UUncertainty Network Risk and Currency Returns
Mykola Babiak * Lancaster University Management School
Jozef Baruník ** Charles University
First draft: December 2020 This draft: January 26, 2021
Abstract
We examine the pricing of a horizon specific uncertainty network risk, extractedfrom option implied variances on exchange rates, in the cross-section of currency re-turns. Buying currencies that are receivers and selling currencies that are transmittersof short-term shocks exhibits a high Sharpe ratio and yields a significant alpha whencontrolling for standard dollar, carry trade, volatility, variance risk premium and mo-mentum strategies. This profitability stems primarily from the causal nature of shockpropagation and not from contemporaneous dynamics. Shock propagation at longerhorizons is priced less, indicating a downward-sloping term structure of uncertaintynetwork risk in currency markets.
Keywords:
Foreign exchange rates, network risk, currency variance, predictability,term structure
JEL:
G12, G15, F31 * Department of Accounting & Finance, Lancaster University Management School, LA1 4YX, UK, E-mail: [email protected] . ** Institute of Economic Studies, Charles University, Opletalova 26, 110 00, Prague, CR and Institute ofInformation Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou Vezi 4,18200, Prague, Czech Republic, E-mail: [email protected] . a r X i v : . [ q -f i n . GN ] J a n Introduction
Countries are connected through a variety of channels including economic activity,trade and financial links among others. Although international connections have beencentral for understanding market and fundamental macroeconomic risks, the literature haslargely ignored the world’s largest financial market - the foreign exchange market. In thispaper, we explore the properties of a variety of network risk measures in the cross-sectionof currencies. We document that an uncertainty network strategy, which buys currenciesreceiving short-term shocks and sells currencies transmitting short-term shocks, generatesa high Sharpe ratio and yields a significant alpha when controlling for popular foreignexchange benchmarks. Also, we find that the long-short portfolios based on currencyconnectedness at longer horizons is less profitable, indicating a downward-sloping termstructure of uncertainty network risk in currency markets. We begin by approximating foreign exchange uncertainty through the risk-neutral ex-pectation of the currency variance. The highly liquid and large foreign exchange volatilitymarket provides an excellent opportunity to synthesize such expectations.
The data forover-the-counter currency options are available on a daily frequency for a large cross-section of countries. A wide variety of strikes and maturities available on the marketallow us to compute the implied variances on exchange rates with precision. Further, theforward-looking nature of currency derivatives, which reflect the expectations of agentsabout future financial and real macroeconomic risks, is distinct from the backward-lookinginformation extracted from historical price and macroeconomic data. We continue our empirical investigation by estimating a dynamic horizon specific net-work among implied variances on exchange rates following the methodology of Barunikand Ellington (2020). The network structure of this paper has several key attributes. First,the connections between two nodes (currencies in our case) are directed, that is, the in-fluence of the currency A on the currency B is not necessarily equal to the impact of the We use the terms network and connectedness interchangeably. We follow the model-free approach of Britten-Jones and Neuberger (2000) and Bakshi, Kapadia, andMadan (2003) to compute spot implied variances on exchange rates from currency option prices. As of June 2019, a daily average turnover was $294 billion and a notional amounts outstanding was$12.7 trillion (BIS, 2019a,b). See, for example, Gabaix and Maggiori (2015), Zviadadze (2017), and Colacito, Croce, Gavazzoni, andReady (2018) for the nature of risks traded in currency markets. UDBRLCADCHF CZKDKK EUR GBP HUFJPY KRWMXNNOK NZDPLN SEK SGDTRY TWDZAR AUDBRL CADCHF CZKDKK EURGBPHUFJPY KRWMXN NOKNZDPLNSEK SGDTRY TWDZAR
Figure 1. Total and causal short-term currency networks: September 30, 2008
The left (right) figure depicts network connections among currency implied variances based on total (causal)connectedness. Total connectedness measures overall dependencies between currency variances includingcontemporaneous and causal effects. Causal connectedness is obtained by removing contemporaneous cor-relations from total connectedness. Arrows denote the direction of connections and the strength of linesdenotes the strength of connections. Grey (black) vertices denote currencies receiving (transmitting) moreshocks than transmitting (receiving) them. The size of vertices indicates the net-amount of shocks. To en-hance readability of plots, the links are drawn if their intensities are greater than a predetermined threshold. currency B on the currency A. Thus, currency connectedness measures of this paper iden-tify novel links, which are not captured by correlation-based measures. Second, we areable to distinguish between short- and long-term connections among idiosyncratic cur-rency variances and hence we shed light on how connectedness from shocks with differentpersistence is being priced in currency markets. Third, the international dependencies arenaturally driven by contemporaneous fluctuations in global markets and causal influencesbetween countries. In our empirical analysis, we are able to isolate causal connected-ness among currency variances by removing contemporaneous correlations. Finally, thenetwork structure is dynamically changing over time, unlike somewhat persistent rela-tionships between countries based on interest rates. Hence, sorting currencies accordingto the network risk measures is not equivalent, for example, to the currency carry trade.For twenty countries studied in our paper, Figure 1 depicts total and causal short-term3
UDBRL CAD CHF CZKDKKEUR GBPHUF JPYKRWMXNNOKNZDPLN SEKSGD TRYTWDZAR AUD BRLCAD CHFCZKDKKEUR GBPHUF JPYKRWMXN NOKNZDPLNSEKSGDTRY TWDZAR
Figure 2. Causal short- and long-term currency networks: August 31, 2011
The left (right) figure depicts short-term (long-term) network connections among currency implied variancesbased on causal connectedness. Causal connectedness is obtained by removing contemporaneous correla-tions from total connectedness. Arrows denote the direction of connections and the strength of lines denotesthe strength of connections. Grey (black) vertices denote currencies receiving (transmitting) more shocksthan transmitting (receiving) them. The size of vertices indicates the net-amount of shocks. To enhancereadability of plots, the links are drawn if their intensities are greater than a predetermined threshold. connections, whereas Figure 2 demonstrates causal links for short- and long-term cases. The plots show connectedness snapshots during two major recessions (the global financialcrisis and the European sovereign debt crisis) and illustrate several interesting features.First, total and causal connections might convey significantly different information aboutshock propagation in the global currency network. Second, the contribution of short-termand long-horizon connectedness is dynamically changing over time. For example, shortlyafter the bankruptcy of Lehman Brothers, the market crashes were strongly driven byshort-term shocks, whereas fears of contagion in Europe reflect mainly long-term risks. Anatural question arises whether this information might be valuable for traders.We find that uncertainty network does predict currency returns. We build monthlyquintile portfolios sorted by the amount of transmitted (received) shocks by each cur-rency to (from) others and also by the difference between transmitted and received shocks. In the empirical investigation, we define short-term as 1-day to 1-week horizon, medium-term as 1-weekto 1-month horizon, and long-term as horizons greater than 1-month. The volatility-related strategies exploit global foreign exchangevolatility (Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a) and currency variance riskpremium (Della Corte, Ramadorai, and Sarno, 2016). These strategies can be explainedby the models of Gabaix and Maggiori (2015) and Colacito, Croce, Gavazzoni, and Ready(2018). We contribute to this literature by showing how connectedness risk from impliedvariances on exchange rates is priced in the cross-section of currency excess returns. Wedemonstrate that network returns stemming from causal nature of shock propagation arevirtually unrelated to the existing strategies. The literature documents the strategies, among many others, based on the carry trade (Lustig andVerdelhan, 2007; Lustig, Roussanov, and Verdelhan, 2011; Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a),momenum (Menkhoff, Sarno, Schmeling, and Schrimpf, 2012b; Asness, Moskowitz, and Pedersen, 2013;Dahlquist and Hasseltoft, 2020), business cycles (Colacito, Riddiough, and Sarno, 2020), and global imbal-ances (Corte, Riddiough, and Sarno, 2016).
6n related work, Mueller, Stathopoulos, and Vedolin (2017) propose a strategy basedon the sensitivity of currencies to the cross-sectional dispersion of conditional foreign ex-change correlation. They construct the conditional correlation from spot exchange ratesas well as using the currency options for the risk-neutral counterpart. They find some in-teresting results about the compensation for exposure to high or low dispersion states. Incontrast, we focus on dependencies in currency implied variances instead of correlationsof spot exchange rates. Furthermore, connectedness measures of our paper are directionalunlike correlation-based proxies in Mueller, Stathopoulos, and Vedolin (2017). Further,we are able to disentable causal from contemporaneous effects in connections betweencurrency variances, which is impossible for correlations in their paper.Our paper is also related to Richmond (2019) who presents a general equilibrium modelexplaining the currency carry trade premia by the country’s position in the global tradenetwork. Unlike his network risk based on trade linkages, we study the market-based net-work from currency implied variances. Further, the currency excess returns sorted on net-work risk measures of our paper are weakly correlated to the standard carry trade. Hence,predictive information of uncertainty network extracted from currency option prices isdistinctive from trade links.Finally, our paper is related to the literature focusing on currency options. Jurek (2014),Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2015) and Chernov, Graveline, and Zvi-adadze (2018) focus on crash risk in currency markets. Although we use currency optionsto synthesize implied variances on exchange rates, our main focus is on the propertiesof network-sorted portfolios, which is different from their papers. Recently, Della Corte,Kozhan, and Neuberger (2020) document a global risk factor in the cross-section of impliedvolatility returns. The key differentiator of our study from their paper is that we study thenetwork risk premia in the cross-section of spot currency excess returns.
This section describes the numerical procedure used to measure currency uncertaintynetwork risk. We begin by approximating foreign exchange uncertainty through the risk-neutral expectation of the currency variance that can be synthesized from the quoted cur-rency options. We then provide a general discussion of the econometric methodology used7o estimate a dynamic horizon specific uncertainty network from the currency option im-plied variances. We finally discuss a variety of currency uncertainty network risk proxiesused in the core analysis.
