Bank Interest Rate Risk Management

Transcription

1 Bank Interest Rate Risk Management Guillaume Vuillemey December 3, 2017 Abstract Empirically, bank equity value is decreasing in the interest rate. Yet (i) many banks do not hedge interest rate risk and (ii) above 50% of hedging banks use derivatives to increase exposure. I model a bank s capital structure, and show that these facts are consistent with optimal hedging under financial frictions. Novel predictions on the characteristics of banks taking long or short interest rate derivative positions are tested, and supported by the data. Therefore, banks derivatives exposures are not necessarily evidence of excessive risk-taking, and can be explained by hedging in the presence of frictions. More broadly, the results challenge the view that hedging and speculative positions can be identified using the comovement between derivatives payoffs and equity value. J.E.L. Codes: G21; G32; E43. Keywords: Interest rate risk; Derivatives; Bank capital structure; Hedging I am grateful to Tim Adam, Vladimir Asriyan, Nicolas Coeurdacier, Andrea Gamba, Denis Gromb, Samuel Hanson, Augustin Landier, Jochen Mankart, Philippe Martin, Bernadette Minton, Erwan Morellec, Christophe Pérignon, Guillaume Plantin, Adriano Rampini, Jean-Charles Rochet, Lukas Schmid, David Thesmar, Vish Viswanathan, and Toni Whited for comments. I also thank seminar participants at Sciences Po, the Bank of France, CREST, ESSEC, HKU, HKUST, HEC Paris, Imperial College, Stockholm School of Economics, ECB, ESMT, HEC Lausanne, the Federal Reserve Board, the Federal Reserve Bank of New York, Indiana University, Cass Business School, WU Vienna, and conference participants at the London Financial Intermediation Theory Network, the EEA Annual Meeting, the OxFIT Conference, the Deutsche Bundesbank/Frankfurt School/IWH/CEPR Workshop Financial intermediaries and the real economy, the 12th Corporate Finance Conference at WUSTL, the Econometric Society Winter Meeting, the WFA Conference, the European Summer Symposium in Financial Markets (evening sessions). I thank the Europlace Finance Institute for financial support. A previous version of this paper was titled Derivatives and Interest Rate Risk Management by Commercial Banks. HEC Paris and CEPR. vuillemey@hec.fr. 1

2 1 Introduction Interest rate risk is a structural feature of banking. Banks expose themselves to interest rate risk by investing in long-duration assets out of short-duration liabilities. Consequently, bank equity value is decreasing in the level of interest rates. This fact has important implications. First, drops in bank capitalization due to interest rate spikes can threaten financial stability and affect macroeconomic outcomes (Holmström and Tirole, 1997; Brunnermeier and Sannikov, 2014). Second, in terms of policy, central banks often respond to financial shocks by cutting interest rates, which de facto recapitalizes banks (Diamond and Rajan, 2012). In this context, understanding banks exposure to interest rate risk, and the extent of interest rate hedging, is critical for monetary and macro-prudential policy. I start by establishing two facts. First, a large fraction of banks do not hedge interest rate risk, even though their exposure to interest rate shocks is sizable. Specifically, more than 90% of US banks do not participate in the interest rate derivative market, which would enable them to reduce fluctuations in their net worth. Second, and perhaps more surprisingly, a large fraction of hedging banks, above 50%, use derivatives to take additional exposure to interest rate increases. Both facts are pervasive over time, and may seem inconsistent with the view that banks are hedging in derivatives markets. For example, Begenau, Piazzesi, and Schneider (2015) rely on related findings to conclude that there is little evidence that these positions are used to hedge other positions such as loans. A possible interpretation is that banks engage in excessive risk-taking due to moral hazard (Fahri and Tirole, 2012). In this paper, I model the capital structure of a bank, and show that all of the above facts are consistent with optimal hedging under financial frictions. I use the model to derive novel predictions on bank characteristics associated both with the sign of hedging positions and with the absence of any hedging. I test these predictions in US data and find strong support for them. I conclude that, in the presence of financial constraints, observed derivatives positions can be explained by hedging. The model reflects important features of banks relative to non-financial firms. Specifically, loans are financed by issuing deposits. When faced with deposit outflows, the bank fills the financing gap by issuing either interbank debt or equity. In 2

3 the model, incentives to engage in interest rate risk management arise from the existence of financial frictions, as in Froot, Scharfstein, and Stein (1993): (i) a collateral constraint limits the bank s ability to raise interbank debt; and (ii) issuing equity is costly. The bank aims to secure funds for states in which the optimal issuance of new loans is large and financing constraints may lead to credit rationing. Risk can be managed either by preserving unused debt capacity or by using derivatives, in the form of interest rate swaps. I study the optimal hedging policy of the bank. To begin with, I show that the equity value of the unhedged bank is decreasing in the interest rate, consistent with stylized facts. Then, I show that financial frictions provide incentives to hedge both increases and decreases in interest rates. Calibrating the model, I find that it quantitatively replicates important moments. In particular, the simulated shares of banks hedging increases or decreases in interest rates are close to their empirical counterparts. Furthermore, a key added value from the model is to deliver new cross-sectional predictions on the relationship between bank-level financial frictions and the sign of hedging positions taken. Which position is taken depends on the friction that is most important for the bank. Intuitively, if it is presently constrained in its access to the debt market, the bank can enhance its credibility to repay debt by hedging against future high-interest-rate-states, in which its equity value will otherwise be low. Doing so, a bank increases its current debt capacity and ability to lend. In contrast, if low interest rates are likely to boost optimal lending in future periods, funds will be most valuable when interest rates drop. Therefore, the bank can use derivatives to hedge interest rate decreases, absent any speculative motive. I test the predictions of the model, and find that they are strongly supported in US data. I use the full sample of US banks filing Call Reports over the 1995Q1-2015Q4 period. The model predicts that banks hedge against interest rate increases when they are financially constrained and have large current funding needs. In the data, I find that more levered banks are more likely to hedge interest rate increases. Using bank fixed effects, I also show that a given bank is more likely to increase hedging of interest rate spikes precisely in quarters when its funding needs are large. Additionally, the model predicts that banks faced with more volatile or more persistent lending 3

4 opportunities are more likely to hedge decreases in interest rates. This prediction is also confirmed in the data, using several measures of volatility and persistence. Therefore, I conclude that reasonably calibrated financial frictions can explain the risk management behavior of banks. An auxiliary prediction of the model is that interest rate derivatives are not used by all banks, also consistent with the data. This is because hedging requires net worth: to take derivatives positions, collateral needs to be pledged. Therefore, the collateral pledged on derivatives is no longer available to obtain interbank debt financing. For constrained banks, financing concerns override hedging concerns, and these banks optimally do not hedge. This result is reminiscent of Rampini and Viswanathan (2010), with one important difference. In their model, hedging always reduces debt capacity. In contrast, hedging positions in my model can increase present debt capacity. This occurs if swap contracts increase the value of the lowest possible future net worth, against which debt is collateralized. I find that this additional mechanism is important to match the data on the type of derivatives positions taken by banks. In spite of this difference, a positive relation between bank net worth and hedging still prevails, in line with empirical evidence (Rampini, Viswanathan, and Vuillemey, 2015): banks with low net worth hedge less, or do not hedge at all. Finally, an alternative explanation of the stylized facts could be that banks engage in excessive risk-seeking due to implicit or explicit guarantees. While I do not seek to explicitly rule out this hypothesis, it is unlikely to be a sufficient explanation. Indeed, I do not observe significant differences in the type of positions taken by small and large banks. Furthermore, all tests of my model s predictions are robust to controlling for bank size and capitalization, that is, two proxies for the moral hazard to which banks may be subject. There are two important implications of my results. First, the results challenge the view that hedging and speculative positions can be identified using the comovement between derivatives payoffs and equity value. Banks in the model may hedge decreases in interest rates, even though their equity value is decreasing in the level of interest rates. This hedging policy is the optimal response to financial frictions, absent any speculative motive or any risk-seeking incentive due to moral hazard. Second, the 4

5 fact that banks do not completely hedge tradable risks should not be considered a puzzle, even though shocks to their net worth have large macroeconomic effects. Overall, my results call for a careful understanding of the financial frictions faced by banks before policy implications are drawn and regulation of exposures is designed. Related literature In an environment where interest rates have been at historically low levels for years, interest rate risk is arguably a first-order concern for banks. Flannery and James (1984) and English, Van den Heuvel, and Zakrajsek (2013) show that the equity value of banks drops when interest rates increase, suggesting that they are exposed to interest rate risk. More recently, Drechsler, Savov, and Schnabl (2017) cast doubt on the idea that exposures to interest rate risk are material, by showing that US banks net interest rate margins are insensitive to interest rate shocks. These findings are not necessarily inconsistent with each other: indeed, it can be that, upon occurrence of an interest rate shock, banks decrease in value but preserve net interest margins by downsizing. Whether interest rate risk can be a major risk for banks remains an open question. 1 A few recent papers seek to explain patterns of interest rate hedging by banks. Begenau, Piazzesi, and Schneider (2015) show that US banks increase their exposure to interest rates through their derivatives positions, and conclude that these banks do not hedge. I document a similar fact, but show that it is consistent with hedging under financial frictions. Relatedly, using data on all interest rate swap transactions, Hoffmann et al. (2017) show that European banks can take long or short positions, and that the observed patterns are consistent with hedging. The role of financial constraints as a barrier to bank risk management is explored in Rampini, Viswanathan, and Vuillemey (2015). However, none of these papers explicitly relates observed derivatives positions to the financial frictions that give rise to risk management incentives. For non-financial firms, a related work by Bretscher, Schmid, and Vedolin (2016) explains observed hedging positions with firm-level financial frictions. The closest paper is by Di Tella and Kurlat (2017), who also explain why observed 1 Regulators have recently paid increasing attention to interest rate risk, both in the US (Bednar and Elamin, 2014) and in Europe (ECB, 2017). 5

