Scrambling for Dollars: International Liquidity, Banks and Exchange Rates∗

Javier Bianchi,† Saki Bigio,‡ and Charles Engel§

September 14, 2021

Abstract

We develop a theory of exchange rate fluctuations arising from financial institutions’ demand for liquid dollar assets. Financial flows are unpredictable and may leave banks “scrambling for dollars.” As a result of settlement frictions in interbank markets, a pre- cautionary demand for dollar reserves emerges and gives rise to an endogenous convenience yield. In our framework, an increase in the volatility of idiosyncratic liquidity shocks leads to a rise in the convenience yield and an appreciation of the dollar—as banks scramble for dollars—while foreign exchange interventions matter because they alter the relative supply of liquidity in different currencies. We present empirical evidence on the relationship between exchange rate fluctuations for the G10 currencies and the quantity of dollar liquidity consistent with the theory.

Keywords: Exchange rates, liquidity premia, monetary policy JEL Classiﬁcation: E44, F31, F41, G20

∗We would like to thank Andy Atkeson, Roberto Chang, Pierre-Olivier Gourinchas, Arvind Krishna- murthy, Martin Eichenbaum, Benjamin Hebert, Oleg Itskhoki, Matteo Maggiori, Dmitry Mukhin, Sergio Rebelo, Hel´ ene` Rey, and Jenny Tang for discussions. As well, we thank various conference participants. †Federal Reserve Bank of Minneapolis, email [email protected] ‡Department of Economics, University of California, Los Angeles and NBER, email [email protected] §Department of Economics, University of Wisconsin, Madison, NBER and CEPR, email [email protected] 1 Introduction

The well-known “disconnect” in international finance holds that foreign exchange rates show little empirical relationship to the supposed economic drivers of currency values, such as interest rates and output (Obstfeld and Rogoff, 2000). More recent work contends that the source of the disconnect is in financial markets (Itskhoki and Mukhin, 2021a). There long has been evidence of time-varying expected excess returns in foreign exchange markets, and, furthermore, it appears that the US dollar is particularly special, as it earns a premium over the rest of the major currencies when measured on historical data (Gourin- chas and Rey, 2007a). To account for the exchange rate disconnect and associated puzzles, the literature has turned to models with currency excess returns as the potential “miss- ing link”. The source or sources of these excess returns, however, remains an unsolved mystery. In this paper, we develop a theory of exchange rate fluctuations arising from the liquidity demand by financial institutions within an imperfect interbank market. We build on two observations on the international financial system. First, U.S. dollars are the dominant foreign-currency source of funding. According to the BIS locational banking statistics, in March 2021, the global banking and non-bank financial sector had cross-border dollar liabilities of over $11 trillion. Second, dollar funding may turn unstable. As documented for example in Acharya et al.(2017), banks are subject occasionally to large funding uncertainty or interbank market freezing that can leave them “scrambling for dollars.” Narrative discussions attribute fluctuations in the US dollar exchange rate to such vicissitudes in the short-term international money markets to. A contribution of our paper is to provide a framework to formally articulate this channel theoretically and to provide empirical evidence consistent with it. We construct a model in which financial institutions, which we simply refer to as banks, have a portfolio of assets and liabilities in two currencies. Banks face the risk of sudden outflows of liabilities. In case a bank ends up short of liquid assets to settle those flows, it needs to find a counterparty, but there may be times when banks may lose confidence in one another—this is the source of interbank market frictions. As insurance against these outflows, banks maintain a buffer of liquid assets—especially dollar liquid assets, in line with the aforementioned observations above on the international financial system. To the extent that uncertainty and the smoothness of interbank markets change over time, this increases the relative demand for currencies which translate into movements in the

1 exchange rate. The theory uncovers how frictions in the settlement of international deposit transactions emerge as a liquidity premium earned by the dollar. The liquidity premium generates a time-varying wedge in the interest parity condition, or “convenience yield,” which plays a pivotal role in the determination of the exchange rate. Critically, the convenience yield is endogenous and depends on the quantity of outside money and policy rates in two currencies, as well as technology parameters such as matching efficiency in the interbank market, and the volatility of banking payments in different currencies. Through this endogenous convenience yield, we link the determination of nominal dollar exchange rates and the dollar liquidity premium to the reserve position of banks in different currencies, funding risk, and confidence in the interbank market. We provide empirical evidence consistent with the theory by relating the banking sector’s balance sheet data to the foreign currency price of U.S. dollars. According to the theory, the financial sector increases its demand for dollar liquid assets—US government obligations, including reserves held at the Federal Reserve for banks in the Federal Reserve system—when funding becomes more uncertain, and in turn, this translates into an appreciation of the dollar. Our analysis shows that indeed the dollar liquidity ratio positively correlates with the relative value of the dollar. Notably, this relationship is robust to controlling for the VIX index, a variable that captures a broad measure of uncertainty, which has been shown to have significant explanatory power. The liquidity ratio is, of course, not an exogenous driver of exchange rates—either in our model or in the real world. However, we show that simulations of a calibrated version of our model implies a positive association between the change in the liquidity ratio and the value of the dollar under multiple driving shocks, including uncertainty shocks, monetary policy shocks, and liquidity demand shocks. Many recent theories have focused primordially on risk-premia or external financing premia to explain excess currency returns and exchange rate movements. Risk premium models explain excess dollar returns as stemming from a greater exposure of currencies other than the dollar to global pricing factors.1 External financing premium models explain excess dollar returns as a funding advantage in dollar liabilities in the presence of limits to international arbitrage. We provide an alternative theory based on a liquidity premium. Our model abstracts, in fact, from risk premia and limits to international arbi-

1See, for example, Lustig et al.(2011, 2014).

2 trage to focus squarely on liquidity. At the center of the model is the idea that funding risk may leave banks scrambling for dollars. On the surface, the model resembles the seminal monetary exchange rate model of Lucas(1982). 2 In that model, two currencies earn a liquidity premium over bonds because certain goods must be bought with corresponding currencies. A money demand equation determines prices in both currencies, and relative prices determine the exchange rate. Our model shares the segmentation of transactions and the exchange-rate determination of Lucas. However, in our model, the demand for reserves in either currency stems from the settlement demand by banks. This distinction is important. First, because our model leads to predictions about the direction of exchange rates as functions of the ratio of reserves to deposits in different currencies, the size and volatility of flows in different currencies and the dispersion of interbank rates in different currencies. Second, because the policy implications are markedly different.

Literature Review

The expected excess return on foreign interest earning assets, or the deviation from “uncovered interest parity (UIP)" is important not just for understanding international pricing of interest-bearing assets, or the expected depreciation (or appreciation) of the currency, but also for the level of the exchange rate. This point is brought out clearly by Obstfeld and Rogoff(2003), which shows how the expected present value of current and future foreign exchange risk premiums affect the current exchange rate in a simple DSGE model. They refer to this present value as the “level risk premium.”3 Potentially, a better understanding of the role of ex ante excess returns can help account for the empirical failure of exchange-rate models (Meese and Rogoff, 1983; Obstfeld and Rogoff, 2000), and the excess volatility of exchange rates (Frankel and Meese, 1987; Backus and Smith, 1993; Rogoff, 1996). Much of the literature has been directed toward explaining the expected excess return as arising from foreign exchange risk. Another branch of the literature has explored limits to capital mobility and frictions in asset markets. A third branch has looked at deviations from rational expectations. A line of research closely related to this paper has been the role of the “convenience yield” in driving exchange rates. 2See also Svensson(1985) and Engel(1992a,b). 3This present value plays a key role in the analysis of Engel and West(2005), Froot and Ramadorai (2005), and Engel(2016). See Engel(2014)’s survey of exchange rates for an overview of the effect of the risk premium on exchange rates.

3 Foreign exchange risk premium. The modeling of failures of uncovered interest parity as arising from foreign exchange risk has a long history. Early contributions include Solnik (1974), Roll and Solnik(1977), Kouri(1976), Stulz(1981), and Dumas and Solnik(1995). Much theoretical work has been devoted toward building models of the risk premium that are consistent with the Fama(1984) puzzle, which finds a positive correlation between the expected excess return and the interest rate differential.4 Bansal and Shaliastovich(2013), Colacito(2009), Colacito and Croce(2011, 2013), Colacito et al.(2018b,a), and Lustig and Verdelhan(2007) examine models with recursive preferences. Verdelhan(2010) presents a model with habit formation to account for the Fama puzzle. Ilut(2012) proposes ambiguity aversion as a solution to the puzzle. Some recent studies, such as Burnside et al. (2011), Farhi and Gabaix(2016), and Farhi et al.(2015), model the risk premium as arising from risks associated with rare events. Recent empirical work sheds light on the nature of the foreign exchange risk premium. Lustig et al.(2014) identify a dollar risk factor that makes the dollar carry trade prof- itable. Hassan and Mano(2019) distinguish between the factors that drive the dollar carry trade and the cross-sectional deviations from uncovreed interest parity. Kalemli- Ozcan¨ (2019)and Kalemli- Ozcan¨ and Varela(2021) note that the factors driving risk premia among advanced economies and emerging markets are different. An important development in the literature has been the modeling of the “safe haven” status of the U.S. dollar, and its association with the “exorbitant privilege” that the U.S. enjoys from the returns on its net foreign asset position. Gourinchas and Rey(2007a,b) report findings that the U.S. earns appreciably higher returns on its foreign investments than foreigners earn on investments in the U.S. Caballero et al.(2008) and Farhi and Maggiori (2018) offer models that help explain this phenomenon in which the U.S. is more able to “produce” safe assets that are valued by foreign investors but earn a lower expected return than foreign assets. Other papers have modeled the U.S. as a country that is willing to assume more risk, perhaps because its internal financial markets are more highly developed, and therefore are willing to hold riskier assets that yield a higher expected return (see, for example, Mendoza et al., 2009, Gourinchas et al., 2010, and Maggiori, 2017). Gourinchas et al.(2019) and Maggiori(2021) provide useful overviews of this literature. Recent research has focused on the interaction between risk-averse investors and habitat investors or noise traders who may not be motivated by risk/return considerations in

4See Tryon(1979) and Bilson et al.(1981) for earlier empirical studies that ﬁnd this relationship. Engel (1996, 2014) surveys empirical and theoretical models.

4 their portfolio choice. These studies include Koijen and Yogo(2020) general model of portfolio choice, the preferred habitat models of Gourinchas et al.(2021), and Greenwood et al.(2020), and the noise trader model of Itskhoki and Mukhin(2021a,b). Limited Capital Mobility. Other models attribute these uncovered interest parity differentials to financial premia earned by foreign currency because of limited market participation as in models in which order flow matters for exchange rate determination also require some frictions in the foreign exchange market (see, for example Evans and Lyons (2002, 2008)). Relatedly, Bacchetta and Van Wincoop(2010) posit that slow adjustment of portfolios can account for the expected excess returns on foreign bonds. An important recent literature has emphasized limited market participation (see the segmented markets models of (Alvarez et al., 2009; Itskhoki and Mukhin, 2021a) or limited international arbitrage (Gabaix and Maggiori, 2015; Amador et al., 2020; Itskhoki and Mukhin, 2021a).) This research has highlighted the role of financial intermediaries in helping to overcome barriers to asset trade, but underscored the balance sheet constraints of such institutions. While our model also brings financial intermediaries to the forefront, we do not introduce the collateral constraints or other limits to lending that has been the subject of these papers. The failure of covered interest parity since the global financial crisis is often interpreted as evidence of limits to arbitrage. See for example Baba and Packer(2009), Du et al.(2018), Avdjiev et al.(2019), Liao(2020), and the survey of Du and Schreger(2021). Our study offers an alternative channel by which deviations from covered interest parity might arise. Deviations from Rational Expectations. A simple alternative story for the UIP deviations is that agents expectations are not fully rational. Empirical studies, such as Frankel and Froot(1987), Froot and Frankel(1989), and Chinn and Frankel(2019) have used survey measures of expectations to uncover possible deviations from rational expectations. Kalemli-Ozcan¨ (2019) and Kalemli- Ozcan¨ and Varela(2021) also find that the component of deviations from uncovered interest parity that is predictable from interest rate differentials may be driven by deviations from rational expectations. Models that incorporate systematically skewed expectations include Gourinchas and Tornell(2004) and Bacchetta and Van Wincoop(2006). Convenience Yield. Our model is closely connected to the recent examination of the “convenience yield”—the low return on riskless government liabilities—and exchange rates. We posit that our model provides one possible channel for the emergence of the conve-

5 nience yield on U.S. government bonds. See Engel(2016), Valchev(2020), Jiang et al. (2020, 2021), Engel and Wu(2018), and Kekre and Lenel(2021) for recent work featur- ing simple, descriptive models of the convenience yield. In particular, Jiang et al.(2021) and Kekre and Lenel(2021) provide equilibrium models in which the appreciation of the dollar during times of global risk is driven by an increase in the convenience yield on the U.S. dollar. Our study might be described as providing the “microfoundations” for the convenience yield that provides a deeper understanding of the factors that might drive changes in the demand for U.S. dollar assets. Special role of the dollar. Our paper is related to the literature on the pre-eminence of the dollar in international financial markets. Rey(2013, 2016) and Miranda-Agrippino and Rey(2020) identify the importance of the US monetary policy as a key global factor in the global financial cycle. Gopinath(2016) and Gopinath et al.(2020) stress the domi- nating role of the dollar unit of account as one key possible channel. Gopinath and Stein (2021) link invoicing patterns and the safety of the dollar as a store of value. We propose a different channel based on the dominance of the dollar in the international financial system and argue that dollar liquidity shocks are a key drivers of exchange rates. The paper is organized as follows. Section2 presents the empirical analysis. Section3 presents the model and Section3 . Section5 presents the calibration of the model and the quantitative results. Section6 concludes. All proofs are in the appendix. 2 Empirical Analysis of Liquidity and Exchange Rates

We begin with a look at the data relating banking sector’s balance sheet and the foreign currency price of US dollars. Our thesis, at its simplest, is that the ﬁnancial sector increases its demand for liquid dollar assets, including US government obligations and reserves held at the Federal Reserve, when funding becomes more uncertain and that this drives movements in exchange rates. Before we describe the data and our analysis, it is useful to provide some institutional background underlying the thesis. Institutional background. The US dollar serves as the leading funding currency. Close to half of all cross-border bank loans and international debt securities are denominated in dollars. In addition to serving as the key international unit of account, the dollar serves a vehicle currency for foreign exchange transactions as well as invoicing currency for global tradel. In fact, 85% of all foreign exchange transactions occur against the US dollar and

6 5 6 40% of international payments are made in US dollars. ˆ, The central role of the US dollar in the international ﬁnancial and monetary system means that every day billions of US dollars circulate within and across countries. As the 2020 BIS working group report (Davies and Kent, 2020, p. 29) puts it:

US dollar funding is channeled through the global ﬁnancial system, involving entities across multiple sectors and jurisdictions. Participants in these markets face ﬁnancial risks typically associated with liquidity, maturity, currency and credit transformation. What makes global US dollar funding markets special is the broad participation of non-US entities from all around the world. These participants are often active in US dollar funding markets without access to a stable US dollar funding base or to standing central bank facilities which can supply US dollars during episodes of market stress.

The other pillar underlying our thesis is that financial institutions are subject to liquidity mismatch. To the extent that banks are not able to liquidate loans or other relatively illiquid assets, the demand of dollars by financial institutions can fluctuate considerably depending on funding conditions and vicissitudes of foreign exchange transactions. We will next argue how these conditions translate in the data into movements in the dollar exchange rates. Data and empirical analysis. We examine the behavior of the dollar against the other nine of the so-called G10 currencies, with special attention given to the euro. The euro area is especially important in our analysis because it encompasses a large economy with a financial system that relies heavily on short-term dollar funding. The other currencies are the Australian dollar, Canadian dollar, Japanese yen, New Zealand dollar, Norwegian krone, Swedish krona, Swiss franc, and the U.K. pound. We look at two sources of data for the US banking system. Detailed data on short- term dollar funding and on liquid dollar assets is not readily available for the global financial system, so we use the US data as a proxy for the dollar-denominated elements of the global banking balance sheets. That is, we presume that foreign banks’ demand for liquid dollar assets responds in a similar way to banks located in the U.S. (including U.S.- based subsidiaries of foreign banks) when faced with uncertainty about dollar funding.

5See Davies and Kent, 2020 and references therein. 6The global status of the US dollar dates to the Bretton Woods agreement, and more recently, this dominance has only deepened (Ilzetski, Reinhart and Rogoﬀ).

7 This approach is also followed by Adrian et al.(2010), a study that aims to show how the price of risk is related to banks’ balance sheets and the expected change in the exchange rate (rather than the level of the exchange rate, which is our concern here), and presents a simple partial-equilibrium model of the banking sector. More precisely, Adrian et al. (2010) focuses on the state of the balance sheet at time t in forecasting et+1 − et, as they are concerned with understanding the expected excess return on foreign bonds between t and t + 1. Our interest is centered on how changes in the balance sheet between t − 1 and t contribute to changes in the exchange rate between t − 1 and t, that is et − et−1. We consider two measures of short-term funding to financial intermediaries. The first is used by Adrian et al.(2010), U.S. dollar financial commercial paper (series DTBSPCKFM from FRED, the Federal Reserve Economic Data website maintained by the Federal Re- serve Bank of St. Louis.) Another major source of short-term funding to U.S. banks is demand deposits, measured by DEMDEPSL from FRED. We construct a variable that measures the level of funding and the response of financial intermediaries to uncertainty about that funding. We look at the ratio of the sum of reserves held at Federal Reserve banks and government (Treasury and agency) securities held by commercial banks (the sum of RES- BALNS and USGSEC from FRED) to short term funding (DTBSPCKFM + DEMDEPSL from FRED). This variable is endogenous in our model, but its movements are a key indi- cator of how the demand for dollars is affected by the financial sector’s demand for liquid assets when uncertainty increases. As dollar funding becomes more volatile for banks, they will increase their ratio of safe dollar assets to liabilities. That in turn will lead to a global increase in dollar demand, leading to a dollar appreciation. Figure1 plots this ratio of liquid government assets holdings to short-term funding of the financial sector.7 During this period, bank reserve balances rose from around 10 billion dollars in August, 2008 to nearly 800 billion dollars one year later, and then continued to climb to a peak of around 2.3 trillion dollars by late 2017 before gradually declining to 1.4 trillion dollars by the end of 2019. However, the liquidity ratio does not show movement anywhere near that magnitude. It is true that it rose during the onset of the global financial crisis, but this movement is not largely driven by the increase in reserves, because demand deposits rose almost proportionately. Mechanically, a large part of the rise in the overall liquidity ratio is driven by a fall in financial commercial paper funding, which works to lower the denominator of the ratio. 7The figure also plots an alternative measure described below.

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2001m1 0 1 2 3 4 ∆ iue1 h ai flqi sest hr-emliabilities short-term to assets liquid of ratio The 1: Figure e t 2003m7 epeetteprmtretmtso h regression: the of estimates parameter the present we 1, Table In ∆( = α x iudt ai AlternativeMeasureofLiquidityRatio Liquidity Ratio t + ) en h cag from “change the means β π 1 2006m1 e t ∆( t − i.e., , LiqRat π LiqRat t ∗ stedffrnebtenya-nya nainrtsin rates inflation year-on-year between difference the is β 2008m7 1 t + ) t oiie uigtmso ihucrany banks uncertainty, high of times During positive. hnw hudas xetapstv relationship positive a expect also should we then , β 2 ( 2011m1 9 π t − π t ∗ t + ) 2013m7 − β 1 3 8 to LiqDepRat h nainvral smeant is variable inflation The t ntevariable the in ” 2016m1 t − 1 2018m7 + t LiqRat x t ; e t t sthe is sthe is (1) hold greater amounts of liquid dollar assets (reserves and Treasury securities) relative to demand deposits, so LiqRatt is higher. That increased demand for safe dollar assets leads to a stronger dollar (an increase in et.)

We also include the lagged level of LiqRatt. This is included because the depreciation of the dollar might depend on lagged as well as current levels of this variable. The regressions we report would have the identical fit if we included current and lagged levels of this variable, instead of the change in the variable and the lagged level. We specify the regression as above for two reasons: First, specifying the regression so that the change in the liquidity variable influences the change in the exchange rate leads to a more natural interpretation. Second, while the current and lagged levels of the variable are highly correlated, which leads to multicollinearity and imprecise coefficient estimates, the change and the lagged level are much less highly correlated. Table1 reports the regression findings for the nine exchange rates. The sample period is February 2001 to July 2020.9 (Data on financial commercial paper starts in January 2001.) With the exception of Japan, the liquidity ratio variable has the expected sign and is statistically significant at the 1 percent level for all exchange rates. The relative inflation variable also has the correct sign for all the currencies and is statistically significant for most countries It is commonplace to look at short-term interest rate movements to account for the effects of monetary policy changes on exchange rates. During much of our sample period, interest rates were near the zero-lower bound, and do not appear to do a good job mea- ∗ suring the monetary policy stance. In the appendix, in Table A1, we include it − it , the interest rate in each of the 9 countries relative to the U.S. as regressors. It is only statistically significant at the 5 percent level for Japan, and none of the major conclusions are altered by its inclusion.

