Shadow Banking and Market Discipline on Traditional Banks
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund
  • | 2 https://isni.org/isni/0000000404811396, International Monetary Fund

We present a model in which shadow banking arises endogenously and undermines market discipline on traditional banks. Depositors' ability to re-optimize in response to crises imposes market discipline on traditional banks: these banks optimally commit to a safe portfolio strategy to prevent early withdrawals. With costly commitment, shadow banking emerges as an alternative banking strategy that combines high risk-taking with early liquidation in times of crisis. We bring the model to bear on the 2008 financial crisis in the United States, during which shadow banks experienced a sudden dry-up of funding and liquidated their assets. We derive an equilibrium in which the shadow banking sector expands to a size where its liquidation causes a fire-sale and exposes traditional banks to liquidity risk. Higher deposit rates in compensation for liquidity risk also weaken threats of early withdrawal and traditional banks pursue risky portfolios that may leave them in default. Policy interventions aimed at making traditional banks safer such as liquidity support, bank regulation and deposit insurance fuel further expansion of shadow banking but have a net positive impact on financial stability. Financial stability can also be achieved with a tax on shadow bank profits.

Abstract

We present a model in which shadow banking arises endogenously and undermines market discipline on traditional banks. Depositors' ability to re-optimize in response to crises imposes market discipline on traditional banks: these banks optimally commit to a safe portfolio strategy to prevent early withdrawals. With costly commitment, shadow banking emerges as an alternative banking strategy that combines high risk-taking with early liquidation in times of crisis. We bring the model to bear on the 2008 financial crisis in the United States, during which shadow banks experienced a sudden dry-up of funding and liquidated their assets. We derive an equilibrium in which the shadow banking sector expands to a size where its liquidation causes a fire-sale and exposes traditional banks to liquidity risk. Higher deposit rates in compensation for liquidity risk also weaken threats of early withdrawal and traditional banks pursue risky portfolios that may leave them in default. Policy interventions aimed at making traditional banks safer such as liquidity support, bank regulation and deposit insurance fuel further expansion of shadow banking but have a net positive impact on financial stability. Financial stability can also be achieved with a tax on shadow bank profits.

1 Introduction

Recent decades have seen rapid growth in financial intermediation by non-bank entities based on a business model that combines highly-leveraged, short-term funding with risky long-term investments such as sub-prime mortgage lending. In the 2008 financial crisis, these “shadow banks” experienced a sudden dry-up in their funding and liquidated their assets. The ensuing turmoil quickly spread to traditional commercial banks which reduced their credit to the private sector. This has contributed to the deepest recession since the Great Depression, raising two important questions.1 First, what circumstances and mechanisms lead to the emergence and expansion of shadow banking? Second, how does shadow banking affect the portfolio and funding strategies of traditional banks?

In this paper, we propose a framework where depositors may withdraw their deposits early in reaction to crises.2 These “early withdrawals” constitute an optimal response to adverse changes in banks’ solvency prospects and become a source of market discipline: traditional banks optimally commit to a safe portfolio strategy to prevent early withdrawals. When commitment is costly, shadow banking emerges as an alternative banking strategy that combines a risky portfolio strategy with early withdrawals in times of crisis. To the best of our knowledge, this paper presents the first model where shadow and traditional banks coexist and interact without regulatory arbitrage or direct contractual linkages.

We bring this theoretical model to bear on the 2008 financial crisis, its transmission to the traditional banking sector, and policy debates on shadow banking and interventions in support of banks. In doing so, we account for two key empirical facts: Shadow banks faced a sudden contraction in funding and the liquidation of their assets caused a fire-sale. Traditional banks did not suffer from withdrawals, experienced a sharp rise in their funding costs, and re-allocated their portfolios towards safe and liquid assets.

We develop our analysis by specifying a closed economy model with households, entrepreneurs, outside investors, and a banking sector. Banks collect deposits from households and choose their portfolios of safe, risky, and liquid assets; households lend to banks on terms that depend on their solvency prospects; entrepreneurs originate assets.3 Following news signals that revise expected asset returns, households decide whether to withdraw their deposits early and banks trade assets with outside investors in a secondary market with cash-in-the-market pricing (Allen and Gale, 1994, 2005).

A key element in the model is the equilibrium relationship between secondary market prices and bank strategies. During early withdrawals, shadow banks liquidate their assets in the secondary market to repay their depositors. Fire-sale externalities then create strategic substitutabilities in banks’ decision to pursue a shadow banking strategy: As the shadow banking sector grows larger, its liquidation causes a deeper fire-sale, reducing the payoff from shadow banking relative to traditional banking, and bringing about an interior equilibrium where shadow and traditional banks coexist.

Analyzing the effects of fire-sales on market discipline yields important insights for vulnerability to financial crises. The key intuition is that low deposit rates strengthen early withdrawal incentives as depositors stand to lose less in terms of interest foregone. Therefore, market discipline on traditional banks is strong whenever deposit rates are low. This is precisely the case when there are no fire-sales: Interest rates on deposits are low and market discipline drives traditional banks to commit to a safe portfolio strategy. With the prospect of a fire-sale, on the other hand, depositors demand higher rates in compensation for liquidity risk.4 This weakens early withdrawal incentives and allows traditional banks to pursue risky portfolios that may leave them in default. In equilibrium, the shadow banking sector expands to a size where its liquidation causes a fire-sale and undermines market discipline through this mechanism.

The model provides novel and important insights for policy design. By the above mechanism, taxation of shadow bank activities is conducive to financial stability. These interventions deter banks from pursuing a shadow banking strategy and can be used to prevent the shadow banking sector from reaching a size where it causes a fire-sale.

We also show that outcomes from policy interventions differ substantially when adjustments in the size of the shadow banking sector are taken into account. For example, asset purchases in the secondary market are effective in alleviating fire-sales at a given shadow bank sector size. However, the ex-ante anticipation of asset purchases fuels further growth of the sector and renders the intervention completely ineffective. This troubles policymakers with time inconsistency issues: once the fire-sale is underway, they find it tempting to intervene.

Policy interventions in support of traditional banks also lead to an expansion in shadow banking but contribute to financial stability. Specifically, we show that collateralized liquidity provision secures traditional banks from liquidity risk and restores market discipline. Deposit insurance guarantees also eliminate liquidity risk but necessitate the replacement of market discipline with bank regulation. These two measures – collateralized liquidity provision and deposit insurance combined with regulatory constraints – have the exact same outcome: they stabilize the traditional banking sector, but in doing so diminish the fire-sale and expand the shadow banking sector. We find that the former effect outweighs the latter so that the net impact on financial stability is positive.

We build upon a growing literature that provides micro-foundations for the existence and growth of shadow banking. The rapid growth of shadow banking in recent decades is well documented (see e.g. Claessens et al., 2012; Pozsar et al., 2013). Gennaioli et al. (2013) emphasize the ability of shadow banks to generate safe assets through securitization. They show that shadow banks become excessively exposed to systemic risk when low probability tail events are neglected by investors. In a similar vein, Moreira and Savov (2017) focus on liquidity transformation whereby shadow banks create money-like assets that become illiquid in times of high uncertainty. Harris et al. (2014), Plantin (2015) and Ordoñez (2017) highlight the role of regulatory arbitrage as a primary cause of shadow banking. In these studies, regulatory constraints restrict intermediation by traditional banks and create opportunities for unregulated shadow banks.

Our contribution to this literature is to show that shadow banking may arise as an equilibrium outcome without any advantages in intermediation technologies or opportunities for regulatory arbitrage. We assume, realistically, that commitment is costly and show that ex-ante identical banks endogenously cluster into traditional and shadow banking strategies, where the former optimally pay a lump-sum cost to commit to a safe portfolio strategy. In this context, commitment costs reflect any costly action undertaken by banks to resolve asymmetric information issues with their depositors, such as providing detailed balance sheet reports, eschewing opaque intermediation processes like securitization, or issuing costly equity with voting rights.

