Assume an asset with a four week orderly liquidation period, and that the four week price change due to general volatility of the risk free yield curve is N(0,σM); the uncertainty on the true asset value at the pre-default valuation is N(0,σV); the liquidation price uncertainty stemming from spread changes (if it is a BBB asset, then the volatility of the BBB-AAA spread) and credit migration risks (the risk that the asset gets downgraded from BBB to e.g. BB with the associated price decline) is N(0,σS). Assuming independent factors, the total uncertainty on liquidation value is *N* - 0, σ2 *M* + σ2 *V* + σ2 *S* . Call σ2 T the variance of the total liquidation value uncertainty of the asset. If the risk tolerance of the central bank has been defined as "preventing with 99% probability that the asset value at liquidation falls short of the last valuation minus the haircut", then haircuts need to be set at σT−1(0.01), where (·) is the cumulative standard normal distribution. If for example σ2 M = 4%; σ2 V = 2%; and σ2 S = 2%, then the adequate haircut achieving a 99% confidence level is 58% since −1(0.01) = 2.33 and σT = √4% + 2% + 2% = 2.83% .

### **<IP_ADDRESS> Collateral Constraints**

The quantity and quality of central bank eligible collateral limits the borrowing potential of banks from the central bank. Limits arise from (i) restricted eligibility (e.g. excluding particularly non-liquid and non-transparent bank asset classes and setting a minimum credit quality for the collateral obligor), (ii) conservative collateral valuation, (iii) haircuts, or (iv) quantitative collateral limits to address concentration and correlation risks (e.g. the share of a certain asset type in a collateral portfolio must not exceed a certain percentage). Assume the bank balance sheet in **Table** 3.10, with two liabilities, household deposits and central bank credit, and two assets, loans and securities. Assume also that the central bank imposes a haircut of h1 on loans to corporates and of h2 on securities, with 1 > h1 > h2 > 0. Collateral value after haircuts (or central bank credit potential) is for loans (1 − h1)L and for securities (1 − h2)S.

The maximum borrowing of this bank from the central bank is the value of loans after haircut and the value of securities after haircut, i.e.: (1 − h1)L + (1 − h2)S. The actual borrowing from the central bank, CB, must not exceed this, i.e.:

$$\mathbf{L} + \mathbf{S} - \mathbf{D} + \mathbf{d} \le (1 - \mathbf{h}\_1)\mathbf{L} + (1 - \mathbf{h}\_2)\mathbf{S}$$

This implies that the bank will hit the collateral constraint when deposit outflows exceed d\* = (1 − h1)L + (1 − h2)S − (L + S − D) = D − h1L − h2S, and could

**Table 3.10** Bank balance sheet to illustrate collateral constraints

default unless it finds alternative funding or is able to fire-sell assets. For example, in the case of the euro area, out of approximately EUR 30 trillion of aggregated bank assets, the value of central bank eligible collateral after haircuts that could be

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# **Chapter 4 Unconventional Monetary Policy**

This chapter introduces the reader to unconventional monetary policy, i.e. monetary policy using instruments going beyond the steering of short-term interest rates as described in the previous chapter. We start by providing the rationale of unconventional monetary policy, i.e. essentially pursuing an effective monetary policy when conventional policies are not able to provide the necessary monetary accommodation because of the zero lower bound. We then discuss negative interest rate policies, and explain why rates slightly below zero have proven to be feasible despite the existence of banknotes. We also discuss possible unintended side-effects of negative interest rates. We continue with a discussion of non-conventional credit operations: lengthening of their duration, the use of fixed-rate full allotment, the widening of the access of counterparties to the central bank's credit operation, targeted operations, credit in foreign currency, and widening the collateral set. Finally, we turn to the purposes and effects of securities purchase programmes. We end the chapter by revisiting the classification of central bank instruments in three categories: conventional, unconventional, and lender of last resort.

### **4.1 Rationale and Definition of "Unconventional" Monetary Policy**

Chapter 3 introduced the basic Wicksellian logic, according to which there is a "neutral" or "non-accelerating" short-term risk-free interest rate i\*, such that if it > it\* ⇒ πt ↓; If it < it\* ⇒ πt↑ or, in words, if the actual short-term risk-free rate is below the neutral level, inflation will increase, while in the opposite case inflation will decrease. In the most basic version, the neutral rate is simply the sum of the expected real rate and the expected inflation rate, i.e. it\* = E(rt) + E(πt). If however the key issue is the funding costs of the real economy, and not just an abstract risk free interest rate, then it is more correct to define the neutral interest rate as: it\* = E(rt) +

© The Author(s) 2021

U. Bindseil and A. Fotia, *Introduction to Central Banking*, SpringerBriefs in Quantitative Finance, https://doi.org/10.1007/978-3-030-70884-9\_4

E(πt) – τ – λ, with τ being a measure of the term spread and λ being a measure of the liquidity and credit risk spreads between the average short-term funding costs of the real economy and the short-term risk-free interest rate. The latter will increase in a financial crisis beyond normal levels and needs to be addressed through an additional easing of monetary policy.

In a **financial crisis,** with the associated economic slowdown, and starting from the low structural growth as prevailing in Japan or Europe, expected growth will easily be zero or negative, also implying low or negative real interest rates. If in addition, credit and liquidity spreads increase by 100 or 200 basis points relative to normal levels, as happened in 2008, and expected inflation is also close to zero, then the neutral interest rate it\* will be negative, meaning that an inflationary impulse will require either negative nominal interest rates, or the combination of zero/negative interest rates with "non-conventional" measures that will exert downward pressure on τ and λ. Downward pressure on τ can be achieved through forward guidance1 (committing to hold rates low for long) and through outright purchase programs of long term fixed rate securities to compress term spreads. Downward pressure on λ can be achieved through so-called credit-easing measures, including purchases of less liquid and more credit risky securities, and strengthening the lender of last resort (LOLR) support to the banking system such as to reduce perceived funding liquidity risks of banks. In this chapter, such non-conventional monetary policies will be discussed, whereby policies relating to the LOLR will be dealt with in Chapter 7.

**Non-conventional monetary policy measures are typically considered to have some potentially negative side effects**, while short-term interest rate policies in positive territory do not. For this reason, non-conventional measures are used only if unavoidable, i.e. when it\* < 0, i.e. when short-term interest rate policies alone are no longer sufficient. Negative side effects are likely to increase with the intensity of measures, such that combining different measures may often be optimal to achieve the adequate overall stance of monetary policy. We can think of each non-conventional measure as having (i) a fixed set up /transition cost (need to analyse, specify, decide, communicate new measure); and (ii) an increasing marginal cost from "distortions" it creates.

It appears that **central banks have assessed the relative costs of the different unconditional measures differently**: for example, the Fed and the Bank of England have not hesitated to conduct large scale asset purchase programs as of 2009 but have not tried negative interest rates. In contrast, the ECB has taken a while before launching a true "quantitative easing" asset purchase program, but did not hesitate to move interest rates into negative territory. Of course, the perceived negative side effects of non-conventional measures always depend on circumstances, i.e. may be different from one jurisdiction to another, or from one episode to another.

The reasoning above assumes that the choice and specification of non-standard measures can basically be mapped into a single number: the additional accommodation needed beyond the zero lower bound. However, one may question this, and

1A more detailed treatment of forward guidance can be found in e.g. Filardo and Hofmann (2014), Campbell Evans et al. (2012), and Swanson (2017).

instead see non-trivial issues in the interaction of non-standard measures that imply that one cannot just add up the accommodation that each measure brings.

### **4.2 Negative Interest Rate Policy (NIRP)**

Four European central banks have applied NIRP in recent years, namely those of Denmark, Switzerland, Sweden and the euro area (for a survey of the implementation of NIRP by these central banks, see e.g. Bech and Malkhozov (2016). In addition, the Bank of Japan introduced NIRP in early 2016. In principle, the rationale for applying negative rates under some circumstances is obvious from the Wicksellian logic above. It could be argued that in 2008, the policy-adequate short-term interest rate would have been as low as between −3 to −5%, i.e. if central banks had been able to implement negative rates at these levels, the crisis would have been more short-lived (avoiding the large scale economic contraction and associated welfare damage) and further non-conventional monetary policies (such as large scale asset purchase programs) with their complexities and side effects would not have been needed. Strong supporters of negative interest rates as an obvious policy tool are for instance Buiter (2009) and Rogoff (2017), who also discuss how to make negative rates possible.

### *4.2.1 Reasons for a Lower Bound*

### **<IP_ADDRESS> Lower Bound Created by the Zero Remuneration of Banknotes**

Deeply negative interest rates should eventually lead to an explosion of the demand for banknotes, as banknotes have a zero remuneration. Indeed, it could be argued that all economic agents (banks, investors, households) can escape negative interest rates by substituting negatively remunerated financial assets with banknotes (which have zero remuneration). This is a powerful and obvious argument against deeply negative interest rates, and the only solution to it would be to discontinue the existence of physical banknotes, e.g. by fully replacing them with central bank digital currency, which could be remunerated negatively when needed. However, critics argue that this would create a tool for central banks to expropriate savers (by imposing negative interest rates, see e.g. Bindseil et al. (2015) and that discontinuing banknotes would also destroy, a la George Orwell's "1984", the freedom provided by anonymous payments. In addition, banknotes are resilient to cyber-attacks and power outages and they score high in terms of financial inclusion, as they do not require even a mobile phone. These arguments prevail for the time being in most countries, and therefore banknotes will continue to limit the scope for negative interest rates to the levels reached over the last few years, i.e. not lower than around −100 basis points. This seems to be the level at which banknote demand could start to have substantial momentum and undermine the effectiveness of negative interest rates. As in the case of a central bank digital currency undermining household deposits with banks seen in section 2.8, a ballooning of central bank money holdings of households would imply that banks would lose large amounts of deposits and become more and more dependent on central bank credit. This would deplete collateral buffers and could put banks under liquidity stress, making it unlikely that bank lending rates will decline, i.e. undermining the effectiveness of NIRP. Banks may not want to pass on negative rates to household deposits to avoid triggering such a run on deposits. However, then, banks' profitability suffers, as discussed further under point 2 below. In principle, the banknote hoarding argument also applies, for example, to banks, who could, in an environment of excess reserves, such as prevailing typically in the negative interest rate countries, start to hoard cash.

In the financial accounts in Table 4.1, we assume that both households and banks started to hoard cash as a consequence of negative interest rate policies. Efficient arbitrage would allow asset holders to fully escape from negative asset remuneration, fully undermining the transmission of negative central bank rates to asset yields. These financial accounts also show the case when the cash hoarding goes so far as to switch back (despite the QE program captured by S) the banks' excess reserves into a liquidity deficit, implying growing needs of central bank credit, eventually creating potential liquidity stress on banks.

**Table 4.1** Banknotes hording under negative interest rate policies of the central bank

### **<IP_ADDRESS> Lower Bound Due to Negative Effects on Profitability of Banks**

It has been argued that negative rates undermine bank profitability and undermine the transmission of negative rates as banks would be unable to pass on negative rates to retail depositors—see e.g. Bech, and Malkhozov (2016, 39–40), or Brunnermeier and Koby (2019)

Most central banks applying NIRP have acknowledged the particular effects of negatively remunerated excess reserves combined with an unwillingness/inability of banks to pass on negative rates to retail deposits through an innovation in their operational framework: so-called **excess reserves tiering systems which exempt a part of the excess reserves from the application of negative interest rates**. The idea is that a tiering system would allow the combination of (i) negative rates still effective at the margin and therefore passed on to money and capital markets, and (ii) the exemption of parts of the excess reserves moderating negative effects on bank profitability that could weaken the effectiveness of NIRP. By disconnecting the two, the transmission of NIRP to bank lending rates could be improved, and the "effective" lower bound, at which further rate cuts are no longer effective in terms of reducing bank lending rates, could be lowered. Still, reserve tiering does not fully eliminate the effects of negative interest rates on bank profitability. One source of bank profitability is the spread between interest rates on sight deposits of banks and interest rates on loans to non-banks, which are of longer duration. As the former turned out to be far less sensitive to NIRP than the latter, the spread between the two declined. Actually this effect is however not specific to NIRP, but also occurs to some extent with low (positive) interest rate policies.

### *4.2.2 Criticism of the Negative Interest Rate Policy*

### **<IP_ADDRESS> Financial Market Functioning Under Negative Interest Rates**

Before the introduction of negative rates, there were some fears over whether money and other key financial markets can function at all with negative interest rates. As also noted by Bech and Malkhozov (2016, 37), steering short term interest rates into negative territory has not been particularly challenging, nor did financial markets change their behaviour in negative territory. One may add that the combination of NIRP and asset purchase programmes also pushed longer term bond yields into negative territory, e.g. for Switzerland, Japan and Germany for the entire risk-free yield curve, even beyond 10 years. Again, there was no indication of negative effects on market functioning.