6 derivative exposures result from optimal hedging. In their model, banks are risk-averse and optimally take losses when interest rates rise, because they expect higher spreads on deposits going forward. Instead, banks in my model are risk-neutral, and become effectively risk-averse due to the existence of explicitly modeled financial frictions. The main advantage of this approach is that cross-sectional differences in banks positions can be directly related to differences in bank-level financial frictions, which are arguably more directly measurable than bank-level differences in risk-aversion. Relatedly, a contribution of my paper relative to Di Tella and Kurlat (2017) is to directly test cross-sectional predictions using data on bank-level exposures. 2 A few papers show that limited interest rate risk hedging by banks matters for the macroeconomy. Purnanandam (2007) and Landier, Sraer, and Thesmar (2015) show that banks exposure to interest rate risk affects the transmission of monetary policy. In Brunnermeier and Sannikov (2016), banks exposures to interest rates leave room for redistributive monetary policy. Finally, Drechsler, Savov, and Schnabl (2016) show that the transmission of monetary policy is well-explained by banks management of interest rate risk, specifically via the pricing of deposits. 2 Stylized facts I start by establishing three stylized facts. I use data on US commercial banks from the Call Reports, described in Appendix A. Stylized fact 1. Bank equity value is decreasing in the level of interest rates. This first fact has been documented in previous empirical work. Flannery and James (1984) and English, Van den Heuvel, and Zakrajsek (2013) find a negative relation between the level of interest rates and the market valuation of bank equity. Furthermore, both papers show that the interest rate sensitivity of bank stock prices correlates with measures of maturity mismatching: banks holding more long-duration assets relative to liabilities face a larger drop in equity value in response to increases in interest rates. 2 Other dynamic models of bank capital structure have focused on different questions. Sundaresan and Wang (2014) study the optimal liability structure of a bank. De Nicolo, Gamba, and Lucchetta (2014), Hugonnier and Morellec (2017) and Mankart, Michealides, and Pagratis (2016) study capital and liquidity requirements. Gornall and Strebulaev (2017) explain the high leverage of banks. 6

7 In the presence of financial frictions, banks may be expected to completely hedge tradable risks, including the interest rate risk that gives rise to Stylized fact 1 (Froot et al., 1993; Froot and Stein, 1998). Instead, the second stylized fact shows evidence of limited hedging by US banks. Stylized fact 2. A large fraction of banks do not hedge interest rate risk in derivatives markets. To show this, I define gross hedging for bank i at time t as Gross hedging i,t = Gross notional amount of interest rate derivatives for hedging of i at t Total assets i,t, (1) where I include all types interest rate derivatives (swaps, futures, etc.) in the numerator. Furthermore, I exploit a breakdown in the US Call Reports between derivatives held for trading or held for hedging, and focus only on the latter; see Appendix A for details. As seen in Table 1, gross hedging is equal to zero at the 90th percentile, implying that a large fraction of banks never use interest rate derivatives. Furthermore, the fraction of banks hedging is monotonically increasing with size. However, even within the top 1% of the size distribution, only 18.6% of banks hedge on average. Relatedly, Rampini et al. (2015) show that non-hedging banks do not have lower on-balance sheet exposure to interest rates and that, conditional on hedging, hedging is incomplete. Therefore, both hedging and non-hedging banks retain exposure to interest rates. Finally, I introduce a third and novel stylized fact. Stylized fact 3. A large fraction of banks use derivatives to hedge interest rate decreases, i.e., are net payers when interest rates increase and their equity value is low. To provide evidence, I define net hedging for bank i at date t as Net hedging i,t = Pay-fixed swaps i,t Pay-float swaps i,t Total assets i,t, (2) where I again include only derivatives held for hedging purposes. As explained in Appendix A, net hedging can only be computed for a subset of banks (21.2 % of 7

8 firm-quarters for which gross hedging is non-zero). A positive (resp. negative) value of net hedging means that a bank is taking a net pay-fixed (pay-float) position, i.e., receives cash flows when interest rates increase (decrease). In Panel A of Table 2, I show the distribution of net hedging, conditional on it being non-zero. In the pooled sample, the mean of net hedging is negative, while the median is negative but close to zero. Therefore, for more than 50% of banks, derivative exposures turn into liabilities when interest rates increase, i.e., when their equity value is already decreasing. Using a factor model, Begenau et al. (2015) also find that the total value of US banks derivatives portfolio declines when interest rates rise. This fact is pervasive over time, as Figure 1 shows. Stylized facts 2 and 3 can be seen as puzzling given Stylized fact 1, and may be interpreted as evidence of risk-seeking. Instead, I show that a model of bank capital structure with financial frictions can simultaneously rationalize all three stylized facts. 3 The model Time t is discrete and the horizon infinite. Each period, the bank s managers take decisions on (i) lending, (ii) financing and (iii) hedging. They are not subject to agency conflicts and maximize the wealth of risk-neutral equity holders. 3.1 Loans and cash flows I start with a description of the bank s cash flows and lending policy. At the beginning of each date t, the bank observes two shocks {z t, r t }. z t is a real shock affecting its cash flows and r t is the risk-free rate between dates t and t + 1. Both shocks are modeled under Assumption 1. Assumption 1. The shocks z t and r t take values in compact sets Z [z, z] and R [r, r], respectively, with r > 0. They jointly follow a first-order Markov process. The conditional distribution at date t of next-period shocks is denoted g (z t+1, r t+1 z t, r t ) and is common knowledge. I denote r the unconditional mean of r t. The bank enters each period t with a stock of loans in place a t. Upon observing {z t, r t }, it realizes net cash flows π (a t, z t, r t ), and pays a proportional tax τ (0, 1) 8

9 on them. π(.) includes interest payments received from outstanding loans, and is net of interest payments made on deposits. Net cash flows satisfy Assumption 2. Assumption 2. π (a t, z t, r t ) is continuous with π (0, z t, r t ) = 0, lim at π a (a t, z t, r t ) = 0 and satisfies (A1.1) π a (a t, z t, r t ) > 0, (A1.2) π aa (a t, z t, r t ) < 0, (A1.3) π z (a t, z t, r t ) > 0 and (A1.4) π r (a t, z t, r t ) 0. It takes the functional form π (a t, z t, r t ) = e γ(r r t) z t a θ t, (3) where θ (0, 1) and γ 0. Assumptions A1.1 and A1.2 ensure the concavity of cash flows in asset size, therefore capturing the decreasing creditworthiness of marginal borrowers (Dell Ariccia and Marquez, 2006). By A1.3, cash flows are larger when real conditions are better. Assumption A1.4 states that net cash flows are lower when the interest rate is higher. Several economic mechanisms motivate this assumption. First, while floatingrate assets reprice in response to an interest rate increase, fixed-rate long-term assets do not reprice immediately and continue to earn a lower interest rate. Instead, shortterm liabilities reprice more quickly, therefore squeezing banks net income. 3 Second, default rates by borrowers holding adjustable-rate loans are higher when interest rates are high, implying that bank cash flows are lower (Campbell and Cocco, 2015). The sensitivity of cash flows to the interest rate is captured by γ. If γ = 0, net cash flows do not depend on r t. Below, we show that a positive cash flow exposure of banks (i.e., γ > 0) is needed to quantitatively explain banks hedging decisions. This being said, even with γ > 0, interest rates have limited effect on net cash flows when they are close to their long-run average, since the unconditional expectation of r r t equals zero. Lending is modeled under the simplifying assumption that a constant share δ (0, 1) of loans matures each period. Incremental bank lending i t is i t = a t+1 (1 δ)a t, (4) 3 Drechsler, Savov, and Schnabl (2016) show that this mechanism is mitigated if banks use market power to delay raising interest rates on deposits. 9

10 where δ < 1 ensures that the average loan maturity exceeds that of one-period debt, modeled below. Thus, the bank engages in maturity mismatching. 3.2 Deposits I turn to the financing of the bank s assets. I start with deposit financing, which has two dimensions. First, the bank enters each period with a fixed stock of deposits d. While simplifying, the assumption of a fixed stock of deposits reflects the fact that retail deposits are publicly insured, thus sticky (see Hugonnier and Morellec, 2017, for a similar assumption). Second, the dynamics of deposits is linked to bank lending. Lending i t is financed by issuing deposits. When it lends, the bank credits the deposit account of a borrower by the principal amount of the loan. Loans received are used by borrowers to make payments, modeled under Assumption 3. 4 Assumption 3. New loans i t are used by borrowers to make payments in period t. These payments are made to agents with deposit accounts located in other banks. Assumption 3 makes it possible to obtain a simple law of motion for deposits. While loans are financed by issuing new deposits, the fact that these deposits are immediately used to make payments implies that they are no longer on a borrower s bank account at t + 1. Furthermore, since payments made out of new loans are directed towards agents with deposit accounts at other banks, these new deposits do not stay on other accounts within the same bank. Instead, when lending i t, the bank needs an amount i t of internal funds to face the corresponding deposit outflow. 5 Therefore, the dynamics of deposits is even though deposits are issued each time loans are made. 6 d t = d, (5) 4 Loans received can also be drawn down to be held in cash. It is irrelevant in this model whether outflows from the bank are held in cash or deposited at other banks. Furthermore, Assumption 3 is easily relaxed by considering that a constant fraction of new deposits stays on deposit accounts within the bank. 5 Another interpretation is that the bank holds reserves on a central bank account, and deposit outflows are associated with reserve outflows. See Appendix D for an explicit model of reserves, together with a reserve requirement. 6 Interest payments on deposits are not modeled explicitly, as they are accounted for in net cash 10