A natural way to measure uncertainty about future exchange rate fluctuations is throughthe expectation of the currency variance. To study a network of such expectations in thecross-section of exchange rates, we obtain spot implied variances from OTC currency op-tions by applying a model-free approach of Britten-Jones and Neuberger (2000) and Bakshi,Kapadia, and Madan (2003). Formalizing the discussion, we use prices of European calland put options expiring at time t + τ to compute the implied variance for an exchangerate k versus the US dollar between two dates t and t + τ : IV ar kt = B k ( t , t + τ ) ∞ (cid:90) F k ( t , t + τ ) C k ( t , t + τ , K ) K dK + F k ( t , t + τ ) (cid:90) P k ( t , t + τ , K ) K dK , (1)where C k ( t , t + τ , K ) and P k ( t , t + τ , K ) denote the prices of call and put contracts at time t with a strike price K and maturity τ , B k ( t , t + τ ) is the price of a country’s bond at time t with maturity τ , F k ( t , t + τ ) is the forward exchange rate of the currency k at time t with maturity τ . To compute the model-free implied variances, we discretize the integralin Equation (1) by adopting call and put option prices interpolated around the τ maturityand considering a range of strike prices for the currency k . Having constructed proxies of forward-looking currency uncertainty, our objective isto define the network for shocks of a specific persistence propagating across the curren-cies. The knowledge of how a shock to a currency j transmits to a currency k definesa directed link at a given period of time. These disaggregate connections between cur-rency pairs then characterize two major types of network risk: a receiver or a transmitterof shocks. Aggregating the information from all pairs provides a system-wide measureof the forward-looking connectedness among foreign exchanges of countries. In contrastto the network literature in finance (Elliott, Golub, and Jackson, 2014; Glasserman and Appendix A provides a detailed description of the estimation procedure used to obtain a dynamichorizon specific uncertainty network risk. j is due to shocks in a variable k . The time-varyingvariance decomposition matrix defines the dynamic network adjacency matrix and is in-timately related to network node degrees, mean degrees, and connectedness measures.Further, a frequency domain view on such a network structure allows us to decompose thenetwork to short-, medium- or long-term network risk (Diebold and Yilmaz, 2014). Algebraically, the adjacency matrix captures all information about the network, and anysensible measure must be related to it. A typical metric used by the wide network literaturethat provides the user with information about the relative importance or influence of nodesis network centrality. For our purposes, we want to measure node degrees that capture thenumber of links to other nodes. The distribution shape of the node degrees is a network-wide property that closely relates to network behavior. As for the connectedness of thenetwork, the location of the degree distribution is key, and hence, the mean of the degreedistribution emerges as a benchmark measure of overall network connectedness.Dynamic horizon specific networks that we work with are more sophisticated thanclassical network structures. In a typical network, the adjacency matrix contains zeroand one entries, depending on the node being linked or not, respectively. In the abovenotion, one interprets variance decompositions as weighted links showing the strength ofthe connections. In addition, the links are directed, meaning that the j to k link is notnecessarily the same as the k to j link, and hence, the adjacency matrix is not symmetric.These measures are the key to our analysis as directional connectedness risk stems directlyfrom asymmetries within the network.We construct a dynamic uncertainty network through the TVP-VAR model estimatedfrom currency implied variances following the methodology of Barunik and Ellington A natural way to characterize horizon specific dynamics (i.e. short- and long-term) of the dynamicnetwork risk is to consider the spectral representation of the approximating model as recently proposed byBarunik and Ellington (2020). p describing the dynamicsas: CIV t , T = Φ ( t / T ) CIV t − T + . . . + Φ p ( t / T ) CIV t − p , T + (cid:101) t , T , (2)where CIV t , T = (cid:16) CIV ( ) t , T , . . . , CIV ( N ) t , T (cid:17) (cid:62) is a doubly indexed N -variate time series of cur-rency variances, (cid:101) t , T = Σ − ( t / T ) η t , T , η t , T ∼ N I D ( I M ) and Φ ( t / T ) = ( Φ ( t / T ) , . . . , Φ p ( t / T )) (cid:62) are the time varying autoregressive coefficients. Note that t refers to a discrete time index1 ≤ t ≤ T and T is an additional index indicating the sharpness of the local approximationof the time series by a stationary process. Rescaling time such that the continuous parame-ter u ≈ t / T is a local approximation of the weakly stationary time-series (Dahlhaus, 1996),we approximate CIV t , T in a neighborhood of u = t / T by a stationary process: (cid:103) CIV t ( u ) = Φ ( u ) (cid:103) CIV t − ( u ) + . . . + Φ p ( u ) (cid:103) CIV t − p ( u ) + (cid:101) t . (3)The TVP-VAR process has a time varying Vector Moving Average VMA( ∞ ) representa-tion (Dahlhaus, Polonik, et al., 2009; Barunik and Ellington, 2020): CIV t , T = ∞ ∑ h = − ∞ Ψ t , T ( h ) (cid:101) t − h (4)where parameter vector Ψ t , T ( h ) ≈ Ψ ( t / T , h ) is a time varying impulse response functioncharacterized by a bounded stochastic process. Information contained in Ψ t , T ( h ) permitsthe measurement of the contribution of shocks in the system. Hence, its transformationsover time will determine the network risk. Since a shock to a variable in the model does notnecessarily appear alone, an identification scheme is crucial in identifying the network. Weadapt the generalized identification scheme of Pesaran and Shin (1998), with its extensionto locally stationary process by Barunik and Ellington (2020).We transform local impulse responses in the system to local impulse transfer functionsusing Fourier transformations. This allows us to measure the horizon specific dynamics ofnetwork based on heterogeneous persistence of shocks in the system. A dynamic repre-sentation of the variance decomposition of shocks from asset j to asset k then establishesdynamic horizon specific adjacency matrix, which is central to our uncertainty network Since Ψ t , T ( h ) contains an infinite number of lags, we approximate the moving average coefficients at h =
1, . . . , H horizons. j are propagated to a currency k at a given point of time u = t / T and a given horizon d i ∈ H = {S , M , L} , is formally defined as: (cid:104) θ ( u , d i ) (cid:105) j , k = (cid:98) σ − kk ∑ ω ∈ d i (cid:32)(cid:20) (cid:98) Ψ ( u , ω ) (cid:98) Σ ( u ) (cid:21) j , k (cid:33) ∑ ω ∈H (cid:34) (cid:98) Ψ ( u , ω ) (cid:98) Σ ( u ) (cid:98) Ψ (cid:62) ( u , ω ) (cid:35) j , j , (5)where (cid:98) Ψ ( u , ω ) = ∑ H − h = ∑ h (cid:98) Ψ ( u , h ) e − i ω h is an impulse transfer function estimated fromFourier frequencies ω of impulse responses that cover a specific horizon d i frequencies. It is important to note that (cid:104) θ ( u , d ) (cid:105) j , k is a natural disaggregation of traditional variancedecompositions to a time-varying and h -horizon adjacency matrix. This is because theportion of the local error variance of the j -th variable at horizon h due to shocks in the k -thvariable is scaled by the total variance of the j -th variable. As the rows of the dynamicadjacency matrix do not necessarily sum to one, we normalize the element in each by thecorresponding row sum: (cid:104)(cid:101) θ ( u , d ) (cid:105) j , k = (cid:104) θ ( u , d ) (cid:105) j , k (cid:44) N ∑ k = (cid:104) θ ( u , d ) (cid:105) j , k . Equation (5) definesa dynamic horizon specific network risk completely. Naturally, our adjacency matrix isfilled with weighted links showing strengths of the connections. The links are directional,meaning that the j to k link is not necessarily the same as the k to j link. In sum, theadjacency matrix is asymmetric, horizon specific and evolves dynamically.To obtain the time-varying coefficient estimates (cid:98) Φ ( u ) , ..., (cid:98) Φ p ( u ) and the time-varyingcovariance matrix (cid:98) Σ ( u ) at a given point of time u = t / T , we estimate the approximatingmodel in Equation (3) using Quasi-Bayesian Local-Likelihood (QBLL) methods (Petrova,2019). Specifically, we use a kernel weighting function, which puts larger weights to thoseobservations surrounding the period whose coefficient and covariance matrices are of in-terest. Using conjugate priors, the (quasi) posterior distribution of the parameters of themodel are available analytically. This alleviates the need to use a Markov Chain MonteCarlo (MCMC) simulation algorithm and permits the use of parallel computing. We pro-vide a detailed discussion of the estimation algorithm in Appendix A. Note that i = √− ∞ ) represen-tation require a truncation of the infinite horizon with a H horizon approximation. As H → ∞ the error disappears (Lütkepohl, 2005). We note here that H serves as an approx-imating factor and has no interpretation in the time-domain. We obtain horizon specificmeasures using Fourier transforms and set our truncation horizon H =100; results are qual-itatively similar for H ∈ {
50, 100, 200 } . An important feature we focus on is a direct causal interpretation of our network riskmeasures. Rambachan and Shephard (2019) provide a general discussion about causalinterpretation of impulse response analysis in the time series literature. In particular,they argue that if an observable time series is shown to be a potential outcome time series,then generalized impulse response functions have a direct causal interpretation. Potentialoutcome series describe the output for a particular path of treatments at time t .In the context of our study, paths of treatments are shocks. The assumptions requiredfor a potential outcome series are natural and intuitive for a time series of currencies: i)they depend only on past and current shocks; ii) series are outcomes of shocks; and iii)assignments of shocks depend only on past outcomes and shocks. The dynamic adjacencymatrix we use above to characterize the currency network risk is a transformation of gener-alized impulse response functions. Therefore, the adjacency matrix and all measures thatstem from manipulations of its elements possess a causal interpretation; thus establishingthe notion of causal dynamic network measures.In computing our measures, we also diagonalize the covariance matrix because our ob-jective is to focus on the causal affects of network connections. The Ψ ( u , d ) matrix embedsthe causal nature of network linkages, and the covariance matrix Σ ( u ) contains contem-poraneous covariances within the off-diagonal elements. By diagonalizing the covariancematrix, we remove the contemporaneous effects and focus solely on causation. Hence,the measures introduced in the next section will be applied to total and causal linkagesdepending on whether we include or exclude contemporaneous correlations.12 .4 Uncertainty Network Risk Measures To evaluate the uncertainty network risk from the estimated model, we use several defi-nitions that focus on aggregate characteristics as well as disaggregate connections betweencurrencies. We focus on measures revealing when an individual currency is a transmitteror a receiver of shocks.First, horizon-specific from-directional network risk, which measures how much of eachcurrency’s j variance is due to shocks of other currencies j (cid:54) = k in the cross-section, isdefined as: F j ←• ( u , d ) = N ∑ k = k (cid:54) = j (cid:104)(cid:101) θ ( u , d ) (cid:105) j , k d ∈ H = {S , M , L} . (6)Second, horizon-specific to-directional network risk, which measures the contribution ofeach currency’s j variance to variances of other currencies in the cross-section, is given by: T j →• ( u , d ) = N ∑ k = k (cid:54) = j (cid:104)(cid:101) θ ( u , d ) (cid:105) k , j d ∈ H = {S , M , L} . (7)One can interpret these measures as dynamic to-degrees and from-degrees that associatewith the nodes of weighted directed network captured by a variance decomposition matrix.These two measures show how other currencies contribute to the risk of a currency j , andhow a currency j contributes to the riskiness of others, respectively, in a time-varyingfashion at a horizon d . One can simply add these measures across all horizons to obtaintotal time-varying measures: F j ←• ( u , T ) = ∑ d ∈{S , M , L} F j ←• ( u , d ) ∧ T j →• ( u , T ) = ∑ d ∈{S , M , L} T j →• ( u , d ) (8)Third, combining two notions of receivers and transmitters of shocks presented above, wedefine a horizon specific net-directional network risk: N j →• ( u , d ) = T j →• ( u , d ) − F j ←• ( u , d ) d ∈ H = {S , M , L , T } . (9)In conclusion, we aim to study the properties of to-, from- and net-directional networkportfolios sorted by the corresponding network risk proxies defined by Equations (6)-(9).Furthermore, each portfolio group is constructed using total and causal network linkages13s discussed in Section 2.3.