### **<IP_ADDRESS> General Counterproductive Effects of Low/Negative Interest Rates**

Finally, a number of critical authors have argued that central banks' low (and by implication, also negative) interest rate policies are ineffective or, at the very least, have major negative side effects that central banks tend to underestimate. These authors also seem to suggest that acknowledging the problem of low interest rate policies could lead to the conclusion that central banks should increase nominal interest rates without delay. The main arguments are as follows.

Bindseil et al. (2015) and others discuss and refute these arguments. The European Systemic Risk Board (2016) and the BIS (2018) have prepared extensive studies on macroprudential issues related to low interest rates. Overall, it seems that problems arise if economic agents deny the new reality of low real and nominal interest rates, and therefore either continue making unsustainable return promises to investors, or try, through unsound risk taking, to generate returns that are unrealistic. Also, if agents did not see the low interest rate environment coming and therefore took positions (or run a business model) that in the low interest rate scenario undermine their solvency, a transition issue arises that needs to be addressed in a way that minimises damage for society while keeping in mind moral hazard issues.

In sum: **negative interest rates may be viewed as an obvious continuation of Wicksellian interest rate policies** when the neutral level of interest rates falls into negative territory, as has become more likely in an environment with low growth potential and high central bank credibility as inflation fighters. In this sense, NIRP could be classified as a conventional monetary policy approach, reducing the need for non-conventional policy measures in the narrow sense with their possible more problematic side effects (such as large-scale asset purchase programmes). That NIRP is effective has been demonstrated by its strong effects on both capital market rates and bank lending rates. At the same time, two lower bound problems have to be acknowledged, namely (i) the one where banknote demand would explode; (ii) the one in which bank profitability would be undermined in such a way that a further lowering of central bank interest rates no longer leads to decreases in bank lending rates, as partially observed in Switzerland. While the former is also determined by storage and insurance costs of banknotes, the latter also depends on the willingness and ability of banks to pass on negative rates to different types of depositors and the amount of excess reserves that banks hold with the central bank at negative interest rates. While the two lower bounds are partially linked (through the decision of banks on whether to pass on negative rates to depositors), they are not necessarily the same. Both lower bounds could be overcome through a discontinuation of banknotes and their full replacement by CBDC—which however is not considered for a number of reasons as banknotes still have specific advantages.

### **4.3 Non-Conventional Credit Operations**

Central banks have taken a variety of measures during the crisis to make their open market operations more supportive. Some of these measures relate to the lender-oflast-resort (LOLR) function, but even those are relevant from the monetary policy perspective. If the zero-lower bound is binding, strengthening the LOLR implies a reduction of funding stress to banks, which reduces pressure on them to deleverage or to increase the role of expensive funding sources. The LOLR therefore contributes to maintain the readiness of banks to provide credit to the economy at a moderate mark up to short-term risk-free rates.

First, central banks have **lengthened the duration of their lending operations**to banks, with the ECB going as far as four-year credit operations. Banks may consider a sequence of short-term borrowings from the central bank as inferior, from a liquidity risk perspective, to one longer-term borrowing operation. Consider three reasons for this: (i) Banks could perceive as uncertain the conditions under which central banks will provide short-term funding in the future (rates, access conditions, etc.). (ii) Even if the central bank commits to keep conditions for short-term access stable, e.g. it commits to full allotment at a given rate for its short-term operations for the next twelve months, banks may, as a matter of principle, find revolving short-term central bank refinancing less certain than twelve-month refinancing. (iii) Banks may be subject to some liquidity regulation, which treats longer-term refinancing from the central bank more favourably.

Second, central banks have replaced auction procedures to allocate central bank credit with **'fixed rate full allotment' (FRFA) operations**. The ECB has done so in October 2008 and ever since then has applied this simpler allotment procedure, which has the following advantages.

allotment tenders is almost equivalent to setting the standing facility rate at the level of the target rate, with the only difference that an open market operation is not continuously open.

Third, central banks have **widened the access of counterparties to their credit operations**. When interbank markets break down, then financial institutions without recourse to central bank credit are in trouble, as they can no longer manage their dayto-day funding needs through credit operations with banks and capital market access. Allowing direct central bank access makes them independent from the functioning of interbank and capital markets.

Fourth, central banks have introduced **"targeted" credit operations** which make favourable lending terms (or access in general) conditional on some desirable behaviour of banks, such as providing more lending to the real economy. The ECB has done this through its so-called TLTRO operations, the Bank of Japan through its "Loan support programme" (LSP) and the Bank of England through its "Funding for lending scheme" (FLS).

Fifth, central banks have started to provide **credit in foreign currency, notably in USD**. The ECB and the Bank of Japan have done so since the end of 2007, based on swap lines established between central banks (see e.g. Goldberg et al. 2010). If USD spot and swap markets are impaired, this ensures that banks have sufficient USD funding to meet their obligations in USD (see Sheets et al. 2018).

**Finally, widening the central bank collateral set applicable to credit operations** is both a monetary policy and a LOLR measure, and will be discussed in more detail in Chapter 6. However, as Bindseil (2013) argues, it is also an unconventional monetary policy measure as it supports the ability of banks to continue providing credit and lowers the intermediation spread between short-term risk-free rates and bank lending rates. At the ZLB, compressing this spread or at least counteracting its increase can be decisive in preventing the economy from gliding into a deflationary trap.

### **4.4 Outright Purchase Programmes**

All major central banks at some stage of the crisis that started in August 2007 established outright purchase programs for financial assets. The following eight objectives of such measures can be identified. The effects (3), (4), (6) and (7) can also be partially achieved through credit operations, but as credit operations are temporary, they may give less confidence to banks that the measure and the effects will be permanent.

### **(1) Reducing long-term risk-free interest rates**

The transmission of monetary policy takes place via longer term rates, as most economic decisions (e.g. building a house or a new factory) depend on longer term rates. Longer term rates can be decomposed into an average of expected short term rates, plus a term premium (according to the expectations hypothesis of the term structure of interest rates). If the zero lower bound constrains reductions in short-term interest rates, then the central bank may want to provide further accommodation by at least reducing term premia through purchases of long-term bonds. This argument has been key to the Fed and the Bank of England programs that started in 2009.

### **(2) Compress credit and liquidity spreads ("market maker of last resort")**

In a financial crisis, risky assets' prices may be depressed due to **asset fire sales and the absence of opportunistic buyers (i.e. buyers who buy whenever they feel an asset has become cheap)**. Moreover, arbitrage between asset classes may no longer work because of high bid-ask spreads, liquidity and capital constraints, systemic uncertainty, and self-fulfilling fears. In such an environment, the central bank can through purchases support depressed assets prices directly ease funding costs and constraints. Of course, central banks should not lower spreads below an adequate risk premium. Assessing what is an appropriate spread is of course challenging, in particular during a crisis.

### **(3) Inject excess reserves to strengthen banks' liquidity buffers**

Large scale outright purchase programmes push the banking system into a liquidity surplus position towards the central bank. This facilitates central bank liquidity management and the control of the overnight rate (which will be close to the deposit facility rate, or to the rate of remuneration of excess reserves). More importantly, a situation of general excess reserves may support financial stability as most banks will feel re-assured in their short-term liquidity position.

### **(4) Inject excess reserves to increase the money supply via the money multiplier**

Excess reserves targets play a role in the "money supply" approach to monetary policy implementation, as promoted in the official communication of the **Bank of Japan** between 2001 and 2016. This approach seems to be in line with traditional monetarist thinking.

### **(5) Absorbing risks from banks' into the central bank balance sheet and easing capital constraints of banks**

The central bank may reduce total risk in banks' balance sheets by buying risky assets from them. Therefore, if banks feel constrained in terms of economic or regulatory capital, outright purchases by central banks may attenuate these constrains and thereby support their lending behaviour and thereby ease monetary conditions. Taking credit risk into the central bank balance sheet, e.g. in the form of purchases of a corporate bond portfolios, implies the need for the central bank to develop relevant expertise on credit risk management for this asset class. Moreover, in case of debt restructurings, the central bank will have to vote in bond holder assemblies, i.e., contribute to decisions which are remote to its core functions, and which entail reputational risks.

### **(6) Substituting banks' illiquid with liquid assets to improve overall liquidity of banks**

Purchasing illiquid assets outright improves liquidity of banks, particularly if these assets were previously not eligible as central bank collateral, or only at a high haircut.

### **(7) Directly supporting through primary market purchases the funding liquidity of banks and/or other firms**

By purchasing in the primary market bonds from issuers (unsecured bank bonds, covered bank bonds, corporate bonds, etc.), the central bank supports directly the funding of these institutions. Central bank purchases of debt of non-financial corporates (NFC), if done in the primary market, directly refinance the real sector and thus can offset the unwillingness of banks to provide their usual lending and liquidity services.

### **(8) Threat to "purchase all real assets in the world" to counter perception of deflationary trap**

Central banks are in principle able to purchase all assets of the world with the money that they can issue without constraints—in particular in a deflationary context. When central banks launch such potentially infinite purchase programs, the other economic agents will become less willing to sell all their assets (including equity, commodities, etc.), and they will thus require higher and higher prices, and hence the purchasing power of the currency will fall. In the case of a credible central bank, this will be anticipated, and the announcement of such a purchase program should immediately defeat deflation.

### **Impact of purchase programmes on yield levels**

There is a growing empirical literature estimating the effects of large-scale asset purchase programmes on the risk-free yield curve and its further transmission to other interest rates and the real economy (a comprehensive recent study covering the programmes of the US, UK, Japan and the euro area is Agostini et al. 2016). Effects on long-term interest rates of recent large-scale asset purchase programmes are generally believed to be in the area of up to 100 basis points. In combination with NIRP, this would mean that these two policies together could achieve reductions of long-term funding rates of up to 200 basis points, which obviously means substantial further easing (NIRP also contributes to reduce long-term rates as expectations on future short-term interest rates decrease). When looking more precisely at the effects of purchase programmes on asset prices and long-term yields, it is important to distinguish between the following three effects (D'Amico and King 2011 were the first to investigate theoretical and empirical aspects of flow vs stock effects of the US Fed's asset purchase programmes):

which investors are able or willing to adjust their stocks, which also depends on who in particular holds the assets (a pension fund vs a bank in its trading book); (iv) the time between the announcement of the programme and its start (more time allows investors to prepare for selling assets and dealers to accumulate stocks waiting for the central bank).

• **Announcement effect**: if asset prices in principle reflect at any moment in time all available information, it can be expected that most of the impact on prices and yields materialises immediately when the central bank announces an asset purchase program. The announcement effect should be an anticipation of the stock effect, and not of the flow effect. The announcement effect will mainly depend on (i) the degree to which the announcement has not been anticipated (for example, when the ECB's PSPP was announced, markets hardly moved as it had been anticipated); (ii) the credibility of the central bank (determined, for example, by its history of meticulously implementing what it promises); (iii) how remote in the future the promised measures are (with non-perfect central bank credibility, more remote measures will have a lesser announcement effect than measures which are relatively nearby), (iv) the clarity of the announcement.

Central bank purchases with too short lead times (after the program's announcement) and at a too high pace distorts markets, in the sense of letting yields temporarily undershoot more than necessary. It also implies that the central bank will over-pay. Buying with too long lead times and with a too low pace unnecessarily delays the desired easing of financial conditions. Interestingly, in the case of limited central bank credibility, stronger flow effects may be desirable as they contribute to a quick price adjustment, i.e. a faster effectiveness of monetary easing, without this implying that the central bank purchases at excessive prices. A less credible central bank should therefore buy at a higher pace and start faster than a credible central bank, which can immediately achieve stock effects.

### **4.5 Distinguishing Between Conventional, Non-Conventional, and LOLR Policies**

Central banks have, despite the presumption that non-conventional measures have negative side effects and conventional measures have not, maintained non-standard measures beyond what is strictly implied by the existence of the ZLB: For example, the Fed's policy normalisation has consisted first in withdrawing accommodation through a number of standard interest rate increases, before reducing its stock of LSAP-securities. This was explained to be preferable because rate hikes would be easier to dose than the impact of changes of securities stocks on the stance of monetary policy. Moreover, it would be easier to switch direction by changes of interest rates than in terms of changes to a securities stock. Also, in 2007 and 2008 central banks undertook various non-standard measures without having yet reached the zero lower bound. Those may have related to LOLR measures, which may be beneficial for society regardless of having reached the ZLB or not (because the LOLR may save viable and solvent, but temporarily illiquid projects).

