A Statistical Measure of Global Equity Market Risk

We construct a new index of global equity market risk (EMR) using market interconnectedness and volatilities. We study the relationship between our EMR and the VIX over the last two decades. The EMR is shown to be a novel approach to measuring global market risk, and an alternative to the VIX. Using data of 20 major stock markets, including G10 economies, we find spikes in our EMR index during the dotcom bubble, the global financial crisis, the European sovereign debt crisis, and the novel coronavirus pandemic. The result shows that the global financial crisis and the COVID-19 induced crisis record the historic highest spikes in financial market risk, suggesting stronger evidence of contagion in both periods.


Introduction
Concerns about monitoring financial market turbulence and risk have intensified in recent times, especially in the heat of the ongoing global pandemic. The reaction of investors and financial markets since the first quarter of 2020, following the spread of the novel coronavirus (COVID-19) from Wuhan, in China to a global pandemic, has heightened the need to find ways of tracking the effect of the COVID-19 induced crisis on financial markets. This paper proposes the construction of a new index of equity market risk (EMR) using market interconnectedness and volatilities. The latter is considered a measure of market uncertainty or fear, which can be proxied via standard deviation of returns. The interconnectedness among markets provides the channels for spillover propagation. Our approach to the construction of the EMR follows the Mahalanobis turbulence measure of [1], and it is distinct from the systemic risk measures in [2]- [8]. The closest benchmark to our measure is the VIX-the Chicago Board of Exchange volatility index, generally used for measuring the level of global market risk.
We study the relationship between our EMR index and the VIX by using daily prices of 20 major stock market from Bloomberg, covering January 2000 to June 2020. The result shows spikes in our EMR index during the dotcom bubble, the global financial crisis, the European sovereign debt crisis, and the novel coronavirus pandemic. We find evidence of a significant relationship between the EMR and the VIX, with both indices providing similar signals about the direction of global market risk.
The organization of the paper is as follows: Section 2 presents the methodology; Section 3 reports the results; and Section 4 concludes the paper.

Methodology
In this section, we briefly present the background to network models. Next, we describe our measure of equity market risk (EMR), and a simple model of the relationship between the EMR and the VIX.

Modeling Interconnectedness
We model interconnectedness among markets via network models. A network model is a convenient class of multivariate analysis that uses graphs to represent statistical models [9]. They are formally represented by ( ) ( ) G is a graph of relationships between variables, θ is the model parameter,  is the space of graphs and Θ is the parameter space. The graph, G, is defined by a set of vertices (nodes/variables) joined by a set of edges (links), describing the statistical relationships between a pair of variables. A typical multivariate multiple regression model is given by A specifies the weight of the relationship.

Measuring Equity Market Risk (EMR)
where EMR captures the average degree of unusual changes in asset returns and their interactions. The index signals widespread market turmoil and can be viewed as a measure of market-level fears (volatilities) that are amplified through interconnectedness.
The above index uses market interconnectedness ( w A ) and volatilities ( S σ ).
The latter can be estimated directly from time series of stock market price/returns.
The market interconnectedness measure ( w A ) is supposed to capture weighted channels of influence among markets. Thus, w A can be estimated by approximating the dynamics in observed multivariate financial time series as in (1) and (2). The commonly discussed methods for estimating w A are: Granger-causality [5]; Lasso regularization [11] [12]; forecast error variance decomposition [6]; and Bayesian graph structural learning [13] [14] [15]. In this study, we derive the weighted w A via a Bayesian graph structural learning approach as in [10] [14].

Modeling Relationship between EMR and VIX
We model the marginal distribution of the EMR as ARIMA (1, 1, 1) and the conditional distribution of the VIX given EMR as ARIMAX (2, 1, 1).

Data Description
We study the relationship between our EMR index and the VIX by using data from the Bloomberg database. The data consists of the daily market indices of 20 countries, including G10 economies, covering January 3, 2000, to June 30, 2020.
We consider only one index per country, which typically contains the stock

Results
We compute daily returns as the log differences of successive daily closing prices.
We obtain monthly estimates of the model parameters and construct the ma- trices Ω and S σ , which are the core components of our EMR index. To improve the efficiency of the estimates of Ω we aggregate monthly estimates in yearly rolling windows of about 240 trading days. We set the increments between successive rolling windows to one month, setting the first window of our study from February 1, 1999, to January 31, 2000, followed by March 1, 1999, to February 29, 2000; the last window is from July 1, 2019, to June 30, 2020. In total, we consider 246 rolling windows. To avoid over smoothing, S σ is instead estimated monthly, that is, using only the last month of each rolling window.
We report in Figure 2  We present in Table 2 the cross-correlation of the first difference of VIX with EMR. The table reports the highest cross-correlation occurring at lag 0, which suggests evidence of a strong positive contemporaneous relationship. Table 3 1 confirms the significant contemporaneous impact of the EMR on the VIX. Table   4 presents the out-of-sample forecast of the monthly VIX for the rest of the year 2020 compared with the realized values for the months of July-September 2020 2 .
The results indicate that not only does the ARIMAX (2, 1, 1) performs better than the benchmark-ARIMAX (1, 1, 1) in terms of the RMSE of the in-sample training set, but also the out-sample forecasts are much closer to the realized observation than the benchmark. In summary, we document evidence of a significant relationship between the EMR and VIX, with both indices providing similar signals about the direction of global market risk. In addition, the EMR improves the prediction of the VIX.

Conclusion
This paper constructs an index of equity market risk (EMR) using market interconnectedness coupled with individual volatilities. The EMR index is shown to be a novel approach to measuring global market risk, and an alternative to the VIX. The empirical application presents a study of the relationship between the EMU and the VIX over the last 20 years. We find evidence of a significant relationship between both indices providing similar signals about the direction of global market risk. The result shows spikes in both indices during the dotcom bubble, the global financial crisis, the European sovereign debt crisis, and the novel coronavirus pandemic. However, the spikes recorded during the global financial crisis and the COVID-19 induced crisis suggest stronger evidence of financial stress and market tension in both periods.