This paper is also related to a recent strand of literature that analyses interactions between traditional and shadow banks. Gornicka (2016) considers traditional banks’ incentives to gain off-balance sheet exposure by extending implicit guarantees to shadow banks. Hanson et al. (2015) present a framework where shadow and traditional banks have access to a common pool of liquidity. They show that traditional banks have a comparative advantage in holding illiquid assets with low fundamental risk when they are protected by deposit insurance guarantees. In a similar environment, Luck and Schempp (2016) find that shadow banking grows excessively large from a social viewpoint due to pecuniary externalities similar to the fire-sale externalities considered here.

We contribute to this literature by deriving an equilibrium where shadow and traditional banks coexist and interact without regulatory arbitrage or (implicit) contractual linkages. By doing this, we provide novel insights about how shadow banking affects market discipline on traditional banks. We also establish a free market benchmark to compare bank regulation and other policy interventions against.

Our modelling choices draw heavily from observations on the 2008 financial crisis. Acharya et al. (2013) document the collapse in the market for asset-backed commercial papers at the onset of the crisis while Gorton and Metrick (2012) and Krishnamurty et al. (2014) show a similar contraction in repo markets. Together, these two markets accounted for the vast majority of funding for shadow banks. Covitz et al. (2013) find that the dry-up in funding for shadow banks was associated with a rise in macro-financial risks such as uncertainty about sub-prime mortgages values. In our model, the definitive characteristic of the shadow banking strategy is its vulnerability to early withdrawals that closely resemble these events. Consistent with Covitz et al. (2013), early withdrawals are triggered by a negative revision in expected asset payoffs.

We propose that fire-sales caused by the liquidation of shadow banks play a key role in the spread of financial instability to traditional banks. Shleifer and Vishny (2011) provide an extensive review of the literature on fire-sales while Krishnamurthy (2010), Merrill et al. (2012), and Mitchell and Pulvino (2012) provide empirical evidence for fire-sales during the financial crisis. Our mechanism is similar to Diamond and Rajan (2005) in that (shadow) bank failures cause contagion by aggravating liquidity shortages in the rest of the financial sector.

Finally, like Diamond and Rajan (2011), we find that illiquidity leads to greater risk-taking by traditional banks. In Diamond and Rajan (2011), this is a consequence of risk-shifting whereas our finding arises due to interactions between deposit rates and market discipline that stem from threats of early withdrawal by depositors. Crucially, our mechanism can generate solvency risk5 where none previously existed, whereas the risk-shifting mechanism in Diamond and Rajan (2005) exacerbates pre-existing risks.

We proceed as follows: Section 2 presents motivating evidence from the 2008 financial crisis. Section 3 describes the core mechanisms of the model in a simple framework without liquidity risk. Section 4 presents the complete model with liquidity risk. Section 5 provides the numerical results. Section 6 conducts policy analysis. Section 7 concludes.

2 Motivating evidence

In this section, we present four key stylized facts about shadow banking and the 2008 financial crisis in the United States.

Fact 1. The shadow banking sector expanded rapidly until its collapse in 2007. The traditional banking sector grew at a slower rate but did not suffer from a collapse.

Figure 1 shows that shadow bank assets expanded from 125% of GDP in 2002 to over 180% at its peak in 2007. This rapid growth came to an end with a collapse during the financial crisis. In 2015, shadow bank assets amounted to 145% of GDP. The traditional banking sector also expanded but at a relatively modest pace with its assets increasing from 88% of GDP in 2002 to 118% in 2015. Although there was no contraction in traditional banking during the crisis, the shadow banking sector remained larger throughout the period 2002-2015.

Figure 1:
Figure 1:

Shadow and traditional bank assets

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Financial assets when available, otherwise total assets. Traditional banks refer to all deposit-taking corporations. Shadow banks refer to all financial corporations that are not classified as central banks, banks, insurance corporations, pension funds, public financial institutions, or financial auxiliaries.Source: Financial Stability Board (2017) and the World Bank

In our model, the size of the shadow banking sector is endogenously determined through free entry and may rise for two reasons: an increase in the liquidity available in secondary markets and a rise in the cost of commitment. Recent financial innovations such as securitization may have increased both the commitment cost and the thickness of secondary markets. In Section 6, we also analyze the role of policy interventions in the expansion of shadow banking. We show that expectations that policymakers will lean against a fire-sale increase the equilibrium size of the shadow banking sector in the same manner as the deepening of secondary markets.6

Fact 2. At the onset of the financial crisis, shadow banks experienced a sudden dry-up of funding and liquidated their assets.

At their peak of $1.2 trillion in July 2007, asset-backed commercial papers (ABCP) were the largest money market instrument in the United States and constituted the main source of funding for shadow banks. Following rising mortgage default rates and the suspension of withdrawals by a number of funds, the market for ABCP contracted by $350 billion in the second half of 2007 and a further $400 billion by the end of 2009 (see Panel A of Figure 2). Faced with this sudden contraction in funding, shadow banks liquidated their asset holdings. Panels B and C show that shadow banks sold $1.5 trillions of debt securities between 2008Q3 and 2010Q1, approximately half of which were mortgage-backed securities.7

Figure 2:
Figure 2:

ABCP markets and shadow bank assets

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Shaded areas indicate US recessions. Shadow banks refer to money market mutual funds, security brokers and dealers, and issuers of asset-backed securities. Mortgage-backed securities refer to agency-and GSE-backed securities.Source: Financial Accounts of the United States

In our framework, shadow banks are vulnerable to early withdrawals which take place following an adverse change in macro-financial fundamentals: a bad news signal that leads to a downward revision in expected asset returns. These withdrawals constitute an optimal response by creditors to bad news rather than self-fulfilling bank-runs in the fashion of Diamond and Dybvig (1983).

During a withdrawal, shadow banks are forced to liquidate their assets in the secondary market. To the extent that this generates an adjustment in secondary market prices, there are consequences for traditional banks even in the absence of implicit guarantees or any other form of direct exposure between the traditional and shadow banking sectors.

Fact 3. Spreads between private debt securities and Treasury bonds increased sharply during the crisis. Securities with higher perceived risk experienced greater increases in spreads.

Figure 3 shows the evolution of yield spreads between corporate bonds (grouped according to their credit rating) and Treasury bonds of comparable maturity. It is notable that spreads peaked in the last quarter of 2008, coinciding with the liquidation of shadow bank assets. Although there was a sharp rise in spreads for every rating category, the increase was greater for lower-rated corporate bonds.

Figure 3:
Figure 3:

Spreads on corporate bonds

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Shaded areas indicate US recessions. AAA (BBB) refers to the spread between Moody’s Seasoned Aaa (Baa) Corporate Bonds and 10-Year Treasury bonds (constant maturity). CCC or below refers to the option-adjusted spread between the Bank of America Merrill Lynch US Corporate C Index and a spot Treasury curve.Source: Moody’s Investor Services, Bank of America Merrill Lynch

Our model generates changes in secondary market prices that are consistent with these movements in spreads. When there is cash-in-the-market pricing, the liquidation of shadow bank portfolios causes a fire-sale and (illiquid) assets trade at a significant discount. Portfolio re-allocation by traditional banks from risky to safe and liquid assets also contributes to the fire-sale. We provide evidence for and further discuss portfolio re-allocation under Fact 4.

Fact 4. Traditional banks re-allocated their portfolios toward safe and liquid assets and were able to increase their liabilities. At the same time, they faced a rise in their funding costs.

Panel A of Figure 4 shows that traditional banks increased their liabilities during the crisis. Although the increase in liabilities was driven by Federal Reserve funding and a rise in deposits protected by deposit insurance guarantees, there was also no decline in large time deposits. Since large time deposits are neither checkable, nor insured, the implication is that traditional banks were perceived to be safe enough to preclude withdrawals even in the absence of government guarantees.

Figure 4:
Figure 4:

Traditional banks during the crisis

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Shaded areas indicate US recessions. Traditional banks refers to “large domestically chartered commercial banks” which are defined as the top 25 domestically chartered commercial banks, ranked by domestic assets. Cash assets and govenment securities include vaullt cash, cash items in process of collection, balances due from depository institutions and Federal Reserve Banks, and liabilities of the U.S. government, U.S. government agencies and U.S. government-sponsored enterprises. The TED spread is calculated as the spread between LIBOR based on US dollars and Treasury Bills. Certificate of deposit and commercial paper spreads are between the secondary market rates of these securities and the Effective Federal Funds Rate. All spreads refer to 3 month maturities.Source: Board of Governors of the Federal Reserve System, the Federal Reserve Bank of St. Louis

Changes in traditional bank portfolios shown in Panel B lend further support to this interpretation. During the crisis, traditional banks re-allocated their portfolios from bank credit (which is risky and illiquid) to cash assets and government securities (which are liquid and safe). Mortgage-backed securities that were at the epicenter of the financial crisis accounted for only 12% of total assets and this did not change significantly over the crisis.

