outflow during a liquidity crisis period. The empirical evidence for this effect is not so extensive, but it does seem to be confirmed.

## 2. Methodology

- ${\mathcal{P}}_{t-1}-{\mathcal{P}}_{t-2}>\gamma {\mathcal{P}}_{t-2}$, where ${\mathcal{P}}_{\tau}={\displaystyle \sum}_{i=1}^{N}{w}_{\tau ,i}{p}_{\tau ,i}$ is the value of the portfolio at time $\tau $. The arbitrary coefficient γ indicates the percentage of profit that would induce a manager to liquidate the portfolio at time $t-1$ in order to reallocate it at time t by solving problem (2)–(4) with a budget ${b}_{t}={\mathcal{P}}_{t-1}.$ If this situation occurs, winning is capitalized and the portfolio weights are updated. In the analysis, we set γ equal to 0.05 since a monthly gain of 5% seems to be high enough to justify a portfolio update, but in general the threshold value of γ is fixed according to the investor’s return expectations: the more an investor is looking for a high yield, the more he/she waits to capitalize the winnings. In other words, a higher threshold implies less frequent portfolio weight updates.
- ${\mathcal{P}}_{t-1}-{\mathcal{P}}_{t-2}<\nu {\mathcal{P}}_{t-2}$, where ν is an arbitrary coefficient indicating the percentage of loss that would induce the manager to liquidate the portfolio and invest in a new portfolio obtained by solving problem (2)–(4). The quantity ν in this analysis is equal to 0.01. As in the case of γ, a higher value of ν highlights a greater willingness of investors to suffer losses and wait for the weights to be updated. In contrast to the previous case, loss is capitalized and the portfolio weight is updated.

## 3. Data

- iShares EDGE MSCI Min Vol USA. This ETF replicates the MSCI USA minimum volatility index, which considers a set of stocks with lower volatility characteristics than the entire US stock market.
- iShares EDGE MSCI USA Momentum Factor. This replicates the MSCI USA momentum index, which allows exposures to stocks with higher prices in the previous time period (6–12 months).
- iShares EDGE MSCI USA Quality Factor. By replicating the performance of the MSCI USA sector neutral quality index, this ETF invests in a portfolio of securities showing fundamental measures that are qualitatively better than the others (for example, a high ROE (Return on equity) or low leverage).
- iShares EDGE MSCI USA Value Factor. This replicates the performance of the MSCI USA enhanced value index, investing in companies undervalued with respect to certain multiples.
- iShares EDGE MSCI USA Size Factor. The reference benchmark is the MSCI USA low size index, which measures the performance of US large and mid-capitalization stocks with relatively smaller average market capitalization.
- iShares Select Dividend. The goal of this ETF is to replicate the Dow Jones US selected dividend index with the aim of being exposed to a group of stocks of companies that have a high dividend-price ratio.

- Consumer price index for all urban consumers (CPI). This measures the average change in prices paid by consumers for a basket of consumer goods and services. It represents the main measure of inflation and is used as a basis for formulating monetary policy interventions and measuring the effectiveness of these measures.
- Real gross domestic product (GDP). This is the typical indicator of the volume of economic activity. It influences the decisions of all economic agents, from policy makers to individuals.
- Effective federal funds rate (FED rate for short). This represents the interest rate at which overnight transactions on federal deposits between financial institutions take place. This rate is a key lever for central banks when implementing decisions about monetary policy.
- CFE (CBOE Futures Exchange)–VIX Index (VIX). This financial index aims to provide a real-time estimate of the expected volatility on the S&P (Standard and Poor) 500 index in the following 30 days and consequently reflects the expectations of investors about the US stock market as a whole.

## 4. Discussion and Results

#### 4.1. Correlation Analysis

#### 4.2. Portfolio Analysis

#### 4.3. Forecasting via a Linear Discriminant Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Details on the results of auto.arima() corresponding to model (9)–(10) in Table 3.

Panel A—January 2014–May 2019 | |||

Smart Beta Price Ratio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (1,0,0) | 0.0364 | 5 |

Mom. ETF | (0,0,1) | 0.6049 | 5 |

Qual. ETF | (0,0,1) | 0.1977 | 5 |

Value ETF | (0,0,1) | 0.3296 | 6 |

Size ETF | (0,0,1) | 0.3894 | 5 |

Div. ETF | (1,0,0) | 0.5892 | 5 |

Smart Beta Log-Return | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (1,0,0) | 0.0410 | 5 |

Mom. ETF | (0,0,1) | 0.6493 | 5 |

Qual. ETF | (0,0,1) | 0.1871 | 5 |

Value ETF | (0,0,1) | 0.2796 | 6 |

Size ETF | (1,0,0) | 0.4782 | 5 |

Div. ETF | (1,0,0) | 0.6235 | 5 |

Panel B—January 2014–May 2020 | |||

Smart Beta Price Ratio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (1,0,0) | 0.4017 | 6 |

Mom. ETF | (2,0,0) | 0.3724 | 6 |

Qual. ETF | (0,0,1) | 0.3802 | 5 |

Value ETF | (1,0,1) | 0.3022 | 6 |

Size ETF | (1,0,0) | 0.5546 | 5 |

Div. ETF | (1,0,0) | 0.5726 | 5 |

Smart Beta Return | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (1,0,0) | 0.5089 | 5 |