11 3.3 External financing: Interbank debt and equity A shortage of internal funds can exist for the bank if current lending is large relative to existing resources. When faced with a shortage of funds, the bank can turn to external sources: either interbank debt or equity. Symmetrically, the bank can lend excess resources in the interbank market or pay them out as dividend. Focusing first on debt, I restrict attention to riskless net interbank debt (i.e., debt minus cash), denoted b. Interbank debt takes the form of discount loans, and can be interpreted either as interbank debt stricto sensu, but also as any form of wholesale funding, such as repo contracts or central bank funding. Upon choosing b t+1 to be repaid at t + 1, the bank obtains b t+1 / (1 + r t ) in reserves at t. If b t+1 < 0, the bank is lending reserves and earns interest r t at t+1. Interests paid on debt are tax-deductible. When repaying b t at t, the bank gets a tax deduction, τb t r t 1 / (1 + r t 1 ). Instead, if it is a lender in the interbank market, the bank pays taxes on interest earned. 7 While the model could be extended to consider risky debt, our focus on riskless debt makes the solution simpler, and is consistent with the fact that short-term bank debt has very low credit risk. 8 A key question is that of the maximum amount of riskless debt that the bank can issue, if its commitment to repay future debt is limited. In Section 3.5, we derive a collateral constraint ensuring that the bank s debt can credibly be riskless. This collateral constraint is one of the two critical financing constraints in the model. Then, after choosing lending and borrowing, the bank may have a shortage of funds. This occurs if the collateral constraint is binding and the bank needs additional funds for lending. These funds are obtained by issuing equity. Symmetrically, a surplus of funds can be distributed as dividend. Denote e t the gross flow of equity between external financiers and the bank, where e t > 0 is a dividend distribution and e t < 0 an equity issue. The second financing constraint in the model is a cost to issuing external equity, as described in Assumption 4. flow π(.). These interest payments are assumed to be withdrawn by depositors each period, so that the stock of deposits does not grow with interest payments. 7 This tax structure ensures that it is never optimal to simultaneously borrow and lend in the interbank market. It is therefore consistent with my focus on net debt. 8 Repurchase agreements and central bank funding are collateralized. In unsecured funding markets, low credit risk comes from the fact that access is restricted to high-grade issuers (Pérignon et al., 2017). 11

12 Assumption 4. The cost of issuing equity, denoted η (e t ), satisfies η (e t ) > 0 if e t < 0, and η (e t ) = 0 otherwise. It satisfies η (e t ) = 1 {et<0} ( η 1 e t ), (6) with η 1 > 0. While neglecting potential fixed costs, the functional form in Assumption 4 is consistent with evidence that underwriting fees increase proportionally with the issue size (Altinkilic and Hansen, 2000; Hennessy and Whited, 2005). The cost of issuing equity can be interpreted as flotation fees or taxes. The friction η (e t ) makes equity issues more costly that debt issues. The two financing constraints i.e., the collateral constraint and the equity issuance cost play a critical role, since they give rise to hedging incentives. Specifically, there may be states where the banks needs funds for lending, but its collateral constraint is binding and it is forced to turn to (more expensive) equity financing. The bank may optimally want to hedge against such states. It is important to note that the existence of both frictions on debt and equity is needed to give rise to effective risk aversion, thus to risk management incentives. If there was no collateral constraint, the bank could always achieve the optimal scale of lending by borrowing with riskless debt. Instead, with a collateral constraint but η 1 = 0, the bank would also achieve the optimal scale of lending by costlessly issuing equity whenever its collateral constraint is binding. In such models, risk management (i.e., transfers of resources across states) would not increase bank value. It is only when both financing frictions are present that the bank sees a rationale for transferring resources to future states in which the collateral constraint may bind and costly equity may be needed. There are two instruments for the bank to manage risk: it can reduce current debt, so as to preserve future debt capacity, or it can use a derivative contract. Discussion on the preservation of debt capacity is postponed to Section 5.1, and we now turn to the explicit modeling of a derivative contract. 12

13 3.4 Interest rate derivatives I introduce a derivative contract, in the form of an interest rate swap, i.e., the most widely used contract in the data. A one-unit swap contract traded at t mandates the payment at t + 1 of a fixed swap rate (known at t) in exchange for the variable rate r t+1 (realized at t + 1). Swaps are provided by risk-neutral dealers, who price them fairly. 9 The swap rate equalizes the present value of the fixed leg and of the floating leg of the contract at inception. At any date t, the swap rate equals the current interest rate plus a (possibly negative) premium p t solving r t + p t = E t [r t+1 r t ]. (7) The notional amount of swap contracts traded at t is denoted s t+1. Whenever s t+1 > 0, the bank has a pay-fixed position, i.e., commits to pay a fixed rate s t+1 (r t + p t ) at t + 1, and to receive a floating rate s t+1 r t+1. Symmetrically, the bank has a pay-float position if s t+1 < 0. When taking a pay-fixed (respectively, pay-float) position, the bank is insuring against increases (decreases) in the interest rate: it is a net receiver of funds on its swap position at t + 1 if r t+1 is high (low), and a net payer otherwise. I restrict attention to risk-free swap contracts, and discuss collateralization in Section 3.5. In derivatives markets, the collateralization of exposures to mitigate counterparty risk is widespread (Duffie, Scheicher, and Vuillemey, 2015). Since swaps are risk-free, p t does not incorporate a default premium. 3.5 Financing constraints Before closing the model, I formulate a necessary constraint for both debt and swaps to be riskless in each period. The existence of a collateral constraint is consistent with the above pricing equations. 9 The distinction between dealers and end-users in derivatives markets is neat. Fleming, Jackson, Li, Sarkar, and Zobel (2012) find that the interest rate derivatives market is concentrated around a few dealers. In the CDS market, where the network structure is best documented, there are 14 dealers concentrating most intermediary activities between hundreds of end-users (Peltonen, Scheicher, and Vuillemey, 2014). Trading relationships not involving at least one dealer are rare. 13

14 I start by defining a new state variable, net worth w t, as w t (a t, b t, s t ) = (1 τ) π (a t, z t, r t ) + (1 δ)a t + τb tr t r t 1 b t + s t (r t (r t 1 + p t 1 )). (8) Net worth corresponds to the bank s available resources after assets and swaps in place have paid off, after maturing debt has been repaid, but before decisions on a t+1, b t+1 or s t+1 have been made. It captures financial slack for the bank at t: for a given amount of lending a t+1, a low net worth w t is associated with a larger need for external funds. Importantly, a negative net worth can be associated with a partial default on interbank debt and/ or swaps, if the bank s continuation value is low enough. For both debt and swap contracts to be risk-free, lenders and swap counterparties require the bank to repay outstanding contracts in all future states. The friction that gives rise to a collateral constraint is stated in Assumption 5. Assumption 5. The bank cannot commit to future lending, financing and hedging decisions. Furthermore, bank owners cannot abscond with part of the existing cash flows or loan stock. Since all contracts have a one-period maturity, Assumption 5 implies that the bank to take debt and swaps at t is limited by the lowest possible value of its net worth at t + 1, net of liquidation costs. I denote by κ [0, 1) the liquidation value of a one-unit loan. κ < 1 follows from Shleifer and Vishny (2011), who show that fire sales are a common response of banks to financial distress. I further assume that the tax benefit of debt is lost in case of liquidation. The collateral constraint writes as which can be rewritten as 0 min{w t+1 (1 κ)a t+1 τb t+1r t 1 + r t }, b t+1 + s t+1 ((r t + p t ) ˆr t ) (1 τ) π (a t+1, z, ˆr t ) + κa t+1, (9) 14

15 where ˆr t, defined as ˆr t (a t+1, s t+1 ) = arg min r t+1 [r,r] w t+1 = arg min r t+1 [r,r] π (a t+1, z, r t+1 ) s t+1 ((r t + p t ) r t+1 ), (10) is the interest rate associated with the lowest realization of net worth at t + 1, for a given a choice of control variables at t. Therefore, by Equation (9), choices of (a t+1, b t+1, s t+1 ) must be such that debt and swap payments do not exceed the lowest possible value of after-tax cash flows at t + 1, plus the liquidation value of loans in place. I highlight two alternative interpretations of Equation (9). First, instead of a collateral constraint imposed by lenders and swap counterparties, Equation (9) can be seen as a capital constraint. Indeed, it ensures that future net worth does not fall below a given threshold. Similarly, regulatory requirements set capital levels so as to prevent bank equity to fall below a threshold with high probability (value-at-risk thresholds). In my case, since default is ruled out, the collateral constraint ensures that the bank still finds it optimal to repay debt in the state where its net worth is the lowest. Total equity value remains positive in this state, due to a positive continuation value after debt repayment. Second, Equation (9) can be interpreted as a liquidity constraint, since it bears on net debt. Since net debt is computed as the difference between gross debt and cash (or liquid assets), any requirement to hold liquid assets tightens the constraint (9). 3.6 Bank value To close the bank s problem, the expression for e t is derived from a flow-of-funds identity, e t = w t a t+1 + b t r t. (11) Equation (11) states that the surplus or shortage of funds after financing and lending decisions have been made is either distributed as dividend (e t > 0) or obtained through equity issuance (e t < 0). I highlight that swap contracts traded at t, that is, s t+1, do not enter flow of funds equations at t, since they mandate future payments only. 15