It is important to note, however, that while traditional banks did not experience a dry-up in funding, there was a significant increase in their funding costs. Panel C shows that the TED spread (for interbank loans) increased to 200 basis points in December 2007 and 335 basis points in October 2008. The spreads on certificates of deposits and commercial papers also increased significantly.

In the following sections, we show that the interaction between market discipline and liquidity risk may account for these observations. In our framework, market discipline stems from the threat of early withdrawal by depositors. To avoid an early withdrawal, traditional banks commit to a minimum recovery rate on deposits (in other words a maximum loss given default) which we refer to as a “no-withdrawal constraint” and respond to bad news by re-allocating their portfolios towards safe assets.

The fire-sale discussed under Fact 3 interacts with this mechanism through two distinct transmission channels. First, since risky assets are discounted to a greater extent than safe assets, it reduces traditional banks’ capacity for portfolio re-allocation. To satisfy the no-withdrawal constraint, traditional banks are then forced to reduce their risky asset holdings ex-ante. This creates an excess return that increases the expected payoff associated with traditional banking. As the extent of the fire-sale is proportionate to the size of the shadow banking sector, equilibrium is achieved through this mechanism. In the next section, we analyze this mechanism and the properties of the equilibrium in a simple model without liquidity risk.

Second, the fire-sale leaves traditional banks illiquid. This drives a wedge between solvency and liquidity, leaving traditional banks vulnerable to self-fulfilling bank-runs as per Diamond and Dybvig (1983). In Sections 4 and 5, we extend the simple model to allow for liquidity risk (i.e. the possibility of bank runs) and show that this undermines market discipline on traditional banks: Depositors demand higher interest rates to compensate for liquidity risk, generating a rise in deposit rates similar to Panel C. Higher deposit rates in turn weaken the threat of early withdrawal, bringing about a relaxation in the no-withdrawal constraint and greater risk-taking by traditional banks. In equilibrium, the shadow banking sector reaches a size that causes financial instability through this mechanism.

3 A simple model

We consider a financial economy populated by four agents: households, banks, entrepreneurs, and outside investors. Events unfold over three time periods (see Figure 5 for a graphical timeline). In the first period, banks collect deposits from households and invest in (safe and liquid) cash and (risky and illiquid) assets originated by entrepreneurs.8

Figure 5:
Figure 5:

Timeline

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

The second period begins with a public news signal that leads to a revision of expected asset returns. With probability q, the signal harbors “bad news” leading to a decline in expected asset returns. After observing the signal, households decide whether to withdraw their deposits early and banks trade assets with outside investors in a secondary market.

In the third period, assets yield a high or low payoff contingent on economic fundamentals. Following bad news in the second period, fundamentals may turn out to be weak with (conditional) probability p, leading to a low payoff from assets. Depending on their investment strategy, banks may then be left with insufficient funds to pay the promised return to their depositors. In this case, they become insolvent under limited liability and a haircut proportionate to their funding shortfall is imposed on deposits.9

The key friction in the model is costly commitment. Specifically, we assume that banks cannot credibly commit to a safe investment strategy unless they pay a commitment cost τ > 0.10 Due to this friction, ex-ante identical banks optimally cluster into two distinct groups according to their investment strategies. An endogenous share γ ∈ [0,1] of banks do not pay the commitment cost and follow a shadow banking strategy that entails high risk-taking. Following bad news, households optimally withdraw their deposits from shadow banks due to concerns about their solvency prospects. Shadow bank assets are then liquidated in the secondary market at an endogenous fire-sale discount and depositors receive the liquidation value. The remaining banks pay the cost τ to credibly commit to a portfolio that is safe enough to prevent an early withdrawal. We refer to this as a ‘traditional banking’ strategy.

It is important to note that early withdrawals are distinct from self-fulfilling (Diamond-Dybvig) bank-runs, as they are an optimal response to a change in banks’ solvency prospects rather than a consequence of strategic complementarities across households. In other words, an early withdrawal takes place when households find it optimal to withdraw their deposits regardless of the extent of withdrawals by other households.

Before explaining these activities in detail, we briefly describe some notational conventions. Table 1 provides a list of variables and parameters. We denote variables that pertain to a shadow (traditional) banking strategy with a superscript ‘SB’ (‘TB’). Deposits, cash and (risky) assets are respectively labelled as (D, M, I). To distinguish between deposits per bank and per household, we denote the latter as d in lower case. The effective return from deposits is contingent on solvency and the time of withdrawal. Upon maturity in the third period, deposits pay an interest rate R when the bank is solvent and a recovery rate V otherwise. When they are withdrawn early, deposits pay a liquidation value θ. Asset prices in the first and second period are labelled as (P1, P2). In period 3, assets yield a payoff σh when fundamentals are strong and σl otherwise. We simplify notation by normalizing the risk-free rate to R* = 1 and the unconditional expectation of asset payoffs to unity such that11

Table 1:

Notation

article image
(1qp)σh+qpσl=1(1)

3.1 Agents and their optimal strategies

3.1.1 Entrepreneurs

In the first period, each bank has access to a separate but ex-ante identical island of entrepreneurs which use capital K to produce assets I1 with a Cobb-Douglas production technology

I1=AKα(2)

where A > 0 is a productivity parameter and α ∈ (0, 1) is the standard Cobb-Douglas elasticity. The representative entrepreneur’s problem can be written as

maxI1,KP1I1K

subject to the production technology (2) where P1 is the asset price.12 This yields a first order condition that can be interpreted as an upward-sloping asset supply schedule

P1=1αA1αI11αα(3)

Due to relationship lending frictions, banks may only purchase assets from entrepreneurs in their own island.13,14 This also constitutes a barrier to entry that gives banks market power over entrepreneurs, allowing them to extract a mark-up

P1I1I1P1=μ>0

In the interest of tractability, we do not explicitly model the bargaining process between banks and entrepreneurs but restrict the mark-up to

μμ¯(1q)(σh1)1(1q)σh(4)

This restriction only serves to simplify the bank’s problem in Section 3.1.4 and we show in Appendix D.2 that it can be relaxed without altering the properties of the equilibrium solution.

3.1.2 Secondary market and outside investors

In the second period, banks and outside investors trade assets in a secondary market. First, we consider the case after good news. Since assets yield a high payoff σh with certainty after good news, there are no early withdrawals. Secondary markets then clear without a fire-sale at a price σh. Whilst trade may take place, the pricing of assets at their expected payoff ensures that it is completely inconsequential to the equilibrium allocation. Therefore, we abuse notation slightly by using the subscript 2 to denote variables after bad news throughout the paper.

Following bad news, the liquidation of shadow banks and portfolio re-allocation by traditional banks leads to an excess supply of assets

I˜=γI1SB+(1γ)(I1TBI2TB)>0

where γ is the share of shadow banks within the banking sector, I1SB is shadow bank assets, and the second term represents the net sale of assets by traditional banks. Outside investors with a lower valuation for assets then become the marginal buyers and the secondary market price is given by

P2=ϕ[(1p)σh+pσl]

where the fire-sale discount ϕ ∈ [0,1] adjusts to clear markets and the term in brackets is the expected payoff conditional on bad news.

Our set up for outside investors is based on Stein (2012). Outside investors begin the second period with an endowment E˜ and allocate their funds between asset purchases in the secondary market and an outside project K˜ that yields a safe payoff g(K˜) where g′ (.) > 0, g″ (.) < 0. Their first order condition equates the expected return between the two investment opportunities and yields an implicit expression for the market-clearing fire-sale discount.

ϕ=1g(E˜ϕ[(1p)σh+pσl]I˜)(5)

Without loss of generality, we can write this as ϕ=f(I˜) where f (.) is a continuous and decreasing function.