Mom. ETF | (2,0,0) | 0.3838 | 6 |

Qual. ETF | (0,0,1) | 0.3846 | 5 |

Value ETF | (1,0,1) | 0.2392 | 7 |

Size ETF | (1,0,0) | 0.5426 | 5 |

Div. ETF | (1,0,0) | 0.5016 | 5 |

**Table A2.**Details on the results of auto.arima() corresponding to model (11)–(12) in Table 10.

Panel A—January 2014–May 2019 | |||

Model (11) for α = 0.1 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (1,0,0) | 0.3616 | 5 |

Financials | (0,0,0) | 0.1370 | 4 |

Model (11) for α = 0.9 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (1,0,0) | 0.1925 | 5 |

Financials | (0,0,0) | 0.6422 | 4 |

Panel B—January 2014–May 2020 | |||

Model (11) for α = 0.1 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (0,0,1) | 0.1889 | 5 |

Financials | (0,0,0) | 0.0960 | 4 |

Model (11) for α = 0.9 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (2,0,0) | 0.1369 | 4 |

Financials | (0,0,0) | 0.6675 | 5 |

**Table A3.**Results of model (A1)–(A2) on smart beta. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 20208.

Panel A | ||||||

Model (A1)–(A2) for ETF price ratio | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 148.43 | −284.86 (0.026) | 4.03 × 10^{−1} (*) | −1.60 × 10^{−3} (-) | 7.30 × 10^{−3} (-) | −2.10 × 10^{−3} (*) |

Mom. ETF | 130.57 | −249.14 (0.034) | 1.38 × 10^{−1} (-) | 3.10 × 10^{−3} (-) | −2.56 × 10^{−2} (-) | −3.00 × 10^{−3} (*) |

Qual. ETF | 135.8 | −259.61 (0.031) | 1.52 × 10^{−1} (-) | 2.80 × 10^{−3} (-) | −2.19 × 10^{−2} (-) | −3.00 × 10^{−3} (**) |

Value ETF | 137.01 | −258.02 (0.031) | 10.77 × 10^{−1} (-) | 2.50 × 10^{−3} (-) | −2.21 × 10^{−2} (-) | −4.00 × 10^{−3} (***) |

Size ETF | 141.24 | −266.48 (0.029) | −5.20 × 10^{−3} (-) | −1.50 × 10^{−3} (-) | 1.24 × 10^{−2} (-) | −2.40 × 10^{−3} (**) |

Div. ETF | 147.95 | −283.9 (0.026) | 3.22 × 10^{−1} (.) | −3.00 × 10^{−4} (-) | −4.70 × 10^{−3} (-) | −2.70 × 10^{−3} (**) |

Panel B | ||||||

Model (A1)–(A2) for ETF price ratio | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 171.49 | −326.98 (0.027) | −4.44 × 10^{−1} (-) | 7.0 × 10^{−4} (-) | 5.70 × 10^{−3} (-) | 2.50 × 10^{−3} (***) |

Mom. ETF | 154.89 | −293.78 (0.034) | 1.57 × 10^{−1} (**) | 1.30 × 10^{−3} (-) | −1.31 × 10^{−2} (.) | −2.60 × 10^{−3} (***) |

Qual. ETF | 159.13 | −304.27 (0.032) | 1.98 × 10^{−1} (***) | 6.00 × 10^{−4} (-) | −7.30 × 10^{−3} (**) | −2.00 × 10^{−3} (**) |

Value ETF | 156.32 | −298.65 (0.033) | 1.96 × 10^{−1} (***) | 1.20 × 10^{−3} (***) | −1.46 × 10^{−2} (***) | −4.00 × 10^{−3} (***) |

Size ETF | 156.5 | −299.01 (0.033) | 2.36 × 10^{−1} (***) | 7.00 × 10^{−4} (-) | −9.80 × 10^{−3} (-) | −3.70 × 10^{−3} (***) |

Div. ETF | 163.03 | −312.06 (0.030) | 2.79 × 10^{−1} (***) | −2.00 × 10^{−4} (-) | −3.90 × 10^{−3} (-) | −3.50 × 10^{−3} (***) |

**Table A4.**Results of model (A1)–(A2) (Panels A and B) applied to smart beta and financial ETF portfolios. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 20209.

