Liquidity Risk, Market Power and Reserve Accumulation Grégory Claeys Chara Papioti Andreas Tryphonides October 2020

Barcelona GSE Working Paper Series Working Paper nº 1206

LIQUIDITY RISK, MARKET POWER AND RESERVE ACCUMULATION

GRÉGORY CLAEYS˚, CHARA PAPIOTI˚˚, AND ANDREAS TRYPHONIDES˚˚˚

Abstract. Using a multi-unit uniform auction model, we combine bidding data from open market operations as well as macroeconomic information to recover the latent distribution of liquidity risk across banks and how it is affected by policy in Chile. We find that unanticipated shocks to foreign reserve accumulation and interest rates have significant effects on aggregate beliefs about a liquidity shock in the near future, while news about reserve accumulation are not effective. These results suggest the presence of a novel informational channel for macroeconomic policy, while we demonstrate that accounting for market power is important for uncovering these effects.

Keywords: Multi-unit auction, Liquidity risk, Reserve Accumulation, Signalling JEL codes: C57, D44, E50, G20 Date: Oct 2020 ˚Bruegel ˚˚Universitat Autonoma de Barcelona and Barcelona GSE ˚˚˚ University of Cyprus

Acknowledgements: We are grateful to Markus Kirchner, Rodrigo Cifuentes and Juan- Francisco Martinez from the Central Bank of Chile for their help and providing access to Open Market Operations data. We also thank seminar participants at Universitat Autonoma de Barcelona, University of Cyprus, Humboldt University of Berlin, Universidad Torcuato Di Tella, IECON Uruguay, ESADE, as well as conference participants at ASSET 2019 and CESC 2019 for their useful comments and input. Papioti acknowledges ﬁnancial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015-0563). 1. Introduction

During and after the 2008 - 2009 financial crisis, major Central Banks engaged in quantitative easing (QE) policies of an unprecedented scale. Between mid 2009 and mid 2013, the U.S. Federal Reserve purchased more than $1000 billion of medium to long term US Treasury securities. At the same time, due to concerns about financial stability and exchange rate pressures as well as the QE induced abundance of liquidity, emerging market countries accumulated foreign currency reserves, a practice that was also prevalent before the crisis as many of these countries transitioned from fixed to floating exchange rates. Indicatively, Mexico accumulated $12.2 billion during 1996-2001, as well as $9.1 billion in 2010-2011. Brazil’s international reserves increased from under $50 billion to over $200 billion during 2003-2008, while Chile increased its reserves to GDP ratio by 4.4% in 2008, reaching 17% in 2011 (Chamon, Hofman, Magud, and Werner, 2019b,c,a). The mere size of these interventions indicates that financial stability has proven to be of prime importance to policy makers. Managing liquidity risk - which is defined as the probability that an institution suffers a large enough shock such that it is unable to meet its liabilities in a timely manner - is one crucial aspect of ensuring stability. Nevertheless, while policy makers can possess information on the aggregate component of this risk and react to developments in the economy that are related to it, the idiosyncratic risk component and its distribution in the financial sector are highly latent as financial institutions have private information about their own exposure in the near future.1 Moreover, they also condition on public information to form beliefs about the distribution of this risk in their industry, which opens up the possibility of asymmetric

1This actually explains the emergence of several measures of liquidity risk after the financial crisis. Examples of measures which are based on balance-sheet data are the Liquidity Coverage Ratio and the Net Stable Funding Ratio introduced by Basel III (BCBS(2013)). Other measures include the LIBOR-OIS spread or other composite financial market liquidity indicators (BoE, 2007). Balance sheet based measures may heavily depend on stress scenarios and can be sensitive to expert categorizations of assets and liabilities while they may not be available at high frequency. On the other hand high frequency market based measures may not disentangle liquidity risk from other risks e.g. Cassola, Hortaçsu, and Kastl(2013) show that CDS data do not seem to explain much of the willingness to pay for liquidity extracted from auctions of the European Central Bank. 2 information between policy makers and financial market participants. From an econometric perspective, the consequence is that learning about this distribution inevitably requires combining economic theory and data at the micro and macro level. This paper develops and applies a structured approach to construct an estimate of the distribution of liquidity risk and to also investigate whether Cen- tral Bank policies such as reserve accumulation or monetary policy do convey information about the aggregate component of this distribution to financial market participants. The latter is important as financial market perceptions about aggregate risk can potentially influence the decision of banks to ex- pand lending and invest in riskier projects.2 If policy makers can influence these perceptions, then this provides an informational channel through which policy can affect the economy. We analyze auction data from open market operations for domestic bonds issued by the Chilean Central Bank during 2002-2012. Since financial institutions have little incentives to reveal their individual risk exposure, we use their bids in these auctions in order to uncover it by exploiting the structure imposed by a theoretical model that closely reflects their features. A bank participating in such an auction will adjust both the quantity of bonds it wishes to buy as well as the interest rate it wishes to receive, according to the state of liquidity it expects to have until the bonds mature. The case of Chile is the natural choice for studying liquidity risk and for providing empirical evidence on the effects of policy. It is one of the aforementioned countries that has historically used reserve accumulation, whether it is for precautionary reasons or otherwise (Cabezas and Gregorio, 2019). In addition, Chile since the 1990’s has adopted inflation targeting, and a standard practice to achieve the desired interest rate is to issue domestic bonds of

2While providing an answer to this question is beyond the scope of the paper, Pflueger, Siri- wardane, and Sunderam(2020) show that in a different context, perceived risk as measured by the price of volatile stocks can affect the macroeconomy by changing the cost of capital. In our context, perceived liquidity risk by banks may well be affected by the quality of the projects they finance, and whether risky firms will eventually repay or not. Of course, our measure encompasses other interpretations as well, such as a run by creditors. 3 different maturities through open market operations. Foreign reserve accumulation and open market operations can be tightly connected in practice as the latter withdraw liquidity from the market to achieve the interest rate target; on the other hand, buying foreign currency is part of the reason why there is excess liquidity in the system. The paper makes several contributions. We build on the empirical literature on the intersection of finance and industrial organization (Kastl, 2017) by showing that identifying liquidity risk requires taking into account both market microstructure and the degree of insurance against this risk. While the non-competitive structure of financial markets is something that has been shown to matter in different settings e.g. in the primary market for liquidity provision by Central Banks (Cassola, Hortaçsu, and Kastl, 2013), we study its implications for learning about liquidity risk from bidding behavior of banks in liquidity draining auctions. We show that even if we control for bid shading that masks a bank’s true valuation for the bond (or equivalently, its cost of giving up liquidity), the latter depends on the probability that the lender of last resort can provide full insurance to the financial sector, which in turn depends on macroeconomic conditions and policy. In this sense, the paper makes a methodological contribution by combining both individual level auction data across time together with macroeconomic time series to identify the latent distribution of liquidity risk. The framework also allows us to investigate the effects of policy on perceived aggregate risk. The paper’s empirical results contribute to the strand of the empirical macroeconomic literature that studies asymmetric information between policy makers and the private sector. More particularly, we find evidence of a novel informational channel of foreign reserve policy; an increase in foreign reserves signals to private financial institutions the expectation of an aggregate liquidity shock in the next period. In addition, we do not find evidence that news about future interventions have significant effects. This could be explained by the costly na- ture of these interventions, as policy actions are more credible than promises. Furthermore, we also find that the financial sector revises its beliefs about aggregate risk downwards after an increase in the nominal rate. This suggests that the private sector understands that a higher interest rate will alleviate 4 some of the pressure on international capital flight, lowering the likelihood of a negative liquidity shock in the next period. Furthermore, the market we consider is oligopsonistic as the number of financial institutions participating in it is small and hence they can affect the market price on the margin. We show that this has implications for existing measures of liquidity risk that are based on observed market returns that do not account for market imperfections, as variation in the interest rates does not only reflect the cost of giving up liquidity, but also the strategic behaviour of banks with market power. We also show that had we ignored the oligopsonistic structure of the primary market of bonds we would not be able to identify the aforementioned signalling effects. The structure imposed by the auction model is therefore important for identification and for understanding the effects of open market operations which are an integral part of the implementation of conventional and unconventional monetary policy.

