FRM Financial Risk Meter for Emerging Markets
FFRM Financial Risk Meter for EmergingMarkets
Souhir Ben Amor ∗ Michael Althof † Wolfgang Karl Härdle ‡§ February 11, 2021
The fast-growing Emerging Market (EM) economies and their improved trans-parency and liquidity has attracted international investors. However, the externalprice shocks can result in a higher level of volatility as well as domestic policy insta-bility. Therefore, an efficient risk measure and hedging strategies are needed to helpinvestors protect their investments against this risk. In this paper, a daily systemicrisk measure, called FRM (Financial Risk Meter) is proposed. The FRM@ EM is ap-plied to capture systemic risk behavior embedded in the returns of the 25 largest EMs’FIs, covering the BRIMST (Brazil, Russia, India, Mexico, South Africa, and Turkey),and thereby reflects the financial linkages between these economies. Concerning theMacro factors, in addition to the Adrian & Brunnermeier (2016) Macro, we includethe EM sovereign yield spread over respective US Treasuries and the above-mentionedcountries’ currencies. The results indicated that the FRM of EMs’ FIs reached itsmaximum during the US financial crisis following by COVID 19 crisis and the Macrofactors explain the BRIMST’ FIs with various degree of sensibility. We then studythe relationship between those factors and the tail event network behavior to buildour policy recommendations to help the investors to choose the suitable market for in-vestment and tail-event optimized portfolios. For that purpose, an overlapping regionbetween portfolio optimisation strategies and FRM network centrality is developed.We propose a robust and well diversified tail-event and cluster risk sensitive portfolioallocation model and compare it to more classical approaches.
JEL Classification:
C30, C58, G11, G15, G21. ∗ Blockchain Research Center, Humboldt-Universität zu Berlin, Germany. International Research Training Group1792, Humboldt-Universität zu Berlin, Germany. Alexander von Humboldt Stiftung. [email protected]. † Blockchain Research Center, Humboldt-Universität zu Berlin, Germany. International Research Training Group1792, Humboldt-Universität zu Berlin, Germany. [email protected] ‡ Blockchain Research Center, Humboldt-Universität zu Berlin, Germany. Wang Yanan Institute for Studiesin Economics, Xiamen University, China. Sim Kee Boon Institute for Financial Economics, Singapore Man-agement University, Singapore. Faculty of Mathematics and Physics, Charles University, Czech Republic.National Chiao Tung University, Taiwan. [email protected]. § Financial support of the European Union’s Horizon 2020 research and innovation program “FIN- TECH: AFinancial supervision and Technology compliance training programme” under the grant agree- ment No 825215(Topic: ICT-35-2018, Type of action: CSA), the European Cooperation in Science & Technology COST Actiongrant CA19130 - Fintech and Artificial Intelligence in Finance - Towards a transparent financial industry, theDeutsche Forschungsgemeinschaft’s IRTG 1792 grant, the Yushan Scholar Program of Taiwan, the CzechScience Foundation’s grant no. 19-28231X / CAS: XDA 23020303 a r X i v : . [ q -f i n . P M ] F e b eywords : FRM (Financial Risk Meter), Lasso Quantile Regression, NetworkDynamics, Emerging Markets, Hierarchical Risk Parity. Highlights • We select the 25 biggest FIs in the BRIMST by market capitalization,• Select Macro variables to reflect the state and impact of the developed andemerging economies,• The FRM is based on Lasso quantile regression designed to capture tail eventco-movements, between the selected EM FIs and the Macro variables,• We use different quantile risk levels to check the robustness of our results.• We found a high positive spillover effects between FIs of the same country andoften negative spillover effects between FIs in between regions.• We examplify EM FIs illustrating high-CoStress and high network centrality.• We give examples of "risk receivers" during the 2020 COVID-19 crisis.• The EMs are strongly influenced by the Macros the Emerging Market Yieldspread, followed the VIX, Moody’s Baa corporate yield spreads, and the shapeof the U.S. Treasury yield curve. However, Emerging market currencies have alesser impact.• We use the FRM technology based results to construct robust tail-event sensi-tive portfolios based on an uplifted HRP approach and compare them to otherclassical approaches.• We analyze the relationship between the FRM network centralities and the port-folio weights and specifically risky concentrations,• The classical portfolio approach leads to high weight concentration. However,the uplifted HRP approach provides better diversification.• The uplifted HRP portfolio overweight low-central FIs and underweight high-central ones.• The Inv λ is less at risk of spill-over effects across EM regions, FIs, and financialsub-sectors. Emerging markets have been commonly acclaimed for providing robust growth potential andoffering investors a higher expected return compared to developed markets. Indeed, due tothe possibility of higher profits and the low level of global equity markets integration, EMs havebeen considered as an investment opportunity for investors, whose aim to build an internationallydiversified portfolio. EMs liquidity and transparency have continuously enhanced [McGuire &Schrijvers (2006); Bunda et al. (2009)]. Moreover, the reputation of EMs, in the framework ofportfolio diversification, has received the attention of international investors, especially after thefinancial crisis that affected mostly developed markets.However, the investment’s benefits of EMs come with additional risks, which are usually notas prominent in developed markets. In fact, EMs are exposed to additional economic, political,and currency risks. Further, the EMs’ economies fast growth and, as consequence, the quickevolution of their structure result in market information being rapidly outdated. Therefore,the existing more traditional methods of risk evaluation may be misleading in EMs, especially2n short and medium investment horizons. Indeed, the existing risk measure methods are notsuited to provide the up to date point of view representing current market structures, if they notsupplemented with the latest market information. Hence, an efficient systemic risk measure isneeded for EM.For that purpose, it is crucial to understand and measure the spillover risk across EMs financialsystem network, which is important for financial risk measurement and portfolio diversification;From the perspective of financial risk measurement, the interdependence among FIs becomesmore important, especially during periods of distress, when losses spread through institutions,rendering the global financial system more vulnerable. In this regard, a systemic crisis thatdisturbs the financial system stability can have serious effects and lead to high losses for theentire economy and society.From the perspectives of risk management and portfolio diversification, the contagion riskacross the FIs from the same market, and across the worldwide markets, leads to a decrease indiversification potential. Hence, understanding the network structure of interdependence amongFIs is crucial to risk managers and portfolio investors, as this can help them design investmentstrategies to reduce dependence risk and thereby increase diversification. Investors are alsointerested in recognizing the FIs that contribute the most (least) risk to their portfolio so thatthey are considered with caution in their portfolio design, especially during financial marketturmoil.To understand the FIs co-movements in EMs, our study examines the effect of Macro factorson the FIs in BRIMST (Brazil, Russia, India, Mexico, South Africa, Turkey) EMs, using dailyinterval data for the period between 2000–2020, with particular focus on the last two years. It iswell known that the economics of the mentioned markets are strongly linked to the US economy[Özatay et al. (2009)], so it is crucial to analyze the effects of US Macro factors on volatility inEMs financial system. It is worth to note that our dataset covers several global crisis periods,allowing us to examine how the EMs Financial systems respond to the different crisis. Afterdetermining the interdependencies among FIs and macroeconomic factors, our research aimsalso to build a robust strategy based on portfolio diversification in EMs.In order to achieve the mentioned goals, our paper seeks to answer the following questions: Canthe US and EMs Macro factors explain BRIMST financial equity indices? Are some categoriesof Macro factors more important than others? What FIs from EMs are the largest (smallest)spillover transmitters (receivers)? What FIs contribute the most (least) risk to total portfoliorisk? What FIs offer greater diversification benefits? And lastly, how the tail spillover effect andportfolio weights change over time, and how they react to the different tail risk levels?Answering all these questions is crucial, as international investors interested in understand-ing the forces behind the interdependence among macroeconomic factors and FIs, to identifypotential risks and rewards and benefit from global diversification. Economic policymakers andregulators in BRIMST are interested in forces behind the co-movement between these marketsto further establish market resilience in EMs.We tackle these questions by employing the Financial Risk Meter (FRM) technology in EMs.The FRM is based on Lasso quantile regression designed to examine tail event co-movementsfinancial securites. The objective is to understand the FIs interconnectedness and represent themin a network topology. Moreover, the FRM indices summarize the systematic risk at a given areaand identify risk factors. Briefly, FRM represents tail event behavior in a financial risk factorsnetwork, which allows us (i) to identify the “stress emitters” and “stress receivers” companies,(ii) to measure the tail dependencies among the FIs and the Macro factors (iii) analyze therisk level in EMs over time. Concerning risk management, (iv) the FRM network is adopted inthe portfolio selection process. More precisely, by interpreting the correlation coefficients of FIsequity indices in the adjacency matrix of the FRM network, an overlapping region between theportfolio optimisation strategies and FRM network is developed, (v) The FRM adjacency matrix3s also adopted to lift the Hierarchical Risk Parity (HRP) approach [propsed by De Prado (2016)]to the quantile level.The remainder of this research is organized as follows. Section 2 presents a brief review of riskmeasurement and risk management methods. The econometric approach is discussed in Section3, as well as a discussion on centrality measures. Section 4 illustrates the proposed portfoliooptimization strategies. The FRM results are analysed in section 5. Section 6 develops theportfolio construction, while section 7 provides policy recommendations before concluding thegeneral scope.The codes are published on indicated by with keyword FRM. The currentlevel of FRM for Emerging markets as well as other channels can be found at http://hu.berlin/frm . The large-scale breakdown of FIs after the global Lehman brother’s crisis in 2008 had causedserious social and economic losses. Therefore, controlling and maintaining financial system sta-bility is one of the principals and the mutual responsibilities of central banks and financial marketregulators around the world. Previous researches have been conducted in the framework of FIsinterconnectedness. In this context, empirical studies commonly focus on resolving this issuein European countries; Betz et al. (2016), Abbassi et al. (2017), and Aldasoro & Alves (2018).Some studies have studied the FIs networks of developed American countries. Among others,Cai et al. (2018); Kreis & Leisen (2018), and Tonzer (2015). Relatively few researchers haveinvestigated bank networks of emerging American countries; Silva et al. (2016); and Berndsen etal. (2018). All of the above studies focused on data from only one country and did not considernetworks between FIs from several countries across the globe.In order to bridge this gap, our study investigates the FIs networks from six among the biggestemerging countries worldwide, as globalization leads to closely linked economic activity amongthese countries in terms of trade and finance. It is well known that FIs are interconnected withinnetworks of several types of financial connections and contracts. The complex links among theseinstitutions, which can be considered as risk factors, can cause systemic risk and results inspillover-effects that deteriorate the network stability and its functioning. These observations onthe joint network dynamics motivated practitioners and researchers to insert tail events into riskmanagement.The Value-at-Risk (VaR) approach Franke et al. (2019) is frequently used to measure marketrisk, by computing the monetary loss of an institution for a given confidence level [Slim et al.(2017)]. However, the VaR measures a tail event probability hosting only one single node, whichdoes not reflect its connection to overall systemic risk. Recently, Adrian & Brunnermeier (2011)& (2016) developed CoVaR to measure the systemic risk [e.g.Fang et al. (2018), Ben Amor etal. (2019)]. The CoVaR measures spillover effects across financial markets by providing the VaRof one market under the condition that the other market is in financial distress as given byits VaR. Consequently, the CoVaR approach can gauge the size of the financial risk spillover.However, it can only capture the extent of risk spillovers for a simple bivariate system and cannotsimultaneously measure the risk spillover effects across multiple financial markets. Härdle et al.