We restrict the fire-sale discount to values within the range ϕ(ϕ_,ϕ¯] by imposing the following restrictions on f (.) (and hence on E˜ and g (.))

ϕ_<f((Aαα)11α((1q)σh1+μ+qσ1)α1α)<ϕ¯(6)

where

ϕ_σl(1p)σh+pσl0ϕ¯min[1,(1q)σh1+μ+qσl(1p)σh+pσl]>ϕ_

The upper bound of this restriction ensures that the fire-sale deepens as the shadow banking sector grows, and is necessary for bringing about an interior equilibrium where shadow and traditional banks coexist.15 The lower bound, on the other hand, only serves to simplify our exposition by eliminating the possibility of an (interior) equilibrium case with slightly different properties to the one we describe in Section 3.2. We discuss the consequences of relaxing the lower bound restriction in Appendix E.

3.1.3 Households

There is a unit continuum of risk neutral households which derive utility only from final period consumption, making their utility maximization problem equivalent to maximizing expected consumption16

E[C]=(1q)Cgh+q(1p)Cbh+qpCbl(7)

where (Cgh, Cbh, Cbl) are respectively consumption under good news and bad news with high and low payoff from assets. In the first period, households allocate their endowment E between deposits in traditional and shadow banks (dTB, dSB) and cash M1 which transfers funds to the next period at zero net return. The first period budget constraint is

dSB+dTB+M1=E(8)

When the second period begins with good news, there are no withdrawals and households simply retain their cash holdings M1. In the third period, consumption is then given by

Cgh=M1+dTBRTB+dSBRSB(9)

where (RTB, RSB) represent the deposit interest rates at traditional and shadow banks.

Following bad news, on the other hand, households observe the decline in expected asset returns and decide whether to withdraw their deposits early. Before we consider this decision, we solve the household’s optimization problem in the first period taking it as given that only shadow banks face an early withdrawal. The second period budget constraint under bad news can then be written as

M2=M1+θSBdSB(10)

where θSB is the liquidation value of shadow bank portfolios. In the third period, traditional banks may be rendered insolvent by a low payoff from assets. Therefore, consumption is contingent on asset payoffs

Cbh=M2+dTBRTB(11)
Cbl=M2+VdTBRTB(12)

with V representing the recovery rate of traditional bank deposits.17

The representative household chooses {dTB, dSB, M1, M2} to maximize its expected consumption (7) subject to (8)-(12). The first order conditions yield the following expressions for interest rates

RTB=1+qp(1V)1qp(1V)(13)
RSB=1+q1q(1θSB)(14)

Observe that the interest rate on traditional bank deposits decreases with a rise in the recovery rate V, while that of shadow banks decreases with a higher liquidation value θSB. Traditional banks may borrow at the risk-free rate RTB = 1 when they guarantee a complete repayment of deposits V = 1 while shadow banks must be completely liquid with θSB = 1.

Early withdrawal decision and market discipline

Following bad news, it is optimal for households to withdraw their deposits when doing so increases their expected consumption (1 – p)cbh + pcbl conditional on bad news. Proposition 1 describes the outcome of the early withdrawal decision.

Proposition 1 For all ϕ > ϕ, it is optimal for households to withdraw their deposits from shadow banks after bad news. Traditional banks may avoid an early withdrawal by committing to a minimum recovery rate

V>V¯1p(1RTB(1p))(15)

In equilibrium, the recovery rate and interest rates on traditional bank deposits are given by

V¯=RTB=1(16)

Proof. Provided in Appendix A

Note that (15) establishes a no-withdrawal constraint for traditional banks. This constraint is the key difference between shadow and traditional banking strategies, and generates similar behaviour to the observations discussed under Fact 4. To satisfy this constraint, traditional banks react to bad news by re-allocating their portfolio from risky to safe assets.

The proposition also shows that the no-withdrawal constraint imposes market discipline on traditional banks, eliminating insolvency risk and reducing their funding costs to the risk-free rate. This is the outcome of a virtuous cycle between deposit rates RTB and the minimum recovery rate V¯. As traditional banks guarantee a minimum recovery rate, households demand less compensation for insolvency risk and deposit rates decrease as per (13). Lower deposit rates in turn strengthen the threat of early withdrawal as households stand to lose less in terms of interest foregone. Therefore, the minimum recovery rate V¯ increases until all solvency risk is eliminated in equilibrium.

It is important to note that this result is contingent on the lack of liquidity risk. In Section 4, we show that even a small risk of a (Diamond-Dybvig) bank run on traditional banks reverses this virtuous cycle, leading to a rise in insolvency risk and funding costs.

3.1.4 Banks

There is a unit continuum of ex-ante identical, risk neutral banks. In the first period, banks collect deposits D from households and purchase assets I1 from entrepreneurs at price P1 as well as holding cash M1. Their first period budget constraint can then be written as

P1I1+M1=D(17)

In the second period, banks trade assets in the secondary market. As discussed in Section 3.1.2, following good news, assets are priced at their expected payoff in the secondary market and trade is inconsequential. Bank profits (in the third period) are given by

Πgh=σhI1+M1DR

Following bad news, banks that are subject to an early withdrawal have a liquidation value

θ=min{1,P2I2+M2D}(18)

and those that are not face the second period budget constraint

P2I2+M2=P2I1+M1(19)

Observe that a fire-sale reduces the liquidation value as well as the maximum cash M2 banks may attain through portfolio re-allocation. When assets yield a high payoff σh in the third period, banks make a profit

Πbh=σhI2+M2DR

while limited liability binds under a low asset payoff σl.18 Under limited liability, banks make zero profits and deposits pay a recovery rate

V=min{1,σlI2+M2DR}(20)

which is proportional to the shortfall of funds.

The commitment cost τ > 0 creates a discontinuity in the optimization problem of banks such that it can be evaluated as a choice between two distinct investment strategies. Under a shadow banking strategy (labelled as ‘SB’), banks do not pay the commitment cost and anticipate an early withdrawal after bad news. Under a traditional banking strategy (labelled as ‘TB’), banks pay the cost τ and commit to satisfying the no-withdrawal constraint (15). We find it convenient to to write the no-withdrawal constraint in terms of the choice variables by combining (15) and (20).19

σlI2TB+M2TBV¯DTBRTB(21)

In the first period, banks adopt the strategy that leads to the highest expected payoff such that shadow banking is preferred when

E[ΠSB]E[ΠTB]τ(22)

where ESB] and ETB] are the expected payoffs associated with shadow and traditional banking. Below, we solve the optimization problem under each strategy and attain an expression for expected payoffs by combining banks’ first order conditions with those of households from Section 3.1.3. In doing so, we take the secondary market price P2 as given.

Shadow banking

Shadow banks choose {I1SB,M1SB,DSB} to maximize their expected profits

E[ΠSB]=(1q)(σhI1SB+M1SBDSBRSB)(23)

subject to the budget constraint (17). Since limited liability binds after an early withdrawal, shadow banks only internalize the payoff after good news. Lemma 1 provides the solution the shadow bank’s problem. It shows that, even without a fire-sale, shadow bank deposits are not paid in full during an early withdrawal (θSB < 1) and funding costs of shadow banks are above the risk free rate (RSB > 1) as a consequence.

Lemma 1 Combining the solution to the shadow bank’s problem given by (35) and (36) with the household first order condition (14) yields the following expressions for (θSB,RSB,E[ΠSB],I1SB)

θSB=(1+μ)P2(1q)σh+q(1+μ)P2<1ϕ[0,1]RSB=σh(1q)σh+q(1+μ)P2>1ϕ[0,1]E[ΠSB]=(1q)μ1+μσhI1SBI1SB=(Aαα)11α((1q)σh1+μ+qP2)α1α

Proof. Provided in Appendix C

Figure 6 provides a graphical representation of the shadow banking strategy. The red line depicts banks demand for deposits, which is inversely related to the asset price P1. As deposits are used to purchase risky assets I1, its downward slope reflects the positive relationship between I1 and P1 given by the asset supply schedule (3). The blue line depicts the supply of deposits by households. It is horizontal at the risk-free rate R = 1 when there is no early withdrawal or θ = 1, but becomes upward sloping when a rise in P1 reduces the liquidation value θ and drives households to require a higher interest rate in compensation as per (14). Under a shadow banking strategy, banks optimally invest up to the intersection of these two curves where the funding costs exceed the risk-free rate.