16 The bank simultaneously chooses lending a t+1, financing b t+1 and hedging s t+1 each period to maximize the expected value of future dividends, discounted by a factor 1/ (1 + r t ) t 1 1 V t0 = E t0 (e t η(e t )) t=t 0 s=t r s, (12) subject to collateral constraint (9) and to the flow identity (11). Bellman equation is The associated V (w t, z t, r t ) = { sup e t η(e t )+ 1 a t+1,b t+1,s t r t } V (w t+1, z t+1, r t+1 ) dg (z t+1, r t+1 z t, r t ). The model satisfies the conditions for Theorem 9.6 in Stokey and Lucas (1989) to apply. 10 Therefore, a solution to Equation (13) exists. Since e t is weakly concave in w t, Theorem 9.8 in Stokey and Lucas (1989) ensures the existence of a unique single-valued policy function {a t+1, b t+1, s t+1 } = Γ (w t, z t, r t ). Before studying the model dynamics, Proposition 1 establishes a property of the equity value function of an unhedged bank, i.e., a bank for which s t+1 = 0 is imposed at any t. Proposition 1. The equity value V t of an unhedged bank is decreasing in the shortterm interest rate r t. Proof. See Appendix B.2. The property established in Proposition 1 is consistent with Stylized fact 1. Intuitively, a bank receives lower cash flows and has a lower continuation value when r t is lower. It also pays a higher interest rate on interbank debt when r t is high, which reduces its debt capacity. Therefore, the bank s equity value is decreasing in the interest rate. (13) 4 Calibration I calibrate the model and solve it numerically. Instead of calibrating parameters in order to match my moments of interest (e.g., the bank s leverage and the sign of swap 10 Appendix C demonstrates that the choice set is nonempty and compact. Furthermore, since e t is continuous with a compact domain, it is also bounded. Finally, Assumption 9.7 in Stokey and Lucas (1989) requires a constant discount factor. It is straightforward to show that Equation (13) can be rewritten with a constant discount factor β (0, 1) through a change of probability measure. 16

17 positions), I set parameter values based on pre-existing work and study the resulting simulated moments. The goal is to know whether realistically calibrated financial frictions can help reproduce stylized facts about bank risk management. 4.1 Financial frictions Two main parameters measure the extent of financial frictions in the model. First, the tightness of the collateral constraint depends on liquidation costs κ. For this parameter, I rely on estimates from the empirical literature. Specifically, Granja, Matvos, and Seru (2015), show that the loss of value in a failed bank represents 28% of its assets. In previous work, James (1991) came to a similar estimate of 30%. In my baseline calibration, I set κ = 0.72 (i.e., ). Furthermore, since the collateral constraint can alternatively be interpreted as a capital constraint, I also show robustness results based on a calibration with κ = This captures the fact that, in Basel banking regulations, book equity cannot fall below 8% of total assets. 11 Second, the costs of issuing external equity are captured by η 1. In the baseline calibration, I set η 1 = 0.059, based on the structural estimate by Hennessy and Whited (2005). One advantage is that their specification of the equity issuance cost is linear, as in my model. One concern, however, is that their estimate is for non-financial firms. Therefore, given the importance of equity issuance costs in the model, I also explore alternative specifications. Whether equity is more or less costly for banks than for non-financial firms is a matter of empirical debate. On the one hand, Gandhi and Lustig (2015) argue that bank stocks are mispriced, especially for large banks, due to government guarantees. If so, equity should be relatively cheaper than for non-financial firms. On the other hand, a number of papers highlight that equity is particularly expensive for banks, due to their role in producing liquid claims (see, for example, DeAngelo and Stulz, 2015). Based on these arguments, I show below simulation results with equity issuance costs both above and below the baseline value, specifically η 1 [0.039, 0.079]. Then, I calibrate several parameters based on data from US banks Call Reports over the period. I set δ = 0.23, which corresponds to the average share of 11 This is a simplification, since regulatory capital requirements include risk weights, which are outside the scope of the model. 17

18 loans and debt securities with a remaining maturity or repricing date below one year, as a percentage of total loans. The constant stock of deposits is set to d = 0.90a, so that the loan-to-deposits ratio at the steady state loan stock a is equal to the data average. 12 Finally, the corporate tax rate is set to τ = 0.35, consistent with the US tax code. 4.2 Shocks I interpret one period in the model as one year in the data. Based on Assumption 1, I model the real and interest rate shocks as AR(1) processes, ln (z t+1 ) = ρ z ln (z t ) + ɛ z,t+1, (15) r t+1 = (1 ρ r ) r + ρ r r t + ɛ r,t+1, (16) where ɛ z,t+1 and ɛ r,t+1 are jointly normal innovations with correlation ρ = I estimate ρ r and σ r from the time series of the 1-year Treasury constant maturity rate over the period (from the FRED database, series DGS1). The autocorrelation at a yearly frequency is ρ r = 0.88, while the standard deviation is σ r = r is the average Treasury rate over that period, equal to Then, I calibrate σ z = For comparison, the estimated volatility of cash flows is estimated to be 0.12 for non-financial firms by Hennessy and Whited (2007). The low volatility of z captures the well-documented fact that cash flows are an order of magnitude less volatile for banks than for non-financial firms, due to diversification (see Gornall and Strebulaev (2017) and Mankart et al. (2016) for direct evidence and related calibrations). 12 Following Strebulaev and Whited (2012), the steady state loan stock is defined as the value of a to which the bank would converge in the absence of shocks. It equals a = [ r ] 1 θ 1 + δ. (14) (1 τ) θ 13 Below, we show the model dynamics with non-zero correlation between ɛ z,t+1 and ɛ r,t+1. 18

19 4.3 Cash flows In the absence of previous guidance on θ and γ, I estimate them from the data. Specifically, I note that cash flows can be re-written, from Equation 3, as ( ) π (at, z t, r t ) ln = (θ 1) ln (a t ) + γ(r r t ) + ln(z t ). (17) a t After transforming Equation 17 in regression form, I estimate ln ( ) Net interest incomeit Loans it = (θ 1) ln (Loans it ) + γ(r r t ) 3 + ν j (GDP ) t j + ξ i + ε it, (18) j=0 where the bank fixed effect ξ i reflects the fact that, in the model, Equation 17 holds within any given bank. 14 Furthermore, when converting Equation 17 into a regression equation, I proxy for ln(z t ) using past innovations to the real GDP (i.e., (GDP ) t = ln(gdp ) t ln(gdp ) t 1 ). 15 In the entire sample of US banks, I obtain an estimate of θ 1 equal to (significant at the 1% level), yielding a value θ = 0.82, which I choose. 16 Finally, the estimated value of γ in the full sample is 0.89 (significant at the 1% level). However, given the importance of γ, I also report below simulation results with alternative values. Using these parameters, summarized in Table 3, I solve for the policy function Γ by value function iteration. The numerical method used is described in Appendix C. To gain further knowledge of the properties of the model, I simulate it. To do so, I draw a series of random shocks {z t, r t } for 10,200 periods. The bank s optimal controls are obtained using the policy function. The first 200 periods are dropped, to ensure convergence to an optimal capital structure from an arbitrary level of initial net worth. Selected moments are collected in Tables 4 and Since the model does not feature additional sources of heterogeneity across banks, cross-sectional variation and within-bank variation should be similar. Empirically, unmodeled factors are likely to introduce additional cross-sectional heterogeneity in cash flows. Bank fixed effects absorbs these unobserved factors, as long as they are time-invariant. 15 Empirically, all the estimates of ν are positive and significant, as expected from the model. 16 Using gross interest income instead of net interest income in Equation 18 would yield a relatively similar for θ, equal to

20 5 Financial frictions and bank leverage The bank can use two instruments to manage interest rate risk: the preservation of debt capacity and interest rate derivatives. I start by focusing on the bank s debt policy, and show that my model with financial frictions can explain important facts about banks leverage. 17 I turn to bank hedging in the next section. 5.1 Preservation of debt capacity Denote by λ t the Lagrange multiplier associated with the collateral constraint (9). The optimal financing policy is given by the first-order condition with respect to b t+1, 1 (1 η e (e t )) λ t = (19) 1 + r t ( 1 1 τr ) t (1 η e (e t+1 )) dg (z t+1, r t+1 z t, r t ), 1 + r t 1 + r t which equalizes the marginal costs and benefits of debt. Equation (19) shows that a marginal unit of debt is valuable both because of its tax benefit and because it enables saving on equity issuance costs. More interesting are the costs of debt. In addition to the interest rate being paid, an additional unit of debt taken at t is more costly for the bank if it is more likely to need equity issuance at t + 1, in order to achieve its optimal policy. This last cost highlights the rationale for risk management in the model. Financial frictions by which (i) debt is capped by a collateral constraint and (ii) issuing equity is costly, make the bank effectively risk averse with respect to next-period cash flows. Increasing debt at t implies that interest payments will absorb a larger share of cash flows at t + 1, i.e., that free cash flows available for lending will be lower. If lending opportunities are large at that date, the bank becomes more likely to resort to equity financing and to pay the issuance cost η(e t+1 ). If this is the case, the bank cannot achieve its optimal scale of lending. Therefore, the debt policy implies that there are benefits from preserving debt capacity: the bank can optimally reduce current interbank debt and forego present lending, so as to benefit from larger internal funds at t + 1 and be better able to lend at that date. 17 The optimal lending policy is given by a standard Euler equation, derived in Appendix B.1. 20