1.1. Related Literature. The market under consideration is one where a Central Bank sells bonds through an auction. The literature on multi-unit auctions was initiated by Wilson(1979), who considered a setting where buyers bid for shares of a divisible good. Bids are continuously differentiable demand schedules, specifying the highest quantity demanded at any price level, while winning bidders pay the market clearing price. Subsequent work modified this framework to better fit realistic features of multi unit auctions in markets for government securities, electricity distribution, electromagnetic spectrum and construction procurement, among others. For example Kastl(2012) characterizes the equilibria in multi unit auctions where bids are k-step functions, i.e bidders can submit up to k price-quantity combinations in order to buy shares of a divisible good. Since the implied bidding function is not differentiable, instead of using calculus of variations techniques like in Wilson(1979), Kastl(2012) uses local optimality conditions to characterise the equilibrium quantity bid at each step for two different price mechanisms. In Kastl(2011), an econometric model is also proposed in order to estimate the marginal val- uations of bidders at each step; Cassola, Hortaçsu, and Kastl(2013) uses the 5 same baseline model in order to study European banks demand for short-term liquidity during the summer 2007 subprime market crisis. Our paper is related to Kastl(2011, 2012); Cassola, Hortaçsu, and Kastl (2013), in the sense that the auction format we wish to replicate is a uniform multi-unit auction where banks submit interest rate-quantity tuples instead of differentiable functions. However, despite the discreteness of quantity bids, since in our dataset we do not observe banks placing more than one bid, we turn to the first order condition with respect to the price, where differentia- bility holds, in order to derive simpler and intuitive results about optimal behaviour. Our model is one where banks are sellers of liquidity and thus the cost of giving up liquidity is one of the main drivers of their bidding behaviour. We consider two theoretical extremes: full insurance against a potential liquidity shock, and no insurance against it. By characterizing the cost of illiquidity in the two extreme cases of insurance, we are able to identify idiosyncratic liquidity risk conditional on state of insurance using a resampling algorithm similar to the one proposed by Kastl(2011); Hortaçsu and McAdams(2010). The paper also relates to the strand of the literature that studies the importance of market power in these auctions. Cassola, Hortaçsu, and Kastl(2013) show that strategic behavior matters for the identification of the cost of funding. Our counterfactual exercises illustrate that market power matters both for identifying liquidity risk as well as the effects of different shocks. On the inefficiencies due to market power in analogous settings, see e.g. Ausubel, Cramton, Pycia, Rostek, and Weretka(2014) and Vives(2011), as well as Ausubel(2004),Ausubel and Cramton(2004) and Kremer and Nyborg(2004) for mechanisms to address the issue. The macroeconomic literature we contribute to relates to the informational effects of reserve accumulation and monetary policy. Starting with Calvo(2006) and the credibility of the lender of last resort policy, the existing literature has focused on the precautionary motives of foreign reserve accumulation (e.g Ob- stfeld, Shambaugh, and Taylor(2010); Bocola and Lorenzoni(2017); Céspedes (2019) ) and counteracting externalities from excessive foreign borrowing (e.g Arce, Bengui, and Bianchi(2019)). To our knowledge, there is not much empirical evidence on the short run effects, if any, of reserve accumulation to 6 beliefs about aggregate risk. Mussa(1981) focused on the signalling effects of foreign exchange interventions to market participants about future monetary policy while Vitale(2003) argues that since foreign reserve accumulation is a costly action, this can reduce the uncertainty about future policy which can stabilise the economy. Dominguez and Frankel(1993) stress that public interventions, whether sterilized or not, can affect the exchange rate through market expectations. Regarding the informational effects of monetary policy, Romer and Romer(2000) show that monetary-policy actions provide signals of the Federal Reserve’s information about inflation, while Nakamura and Steins- son(2018) find that unexpected monetary tightening raises expected output growth, which provides further evidence on the existence of an informational channel of monetary policy actions. Melosi(2016) stresses the signalling effect of monetary policy by conveying the central bank’s view about the macroeconomy to price setting firms. The rest of the paper is organized as follows: Section 2 provides some institutional details regarding open market operations while Section 3 describes the auction dataset. Section 4 provides the auction model while Section 5 characterizes optimal behavior. Sections 6 and 7 discuss the estimation methodology, the aggregate model and the resulting estimates of the distribution of liquidity risk. Section 8 discusses the identified effects and the counterfactual exercise.

2. Institutional Details

Like other Central Banks, the Central Bank of Chile (BCCh) can inject money in the economy via repo operations. However this is not their main tool for liquidity management. Because of large capital inflows in the past and large foreign exchange reserves in the 1990s (before the adoption of a fully flexi- ble foreign exchange rate regime in 1999), the BCCh controls the quantity of money in the Chilean market mostly by draining liquidity from the financial sector. Therefore, the BCCh conducts open market operations by issuing promissory notes and bonds at various maturities through auctions, performed 7 weekly (or sometimes bi-weekly).3 The BCCh determines the maximum number of participants that can be different in every auction: potential market participants include twenty-three banks, four pension fund administrators, the unemployment fund administrator, three insurance companies and four stock brokers. Among the bonds sold by the Central Bank, PDBC (the short term notes) are the most heavily used to manage and regulate the quantity of money in circulation in the financial system within a given month or from one month to the next. The auction schedule for these notes is announced on a monthly basis. The program planning takes into account the liquidity demand expectations, maturing debt from previous periods, strategies for complying with reserve requirements and seasonal factors affecting liquidity in that period.4 The PDBC auctions are mainly uniform multi-unit auctions where banks bid for a discrete amount of bonds they wish to buy along with the minimum interest rate they would accept. The Central bank then decides the cutoff interest rate awarded to all winners of bonds by ranking interest rate bids from smallest to highest and assigning bonds up to capacity. During the period relevant to our paper, the BCCh retained the option to award a different amount than scheduled, which in the case of bonds is ˘20% of the amount auctioned, or to declare unilaterally the auction as deserted because the rates asked by the banks are too high.

3. Bidding Data

The paper focuses on the 30 day discount promissory notes (PDBC30) for several reasons. First, in the absence of liquidity risk, the shorter maturity of these bonds makes them a better substitute for cash or reserves at the central bank. Second, we can safely assume that in this market there is no inﬂation premium asked, as would be the case with longer maturity bonds. Third, short term bonds are more likely to be kept to maturity and not resold in secondary markets.

3Bonds sold include short-term notes due in 30 to 360 days, nominal bonds with maturities of 2, 5 and 10 years and inflation-indexed bonds with maturities of 5 and 10 years. 4For more details, please refer to the liquidity management reports regularly published by the Central Bank e.g. BCCh(2012) available here. 8 Moreover, Central Bank short term debt is considered as a safe asset as it is much less likely to carry a default premium.5 Banks’ liquidity risk should therefore be the main driver of the positive risk premium asked in these auctions, while we will also take into account the possibility that due to a severe aggregate liquidity shock, the Central Bank may not be able to act as a lender of last resort due to public insolvency or fear of hyperinflation. Finally, a more practical reason is that 30-day PDBC auctions are the most frequent open market operations of the BCCh, with the largest number of participants. Our dataset contains all bidding information for the open market operations of the BCCh from September 2002 to August 2012. The information includes the total volume of bonds allotted by the Central Bank in each auction, the marginal (or cut-off) rate, the bidders’ identities and the rates and quantities of bonds asked by each bidder.6

# of Auctions: 934 1st Quartile 2nd Quartile 3rd Quartile # Bids per auction 5 6 8 Interest Rate Bids 2.1199 3.5155 5.0328 Quantity Bids (Billion) 6.2 16 40 Total Vol. Demanded (Billion) 95 140 183 C.B. Supply (Billion) 90 200 300 Total Vol. Assigned (Billion) 60 90 200 Equilibrium Rate 1.8270 3.7836 5.12 # Bids per Month 38.75 48.5 59.25

Due to the high frequency of these auctions and the relatively thin market for bonds, the number of participants in each auction is small. We will deal with this by pooling data of diﬀerent auctions as is typically done in the literature

5As noted in Reis(2015), the Central Bank is just one of many government agencies, and thus solvency of the Central Bank cannot be separated from the solvency of the government. In this sense, Central Bank issued securities can be comparable to government issued debt that also carries low default risk. Demand for the latter is high as it is valued for its safety and liquidity, at least in the case of the US Treasury Debt (Krishnamurthy and Vissing- Jorgensen, 2012), while similar reasoning could apply for government debt issued by other countries (Du, Im, and Schreger, 2018). 6This dataset is not publicly available and usually only the information on the total volume allotted and marginal rates are available on the BCCh web site. 9 (see e.g. Hortaçsu and Kastl(2012a)). More information on sample selection can be found in the Appendix.

4. The Model

We consider a primary market for liquidity, where ﬁnancial institutions, mainly banks, give up liquidity by buying bonds from the Central Bank through an auction. Below we outline the model environment and the underlying assumptions about primitives, while we later turn to analyzing optimal bidding behaviour and equilibrium outcomes.

4.1. Assumptions.

A1: Market Participants, Endowments and Ownership. The market is populated by N risk neutral ﬁnancial institutions, the buyers, and a Central Bank (CB), the seller. N is common knowledge, and N ď N¯, where N¯ is the maximum number of potential market participants, exoge- nously ﬁxed by institutional constraints. The buyers are of two types: banks and non banks, with NB and NO members respectively, also common knowledge, where NB ` NO “ N.

Each buyer i P r1,Ns is endowed with mi monetary units of net external funds; in the case of banks, these funds may be borrowed from creditors and are net of any other investments, excluding the bonds in the auction under consideration. We deﬁne a bank to be illiquid in the next period if its immediate net liabilities are larger than or equal to mi. The CB issues Q P r0, Q¯s bonds with nominal value equal to one, which are sold through an auction.