We highlight that the key regressor, ∆LiqRatt, is not simply a market price. That is, these regressions “explain” exchange rate movements but are not relying on other market prices to do the job. It is the balance sheet variables that play the pivotal role. We argue that uncertainty about funding drives the balance sheet variables, but what if we were to include a direct measure of uncertainty in the regressions? Many asset-pricing studies have used VIX to quantify market uncertainty, and VIX has power in explaining the movements of many asset prices. However, VIX does not directly measure uncer-

9Australia and New Zealand’s sample end in May 2020 because of availability of inﬂation data.

10 tainty about dollar funding for banks. Indeed, VIX might measure some dimensions of uncertainty, but it might also be capturing global risk, and global risk might be driving the dollar, as in the model of Farhi and Gabaix(2016). In Table2, we have included the change in VIX along with the other variables. As expected, VIX has positive coefficients in all cases (except Japan) and is statistically significant. An increase in VIX is associated with an appreciation of the dollar. However, the introduction of this variable does not reduce the significance of the liquidity ratio variable, for any of the countries, and for most only has a small effect on the magnitude of the coefficient. This suggests that the uncertainty that is quantified by the VIX does not include all of the forces that drive the liquidity ratio and lead to its positive association with dollar appreciation. (Appendix table A2 also includes the interest rate differential, and as in Table A1, we see that its inclusion has little influence on the findings and its effect is mostly small and insignificant.) It is important to note that the liquidity ratio is not an exogenous driver of exchange rates—either in our model or in the real world. We will show that a calibrated version of our model implies a positive association between the change in the liquidity ratio and the value of the dollar under multiple driving shocks, including uncertainty shocks, monetary policy shocks and liquidity demand shocks. We can, however, use instrumental variables to isolate the effects of uncertainty on the liquidity ratio, and its transmission to exchange rates. To that end, Table3 uses two measures of uncertainty as instruments for the liquidity ratio: the cross-section standard deviation at each time period of the inflation rates of the G10 countries, and the cross- section standard deviation of the rates of depreciation for these currencies. The findings are largely the same as in Table2. For most of the countries, the magnitude of the effect of the liquidity ratio on the exchange rate is increased, and for some (such as Canada), the statistical significance greatly increases. The model still fits poorly for the Japanese yen, and we now find statistical significance of the liquidity ratio on the Swiss franc exchange rate. In appendix Table A3, we take the alternative tack of including VIX as an instrument for the liquidity ratio. It is possible for VIX to be a valid instrument that is uncorrelated with the regression error, even though it is statistically significant when included in the regression separately (as in Tables2 and3) if the other forces that drive the liquidity ratio are uncorrelated with VIX. That is, Table A3 reports the influence of the liquidity ratio on exchange rates when the liquidity ratio is driven by VIX and other measures of un-

11 certainty, while any other forces that might influence the exchange rate are relegated to the regression error. VIX is a valid instrument if it is uncorrelated with those forces. If one takes such a stance, then the estimates reported in Table A3 reveal a strong channel of uncertainty on exchange rates working through the liquidity ratio. We consider next an alternative measure of the liquidity ratio that includes “net financing” of broker-dealer banks. This is a measure of “funds primary dealers borrow through all fixed-income security financing transactions,” as described in Adrian and Fleming(2005). We include this as another source of short-term liabilities, similar to how net repo financing is included as a measure of short-term liabilities in the liquidity ratio calculated by the International Monetary Fund (Global Financial Stability Report, 2018). Figure1 also plots this alternative measure of the liquidity ratio, which is smaller than our baseline measure because it includes another class of liabilities of the banking system in the U.S. Tables A4, A5, A6, and A7 in the appendix are analogous to Tables1,2,3, and A3, respectively. The conclusions using this alternative measure are virtually unchanged quali- tatively, though of course the numerical values of the estimated coefficients are different. The liquidity ratio is still highly statistically significant for all currencies, except for Japan, and, in the case of the instrumental variable regressions, Switzerland. In Table4 we consider another dimension along which greater volatility in funding might affect banking behavior, which in turn can influence the exchange rate. In this table, we look at how the volatility of intra-day Fed funds rates are associated with the value of the dollar. We measure the volatility for a given month by taking the difference within each day between the high and the low transactions rates for that day, then taking the maximum of those differences over the month. Table4 reports instrumental variables regressions in which the instruments are the same as for Table3, and as in that table, we include VIX as a separate explanatory variable. We find that as the Fed funds volatility increases, the dollar tends to appreciate. This pattern holds across all dollar exchange rates, except for the Japanese yen and Swiss franc. In Table A8, as in Table A3, we include VIX as an instrument for the Fed funds volatility but not as an explanatory variable for the exchange rate, and we find a strong relation between the volatility measure and the value of the dollar.

12 Table 1: Exchange Rates and Liquidity Ratio Feb. 2001 – July 2020 EUR AUS CAN JPN NZ NWY SWE SWZ UK

∆(LiqRatt) 0.225*** 0.243*** 0.131*** -0.152*** 0.295*** 0.189*** 0.213*** 0.145*** 0.165*** (4.525) (3.597) (2.643) (-3.036) (4.302) (3.023) (3.555) (2.654) (3.320) ∗ πt − πt -0.542*** -0.422** -0.412* 0.008 -0.718*** -0.117 -0.492** -0.666*** -0.390** (-3.718) (-2.226) (-1.928) (0.055) (-3.757) (-0.803) (-2.521) (-2.803) (-2.114)

LiqRatt−1 0.011** 0.006 0.007 0.002 0.009 0.010* 0.006 0.005 0.009* (2.425) (1.065) (1.578) (0.315) (1.508) (1.763) (1.145) (0.990) (1.735) Constant -0.012*** -0.004 -0.006* -0.001 -0.009** -0.007* -0.009** -0.017*** -0.006 (-3.452) (-1.053) (-1.832) (-0.108) (-2.095) (-1.653) (-2.069) (-3.169) (-1.597) N 234 232 234 234 232 234 234 234 234 2 adj. R 0.11 0.05 0.03 0.03 0.10 0.03 0.05 0.04 0.04 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

Table 2: Exchange Rates and Liquidity Ratio and VIX Feb. 2001 – July 2020 EUR AUS CAN JPN NZ NWY SWE SWZ UK

∆(LiqRatt) 0.195*** 0.163*** 0.086* -0.137*** 0.232*** 0.139** 0.173*** 0.129** 0.140*** (3.999) (2.798) (1.909) (-2.741) (3.674) (2.358) (3.017) (2.337) (2.831) ∗ πt − πt -0.427*** -0.185 -0.277 -0.016 -0.519*** -0.032 -0.415** -0.595** -0.304* (-2.950) (-1.130) (-1.436) (-0.112) (-2.941) (-0.235) (-2.234) (-2.491) (-1.660)

∆VIXt 0.001*** 0.004*** 0.002*** -0.001** 0.003*** 0.002*** 0.002*** 0.001* 0.001*** (3.881) (9.339) (7.513) (-2.250) (6.909) (5.934) (5.153) (1.928) (3.116)

LiqRatt−1 0.011** 0.007 0.007* 0.002 0.009* 0.010* 0.007 0.005 0.008 (2.492) (1.427) (1.829) (0.334) (1.696) (1.892) (1.390) (1.063) (1.617) Constant -0.011*** -0.005 -0.006* -0.001 -0.009** -0.007* -0.008** -0.016*** -0.005 (-3.277) (-1.504) (-1.943) (-0.212) (-2.210) (-1.711) (-2.128) (-2.973) (-1.459) N 234 232 234 234 232 234 234 234 234 2 adj. R 0.16 0.31 0.22 0.05 0.25 0.16 0.15 0.05 0.07 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

13 Table 3: Exchange Rates and Liquidity Ratio Instrumental Variable Regression: StDev(Inf) and StDev(XRate) instrument for ∆(LiquidityRatio) Feb. 2001 – July 2020 EUR AUS CAN JPN NZ NWY SWE SWZ UK

∆(LiqRatt) 0.450*** 0.593*** 0.441*** -0.306** 0.507*** 0.526*** 0.473*** 0.019 0.717*** (3.150) (3.182) (3.274) (-2.266) (2.804) (2.831) (2.800) (0.121) (3.547) ∗ πt − πt -0.566*** -0.444** -0.496** 0.051 -0.658*** -0.242 -0.595*** -0.476 -0.981*** (-3.339) (-2.111) (-2.149) (0.328) (-3.250) (-1.370) (-2.728) (-1.640) (-3.034)

LiqRatt−1 0.014*** 0.013** 0.013** -0.002 0.013** 0.017** 0.011* 0.004 0.024*** (2.912) (2.119) (2.589) (-0.255) (2.123) (2.596) (1.897) (0.733) (2.922)

∆VIXt 0.001*** 0.003*** 0.002*** -0.001* 0.003*** 0.002*** 0.002*** 0.001** 0.000 (2.693) (6.879) (5.423) (-1.680) (5.659) (4.278) (3.971) (2.049) (0.856) Constant -0.015*** -0.010** -0.011*** 0.003 -0.012*** -0.013** -0.013*** -0.013* -0.017*** (-3.641) (-2.263) (-2.859) (0.433) (-2.645) (-2.528) (-2.708) (-1.834) (-2.896) N 234 232 234 234 232 234 234 234 234

t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

Table 4: Exchange Rates and Measure of Intraday Fed Funds Rate Dispersion Instrumental Variable Regression: StDev(Inf) and StDev(XRate) instrument for ∆(MaxDift) EUR AUS CAN JPN NZ NWY SWE SWZ UK

∆(MaxDift) 0.013** 0.016** 0.011*** -0.006 0.013* 0.013** 0.014** 0.000 0.018*** (2.166) (2.358) (2.707) (-1.486) (1.908) (2.165) (2.067) (0.049) (2.725) ∗ πt − πt -0.493* -0.345 -0.046 0.028 -0.671** -0.003 -0.696* -0.518 -0.370 (-1.811) (-1.161) (-0.195) (0.201) (-2.241) (-0.014) (-1.968) (-1.249) (-1.217)

MaxDift−1 0.005* 0.006* 0.004* -0.005** 0.007* 0.005 0.007* 0.001 0.008** (1.680) (1.667) (1.736) (-2.302) (1.781) (1.507) (1.846) (0.222) (2.224)

∆VIXt 0.001** 0.004*** 0.002*** -0.001* 0.003*** 0.002*** 0.002*** 0.001** 0.000 (1.977) (4.636) (4.332) (-1.660) (3.867) (2.851) (2.803) (2.222) (0.260) Constant -0.013* -0.011* -0.007 0.008* -0.015** -0.009 -0.019** -0.013 -0.013* (-1.821) (-1.686) (-1.618) (1.673) (-2.013) (-1.351) (-2.003) (-1.085) (-1.859) N 187 187 187 187 187 187 187 187 187

t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01

14 3 A Model of Banking Liquidity and Exchange Rates

We present a dynamic equilibrium model of global banks that intermediate international financial flows and are subject to idiosyncratic liquidity shocks. The model has two countries, the EU and the US, with corresponding currencies. To fix ideas, we think about the euro as the domestic currency and the dollar as the foreign currency. In each country, there is a continuum of households and a central bank that sets monetary policy. Produc- tion of the single tradable good is carried out globally by multinationals. We assume the law of one price. 3.1 Environment

Timing. Time is discrete and there is an inﬁnite horizon. Every period is divided in two sub-stages: a lending stage and a balancing stage. In the lending stage, banks make their equity payout, Divt, and portfolio decisions. In the balancing stage, banks face liquidity shocks and re-balance their portfolio. Notation. We use “‘asterisk” to denote the foreign currency (i.e., the “dollar”) variable and “tilde” to denote a real variable. The vector of aggregate shocks is indexed by X. The exchange rate is deﬁned as the amount of euros necessary to purchase one dollar—hence, a higher e indicates an appreciation of the dollar. Preferences and budget constraint. Payouts are distributed to households that own bank shares and have linear utility with discount factor β. Banks’ objective is to maximize share- holders’ value and therefore they maximize the net present value of dividends:

∞ X t β · Divt. (2) t=0

Banks enter the lending stage with a portfolio of assets/liabilities and collect/make associated interest payments. The portfolio of initial assets is given by liquid assets mt, both in euros and dollars, and corporate loans, bt, denominated in consumption goods. Note that we refer to liquid assets as “reserves”, for simplicity, but they should be understood as capturing also government bonds—the important property, as we will see, is that these are assets that can be used as settlement instruments. On the liability side, banks issue demand deposits, dt, discount window loans, wt, and net interbank loans, ft (if the bank has borrowed funds, ft is positive, and vice versa), again in both currencies. Deposits and

15 interbank market loans have market returns given by id and ¯if while central banks set the m w corridor rates for reserves and discount window, respectively i and i . A yield it paid in period t is pre-determined in period t − 1. Meanwhile, Rb is the real return on loans. The bank’s budget constraint is given by

∗ mt+1 − dt+1 ∗ ∗ ∗ ∗ b ∗ m,∗ ∗ d,∗ Pt Divt + + bt+1Pt + mt+1 − dt+1 ≤ Pt btRt + mt (1 + it ) − dt (1 + it ) et m d f w ∗ f,∗ ∗ w,∗ mt(1 + it ) − dt(1 + it ) + ft(1 + it ) + wt(1 + it ) + ft (1 + it ) + wt (1 + it ) − . (3) et

Withdrawal shocks. In the balancing stage, banks are subject to random withdrawal of deposits in either currencies. As in Bianchi and Bigio(2021), withdrawals have zero mean— hence deposits are reshuffled but preserved within the banking system. In addition, we assume that the distribution of these shocks is time-varying: As a way to capture the special role of the dollar in wholesale funding and other volatile funding sources of funding, as discussed in Section2. In particular, withdrawal shocks, denoted by ω, have a CDF denoted by Φt—PDF denoted by φt. The inflow/outflow of deposits across banks generates, in effect, a transfer of liabilities. We assume that these transfer of liabilities are settled using reserves of the corresponding currency. Importantly, reserves must remain positive at the end of the period. We denote j j by s the euro surplus of a bank facing a withdrawal shock ωt . This surplus is given by the amount of euro reserves a bank brings from the lending stage minus the withdrawals of deposits: j j st = mt+1 + ωt dt+1. (4)

We omit the subscript of bank choices in deposits and reserves, because it is without loss of generality that all banks make the same choices in the lending stage. If a bank faces a negative withdrawal shock, lower than ω˜ ≡ −m/d, the bank has a deﬁcit of reserves. Conversely, if the withdrawal shock is larger than ω˜, the bank has a surplus. Notice that if m = 0, the sign of the surplus has the same sign as the withdrawal shock. A higher liquidity ratio makes more likely that the bank will be in surplus. Similarly, we have the following surplus in dollars

j,∗ ∗ j,∗ ∗ st = mt+1 + ωt dt+1. (5)

16 Interbank market. After the withdrawal shocks are realized, there is a distribution of banks in surplus and deficits in both currencies. We assume that there is an interbank for each currency in which banks that have a deficit in one currency borrow from banks that have a surplus in the same currency. These two interbank market behave symmetrically, so it suffices to show only how one of the markets work.10 We model the interbank market as an over-the-counter (OTC) market, in line with institutional features of this market (see Ashcraft and Duffie, 2007; Afonso and Lagos, 2015). Modeling the interbank market using search and matching is also natural considering that the interbank market is a credit market in which banks on different sides of the market— surplus and deficit—must find a counterpart they trust. As a result of the search frictions, only a fraction of the surplus (deficit) will be lent (borrowed) in the interbank market. We assume, in particular, that each bank gives an order to a continuum of traders to either lend or borrow, as in Atkeson et al.(2015). A bank with surplus s is able to lend a fraction Ψ+ to other banks. The remainder fraction is kept in reserves. Conversely, a bank that has a deficit is able to borrow a fraction Ψ− from other banks, and the remainder deficit is borrowed at a penalty rate iw. The penalty rate can be thought of as the discount window rate or as an overdraft-rate charged by correspondent banks that have access to the Fed’s discount window. The fractions Ψ+and Ψ− depend on the abundance of reserve deficits relative to surpluses. Assuming a constant returns to scale matching function, the probabilities depend entirely on market tightness, defined as

− + θt ≡ St /St (6)

+ 1 j − 1 j where St ≡ 0 max st , 0 dj and St ≡ − 0 min st , 0 dj denote the aggregate surplus ´ ´ and deficit, respectively. Notice that because m ≥ 0 and E(ω) = 0, we have that in equilibrium θ ≤ 1. That is, there is a relatively larger mass of banks in surplus than deficit. The interbank market rate is the outcome of a bargaining problem between banks in deficit and surplus, as in Bianchi and Bigio(2021). There are N trading rounds, in which banks trade with each other. If banks are not able to match by the N trading rounds, they deposit the surplus of reserves at the central bank or borrow from the discount window.

10We are assuming in the background an extreme form of segmented interbank markets: penalties in dollars and euros are independent because dollar surpluses cannot be used to patch Euro deﬁcits and vice versa. This assumption can be relaxed to some extent but some form of segmentation of asset markets is necessary to obtain liquidity premia. See the discussion below.

17 Throughout the trading, the terms of trade at which banks borrow/lend, i.e., the interbank market rate, depend on the probabilities of finding a match in a future period. We denote f by i the average interbank market rate at which banks trade on average. Ultimately, we can define a real penalty function χ that captures the benefit of having a real surplus or deficit s˜ upon facing the withdrawal shock as follows:  + 0 0 χ (θ; X,X )˜s if s˜ ≥ 0, χ(θ, s˜; X,X ) = (7) χ−(θ; X,X0)˜s if s˜ < 0

where χ+ and χ−are given by

− 0 w 0 m 0 − w 0 f 0 χ (θ; X,X ) = [R (X,X ) − R (X,X )] − Ψ (θ)[R (X,X ) − R (X,X )] (8) + 0 + f 0 m 0 χ (θ; X,X ) = Ψ (θ)[R (X,X ) − R (X,X )]. (9)

y 0 1+iy(X) In these expressions, R (X,X ) ≡ 1+π(X,X0) , denote the realized gross real rate of an as- 0 P (X0) set/liability y and π(X,X ) ≡ P (X) − 1 denotes the inflation rate when the initial state is X and the next period state is X0. When it does not lead to confusion, we streamline the h y i 0 ¯y 1+i (X) argument (X,X ) in these expressions. We also denote by R ≡ E 1+π(X,X0) |X as the expected real rate—recall that the nominal rate is pre-determined but the ex-post real return depends on the inflation rate. Equation (8) reflects that a bank that borrows from the the interbank market or from the discount window, obtains the interest on reserves—hence the cost of being in deficit is given by Rf − Rm in the former and Rw − Rm in the latter. By the same token, (9) reflects that the benefit from lending in the interbank market in case of surplus is Rf − Rm. Figure2 present a sketch of the timeline of decisions within each period. We next turn to describe the bank optimization problem. Banks’ problem. The objective of a bank is to choose dividends and portfolios to max- imize2 subject to the budget constraint. Critically, when choosing the portfolio, banks anticipate how withdrawal shocks may lead to a surplus or deficit of reserves and associated costs and benefits. We express the bank’s optimization problem in terms of real portfolio holdings {˜b, m˜ ∗, d˜∗, d,˜ m˜ } and the real returns. For example, we define for example m˜t ≡ mt/Pt−1. The problem can be expressed recursively as follows:

18 FX openFX close

ω Portfolio OTC Settlements Average Choices Shock Market Market Payouts (m∗, b∗, d∗, m, b, d) R,¯ Ψ+, Ψ− (χ+, χ−)

Figure 2: Timeline

Problem 1. The recursive problem of a bank is

0 0 v (n, X) = max Div + βE [v (n ,X ) |X] (10) {Div,˜b,m˜ ∗,d˜∗,d,˜m˜ } subject to the budget constraint:

∗ ∗ Div+˜b +m ˜ +m ˜ = n + d˜+ d˜ , (11) where the evolution of bank networth is given by,

n0 = Rb(X)˜b + Rm(X,X0)m ˜ + Rm,∗(X,X0)m ˜ ∗ − Rd(X,X0)d˜− R∗,d(X,X0)d˜∗ | {z } Portfolio Returns ∗ ∗ ∗ ∗ ∗ 0 0 + χ (θ (X), m˜ + ω d˜ ; X,X ) + χ(θ(X), m˜ + ω d˜; X,X ) . (12) | {z } Settlement Costs

The variable n represents the banks’ net worth at the beginning of the period.11 The bank n o then chooses the portfolio ˜b, m˜ ∗, d˜∗, d,˜ m˜ and its dividends Div to maximize the future value. The evolution of n depends on the realized return on assets, but also on the realized settlement costs. Because of the linearity of the banks’ payoﬀs and the objective function, the value function is linear in net worth. An equilibrium with a ﬁnite supply of

11To obtain (12), we use the deﬁnition of χ as expressed in (7)-(9) and the real returns. Implicit in the law of motion for net worth is the convention that a bank that faces a withdrawal covers the associated interest payments net of the interest on reserves.