(2016) developed the Tail Event NETwork (TENET) risk approach by generalizing the CoVaR tobe able to accommodate all system nodes as risk factors. TENET applies the quantile regressionmethod on a set of network nodes stock market information and macroeconomic variables in arolling window approach. The innovation of TENET model is to employ dimensional reduction(in a semi-parametric setting) by using the Lasso in a quantile regression framework Tibshirani(1996). Also, Chen et al. (2019) propose a Tail Event driven Network Quantile Regression4TENQR) to adress the interdependence, dynamics and riskiness of financial institutions. TheTENQR captures risk dynamics within a panel quantile autoregression in a network topology,quantified through a time-varying adjacency matrix. TENQR technique is evaluated using theSIFIs (systemically important financial institutions). To extend the TENET further, Mihoci etal. (2020) developed an improved systemic risk measure that summarizes the high-dimensionaltail stress into a single real value indicator, called the Financial Risk Meter (FRM). The FRMis computed as an average of each node and at each time window selected penalization terms.The FRM level contains fundamental information about the active set of influential neighboringnodes and about the contributors to systemic risk.Yu et al. (2019) compared the proposed FRM to other measures for systemic risk, such as GoogleTrends, SRISK, and VIX. They found a Granger causality between the FRM and these measures,which confirmed the validity of the FRM as an efficient systemic risk measure. Mihoci et al.(2020) applied the FRM to measure the dependencies between FIs and Macro factors exploitingtail event information. They built two FRM indices namely FRM@Americas and FRM@Europe,but also on bond yield and credit default swap spread co-movements.So far, the FRM applications took into consideration financial securities from one economicregion and its domestic Macro factors. In our study, we extend the application of FRM for EMs,and we will consider the domestic Macro factors from the BRIMST as well as those from the US(proposed by Adrian & Brunnermeier (2016) to represent both the domestic and foreign state ofthe economy.In many portfolio construction approaches, the correlation between financial assets representsthe basis for portfolio selection. To exemplify, Markowitz (1959) developed the modern portfoliotheory (MPT). He found that when correlation between assets is not perfect, a diversified portfoliocan be constructed. Therefore, to reach efficient diversification, investors should select anti-correlated assets and verify and ensure the satisfaction of this condition over time. But thecorrelation structure among assets changes over time and evolves especially during crises periods.For that reason, the Markowitz theory is usually oriented to select the most stable assets such asthe industrial assets, and hence, the Markowitz optimal portfolio is often composed of a limitedand invariant set of assets. The second weakness of Markowitz’s MPT [Markowitz (1959)] isrelated to the large estimation errors of the expected returns vector [Merton (1980)], as well asof the covariance matrices [Jobson & Korkie (1980)]. Hence, a robust method for the modellingof the dynamic interconnectedness of assets is needed to support the MPT and guide investorsin building an efficient and well diversified portfolio. In this regard, recent researchers designedfinancial markets in networks (FMN), where assets are represented by nodes and the correlationof returns are represented by links that relate these nodes [Chi et al. (2010); Diebold & Yılmaz(2014); Peralta & Zareei (2016)]. Despite the originality and interesting results provided by thisnetwork approach, the majority of its applications are descriptive and lack concrete applicationsin the portfolio optimization procedure. Recently, Pozzi et al. (2013) extracted the dependencystructure of financial equities from the network approach to build a diversified portfolio to reduceinvestment risk. This procedure visualized portfolio selection directly over the FMN design.Peralta & Zareei (2016) used the FMN as a powerful device to enhance the portfolio optimizationprocedure by selecting a set of assets according to their network centrality.However, the adopted spillover and financial network methods focused on estimating the riskspillover based on the return distributions’ first two conditional moments, thereby ignoring highermoments of the distribution (i.e. right and left tails). Indeed, the existing spillover measuresconcentrate only on the mean and variance of the distributions, which may underestimate the realspillover effects among FIs in tail-events, since they do not take into consideration the extremerisk spillover across financial markets. More specifically, previous researchers investigated mean-to-mean effect, assuming that investors are mean–variance optimizers. But ideally, a portfolioselection decision should be based on the entire return distribution estimation, since investors are5ore risk averse to the extreme downside risk related to the tails of the distribution. To overcomethis limitation, we adopt the FRM to measure the joint tail events. In fact, the FRM has theadvantage to represent instantaneously the co-movements and the dynamics of high-dimensionnetworks. Furthermore, the FRM can display the hidden interdependency structures betweenthe financial network’s nodes. Given that the functions of complex FMN are reflected in therisk evaluation and portfolio optimization [Haluszczynski et al. (2017); Wang et al. (2018)], wecontribute to this line of research by investigating the extent to which the underlying structureof this financial market tail event network can be used as an effective tool in enhancing theportfolio selection process. For that purpose, we establish a bridge between the FRM networkand portfolio optimisation strategies. More precisely, we study the relationship between optimalportfolio weights and the FIs’ centrality in the FRM network.The HRP is a second approach that aims to overcome the shortcomings of MPT. The Hierar-chical risk parity (HRP) is firstly proposed by López de Prado (2016) as a risk parity allocationalgorithm. The HRP applies machine learning and graph theory to extrapolate a hierarchicalimplementation of an inverse-variance allocation with weights computed among the formed corre-lated asset return clusters. By substituting the covariance structure with a hierarchical structureof clusters, HRP fully benefits from the covariance matrix information and improves the stabil-ity of the weights. López de Prado (2016) shows that the HRP achieves higher risk-adjustedreturn and lower out-of-sample volatility than inverse-variance allocations. Despite its appealingfeatures, the application of HRP is still rare, with increased interest Lohre et al. (2020), Jain &Jain (2019), and Raffinot (2017) test the performance of HRP in a multi-asset allocation. Fur-thermore, Raffinot (2017) and Alipour et al. (2016) evaluate the performance of variant HRP.However, the HRP focusses on the covariance matrix in computing the weights, consequently, itoffers only a conditional mean view of the assets’ connectedness, which may underestimate thereal spillover effects among assets in tail events. Taking into consideration tail spillover effectsusing quantile regression methods seeks to broaden this view, by providing a complete descriptionof the stochastic relationship between assets and offering more robust and more efficient esti-mation Ma & Pohlman (2008). In this context, our paper contributes to the growing literatureby employing an uplifted HRP based on FRM as a quantile regression method. More precisely,our research introduces two modifications to the HRP algorithm of López de Prado (2016): (i)Replace the covariance matrix with the FRM-adjacency matrix. Indeed, the HRP assign portfo-lio weights without the necessity to invert the covariance matrix. This propriety of HRP makesthe replacement of the symmetric covariance matrix with the non-symmetric adjacency matrixpossible. (ii) The HRP uses the inverse variance of each asset to calculate the optimal weights,and we replace the variance with the FRM’s Lasso penalization parameter ( λ ). By introducingthese two modifications, we build the uplifted HRP based on the FRM technology. The HRPportfolio is then benchmarked against the classical HRP, the Minimum variance (MinVar) andinverse variance portfolio (IVP) optimization methods.Our research investigates a comprehensive set of risk measurement and portfolio optimizationin EMs, contributing a new dimension to the existing literature as follows:(1) The FRM based Lasso quantile regression yields novel insights into the co-movement amongFIs and US and EM domestic macroeconomic risk factors. The risk of spillover and its directionare also quantified and visualized, as well as how spillover risk evolves during the financial crisis.Therefore, our research paper is among a few investigating the risk spillover across BRIMSTEMs. To the best of our knowledge, there is a considerable gap in financial literature on thesubject of tail risk spillover among US Macro factors and FIs in EMs. This paper is an attemptto bridge this gap by building our research on the above subject. For that purpose, we cover asample of the biggest six emerging countries from different areas around the world which offers a6easonable basis for comparisons at country and regional market levels. Most of the earlier studiesfocus on fewer countries, mostly in one region. In addition, we adopt a sample that containsthe largest FIs from each country as well as the domestic and US Macro factors to build ourFRM network. Hence, our study investigates a comprehensive set of emerging equity markets,contributing a new dimension to the literature on international equity market co-movement thathas traditionally focused on developed markets.(2) Examine the existence of portfolio diversification benefits for foreigners investing in EMs.Several studies have been conducted to examine the existence of portfolio diversification benefitsin less correlated financial markets; Abid et al. (2014); and Saiti et al. (2014). However, few otherstudies investigate this subject from the perspective of an EMs framework [Arreola Hernandezet al. (2020)].(3) This paper sheds light on the connection between the portfolio optimisation approaches andthe financial tail event networks. Our strategy aims to simplify the portfolio selection procedureby targeting a set of assets within a certain range of network centrality. As far as we are aware,Arreola Hernandez et al. (2020) is the only paper that attempts to take advantage of the topologyof the tail event financial market network for investment purposes. They applied a directionalspillover index, the tail-event driven network (TENET), and nonlinear portfolio optimizationmethods on the bank returns from emerging and developed America.(4) While most previous studies focus on portfolio optimization based on mean variance esti-mation, there is a lack of empirical literature on quantile estimation (both tails of distribution).This study attempts to fill this gap by extending the classical HRP approach to build an optimalportfolio based on tail information provided by the FRM quantile regression.(5) Finally, it is worth mentioning that our results are robust to different estimation timewindows, market situations (pre-crisis, crisis, and post crisis), and tail risk levels. We arguethat our proposed FRM based portfolio optimisation approach benefits from the tail event co-movement and asset clustering making more useful use of fundamental tail-event information,resulting in an efficient risk portfolio selection strategy. This practice is unique to this researchand remains an important contribution to the literature on risk measurement and internationalportfolio diversification. This section describes the genesis and framework of FRM. For that purpose, we lay down thefundamental structure and the background of our systemic risk analysis. More precisely, we willstate the systemic risk measure models that have been evaluated to lead to an augmented riskmeasure, the FRM.
The Value at Risk (VaR) and the expected shortfall (ES) are traditional risk measures. Theycompute the risk of a given financial institution based either on company characteristics or byintroducing macroeconomic variables as a proxy for the state of economy. To exemplify, the VaRof a financial institution i at a quantile level τ is given by the following equation: P ( X i,t ≤ V aR i,t,τ ) def = τ, τ ∈ (0 , (1)Where X i,t represent the log return of financial institution i at time t .7drian & Brunnermeier (2011) proposed the Conditional Value at Risk (CoVaR), the CoVaRconsiders the spillover effects and the macro economy state. The CoVaR of F I j given X i atquantile level τ at time t is given as follow: P ( X j,t ≤ CoV aR j | i,t,τ | R i,t ) def = τ, (2)Where M t − denotes the vector of macro state variables, and R i,t indicates the information setthat involves the event of X i,t = V aR i,t,τ and M t − . The CoVaR is estimated in two steps oflinear quantile regression assuming the following equations: X i,t = α i + γ (cid:62) i M t − + ε i,t , (3) X j,t = α j | i + β j | i X i,t + γ (cid:62) j | i M t − + ε j,t , (4) F − ε i,t ( τ | M t − ) = 0 and F − ε j,t ( τ | M t − , X i,t ) = 0 (5)Where β j | i in Equation(4) defines the sensitivity of log return of a company j to variation in tailevent log return of a company i . Step 1: Estimate the
V aR of an institution i ( V aR i ) To estimate the
V aR i , Adrian & Brunnermeier (2011) apply the quantile regression of log returnof company i on macro state variables. The estimated equation is defined as follow: (cid:100) V aR τi,t = (cid:98) α i + (cid:98) γ (cid:62) i M t − , (6) Step 2: Estimate the
CoV aR
The CoVaR is computed by integrating the
V aR τi,t in Equation (6) into the CoVaR equation asfollow: (cid:100)
CoV aR τj | i,t = (cid:98) α j | i + (cid:98) β j | i (cid:100) V aR τi,t + (cid:98) γ (cid:62) j | i M t − . (7)Where the coefficient βj | i of Equation (7) indicates the degree of interconnectedness.Therefore, the risk of a financial company j is computed through a V aR and macro state ofcompany i . By setting j equal to the return of a system, and i to be the return of a financialcompany i , we obtain the contribution CoVaR that illustrates how a company i effects the restof the financial system. Or, by setting j to be a financial company and i to be a financial system,we obtain exposure CoV aR , which characterizes how a single institution is exposed to the overallrisk of a system.Recently, Härdle et al. (2016) developed the tail-event driven network (TENET). The TENETis a risk approach generalizes CoVaR by joining systemic interconnectedness between FIs andaccommodates all system nodes as risk factors. It is estimated based on quantile regressions andillustrated by three steps.