Panel A—January 2014–May 2019 | ||||||

Model (A1)–(A2) for α = 0.1 | ||||||

Portfolio | ML | AIC (Res. SE) | GDP | CPI | FED | VIX |

Smart betas | 126.93 | −241.87 (0.036) | −4.56 × 10^{−2} (-) | 6.70 × 10^{−3} (-) | −5.12 × 10^{−2} (.) | −2.70 × 10^{−3} (*) |

Financials | 37.17 | −62.34 (0.142) | 3.52 × 10^{−1} (-) | −1.64 × 10^{−3} (-) | 1.60 × 10^{−2} (-) | −6.03 × 10^{−3} (-) |

Model (A1)–(A2) for α = 0.9 | ||||||

Portfolio | ML | AIC (Res. SE) | GDP | CPI | FED | VIX |

Smart betas | 120.05 | −228.1 (0.040) | 7.69 × 10^{−1} (-) | 4.70 × 10^{−3} (-) | −4.01 × 10^{−2} (-) | −2.40 × 10^{−3} (.) |

Financials | −45.15 | 102.34 (0.504) | 8.82 × 10^{−1} (*) | −1.54 × 10^{−1} (*) | 1.13 × 10^{−1} (*) | 2.03 × 10^{−2} (-) |

Panel B—January 2014–May 2020 | ||||||

Model (A1)–(A2) for α = 0.1 | ||||||

Portfolio | ML | AIC (SE) | GDP | CPI | FED | VIX |

Smart betas | 145.46 | −274.92 (0.038) | 8.12 × 10^{−2} (*) | 1.30 × 10^{−3} (***) | −1.32 × 10^{−2} (***) | −1.60 × 10^{−3} (***) |

Financials | 49.24 | −86.47 (0.132) | 1.18 × 10^{−1} (-) | 2.62 × 10^{−3} (-) | −1.83 × 10^{−2} (-) | −5.84 × 10^{−3} (*) |

Model (A1)–(A2) for α = 0.9 | ||||||

Portfolio | ML | AIC (SE) | GDP | CPI | FED | VIX |

Smart betas | 136.24 | −256.47 (0.043) | 1.29 × 10^{−1} (*) | 1.30 × 10^{−3} (-) | −1.52 × 10^{−2} (.) | −1.70 × 10^{−3} (**) |

Financials | −85.57 | 187.14 (0.770) | 34.98 × 10^{−1} (.) | −6.44 × 10^{−2} (.) | −1.41 × 10^{−1} (-) | 1.62 × 10^{−1} (***) |

**Table A5.**Details on the results of auto.arima() corresponding to model (A1)–(A2) in Table A3.

Panel A—January 2014–May 2019 | |||

Smart Beta Price Ratio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (0,0,0) | 0.0309 | 5 |

Mom. ETF | (0,0,0) | 0.4397 | 5 |

Qual. ETF | (0,0,0) | 0.2523 | 5 |

Value ETF | (2,0,0,) | 0.4196 | 7 |

Size ETF | (0,0,1) | 0.1295 | 7 |

Div. ETF | (0,0,0) | 0.5715 | 5 |

Panel B—January 2014–May 2020 | |||

Smart Beta Price Ratio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Min. Vol. ETF | (1,0,0) | 0.2608 | 7 |

Mom. ETF | (2,0,0) | 0.2378 | 7 |

Qual. ETF | (0,0,1) | 0.3344 | 6 |

Value ETF | (0,0,1) | 0.1484 | 6 |

Size ETF | (1,0,0) | 0.4414 | 6 |

Div. ETF | (1,0,0) | 0.3968 | 6 |

**Table A6.**Details on the results of auto.arima() corresponding to model (A1)–(A2) in Table A4.

Panel A—January 2014–May 2019 | |||

Model (A1)–(A2) for α = 0.1 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (0,0,0) | 0.2593 | 5 |

Financials | (0,0,0) | 0.0804 | 5 |

Model (A1)–(A2) for α = 0.9 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (0,0,0) | 0.1647 | 5 |

Financials | (0,0,0) | 0.5326 | 5 |

Panel B—January 2014–May 2020 | |||

Model (A1)–(A2) for α = 0.1 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (1,0,1) | 0.1078 | 7 |

Financials | (0,0,0) | 0.0493 | 5 |

Model (A1)–(A2) for α = 0.9 | |||

Portfolio | Arima (p,d,q) | p-Value | Model Degree of Freedom |

Smart betas | (2,0,0) | 0.0856 | 7 |

Financials | (0,0,1) | 0.1614 | 7 |

## Appendix B

**Table A7.**Correlation between optimal weights of smart beta (a) and ETFs (b) and macroeconomic variables for α = 0. Significant correlations (5% significant level) are highlighted by gray color.

(a) | (b) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Min. Vol | Mom. | Qual. | Value | Size | Div. | DIA | IHI | IXG | IYF | IYG | SPY | ||

GDP | 0.597 | −0.2329 | 0.1596 | −0.3022 | 0.4931 | −0.0262 | GDP | 0.0782 | −0.3311 | 0.5027 | 0.1839 | 0.4329 | −0.0912 |

VIX | 0.5253 | −0.1325 | 0.0072 | −0.0432 | 0.2198 | −0.0412 | VIX | 0.0106 | 0.0955 | −0.1065 | −0.2016 | −0.3073 | −0.1219 |

CPI | 0.8726 | −0.3416 | 0.1522 | −0.3433 | 0.521 | 0.0678 | CPI | −0.0612 | −0.2061 | 0.3833 | 0.0341 | 0.2382 | −0.0492 |

FED | 0.557 | −0.2653 | 0.0459 | −0.3073 | 0.5318 | 0.121 | FED | −0.0267 | −0.1767 | 0.4603 | 0.1304 | 0.3213 | −0.0817 |

**Table A8.**Correlation between optimal weights of smart beta (a) and ETFs (b) and macroeconomic variables for α = 0.9. Significant correlations (5% significant level) are highlighted by gray color.

