Market Liquidity: A Primer - Brookings Institution

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Market Liquidity: A PrimerJune 2015 l The Brookings InstitutionDouglas J. Elliott, Fellow, Economic StudiesIntroductionOverview and recommendationsUS financial markets are critical to thefunctioning of our entire economy, providingmore credit, for example, than banks do. Ourunusually large financial markets have been anAmerican competitive advantage for years,providing a cost-effective means of matchinginvestors with worthy companies and projects.Therefore, the current debate about whethermarket liquidity is drying up is an importantone, since the ability to buy and sell securities iscentral to market functioning. This primerprovides an introduction to the issues byaddressing the following questions.Market liquidity refers to the ability of buyersand sellers of securities to transact efficientlyand is measured by the speed with which largepurchases and sales can be executed and thetransaction costs incurred in doing so. These What is market liquidity?Why do we care about it?Has it actually declined?What do the recent bouts of marketvolatility mean?Why would we expect market liquidityto be down?Will market liquidity decline further?What factors might offset tighteningliquidity?What should be changed to improvemarket liquidity?Before going systematically through thesequestions, the following section provides anoverview and recommendations.costs include both the explicit commission orbid/ask spread and the, often larger, loss frommoving the market price by the act of makingthe bid or offer for a large block. This lattereffect ties market liquidity to price volatility, astransaction volumes lead to bigger pricemovements when markets are illiquid.We care about market liquidity because itaffects the returns for investors, such as thosesaving for retirement or college, and the coststo corporations, governments, and otherborrowers. Further, illiquid markets are morevolatile. At the extreme, volatility can helptrigger or exacerbate financial crises. Even theaverage level of volatility matters, as it isfactored into the interest rates demanded byinvestors and paid by borrowers.Market liquidity is a complicated issue in partbecause it is not clear what is happening tounderlying liquidity. Pretty much everyoneagrees that markets are less liquid than theywere in the run-up to the financial crisis, but itis not clear that this is a problem, since thoseliquidity levels were unsustainable andevaporated quickly under stress. The harderparts are to compare liquidity to an optimalsustainable level and to project liquidity intothe future. There is no agreement on either theoptimum level or the future course of marketliquidity.

Market Liquidity: A PrimerBrookings 2015at least four incidents in the last couple of yearsin which markets showed extreme volatility thatmay have been exaggerated by lower liquidity,such as the “taper tantrum” in the bondmarkets. It is difficult to know if these areisolated incidents or the tip of a dangerousiceberg. On the other hand, there are a numberof indicators, such as average bid/ask spread,that do not show signs of a less liquid market,so while there appears to have been an overalldecrease in liquidity, the evidence isambiguous.Despite the uncertainties, policymakers areright to take this issue seriously and to worryabout the risks. There appears to have been adecline in underlying liquidity in the marketsand this seems highly likely to worsen to someextent. There are numerous factors at work,including the evolution of the structure offinancial markets and the effects of unusualeconomic conditions, especially extremely loosemonetary policies and massive direct centralbank purchases of bonds. I also believe we haveovershot in our regulations in a way that willcramp market liquidity excessively, producingmore social costs than the benefits of greaterfinancial stability. To be clear, most of what hasbeen done is positive; it is a matter ofrecalibrating the details to reduce the socialcosts while keeping the core benefits.Unfortunately, this cost-benefit analysis iscomplex and still subjective at this point, in partbecause so much of what is happening toliquidity remains ambiguous and the largesteffects are likely to be in the future.Thus, the effects we have seen already are notdeeply worrisome on their own. The biggerissue is the probability that market liquidity willconsiderably worsen going forward. First, thevery loose monetary policies of central banksaround the world appear to have providedconsiderable support for market liquidity whilealso holding down price volatility. Whenmonetary policies eventually tighten, marketliquidity is likely to be more of a problem.Second, banks and large dealers are almostcertain to cut back further on their liquidityprovision and to raise their prices over the nextcouple of years. Many of the rules that increasetheir costs are only now being finalized or arebeing phased in over time. Further, dealersknow they will lose customers if they make onebig move, rather than spreading the pain overmultiple years, especially if their competitorstake smaller steps.Whatever the overall conclusions aboutregulation, it is clear that the cumulative effectsof a series of regulations have made it moredifficult and expensive for banks and largesecurities dealers to act as market makers.(These rules include the liquidity coverage ratio,the net stable funding ratio, the supplementaryleverage ratio, various changes to the capitalrules under the Basel capital accords, theVolcker Rule, and others.) Smaller dealers,hedge funds, and similar firms will pick up someof the slack as the large dealers pull back, butthere are real limitations on their ability to doso cost-effectively. The markets can also adapt,such as by moving to agency rather thanprincipal models and by embracing electronicmarkets, but, again, there are some seriouslimits on how far these moves can go.In sum, there are good reasons to worry aboutmarket liquidity and to believe thatpolicymakers may have unintentionallyovershot. However, the disaster scenarios thatsome suggest do not seem plausible, nor doesany regulatory overshoot mean that we have toredo financial reform in major ways. This is amatter of taking the issue seriously andrecalibrating a series of technical measures toreduce the damage to market liquidity withoutincreasing the risks to financial stability in anysignificant way. At this point, the key is to revisitthe various key regulations and to seriouslyThe net result should logically be decreasedliquidity and we have already seen much lowersecurities inventories held for market-makingpurposes by dealers along with some othersigns of lessened liquidity. There have also been2