Step 1: Estimate the
V aR of each FI
The
V aR for the studied FIs is estimated using the linear quantile regression based on Equa-tions (3 and 6).
Step 2: The network analysis
The TENET estimates the non-linear relationship among FIs, and takes more institutions intoconsideration to compute the tail-driven risk interdependencies. Therefore, we have: X j,t = g ( β (cid:62) j | R j R j,t ) + ε j,t , (8) (cid:100) CoV aR
T ENETj | ˜ R j,t,τ def = (cid:98) g ( (cid:98) β (cid:62) j | ˜ R j ˜ R j,t ) , (9)8 D j | ˜ R j def = ∂ (cid:98) g ( (cid:98) β (cid:62) j | R j R j,t ) ∂R j,t | R j,t = ˜ R j,t = (cid:98) g (cid:48) ( (cid:98) β (cid:62) j | ˜ R j ˜ R j,t ) (cid:98) β j | ˜ R j (10)where R j,t def = { X _ j,t , M t − , B j,t − } denotes the set of information. Note that X _ j,t def = { X ,t X ,t ..., X ,t } is a set of { k − } independent explanatory variables, such as the log returns ofall the FIs expected the F I j , and k here denotes the number of FIs. The term B j,t − characterizesthe internal factors of the institution j .The parameters are defined through the term β j | R j def = { β j | _ j , β j | M , β j | B j } (cid:62) . The (cid:100) CoV aR
T ENET stands for tail-event driven network risk by means ofa single-index model (SIM) model and is estimated by plugging in
V aR of institution i at level j | Rj, t , defined in Equation (6) into Equation (8).Note that β j | R j def = { β j | _ j , β j | M , β j | B j } (cid:62) and R j,t def = { (cid:100) V aR τi,t , M t − , B j,t − } . (cid:98) g ( • ) characterizethe non-linear relationship among them. The parameter D j | ˜ R j represents the gradient evaluatingthe marginal effect of covariates measured at R j,t = ˜ R j,t . The component wise expression is givenby: (cid:98) D j | ˜ R j ≡ { (cid:98) D j | _ j , (cid:98) D j | M , (cid:98) D j | B j } (cid:62) . Specifically, (cid:98) D j | _ j permits to quantity spillover effects throughthe FIs and to illustrate their evolution as a network system. The TENET network represents aset of FIs links. The estimation results of this interconnectedness can be summarized in weightedform of an adjacency matrix. Note (cid:98) D sj | i an element in (cid:98) D sj | _ j at estimation window s for the F I j given another F I i (where i is an element in the other F I j ). Therefore, the adjacency matrixincludes absolute values of (cid:98) D sj | i (in upper triangular matrix) and (cid:98) D si | j (in lower triangular matrix),where (cid:98) D sj | i denotes the influence from a F I i to a F I j and (cid:98) D si | j is the influence from F I j to F I i .The adjacency matrix A s = ( k ∗ k ) is represented in the following equation, where for each timewindows s only one adjacency matrix is estimated. A s = I I I · · · I k I | (cid:98) D s | | | (cid:98) D s | | · · · | (cid:98) D s | k | I | (cid:98) D s | | | (cid:98) D s | | · · · | (cid:98) D s | k | ... ... ... ... . . . ... I k | (cid:98) D sk | | | (cid:98) D sk | | | (cid:98) D sk | | · · · (11)The matrix A s summarizes the overall connectedness between variables at time window s , and I i denotes the name of F I i . Step 3: Systemic risk contributions
The objective here is to measure the systemic risk relevance of a specific FI by its total in- andout-connections, weighted by market capitalization. Hence, we define the Systemic Risk ReceiverIndex ( SRR ) for a F I j at time windows s as follow: SSR j,s def = M C j,s (cid:40) (cid:88) i ∈ k INs ( | (cid:98) D sj | i | .M C i,s ) (cid:41) (12)The Systemic Risk Emitter Index ( SRE ) for a F I j at time windows s is given by the followingequation: SSE j,s def = M C j,s (cid:40) (cid:88) i ∈ k OUTs ( | (cid:98) D si | j | .M C i,s ) (cid:41) (13)9ere k INs and k OUTs denotes the set of FIs linked with the
F I j at time windows s by incomingand outgoing links respectively. M C i,s is the market capitalisation of FI i at the starting pointof time windows s . | (cid:98) D sj | i | and | (cid:98) D si | j | are absolute partial derivatives derived from Equation (10)which represents row (incoming) and column (outgoing) direction connection of F I j . Thus both SRR j , s and SRE j,s take into consideration the
F I j and its linked FI’ market capitalization aswell as its connectedness within our network. For more details see Härdle et al. (2016). The FRM is a systemic risk measure based on the penalization parameter ( λ ) of a linear quantileLasso regression using moving-window approach. In this section we present the methodologyand algorithm that constitutes the FRM technology. Since the penalization parameters ( λ ) arecomputed based on an L -norm (Lasso) quantile linear regression, this method is introducedfirst. Linear Quantile Lasso Regression Model
The FRM aims to simultaneously capture all interdependencies in one single number basedon the log return of FIs and a set of macroeconomic variables that illustrate the state of theeconomy. Consider J the number of FI under consideration, where j ∈ { , ..., J } . Therefore p = J + M − denotes the number of covariates. Let t = { , ..., T } be the time point, where T denotes the total number of daily observations. Suppose s is the index of time windows, where s ∈ { , ... ( T − ( n − } and n is the length of windows size.The Linear quantile Lasso regression for return series X is defined as follow: X sj,t = α sj,t + A s (cid:62) j,t β sj + ε sj,t , (14)Where A s (cid:62) j,t def = (cid:20) M st − X s − j,t (cid:21) , M st − is the M dimensional vector of macro variables, X s − j,t is the p − M dimensional vector of log returns of all other FIs except F I j at time t and in movingwindow s , β sj is a p × vector defined for moving window s and α sj is a constant term.The regression is performed using an L -norm penalisation given a parameter λ j , definedby Tibshirani (1996) as the least absolute shrinkage and selection operater (Lasso). Accordingto Bassett Jr & Koenker (1978), the current company’s λ j are estimated by a modification ofLasso in a quantile regression [see Belloni & Chernozhukov (2011) and Li & Zhu (2008) for moredetails), where the optimization is solved as follow: min α sj ,β sj (cid:110) n − s +( n − (cid:88) t = s ρ τ ( X sj,t − α sj − A s (cid:62) j,t β sj ) + λ sj (cid:107) β sj (cid:107) (cid:111) (15)with the following check function: ρ τ ( u ) = | u | c | τ − I { u< } | (16)where c = 1 corresponds to quantile regression employed here and c = 2 corresponds to expectileregression. Penalization Parameter λ The formula of Lasso’s penalization parameter λ can be estimated in a linear regression context,following the work of Osborne et al. (2000) Treating λ as a fixed value in the objective function10f the penalized regression: f ( β, λ ) = (cid:40) n (cid:88) i =1 ( y i − X (cid:62) i β ) + λ p (cid:88) i =1 | β j | (cid:41) (17)Here f ( β, λ ) is convex in the parameter λ . In addition, with diverging β we note that f ( β, λ ) →∞ . Consequently, the function f ( ., λ ) admits at least one minimum, which is attained in (cid:98) β ( λ ) [Osborne (1985)] if and only if the null vector ∈ R p is an element of sub-differential: ∂f ( β, λ ) ∂β = − X (cid:62) ( Y − Xβ ) + λu ( β ) (18)Were: u ( β ) = ( u ( β ) , ..., u p ( β )) (cid:62) is defined as u j ( β ) = 1 if β j > u j ( β ) = − if β j < u j ( β ) ∈ [ − , if β j = 0 Therefore, considering that f ( β, λ ) admit a minimum in (cid:98) β , the following equation has to besatisfied: − X (cid:62) { Y − X (cid:98) β ( λ ) } + λu ( (cid:98) β ( λ ) (19)The estimator of the vector of parameters β here is a function of the penalization parameter λ .This dependency follows from the formulation of the penalized regression method and its objec-tive function Equation (17). Following this method we select first the penalization parameter λ and then search for β λ that minimizes Equation (17). Given that u ( β )) (cid:62) β = (cid:80) pj =1 | β j | = (cid:107) β (cid:107) ,where (cid:107) . (cid:107) represents here L -norm of a p -dimensional vector, Equation (19) can be rewrittenin the following formula: λ = { Y − Xβ ( λ ) } (cid:62) Xβ ( λ ) (cid:107) β (cid:107) (20)Looking to formula Equation (20), we can define the elements that influence the value of λ andits dynamics when treated in a time- varying framework. The following three elements affect thesize of λ :1. The size of residuals of the model;2. The absolute size of the coefficients of the model, (cid:107) β (cid:107) ;3. The singularity of a matrix X (cid:62) X .The formulae for the penalization parameter λ in a quantile regression problem Equations (15and 16)can be derived following Li & Zhu (2008): λ = ( θ ) (cid:62) X (cid:98) β ( λ ) (cid:13)(cid:13)(cid:13) (cid:98) β ( λ ) (cid:13)(cid:13)(cid:13) (21)Where θ = ( θ , ..., θ n ) (cid:62) satisfies the following conditions: θ i = τ if Y i − X (cid:62) i (cid:98) β ( λ ) > − (1 − τ ) if Y i − X (cid:62) i (cid:98) β ( λ ) < ∈ ( − (1 − τ ) , τ ) if Y i − X (cid:62) i (cid:98) β ( λ ) = 0 L loss function and an L -norm penalty term, the optimizationproblem is an L -norm quantile regression estimation problem. The selection of the penalizationparameter λ sj is fundamental. There are different possibilities to choose λ sj , for example withthe Generalized Approximate Cross- Validation criterion (GACV) or the Bayesian InformationCriterion ( BIC ) . In this regards, Yuan (2006) conducted simulations and concluded that GACVoutperforms BIC in terms of statistical efficiency. Hence, we estimate λ sj with the GACV criterionin the FRM model and set λ sj as the solution of the following minimization problem: min GACV ( λ sj ) = min (cid:80) s +( n − t = s ρ τ ( X sj,t − α sj − A s, (cid:62) j,t β sj ) n − df (22)where df stands for the estimated effective dimension of the fitted model. The advantage ofGACV is that it can be also adopted for p > n , which can be crucial for the FRM if the movingwindow size is small. For further details see Mihoci et al. (2020). Financial Risk Meter (FRM)