21 Incentives to preserve debt capacity are illustrated in Figure 2, which plots policy functions at the steady state loan stock. As seen in Panel A, optimal lending rises with the real shock. Additional lending is funded primarily through the issuance of interbank debt. The bank levers up, and eventually issues equity only for the highest values of z, when its collateral constraint binds and it still needs financing. Furthermore, when the profitability of lending is lower (low z), the opportunity cost of preserving debt capacity is low. Thus, the bank preserves significant debt capacity. Instead, when lending is highly profitable (high z), the bank exhausts its debt capacity, i.e., operates with high leverage. Thus, in the region where equity is not issued, bank leverage is procyclical. Using simulations, Table 4 shows how the preservation of debt capacity changes with the properties of the shocks. When lending opportunities are more volatile (σ z or σ r are high), the bank preserves higher unused debt capacity, i.e., operates with a lower leverage. Indeed, it is more likely to face large lending opportunities which are costly to forego. Furthermore, when shocks are more persistent (ρ z or ρ r are high), the bank also operates with lower leverage. This is because good shocks are more likely to be followed by other good shocks. Therefore, optimal lending may be large several periods in a row. Again, since foregoing these opportunities is costly, the bank chooses to maintain a lower leverage ex ante. 18 Furthermore, the bank preserves more debt capacity when equity issuance costs are higher (see Table 5). 5.2 Empirical evidence I show that the model s predictions about leverage are supported by the data. To do so, I formulate a first hypothesis, which follows directly from Section 5.1. Hypothesis 1. Banks faced with more volatile or more persistent lending opportunities operate with a lower leverage. To test Hypothesis 1, I construct bank-level measures of shocks. While the interest rate r t is the same for all banks at any given date t, I interpret all differences in lending across banks at any t as coming from realizations of the real shock z t. As 18 These properties are not specific to our model of banks, and also arise in models of non-financial firms such as Hennessy and Whited (2005). 21

22 such, z t incorporates the effect of local economic conditions or of bank-level differences in lending technologies. Instead of directly measuring z t at the bank-level (which is unobservable), I proxy for it using observed lending patterns. Specifically, I proxy for the volatility and persistence of z at the bank level by the volatility and persistence of observed lending. The idea behind this approach, consistent with the model, is that banks facing more volatile (resp. more persistent) shocks should also have more volatile (resp. more persistent) observed lending policies. Therefore, for each bank i, I construct a vector Loans i,t = log(loans i,t ) log(loans i,t 1 ). Then, the volatility σ i and the persistence ρ i are proxied respectively by the standard deviation and the first-order autocorrelation of Loans i,t. In Panel A of Table 6, I estimate a pooled OLS regression of banks book leverage on σ i and ρ i. Across all specifications, I confirm that banks facing more volatile or more persistent lending opportunities operate with a lower leverage, consistent with the model and with Hypothesis 1. A potential concern with these cross-sectional estimates is that measures of lending growth volatility may correlate with unobserved bank characteristics, which may themselves explain bank leverage. Ideally, the inclusion of bank fixed-effects should help address this concern. However, it requires constructing time-varying measures of shocks. 19 For any bank i, I proxy for the volatility of future lending growth opportunities with past lending growth. 20 The idea is that past and future shocks to lending opportunities around any date t are likely to be positively autocorrelated for any bank i. Specifically, I use both the squared and the absolute value of realized loan growth over the past eight months as a measure of time-varying lending growth volatility. Since large funding needs arise when lending opportunities are good, I also estimate regressions with realized loan growth, which can be either positive or negative. Panel B of Table 6 shows the relationship between book leverage and time-varying 19 In Panel A of Table 6, σ i and ρ i are bank-specific and time-invariant. Therefore, in a within-bank regression, they would be absorbed by bank fixed effects. 20 Unfortunately, due to the low frequency of the data (quarterly) and the relatively shock panel, I cannot construct time-varying measures of lending growth persistence at the bank-level. 22

23 measures of lending volatility, after including bank fixed effects. I find a negative and significant relation across all specifications. Therefore, a given bank preserves larger unused debt capacity at times its lending opportunities are more volatile. This gives further support for the model s predictions. 6 Financial frictions and bank hedging I turn to the hedging policy. I show that banks have incentives to hedge both increases and decreases in the interest rate, absent any speculative motive. The model s predictions are tested in the next section. 6.1 Intra-temporal trade-offs: Hedging motives I turn to interest rate risk management using derivatives. The hedging policy is obtained by deriving V t with respect to s t+1, 1 [rt+1 (r t + p t (r t )) ] (1 η e (e t+1 )) dg (z t+1, r t+1 z t, r t ) (20) 1 + r t = λ t s t+1 [ (1 τ) π (at+1, z, ˆr t (a t+1, s t+1 )) s t+1 ((r t + p t (r t )) ˆr t (a t+1, s t+1 )) ]. As seen on the left-hand side, hedging derives value because equity issuance is costly. Swaps allow saving on issuance costs, by transferring funds at t+1 from states where no equity will be issued to states where external equity financing would, absent hedging, be needed. The main difference between interest rate derivatives and the preservation of debt capacity is that derivatives provide state-contingent payoffs at t + 1. Instead, when preserving debt capacity, the bank chooses to keep funds that will be available in all future states. Equation (20) shows that there would be no ex ante benefit from hedging in the absence of issuance cost. Indeed, with η 1 = 0, the integral would simplify to zero, using the pricing equation (7). In this case, the bank would always operate at the optimal lending scale by issuing equity whenever it needs external financing. Next, I examine the expected benefits from hedging (left-hand side of Equation 20) in more details. To understand what derivative positions are taken, I identify 23

24 whether future states in which the bank may be constrained and need equity financing are associated with a high or with a low realization of r t+1. There are three channels through which the bank is exposed to interest rate changes. A first channel is through the cost of debt financing. When r rises, the bank s debt capacity is reduced, since the amount it can borrow is capped by the collateral constraint. Consequently, there is a risk that the cost of debt financing is high in states where lending opportunities are large. This may cause these opportunities to be foregone, thus giving rise to a financing motive for risk management: the bank wants to hedge against interest rate increases. In the swap market, the financing motive is addressed by taking pay-fixed positions (s t+1 > 0), which pay off when the next-period interest rate is high. A second channel through which the bank is exposed to the interest rate is via the discount factor, 1/ (1 + r t ). It r t is high, foregoing present dividends is more costly for equity holders. Therefore, a larger share of internal funds is distributed, and optimal lending is lower. Since optimal lending is larger when 1/ (1 + r t ) is larger, the discount motive provides incentives to hedge against decreases in interest rates. The third channel arises from the sensitivity of cash flows to r t. Provided γ > 0, optimal lending is larger when r t is lower, giving rise to an investment motive for risk management: the bank wants more funds in states where r t+1 is lower, to meet higher lending opportunities. As with the discount motive, risk management driven by the investment motive requires hedging against decreases in interest rates. 6.2 Inter-temporal trade-offs: Hedging and debt capacity Even though all cash flows associated with swaps are realized at t + 1, hedging future states has a cost or benefit at t. An intertemporal trade-off arises due to the fact that both debt and swaps are collateralized. Thus, hedging can reduce or increase the bank s current debt capacity. Which effect prevails depends on the sign of the righthand side term in Equation 20. Proposition 2 establishes a relation between hedging and debt capacity. Proposition 2. A derivative position s t+1 0 may either increase or decrease the 24

25 bank s debt capacity at date t. Denote D (a t+1, s t+1, ˆr t ) = (1 τ) π (a t+1, z, r) (1 τ) π (a t+1, z, ˆr t ) + s t+1 (r t + p t ˆr t ). The bank s debt capacity increases whenever D (.) < 0 and decreases otherwise. Proof. See Appendix B.3. Intuitively, D (.) is the difference between the bank s debt capacity if it does not hedge (s t+1 = 0) and if it hedges (s t+1 0). Whether the bank s debt capacity increases or decreases depends on the effect of future swap payments on the lowest possible value of its net worth. If swap payments are such that the bank receives cash flows in states where its net worth is low, taking such positions may enhance its future ability to repay debt, thus increase its current debt capacity. If swap positions do not provide such a hedge against future cash flows, the bank s debt capacity will decrease. Indeed, future net worth that is pledged as collateral for swaps will no longer be pledgeable to obtain debt. I stress implications of Proposition 2. In case γ = 0, I obtain Corollary 1. Corollary 1. If γ = 0, any swap position s t+1 0 reduces the bank s debt capacity at date t. Corollary 1 can be understood by noting that, if γ = 0, cash flows π(.) do not depend on the realized interest rate. Therefore, derivatives cannot be used to offset low cash flows from loans, thus to increase the lowest possible realization of future net worth. Whenever the bank takes a swap position, regardless of its sign, it has to pledge collateral, which can no longer be used to obtain debt financing. The case where γ = 0 is therefore of interest: since the choice of a swap position cannot be guided by the desire to increase current debt capacity, it is a useful benchmark to assess the quantitative relevance of this hedging motive relative to other motives. Next, Proposition 2 also allows obtaining predictions related to the type of swap positions taken. Corollary 2. For any γ 0, pay-float swap positions s t+1 < 0 reduce the bank s debt capacity. 25

