Banking and Trading Arnoud W.A. Boot

Lev Ratnovski

University of Amsterdam and CEPR

International Monetary Fund

September 15, 2014

Abstract We study the interaction between relationship banking and the arm’s length marketbased activities of banks, which we call trading. Relationship banking is not scalable, with high franchise value, and a long-term focus. Trading is scalable, with lower margins (hence capital constrained), and short-term. A bank can use the franchise value of its relationships to pro…tably expand the scale of trading. However, we show that a bank may allocate too much capital to trading ex post, compromising its ability to build relationships ex ante. This e¤ect is reinforced when a bank can use trading for risk-shifting. Overall, combining relationship banking and trading o¤ers bene…ts at a low scale of trading, but distortions may dominate when trading is unbridled. The analysis suggests that trading by banks, while benign historically, might have become distortive as …nancial markets have deepened. It speaks to fundamental questions about the stability of bank business models going forward, and helps assess recent initiatives aimed at structural reforms in banking.

Contact: [email protected], [email protected]. We thank Tobias Adrian, Stijn Claessens, Giovanni Dell’Ariccia, Giovanni Favara, Thomas Gehrig, Hans Gersbach, Linda Goldberg, Mark Flannery, Jan-Pieter Krahnen, Luc Laeven, Christian Laux, Richhild Moessner, Marco Pagano, George Pennacchi, Joel Shapiro, Kostas Tsatsaronis, and participants of the ETH-NYU Law Conference on “Tackling Systemic Risk”, Basel Committee-CEPR Conference on “Banks – How Big is Big Enough”, Toulouse Conference “Risk Management and Financial Markets”, Oxford Financial Intermediation Conference, FDIC Annual Conference “Financial Services in the Current Environment”, Swiss Winter Conference on Financial Intermediation, and New York Fed-LBS conference on “Global Banks,” the Bonn conference on “The Structure of Banking Systems,” and of seminars at IMF and BIS for helpful comments. The views expressed are those of the authors and do not necessarily represent those of the IMF.

1

1

Introduction

The recent crisis has reopened the debate on the merits of combining traditional commercial banking with market-based activities. This paper sheds light on a novel interaction between relationship banking and arm’s length market-based activities of banks, which we call “trading.” We focus on di¤erences in the time horizon and scalability of relationship banking and trading, and study the desirability of combining them. We show that the deepening of …nancial markets might have led to time inconsistency problems in capital allocation, where banks engage in too much trading at the expense of relationship banking to the detriment of shareholder value. The results help explain existing evidence on diseconomies of scope in banking. Relationship banking involves private information and repeated interactions with an established set of customers. This makes relationship banking a long-term and not easily scalable activity. Trading does not rely on private information. When markets are deep enough, a bank can go in and out of positions rapidly. This makes trading relatively short-term and scalable. In modern banks, trading covers activities such as proprietary trading, investing in securitized debt, carry trade, market-making, originating loans based solely on hard information (typically mortgages), and all other activities that do not rely on relationship-speci…c private information. The increased involvement of banks in trading re‡ects a fundamental change in the nature of arm’s length …nance over the last decades. Financial innovation and the deepening of …nancial markets, both artifacts of information technology, have improved the tradability of previously non-marketable assets. As a result, much of banks’arm’s length exposures have become shortterm and scalable, i.e. trading-like. Indicators of the shift of U.S. banks towards trading include an increase in the ratio of trading assets and securities to loans on bank balance sheets from 30% in early 1990-s to 60% in 2013, and an increase of non-interest income as a share of bank revenues from 35% to 50% over the same period (NY Fed, 2014).1 The Liikanen Report (2012) points at similar developments in Europe. The interaction between relationship banking and trading is substantially di¤erent from that between lending and underwriting, which was the focus of the literature on the merits of the 1

The two measures are proxies and thus not perfect. For example, in balance sheet composition, some loans may be arm’s length, while in income some non-interest earnings may come from relationships.

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Glass-Steagall Act in the U.S. and universal banking in Europe (Kroszner and Rajan, 1994; Puri, 1996; Schenone, 2004; Fang et al., 2010). A key question in that literature is whether there are positive or negative synergies between two bank activities that use private information: lending and underwriting. We instead contrast relationship banking to an activity that does not rely on private information –trading. There is no scope for informational spillovers, so we identify di¤erent relevant interactions. The key contribution of this paper is to show that trading by a bank, while sometimes pro…table, may also undermine the bank’s relationship franchise. The main channel for that is the time inconsistency problem in bank capital allocation: a bank may allocate too much capital to trading ex post, which undermines its ability to build relationships with customers ex ante. This e¤ect is reinforced when a bank can use trading for risk-shifting. Overall, a bank may trade too much and in too risky a fashion, compared to what is ex ante optimal for its shareholders. The source of the time inconsistency problem in our model deserves further elaboration. We highlight the type of relationship banking activity where banks provide their customers with funding insurance (i.e. guarantees for the availability of funding), such as credit lines and other loan commitments. The funding insurance role of relationship banks, while sometimes overlooked in the literature, is empirically very important. Kashyap et. al. (2002) show that borrowing under credit lines represents up to 70% of bank lending. Bolton et al. (2013) show that relationship banks o¤er favorable terms to borrowers during crises in return for higher interest rates in normal times, indicating additional, informal funding insurance arrangements.2 The funding insurance activity of relationship banks has two properties. First, the upfront fees paid for funding insurance lead to ‘front-loaded’ (received in advance) income for banks. Second, banks have considerable discretion as to whether or not to honor lending commitments (Demiroglu and James, 2011). Combined, these two features invite time inconsistency problems. In particular, banks may opportunistically shift capital to trading once the upfront revenue from the funding insurance business has been collected. This may undermine their ability to honor 2 Bae et al. (2002) show that banks in Korea o¤ered their relationship customers better credit terms than those available in the spot market during the 1997-98 crisis. Beck et al. (2013) show that relationship banks constrained credit less than transactions-oriented banks in Eastern Europe during the 2008 credit crisis. Petersen and Rajan (1994) originated this strand of empirical literature by showing that banks o¤er their relationship customers better availability of credit.

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lending commitments, and the relationship banking business may su¤er. The outline of the analysis is as follows. We start by describing the synergy between relationship banking and trading. Relationship banking has information-based rents that generate implicit capital, yet is not readily scalable. The trading activity is scalable but has lower margins, so can be capital constrained. Accordingly, relationship banks may expand into trading in order to pro…tably use their ‘spare’capital.3 Opening up banks to trading, however, creates distortions. The main distortion is a time inconsistency problem in capital allocation, which leads to excessive trading. A complementary distortion is the risk-shifting that may be facilitated by trading. The distortions intensify when …nancial markets are deeper, allowing larger trading positions, and when returns on relationship banking are lower. (Indeed, key to our results is that future bank pro…ts are insu¢ cient to deter opportunistic behavior; cf. Keeley, 1990.) Both trends –deeper …nancial markets and less pro…table relationship banking –were present in the last decades. This explains why trading by banks, while benign historically, might have recently become distortive from the perspective of its impact on the bank’s relationship franchise. The interaction between banks and …nancial markets is a rich research area. Some papers study how the expansion of …nancial markets a¤ects the nature of banking –by impacting the depth of relationships (Boot and Thakor, 2000) or making banking more cyclical (Shleifer and Vishny, 2010). Other papers study the reverse e¤ects – how banks’ involvement in …nancial markets a¤ects the markets’ liquidity (e.g., Brunnermeier and Petersen, 2009). Our paper is related to the former, but focuses on the time inconsistency problem in bank capital allocation, driven by di¤erences in the time horizon and scalability of relationship banking and trading. The focus on time inconsistency di¤erentiates this paper also from related work by Kahn and Winton (2004) and Boot and Schmeits (2000) who study risk spillovers, and Boyd et al. (1998) and Freixas et al. (2007) who focus on the abuse of the safety net.4 The notion that trading activities can destabilize banks goes back at least to the UK Barings 3

This liability-side synergy is akin to the assertions of practitioners that one can “take advantage of the balance sheet” of the bank. 4 More generally, our paper relates to the literature on internal capital markets (Williamson, 1975; Donaldson, 1984). Conglomeration may relax the …rm’s overall credit constraint (Stein, 1997) but can lead to distortions, e.g. divisional rent-seeking (Rajan et al., 2000). In our model conglomeration also relaxes credit constraints, but leads to a speci…c distortion: a misallocation of capital by headquarters due to the time inconsistency problem in combining long-term relationship banking with short-term trading activities.