The FRM is estimated using a regression analysis as explained above and select the λ j for eachFI j using GACV. The distribution of the λ sj in a moving window gives important informationon the network dependencies among the financial nodes. The standard FRM is defined as theaverage of λ j over the set of J FIs for all windows. It is formally written as follows:
F RM = J − J (cid:88) j =1 λ j (23)Note that the distribution of λ j provides details regarding the overall market movement andgives information to macro-prudential decision makers about the network dynamics. Therefore,the FRM is adopted to identify the high joint tail event risks resulting from each FI. Moreprecisely, the FI with high λ j exhibits common high stress levels as the FI at the origin of thecrisis. Therefore, this FI is considered as having high ”co-stress”. The FRM network illustrates the tail event interaction between the selected FIs based on the adja-cency matrix A s (11). To describe the topology of the FRM networks, we focus on node centralitymeasures, specifically, eigenvalue centrality, degree centrality, indegree, outdegree, betweennessand closeness. The concept of centrality aims to measure the impact and the importance of agiven node in the network. Eigenvector centrality
Bonacich (1972) proposed the so-called eigenvector centrality, which evolved to be a standardmeasure in network analysis.Consider a network G = { N, ω } , constituted by a set of links ω that connecting pairs of nodesand a set of nodes N = 1 , , ..., N . If there is a connection between two nodes i and j , we denoteit as ( i, j ) ∈ ω . The network connections are defined by a simplified version of (11): the ( N × N ) A si,j = [ β sij ] whose element β ij (cid:54) = 0 whenever ( i, j ) ∈ ω , s is the rolling window. A s = β , · · · β ,n β , · · · β ,n ... ... . . . ... β n, β n, · · · (24)Based on the Bonacich (1987); (1972) definition, the eigenvector centrality of node i , v i , isexpressed as the proportional sum of its neighbors’ centralities. This concept has been extendedby Bonanno et al. (2004) suggesting that for the weighted networks, v i is relative to the weightedsum of neighbors’ centralities of node i with the adjencency matrix A si,j that illustrates thecorresponding weighting factors. Therefore, the eigenvector centrality of node v i is calculated asfollows: v i = δ − (cid:88) j β sij v j (25)where δ here denotes the eigenvalue. A large value of v i means that the node i is highly central,implying that node i is linked either to several other nodes or is linked to a few highly centralnodes. The Equation (25) can be rewritten in matrix terms. We have δv = Av specifying thatthe centrality vector v is defined by the eigenvector of A s corresponding to the largest eigenvalue δ . In general, each eigenvector of A s is a solution to Equation (25). Nevertheless, the centralityvector matching to the largest network component is specified using the eigenvector correspondingto the largest eigenvalue Bonacich (1972) and Peralta & Zareei (2016). Closeness
Closeness is a centrality measure proposed by Freeman (1978). It highlights the distance ofa given vertex to the rest of vertices in the network. It can be considered as duration of theinformation spread from one vertex to another. Closeness of a given vertex j is defined as follow: Closeness j = (cid:88) Ni =1 d ( i,j ) (26)Where d ( i,j ) measure the distance between a vertix i and another vertix j in the network. Betweenness
Betweenness is a centrality measure, which computes the number of times a vertex lies on theshortest path between other vertices in the network [V`yrost et al. (2019)].Suppose we need to compute this measure for a vertex j ∈ ω , for any two distinct vertices otherthan j , named l and k ∈ ω , the number of shortest paths between l and k is n l,j,k ∈ N andthe total number of shortest paths between l and k is n l,k ∈ N , the betweenness for j is thencomputed as follow: Betweenness j = (cid:88) l (cid:54) = j (cid:54) = kl,j,k ∈ ω n l,j,k n l,k (27)13 egree centrality Degree centrality captures total connectedness in the network, it is given by the followingequation: D = N (cid:88) j =1 N (cid:88) i =1 ( β sj,i ) (28)where ( β sj,i ) = (cid:40) if β sj,i (cid:54) = 00 if β sj,i = 0 Indegree:
Indegree computes the number of inflows, or in this paper the number of other FIsinfluencing one node. Indegree of
F I j is defined as: Ind j = N (cid:88) i =1 ( β sj,i ) (29)Here the F I j can be considered as a risk receiver. Outdegree:
Outdegree computes the number of out-going links, or in this paper the number ofother FIs influenced by the node. Outdgree of
F I i is defined as: Outd i = N (cid:88) j =1 ( β sj,i ) (30)Here the F I i can be considered as a "risk emitter". Consider N risky assets with a vector of expected returns µ , and covariance matrix, Σ = [ σ ii ] .The problem is to define the optimal portfolio weights vector w , which minimizes the portfoliovariance subject to w (cid:62) = where represents a column vector whose components are equivalentto one Markowitz (1959). This approach is usually defined as minimum-variance or shortly M inV ar . Formally the problem is stated as follow: min w σ p = w (cid:62) Σ w ; subject to w (cid:62) = (31)The solution of Equation (31) is: (cid:98) w ∗ minv = (cid:62) Σ − − (32) The HRP is a risk-based portfolio optimization approach that diversifies portfolios without impos-ing a positive-definite return covariance matrix [López de Prado (2016)]. The algorithm employs14achine learning methods and graph theory to classify the underlying hierarchical correlationstructure of the assets, which allow clusters of similar assets to compete for capital allocation.The algorithm of the HRP approach can be broken down into three main steps: tree clustering,quasi-diagonalization, and recursive bisection. In the following, we explain each step in detail.
Step 1- Hierarchical Tree Clustering
This step involves breaking down the assets of theconsidering portfolio into different hierarchical clusters exploiting a Hierarchical Tree Clusteringalgorithm. The clusters are formed as follows:1. For N stock returns compute the correlation matrix [see Härdle & Simar (2019)(p 431-442)],which gives an N × N matrix Ω of these correlations ρ , ρ = { ρ i,j } i,j =1 ,...,N Where ρ i,j is the correlation coefficient between a pair of assets { i, j } ρ [ X i , X j ]
2. Transform the correlation matrix to a correlation-distance matrix D , where for d : ( X i , X j ) ⊂ B → R ∈ [0 , d i,j = d [ X i , X j ] = (cid:114)
12 (1 − ρ i,j ) (33)3. Compute a new distance matrix ˜ d where, by taking the Euclidean distance among columns ina pair-wise manner, the augmented distance matrix is given as follow: ˜ d = ˜ d [ d i , d j ] = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) n =1 ( d n,i − d n,j ) (34)where ˜ d i,j : ( d i , d j ) ⊂ B → R ∈ [0 , √ N ] .Note that for two assets i and j , D i,j represents the distance between them, however ˜ d i,j repre-sents the closeness in similarity of { i, j } with the rest of the portfolio. More precisely, a lower ˜ d i,j indicates that the assets i and j are similarly correlated to the rest of stocks in the portfolio.4. Form the assets clusters in a recursive manner based in Equation(34). The set of clusters isdonated by U and the first formed cluster ( i ∗ , j ∗ ) is define as: U [1] = ( i ∗ , j ∗ ) = argmin ( i, j ) i (cid:54) = j { ˜ d i,j } (35)5. Update the distance matrix d by computing the distances of other assets from the newlyformed cluster U (1) using single linkage clustering. Therefore, for any asset i outside of U (1) ,the distance from U (1) is updated as follows: d i,U [1] = min (cid:2) { ˜ d i,j } j ∈ U [1] (cid:3) (36)Thereby, the algorithm recursively forms assets clusters and updates the distance matrix untilwe are left with one giant cluster of all stocks. Finally, the clusters are visualised in a dendrogram.See Härdle & Simar (2019) (p. 363-393) for more details. Step 2- Quasi Diagonalisation or Matrix Seriation
The Quasi-Diagonalisation of the covariance matrix or Matrix seriation is adopted to rearrangethe data to clearly represent the inherent clusters. Using the hierarchical clusters from theprevious step, the columns and rows of the covariance matrix are reorganized so that similarassets are placed together and dissimilar assets are placed far apart. More precisely, the larger15ovariances are positioned along the diagonal and smaller ones around this diagonal and sincethe off-diagonal elements are not completely zero. This is named a quasi-diagonal covariancematrix.