19 loans, deposits and reserves requires that Rb (X) = 1/β ≥ Rm (X,X0) .12 The next lemma characterize the solution of the bank under this result. Lemma (1) summarizes these results: Lemma 1. The solution to (10) is v (n, X) = n, and the optimal portfolios solve

? b 0 ∗,d 0 ˜∗ b ∗,m ∗ Π (X) = max EX0 R (X,X ) − R (X,X ) d − R (X) − R (X) m˜ {m,˜ d˜∗,d,˜m˜ } + Rb(X,X0) − Rd(X,X0) d˜− Rb(X,X0) − Rm(X,X0) m˜ h ∗ ∗ 0 ∗ ∗ ˜∗ 0 i h 0 ˜ 0 i + Eω∗ χ (θ (X,X ), m˜ + ω d ; X,X ) + Eω χ(θ(X,X ), m˜ + ω d; X,X ) . (13)

In equilibrium, Π(X) = 0.

Central to this optimization problem are the liquidity costs, as captured by χ and χ∗. Deposits in either currency have direct interest costs given by the real returns on deposits, but also affect indirectly the banks’ settlement needs. Reserves in each currency yield direct real returns, correspondingly, but have the additional indirect benefit of leading to higher average positions in the interbank market. It is important to note that the bank problem is homogeneous of degree one: As a result, the scale of dollar and euro deposits of each bank is indeterminate for an individual bank. The liquidity ratio and the leverage ratio, however, are not. In effect, the kink in the liquidity cost function creates concavity in the bank objective, generating interior solutions for the ratios.13 Thus, liquidity risk generates an endogenous bank risk-averse behavior, which will be critical for the determination of the exchange rate, as will become clear below. Importantly, at the individual level, the liquidity ratio and the leverage ratio are determined. This is not the case, however, for dividends and the scale of the portfolio, which are are determined only at the aggregate level. In particular, {n, Div} must guarantee that the quantity of loans be consistent with Rb = 1/β and zero expected profits, Π(X) = 0. Non-Financial Sector. This section presents the description of the non-financial block: This block is composed of a representative household, one in each country. Households

12If the return on loans were lower than 1/β, banks would not invest in loans. Conversely, if the return on loans were higher than 1/β,banks would inject infinite equity in the bank. If Rm > 1/β banks would issue equity to buy reserves and the bank’s value would be indeterminate. 13This behavior is analogue to the behavior of productive firms with Cobb-Douglas technologies: firms earn zero-profits, their production scale is indeterminate, but the ratio of production inputs is determined in equilibrium as a function of relative factor prices.

20 supply labor and save in deposits in both currencies. Firms are multinationals which use labor for production and are subject to working capital constraints, giving rise to a demand for loans. This block delivers an endogenous demand schedule for loans, and deposit demands in both currencies. To keep the model simple, we make assumptions so that the eﬀective loan demand and deposit supplies are static. As we show in AppendixC, we obtain the following schedules:

¯b b b Rt+1 = Θ (Bt) , < 0, Θt > 0, (14) ¯∗,d ∗,d ∗,s ς∗ ∗,d Rt+1 = Θ (Dt ) , ς > 0, Θ > 0, and (15) ¯d d s ς d Rt+1 = Θ (Dt ) , ς > 0, Θ > 0, (16) where is the semi-elasticity of credit demand and {ς, ς∗} are the semi-elasticities of the deposit supplies with respect to the real returns. These parameters are linked to the production structure and preference parameters in the micro-foundation. m Government/Central Bank. Both central banks choose the rates for reserves it and dis- w ∗ count window it . Central banks in each country also set the supply of reserves Mt+1,Mt+1 . To balance the payments on reserves and the revenues from discount window loans, we assume that central banks use lump sum taxes/transfers rebated to households from the same country. Because households have linear utility in the tradable consumption good, these lump sum taxes only aﬀect the level of consumption, but have no other implications. Using Wt to denote the discount window loans, we have the following budget constraint for the domestic economy,

m w Mt + Tt + Wt+1 = Mt−1(1 + it ) + Wt(1 + it ). (17)

An identical budget constraint holds for the foreign economy. In both countries Tt adjusts to passively to balance the budget constraint, given the monetary policy choices. 3.2 Equilibrium

We study recursive competitive equilibria where all variables are indexed by the vector of aggregate shocks, X. We consider shocks to the nominal interest rates on reserves, the deposit supply, and the volatility of withdrawals. Without loss of generality, we restrict to a symmetric equilibrium, in which all banks choose the same portfolios.

Deﬁnition 1. Given central bank policies for both countries {M(X), im(X), iw(X),W (X)},

21 {M ∗(X), im,∗(X), iw,∗(X),W ∗(X)} a recursive competitive equilibrium is a pair of price level functions {P (X),P ∗(X)}, exchange rates e(X), real returns for loans, Rb(X), nomi- f nal returns for deposits {id(X), id,∗(X)}, an interbank market rate i (X), market tightness θ(X), bank portfolios {d(X), d∗(X), m(X), m∗(X), ˜b(X)}, interbank and discount window loans {f(X), f ∗(X), w(X), w∗(X)} and aggregate quantities of loans {B˜(X)} and deposits {D(X),D∗(X)} such that:

(i) Households are on their deposit supply and ﬁrms are on their loan demand. That is, equations (14)-(16) are satisﬁed given real returns and quantities {B˜(X),D(X),D∗(X)}.

(ii) Banks choose portfolios {d˜(X), d˜∗(X), m˜ (X), m˜ ∗(X), ˜b(X)} to maximize expected profits, as stated in (13).

(iii) The law of one price holds P (X) = P ∗(X)e(X).

(iii) All market clear: The deposit markets: d˜(X) = Ds,(X) and d˜∗(X) = Ds,∗(X). The reserve markets: m˜ (X) P (X) = M(X) and m˜ ∗(X)P ∗(X) = M ∗(X). The loans market: ˜b(X) = B(X). The interbank market: Ψ+(X)S+ = Ψ−(X)S−.

(vi) Market tightness θ(X) is consistent with the portfolios and the distribution of withdrawals while the matching probabilities {Ψ+(X), Ψ−(X)} and the fed funds rate f i (X) are consistent with market tightness θ.

Combining both equations in reserve-market clearing conditions, and the law of one price and the deposit clearing conditions, we arrive at a condition for the determination of the nominal exchange rate:

P (X) M(X)/m˜ (X) M(X)/D (X) /m˜ (X)/D (X) e(X) = = = . P ∗ (X) M ∗(X)/m˜ ∗(X) M ∗(X)/D (X) /m˜ ∗(X) (18)

Condition (18) is a Lucas-style exchange rate determination equation, but rather than following from cash-in-advance constraints these follow from bank portfolio decisions link- ing liabilities with reserve assets. Given a real demand for reserves in euro and dollars that emerge from the bank portfolio problem (13), the dollar will be stronger (i.e., higher e) the larger is the nominal supply of euro reserves relative to dollar reserves. Similarly, for given nominal supplies of euro and dollar reserves, the dollar will be stronger the larger is the demand for real dollar reserves.

22 The novelty here relative to the canonical model is that liquidity factors play a role in the real demand for currencies, and hence aﬀect the value of the exchange rate. We turn next to analyze this mechanism. 4 Characterization

Liquidity Premia and Exchange Rates. To understand how liquidity aﬀects exchange rates, we inspect the portfolio problem (13). Using (7), we write the expected proﬁts of a bank with portfolio (m, m∗, d, d∗) as follows:

∗ b ∗,d ∗ b ∗,m ∗ Π (X) = R¯ − R¯ d˜ − R¯ − R¯ m˜ + (19) R¯b − R¯d d˜− R¯b − R¯m m˜ + −m∗/d∗ ∞ χ∗,−(θ) (m ˜ ∗ + ω∗d˜∗)dΦ∗(ω∗) + χ∗,+(θ) (m ˜ ∗ + ω∗d˜∗)dΦ∗(ω∗)+ ˆ−1 ˆ−m∗/d∗ −m/d ∞ χ−(θ) (m ˜ + ωd˜)dΦ(ω) + χ+(θ) (m ˜ + ωd˜)dΦ(ω). ˆ−1 ˆ−m/d

The ﬁrst two lines are the excess returns on loans relative to deposits and reserves. The last two lines represent the expected liquidity costs in both currencies. The ﬁrst-order condition with respect to m∗ is

¯b ¯m,∗ ∗ ∗ ∗ +,∗ ∗ ∗ ∗ −,∗ ∗ ∗ ∗ R − R = (1 − Φ (−m /d ))χ (θ ) + Φ(−m /d )χ (θ ) = Eω∗ [χm∗ (s ; θ )] . (20)

At the optimum, banks equate the expected marginal return of making loans, R¯b, with the marginal return on reserves. The latter is given by the interest on reserves R¯m plus a random marginal liquidity value. If the bank ends up in surplus, which occurs with probability 1 − Φ(−m∗/d∗), the marginal value is χ+,∗ and if the bank ends up in deﬁcit which occurs with probability Φ(−m∗/d∗), the marginal value is χ−,∗. Given {χ+,∗, χ−,∗} a higher ratio of reserves to deposits is associated with a smaller R¯b − R¯m premium. We label this diﬀerence as the bond premium, BP ≡ R¯b − R¯m. We have an analogous condition for m:

¯b ¯m + − R = R + (1 − Φ(−m/d))χ (θ) + Φ(−m/d)χ (θ) = Eω [χm (s; θ)] . (21)

23 Combining (20) and (21), we obtain a condition that relates the diﬀerence in the real return on reserves on both currencies:

¯m ¯m,∗ ∗ ∗ R − R = Eω∗ [χm∗ (s ; θ )] − Eω [χm (s; θ)] . (22) | {z } LP

We label this difference, the dollar liquidity premium (LP), LP ≡ R¯m − R¯∗,m. Using that 1 + π = (1 + π∗)e0/e and the law of one price, we express (22) as: 1 m m,∗ et+1 Et 1 + it − (1 + it ) · = LP. (23) 1 + πt+1 et This is an uncovered interest parity condition (UIP) adjusted by a liquidity premium. Absent a liquidity premia, (23) would reduce to a canonical UIP that simply equates, to a first order, the difference in nominal returns to the expected depreciation. However, whenever the marginal liquidity value of a dollar is larger than the marginal liquidity value of a euro (i.e.,. when the LP is positive), and the nominal rates are equal, this implies that the dollar must be expected to depreciate. This way, the dollar reserve delivers a lower expected real return compensating for the higher liquidity value. Because banks are risk neutral, there is no risk premiumand the UIP emerges entirely through liquidity. Finally, we have the first-order conditions with respect to deposits in both currencies:

¯d ¯m R = R + Eω [χm (s; θ)] + Eω [χd (s; θ)] and (24) ¯d,∗ ¯m,∗ ∗ ∗ ∗ ∗ R = R + Eω∗ [χm∗ (s ; θ )] + Eω∗ [χd∗ (s ; θ )] . (25)

The four ﬁrst-order conditions, together with the bank’s budget constraint, characterize the bank’s portfolio decision. Couple with the demand-supply system, this equations fully characterize the equilibrium. Funding and Payment Volatility shocks. We now provide a theoretical characterization of how the exchange rate and the liquidity premia vary with shocks. In this section, we assume that the supply of deposits are perfectly inelasticto sharpen the results, while not altering the essence of the model. Here we present exact formulas for permanent and iid shocks, and relegate approximate results for more general AR(1) shocks to AppendixB which contains all the proofs. We focus on shocks to the dollar. The same shocks to the euro interbank market will carry the opposite eﬀect on the exchange rate and the LP. Throughout the results, a useful

24 object for the characterization is the liquidity ratio which, deﬁned in terms of aggregates, M/P is given by µ ≡ D . A critical object in the characterization is the derivative of the LP with respect to θ∗:

∗ ∗ ∗,+ ∗ ∗ ∗,− LPθ∗ = (1 − Φ (−µ )) · χθ∗ + Φ (−µ ) · χθ∗ > 0.

We proceed to discussing the eﬀects of various shocks. First, we characterize the eﬀects shocks to the supply of dollar deposits.

Proposition 1 (Deposit scale eﬀects). Consider a shock that increases the real supply for dollar deposits, D∗. 1) If the shock is temporary and (iid) it appreciates the dollar, reduces the dollar liquidity ratio, and increases the premia:

∗ ∗ ∗ d log e d log P −LPθ∗ · θµ · µ ∗ = − ∗ = b ∗ ∗ ∈ [0, 1), d log D d log D R − LPθ∗ · θµ · µ

∗ b d log µ R ∗,m ∗ = − b ∗ ∗ ∈ [−1, 0), and dBP = dLP = R d log e. d log D R − LPθ∗ · θµ · µ

2) If the shock is permanent (random walk) it appreciates the dollar one for one and is neutral on the liquidity ratio and the premia:

d log e∗ d log P ∗ = − = 1, and dµ∗ = dBP = dLP = 0. d log D∗ d log D∗

Proposition1 establishes that a higher supply of dollar deposits appreciate the dollar. The logic is simple: a higher amount of real dollar deposits increases the demand for real dollar reserves. Given a ﬁxed nominal supply, we must have an appreciation of the dollar. There is an important distinction between temporary and permanent shocks. When the shock is temporary, the exchange rate is expected to revert to the initial value in the following period. Given nominal rates, this reduces the expected real return of holding dollar reserves. As a result, the demand for real dollar reserves fall. In equilibrium, the increase in the scarcity of dollar reserves leads to an increase in the LP. Overall, we then have that in response to an increase in the supply of dollar deposits by households, the dollar appreciates, the dollar liquidity ratio falls, and the LP increases.

25 When the shock is permanent, the effect on the exchange rate is expected to be permanent. Absent any expected depreciation effects, the dollar liquidity ratio remains constant. As a result, there are no permanent effects on the LP.14 Next, we characterize the effects of a rise in payment dispersion of dollar deposits. For that purpose, it is useful to index Φ by a parameter that captures the payment dispersion, σ. We also define notation for the mass of surplus payment shocks:

∞ δ∗ (µ∗, σ∗) ≡ ωφ (ω; σ∗) dω. ˆµ∗

We maintain the following assumption:

∗ ∗ ∗ Assumption 1. Any right tail of Φ is increasing in σ , i.e. δσ∗ > 0 for any µ .

This assumption is enough to characterize the eﬀects of payment volatility shocks.

Proposition 2 (Volatility eﬀects). Consider a shock that increases the size of dollar payment shocks, σ∗. 1) If the shock is temporary (iid) it appreciates the dollar, raises the dollar liquidity ratio and raises the :

∗ ∗ ∗ ∗ d log e d log µ d log P LPσ∗ σ 1 − θ δσ∗ ∗ = ∗ = − ∗ = b ∗ ∈ 0, · ∗ , and d log σ d log σ d log σ R − LPµ∗ · µ 1 + θ µ

dBP = dLP = R∗,md log e.

2) If the shock is permanent (random walk). Then, the shock appreciates the dollar, raises the liquidity ratio:

∗ ∗ ∗ d log e d log µ d log P 1 − θ δ ∗ = = − ∈ 0, · σ , and dBP = dLP = 0. d log σ∗ d log σ∗ d log σ∗ 1 + θ µ∗

Proposition2 presents a central result. In response to a rise in the volatility of dollar payments, the dollar appreciates, and there is an increase in the dollar liquidity ratio and the LP. Intuitively, with a larger dollar volatility, banks demand greater real dollar reserves. With the nominal supplies given, this must lead to an appreciation of the dollar.

14This neutrality result hinges on the assumption of constant returns to scale in the interbank matching technology. As banks scale up proportionally dollar deposits and reserves, given the same real returns on dollar and euro reserves, the original liquidity ratio remains consistent with the new equilibrium.

26 Again, there is a relevant distinction between temporary or permanent shocks. When the shock is temporary, the expected depreciation of the dollar reduces the expected real return of holding dollar reserves. Given the nominal rates, this implies that the LP must be larger in equilibrium for (23) to hold. When the shock is permanent, the volatility shock appreciates the dollar without any effects on the liquidity ratio or the LP. The result in Propositions1 and2 highlight that episodes of large supply of dollar deposits and high dollar volatility and go hand in hand with appreciations of the dollar. The proposition characterizes the effects of payments volatility under iid shocks and random-walk shocks and perfectly inelastic deposit supplies, but the effects hold for mean- reverting processes and general elasticities. Based on a calibration to be described below, Figure3 shows endogenous variables as a function of the volatility of the withdrawal of dollar deposits.15 In line with the results of Proposition2, we can see that a higher volatility appreciates the dollar (panel [a]) and lowers the expected return on dollar bonds relative to euros (panel [b]), reflecting the larger dollar liquidity premium. In addition, we can see a rise in the differential rate on deposits (panel [e]). That is, the rate on euro deposits increases relative to the dollar rate as the rise in volatility makes euro deposits more attractive. Furthermore, there is a rise in the loan rate because higher volatility increases, in effect, the liquidity frictions and reduces the demand for loans. Finally, we also see an increase in the dollar liquidity ratio concomitantly with a reduction in the euro liquidity ratio. The latter occurs because a higher lending rate makes euro reserves relatively less attractive (in the absence of any shocks to the euro market). Another message from the propositions is that the convenience yield of the dollar is linked to the payment volatility of the dollar, but not the size of dollar per se. To see this point, observe that the effects of permanent shocks in Propositions1 and2 are akin to comparative statics about steady states. Whereas Proposition1 shows that a permanent increase in dollar funding has no effects on the LP, the payment volatility does have effects. The results on the long term effects of both shocks also relate to the exorbitant privilege of the dollar. Funding shocks scale the demand for dollar reserves and dollar penalties. Thus, a greater use of the dollar in international funding increases the fiscal revenues. In turn, a greater dollar payment volatility appreciates the value of dollar reserves, offsetting the effect on the US budget constraint. By contrast, if the scale of deposits and reserves

15In Figure3, we consider a version of the model where the only shock is a Markov process for volatility of dollar withdrawals, which is solved fully non-linear, as described in Appendix .

27 is the same in both countries, but dollar funding is more volatile, the US can fund its liabilities at a lower cost. Policy Intervention: Interest Policy. We now study how monetary policy aﬀects the exchange rate. We start by considering the eﬀect of a change in the nominal policy rates.

Proposition 3 (Eﬀects of Changes in Policy Rates). Consider a temporary change in the dollar interest rate on reserves, i∗,m while holding ﬁxed the policy spread (i∗,w − i∗,m). 1) If the shock is temporary (iid) it appreciates the dollar, raises the liquidity ratio and reduces the premia:

d log e d log µ d log P ∗ R∗,m ∗,m = ∗,m = − ∗,m = b ∗ ∈ (0, 1] and d log (1 + i ) d log (1 + i ) d log (1 + i ) (R − LPµ∗ · µ ) dBP = −Rm (d log (1 + im) − d log e) < 0. 2) If the shock is permanent (random walk), it appreciates the dollar, increases the liquidity ratio and reduces all premia:

d log e d log µ∗ d log P ∗ 1 ∗,m = ∗,m = − ∗,m = − ∗ > 0 and d log (1 + i ) d log (1 + i ) d log (1 + i ) LPµ∗ · µ dBP = dLP = −Rmd log (1 + im).