We start our empirical investigation by collecting daily OTC option implied volatili-ties on exchange rates versus the US dollar from JP Morgan and Bloomberg. FollowingDella Corte, Ramadorai, and Sarno (2016) and Della Corte, Kozhan, and Neuberger (2020),we consider a sample of the following 20 developed and emerging market countries: Aus-tralia, Brazil, Canada, Czech Republic, Denmark, Euro Area, Hungary, Japan, Mexico, NewZealand, Norway, Poland, Singapore, South Africa, South Korea, Sweden, Switzerland, Tai-wan, Turkey, and United Kingdom. The data cover the sample period from January 1996to December 2013. The cross-section of currencies starts with 10 countries at the beginningand gradually increases over time, with implied volatilities on all exchange rates beingavailable from 2004 until the end of the sample in 2013. We synthesize spot implied variances using a model free approach of Britten-Jonesand Neuberger (2000), which requires currency option prices for a range of strike prices.Quotes for OTC currency options are expressed in terms of Garman and Kohlhagen (1983)implied volatilities for selected combinations of plain-vanilla options (at-the-money, 10 and25 delta put and call options). We recover strike prices from deltas and option prices fromimplied volatilities by employing interest rates from Bloomberg and spot and forwardexchange rates from Barclays and Reuters via Datastream. Using this recovery procedure,we obtain plain vanilla European calls and puts for exchange rates versus the US dollar fora range of maturities: 1 month, 3 months, 6 months, 12 months, and 24 months.Since our investment strategy is carried out at the monthly frequency, it is natural toassume that traders prefer to employ the 1-month spot implied variances on exchangerates for detecting uncertainty network risk instead of using data for longer maturities.We therefore work with the spot 1-month variances on currencies in our empirical anal-ysis. Further, we construct currency connectedness measures using the variances at thedaily frequency to increase the number of observations in our estimation procedure andultimately to better capture the dynamic nature of uncertainty network risk. We then fil- We greatly appreciate help of Roman Kozhan with the currency option data.
We retrieve daily bid, mid, and ask spot and forward exchange rates versus the USdollar from Barclays and Reuters via Datastream. We further obtain daily nominal interestrates for domestic (the US in our case) and foreign countries from Bloomberg. The coreempirical analysis is conducted at the monthly frequency and hence we sample end-of-month observations of all time series. We match exchange and interest rate data withcurrency option data for the cross-section of 20 countries and the sample period fromJanuary 1996 to December 2013 as described above.
We denote spot and forward exchange rate of foreign currency k at time t as S kt and F kt .Exchange rates are expressed in units of foreign currency per US dollar. Thus, an increasein S kt indicates a depreciation of the foreign currency. Following Menkhoff, Sarno, Schmel-ing, and Schrimpf (2012a), we define one-period ahead excess return to a US investor forholding foreign currency k at time t as rx kt + = i kt − i t − ∆ s kt + ≈ f kt − s kt + , (10)in which i kt and i t represent the risk-less rates of the foreign country k and the US, ∆ s kt + is the log change in the spot exchange rate, f kt and s kt + denote the log spot and forwardrates. Under covered interest rate parity (CIP), the interest rate differential i kt − i t is equal toforward discount f kt − s kt . Thus, the approximation in Equation (10) states that the excesscurrency return equals the difference between the current forward rate and future spotrate. The early literature documented that CIP held even for very short horizons (Akram,Rime, and Sarno, 2008), while recent evidence has shown CIP deviations in the post globalfinancial crisis period (Du, Tepper, and Verdelhan, 2018; Andersen, Duffie, and Song, 2019).We demonstrate that the profitability of uncertainty network strategies studied in our pa-per stems primarily from spot exchange rate predictability. Therefore, our key results donot depend on the validity of the CIP condition.15 .4 Uncertainty Network Portfolios The measures of network connectedness among exchange rate implied variances cap-ture multiple risks that could be important for investors forming currency portfolios. First,unlike the previous literature focusing on the correlation risk in currency returns, thenetwork risk proxies of our paper can identify the causal nature of network linkages byremoving the contemporaneous effects. Thus, we are able to detect novel risks originatingfrom the causal propagation of shocks in the cross-section of exchange rates. Second, us-ing individual connections between exchange rates, we can quantify the aggregate amountof shocks that a particular currency transmits to or receives from others. Similarly, wecan compute the net-directional connectedness measure by taking the difference betweenshocks that are transmitted and received. Third, a large strand of the literature studies therole of shocks with different persistence. For instance, long-term fluctuations in expectedgrowth and volatility of cash-flows (Bansal and Yaron, 2004) have played a central role forunderstanding equity, bond, and currency returns. Our econometric methodology allowsus to disentable the effect of a horizon specific network risk. We therefore can shed lighton the term structure of forward-looking uncertainty connectedness in the cross-section ofcurrencies. In sum, we construct a battery of portfolios based on a variety of network con-nectedness measures to quantitatively evaluate which network risks are priced in currencymarkets.Specifically, at the end of each time period t (the last day of the month in the coreanalysis), we sort currencies into five portfolios using one of network measures constructedand described in Section 2.4. The first quintile portfolio P comprises 20% of all currencieswith the highest values of a particular network characteristic, whereas the fifth quintileportfolio P contains 20% of all currencies with the lowest values. Each P i is an equallyweighted portfolio of the corresponding currencies. We next form a long-short strategythat buys P and sells P .We report the results for five quintile portfolios and a long-short strategy sorted by(i) short- ( S ), medium- ( M ), and long-term ( L ) as well as total ( T ) net-directional con-nectedness constructed from total (contemporaneous and causal) and only causal (exclud-ing contemporaneous) linkages. The corresponding zero-cost strategies are denoted by N ( H ) where H ∈ {S , M , L , T } . We additionally dissect the sources of profitability of16et-directional network strategies by solely looking at the risk of being a transmitter or areceiver of shocks. In particular, we construct the portfolios based on (ii) to-directional and(iii) from-directional connectedness measures. Similarly to the portfolios in (i), we reportthe results for all horizons considered, but for the sake of a convenient illustration we fo-cus on the case with causal linkages. The respective to-directional and from-directionallong-short portfolios are denoted by T ( H ) and F ( H ) where H ∈ {S , M , L , T } . We compare the performance of network-sorted portfolios to standard investment strate-gies from the existing literature. Following Lustig, Roussanov, and Verdelhan (2011), webuild a portfolio that is the average of all currencies available in a particular time period.The resulting returns are equivalent to borrowing money in the US and investing in globalmoney markets outside the US. This zero-cost strategy is commonly called the dollar riskfactor or the dollar portfolio (dol). Further, we sort all currencies available at time t intofive quintile portfolios on the basis of their interest rate differential (or forward premia)relative to the US. The first quintile portfolio P comprises 20% of all currencies with thehighest interest rates, whereas the fifth quintile portfolio P contains 20% of all currencieswith the lowest interest rates. The difference between P and P is called the carry tradestrategy (car), which is equivalent to borrowing money in low interest rate countries andinvesting in high interest rate countries. We create a tradable strategy taking into account past realized volatility of currencies inthe spirit of Menkhoff, Sarno, Schmeling, and Schrimpf (2012a). At the end of each month t , we compute the square root of the sum of squared daily log exchange rate returns duringthe current month. We sort all currencies available at time t into five quintile portfolios onthe basis of their monthly realized volatility. The first quintile portfolio P comprises 20%of all currencies with the highest volatility, whereas the fifth quintile portfolio P contains20% of all currencies with the lowest volatility. The difference between P and P is calledthe volatility strategy (vol), which is equivalent to selling low volatility risk countries andbuying high volatility risk countries. The results of quintile and zero-cost portfolios sorted on to- and from-directional connections with totallinkages are available upon request. .7 Variance Risk Premium Portfolios We construct an investment strategy reflecting the costs of insuring currency variancerisk that has been recently proposed by Della Corte, Ramadorai, and Sarno (2016). Atthe end of each month t , we compute the volatility risk premium (vrp) for each currency,that is, the difference between expected realized volatility and implied volatility over thenext month. We sort all currencies available at time t into five quintile portfolios on thebasis of their monthly vrp. The first quintile portfolio P comprises 20% of all currencieswith the highest vrp, whereas the fifth quintile portfolio P contains 20% of all currencieswith the lowest vrp. The difference between P and P is called the volatility risk premiastrategy, which is equivalent to selling high insurance-cost currencies and buying lowinsurance-cost currencies. We form an tradable strategy linked to the past performance of currencies as initiallyproposed by Menkhoff, Sarno, Schmeling, and Schrimpf (2012b). Recently, Dahlquist andHasseltoft (2020) further connect currency returns to past trends in fundamentals includingeconomic activity and inflation. At the end of each month t , we compute the average ofcurrency excess returns over the last six months. We sort all currencies available at time t into five quintile portfolios on the basis of their trend. The first quintile portfolio P comprises 20% of all currencies with the highest average returns, whereas the fifth quintileportfolio P contains 20% of all currencies with the lowest average returns. The differencebetween P and P is called the momentum strategy (mom), which is equivalent to sellingpast losers (or worst performing currencies) and buying past winners (or best performingcurrencies). Table 1 reports summary statistics of the excess returns of the five quintile portfolios ( P i : i =
1, . . . , 5 ) and the long-short investment strategy buying P and selling P . Fur- Della Corte, Ramadorai, and Sarno (2016) work with the one-year volatility risk premium. We decideto switch to the monthly horizon to ensure that the volatility risk premium strategy employs one-monthimplied volatilities on exchange rates consistent with network connectedness portfolios. Our results remain quantitatively similar for other lags over which the past performance is evaluated. N ( S ) portfolios are 5.53% and 6.43% per annum for totaland causal linkages, which are statistically different from zero at the 5% and 1% levels,respectively. The “fx (%)” and “ir (%)” rows further indicate that this predictability ofthe cross-sectional network strategy based on total connections is partially driven by pre-dicting the interest rate differential. This result is expected in light of the prior literature(Menkhoff, Sarno, Schmeling, and Schrimpf, 2012a) documenting a link between globalforeign exchange volatility, which is strongly reflected in contemporaneous covariances ofnetwork connectedness, and the carry trade strategy, which is entirely driven by the for-ward premium across countries. In contrast, the spread between P and P portfolios,which are constructed from the causal nature of network linkages, is largely driven bypredicting the spot exchange rates. For instance, Panel B shows that the spread in theexchange rate component of the excess returns of N ( S ) is almost twice-as-large comparedto the one reported in Panel A (4.33% versus 2.28% per annum), whereas the spread in theinterest rate differential substantially shrinks (from 3.24% to 2.10% per annum). Also, themonotonicity in the forward premium does not hold as we move from P to P portfolios.Second, the risk-adjusted performance of long-short portfolios deteriorates with thehorizon of net-directional network risk. Using total network connectedness, the annualizedSharpe ratio of our network strategies gradually declines from 0.65 to 0.50 and 0.32 whenusing medium- and long- instead of short-term connections. The causal network zero-costportfolios experience a steeper decline in the annualized Sharpe ratio from 0.80 to 0.47and 0.39 when moving from short- to medium- and long-term horizons. Interestingly, the N ( T ) portfolio based on causal linkages exhibits the annualized Sharpe ratio of 0.66 andthe average return of 4.90% per annum, which is statistically different from zero at 1% level.Overall, the performance of horizon specific network portfolios indicates the downward-sloping term structure of uncertainty network risk in the cross-section of exchange rates.This finding extends the results of the existing literature on the price of uncertainty risk inequity markets (Dew-Becker, Giglio, Le, and Rodriguez, 2017).19 able 1. Net-directional Network Portfolios This table presents descriptive statistics for quintile ( P i : i =