While the LOLR will only be discussed in Chapter 6, it is interesting here to try to put order into the three types of policy objectives determining the specification of central bank market operations. The following chart puts the LOLR into context with conventional and non-conventional monetary policies and assigns central bank financial operations and instruments to any of the three (overlapping) areas (Fig. 4.1).

**Fig. 4.1** Instruments and types of central bank policies

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# **Chapter 5 Financial Instability**

### **5.1 Liquidity, Asset Prices, and Default**

Financial crises always encompass liquidity crisis, and liquidity crises typically have a funding liquidity and a market liquidity component. In a *funding* **liquidity crisis**, the problem is credit availability: debtors cannot obtain credit to roll over their liabilities, as money and capital markets freeze. In a *market* **liquidity crisis**, bid-ask spreads and asset fire-sale discounts increase, reducing the total liquidity obtainable through asset liquidation.

Liquidity crises may lead to the inability of debtors to fulfil their contractual obligations and hence to their default, with additional economic damage. Financial crises are typically triggered by **strong downward revisions of asset values** (Bagehot 1873, Kindleberger and Aliber 2011). Asset price changes can be driven by any major unexpected news the economy, e.g. a natural disaster, a pandemic, an unexpected change of government and of economic policies, the outbreak of a war, the rise of a new major technology, a strong re-assessment of the inflation and thus central bank interest rate outlook, a strong collective re-assessment of the prospects of an asset class (e.g. burst of a housing bubble) etc. can all heavily impact on few, or various asset classes. For example, the outbreak of Covid-19 and the uncertain perspective of a return to the pre-Covid19 normal has reduced the value of assets owned by the international travel and tourist industry (cruise ships, airplanes, hotels, etc.).

**Strong declines of asset values have various negative effects on economic agents.** Solvency declines, which undermines the ability to access funding sources and the willingness and the ability to undertake risky projects. In the case of banks, a decline in solvency puts at risk compliance with capital adequacy regulations, adding urgency to deleveraging through the shrinking of lending or through asset fire sales. In the case of households, lower wealth reduces consumption, which will have recessionary effects.

**Table 5.1** Balance sheet of an indebted firm

The mechanisms of liquidity crises have been similar across time. Already Thornton (1802) noticed the problem of liquidity hoarding and bank runs, and how they relate to a lack of trust.

Bagehot (1873, Chapter VI "Why Lombard Street Is Often Very Dull, and Sometimes Extremely Excited") argues that while liquidity crises are caused by heterogenous exogenous events, their mechanics, once having been triggered, are similar:

Any sudden event which creates a great demand for actual cash may cause, and will tend to cause, a panic in a country where cash is much economised, and where debts payable on demand are large. …. Such accidental events are of the most various nature: a bad harvest, an apprehension of foreign invasion, the sudden failure of a great firm which everybody trusted, and many other similar events, have all caused a sudden demand for cash. And some writers have endeavoured to classify panics according to the nature of the particular accidents producing them. But little, however, is, I believe, to be gained by such classifications. There is little difference in the effect of one accident and another upon our credit system. We must be prepared for all of them, and we must prepare for all of them in the same way—by keeping a large cash reserve.

Understanding the logic of liquidity crises is a precondition for understanding the role of the central bank in stopping the escalation of liquidity crises and in addressing their economic consequences.

To illustrate how default probabilities increase in financial crises when assets values decline, consider in **Table** 5.1 the balance sheet of a leveraged financial or non-financial corporate with ε being a random variable impacting asset values.

Asset values can be thought of as being subject to periodic random shocks ε. Assuming simplistically that ε is N(0, σε) **then the probability of default (PD)** of a company, in the sense of the probability that its asset values will fall below the value of debt in the next time period, could be estimated as (-(·) is the cumulative standard normal distribution):

$$PD = P(E + \varepsilon < 0) = P(A + \varepsilon < D) = \Phi\left(-\frac{A - D}{\sigma\_{\varepsilon}}\right)$$

This formula for the probability of default is however a strong simplification, since asset values are not normally distributed, variables such as σε are not directly observable, time is continuous, and there is no unique horizon to consider. Moreover, default does not need to occur exactly when A + ε touches D, since default is eventually triggered by illiquidity. Merton's structural credit model (Merton 1974) is a more sophisticated version of this basic default probability model.

**Table 5.2** Annual default probability of rated debtors according to Standard and Poor's S&P 2020

**Rating agencies provide comprehensive statistics on how rated debtors performed**. For example, Standard & Poor's regularly publishes a statistical default study (e.g. S&P 2020). According to **Table** 5.2, the following annual default rates applied in the period 1981 to 2019.

The differences between good years (minimum) and bad years (maximum) are significant. The time series of annual default shows a sort of financial cycle with peaks in default frequency around 1991, 2001, and 2008. Creditors do not only care about the **probability of default**, but also about **what losses occur in case of default**. If all debtors rank pari passu, and if default occurs exactly when A + ε = D, then the "Loss-given default" (LGD, = 1 – recovery ratio) should be zero (the recovery ratio should be 1). However, evidence collected by rating agencies suggests that LGDs are on average around 50% (depending also on the debt instrument). This can have two explanations: (i) default often occurs only when A is already clearly below D (i.e. creditors did not realize that the company had negative equity, or there was no debt redemption date); (ii) the default event itself is costly, as default implies that organisational and human capital is destroyed and specific assets are liquidated at fire-sale prices (e.g. a sophisticated machine is sold at its raw material value, minus the costs of removing, transporting and dismantling the machine).

**The corporate finance literature provides estimates of the costs of default between 10 and 44%** (e.g. Glover 2016; Davydenko et al. 2012). The cost of default is one key reason for **central banks trying to prevent defaults of sound companies due to illiquidity**. We will see below that in settings of asymmetric information which are typical for the high uncertainty prevailing in financial crisis, credit and asset markets can break down such that illiquidity and default can occur even for firms which are solvent and viable.

The "credit channel" literature has analysed since the 1980s how high credit riskiness and low equity have been identified for a while as an issue for monetary policy transmission. According to this literature, low equity implies higher agency costs in the lending between banks and corporates. Lower bank equity implies higher agency costs between holders of bank liabilities and banks. Higher agency costs result from a deterioration of the alignment of incentives between debt and equity owners when equity levels fall (e.g. Holmström and Tirole 1997). In addition, debtors with insufficient equity will attempt to restore their creditworthiness by aiming at deleveraging, causing economic contraction and deflationary tendencies.

To sum up, it is important to distinguish the following four key concepts for troubled debtor:

An indebted corporate can default despite being solvent. It may be illiquid because a systemic liquidity crisis situation made all possible lenders stop lending, and it has to refinance a maturing loan or debt instrument. A corporate may be insolvent without yet defaulting, because no debt payment is due. Eventually, a corporate which is clearly insolvent will also end up being illiquid and default because it does not make sense for creditors to give fresh loans to an insolvent debtor.

### **5.2 Conditional and Unconditional Insolvency, and Bank Runs**

Debtors that are **solvent conditional on the access to funding** may become insolvent if funding constraints force them to undertake fire sales of some of their assets at some specified horizon. This relates to problems explained in Sect. 1.2 that fair book values of assets are normally higher than their (short-term) liquidation value. The following assumes that the fair value of the assets of an indebted company is 1, and that these assets are ordered from the most liquid to the least liquid (x-axis). Assume further that L(x) is the liquidity generated by liquidating at some time horizon, say one week, the share x of assets that is most liquid. It follows that L(0) = 0 (if no assets are sold, no liquidity is generated), L(1) ≤ 1 (a sale of all assets provides at the very maximum their book value), dL(x)/dx ≥ 0 (the liquidity generated cannot decrease with more assets being sold) and d2L(x)/d2x ≤ 0 (each new unit of asset sold does not provide more liquidity than the previous one, as assets were ranked from the most to the least liquid). In other words, the liquidity generation can be described by a concave monotonously non-decreasing function of x. We can also define similarly the fire-sale loss function F(x) = x – L(x) which indicates the fire-sale losses generated by selling a share of assets x, starting with the most liquid assets. The function f(x) = dF(x)/dx is the marginal fire- sales loss function indicating the size of fire-sale losses resulting from selling the asset ranked at x, and q(x) = dL(x)/dx = 1 – f(x) is the marginal liquidity generated by selling the asset ranked x.

**Figure** 5.1 illustrates the **simple case when f(x) = x, and therefore q(x) = 1 – x, F(x) = x**2**/2 and L(x) = x–x**2**/2**. For example, if the company needs to generate a liquidity of L1 in order to meet a due payment to its bondholders, then it needs to solve the quadratic equation L1 = x –x2/2 or x2/2 – x + L1 = 0. The relevant solution to this problem is x = 1–(√1 − 2*L*1). For example, if L1 = 0.4, then the required

**Fig. 5.1** Liquidity generation and losses due to fire sales—example with F(x) = x

fire sales are x = 0.55. **Figure** 5.1a draws the marginal fire sale loss function f(x) for the assets x ordered from the most liquid to the least liquid. **Figure** 5.1b shows the total fire-sale loss F(x) and total liquidity generated L(x) for f(x) = x, again for liquidity-ordered assets. **Figure** 5.1c shows the necessary fire sales of assets as a function of the liquidity to be generated, i.e. x = L−1(L1). Last but not least, **Fig.** 5.1d illustrates the concrete case when the firm needs to generate the cash flow of 0.4, whereby a share of 0.55 of the assets need to be sold.

Assume the company has equity E, and has to generate through fire sales a cash flow at a certain time horizon T. For instance, because it needs to repay a debt instrument at maturity and is unable to roll it over or find another form of financing. For every time horizon T the function LT(x) is non-decreasing and therefore invertible, so that we can write the inverse function xT(L), describing the share of assets that must be liquidated for generating liquidity of a specified size L. Then, the company is solvent conditional on the need to generate a cash flow L1 at time horizon T = 1 if, and only if, the fire-sales losses caused by the liquidation are lower than the equity of the company:

$$\mathbf{F}\_1(\mathbf{x}\_1(\mathbf{L}\_1)) < \mathbf{E}$$

We expect that the higher the time horizon, the lower the losses: if T1 > T2, then for every x, we expect that fT2(x) ≥ fT1(x). To distinguish fire-sale losses due to time constraints from fire-sale losses due to asset specificity, one could say that the marginal fire-sale losses **due to asset specificity** (fAS) are the marginal fire-sale losses without any time pressure, i.e.

$$f\_{AS}(\mathbf{x}) = \lim\_{T \to \infty} f\_T(\mathbf{x})$$

Consequently, the marginal fire-sale losses due **exclusively to time pressure** (fT,P) can be defined as the difference between the marginal loss function and the asset-specificity related marginal loss function.

$$f\_{T,P}(\mathbf{x}) = f\_T(\mathbf{x}) - f\_{AS}(\mathbf{x})$$

**Figure** 5.2 provides an example of marginal fire sale loss functions for the same company at different time horizons.

**To ensure funding stability, the bank (or any debtor) should ideally be able to ensure liquidity and solvency at all time horizons**. For example, if one debt position of an amount L1 matures in 1 week, and another of L2 in four weeks, then both conditions

$$\mathcal{F}\_{\text{1Week}}(\mathbf{x}\_{\text{1Week}}(\mathbf{L}\_1)) \precleftarrow \to \text{and} \\ \mathcal{F}\_{\text{4Week}}(\mathbf{x}\_{\text{4Week}}(\mathbf{L}\_1 + \mathbf{L}\_2)) \precleftarrow$$

should be fulfilled in order to ensure that the debtor is stable and can communicate to its current or potential future creditors that it will be fine anyway (even conditional on no roll-over of funding). Chapter 6 will provide a more precise model in which, for a specific functional form of L(x) and F(x), precise conditions for funding stability will be derived.

For the moment, it is useful to retain that (i) solvent debtors can be sub-classified into those which are solvent regardless of assumptions taken with regards their ability to roll over some debt instruments maturing at some horizon, and those which are solvent only conditional on accessing fresh funding; (ii) the latter may create multiple equilibrium situations, as further explained in chapter 6; (iii) both a negative asset value and a deterioration of asset liquidity can push a lender from being unconditionally solvent into being solvent only subject to funding renewal.