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Bank trading disaster in Singapore in 1995. But the concerns were especially vivid during the recent …nancial crisis. Many European universal banks su¤ered dramatic losses on their securitized debt holdings or exposures to sponsored investment vehicles (UBS in Switzerland is a good example; UBS, 2008). Similarly in the U.S., investments in securitized debt have back…red in both investment banks (Bear Stearns, Lehman Brothers and Merrill Lynch) and commercial banks (Washington Mutual and Wachovia). Consistent with these examples, the empirical literature con…rms that trading activities add risk to the overall risk pro…le of banks.5 This paper highlights that trading may undermine banks also in a more subtle way: by diverting resources away from the relationship banking activity. An example illustrating this point is that following the creation of Bank of America Merrill Lynch, analysts and regulators expressed worry that the trading exposures of Merrill Lynch may have become “a drain on the resources” of Bank of America. Probably in response to such concerns, some major universal banks have recently announced plans to retrench to their core commercial banking business.6 The empirical literature con…rms, perhaps strikingly, that trading activities may reduce the value of banks. Laeven and Levine (2007) show that banks that combine lending and nonlending activities lose value compared to engaging in these activities separately. Similar evidence is o¤ered by Stiroh (2004), Mercieca et al. (2007), Schmidt and Walter (2009), and Baele et al. (2007). Our model is the …rst one to explain this dynamics. We show that banks may ine¢ ciently divert resources to trading as a result of time inconsistency and risk shifting problems to the detriment of relationship-oriented activities and overall bank pro…tability. The paper is organized as follows. Section 2 sets up the model. Section 3 shows the synergy between relationship banking and trading. Section 4 describes the time inconsistency problem in capital allocation, including its interaction with risk shifting. Section 5 discusses robustness issues and empirical and policy implications. Section 6 concludes. 5

De Jonghe (2010), Demirguc-Kunt and Huizinga (2010), Brunnermeier et al. (2012) and DeYoung and Torna (2012) show that banks’non-lending activities are riskier than lending. De Jonghe (2010) and Brunnermeier et al. (2012) show that within the non-lending activities trading is the riskiest. Fahlenbrach et al. (2012) show that banks with more exposure to trading securities had higher losses during the 1998 and 2008 crises. 6 Cf. “BofA Said to Split Regulators Over Moving Merrill Derivatives to Bank Unit,”Bloomberg, 11/18/2011; “Credit Suisse to Accelerate E¤ort to Get Rid of Unwanted Assets,” Bloomberg, 1/7/2014. Another example of a drain on bank resources is occasional sizable losses in rogue proprietary trading, e.g. in Société Générale in France in 2008 or in JP Morgan’s treasury operations in 2012 (“The London Whale”).

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2

Model

We model two activities. One is relationship banking (“banking”); the other is arm’s length market-based operations (“trading”). Banking relies on an endowment of private information about an established set of customers; trading does not. We argue that this observation alone su¢ ces to highlight a range of distinctions between the two. The information endowment makes banking pro…table (hence, we assume, not credit constrained), yet not easily scalable. Securing the value of information requires multiple interactions with customers, so banking is long-term in nature. In contrast, since it does not rely on an information endowment, trading is scalable, less pro…table per unit (and hence, we assume, credit constrained), and short-term oriented. We use these distinctions to study synergies and con‡icts between banking and trading.

2.1

Credit Constraints

Before describing the banking and trading activities, we introduce a key modeling feature: the presence of credit constraints. We build on Holmstrom and Tirole’s (1997) formulation that limits a …rm’s leverage based on the owner-manager’s incentives to engage in moral hazard. Assume that the owner-manager can choose to run the bank normally or engage in moral hazard. She will run the bank normally when:

bA,

where

(1)

is the shareholder return when assets are employed for normal business, and bA is the

shareholder return to moral hazard: the initial investment A multiplied by the conversion factor b, 0 < b < 1, of assets into private bene…ts. When the incentive compatibility constraint (1) is not satis…ed, the owner-manager engages in moral hazard and the bank becomes worthless for creditors. Anticipating that, creditors would not provide funding. Thus, (1) can be seen as describing a leverage-related credit constraint for a bank.7 7 The moral hazard payo¤ bA can represent savings on abstaining from the owner-manager’s e¤ort, limits on the pledgeability of revenues (Holmstrom and Tirole, 1998), or the possibility of absconding (Calomiris and Kahn, 2001). We let the form of moral hazard be identical for standalone banking, trading, and the conglomerated bank. In practice, the relative importance of moral hazard in banking vs. trading is ambiguous. While the relationship banking assets are opaque and hence hard to monitor by the providers of funds, the trading assets may be more liquid and hence have higher transformation risk (Myers and Rajan, 1998).

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2.2

Banking

We model relationship banking as a pro…table, yet not scalable business, which therefore is not credit constrained. A bank operates in a risk-neutral economy with no discounting. It has no explicit equity and has to borrow in order to lend. Bank creditors require a zero expected return. In the base model all activities are risk-free, so the interest rate on bank borrowing is zero. There are three dates: 0, 1 and 2. At date 0 the bank is endowed with private information on a mass R of customers. Thanks to this information, the bank can earn rents in two ways:

First, the bank has a …xed pro…t R0 , coming from services (e.g. payment services) to its customers. Second, the bank has an opportunity to serve its customers’funding needs. Each customer needs to borrow 1 unit at date 1, to repay at date 2 (with certainty). When a bank lends, the information endowment enables it to collect informational rents r per customer. The total rents are then rR, where R

R is the volume of relationship banking activity –the

amount that the bank borrows from the market and lends to its customers.

The bank collects pro…ts R0 and rR (the latter in the form of higher interest rates) at date 2. Under this setup, the leverage constraint (1) for the bank takes the form:

R0 + rR

bR,

(2)

where the left hand side is the bank’s pro…t in normal operations, and the right hand side is the moral hazard payo¤. We assume that this constraint is satis…ed, including at R = R, implying spare borrowing capacity in banking:

R0 + rR > bR.

(3)

Note two assumptions implicit in this characterization of relationship banking. The …rst assumption is that the information endowment is …xed, so that relationship banking is not easily 7

scalable. Indeed, expanding the relationship banking customer base may be costly because of adverse selection (Dell’Ariccia and Marquez, 2006) or di¢ culties in processing large amounts of soft information (Stein, 2002). The second assumption is that a relationship bank has some informational monopoly over its borrowers, enabling it to earn rents. This might be related to past investments by the bank and its customers in their relationships, and/or to the advantages of proximity or specialization in local markets. The time and proximity elements involved in building relationships also provide an additional explanation for the lack of scalability.