26 When the bank hedges decreases in the interest rate by holding pay-float swaps (s t+1 < 0), ˆr t = r > r t + p t. Therefore, D > 0 and the bank s debt capacity is reduced. Intuitively, cash flows π (.) are low in states where the bank is a net swap payer (when r t+1 is high). Thus pledgeable cash flows to debt holders are reduced when pay-float swap positions are taken. Finally, when the bank hedges increases in the interest rate by taking pay-fixed swaps (s t+1 > 0), D can be either positive or negative, due to two opposite forces. First, cash flows π (.) reach their lowest level when r is realized. Second, the bank receives swap payments in such states. Whether cash flows after swap payments are higher or lower when r is realized depends on the size of the swap position s t+1 and on γ. To conclude, the choice of hedging positions is driven not only by expectations about future financing needs, i.e., by the financing, discount and investment motives described in Section 6.1. It is also driven by concerns related to the bank s present debt capacity. The co-existence of opposite incentives to engage in risk management yields an important result: both increases and decreases in the interest rate can be hedged, in the absence of speculative motive for trading. This result can be illustrated using simulations of the model. With the baseline calibration, conditional on using swaps, 34.3% of the positions taken are pay-fixed (Table 4, Column 1). This numbers is reasonably close to the data (45.9% of pay-fixed positions in the pooled dataset), even though I did not use any moment related to hedging positions in order to calibrate the model. Furthermore, as Table 5, this result holds for alternative values of equity issuance cost (η 1 ), of liquidation costs (κ) and of the interest rate sensitivity of cash flows (γ). 7 Hedging policy: Empirical evidence In this section, I show that my model with financial frictions can explain why a large fraction of banks use derivatives to increase their interest rate exposure. I obtain predictions on bank characteristics associated with long or short positions in interest rate derivatives. I test these predictions in US data and find strong support for them. 26

27 7.1 Model predictions: Hedging interest rate increases I start by focusing on bank characteristics associated with hedging of interest rate increases. To predict how hedging positions change with the nature of lending opportunities, I vary parameters capturing the volatility and persistence of the real shock (σ z and ρ z ), of the interest rate (σ r and ρ r ) and the correlation between shocks (ρ). In Columns 2 to 11 of Table 4, the predicted capital structure impact of variation in these parameters is summarized. There are two reasons for banks to hedge against increases in interest rates: either to address the financing motive for risk management, or to increase present debt capacity. The relative role of these two forces can be disentangled by varying γ. Indeed, when γ = 0, the bank cannot use derivatives to increase present debt capacity. Therefore, the only reason why pay-fixed positions may be taken is because of the financing motive. To obtain guidance on the relative role of these incentives, I compare simulations with γ = 0.89 (baseline case, in Panel A) and with γ = 0 (Panel B). A comparison of the two panels in Table 4 shows that the financing motive is quantitatively unimportant. I find that pay-fixed swaps represent only 3.0% of derivatives positions taken when γ = 0 (Column 1). Since the use of pay-fixed swaps when γ = 0 is driven only by the financing motive, I conclude that it is not a quantitatively strong motive to engage in hedging. This result is unlikely to be due to miscalibration, since it is true for a wide set of parameters, as seen in Columns 2 to 11. One reason why the magnitude of the financing motive is limited is that the bank partially benefits from a natural hedge: both its debt capacity and its lending opportunities are decreasing in r. Thus, it has a high debt capacity at times large lending driven by low realizations of r is optimal. In contrast, pay-fixed swaps are used far more often when γ > 0, keeping other parameters constant. With the baseline calibration, pay-fixed swaps represent 34.3% of all derivative positions (Panel A, Column 1). The only difference with the previous case is that, with γ > 0, derivatives can be used to increase present debt capacity. Again, this is true for a wide range of parameters (Columns 2 to 11). Therefore, I conclude that the main reason why banks hedge against increases in the interest rate is not to hedge future debt costs (the financing motive), but to increase their present debt 27

28 capacity. Taking pay-fixed swap positions is needed when current funding needs are large relative to net worth, and current debt constraints are binding. Swaps alleviate these current constraints by enhancing the bank s ability to credibly repay future debt. 7.2 Model predictions: Hedging interest rate decreases I turn to the reasons why banks hedge decreases in interest rates. They can do so to address the discount motive or the investment motive for risk management, described in Section 6.1. First of all, when γ = 0, the investment motive for risk management does not exist, and pay-float positions are taken only because of the discount motive. Since pay-float positions are used to a sizable extent when γ = 0, I conclude that the discount motive is quantitatively important for a wide range of parameters. Second, for a given value of γ, I investigate which properties of the shocks are associated with a greater propensity to hedge against interest rate decreases. The main result is that pay-float swaps tend to be used more often when future funding needs are likely to be large. This can be the case either because shocks have a higher standard deviation or are more persistent. These effects are seen in Table 4. Higher values of σ z (Column 3), ρ z (Column 5), σ r (Column 7), ρ r (Column 9) are associated with a larger proportion of pay-float swaps being used. There are two main explanations for this result. First, when taking a pay-fixed position to increase its present debt capacity, the bank commits to possibly large swap payments in some future states. If financing needs are large next period, the bank is likely to need external financing, and possibly to lend less than is optimal. This risk becomes more important if future funding needs are more unpredictable or if large present funding needs tend to be followed by other large needs for funds. Therefore, as future funding needs become more uncertain, it becomes more costly for the bank to use derivatives to address present funding needs. Therefore, it tends to use less pay-fixed swaps. Second, when shocks to interest rates are more volatile or more persistent, it becomes more likely that future lending opportunities are driven by low realizations of interest rates. The investment motive becomes stronger, and the bank optimally wants more funds in future states associated with a low interest rate. This section yields novel predictions. Consistent with the data, banks can hedge 28

29 both increases and decreases in the interest rate, in the absence of incentives to speculate. The choice of pay-fixed positions (insurance against interest rate increases) is driven to a large extent not by future financing needs but by present needs for funds. Pay-float positions (insurance against interest rate decreases) are more likely to be taken by banks that have more unpredictable future funding needs. 7.3 Empirical tests To assess the ability of my model with frictions to explain US data on interest rate hedging, I test two hypotheses and find strong support for each of them. Hypothesis 2. Banks faced with more volatile or more persistent lending opportunities are more likely to hedge decreases in interest rates with derivatives. Hypothesis 3. Banks hedge increases in interest rates with derivatives to alleviate current financing constraints, i.e., when current financing needs are large relative to net worth. These hypotheses follow directly from results in Sections 7.1 and 7.2. To test Hypothesis 2, I use banks net hedging (defined in Equation 2) as dependent variable. This ensures a close mapping between the model and the tests. 21 independent variables, I use the measures of volatility and persistence of lending opportunities constructed in Section 5.2, i.e., σ i and ρ i. The model predicts that both measures are associated with more negative net hedging, i.e., banks with more uncertain future lending opportunities are more likely to take additional exposure to interest rates when hedging using derivatives. In Panel A of Table 7, I report results on two separate samples: one including only banks with non-zero net hedging, and one also including all observations with zero net hedging. 22 As I find that all coefficient signs are consistent with the model s predictions, regardless of the sample used, and statistically significant. Overall, these regressions confirm the ability of the model to 21 Appendix A shows that, along a number of dimensions, banks for which net hedging is non-zero are very similar to hedging banks for which net hedging cannot be computed. 22 Even though the model can explain why banks choose zero net hedging (see Section 8), the absence of any hedging can also arguably be explained by (unmodeled) fixed costs. This gives us a rationale for using two samples. 29

30 explain the data: banks that increase their interest rate exposure using derivatives are those predicted to do so by the model. Then, I test whether Hypothesis 2 also holds after inclusion of bank fixed effects. This arguably provides a stringent test of the model, as I test whether a given bank is more likely to take pay-float positions at times it is exposed to more volatile shocks. I use the same time-varying measures of volatility as in Section 5.2, and report estimates in Panel B of Table 7. Consistent with the cross-sectional findings, I find that higher measures of loan growth volatility are associated with more negative net hedging, i.e., a greater tendency of banks to take additional exposure to interest rates via derivatives. The fact that these estimates are robust to the inclusion of bank fixed effects provides support to the model s predictions. Again, estimates are significant regardless of whether the sample includes observations with zero net hedging or not. Finally, I test Hypothesis 3. In Table 8, I start by showing that more constrained banks are more likely to have positive net hedging, i.e., to hedge against increases in interest rates. I use four measures of financial constraints and find that banks that are less constrained, as captured by a larger size, a higher net income, and higher dividends, and more likely to have negative net hedging (Panel A). These estimates are consistent with the model s predictions. Next, I test more directly Hypothesis 3 by focusing on quarters in which banks actively take on additional pay-fixed positions. I identify these positions in the data as increases in net hedging in a given quarter. However, since such increases can be due to the maturation of existing pay-float positions, I also require that gross hedging goes up in this quarter. This method identifies active changes towards larger pay-fixed positions. I construct a dummy variable equal to one when such pay-fixed positions are taken, and zero otherwise. My model predicts that pay-fixed positions are taken to alleviate current financing constraints, at times when large increases in debt are needed. In Panel B of Table 6, I regress this dummy variable on changes in book leverage within a given quarter. Therefore, this regression provides a direct test of the model s prediction. I find that large increases in leverage are associated with active decisions to take pay-fixed swap positions. The estimate is significant at the 1% level. Furthermore, it is robust to the inclusion of both time and bank fixed effects. 30