(a) | (b) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Min.Vol | Mom. | Qual. | Value | Size | Div. | DIA | IHI | IXG | IYF | IYG | SPY | ||

GDP | 0.1475 | −0.5254 | −0.2013 | 0.2965 | 0.317 | −0.432 | GDP | 0.2851 | −0.3889 | −0.0453 | −0.0493 | 0.0127 | −0.1147 |

VIX | 0.0092 | −0.0707 | −0.0893 | 0.2619 | −0.0239 | −0.1293 | VIX | −0.0706 | −0.1148 | 0.3874 | −0.0259 | −0.1987 | −0.1485 |

CPI | 0.1146 | −0.5371 | −0.2803 | 0.4127 | 0.2864 | −0.3635 | CPI | 0.2608 | −0.2706 | −0.0303 | −0.0924 | −0.0503 | −0.2126 |

FED | 0.1463 | −0.4599 | −0.2523 | 0.2099 | 0.4195 | −0.3926 | FED | 0.3131 | −0.2782 | −0.0417 | −0.0827 | −0.0761 | −0.1443 |

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1 | Transaction costs of 1% are considered only when the monthly rehedging occurs. |

2 | Tables in the paper consider the period January 2014–May 2020; two periods are considered when there are differences to be highlighted and they are respectively January 2014–May 2019 and January 2014–May 2020. Tables referring to the first period are shown in the supplementary material of the paper. |

3 | Significance codes: p-value ≤ 0.001 (***); (**) 0.001 < p-value ≤ 0.01; (*) 0.01 < p-value ≤ 0.05; (·) 0.05 < p-value ≤ 0.1; (-) 0.1 < p-value ≤ 1. |

4 | |

5 | The ticker symbols of financial ETFs considered in the analysis are DIA, IXG, IHI, IYF, IYG, and SPY. |

6 | Significance codes: p-value ≤ 0.001 (***); (**) 0.001 < p-value ≤ 0.01; (*) 0.01 < p-value ≤ 0.05; (·) 0.05 < p-value ≤ 0.1; (-) 0.1 < p-value ≤ 1. |

7 | In Figure 6, the symbols represent the observed tilting variable (rehedging-triangle or not rehedging-circle) according to the LDA prediction at time t − 2, which provides the forecast of the tilting variable at t −1. The accuracy of this prediction is assessed at time t. |

8 | Significance codes: p-value ≤ 0.001 (***); (**) 0.001 < p-value ≤ 0.01; (*) 0.01 < p-value ≤ 0.05; (·) 0.05 < p-value ≤ 0.1;(-) 0.1 < p-value ≤ 1. |

9 | Signif. codes: p-value ≤ 0.001 (***); (**) 0.001 < p-value ≤ 0.01; (*) 0.01 < p-value ≤ 0.05; (·) 0.05 < p-value ≤ 0.1; (-) 0.1 < p-value ≤ 1. |

**Figure 1.**Monthly net asset value (NAV) in US dollars (

**a**) and log returns (

**b**) of smart beta products from January 2014 to May 2020. Source: Datastream.

**Figure 2.**Monthly data as a function of the time period January 2014–May 2020: FED (effective federal funds rate) and VIX (volatility index) (percent) gross domestic product (GDP) and consumer price index (CPI) (US dollars). Source: Datastream.

**Figure 3.**Portfolio optimal weights for α equal to 0.1 (

**a**), 0.5 (

**b**), 0.9 (

**c**), and portfolio performances (budget expressed in US dollars) for all strategies considered (

**d**) from January 2014 to May 2020. Source: our elaboration on data.

**Figure 4.**Bivariate scatter plots below the diagonal, histograms on the diagonal, and the Pearson correlation above the diagonal. Points represent the grouping variables G (red), L (black), and N (white). Source: our elaboration on data. Period: January 2014–May 2020. p-value ≤ 0.001 (***).

**Figure 5.**Box plots of smart beta returns for different tilting investment variables (G, L, N), gain threshold values (from top to bottom: 2.5%, 5%, and 7.5%), and values of alpha (from left to right: 0.1, 0.5, 0.9). Source: our elaboration on data. (**) 0.001 < p-value ≤ 0.01; (*) 0.01 < p-value ≤ 0.05.

**Figure 6.**The solid line shows the true value of the portfolio budget while upper and lower bounds are computed as 5% and 1% of the portfolio budget at time t. Upper panel α = 0.1, middle panel α = 0.5, and bottom panel α = 0.9. Source: our computation on data. Period: January 2014–May 2020.

**Table 1.**Exchange traded funds (ETF) return summary statistics. Summary statistics for exchange traded funds: mean, standard deviation, excess of kurtosis, and skewness. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 2020.