2.3

Trading

We model trading as a scalable but less pro…table business, which consequently is credit constrained. For T units invested at date 1, trading produces at date 2 net returns tT for T

S

and 0 for each unit exceeding S, thus tS for T > S. Trading is less pro…table per unit than banking since it does not bene…t from the informational endowment:8

t < r.

(4)

And the low pro…tability of trading makes it credit constrained:

t < b,

(5)

implying that the leverage constraint (1) does not hold when trading is a standalone activity. This implies that, in this model, standalone trading is impossible, despite the opportunity to pro…tably invest up to S units. This simplifying assumption should not be taken literally. It only means that –consistent with practice –most trading operations require a substantial equity commitment.9 8

Trading by banks is indeed less pro…table per unit than lending. In 2000-2007, the average return on banks’ trading assets was 2% (during the crisis, the return was negative). In the same period, the average bank net interest margin was 3.25% (NY Fed, 2014) and the average cost of bank funding 3% (according to the Federal Home Loan Bank of San Francisco Cost of Funds Index), making the gross return on lending 6%. Another useful observation is that most trading by banks is volume-based (e.g., carry trade); it is di¤erent from –and has a lower return – than proprietary strategy-based trading by hedge funds. Overall, one could characterize relationship banking as a high-margin-low-volume operation and trading by banks as a low-margin-high-volume operation. 9 For example, hedge funds (or other independent trading houses) are normally partnerships with equity commitments by the partners that facilitate substantial recourse. Note that we do not consider external equity. A lack of access to external equity can be rationalized in a standard way through agency costs.

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The parameter S captures the scalability of trading. In a richer model one could think of decreasing returns to scale in the context of a Kyle (1985) framework, where the average return of an informed trader falls in the size of her trade, but less so when the mass of liquidity traders is larger. Thus, S is higher –trading is more scalable –in deeper …nancial markets, or for higher …nancial development. The timeline for the benchmark model is summarized in Figure 1. The bank maximizes shareholder wealth and prefers banking to trading when indi¤erent.10

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Synergy between Banking and Trading

Our model implies a natural bene…t to combining banking and trading: it links a business that has borrowing capacity but lacks scalability (the relationship bank) with a business that has investment opportunities but is subject to credit constraints (trading). Under conglomeration, at date 1, the bank maximizes its pro…t:

C

= R0 + rR + tT ,

(6)

subject to the joint leverage constraint (use (1)):

R0 + rR + tT

where T

b(R + T ),

(7)

S. Comparing this to (3) and (5) shows that conglomeration allows using the

borrowing capacity of the relationship bank to fund some trading. In allocating the borrowing capacity between relationship banking R and trading T the bank chooses to serve all banking customers …rst (R = R) before allocating any funds to trading, because banking is more pro…table: r > t. The maximum amount of trading that a bank can 10

An approach alternative to shareholder value maximization would be to focus on the incentive problems of managers (e.g. as in Acharya et al., 2013). While managerial incentive issues in banks are undoubtedly important, understanding the distortions that can be caused by shareholder value maximization alone remains critical, particularly when such distortions may have worsened recently due to external factors, such as information technology-linked increases in …nancial development.

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support, Tmax (assuming Tmax

S), is given by (7) set to equality, with R = R:

Tmax =

R0 + R(r b t

b)

.

(8)

Since it is never optimal to trade at a scale that exceeds S, T = min fS; Tmax g. This means that when the scalability of trading is small – S is low – the bank covers all pro…table opportunities in trading. When trading is more scalable –S is high –the bank covers trading opportunities Tmax < S and abstains from the rest. Proposition 1 (Synergy between banking and trading) The relationship bank can use its implicit capital to expand the scale of credit-constrained trading. In equilibrium, the bank serves all relationship banking customers, R = R, and allocates the rest of its borrowing capacity to trading, as long as trading is pro…table: T = min fS; Tmax g. Proposition 1 is a benchmark that explains why banks may choose to engage in trading and speci…es the …rst-best allocation of borrowing capacity between the two activities. Next we study ine¢ ciencies that may arise in combining banking and trading.

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Time Inconsistency of Bank Capital Allocation

The previous section has outlined the synergy between banking and trading –the use of ‘spare’ capital of the relationship bank to expand trading. We now turn to the cost of conglomeration: the time inconsistency problem in bank capital allocation, driven by a con‡ict between the long-term nature of relationship banking and short-term trading. Relationship banking is long-term because it involves repeated interactions with customers, with returns distributed over time. We model this intertemporal nature by letting a bank o¤er funding commitments to customers. The commitments take the form of credit lines that cover future funding needs in return for ex ante fees that customers pay to the bank. A rationale for credit lines is that customers may be constrained in how much they can pay to the bank ex post, due to moral hazard at the borrower level (Holmstrom and Tirole, 1998; Bolton et al., 2013; Acharya et al., 2014). 10

The main analysis takes a simpli…ed credit line contract as given, and analyzes its consequences. A salient feature of credit line contracts is that the lender has discretion on whether to honor lending commitments. In Section 5.1 we show that such a contract arises as an optimal response to two distortions: limited pledgeability of assets, which restricts the borrower’s ability to repay the bank in some states of the world, and moral hazard at the borrower level, which prevents fully committed (i.e. non-discretionary) funding arrangements. By making some payments ex ante, customers reduce the amount that they owe the bank ex post. As a result, returns to banking, although higher than returns to trading overall, might ex post be lower. This may distort capital allocation: once credit line fees have been collected, a bank might obtain incentives to allocate capital to trading, leaving itself with insu¢ cient borrowing capacity to fully serve the credit lines. Anticipating that, customers would reduce the credit line fees that they are willing to pay ex ante, lowering the bank’s overall pro…t and borrowing capacity. The emphasis on the “funding insurance” role of relationship banks is a key feature of our model. It contrasts with another common view on relationship banking that focuses on the informational capture e¤ects. Section 5.2 discusses the relevance of our modeling approach. It also highlights that a credit line is just one example of a variety of arrangements through which relationship banks provide funding insurance to customers.

4.1

Setup

Assume that while a bank can generate a return r on covering its customers’ future funding needs, it can only capture

r through interest rates charged on the actual lending between

dates 1 and 2. The remaining (r

) can be captured at date 0 as a credit line fee. Importantly,

a bank cannot commit to cover the future liquidity needs of customers. That is, we let the bank have discretion to refuse lending in the future if it has no borrowing capacity left to lend under the credit line. The timeline incorporating the credit line arrangement is shown in Figure 2.11 11

Here, for simplicity, we set the probability that the credit line will be used equal to one. In practice, the probability is usually less than one (see Section 5.1): the …rm only draws on the credit line in states where external circumstances make spot markets “too expensive.”

11

At date 1, the bank chooses R and T to maximize its pro…t:

C

where (r

= R0 + (r

)Ranticipated + R + tT ,

(9)

)Ranticipated is the total credit line fee received by the bank at date 0, Ranticipated is

the volume of borrowing that customers expect to get under the credit line and R is the actual borrowing under the credit line. The bank maximizes (9) subject to the IC constraint:

R0 + (r

)Ranticipated + R + tT

b (R + T ) .