**Bank runs have been a major issue at least since the nineteenth century**, with particularly devastating episodes in the early 1930s, leading to the general introduction of deposit insurance schemes. More recent runs occurred in the UK (Northern Rock in 2008), and in Greece and Cyprus during the euro sovereign debt crisis. The latter runs mainly materialised through electronic transfers of deposits to accounts with non-domestic euro area banks, i.e. without queues in front of the banks to withdraw cash. Bank runs have been extensively modelled in the economic literature, such as in Diamond and Dybvig (1983) and Rochet and Vives (2004). The particularity of bank runs is their self-fulfilling property: once a run on a bank starts, it can lead to the default of the bank, confirming the individual wisdom of those who were first in the queue to withdraw their money. We will illustrate in chapter 7, with a very simple but powerful strategic bank run model, that a bank can essentially be in three states in terms of stability of its short-term liabilities:

**A switch from state a) to state b) can occur if: (i) asset liquidity deteriorates; and/or (ii) asset values decline, implying a decline of equity. A switch from state a) or b) to state c) will occur if asset values fall such that equity becomes negative.** In crisis situations, both factors tend to materialise with higher probability than usual, in particular for banks with more limited solvency and liquidity buffers, or with an unlucky asset concentration, i.e. an asset concentration towards those assets which by bad luck suffer from value losses and a decline in liquidity. This will be taken up in chapter 6 and in particular in the model of Sect. 6.5.

### **5.3 Illiquidity in Credit and Dealer Markets**

In this section we attempt to shed further light on the mechanisms leading to illiquidity in credit and asset markets.

### *5.3.1 Credit Markets*

**Information asymmetries can lead to a freeze of credit markets** and related economic damage (e.g. Stiglitz and Weiss 1981; Bolton and Freixas 2006). The following basic model of a credit market freeze is based on Flannery (1996). It is assumed that entrepreneurs who would like to borrow are either "Good" or "Bad", i.e., will repay or not, respectively. The proportion of Good entrepreneurs is g, while the proportion of Bad entrepreneurs is (1 – g). The entrepreneur needs a unit bank loan to finance her project. At the end of the period, Good entrepreneurs' projects will be worth VG > 1, which suffices to repay loans, assuming that the lending interest rate was not higher than VG – 1.

**Banks are imperfect in assessing loan applicants**. The creditworthiness will be assessed correctly with a probability p. The bank obtains either a signal, *SG* (good borrower) or a signal *SB* (bad borrower). If the borrower is actually good, then with probability p > g, a good borrower signal (*SG*) is captured, and the bank may lend. With probability (1 – p), the bad borrower signal (*SB*) is received, and no loan will be provided. If the borrower is actually bad, then the bad borrower signal will be received by the bank with probability p, and the good borrower signal with probability (1 – p). With the help of Bayes' Law (shown below explicitly only for the first of the four cases), this allows the calculation of the probabilities of all combinations of signal and actual quality of entrepreneurs:

$$P(S\_G|G) = \frac{P(S\_G \cap G)}{P(G)} \Rightarrow P(S\_G \cap G) = pg$$

$$P(S\_B|B) = p(1-g)$$

$$P(S\_G|B) = (1-p)(1-g); P(S\_B|G) = (1-p)g$$

Banks are assumed to be competitive and to have zero profits, implying that **profits from lending to good borrowers must on average compensate the credit losses due to bad borrowers**. Let j\* be the interest rate that a bank needs to set to have zero expected profits. As lending occurs only if a good signal is obtained, non-zero pay offs occur only in two out of the four possible cases. The zero-profit condition can therefore be formulated as follows for a bank that itself pays zero interest on its liabilities:

$$j^\* \cdot gp + (-1)(1 - g)(1 - p) = 0$$

$$\Rightarrow j^\* = \frac{(1 - g)(1 - p)}{gp}$$

Lending, i.e., an active credit market, will take place as long as VG – 1 ≥ j\*. Otherwise, even good entrepreneurs would make losses and will therefore better not launch their projects. For example, writing j\* as j\*(g,p): j\*(0.5, 0.5) = 100%, j\*(0.5, 0.75) = 33%; j\*(0.5, 0.9) = 11%; j\*(0.7, 0.9) = 4.8%, etc.

**The model illustrates three causes for a break-down of credit markets**: first, a decline of g, the share of Good entrepreneurs; second, a decrease of VG, the project return of Good entrepreneurs; third, a decrease of p, the power of the banks' screening technology. All three effects are associated with a negative economic shock. The model thereby explains why economic shocks trigger credit market crises, and moreover why these effects can be so abrupt: according to the model, a minor further parameter deterioration can make the market collapse, because the critical condition no longer holds. But even before such a complete funding market breakdown, economic deterioration is already felt in the form of an increasing equilibrium lending rate j\*. This matters at the zero lower bound, when the central bank can no longer compensate such effects through a lowering of its interest rates.

### *5.3.2 Dealer Markets*

Assume a dealer market in which dealers commit bid and ask prices for some standard quantity q. The bid-ask spread, which measures asset liquidity, typically increases in financial crises. The following simple model takes up basic elements of Kyle (1985) to explain why asset liquidity in a dealer market will deteriorate in a financial crisis. The model assumes that:

• the **fair value of the asset, At**, changes every day according to At = At–1 + εt with εt being a symmetric random variable with expected value 0.

Bid-ask spreads and trading volumes will reflect the existence of insider information, as captured by σ2 ε . The competitive market maker will set z such that expected profits are zero. Expected profits have two components: the profit of market makers generated by the noise traders is z·W(z). If *f*ε(*x*)is the density function of asset value innovations εt, then the expected losses of the market maker due to insiders are:

$$\left[\int\_{-\infty}^{-\frac{\varepsilon}{2}} \left(\mathbf{x} + \frac{\mathbf{z}}{2}\right) f\_{\varepsilon}(\mathbf{x}) d\mathbf{x} + \int\_{\frac{\varepsilon}{2}}^{\infty} \left(\mathbf{x} - \frac{\mathbf{z}}{2}\right) f\_{\varepsilon}(\mathbf{x}) d\mathbf{x}\right] q^{\frac{1}{2}}$$

The competitive equilibrium bid-ask spread z is the one in which the expected profits of the market maker are zero, with the profits extracted from noise traders compensating exactly the expected losses due to insiders. The model illustrates the empirical pattern that information intensity and price volatility of assets reduce asset liquidity, as measured by bid-ask spread, for example. It thereby also explains the strong increases of bid-ask spreads in financial crises, which tend to be characterised by an intensive news flow and high uncertainty.

### **5.4 Increasing Haircuts and Margin Calls**

**Financial exposures are often protected by collateral**, also called "margin", such as in particular in the following three cases: (i) Interbank repo operations (i.e. collateralised interbank lending); (ii) Lending of banks to non-banks: e.g. mortgage loans; loans of banks to corporates, to hedge funds, etc. (iii) derivatives transactions, be they via central clearing counterparties (CCPs), or "over-the counter" (OTC), i.e. directly between counterparties.

The **unexpected request of large amounts of additional collateral ("margining")** can trigger liquidity crises and depreciate asset values via forced fire sales because margin requirements limit leverage for investors that use as collateral the assets they invest in. Large marginal calls of counterparties were the eventual trigger of the defaults of Lehman brothers in September 2008. Margin calls of CCPs in March 2020 related to the outbreak of the Covid-19 pandemic also created significant stress on banks, but remained manageable (see Huang and Takats 2020).

In financial crises, margin requirements tend to increase substantially. If the cash investor (or a CCP) wants to maintain the probability of a loss conditional on counterparty default at a certain confidence level, then it needs to increase haircuts whenever volatility increases, liquidity decreases, or the desired protection level increases. Haircuts are often set such as to limit the probability of a loss at a certain confidence level in the scenario of collateral liquidation due to borrower default. **The haircut is thus calculated on the basis of the following factors:**

We obtain therefore the following adequate haircut h for a daily price volatility σ, a liquidation horizon of T days, and a confidence level β to avoid a loss in case counterparty default and collateral liquidation:

$$h = \Phi^{-1}(\beta)\sigma\sqrt{T}$$

**In a financial crisis, haircuts will increase because** (i) cash providers may seek a higher confidence level of protection as their capital buffers may have suffered due to the crisis, implying a need to reduce overall risk taking: β ↑⇒ Φ−1(β) ↑; (ii) asset price volatility increases: σ ↑; (iii) it takes longer to liquidate assets without the liquidation having additional price effects: T↑. Increases in haircuts will mean less leverage, and thus, for a given level of capital, the need for fire sales with various potential second round effects.

### **5.5 Interaction Between Crisis Channels**

The various crisis channels described in this chapter interact and create vicious circles of deteriorating solvency, liquidity and default leading potentially to an economic meltdown:

• Asset value declines **may lead to immediate insolvency**, which normally will lead to an inability to roll over funding, and eventual default.

Once lending to the real economy tightens, a recession can occur and will lead to additional losses via renewed asset price declines and impairment of banks' loan portfolios. The resulting dynamics may call for external circuit breakers, including in particular the central bank. Central banks are not subject to liquidity constraints and can in theory provide unlimited liquidity to banks. Thereby, they can suppress the liquidity part of any vicious crisis circle, making the central bank (unwillingly) the key player deciding on the fate of banks and other leveraged entities.

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# **Chapter 6 The Central Bank as Lender of Last Resort**

In this chapter we review the function of the central bank as lender of last resort (LOLR), starting from the understanding of financial crises developed in the previous chapter. We recall long-established LOLR principles: proactive lending, inertia of the central bank risk control framework, and risk endogeneity. Because of its systemic role, a central bank should not tighten its collateral framework in a crisis, as restrictive policies are likely to not only increase the overall damage done by a crisis to society, but to even increase central bank losses. We explain in more detail the main reasons why a central bank should act as LOLR: prevent negative externalities from fire sales; its unique status as institution with unlimited liquidity; its status as a risk-free counterparty making others accept to deliver collateral to it even at high haircuts; and its mandate to preserve price stability. We distinguish three different forms of LOLR: elements built into the regular operational framework; readiness to relax parameters in a crisis; and provision of emergency liquidity assistance to individual firms. We then discuss what could be the optimal propensity of a central bank to engage in LOLR activities and outline possible trade-offs. Last but not least, we develop a bank-run model which highlights the role of asset liquidity and central bank eligible collateral. We calculate through a model variant with binary asset liquidity and uniform central bank collateral haircut, but then also introduce a model variant with continuous asset liquidity and haircuts.

© The Author(s) 2021 U. Bindseil and A. Fotia, *Introduction to Central Banking*, SpringerBriefs in Quantitative Finance, https://doi.org/10.1007/978-3-030-70884-9\_6

### **6.1 Principles and Rationale for the Central Bank Acting as Lender of Last Resort**

### *6.1.1 Origin and Principles of LOLR*

While large-scale and successful LOLR measures of central banks can be traced back to at least 1763 (e.g. Bindseil 2019), today's thinking on the LOLR function is still strongly inspired by nineteenth century experience, and in particular Walter Bagehot's *Lombard Street* of 1873 (see also e.g. Goodhart 1999; Goodhart and Illing 2002). Consider three key insights of nineteenth century experience which still appear valid today.

**Lend pro-actively while preserving the safety of the central bank**. In a hearing of the Lords' Committee in 1832, Bank of England director Jeremiah Harman summarised the Bank's actions in the panic of 1825 as follows (see Bagehot 1873):

We lent… by every possible means, and in modes that we never had adopted before; we took in stock of security, we purchased Exchequer bills, we made advances on Exchequer bills, we not only discounted outright, but we made advances on deposits of bills to an immense amount; in short, by every possible means consistent with the safety of the Bank;… seeing the dreadful state in which the public were, we rendered every assistance in our power.

Harman presents the Bank of England's action as having been *creative and proactive*, i.e. to have innovated to find the best ways to support funding liquidity of financial institutions, the only constraint to creativity being the need to preserve the "safety of the Bank", i.e. limit additional risk taking.

**Inertia of risk control framework.** Bagehot (1873) himself advises the Bank of England that, in a crisis, it should maintain its risk control framework broadly unchanged, and not tighten it similarly to private lenders as a reaction to a worsened asset quality and liquidity, as well as higher volatility, etc.:

If it is known that the Bank of England is freely advancing on what in ordinary times is reckoned a good security and on what is then commonly pledged and easily convertible, the alarm of the solvent merchants and bankers will be stayed. But if securities, really good and usually convertible, are refused by the Bank, the alarm will not abate, the other loans made will fail in obtaining their end, and the panic will become worse and worse.

Bagehot refers to various episodes in which the Bank of England did not follow this principle and ended up making the crisis worse than it would have needed to be.

**Risk Endogeneity**. Bagehot argues that supportive liquidity provision could be necessary *to minimize* the Bank of England's eventual own financial risks, because it would be the only way to prevent a financial meltdown with unavoidable large losses also for the Bank of England:

(M)aking no loans as we have seen will ruin it (Bank of England); making large loans and stopping, as we have also seen, will ruin it. The only safe plan for the Bank is the brave plan, to lend in a panic on every kind of current security, or every sort on which money is ordinarily and usually lent. This policy may not save the Bank; but if it does not, nothing will save it.