Panel A | |||||

ETF Name | Benchmark | Mean | SD | Kurt. | Skew. |

1 ISHARES EDGE MSCI MIN VOL USA | MSCI USA Minimum Volatility Index | 0.008 | 0.027 | 0.83 | −0.54 |

2 ISHARES EDGE MSCI USA MOMENTUM FACTOR | MSCI USA Momentum Index | 0.01 | 0.036 | 1.62 | −0.87 |

3 ISHARES EDGE MSCI USA QUALITY FACTOR | MSCI USA Sector Neutral Quality Index | 0.007 | 0.034 | 0.94 | −0.51 |

4 ISHARES EDGE MSCI USA VALUE FACTOR | MSCI USA Enhanced Value Index | 0.005 | 0.039 | 1.29 | −0.62 |

5 ISHARES EDGE MSCI USA SIZE FACTOR | MSCI USA Risk Weighted Index | 0.007 | 0.034 | 1.83 | −0.50 |

6 ISHARES SELECT DIVIDEND | Dow Jones U.S. Selected Dividend Index | 0.005 | 0.029 | 1.09 | −0.57 |

Panel B | |||||

ETF Name | Benchmark | Mean | SD | Kurt. | Skew. |

1 ISHARES EDGE MSCI MIN VOL USA | MSCI USA Minimum Volatility Index | 0.001 | 0.033 | 2.90 | −1.02 |

2 ISHARES EDGE MSCI USA MOMENTUM FACTOR | MSCI USA Momentum Index | 0.01 | 0.040 | 1.77 | −0.79 |

3 ISHARES EDGE MSCI USA QUALITY FACTOR | MSCI USA Sector Neutral Quality Index | 0.008 | 0.040 | 1.54 | −0.51 |

4 ISHARES EDGE MSCI USA VALUE FACTOR | MSCI USA Enhanced Value Index | 0.004 | 0.047 | 3.25 | −1.15 |

5 ISHARES EDGE MSCI USA SIZE FACTOR | MSCI USA Risk Weighted Index | 0.006 | 0.044 | 4.74 | −1.03 |

6 ISHARES SELECT DIVIDEND | Dow Jones U.S. Selected Dividend Index | 0.003 | 0.040 | 7.99 | −1.94 |

**Table 2.**Summary statistics of macroeconomic and financial time series from January 2014 to May 2020: FED and VIX (percent) GDP and CPI (US dollars). Source: Datastream.

Mean | SD | Kurt. (Excess) | Skew. | |
---|---|---|---|---|

GDP | 17,876.56 | 829.53 | −1.19 | 0.07 |

FED | 0.92 | 0.83 | −1.27 | 0.52 |

CPI | 245.12 | 7.80 | −1.42 | 0.32 |

VIX | 16.15 | 6.83 | 17.72 | 3.71 |

**Table 3.**Results of model (9) and (10) on smart beta. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 20203.

Panel A | ||||||

Model (9)–(10) for ETF Price Ratio | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 150.1 | −288.32 (0.025) | 3.95 × 10^{−1} (**) | −2.67 × 10^{−3} (-) | 1.52 × 10^{−2} (-) | −2.34 × 10^{−3} (**) |

Mom. ETF | 135.1 | −285.13 (0.034) | 3.28 × 10^{−1} (*) | −1.38 × 10^{−3} (-) | 5.26 × 10^{−13} (-) | −3.53 × 10^{−3} (***) |

Qual. ETF | 153.6 | −295.24 (0.034) | 2.77 × 10^{−1} (*) | −5.34 × 10^{−4} (-) | 4.75 × 10^{−4} (-) | −2.91 × 10^{−3} (***) |

Value ETF | 137.5 | −263.08 (0.030) | −1.30 × 10^{−1} (-) | 3.17 × 10^{−3} (-) | −2.34 × 10^{−2} (*) | −3.29 × 10^{−3} (***) |

Size ETF | 140.8 | −269.55 (0.028) | 3.71 × 10^{−1} (**) | −2.20 × 10^{−3} (-) | 1.11 × 10^{−2} (-) | −2.93 × 10^{−3} (***) |

Div. ETF | 148.7 | −285.42 (0.026) | 3.19 × 10^{−1} (*) | −1.30 × 10^{−3} (-) | −3.66 × 10^{−3} (-) | −262 × 10^{−3} (**) |

Model (9)–(10) for ETF log-returns | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 150.51 | −289.03 (0.025) | 1.27 × 10^{−1} (-) | −2.10 × 10^{−3} (-) | 1.56 × 10^{−2} (-) | −2.35 × 10^{−3} (**) |

Mom. ETF | 134.27 | −256.53 (0.032) | 4.49 × 10^{−2} (-) | −5.37 × 10^{−3} (-) | 3.63 × 10^{−3} (-) | 3.60 × 10^{−3} (***) |

Qual. ETF | 138.46 | −264.91 (0.03) | 3.40 × 10^{−3} (-) | 1.51 × 10^{−4} (-) | −8.73 × 10^{−5} (-) | −2.92 × 10^{−3} (***) |

Value ETF | 138.23 | −262.45 (0.03) | −1.73 × 10^{−1} (-) | 3.49 × 10^{−3} (.) | −2.55 × 10^{−2} (*) | −3.30 × 10^{−3} (***) |

Size ETF | 141.13 | −270.26 (0.03) | 8.15 × 10^{−2} (-) | −1.21 × 10^{−3} (-) | 8.96 × 10^{−3} (-) | −3.60 × 10^{−3} (***) |