1, . . . , 5 ) and long-short portfolios ( N ( · )) sorted by short- ( S ) , medium- ( M ) , and long-term ( L ) as well as total ( T ) net-directional connectednessextracted from total (Panel A) and causal (Panel B) linkages. The portfolio P ( P ) comprises currencies withthe highest (lowest) network characteristic. The long-short portfolio buys P and sells P . Mean, standarddeviation, and Sharpe ratio are annualized, but t-statistic of mean, skewness, kurtosis and the first-orderautocorrelation are based on monthly returns. We also report the annualized mean of the exchange rate(fx = − ∆ s k ) and interest rate (ir = i k − i ) components of excess returns. The t-statistics are based on Neweyand West (1987) standard errors with Andrews (1991) optimal lag selection. The sample is from January 1996to December 2013. Panel A: Total linkages P P P P P N ( S ) P P P P P N ( M ) mean (%) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − P P P P P N ( L ) P P P P P N ( T ) mean (%) 0.05 0.41 2.54 1.94 2.81 2.76 − − − − − − − − − − − − − − − − − − − − − − − − − − − Panel B: Causal linkages P P P P P N ( S ) P P P P P N ( M ) mean (%) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − P P P P P N ( L ) P P P P P N ( T ) mean (%) − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − able 2. To- and From-directional Network Portfolios: Causal Linkages This table presents descriptive statistics for quintile ( P i : i =
1, . . . , 5 ) and long-short portfolios ( T ( · ) and F ( · )) sorted by short- ( S ) , medium- ( M ) , and long-term ( L ) as well as total ( T ) to-directional(Panel A) and from-directional (Panel B) connectedness extracted from causal linkages. The portfolio P ( P ) comprises currencies with the highest (lowest) network characteristic. The long-short portfolio buys P andsells P . Mean, standard deviation, and Sharpe ratio are annualized, but t-statistic of mean, skewness, kur-tosis and the first-order autocorrelation are based on monthly returns. We also report the average networkcharacteristic (net), the annualized mean of the exchange rate (fx = − ∆ s k ) and interest rate (ir = i k − i )components of excess returns. The t-statistics are based on Newey and West (1987) standard errors withAndrews (1991) optimal lag selection. The sample is from January 1996 to December 2013. Panel A: To-directional network portfolios P P P P P T ( S ) P P P P P T ( M ) mean (%) 0.04 0.00 − − − − − − − − − − − − − − − − − − − − − − − − − − − P P P P P T ( L ) P P P P P T ( T ) mean (%) − − − − − − − − − − − − − − − − − − − − − − − − − − − Panel B: From-directional network portfolios P P P P P F ( S ) P P P P P F ( M ) mean (%) 0.85 1.97 1.85 1.49 1.51 0.67 0.57 1.47 1.86 2.41 1.98 1.41t-stat 0.33 0.75 0.77 0.71 0.72 0.43 0.21 0.56 0.86 1.13 0.92 0.85fx (%) − − − − − − − − − − − − − − − − − − − P P P P P F ( L ) P P P P P F ( T ) mean (%) 1.48 1.19 0.98 2.46 2.07 0.59 − − − − − − − − − − − − − − − − − − − − − − Table 2 presents the performance statistics of the excess returns sorted on to-directional(Panel A) and from-directional (Panel B) connectedness extracted from causal linkages.The table shows the results for horizon-specific network risk measures.For the to-directional case, the spread between the excess returns of P and P port-folios is increasing in the horizon and is statistically significant at the 1% level for allcases. Also, one can generally observe a monotonic pattern in the average excess returnsof quintile portfolios, particularly for the cases of longer-term and total-horizon networkrisks. Consequently, the long-short currency portfolios based on the amount of transmittedshocks have the annualized Sharpe ratios ranging from 0.74 to 0.83 for the short- and long-term horizon connectedness. All zero-cost investment strategies display a positive skew oftheir excess returns. Interestingly, this performance primarily stems from the exchange ratepredictability, while the interest rate differential contributes less. For the from-directionalcase, the results indicate no clear patterns in the performance statistics of currency net-work strategies taking into account the information about the received shocks. Althoughthe average excess returns of quintile and long-short portfolios tend to be positive, theyremain insignificant at all conventional confidence levels. This ultimately leads to much22maller Sharpe ratios compared to those from other strategies.Overall, the results presented in Table 2 suggest that the impact of a particular currencyon exchange rates of other countries is the key to understanding the profitability of net-directional network portfolios. Specifically, we document that the currencies transmittingmore shocks to others (in the net or total amount) tend to appreciate, leading to lowercurrency risk premia. Unlike the carry trade strategy, we demonstrate that the strongertransmitters of causal shocks do not necessarily have the lowest interest rates. By connect-ing currency returns to uncertainty network risk extracted from currency option data, weshed light on the novel risk that drives international asset prices above and beyond theexisting risks capturing macroeconomic country-specific conditions and trade connectionsamong others. We now study the relationship between the network long-short portfolios and existingbenchmarks. We begin by reporting the summary statistics of the standard dollar, carrytrade, volatility, variance risk premium, and momentum strategies as well as an equallyweighted average of all currency benchmarks in Table 3. The carry and momentum strate-gies exhibit the highest Sharpe ratios of 0.69 and 0.38, with the former having a statisti-cally significant mean excess return. However, both have a negative skewness, indicatingthe possibility of large losses. The last column shows limited diversification gains fromequally combining all strategies as indicated by a tiny increase in the Sharpe ratio and anegative skewness of the “1/N” portfolio.Next we examine how well the benchmark strategies can explain the network portfolios.We perform a two-step analysis. First, we compute the sample correlations between theexcess returns of different strategies. Second, for the zero-cost network portfolios, werun contemporaneous regressions using their monthly returns as dependent variables andbenchmark strategies as independent variables. Tables 4 and 5 report the results of thetwo-stage procedure for investment strategies constructed in Sections 4.1 and 4.2.Several observations in Table 4 are worth discussing. First, the currency portfolio re-turns sorted on total network connectedness tend to be more correlated with benchmarkstrategies than those based on causal linkages. Also, the negative correlations of total23 able 3. Benchmark Strategies: Summary Statistics
This table presents descriptive statistics (Panel A) and correlations (Panel B) between dollar (dol), carry trade(car), volatility (vol), volatility risk premium (vrp), momentum (mom) strategies and an equally weightedaverage of all currency benchmarks (1/N). Mean, standard deviation, and Sharpe ratio are annualized, butt-statistic of mean, skewness, kurtosis and the first-order autocorrelation are based on monthly returns. Thet-statistics are based on Newey and West (1987) standard errors with Andrews (1991) optimal lag selection.The sample is from January 1996 to December 2013.
Panel A: Benchmark strategies dol car vol vrp mom 1/Nmean (%) 1.60 7.29 2.28 1.66 3.64 3.30t-stat 0.75 2.58 1.04 0.84 1.66 2.55Sharpe 0.20 0.69 0.25 0.21 0.38 0.71std (%) 8.16 10.52 8.98 8.04 9.49 4.62skew − − − − − Panel B: Correlations dol car vol vrp mom 1/Ndol 1.00 0.33 0.61 − − − − − − − − − − (causal) connectedness portfolios with vol and vrp (car and vol) provide the scope for di-versification benefits. Second, the strength of the correlations directly translates into thesignificant coefficients for car, vol and vrp (Panel B for total linkages). We can conclude thatthe significant portion of the excess returns obtained from total connectedness measuresreflects interest rate differentials and global components of realized and implied currencyvariances. Third, once the contemporaneous effects are removed, none of the four factorsappear to be significant (the right part of Panel B). Also, the predictive power of the fourbenchmarks for network portfolios dramatically drops as measured by the adjusted R ,which range from 9.36% to 31.05% for total linkages and are around 2% for causal link-ages. Fourth, the resulting alphas for N ( S ) are economically and statistically significantfor both network risk measures. For instance, the annualized alphas of 5.00% and 6.58%are close to the average returns of 5.53% and 6.43% for the corresponding N ( S ) strate-gies, that is, less than 10% of the network returns are explained by the four benchmarkstrategies. Panel C in Table 4 shows that the inclusion of the dollar slightly reduces theestimated alphas and increases the adjusted R statistics, but the significance of constants24emains unchanged.Table 5 replicates the analysis separately for the currency returns sorted on to-directionaland from-directional causal connections. It demonstrates that the transmitted shocks in theglobal network of currencies play the key role. Indeed, all T ( H ) : H ∈ {S , M , L , T } gen-erate highly significant performance, both economically and statistically, which cannot beunderstood through the lens of the benchmarks.We further investigate the diversification benefits of our network portfolios. For theease of the presentation, we focus on short-term net-directional network cases (both totaland causal). We implement a naive strategy combining the network portfolio and one ofthe benchmarks with 0.5-0.5 weights. Table 6 report the results. For the strategies based ontotal connectedness, the resulting Sharpe ratios become considerably higher relative to theindividual benchmarks, with the increase ranging from 26% for car and to well above 200%for dol and vrp. As can be expected from the correlation analysis, the causal connectednessportfolio leads to larger levels and differences in Sharpe ratios. For instance, the allocationin N ( S ) and car generates the ratios of 1.08, which is 56.52% higher than the original carrytrade.Finally, we now provide the allocation analysis of selected portfolios: N ( S ) (causallinkages), car, vol, vrp, and mom. Table 7 reports the fraction of months each investmentstrategy goes long (the “Buy” columns) or short (the “Sell” columns) in each currency. Wealso compute the fraction of months when the currency position in N ( S ) is different fromthe currency allocation in the benchmarks (the “Diff” columns). The bottom row showsthe average fraction of “Diff” statistics across the currencies.Table 7 demonstrates significant differences across strategies and countries. For in-stance, the network strategy on average buys or sells alternative currencies in 40%, 46%,47%, and 44% of the time relative to the carry, volatility, variance risk premium, and mo-mentum strategies. The countries whose allocations differ most in their distributions rel-ative to N ( S ) are Japan and Switzerland for the carry trade, South Africa and Mexico forvolatility, Mexico and South Africa for variance risk premium, Japan and South Africa formomentum. Most notably, if we sort the currencies according to interest rate differentials,we would have bought South African rand (ZAR) in 92% of months and would have al-25 able 4. Net-directional Network Portfolios and Benchmark Strategies This table presents correlations (Panel A) and a contemporaneous regression (Panels B and C) of the monthlyreturns of net-directional network portfolios ( N ( H ) : H ∈ {S , M , L , T } ) on benchmark strategies - dollar(dol), carry trade (car), volatility (vol), volatility risk premium (vrp), and momentum (mom). Constantsreported in the “alpha (%, annual)” row are expressed in percentage per annum. The numbers in rows with agrey font are t-statistics of the estimates. The t-statistics are based on Newey and West (1987) standard errorswith Andrews (1991) optimal lag selection. The last two rows report adjusted R values (in percentage) andthe number of observations. The sample is from January 1996 to December 2013. Panel A: Correlations with trading strategies
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) dol − − − − − − − − − − − − − − − − − − − − Panel B: Returns of network portfolios on benchmark strategies (without dollar)
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) alpha (%, annual) 5.00 2.28 1.37 3.53 6.58 3.06 2.28 4.462.29 1.18 0.55 1.66 3.35 1.63 1.32 2.39car 0.16 0.38 0.32 0.24 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) Panel C: Returns of network portfolios on benchmark strategies (with dollar)