31 Therefore, a given bank is more likely to take pay-fixed positions precisely in quarters when it also increases leverage. I thus fail to reject Hypothesis 3. To summarize, I find empirical support for the novel predictions of the model. I conclude that a model of bank capital structure with financial frictions can explain otherwise puzzling stylized facts on bank s derivatives positions: the fact that a large fraction of banks use derivatives to increase exposure to interest rates is consistent with hedging. Additionally, financial frictions also have the potential to explain observed leverage patterns. 7.4 Alternative hypothesis: moral hazard I briefly discuss an alternative interpretation of the stylized facts. The existence of a large fraction of banks increasing their interest rate exposure with derivatives can be interpreted as evidence of risk-seeking. The main reason why overly risky decisions may be taken by banks is arguably the existence of moral hazard arising from implicit or explicit guarantees (Fahri and Tirole, 2012). Theories based on moral hazard can in principle explain the coexistence of banks with long and short interest rate positions: only banks protected by guarantees should use derivatives to increase exposure to interest rates. While my goal is not to test the view that excessive risk-seeking is explaining banks derivatives positions, I argue that this explanation is unlikely to be sufficient. First, it is not the case that banks increasing interest rate exposure through derivatives are mostly the largest or the less capitalized ones, which are more subject to moral hazard. In Panel B of Table 2, I break down the distribution of net hedging by quintiles of size and equity/assets. For these two measures of moral hazard, I observe that mean net hedging is negative for all quintiles. Furthermore, large net pay-float swap positions (at the 5th or 10th percentiles of net hedging) are not concentrated among large or poorly capitalized banks. Overall, the distribution of net hedging is extremely similar across size and capitalization quintiles. There is also no marked difference when focusing on the 1% largest institutions. Therefore, proxies for moral hazard are unlikely to be the main explanatory variables. Second, I ensure that all tests of the model s predictions are robust to the inclusion 31

32 of proxies for moral hazard, namely bank size and the ratio of book equity to total assets. These alternative specifications are reported in Tables 6 and 7. As can be seen, all estimates remain of similar magnitude and equally significant. 23 Therefore, regardless of bank size and capitalization, the nature of financial frictions facing banks is an important determinant of the hedging positions taken. While we cannot rule out the existence of any moral hazard, our findings suggest that moral hazard is not a sufficient explanation for Stylized fact 3. 8 Coexistence of hedging and non-hedging banks I finally discuss an auxiliary result of the model: banks do not always use interest rate derivatives. This prediction is consistent with Stylized fact Non-hedging banks Even though they enable banks to better achieve their optimal lending policy, derivatives in the model are not used in all periods, even in the absence of any fixed cost. With the baseline calibration, derivatives are used in 53.1% of the periods (Table 4, Column 1). This is due to the fact, discussed in Section 6.2, that the use of derivatives may reduce a bank s debt capacity. Indeed, both debt and swap contracts involve promises to future payments and, as such, need to be collateralized. Consequently, any unit of collateral pledged on derivatives is no longer available to obtain debt financing, under the conditions stated in Proposition 2. A trade-off between hedging and financing also appears in the seminal work by Rampini and Viswanathan (2010). However, there is one important difference. In their model, debt contracts are state-contingent: repayments in a given state depend on net worth in this state. In my model, debt contracts are not state-contingent: the ability to borrow at t depends on the lowest possible value of net worth at t + 1 across all states. This difference has an important implication: while hedging in Rampini and Viswanathan (2010) always reduces debt capacity, hedging positions in my model can increase debt capacity. Indeed, hedging may raise the value of the lowest possible 23 In unreported regressions, we also check that our results are robust to the inclusion of size and equity/assets separately. 32

33 realized net worth at t + 1. This difference matters quantitatively since, as Section 7.1 shows, the possibility to increase debt capacity is the main reason explaining why banks hedge against interest rate increases. Furthermore, this prediction is supported by the data (Section 7.3). 8.2 Net worth and hedging What determines a bank s decision to hedge or not? I show that the model yields a positive relation between bank net worth and hedging. This relation prevails even though derivatives may be used by constrained banks to increase their debt capacity. To study the relation between net worth and hedging, I simulate a panel of heterogeneous institutions. Heterogeneity is introduced in the real shock. Each bank i has a permanent level of the real factor z i and receives each period an aggregate transitory shock z t, similar to that described in Equation 15. The date-t realization of the real shock for bank i, z i,t, is z i,t = z i + z t, (21) where heterogeneity can arise from permanent differences in location, in managerial ability, or in business models. I simulate a panel of 1,000 banks, each of them with a value of z i drawn from a normal distribution with mean zero and standard deviation 0.1. This panel is simulated for 300 periods in which each bank receives aggregate shocks { z t, r t }. For each of them, the first 200 periods are dropped, yielding 100,000 bank-period observations. I draw a distinction between the decision to hedge and the extent of hedging. I measure the decision to hedge using a dummy variable that takes value one at t if a bank chooses non-zero derivatives s t+1, and the decision to hedge as the absolute value of s t+1 normalized by a t I regress these variables on net worth w t and show estimated coefficients in Table 9. Using a probit estimation (Column 1), I find a positive and significant relation between the decision to hedge and net worth in the pooled sample. The same is also true in a specification with bank fixed effects (Column 2): a given bank is more likely to use derivatives at times its net worth is higher. Turning to the extent of hedging, a 24 A bank is considered to use zero derivatives if s t+1 < 0.001a. 33

34 positive and significant relation is also observed. This is the case in the cross-section, where I estimate both a pooled OLS (Column 4) and a Tobit (Column 5) model. Estimates with bank fixed effects (Column 5) also indicate that a given bank takes larger swap positions at times its net worth is higher. This positive relation between net worth and hedging holds for both γ = 0.89 (Panel A) and γ = 0 (Panel B). These results are driven by a dynamic trade-off between financing and hedging. This trade-off clearly exists when γ = 0 (see Corollary 1), but also when γ > 0. In this case, a bank can increase present debt capacity by increasing hedging, but it also commits to larger swap payments in some future states. Therefore, increasing hedging at t may reduce net worth and financing capacity at t + 1, implying that future lending opportunities may be foregone. This problem gets worse for banks with a low net worth. Consequently, even though constrained banks may hedge to increase present debt capacity, the extent of hedging remains limited, due to the fear that financing constraints may worsen in the future. This yields a positive and significant correlation between net worth and hedging both across and within banks. To summarize, while banks may use interest rate derivatives to alleviate present financing constraints, a positive relation between net worth and hedging prevails in the model. This result is consistent with empirical evidence about banks hedging of interest rate risk (see Rampini, Viswanathan, and Vuillemey, 2015). 9 Conclusion While bank net worth is decreasing in the level of interest rates, data show that (i) many banks do not hedge interest rate risk, and (ii) a large fraction of hedging banks use derivatives to take additional exposure to interest rate increases. I build a model of bank capital structure, and show that these patterns are consistent with optimal risk management under financial frictions. Furthermore, I show that novel predictions about the characteristics of banks hedging positions are supported by the data. I conclude that stylized facts are not necessarily evidence of speculative behavior or of excessive risk-seeking. My findings call for further empirical research on risk management in banking. One implication of my results is that, in the presence of financial frictions, hedging 34

35 and speculative positions cannot be readily identified using the comovement between derivatives payoffs and equity value. Another implication is that financial constraints can be a major impediment to risk management. More work is needed, however, to document the extent to which financial frictions can explain actual hedging patterns in banks, relative to other theories. Finally, an open question is whether models with financial frictions can explain other facts about bank capital structure, such as the dynamics of capital ratios or bank lending. References Altinkilic, O. and R. S. Hansen (2000). Are there economies of scale in underwriting fees? Evidence of rising external financing costs. Review of Financial Studies 13, Bednar, W. and M. Elamin (2014). Rising interest rate risk at U.S. banks. Economic Commentary 12. Begenau, J., M. Piazzesi, and M. Schneider (2015). Banks risk exposures. Working Paper. Bretscher, L., L. Schmid, and A. Vedolin (2016). hedging. Working Paper. Interest rate risk and corporate Brunnermeier, M. K. and Y. Sannikov (2014). A macroeconomic model with a financial sector. American Economic Review 104, Brunnermeier, M. K. and Y. Sannikov (2016). The I theory of money. Working paper. Campbell, J. Y. and J. F. Cocco (2015). A model of mortgage default. Journal of Finance 70, De Nicolo, G., A. Gamba, and M. Lucchetta (2014). Microprudential regulation in a dynamic model of banking. Review of Financial Studies 27, DeAngelo, H. and R. M. Stulz (2015). Liquid-claim production, risk management, and bank capital structure: Why high leverage is optimal for banks. Journal of Financial Economics 116, Dell Ariccia, G. and R. Marquez (2006). Lending booms and lending standards. Journal of Finance 5, Di Tella, S. and P. Kurlat (2017). Why are bank balance sheets exposed to monetary policy? Working paper. 35