Figure 6:
Figure 6:

Shadow banking strategy

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: The deposit demand curve is attained by combining (3) and (17). Deposit supply stems from (14) and (35).

Traditional banking

Traditional banks choose {I1TB,I2TB,M1TB,M2TB,DSB} to maximize their expected profits

E[ΠTB]=(1q)(σhI1TB+M1SB)+q(1p)(σhI2TB+M2TB)(1qp)DTBRTB

subject to the budget constraints (17), (19) and the no-withdrawal constraint (21).20

Lemma 2 provides the solution to the traditional bank’s problem.21 It shows that, following bad news, traditional banks liquidate their risky asset holdings in the secondary market and re-allocate their portfolio toward cash to satisfy the no-withdrawal constraint. 22 Since the terms of trade between risky assets and cash depend on the secondary market price P2, the no-withdrawal constraint also reduces banks risky asset purchases I1 in period 1 in line with P2. Since the funding costs of traditional banks remain at the risk-free rate (RTB = 1), this creates an excess return that contributes to traditional bank profits (see Figure 7 for a graphical representation).

Figure 7:
Figure 7:

Traditional banking strategy

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: The deposit demand curve is attained by combining (3) and (17). Deposit supply stems from (14) and (35). The no-withdrawal constraint is given by (21), (24), and (25).

Lemma 2 Under the restrictions μμ¯,ϕ>ϕ_, the solution to the traditional bank’s problem is

I1TB=(Aαα)11αP2α1αI2TB=0(24)
M2TB=P2I1TB+M1TB(25)E[ΠTB]=(1q)(σhP2)I1TB

where M1TB0 is indeterminate, (RTB,V¯) are given by (16), and

E[ΠTB]P2<0P2>ασh

Proof. Provided in Appendix D. ▪

3.2 Equilibrium

We solve for a rational expectations equilibrium such that all optimality conditions and constraints of banks, households, entrepreneurs and outside investors are satisfied, expectations are confirmed, and deposit and secondary markets clear such that

γDSB=dSB(1γ)DTB=dTBϕ=f(γI1SB+(1γ)(I1TBI2TB))

where γ ∈ [0,1] is the share of banks that adopt a shadow banking strategy.

Furthermore, an equilibrium is characterized as ‘interior’ when traditional and shadow banks coexist such that γ falls within the range 0 < γ < 1. In an interior equilibrium, banks are indifferent between shadow and traditional banking strategies such that (22) holds with equality

E[ΠSB]=E[ΠTB]τ(26)

and may be interpreted as a free entry condition that determines γ in equilibrium.

We proceed as follows in our description of the equilibrium solution: Section 3.2.1 builds up on the intuition provided about bank strategies above by discussing their interaction with fire-sales. We focus on fire-sales due to their role in creating strategic substitutabilities in banks’ decision to enter shadow banking, which is crucial for bringing about an interior equilibrium. Section 3.2.2 then provides the conditions under which an interior equilibrium arises and discusses the implications of a rise in the costs of commitment and a deepening of secondary markets.

3.2.1 Fire-sales and bank strategies

There is a two-way interaction between fire-sales and bank strategies. On the one hand, entry into shadow banking increases the excess supply of assets in the secondary market and exacerbates fire-sales.23 On the other hand, fire-sales reduce the expected payoff from a shadow banking strategy relative to traditional banking and deter entry into the shadow banking sector.

Figure 8 demonstrates the mechanism behind this. As shown in Panel A, a fall in the secondary market price P2 reduces shadow banks liquidation value during an early withdrawal. This leads to an upward pivot in the deposit supply curve, raising shadow banks funding costs and reducing the expected payoff from shadow banking.

Figure 8:
Figure 8:

Fall in the secondary market price

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: The deposit demand curve is attained by combining (3) and (17). Deposit supply stems from (14) and (35). The no-withdrawal constraint is given by (21), (24), and (25).

In contrast, a fall in P2 leads to a rise in the expected payoff from traditional banking.24 This is because it worsens the terms of trade between risky assets and cash after bad news, tightening the no-withdrawal constraint. As shown in Panel B, an inward shift in the constraint forces traditional banks to reduce investment I1 in risky assets in the first period. Since traditional banks funding costs remain at the risk-free rate, this increases the excess return and expected payoff from traditional banking.

These interactions constitute a fire-sale externality since banks do not internalize the impact of their entry into shadow banking on the profitability of other banks. Moreover, since profitability affects entry incentives, fire-sale externalities create strategic substitutabilities in banks decisions to adopt a shadow banking strategy. As the shadow banking sector grows, the fire-sale discount on risky assets gets larger (i.e. ϕ falls), reducing the payoff from shadow banking relative to traditional banking until we reach an equilibrium sector size where banks are indifferent between the two strategies.

3.2.2 Interior equilibrium

Proposition 2 provides the conditions for an interior equilibrium. It shows that there will be an interior equilibrium when commitment costs fall within the range τ(τ_,τ¯).

Proposition 2 There is an interior equilibrium under the parameter restrictions μμ¯,ϕ(ϕ_,ϕ¯),α<0.5,p<1α and τ(τ_,τ¯) where

τ_{κ[σh1q(1(1q)σhq)α1αμ1+μσh(1μ1+μ(1q)σh)α1α]forμ<(pq)(σh1)qp(1q)σh(pq)(σh1)κ(σhϕ¯[(1p)σh+pσl])(ϕ¯[(1p)σh+pσl])α1ακμσh1+μ((1q)σh1+μ+qϕ¯[(1p)σh+pσl])α1αotherwiseτ¯{κ(σh)11α[(1α)αα1αμ1+μ(1q1+μ+qα)α1α]forσh11qp(1α)κ[σh1qp(1(1qp)σhpq)α1αμ1+μσh((1q)σh1+μ+1(1qp)σhp)α1α]otherwise

such that κ(1q)(Aαα)11α,τ¯>τ_>0.

Proof. Provided in Appendix 2 ▪

Finally, we briefly discuss the properties of this equilibrium. In an interior equilibrium, the free entry condition (26) pins down the size of the shadow banking sector γ and fire-sale discount ϕ through the interactions described in Section 3.2.1. Figure 9 illuminates the mechanism behind this by demonstrating the comparative statics of a rise in the commitment cost and a deepening of secondary markets under a numerical example.

Figure 9:
Figure 9:

Numerical example

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: The numerical example correponds to the calibration A = 1, α = 1/3, q = p = σl = 1/2, μ=μ¯, and τ = 0.0825. We parameterize the fire-sale function f (.) according to Appendix H and calibrate E˜ to get ϕ as a lower bound and set κ = 10 ((1 – p) σh + l)–1. Panel A and B respectively display the effects of a small rise in τ and E˜.

Observe that the expected payoff schedule for traditional (shadow) banking is upward (downward) sloping in γ in line with the intuition from Section 3.2.1. The equilibrium share of shadow banks is at the point where these two schedules intersect as per (26). We also plot this as a vertical bar along with the fire-sale schedule in order to deduce the equilibrium fire-sale discount.

Panel A shows that a rise in the cost of commitment τ causes a downward shift in the expected payoff schedule for traditional banking. At a given sector size γ, this makes shadow banking relatively profitable and leads to further entry into the sector. This in turn increases the excess supply of assets in the secondary market following bad news and exacerbates the fire-sale. Therefore, in equilibrium, a rise in the commitment cost increases the size of the shadow banking sector and the vulnerability of the economy to fire-sales.

Panel B shows the effects of an increase in the liquidity available in the secondary market due to a rise in the endowment of outside investors. At any given sector size γ, this reduces the fire-sale discount (i.e. a rise in ϕ) bringing about an upward shift in the fire-sale schedule. Consequently, the expected payoff schedule for traditional (shadow) banking shifts down (up) in line with Section 3.2.1, and there is entry into shadow banking until the new schedules intersect at a larger sector size.