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) alpha (%, annual) 4.36 1.83 0.80 2.92 6.31 3.17 2.48 4.442.14 0.92 0.36 1.54 3.48 1.66 1.39 2.40dol − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) able 5. To- and From-directional Network Portfolios and Benchmark Strategies This table presents correlations (Panel A) and a contemporaneous regression (Panels B and C) of the monthlyreturns of to- and from-directional network portfolios ( T ( H ) and F ( H ) : H ∈ {S , M , L , T } ) on benchmarkstrategies - dollar (dol), carry trade (car), volatility (vol), volatility risk premium (vrp), and momentum(mom). Constants reported in the “alpha (%, annual)” row are expressed in percentage per annum. Thenumbers in rows with a grey font are t-statistics of the estimates. The t-statistics are based on Newey andWest (1987) standard errors with Andrews (1991) optimal lag selection. The last two rows report adjusted R values (in percentage) and the number of observations. The sample is from January 1996 to December 2013. Panel A: Correlations with trading strategies
Causal linkages T ( S ) T ( M ) T ( L ) T ( T ) F ( L ) F ( M ) F ( L ) F ( T ) dol − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Panel B: Returns of network portfolios on benchmark strategies (without dollar)
Causal linkages T ( S ) T ( M ) T ( L ) T ( T ) F ( L ) F ( M ) F ( L ) F ( T ) alpha (%, annual) 6.14 6.39 6.53 6.05 1.67 2.13 0.44 3.343.20 3.15 3.52 3.07 1.03 1.20 0.27 1.99car − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) Panel C: Returns of network portfolios on benchmark strategies (with dollar)
Causal linkages T ( S ) T ( M ) T ( L ) T ( T ) F ( L ) F ( M ) F ( L ) F ( T ) alpha (%, annual) 5.81 6.07 6.36 5.74 1.29 1.54 0.12 2.843.20 3.20 3.53 3.09 0.77 0.96 0.07 1.81dol − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) able 6. Benchmark Strategies: Diversification Gains This table presents the impact of adding short-term net-directional strategy ( N ( S )) to benchmark strategies- dollar (dol), carry trade (car), volatility (vol), volatility risk premium (vrp), and momentum (mom). Weconstruct a naive 50%-50% portfolio of N ( S ) and one of benchmark strategies. The “1/N” column presentsthe statistics of an equally weighted portfolio of all benchmarks and a network strategy. Panel A (B) reportsthe results for the case of total (causal) linkages. Mean, standard deviation, and Sharpe ratio are annualized,but t-statistic of mean, skewness, kurtosis and the first-order autocorrelation are based on monthly returns.The t-statistics are based on Newey and West (1987) standard errors with Andrews (1991) optimal lag selec-tion. The last row in each panel shows the percentage increase in the Sharpe ratio of a diversified portfoliorelative to the original benchmark strategy. The sample is from January 1996 to December 2013. Panel A: Including short-term net-directional strategy: total linkages dol car vol vrp mom + N ( S ) − − − − − ∆ Sharpe 290.00 26.09 172.00 223.81 76.32 35.21
Panel B: Including short-term net-directional strategy: causal linkages dol car vol vrp mom + N ( S ) − − − − − − ∆ Sharpe 300.00 56.52 212.00 223.81 102.63 54.93 ways kept Japanese yen (JPY) in the short position. In contrast, our causal net-directionalconnectedness strategy buys and sells JPY in 27% and 22% of the time and ZAR - 33% and25%.
Given the availability of the daily network connectedness data, it is reasonable to askwhether the profits of network strategies are sensitive to the frequency of rebalancing.We therefore construct horizon specific net-directional network portfolios (total and causallinkages) and the benchmark strategies at the daily and weekly frequencies. Specifically, we28 able 7. Allocation Analysis for the Network Portfolio and Benchmark Strategies
This table presents an allocation analysis of a short-term net-directional network portfolio based on causallinkages ( N ( S )) and carry trade (car), volatility (vol), volatility risk premium (vrp), and momentum (mom)strategies. The “Buy” and “Sell” columns report the fraction of months each currency belongs to the longand short positions of portfolios considered. The “Diff” column for each benchmark strategy reports thefraction of months the position for a particular currency is different from the one in N ( S ) . The bottom rowreports the average fraction across the currencies. The sample is from January 1996 to December 2013. N ( S ) car vol vrp momBuy Sell Buy Sell Diff Buy Sell Diff Buy Sell Diff Buy Sell DiffAustralia 0.25 0.15 0.00 0.00 0.40 0.26 0.02 0.53 0.28 0.10 0.53 0.25 0.16 0.57Brazil 0.41 0.06 0.67 0.00 0.26 0.23 0.07 0.47 0.12 0.36 0.53 0.35 0.10 0.38Canada 0.10 0.25 0.00 0.07 0.37 0.02 0.54 0.60 0.24 0.08 0.47 0.17 0.20 0.52Czech Republic 0.06 0.12 0.00 0.07 0.23 0.17 0.01 0.28 0.17 0.08 0.29 0.09 0.09 0.23Denmark 0.00 0.15 0.00 0.10 0.20 0.05 0.07 0.25 0.07 0.08 0.28 0.09 0.16 0.33Euro Area 0.01 0.17 0.00 0.26 0.24 0.05 0.06 0.27 0.09 0.08 0.30 0.06 0.15 0.25Hungary 0.13 0.16 0.53 0.00 0.48 0.34 0.00 0.48 0.16 0.20 0.42 0.24 0.10 0.38Japan 0.27 0.22 0.00 1.00 0.78 0.24 0.13 0.50 0.26 0.23 0.58 0.16 0.40 0.65Mexico 0.59 0.12 0.36 0.00 0.59 0.09 0.36 0.71 0.11 0.40 0.67 0.27 0.25 0.56New Zealand 0.13 0.20 0.24 0.00 0.38 0.40 0.02 0.55 0.34 0.18 0.54 0.27 0.14 0.51Norway 0.04 0.20 0.08 0.00 0.29 0.19 0.04 0.40 0.23 0.13 0.51 0.13 0.12 0.40Poland 0.27 0.09 0.18 0.00 0.34 0.40 0.03 0.52 0.21 0.24 0.55 0.26 0.11 0.40Singapore 0.07 0.27 0.00 0.48 0.36 0.00 0.66 0.40 0.10 0.04 0.34 0.04 0.15 0.38South Africa 0.33 0.25 0.92 0.00 0.64 0.59 0.14 0.80 0.21 0.52 0.67 0.31 0.27 0.60South Korea 0.40 0.00 0.00 0.00 0.40 0.02 0.27 0.45 0.07 0.20 0.44 0.11 0.08 0.38Sweden 0.06 0.16 0.00 0.12 0.28 0.19 0.02 0.34 0.23 0.12 0.47 0.10 0.16 0.40Switzerland 0.06 0.30 0.00 0.98 0.69 0.19 0.03 0.49 0.27 0.10 0.59 0.14 0.25 0.53Taiwan 0.09 0.35 0.00 0.41 0.33 0.00 0.71 0.37 0.10 0.09 0.43 0.05 0.27 0.43Turkey 0.17 0.20 0.55 0.00 0.38 0.12 0.10 0.39 0.08 0.23 0.34 0.19 0.11 0.39United Kingdom 0.11 0.14 0.01 0.06 0.32 0.01 0.27 0.33 0.21 0.09 0.39 0.18 0.16 0.48Average 0.40 0.46 0.47 0.44 use the daily observations of currency connectedness from the core analysis and sample thedaily or end-of-week observations to construct long-short portfolios. The realized volatilityand variance risk premium are computed on the rolling one-month window, while thecurrency momentum is computed over the rolling six-month horizon. For daily and weeklyfrequencies, Tables 8 and 9 report summary statistics of the long-short portfolios (PanelsA and B), regression outputs with the network excess returns as dependent variables andbenchmarks as independent variables (Panel C).Several interesting observations emerge from this investigation. First, the Sharpe ra-tios of short-term (long-term) network portfolios sorted on total connectedness decline(increase) with more frequent rebalancing, whereas the medium-term and total connec-tions are priced similarly. Second, the Sharpe ratios of the causal network strategies forall horizons substantially increase for weekly and especially daily frequency: 0.99, 0.57,0.57, 0.64 (weekly) and 1.13, 0.72, 0.71, 0.84 (daily) for N ( H ) : H ∈ {S , M , L , T } , re-spectively. Thus, we similarly document the downward-sloping term structure of causal29 able 8. Daily Frequency
This table presents a robustness analysis of currency strategies implemented on a daily frequency. It reportsdescriptive statistics of net-directional network portfolios (Panel A) and benchmark strategies (Panel B), anda contemporaneous regression (Panel C) of the daily returns of net-directional network portfolios ( N ( H ) : H ∈ {S , M , L , T } ) on benchmark strategies - dollar (dol), carry trade (car), volatility (vol), volatility riskpremium (vrp), and momentum (mom). In Panels A and B, mean, standard deviation, and Sharpe ratioare annualized, but t-statistic of mean, skewness, kurtosis and the first-order autocorrelation are based ondaily returns. In Panel C, constants reported in the “alpha (%, annual)” row are expressed in percentageper annum. The numbers in rows with a grey font are t-statistics of the estimates. The t-statistics are basedon Newey and West (1987) standard errors with Andrews (1991) optimal lag selection. The last two rowsreport adjusted R values (in percentage) and the number of observations. The sample is from January 1996to December 2013. Panel A: Performance of network portfolios
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) mean (%) 4.22 4.72 4.46 4.57 9.91 6.13 5.73 7.02t-stat 2.04 2.34 2.10 2.14 4.96 3.14 3.02 3.65Sharpe 0.47 0.50 0.47 0.47 1.13 0.72 0.71 0.84std (%) 9.02 9.41 9.50 9.65 8.81 8.51 8.07 8.41skew − − − − − − − − − Panel B: Performance of benchmark strategies dol car vol vrp mommean (%) 1.62 7.23 1.86 6.13 0.40t-stat 0.85 2.87 0.84 2.86 0.17Sharpe 0.21 0.65 0.18 0.65 0.04std (%) 7.64 11.14 10.47 9.38 10.49skew − − − − − Panel C: Returns of network portfolios on benchmark strategies
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) alpha (%, annual) 3.68 4.62 3.95 4.64 10.36 6.83 5.71 7.382.04 2.64 2.09 2.50 5.35 3.46 3.07 4.00dol − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) able 9. Weekly Frequency This table presents a robustness analysis of currency strategies implemented on a weekly frequency. Thetable reports descriptive statistics of net-directional network portfolios (Panel A) and benchmark strategies(Panel B), and a contemporaneous regression (Panel C) of the weekly returns of net-directional networkportfolios ( N ( H ) : H ∈ {S , M , L , T } ) on benchmark strategies - dollar (dol), carry trade (car), volatility(vol), volatility risk premium (vrp), and momentum (mom). In Panels A and B, mean, standard deviation,and Sharpe ratio are annualized, but t-statistic of mean, skewness, kurtosis and the first-order autocorrelationare based on weekly returns. In Panel C, constants reported in the “alpha (%, annual)” row are expressed inpercentage per annum. The numbers in rows with a grey font are t-statistics of the estimates. The t-statisticsare based on Newey and West (1987) standard errors with Andrews (1991) optimal lag selection. The lasttwo rows report adjusted R values (in percentage) and the number of observations. The sample is fromJanuary 1996 to December 2013. Panel A: Performance of network portfolios
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) mean (%) 4.09 3.60 3.22 4.13 8.70 4.78 4.61 5.35t-stat 1.90 1.83 1.59 2.03 4.29 2.66 2.79 2.99Sharpe 0.49 0.42 0.36 0.47 0.99 0.57 0.57 0.64std (%) 8.41 8.57 8.89 8.78 8.76 8.43 8.09 8.35skew − − − − − − − − − Panel B: Performance of benchmark strategies dol car vol vrp mommean (%) 1.61 7.38 0.31 2.58 2.51t-stat 0.83 2.96 0.17 1.02 1.13Sharpe 0.20 0.70 0.04 0.24 0.24std (%) 7.87 10.59 7.80 10.66 10.47skew − − − − − − − Panel C: Returns of network portfolios on benchmark strategies
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) alpha (%, annual) 3.20 2.59 2.18 3.37 7.77 4.23 3.05 4.451.62 1.48 1.15 1.95 3.70 2.09 1.81 2.50dol − − − − − − − − − − − − − − − − − − − − − − − − − − − − R ( % ) N ( L ) and 3.20% and 3.68% per annum for N ( S ) with weekly and daily fre-quencies, respectively. The alphas of strategies employing causal connectedness stronglyincrease in magnitude and become highly significant, both economically and statistically:5.71% per annum (a t-stat of 3.07) for N ( L ) and 10.36% per annum (a t-stat of 5.35) for N ( S ) with the daily rebalancing.In sum, the profits of network strategies strongly increase when we move to daily port-folio constructions. The improvement is especially pronounced for causal connectedness.Hence, unlike most of the standard benchmark strategies, investing in currencies based onthe network risk information is profitable regardless of trader’s investment horizons. We perform two additional robustness checks. First, we report the performance statis-tics for the network excess returns net of transaction costs. Since the bid-ask quotes forexchange rates are available from Barclays and Reuters, we incorporate those into the cur-rency excess returns following Menkhoff, Sarno, Schmeling, and Schrimpf (2012b). It isworth noting that the bid-ask data are for quoted spreads and not effective spreads. Lyonset al. (2001) suggest that the bid-ask spread data from Reuters are based on the indicativespreads and, therefore, might be too high relative to actual effective ones. Following theexisting literature (see, for example, Goyal and Saretto (2009), Menkhoff, Sarno, Schmeling,and Schrimpf (2012a, 2017), and Colacito, Riddiough, and Sarno (2020) among others), weemploy 50% of quoted bid-ask spreads in our calculations. Please see Appendix C for more details on how we account for transaction costs. Gilmore and Hayashi (2011) suggest that the effective bid-ask spreads could be even lower than 50%,while Cespa, Gargano, Riddiough, and Sarno (2019) suggest a 25% rule for the data from 2011. able 10. Transaction Costs This table presents descriptive statistics for long-short net-directional (Panel A), to-directional and from-directional (Panel B) network portfolios adjusted for transaction costs. Mean, standard deviation, and Sharperatio are annualized, but t-statistic of mean, skewness, kurtosis and the first-order autocorrelation are basedon monthly returns. The t-statistics are based on Newey and West (1987) standard errors with Andrews(1991) optimal lag selection. The sample is from January 1996 to December 2013.