(10)

Note that in equilibrium, Ranticipated = R: This follows because customers correctly anticipate the bank’s ability and willingness to lend under the commitment. Recall that in the benchmark case in Section 3, the bank always covers the customers’ liquidity needs …rst, because the return on the banking activity exceeds that on trading (r > t). However, when part of the return to banking is obtained ex ante as credit line fees, a time inconsistency problem may distort capital allocation. As long as the ex post return to banking is su¢ ciently high,

t, the …rst best allocation persists. Yet, when

< t, the bank chooses

to allocate the borrowing capacity to trading up to its maximum pro…table scale S …rst, and only then to give the remainder to banking. This constitutes a time inconsistency problem in bank capital allocation. In what follows, we analyze its consequences.12 The severity of the time inconsistency depends on the scalability of trading and the returns on relationship banking. When the scalability of trading S is low and/or the pro…tability of relationship banking r and R0 is high (making Tmax higher, see (8)), so that S

Tmax , the bank

can cover all liquidity needs of customers R even after allocating S to trading. Then, trading does not trigger time inconsistency. However when the scalability of trading S is high and/or the return to relationship banking r and R0 is low, so that S > Tmax , the relationship banking 12

As Section 5.1 shows, the bank will always charge the maximum feasible ex-post payment (maximum that does not trigger moral hazard at the borrower level) to minimize the time inconsistency problem. Note also that in a multi-period setting the future value of the relationship business could mitigate the time inconsistency problem in credit lines. In the model this could be captured by > that incorporates future pro…ts. Our results are therefore more likely to hold when future pro…ts are insu¢ cient to prevent opportunistic behavior (Keeley, 1990), i.e. is small. See Edwards and Mishkin (1995) and DeYoung and Rice (2004) for a discussion of declining pro…tability in relationship banking.

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activity is credit constrained ex post. The borrowing capacity that remains after the bank has allocated S to trading is insu¢ cient to cover all the funding needs of customers. Relationship banking customers correctly expect Ranticipated < R, so they pro-rate the credit line fee that they are willing to pay ex ante: (r

)Ranticipated < (r

)R. The pro…ts of the relationship

bank on lending to customers fall from rR to rRanticipated . We can now calculate the equilibrium capital allocation. Cutting down on relationship banking R to accommodate more trading conserves capital at a rate b but undermines bank pro…tability at a rate r (see the leverage constraint (1)). When r is small, such that r < b, cutting down on banking frees up more capital than is lost in pro…ts (the leverage constraint becomes slack). Hence as S starts exceeding Tmax , trading expands while banking contracts smoothly. We solve (10) set as equality to get the equilibrium allocation:

R =

T

=

When r is high, such that r

8 > <

R0 S(b t) , b r

for Tmax < S

> : 0, for S > R0 b t 8 > < S, for Tmax < S > :

R0 b t,

for S >

R0 b t

R0 b t

,

(11)

.

R0 b t

b, cutting down on banking leads to a loss in pro…ts that

exceeds (or equals) the capital freed up. This makes the leverage constraint (10) more instead of less binding, reducing the capital available to trading. No smooth reduction in the banking activity can help; in equilibrium, relationship banking unravels. For S > Tmax , the allocation is:

R = 0, T

=

(12)

R0 . b t

Overall, higher relationship banking rents r make it less likely that the time inconsistency problem is binding (only for a higher S, because Tmax is higher). But the consequences of time inconsistency once it is binding become more severe, because the commitment-oriented credit line business that produces rents r is more important for overall bank pro…t.

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Figure 3 shows the volume of the relationship banking activity R under time inconsistency as a function of trading opportunities S. It can help interpret some fundamental changes in the …nancial sector. Historically, with low scalability of trading and relatively pro…table relationship banking, time inconsistency was not binding because the implicit capital of a relationship bank could accommodate all trading. Yet more recently, due to developments in information technology, the scalability of trading S has increased and the pro…tability r and R0 of relationship banking has declined (so that Tmax has fallen). Banking is now unable to support the increased volume of trading without triggering detrimental time inconsistency. In Figure 3, this means a shift from case r

b to case r < b and a shift of S to the right.13

Figure 4, panels A and B, illustrates the comparative statics of R and T for the cases r < b and r

b, without (as in Section 3) and with time inconsistency (this section).

Proposition 2 (Time inconsistency of capital allocation) For

< t, when the pro…tabil-

ity of banking (r and R0 ) is low and trading is su¢ ciently scalable, so that S > Tmax , the bank allocates insu¢ cient capital to serving the future funding needs of its customers: R < R. Anticipating this, customers pay lower credit line fees ex ante, and the bank’s relationship franchise su¤ ers.

Proposition 2 is a key result of our model. It provides an important lesson for the industry structure of banking. With limited trading opportunities and relatively pro…table banking activities, there are synergies to combining relationship banking and trading. However, more scalable trading coupled with less pro…table banking can undermine commitments that are essential for relationship banking. Combining the activities may become costly. 13

The di¤erences between the cases r b and r < b can also be related to industry structure. In a more protected banking system with high r, trading initially poses no problem: banks can safely accommodate a lot of trading (Tmax is high). However, once substantial trading opportunities arise, the relationship banking arrangements might collapse rapidly. In a more competitive banking system where r < b, less trading can be accommodated, but a more smooth transition takes place once trading opportunities increase.

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4.2

The Costs of Conglomeration

We can now derive bank pro…ts. Recall that the cumulative pro…t of banking and trading as standalone activities is: S

= R0 + rR,

(13)

where R0 + rR is the pro…t of a standalone relationship bank (see (3)), and the pro…t of standalone trading (which is not viable) is zero. The pro…t of a bank that engages in trading depends on the scalability of trading, S. For S

Tmax , time inconsistency is not present, and the pro…t is increasing in S:

C

= R0 + rR + tS.

(14)

For S > Tmax , time inconsistency distorts capital allocation, and pro…t is decreasing in S. Following the two cases in Section 4.1, for r < b, a bank substitutes trading for banking smoothly, and its pro…t decreases in a continuous way (use (11)):

C

For r

=

8 > < R0 +

rR0 Sb(r t) , b r

for Tmax < S

> : R0 + t R0 , for S > R0 =(b (b t)

R0 =(b

t)

.

(15)

t)

b, the banking activity collapses for S > Tmax , leading to an immediate drop in pro…t:

C

= R0 + t

R0 , for S > Tmax . (b t)

(16)

Figure 4, panels C and D, illustrates bank pro…ts for the cases r < b and r

b, respectively.

Overall, from (14) through (16), in comparison to stand-alone banking, trading increases bank pro…t initially (for S

Tmax ) but can lead to a loss of pro…t for S > Tmax . When a bank fully

substitutes trading for relationship banking, so that R = 0 and T = R0 =(b less than the standalone pro…t

S

t), its pro…t

C

is

(compare (13) and (16)) when the value of the relationship

banking business linked to funding commitments, rR, is su¢ ciently high:

rR > t

R0 . (b t)

15

(17)

The results can be summarized as follows:

Proposition 3 (Pro…ts under time inconsistency) For

< t, the e¤ ect of conglomeration

on bank pro…t is inverse U-shaped in trading opportunities S. For low S, S

Tmax , time

inconsistency is not present, and pro…t increases as a bank trades more. For higher S, S > Tmax , pro…t falls with additional trading as the time inconsistency problem intensi…es. There exist parameter values such that beyond a certain scale of trading, banks that do not trade generate higher pro…ts than banks that trade.

Proposition 3 con…rms that trading at a large scale can be detrimental for banks.