Proposition3 establishes that in response to an increase in the US nominal rate, the dollar appreciates, the liquidity ratio increases and the liquidity premium falls. The appreciation of the dollar follows a standard effect: a higher nominal rate leads to a larger demand for dollars, which in equilibrium requires a dollar appreciation. Absent liquidity premia, the difference in nominal returns across currencies would be exactly offset by the expected depreciation of the dollar, following the current revaluation. With a liquidity premium, the expected depreciation is not one for one: given the larger abundance of dollar reserves, there is a decrease in the marginal value of dollarreserves together with reduction in dollar liquidity premium. This result breaks the tight connection between interest-rate differentials and expected depreciation, and thus explains the forward-premium (Fama) puzzle. Policy Intervention: Open-market operations. Finally, we consider the effects of open market operations. In the model description, expansions in M are implemented with transfers—helicopter drops. In practice, central banks conduct open-market operations

28 purchasing assets and issuing central bank liabilities. We now modify the model and assume that the central bank buys private securities which we denote by St. To do so, we assume that securities are supplied inelastically and that households securities are perfect substitutes for private deposits.16 Thus, the demand for deposits and securities jointly given by: us d ¯d ς Dt + St = Θt Rt+1 ,

and analogously for dollar securities. We denote the holdings of securities by the central g d g bank as St . The government budget constraint is modiﬁed by adding 1 + it St to the sources of funds and St+1 to the uses. We have an analogue equation for the euro. Next, we characterize the eﬀects of an open-market purchase of securities in the US.

Proposition 4 (Eﬀects of Open-Market Operations). Consider an open-market operation in ∗,g ∗ the US starting from a central bank balance sheet where securities St are fraction the Γ of real ∗ ∗,g ∗ ∗ reserves, Pt · St = Γ · Mt . 1) If the shock is temporary (iid) it depreciates the dollar, increases the liquidity ratio and lowers the premia: ∗ ∗ ∗ ∗ d log e d log P LPθ∗ θµ∗ µ (1 − µ ) ∗,g = − ∗,g = b ∗ ∗ ∗ ≤ 0, d log St d log St R − LPθ∗ θµ∗ µ (1 − Γ )

∗ ∗ ∗ b d log µ µ (1 − µ ) R ∗,m ∗ ∗,g = b ∗ ∗ ∗ ∈ [0, 1] , and dBP = dLP = −R d log P ≤ 0. d log St R − (1 − Γ ) LPθ∗ θµ∗ µ

2) If the shock is permanent (random walk), it depreciates the dollar and is neutral on the liquidity ratio and the premia:

d log e d log P ∗ Γ∗ = − = − (1 − µ∗) ≤ 0, d log S∗,G d log S∗,G 1 − Γ∗ and 16In practice, households hold securities indirectly through pension funds, non-ﬁnancial institutions, etc. For simplicity, we assume that securities and deposits are perfect substitutes for households.

29 (a) Liquidity (b) Real Rates on Reserves

Figure 3: Equilibrium solution for a range of values of volatility

Connecting the theoretical results with the empirical results. Recall from Section2 that there is a positive relationship between the dollar liquidity ratio and the appreciation of the dollar, once controlling for monetary policy. Thus, we are interested in explaining this co-movement through the eﬀects of the unobservable shocks, shocks to the dollar funding and the payment volatility, There is an important diﬀerence in the prediction of funding shock and the payment volatility shocks. Whereas funding shocks lead to a reduction in the liquidity ratio, the payment volatility shock leads to an increase in the liquidity ratio. These results suggest that to reconcile the empirical observations, volatility shocks ought to play an important role in driving the exchange rate. The previous results were shown for two extreme scenarios, iid and permanent shocks. In section5, we show that a calibrated version of the model with more general processes

30 for both shocks is indeed consistent with the empirical results. Here, we are interested in theoretically characterizing how, for AR(1) processes different parameters affect the regression coefficients that we obtain in Section2. In particular, we assume that all exogenous process for x follow a log AR(1) process of the form:

x x x x ln (xt) = (1 − ρ ) ln (xss) + ρ · ln (xt−1) + Σ εt (26)

where ρx is the mean-reversion rate of x and Σx its volatility. We have the following formal result for the regression coeﬃcient of the exchange rate against the dollar liquidity ratio, e x x βµ∗ given the {ρ , Σ } and diﬀerent equilibrium objects.

Proposition 5. The regression coeﬃcient of the change in the liquidity against the change in the liquidity ration is the weighted average:

e X βµ∗ = βx · wx x∈{σ∗,D∗}

where βx represents the regression coeﬃcient if only the shock x is activated:

1 − ρD∗ Rb ∗ ∗ βσ = 1 and βD ≈ ∗ ∗ < 0. LPθ∗ θµ∗ µ

In turn, the weights are given by:

µ∗ x2 x x · Σ w ≈ ∗ 2 P µ x y∈{σ∗,D∗} x · Σ

µ where x is the elasticity of the liquidity ratio with respect to x:

b D∗ µ∗ R 1 − ρ µ∗ LPθθσσ D∗ = − b D∗ ∗ ∗ and σ∗ = b σ∗ ∗ ∗ . R (1 − ρ ) − LPθ∗ θµ∗ µ R (1 − ρ ) − LPθ∗ θµ∗ µ

The takeaway from the proposition is to drive the positive regression coeﬃcient in the data, dollar payment shocks must be suﬃciently important as given by the geometric weights wx.

31 4.1 Risk premia

In this section, introduce risk premia to the framework. To squarely focus on liquidity, we assumed risk-neutral banks. In doing so, we have abstracted from the “risk premia view” on exchange rates that postulates that the dollar appreciates in bad times, and as a result, episodes of high volatility coupled with risk aversion imply that the dollar appreciates in times of high uncertainty.17 Here we extend our framework to allow for risk premia and establish a connection between risk premia and liquidity premia. We consider that banks expectations about X0 0 e are now exogenous stochastic discount factor Λt(X ). Then, let ∆ ≡ e0 denote the realized devaluation of the euro. Modifying the bank objective accordingly, we obtain the following modiﬁed UIP that replaces (23):

∗ ¯∗ ¯ UIPt ≡ Eω χm(m ¯ , d , ω) − EX [∆] Eω χm(m, ¯ d, ω) +ζt (27) | {z } ≡DLP∗ where the dollar safety premium is given by:

0 Λ(X ) ∆ m ¯ ζt ≡ −COV 0 , EX [∆] 1 + i + Eω χm(m, ¯ d, ω) . EX [Λ(X )] EX [∆]

Relative to (23), there is now an additional dollar “safety premium” term driving the diﬀerence in expected returns between the dollar and liquid euro assets. The safety premium regards the covariance between the aggregate state and the devaluation of the euro . If the dollar tends to appreciate in bad times (i.e., when equity returns are low), this also raises the DLP. These two premiums interact in an important. In the economy where persistent payment volatility shocks are prevalent, the dollar appreciates in bad times because of a higher overall liquidity premium, which occurs at the same time as the safety premium increases. Our baseline model does not include an explicit forward market. To speak to the observed deviations from covered interest parity (CIP), we now allow for a forward market.

17Notice, however, that Maggiori(2017) uncovers a modelling challenge regarding the risk premia view. He shows in a model of ﬁnancial intermediation that the dollar tends to appreciate in bad times as US households are optimally more exposed to aggregate risk and therefore face larger losses in bad times relative to the rest of the world. He refers to this phenomenon as the “reserve currency paradox”. Farhi and Gabaix (2016) are able to obtain an appreciation of the dollar in bad times by positing a diﬀerential exposure of the tradable and non-tradable sectors in a disaster.

32 For that, we consider a perfectly competitive market for forwards.18 A forward traded at time t promises to exchange one dollar for eˆt,t+1 in the lending stage in the following period. The ﬁrst-order condition with respect to the quantity of forwards purchased yields

1 1 1 0 = βEt ∗ − , (28) Pt+1 eˆt,t+1 et+1

Under risk neutrality, we obtain the canonical result that the forward rate equals the expected exchange rate.Let us examine the government bond based CIP deviation on the reserve asset: e CIP = (1 + im) − (1 + im,∗) = DLP∗. eˆ (29) Replacing the price condition for the forward (28) in (29), we obtain:

0 ∗ Λ(X ) ∆ ¯ CIP = DLP − COV 0 , EX [∆] χm(m, ¯ d, ω) EX [Λ(X )] EX [∆]

Thus, notice that under risk-neutrality, there is no wedge between the CIP and UIP. Both may be positive given liquidity factors. With a risk premium, relative to the UIP which carries a risk premium that adjusts all the returns to the Euro, the CIP carries a risk- premium term associated only with respect to the liquidity service of the Euro, the return not that is not covered by issuing the forward. As a result, a wedge between the two conditions emerges, CIP = UIP + ξt.

0 Λ(X ) ∆ m ξt ≡ COV 0 , EX [∆] (1 + i ) . EX [Λ(X )] EX [∆]

All in all, both deviations of both premia from zero are informative about the liquidity premia, and the deviation between both premia are informative about risk aversion. It is also worth highlighting that an alternative measurement of CIP deviation is con- ducted using the interbank market rate (Libor), rather than the rate on bonds. A prediction of our model is that the two deviations from CIP, the one with the government bond and the interbank market rate, are tightly linked. We should also note that although the probability twist Λ(X0) may be interpreted as risk premium, from the lens of the model, it may be interpreted as model misspeciﬁcation. Furthermore, the wedge in ξt may also

18Since there are no aggregate shocks in the balancing stage, it is equivalent to price the forward in the lending or in the balancing stage.

33 result from regulatory constraints on the forward market. An interesting observation made by Du et al.(2018) is that low interest rate countries have had positive CIP deviation since the global ﬁnancial crisis—i.e., low interest rate currencies deliver a high risk-free excess return relative to the high interest rate currencies. This pattern contrasts sharply with the carry trade phenomenon by which low interest rate countries have a negative UIP deviation—i.e. low interest rate countries deliver a low expected return relative to high interest rate currencies. As we showed earlier, an increase in the nominal interest rate reduces the liquidity premium. Our model is thus consistent with this observation.19 Real exchange rates. [TBA]

5 Quantitative Analysis

We now explore a quantitative version of the model. We use the euro-dollar exchange rate, liquidity ratios, policy variables and observed premia to calibrate the model and estimate the shocks the model needs to explain the data. We then study the importance of the dollars-payment volatility to account for movements in the exchange rate. We replace the {∗} notation for i ∈ {us, eu}. Later, we will also introduce other currencies into the analysis. 5.1 Additional Features

Before we proceed to the description of the calibration and estimation, we modify the model to incorporate other features that will help the model to quantitatively account for the data. Limited Equity. In the baseline model we presented, banks have unlimited access to equity financing. This is because banks have linear utility and face no lower bound on div- idend payments. This assumption ensures that the return on the illiquid asset is fixed at 1/β, the return on bank equity. This simplifies the analytical characterization of how

19An alternative explanation provided by Amador et al.(2020) is related to limits to international arbitrage and central bank policies of resisting an appreciation at the zero lower bound. The policy of keeping an exchange rate temporarily depreciated together with a lower bound on the nominal rate, created excess returns on domestic currency assets. The excess returns, in turn, generated capital inﬂows, which had to be oﬀset with central bank purchases of foreign reserves. See Kalemli-Ozcan¨ and Varela(2021) for a comprehensive empirical analysis of UIP and CIP deviations for emerging and advanced economies.

34 shocks affect movements in the exchange rate. However, the assumption is unrealistic because the loans rate is fixed, so all the movements in the bond premium follow from changes in the real expected return to dollar reserves. To enrich the quantitative model and deliver endogenous variations in the loan rate, we assume that banks face limited equity financing. Specifically, we assume that banks do not accumulate equity: banks payout their realized previous-period as dividends (or raise equity to finance their losses). Since average profits must equal zero in equilibrium, this implies that averaging across banks, the following must hold. us eu us eu bt + mt + mt = dt + dt . (30)

Hence, the loans rate is no longer fixed, but determined by the supply of loans given the portfolio choices of the bank. Open-Market Operations. Given that we interpret M as capturing both central bank and government liabilities, what is relevant here are the so-called unconventional open market operations, which have been especially prevalent since the 2008 financial crisis. We assume that all of the increase in the money supply away from steady state is through the purchase us us us us of securities. Thus, from the budget constraint we have that: Mt − Mss = − (St − Sss ). We use analog equations for the Euro. Thus, in this section, we assume households also holds securities as in xxx. CIP-UIP wedge. In the data, the deviation from UIP exceeds by some margin the deviations from CIP (see e.g. Kalemli-Ozcan¨ and Varela 2021). In our baseline model with risk neutrality, UIP and CIP coincide. We also showed in Section 4.1, that the model can be extended to allow for risk aversion, biased expectations and regulatory constraints to account for that wedge. In what follows, we leave open an explicit interpretation and use a simple “risk-premium wedge”, denoted by ξt, to express the difference between the UIP and CIP deviations, UIPt = CIPt + ξt. 5.2 Calibration and Estimation

We calibrate externally and internally a subset of parameters and estimate the auto-correlation and variances of diﬀerent shocks to the model. For the distributions of the payment shocks, ω, we adopt a parametric form. We assume that these shocks are distributed as two sided exponential—formally known as Laplace distributions—in each country at each

35 date.20 Each possible distribution is indexed by a single dispersion parameter, σ, and the distribution is centered at zero. Shocks processes. We deduce shocks to the payment volatility, funding and the risk-premium wedge. In particular, at each date, there are distributions of payment shocks to Dollar and i d,i Euros, σt for i ∈ {us, eu} and to the scale of deposits Θt for i ∈ {us, eu} and the wedge in the UIP condition, ξt. In addition, we assume that there are shocks to the interest on m,i i reserves, it and to the money supply, Mt . i d,i m,i i , x ∈ ξt, σ , Θ , i ,Mt i∈{us,eu} , Overall, we have 9 shocks and 18 parameters. m,i i From the model’s perspective, it ,Mt i∈{us,eu}, are exogenous policy variables that have observable counterparts so their processes can be directly estimated using (26). By n us d,us eu d,euo contrast, from the model’s perspective, ξt, σt , Θt , σt , Θt are exogenous and we deduce these using the Kalman filter and estimate the parameters in (26), as we explain m,i i below. We have a total of 4 exogenous policy variables, it ,Mt i∈{us,eu}, and 5 exogenous n us d,us eu d,euo shocks, ξt, σt , Θt , σt , Θt . Data Counterparts. We use monthly data from 2001m1 to 2016m12. The data series em- ployed as counterparts are presented in Figure9. As the data counterparts for the interest m,i i on reserves, it , and the money supply, Mt , we use the 3-month US and German govern- m,us m,eu ment bond rate as the data counterparts for it and it —see panels (c) and (d). In the data, the exchange rate is stationary whereas the data features a slight inflation differential. Since the model is stationary, we divide each bond rate by the average annual inflation m,i i in the corresponding area to construct it . For the counterpart of money supply Mt , as in Section2, we use the stock of reserves and government bonds in the financial system. We then scale of the data for the Euro money supply to match the average Euro-Dollar rate. m,i i m,i im,i im,i With this data it ,Mt i∈{us,eu} we estimate iss , ρ , Σ for i ∈ {us, eu} . The estimated parameter values are given presented in Table B1. n us d,us eu d,euo To obtain the filtered time series for ξt, σt , Θt , σt , Θt , we use data counterparts ∗ for {CIP, BP, µ, µ , e}. For the data counterpart of CIPt, we use the mid-point quotes for spot and forward exchange rates from Bloomberg and the nominal rates on the rates on reserves, as data counterparts of the terms in equation (23). For the reference period, we

20This distribution is convenient because it allows for a continuum of shocks, but renders closed form solutions for the conditional expectations below a threshold, thereby allows to compute the reserve deﬁcit probability analytically which is useful in computations.

36 obtain a value of 12 basis points, in line with Du et al.(2018). For the data counterpart of BP we use a measure of liquidity proposed by Stock and Watson(1989) and Friedman and Kuttner(1993), the commercial paper spread, the difference between the three month spread between the AAA commercial paper and the 3 month T-Bill, our analogue for the policy rate. For the data counterpart of µ∗, we employ the same series as in Section2. We construct the same series for the Euro area in µ. As we noted in Figure1, the liquidity ratio may be often be greater than one, because we take the liquid assets of all the financial system as whole, but deposits are liabilities of the banking system, a subset of financial system. For the regressions, this does not matter since we conduct the regression in changes. Here, we have to scale the liquidity ratio. For that, we scale the ratios to so that the steady-state targets for µus and µeu are a about 0.2, a figure consistent with the data liquidity targets in Bianchi and Bigio(2021) based on Call Reports data. The scaled liquidity ratios are plotted in panel (b) in Figure fig:data in the Appendix. External Calibration. We set the matching efficiency parameters and the values for the semi-elasticities of the loan demand and deposit supply schedules to the values from Bianchi and Bigio(2021), treating the Euro area and the US funding supplies as symmetric. Likewise, to we set iw = im + 10%, in line with the penalty they obtain. The loan demand scale Θb can be set to one as a normalization. Internal Calibration. We proceed next to describe the parameterization of the steady-state us eu d,us d,eu values of the shock processes σss , σss , ξss, Θss , Θss , which is set to match data tar- d,us d,eu gets. We assume symmetry in the funding scales at steady state, Θss = Θss . We are thus left with four parameters to match four steady-state moments. To have stationarity of prices, and the exchange rate, we assume a steady state with constant prices and exchange rates, and accordingly, adjust the rates observed in the data.21 Thus, we think of m,us m,eu m,us m,eu the steady-state real rates as given by {Rss ,Rss }={1 + iss , 1 + iss }. The four targets us eu are {CIPss, BPss, µss , µss } which correspond to the averages between the years the period 2002:2005. We calibrate the steady-state values to the time-series averages of that period, rather than estimating these parameters because the early years of the Euro and the post 2008 where periods of substantial instability. Kalman filter estimates would instead obtain values close to the historical average which we want to avoid.

21This is without loss of generality, as we can scale all nominal variables by inﬂation without real implications.

37 We adopt a sequential procedure to calibrate the four parameters. First, we solve us us us for σ from BPss = E [χm (µss , σss )]. Second, we use the steady-state version of equa- eu eu eu tion 23 to compute the CIP deviation and obtain σ . That is,CIPss = E [χm (µss , σ )] − us us us eu E [χm (µss , σss )] . Given the targets for µss , µss , we obtain a calibration calls for dollar volatil- us eu 22 ity σss , which is almost four times the σss in steady state. Third, the average risk-premium, m m,∗ ξss, is obtained as UIPss = CIPss + ξss where to measure the UIPss = Rss − Rss , we use 23 d,us the average realized return diﬀerential over the reference period. Finally, we obtain Θss from the bank’s budget constraint and the target for the liquidity ratio. To do so, notice b m,us − that we can ﬁrst obtain b = Θ (Rss + BPss) from loan market clearing. Similarly, we us eu d,us d,eu can obtain dss and dss from their corresponding clearing conditions. Setting Θss = Θss , us eu us eu and replacing {bss, dss , dss , µss , µss } into the bank’s budget constraint, (30), we obtain a d,us single equation for Θss :

m,us − d,us eu d,eu1/ςd,eu us d,us1/ςd,us (Rss + BPss) = Θss (1 − µss ) Rss + (1 − µss ) Rss (31)

d,eu d,us where Rss and Rss are the steady-state versions of (24-25). eu us Finally, we note that the values of Mss and Mss can be normalized. For simplicity, we eu us set Mss to obtain a price in euros equal to one and Mss to generate a level of the exchange rate equal to the one in the data. Combining the money demand equations implied by the model: M eu µus · Rd,us1/ςd,us eeu,us = ss ss ss . ss us eu d,eu1/ςd,eu (32) Mss µss Rss Filtering of shocks. Once we set the calibrated steady-state values of the shock processes and the other parameters, we we run a Kalman filter on the linearized version of the model to filter the shocks and estimate their persistence and volatility using Bayesian maximum likelihood. It is instructive to sketch what series are informative about what series in the filter- us ing step. First, the bond premium is informative about the payments volatility σt as in us us the steady state since, BPt = E [χm (µt , σ )] . Likewise, the CIP deviation is informa- eu eu eu tive about σt since CIPt = E [χm (µt , σt )] − BPt. Within a transition, the values of {ρx} and the internal structure lead to a forecast of the expected future exchange rate,

22 Note that to reconcile an average BPss > CIPss, the calibration calls for more volatile payments in us eu dollars: σss is almost four times the σss in steady state. This is consistent with our notion that dollar ﬂows are larger do to the extensive use of dollars in international transactions. 23? provide a comprehensive comparative analysis of the diﬀerent ways to measure UIP.