In other words, the riskiness of exposures would itself be endogenous to the central bank measures. Liberal central bank lending could imply lower central bank financial risk taking than tight risk controls, turning upside down the logic of private lenders.

### *6.1.2 Why Should Central Banks Be Lenders of Last Resort?*

We identify five reasons for a central bank to act as lender of last resort in a financial crisis.

### **<IP_ADDRESS> Negative Externalities of Funding Liquidity Stress**

Public authorities may intervene in markets in case of negative externalities. A major negative externality of bank stress relates to the fire sale spiral induced by liquidity problems of individual banks. If banks are forced to sell assets to generate liquidity, these sales likely depress market prices. In turn, this generates renewed solvency and liquidity stress for banks, possibly triggering further fire sales, etc. Central bank loans which reduce the need for asset fire sales can prevent such a downward spiral. Asset fire sales are not the only form of negative externalities of bank funding stress and illiquidity-induced default. Other negative externalities are, for example, the contagion of depositors' fears if they observe a bank run, possibly leading to further bank runs such as observed in the early 1930s.

### **<IP_ADDRESS> Central Banks Have Unlimited Liquidity (in a Paper Standard)**

Unlike leveraged private entities, a central bank is not threatened by illiquidity in the currency it issues. Modern central banks are endowed with the monopoly and freedom to issue legal tender. It is therefore opportune that, in case of a liquidity crisis when all financial and non-financial institution tend to hoard liquidity, central banks remain willing to lend and to hold illiquid assets outright or as collateral. This is unrelated to negative externalities, and even if a central bank were purely profitoriented, its unique access to liquidity justifies lending and purchases of illiquid assets in a crisis.

### **<IP_ADDRESS> Haircuts Are a Particularly Effective Risk Mitigation Tool for Central Banks**

**Haircuts are an effective tool if the collateral provider is more credit risky than the cash investor**. In contrast, haircuts are less effective if cash provider and collateral provider are equally credit risky since the implied protection of the cash provider is at the expense of the collateral provider (Ewerhart and Tapking 2008). Therefore, simply increasing haircuts in symmetric interbank repo markets is not an adequate solution to provide more risk protection, while it is for asymmetric relationships, such as the one between a prime bank lending to a hedge fund. From the perspective of the collateral provider, a central bank is a risk-free counterparty as it cannot default and will always return pledged collateral. Central bank credit against illiquid collateral can be well-protected through high haircuts, without the collateral provider feeling unduly exposed. Against any other cash provider, i.e. against any credit risky cash provider, the collateral provider would likely be unwilling to accept the exposure implied by high haircuts.

### **<IP_ADDRESS> Central Banks May Have Superior Information**

A central bank may have, as bank supervisor, better information on the credit worthiness of banks in need of liquidity, compared with other market participants. Moreover, as a public entity not competing with banks, banks may be willing anyway to share private information with a central bank to establish their creditworthiness. In contrast, banks may be unwilling to reveal private information to competitors or private investors, even if this is made a pre-condition to obtaining funding from them. This may be particularly relevant when decisions need to be taken urgently.

### **<IP_ADDRESS> LOLR as an Unconventional Monetary Policy at the ZLB**

Taking LOLR measures may be decisive for a central bank to achieve its mandate to maintain price stability and to prevent the economy from falling into a deflationary trap. LOLR measures can prevent bank intermediation spreads from increasing in a crisis situation, which may be essential from a monetary policy perspective if the central bank has exhausted conventional monetary policy because of the zero lower bound (ZLB) on interest rates. This will be illustrated further by the model in Sect. 6.3.

### **6.2 Forms and Propensity to Act as LOLR**

### *6.2.1 Forms of LOLR*

**The central bank LOLR function can take three forms**: (a) LOLR built into the regular operational framework of the central bank; (b) LOLR added through changes of the framework and additional LOLR operations for all banks in crisis times; (c) emergency liquidity assistance to individual banks or, more rarely, even to non-banks. We consider these three one after the other.

### **LOLR built into the regular operational framework**

The following elements determine the LOLR content of the regular operational framework.

### **Readiness of central banks to add LOLR content to the operational framework in crisis times**

The impact of the LOLR on bank behaviour will not be limited to the LOLR content of the operational framework in normal times. What matters as well is the bank's liquidity in a scenario of financial market stress. Anticipating this case also includes building expectations on the readiness of the central bank to adjust the abovementioned parameters that determine the LOLR content of the operational framework. Expectations will be determined by historical experience and forward-looking central bank communication.

### **Readiness of central banks to provide emergency liquidity assistance (ELA) to individual banks**

ELA can be defined as a non-rule based LOLR activity for the benefit of individual banks. Of course, ELA also needs to take place within some legal framework, within the mandate of the central bank and ideally in a consistent manner. Limitations to ELA provision can result from:

(i) ELA collateral requirements (normally ELA collateral sets should be wider than the standard collateral set). (ii) Pricing of ELA, i.e. what surcharge relative to monetary policy credit operations is imposed (some surcharge is typically applied). (iii) Relevance to preserve systemic financial stability may be a precondition for granting ELA. The higher the hurdle set by the central bank in declaring a systemic financial stability interest before granting ELA, the less a bank can rely ex-ante on it, in particular if a bank is small. (iv) Limitations on the duration of ELA (ELA is typically assumed to be of limited duration). (v) Possible requirement that ELA is only granted if the central bank is protected in addition by a government guarantee. Beyond additional risk protection, this may be considered useful as it requires an elected government to confirm its backing of ELA operations (but it should not delay very urgent and obvious ELA provision by the central bank). (vi) ELA counterparty set: While normal central bank credit is only granted to banks, ELA could also be granted to any other financial corporate (or in theory even to any debtor).

### *6.2.2 Overall Propensity of a Central Bank to Act as LOLR*

It is conceptually useful to first consider two extreme LOLR choices of the central bank.

Central bankers believe that the optimal LOLR is in **between these two extremes**. The LOLR strengthens the ability of the financial system to provide maturity and liquidity transformation as services to society. At the same time, putting some limits to the LOLR role is beneficial for society, to have some protection against information asymmetries and moral hazard, to avoid relying excessively on the abilities of supervisors and auditors, and generally to preserve stronger incentives to maintain funding market access and thereby market discipline. Proponents of a tight approach may argue that a supportive LOLR will lead to as many financial crises as a very tight one, but crisis will be messier because when they occur the financial leverage will be much higher ("four-wheel vehicles make you get stuck in areas which are more difficult to access when you need to be rescued").

Assume for a moment that **we capture in the unit interval [0,1] the supportiveness of the LOLR framework** of a central bank and let the most restrictive framework described above be represented by 0 and the most forthcoming framework by 1 (it is of course a simplification to assume that designing the LOLR framework is a one-dimensional problem). One can map the LOLR unit interval into at least five effects, which should not be expected to be identical, although often this seems to be implicitly assumed:

(5) **Market discipline and funding market functioning** can be thought of as either falling monotonously, or as mirroring the financial fragility curve, i.e. it would benefit from some moderate LOLR, but is undermined if the LOLR is excessive. Section 6.5 shows that when asset liquidity deteriorates after an exogenous shock, then the LOLR can preserve funding market access for solvent banks, but not for insolvent banks, while a restrictive LOLR will imply a run also on solvent banks. In this sense a more supportive LOLR can allow for a more effective market mechanism than a very restrictive one.

### **Moral hazard and central bank losses**

A popular theme in papers on the LOLR is moral hazard, but the concept often remains vague. One pragmatic view is that moral hazard only materializes in the context of the LOLR if the central bank faces actual losses from its credit operations. This interpretation also has the advantage that it would reduce the complexity of the LOLR design problem by one dimension and map something vague and complex (moral hazard) into something concrete and more measurable (central bank risk taking—even if complicated by endogeneity). If central banks are worried about moral hazard, they could tighten risk control measures (in normal times, to not be pro-cyclical) so that the probability of central bank credit losses declines even further.

### **Excessive stigmatization of the LOLR?**

Sometimes central banks worry that banks attach excessive stigma to recourse to the LOLR. For example, recourse to the Discount Window is considered to remain stigmatized in the US although the Fed has wanted to change this since 2002 (Armantier et al. 2015). Also, in a number of credit open market operations of central banks during the financial crisis, aversion of banks to participate materialized so that the accommodation that the operations aimed at could not be achieved. Excessive stigmatization seems to go in the opposite direction of moral hazard. Central banks should therefore have tools in hand to adjust *in both directions* the willingness of banks to come to LOLR operations.

### **6.3 Central Bank Collateral as a Key LOLR Parameter in a Simple Bank Run Model**

In this section we will integrate the LOLR in a bank-run model. In Sect. 6.3.1 we will introduce a bank-run model in which a bank owns two kinds of assets, a completely liquid asset and a completely illiquid asset. In Sect. 6.3.2, asset liquidity will be described by means of a power function, which allows additional insights.

### *6.3.1 A Bank Run Model with Binary Levels of Asset Liquidity*

Throughout this section, we consider the stylized bank balance sheet in **Table** 6.1. The total length of the balance sheet has been set to unity. Assets are grouped into two homogeneous classes in terms of asset liquidity and fire-sale discounts (- ∈[0,1]). There are three types of liabilities, equity, long-term debt, and short-term deposits (with e ∈[0,1], t ∈ [0,1] and d ∈ [0,0.5]).

The stylized balance sheet is sufficient to capture one key issue of banking: how to ensure the confidence of short-term depositors of the bank such that they do not easily withdraw deposits because of perceived credit risk, triggering self-fulfilling destructive dynamics ending in bank default. Confidence can be sustained by two means. *First*, the bank may limit the role of short-term funding. However, in general, households and institutional investors prefer to hold short-term debt instruments over long-term debt instruments and equity and request a higher return rate on the latter two types of claims, so that long-term debt and capital is associated with higher funding costs for the bank. *Second*, the bank may aim at holding sufficient amounts of liquid assets, both in the sense of being able to liquidate these assets in case of need, and to pledge them with the central bank at favourable haircuts. However, on average, liquid assets generate lower returns than illiquid ones. We now consider the representative bank in more detail.

### **<IP_ADDRESS> Asset Liquidity and Central Bank Collateral Treatment**

Assume two types of assets with extreme liquidity properties:

At the same time, it is assumed that, when accepting bank assets as collateral, the central bank applies a homogeneous haircut h on all assets. In other words, the central bank haircut and collateral framework is not sensitive at all to asset liquidity.

**Table 6.1** A stylised bank balance sheet to analyse funding stability of a bank

**Fig. 6.1** Liquidity generation in a binary level of liquidity. Left: by liquidating all assets. Centre: by pledging all assets with the central bank. Right: liquidity-maximising combination

We summarise these assumptions in **Fig.** 6.1. Obviously in this case it never makes sense to fire-sell the illiquid assets as this would generate no liquidity but maximum losses. The illiquid assets should instead be pledged with the central bank. At the same time, to generate maximum liquidity, it makes sense to sell the liquid assets and to not pledge them.

### **<IP_ADDRESS> Bank Liabilities**

There are four types of liabilities: (i) *Short*-*term liabilities* are equally split to two ex-ante identical depositors; (ii) *Long*-*term debt* does not mature within the period considered and is ranked *pari passu* with short-term debt in case of liquidation of the bank; (iii) *Equity* is junior to all other liabilities and is also a stable funding source; (iv) *Central bank borrowing* is zero initially but can substitute for outflows of short-term liabilities in case of need.

### **<IP_ADDRESS> Timeline**

The model is based on the following timeline:

	- (i) The bank substitutes lost deposit with central bank credit, assuming the bank has sufficient eligible collateral.

### **<IP_ADDRESS> Equilibrium**

We use a *Strict Nash No*-*Run (SNNR)* equilibrium concept. The decision set of depositor i (i = 1,2) from which he will choose his decision Di is {Ki, Ri}, where "K" stands for "keeping" deposits and "R" stands for "run". The payoff function of depositor i is: Ui = Ui(D1, D2). The strategic game is symmetric, i.e. U1(K1, K2) = U2(K1, K2), U1(K1, R2) = U2(R1, K2), U1(R1, K2) = U2(K1, R2), U1(R1, R2) = U2(R1, R2). This allows us to express in the rest of the model conditions only with reference to one of the two players, say depositor 1.