4.3

Risk Shifting

In the model so far, relationship banking and trading were risk-free. We now extend the model to explore how risk in the trading activity can reinforce the time inconsistency e¤ects. Shareholders of a leveraged …rm may have incentives for risk shifting when risk is not priced at the margin in the …rm’s cost of funding (Jensen and Meckling, 1978). This is a standard corporate …nance ine¢ ciency that arises when funds are attracted before an investment decision is made to which shareholders cannot commit. Such a lack of commitment, and hence incentives for risk-shifting, can be expected in the context of banks too.14 Yet the scope to generate risk in a traditional relationship bank might be limited. For example, a bank that has a portfolio of loans with independently distributed returns may have minimal idiosyncratic risk thanks to the law of large numbers, while the aggregate risk (such as interest rate risk) might be absorbed by the bank’s charter value if it is su¢ ciently high thanks to relationship rents. In contrast, banks might be able to use trading to generate highly skewed returns (for example, from large undiversi…ed positions) and perform risk-shifting through trading. 14

The pricing of risk might be more distorted in banks than in non-…nancial …rms, due to the safety net (deposit insurance and “too big to fail” guarantees) and the use of secured or senior funding that is e¤ectively subsidized by preexisting creditors (Gorton and Metrick, 2011; Brunnermeier and Oehmke, 2011; Huang and Ratnovski, 2011). Also, it might also be easier for banks than for industrial …rms to opportunistically change their risk pro…le thanks to a higher liquidity of their assets (see Myers and Rajan, 1998). We do not need these additional e¤ects in our model, but if present they would strengthen our results.

16

In this section, we examine the setup where relationship banking is safe, while trading can generate skewed returns. Although this pattern need not hold always, it corresponds closely to the experience from the run-up to the recent crisis when banks used trading to generate extra return “alpha”by taking bets that were safe in most states of the world, but occasionally led to signi…cant losses (Acharya et al., 2010). For this reason, it is instructive to analyze its consequences of such a setup for the interaction between relationship banking and trading activities within a single …rm. Assume that a bank can choose between the safe trading strategy considered before and a risky trading strategy, which for T units invested generates a gross return (1 + t + ) T with probability p, and 0 with probability 1

p (up to the maximum scale of trading S). The binary

return of risky trading is a simpli…cation, representing highly skewed outcomes. We assume that risky trading has a per unit NPV that is lower than that of safe trading, yet positive: 0 < p (1 + t + )

1 < t.

(18)

Holding the cost of debt …xed, risky trading o¤ers a higher return to shareholders than safe trading (creating incentives for risk shifting), yet not as high as to make the leverage constraint (1) not binding: t < p(t + ) < b.

(19)

The choice of the trading strategy by the bank is not veri…able. In the model so far, the interest rate on bank borrowing was zero because the bank was riskfree. Yet, with risky trading, a bank may be unable to repay its creditors in full. Accordingly, creditors ex ante set the interest rate i based on their expectations of the bank’s future trading strategy. For simplicity, we assume that if multiple equilibria are possible, the creditors set the lower rate. The bank chooses safe trading when indi¤erent. We focus on the richest case that combines time inconsistency ( < t) and a smooth contraction of banking (b > r). We can prove the following result: Proposition 4 (Risk shifting) When the pro…tability of banking (r and R0 ) is low and trading is su¢ ciently scalable (S is high), the bank engages in risky trading as a form of risk shifting. 17

The bank internalizes the costs of risk shifting through higher borrowing costs and lower borrowing capacity. Risk shifting reduces bank pro…ts, makes the bank more credit-constrained, and in this way damages the bank’s relationship franchise (its ability to serve relationship customers).

Proof. See Appendix A. The Proposition implies that a bank only uses trading for risk shifting when the pro…tability of relationship banking is relatively low, while the scalability of trading is high. The intuition is that, from the perspective of bank shareholders, risky trading has a …xed cost – the loss of the relationship banking franchise value in bankruptcy with probability (1 of risky trading –the additional return

p). The bene…ts

–are proportional to the scale of trading. Hence, the

scale of risky trading has to be high enough to compensate for putting the relationship banking franchise at risk. Note that the shift to risky trading has two negative e¤ects. First, it has a direct negative impact on bank pro…ts because risky trading has a lower NPV than safe trading. Although shareholders do not internalize this ex post, they internalize it ex ante through higher interest rates on bank borrowing. Second, lower bank pro…ts reduce bank borrowing capacity, and this further undermines bank pro…tability. It is also useful to note that the two ine¢ ciencies –time inconsistency of capital allocation and risk shifting – can reinforce each other, so that the presence of one makes the presence of the other more likely. E¤ect 1. Risk shifting makes time inconsistency in capital allocation more likely by increasing the ex post return on risky trading. There exist parameter values: t <

< p(t + ), such

that there is no time inconsistency in the absence of risk shifting (t < ), but it is present under risk shifting ( < p(t + )). E¤ect 2. Time inconsistency induces risk shifting by increasing the equilibrium scale of trading. This follows from Propositions 2 and 4. Risky trading is only optimal when the scale of trading is su¢ ciently high, and time inconsistency increases the equilibrium volume of trading. E¤ect 3. Time inconsistency induces risk shifting by reducing the relationship bank’s franchise value. The incentives for risk shifting are countered by the risk of a loss of the bank’s 18

relationship franchise value with probability (1

p). Time inconsistency reduces that franchise

value, and hence lowers the cost of risk shifting. We o¤er in Appendix B an extension of the model that captures this e¤ect.

5

Discussion

This section discusses some modeling features, and summarizes empirical and policy implications.

5.1

The Discretionary Credit Line Contract

This section presents a stylized model to microfound the discretionary credit line contract in Section 4. We establish why a credit line may add value and explain the discretion feature. Consider the following stylized framework. At date 0, the …rm is endowed with a project that can produce X at date 2. At date 1, there are two states of the world, H and L (with probabilities ph = 1

pl and pl , respectively). In state L, the …rm needs to invest an additional

1 unit for the project to succeed and realize X for sure (otherwise it realizes zero); in state H no such investment is required and X will be realized. The investment that might be needed at date 1 is externally funded. The amount

< X is pledgeable to …nanciers. We let 1 <

< X,

so that project continuation is optimal. Assume that the …rm has a relationship with one bank and cannot borrow elsewhere. Hence, the bank has all the bargaining power vis-à-vis the …rm. We introduce the following friction. The amount

< X is pledgeable only if the bank is engaged with the …rm starting from date

0. (For example, the bank needs to continuously maintain and monitor the …rm’s transaction accounts to collect any repayments.) This friction implies that at date 0 the bank can make the …rm a ‘take-it-or-leave-it’o¤er, e¤ectively forcing the …rm to buy a credit line for the potential funding need of 1 at date 1. The bene…t of the credit line to the bank is that it can extract higher rents from the …rm. To see this, note that without a credit line the bank can only charge the …rm

upon lending 1 in state L, leaving the …rm with rents X

. With a credit line, the

bank contracts at date 0 and can ask for extra compensation such that in the expected value 19

sense the …rm is giving up all rents from state L. The bank cannot ask for more than that, because otherwise the …rm would be better o¤ without a credit line, abandoning the project in state L. Under a credit line, the bank charges the …rm an unconditional fee F payable in both states,15 and a repayment obligation

F upon the borrowing of 1 at date 1. The …rm’s

participation constraint is given by:

(1

where (1

pl )X + pl (X

(

F ))

pl )X is the …rm’s pro…t in state H, pl (X

F = (1

(

pl )X,

(20)

F )) is the expected repayment on

borrowing in state L, and F is the credit line fee. Thus,

F = (X

)

In total, the …rm pays the bank F in state H and

pL . 1 pL in state L.