Div. ETF | 148.84 | −285.68 (0.025) | 5.056 × 10^{−2} (-) | −7.16 × 10^{−4} (-) | 4.03 × 10^{−3} (-) | −3.00 × 10^{−3} (***) |

Panel B | ||||||

Model (9)–(10) for ETF Price Ratio | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 171.5 | −328.91 (0.027) | −4.66 × 10^{−1} (-) | 7.24 × 10^{−4} (-) | 5.59 × 10^{−3} (-) | −2.48 × 10^{−3} (***) |

Mom. ETF | 154.7 | −295.49 (0.033) | 1.74 × 10^{−1} (***) | 1.32 × 10^{−3} (.) | −1.36 × 10^{−2} (*) | −2.24 × 10^{−3} (***) |

Qual. ETF | 159.0 | −306.11 (0.031) | 2.06 × 10^{−1} (***) | 7.09 × 10^{−4} (*) | −8.60 × 10^{−3} (**) | −2.26 × 10^{−3} (***) |

Value ETF | 156.2 | −298.49 (0.033) | 1.78 × 10^{−1} (***) | 1.31 × 10^{−3} (*) | −1.48 × 10^{−2} (**) | −4.01 × 10^{−3} (***) |

Size ETF | 155.9 | −299.98 (0.033) | 2.02 × 10^{−1} (***) | 8.53 × 10^{−4} (-) | −1.04 × 10^{−2} (-) | −3.48 × 10^{−3} (***) |

Div. ETF | 162.8 | −313.71 (0.029) | 2.57 × 10^{−1} (***) | 1.39 × 10^{−4} (-) | −4.41 × 10^{−3} (-) | −3.38 × 10^{−3} (***) |

Model (9)–(10) for ETF log-returns | ||||||

Smart Beta | ML | AIC (SE) | GDP | CPI | FED | VIX |

Min. Vol. ETF | 171.03 | −330.06 (0.027) | −7.03 × 10^{−3} (-) | 3.08 × 10^{−4} (-) | −3.12 × 10^{−4} (-) | −2.43 × 10^{−3} (***) |

Mom. ETF | 155.18 | −296.36 (0.033) | −7.27 × 10^{−2} (.) | 1.53 × 10^{−34} (*) | −1.07 × 10^{−2} (.) | −2.98× 10^{−3} (***) |

Qual. ETF | 160.16 | −308.32 (0.031) | −4.10 × 10^{−2} (-) | 9.21 × 10^{−4} (.) | −5.79 × 10^{−3} (-) | −2.48 × 10^{−3} (***) |

Value ETF | 155.94 | −295.88 (0.033) | 1.31 × 10^{−1} (-) | 1.39 × 10^{−3} (***) | −1.48 × 10^{−2} (**) | −4.41 × 10^{−3} (***) |

Size ETF | 156.96 | −301.91 (0.032) | −3.35 × 10^{−2} (-) | 8.68 × 10^{−4} (-) | −5.85 × 10^{−3} (-) | −3.79 × 10^{−3} (***) |

Div. ETF | 161.61 | −311.22 (0.031) | 2.47 × 10^{−2} (-) | −1.85 × 10^{−4} (-) | 7.20 × 10^{−4} (-) | −3.68 × 10^{−13} (***) |

**Table 4.**Pearson correlation between smart beta returns and macroeconomic time series. Significant correlations (5% significance level) are highlighted in gray. Period: January 2014–May 2020.

Min.Vol | Mom | Qual | Value | Size | Div | |

GDP | −0.04861 | −0.05188 | −4.25 × 10^{−2} | −0.06089 | −0.07953 | −0.08971 |

VIX | −0.30084 | −0.30944 | −0.27677 | −0.41363 | −0.3344 | −0.39969 |

CPI | −0.04145 | −0.01419 | −0.00572 | −0.08593 | −0.05696 | −0.14804 |

FED | −0.01485 | −0.0462 | −0.04122 | −0.06181 | −0.05554 | −0.06757 |

**Table 5.**Correlation between smart beta and VIX for $\alpha $ = 0. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 2020. Highlighted values are those significantly different from zero—Pearson correlation.

Panel A | ||||||

$\mathbf{\psi}$ | MinVol | Mom | Qual | Value | Size | Div |

0.6 | 0.21 | 0.23 | 0.26 | −0.34 | −0.34 | −0.18 |

0.7 | 0.20 | −0.23 | 0.09 | 0.21 | 0.004 | −0.11 |

0.8 | 0.21 | −0.06 | 0.10 | 0.10 | −0.09 | −0.09 |

0.9 | −0.11 | 0.14 | −0.14 | −0.18 | −0.14 | 0.04 |

Panel B | ||||||

$\mathbf{\psi}$ | MinVol | Mom | Qual | Value | Size | Div |

0.6 | −0.047 | 0.011 | 0.1803 | −0.2054 | −0.2808 | 0.122 |

0.7 | 0.2435 | −0.095 | 0.1768 | 0.0426 | 0.0563 | −0.088 |

0.8 | 0.5037 | −0.1706 | 0.1625 | −0.0475 | 0.1255 | −0.0362 |

0.9 | 0.0681 | 0.0871 | −0.0489 | −0.2188 | −0.198 | 0.0324 |

**Table 6.**Correlation between smart beta and VIX for all values of α. Highlighted values are those significantly different from zero—Pearson correlation. Period: January 2014–May 2020.