A *Strict Nash equilibrium* is defined as a strategic game in which each player has a unique best response to the other players' strategies (see Fudenberg and Tirole 1991, 11). A *Strict Nash No*-*Run (SNNR) equilibrium* in the run game is therefore one in which the "no-run" choice dominates the "run" choice regardless of what the other depositors decide, i.e. an SNNR equilibrium is defined by:

$$\mathbf{U}\_1(\mathbf{K}\_1, \mathbf{K}\_2) > \mathbf{U}\_1(\mathbf{R}\_1, \mathbf{K}\_2) \cap \mathbf{U}\_1(\mathbf{K}\_1, \mathbf{R}\_2) > \mathbf{U}\_1(\mathbf{R}\_1, \mathbf{R}\_2)$$

A strict run equilibrium applies if U1(R1, K2)>U1(K1, K2) ∩ U1(R1, R2)>U1(K1, R2), and a multiple equilibrium case arises if U1(K1, K2)>U1(R1, K2) ∩ U1(R1, R2) > U1(K1, R2).

To identify the *cheapest sustainable funding structure*, we now define as a liquidity-stress strategy (LSS) of a bank a mapping of the assets of the bank into either their use as fire-sale reserves or as collateral for recourse to the central bank. In the chosen simple case, the choice of the LSS is trivial for the bank: liquid assets should be fire-sold, and illiquid assets should be pledged as collateral. This keeps liquidity generation capacity at a maximum and fire- sale losses at the minimum (zero). It is shown below that an SNNR applies if the liquidity generating power of the bank assets is at least equal to the deposits of one depositor, and equity is non-negative:

$$\mathbf{L} = A + (\mathbf{l} - A)(\mathbf{l} - \mathbf{h}) \ge \mathbf{d} \quad \text{and} \quad \mathbf{e} \ge \mathbf{0}$$

There are two possible states for solvency: either e ≥ 0 (the bank is solvent), or e < 0 (the bank is insolvent). The liquidity condition can be divided into three cases: either liquidity is enough to pay out both depositors, or only one depositor, or neither. **Table** 6.2 shows the depositor's equilibrium decision for the resulting six cases (because of symmetry it is the same for both players).

If equity is negative, run is always the equilibrium choice. If equity is positive a no-run equilibrium can be ensured if liquidity is sufficient to pay out one depositor. In establishing the payoffs in case of liquidation, we take a simplifying assumption, namely that the central bank, liquidating the assets pledged by the bank after default, recovers exactly the liquidity it had provided through the pledge, while the security buffer granted by the haircut is completely depleted, i.e. equal to L = - + (1 – -)(1 – h). Under this assumption payoffs can be calculated in scenarios of liquidity-induced defaults by just taking the pay-outs before the moment of default.

### **<IP_ADDRESS> Positive Equity**

First, we analyse cases (1) to (3), in which equity is positive.

**Table 6.2** Equilibrium decision of depositors depending on liquidity and solvency of the bank

**Table 6.3** Pay-offs to depositors if L ≥ 2d and e ≥ 0

**Table 6.4** Pay-offs to depositors if d ≤ L < 2d and e ≥ 0

**Table 6.5** Bank's balance sheet at the moment of default in the non-equilibrium run scenario L = - + (1 – -)(1 – h)

**Table 6.6** Pay-offs to depositors if L < d and e ≥ 0

All deposits that could not be withdrawn, as well as all long-term claims and equity are lost because following our assumption regarding collateral liquidation by the central bank, the liquidation of the pledged assets will suffice to just repay central-bank funding.

(3) **Table** 6.6 shows pay-offs if liquidity is insufficient to pay out even one depositor, but equity is still non-negative. Two equilibriums emerge: one in with both depositors stay with their deposits, and one, inferior, in which they both run, causing default and the related losses.

Running is now an equilibrium because if the other depositor runs, and you don't, then you end up with zero value as the liquidation of the bank will lead to a zero recovery ratio. In contrast, if you are the one who runs and the other doesn't, then you recover L > 0, while if you also do not run, you recover only L/2.

### **<IP_ADDRESS> Negative Equity**

If equity is negative, the bank will eventually be closed and its assets will be liquidated by the regulator. By definition, the recover ratio r with negative equity will be 0 < r < 1 of their claims, while 1 – r will be the loss-given default. Depositors can try to withdraw their deposits without losses before liquidation and the more a depositor withdraws, the more the losses will be dumped on the other creditors who will have a higher loss-given-default. As the game is symmetric, there will be a unique run equilibrium. We call r = 1 + e (with e < 0) the recovery ratio in the absence of runs, i.e. the recovery ratio that would be equally applied to short-term and long-term creditors in this case. We call r' the recovery ratio for the remaining creditors if only one depositor runs, and r" the one if both depositors run. Clearly, 0 ≤ r" ≤ r' ≤ r ≤ 1.

	- If no depositor runs, both depositors will recover a share r of their deposit after liquidation.
	- If one depositor runs, one will completely recover her deposit and the other deposit a share r' after liquidation.
	- If both run (equilibrium solution), both will recover all their deposits, all losses will fall on the long-term creditor.
	- If none run: they will recover a share r of their deposit after liquidation, which is exactly the same as in case 4
	- If only one runs: she will recover her full deposit, while the other depositor will recover a share r' after liquidation
	- If both run: there is not enough liquidity to withdraw all deposits. They will withdraw an amount equal to the whole disposable liquidity and divide them proportionally on each's share of the total deposit. The long-term creditor will lose all his capital, as we assumed that the sale of the bank's asset will not provide any further amount beyond what is provided by the central bank

**Table 6.7** Pay-offs to depositors if L ≥ 2d and e < 0

.

**Table 6.8** Pay-offs to depositors if d ≤ L < 2d and e < 0

**Table 6.9** Pay-offs to depositors if L < d and e < 0

**Table 6.10** Utility of depositor 1 depending on own and depositor 2's decisions: U1(D1D1)

In **Table** 6.10 we summarise the results showing the payoffs for player 1. The shaded area indicates where the bank is liquidated. The areas with bold larger font are the equilibrium solutions of the run game.

### **<IP_ADDRESS> Central Bank Collateral Easing to Restore Financial Stability in a Financial Crisis**

**Asset liquidity** (as captured by the parameter -) **and asset values can change over time, and also the central bank may change haircuts over time**. For example, **Fig.** 6.3 (from Dötz and Weth 2019, 12) illustrates how asset liquidity fluctuates over time. Moreover, asset values can change, which is reflected in a change of equity. For example, if initial equity is 0.2 and total assets 1, then a decline of asset values by more than 20% depletes equity and therefore pushes the bank into a single run equilibrium. Asset value deterioration also leads to a deterioration of the liquidity condition, as it leads to a shrinkage of liquidity relative to short term debt.

To what extent **can the collateral framework of the central bank as captured by h make a difference for funding stability?** First, obviously h does not impact on solvency. Therefore, whenever e < 0 we are unavoidably in the case of the single bank run equilibrium. Central banks should not combat financial instability due to negative equity with collateral policies. However, h can make the difference for meeting the liquidity condition. We can calculate the maximum haircut compatible with a single no-run equilibrium from the condition of sufficient liquidity. Therefore, it is easy to show that decreases of h can compensate both unexpected asset value declines and a drop of the share of liquid assets in terms of restoring the condition L > d, assuming that equity remained positive.

### **Cheapest stable funding structure**

If the bank does not fear a deterioration of asset values or a decline of the share of liquid assets, it will choose **the cheapest liability structure that is just supporting a single no-run equilibrium**. If cost of equity is higher than the cost of long-term debt, the bank will, in the model above, however not issue any equity and achieve its competitive stable funding structure only through sufficient long-term funding. This is the limitation of the simplest model setting: it does not allow modelling equity as a safeguard against fire-sale losses, and therefore does not contribute to a better understanding of the full liability structure including equity. Consider the following example: if - = 0.3 and h = 0.8 then the funding structure maximising short term deposits is the one in which d = 0.44, i.e. t + e = 0.12 and e ≥ 0. For any (it, ie) with it < ie, the cheapest stable funding structure will be e = 0, d = 0.88, and t = 0.12.

### **<IP_ADDRESS> Collateral Policies as Monetary Policies at the ZLB**

**Broadening the collateral set in a liquidity crisis may be a key monetary policy measure, in particular when conventional monetary policy has hit the zero lower bound**. The simple bank- run model above allowed us to show that when asset liquidity deteriorates, then banks need to move to a more expensive bank liability structure. If they do so quickly enough, the bank-run equilibrium may not materialize. But a more expensive capital structure means that the spread between the short-term risk-free interest rate (controlled by the central bank) and the actual bank financing costs, and thus bank lending rates, increase. The central bank could maintain financial conditions unchanged by lowering the short-term risk-free interest rate. This is however not an option if the zero lower bound has been reached. The central bank **could therefore broaden its collateral framework so as to make the old, cheap bank liability structure stable again.**

We illustrate the last point further with the bank balance sheet in Table 6.11.

This bank has stable short term funding if L = - + (1 – -)(1 – h) ≥ d/2 ⇒ d\* = 1 – h + h-, with d obviously capped at 1. Assume that the financial conditions ("FC") are equal to bank lending rates, and these are equal to the average funding costs of the

**Table 6.11** Effectiveness of collateral policies at the zero lower bound

banks, as banks would be perfectly efficient, i.e. would not have any administrative costs and be within a competitive sector. Also assume that the funding costs of shortterm deposits is equal to i, the risk-free short-term interest rate, which is the monetary policy interest rate set by the central bank. Assume that the cost of long-term funding is equal to the sum of the short-term risk-free lending rate, i, plus the term premium, ω. Therefore: FC = id + (i + ω)(1 – d) = i + ω – ωd.

By substituting the highest possible share of short-term deposits which ensures stable funding, we obtain1: FC = i + ω – ω2(1 – h + h-) ⇔ FC = i + ω (2 h(1 – -) – 1). In words: financial conditions tighten (funding costs increase) with (i) shortterm risk-free interest rates, (ii) the equity risk premium, (iii) the haircuts imposed by the central bank, (iv) the share of illiquid assets. Therefore, when the zero lower bound is reached for conventional monetary policy, then haircuts can contribute to achieve the adequate monetary policy (i.e. the adequate financial conditions), and **a decrease in haircuts can be a measure necessary to compensate for an increase of the equity risk premium or a deterioration of asset liquidity**. Of course, such a lowering of haircuts for monetary policy purposes should not imply that the role of haircuts to protect the central bank from losses should be forgotten, i.e. the central bank will face a trade-off between monetary policy objectives and risk objectives.

### *6.3.2 The Model with Continuous Asset Liquidity*

In this model variant (Bindseil 2013), one assumes that (i) **assets are continuous in terms of liquidity properties**, that (ii) they are equally ranked from both the fire-sale loss and central bank haircut perspectives, and that (iii) both haircuts and fire-sale discounts have the functional form across the assets of a power function, i.e. haircuts are h(x) = xδ and marginal fire sale losses are f(x) = xθ with δ > 0 and θ > 0, i.e. θ summarises the liquidation cost function and δ the central bank haircut function. This continuous approach to asset liquidity and central bank haircuts has a number of advantages: (i) it allows us to differentiate between the roles of equity and long-term debt; (ii) it is more realistic than the assumptions taken on assets so far; (iii) the power function is tractable in the context of our model.

Consider first the case when the central bank does not at all act as lender of last resort, i.e. the only source of liquidity generation in the case of a run is to fire-sell assets. If a certain share x of the bank's assets has to be sold, then the fire-sale discounts will have to be booked as a loss and reduce equity. Assuming that the bank starts with the most liquid assets and sells a share x of total asset, the total fire sale loss will be:

$$F(\mathbf{x}, \theta) = \underset{\mathbf{s} = \mathbf{0}}{\overset{\mathbf{x}}{\cdot}} \mathbf{s}^{\theta} \mathrm{d}s = \frac{\mathbf{x}^{\theta + 1}}{\theta + 1}$$

1Note that the multiplier of ω will be positive for the assumption that 2d\* < 1. Indeed 2d\* = 2(1 – h + h-)<1 ⇒ –2 h(1 – -) < –1 ⇒ h(1 – -) > 1/2.

As mentioned in Chap. 5, empirical estimates of default costs in the corporate finance literature vary between 10% and 44%. This cost can be interpreted as the liquidation cost of assets, captured in the parameter θ. Liquidation of all assets will lead to a damage of F(1) = 1/(1 + θ), and sales proceeds (cash generated) will be 1 – F(1) = θ/(1 + θ). Consequently, θ can be calculated as θ = (1 – F(1))/F(1). If default cost is 10%, this would mean that θ = 9, and if default cost is 44%, then θ = 1.27. For a value of default costs in the middle of the empirical estimates of say 25%, one obtains θ = 3.