Market Liquidity: A PrimerBrookings 2015One of the major effects of this marketstructure is that the great majority of bonds arebought and sold through dealers rather thantraded on exchanges, since there is not enoughtransaction volume to support exchange tradingof each of the individual bonds. These dealersdo not normally charge a commission, but arepaid through their expected profits frombidding for bonds at one price and offering tosell them at a higher one. The “bid/ask” spreadbetween the two quotes can be viewed asconsisting of two parts. A portion is theequivalent of a commission and is necessary tocover expenses and provide a reasonable profitfor helping customers to execute transactions.The second part compensates dealers for therisk that they will lose money on a transactionby buying too high or selling too low, as well ascovering the costs of holding a securitiesinventory to facilitate transactions, includingthe necessary levels of capital and liquidity toback their inventories. Therefore, one of thesignificant measures of market liquidity is theaverage bid/ask spread, since it represents animportant transaction cost.review the costs and benefits of the choicesthat were made about the details.What is market liquidity?In financial terms, the “liquidity” of any assetrefers to the combination of the degree of easewith which it can be sold (or bought) in a timelymanner and the level of costs associated withthat sale, either in terms of transactions costsor the acceptance of a lower price in order tofind a buyer in a reasonable time. Houses arerelatively illiquid assets, since they can takemonths to sell, there are quite substantialtransaction costs, and, depending on marketconditions, the seller may have to take a hit tomove the house in a reasonable time period. Onthe other hand, a US Treasury bond is highlyliquid. It can easily be sold within hours,transaction costs are minimal, and there aremany potential buyers who are willing to payroughly the bond’s theoretical market value.Recent concerns about “market liquidity” referto the functioning of markets for purelyfinancial assets, particularly bonds issued byboth governments and corporations, alsoknown as “fixed income” instruments since theypromise a fixed set of payments to the owner.Sometimes these discussions have broadenedout to reference derivatives based on thesebonds or the related markets in foreigncurrencies and commodities.Why do we care about it?Most of the credit provided to businesses andhouseholds in this country is ultimately suppliedthrough financial markets. (This is a contrastwith the rest of the world, where creditprimarily ends up on bank balance sheets). Thesuppliers of credit are insurers, pension funds,mutual funds, individual investors, and others.The ultimate sources of all these funds arehouseholds who rely on their returns fromthese securities to provide funding forretirement, educational expenses, and otherneeds. So the functioning of these markets hassignificant impacts on the economy as a whole.When liquidity declines, there are a series ofeffects:It is important to understand that the fixedincome market is quite different from the stockmarkets with which most people are morefamiliar. There is usually one type of commonstock for each public company (occasionallytwo); whereas firms and governments issuemany distinct bonds each. They differ inmaturity, interest rate, and other materialfeatures, so that they are not inter-changeable,even though they are affected by somecommon factors, particularly those related tothe creditworthiness of the issuer.3