38 h us d,us eu d,eui E et+1|ξt, σt , Θt , σt , Θt . Thus, given the policy rates, the observed exchange rate is informative about the risk premium, ξt:

m,eu m,us E [et+1] (1 + it ) − (1 + it ) = CIPt + ξt. (33) et

n d,us d,euo Finally, the liquidity ratios are informative about Θt , Θt . Since we observe the se- i i ries for {µt,Mt }i∈{us,eu}, we the time-series version of the subsystem of equations in (31- 32). The priors and posteriors are given in Table (B1). Using the parameter estimates, we conduct Monte Carlo simulations to generate moments in the model and compare with the data counterparts. As Table B1 shows, the model matches targeted steady state moments. In terms of untargeted moments, the model matches the persistence and standard deviation well, particularly for the liquidity ratios and the exchange rate. 5.3 Results

The shocks that that we infer from the filtering exercise are presented in Figure4. There are some noticeable features. First, as we can observe from panel (a), the payments volatility us in σt is close to steady state prior to 2007, but begins experiencing substantial volatility during the financial crisis and remains higher past that period. Indeed, recall from Figure us fig:data that BP spikes during the financial crisis, while the US liquidity ratio µt remains consistently high after the financial crisis. To reconcile a higher liquidity ratio with a BP us that returns to normality, the model needs a persistently higher σt .

39 us eu us (a) σt (b) σt (c) Θt

0.24 1.35 0.4 0.22

0.2 1.3 0.35 0.18 1.25

0.3 0.16 1.2

0.14 1.15 0.25 0.12 1.1

0.2 0.1 1.05

0.08 1 0.15 0.06 0.95 2002-07 2005-01 2007-07 2010-01 2012-07 2015-01 2002-07 2005-01 2007-07 2010-01 2012-07 2015-01 2002-07 2005-01 2007-07 2010-01 2012-07 2015-01 eu (d) Θt (e) ξt

1.7 200 1.65

1.6 0

1.55 -200 1.5

1.45 -400

1.4 annualize bp annualize -600 1.35

1.3 -800

1.25 -1000 1.2

2002-07 2005-01 2007-07 2010-01 2012-07 2015-01 2002-07 2005-01 2007-07 2010-01 2012-07 2015-01

Figure 4: Filtered Shocks Note: The ﬁgure plots the series obtained using the Kalman ﬁlter.

eu In panel (b) we observe that the estimate for Euro payments volatility σt is substantially more stable. There are couple of dips and reversals, around 2008 and the 2012 Euro. eu The model infers this rapid changes σt from the spike in the CIP deviations. Panel (c) shows the path of the scale of dollar funding, which only shows secular trends, first showing a decline in dollar funding up to 2008 and a subsequent increase. Likewise, panel (d) shows low frequency movements in the funding in Euros, showing two waves, one that preceded the crisis, and a second one, that shows increase Euro funding after the 2008 financial crisis, but a consistent decline following the debt crisis. The risk premium wedge presented in Panel (e) shows an interesting pattern: the first years of the inception of the Euro, the premium is consistently negative, but converging to zero—this can be do to a consistent bias in the exchange rate projection of our model, as we explained earlier. Dur- ing the crisis, the risk premium experiences a dip with a quick reversal thereafter. From (33), a more positive premium is associated with a weaker Euro, so the risk premium was a force toward a strong Euro during the crisis, but a weaker Euro entering the Sovereign debt crisis. Model Validation. A second validation exercise that we conduct is to us the simulated data to run the same regressions in the model as we did with the data in Section2. We

40 extend our analysis to multiple currencies by assuming that all currencies but euro and dollars have measure zero in the portfolio.24 Thanks to the fact portfolio holdings are inﬁnitessimal for the additional currencies, the equilibrium allocations do not change, but now we can also compute the exchange rates and rates of return for all other currencies. We calibrate accordingly the interest rate processes and assume for simplicity that the withdrawal volatility is the same as for euros. Table5 shows the results. The key takeaway is that the simulated data based regression delivers a positive coeﬃcient on the liquidity ratio, just like we found in the data.

Table 5: Regression Coefficients with Simulated Data for Multiple Currencies

Eur AUS CAN JPN NZ NWY SWE SWZ UK ∆(LiqRatiot) 0.093 0.114 0.139 0.138 0.124 0.101 0.118 0.133 0.123 (0.063) (0.119) (0.057) (0.057) (0.085) (0.061) (0.072) (0.057) (0.066) co us ∆(πt − πt ) 0.196 0.240 0.186 0.187 0.219 0.196 0.210 0.187 0.201 (0.109) (0.195) (0.100) (0.103) (0.138) (0.108) (0.127) (0.102) (0.115) LiqRatio(−1) 0.005 0.008 0.006 0.006 0.007 0.006 0.007 0.006 0.007 (0.022) (0.038) (0.020) (0.020) (0.028) (0.022) (0.026) (0.020) (0.023) Constant 0.008 0.011 0.010 0.009 0.011 0.009 0.011 0.009 0.010 (0.034) (0.058) (0.031) (0.031) (0.043) (0.034) (0.039) (0.031) (0.036) N 234 234 234 234 234 234 234 234 234 2 adj. R 0.038 0.023 0.055 0.053 0.034 0.042 0.038 0.053 0.043 standard deviation in parenthesis (4273 dataset simulations of 234 months)

How Important is Liquidity Risk? In this section, we show that liquidity risk plays a central role in driving exchange rate ﬂuctuations. We start by providing an variance decomposition on the linearized version of the model. The results are presented in Figure5. Each panel reports the % contribution of each exogenous variable on a variable of interest. In Panel (a) we show that the contribution of dollar payment shocks accounts fro about 15 of the volatility of the Euro-Dollar exchange rate. If we consider the funding shocks, which are germane to our theory, these new variables account for 25 of the variation in the exchange rate. The policy variables, interests and the money supply, account for about one third of the variation, and the rest, about 40 is accounted for by the wedge in the UIP, an unexplained residual from the perspective of our model. Panel (b) moves to explain the contribution of shocks on the BP. The dollar payment volatility accounts for 90% of the volatility in the bond premium. Among the other shocks, shocks that directly aﬀect the liquidity ratio, M us and Θd,us account for the

24Technically, we assume that the supply of deposits approaches zero at the limit for all other currencies.

41 rest of the volatility. Finally, the model attributes about two thirds of the volatility in the CIP. In turn, the Euro volatility is responsible for only one sixth of the volatility.

(a) Euro-Dollar (b) BP (b) CIP < 1% 15% < 1%8% 3% 4% < 1% 2% 2% < 1%8% < 1% 5% < 1% < 1% 3% < 1% 2% 37% 7% 3%

52%

25% 31% 2% 6% 84%

Figure 5: Variance Decomposition of Shocks

We now analyze the contribution of all shocks in explaining the level the exchange rate. Figure6 presents the results. The horizontal bar represents a particular date and the solid curve the deviation of the Euro-Dollar exchange rate with respect to the steady state. The vertical bars account for the contribution of different shocks, grouped together into two liquidity (payment shocks), the two funding shocks, the policy shocks (changes in rates and the money supply) and the risk premium. To compute the contribution of each shock, we turn-off the shock in each period by setting the value equal to the steady state. A key takeaway is that since the 2008 financial crisis, the liquidity shock has played a prominent role in accounting for the level of the exchange rate.

42 30

0 % Deviation from Steady State Steady from Deviation %

-10

-20

-30

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

Figure 6: Variance Decomposition Note: The ﬁgure reports the contribution of diﬀerent groups of shocks in explaining the exchange rate.

To zoom in the the role of volatility, we show the time-series of the exchange rate and the CIP absent the volatility shock in Figure7. Consistent with the analysis above, the counterfactual shows a much lower deviation from CIP (both measured with reserves and interbank market), a more depreciated dollar.

(a) Log Euro-Dollar (b) CIP (reserves) (d) CIP (interbank)

250 250

1.2

200 200

1.1 150 150

1 100 100

50 0.9 50

0.8 0

-50

-50 0.7

-100

-100 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

Figure 7: Counterfactual w/o σus Note: The ﬁgure reports the predicted series of the model with all shocks and with all shocks us except for σt .

43 The March 2020 “Dash for Cash.” A revealing case study of liquidity demand for dollars is the sudden turmoil in markets at the onset of the COVID crisis in March 2020, often referred to as the “dash for cash”. As governments began to institute lockdowns in late February, and the severity of the COVID crisis began to come into focus in early March, there was naturally a “flight to safety” as investors fled from equities and other risky assets into safer government-backed securities. Quickly, however, the nature of the financial stress changed character, as agents fled even from assets normally considered to be safe, such as longer-term U.S. Treasury bonds, into cash and other very liquid assets such as T-bills. Even though the Fed cut rates on March 3, medium- and long-term Treasury rates began to rise on March 13th, as evidenced in Table B1. Yields on all bonds with maturities of five years or longer rose substantially through March 18th. At the same time, interest rates on the shorter end of the spectrum, for highly liquid T-bills fell during this time period. Table B1 clearly shows this drop for maturities less than one year. Very late on March 18th, the Federal Reserve announced a plan that was step toward dealing with the problem, by offering to buy illiquid bonds held by money market funds. This action slowed the dash toward cash, but the markets were not calmed until early Monday morning, March 23rd, when the Fed announced a plan to buy large amounts of Treasury bonds and agency securities. After that move, the flight toward liquidity slowed, and the increase in long term bond rates ceased. Consistent with the theory, Figure6 shows the behavior of the dollar/euro exchange rate during this period. The dollar sharply appreciated between March 9th and March 19th, during the strongest surge of liquidity demand. The dollar subsequently began to depreciate after the Fed announced its moves to ease the liquidity shortage. 6 Conclusions

We developed a theory of exchange rate determination emerging from financial institutions’ demand for liquid dollar assets. Periods of increased funding volatility generate a “scrambling for dollars” effect that raise liquidity premia and appreciates the dollar. In line with the theory, we document, based on banks’ balance sheet, that a higher liquidity ratio is associated with a stronger dollar. We also use the model as a quantitative labo- ratory to decompose the different forces driving exchange rate fluctuations. The analysis

44 suggests funding volatility is a key factor.

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51 Online Appendix

(Additional Tables)

52 Table A1: Relationship of Exchange Rates and Banking Liquidity Ratio Feb. 2001 – July 2020 Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRatt) 0.220*** 0.233*** 0.148** -0.141*** 0.277*** 0.184*** 0.215*** 0.140** 0.163*** (4.422) (3.359) (2.557) (-2.830) (3.947) (2.914) (3.561) (2.557) (3.254) ∗ πt − πt -0.546*** -0.408** -0.653*** 0.050 -0.729*** -0.118 -0.504** -0.649*** -0.394** (-3.750) (-2.136) (-2.656) (0.343) (-3.815) (-0.807) (-2.528) (-2.737) (-2.129) ∗ ∆(it − it ) -1.284 -0.491 -0.093 -1.793** -1.114 -0.513 0.249 -1.541* -0.269 (-1.486) (-0.599) (-0.083) (-2.336) (-1.245) (-0.638) (0.311) (-1.663) (-0.351) LiqRatt−1 0.010** 0.006 0.002 -0.001 0.008 0.010 0.006 0.004 0.009* (2.141) (0.940) (0.319) (-0.142) (1.279) (1.643) (1.181) (0.775) (1.705) Constant -0.011*** -0.004 -0.007* 0.002 -0.008* -0.006 -0.009** -0.016*** -0.005 (-3.239) (-0.950) (-1.912) (0.349) (-1.892) (-1.560) (-2.081) (-2.980) (-1.571) N 234 232 148 234 232 234 234 234 234 adj. R2 0.11 0.05 0.05 0.05 0.10 0.03 0.05 0.05 0.04

53 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 Table A2: Relationship of Exchange Rates and Banking Liquidity Ratio with VIX Feb. 2001 – July 2020 Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRatt) 0.192*** 0.155** 0.073 -0.123** 0.208*** 0.137** 0.174*** 0.126** 0.138*** (3.922) (2.592) (1.378) (-2.477) (3.230) (2.291) (3.012) (2.291) (2.769) ∗ πt − πt -0.432*** -0.174 -0.421* 0.027 -0.530*** -0.033 -0.419** -0.589** -0.308* (-2.989) (-1.054) (-1.893) (0.191) (-3.016) (-0.240) (-2.210) (-2.468) (-1.676) ∆VIXt 0.001*** 0.004*** 0.003*** -0.001** 0.003*** 0.002*** 0.002*** 0.001* 0.001*** (3.794) (9.317) (6.169) (-2.538) (7.018) (5.896) (5.133) (1.683) (3.110) ∗ ∆(it − it ) -1.075 -0.404 0.456 -1.994*** -1.404* -0.289 0.096 -1.285 -0.266 (-1.277) (-0.578) (0.455) (-2.615) (-1.725) (-0.385) (0.126) (-1.374) (-0.353) LiqRatt−1 0.010** 0.007 0.004 -0.001 0.008 0.010* 0.007 0.004 0.008 (2.240) (1.299) (0.686) (-0.177) (1.390) (1.808) (1.385) (0.871) (1.587) cons -0.011*** -0.005 -0.006* 0.002 -0.008* -0.006 -0.009** -0.015*** -0.005 (-3.095) (-1.399) (-1.881) (0.291) (-1.941) (-1.648) (-2.095) (-2.837) (-1.433) N 234 232 148 234 232 234 234 234 234 54 adj. R2 0.16 0.31 0.25 0.07 0.26 0.15 0.15 0.05 0.07 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 Table A3: Relationship of Exchange Rates and Banking Liquidity Ratio Instrumental Variable Regression: ∆(VIX), StDev(Inf) and StDev(XRate) instrument for ∆(LiquidityRatio) Feb. 2001 – July 2020 Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRatt) 0.604*** 1.099*** 0.688*** -0.380*** 0.888*** 0.832*** 0.710*** 0.153 0.799*** (4.231) (4.663) (4.421) (-2.904) (4.357) (4.087) (3.961) (1.058) (4.249) ∗ πt − πt -0.717*** -0.892*** -0.732*** 0.094 -0.985*** -0.450** -0.778*** -0.673** -1.094*** (-4.120) (-3.245) (-2.627) (0.596) (-4.176) (-2.234) (-3.217) (-2.465) (-3.519) LiqRatt−1 0.016*** 0.018** 0.016*** -0.003 0.017** 0.023*** 0.012** 0.005 0.026*** (3.063) (2.177) (2.618) (-0.466) (2.349) (2.862) (1.972) (0.951) (3.208) Constant -0.018*** -0.014** -0.015*** 0.005 -0.017*** -0.017*** -0.017*** -0.017*** -0.019*** (-4.085) (-2.325) (-3.065) (0.723) (-3.049) (-2.932) (-3.068) (-2.618) (-3.219) N 234 232 234 234 232 234 234 234 234 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 55 Table A4: Relationship of Exchange Rates and Alternative Measure of Banking Liquidity Ratio Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRat2t) 0.098*** 0.109*** 0.069*** -0.012 0.123*** 0.101*** 0.088*** 0.079*** 0.103*** (3.718) (3.073) (2.675) (-0.450) (3.388) (3.100) (2.789) (2.768) (4.049) ∗ πt − πt -0.511*** -0.406** -0.421* -0.082 -0.673*** -0.126 -0.440** -0.640*** -0.346** (-3.480) (-2.135) (-1.944) (-0.540) (-3.519) (-0.849) (-2.253) (-2.731) (-2.065) LiqRat2t−1 0.006** 0.005 0.005* 0.004 0.005 0.006* 0.004 0.003 0.004 (2.118) (1.415) (1.833) (1.217) (1.388) (1.796) (1.377) (1.129) (1.280) Constant -0.006*** -0.001 -0.002 -0.002 -0.003 -0.001 -0.005* -0.014*** -0.001 (-2.699) (-0.257) (-1.249) (-0.659) (-1.414) (-0.496) (-1.765) (-3.175) (-0.311) N 234 232 234 234 232 234 234 234 234 adj. R2 0.09 0.04 0.04 -0.00 0.07 0.04 0.04 0.04 0.06 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 56 Table A5: Relationship of Exchange Rates and Alternative Measure of Banking Liquidity Ratio with VIX Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRat2t) 0.090*** 0.088*** 0.059** -0.008 0.106*** 0.088*** 0.078*** 0.074*** 0.096*** (3.501) (2.933) (2.544) (-0.283) (3.233) (2.911) (2.625) (2.616) (3.833) ∗ πt − πt -0.393*** -0.188 -0.299 -0.112 -0.480*** -0.052 -0.379** -0.575** -0.274* (-2.721) (-1.163) (-1.543) (-0.742) (-2.748) (-0.373) (-2.052) (-2.451) (-1.655) ∆VIXt 0.001*** 0.004*** 0.002*** -0.001** 0.003*** 0.002*** 0.002*** 0.001** 0.001*** (4.215) (9.677) (7.735) (-2.589) (7.238) (6.145) (5.446) (2.126) (3.330) LiqRat2t−1 0.006** 0.006* 0.005** 0.004 0.006* 0.007* 0.005* 0.003 0.003 (2.224) (1.860) (2.137) (1.259) (1.656) (1.951) (1.663) (1.220) (1.268) Constant -0.005** -0.001 -0.002 -0.003 -0.003 -0.001 -0.004* -0.013*** -0.000 (-2.372) (-0.660) (-1.235) (-0.829) (-1.466) (-0.502) (-1.698) (-2.931) (-0.277) N 234 232 234 234 232 234 234 234 234 adj. R2 0.15 0.32 0.23 0.02 0.24 0.17 0.15 0.06 0.10

57 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 Table A6: Relationship of Exchange Rates and Alternative Measure of Banking Liquidity Ratio Instrumental Variable Regression: StDev(Inf) and StDev(XRate) instrument for ∆(LiquidityRatio) Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRat2t) 0.362*** 0.491*** 0.352*** -0.239* 0.411** 0.450*** 0.373** 0.018 0.604*** (2.758) (2.791) (3.033) (-1.872) (2.579) (2.625) (2.593) (0.146) (2.674) ∗ πt − πt -0.607*** -0.548** -0.403 0.100 -0.698*** -0.324 -0.552** -0.490 -1.149** (-2.991) (-2.065) (-1.573) (0.483) (-2.996) (-1.496) (-2.347) (-1.630) (-2.436) LiqRat2t−1 0.008** 0.008* 0.007** 0.000 0.007* 0.011** 0.007* 0.003 0.011** (2.318) (1.966) (2.124) (0.091) (1.832) (2.310) (1.795) (1.087) (1.989) ∆VIXt 0.001*** 0.004*** 0.002*** -0.001* 0.003*** 0.002*** 0.002*** 0.001** 0.001 (2.617) (6.194) (5.240) (-1.693) (5.402) (4.018) (3.955) (2.146) (0.859) Constant -0.007*** -0.002 -0.004 0.002 -0.005* -0.003 -0.007** -0.011* -0.004 (-2.717) (-0.832) (-1.636) (0.484) (-1.839) (-1.137) (-2.146) (-1.945) (-1.228) N 234 232 234 234 232 234 234 234 234 58 t statistics in parentheses. * p < 0.1, ** p < 0.05, *** p < 0.01 Table A7: Relationship of Exchange Rates and Alternative Measure of Banking Liquidity Ratio Instrumental Variable Regression: ∆(VIX), StDev(Inf) and StDev(XRate) instrument for ∆(LiquidityRatio) Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(LiqRat2t) 0.471*** 0.822*** 0.484*** -0.296** 0.651*** 0.649*** 0.507*** 0.105 0.686*** (3.276) (3.327) (3.420) (-2.251) (3.250) (3.135) (3.076) (0.882) (3.036) ∗ πt − πt -0.774*** -1.007*** -0.552* 0.171 -1.023*** -0.527** -0.678** -0.677** -1.315*** (-3.466) (-2.679) (-1.734) (0.788) (-3.468) (-1.978) (-2.466) (-2.353) (-2.760) LiqRat2t−1 0.008** 0.010 0.008* -0.001 0.008 0.013** 0.007 0.003 0.012** (2.177) (1.597) (1.831) (-0.120) (1.598) (2.219) (1.559) (1.149) (2.074) Constant -0.009*** -0.002 -0.005 0.004 -0.007* -0.004 -0.008** -0.015*** -0.004 (-2.938) (-0.601) (-1.641) (0.780) (-1.841) (-1.240) (-2.235) (-2.662) (-1.300) N 234 232 234 234 232 234 234 234 234 t statistics in parentheses. p < 0.1 p < 0.05 p < 0.01 59 * , ** , *** Table A8: Relationship of Exchange Rates and the Measure of Intraday Fed Funds Rate Volatility Instrumental Variable Regression: ∆(VIX), StDev(Inf) and StDev(XRate) instrument for ∆(MaxDift) Euro Australia Canada Japan New Zealand Norway Sweden Switz U.K. ∆(MaxDift) 0.019*** 0.030*** 0.018*** -0.009** 0.025*** 0.020*** 0.023*** 0.006 0.019*** (2.964) (3.376) (3.536) (-2.125) (3.034) (2.951) (2.899) (1.195) (3.110) ∗ πt − πt -0.710** -0.822* -0.168 0.047 -1.091*** -0.108 -1.006** -0.916** -0.394 (-2.364) (-1.947) (-0.547) (0.325) (-2.821) (-0.480) (-2.320) (-2.168) (-1.326) MaxDift−1 0.008** 0.013*** 0.007** -0.006*** 0.013*** 0.008** 0.011** 0.004 0.008** (2.409) (2.697) (2.550) (-2.808) (2.786) (2.222) (2.577) (1.291) (2.505) Constant -0.020** -0.021** -0.013** 0.011** -0.026*** -0.014* -0.029*** -0.025** -0.013** (-2.502) (-2.359) (-2.317) (2.099) (-2.794) (-1.962) (-2.626) (-2.133) (-2.063) N 187 187 187 187 187 187 187 187 187 t statistics in parentheses. p < 0.1 p < 0.05 p < 0.01