Step 3- Recursive Bisection
The final recursive bisection step implicates assigning actual portfolio weights to the assets inthe portfolio.1. Assign a unit weight to all assets, W i = 1 , ∀ i = 1 , ...N
2. Bisect each cluster into two sub-clusters in a top-down manner, it means by starting with thetopmost cluster, so, for each cluster, we obtain a left and right sub-cluster.3. Calculate the variance for each of these sub-cluster V , = w (cid:62) Σ w ; Σ is the covariance matrix (37)where, w = diag [Σ] − tr[diag [Σ] − ] (38)Since we are dealing with a quasi-diagonal matrix, the algorithm uses the property of the portfoliothat the inverse-variance allocation is optimal for a diagonal covariance matrix. Hence, we adoptthe inverse-variance allocation weights when computing the variance for sub-clusters.4. Calculate the weighting factor based on the quasi-diagonalised covariance matrix α = 1 − V V + V , so that ≤ α i ≤ α = 1 − α (39)5- Update the weights w and w for both sub-clusters: w (cid:48) = α · w ; w (cid:48) = α · w (40)6- Execute recursively steps 2-5. The algorithm stops when we have a single asset for eachcluster and then the weights are assigned to all assets in the portfolio.Since the weights are assigned in a top-down manner, only assets within each cluster competefor allocation for the final portfolio, rather than competing with all the assets in the portfolio,see V`yrost et al. (2019). The classical HRP approach focused on calculating the optimal portfolio weights based on thevariance and the covariance matrix, in other words it is focused on estimating the spilloverrisk on the mean-variance levels, ignoring analysis of other quantile levels. To overcome thislimitation, we extend the classical HRP based on FRM in order to take into consideration thehidden interdependency structures among the FIs tail quantile level when optimizing the EMsportfolio. The basic idea is to use the FRM output in order to compute the portfolio weightsbased on quantile level analysis. In the classical approach, these weights are computed usingthe variance (which is a measure of individual risk) and the covariance matrix that measuresthe relationship between a pair of assets in the mean-variance level. To improve on this, the16RM provides the penalty parameters ( λ j ) as a measure of individual risk of each company orFI at the quantile level. In addition, the FRM provides also the adjacency matrix to estimatethe interconnectedness between each pair of the studied FIs ( β i,j ) . By analogy, for the upliftedHRP approach based FRM, we replace the covariance matrix with the adjacency matrix. Recallthat the diagonal elements of the adjacency matrix are zero (Equation (24)), which allow usto introduce the vector of penalty parameters ( λ j ) in the diagonal of the adjacency matrix toreplace the variance in the classical approach. Following this reasoning, the uplifted HRP stepswill be as follows: Step 1- Hierarchical Tree Clustering :1. Calculate the the FRM adjacency matrix A s (Equation (24)) Where β i,j is the degree ofconnectedness between a pair of assets { i, j }
2. Introduce the vector of ( λ j ) as diagonal elements in the adjacency matrix A s , so we obtainthe new adjacency matrix ˜ A ˜ A s = λ β , · · · β ,n β , λ · · · β ,n ... ... . . . ... β n, β n, · · · λ n (41)3. Replace the covariance matrix in the HRP algorithm with the adjacency matrix ˜ A s .4. Transform ˜ A s to an adjacency-distance matrix D , D i,j = D [ X i , X j ] = (cid:113) (1 − β i,j ) if | β i,j | < max D [ X i , X j ] otherwise. (42)5. Compute a new distance matrix ˜ D ˜ D = ˜ D [ D i , D j ] = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) n =1 ( D n,i − D n,j ) (43)6. Form the assets clusters in a recursive manner based in Equation(43). The set of clusters isdonated by ˜ U and the first formed cluster ( i ∗ , j ∗ ) is define as: ˜ U [1] = ( i ∗ , j ∗ ) = argmin ( i, j ) i (cid:54) = j { ˜ D i,j } (44)7. Update the distance matrix D by computing the distances of other assets from the newlyformed cluster ˜ U (1) as follows: D i, ˜ U [1] = min (cid:2) { ˜ D i,j } j ∈ U [1] (cid:3) (45) Step 2- Quasi Diagonalisation or Matrix Seriation :The Quasi-Diagonalisation of the adjacency matrix or Matrix seriation is adopted to rearrangethe data to represent clearly the inherent clusters (as explained in the classical approach).17 tep 3- Recursive Bisection :1. Assign a unit weight to all assets, W i = 1 , ∀ i = 1 , ...N
2. Bisect each cluster into two sub-clusters.3. Calculate the variance for each sub-cluster ˜ V , = ˜ w (cid:62) ˜ A ˜ w ; ˜ A is the adjacency matrix (46)where, ˜ w = diag [ ˜ A ] − tr[diag [ ˜ A ] − ] (47)4. Calculate the weighting factor based on the quasi-diagonalised adjacency matrix. ˜ α = 1 − ˜ V ˜ V + ˜ V , so that ≤ ˜ α i ≤ α = 1 − ˜ α (48)5- Update the weights ˜ w and ˜ w for both sub-clusters: ˜ w (cid:48) = ˜ α · w ; ˜ w (cid:48) = ˜ wα · w (49)6- Execute recursively steps 2-5, the algorithm stops when we have a single asset for eachcluster and then the weights are assigned to all assets in the portfolio. In this paper we study the largest 25 Emerging Market financial institutions (FIs) by marketcapitalisation at any given point in time, with focus on the BRIMST FIs. We compile a databaseof daily price levels in as well as market capitalisations in U.S. Dollars from Bloomberg, and selectthe biggest financial institutions on a daily basis from the most liquid local EM equity marketindices. On any given trading day in consideration, we take the price returns of those biggest j = 25 FIs over an estimation window s = 63 business days.As for the macroeconomic data, we follow Adrian & Brunnermeier (2016) concerning thedeveloped market specific risk factors, to repeat, returns in US REITs, S&P 500 index, U.S. threemonths treasury bill rates, the spread between 3 months and 10 year U.S. treasury rates, thespread between BAA rates corporate bonds by the rating agency Moody’s to U.S. treasury bonds,and the implied volatility index VIX based on outstanding options on the S&P 500 equity index.We add the following EM specific macroeconomic risk variables: The J.P. Morgan EmergingBond Index Global Sovereign Spread index tracking the Emerging Bond Index yields over thebenchmark U.S. Treasury bonds and thereby representing the risk compensation demanded frominvestors when investing in EM sovereign bonds, as well as the respective countries currencyversus the U.S. dollar cross. Figure 1 depicts the FRM@EM from April 2000 to June 2020. Clearly observable are the periodsof distress in the global financial system around 2008, 2012, more recently in 2020 but also during18he EM specific market distress periods such as 2002 (Argentina) and 2013 (following the FederalReserve Board’s Open Market Committee forward guidance).Figure 2 introduces one of the FRM’s tool: FRM is not just the mean but actually a distributionof λ sj as well as the adjacency matrix containing all (Lasso penalised) β sj between FIs, and alsobetween FIs and macroeconomic risk variables. As an example of the information contained, wemark some of the more extreme maxima in Figure 2. For example, into the crisis, Standard BankGroup (SBK SJ) had a very high λ sj reading, indicating the bank was a "risk receiver", thus atrisk to be impacted by spill over effects.But we can also have a more detailed look at the adjacency matrix containing the β sj betweenFIs and macroeconomic risk variables, with two examples shown for 20200429 in Figure 4 aswell as post crisis 20200630 in Figure 5, both estimated at τ = 0 . , and colour scaled in blue(low/negative) to high (red/positive). Observable are the high dependencies between countries ofthe same region ( B S for Brazil, R M for Russia, I S for India, M F for Mexico, S J for South Africaand T I for Turkey), and often negative relationships (adjusted for co-movements with macroe-conomic risk variables) in between regions. However, there are also detectable co-dependencies,which necessitate closer inspection from both investors as well as regulators. For example inFigure 4 South Africa’s Sanlam Ltd. (SLM SJ)’s returns are explained to a significant extend byBajaj Finance Ltd. of India (BAF IS), as both financial services companies, providing an assort-ment of financial services. Similarly, Banco BTG Pactual (BPAC11 BS)’s returns are explainednot only by other Brazilian FIs, but also significantly by Russia’s VTB Bank (VTBR RM), bothoperating in more banking and investment banking related markets. Clearly, sub-sector depen-dencies across EM FIs are to be considered to prevent risk clusters.Another component to consider is the impact from macroeconomic risk variables’ changes on FIsreturns. As can be seen on both adjacency matrices, the "classic" macroeconomic risk variablesdo have an impact, however, to a large extend, the Emerging Market Sovereign Yield Spread toU.S. Treasuries (JPEGSOSD) is the main influence. In fact, this is true across from τ = 0 . to τ = 0 . . In Figure 3 we show the smoothed (rolling seven day mean) of the share of FIsimpacted by the respective macroeconomic risk variable. The Emerging Market Yield spread isthe dominant driver, followed by more general market risk measures such as the VIX, Moody’sBaa corporate yield spreads, and the shape of the U.S. Treasury yield curve. Emerging marketcurrencies as a cross versus the U.S. Dollar have a lesser impact. It is mostly one or two EMcurrencies having an overall impact, and not only on domestic banks. In Figure 5 for example,the Brazilian Real (USDBRL) has marginal negative return contribution to Bajaj Finance Ltdin India (BAF IS), and positive return contribution to Brazilian Itausa SA (ITSA4 BS) andMexican Grupo Elektra SAB (ELEKTRA MF).In Figure 7, we show the time series of FRM against various centrality measures. We observethat Betweenness, Eigenvector have similar trends as the FRM EM series, especially during thecrisis period of March to May 2020. On the other hand, Closeness centrality drops sharply intothe crisis period. When the FRM rises, the number of β sij equal to zero increase, increasing withit the distance between the vertices. The average length of one node (one FI in our case) andall other FIs increases, thereby sharply reducing Closeness centrality. Betweeness as a centralitymeasure of a vertex within a network rises when the FRM rises, as information flow has a veryhigh probability to pass between some central FIs, indicating a concentration of risk aroundcertain FIs which we call "risk emitters". Similarly, Eigenvector centrality is a measure of anFIs influence in an observed financial system network. Central FIs Eigencentrality rises sharplyaround a crisis period, as the FRM increases in value. In- and Out-degree centrality drop whenFRM rises, since the edges or connections between FIs have reduced sharply, to mostly thenetwork’s risk emitters. We can conclude therefore, that a close inspection if the distributionof λ sj as well as the detailed information within the adjacency matrix across a range of τ areparticularly important. In Section 6 we aim to make use of this richness of information for the19onstruction of more robust, tail-event network behaviour attentive portfolios. As an indication,Figure 6 shows a network graph with edges between the 25 largest FIs, estimated at τ = 0 . on20200429. We highlight one exemplary bank, and its edges stemming from the adjacency matrix.Condensing such information into clusters of risk as outlined above is the focus of the followingportfolio construction discussion. In Figure 3 we depict the macroeconomic risk variables influ-ence over time. Clearly, the EM Sovereign Yield Spread to U.S. Treasury Bonds is the prominentmacroeconomic risk variable not only during tail events but also at "normal" times estimatedat τ = 0 . . We can conclude that most EM risk is rapidly priced into yield spreads, and thenconsequently impacts financial institutions. This link between the sovereign and banks needs tobe considered for investors and policy makers as well. Other strongly influencing risk variablesare more expected such as the VIX, the US yield curve shape, and Corporate Bond yield spreads.However, EM currency fluctuations only have a marginal effect in tail events, on some of the FIs.With the increase of debt issuance in local currency denominated debt, the risk of mismatchesversus developed market currencies has diminished overall, and has lower influence on EM FIs.Figure 1: FRM@EMs Time series20igure 2: FRM@EM Boxplot, with mean and maximum at τ = 0 . a) τ = 0 . (b) τ = 0 . (c) τ = 0 . (d) τ = 0 . Figure 3: Macro variables’ most frequent marginal return contribution across time with 7-daymoving average: EM Sovereign Spread, VIX, U.S. 3mth to 10yr yield spread, Moody’sBAA Corporate Yield Spread, S&P 500 Index22 S R M I S B S I S B S B S I S T II S B S I S I S B S B S B S M FS J I S B SS J M FS J B S R M t a u = . HDFCB IS SBER RM HDFC IS ITUB4 BS KMB IS BBDC3 BS BBDC4 BS ICICIBC IS QNBFB TI SBIN IS SANB11 BS BAF IS AXSB IS BBAS3 BS ITSA4 BS B3SA3 BS ELEKTRA MF FSR SJ BJFIN IS BBSE3 BS SBK SJ GFNORTEO MF SLM SJ BPAC11 BS VTBR RM REIT SPX USGG3M VIX USGG3M10YR MOOD BAA SPD JPEGSOSD USDBRL USDRUB USDINR USDMXN USDTRY I S H D F C B I S . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . - . - . - . . . . . . R M S B E R R M . . . . . . . . . . . . . . . . . . . - . . . . . . . . . - . . - . . . . . . . I S H D F C I S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . B S I T U B B S . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . - . . . . . . . I S K M B I S . . . . . . . . - . . . . . . . . . . . . . . . . . . . - . . . - . . . . . . . B S BB D C B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . B S BB D C B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . - . . . . . . I S I C I C I B C I S . . . - . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . T I Q N B F B T I . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I S S B I N I S . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . - . . . . . . B S S A N B B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I S B A F I S . . . . . . . . - . . - . . - . - . . . . . . . . . . . . - . - . . . . . . . . . . . I S A X S B I S . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . - . - . . . . . . B S BB A S B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . B S I T S A B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . B S B S A B S . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . - . . - . . . . . . M F E L E K T R A M F . . . . . . . . . . - . . . . . . . . . . . . . . . . . - . - . - . . - . . . . . . S J F S R S J . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . - . . - . . . . . . I S B J F I N I S . . . . . . . . . . . . . . . . . - . . - . . . . - . - . . . - . . . . - . . . . . . B S BB S E B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . - . . . . . . S J S B K S J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . M F G F N O R T E O M F . . - . . . . . . . . . . . . . . . . - . . . . . . . . . . - . . . - . . . . . . S J S L M S J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B S B P A C B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . R M V T B R R M . . . . . . . . . . . . . - . . . . . . . . . . . . . . . - . - . - . - . . . . . . R E I T . . . . . . . . . . . . . . . . . . . . . . . . . S P X . . . . . . . . . . . . . . . . . . . . . - . . . . U S GG M . . . . . . . . - . . . . . . . . . . . . . . . . - . V I X . . . . . . . . . . . . . . . - . . . . . . . . . - . U S GG M Y R . . . . . . . . - . . . . . . . - . . . . . . . . . . M OO D B AA S P D - . . . . . . . . . . . . . . . . . - . . . . . . . . J P E G S O S D . . . . . . . . . . . . . . . . . - . . . - . . . - . . U S D B R L . . - . . . . . . . . . - . . . . . . . . . - . - . . . . U S D R U B . . . . . . . . . . . . . . . . . - . - . . . . . . . U S D I N R . . . . . . . . . . . . . . . . . . . . . . . - . . U S D M X N . . - . . . . . . . . . . . . . . . . . . . . . . . U S D T R Y - . . . . . . . . - . . . . . . . . . . - . . . . - . . . F i g u r e : A d j a ce n c y m a t r i x e s t i m a t e d a t τ = . o n S R M B S I S I S B S B S I S I S T II S B S B S B S B S I S M F I SS J B S M F B SS J S J R M t a u = . HDFCB IS SBER RM ITUB4 BS HDFC IS KMB IS BBDC3 BS BBDC4 BS ICICIBC IS BAF IS QNBFB TI SBIN IS B3SA3 BS SANB11 BS BBAS3 BS ITSA4 BS AXSB IS ELEKTRA MF BJFIN IS FSR SJ BPAC11 BS GFNORTEO MF BBSE3 BS SBK SJ SLM SJ VTBR RM REIT SPX USGG3M VIX USGG3M10YR MOOD BAA SPD JPEGSOSD USDBRL USDRUB USDINR USDMXN USDTRY I S H D F C B I S . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . R M S B E R R M . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . B S I T U B B S . . . - . . . . . - . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . I S H D F C I S . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . - . . . . . . . . I S K M B I S . . . . . . . . . - . . . . . . . . . . . . . . - . . . . . . . . - . . . . . . B S BB D C B S . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . - . . . . . . B S BB D C B S . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . - . . . . . . I S I C I C I B C I S . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . - . - . . . . . . . I S B A F I S . . . . . . . . . . . . . - . . - . . . . . . . . . . . . . . - . . - . . . . . . T I Q N B F B T I . . . . . . . . . . . . - . . . . . - . . . - . . . . . . . . - . . - . . - . . . . . I S S B I N I S . . - . . . . . . . - . . . . . . . . . . . . . . . . . . . . . - . - . . . . . . B S B S A B S . . . . . . . . - . . . . . . . . . . . . - . . . . . . . . . - . . - . . . . . . B S S A N B B S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . . . . . . B S BB A S B S . . . . . . . . . . . . . . . . . . . . - . . . . . . . . - . . . . . . . . . B S I T S A B S . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . - . - . . . . . . I S A X S B I S - . . . . . . . . . . . . . . . . . . . . - . . - . . . . . . . . . - . . . . . . M F E L E K T R A M F . . . . . . . - . . . . . . . . . . . . . . . - . . . - . . . . . . . . . . . . I S B J F I N I S . . . . . . . . . - . . - . . . . . . . - . . . . - . . . . . - . - . . . . . . . . . S J F S R S J . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . - . . . . . . . B S B P A C B S . . . - . . . . - . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . M F G F N O R T E O M F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . B S BB S E B S . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . - . . - . . . . . . S J S B K S J . . . . . . . . - . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . S J S L M S J . . - . . . . . . . . . . - . . . . . . . . . - . . . . . . . - . - . . - . . . . . . R M V T B R R M . . . . . . . . . . . . . . . . . - . . . . - . . . . . . . . . . . . . . . . R E I T . . . . . . . . . . . . . . . . . - . . . . . . . . S P X . . . . . - . . . . . . . . . . . . . . . - . . . . . U S GG M . . . . . . . . . - . . . . - . . . . . . . - . . . . . V I X . . . . . . . . . . . . . - . . . . . . - . . . . . . U S GG M Y R . . . . . . . . . - . . . - . . . . . - . . . . . . - . . M OO D B AA S P D . . . - . . . . . - . - . . . . . . - . . . - . . . . . . . J P E G S O S D . . . . . . . . - . . . . . . . . . . . . . . . - . . U S D B R L . . . - . . . . . . . . . . . . . . . . . - . . - . . . U S D R U B . . . . - . . . . . . . . - . . . . . - . . - . . . - . . . U S D I N R - . . . - . . . . . . . . . . . . . . . . . . . . . . U S D M X N . . . . . - . . . . . . . . . . . . . . . . - . . . . U S D T R Y . . . . . . . . - . . . . . . . . . . . - . . . - . - . . F i g u r e : A d j a ce n c y m a t r i x e s t i m a t e d a t τ = . o n a) FRM and Betweenness (b) FRM and Closeness(c) FRM and Eigenvector (d) FRM and In-degree(e) FRM and Out-degree Figure 7: FRM at τ = 0 . and Centrality measures The portfolio is constructed using daily FIs prices for the period between 20200101 and 20200630(which yields 130 observations of the 25 biggest EM FIs).The performance of the uplifted portfolio approaches (Inv λ and upHRP) are benchmarked againstthe minimum-variance portfolio (MinVar), the inverse variance (IVP), and the classical HRPapproaches. The MinVar weights are simply computed based on Equation (32), and often lead toconcentration in low-volatility assets. The IVP strategy can be considered as a naive risk paritystrategy (Equation (38)), since it is agnostic with respect to asset correlations. Thereafter,26n overlapping region between portfolio optimisation startegies and FRM network centrality isdeveloped.As shown in the FRM network (Figure 6), all FIs are treated as potential substitutes withoutspecifying any hierarchical structure among them. Therefore, tree structures that integratehierarchical relationships are needed. For that purpose, the upHRP algorithm aims to build andmake use of a clustered adjacency matrix. Matrix seriation
For the classical approach, the steps of the HRP are applied using the price time series of the25 biggest FIs selected during the studied period. Figure 8 presents the estimation results: (a) Unclustered Correlations (b) clustered Correlations
Figure 8: Classical HRP: Matrix SeriationFigure 8a shows the original or the unclustered correlation matrix. Figure 8b illustrates thecorrelation matrix after reordering in clusters using the hierarchical tree clustering, also calledclustered correlation matrix. This matrix then serves as the input for the asset allocation proce-dure. As shown in Figure 8b the reorganizing results group similar FIs together and the dissimilarfurther away in a new correlation matrix, which helps to construct more meaningful asset allo-cation decisions and building more risk diversified portfolios. More precisely, from Figure 8b thelighter-colored squares (indicating a higher correlation coefficient) are all concentrated aroundthe diagonal matrix. 27 a) Unclustered Adjacency matrix (b) clustered Adjacency matrix
Figure 9: Uplifted HRP: Matrix SeriationFor the uplifted approach, the quasi-diagonalization step is applied to the adjacency matrixdated on 20200630 with τ = 0 . . In this stage, the algorithm aims at reordering the adjacencymatrix by placing similar assets together. As we can notice from Figure 9b the closer assets(similar given ( β )) are placed together forming assets clusters. Tree clustering or Dendrogram
Figure 10: Classical HRP: EM@DendrogramThe clusters are visualised in the form of a cluster diagram called a dendrogram. Figure 1028llustrates the hierarchical clusters for our FIs data, where the x-axis indicates the name of theFIs in the studied portfolio and the y-axis measures the distance between the two merging FIs.The key to interpreting a dendrogram is to focus on the height at which any two FIs are joinedtogether. In Figure 10, we can see that Bajaj Finance (BAF IS) and Bajaj Finserv Ltd (BJFINIS) (plotted with green) are expectedly most similar, as the height of the link that joins themtogether is the lowest. Note that both of them are same ultimate parent Indian financial servicescompanies focused on insurance. The dendrogram also detects that the most similar FIs mostoften belong to the same market (FirstRand Ltd (FSR SJ) and Standard Bank Group Ltd (SBKSJ) plotted with red), or a similar financial sector. Finally, the highest cluster represents thegiant cluster that joins all the FIs formed clusters together.Figure 11: Uplifted HRP: EM@DendrogramBased on the FRM clustered adjacency matrix, Figure 11 indicates that the tree clusteringarchitecture differs from the previous one (Figure 10). Indeed, here State Bank of India (SBINIS) is a perturbation of BJFIN IS, and consequently the two assets present the first formed clustersince they are the most similar FIs. Also, Grupo Financiero Banorte (GFNORTEO MF) andVTR Bank (VTRB RM) are perturbations of Sberbank of Russia (SBER RM), hence these threeFIs are clustered together, forming a cluster of two different markets, contrary to the previousdendrogram, Figure 10, where the most similar FIs belong the same market.