Market Liquidity: A PrimerBrookings 2015conditions. At other times, volatility is affectedby changes in bid/ask spreads or otherelements of liquidity. When it is more expensiveor harder to trade, then fewer traders arewilling or able to step in when prices move outof line by modest amounts, allowing prices toswing more widely.Direct transaction costs for investors rise. Insome cases, external factors, such as increasesin regulatory requirements for trading, directlypush bid/ask spreads higher, which raisestransactions costs for investors, which is oneaspect of liquidity. Further indirect effects resultfrom cutting transaction volumes, which mayalso lengthen the time necessary to complete atransaction.Whatever the cause of increased volatility, itgenerally reduces the return for investors whoare buying or selling in any significant size, astheir initial purchases or sales will move themarket price further in the wrong direction forthem.In other cases, the causality runs in the otherdirection, and markets initially become lessliquid in some other way, such as through a risein the volatility of price movements. Bid/askspreads would then usually increase as well, forseveral reasons. Transaction volumes wouldtend to fall, so the dealer’s fixed costs would bespread over fewer transactions, raising the costper transaction. Further, risk premiums wouldrise as well to cover the higher price volatility,as may also be true of the capital and liquiditycharges, at least if illiquidity persists.There is greater potential for financial crises.Illiquidity in financial markets can help trigger orexacerbate a financial crisis by creating actualor paper losses at banks or other financialinstitutions. If a bank needs to raise cashquickly, perhaps to meet deposit outflows inthe event of a loss of confidence in thatinstitution, they will likely need to sellsecurities, especially if they have an excessivemismatch between the maturities of theirassets and liabilities. In illiquid markets, thiswould require “fire sales” in which the selleraccepts a significantly lower price in order toget cash quickly. In addition to the direct loss tothe troubled institution, which may threaten itssolvency, rapid declines in securities prices canaffect other institutions, either because theytoo need to sell or because they use “mark tomarket” accounting for their assets andtherefore paper losses directly affect theircapital positions.Whatever the derivation of the highertransaction costs, they flow through to lowerreturns for investors when they buy or sell theinstrument.Volatility of prices increases. The biggestfactors moving securities prices are those thataffect perceptions of their fundamental value,such as good or bad news about a firm’screditworthiness or an overall move in interestrates. However, the rapidity and extent of pricemovements is also influenced by marketliquidity. If there are many potential buyers andsellers and they can transact quickly, easily, andcheaply, then price movements tend to besmoother as news events are factored intoprices quickly based on the market consensusabout their significance. Similarly, if a marketparticipant wants to buy or sell a large block ofbonds, they can do so without greatly movingthe price.Bond prices fall as Investors demand higherliquidity risk premiums. When investors decidethe minimum interest rate they will accept on abond, they take account of multiple factors.First, they need a base return that compensatesthem for giving up the use of their funds untilthe maturity of the bond, often known as the“time value of money.” Second, they need to becompensated for credit risk, the possibility thatthey will not be repaid in full. Third, they maycharge an interest rate risk premium to reflectAs with transaction costs, sometimes volatilitydirectly changes, perhaps due to higheruncertainty about economic or monetary policy4

Market Liquidity: A PrimerBrookings 2015the potential for a decline in value if interestrates rise. Fourth, they will charge a “liquiditypremium” based on the degree of difficulty orcost they will encounter if they decide to selltheir investment early. (On top of these basicelements, there may be others, such as foreignexchange risk premiums, depending oncircumstances.)global financial crisis of 2007-9. However, thereis a great deal of controversy about the extentto which this has occurred and whether itrepresents a bad thing or a return to normalconditions after an unsustainably high degree ofliquidity in markets. (Fender at. al., 2015,provides a good overview of the changes sincethe crisis.)If markets become less liquid, then investorsover time should increase the liquidity riskpremium that they demand, raising their overallrequired interest rate. This would cause theprice of existing bonds to fall, since lower pricesare needed to raise the effective interest rateon the amount invested.The picture looks different depending on whichaspect of liquidity one focuses on and whichmarkets one considers.Level of dealer inventories. This is not a directmeasure of liquidity, but rather an indicator ofthe potential for dealers to provide liquidity.Large inventories make it easier for a dealer tosupply bonds if customers desire them. Theyalso tend to be an indicator of the willingnessand desire of dealers to make markets. With theexception of government bonds at that nationallevel, dealer inventories are down pretty muchacross the board in the last few years. Thedecline was very substantial in many types ofbonds, particularly corporate bonds. Sovereignbonds have shown little decline, but a large partof this is likely due to new regulatoryrequirements and other pressures to hold largevolumes of government bonds at the banks andmajor dealers. The chart below provides somedata on inventories from Fender, et. al., 2015.Capital raising becomes more expensive.Similarly, an increased liquidity risk premiummeans investors would demand higher interestrates when businesses and governments issuenew bonds. This would directly flow through asa cost to borrowers, including householdswhose borrowing is financed indirectly throughfinancial markets, such as is true for mostmortgages.Has market liquidity actuallydeclined?Almost everyone believes that market liquidityhas fallen overall since the period prior to the5