A2: Information. Buyers’ Signals: Before the auction takes place, each buyer privately learns si P r0, 1s, to be interpreted as their individual probability of becoming illiquid in the next period conditional on all available common information (Ω) up to the beginning of the auction, including any policy actions taken by the CB. 10 It is common knowledge that signals tsiuiPr1,Ns are independent conditional on public information.7 More particularly,

si „ FGp. | Ωq iid where FGp. | Ωq is the perceived distribution of risk among buyers of type G “ tbanks, non banksu, which is atomless on [0,1] with a positive and continuous density fGp. | Ωq, almost everywhere. Bond supply by the CB: The density of Q conditional on Ω, is also common knowledge.8

A3: Auction Format and Market Structure. The CB sells bonds through a uniform multi-unit auction. After the CB announces the amount of bonds it wishes to sell in the auction, buyers submit a two dimensional bid specifying the share qi of total volume Q of bonds that they wish to buy, which is restricted to take discrete values, along with bi, their corresponding minimum acceptable interest rate.

Uniform Interest Rate Auction: Given the bids pbi, qiqiPr1,Ns, the CB ranks the interest rate bids from the lowest to the highest: b1 ď b2 ď ... ď bN . Denote the corresponding shares asked by q1, q2, ..., qN . The buyer with b1 is assigned q1, then the buyer with b2 is assigned q2, and so on and so forth, until Q bonds have been assigned. All winning buyers receive the same interest rate once the bonds mature -to be referred to as the equilibrium cutoﬀ bid Bc-, which is deﬁned as the highest winning interest rate bid.

Since banks can potentially diﬀer in two dimensions, mi and si, rationing is (ex-ante) possible. Thus, a buyer having placed the equilibrium cutoﬀ interest rate receives a share of

c qi qi ” 1 ´ qj qi ` ql l:bl“bi ˜ j:b ăb ¸ ÿj i ř 7 By conditional independence, we mean that Prpsi ď ui, sj ď uj | Ωq “ Prpsi ď ui | Ωq Prpsj ď uj | Ωq. 8In the Appendix, we plot the planned versus the realized supply of bonds in the BCCh auctions. After the Lehmann Brothers episode, supply uncertainty measured as the difference between the planned and realized supply, increased. 11 bonds. In the case of ties, the cutoff bidders proportionally share the residual c demand they are facing, while when buyer i is the only cutoff bidder, qi is equal to the residual supply of bonds it faces, qc 1 q . The quantity i “ ´ j:bj ăbi j received by buyer i in equilibrium is therefore summarized as follows: ř

c 0 if bi ą B ˚ c c qi pbi, qiq “ $qi if bi “ B c &’qi if bi ă B ’ A4: Action Set. %

Each buyer i submits a pair of an interest rate bid (bi) and a quantity bid (qi)

from the action set Ai.

Ai “ tpnot bidq Y tpbi, qiq : bi ě 0, qi P r0, 1suu

Whereas in Kastl(2011), Kastl(2012), Hortaçsu and Kastl(2012b), and Cas- sola, Hortaçsu, and Kastl(2013), banks are allowed to submit several steps of an implied demand function for bonds, we assume that buyers are allowed to submit only one interest rate-quantity bid in the auction. Although allowing for more steps is entirely reasonable within some contexts, we have not observed this kind of behaviour in our dataset.

A5: Maturity Structure for endowments and Costs. Maturity Structure: If the liquidity shock does not materialize for bank i, the bank is sure to be able to repay its creditors once the bonds mature, as its liabilities in mi mature after the bonds bought in the auction do. If however the bank suffers the liquidity shock, its liabilities mature before the bonds do; in this case, a bank that has successfully bid in the auction will not be able to repay its creditors, at least partly. Costs: There are no transaction costs associated with participating in the auction. On the contrary, there is an opportunity cost: if a financial institution does not participate in the auction, it can deposit its endowment mi in return for an interest rate ι. Additionally, there is a cost associated to the event that a buyer is hit by the liquidity shock and cannot repay its creditors with mi in its totality or 12 in part. We consider two cases: one where there is no insurance against the liquidity shock, and one when there is full insurance. In the first case, we assume limited liability of the buyer, that makes a loss equal to the amount of money not repaid in case it becomes illiquid.9 Therefore, the expected cost of having bought an amount of bonds say qiQ in the case in which a liquidity shock materializes is: siqiQ. In the second case, the CB acts as Lender of Last Resort, lending money to institutions in need at an interest rate d.

A6: Timing.

We divide the game in four stages: In stage 0, each buyer learns mi and si and then bids in the auction in stage 1. In stage 2 the liquidity shock materializes (or not) while in stage 3 the bonds mature. Therefore, an institution that had a liquidity shock might not be able to meet its obligations to its creditors. The timing can be seen in the graph below.

5. Equilibrium Bidding Strategies

In our setting, a bidder can win some, all or none of the bonds they bid for, depending on the relative ranking of their interest rate bid in comparison to the rest. When the bidder’s interest rate is smaller than the cutoff bid, the bidder is assigned the total quantity of bonds that it asked for; when the bidder’s interest rate is the cutoff bid, the bidder is assigned the residual supply of bonds, or a proportion of it, when there are more than one cutoff bidders. Finally, if the bidder’s interest rate is higher than the cutoff bid, the bidder is not assigned any bonds, and thus has a payoff from participating in the auction equal to zero.

9See for example Blum(2002), Jensen and Meckling(1976), Sinn(2001). 13 Therefore, each bidder i chooses pbi, qiq to maximize its expected payoﬀs pΠiq subject to the constraint that total investment should be less than or equal to

its endowment mi:

max EpΠi | siq bi,qi

s.t. qiQ ď mi

We ﬁrst consider the case when there is no insurance against the liquidity shock and then present the case when the Central Bank can act as a Lender of Last Resort when the liquidity shock materializes. Since bidders are risk neutral, the utility function representing their preferences over wealth is linear.

5.1. No insurance against the shock. When there is no insurance against the liquidity shock, we assume that bidders have limited liability: should the financial institution become illiquid, then it can unilaterally declare default on its debt obligations towards its creditors. It may lose ownership of the government bonds, which are transferred to its creditors as part of the settlement. In this case, the institution’s loss is equal to the nominal value of the bonds, ‹ qi Q, which is exactly the cost of illiquidity. When the shock does not materialize, the bidder’s payoff is equal to the value of the bonds it was allotted plus interest. Expected payoffs are therefore given by:

c c c c 1 ` bi ´siEpqi | bi “ B , siq ` p1 ´ siqEpqi | bi “ B , siq ´ 1 1 ` ι „ ˆ ˙ where we use that the total quantity of bonds supplied by the Central Bank c c is exogenous, i.e. EpQ | si,B ą biq “ EpQ | si,B “ biq “ EQ. Whether the liquidity shock materializes or not, a winning bidder incurs a cost equal to the value of the bonds bought; additionaly, when the shock does 14 not materialize, the bidder receives a return equal to the value of the bonds bought plus interest, where the opportunity cost is taken into consideration. Bidder i is the cutoﬀ bidder under two conditions: a) the total demand of bonds of all bidders with a lower interest rate is smaller than supply, and b) its demand for bonds (or the total demand for bonds of all bidders with an

interest rate equal to bi, if i ties in interest rate with others) added to the total demand of bidders with a lower interest rate surpasses the CB’s supply. Similarly, i will not be the cutoﬀ bidder but will win the whole amount of bonds demanded under one condition: the residual supply from bidders with a lower or equal interest rate than i is greater than i’s demand (or the total

demand of everyone with a bid equal to bi, in case of ties). Thus, the above

probabilities depend on both bi and qi, as

c PpB ą bi | siq “ P qj ` qi ă 1 | si ˜j:b ďb ¸ ÿj i and

c PpB “ bi | siq “ P ql ă 1, qj ` qi ě 1 | si ¨ ˛ l:b ăb j:b ďb ÿl i ÿj i ˝ ‚ Since the choice of quantity is restricted to lie in a ﬁnite set, the derivation of the optimality condition with respect to quantity should use local perturbation arguments like in Kastl(2012). We choose to focus on the optimality condition with respect to the interest rate, that is a continuously diﬀerentiable variable, and has an intuitive interpretation.

The ﬁrst order condition with respect to bi is: c c c BPpB ą bi | siq 1 ` EpB | B ą bi, siq 0 “ qi p1 ´ siqr ´ 1s ´ si Bb 1 ` ι i „  c c c p1 ´ siq BEpB | si,B ą biqq `PpB ą bi | siq qi p1 ` ιq Bbi c BPpbi “ B | siq c c 1 ` bi ` Epqi | si, bi “ B qpp1 ´ siqr ´ 1s ´ siq Bbi 1 ` ι c `Ppbi “ B | siq ˆ c c p1 ´ siq c c BEpqi | si, bi “ B q 1 ` bi ˆ Epqi | si, bi “ B q ` pp1 ´ siqr ´ 1s ´ siq 1 ` ι Bb 1 ` ι „ i  15 In order to interpret this condition, we decompose conditional expectations follows:10

“ Encp.q ` Ecp.q ` Enbp.q

c c ´1 BPpB ąbiq BEcpqi q Deﬁning ci :“ qi ` 1 and after a few manipulations, we can Bbi Bbi rewrite the ﬁrst order condition´ ¯ as follows:

si`ι c ´1 c bi ´ ci 1 BE pq q BE pB q (1) 1´si “ ´ c i E pq q ` q nc b b Bb c i i Bb i i ˆ i ˙ ˆ i ˙ Equation (1) characterizes the behaviour of bidders in a Bayes Nash equilibrium. There are a few observations worth making. First, ci is the ratio of the marginal cost of participating in the auction, and the marginal cost in the cut- oﬀ case. Thus, the left hand side (LHS) of (1) can be thought of as a markup asked by Bank i as a percentage of the interest rate asked, within a stochastic environment. Correspondingly, the RHS represents the market power of bank i.