60 * , ** , *** Table B1: U.S. Treasury Yields, March 2020 Date 1 Mo 2 Mo 3 Mo 6 Mo 1 Yr 2 Yr 3 Yr 5 Yr 7 Yr 10 Yr 20 Yr 30 Yr 3/2/2020 1.41 1.27 1.13 0.95 0.89 0.84 0.85 0.88 1.01 1.1 1.46 1.66 3/3/2020 1.11 1.05 0.95 0.83 0.73 0.71 0.72 0.77 0.91 1.02 1.44 1.64 3/4/2020 1 0.87 0.72 0.68 0.59 0.67 0.68 0.75 0.9 1.02 1.45 1.67 3/5/2020 0.92 0.83 0.62 0.53 0.48 0.59 0.61 0.67 0.81 0.92 1.34 1.56 3/6/2020 0.79 0.64 0.45 0.41 0.39 0.49 0.53 0.58 0.69 0.74 1.09 1.25 3/9/2020 0.57 0.52 0.33 0.27 0.31 0.38 0.4 0.46 0.56 0.54 0.87 0.99 3/10/2020 0.57 0.55 0.44 0.43 0.43 0.5 0.58 0.63 0.73 0.76 1.16 1.28 3/11/2020 0.42 0.42 0.42 0.4 0.4 0.5 0.58 0.66 0.78 0.82 1.13 1.3 3/12/2020 0.41 0.33 0.33 0.37 0.39 0.5 0.58 0.66 0.82 0.88 1.27 1.49 3/13/2020 0.33 0.3 0.28 0.38 0.38 0.49 0.58 0.7 0.89 0.94 1.31 1.56 3/16/2020 0.25 0.25 0.24 0.29 0.29 0.36 0.43 0.49 0.67 0.73 1.1 1.34 3/17/2020 0.12 0.18 0.19 0.24 0.3 0.47 0.54 0.66 0.91 1.02 1.45 1.63 3/18/2020 0.04 0.03 0.02 0.08 0.21 0.54 0.66 0.79 1.08 1.18 1.6 1.77 61 3/19/2020 0.04 0.04 0.04 0.06 0.2 0.44 0.53 0.66 1 1.12 1.56 1.78 3/20/2020 0.04 0.05 0.05 0.05 0.15 0.37 0.41 0.52 0.82 0.92 1.35 1.55 3/23/2020 0.01 0.04 0.02 0.08 0.17 0.28 0.31 0.38 0.63 0.76 1.12 1.33 3/24/2020 0.01 0.01 0.01 0.09 0.25 0.38 0.44 0.52 0.75 0.84 1.19 1.39 3/25/2020 0 0 0 0.07 0.19 0.34 0.41 0.56 0.77 0.88 1.23 1.45 3/26/2020 0.01 0.01 0 0.04 0.13 0.3 0.36 0.51 0.72 0.83 1.2 1.42 3/27/2020 0.01 0.03 0.03 0.02 0.11 0.25 0.3 0.41 0.6 0.72 1.09 1.29 3/30/2020 0.04 0.07 0.12 0.12 0.14 0.23 0.29 0.39 0.57 0.7 1.1 1.31 3/31/2020 0.05 0.12 0.11 0.15 0.17 0.23 0.29 0.37 0.55 0.7 1.15 1.35 Source: U.S. Department of Treasury Figure 8: Dollar per Euro Exchange Rate, March 2020 1.14 1.12 1.1 1.08 1.06

3/2/2020 3/6/2020 3/10/2020 3/14/2020 3/18/2020 3/22/2020 3/26/2020 3/30/2020

(a) Log Euro-Dollar (b) Liquidity Ratio (c) Money Supply

106

1.15 0.5 3 1.1 0.45 1.05 2.5 1 0.4

0.95 0.35 2 0.9 0.3 0.85

0.8 0.25 1.5 0.75 0.2 0.7 1 0.65 0.15 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 (d) Policy Rates (e) Premia

300 500

250 400 200

150 300 100

50 200 0

-50 100 -100

-150 0 -200

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

Figure 9: Data series

62 Table B1: Calibrated Parameters Parameter Value Description Target

External Calibration us eu Mss /Mss 0.6841 relative money supply normalized to exchange rate levels Θb 1 global loan demand scale normalization -35 loan demand elasticity Bianchi and Bigio(2021)/symmetry ςus = ςeu 35 US/EU deposit demand elasticity Bianchi and Bigio(2021)/symmetry λus = λeu 7.9 US/EU interbank market matching eﬃciency Bianchi and Bigio(2021) ιus=ιeu 10 US policy corridor spread Bianchi and Bigio(2021) ηus = ηeu 1/2 benchmark/symmetry Steady-State Estimation of Financial Variables d,∗ Θss 1.0026 US deposit demand scale To match steady-state moments d Θss 1.0026 EU deposit demand scale symmetry us σss 76.8541 average US payment shock steady-state moment targets eu σss 21.9133 average US payment shock steady-state moments targets

ξss -62.5845 average CIP and UIP wedge steady-state moments targets Estimates of US and EU policy variables m,us E (it ) 0.9957 annualized US interest on reserves data m,us Σ(it ) 0.0026 std annual US policy rate data m,us ρ (it ) 0.9777 autocorrelation annual US policy rate data m,eu E (it ) 1.0027 average annual Euro policy rate data m,eu Σ(it ) 0.0017 std annual Euro policy rate data m,eu ρ (it ) 0.9894 autocorrelation annual Euro policy rate data us E (ln Mt ) 6.7255 average monthly US liquid assets stock ($ Billion) data us Σ (ln Mt ) 0.0281 std monthly US liquid assets stock data us ρ (ln Mt ) 0.9894 autocorrelation monthly US liquid assets stock data us E (ln Mt ) 6.7255 average monthly US liquid assets stock ($ Billion) data us Σ (ln Mt ) 0.0281 std monthly US liquid assets stock data us ρ (ln Mt ) 0.9894 autocorrelation monthly US liquid assets stock data

63 Table B1: Estimated Process Parameter Prior Fam Prior Mean Post. Mean Post. Interval us Σσ Inv. Gamma 0.150 0.100 0.091-0.108 eu Σσ Inv. Gamma 0.150 0.124 0.115-0.134 us ΣΘ Inv. Gamma 0.030 0.017 0.015-0.018 eu ΣΘ Inv. Gamma 0.020 0.015 0.014-0.017 Σξ Inv. Gamma 0.000 0.001 0.000-0.001 us ρσ Beta 0.930 0.989 0.986-0.991 eu ρσ Beta 0.930 0.954 0.931-0.973 us ρΘ Beta 0.985 0.991 0.989-0.992 eu ρΘ Beta 0.980 0.989 0.988-0.991 ρξ Beta 0.980 0.980 0.979-0.982

64 Table B1: Model and Data Moments

Moment Data Simulated Moment Steady-State Targets Mean of CIP 13.407 12.980 Mean of BP 142.005 142.829 Mean of µus 0.226 0.235 Mean of µeu 0.222 0.230 Mean of e 1.251 1.274 Untargetted Moments Std. of CIP 42.672 139.150 Std. of BP 74.811 107.779 Std. of µus 0.133 0.037 Std. of µeu 0.025 0.043 Std. of e 0.167 0.180 AR(1) Coef. of CIP 0.801 0.936 AR(1) Coef. of BP 0.917 0.950 AR(1) Coef. of µus 0.989 0.957 AR(1) Coef. of µeu 0.986 0.956 AR(1) Coef. of e 0.963 0.953

65 Appendix not for Publication

(Proofs and Algorithms)

66 A Expressions for {Ψ+, Ψ−, χ+, χ−}

Here we reproduce formulas derived from Proposition 1 in Bianchi and Bigio(2017). This proposition gives us the formulas for the liquidity yield function and the matching probabilities as functions of the tightness of the interbank market. Given θ, the market tightness after the interbank-market trading session is over:   1 + (θ − 1) exp (λ) θ > 1  if ¯  θ = 1 if θ = 1 .  −1  (1 + (θ−1 − 1) exp (λ)) if θ < 1

Trading probabilities are given by   −λ −λ −1 + 1 − e if θ ≥ 1 −  1 − e θ if θ > 1 Ψ = , Ψ = . (34) −λ −λ θ 1 − e if θ < 1 1 − e if θ ≤ 1

The parameter λ captures the matching eﬃciency of the interbank market. A reduced- form bargaining parameter is obtained as:

 η θ θ¯ − 1 (exp (λ) − 1)−1 θ > 1  θ−1 θ if  φ ≡ η if θ = 1 .  ¯ ¯ η  θ(1−θ)−θ θ¯ −1  θ¯(1−θ) θ − 1 (exp (λ) − 1) if θ < 1 where η is a parameter associated with the bargaining power of banks with reserve deficits in each trade—a Nash bargaining coefficient. Hence, φ is an effective bargaining weight. The average interbank rate is:

Rf = (1 − φ)Rw + φRm

The slopes of the liquidity yield function are given by

θ¯η θηθ¯1−η − θ θ¯η θηθ¯1−η − 1 χ+ = (Rw − Rm) χ− = (Rw − Rm) . θ θ¯ − 1 and θ θ¯ − 1 (35)

67 These formulas are consistent with:

+ + f m − − f m − w m χ = Ψ R − R and χ = Ψ R − R + 1 − Ψ (R − R ) .

68 B Proofs

B.1 Proof of Lemma1

Substitute the budget constraint (11) into (12). We obtain:

n0 = Rb(X)n − Rb(X)Div − Rb (X) − Rm(X,X0) m˜ − Rb (X) − Rm,∗(X,X0) m˜ ∗ + Rb (X) − Rd(X,X0) d˜+ (Rb (X) − R∗,d(X,X0))d˜∗ h ∗ ∗ ∗ ∗ ˜∗ i h ˜ i + Eω∗ χ (θ (X), m˜ + ω d ) + Eω χ(θ(X), m˜ + ωd) . and thus: n0 = Rb(X)n − Rb(X)Div + Π (X,X) , where

Π(X,X0) = Rb (X) − Rm(X,X0) m˜ − Rb (X) − Rm,∗(X,X0) m˜ ∗ + Rb (X) − Rd(X,X0) d˜+ (Rb (X) − R∗,d(X,X0))d˜∗ h ∗ ∗ ∗ ∗ ˜∗ 0 i h ˜ 0 i + Eω∗ χ (θ (X), m˜ + ω d ,X ) + Eω χ(θ(X), m˜ + ωd, X ) .

Conjecture that v (n, X) = n. Then, substituting v (n0,X0) = n0, into (10) we obtain:

b b 0 v (n, X) = max Div + βE R (X)n − R (X)Div + Π (X,X ) |X . {Div,˜b,m˜ ∗,d˜∗,d,˜m˜ }

Note that if βRb(X) 6= 1,dividends are either ∞ or −∞. Thus, in equilibrium, it must be the case that Rb(X) = 1/β. As a result,

0 v (b, X) = max Div + βE [1/βn − 1/βDiv + Π (X,X ) |X] {Div,˜b,m˜ ∗,d˜∗,d,˜m˜ } ∗ = n + β max Π (X) . (36) {˜b,m˜ ∗,d˜∗,d,˜m˜ } where

∗ 0 Π (X) = max E[Π (X,X )]. {˜b,m˜ ∗,d˜∗,d,˜m˜ }

Next, consider the ﬁrst-order conditions for {m˜ ∗, d˜∗, d,˜ m˜ }. It must be the case that

69 in equilibrium, the following are true, otherwise the bank would make inﬁnite proﬁt or losses: ∗ ˜∗ ˜ b d d :Πd m˜ , d , d, m,˜ X ≡ R (X) − R (X) − Eω [χd˜] = 0.

∗ ∗ ∗ b ∗,d ∗ ∗ ˜ ˜ d :Πd m˜ , d , d, m,˜ X ≡ R (X) − R (X) − Eω χd˜ = 0.

∗ ˜∗ ˜ b m ∗ m :Πm m˜ , d , d, m,˜ X ≡ R (X) − R (X) − Eω [χm] = 0.

∗ ∗ ˜∗ ˜ b ∗,m ∗ m :Πm∗ m˜ , d , d, m,˜ X ≡ R (X) − R (X) − Eω [χm˜ ] = 0. Next, observe that Π m˜ ∗, d˜∗, d,˜ m,˜ X is homogeneous of degree 1 in {m˜ ∗, d˜∗, d,˜ m˜ }. Hence, by Euler’s Theorem for Homogeneous Functions:   d    ∗  ∗ h i  d  Π (X) = max Πd Πd∗ Πm Πm∗ ·   , {˜b,m˜ ∗,d˜∗,d,˜m˜ }    m    m∗ and thus Π∗ (X) = 0. As a result, replacing this result in (36), we verify the conjecture that v (n, X) = n.

70 B.2 Preliminary Observations

We ﬁrst derive the results for a binomial distribution of withdrawal shocks and then gen- eralize the results. Useful Derivatives: Binomial Distribution. To produce theoretical results we ﬁrst elabo- rate some observations. First, recall that the market tightness in both currencies is always lower than one: δ − µ θ = < 1. δ + µ (37) For this reason, we the tightness is increases in the size of the dispersion shocks:

∂θ 1 δ − µ 1 1 θ ≡ = − = (1 − θ) > 0. δ ∂δ µ + δ (µ + δ) (µ + δ) µ + δ (38)

Also, note that: ∂θ −1 δ − µ 1 1 θ ≡ = − = − (1 + θ) < 0. µ ∂µ δ + µ δ + µ δ + µ δ + µ (39)

Moreover, the penalties are increasing in tightness.

∂χ+ ∂χ+ ∂θ ∂χ− ∂χ− ∂θ = > 0, = > 0. ∂δ ∂θ ∂δ and ∂δ ∂θ ∂δ

We also know that the penalty rates are increasing in tightness, and hence:

∂χ+ ∂χ+ ∂θ ∂χ− ∂χ− ∂θ = < 0, = < 0. ∂µ ∂θ ∂µ and ∂µ ∂θ ∂µ

Useful Derivatives: Continuous Distribution. Consider now the case of a continuous distribution for withdrawal shocks, F (ω; σ) where σ is an index. We assume symmetry: F (−ω) = F (ω). Market tightness in this case, is given by:

µ − −∞ ωf (ω) dω − µ θ = ´∞ . −µ ωf (ω) dω + µ ´ Since the function is symmetric, we have that:

∞ µ ∞ ∞ ωf (ω) dω = ωf (ω) dω + ωf (ω) dω = ωf (ω) dω. ˆ−µ ˆ−µ ˆµ ˆµ

71 We also have that: µ ∞ − ωf (ω) dω = ωf (ω) dω. ˆ∞ ˆµ We have shown the following Lemma.

Lemma 2. The tightness can be expressed more conveniently as:

∞ µ ωf (ω; σ) dω − µ θ = ´ ∞ . µ ωf (ω; σ) dω + µ ´ We consider the family of distributions that satisﬁes the following:

Condition 1. The distribution has increasing right tales with respect to σ:

∞ ∂f (ω; σ) ω dω > 0 for any µ > 0. ˆµ ∂σ

It is convenient to deﬁne:

∞ δ (µ, σ) ≡ ωf (ω; σ) dω, ˆµ

such that by Lemma2, we obtain

δ (µ, σ) − µ θ = . δ (µ, σ) + µ

The expression is identical to (37). By the condition above, we we have that:

δσ (µ, σ) > 0

and

δµ (µ, σ) < 0.

Next, we compute the derivative of the interbank-market tightness with respect to µ. We obtain: ! 1 + θ 1 1 θµ = − − µf (µ; σ) dω · θ · ∞ − ∞ .(40) δ + µ µ ωf (ω; σ) dω − µ µ ωf (ω; σ) dω + µ ´ ´ 72 where the term in parenthesis is positive since µ > 0. Thus, we have shown that:

Lemma 3. The market tightness is decreasing in the liquidity ratio, θµ < 0.

Next, we compute the derivative of the market tightness with respect to σ. We have that:

θσ = θδ · δσ.

Now, notice that: 1 − θ θ = . δ µ + δ (41)

Thus, we have the following preliminary Lemma:

Lemma 4. The derivative of the market tightness with respect to the payment dispersion is given by:

1 − θ θ = · δ > 0. σ µ + δ σ (42) As a consequence of the previous results, we have that with continuous distributions,

Lemma 5. The liquidity coeﬃcients have the following derivatives:

∂χ+ ∂χ+ ∂θ ∂χ− ∂χ− ∂θ = < 0, and = < 0. ∂µ ∂θ ∂µ ∂µ ∂θ ∂µ

and ∂χ+ ∂χ+ ∂θ ∂χ− ∂χ− ∂θ = > 0, and = > 0. ∂σ ∂θ ∂σ ∂µ ∂θ ∂σ

Diﬀerential Form of Money-Market Equilibrium. Consider the equilibrium in the money demand for Euros, M = µD, P (43) The diﬀerential form of this equation is:

M dM dP dµ dD − = dµD + µdD = µD + µD . P M P µ D

Using (43) we obtain: d log M − d log P = d log µ + d log D, (44)

73 and likewise for dollars:

∗ ∗ ∗ ∗ d log M − d log P = d log µ + d log D . (45)

Diﬀerential Form of Exchange Rate. We have that

P e = P ∗

Then, dP P dP ∗ de = − · . P ∗ P ∗ P ∗ We re-arrange terms to obtain:

∗ d log e = d log P − d log P . (46)

Diﬀerentials of the liquidity premia. We have that:

d [χ (s; θ)] LP ≡ Eω = (1 − F (µ, σ)) · χ+ + F (µ, σ) · χ− > 0. θ dθ θ θ Then, we deﬁne:

∂ log (δ) 1 − θ ∂ log (δ) LP ≡ LP · θ · δ = LP · θ δ = LP δ . σ θ δ σ θ δ ∂σ θ µ + δ ∂σ (47) and LPµ ≡ LPθ · θµ = LPθ. (48)

Similar expressions hold for the dollar market.

74 B.3 Proof of Proposition1

In this appendix, we proof the corresponding proposition for the euro market. The results in the body of the paper, follow by symmetry. Here, we let BP denote the euro bond premium and LP the euro liquidity premium.

Proposition 6. [Scale Eﬀects] Part 1. Consider a temporary (i.i.d.) shock that increases the demand for Euro deposits. Then, the shock appreciates the Euro and lowers the liquidity ratio:

b d log e LPθθµµ d log µ R = b ∈ (−1, 0] and = − b ∈ (−1, 0]. d log D R − LPθθµµ d log D R − LPθθµµ

d log e m Hence, d log µ > 0. Furthermore d (LP) = −d (BP) = R d log e. Part 2. Consider a permanent shock that increases the demand for Euro deposits. Then,

d log e d log µ = −1 and = 0, d log D d log D

d log e Hence, d log µ = ∞. Furthermore, d (LP) = d (BP) = 0.

To produce the proof, we employ the implicit function theorem. Since Rb is a constant, the equilibrium condition for Euro reserve holdings yields

P Rb = (1 + im) + LP. E [p (X0)] The price level appears in the real rate on reserves, on the value of real balances, and the real penalties {χ+, χ−}. We use this condition to obtain all the eﬀects. Temporary i.i.d. funding shocks. From the liquidity premium, compute the total diﬀer- ential with respect to D. We obtain:

m P dP dP (1 + i ) + LP + (LPθθµ)(µP dP + µDdD) = 0. (49) E [p (X0)] P P

Observe that: dD µ dD = −µ = −µd log D. D D and that: ∂µ dP = −µd log P. ∂P

75 Hence, b R − LPθθµµ d log P = LPθθµµ · d log D.

Thus, we obtain that: d log P LPθθµµ −1 ≤ = b ≤ 0 d log D R − LPθθµµ + − where the sign follows from χθ , χθ > 0 and θµ < 0. Next, using (46), we have that:

d log e = d log P.