In principle, the payments from the …rm to the bank can be made more loaded on state H through a higher credit line fee F and a lower repayment on borrowing

F . But when the

bank has discretion on whether to make good on lending commitments (a feature that we will explain momentarily), the payments schedule above is the only one possible. Indeed, whenever the total payment to the bank in state L is less than , the bank can ex post renege and demand from the …rm a higher payment –up to X

, and the …rm will agree in order to maintain pro…ts

. Thus the ex ante agreed payment in state L has to be

–the highest possible.

We now motivate the need for the bank’s discretion. Assume that after the credit line is contracted, but before borrowing, the …rm can choose (at no cost) to transform the assets such that they are no longer pledgeable to the bank. Then, the bank will never o¤er a nondiscretionary credit line; absent discretion the …rm would always transform assets. Allowing the bank discretion makes the credit line feasible. As a …nal step, consider the e¤ect of trading opportunities for the bank. Assume that, at date 1 in state L, the bank with probability

gets an opportunity to invest 1 unit in trading

15

In a richer model where a …rm has some initial monetary endowment we could have had a fee being paid at date 0.

20

to produce t, where

< t < X (with probability 1

no trading opportunity exists). When

the bank invests in trading, it cannot simultaneously lend to the …rm. If the bank could commit to lend to the …rm through a non-discretionary credit line (and abstain from trading), it would have done so because lending is more pro…table than trading (t < X). But the credit line can only be discretionary (see above). Discretion makes the bank choose ex post to trade whenever such an opportunity is present because can only lend to the …rm with probability pL (1

< t. The bank then

). This reduces the equilibrium credit line

fee to: F 0 = (X and expected bank pro…t to: ((X

5.2

1)(1

)

1

pL (1 ) < F; pL (1 )

) + (t

1) ) pL < (X

(21) 1)pL .

Front-loaded Income in Relationship Banking

We now highlight the relevance of our approach to modeling relationship banking as a funding insurance activity. Relationship banking involves repeated interactions; it is inherently an intertemporal arrangement. We focused on a particular intertemporal arrangement – funding insurance – where a bank obtains upfront credit line fees but needs to honor future lending commitments, resulting in front-loaded income from relationships. There is ample evidence that banks play a substantial role in providing funding insurance to customers. Kashyap et al. (2002) show that credit lines and other loan commitments constitute up to 70% of bank lending, and this high share is unique to banks among other …nancial intermediaries. Berger and Bouwman (2009) con…rm these …ndings, showing that credit lines represent about a half of banks’liquidity exposures. Moreover, our credit line setup can be seen as a formalization of a variety of circumstances where relationship banking involves future funding commitments. A notable example is “local banking.” A bank’s local market presence facilitates the funding of customers, especially in times of economic stress (Petersen and Rajan, 1994; Bae et al., 2002; Beck et al., 2013). In return, such “local”banks can charge customers higher fees in normal times (Bolton et al.,2012), leading to front-loaded income similar to that in more formal credit line arrangements.

21

A salient feature of credit lines is that banks have discretion whether to honor lending commitments. Discretion serves two purposes. First, the ability of a bank to call a loan (or refuse to lend under a credit line) may improve incentives of borrowers to behave prudently (Bolton and Scharfstein, 1990; Acharya et al., 2014). Second, the discretion o¤ers protection to a bank in case it becomes …nancially constrained (Boot et al., 1993). There are multiple ways for banks to exercise discretion even for formal credit lines. Credit lines typically include a material adverse change (MAC) clause, often de…ned so broadly as to give banks a signi…cant option to renege. Credit lines also have …nancial collateral (“borrowing base”) covenants, often set deliberately tight, so that their breaches give banks an opportunity to renegotiate or revoke the credit line (Demiroglu and James, 2011).16 Clearly, other commitments, such as to the bank’s local markets, are informal, and so straightforward to renege upon. Our emphasis on funding insurance as a critical feature of relationship banking contrasts with the commonly postulated approach of describing long-term relationships between a bank and its customers based on information capture. There, borrowers are subsidized initially, while hold-up allows the bank to recoup the subsidies later (Petersen and Rajan, 1995). This leads to back-loaded income, making relationship banking highly attractive ex post. Observe that the ex post rents in banking have declined in the recent past as higher competition and more easily available borrower information have eroded the informational advantage of banks (Keeley, 1990). Accordingly, the informational capture aspect of relationship banking may have become less important.

5.3

Commitment Issues

Our analysis makes an important assumption that banks cannot commit to only trade at a scale that does not trigger time inconsistency. A rationale for this assumption is that the veri…cation of such a private commitment might be di¢ cult. Banks may always have to engage in some trading to support their lending activity (for example, to hedge lending exposures), so 16

Chava and Roberts (2008) estimate that a third of borrowers fall into a violation of contractual clauses of a credit line at some point during a ten-year period. Su… (2009) and Acharya et al. (2014) verify that credit lines are indeed actively revoked in response to covenant violations. Evidence from the recent …nancial crisis (Ivashina and Scharfstein, 2010; Huang, 2009) con…rms that banks’ own …nancial conditions were a factor a¤ecting the extension of credit even under formal, precommitted lines of credit.

22

committing not to trade at all may be prohibitively costly. And once banks trade, trading may take many forms, especially during times of rapid …nancial innovation. This makes it hard to write ex ante contracts on the volume or nature of trading.17 Another issue is whether the distortion in the internal allocation of capital, i.e. the time inconsistency problem, could be mitigated by returning the capital back to shareholders (e.g. via dividends or share buybacks). The answer is no. To understand this, note that in the normal course of business a relationship bank always maintains ‘unused’capital (spare borrowing capacity) in order to cover future funding needs of customers. The model highlights that such capital can be misallocated to trading, making the bank unable to ful…ll its relationship commitments. If this unused capital was returned to shareholders, the bank would still be unable to make good on its lending commitments. So the relationship franchise would su¤er in either case. The problem of time inconsistency between short- and long-term activities that is the focus of this paper is reminiscent of the literature that shows how trading at an intermediate stage in dynamic models of …nancial intermediation might undermine commitment. For example, Jacklin (1987) shows that trading possibilities may undermine the intertemporal smoothing between early and late consumers in the Diamond and Dybvig (1983) analysis, while Bhide (1993) points at a lack of shareholder discipline under di¤used (and liquid) ownership.

5.4

Empirical Implications

The analysis leads to several implications that are useful for understanding the dynamics of banks that engage in trading: Relationship banks are tempted to use their balance sheet (i.e. ability to borrow) for scalable trading opportunities. While trading at a limited scale can enhance bank profitability and franchise value, sizable trading can reduce pro…ts and damage the relationship banking franchise. 17

Recently some banks have announced plans to retrench to their core commercial banking operations. The desire to so is consistent with the message of our model. Whether this represents a credible commitment is unclear. For example, banks may retrench in periods when …nancial markets are subdued (t is low, such that t < and there is no time inconsistency). But banks could reverse course and reengage in large-scale trading once …nancial markets become more buoyant (making t > ).