Market Liquidity: A PrimerBrookings 2015transactions in many market segments, whichmay indicate that investors have found theneed to break up their transactions due to theinability or high cost of moving large blocks in asingle transaction. However, there could beother factors leading to this reduction in size,including the rise of trading strategiesemploying frequent trades in smaller sizes to tryto profit from fleeting arbitrage opportunities.Bid/ask spreads. On the other hand, there hasbeen much less of a movement in bid/askspreads. On this basis, one would not presumethere were any concerns about a decline inmarket liquidity, except in certain marketsegments which were already less liquid. Thechart above, also from Fender, shows corporatebid/ask spreads over time.Volatility on “normal” days. Price movementshave been relatively calm for the most part,again not indicative of a current problem withliquidity.Time to completion of transactions. Althoughthere do not appear to be good measures,anecdotal evidence suggests some slowingdown of the disposition or acquisition of largepositions. This would be consistent with smalleraverage transaction sizes.Bouts of extreme volatility. On the other hand,there have been a few incidents, described inthe next section, in which price movementshave been extreme enough to trigger fears thatmarkets have indeed become less liquid.Overall, the decline in liquidity has been mostmarked in riskier market segments, asdemonstrated above in the charts from Fender,which showed little or no decline in the liquidityof government bond markets of developedAverage size of transactions. There hasgenerally been a decrease in the size of6

Market Liquidity: A PrimerBrookings 2015economies, but noticeable declines in liquidityof corporate bond markets. This results bothfrom factors specific to such markets (includingchanges in regulatory requirements that raisethe required level of capital for banks anddealers holding some instruments) and as areflection of an overall “flight to quality” bymany bond investors after the financial crisis, aswell as from various other factors.after the testimony. This may not seem large,but is quite a sharp move for a governmentbond market.October 15, 2014 Treasury market rally. Thisincident is so complex, and the causes sounclear, that it still does not have a singlenickname. A variety of factors led to a rise inTreasury market prices roughly equal to theentirety of the taper tantrum within about onehour, with prices subsequently gyrating stronglyover the remainder of the day. US authoritieswill soon conclude a study of this episode thatmay shed more light on the underlying causes.Similarly, the IMF’s Global Financial StabilityReview of October 2014 highlighted concernsabout market liquidity particularly in the highyield bond and emerging market bond areas.Swiss Franc revaluation. For several years, theSwiss central bank held down the value of theSwiss franc versus the euro, in order to mitigatea loss of competitive position by Swissexporters versus those in the eurozone. Thisrequired the Swiss to buy large sums of euros inexchange for francs. Eventually, the holdings ofeuros grew very large, as did the likelihood thatthe central bank would eventually have to takea loss on these holdings, in part due to theanticipated advent of Quantitative Easing by theEuropean Central Bank. As a result, in Januaryof 2015, the Swiss National Bank gave up andallowed the Swiss franc to rise, switching to apolicy of intervening sporadically if marketforces appeared to be excessive. This retreat bythe central bank caused the Swiss franc to rise30% in the first 13 minutes, with knock-oneffects in other foreign exchange markets. (Thefranc gave back some of these gains over thecourse of the day, but most of the initial impactremained.) Some observers believe that thespeed and extent of the initial price movementwould have been considerably less in moreliquid markets. This is hard to judge asdeveloped economies rarely undertake this kindof capping of foreign exchange rates anymoreand therefore it is difficult to compare withother instances where such a cap wasunexpectedly withdrawn.What do the recent bouts ofmarket volatility mean?There have been at least four occasions in thelast several years that may indicate a greatervulnerability of fixed income markets to periodsof excessive volatility as a result of regulatoryand other market changes 1. As noted below,however, each of the episodes were associatedwith major news events, often related tocentral bank activities, that make it difficult topin down what portion of the volatility was“excessive” and what was a reasonableresponse to fundamentals. These are:The “taper tantrum”. When then-ChairmanBernanke testified before Congress in May 2013that the Fed might “taper” off its purchases ofbonds in the markets more quickly than some inthe markets had expected, there was a quickmovement down in government bond prices,which carried over to most other categories ofbonds, which generally price on the basis of aninterest rate spread over the government bondrates. The 10-year Treasury bond saw its marketprice fall about 3% in the course of two days,with most of this occurring in the first few hours1Some analysts who have reviewed this paper in draftformat have nominated additional examples, such as sharpprice movements in Japan connected with theannouncement of changes in the investment strategy ofthe government pension plan.7