Alpha | Min Vol | Mom | Qual | Value | Size | Div |
---|---|---|---|---|---|---|

0 | 0.5037 | −0.1706 | 0.1625 | −0.0475 | 0.1255 | −0.0362 |

0.1 | 0.5253 | −0.1325 | 0.0072 | −0.0432 | 0.2198 | −0.0412 |

0.2 | 0.5186 | −0.136 | 0.0349 | −0.0588 | 0.2617 | −0.042 |

0.3 | 0.5483 | −0.2651 | 0.061 | −0.0126 | 0.4809 | −0.04 |

0.4 | 0.5241 | −0.2562 | 0.06 | −0.1881 | 0.4376 | 0.1365 |

0.5 | 0.4093 | −0.3041 | 0.0319 | 0.0194 | 0.0806 | 0.4312 |

0.6 | 0.3629 | −0.243 | 0.049 | −0.1942 | 0.0964 | 0.4776 |

0.7 | 0.0643 | −0.3192 | 0.0112 | 0.6579 | −0.0584 | −0.2915 |

0.8 | −0.0357 | −0.08 | −0.0757 | 0.2796 | −0.0211 | −0.1048 |

0.9 | 0.0092 | −0.0707 | −0.0893 | 0.2619 | −0.0239 | −0.1293 |

1 | 0.1238 | −0.2901 | −0.0476 | −0.0255 | 0.3633 | −0.1402 |

**Table 7.**Correlation between ETF returns and macroeconomic time series. Significant correlations (5% significant level) are highlighted in gray. Period: January 2014–May 2020.

DIA | IHI | IXG | IYF | IYG | SPY | |
---|---|---|---|---|---|---|

GDP | −0.03156 | −0.04831 | −0.02229 | −0.02914 | −0.01632 | −0.04125 |

VIX | 0.3333175 | −0.1644 | −0.50249 | −0.45831 | −0.42538 | −0.29939 |

CPI | −0.0385 | 0.002127 | −0.09718 | −0.07781 | −0.05002 | −0.00824 |

FED | −0.03401 | −0.02914 | −0.02799 | −0.02576 | −0.03111 | −0.03512 |

**Table 8.**Correlation between financial ETFs and VIX for α = 0. Period: January 2014–May 2020. Significant correlations (5% significant level) are highlighted in gray.

Ψ | DIA | IHI | IXG | IYF | IYG | SPY |
---|---|---|---|---|---|---|

0.6 | −0.03 | 0.15 | −0.15 | −0.11 | −0.10 | 0.05 |

0.7 | 0.06 | −0.2401 | 0.10 | 0.17 | 0.03 | 0.18 |

0.8 | 0.02 | −0.01 | −0.06 | −0.09 | −0.01 | −0.03 |

0.9 | 0.06 | −0.19 | −0.22 | −0.03 | 0.15 | 0.2359 |

**Table 9.**Correlation between financial ETFs and VIX for all values of α. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 2020. Significant correlations (5% significant level) are highlighted in gray.