Panel A: Net-directional network portfolios
Total linkages Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) mean (%) 4.60 3.26 1.86 3.45 5.37 2.37 1.70 3.88t-stat 2.06 1.44 0.81 1.52 3.19 1.46 1.14 2.40Sharpe 0.54 0.39 0.21 0.41 0.66 0.33 0.24 0.52std (%) 8.49 8.29 8.71 8.43 8.07 7.21 7.07 7.43skew − − − − − − − − − Panel B: To- and from-directional network portfolios
Causal linkages Causal linkages T ( S ) T ( M ) T ( L ) T ( T ) F ( S ) F ( M ) F ( L ) F ( T ) mean (%) 5.03 5.46 5.56 4.93 − − − − − − − − − − − − − − − − Table 10 reports summary statistics of the excess returns of currency network portfoliosadjusted for transaction costs. Comparing with the results shown in Tables 1 and 2, theSharpe ratios of the N ( S ) portfolios based on total and causal connectedness decline from0.65 and 0.80 to 0.54 and 0.66, respectively. The long-short to-directional network portfo-lios experience a comparable drop in their performances, with the Sharpe ratios rangingfrom 0.61 to 0.70. Hence, although the returns are somewhat lower after accounting fortransaction costs, the network portfolios still exhibit both economically and statisticallysignificant performance.Second, we divide the whole sample into half and look at the performance of the net-work strategies for the two subperiods (1996-2004 and 2005-2013). The availability and theamount of currency options vary over time and countries, with sparser data for the firsthalf of the sample. As can be expected, with less precise estimates of the forward-lookingcurrency variances, trading network connectedness is less profitable from 1996 to 2004,33 able 11. Subsamples This table presents a robustness analysis of currency strategies for the subsamples from January 1996 toDecember 2004 and from January 2005 to December 2013. The table reports descriptive statistics of net-directional network portfolios from total (Panel A) and causal (Panel B) connectedness. Mean, standarddeviation, and Sharpe ratio are annualized, but t-statistic of mean, skewness, kurtosis and the first-orderautocorrelation are based on monthly returns. The t-statistics are based on Newey and West (1987) standarderrors with Andrews (1991) optimal lag selection.
Panel A: Total linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) mean (%) 5.22 3.73 0.60 3.76 5.83 4.59 4.89 4.92t-stat 1.51 0.98 0.16 1.01 2.04 1.86 1.99 1.91Sharpe 0.57 0.39 0.06 0.40 0.73 0.65 0.65 0.67std (%) 9.08 9.44 9.73 9.40 7.96 7.02 7.54 7.39skew − − − − − − − − Panel B: Causal linkages N ( S ) N ( M ) N ( L ) N ( T ) N ( S ) N ( M ) N ( L ) N ( T ) mean (%) 6.07 0.30 0.61 2.39 6.79 6.51 4.83 7.37t-stat 2.25 0.14 0.30 1.00 3.27 2.83 2.28 3.52Sharpe 0.69 0.04 0.08 0.28 0.92 0.99 0.75 1.18std (%) 8.74 7.74 7.64 8.45 7.42 6.57 6.43 6.25skew 0.44 0.09 0.04 − − − − − − T ( S ) T ( M ) T ( L ) T ( T ) T ( S ) T ( M ) T ( L ) T ( T ) mean (%) 6.64 6.38 7.10 6.40 5.57 6.65 6.13 5.59t-stat 2.56 2.24 2.72 2.29 2.67 3.11 2.98 2.81Sharpe 0.73 0.69 0.77 0.69 0.76 0.97 0.93 0.83std (%) 9.15 9.28 9.22 9.23 7.34 6.85 6.58 6.71skew 0.07 0.10 0.18 0.09 0.86 0.58 0.38 0.85kurt 3.57 3.52 3.67 3.48 8.70 4.19 3.79 5.53ac1 − − − − − − − − though the short-term net-directional and especially to-directional portfolios still exhibithigh Sharpe ratios. With more data available from 2005 to 2013, the network strategiesgenerate the Sharpe ratios ranging from 0.65 to 1.18. In particular, the risk-adjusted returnof the causal N ( S ) portfolio is 0.92 with a skewness of 1.78 in the second half.34 Asset Pricing
This section presents the cross-sectional asset-pricing tests performed on the excessreturns of network portfolios. Motivated by the previous results, we focus on two sets oftest portfolios: the cross-section of currency returns sorted by the short-term net-directionalconnectedness extracted from total and causal linkages.
Cross-sectional asset pricing tests are based on a stochastic discount factor (SDF) ap-proach (Cochrane, 2005). In our application, we adopt the setting of Lustig, Roussanov,and Verdelhan (2011) for the network portfolios of our paper. In the absence of arbitrageopportunities, the excess returns rx jt + of a portfolio j have a zero price and satisfy thefollowing Euler equation: E t (cid:16) M t + rx jt + (cid:17) =
0, (11)in which M t + is the SDF. Following a common approach in the literature, we consider thelinear specification of M t + : M t + = − b (cid:48) ( f t + − µ f ) , (12)in which f t + is the vector of pricing factors, b is the vector of SDF loadings, µ f is the vectoron factor means. Combining Equations (11) and (12), one can obtain a beta pricing model E t (cid:16) rx jt + (cid:17) = λ (cid:48) β j , in which λ is the vector of the factor risk prices, and β j is the vectorof the risk quantities. The latter are also the regression coefficients of excess returns rx jt + on the risk factors f t + . Further, the SDF loadings and factor risk prices are related to eachother via the equation λ = Σ f b , where Σ f = E t (cid:2) ( f t + − µ f )( f t + − µ f ) (cid:48) (cid:3) is the variance-covariance matrix of the risk factors.We test a variety of linear factor models for the cross-section of network portfolios.For each model specification, we estimate the factor loadings via the one-step generalizedmethod of moments (GMM) with the identity weighting matrix (Hansen, 1982). We simul-taneously estimate the unknown factor means by adding the corresponding restrictions toa set of moments for the pricing errors. Since we are interested in testing whether a partic- Other prominent examples considering a linear SDF specification include Menkhoff, Sarno, Schmeling,and Schrimpf (2012a), Della Corte, Ramadorai, and Sarno (2016), Colacito, Riddiough, and Sarno (2020) andDella Corte, Kozhan, and Neuberger (2020) among many others. λ = Σ f b and calculatetheir standard errors using the Delta method. The t-statistics of b (cid:48) s and λ ’s are based onNewey and West (1987) standard errors with Andrews (1991) optimal lag selection. Weevaluate the fit of the linear pricing models by using three statistics: the cross-sectional R ,root mean squared pricing error (RMSE), and the Hansen and Jagannathan (1997) distance ( HJ dist ) . We further calculate the simulated p-values for testing the null hypothesis that thepricing errors equal zero, i.e. HJ dist equals zero. Following Jagannathan and Wang (1996)and Kan and Robotti (2008), we obtain the simulated p-values by using a weighted sum ofindependent random variables from χ ( ) distribution. Before performing formal cross-sectional asset pricing tests, we investigate whether av-erage network excess returns stemming from the predictability of short-term net-directionalconnectedness can be associated with a small group of risk factors. Following Lustig, Rous-sanov, and Verdelhan (2011), we conduct a principal component (PC) decomposition of thecurrency returns sorted on network risk measures extracted from total and causal linkage.Further, we study the correlations of the principal components with the correspondinglong-short network portfolios and benchmark strategies.Table 12 presents the results. There are several common and distinctive features ofthe two cross-sections. First, the PC loadings indicate a strong factor structure in bothgroups of currency portfolios. The first principal component (PC1) accounts for most ofthe time-variation in quintile portfolios and has similar loadings across the five portfolios.The second principal component (PC2) in turn displays a pronounced monotonic patternin loadings as we move from P to P : the increasing and decreasing tendencies for totaland causal linkages. Second, the first two principal components explain around 84% and86% of the common variation in network portfolios with total and causal connections.Further, they exhibit similar correlations with the risk factors. PC1 is perfectly correlatedwith the dollar factor in both cases. PC2 exhibits the strongest correlation with the long-short network portfolio, though the relationship is of the opposite sign in the two cases. Appendix B provides a detailed description of the GMM estimation and test statistics. able 12. Principal Components: Short-term Net-directional Network Portfolios This table presents the loadings of principal components ( PCi : i =
1, . . . , 5 ) for quintile portfolios ( P i : i =
1, . . . , 5 ) sorted by short-term net-directional connectedness extracted from total (Panel A) and causal (PanelB) linkages. Each panel also reports correlations of the principal components with a long-short networkportfolio ( N ( S )) and benchmark strategies - dollar (dol), carry trade (car), volatility (vol), volatility riskpremium (vrp), and momentum (mom). The sample is from January 1996 to December 2013. Panel A: Total linkages
PC loadings Correlations P P P P P CV N ( S ) dol car vol vrp momPC1 0.53 0.52 0.44 0.38 0.32 76.03 − − − − − − − − − − − − − − − − − − − − − − − − − − − Panel B: Causal linkages