Holding nominal reserve balances and real deposits ﬁxed, we obtain that (44) becomes

d log µ = − (d log D + d log P ) hence: LPθθµµ d log µ = − 1 + b d log D < 0. R − LPθθµµ

The diﬀerential of the liquidity premium is then:

d (LP) = LP · d log P − LPθθµ (µd log P + µd log D) b m LPθθµµ LPθθµµ = R − R b d log D − LPθθµµ 1 + b d log D R − LPθθµµ R − LPθθµµ b b m LPθθµµ R = R − R b − LPθθµµ b d log D R − LPθθµµ R − LPθθµµ

m LPθθµµ = −R b d log D. R − LPθθµµ

This veriﬁes the result since: d (LP) = −Rmd log P

Re-arranging, we obtain:

Rm Rm d log (LP) = − d log P = − d log e. LP LP

Likewise, the bond premium satisﬁes:

d (BP) = −d log (LP) = Rmd log e,

76 and re-arranging: Rm dP Rm d log (BP) = = d log e. BP P BP Temporary dollar funding shocks. The statement in the body of the paper, which refers to the dollar, follows by symmetry. Namely, if we shock dollar loans:

∗ ∗ ∗ ∗ LPθ∗ θµ∗ µ ∗ −1 ≤ d log P = b ∗ ∗ ∗ d log D ≤ 0. R − LPθ∗ θµ∗ µ

In this case, the eﬀect on the exchange rate is:

∗ ∗ ∗ LPθ∗ θµ∗ µ 0 ≤ d log e = − b ∗ ∗ ∗ d log D ≤ 1. R − LPθ∗ θµ∗ µ

Thus, for the dollar R∗,m d log (LP∗) = d log e ≥ 0, LP∗ and likewise the d log (LP∗) = d log (BP∗) ≥ 0.

Finally, replacing the expressions we used we obtain the coeﬃcient that relates the change in the exchange rate to the change in the liquidity ratio:

d log e LP θ µ β∗ ≡ = − θ µ > 0. d log µ Rb

Permanent Euro funding shocks. From the liquidity premium, compute the total diﬀeren- tial with respect to D. Because the shock is permanent, expected inﬂation is zero. Hence, optimality in reserves yields:

LPθθµ (−µd log P + µd log D) = 0. (50)

Thus, d log P = −d log D,

and, thus, d log e = −d log D.

From the result above, we know then that the liquidity ratio does not change when the

77 shock is permanent, d log µ = 0.

From here we conclude that: d (BP) = d (LP) = 0.

Thus, the same is true if the increase is in dollar funding. From the liquidity premium, we compute the total diﬀerential with respect to D. Be- cause the shock is permanent, expected inﬂation is zero. Hence, optimality in reserves yields: LPθθµ (−µd log P + µd log D) = 0. (51)

As a result: d log P = −d log D, and thus, d log e = −d log D.

In the permanent shock case, we cannot obtain a regression coeﬃcient because the variation on the liquidity ratio is non-existing. Permanent dollar funding shocks. We obtain, by symmetry, a similar results for dollars:

d log P = −d log D∗, but in this case: d log e = −d log P ∗ = d log D∗.

Hence, in both cases, d log µ = d log µ∗ = 0. Since the liquidity ratio is constant and inﬂation is constant, the eﬀects on premia are constant. Generalization to mean reverting shocks. Assume now that the shock has a mean reversion rate α. Then, for small perturbations, we have:

d [p (X0)] dP E = α . E [p (X0)] P

dP Then, in the previous proofs, we have that the term P should be replaced by terms:

dP (1 − α) . P

78 From the liquidity premium, compute the total diﬀerential with respect to D. We obtain: b ∼ R (1 − α) − LPθθµµ d log P = −LPθθµµ · d log D. (52)

from which we obtain the more general formula:

d log P d log e ∼ LPθθµµ = = b . (53) d log D d log D R (1 − α) − LPθθµµ which holds exactly when α ∈ {0, 1}. Then,

∗ ∗ ∗ LP ∗ θ ∗ µ d log P d log e ∼ θ µ ∗ = − ∗ = b ∗ ∗ d log D d log D R (1 − α) − LPθ∗ θµ∗ µ

Thus, the liquidity ratio varies by:

d log µ∗ d log P ∗ Rb (1 − α) ∗ = − 1 + ∗ = − b ∗ ∗ . (54) d log D d log D R (1 − α) − LPθ∗ θµ∗ µ

79 B.4 Proof of Proposition2

Proposition 7. [Volatility Eﬀects] Part 1. Consider a temporary (i.i.d.) shock that increases the demand for Euro payment volatility. Then, the shock appreciates the Euro and lowers the liquidity ratio: d log e d log µ −LPθθσσ = − = b ≤ 0. d log σ d log σ R − LPθθµµ

d log e m Hence, d log µ = −1. Furthermore d (LP) = −d (BP) = R d log e. Part 2. Consider a permanent (random walk) shock that increases the demand for Euro deposits. Then, d log e d log µ 1 − θ δ = − = − · σ . d log σ d log σ 1 + θ µ

d log e m Hence, d log µ = −1. Furthermore d (LP) = −d (BP) = R d log e.

Again, the starting point of the proof is the bond premium:

P Rb = (1 + im) + LP. E [p (X0)]

Recall that the price level level appears in the real rate on reserves, on the value of real + − balances, and the real penalties {χ , χ }. Also, note that we already established that θσ > 0. Temporary Euro Payment Shocks. From the liquidity premium, we have that:

m P dP dP (1 + i ) + LP + LPθθσdσ + LPθθµdµ = 0. (55) E [p (X0)] P P

Holding nominal reserve balances and real deposits ﬁxed, we obtain that

M dP dP dµ = = −µ = −µd log P P 2 D P (56)

Thus, re-arranging (55) we obtain:

LPσσ d log P = − b d log σ < 0. R − LPθθµµ

b m where we used R = R + LP and the sign follows from θσ > 0 and θµ < 0. Next, using

80 (46), we have that: d log e = d log P.

Next, we obtain the eﬀect on the liquidity ratio,

LPσσ d log µ = −d log P = b d log σ > 0. R − LPθθµµ

Consider next the liquidity premium. We have that

d (LP) = −Rmd log P.

Likewise the euro bond premium is:

d (BP) = −d (DLP) = Rmd log P.

Now, we want to bound the derivatives. We know that:

LPσσ LPθθδδσσ b d log σ = b d log σ. R − LPθθµµ R − LPθθµµ

We obtain the following bounds: LPθθδδσσ 1 + θ δσσ b ∈ 0, R − LPθθµµ 1 − θ µ

To obtain the bound, we used that Rb > 0, and from (40), we used that:

1 + θ −θ > , µ µ + δ

and ﬁnally, we used (41). Temporary Dollar Payment Shocks. The statement in the body of the paper follows by symmetry. Namely, if we shock dollar volatility:

∗ ∗ −LPσ∗ σ ∗ d log P = b ∗ ∗ d log σ ≤ 0. R − LPθ∗ θµ∗ µ

81 Since the money supply and deposits are ﬁxed:

∗ ∗ ∗ ∗ LPθ∗ θσ∗ σ ∗ d log µ = −d log P = b ∗ ∗ d log σ ≥ 0. R − LPθ∗ θµ∗ µ

In this case, the eﬀect on the exchange rate is:

∗ ∗ LPθ∗ θσ∗ σ ∗ d log e = b ∗ ∗ d log σ ≥ 0. R − LPθ∗ θµ∗ µ

Thus, for the dollar

d (DLP) = d (BP) = R∗,md log e ≥ 0,

Permanent Euro Payment Shocks. From the liquidity premium compute the total differential with respect to σ. Because the shock is permanent, expected inflation is zero. Hence, again from the definition of the bond premium:

LPθ (−θµµd log P + θσσd log σ) = 0. (57)

Thus, clearing the equation above we obtain:

θ d log e = d log P = σ · σ · d log σ < 0. θµµ

Thus, the eﬀect on the liquidity ratio is:

θ d log µ = − σ · σ · d log σ. θµµ

We know then that the liquidity ratio must not change. Thus, in this case,

d log θ = 0, and as a result: d (EBP) = d (DLP) = 0.

82 Using ((41)) we obtain a bound to the derivative term:

θ θ δ 1−θ δ 1 − θ δ σ · σ = δ σ σ ≥ − µ+δ σ σ = − σ σ. θ µ θ µ 1+θ µ 1 + θ µ µ µ δ+µ

1+θ where we again used that −θµ > δ+µ . Permanent Dollar payment shocks. A similar condition holds for the dollar payment shock. Thus:

∗ ∗ ∗ ∗ (1 − θ ) θσ∗ ∗ ∗ d log P = −d log µ = −d log e = ∗ · ∗ σ d log σ < 0, (1 + θ ) θµ∗ µ

We know then that the liquidity ratio must not change in this case,

d log θ∗ = 0, and as a result: d (BP∗) = d (DLP) = 0.

∗ 1−θ δσ We obtain the same bound as the one for the Euro: d log P ≥ − 1+θ µ σ. Generalization to mean reverting shocks. As in the example of the previous shock, we only need do adjust the terms that are aﬀected by inﬂation after a shock. In this case,

−LPθθσσ d log e ≈ b d log σ < 0 (58) R (1 − α) − LPθθµµ

LPθθσσ d log (µ) ≈ b d log σ > 0 (59) R (1 − α) − LPθθµµ Finally, Rm (1 − α) d (LP) ≈ b LPθθσdσ. R (1 − α) − LPθθµµ Again, the we can verify the consistency of the solution.

83 B.5 Proof of Proposition3 (Interest Rate Pass-Through)

Proposition 8. [Scale Eﬀects] Part 1. Consider a temporary (iid) shock that increases the Euro nominal policy rate. Then, the shock appreciates the Euro and lowers the liquidity ratio:

d log e d log µ Rm −1 ≤ m = − m = − b < 0. d log (1 + i ) d log (1 + i ) (R − LPθθµµ)

d log e m d log e m Hence, d log µ = −1. Furthermore d (ELP) = −d (EBP) = R 1 − d log(1+im) d log (1 + i ) . Part 2. Consider a permanent (random walk) shock that increases the demand for Euro deposits. Then, d log e d log µ = − . d log (1 + im) d log (1 + im)

d log e m Hence, d log µ = −1. Furthermore d (ELP) = d (EBP) = R d log e.

The gist of the proof is similar to the ones in the previous Propositions. Temporary shocks to Euro Policy Rate. From the bond premium in Euros, we obtain that:

0 P m m E [p (X )] dP dP dP d (1 + i ) + (1 + i ) + LP − E [χθ] θµµ = 0. (60) E [p (X0)] P P P P

Re-arranging terms we obtain:

P d (1 + im) E[P (X0)] d log (P ) = − b . (R − LPθθµµ)

Multiplying and dividing by (1 + im) we obtain:

m R m d log (P ) = − b d log (1 + i ) . (R − LPθθµµ)

We know that: m b R ≤ R and LPθθµµ≤ 0.

Thus, m R m −1 ≤ d log (P ) = − b d log (1 + i ) < 0. (R − LPθθµµ) with equality only under satiation.

84 Next, using: dP dµ = −µ , P we obtain: d log (µ) d log (P ) = − ∈ (0, 1]. d log (1 + im) d log (1 + im)

Now consider the eﬀect on the dollar liquidity premium. We have that:

d [LP] = Rmd log (1 + im) − Rmd log (P ) = Rm (log (1 + im) − d log (P )) m m R m m = R 1 − b d log (1 + i ) ∈ [0,R ). R − LPθθµµ

Hence, we have that:

dLP dLP −1 d log (P ) dLP −1 d log (e) = − 1 + = − 1 − ≤ 0, d log (1 + im) Rm log (1 + im) Rm log (1 + im)

d log(P ) because log(1+im) ≥ −1 with strict inequality away from satiation. Thus, we also have that:

m LP − LPθθµµ m m dBP = −dLP = −R b d log (1 + i ) ∈ (−R , 0] R − LPθθµµ

m dBP R LP − LPθθµµ m = − b ≤ 0. d log (1 + i ) BP R − LPθθµµ

Temporary shocks to Dollar Policy Rate. In this case, by symmetry:

R∗,m log (P ∗) = − d log (1 + i∗,m) . ∗,b ∗ ∗ ∗ R − LPθ∗ θµ∗ µ

Then, R∗,m d log (e) = d log (µ∗) = −d log (P ∗) = ∈ (0, 1] ∗,b ∗ ∗ ∗ R − LPθ∗ θµ∗ µ and equal to 1 under satiation. Furthermore:

dBP = dLP = −R∗,m (d log (1 + im) − d log (P ∗)) ∈ [0,Rm)

with equality at satiation.

85 Permanent Eﬀects. Consider now a permanent eﬀect. Then, expected prices respond with innovation. Thus, m d (1 + i ) − E [χθ∗ ] θµµd log P = 0.

Thus, clearing the condition yields:

d (1 + im) d log P = . E [χθ∗ ] θµµ

Multiplying and dividing by (1 + im) yields:

d log (1 + im) d log P = > 0. E [χθ∗ ] θµµ

Then, for the liquidity ration and the exchange rate:

d log e = d log P = −d log µ.

Finally, dBP = −dLP = Rmd log (1 + im) .

Likewise, for dollars: ∗,m ∗ d log (1 + i ) d log P = ∗ ∗ > 0. E [χθ∗ ] θµ∗ µ Then, for the liquidity ration and the exchange rate:

d log e = d log µ∗ = −d log P ∗ < 0.

Finally, dBP = dLP = −Rmd log (1 + im) .

B.6 Proofs of Proposition4 (Open-Market Operations)

Preliminary Observations. We now consider a purchase of securities with reserve. In G particular, we let the domestic central bank hold private loans in the amount St . The central banks’ budget constraint in this case is modiﬁed to:

d G G m w Mt + Tt + Wt+1 + 1 + it · Pt−1St−1 = Pt · St + Mt−1(1 + it ) + Wt(1 + it ).

86 As in earlier proofs, we avoid time subscripts. Next, we study the eﬀects of a one time increase in SG while holding the path of transfers the same. We evaluate the role of random purchases of loans ﬁnanced with reserves. We obtain:

dP dSG dM = SGdP + P dSG = PSG + PSG . P Sg

Any operating losses are ﬁnanced with transfers. Assuming that a fraction Γ central bank’s balance liabilities are ﬁnanced with loans, we have ΓM = PSG, dividing both sides by M

dM PSG dP dSG dP dSG = + = Γ + . M M P SG P SG

In logarithms: G d log M = Γ d log P + d log S . (61)

The equation has the interpretation that the increase in the money supply needed to finance the open-market operation, in nominal terms has to compensate for the increase in the price level. This is because the operation is defined in real terms. Next, since the households has an inelastic supply of savings it must be that dD = −dS,and since the supply of the security is fixed −dS = dSG.Hence,

dD = dSG.

Next, we express the change in the liquidity ratio in its diﬀerential form:

dM dP dD dµ = µ − + M P D dM dP dSG = µ − + M P D dM dP SG dSG = µ − + M P D SG dM dP M dSG = µ − + Γ . M P PD SG

We used the conditions above. Rearranging the last line:

dµ = µ d log M − d log P + Γµd · d log SG = µ (d log M − d log P ) − Γµ2d log SG.

87 Using (61), we obtain:

1 dµ = µ (d log M − log P ) − Γµ2 d log M − d log P Γ = µ (1 − µ) d log M − µ (1 − Γµ) log P. (62)

A temporary Euro open-market operation. From the liquidity premium, we have that:

m P dP dP (1 + i ) + LP + LPθθµdµ = 0. (63) E [p (X0)] P P Substituting (62) and collecting terms we obtain:

m (R + LP − LPθθµµ (1 − Γµ)) d log P + LPθθµµ (1 + µ) d log M = 0. (64)

Equations (63-64) yields a system of two equations and two unknowns:       b R − LPθθµµ (1 − Γµ) LPθθµµ (1 − µ) d log P 0     =   . −Γ 1 d log M Γd log SG

Then, we solve for d log M in (64):

b R − LPθθµµ (1 − Γµ) d log M = d log P. (65) −LPθθµµ (1 − µ)

Using (61), we have:

Rb − LP θ µ (1 − Γµ) θ µ d log P = Γ d log P + d log SG −LPθθµµ (1 − µ)

We can now clear the expression for the change in the price level using (61):

Hd log P = d log SG,

b H ≡ R −LPθθµµ(1−Γ) where −LPθθµµ(1−µ) . Thus, we have that:

d log P −LPθθµµ (1 − µ) G = b ≥ 0, d log S R − LPθθµµ (1 − Γ)

88 where the bounds follow because θµ < 0, and 1 − Γµ > Γ (1 − µ), Thus, we know that the OMO produces inﬂation. Next, from (65), we have that the increase in the money supply is:

b d log M R − LPθθµµ (1 − Γµ) G = b ≥ 1. d log S R − LPθθµµ (1 − Γ)

From the liquidity ratio, (62), then we conclude that:

b b d log µ R − LPθθµµ (1 − Γµ) −LPθθµµ (1 − µ) µ (1 − µ) R G = µ (1 − µ) b −µ (1 − Γµ) b = b ∈ [0, 1] . d log S R − LPθθµµ (1 − Γ) R − LPθθµµ (1 − Γ) R − (1 − Γ) LPθθµµ

Thus, the liquidity ratio increases, but less than one for one. In turn, the exchange rate, follows immediately from movement in the price:

d log e d log P −LPθθµµ (1 − µ) G = G = b ≥ 0. d log S d log S R − LPθθµµ (1 − Γ)

As before, the dollar liquidity premium and the Euro bond premia satisfy:

m −LPθθµµ (1 − µ) G dBP = dLP = R d log P = b d log S > 0. R − LPθθµµ (1 − Γ)

A temporary Dollar open-market operation. Now consider a purchase in the US. By symmetry we have the following three conditions:

∗ ∗ ∗ ∗ d log P −LPθ∗ θµ∗ µ (1 − µ ) G,∗ = b ∗ ∗ ∗ ≥ 0, d log S R − LPθ∗ θµ∗ µ (1 − Γ )

b ∗ ∗ ∗ ∗ d log M R − LPθ∗ θµ∗ µ (1 − Γ µ ) G = b ∗ ∗ ∗ ≥ 1. d log S R − LPθ∗ θµ∗ µ (1 − Γ ) d log µ µ (1 − µ) Rb G = b ∈ [0, 1] . d log S R − (1 − Γ) LPθθµµ

Now, in the exchange rate in this case follows:

∗ ∗ ∗ ∗ ∗ LPθ θµ∗ µ (1 − µ ) d log e = −d log P = b ∗ ∗ ∗ ≤ 0. R − LPθ∗ θµ∗ µ (1 − Γ )

89 Finally, the excess-bond premium and the dollar liquidity premium is:

dBP = dLP = −R∗,md log P ∗ ≤ 0.

A permanent Euro open-market operation. Since policy rates are ﬁxed, from the liquidity premium we have:

LPθθµdµ = 0, (66)

for purchase. Thus, from (62) we have that:

0 = µ (1 − µ) d log M − µ (1 − Γµ) log P

We combine this equation with (61) to obtain the system:       −µ (1 − Γµ) −µ (1 − µ) d log P 0     =   . −Γ 1 d log M Γd log SG

Then, we solve for d log M using the ﬁrst row,

(1 − Γµ) d log M = d log P. (1 − µ)

Then, replacing in the second row:

Γ d log P = (1 − µ) d log SG. 1 − Γ

Back in the previous equation:

(1 − Γµ) d log M = Γ . 1 − Γ

The exchange rate then permanently devalues by

d log e d log P Γ = = (1 − µ) . d log SG d log SG 1 − Γ

90 and

dEBP = dLP = 0.

A permanent US open-market operation. Again proceeding by symmetry

d log e d log P ∗ Γ∗ = − = − (1 − µ∗) . d log S∗,G d log S∗,G 1 − Γ∗ Then, we have that: d log M ∗ (1 − Γ∗µ∗) = Γ∗ . d log S∗,G 1 − Γ∗ Finally, the premia satisfy:

d log µ∗ = dBP = dLP = 0.