Second, recall that when the bank is the cutoﬀ bidder, its bid bi is equal to the equilibrium bid, while the quantity it receives is the residual supply of bonds

at bi. Conversely, when the bank is not the cutoff bidder it receives the full quantity it bid for, and an interest rate that is higher than its own bid. The c BEncpB q term Ecpqiq`qi is the price effect on the total revenue of an infinitesimal Bbi change in the interest rate asked; the cutoff and non cutoff case effects are its c BEcpqi q first and second components respectively. The term bi is the quantity Bbi effect on the total revenue of an infinitesimal change of the interest rate asked in the cutoff case. Since the quantity effect in the non cutoff case is zero, the term is equal to the quantity effect overall. Consequently, the RHS of (1) can

10For what follows, further derivations are provided in the Appendix. 16 be thought of the interest rate elasticity of residual supply, within our setting which essentially is one of an oligopsony. In the absence of market power, or as N Ñ 8, the right hand side (RHS) of

(1) becomes zero, while limNÑ8 ci “ 1. Therefore, (1) implies that

si ` ι bi “ 1 ´ si

Note that when furthermore si “ 0, the bidder’s interest rate becomes equal

to ι, the interest rate that the bidder would have received by depositing qi instead. The above has important implications regarding the identification of risk. With perfect competition, high interest rates are directly translated as high risk. When banks have market power, this is not the case anymore, as the bid will also contain a mark-up, which also depends on risk. We will revisit this issue when we deal with policy evaluation. An additional key observation is related to the risk premium bank i asks, which is defined as the difference between the interest rate asked under uncertainty

and the interest rate asked when there is no liquidity risk, bipsiq ´ bip0q. Since bip0q P r0, 8q, then as the probability of having the liquidity shock goes to 1, the bank asks for an inﬁnite risk premium: limsiÑ1 bipsiq “ 8.

5.2. Full Insurance - Lender of Last Resort. Emergency liquidity assistance can help a bank by providing full insurance in the case in which there is a liquidity shock. A bank in need of liquidity can borrow from the LLR and repay its creditors, obviating the need to default on its debt. This is of course true under the assumption that the central bank (the government) can credibly commit to provide such assistance. In our setting, under the presence of an LLR, a bidder that suffered a liquidity shock in period 2 can still receive the full amount of profits when bonds mature in period 3, but will have to repay the LLR at some interest rate, which we set to be equal to the monetary policy rate. This leads to a slightly different payoff structure and optimality conditions. 17 Moreover, since the Central Bank sets aggregate supply depending on its expectations of aggregate conditions conditional on Ω, Q does not directly depend on future interest rates, and thus Q and d are conditionally independent: EpdQ|Ωq “ EpQ|ΩqEpd|Ωq. Bidder i’s expected payoff conditional on current information becomes as follows:11

c EpΠi | siq “ PpB ą bi | siqEpQq ˆ c c c c EpB | si,B ą biq ´ Epdq 1 ` EpB | si,B ą biq siqir ´ 1s ` p1 ´ siqqi ´ 1s 1 ` ι 1 ` ι c “ `PpB “ bi | siqEpQq ˆ c c bi ´ Epdq c c 1 ` bi siEpqi | si,B “ biqr ´ 1s ` p1 ´ siqEpqi | si,B “ biqr ´ 1s 1 ` ι 1 ` ι „  The probabilities of being the cutoff bidder or not are defined in the same way as in the no insurance case. Additionaly, the revenues when the shock does not materialize are identical to the ones in the no insurance case. When the liquidity shock does materialize, the costs are different from before; here, an illiquid bank can always borrow money from the LLR. Thus, it will be able to repay its creditors, but it will also need to repay the LLR, making a per unit revenue in the case that the shock materializes equal to the difference between the received interest rate and the LLR interest rate, taking into consideration the opportunity cost. Using the same definitions as in the previous section, the first order condition with respect to bi is: c ´1 c bi ´ psip1 ` Epdqq ` ιqci 1 BEcpqi q BEncpB q (2) “ ´ E pq q ` q b b Bb c i i Bb i i ˆ i ˙ ˆ i ˙ Notice that since the right hand side of equations (1) and (2) are identical, the premium asked behaves differently in the LLR case: even when si tends to 1, full insurance implies that the premium asked remains finite. Moreover, when N Ñ 8,(2) becomes

bi “ sip1 ` Epdqq ` ι

11A more detailed form of the expected payoff along with the corresponding first order condition can be found in the Appendix. 18 Again, there is a proportional relationship between risk and return. The main difference from the no insurance case is that now whether high returns are indicative of risk depends on expectations of policy response, which determines the cost of illiquidity. For example, if the Central Bank announces that it will reduce policy rates in anticipation of a liquidity shock, then bond returns are not going to rise as much. Equivalently, high returns are not directly translated to high risk if there is a negative shock to Epdq. The same interpretation holds when we account for market power.

6. Identifying Idiosyncratic Liquidity Risk

Despite the non-trivial prediction problem that the banks have to solve, liq-

uidity risk can be non-parametrically identiﬁed from observations on pbi, qiq.

More speciﬁcally, (1) and (2) can be re-expressed in terms of si as follows:

NI qipS1,i ´ ιS4,iq ` pbi ´ ιqS2,i ` S3,i (3) si “ qipS1,i ` S4,iq ` pbi ` 1qS2,i ` S3,i

FI qipS1,i ´ ιS4,iq ` pbi ´ ιqS2,i ` S3,i (4) si “ p1 ` Epdqq pS2,i ` qiS4,iq NI FI where si and si are the identifying conditions in the case of no insurance and that of full insurance (lender of last resort), respectively. The objects 12 S1,i,S2,i,S3,i,S4,i have an intuitive interpretation:

c c c B c c B fpB |siqdB B :B ąbi S1,i :“ ´ ´ Bbi ¯ measures the ability of the bank to affect the expected equilibrium price by infinitesimally changing the bid in the states in which it is not a cutoff bidder

c c c B c c qi fpqi |siqdqi qi :B “bi S2,i :“ ´ ´ Bbi ¯

12 All elements on the right hand side are indeed functions of si, but we can estimate them non-parametrically. 19 measures the ability of the bank to inﬂuence the expected quantity of bonds it receives in the states in which it is the cutoﬀ bidder,

c c c S3,i :“ qi fpqi |si,S´iqdqi ˆ c c qi :B “bi is the expected quantity received when the bank is the cutoﬀ bidder, and

c c B c c fpB |siqdB B :B ąbi S4,i :“ ´ ´ Bbi ¯ measures the marginal eﬀect of the bank’s bidding behavior on the probability of not being a cutoﬀ bidder.

As long as S1,i,S2,i,S3,i,S4,i are non-parametrically identified from the data, then liquidity risk is also non-parametrically identified. Note that these quantities depend on the distributions of the equilibrium interest rate and quantity received by bank i. In order to estimate these distribu- NI FI tions, and hence derive the estimates for si and si , we employ a resampling algorithm similar to Kastl(2011); Hortaçsu and McAdams(2010). This procedure draws from the pool of observed bids different realizations of residual supply faced by bank i, and therefore all the possible realizations of c ‹ the auction outcomes, pB , qi , "Cutoff/No Cutoff"q. Due to the relatively small number of market participants in each auction, we pool auctions that occurred during the first 15 days of the each month and after the 20th day of the previous month. Not pooling the data in different auctions would produce noisier estimates, while by pooling we invoke the im- plicit assumption that the main unobservable driver of bidding, liquidity risk, is constant across the auctions we pool together. Given that public information (Ω) evolves at a monthly frequency e.g. interest rate decisions by the CB, we deem that this assumption is innocuous. Moreover, banks that participate more than once within a month are consid-

ered as diﬀerent banks. Since we assume that liquidity risk psiq is a draw from a common distribution, including observations from bank i in the same data-set amounts to including an additional realization from this distribution.

20 Eﬀectively, including an additional observation gives a better approximation for the expected equilibrium price and quantities. Moreover, since the number of participants varies across auctions, we resample with a weighting scheme that gives higher weight on observations from auctions in which the number of participants is closer to the number of participants in the auction that bank i places its bids. We thus employ the following resampling algorithm: Algorithm. Given a pool of T auctions, and a bank i that belongs to group G

with size Ng ď N where N is the number of participant in the auction where bank i participates. For m “ 1..M :

(1) Fix bid vector by bank i: pqi, biq.