In **Fig.** 6.2 below, we illustrate this approach by showing the distribution between liquidity generation and asset fire-sale losses under the assumption of a power function of fire-sale costs and ranked asset from the most to the least liquid, **for the range of the empirical estimates of costs of default**, i.e. for 10% (implying θ = 9) and 44% (implying θ = 1.25). Moreover, we show the power function to replicate the estimated recovery ratio in the case of the Lehman Brothers of 28%, i.e. default costs of 72% under the assumption that before default, Lehman had zero equity (Fleming and Sarkar 2014a, b).

That asset liquidity is continuous, and that it fluctuates over time, has been described empirically in the finance literature, such as recently in Dötz and Weth (2019), who also argue that liquidation will be carried out in a liquidity pecking order style and that marginal liquidation costs should be expected to increase in redemptions. They construct a sample of corporate bond fund asset liquidity data covering the 80 months before June 2016, referring to around 700 thousand security holdings positions. Price and liquidity information are added to each such position. The liquidity measure consists in monthly averages derived from daily bid-ask spreads. **Figure** 6.3 shows continuous portfolio liquidity, put at any moment in time into a "liquidity pecking order" (i.e. securities ranked from the most to the least liquid). Obviously, the least liquid assets held by a corporate bond fund will still be more liquid than many other bank assets (e.g. loan portfolios). Still, it nicely illustrates the idea of continuous asset liquidity and the changes of asset liquidity over time.

Assume that in the case of a bank run, the bank does whatever it takes in terms of asset liquidation to avoid **illiquidity induced default**. The total amount of liquidity that the bank can generate through asset fire sales is θ/(θ + 1). Therefore, illiquidity

**Fig. 6.2** Three representations of fire sale losses and liquidity generation assuming that marginal fire sale losses f(x) are a power function with exponent theta

**Fig. 6.3** Liquidity structure of corporate bond funds, according to Dötz and Weth (2019, 12)

induced default will materialise only if deposit withdrawals eventually exceed this amount. Two default triggering events need to be considered. Indeed, even if the bank has survived a liquidity withdrawal, it may afterwards be assessed as insolvent and thus be liquidated at the request of the bank supervisor. As noted above, for a given liquidity withdrawal x, the fire-sale related loss is xθ+1/(θ + 1) (Fig. 6.3).

**Default due to insolvency** occurs if this loss exceeds initial equity, i.e. e < xθ + 1/(θ + 1).2 It can be shown (see e.g. Bindseil 2013, proposition 2) that a single no-run equilibrium exists if and only if (and assuming again the bank liability structure shown at the beginning of Sect. 6.5) both a liquidity and a solvency condition are fulfilled:

$$
\theta/(\theta+1) \ge \mathbf{d} \quad \text{and} \quad \mathbf{e} \ge \mathbf{d}^{\theta+1}/(\theta+1).
$$

The liquidity condition is similar to the discrete case: to ensure financial stability in the case of absence of central bank credit, the liquidity generating capacity of the bank needs to correspond at least to the deposits of one of the two depositors. The solvency condition expresses an aspect that could not be captured in the discrete case: the financial damage suffered by generating through fire sales the liquidity needed to pay out one of the two depositors must not exceed the bank's equity.

What is the **cheapest sustainable liability structure** in this model? For given θ, competing banks will always go to the limit in terms of the cheapest possible liability structure as determined by the conditions in the strategic depositor game, such that the no-run equilibrium is still maintained as an SNNR equilibrium. Assume that the

2Note that it is assumed that equity is never sufficient to absorb the losses resulting from a bank default, i.e. it is assumed that e ≤ 1/(θ + 1). Of course, one could also calculate through the opposite case, but it is omitted here as it does not seem to match reality.

cost of remuneration of the three asset types are re for equity, rt for term funding, and 0 for short term deposits. Also assume that re > rt > 0. In this setting what will the composition of the banks' liabilities be? The objective of choosing a liability composition will be to minimize the average overall remuneration rate subject to maintaining a stable short-term funding basis. The two minimum conditions to be fulfilled are θ/(1 + θ) = d and e = d(1+θ) /(1 + θ). These conditions can be solved for a unique optimum e\*, d\*, and hence also for the average necessary remuneration rate of bank liabilities t\* being rt + e\* re.

### **If the central bank offers collateralised credit**

Now consider **the case in which also the pledging of collateral with the central bank is possible**. To obtain outcomes in which the banks rely both on fire sales and haircuts in their liquidity stress strategy, we obviously need δ > θ (otherwise it is always superior to only pledge and never to fire-sale). It can be shown in the nontrivial case that the bank's liquidity stress strategy will always foresee the share z of most liquid assets to be fire-sold, while the rest, the 1 – z less liquid assets, will be pledged with the central bank. The condition for an SNNR is provided in proposition 5 of Bindseil (2013):

*Let z in [0,1] determine which share of its assets is foreseen by the bank to be used for fire sales (i.e. the less liquid share 1* – *z of assets are foreseen for pledging with the central bank). Let F* = *F(z) be the fire*-*sale losses from fire selling the z most liquid assets and let L* = *L(z) be the total liquidity generated from fire*-*selling the most liquid assets z and from pledging the least liquid assets (1* – *z). Then a single no run equilibrium exists if and only if*

$$\exists z \in [0,1] : L = L(z) = \frac{\delta}{\delta + 1} + \frac{z^{\delta + 1}}{\delta + 1} - \frac{z^{(\theta + 1)}}{\theta + 1} \ge d\theta$$

*and*

.

$$F = F(z) = \frac{z^{(\theta+1)}}{\theta+1} \le e$$

In contrast to the discrete model variant, this variant allows us to explain the full capital structure, including the distinction between long-term debt and equity, and the roles of these two funding sources can be shown to depend on the relative cost of the two and the relative size of δ compared to θ.

The model and its solution are illustrated in **Fig.** 6.4. The vertical line z separates the liquidity-ranked asset space into the part that will be fire-sold (assets on the left of z, i.e. most liquid assets) and the part that will be pledged (assets on the right of z, i.e. least liquid assets). The bank foresees in its liquidity stress strategy to fire-sale the assets [0,z] and to incur fire-sale losses of F, and generate liquidity through fire sales equal to L1. Moreover, in this strategy the bank pledges the assets [z,1] and generates through this liquidity equal to L2. Therefore, total liquidity generated (which must

**Fig. 6.4** Liquidity generation and fire sales in a model of continuous asset liquidity. Left: using fire sales, centre: using pledging at the Central Bank, right: using box

**Table 6.12** Bank financed only by short term debt and equity

at least be equal to the deposits of one depositor) is L1 + L2 and total fire-sale losses are F (which must not exceed equity e).

To calculate the relevant surfaces as illustrated in Fig. 6.4, we only need to apply that the integral of xa in [0,z] is equal to z(a+1) /(a + 1). L1 is the surface above the fire-sale loss function up to z (the liquidity generated from fire sales); F is the surface below the fire-sale loss function up to z (the losses generated from the sales), and L2 is the surface above the haircut function between z and 1 (the liquidity generated by pledging assets).

$$L\_1 = z - B = z - \frac{z^{\theta + 1}}{\theta + 1} \\ F = \frac{z^{\theta + 1}}{\theta + 1} \\ L\_2 = 1 - \frac{1}{\delta + 1} - \left( z - \frac{z^{\delta + 1}}{\delta + 1} \right)^2$$

**Example**: assume that banks' liabilities consist only of equity and short-term debt, such as in Table 6.12.

Also assume that initially θ = 1.4, δ = 0.5 and e = 0.2 so that each depositor has deposits of 0.4. One can now calculate that with z = 0.5, one obtains liquidity generating power L = 0.49 and associated fire-sale losses F = 0.08 (this is easily done by putting the formulas of the surfaces L1, F, L2 from the chart above into Excel). This allows for a single no-run funding equilibrium.

### **6.4 Conclusions**

The following five key conclusions can be drawn from the simple bank run model (and taking various simplifying model assumption as described, including that banks and depositors have the same perfect information on asset values, asset liquidity, and bank liability composition): First, both asset value and asset liquidity deterioration can trigger a run. Second, insufficient liquidity leads to multiple equilibria, while negative equity always implies a run. Therefore, the LOLR (i.e. captured in this simple setting by central bank collateral haircuts) will never stop a run if equity is negative. However, LOLR action can restore a single no-run equilibrium when only an asset liquidity deterioration and/or an asset value deterioration occurs, as long as equity remains positive. Third, tightening collateral rules can destabilize banks by pushing them into the multiple equilibrium case. Fourth, if an asset liquidity deterioration pushes banks into the multiple equilibrium case without the run equilibrium materializing, banks will be incentivized to adjust their capital structure so as to restore the single no-run equilibrium case. This typically leads to a more expensive capital structure, i.e. to more expensive bank intermediation and hence, everything else equal, a tightening of monetary and financial conditions. If monetary policy has reached the zero lower bound, increasing collateral availability can be an effective monetary policy tool. Fifth, if banks through competition and myopic behaviour tend to converge to the cheapest sustainable liability structure, then very small shocks on asset value and asset liquidity can destroy funding stability. It may therefore be useful to impose liquidity regulation on banks.

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# **Chapter 7 International Monetary Frameworks**

In this chapter we turn to representing flows of funds in alternative international monetary frameworks, and what global liquidity these different frameworks provide. We first recall some arguments in favour of and against fixed exchange rate systems. We then introduce two international monetary arrangements of the past which imply fixed exchange rates, namely the gold standard and the Bretton Woods system, and recall why both eventually failed. We then turn to three international monetary frameworks in the context of the current paper standard, i.e. fixed exchange rate systems, flexible exchange rate systems, and the European monetary union. We explain the role of an international lender of last resort and related solutions, and how these allow for more leeway in running fixed exchange rate systems. We also show how banks and central bank balance sheets are affected by international flows of funds and the balance of payments. Finally, we briefly review recent developments of foreign currency reserves, being the key central bank balance sheet position in this context.

### **7.1 Why Do Fixed Exchange Rates Persist?**

This chapter introduces the flow of funds mechanics of various international monetary frameworks. It will be shown how the frameworks absorb capital and current account imbalances and see what limits the systems may encounter.Most of the sections of this chapter are devoted to forms of fixed exchange rate systems. International monetary frameworks often aimed at supporting fixed exchange rate systems, or they imply by construction fixed rates, like the international gold standard. Under fixed exchange rates, the central bank loses its otherwise unconstrained LOLR powers in its domestic currency. In addition, with fixed exchange rates, central banks lose the power to do independent monetary policy, as monetary policy will be determined by the need to be consistent with the fixed exchange rate. Obstfeld and Rogoff (1995) have therefore concluded that "for most countries, it is folly to recapture the lost innocence of fixed exchange rates". Why do countries or central banks want to have fixed exchange rates at all, or bind themselves to gold so as to lose parts of their freedom and power, both in terms of monetary policy, and as a lender-of-last resort? For example, why has the EU launched the euro project and why has China shown so much commitment over the last decades to keep its exchange rates relatively stable by allowing its foreign reserves to fluctuate considerably?

Mainly four reasons for fixed exchange rates are still sometimes considered valid:

For these reasons, countries have often chosen to try to fix their currency to a precious metal or to other currencies. Different forms of fixed currencies systems exist, as will be explained below (e.g. peg to metal; unilateral peg to another currency; multilateral agreement like the Bretton woods framework; EMS; monetary union). In the following sections, four international monetary arrangements will be introduced. First, frameworks of the past: the gold standard, which had its height from around 1875 to 1914, and second the Bretton Woods System, which regulated the international monetary relation from 1945 to 1971 and was officially dissolved in 1976. Then, arrangements that are still in place today: fixed exchange rates and flexible exchange rates in the context of a paper standard.

### **7.2 Past International Monetary Frameworks**

### *7.2.1 The Gold Standard*

To represent the flow of funds under an international gold standard, assume two countries i = {1,2}, and the following for each sector:

Cross border economic flows (financial flows and those relating to trade) are captured in the balance of payment, which records the economic transactions of a country within the international context. For a detailed description of the international accounting standards of the balance of payment, see the IMF Balance of Payments Manual (IMF 2009). Here, the flow of funds analysis will be limited to basic trade and financial flow transactions. For example, the following two events, which also affect the international accounts of a country, will not be captured:

country winning a war imposes an indemnity to the country that lost it; a country grants debt relief to another.

Table 7.1 shows basic capital and current account transactions defined as follows. **Capital transactions (ca)**: We define a capital transaction as one in which net financial claims between sectors of countries change without transfer of real goods (other than gold). In the gold standard, they are settled in central bank gold. We assume that household 2 is behind the capital move, namely that household 2 opens an account with bank 1 and then shifts a part of her bank deposit from country 2 into country 1. There are different ways of imagining how these transfers can concretely take place. Capital account transactions do not change the net foreign position of a country, i.e. having a net credit or a net debt towards the rest of the world. But in a fixed exchange rate system, they typically change the net cross-border position of the private sector, with opposite changes of the public sector as represented by the central bank.