It is important to note that the fire-sale discount returns to its initial value at the new equilibrium. As such, a thicker secondary market for assets increases the size of the shadow banking sector but does not alleviate the fire-sale. This finding stems from an essential property of the interior equilibrium: the fire-sale discount is implicitly determined by the free entry condition (26). Note that these results also apply to policy interventions aimed at alleviating fire-sales such as asset purchases and bailouts of financial institutions that would have contributed to the excess supply of assets. We analyze policy interventions in greater detail in Section 6.

4 A model with liquidity risk

In this section, we extend the simple model to a setting with liquidity risk and a richer asset space. We introduce liquidity risk by allowing for the possibility of bank-runs as in Diamond and Dybvig (1983). Bank-runs are distinct from early withdrawals in that they are driven by self-fulfilling expectations in the presence of a liquidity shortfall (θ < 1) in bank balance sheets, rather than an optimal response to bad news.

We also expand the asset space to allow for three different asset types, liquid, safe, and risky, which we respectively denote with (λ, s, r). The risky asset is identical to the (non-cash) asset in the simple model. The safe and liquid assets both yield a unit payoff with certainty, but they differ in that the safe asset matures in period 3 while the liquid asset yields its payoff in period 2.

The richer asset space serves two purposes. First, it allows us to consider the conditions under which safe assets are endogenously liquid due to secondary markets. A priori, it is not clear whether safe assets would be subject to a fire-sale since purchases by traditional banks (which re-allocate their portfolios towards safe assets) may offset sales by shadow banks.

Second, a richer asset space permits us to consider the interplay between fire-sales, market discipline, and liquidity risk. On the one hand, without a fire-sale on safe assets, portfolio re-allocation required to satisfy the no-withdrawal constraint increases traditional banks liquidation values (θTB = 1) and reduces their vulnerability to bank-runs. On the other hand, when there is a fire-sale on safe assets, liquidity risk leads to a rise in bank funding costs and undermines market discipline such that traditional banks may also take on solvency risk.

In the interest of brevity, we only describe the aspects of the model that differ from Section 3.25 The remainder of the section is organized as follows: Section 4.1 provides further details about bank-runs. Section 4.2 extends secondary markets and outside investors to a framework with multiple assets. Finally, Section 4.3 presents analytical results pertaining to the relationship between the fire-sale on safe assets, market discipline, and liquidity risk.

4.1 Bank-runs

When a bank has a liquidity shortfall θ < 1, sequential service in withdrawals leads to the emergence of a bank-run equilibrium where households find it optimal to withdraw their deposits given that everyone else is withdrawing. We resolve this multiplicity with the use of sunspots. Specifically, we let εξ be be a random variable drawn from a uniform distribution on the unit interval at the beginning of period 2 and ξ¯[0,1] a constant threshold. When ɛξξ¯, household sentiments coordinate on a bank-run. Provided that there is a liquidity shortfall, these sentiments become self-fulfilling and the bank suffers from a run. Since εξ is uniformly distributed on a unit interval, the probability of a bank run is then simply given by

ξ={0forθ=1ξ¯forθ<1

Only traditional banks are affected by bank-runs as there is no liquidity shortfall after good news and shadow banks face an early withdrawal after bad news.26 Moreover, we assume that the sunspot realization is idiosyncratic to each bank such that ξ may be interpreted as the share of traditional banks liquidated in a bank-run.27

Finally, note that it is straightforward to introduce a more sophisticated specification for the bank-run probability. We adopt this simple framework as it permits us to isolate the role of banks’ portfolio strategies and fire-sales in bringing about a liquidity shortfall in the first place. In Appendix J, we approximate the global games solution of Goldstein and Pauzner (2005) by depicting ξ¯ as a negative function of the liquidation value θ. The qualitative results from our mechanism remain the same.

4.2 Secondary market

We extend the secondary market described in Section 3.1.2 to a framework with multiple assets. Since the liquid asset yields its payoff in period 2, only safe and risky assets are traded in the secondary market. The excess supply of each asset is given by the expressions

I˜(i)=γI1SB(i)+(1γ)(I1TB(i)(1ξ)I2TB(i))0i{s,r}

and sold to a common set of risk neutral outside investors.

Outside investors allocate their endowment E˜ between purchases of safe and risky assets and an outside project K˜ that yields a safe payoff g(K˜) where g′ (.) > 0, g″ (.) < 0. Their first order conditions indicate that there is a common fire-sale discount ϕ ∈; [0,1] for the two assets, and it is defined implicitly by the expression

ϕ=1g(E˜ϕ[(1p)σh+pσl]I˜(r)ϕI˜(s))(27)

which suggests that a rise in the total excess supply of safe and risky assets leads to a decrease in ϕ. Crucially, however, an asset type is only subject to the fire-sale discount when it is in excess supply such that its marginal buyer is an outside investor. Therefore, secondary market prices depend on an asset s own excess supply such that

P2(s)={1forI˜(s)=0ϕotherwiseP2(r)={(1p)σh+pσlforI˜(i)0ϕ[(1p)σh+pσl]]otherwise

4.3 Analytical results

Proposition 3 shows that the fire-sale on safe assets plays a definitive role in shaping the relationship between market discipline and liquidity risk.

Proposition 3 In equilibrium, traditional banks commit to a minimum recovery rate

V¯=1qpξ(1θTB)1q(1ξ(1θTB))(28)

and the interest rate on their deposits is given by

RTB=1+q1qξ(1θTB)(29)

With P2(s) = 1, traditional banks have a liquidation value θTB = 1. This leads to ξ = 0 and (28), (29) yield

V¯=RTB=1

With sufficiently low P2 (s) < 1, traditional banks have a liquidity shortfall θTB < 1 which leads to ξ > 0. (28), (29) then yield

V¯<1<RTB

Proof. Provided in Appendix G

When there is no fire-sale on safe assets such that P2 (s) = 1, satisfying the no-withdrawal constraint

σlI2TB(r)+I2TB(s)V¯RTBDTB(30)

leads to a liquidation value

θTB=min[1,P2(r)I2TB(r)+P2(s)I2TB(s)DTB]=1(31)

since risky assets never trade at a price below their payoff under weak fundamentals (P2 (r) ≥ σl). This in turn eliminates the possibility of bank-runs such that ξ = 0. Without liquidity risk, market discipline through the threat of an early withdrawal ensures that there is also no solvency risk (V¯=1) and traditional banks collect deposits at the risk-free rate as in Section 3.1.3.

With a sufficiently large fire-sale on safe assets, on the other hand, traditional banks have a liquidity shortfall (θTB < 1) despite satisfying the no-withdrawal constraint. This leaves traditional banks vulnerable to bank-runs and creates a vicious cycle between rising deposit interest rates and solvency risk. Households anticipate that they may not be repaid fully in the event of a bank run and demand higher deposit rates as compensation. The rise in deposit rates in turn weakens the threat of early withdrawal as households stand to lose more in terms of interest foregone. This leads to a decline in the minimum recovery rate V¯, allowing traditional banks to take on solvency risk. Finally, the increase in solvency risk brings about a further rise in deposit rates and completes the vicious cycle.

In the next section, we present numerical results that highlight the role of a large shadow banking sector in causing a fire-sale on safe assets, which leaves traditional banks illiquid and undermines market discipline through the mechanism described above.

5 Numerical results

This section provides numerical results from the model with liquidity risk. It proceeds as follows: Section 5.1 describes the calibration which targets the United States over the recent financial crisis. Section 5.2 presents and discusses the results from a numerical simulation.

5.1 Calibration

Table 2 reports the calibrated parameters. The payoff of risky asset under weak fundamentals is set to σl = 0.21 in line with the average recovery rate from junior debt and σh is backed out using the normalization of expected payoffs given in (1).28 The probabilities (q, p) are calibrated to the frequency of recessions and deep recessions (conditional on a recession) in the United States.29

Table 2:

Calibration

article image

Regarding the entrepreneurs, we calibrate the output elasticity of capital to the standard Cobb-Douglas value α = 1/3 and normalize productivity to A = 1. For the mark-up, we adopt a parameterization that approximates monopolistic competition

μ=υ1αα

where υ ∈ [0,1] represents banks’ market share. We then calibrate the mark-up to a value consistent with the 5-bank asset concentration in the United States over 2007-2010.