Market Liquidity: A PrimerBrookings 2015Volatility of European government bonds inearly 2015. Prices of government bonds in thecore of Europe fluctuated sharply in the firsthalf of 2015, with a cumulative move in German10-year government bond prices of 7-8% frompeak to trough. Within this overall trend, therewere fairly rapid moves on some days. Someascribe the sharpness of the moves tounderlying liquidity problems, although theargument is less strong than in the case of thetwo incidents involving US Treasury bonds.which can push liquidity in either directiondepending on the particular circumstances.The first compelling reason is that marketliquidity in the US was greater in the run-up tothe financial crisis than it had ever been, quitesubstantially so in many markets. In part, therewas clearly a self-reinforcing cycle of increasedliquidity leading to lower liquidity risk premiumsdemanded by dealers and investors, leading tostill more liquidity. A second major componentwas a belief in the “great moderation,” thatcentral banks had determined how tosubstantially reduce volatility in the economyand consequently in financial markets. Lowervolatility begets greater liquidity as dealers andinvestors become more willing to take positionswithout fear of excessive losses. Both of thesefactors have vanished or reversed, helping toexplain the lower liquidity levels today.Outside these markets, there was also the“flash crash” in the stock markets in May 2010and smaller versions since.At the end of the day, it is difficult to tell howmuch meaning to ascribe to these events. It iscertainly possible that they represent the tip ofthe iceberg and that once we return to morenormal economic and monetary conditions,these types of volatility events will be morefrequent and potentially much more painful.However, it is also dangerous to generalize toomuch from a few data points. One couldcertainly argue that at least some of the eventsmerely showed the market reacting sensiblyand swiftly to new economic news, such as thewithdrawal of the Swiss central bank as aprovider of massive artificial support to theeuro or the news about the Fed’s intentions forits future bond purchases.The second compelling reason is that thedealers who have dominated fixed incomemarket making are virtually all subject to awhole set of new regulations that make it moredifficult and more expensive to provide thatservice. It would be surprising if such a distinctdeterioration in their business position did notlead to a significant retrenchment and repricingof their liquidity provision to the markets. Asnoted earlier, dealer inventories in mostmarkets have come down quite markedly, inline with this expectation.It is probably best to view these incidents as redflags, and indicators of the degree to whichvolatility might become more normal, ratherthan drawing stronger conclusions from thislimited set of data points.There are quite a number of new regulationsthat have a significant impact on the cost ofdoing business as a market maker:Basel III capital accord. The Basel Committeeon Banking Supervision is the globalcoordinating body for bank regulators. Althoughit cannot directly bind national governments, itsrules are virtually always adopted, sometimeswith modifications. The Basel Committeepromulgated the third version of the BaselCapital Accords after the financial crisis andthey are well along the phase-in process today.The latest version significantly raises theWhy would we expect marketliquidity to be down?There are two broad and compelling reasons toexpect market liquidity to have declined,especially for securities that were already lessliquid. In addition, there are a number of otherfactors at work that are of lesser significance or8