(2) Draw NG ´ 1 bid vectors pqj, bjq from the pool of bids with j“1,...,NG´1 Nτ Ń Kp h q weight wm “ Nτ Ń where K is the Normal kernel with band- τ“1..T Kp h q 13 width h. ř (3) Construct the residual supply curve (share) at each interest rate bid: r qi pbq “ 1 ´ j‰i qjpbq. (Aggregate horizontally) (4) Compare pq , b q with qr(b) to compute the equilibrium price, Bc, and iř i i c ‹ then compute quantity received by bank i at B , qi . Bank i is the c cutoff bidder if bi “ B . (5) Go to 2.

c ‹ The above algorithm generates a sample tBm, qi,m, um“1..M , whose empirical

distribution can be used to compute S1,i, S2,i, S3,i and S4,i and therefore si.

Partial derivatives in S1,i, S2,i, S3,i and S4,i are computed by perturbing bi by an arbitrarily small step and computing the corresponding numerical derivative of each integral e.g : 1 1 1 Sˆ “ 1rBc pq , b ` q ą b ` qs ´ 1rBc pq , b q ą b s 4,i M m i i i M m i i i ˜ m“1..M m“1..M ¸ ÿ ÿ

13 ´0.2 We set h “ 2σNτ N where σNτ is the standard deviation of the number of auction participants. 21 c where tBmpqi, bi ` qum“1..M is the sample generated by repeating the resampling procedure above with the perturbed demand for bank i.14 In Figure1, we plot the realization of a few residual supply curves faced by a speciﬁc bank according to the algorithm outlined above. Interacting the bank’s demand with these alternative residual supplies eﬀectively generates a sample of possible equilibrium prices and quantities.

Figure 1. Random Curve Realizations

Note that when the rest of the banks demand too many bonds given the quantity the Central Bank wishes to supply, it is possible that a bank is facing no excess supply for bonds above some level of interest rate. In Figure2, we provide a graphical example of the resulting equilibrium interest rate and quantity assigned to bank i in step p4q of the algorithm.

? 14In the application, we set the M “ 500000 and “ κ eps where eps is machine precision error. We have used different step sizes, which are multiples κ P p1, 10, 100q of eps “ 2´52, and give similar results. In the aggregate model, we employ the average of the results obtained from the different step sizes. 22 Define the residual demand of bonds that bank i faces at the interest rate bj as 1 ´ Qj. Suppose that bank i bids pb, qq. Since its interest rate bid is higher

than bk, the highest interest rate at which there is positive residual supply of bonds, bank i does not receive any bonds in equilibrium. On the contrary, for

any interest rate bid smaller than bk, the bank will be awarded some bonds. If the quantity bid by bank i is smaller than the residual supply it faces at the interest rate it bid for, then it will be awarded the total quantity asked, and it will receive a higher interest rate than its own bid. An example of this 1 1 1 is the bid pb , q q in Figure 2, where the bank is awarded pbk´1, q q. However, when the bank asks for a quantity that is higher than the residual supply it faces at the interest rate it bid for, it will be the cutoﬀ bidder. This is exactly the case for bid pb2, q2q, where the bank is awarded the residual supply at the 2 previous step, 1 ´ Q1, and the equilibrium interest rate is b .

bi b

bk´1 b1 bk´2 . . .

b2 b2

q1 q q2 ...... 1 ´ Qk´11 ´ Qk´2 1 ´ Q2 1 ´ Q1 1 Q

A detailed derivation of the equilibrium interest rate and quantity for all possible cases can be found in the Appendix, along with more graphical examples, including the cases in which there is positive residual supply of bonds for bank i at any level of the interest rate. 23 In the rest of the paper, we employ the predictions of the model in order to estimate the distribution of liquidity risk in the case of Chile. In the empirical model we will use, we have to nest the two cases, (3) and (4). Since whether there is insurance or not depends on the credibility of the Lender of Last Resort, and in turn this depends on macroeconomic factors, in the next section we consider an aggregate model which will allow us to take them into consideration.

7. Estimation using Micro and Macro Data

Since we are interested in the evolution of liquidity risk, from now on, we consider a sequence of auctions, and thus index idiosyncratic risk also by time ptq. Note also that we have not imposed any restriction on the relationship between the distribution function of liquidity risk Ft and its past, Ft´l,lą0. In what follows, we adapt the previous analysis in order to build a model that can provide empirically plausible measures of liquidity risk that are are consistent with the time dimension of risk. In particular, from the earlier discussion of the identiﬁcation of individual NI FI liquidity risk, it is clear that si,t and si,t can only provide estimates of liquidity risk at the extreme points of full and no insurance. Moreover, there is no clear ranking between the two measures e.g these measures could be an upper or a lower bound to the true measure of individual liquidity risk. We instead combine the two measures by acknowledging that while in many countries the CB can constitutionally act as a lender of last resort to a particular bank that faces liquidity problems, the credibility of the LLR depends on macroeconomic conditions, especially in the presence of aggregate risk.15 This creates uncertainty on the degree of insurance a bank is facing, which we account for by modelling the probability that when the bank receives the shock, the CB will not be able to act as a lender of last resort, pt, common across i.

15See for example Calvo(2006), where it is argued that in Emerging Market countries, the effectiveness of the lender of last resort may be compromised in sudden stop episodes or in cases in which there is a bank run with private sector liquidity held in foreign exchange. 24 In this case, the expected payoff of bank i, which can be found in the Appendix, is a combination of the expected payoffs in the no insurance and full insurance

case. Taking the ﬁrst order condition with respect to bi, the corresponding

expression for si,t is as follows:

qi,tpS1,i,t ´ ιtS4,i,tq ` biS2,i,t ` S3,i si,t “ p1 ` ptdtq pS2,i,t ` qiS4,i,tq ` p1 ´ ptq pqi,tS1,i,t ` bi,tS2,i,t ` S3,i,tq sNI (5) “ i,t dt´ιt 1 1 ` pt 1 d FI ´ 1 ` t si,t ˆ ˙ NI FI which nests si,t and si,t when pt “ 0 and pt “ 1 respectively. NI FI It is clear that in order to estimate si,t, we need to obtain si,t , si,t and pt. The ﬁrst two components can be readily estimated using the algorithm described in the previous section. As for pt, it is likely to be an endogenous variable that reacts to past and prospective macroeconomic conditions, including the distribution of risk. To estimate it, we consider an empirical model of the interaction of aggregate risk, policy and other macroeconomic variables. This framework will also allow us to evaluate the eﬀects of policy and other shocks on risk.

Recall that si,t „ Ftp. | Ωq and thus aggregate shocks can affect all cross iid sectional moments of si,t. We define aggregate risk to be the first moment, St :“ si,tdFtpsi,t|Ωq, which is the average perceived probability of a liquidity shock´ in the next period conditional on any public information, including the Central Bank’s actions. To facilitate aggregation and matching with observable moments, we perform a series of approximations. We first obtain a a first order approximation to (5), which yields that:

2 d ´ ι s « sNI ´ p sNI t t p2 ´ sFI q ´ 1 i,t i,t t i,t 1 ` d i,t ˆ t ˙ ` ˘2 d ´ 2ι ´ 1 d ´ ι “ sNI ´ p sNI t t ` t t sFI i,t t i,t 1 ` d 1 ` d i,t ˆ t t ˙ ` ˘ Aggregating si across banks and keeping moments up to second order, yields that aggregate risk can be linked to the distribution of the no-insurance measure as follows:

25 NI 2ιt ` 1 ´ dt NI NI 2 (6) St “ St ` pt Vt ` St 1 ` dt ˆ ´ ¯˙ NI NI NI NI NI NI 2 ` NI˘ where St “ si dFtpsi q, Vt “ psi ´ St q dFtpsi q . ´ ´ Following the literature on the role of international reserves in easing domestic illiquidity as well as dealing with external drains and enhancing the credibility of the lender of last resort, (Kaminsky and Reinhart, 1999; Calvo, 2006;

Obstfeld, Shambaugh, and Taylor, 2010), we model pt as follows: α Rt Rt´1 pt “ α P p0, 1q 1 `´ R¯t Rt´1 ´ ¯ where Rt is the level of foreign currency reserves and α ě 0 determines how elastic the probability is to changes in the growth rate of foreign currency reserves. A rapidly declining stock of foreign exchange reserves may put downward pressure on this probability; the higher α is, the stronger is the eﬀect on this probability.

From (6) it becomes clear that in order to estimate St we need to estimate α, as everything else on the RHS of (6) is either observable or has already been estimated. Since (6) has two unknowns, St and α, we introduce additional information through a small scale macroeconomic model with minimal structure. This will generate enough restrictions to estimate α, as well as the eﬀects of unpredictable shocks on aggregate risk, amongst other macro variables. Importantly, by estimating α, and therefore pt, we can recover the latent distribution of liquidity risk.