In the secondary market, we parameterize the payoff function for outside investments to g(K˜)=z1ln(K˜) where z is a constant such that (27) becomes

ϕ=zE˜1+κ[((1p)σh+pσl)I˜(r)+I˜(s)]

Our calibration strategy for (E˜,z), targets the rise in the yield spreads between AAA-rated seasoned corporate bonds and the effective Federal Funds Rate during the financial crisis. Specifically, we back out a percentage decline in bond prices ΔP^/P^ from the difference between the peak spread in November 2008 and the average spread for the pre-crisis period between January 2016 and June 2017. We then calibrate (E˜,z) to generate an equilibrium fire-sale discount that matches the decline in the price of safe assets under bad news to ΔP^/P^.30

Similarly, the calibration for the bank-run probability ξ¯ targets TED spreads at the peak of the financial crisis. To do this, we construct a hypothetical interest rate for traditional bank deposits collected after bad news but before the realization of bank-runs. With risk neutral households, this is given by

R2TB=11ξ¯1ξ¯θTB1p(1V¯)(32)

and we calibrate ξ¯ to match R2TB to the average TED spread in the last quarter of 2008.31

Finally, the calibration for the commitment cost τ targets the pre-crisis ratio of shadow bank assets to total bank assets, which is approximately 63%. Accordingly, we set τ = 0.11 which amounts to just below 7% of the value of a traditional bank’s assets in period 1.

5.2 Results

Figure 10 plots the numerical solution across a range γ ∈ [0,1] of shadow banking sector sizes. The equilibrium sector size is denoted by the vertical bar labelled γ* where traditional and shadow banking strategies yield the same expected payoff. The first panel of Figure 10 indicates that there is a unique and globally stable interior equilibrium where shadow and traditional banks coexist. This reflects the strategic substitutabilities between banking strategies described in Section 3.2.

Figure 10:
Figure 10:

Numerical results

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Expected payoffs are inclusive of the commitment cost τ. Total assets in period 1 and 2 are respectively defined as I¯1Σi{λ,s,r}I1(i) and I¯2Σi{s,r}P2(i)I2(i). Panel 7 plots (I¯2TB)1P2(s)I2TB(s)(I¯1TB)1Σi{λ,s}I1TB(i) for safe and liquid assets, and (I¯2TB)1P2(r)I2TB(r)(I¯1TB)1I1TB(r) for risky assets. Panel 8 plots I1TB(i)/I¯1 respectively for i = {λ, s, r} and Panel 9 does the same for shadow banks.

Entry into shadow banking exacerbates the fire-sale after bad news (see Panel 2). This increases the interest rates on shadow bank deposits (see Panel 4) and reduces the expected payoff from shadow banking. In contrast, traditional bank profits rise as the fire-sale increases the excess return from the binding no-withdrawal constraint. Therefore, the shadow banking sector grows until banks are indifferent between the two strategies at the equilibrium sector size γ*.

The bar labelled γ¯ denotes the threshold sector size above which safe assets suffer from a fire-sale. Under our calibration, high commitment costs lead to γ*>γ¯ such that there is a fire-sale on safe assets in equilibrium. Panel 3 shows that this reduces the liquidation value of traditional banks, leaving them vulnerable to bank-runs. As explained in Section 4.3, liquidity risk increases deposit rates RTB (see Panel 4) and undermines market discipline on traditional banks. Accordingly, Panel 5 shows that the minimum recovery rate V¯ declines below full repayment for γ>γ¯ such that traditional banks have insolvency risk in equilibrium.

Note that, with lower commitment costs, the expected payoff from traditional banking is higher at all sector sizes such that we may have a smaller shadow banking sector γ*γ¯ in equilibrium and no fire-sale on safe assets.32 Panel 3 shows that traditional banks would then have no liquidity shortfall and hence no vulnerability to bank-runs. Without liquidity risk, traditional banks borrow at the risk-free rate and the no-withdrawal constraint imposes market discipline with V¯=1 such that traditional banks have no solvency risk either.

Panel 6 shows the evolution of shadow and traditional bank deposits across γ. As γ rises and the fire-sale deepens, shadow banks respond to the rise in their funding costs by reducing their deposits. Traditional banks, on the other hand, expand their balance sheets due to the relaxation of the no-withdrawal constraint. This is consistent with observations in Fact 1 of Section 2 which indicate that traditional bank balance sheets expanded at a faster rate in 2002-07 when the shadow banking sector was growing rapidly than in 2008-15 when the shadow banking sector was stagnant. 33

Panel 7 demonstrates portfolio re-allocation by traditional banks after bad news. When the shadow banking sector is small and market discipline intact, traditional banks re-allocate up to 18% of their portfolio from risky assets to safe and liquid assets in order to prevent an early withdrawal. As the shadow banking sector grows and market discipline is undermined, traditional banks reduce the extent of their re-allocation away from risky assets and eventually the direction is reversed. In equilibrium, there is a re-allocation of approximately 4% away from risky assets. Compared to the observations in Fact 4 of Section 2, this predicts the direction of portfolio re-allocation correctly, but falls somewhat short of the observed amount of 6% to 8% (see Figure 4). It is possible that this discrepancy reflects the impact of macroprudential regulation and additional liquidity provided by the Federal Reserve.

Finally, Panels 8 and 9 display the investment strategies of traditional and shadow banks in period 1. When there is no fire-sale on safe assets (γ<γ¯), market discipline imposed by the no-withdrawal constraint forces traditional banks to behave as if they internalize asset payoffs under weak fundamentals. Therefore, they devote an equal share of their investment to each asset type. In response to a fire-sale on safe assets, traditional banks increase their holdings of liquid assets to prop up their liquidation value. While this increases the funds available to them after bad news, their reaction is not strong enough to keep them solvent under weak fundamentals because of the decline in the minimum recovery rate V¯ (as shown in Panel 5).

Shadow banks, on the other hand, skew their investment towards risky assets since they only internalize the states after good news due to limited liability. Without the ability to commit, they also take their borrowing costs RSB as given and hence do not change their asset composition in response to fire-sales.

6 Policy analysis

In this section, we consider policy interventions aimed at fostering financial stability. The numerical results in Section 5.2 demonstrate that, with sufficiently high commitment costs, the shadow banking sector expands to a size that is systemically risky in the sense that its collapse gives rise to both liquidity and solvency risk for traditional banks.

We consider three distinct interventions that aim to offset this. Section 6.1 considers asset purchases in the secondary market. Section 6.2 evaluates interventions to secure the traditional banking sector such as liquidity support, deposit insurance guarantees and macroprudential regulation. Finally, Section 6.3 proposes a tax on shadow bank profits.

6.1 Asset purchases

The government may lean against fire-sales by purchasing assets in the secondary market. In the interest of simplicity, we consider an intervention that purchases solely safe assets such that their excess supply becomes

I˜(s)=γI1SB(s)+(1γ)(I1TB(s)(1ξ)I2TB(s))IAP

where IAP > 0 refers to asset purchases by the government.34 Figure 11 shows the outcome of an intervention that aims to eliminate the fire-sale on safe assets by setting IAP to absorb their excess supply at the equilibrium sector size γ*.

Figure 11:
Figure 11:

Asset purchases

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Expected payoffs are inclusive of the commitment cost τ. AP refers to the outcome under asset purchases.

The outcome of this intervention is identical to the deepening of secondary markets considered in Section 3.2.2. It reduces the fire-sale discount (i.e. increases ϕ) at any given sector size and is successful in eliminating the fire-sale on safe assets at the initial equilibrium γ*. In other words, the intervention shifts the fire-sale threshold γ¯ to γ* (see Panel B). However, by alleviating the fire-sale, the intervention also reduces the funding costs of shadow banks and increases the expected payoff from shadow banking relative to traditional banking (see Panel A). This leads to entry into the shadow banking sector until we reach the new equilibrium sector size γAP*>γ*.

It is not a coincidence that in the new equilibrium γAP*, the fire-sale discount (and all variables other than the sector size) takes the same value as in the initial equilibrium γ*. This is because, as explained in Section 3.2.1, the fire-sale discount is implicitly determined by the free entry condition (26). Our analysis then leads to a crucial insight: policy interventions in secondary markets are ineffective when there is free entry into shadow banking. This is also true in the opposite direction. For example, a tax on secondary market transactions deepens the fire-sale and reduces shadow bank profits at a given sector size, but both of these effects are reversed by exit from the shadow banking sector.