23

Financial development has undermined banking through two channels: trading became more scalable, while relationship banking less pro…table (due to higher competition and more widely available customer information). Both increase bank incentives to overallocate capital to trading. Relationship banking activities that involve funding commitments su¤er most from the over-allocation of capital to trading. These include credit lines, as well as commitments to the local market to o¤er funding in periods of economic stress. In the presence of time inconsistency problems, such activities become less valuable and may no longer be viable. Banking as a whole then becomes more transactional and less pro…table. A broad implication from the increased severity of the time inconsistency problem is that combining banking and trading (the traditional European universal bank model shared by some U.S. conglomerates) might have become less sustainable. Universal banks have historically combined a sizable relationship banking activity with a much smaller transactionbased activity. Now banks might allocate too many resources to trading, leading to lower pro…ts and higher risk. Anticipating a bank’s ability to honor commitments has become more important. Credit ratings may in part re‡ect the risk bearing capacity that is not …lled up opportunistically. Hence, borrowers may anticipate that banks bene…tting from a higher rating are better able to deliver on (implicit) guarantees of future funding. This could help explain the importance of ratings in the …nancial services industry. The deepening of …nancial markets allows banks to engage in trading on a larger scale, increasing bank size and complexity (the diversity of activities). Also, banks may become more exposed to boom-and-bust patterns in …nancial markets – through cyclical returns to trading, which may moreover drive a cyclical time inconsistency problem (binding for high returns to trading, during …nancial market upswings). Therefore, the problems of banks’market-based activities, procyclicality and “too-big-to-fail” are interrelated.

24

5.5

Policy Implications

Our analysis is partial equilibrium, so we need to be cautious about drawing strong implications. Still, we have highlighted speci…c distortions: over-allocation of capital to trading and the use of trading for risk shifting. Accordingly, we can consider how some regulatory proposals that deal with trading by banks may help correct these distortions. Capital charges on banks’ trading assets. When bank capital is costly, increased capital charges on trading assets can discourage excess trading by reducing the return to trading. Once the return to trading t falls below the ex post return to banking , the time inconsistency problem in capital allocation disappears. Restricting trading. When banks trade too much, restricting trading may increase bank value (Proposition 3) and prevent the use of trading for risk shifting (Proposition 4). Note that banks can always be allowed to trade on a limited scale (below Tmax in (8)) as that does not divert resources away from relationship banking, nor does it induce risk shifting. Banks could use this trading capacity to support market-based activities that are inherent to lending, such as hedging. Segregation. Segregating trading into a separate subsidiary might reduce risk spillovers from trading to the relationship bank. However segregating trading cannot resolve the time inconsistency problem: a bank may still over-allocate capital to the trading subsidiary (i.e. trade too much), undermining its relationship banking franchise. A prohibition of trading activities might then be warranted. Our analysis also o¤ers insights into some additional policy questions: Can trading move to “the shadow”? A common argument against stronger bank regulation is that trading may move to the unregulated sector and become an even greater concern. Our analysis suggests that such a migration of trading activities would not necessarily occur. Trading by banks is a concern because the banks’franchise value creates borrowing capacity and induces trading at possibly too large a scale. Stand-alone trading would be capital constrained and have lower scale, hence, as a rule, pose lower risks to …nancial stability. Investment banking: standalone or within bank groups? 25

The crisis has highlighted the

instability of standalone investment banks. Our analysis suggests that this may be because investment banks have e¤ectively become trading operations (at the expense of underwriting and other relationship-based activities) and through leverage managed to operate at a scale that was not compatible with the capital constraints that trading should face.18 Investment banks may have been able to operate at such scale thanks to perceived government guarantees, or because the market overlooked their transition from a relationship-oriented to a tradingdominated business. When markets started doubting the viability of standalone investment banks, the U.S. authorities adopted two approaches to rescuing them. One approach was to merge investment banks with commercial banks (as in the case of Bear Stearns and Merrill Lynch). The other approach was to grant them commercial bank licenses and hence access to discount window and other central bank facilities (for Goldman Sachs and Morgan Stanley). Our analysis suggests that while a merger with a commercial bank might be a quick way to inject implicit equity and stabilize the trading operation, in the medium term such conglomeration may be detrimental due to the time inconsistency and risk shifting problems which can destabilize the merging commercial bank. Dynamic approach to bank regulation. Regulatory frameworks that focus on current bank balance sheet conditions only may insu¢ ciently capture the aspects of bank activities highlighted in this paper. Regulation should be forward-looking, and focus on ensuring the banks’ ability to expand the balance sheet to serve relationship banking customers, particularly during downturns. Counter-cyclical capital bu¤ers may be a step in that direction (Drehmann et al., 2011). Also, regulation should be cognizant of the fact that, with deep …nancial markets, banks may rapidly allocate capital to high-risk activities.

6

Conclusion

The paper studies the interaction between relationship banking and trading (arm’s length market-based activities of banks). The increased involvement of banks in trading is a fun18

Some investment banks also have lost equity commitments intrinsic to partnerships by abandoning their partnership structure (Morrison and Wilhelm, 2008).

26

damental change a¤ecting the industry over the last decades. We show that banks can use their franchise value to pro…tably expand the scale of trading. However, banks may over-allocate capital to trading, due to a time inconsistency problem, undermining their relationship banking franchise. This e¤ect is reinforced when banks can use trading for risk-shifting. The distortions are more likely when the scalability of trading is high and the pro…tability of banking is low. These factors were in play in recent decades, as information technology led to deeper …nancial markets and weakened the grip of banks over their relationship borrowers. As a result, trading by banks, while benign historically, might have recently become distortive. The results of the analysis appear consistent with the evidence that large-scale trading by banks was a source of vulnerability in the run-up to the recent crisis. On a fundamental level, the paper asks questions about the essence of the modern banks’ business model. One could argue that recent …nancial development has led to an “identity crisis” in banking. Many of activities that create franchise value for banks involve long-term commitments to customers. Particularly, the funding guarantees – in the form of credit lines or less formal arrangements – may be core to the business of banking. But access to transactional, opportunistic market-based activities may undermine such commitments. Our partial equilibrium analysis cannot give absolute answers, but it points to a range of distortions that bank managers and regulators could address to strengthen bank business models and maintain the franchise value and stability of banks.

27

A

Proof of Proposition 4

We …rst derive conditions under which the bank shifts to risky trading. To do so, start by assuming that no default risk is priced in. This allows us to derive the bank’s risk choice. Consider the payo¤ to risky trading under i = 0. When a bank invests R in relationship banking and T in risky trading, it obtains at date 2 from relationship banking R0 + (1 + r)R and from trading (1 + t + ) T with probability p and 0 otherwise. The bank has to repay creditors R + T . It can do so in full when trading succeeds. When trading produces a zero return, the bank has su¢ cient returns to repay in full only if R0 + (1 + r)R

R + T , corresponding to an

upper bound on the volume of trading:

T

R0 + rR.

(22)

When (22) holds, bank debt is safe, and shareholders fully internalize the losses from risky trading. Their payo¤:

Risky jT R0 +rR

= R0 + rR + p (1 + t + ) T

is lower than the payo¤ with safe trading:

Risky jT R0 +rR

<

C,

T.

where

(23)

C

is given in (6),

because risky trading has a lower NPV than safe trading (see (18)). Hence, bank shareholders never choose risky trading if they fully internalize possible losses. We thus focus on the case T > R0 + rR. Here, there is default risk: a bank cannot repay creditors in full when the risky trading strategy produces a zero return. The shareholders’ return is: Risky

= p (R0 + rR + (t + ) T ) .

The bank chooses risky trading when

Risky

>

C.

(24)

From (6) and (24), this holds when the

scale of trading exceeds a threshold TRisky such that:

T > TRisky =

(R0 + rR) (1 p) . p(t + ) t

28

(25)

To express TRisky in exogenous variables, note that when a bank allocates T to trading, the borrowing capacity left to banking (from (10)) is:

R=

R0

T (b b r

t)

.

(26)

Substituting this into (25) gives:

TRisky =

r (p

R0 (1 p) p)) =b + (p

b(1

t(1

p))

.