Market Liquidity: A PrimerBrookings 2015intended to ensure that banks and majordealers have high levels of liquid assets to meetpotential demands for funds in a crisis and thattheir overall business models do not have anexcessively large mismatch between thematurity of their assets and their liabilities. (SeeElliott, 2014, for a primer on bank liquidityrequirements, which also apply to the majordealers.) The Liquidity Coverage Ratio (LCR) is astylized stress test to ensure that a bank has theability to handle a 30-day liquidity crisis in themarkets. Under this test, assets which arelonger-term or less liquid effectively need to befunded by longer-term liabilities, which tend tobe more expensive. This raises the cost ofholding inventories of most bonds.amount of capital required by banks and majorsecurities dealers, which makes it moreexpensive for them to do business. (See Elliott,2010 for a primer on bank capital.)Basel 2.5. The capital required for assets held ina bank’s “trading book” was considerably lowerunder Basel II than was the case for otherassets, such as securities that were intended tobe held to maturity. After the financial crisis,there was such a strong consensus that thesecapital levels needed to be raised sharply, thatnew rules were put in place to modify Basel II inthis area even before the Basel III accord wasagreed. (Hence, the nickname of Basel 2.5.)Trading book assets now require multiples ofthe capital previously mandated, representingone of the sharpest percentage changes incapital requirements. In addition, the BaselCommittee is currently conducting a review ofthese requirements and there is an expectationof still further increases.Net stable funding ratio. The second liquidityrelated requirement in Basel III is a ruleintended to ensure that banks and majordealers do not have an excessive mismatchbetween the maturity of their liabilities and thatof their funding. This produces a similar effectto the LCR, by raising the cost of funding forlonger-term instruments, such as most bonds.Leverage ratio. The Basel Committee alsoconcluded that its core approach, which usesrisk weightings so that more capital is requiredfor riskier assets and less for safer ones, was toosubject to gaming or error when used on astand-alone basis. Therefore, a “leverage ratio”has been adopted as well which, in essence,requires the same level of capital for all assets,regardless of risk. Banks must meet the higherof the capital levels required by the riskweighted approach and that calculated by theleverage ratio. In the US, regulators wentfurther and established a “SupplementaryLeverage Ratio” (SLR) for the largest banks thatis higher still. The SLR has particular impact ontrading, since most of the instruments that aretraded, or are used to hedge trading positions,involve securities with very low credit risk.These have correspondingly low capitalrequirements under the risk-based rules, but donot receive any benefit under the leverageratio.Single counterparty credit limits. The DoddFrank Act required that the rules be tightenedon the amount of credit exposure that thelargest banks and their affiliates could take toany one counterparty. Bonds in dealerinventories count against this limit as do manyof the instruments used by dealers to hedgetheir risk of holding those inventories. Thetighter requirements mean that the largestbanks have to ration their credit exposuresmore than they did, which adds an opportunitycost when dealing activity uses up some of thisroom under the exposure limits.The Volcker Rule. Banks and their affiliates arenow prohibited from engaging in “proprietarytrading”. As I, and others, have written aboutextensively, there is no clear meaning to theterm and therefore dealers have a strongincentive to cut back on some of their marketmaking that might be misinterpreted asproprietary trading. In particular, dealerLiquidity coverage ratio. Basel III also includestwo completely new requirements that are9

Market Liquidity: A PrimerBrookings 2015inventories that rise too much or too qui

securities dealers to act as market makers. (These rules include the liquidity coverage ratio, the net stable funding ratio, the sup

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During the liquidity crisis, observed funding and market liquidity mutually reinforce one another. A small negative shock to the economy might be amplified through this mechanism and result in a sudden drying-up of the liquidity. During the financial crisis, policy interventions are expected to alleviate the liquidity crunch.

Principi basilari comuni alle guidelines, principles in materia di Liquidity Risk Management (LRM): definizione rischio/rischi di liquidità (funding, market, contingency liquidity risk); determinazione di un livello di liquidity risk appetite e liquidity risk tolerance; presenza di una, policy per la gestione della liquidità (Liquidity policy, Funding Liquidity policy, Collateral .

Use of capital requirements creates regulatory arbitrage 3. The degree to which regulations act as . Assumes that there are no systemic liquidity needs “A Theory of Bank Liquidity Requirements” . Liquidity only part of the new regulatory toolkit Are liquidity and capital regulations complements? Substitutes? Liquidity regulation is .

liquidity are still unhedged against market liquidity risk. Therefore, asset pricing models with perfect liquid markets implys fallacious hedges. In models that do not account for liquidity and liquidity risk, all these components would be summarised as model risk leading to higher P&L-volatility.

Our analysis is careful in distinguishing the funding liquidity and market liquidity chan-nels. Research has demonstrated the role of liquidity risk in international investments and has shown that liquidity risk as a priced local factor may lead to valuation differentials (see for exampleBekaert, Harvey, and Lundblad(2007) andLee(2011)).