PC loadings Correlations P P P P P CV N ( S ) dol car vol vrp momPC1 0.45 0.50 0.47 0.47 0.34 77.62 − − − − − − − − − − − − − − − − − − − − − − − In relative terms, under the causal linkages, the correlation of PC2 with N ( S ) is stronglydominant, whereas PC2 from total connectedness almost equally correlate with N ( S ) andthe carry trade factor. Finally, the starkest difference between the two portfolio groups isrelated to loadings of the third principal component (PC3), which show no visible patternin Panel A but display the monotonicity in Panel B. In the latter case, PC3 strongly relatesto the network and carry trade risk factors.Overall, the results, presented in Table 12, suggest that the network-sorted portfoliosindeed can be summarized by a small number of risk factors. We can approximate thefirst using the average returns across spot currency portfolios and interpret it as a “level”factor. We can approximate the second using the spread between P and P portfolios andinterpret it as a “level” factor. In case of causal connectedness portfolios, the results aresuggestive of an additional “level” factor, which is strongly correlated with the carry trade. We now turn to the formal investigation of our network portfolios following the method-ology outlined in Section 5.1. Motivated by the principal component analysis in Section37.2, we consider (A1) a variety of two-factor linear models for the cross-section of excessreturns sorted on total connectedness and (A2) a variety of two- and three-factor linearmodels for the test excess returns sorted on causal connectedness. In particular, the two-factor SDFs has dol as the first factor plus a second factor, including car, vol, vrp, mom,and N ( S ) . For the three-factor SDFs, we start with the two factors, dol and N ( S ) , andthen consider various third factors, including car, vol, vrp, and mom.Table 13 presents the asset pricing results for all models considered, with Panel Ashowing the specifications in (A1), and Panels B and C reporting the frameworks in (A2).The results in Panel A indicate that none of the SDF loadings and risk prices for benchmarkrisk factors are statistically significant at 5% level. In contrast, we document the positiveand statistically significant loading (a t-stat of 2.38) and price (a t-stat of 2.15) of the networkrisk factor. In particular, the GMM estimate of λ N ( S ) is 0.47% per month. Since the networkfactor is actually tradable, we can apply the Euler equation to the factor excess returns andderive that its price of risk must be equal to the average excess return. Using statisticsreported in Table 1, we verify that this no-arbitrage condition indeed holds: the monthlyaverage return of 0.46% is close to the estimated price of 0.47%. Regarding the dollar factor,its SDF loading and price of risk are insignificant at conventional confidence levels (a t-statof 1.21 for b dol and a t-stat of 0.66 for λ dol ). Moreover, the estimated λ dol matches factor’saverage excess return of 0.13% per month as reported in Table 3. Even though the dollarfactor does not help to explain the average excess returns, it serves as a constant capturingthe common mispricing in the cross-sectional regression.In terms of the model fit, the two-factor SDFs combining the dollar and other bench-mark risk factors produce similar performances, capturing from 32.61% to 47.06% of totalvariance in the cross-sectional returns and yielding RMSEs from 0.11% to 0.14%. One alsocannot reject any of these linear model based on HJ dist because the simulated p-values arefar above 50% in all cases. At the same time, the SDF specification comprising the dollarand network risk factors outperforms other models by a large margin. For instance, itgenerates more than twice-as-large cross-sectional R of 97.18% and more than three timessmaller RMSE and HJ dist of 0.03% and 0.05.In sum, the benchmark risk factors from the existing literature have a hard time ex-38 able 13. Pricing Short-term Net-directional Network Portfolios This table presents cross-sectional asset pricing results. We price quintile portfolios ( P i : i =
1, . . . , 5 ) sorted by short-term net-directional connectedness. In Panels A and B, we construct two-factor linear SDFswith the dollar (dol) factor plus a second factor, including carry trade (car), volatility (vol), volatility riskpremium (vrp), momentum (mom), and short-term net-directional network ( N ( S )) factors. In Panel C, weconstruct three-factor linear SDFs with dol, N ( S ) plus a third factor, including car, vol, vrp, and mom. Eachpanel reports one-step GMM estimates of factor loadings ( b ) and prices of factor risks ( λ ) . Goodness-of-fitstatistics include the R and root mean squared pricing error (RMSE) (both are expressed in percentage), andthe Hansen and Jagannathan (1997) distance ( HJ dist ) with simulated p-values in parentheses. The p-valuesare for the null hypothesis that the pricing errors are equal to zero. The remaining numbers in rows with agrey font are t-statistics of the estimates, which are based on Newey and West (1987) standard errors withAndrews (1991) optimal lag selection. The sample is from January 1996 to December 2013. Panel A: Total linkages: two-factor models
SDF Risk loadings prices
Model fit b dol b f λ dol λ f R (%) RMSE (%) HJ dist dol + car − − ( ) dol + vol − − ( ) dol + vrp 0.01 − − − − ( ) dol + mom 0.02 0.26 0.12 1.94 32.61 0.13 0.160.35 1.57 0.40 1.57 ( ) dol + N ( S ) ( ) Panel B: Causal linkages: two-factor models dol + car − − ( ) dol + vol − − ( ) dol + vrp 0.02 − − − − − ( ) dol + mom 0.02 0.43 0.12 3.19 32.04 0.17 0.300.25 1.94 0.31 1.94 ( ) dol + N ( S ) ( ) Panel C: Causal linkages: three-factor models
SDF Risk loadings prices
Model fit b dol b f b N ( S ) λ dol λ f λ N ( S ) R ( % ) RMSE (%) HJ dist dol + car + N ( S ) − − ( ) dol + vol + N ( S ) − − ( ) dol + vrp + N ( S ) − − − − ( ) dol + mom + N ( S ) ( ) R statistics drop dramatically to 4.90%and 1.79% with car and vol as a second factor or become negative -9.82% in case of vrp.Interestingly, the performance of the linear model with dol and mom remains the same,though the evidence on priced momentum risk is weak. In contrast, the factor loadingand price of network risk become statistically significant at 1% level (a t-stat of 2.89 for b N ( S ) and a t-stat of 2.92 for λ N ( S ) ). The model involving the network factor also displaysstronger explanatory power as measured by higher R (40.72%) and generates the lowerpricing errors as measured by lower RMSE (0.16%) and HJ dist (0.25).In Panel C in Table 13, we extend the model with dol and N ( S ) to the three-factorspecification with car, vol, vrp or mom. The inclusion of an additional factor generallyleads to higher R , lower RMSE and HJ dist statistics relative to the original two-factor SDF.Most importantly, the network risk remains strongly priced in all specifications. Further,consistent with the principal component decomposition, the best-performing three-factormodel includes the dollar, carry trade and network risk factors. Overall, the results empha-size the importance of the network risk factors extracted from total and especially causalcurrency connectedness in explaining the novel cross-sections of currency returns. Theseexcess returns cannot be understand through the lens of the benchmark factors. We further estimate the sensitivity of the excess returns of quintile portfolios ( P i : i =
1, . . . , 5 ) to the network risk. Table 14 reports the outputs of a contemporaneous regression40 able 14. Net-directional Network Portfolios: Factor Betas This table presents a contemporaneous regression of monthly excess returns of each quintile portfolio ontwo risk factors - the dollar and short-term net-directional network portfolios (Panel A), and on three riskfactors - the dollar, carry/volatility, and short-term net-directional network portfolios (Panel B). Constantsreported in the “alpha (%, annual)” row are expressed in percentage per annum. The numbers in rows witha grey font are t-statistics of the estimates, which are based on Newey and West (1987) standard errors withAndrews (1991) optimal lag selection. The last row in each panel shows the adjusted R (in percentage). Thesample is from January 1996 to December 2013. Panel A: Short-term net-directional portfolios
Total linkages Causal linkages P P P P P P P P P P alpha (%, annual) 0.34 − − − − − − − − − − − − β dol β N ( S ) − − − − − − R ( % ) Panel B: Short-term net-directional portfolios: causal linkages car vol P P P P P P P P P P alpha (%, annual) 1.49 − − − − − − − − − − β dol β f − − − − − − − − − − β N ( S ) − − − − R ( % ) of excess returns of each quintile portfolio on the dollar and network risk factors (Panel A)and on the dollar, carry trade/volatility, and network risk factors (Panel B).For the currency returns sorted on total network connectedness, the estimated alphasare statistically insignificant. The β dol coefficients are statistically indistinguishable fromone. The β net coefficients display a pronounced monotonicity when we move from P to P , increasing from -0.47 (a t-stat of -19.84) to 0.53 (a t-stat of 22.18). The two factorscapture a lot of variation of quintile portfolios ranging from 67.35% for P to 94.26% for P .For the currency returns sorted on causal network connectedness, the right part of PanelA and Panel B report the outputs for the two- and three-factor regressions. The results sug-41est that the first, fourth and fifth portfolios have statistically significant alphas. The sizeand the degree of significance of alphas are generally reduced when we include additionalrisk factors, with the impact being particularly large for the inclusion of the carry trade.This can be explained by the observation that the exposure to the volatility factor is sta-tistically insignificant for all portfolios, while the beta estimates for the carry trade aresignificant for three excess returns. The the goodness of fit and slope coefficients of thenetwork risk factor remain largely unchanged for the two- and three-factor regressions.Overall, the results of the time-series regressions are consistent with the cross-sectionalregressions. Specifically, they reinforce the conclusion that the dollar and network riskfactors fully explain the sources of risk in the cross-section of total and causal connect-edness portfolios, while the inclusion of the carry trade into a set of factors improves therepresentation of risks in causal connectedness portfolios. We show that connectedness risk among implied variances on exchange rates predictscurrency returns. A long-short portfolio strategy, which buys currencies receiving short-term shocks and sells currencies transmitting short-term shocks, generates a high Sharperatio and yields a significant alpha when controlling for popular foreign exchange bench-marks. Trading currency connectedness at longer horizons is less profitable, indicatinga downward-sloping term structure of uncertainty network risk in currency markets. Arisk factor - an uncertainty network strategy - fully explains the cross-sectional variationof network-sorted excess returns, which cannot be understood through the lens of the ex-isting risk factors - dollar, carry trade, volatility, variance risk premium and momentum.In robustness checks, we show that the performance of network portfolios in terms of risk-adjusted (Sharpe ratios) and benchmark adjusted (estimated alphas) actually improveswhen the strategies are implemented at daily or weekly frequencies. The significance ofmonthly network excess returns is robust to transaction costs and subperiods.Overall, the results of our paper provide new insights into the sources of currencypredictability. We do not provide possible explanations for the returns and hence develop-ing a formal theoretical model, which can rationalize our empirical findings, remains anopen question. The common international linkages based on trade and cash-flow channels,42hich have been proposed to mainly explain the carry trade, are unlikely to capture thenetwork returns, which remain uncorrelated with the popular currency factors. We leavesuch an interesting and important avenue for future research.43 eferences