(27)

TRisky is a key threshold of this model; it shows the minimum scale of trading that induces risk shifting. We can show the following: Observation 1. @TRisky =@ upside

< 0 and @TRisky =@p < 0: when risky trading has a higher

and/or a higher probability of success p, the switch to risky trading occurs at a lower

scale of trading. Proof. By di¤erentiation:

@TRisky =@ = bp (r

b)

R0 (1 p) b(1 p)) + b(p

( r (p

t(1

p)))2

< 0,

and: @TRisky =@p = b R0

( r (p

b(1

r b p)) + b(p

t(1

p)))2

< 0,

because r < b. Observation 2. @TRisky =@R0 > 0 and @TRisky =@r > 0: when the value of the relationship banking franchise is higher, the switch to risky trading occurs at a higher scale of trading. Proof. Also by di¤erentiation:

@TRisky =@R0 =

r (p

(1 p) p)) =b + (p

b(1

t(1

p))

> 0.

This is positive because:

r (p

b(1

p)) =b + (p

t(1

p)) = [(b

29

r)p + (1

p)b(r

t)] =b > 0.

And:

@TRisky =@r = bR0 (1

p)

p b(1 p) b(1 p)) + b(p

( r (p

t(1

p)))2

> 0.

Which holds because:

p

b(1

p) > p

p(t + )(1

p) = p (p(t + )

t) > 0.

We can now derive the interest rates, capital allocation and pro…ts under risky trading. To limit attention to the more insightful case, we let:

Tmax < TRisky <

R0 (p(1 + t + )

b

1)

.

(28)

This ensures that (i ) risk shifting only occurs when the time inconsistency problem in capital allocation is already present and (ii ) implicit capital is su¢ cient to support the scale of trading necessary for risk shifting. For S

TRisky , the bank chooses the safe trading strategy with the allocation of borrowing

capacity given by (11); for S > TRisky , the bank chooses risky trading. Note that when there is time inconsistency in the capital allocation under safe trading ( < t), it also exists under risky trading because the ex post return to shareholders under risky trading is higher than that under safe trading, i.e.

< t < p(t + ) (see (19)). Therefore, under risky trading, the bank

will also …rst allocate the borrowing capacity to trading before using the remainder for banking. The interest rate i follows from the creditors’zero-pro…t condition:

(R + T ) = p (R + T ) (1 + i) + (1

p)(R0 + (1 + r)R),

(29)

where (R + T ) is the amount borrowed by the bank, (R + T ) (1 + i) is the debt repayment when risky trading succeeds with probability p, and (R0 + (1 + r)R) is the value of bank assets transferred to creditors in bankruptcy with probability (1

30

p). The capacity R left for the

banking activity after allocating T to trading follows from the IC condition:

p(R0 + (r

i)R + (t +

i)T )

b(T + R),

(30)

where the left hand side is the expected payo¤ to bank shareholders in normal operations, and the right hand side is the moral hazard payo¤. Similar to (11), use (29) and (30) as equality to obtain the allocation of borrowing capacity:

R =

T

=

8 > <

R0 S((1 p)+b p(t+ )) , b r

> R0 : 0, for S > b (p(1+t+ 8 > < S, for TRisky < S > :

R0 b (p(1+t+ ) 1) ,

for TRisky < S

R0 b (p(1+t+ ) 1)

,

(31)

) 1) R0 b (p(1+t+ ) 1)

for S >

.

R0 b (p(1+t+ ) 1)

Similarly to (15), bank pro…ts are:

Risky

=

8 > < p R0 + > : p R0 +

rR0 Sb(r (p(1+t+ ) 1)) b r (p(1+t+ ) 1)R0 b (p(1+t+ ) 1)

,

,

for S >

31

for TRisky < S R0 b (p(1+t+ ) 1)

R0 b (p(1+t+ ) 1)

.

(32)

B

Extension: Time Inconsistency, Franchise Value, and Risk Shifting

We o¤er an extension capturing the fact that the presence of time inconsistency may induce risk shifting because it lowers the bank’s franchise value (E¤ect 3 in Section 4.3). To show this interaction, we need to enrich the model in order to isolate the e¤ect of time inconsistency on franchise value. Assume that the presence of the time inconsistency problem is uncertain at date 0. At date 0 let there be a probability time inconsistency; and a probability 1 is present. A di¤erent probability (1

that at date 1, t = tLow < , so that there is no that t = tHigh > , and hence time inconsistency

) of the presence of time inconsistency a¤ects a bank’s

franchise value, and through it the threshold value TRisky (see (27)). Consider the realization t = tLow at date 1, so that time inconsistency is present. Then, at date 1, the payo¤ to the bank’s shareholders from safe trading is:

C

The expression

C

= R0 + (r

is similar to

arises with probability 1

)

C

R + (1

)Ranticipated + R + tT .

(33)

in (9), except for the second term. Time inconsistency now

rather than with probability 1, hence compared to (9) the credit

line fees (and the bank’s franchise value) are higher by (r

)

R

Ranticipated .

Similarly, the payo¤ from risky trading is:

R

= p R0 + (r

)

R + (1

Setting R = Ranticipated and equating

C

)Ranticipated + R + (t + ) T .

and

R

(34)

provides the threshold for a switch to risk

shifting (similar to (25)):

TRisky =

Observe that a lower

R0 + (r

) R R + rR (1 p (t + ) t

p))

.

(35)

–meaning more time inconsistency –lowers TRisky , hence risk shifting

becomes more likely when time inconsistency intensi…es.

32

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Figure 1. The timeline.

Date 0  Bank is endowed with a customer base R. Date 1  Bank customers have liquidity needs;  Bank chooses the allocation of borrowing capacity between relationship banking (covering customers’ liquidity needs) R (to earn r) and trading T (to earn t);  Bank borrows and can divert resources to moral hazard, which would obtain a payoff b(R+T);  Bank lends to customers and engages in trading. Date 2  Returns are realized, everyone is repaid.

Figure 2. The timeline with a credit line arrangement.

Date 0  Bank is endowed with a customer base R ;  Bank collects credit line fees (r – ρ)Ranticipated. Date 1  Bank customers have liquidity needs;  Bank chooses the allocation of borrowing capacity between relationship banking (covering customers’ liquidity needs) R (to earn ρ) and trading T (to earn t);  Bank borrows and can divert resources to obtain a moral hazard payoff b(R+T);  Bank lends to customers and engages in trading. Date 2  Returns are realized, everyone is repaid.

Figure 3. Relationship banking allocation R as a function of trading opportunities S.

Case 1, r < b

Optimal banking: R = R.

Reduced banking: R <..R

Tmax(r)

Case 2, r ≥ b

No banking: R=0

S

R0/(b-t)

Optimal banking: R = R.

No banking: R=0 R0/(b-t)

Tmax(r)

S

Figure 4. The volumes of relationship banking and trading activities (R and T) and bank profits Π.

R, T

R, T R

R

R

R0/(b-t)

R

R0/(b-t)

T Tmax T

T R

S Tmax

R0/(b-t)

R0/(b-t)

Panel A: R and T for r
S

Tmax

Panel B: R and T for r≥b

Π

Π

R0+r R +tTmax

R0+r R +tTmax R0+r R

R0+r R

R0+t R0/(b-t)

R0+t R0/(b-t)

S Tmax

R0/(b-t)

R0/(b-t)

Panel C: Π for r
Panel D: Π for r≥b

Tmax

No time inconsistency (as in Section 4); With time inconsistency (this section; ρ
S