7.1. The Aggregate Model. We employ a log-linear Gaussian state space model at monthly frequency: ˜ ˜ Xt “ AXt´1 ` Bt ˜ ¯ where Xt is in log-deviations from the long-run average X. These states sum- marize the dynamics of the economy as well as the potential information available to the policy makers on a monthly basis, and can either be directly observed or potentially recovered from the data. 26 ˜ ˜ ˜ ˆ ˜ NI ˜ ˜ We let Xt “ Zt, ∆Rt, St, Vt , dt, ˜ιt, Qt where Zt is a factor that summarizes the macroeconomic´ state and is computed¯ as the first principal component of ˜ ˜ employment, inflation and the real exchange rate ; ∆Rt and Qt are the growth ˜ rate of foreign reserves and the supply of 30-day bonds respectively ; dt are ˆ aggregate expectations of next period’s interest rate ; St is aggregate risk and ˜ NI Vt is the variance of the distribution of liquidity risk in the no-insurance case. ˜ We next illustrate how we link the set of states (Xt) to the set of measure- ˜ ˜ ˜ ÑI ˜ NI ˜ 1617 ments Yt “ Zt, ∆Rt, St , Vt , ˜ιt, Qt . Consistent with the macroeconomic model, we adopt a log-linearized version of (6) as follows: ` ˘

¯NI ¯ ˜ S 2¯ι ` 1 ´ d ¯NI ˜NI St “ 1 ` 2¯p S St ¯ 1 ` d¯ S ˆ ˙ ¯ ¯ p¯ 2¯ι ` 1 ´ d ¯ NI ¯NI 2 2 dp1 ` ¯ιq ˜ ` V ` S p˜t ´ dt ´ ¯ι˜ιt ¯ 1 ` d¯ 2¯ι ` 1 ´ d¯ 1 ` d¯ S ˆ ˆ ˆ ˙˙˙ ¯ ´ ` ˘ ¯ p¯ 2¯ι ` 1 ´ d ¯ NI ˜ NI (7) ` V Vt S¯ 1 ` d¯ p˜t “ αp1 ´ p¯q∆R˜t

Therefore, the link of unobservables to observables can be parameterized by a matrix H, which is the identity matrix apart from the third row, which parameterizes (7). ˜ ˜ Xt “ HpαqYt

Since H is invertible, the reduced form model that is taken to the data is a ﬁrst order vector autoregression with cross equation restrictions implied by a

16 ˜ ˜ To make estimation feasible, we set dt “ ˜ιt as dt and ˜ιt are strongly correlated leading to near-collinearity. In the Appendix, we plot the interest rate and two measures of expectations of one month ahead interest rate which confirm this strong co-movement. Nevertheless, the long run averages are different, and reflect the (constnat) spread between the rate at which banks borrow from the central bank and the benchmark policy rate rate, d¯“ ¯ι`0.25. 17 ÑI ˜ NI NI We compute pSt , Vt q by computing the mean and variance of si . Since the estimates NI of individual risk may fall outside p0, 1q, we apply the monotone transformation si “ NI exppsraw,iq NI . 1èxppsraw,iq 27 combination of structural and measurement relationships in pB,Hq:

˜ ´1 ´1 ˜ ´1 Yt “ H AH Yt´1 ` H Bt ˜ (8) “ CYt´1 ` Dt ˜ (9) “ CYt´1 ` ut

18 where t „ Np0, Σq with Σ a diagonal matrix. Apart from the moments of the distribution of liquidity risk in the case of no insurance, the rest of the data employed as measurements are Consumer Price

Index Inﬂation for πt, the real broad eﬀective exchange rate expressed in terms of foreign currency for petq, and the 30 day interest rate swaps as a measure 19 of dt.

7.1.1. Identiﬁcation. The parameters of interest are the elements of B and α, which are identiﬁed via solving

´1 1 ´1 (10) H BΣB H “ Σu Besides α, identifying B will allow us to trace the eﬀect of aggregate shocks to the states in Xt, and in particular, the response of aggregate risk to these 20 1 shocks. We deﬁne t “ pz,t, R,t, Rnews,t, s,t, ι,t, q,tq , that is, the structural shocks we seek to identify are a demand shock z,t, a non-anticipated and an anticipated (one period ahead) shock to the growth rate of foreign reserves,

(R,t, Rnews,t) , a shock to aggregate risk s,t, a shock to the nominal interest rate ι,t, and a shock to the supply of bonds, q,t. The demand shock is identified by ordering it first, that is, macroeconomic conditions do not contemporaneously react to financial conditions. Foreign reserves are allowed to contemporaneously respond to their own shock and to

x,t. The latter is justiﬁed as foreign reserve policy may respond to exchange

18We have tested for autoregression lags up to 8th order. The Schwarz criterion selects one lag, which we adopt. 19We describe the macro dataset and variable transformations in the Appendix. 20As is standard in the macroeconometric literature, the covariance matrix of the VAR residuals is matched with DD1 to obtain estimates of the underlying components of H´1, B and Σ. It can be shown that the restrictions employed imply satisfying the order condition, that is, there are 22 independent identifying restrictions for the 22 unknown parameters. 28 rate developments. We allow for some anticipation to changes in foreign reserves as announced policies usually specify gradual changes in the stock of reserves. Given the uncertainty that exists around the actual implementation i.e. timing and volume, we deem that allowing for one period is a reasonable approximation. The news shock is identiﬁed by imposing a zero restriction on the contemporaneous response of reserves to the news shock and a zero restric-

tion on the contemporaneous response of Qt to the news about the future, as

Qt is a monetary policy operation that aims to absorb existing liquidity in the ˜ NI system. Aggregate liquidity risk and Vt are allowed to contemporaneously respond to all shocks but Q,t and x,t. The rationale behind these restrictions is that individual beliefs about aggregate risk depend on the macroeconomic conditions as signalled by interest rate changes and reserve accumulation rather that current shocks to unemployment, inflation or the real exchange rate. In turn, the nominal interest rate is allowed to contemporaneously respond to shocks to macroeconomic conditions, which is consistent with a Taylor rule interpretation of monetary policy. Therefore, the sets of restrictions used are as follows: 1 0 0 0 0 0 ¨ ‹ 1 0 0 0 0 ˛ 0 ‹‹ 1 ‹ 0 B “ ˚ ‹ ˚ 0 ‹‹‹‹ 0 ‹ ˚ ‹ ˚ ‹ ˚ ‹ 0 0 0 1 0 ‹ ˚ ‹ ˚ ‹‹ 0 0 ‹ 1 ‹ ˚ ‹ ˝ ‚ Inference is based on 15000 posterior draws using the Gibbs sampler, dis- carding 2000 draws as burn in. We use the standard prior of independent ´1 Normal-Wishart, with C „ Np0,Inyq and Σu „ W ishartpIny , nyq, the de- 21 grees of freedom being equal to the number of observables (ny). Given the posterior draws of pvecpCqT , vecpBqT , αqT , we are able to recover both the distribution of liquidity over time, as well as the identified effects of shocks to the

21See Koop and Korobilis(2010) for details on the implementation and the posterior, which has the same Normal - Wishart form as the prior. For every posterior draw of Σu, we obtain ´1 1 ´1 a draw for pα, B, Σq using GMM based on the moments: vechpH BΣB H ´ Σuq “ 0. 29 mean of this distribution. The approach of this paper can be easily extended to identify the eﬀects to higher moments, however we deem that the ﬁrst cross sectional moment is more precisely estimated.

7.1.2. Distribution Estimates. While we discuss the identified effects of shocks on aggregate risk in the next section, we first present the kernel density estimate of the distribution of si. We construct the latter using the median value of α given the draws from its posterior distribution.22

Figure 2. Evolution of the distribution of Liquidity Risk during the Global Financial Crisis

In Figure2, we focus on the period during which the global ﬁnancial crisis unravelled. Our measure captures the sudden shift in the distribution of risk towards one in the month prior to the Lehmann Brothers collapse, as well as

22A plot of this distribution can be found in the Appendix. More than 80% of the probability mass is at values that imply an inelastic credibility pptq to reserve accumulation. This suggests that variation in pt that is due to reserve accumulation is not the driver of the identiﬁed eﬀect of reserve accumulation on St. 30 interesting shifts before and after this event. In the next section, we discuss further the role of policy in the dynamics of this distribution.

8. Does policy convey information about aggregate risk?

As we have seen from the analysis earlier on, from the perspective of the individual bank, bidding in the bond auction requires forming a belief about the distribution of liquidity risk in the ﬁnancial sector conditional on all available

public information (Fbankp. | Ωtq). If there was perfect information on liquidity risk in the economy, rational banks’ forecast errors would not be sensitive to re- ﬁnements of the information set due to e.g. policy actions and announcements as this information would already be incorporated in their forecast. To see this, suppose that the signals received are related to the state of aggregate liquidity in the following way: ˜ si,t “ λi,tSt ` ξi,t ˜ ‹ St “ St ` ηt

2 2 2 where ξi,t „ Np0, σξ q, ηt „ Np0, σηq, λi,t „ Np1, σλq, pξt, λi,t, ηtq are mutually iid iid iid ‹ independent and St is the state of aggregate liquidity risk under complete information. The distribution of si,t conditional on public information is therefore ˜ 2 equal to NpSt, σξ q.