Finally, is important to stress that entry into shadow banking is driven by the ex-ante anticipation of asset purchases. In period 2, the size of the shadow banking sector is already determined and a surprise asset purchase scheme would be successful in alleviating the fire-sale on safe assets and reducing both liquidity and solvency risk. This may potentially lead to time inconsistency issues as policymakers would naturally find it tempting to intervene once the fire-sale is underway.

6.2 Interventions to secure traditional banks

Recall from Fact 4 in Section 2 that traditional bank liabilities sharply increased in the second half of 2008. This increase was driven by Federal Reserve funding and a rise in deposits protected by deposit insurance guarantees. In this section, we show that interventions of this form may effectively ringfence the traditional banking sector from the financial instability caused by shadow banks. In equilibrium, these interventions increase the size of the shadow banking sector but have a net positive impact on financial stability.

To begin with, we consider a liquidity provision scheme that permits traditional banks to borrow L from the central bank at the risk-free rate in the second period. The liquidation value of traditional banks can then be written as

θTB=min[1,P2(r)I1TB(r)+P2(s)I1TB(s)+I1(λ)+LDTB]

and two conditions must be satisfied for liquidity provision to eliminate both solvency and liquidity risk. First, the amount of liquidity provided should be sufficient to offset liquidity shortfalls such that

LL_DTBP2(r)I1TB(r)P2(s)I1TB(s)I1(λ)

This eliminates the possibility of bank-runs (ξ = 0) and liquidity risk.

Second, liquidity borrowed from the central bank should be fully collateralized.35 This rules out default on the central bank and makes depositors effectively junior to the central bank. The no-withdrawal constraint then becomes

σlI2TB(r)+I2TB(s)L+V¯DTBRTB

such that market discipline is not adversely affected by liquidity provision. The results from Proposition 3 then follow through: in the absence of liquidity risk, the threat of an early withdrawal eliminates solvency risk (V¯=1) and interest rates on traditional bank deposits fall to the risk-free rate (RTB = 1).

Figure 12 shows the equilibrium outcome under liquidity provision in comparison to the baseline results in Section 5.2. Panel B shows that, in addition to ringfencing the traditional banking sector, liquidity provision eliminates the fire-sale on safe assets. This is because, with P2 (s) < 1, traditional banks take advantage of an arbitrage opportunity by borrowing from the central bank at the risk-free rate and purchasing safe assets at a discount.36 They continue to do this until there is no excess supply of safe assets such that P2 (s) = 1. Panel D shows that these purchases also increase the extent of portfolio reallocation towards safe and liquid assets.

Figure 12:
Figure 12:

Liquidity Provision

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Expected payoffs are inclusive of the commitment cost τ.

By alleviating the fire-sale, liquidity provision indirectly reduces the funding costs of shadow banks. This in turn increases the expected payoff from shadow banking, leading to further entry into the sector (see Panel A) and increases its size in equilibrium. However, this does not have any adverse effects on traditional banks.

Without explicitly conducting welfare analysis, we also find it worth pointing out that, in equilibrium, liquidity provision increases household consumption both under weak fundamentals (Panel E) as well as in terms of its unconditional (period 1) expectation (Panel F).37 This is because, by preventing the fire-sale on safe assets, the intervention alleviates the destabilizing impact of the expansion in shadow banking. Therefore, the gains from securing the traditional banking sector are not offset by the rise in the size of the shadow banking sector.

Finally, note that the exact same outcome may be achieved by combining deposit insurance guarantees with bank regulation that imposes a minimum recovery rate V¯=1. This intervention differs from (collateralized) liquidity provision only with respect to the role of market discipline. As explained above, liquidity provision is designed to take advantage of market discipline while securing traditional banks from liquidity risk. In contrast, eliminating liquidity risk through deposit insurance also removes the no-withdrawal constraint on traditional banks since depositors never stand to take a loss. Market discipline must then be replaced with bank regulation to prevent risk-shifting.

6.3 Tax on shadow bank profits

The last policy intervention we consider is the taxation of shadow bank profits with the purpose of deterring entry into the sector. This can be considered as a Pigouvian tax since entry into shadow banking imposes a negative externality on the remainder of the financial sector through its contribution to fire-sales. We consider a constant tax T such that the free entry condition becomes

E[ΠSB]T=E[ΠTB]τ

Figure 13 shows the outcome under a tax level 0 < T < τ that reduces the equilibrium sector size of shadow banks to the fire-sale threshold γ¯. Panel A shows that the tax shifts down the expected payoff schedule for shadow banking and brings about an equilibrium without a fire-sale on safe assets. Since this is a tax on profits, the equilibrium allocation at any given sector size γ is identical to the numerical results in Figure 10. Therefore, reducing the equilibrium sector size to γ¯ eliminates liquidity risk in the traditional banking sector which strengthens market discipline such that there is also no solvency risk.

Figure 13:
Figure 13:

Tax on shadow bank profits

Citation: IMF Working Papers 2017, 285; 10.5089/9781484335376.001.A001

Note: Expected payoffs are inclusive of the commitment cost τ and tax T.

Note that the effect of a tax on shadow banks is equivalent to a decrease in the commitment cost τ. Therefore, when taxing shadow banks is not feasible, the same outcome can also be achieved with a transfer to traditional banks.38 Alternatively, shadow bank profits may be reduced indirectly with a tax on their liabilities DSB. In a less stylized set up, this would correspond to a tax on the funding instruments most widely used by shadow banks such as ABCPs.

7 Conclusion

We have presented a model of the financial sector in which shadow banking emerges endogenously as an alternative banking strategy. A key aspect of the model is that depositors re-optimize in response to revisions in expectations about asset returns. To prevent early withdrawals by their depositors, traditional banks optimally commit to a safe portfolio strategy while shadow banking strategies combine high risk-taking with the prospect of an early liquidation after bad news.

Two important insights emerge as a consequence. First, costly commitment and fire-sale externalities bring about an equilibrium where ex-ante identical banks optimally cluster into shadow and traditional banking strategies. The size of the shadow banking sector increases in the cost of commitment and the availability of liquidity in the secondary market. Second, when the shadow banking sector is large, the liquidation of shadow banks leave traditional banks susceptible to liquidity risk. This increases deposit rates offered by traditional banks and weakens market discipline on them, engendering greater risk-taking and a rise in solvency risk.

The model also naturally provides novel insights for policy design. First, we show that policy interventions have significantly different implications when the adjustment on the size of the shadow banking sector is taken into account. Secondary market interventions, such as asset purchases by the government, are effective in alleviating the fire-sale ex-post, but their ex-ante expectation fuels further growth of the shadow banking sector in a manner that exactly offsets these gains. We conjecture that this may lead to time inconsistency issues for policymakers who may find it tempting to intervene once the fire-sale is underway.

Second, we find that policy interventions aimed at securing the traditional banking sector, similar to those implemented by the Federal Reserve during the 2008 financial crisis, also lead to an expansion in shadow banking when anticipated, but have a net positive impact on financial stability. Third, our findings regarding the destabilizing consequences of a large shadow banking sector lend support to taxation of shadow bank profits (or equivalently a transfer to traditional banks) with the purpose of reducing the size of the shadow banking sector to a level compatible with financial stability.

Finally, it is worth noting that the mechanism considered in this paper is more general than its application to shadow banking and the 2008 financial crisis. In economies without credible deposit insurance guarantees and strict enforcement of banking regulation, financial intermediation strategies that combine high risk-taking with an unstable funding structure may exist within the commercial banking sector. One clear example of this is the United States during the National Banking Era of 1863-1914. In this period, there was no central bank to act as a regulator or a public backstop for the banking sector, and commercial banks were subject to frequent bouts of early withdrawals (Gorton and Tallman, 2016). Another potential application of this framework is emerging market economies in contemporary times.

Shadow Banking and Market Discipline on Traditional Banks
Author: Mr. Anil Ari, Matthieu Darracq-Paries, and Christoffer Kok