**Current account transactions (cu):** Here we assume that household 1 sells a real asset to household B (alternatively, the transaction could also take place between corporates). Household 2 instructs her bank to credit the account of household 1 with bank 1. Again, this transaction can be implemented in different ways. At the end, it will impact on accounts as shown in Table 7.1. Current account transactions normally change the **net foreign position** of a country, i.e. the difference between foreign assets and foreign liabilities of a country.

Table 7.2 shows these accounts in the form of an asset liability matrix (see Table 7.2), such as introduced in Sect. 2.2.1.

If for example ca = – cu, then under fixed exchange rates (including the gold standard), the central bank gold (or foreign exchange) reserves do not change. This means that in net terms, capital flows exactly finance the net transfer of goods. There are single transactions that represent both types of transactions at once, for example: a machine is exported from corporate 1 to corporate 2, but not paid yet, such that a financial claim from corporate 1 to corporate 2 is created at the same time when the real good passes the border. This transaction takes place entirely in the corporate sector balance sheets. In the accounts below (Table 7.3), the value of the transaction is denoted by X, with **X = cu = – ca**.

**Capital and current account balances can have various reasons**:

**Table 7.1** Two countries' financial accounts, gold standard

**7.2**Twocountries'financialaccounts,goldstandard,matrixrepresentation

**Table 7.3** A balance of payment equilibrium within the corporate sectors of two countries

by numerous emerging market crises (Mexico in the 1980s, Thailand, Indonesia and Russia in the late 1990s), and also in the euro area crisis.

Generally, observers may find current and capital accounts of specific countries in specific periods, as well as cumulated external positions of these countries, as something economically sensible and welfare improving, or as reflecting undesirable imbalances with the potential for financial destabilisation and corresponding welfare damage. For example, the large short-term foreign indebtedness of Germany that built up in the second half of the 1920s and that created the subsequent run on Germany in 1931 was assessed early as problematic (see e.g. annual reports of the Reichsbank in the 1920s). Similar cases were often observed until recently with emerging market economies. During his entire Presidency, US President Trump criticised German current account surpluses as unnatural and problematic, while the Bundesbank defended them as reflecting the German age pyramid and hence the need for the German society as a whole to save through the temporary accumulation of a net external claim.

### **Alternatives to settlement in gold**

Returning now to the initial case in which the balance of payment is not balanced, i.e. ca = – cu, it should be noted that the resulting **net claim does not necessarily need to be settled through a physical gold transfer**, but it can also be settled through the creation of a foreign reserves claim of central bank 1 towards central bank 2. One could for example imagine that also under the gold standard, the two central banks have a settlement agreement in which cross border bank transfers are settled through a counterbalancing central bank position, up to a certain limit beyond which settlement in gold is required. This case is illustrated in the accounts in Table 7.4.

Table 7.5 shows still another alternative, namely that a claim of central bank 1 towards commercial bank 2 is created. For both central banks, this could make a difference: from the perspective of central bank 1, it could mean problems in case of a need to liquidate these claims when they are needed for interventions. For central bank 2, it could cause domestic financial stability issues, requiring it to act as LOLR to its domestic debtors when central bank 1 liquidates its foreign reserves for intervention purposes.

Returning to the base case that the international transactions are settled in gold and affect central bank gold reserves, two limits may eventually become binding:

**Table 7.4** Central banks' accounts in gold standard with claims on gold instead of shipments

**Table 7.5** Accounts of financial sectors if deposits with foreign banks replace gold shipment

first, the **limit with regard to the share of banknotes that needs to be covered by gold reserves**, according to the central bank law. For example, the Reichsbank was subject, according to its mandate established by the Dawes Plan in 1924, to a 40% gold coverage ratio for banknotes. Second, assuming that gold coverage ratios have been given up, when **gold reserves are fully exhausted**, such as in the case of the Reichsbank in July 1931. Then, eventually the gold convertibility has to be given up.

### **Performance of the gold standard**

The gold standard worked fine during the period 1876–1914, but poorly in the interwar period. The poor interwar performance of the gold standard is explained by Eichengreen (1995) with the global scarcity of gold and non-collaborative international behaviour in the context of larger capital flow volatility due to the political and financial instabilities. The interwar gold standard came after the WWI experience that one cannot rely on a universal commitment of Governments to maintain a gold parity. For example, Germany devalued in 1924 by a factor of 109, and only very few like the US and the UK did not. Similarly, the belief in Governments to repay their debt was shaken (e.g. the Russian mega-default of 1917); moreover, the inter-war period was characterised by unsolved problems of international debt imbalances (war and reparation debt) and political instability, materialising in the rise of fascism and communism. Finally, the willingness of central banks to collaborate, e.g. through inter-central bank loans, was insufficient (while an international LOLR like the IMF was still missing). With increasing uncertainties after the outbreak of the global financial crisis in 1929, central banks were even keener to each hold sufficient gold reserves to be protected against future outflows, implying that on average central banks kept interest rates too high in their competition for the global gold stock, triggering deflation and depression.

### *7.2.2 The Bretton Woods System*

The Bretton Woods system was set up in 1944 and included establishing the IMF and the World Bank. The related convertibility promises were given up in 1971 and 1974, although the role of the IMF as international LOLR has continued until today. The Bretton Woods system was a fixed exchange rate system in which the US committed to fix the value of the USD in gold (and to ensure convertibility), while the others promised to fix the price of their currency in USD (and to defend these fixed exchange rates). Therefore, in principle the US Fed needed gold as a reserve asset, while the other members needed USD. The accounts shown in Table 7.6 illustrates this situation, focusing on the case of flows handled through the foreign reserves of a non-US country. If country 2 is Germany, then in the Bretton Woods era (ca + cu) < 0, i.e. Germany had balance of payment surpluses and the Bundesbank accumulated foreign reserves.

Bordo (1993) notes that a fixed rate system of the **Bretton Woods type was subject to the following three problems**:

**Table 7.6** Two countries' financial accounts under the Bretton Woods system

of U.S. dollars depended on the U.S. balance of payments, which in turn was related to the vagaries of government policy and the confidence problem."

• **Confidence**: "as in the interwar period, involved a portfolio shift between dollars and gold. As outstanding dollar liabilities held by the rest of the world monetary authorities increased relative to the U.S. monetary gold stock, the likelihood of a run on the "bank" increased. The probability of all dollar holders being able to convert their dollars into gold at the fixed price declined."

This led surplus countries to prefer hoarding gold instead of holding USD, i.e. a run on the US Fed's gold holdings took place by the surplus countries who made use of the US Fed's obligation to convert the surplus countries' USD into gold. Another way to describe the Bretton Woods problem is the **Triffin Dilemma** (Triffin 1960). In the words of the IMF (IMF 2020b):

If the United States stopped running balance of payments deficits, the international community would lose its largest source of additions to reserves. The resulting shortage of liquidity could pull the world economy into a contractionary spiral, leading to instability. If U.S. deficits continued, a steady stream of dollars would continue to fuel world economic growth. However, excessive U.S. deficits (dollar glut) would erode confidence in the value of the U.S. dollar. Without confidence in the dollar, it would no longer be accepted as the world's reserve currency. The fixed exchange rate system could break down, leading to instability.

In principle, the solution of this problem was supposed to be the "SDR", the special drawing rights of the IMF, which created additional international liquidity.

### **Table** 7.7 **shows the case if the other country experiences balance of payment surpluses and the central banks prefer to accumulate gold instead of USD**

The accounts in Table 7.6 had suggested that the US Fed's balance sheet was not really affected by the balance of payment deficits of the US—however this was not true, as highlighted by the Triffin Dilemma: Central bank 2 can also exchange its foreign reserves against gold, and this is indeed what central banks of balance of payment surplus countries tended to do. For example, the Bundesbank (until 1957 its predecessor, the Bank Deutscher Länder) had no gold reserves in 1949, but more than 3000 tons at the end of the Bretton Woods era in 1974. The desire to accumulate not only US dollars, but also gold, may have been understandable, since holding very large amounts of US dollars exposed a central bank to risks that the US eventually devalues—which it actually did. A bank run logic applies. The mechanics of the self-fulfilling prophecy works as follows in this case: if surplus countries start to doubt the ability or willingness of the US to defend the peg of the USD to gold, they are incentivized to start hoarding gold instead of USD, and eventually this leads the US to run out of gold reserves, forcing it to devalue the USD against gold, validating the fears of the countries who "ran" on the USD. Of course, at an early stage, the US could have tried to defend the peg by restrictive monetary policies which would have triggered capital inflows exceeding the negative effects of current accounts. But this would have had significant economic costs in the view of the responsible policy makers, and therefore did not take place. In the financial accounts, the tendency of surplus countries to hoard gold is reflected as follows, assuming the case that country

2 (Germany) has surpluses and converts these completely into gold (Table 7.7).

In retrospect, it appears that the Bretton Woods system could have worked if:

**Table 7.7** Bretton Woods financial accounts if surplus countries hoard gold instead of USD

There were several changes of pegs before the eventual dismissal of the Bretton Woods system in 1974.

### **7.3 International Monetary Frameworks of the Present**

After the collapse of the Bretton Woods system in 1971, and its official dissolution with the Jamaica Accords of 1976, a "paper standard" emerged in which currencies were no longer pegged to gold. While the biggest economies opted for flexible, floating exchange rates, the smaller economies mostly fixed their exchange rate to that of the US or of the nearest large economy. The western European countries tried to develop arrangements for limiting the fluctuations of the currencies between each other, whereas the German Mark emerged as the anchor. Thereafter these agreements led to the creation of the euro.

### *7.3.1 Fixed Exchange Rate System—Paper Standard*

The following financial accounts show a fixed exchange rate system in a paper standard. Country 1 is a large country, which does not care about the exchange rate, while country 2 is a smaller country that does care and that ensures that its currency is pegged 1:1. For example, country 1 could be the euro area and country 2 Bulgaria. To be able to defend the currency peg, the central bank of country 2 needs foreign reserves. In the accounts below, we assume that these have a level **FR** and are held in the form of deposits in currency 1 with Bank 1, and that they originated in the past from capital account inflows into country 2, which still materialize in the accounts in the form of lending of Bank 1 to corporate 2 of FR.

Table 7.8 illustrates the example of a **current account transaction in which a household in country 1 sells a car and exports it to a household in country 2**.

### **Exhausted central bank foreign reserves and the ILOLR**

What if central bank 2's foreign reserves are exhausted, i.e. if ca + cu > FR? Eventually, the central bank and the government of country 2 have to restore macroeconomic conditions that stop and revert the flows that led to this situation (e.g. increase interest rates, strengthen the competitiveness through supply side reforms, etc.). However, such measures typically require some time to be effective. There are two short term options: either **Central bank 2 finds an** *international* **LOLR to replenish its foreign reserves**, or central bank 2 "defaults" on its promise to fix the exchange rate. In the latter case, the system moves towards a variable exchange rate system.

An international LOLR can take two forms: a direct lending between central banks, or through an intermediary like the IMF. Table 7.9 shows the latter case in a stylized way. **The intermediary (which we call "IMF") takes a loan from central**

**Table 7.8** Two countries' financial accounts in paper standard with fixed exchange rates

**bank 1 to obtain currency 1 and provides this as credit to central bank 2**. The loan is assumed here to exactly close the gap of missing foreign reserves to stem the outflow due to current and capital account deficits.

The intermediary could also have initially, when founded as an international institution, created a sufficient balance sheet to accommodate such loans. This is displayed in Table 7.10, where the IMF balance sheet is initially based on paid-in capital. This paid in capital is "invested" by the IMF in the form of deposits with the central banks.

**Table 7.9** Financial accounts, fixed exchange rate, with IMF providing additional foreign reserves obtained by credit line

**Table 7.10** Financial accounts, fixed exchange rate, with IMF providing additional foreign reserves obtained by pre-paid capital

### **Devaluation and settlement of the implied negative central bank capital**

A devaluation by a central bank under a fixed exchange rate system could in some sense be compared to a default of a commercial bank when it is no longer able to pay back deposits when these are withdrawn, as developed in Chap. 6. Based on this analogy, one could also aim at a model of runs on currencies, in which the no-run conditions would depend both on liquidity (quantity and liquidity of foreign exchange reserves of the central bank) and on "fundamentals" being the (maybe somewhat less obvious) analogue to solvency in the bank run model.

**Table 7.11** Central bank accounts when currency 2 is devalued

Table 7.11 shows the case of an appreciation of currency 2. This is what happened to Germany during Bretton Woods (but it would be similar in a paper standard).