It can be shown that given the private signal si,t and public information Ωt, ‹ the best individual forecast about St is given by

‹ λi,t ˜ E pSt | si,t, Ωtq “ 2 2 2 ξi,t ` St λi,tση ` σξ while the average forecast across banks is

‹ ˜ E pSt | Ωtq “ St

If Ωt is the complete information set, then new information will affect neither ˜ S nor si,t. If Ωt is incomplete, then any refinement to Ωt will affect financial market perceptions of aggregate risk, and as we already mentioned, this intro- duces the possibility of signalling effects. Of course, policy actions may as well have direct effects on the possibility of an aggregate liquidity shock in the next 31 period. An advantage of the aggregate model we employed is that estimates ˜ in pB,Hq can be used to both identify the latent St as well as the effects of different aggregate shocks. We will next discuss in more detail the empirical results as well as the role of market power in masking the identification of the effects. Below, we plot the impulse response function of aggregate risk with respect to the identified reserve accumulation shock and the corresponding news shock. Aggregate risk responds in a significant way to unpredictable changes in foreign reserves. An increase in the growth rate of foreign reserves by 1% is associated with a median 5.8% increase in expected aggregate risk.

In light of the above insights, we interpret our result as evidence of an informational channel of foreign reserve policy. A positive shock to foreign reserve accumulation by the Central Bank can impact aggregate risk in two opposite ways. First, an increase in the growth rate of reserves implies a permanent positive shock to the level of reserves, which may act as a deterrent to possible 32 self fulfilling speculative attacks, leading to a fall in the probability of an ex- ternally induced liquidity shock in the next period. On the other hand, reserve accumulation may signal to private financial institutions that the Central Bank has information of an increase in the average probability of a liquidity shock in the next period. A net positive effect implies that the signalling channel dominates.23 Figure3 presents the ratio of the actual level of reserves to a measure of the efficient level for several emerging market economies; reserve accumulation efforts during and after the crisis are consistent with striving to achieve a ratio above or close to one.

Figure 3. The ARA metric developed by the International Mone- tary Fund (IMF) estimates the efficient level of reserves that balances the benefits of holding reserves (e.g. reduced likelihood of Balance of Payment crises, financial stability, policy autonomy) against the corresponding opportunity costs (lower returns) . A ratio of reserves to ARA between 1.5 and 1 is considered adequate. Source: IMF, Assessing Reserve Adequacy.

In Figure4 we plot the corresponding ratio for Chile against the actual changes in reserves from 2000 to 2018. As can be observed, when the ratio falls below

23Reserve accumulation is indeed a signal for future liquidity issues as it is considered a precautionary measure against such crises, see e.g. Obstfeld, Shambaugh, and Taylor(2010); Bocola and Lorenzoni(2017); Céspedes(2019). In fact, Obstfeld, Shambaugh, and Taylor (2010) emphasize that the need for ample reserves is higher in ﬁnancially open economies. In episodes of internal and external liquidity drains and concerns about public insolvency due to large-scale bailouts, there might be no other means for managing the exchange rate. 33 one, as in e.g. 2006-2007 and in 2010, there is a subsequent increase of reserve accumulation. We interpret this a sign of commitment by the Central Bank to maintain ﬁnancial stability.

Figure 4. Reserve Accumulation

Furthermore, in Figure5 we can see that there is no conclusive evidence of an effect of news about increments to reserves in the next period. One interpretation of this finding is that actual increases in reserves are more credible than announcements of future interventions as the former incur an actual cost. In- deed, there are cases where announcements of future interventions were only partially carried through in Chile. For example, while BCCh announced in April 2008 a series of interventions that were meant to last until the end of the year, they were prematurely ended with the Lehmann episode (Chamon, Hofman, Magud, and Werner, 2019c). This can rationalize the absence of evidence on a significant effect of news on perceived risk. On the possible informational effects of monetary policy, there is posterior evidence that an increase in the nominal rate decreases aggregate risk. A potential explanation of this fact is that a rise in the nominal interest rate will put downward pressure on capital flight leading to reduced liquidity risk in the next period. This implies that although a rise in the nominal rate might initially signal the expectation of a shock in the next period, the rise in the rate makes it less likely, leading to an34 eventual fall in St. 35 8.1. Counterfactual with Competitive Market. As we already mentioned earlier on, explicitly taking into account market power is essential for describ- ing the strategic behavior of banks in the auction. We next compare the ˜ estimated St to a counterfactual measure where the strategic component of bidding behavior is ignored, as well as how this changes the empirical results regarding the effects of policy shocks. Recall that in the absence of market power no bank can affect the equilibrium NI,0 price and quantity and thus equations (1) and (2) can be written as si “ bi and sFI,0 “ bi respectively. We can therefore infer the counterfactual 1`bi i 1Èd measure of aggregate risk by performing the same estimation as before, using NI,0 FI,0 si and si . In Figure5, we plot the recovered time series for aggregate risk (in log deviation from the mean units), against the counterfactual.

Figure 5. Aggregated Risk with/without Market Power

The shaded areas in Figure5 indicate periods for which the Central Bank announced and implemented official foreign exchange interventions. The first 36 series of interventions were announced in April 2008, but as we already mentioned, they ended in September 2008. The second series of interventions were implemented during the year 2011, were the Central Bank scheduled daily purchases of $50 billion worth of foreign currency, strengthening the liquidity position to 17% of GDP. Chamon, Hofman, Magud, and Werner(2019c), using daily and intra-day data find that announcements have clear cut effects on the level, trend and volatility of the exchange rate while they bring mixed evidence on the effects of the actual intervention. Eyeballing Figure5, we can see that some of the increases in perception of aggregate risk do coincide with these periods, partially justifying the more formal evidence using the identified impulse responses, where the latter also capture other occasions in which changes in reserves did not involve an official policy response to liquidity issues. On the contrary, while we observe increases in the counterfactual measure during the periods of official interventions, the effect is much smaller, especially in 2008. To have a quantitative sense of whether employing the counterfactual measure would lead to distorted inference regarding the effects of foreign reserve interventions on aggregate risk, we repeat the VAR exercise using the counterfactual measures for si. Below we plot the counterfactual impulse response identified using the same restrictions. Employing a measure that conflates risk with market power leads to incorrectly inferring that there is no conclusive evidence of a contemporaneous effect of foreign reserve policy as the credible set puts positive probability both on a negative and a positive effect, while the median effect is much weaker than in the case in which market power is accounted for. An analogous statement can be made about the effect of a monetary policy shock. What is also important is to acknowledge the fact that the counterfactual measure employed has no estimation error, which implies that mis-measuring the effect is not due to random noise but rather due to information masked by strategic bidding. On the contrary, while our estimates entail noise due to the resampling estimation of individual risk, we can still infer the sign of the impact effect of policy.

37 38 9. Conclusion

In this paper we have built a structural model of a uniform multi unit auction where banks facing the probability of becoming illiquid place interest rate and quantity bids to sell liquidity to the Central Bank. We combined bidding data from auctions of short term notes issued by the Central Bank of Chile and macroeconomic data to uncover the financial institutions’ liquidity risk at monthly frequency. We have shown that identification depends on strategic behavior due to imperfect competition as well as on the credibility of the lender of last resort. Beyond providing estimates of the distribution of liquidity risk, we undertook a structural study of the effects of unpredictable shocks and news shocks about reserve accumulation, a popular tool amongst emerging economies to promote financial stability. We find that actual interventions have statistically significant effects while there is no evidence on news effects. This has implications for policy design. While in the recent years Central Banks have placed a lot of emphasis on communication strategies regarding present and future policy, when actions are costly, actual policy may have significantly stronger effects than mere announcements.

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42 10. Appendix

10.1. Evidence on Supply Uncertainty. Note that while positive differ- ¯ ences between Qt and Qt are expected due to demand falling below supply, positive differences signify that supply is not fixed ; it can increase to accom- modate additional demand. Dashed lines indicate the US sub-prime crisis and the Lehman Brothers collapse respectively.

Figure 6

10.2. Details about the optimization problem of bank i. Below we cal- culate the expected cost of being hit by the illiquidity shock when there is no

insurance against it, TCi.

c c c c TCi “ siPpB ą bi | siqqiEpQq ` siPpB “ bi | siqEpqi | bi “ B , siqEpQq where the first component of the sum is the cost when firm i is not the cutoff NC C bank, TCi and the second component is the cost when it is, TCi . The 43 derivative of TCi with respect to bi is then: c BTCi BPpB ą bi | siq “ si qiEpQq ` Bbi Bbi c BPpB “ bi | siq c c si Epqi | bi “ B , siqEpQq Bbi c c “ c BEpqi | bi “ B , siq `PpB “ bi | siq EpQq Bbi where the first component of the sum is the derivative of the cost‰ when firm i is NC 1 not the cutoff bank pTCi q , and the second component is the corresponding C 1 derivative when it is, pTCi q . 1 pTCiq Then ci “ C 1 is the ratio of the expected marginal cost of being hit pTCi q by the liquidity shock to the expected marginal cost of being hit by the liquidity shock when being the cutoff bidder. When we decompose conditional expectations as Ep.|siq “ Encp.q ` Ecp.q ` Enbp.q , we can rewrite it as c c ´1 BPpB ąbiq BEcpqi q ci :“ qi `1. Note that the expected cost of being hit by the Bbi Bbi illiquidity shock´ when there¯ is full insurance against it, TCF I i, can be written as: Epdq TC “ TC 1 ` F I i i 1 ` ι 1 „  pTCF Iiq and thus ci also satisfies ci “ C 1 . pTCF Iiq

10.3. Optimization problem of the Bank with uncertainty in LLR. The expected payoﬀ in this case is

c 1 c c “ PpB ą bi | siqEpQq p1 ´ siqqi pEpB | B ą bi, siq ` 1q ´ 1 1 ` ι ˆ ˙ “ 1 c c `siqi p EpB ´ d | B ą bi, siq ´ 1 ´ p1 ´ pq 1 ` ι ˆc ˆ ˙ ˙ `PpB “ bi | siqEpQq ˆ ‰

c c 1 ` bi p1 ´ siqEpqi | bi “ B , siq ´ 1 1 ` ι ˆ ˙ “ c c bi ´ Etd `siEpqi | bi “ B , siq p ´ 1 ´ p1 ´ pq 1 ` ι ˆ ˆ ˙ ˙ ‰ The ﬁrst order condition in this case yields that: 44 c c c c BEpB | B ą biq BEpqi | B “ biq c c p1 ´ sip1 ´ pqq qi ` bi ` E pq | B “ biq Bb Bb i ˆ i i ˙ c c c c c c BPpB ą biq BEpqi | B “ biq BPpB ą biq BEpqi | B “ biq ´ι qi ` ´ sip1 ` pEtdq qi ` Bb Bb Bb Bb ˆ i i ˙ ˆ i i ˙ “ 0

Rearranging in terms of si gives equation (5).