Panel A | ||||||

Alpha | DIA | IHI | IXG | IYF | IYG | SPY |

0 | 0.0995 | 0.053 | −0.103 | −0.3962 | −0.2128 | −0.0946 |

0.1 | 0.0327 | 0.1385 | 0.0711 | −0.1853 | −0.18 | −0.1826 |

0.2 | 0.0978 | 0.0487 | 0.1283 | −0.0675 | −0.0704 | −0.1373 |

0.3 | 0.006 | 0.0727 | −0.0116 | −0.2549 | −0.1106 | −0.0619 |

0.4 | 0.0882 | 0.093 | −0.0941 | −0.2784 | 0.0562 | −0.1804 |

0.5 | −0.0466 | 0.0741 | −0.0367 | −0.1762 | 0.0155 | −0.0315 |

0.6 | −0.0924 | 0.1086 | −0.0078 | −0.1825 | 0.1591 | −0.0775 |

0.7 | −0.2067 | 0.0235 | 0.3792 | 0.039 | −0.212 | 0.0066 |

0.8 | 0.2454 | 0.1213 | 0.3473 | 0.145 | −0.227 | −0.4488 |

0.9 | −0.0733 | 0.0053 | 0.2822 | 0.051 | −0.2772 | −0.1419 |

1 | 0.1463 | 0.086 | −0.0858 | 0.3851 | 0.1197 | 0.1329 |

Panel B | ||||||

Alpha | DIA | IHI | IXG | IYF | IYG | SPY |

0 | 0.02 | −0.01 | −0.06 | −0.09 | −0.01 | −0.03 |

0.1 | 0.01 | 0.09 | −0.11 | −0.20 | −0.3073 | −0.12 |

0.2 | 0.14 | −0.02 | 0.19 | 0.2493 | 0.003 | −0.16 |

0.3 | −0.19 | −0.08 | 0.8062 | −0.194 | −0.2538 | −0.22 |

0.4 | −0.13 | −0.09 | 0.8041 | −0.22 | −0.16 | −0.2319 |

0.5 | −0.21 | −0.10 | 0.802 | −0.19 | −0.17 | −0.14 |

0.6 | −0.2422 | −0.08 | 0.8037 | −0.22 | −0.12 | −0.17 |

0.7 | −0.26 | −0.08 | 0.6897 | −0.05 | −0.2395 | −0.09 |

0.8 | 0.01 | −0.02 | 0.492 | −0.03 | −0.2008 | −0.32 |

0.9 | −0.07 | −0.11 | 0.3874 | −0.03 | −0.20 | −0.15 |

1 | 0.02 | 0.004 | −0.02 | 0.12 | 0.02 | 0.02 |

**Table 10.**Results of model (11)–(12) (Panels A and B) applied to smart beta and financial ETF portfolios. Panel A includes data from January 2014 to May 2019, Panel B from January 2014 to May 20206.

Panel A—January 2014–May 2019 | ||||||

Model (11)–(12) for α = 0.1 | ||||||

Portfolio | ML | AIC (Res. SE) | GDP | CPI | FED | VIX |

Smart betas | 128.8 | −245.7 (0.034) | 8.10 × 10^{−2} (-) | 3.00 × 10^{−3}(-) | −2.48 × 10^{−2} (-) | −3.17 × 10^{−3}(**) |

Financials | 37.17 | −64.33 (0.1410) | 3.47 × 10^{−1} (-) | −1.61 × 10^{−3}(-) | 1.57 × 10^{−2} (-) | −5.93 × 10^{−3} (-) |

Model (11)–(12) for α = 0.9 | ||||||

Portfolio | ML | AIC (Res. SE) | GDP | CPI | FED | VIX |

Smart betas | 121.55 | −231.1 (0.038) | 1.69 × 10^{−1} (-) | 1.40 × 10^{−3} (-) | −1.52 × 10^{−2} (-) | −2.89 × 10^{−3}(**) |

Financials | −45.98 | 101.96 (0.5067) | 7.22 × 10^{−1} (*) | −1.26 × 10^{−1} (*) | 9.33 × 10^{−1} (*) | 2.13 × 10^{−2} (-) |

Panel B—January 2014–May 2020 | ||||||

Model (11)–(12) for α = 0.1 | ||||||

Portfolio | ML | AIC (SE) | GDP | CPI | FED | VIX |

Smart betas | 143.62 | −275.23 (0.039) | 1.39 × 10^{−1} (**) | 1.89 × 10^{−3}(*) | −1.97 × 10^{−2} (**) | −2.06 × 10^{−3} (***) |

Financials | 49.22 | −88.44 (0.131) | 1.12 × 10^{−1} (-) | 2.62 × 10^{−3}(-) | −1.84 × 10^{−2} (-) | −5.74 × 10^{−3} (*) |

Model (11)–(12) for α = 0.9 | ||||||

Portfolio | ML | AIC (SE) | GDP | CPI | FED | VIX |

Smart betas | 135.8 | −257.59 (0.043) | 1.65 × 10^{−1} (***) | 1.41 × 10^{−3}(-) | 1.73 × 10^{−2} (*) | −1.76 × 10^{−3}(**) |

Financials | −93.17 | 198.34 (0.839) | 37.32 × 10^{−1} (*) | −7.39 × 10^{−2}(*) | −8.51 × 10^{−2} (-) | 1.35 × 10^{−1} (***) |

**Table 11.**First linear discriminant analysis (LDA) coefficient (proportion of trace) for different values of gain threshold and α.

Gain Threshold | α = 0.1 | α = 0.5 | α = 0.9 |
---|---|---|---|

1% | 0.9178 | 0.8123 | 0.9178 |

2.5% | 0.9148 | 0.7689 | 0.9105 |

5% | 0.9376 | 0.9346 | 0.9168 |

7% | 0.6307 | 0.6307 | 0.5796 |

10% | 0.6307 | 0.6307 | 0.6655 |

No Lag | 1 Month Lag | 2 Month Lag | |
---|---|---|---|

α = 0.1 | 0.88 | 0.71 | 0.81 |

α = 0.5 | 0.91 | 0.75 | 0.85 |

α = 0.9 | 0.87 | 0.73 | 0.81 |

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## Share and Cite

**MDPI and ACS Style**

Foglia, M.; Recchioni, M.C.; Polinesi, G.
Smart Beta Allocation and Macroeconomic Variables: The Impact of COVID-19. *Risks* **2021**, *9*, 34.
https://doi.org/10.3390/risks9020034

**AMA Style**

Foglia M, Recchioni MC, Polinesi G.
Smart Beta Allocation and Macroeconomic Variables: The Impact of COVID-19. *Risks*. 2021; 9(2):34.
https://doi.org/10.3390/risks9020034

**Chicago/Turabian Style**

Foglia, Matteo, Maria Cristina Recchioni, and Gloria Polinesi.
2021. "Smart Beta Allocation and Macroeconomic Variables: The Impact of COVID-19" *Risks* 9, no. 2: 34.
https://doi.org/10.3390/risks9020034