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Abstract
This appendix presents supplementary details not included in the main body of thepaper. ontents A. Estimation of the time-varying parameter VAR model 50B. Asset Pricing Tests 52C. Transaction Costs 54 Estimation of the time-varying parameter VAR model
Let
CIV t be an N × p lags: CIV t , T = Φ ( t / T ) CIV t − T + . . . + Φ p ( t / T ) CIV t − p , T + (cid:101) t , T , (A.1)where (cid:101) t , T = Σ − ( t / T ) η t , T , η t , T ∼ N I D ( I M ) and Φ ( t / T ) = ( Φ ( t / T ) , . . . , Φ p ( t / T )) (cid:62) are the time varying autoregressive coefficients. Note that all roots of the polynomial, χ ( z ) = det (cid:16) I N − ∑ Lp = z p B p , t (cid:17) , lie outside the unit circle, and Σ − t is a positive definitetime-varying covariance matrix. Stacking the time-varying intercepts and autoregressivematrices in the vector φ t , T with CIV (cid:62) t = ( I N ⊗ x t ) , x t = (cid:16) x (cid:62) t − , . . . , x (cid:62) t − p (cid:17) and denotingthe Kronecker product by ⊗ , the model can be written as: CIV t , T = CIV (cid:62) t , T φ t , T + Σ − t / T η t , T (A.2)We obtain the time-varying parameters of the model by employing the Quasi-BayesianLocal-Likelihood (QBLL) approach of Petrova (2019). The estimation of Equation (A.1) re-quires re-weighting the likelihood function. The weighting function gives higher propor-tions to observations surrounding the time period whose parameter values are of interest.The local likelihood function at time period k is given by:L k (cid:0) CIV | θ k , Σ k , CIV (cid:1) ∝ (A.3) | Σ k | trace ( D k ) /2 exp (cid:26) − ( CIV − CIV (cid:62) φ k ) (cid:62) ( Σ k ⊗ D k ) ( CIV − CIV (cid:62) φ k ) (cid:27) The D k is a diagonal matrix whose elements hold the weights: D k = diag ( (cid:36) k , . . . , (cid:36) kT ) (A.4) (cid:36) kt = φ T , k w kt / T ∑ t = w kt (A.5) w kt = ( √ π ) exp (( − )(( k − t ) / H ) ) , for k , t ∈ {
1, . . . , T } (A.6) ζ Tk = (cid:32) T ∑ t = w kt (cid:33) − (A.7)where (cid:36) kt is a normalised kernel function. w kt uses a Normal kernel weighting function.50 Tk gives the rate of convergence and behaves like the bandwidth parameter H in (A.6).The kernel function puts a greater weight on the observations surrounding the parameterestimates at time k relative to more distant observations.We use a Normal-Wishart prior distribution for φ k | Σ k for k ∈ {
1, . . . , T } : φ k | Σ k (cid:118) N (cid:16) φ k , ( Σ k ⊗ Ξ k ) − (cid:17) (A.8) Σ k (cid:118) W ( α k , Γ k ) (A.9)where φ k is a vector of prior means, Ξ k is a positive definite matrix, α k is a scale param-eter of the Wishart distribution ( W ), and Γ k is a positive definite matrix.The prior and weighted likelihood function implies a Normal-Wishart quasi poste-rior distribution for φ k | Σ k for k = {
1, . . . , T } . Formally, let A = ( x (cid:62) , . . . , x (cid:62) T ) (cid:62) and Y = ( x , . . . , x T ) (cid:62) , then: φ k | Σ k , A , Y (cid:118) N (cid:18)(cid:101) θ k , (cid:16) Σ k ⊗ (cid:101) Ξ k (cid:17) − (cid:19) (A.10) Σ k (cid:118) W (cid:16)(cid:101) α k , (cid:101) Γ − k (cid:17) (A.11)with quasi posterior parameters (cid:101) φ k = (cid:16) I N ⊗ (cid:101) Ξ − k (cid:17) (cid:104)(cid:16) I N ⊗ A (cid:62) D k A (cid:17) ˆ φ k + ( I N ⊗ Ξ k ) φ k (cid:105) (A.12) (cid:101) Ξ k = (cid:101) Ξ k + A (cid:62) D k A (A.13) (cid:101) α k = α k + T ∑ t = (cid:36) kt (A.14) (cid:101) Γ k = Γ k + Y (cid:48) D k Y + Φ k Γ k Φ (cid:62) k − (cid:101) Φ k (cid:101) Γ k (cid:101) Φ (cid:62) k (A.15)where (cid:98) φ k = (cid:0) I N ⊗ A (cid:62) D k A (cid:1) − (cid:0) I N ⊗ A (cid:62) D k (cid:1) y is the local likelihood estimator for φ k . Thematrices Φ k , (cid:101) Φ k are conformable matrices from the vector of prior means, φ k , and a drawfrom the quasi posterior distribution, (cid:101) φ k , respectively.The motivation for employing these methods are threefold. First, we are able to esti-mate large systems that conventional Bayesian estimation methods do not permit. This istypically because the state-space representation of an N -dimensional TVP VAR ( p ) requiresan additional N ( + N ( p + )) state equations for every additional variable. Conven-tional Markov Chain Monte Carlo (MCMC) methods fail to estimate larger models, which51n general confine one to (usually) fewer than 6 variables in the system. Second, the stan-dard approach is fully parametric and requires a law of motion. This can distort inferenceif the true law of motion is misspecified. Third, the methods used here permit direct esti-mation of the VAR’s time-varying covariance matrix, which has an inverse-Wishart densityand is symmetric positive definite at every point in time.In estimating the model, we use p =2 and a Minnesota Normal-Wishart prior with ashrinkage value ϕ = p = {
2, 3, 4, 5 } ; shrinkage values, ϕ = { } ; and values tocentre the coefficient on the first lag of each variable, {
0, 0.05, 0.2, 0.5 } . Network measuresfrom these experiments are qualitatively similar. Notably, adding lags to the VAR andincreasing the persistence in the prior value of the first lagged dependent variable in eachequation increases computation time. B Asset Pricing Tests
The standard Euler equation implies that the excess returns rx jt + of a portfolio j satisfythe equation: E t (cid:16) M t + rx jt + (cid:17) =
0, (B.16)in which M t + is the stochastic discount factor (SDF). We assume that the SDF is a linearfunction of a set of risk factors f t + and is defined as follows: M t + = − b (cid:48) ( f t + − µ f ) . (B.17)Notice that we employ a de-meaned version of the SDF to avoid the issue related to anaffine transformation of the factors (Kan and Robotti, 2008).We are interested in testing the perfomance of the linear pricing models defined byEquations (B.16)-(B.17). To do so, we estimate factor loadings using the generalized methodof moments (GMM) (Hansen, 1982). Substituting (B.17) into (B.16), we obtain the fol-lowing N moment conditions E t (cid:0) [ − b (cid:48) ( f t + − µ f )] rx t + (cid:1) = N , where rx t + is the N -52imensional vector of test asset excess returns. We simultaneously estimate the unknownvector of factor means µ f . Thus, GMM moment conditions also include the set of k restric-tions E t (cid:0) f t + − µ f (cid:1) = k , where k denotes the number of factors in the SDF specification.Therefore, we have the following population moment conditions: E t [ g t + ( θ )] = E t [ − b (cid:48) ( f t + − µ f )] rx t + f t + − µ f = N + k ,where θ = ( b (cid:48) , µ (cid:48) ) (cid:48) is the vector of parameters. The sample moment conditions are thendefined as: ¯ g T ( θ ) = ¯ g T ( θ ) ¯ g T ( θ ) = T T ∑ t = (cid:2) − b (cid:48) ( f t + − µ f ) (cid:3) rx t + T T ∑ t = (cid:2) f t + − µ f (cid:3) .We implement a one-stage GMM estimation with the prespecified weighting matrix con-sisting of the identity matrix I N for the first moment conditions and a large weight assignedto the remaining restrictions. Standard errors are computed based on a heteroscedas-ticity and autocorrelation consistent (HAC) estimate of the long-run covariance matrix S = ∞ ∑ j = − ∞ E [ g ( θ ) g ( θ ) (cid:48) ] by the Newey and West (1987) procedure with Andrews (1991)optimal lag selection.We now evaluate the performance of linear pricing models in explaining the cross-section of network portfolios. We construct the cross-sectional R , root mean squaredpricing error (RMSE), and the Hansen and Jagannathan (1997) distance ( HJ dist ) . Hansenand Jagannathan (1997) provide two nice illustrations of HJ dist . First, it is the maximumpricing error of a portfolio with a unit second moment. Second, it measures the minimumdistance between the proposed SDF and the set of admissible SDFs. Thus, tests of thelinear SDFs defined by Equation (B.17) boil down to testing the null hypothesis that thepricing errors equal zero, i.e. HJ dist equals zero. Formally, the Hansen and Jagannathan(1997) distance is defined as: HJ dist = (cid:114) min θ ¯ g T ( θ ) (cid:48) G − T ¯ g T ( θ ) , (B.18)in which G T is the sample second moment matrix of the test excess returns, that is, G T = T T ∑ t = rx t + rx (cid:48) t + . One can obtain HJ dist by applying the one-stage GMM estimation with the53eighting matrix equal to G − T . The advantage of this definition is that G − T is independentof the optimal parameters and hence this allows the comparison between different SDFspecifications (Hansen and Jagannathan, 1997). The disadvantage of this approach is that G − T is not optimal in the sense of Hansen (1982) and hence HJ dist is not asymptotically arandom variable of χ ( N − k ) distribution. Instead, the sample HJ dist follows a weightedsum of χ ( ) random variables (see Jagannathan and Wang (1996) and Kan and Robotti(2008) for specification tests using gross and excess returns, respectively). Therefore, wecalculate the simulated p-values for HJ dist based on this statistic. C Transaction Costs
We use time-varying quoted bid-ask spreads to compute the currency excess returns ad-justed for transaction costs. Following Menkhoff, Sarno, Schmeling, and Schrimpf (2012b),we take into account the whole cycle of each currency in the short or long positions from t − t +
1. When the investor buys the currency at time t and sells at time t +
1, he paysthe corresponding bid-ask costs each period. In our notations, the excess returns of long ( l ) and short ( s ) positions are respectively rx lt + = f bt − s at + and rx st + = − f at + s bt + . If theinvestor buys the currency at time t but decides to keep it in the portfolio at time t + rx lt + = f bt − s t + and rx st + = − f at + s t + .If the currency, which belongs to the portfolio at time t and is sold at time t +
1, was al-ready in the current portfolio at time t −
1, then the excess returns rx lt + = f bt − s at + and rx st + = − f at + s bt +1