10.4. Computing Equilibrium Prices: Further Details. Given the demand for bonds of N ´ 1 bidders, and the supply by the Central Bank, we construct the excess supply faced by bank i, in order to derive all possible equilibrium prices in this market, and the corresponding equilibrium quantities assigned.

Let pbi, qiq be bank i’s bid vector. We rank the interest rate bids of all but bank i from the lowest to the highest: b1 ď b2 ď ... ď bN´1, and denote their corresponding shares asked by q1, q2, ..., qN´1. j Deﬁne Qj ” ql with j P r1,N ´ 1s. Note that although qj ă 1@j, it can well l“1 be that Qj ąř1, i.e. the Nth bank is facing no excess supply for bonds above some level of interest rate. We distinguish between two cases. The ﬁrst, is when there exists some bank k P r1,N ´ 1s at which the residual supply of bonds becomes zero and thus bank i is not guaranteed a positive allocation of bonds. The second, is when the CB’s supply of bonds is high enough such that there is positive residual supply at any level of interest rate, and thus bank i is guaranteed to receive at least part of its demanded quantity in equilibrium.

10.4.1. No Excess Supply after some level of interest rate. Take rk to be the interest rate above which there is no excess supply of bonds for bank i, i.e:

Qk´1 ă 1 ď Qk, where k P r1, ..., N ´ 1s. In this case, the equilibrium interest 45 rate received by all winning banks is:

bk if bi ą bk

$ either bm´1 ă bi “ bm and Qm´1 ă 1 ď Qm ` qi c ’ B pbi, qiq “ ’bi if $ or b ă b ă b and Q ă 1 ď Q ` q ’ m´1 i m m´1 m´1 i ’ ’ & & or bi “ b1 “ ... “ bm and Qm ` qi ě 1 ’ ’ ’bm`h if b%’m´1 ď bi ă bm and Qm´1`h ` qi ă 1 ď Qm`i ` qi ’ ’ 24 with 1 ď m ď%’k and h “ 0, 1, ..., k ´ m. In the ﬁrst case bank i does not receive any shares in equilibrium, whereas it is assigned their total shares asked, qi, in the third case. When the equilibrium interest rate is bi, the bank is assigned their residual demand (or a proportion of it in case of ties), which is given by:

1 ´ Qm´1 if bm´1 ă bi ă bm and Qm´1 ă 1 ď Qm´1 ` qi

c $ qi q b , q p1 ´ Qm´1q if bm´1 ă bi “ bm and Qm´1 ă 1 ď Qm ` qi p i iq “ ’ qi` ql ’ l:bl“b ’ ř & qi if bi “ b1 “ ... “ bm and Qm ` qi ě 1 qi`Qm ’ ’ In ﬁgure7 below,%’ we consider the case where bank i places an interest rate bid b that is equal to the interest rate bid by another bank. Given the residual supply of bonds, b is going to be the equilibrium cutoﬀ rate when bank i has asked for quantity share q. In this case, the two banks share the residual q demand at the previous step. More precisely, bank i receives p1 ´ Qk 1q. q`q2 ´

24 Deﬁne b0 ” 0 and Q0 ” 0 for completeness of notation. 46 bi

bk´1

. . .

b “ b2

q ... 1 ´ Qk´11 ´ Qk´2 1 ´ Q21 ´ Q1 1 Q

Figure 7. Weakly Positive Residual Supply

10.4.2. Excess Residual Supply. In this case the CB’s supply of bonds is too high so that there is positive residual supply for bank i for any level of interest

rate i.e.: QN´1 ă 1. The equilibrium cutoﬀ interest rate is:

bN´1, if bi ă bN´1 and 1 ´ QN´1 ą qi

$ either bi ą bN´1 ’ ’ $ or b ă b “ b and Q ă 1 ď Q ` q c ’ m´1 i m m´1 m i B pbi, qiq “ ’bi if ’ ’ ’ or b ă b ă b and Q ă 1 ď Q ` q &’ &’ m´1 i m m´1 m´1 i or bi “ b1 “ ... “ bm and Qm ` qi ě 1 ’ ’ ’ ’ ’bm`h if b%’m´1 ď bi ă bm and Qm´1`h ` qi ă 1 ď Qm`h ` qi ’ ’ with m ď N %’´ 1 and h “ 0, 1, ..., N ´ 1 ´ m. In the ﬁrst and third cases the bank receives exactly qi, whereas in the second case it is assigned its residual supply (or a proportion of it in case of ties), which is the same as in before. In this case the bank i is always guaranteed a strictly positive quantity of bonds in equilibrium. Figure8 below illustrates examples of the possible scenarios when there exists positive residual supply for bank i at any level of interest rate. When bidding 47 1 1 1 pb , q q where bk´3 ă b ă bk´2 and q P p1 ´ Qk´1, 1 ´ Qk´2q, the bank receives

their full quantity asked at the cutoﬀ interest rate bk´1. Instead, the bank’s interest rate is the cutoﬀ in both the cases in which they bid b and b2. In the

ﬁrst case the bank receives 1 ´ Qk´1 since b ą bk´1 and q ą 1 ´ Qk´1; in the 2 2 second they receive 1 ´ Q1 since b P pb1, b2q and q ą 1 ´ Q1.

bi b

bk´1

bk´2 b1 b k´3. . .

b2 b2

q1 q q2 ...... 1 ´ Qk´11 ´ Qk´2 1 ´ Q12 ´ Q1 1 Q

Figure 8. Strictly Positive Residual Supply

48 10.5. Dataset Description and Transformations.

10.5.1. Auction Dataset. The dataset contains all bidding information for the open market operations of the BCCh from September 2002 to August 2012. The information includes the total volume of bonds allotted by the Central Bank in each auction, the marginal rate, the bidders’ identities and the rates and quantities of bonds asked by each bidder. For the structural estimation, we pool auctions that occured after the 20th day of month t and the ﬁrst 15 days of the month t ` 1, in order to measure liquidity risk of month t ` 1. 49

Figure 9 10.5.2. Macro Data.

Table 1. Sources and Transformations

Variable Data Source Final Transformation Ut Unemployment Rate: Aged 15 and Over: All Persons for Chile, Percent, Monthly, S.A FRED (OECD) et Real Broad Effective Exchange Rate for Chile, Index 2010=100, Monthly FRED (OECD) Pt Consumer Price Index (2005=100) FRED (OECD) Pt πt CPI Inflation – lnp q Pt´1 Rt International Reserves at the C.B, USD Thomson Reuters (BCCh) ∆lnpRtq dt MPR expectations within the next two months ‹ Expectations Survey - BCCh 1 dt 3m, 6m, 12 m SWAP rates ‹ BCCh 3 p0.7SWAP90t ´ 0.5SWAP180t ` SWAP360tq NI NI St Aggregate Risk (NI) Own Calculations lnp i“1..N sî,t q NI Risk Variance (NI) Own Calculations lnpVˆ arpsˆNI qq Vt ř i,t

50 ‹ There are several measures of interest rate expectations. A market based measure can be constructed based on swaps, yet 30 Day swaps (SWAP30) do not explicitly exist as instruments. As in Becerra, Claeys, and MartÃnez(2016), we compute the implied rate using a combination of 3 month, 6 month and 1 year swap rates. Nevertheless, 3 month swap rates are not available before May 2015, so we impute expectations for the one month ahead interest rate using the "within two month" measure of interest rate expectations survey of the BCCh. 10.6. Comparing Expectations Measures. Below we plot the current monthly interest rate, the constructed 30-day measure based on SWAPS post May 2015 and the within two month interest rate expectations BCCh survey pre May 2015, and the average of within two month and current month interest rate expectations of the BCCh survey. As evident they are all strongly correlated.

Figure 10

51 10.7. Estimate of α and contribution to impact eﬀect.

Figure 11

52 10.8. Additional Impulse Responses.

53 54 10.9. Additional Impulse Responses - Competitive Outcome.

55 56 57