Calculate allocation through recursive bisection
Table 1 and Figure 12 specify further explanation regarding the weight’s allocation of thestudied FIs according to the adopted strategies.29able 1: Weight allocations of FIs: Classical approach vs uplifted approach
EM@ FIs MinVar IVP HRP Inv λ upHRP HDFCB.IS.EQUITY 0.002 0.045 0.017 0.039 0.051SBER.RM.EQUITY 0 0.054 0.020 0.021 0.018ITUB4.BS.EQUITY 0 0 0 0.019 0.074HDFC.IS.EQUITY 0 0.032 0.007 0.065 0.023KMB.IS.EQUITY 0.062 0.038 0.011 0.036 0.003BBDC3.BS.EQUITY 0 0 0 0.055 0.074BBDC4.BS.EQUITY 0.002 0.005 0.002 0.017 0.030ICICIBC.IS.EQUITY 0.002 0.027 0.005 0.022 0.072QNBFB.TI.EQUITY 0.008 0.009 0.006 0.084 0.001SBIN.IS.EQUITY 0 0.035 0.010 0.030 0.044B3SA3.BS.EQUITY 0 0 0 0.040 0.053SANB11.BS.EQUITY 0 0.024 0.007 0.029 0.107BBAS3.BS.EQUITY 0 0.016 0.025 0.032 0.012BAF.IS.EQUITY 0 0.021 0.014 0.056 0.012ITSA4.BS.EQUITY 0.001 0 0 0.018 0.024AXSB.IS.EQUITY 0 0.018 0.009 0.041 0.056ELEKTRA.MF.Equity 0.866 0.503 0.787 0.032 0.073FSR.SJ.EQUITY 0.001 0.033 0.018 0.062 0.016SBK.SJ.EQUITY 0 0.029 0.011 0.074 0.034BBSE3.BS.EQUITY 0 0 0 0.052 0.041BJFIN.IS.EQUITY 0.004 0.026 0.012 0.025 0.035GFNORTEO.MF.Equity 0.044 0.035 0.017 0.037 0.004BPAC11.BS.EQUITY 0 0.010 0.003 0.032 0.016SLM.SJ.EQUITY 0.007 0.039 0.018 0.045 0.084VTBR.RM.EQUITY 0 0 0 0.037 0.04330 H D F C B I S E Q U I TY S B E R R M E Q U I TY I T U B B S E Q U I TY H D F C I S E Q U I TY K M B I S E Q U I TY BB D C B S E Q U I TY BB D C B S E Q U I TY I C I C I B C I S E Q U I TY Q N B F B T I E Q U I TY S B I N I S E Q U I TY B S A B S E Q U I TY S A N B B S E Q U I TY BB A S B S E Q U I TY B A F I S E Q U I TY I T S A B S E Q U I TY A X S B I S E Q U I TY E L E K T R A * M F E Q U I TY FS R S J E Q U I TY S B K S J E Q U I TY BB S E B S E Q U I TY B J F I N I S E Q U I TY G F N O R T E O M F E Q U I TY B P A C B S E Q U I TY S L M S J E Q U I TY V T B R R M E Q U I TY HPR Weights MinVar Weights IVP Weights (a) Optimal weights MinVar, IVP, HRP H D F C B I S E Q U I TY S B E R R M E Q U I TY I T U B B S E Q U I TY H D F C I S E Q U I TY K M B I S E Q U I TY BB D C B S E Q U I TY BB D C B S E Q U I TY I C I C I B C I S E Q U I TY Q N B F B T I E Q U I TY S B I N I S E Q U I TY B S A B S E Q U I TY S A N B B S E Q U I TY BB A S B S E Q U I TY B A F I S E Q U I TY I T S A B S E Q U I TY A X S B I S E Q U I TY E L E K T R A * M F E Q U I TY FS R S J E Q U I TY S B K S J E Q U I TY BB S E B S E Q U I TY B J F I N I S E Q U I TY G F N O R T E O M F E Q U I TY B P A C B S E Q U I TY S L M S J E Q U I TY V T B R R M E Q U I TY mHPR Weights InvLambda Weights (b) Optimal weights Inv λ , upHRP Figure 12: Weights allocations of FIs: classical approach vs uplifted approach31able 1 and Figure 12 specify further explanation regarding the weight’s allocation of thestudied FIs according to the adopted strategies. Several points are worth noting. First, forboth classical and uplifted approaches, IVP, HRP, Inv λ and upHRP and similarly allocate theirassets. This result can be explained by the fact that they are based on the inverse variancemethod to compute their weights. However, while IVP and Inv λ show more stable weightdistributions, HRP and upHRP adjust their weight allocation more frequently, since it also takesinto consideration the correlation (for HRP)/ tail connectedness (for upHRP) and the assetclusters when assigning weights. Second, results also show that the MinVar allocations have themost concentrated weight allocation. While it accorded zero allocation to certain FIs (i.e SBINIS, B3SA3 BS, etc.), it significantly overweights others (ELEKTRA MF). Third, the MinVar,IVP and HRP classical approaches over-weighted the Mexican ELEKTRA MF (0.866, 0.503,0.787, respectively). In this case, any distress situation affecting this FI will have a great impacton those concentrated portfolios. Contrary, these extreme weight concentrations disappear withthe Inv λ and upHRP uplifted approaches, where the weights are wider distributed among FIs,providing a well-diversified portfolio.Table 2: Weight’s allocation of EMs: Classical approach vs Uplifted approach EM@ FIs Nbr of FIs MinVar IVP HRP Inv λ upHRPIndia (IS) Brasil (BS)
Mexico (MF)
Russia (RM)
Turkey (TI)
S.Africa (SJ) λ allocates 29% of weights to the Brazilian and Indian FIs, while, the upHRP allocates 29.4%to Indian FIs and 43.1% to the Brazilian FIs since this last market has the highest number ofFIs in the selection of the 25 biggest EM@FIs. Moreover, the upHRP allocates only 0.1% for theTurkish FIs, since this market contributes only by one FI. To recapitulate, the upHRP appearsto find a compromise between classical concentrated weight approaches and the Inv λ strategy.The classical strategies can concentrate weights on a few FIs, leading to vulnerabilities. The Inv λ evenly assigns weights across all FIs, ignoring the correlation structure. This makes it exposedto systemic shocks. However, the upHRP finds a compromise between diversifying across all FIsand diversifying across clusters, which makes it more resistant against both types of shocks.32able 3: Backtesting on EM FIs data: Classical approach vs uplifted approach Mean Std Sharpe ratio Effective nMinVar
IVP
HRP τ = 5% Inv λ upHRP τ = 10% Inv λ upHRP τ = 25% Inv λ upHRP τ = 50% Inv λ upHRP λ , upHRP) 33 a) Backtesting on EM asset data: uplifted ap-proach (Inv λ λ λ λ λ ) vs Optimal weights (Penalisation parameter ( λ , Inv λ
5% weights, upHRP 5% weights)
Figure 14: Impact of Penalisation parameter ( λ ) and FRM on optimal weights and portfolioreturns Backtesting results
For the backtesting, we adopt a 30 days rebalancing period. Table 3 reca-pitulates the out-of-sample performance of the studied approaches.For the classical approaches, the mean and the volatility of the HRP are 0.003 and 0.049, re-spectively. While the MinVar delivers the highest return value (0.004) and the highest volatility(0.081), the IVP provides approximately the same portfolio return (0.003) with higher volatility(0.0712). Therefore, the HRP balances out both return and volatility most efficiently providingthe best approach in terms of Sharpe ratio (0.072 compared to 0.052 and 0.056 of the MinVarand IVP portfolios, respectively).For the uplifted approach (with τ = 5% ), the return and volatility of the upHRP approach are0.052 and 0.188, respectively. While the Inv λ offers a higher return (0.076), and approximatelythe same volatility, the Inv λ provides a risk-return balance registering the best Sharpe ratio value(0.412 compared to 0.276 of the upHRP portfolio).Besides the sharp ratio, we measured the portfolio diversification effects using the Effective NEf f ect ( N ) measurement, proposed by Strongin et al. (2000) as one of allocation concentrationand defined as follow: Ef f ect ( N ) = 1 (cid:80) Nj =1 (cid:98) ω j,t (50)34he portfolio has high concentration risk if Ef f ect ( N ) is close to one.Comparing the classical and the uplifted approaches(with τ = 5% ), it seems that the upHRPimproves the Sharpe ratio of the HRP approach (0.072 and 0.276, respectively). Moreover,the HRP has discarded five EMs in favor of one single market, with ( Ef f ect ( N ) = 1 . , seeTable 2). Therefore, the HRP’s portfolio is deceitfully diversified, since any distress situationaffecting this market will have a greater negative impact on HRP’s portfolio than the upHRP,which allocates only 6.9% to the Mexican market and providing more diversification across EMs( Ef f ect ( N ) = 16 . , see Table 3).In fact, the main innovation of upHRP strategy is to apply the HRP algorithm based on taildependence clustering instead of the standard correlation-based clustering. First, this strategyis motivated by the fact that the correlation coefficients can change drastically during financialcrises due to contagion effects, and such a crisis can spill over quickly. Consequently, diversifyinga portfolio based on correlation clusters may be a failing strategy without attention to tail events.Second, while the correlation matrix illustrates the relationship based on the mean-variance ofthe distribution, the FRM measures the co-movements in extreme events based on both tail ofthe distribution. To exemplify, lower tail dependence is associated with the capacity to diversifyduring crises, which can improve the tail risk management of a given strategy. More precisely,the main idea is to cluster the studied FIs illustrating a high probability to experience extremelynegative events contemporaneously. Finally, empirical results show that the proposed upliftedapproach has the potential to compete with the classical portfolio optimization approach byproviding desirable diversification properties, especially if this hierarchical risk parity strategyis based on the tail dependence coefficient (provided by the FRM adjacency matrix), which is abenefit to tail risk management.Besides, for the uplifted approach, we take into consideration four tail risk levels (5%, 10%,25%, 50%). The results in Table 3 indicate that: The Ef f ectN is an increasing function of thetail risk level, therefore for a higher tail risk level, more diversification is needed to guaranteestable portfolio returns. Comparing the Inv λ strategy with the upHRP, we noticed that the Inv λ strategy needs to include a higher number of assets compared to the upHRP (i.e for τ = 10% , Ef f ect ( N ) = 21 . and 16.24, respectively) to provide the same portfolio performance level (approximately the same sharp ratio=0.4). On the one hand, this result can be explained bythe fact that for the upHRP, only FIs within the same cluster compete for portfolio allocationrather than competing with all the FIs in the portfolio, which avoids the redundancy in theFIs included in the optimal portfolio. On the other hand, the larger number of FIs increasesestimated parameters, which also can increase the risk of estimation error and thus biased results.For that purpose, for relatively similar values of sharp ratio (portfolio efficiency), the upHRP ispreferred over the Inv λ .Figure 13 illustrates the backtesting on EM’s FIs with 30 days rebalancing period. It plotsthe portfolio turnover for the first half of 2020 according to the classical strategies (MinVar, IVP,and HRP) and the uplifted strategies (Inv λ and upHRP), where the classical approaches show arelative stability in returns compared to the uplifted approaches since, the classical approachesconcentrate their weights on one Mexican FI (see Table 2), which limited their profit (increase inportfolio returns in mid-February and in June) and also limit their loss (end of March ) duringthe COVID 19 crisis. However, this is a trade-off between historical stability and very highconcentration risk. Any idiosyncratic risk impact would lead to substantial portfolio losses.Figure 14a plots the backtesting on EM’s FIs with 30 day rebalancing period for the upliftedstrategies, taking into consideration the adjacency matrices with different tail risk levels (5%,350%, 25%, and 50%). The Figure indicates that all portfolio returns follow the same trend, despitethe difference in portfolio composition, since the biggest 25 FIs that compose the adjacencymatrices vary over time and the risk levels.Indeed, by comparing this portfolio return’s trend with the FRM plot (Figure 14b) during thesame period, we noticed a negative relationship, where the high-risk period plotted by the FRMis translated by a decrease in the portfolio returns (March). Moreover, Figure 14c indicates thatthe uplifted approach underweights the high risky FIs (with high penalization parameter λ ) andoverweight the less risky FIs, specially for the Inv λ , since the upHRP takes into considerationthe tail co-movement between the FIs. 36 .1 Bridging optimal portfolio weights and network centrality (a) τ = 0 . (b) τ = 0 . (c) τ = 0 . (d) τ = 0 . Figure 15: Network graphs with the size of nodes representing total degree centrality on 20200630Figure 15 plots the FRM Network with the size of nodes representing total degree centrality(indegree and outdegree). The network information is provided by the adjacency matrices esti-mated at τ equal to 0.05, 0.10, 0.25, and 0.50, with dates 20200630. The Figure shows that thetotal degree centrality (both in and out) increase and become more visible with the increasingof tail risk level τ . By interpreting the adjacency matrix of the network, an overlapping regionbetween the portfolio theory and network theory can be established.Indeed, according to Figure 15a, Itau Unibanco (ITUB4 BS) is a highly central FI followed byBanco Santander Brasil (SANB11 BS) and Banco Bradesco SA (BBDC4 BS). Therefore, the37razilian market illustrates high-CoStress compared to the other markets (see also the boxplot,Figure 2. Contrary to that, QNB Finansbank (QNBF TI) and Housing Development FinanceCorp (HDFC IS) are low-central FIs.Regarding ELEKTRA MF, the networks (Figure 15) indicate that this FI is only a risk emitter(high out-degrees with zero in-degree), which makes this FI insensitive to the other FIs shocks.Moreover, the classical portfolio approaches overweight ELEKTRA MF (see Table 1). Therefore,at least historically, ELEKTRA MF had more stable return patterns, but as a risk emitter, anychange in risk perception concerning this FI would rapidly spill over into the entire market.Moreover, referring to Table 4, the results of the optimal weight indicate that the Inv λ portfoliois insensitive to the centrality degree. It often allocates similar weights for the high and lowcentral FIs (i.e. BBDC4 BS and GFNORTEO MF, see Figure (15d). This result is due to thefact that this portfolio approach allocates weights based only on the individual tail risk level Λ ,varying by tail risk levels τ but is insensitive to the network centrality degree.Referring to Table 4 and Figure 15, results from different τ levels indicate that the upHRP ap-proach underweights the high central FIs (i.e. BBDC4 BS (for τ =5% and τ =50%), ICICIBCIS (for τ =10%), ITUB BS and HDFC IS (for τ =25%)). Contrary to that, the upHRP strategyoverweights the low central FIs (i.e. ELEKTRA MF and SLM SJ (for τ =5%), ELEKTRA MF(for τ =10%), GFNORTEO MF and B3SA3 BS (for τ =25% and τ =50%). These findings are inline with previous research, studying the relationship between the optimal portfolio weights andnetwork centrality in the mean-variance level. In fact, our findings are consistent with Peralta& Zareei (2016) and Pozzi et al. (2013) in establishing that "optimal portfolio strategies shouldoverweigh low-central assets and underweight high-central ones". Nonetheless, they contradictthe findings of V`yrost et al. (2019), who argue that “asset weights must be ordered in the sameway as the reciprocal of asset centrality in a given network”.Indeed, the sensibility of the Adjacency matrix to different tail risk levels τ results in centralitydegree variation as well as FIs clustering (see Figures 11 and 16), which leads to upHRP optimalweights variation through different τ for all FIs.In fact, the low concentration of weights for the uplifted strategies (Inv Λ and upHRP) as well asthe sensitivity of weights to different tail risk levels, don’t allow us to conclude about the best FIthat offers greater diversification benefits in EMs (contrary to the classical approach that con-centrates weights on ELEKTRA MF). However, uplifted strategies overweight the Brasilian FIs(for different τ ). Moreover, by contributing with the highest number of FIs to the basket of the25 biggest FIs in EMs (see Table 2), the Brazilian market seems more interesting to internationalinvestors, besides the need of diversification advantages provided by the other EMs. Besides the outlined portfolio construction in Section 6, we want to derive some recommenda-tions for both policy makers as well as investors.In Section 5 we have shown that particular EM FIs are not only influenced by FIs and macroeco-nomic risk variables from the same region, but also have tail-event co-movement dependencies toother EM geographic regions and currencies, the latter all the more important at lower levels of τ = 0 . or τ = 0 . . Secondly, a dominant driver of EM FIs returns is the fiscal and economicalstance of EM Sovereign bond issuers. If the risk perception for sovereign issuers increases, thereturn of EM FIs is impacted across the board and across the multivariate distribution. Lastly,co-movements can be detected within similar economic sub-sectors across geographical regions,which is also apparent in the portfolio construction exercises in Section 6. We recommend in-vestors to analyse concentrations across economic sub-sectors across regions, as well as to analysedependencies between equity investments and bond investments in EM, given the clear linkages38etween the sovereign and the domestic EM FIs. Lastly, currency fluctuations have marginalreturn contributions in tail-events especially. In so far as EM FI investments are unhedged intodeveloped market currencies, there is risk of a compound effect on returns.As for the portfolio construction, the "classical" approaches show a relative stability in portfo-lio returns compared to the uplifted approaches, which limited their profit and also limit theirloss during the COVID 19 crisis. But, this often comes at the cost of high concentration risks,exposing the portfolio to idiosyncratic tail events. The uplifted portfolio approaches show morevolatile turnover with higher loss and profit, but are much better diversified, preventing sizeablerisk clusters. Even though a risk averse investor might opt for the classical approach at first,closer inspection of tail risk behaviour and concentration risk should let the investor prefer theuplifted portfolio.EM policy makers can derive important recommendations from our analysis. Firstly, coordina-tion between EM regulatory bodies is of importance in order to mute EM FI fluctuations. Here,particular attention should be put on linkages in same sub-sector operating FIs across regions,which will be increasingly more important as globalisation continues. Secondly, fluctuations inrisk perception of the sovereign issuer has an immediate impact on EM FI returns. Regulatorybodies are therefore advised to preemptively verify sound capitalisation of their domestic banks,even if the sovereign issuers distress is stemming from another geographic region. This is furtheramplified by a tail-event leading to a weakening of an EM’s currency versus for example the U.S.Dollar. Global financial linkages, we show, between EM FIs lead to spill-over effects. Overall, inorder to protect the domestic as well as global EM economies and their FIs, EM regulatory bod-ies should continuously work towards closer coordination between Emerging Market economies.This will help increase robustness versus developed market economies distress, of which more islikely to come, as developed markets fight with difficult fiscal situations and low growth patterns.But also, closer coordination between countries will prevent spill-over effects from one geograph-ical region’s FIs onto others, thereby increasing the attractiveness of the asset class further. Ourapproach via dendograms rapidly indicates any such risk clusters, and can be updated at easeand frequently during financial crises. In this study, we examine the co-movements of EM FIs across six geographical regions, withaim to analyse within EM country co-dependencies in tail events but importantly also betweenregions as well as FIs of the same sub-sector across regions. We also analyse the importantmacro-economical risk variables impacting EM FIs and conclude that in addition to the devel-oped market variables suggested by Adrian & Brunnermeier (2016), EM specific macroeconomicrisk variables have significant explanatory power. This can be used to construct more robusttotal asset class portfolio allocation and supplies EM regulatory bodies with detailed informationon co-dependencies for better and faster stabilisation measures during periods of distress.We also propose a novel asset allocation method – Hierarchical Risk Parity based the tail eventinformation from the FRM technology, allowing us to extend this approach to the quantile leveland replace the covariance matrix with the rich information contained in the FRM adjacencymatrix. We applied this proposed approach to a portfolio of the biggest 25 FIs in EM, and our re-sults show that uplifted strategies provide appropriate diversification properties. In comparison,the Inv λ portfolios tend to be too static and the classical approaches result in too concentratedportfolios. Bridging optimal portfolio weights and network centrality, we conclude that the Inv λ insensitive to the network centrality degree. 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Table 4: Sensitivity of the optimal weights of the uplifted strategies to different tail risk levels τ FIs
5% 10% 25% 50%Inv ( λ ) upHRP Inv ( λ ) upHRP Inv ( λ ) upHRP Inv ( λ ) upHRPHDFCB.IS.EQUITY 0.039 0.051 0.030 0.017 0.021 0.320 0.038 0.027SBER.RM.EQUITY 0.021 0.018 0.021 0.086 0.053 0.052 0.037 0.031ITUB4.BS.EQUITY 0.019 0.074 0.023 0.069 0.020 0.026 0.026 0.066HDFC.IS.EQUITY 0.065 0.023 0.038 0.034 0.022 0.047 0.026 0.027KMB.IS.EQUITY 0.036 0.003 0.026 0 0.027 0.052 0.022 0.031BBDC3.BS.EQUITY 0.055 0.074 0.041 0.050 0.032 0.022 0.045 0.069BBDC4.BS.EQUITY 0.017 0.030 0.027 0.043 0.020 0.039 0.030 0.027ICICIBC.IS.EQUITY 0.022 0.072 0.022 0.026 0.024 0.023 0.034 0.052BAF.IS.EQUITY 0.056 0.012 0.032 0.049 0.044 0.014 0.057 0.033QNBFB.TI.EQUITY 0.086 0.001 0.023 0.001 0.029 0.026 0.028 0.046SBIN.IS.EQUITY 0.030 0.044 0.036 0.049 0.025 0.024 0.022 0.066B3SA3.BS.EQUITY 0.040 0.053 0.060 0.030 0.030 0.114 0.030 0.054SANB11.BS.EQUITY 0.029 0.107 0.033 0.059 0.025 0.050 0.028 0.053BBAS3.BS.EQUITY 0.032 0.012 0.026 0.039 0.019 0.018 0.026 0.039AXSB.IS.EQUITY 0.018 0.024 0.029 0.083 0.044 0.024 0.042 0.019ITSA4.BS.EQUITY 0.041 0.056 0.128 0.033 0.091 0.042 0.068 0.045ELEKTRA..MF.Equity 0.032 0.073 0.031 0.043 0.058 0.016 0.046 0.044BJFIN.IS.EQUITY 0.025 0.035 0.043 0.052 0.039 0.056 0.035 0.024FSR.SJ.EQUITY 0.062 0.016 0.062 0.051 0.021 0.057 0.052 0.023BPAC11.BS.EQUITY 0.032 0.016 0.051 0.030 0.060 0.018 0.058 0.033BBSE3.BS.EQUITY 0.037 0.004 0.032 0.007 0.041 0.010 0.042 0.029GFNORTEO.MF.Equity 0.052 0.041 0.044 0.024 0.090 0.076 0.030 0.070SBK.SJ.EQUITY 0.074 0.034 0.052 0.050 0.046 0.073 0.070 0.039SLM.SJ.EQUITY 0.045 0.087 0.049 0.029 0.050 0.078 0.048 0.018VTBR.RM.EQUITY 0.037 0.043 0.042 0.044 0.058 0.021 0.058 0.038 a) τ = 0 . (b) τ = 0 . (c) τ = 0 . Figure 16: Sensitivity of the upHRP’ dendrogram of to different tail risk levels τ Financial Institutions Emerging Market
ITUB4 BS Itau Unibanco Holding SA BRAZILBBDC3 BS Banco Bradesco SA BRAZILBBDC4 BS Banco Bradesco SA BRAZILB3SA3 BS B3 SA - Brasil Bolsa Balcao BRAZILSANB11 BS Banco Santander Brasil SA BRAZILBBAS3 BS Banco do Brasil SA BRAZILITSA4 BS Itausa SA BRAZILBBSE3 BS BB Seguridade Participacoes SA BRAZILBPAC11 BS Banco BTG Pactual SA BRAZILHDFCB IS HDFC Bank Ltd INDIAHDFC IS Housing Development Finance Co INDIAKMB IS Kotak Mahindra Bank Ltd INDIAICICIBC IS ICICI Bank Ltd INDIASBIN IS State Bank of India INDIABAF IS Bajaj Finance Ltd INDIAAXSB IS Axis Bank Ltd INDIABJFIN IS Bajaj Finserv Ltd INDIAELEKTRA MF Grupo Elektra SAB DE CV MEXICOGFNORTEO MF Grupo Financiero Banorte SAB d MEXICOSBER RM Sberbank of Russia PJSC RUSSIAVTBR RM VTB Bank PJSC RUSSIAFSR SJ FirstRand Ltd SOUTH AFRICASBK SJ Standard Bank Group Ltd SOUTH AFRICASLM SJ Sanlam Ltd SOUTH AFRICAQNBFB TI QNB Finansbank AS TURKEY
Macroeconomic Risk Factors