B.7 Proof of Proposition5

The proof is immediate. Now consider a univariate linear regression of ∆ log e∗ against ∆ log µ∗. The regression coeﬃcient is expressed as:

∗ ∗ e∗ CoV (∆ log e , ∆ log µ ) γ ∗ = . µ V ar (∆ log µ∗) (67)

Provided that shocks are orthogonal,

∗ ∗ ∗ ∗ d log e d log µ ∗ 2 d log e d log µ ∗ 2 CoV (∆ log e∗, ∆ log µ∗) ≈ · Σσ + · ΣD d log σ∗ d log σ∗ d log D∗ d log D∗ ∗ 2 ∗ 2 e∗ d log µ σ∗ e∗ d log µ D∗ = γ ∗ ∗ · · Σ + γ ∗ ∗ · Σ . µ ,σ d log σ∗ µ ,D d log D∗

The variance of the change in the liquidity ratio is:

∗ 2 ∗ 2 d log µ ∗ d log µ ∗ V ar (∆ log µ∗) = Σσ + · ΣD . d log σ∗ d log D∗ From (58) and (59), we have that:

∗ ∗ e d log e /d log σ ∗ γ ∗ ≡ ≈ 1 σ . µ,σ d log µ∗/d log σ∗ considering small deviations on

91 Likewise, from (53) and (54), we obtain that:

∗ ∗ b e∗ d log e /d log D D∗ R ∗ γµ∗,D∗ ≡ ∗ ∗ ≈ 1 − ρ ∗ ∗ for a small devion in D . d log µ /d log D LPθ∗ θµ∗ µ

Thus, substituting the term combining back into (67)

2 2 e∗ d log µ∗ σ∗ e∗ d log µ∗ D∗ γµ∗,σ∗ · ∗ · Σ + γµ∗,D∗ ∗ · Σ e∗ d log σ d log D γµ∗ = ∗ 2 ∗ 2 d log µ σ∗ d log µ D∗ d log σ∗ Σ + d log D∗ · Σ D∗ b σ∗ 1 − ρ R D∗ = w + ∗ ∗ w . LPθ∗ θµ∗ µ

Where: ∗ 2 ∗ 2 ∗ 2! ∗ d log µ ∗ d log µ ∗ d log µ ∗ wσ = · Σσ / Σσ + · ΣD d log σ∗ d log σ∗ d log D∗ and ∗ 2 ∗ 2 ∗ 2! ∗ d log µ ∗ d log µ ∗ d log µ ∗ wD = · ΣD / Σσ + · ΣD . d log D∗ d log σ∗ d log D∗

92 C Microfoundations for Deposit Supplies and Loan Demands

We provide here the microfoundations under the assumption that there are no aggregate shocks. h d G Household Problem. Deﬁne the household net worth e = 1 + it D + 1 + it G + h h qt + rt Σ − Tt , as the right-hand side of its budget constraint, excluding labor income. Then, substitute ch from the budget constraint and employ the deﬁnition eh. We obtain the following value function:

1+ν h h d d g g h h Vt G ,D, Σ = max U c + U (c ) − + e {cd,cg,h,G0,D0,Σ0} 1 + ν z h − P cg + P cd + D0 + G0 + q Σ0 + t t t t Pt h 0 0 0 +βVt+1 (G ,D , Σ )

cd ≤ 1 + id D cg ≤ D . subject to t Pt and Gt Step 1 - deposit and bond-goods demand. The step is to take the ﬁrst-order conditions for d g c , c . Since {G, D} enter symmetrically into the problem, we express the formulas in terms of x ∈ {d, g}, an index that corresponds to each asset. From the ﬁrst-order conditions D G with respect to Pt and Pt , we obtain that: x x −1 x X c (X, t) = min (Ucx ) (1) ,Rt · for x ∈ {d, g} . (68) Pt−1

The expression shows that the deposit- and bond-in-advance constraints bind if the marginal utility associated with their consumption is less than one. Note that

x x ¯ ¯γ −γx Ucx X = X x for x ∈ {d, g} , (69)

¯ marginal utility is above 1, for X/Pt < X. Then, the marginal consumption as a function of real balances is:  x ¯ x Rt X/Pt < X cX/Pt (X, t) = for x ∈ {d, g} 0 otherwise

93 We return to this conditions below to derive the demand for deposits and bonds by the non-ﬁnancial sector. Step 2 - labor supply. The ﬁrst-order condition with respect to labor supply yields a labor supply that only depends on the real wage:

ν ht = zt/Pt. (70)

Step 3 - deposit and bond demand. Next, we the derive deposit demand and T-Bill de- 0 0 mand. By taking ﬁrst-order conditions with respect to D /Pt and G /Pt, the real balances of deposits and bonds.

h x x h h ∂Vt+1 ∂U ∂c ∂U ∂c 1 = β 0 = β x · 0 + h · 0 for x ∈ {d, g} . ∂ (X /Pt) ∂c ∂ (X /Pt) ∂c ∂ (X /Pt)

The first equality follows directly from the first-order condition and the second uses the envelope Theorem and the solution for the optimal consumption rule. If we shift the period in (68), by one period, the first-order condition then becomes:  ∂U x x ¯ 1  ∂cx Rt X/Pt < X = for x ∈ {d, g} . β x Rt otherwise

Finally, once we employ the deﬁnition of marginal utility, we obtain:

 x ¯γ x −γx x ¯ 1  X (Rt X/Pt) Rt X/Pt < X = for x ∈ {d, g} . β x Rt otherwise

Inverting the condition yields:

 x 1 −1 ¯ 1/γ x γx x Xβ (Rt ) Rt < 1/β X/Pt = for x ∈ {d, g} . ¯ x [X, ∞) Rt = 1/β

Thus, we have that

x 1 Θx = X¯ β1/γ x = − 1 x ∈ {d, g} . t t and γx for

Next, we move to the ﬁrm’s problem to obtain the demand for loans.

94 Firm Problem. In the appendix, we allow the ﬁrm to save in deposits whatever it doesn’t spend in wages. From ﬁrm’s problem, if we substitute the production function into the objective we obtain:

h α b d d d Pt+1r = max Pt+1At+1h − 1 + i B + 1 + i B − ztht t+1 d t t+1 t+1 t+1 t+1 Bt+1≥0,xt+1,ht≥0

d subject to ztht ≤ Bt+1. Observe that

α b d d d Pt+1At+1ht − 1 + it+1 Bt+1 + 1 + it+1 Bt+1 − ztht α b d d = Pt+1At+1ht − ztht − it+1 − it+1 Bt+1 + ztht .

b d Step 4 - loans demand. Since it+1 ≥ it+1, then it is wihout without loss of generality, that d the working capital constraint is binding, ztht = Bt+1. Thus, the objective is

α b Pt+1At+1ht − 1 + it+1 ztht.

The ﬁrst-order condition in ht yields

α b Pt+1αAt+1ht = 1 + it+1 ztht.

Dividing both sides by Pt, we obtain

Pt+1 α b zt αAt+1ht = 1 + it+1 ht. Pt Pt Next, we use the labor supply function (70), to obtain the labor demand as a function of the loans rate:

α Pt+1 α b ν+1 b αAt+1ht αAt+1ht = 1 + it+1 ht → Rt = ν+1 . (71) Pt ht

Once we have the wage bill, and the fact that the working capital constraint is biding,

1 d d ν+1 Bt+1 ztht ν+1 Bt+1 = ht = ht → ht = . (72) Pt Pt Pt

95 Thus, we can combine (71) and (72) to obtain the demand for loans:

−1 α d d ν+1 d b b Bt+1 Bt+1 Bt+1 b Rt = αAt+1 → = Θt Rt+1 (73) Pt Pt Pt

Thus, the coeﬃcients of the loans demand are b ν + 1 Θb = (αA )− b = . t t+1 and α − (ν + 1)

Step 5 - deposit and bond demand. We replace the loans demand (73) into (72), to obtain the labor market equilibrium:

1 1 α−(ν+1) 1 b α−(ν+1) ht = Rt+1 . αAt+1

We replace (72) into the production function to obtain:

α α 1 α−(ν+1) α 1 α−(ν+1) (ν+1) α b α−(ν+1) ν+1−α b α−(ν+1) yt+1 = At+1 Rt+1 → yt+1 = At+1 Rt+1 · αAt+1 α

The proﬁt of the ﬁrm is given by:

(ν+1) α ν+1 α h b h ν+1−α − α−(ν+1) − α−(ν+1) b α−(ν+1) rt+1 = yt+1 − Rt+1Bt+1 → rt+1 = At+1 α − α · Rt+1 .

The asset price qt then is determine as:

X s h qt = β rs . s≥1

With this, we conclude that output, hours and the ﬁrm price are decreasing in current (and future) loans rate. Note that throughout the proof we use the labor market clearing condition. Then, clearing in the loans and deposit markets, by Walras’s law, implies clearing in the goods market. Once we compute equilibria taking the schedules as exogenous in the bank’s problem, it is possible to obtain output and household consumption from the equilibrium rate. The equivalence table from the structural parameters to the reduced form parameters is:

96 Reduced Θx x Θb b

x − ν+1 ¯ 1/γ 1 ( α−(ν+1) ) ν+1 Structural Xtβ γx − 1 (αAt+1) α−(ν+1)

Table B1: Structural to Reduced form Parameters

D Computational Algorithms

Steady State: equilibrium conditions. We now solve the steady-state of the model where n = 0 every period. Solving for steady-state equilibrium requires requires to solve for ∗,d d b ∗ 11 variables, three interest rates, R ,R ,R , three prices {P,P , e} and ﬁve quantities {m∗, m, d, d∗, b∗}. Prices given quantities are given by:

∗,d ∗,d ∗ −ς∗ R = Θ (d ) (74)

d d −ς R = Θ (d) (75) and b b ∗ R = Θ (b ) . (76)

The two prices are given by (a) the equilibrium in the market for real dollar reserves is,

M ∗ m∗ = . P ∗ (77) and euro reserves is for euro reserves: M m = . P (78)

In turn, the exchange rate is obtained via the law of one price:

e = P/P ∗.

Finally, we have four ﬁrst-order conditions and the budget constraint to pin down the quantities:

97 a) the dollar liquidity premium:

m,∗ ∗ m R + E [χm∗ ] = R + E [χm]

b) The excess bond premium:

b m,∗ ∗ R = R + E [χm∗ ] (79)

c-d) And the deposit premia

d,∗ ∗ m,∗ ∗ R + E [χd∗ ] = R + E [χm∗ ] (80)

Dollar deposit-liquidity premium:

d,∗ ∗ m,∗ ∗ R + E [χd∗ ] = R + E [χm∗ ] (81)

Finally, we have the budget constraint corresponding to n = 0, given by:

b + m + m∗ = d + d∗.

This is a system of 11 equations and 11 unknowns. Next, solve the model in ratios. Steady State: solving the model in ratios. At the individual bank level, only the relative portfolios are determined, but not the scale. The scale of deposits, loans, and reserves are determined by their relative supply schedules. This allows us to solve the model in terms of ﬁnancial ratios. Given returns of all assets and liabilities, banks optimal portfolio will give rise to liquidity ratios in both currencies, which we deﬁne as follows:

m m∗ µ ≡ µ∗ ≡ . d and d∗

We deﬁne the ratio of real euro to dollar funding as:

d υ = . d∗

Thus, once we obtain {d∗, ν, µ, µ∗} we obtain {m, m∗, d} from these three equations. The Libor and Euribor markets, are functions of these ratios, but not of the scale. Mul-

98 tiplying and dividing by d the market tightness in the Euribor market is

µ − δ θ = max , 0 µ + δ

and in the Libor market: µ∗ − δ∗ θ∗ = , 0 . µ∗ + δ∗ Thus we have introduced three ratios. Notice that once we obtain {µ, µ∗} we obtain {d, d∗}. Once we obtain υ, we have {e−1} . Finally, we express the bank’s budget constraint in terms of this ratios:

∗ ∗ ∗ b = (υ (1 − µ) + (1 − µ )) d . (82)

These are convenient expressions to understand the equilibrium that follows. b d ∗,d We substitute out R ,R ,R and work directly with the market clearing conditions. We replace the budget constraint, we can get rid of loans. If we substitute the ratios {µ∗, µ, υ, d∗} into the equilibrium conditions and, thus only have one quantity variable d∗, and the rest of the system is expressed in ratios. Steady State: autonomous sub-system. We solve for {µ∗, µ, υ, d∗} using: Excess bond premium:

b ∗ ∗ ∗,m ∗ Θ ((υ (1 − µ) + (1 − µ )) d ) = R + E [χm (µ )] . (83)

The dollar liquidity premium:

m ∗,m ∗ R + E [χm (µ)] = R + E [χm (µ )] .

The euro funding premium:

d ∗ −d ∗,m ∗ Θ (υd ) + E [χd (µ)] = R + E [χm (µ )] (84)

99 Dollar funding premium:

∗ ∗,d ∗ −d ∗ ∗,m ∗ Θ (d ) + E [χd∗ (µ )] = R + E [χm∗ (µ )] (85)

Steady State: Solving the rest of the model. With the solution to {µ∗, µ, υ, d∗} we obtain {m∗, m, d} using:

∗ ∗ ∗ ∗ m = µd, m = µ d , and d = νd .

We deﬁne the ratio of real euro to dollar funding as:

d υ = . d∗

Then, we obtain prices from:

M P = µυd∗ and M ∗ P ∗ = µ∗d∗ and ﬁnally the exchange rate from P e = . P ∗ Calibration of Steady State. We use the following targets:{EBP, DLP, µ∗, e} to obtain the ∗ d d∗ b following parameters: σss, σss, Θ , Θ . We treat Θ = 1 as a normalization and we ﬁx, ∗ Θd = Θd by symmetry. Thus, we only need to obtain three parameters. From

∗ EBP = E [χm (µ )]

∗ ∗ ∗ we solve for σss directly, taking µ as given. From here we obtain, E [χd∗ (µ )]. We obtain d∗ from:

∗,m ∗ ∗ 1/ς∗ R + [χ ∗ (µ )] − [χ ∗ (µ )] d∗ = E m E d , Θ∗,d

by assuming a value for Θ∗,d.

100 Then, we obtain {ν, µ} using the {EBP} by jointly solving

m 1/ς∗ R + E [χm∗ (µ)] − E [χd∗ (µ)] υ = ∗,m ∗ ∗ R + E [χm∗ (µ )] − E [χd∗ (µ )] and

b ! 1 (R∗,m + EBP)1/ µ = 1 − − (1 − µ∗) . υ d∗

Combining the solution, we obtain an exchange rate. From, here we search for the ∗ values of Θd to obtain e M µ∗ e = M ∗ µν Finally, after solving for µ, we obtain σ from:

∗ E [χm (µ)] = E [χm (µ )] − DLP.

D.1 Algorithm to solve for transitions

Transitions. We can consider the previous section as a steady-state version of the model, if prices are ﬁxed in both currencies, then policy rates are actual real rates. In this section we consider what happens one period before the steady state, we call that period t = 0. Assume that the nominal policy rates are given:

∗,a (1 + i ) for a ∈ {m, w} ,

for the US and for the EU: a (1 + i ) for a ∈ {m, w}·

The real rates now satisfy:

(1 + i∗,a) R∗,a = a ∈ {m, w} , (1 + π∗) for

for the US and for the EU:

101 (1 + ia) Ra = · (1 + Ω) a ∈ {m, w}· (1 + π) for Where now we have that: p (1 + π∗) = ss p0 and e (1 + Ω) = ss . e0

The values {pss, ess} are solved from the steady state solution. m ∗,m w ∗,w Transitions. Step 1. Conjecture {R0 ,R0 ,R0 ,R0 } . Solve for the liquidity ratios in Dol- ∗ d ∗,d lars and Euro µ, µ ,R ,R using:

1 1 Rd + ω χ+ (µ) − χ− (µ) = R∗,d + ω∗ χ∗,+ (µ∗) − χ∗,− (µ∗) 2 2 1 1 Rm + χ+ (µ) + χ− (µ) = R∗,m + χ∗,+ (µ∗) + χ∗,− (µ∗) 2 2

1 Θb ((υ (1 − µ) + (1 − µ∗)) d∗) = R∗,d + ω∗ χ+ (µ) − χ− (µ) 2

1 1 Rm = R∗,d + ω χ+ (µ) − χ− (µ) − χ+ (µ) + χ− (µ) . 2 2

d ∗,d ∗ Step 2. Given the solutions to R ,R , solve {d , υ} using:

Rd 1/ς d = Θd

∗ Rd 1/ς R∗,d −1/ς υ = . Θd Θ∗,d

Step 3. Solve for prices and the exchange rate using the solutions:

M µυd∗ = pss

102 e µ∗d∗ = M ∗ pss ∗ −1 pss = e pss.

Step 4. Update values for real policy rates:

(1 + i∗,a) R∗,a = a ∈ {m, w} , (1 + π∗) for for the US and for the EU:

(1 + ia) Ra = · (1 + Ω) a ∈ {m, w}· (1 + π) for Where now we have that: p (1 + π∗) = ss p0 and e (1 + Ω) = ss . e0

D.2 Algorithm to obtain a Global Solution

The algorithm to obtain the global solution to the model follows the algorithm to produce transitions. First, we define s ∈ S = {1, 2, 3,...,N s} to be a finite set of states. We let s follow a Markov process with transition matrix Q. Thus, s0 ∼ Q (s) . The state now affects the ∗,m m ∗,w w ∗ d ∗,d ∗,m parameters of the model. That is, at each period, δ, λ, i , i , i , i ,M,M , Θ , Θ , Θ are all, potentially, functions of the state s. The algorithm proceeds as follows. We define a “greed” parameter ∆greedand a tolerance parameters εtol, and construct a grid for S. We conjecture a price-level functions ∗ p(0) (s) , p(0) (s) which produces a price levels in both currencies as a function of the state. ∗ ∗ ∗ As an initial guess, we propose to use p(0) (s) = pss, and p(0) (s) = pss setting the exchange rate to its steady state level in all periods. We proceed by iterations, setting a tolerance count tol to tol > 2 · εtol.

Outerloop 1: Iteration of price functions. We iterate price functions until they converge. ∗ Let n be the n − th step of a given iteration. Given a p(n) (s) , p(n) (s), we produce a ∗ tol new price level functions p(n+1) (s) , p(n+1) (s) if tol > ε .

103 Innerloop 1: Solve for real policy rates. For each s in the grid for S, we solve for

{Rm (s) ,R∗,m (s) ,Rw (s) ,R∗,w (s)} .

Let j be the j − th step of a given iteration. Conjecture values

n m ∗,m w ∗,w o R(0) (s) ,R(0) (s) ,R(0) (s) ,R(0) (s)

m ∗,m w ∗,w —we propose {Rss,Rss ,Rss,Rss } as an initial guess. We then update

n m ∗,m w ∗,w o R(j) (s) ,R(j) (s) ,R(j) (s) ,R(j) (s)

until we obtain convergence:

∗ d ∗,d 2.a Given this guess, we solve for the liquidity ratios in Dollars and Euro µ, µ ,R ,R as a function of the state using:

1 1 Rd + ω χ+ (µ) − χ− (µ) = R∗,d + ω∗ χ∗,+ (µ∗) − χ∗,− (µ∗) 2 2 1 1 Rm + χ+ (µ) + χ− (µ) = R∗,m + χ∗,+ (µ∗) + χ∗,− (µ∗) 2 2

1 Θb ((υ (1 − µ) + (1 − µ∗)) d∗) = R∗,d + ω∗ χ+ (µ) − χ− (µ) 2

1 1 Rm = R∗,d + ω χ+ (µ) − χ− (µ) − χ+ (µ) + χ− (µ) . 2 2

d ∗,d ∗ 2.b Given the solutions to R (s) ,R (s) , solve {d , υ} using:

Rd 1/ς d = Θd

∗ Rd 1/ς R∗,d −1/ς υ = . Θd Θ∗,d

104 2.c Given {d∗ (s) , υ (s)} we solve for prices {p, p∗, e} using:

M µυd∗ = p e µ∗d∗ = M ∗ p p∗ = e−1p.

2.d Finally, we update the real policy rates. For that we construct the expected inﬂation in each currency: P 0 ∗ 0 Q (s |s) p (s) [π∗] = s ∈S (n) E p∗ (s) and P Q (s0|s) p (s) [π] = s0∈S (n) . E p (s) We then update the policy rates by:

(1 + i∗,a) R∗,a = a ∈ {m, w} (j+1) (1 + π∗) for

and (1 + ia) Ra = a ∈ {m, w} . (j+1) (1 + π) for

2.e Repeat steps 2.a-2.d, unless

n m ∗,m w ∗,w o R(j) (s) ,R(j) (s) ,R(j) (s) ,R(j) (s)

is close to n m ∗,m w ∗,w o R(j+1) (s) ,R(j+1) (s) ,R(j+1) (s) ,R(j+1) (s) .

If the real policy rates have converged, update prices according to

∗ greed ∗ greed ∗ p(n+1) (s) = ∆ p + 1 − ∆ p(n) (s)

and greed greed ∗ p(n+1) (s) = ∆ p + 1 − ∆ p(n) (s)

105 and proceed back to the outerloop.

106