36 Diamond, D. and R. G. Rajan (2012). Illiquid banks, financial stability, and interest rate policy. Journal of Political Economy 120, Drechsler, I., A. Savov, and P. Schnabl (2016). The deposit channel of monetary policy. Quarterly Journal of Economics (forthcoming). Drechsler, I., A. Savov, and P. Schnabl (2017). Banking on deposits: Maturity transformation without interest rate risk. Working paper. Duffie, D., M. Scheicher, and G. Vuillemey (2015). Central clearing and collateral demand. Journal of Financial Economics 116, ECB (2017). Sensitivity analysis of irrbb - stress test English, W. B., S. J. Van den Heuvel, and E. Zakrajsek (2013). Interest rate risk and bank equity valuations. Working Paper (26). Fahri, E. and J. Tirole (2012). Collective moral hazard, maturity mismatch, and systemic bailouts. American Economic Review 102 (1), Flannery, M. and C. James (1984). The effect of interest rate changes on the common stock returns of financial institutions. Journal of Finance 39, Fleming, M., J. Jackson, A. Li, A. Sarkar, and P. Zobel (2012). An analysis of OTC interest rate derivatives transactions: Implications for public reporting. Federal Reserve Bank of New York Staff Reports 557. Froot, K. A., D. S. Scharfstein, and J. C. Stein (1993). Risk management: Coordinating corporate investments and financing policies. Journal of Finance 5, Froot, K. A. and J. C. Stein (1998). Risk management, capital bugdeting and capital structure policy for financial institutions: an integrated approach. Journal of Financial Economics 47, Gandhi, P. and H. Lustig (2015). Size anomalies in u.s. bank stock returns. Journal of Finance 70, Gornall, W. and I. Strebulaev (2017). Financing as a supply chain: The capital structure of banks and borrowers. Journal of Financial Economics (forthcoming). Granja, J., G. Matvos, and A. Seru (2015). Selling failed banks. Journal of Finance (forthcoming). Hennessy, C. and T. Whited (2005). Debt dynamics. Journal of Finance 60, Hennessy, C. and T. Whited (2007). How costly is external financing? Evidence from a structural estimation. Journal of Finance 62,

37 Hoffmann, P., S. Langfield, F. Pierobon, and G. Vuillemey (2017). Who bears interest rate risk? Working paper. Holmström, B. and J. Tirole (1997). Financial intermediation, loanable funds, and the real sector. Quarterly Journal of Economics 112, Hugonnier, J. and E. Morellec (2017). Bank capital, liquid reserves and insolvency risk. Journal of Financial Economics 125, James, C. (1991). The losses realized in bank failures. Journal of Finance 46 (4), Landier, A., D. Sraer, and D. Thesmar (2015). Bank exposure to interest rate risk and the transmission of monetary policy. Working Paper. Mankart, J., A. Michealides, and S. Pagratis (2016). Bank capital buffers in a dynamic model. Working Paper. Peltonen, T., M. Scheicher, and G. Vuillemey (2014). The network structure of the CDS market and its determinants. Journal of Financial Stability 13, Pérignon, C., D. Thesmar, and G. Vuillemey (2017). Journal of Finance (forthcoming). Wholesale funding dry-ups. Purnanandam, A. (2007). Interest rate derivatives at commercial banks: An empirical investigation. Journal of Monetary Economics 54, Rampini, A. and S. Viswanathan (2010). Collateral, risk management and the distribution of debt capacity. Journal of Finance 65, Rampini, A., S. Viswanathan, and G. Vuillemey (2015). Risk management in financial institutions. Working Paper. Shleifer, A. and R. W. Vishny (2011). Fire sales in finance and macroeconomics. Journal of Economic Perspectives 25, Stokey, N. and R. Lucas (1989). Recursive Methods in Economic Dynamics. Harvard University Press. Strebulaev, I. and T. Whited (2012). Dynamic models and structural estimation in corporate finance. Foundations and Trends in Finance 6, Sundaresan, S. and Z. Wang (2014). Bank liability structure. Working Paper. Tauchen, G. (1986). Finite state Markov-chain approximations to univariate and vector autoregressions. Economic Letters 20,

38 Table 1 Gross interest rate hedging and trading US data This table provides descriptive statistics on gross interest rate hedging and trading by US banks. Gross hedging is defined in Equation (1). Gross trading is defined similarly using derivatives held for trading purposes. The upper panel shows the distribution of these variables in the pooled sample. The lower panel shows a breakdown of hedging and trading exposures in each size quintile and for the 1% largest banks. For each of these quintiles/percentile, I report the average number of banks, the fraction of banks holding interest rate derivatives for hedging and trading purposes, as well as the average gross hedging and trading (conditional on being non-zero). The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. Mean Med. 75th 90th 95th 98th 99th Max Gross hedging / Assets Gross trading / Assets Size quintiles 1st 2nd 3rd 4th 5th Top 1% All sample Number of banks 1,564 1,563 1,563 1,563 1, ,819 Fraction hedging Extent of hedging (conditional) Fraction trading Extent of trading (conditional)

39 Table 2 Net interest rate hedging US data This table provides descriptive statistics on net interest rate risk hedging by US banks. Panel A describes net hedging, defined in Equation (2). I show its distribution in the pooled sample, and by quintiles of size of book equity/assets. I additionally report the distribution of net hedging for the 1% largest banks. Panel B compares the size distribution of banks with non-zero net hedging with the size distributions of all hedging banks (i.e., non-zero gross hedging) and of all sample banks. The first two distributions are not identical, due to the fact that net hedging cannot be computed for all banks (see Appendix A). Size is defined as the logarithm of total assets. The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. Panel A: Hedging positions 5th 10th 25th Med. Mean 75th 90th 95th Net hedging / Assets (cond.) By size quintiles: 1st quintile (small) nd quintile rd quintile th quintile th quintile (large) Top 1% By equity/assets quintiles: 1st quintile (low equity) nd quintile rd quintile th quintile th quintile (high equity) Panel B: Size distribution of hedging banks 5th 10th 25th Med. Mean 75th 90th 95th Gross hedging > Net hedging All banks

40 Table 3 Calibration This table contains the calibrated values of the model parameters. Parameter Description Value Source δ Per-period share of maturing loans 0.23 Call Reports θ Concavity of operating cash flows 0.82 Estimated η 1 Cost of equity financing 0.09 Hennessy and Whited (2005) τ Corporate tax rate 0.35 US tax code κ Liquidation value of an asset unit 0.72 Granja et al. (2015) γ Interest rate sensitivity of cash flows 0.89 Estimated d Stock of deposits 0.90 a Call Reports ρ z Persistence of z σ z Standard deviation of z r Unconditional mean of r FRED ρ r Persistence of r 0.88 FRED σ r Standard deviation of r FRED 40

41 Table 4 Choice of derivative positions Simulated data This table shows moments simulated from the model, including statistics about the use of interest rate swaps, the average ratio of total debt to assets ((b t+1 + d)/a t+1 ) and the percentage of interbank debt capacity used. When the bank is a net lender in the interbank market (b t+1 < 0), the percentage of debt capacity used is zero. Column (1) corresponds to the baseline calibration, given in Table 3. Columns (2) to (11) correspond to alternative parameter values, respectively σ z {0.005, 0.05}, ρ z {0.2, 0.85}, σ r {0.004, 0.02}, ρ r {0.1, 0.9} and ρ { 0.6, 0.6}. When changing one parameter, all others are kept at their baseline value. Panel A and B are, respectively, for γ = 0.89 and γ = 0. For each set of parameters, I solve for the policy function and simulate the model for 10,200 periods in which the bank receives stochastic real and interest rate shocks {z t, r t }. The first 200 periods are dropped before moments are calculated. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) σ z ρ z σ r ρ r ρ Baseline Low High Low High Low High Low High Low High Panel A: γ = Frequency of swaps use Frequency of pay-fixed swaps Frequency of pay-float swaps Share of pay-fixed swaps Average total debt/assets Average % debt capacity used Panel B: γ = 0 7 Frequency of swaps use Frequency of pay-fixed swaps Frequency of pay-float swaps Share of pay-fixed swaps Average total debt/assets Average % debt capacity used

42 Table 5 Simulated moments with alternative parameterizations This table shows moments simulated from the model, using alternative parameterizations for the main financial frictions (η 1 and κ) and for the interest rate sensitivity of cash flows (γ). I report statistics about the use of interest rate swaps, the average ratio of total debt to assets ((b t+1 + d)/a t+1 ) and the percentage of interbank debt capacity used. When the bank is a net lender in the interbank market (b t+1 < 0), the percentage of debt capacity used is zero. Columns (1) to (6) correspond respectively to parameter values η 1 {0.039, 0.079}, κ {0.55, 0.92}, and γ {0.8, 0.95}. When changing one parameter, all others are kept at their baseline value (see Table 3). For each set of parameters, I solve for the policy function and simulate the model for 10,200 periods in which the bank receives stochastic real and interest rate shocks {z t, r t }. The first 200 periods are dropped before moments are calculated. (1) (2) (3) (4) (5) (6) η 1 κ γ Low High Low High Low High 1 Frequency of swaps use Frequency of pay-fixed swaps Frequency of pay-float swaps Share of pay-fixed swaps Average total debt/assets Average % debt capacity used

43 Table 6 Shocks and bank leverage US data The table uses data on US banks to study the relationship between the structural properties of shocks to lending opportunities and banks book leverage. In Panel A, I regress book leverage on the volatility σ i and persistence ρ i of loan growth in logs. Book leverage for any bank i at date t is defined as 1 Book equity i,t /Total assets i,t. Denoting Loans i,t = log(loans i,t ) log(loans i,t 1 ), I define σ i and ρ i for any bank i respectively as the standard deviation and first-order autocorrelation of Loans it. In Panel B, I regress book leverage on time-varying measures of lending growth volatility. Denoting Loans i,t,t 8 = log(loans i,t ) log(loans i,t 8 ) (i.e., loan growth over 8 quarters) I use the squared value, the absolute and the signed value of Loans i,t,t 8 as measures of volatility. All regressions in Panel B include bank fixed effects. Standard errors are in parentheses., and denote respectively statistical significance at the 10%, 5% and 1% levels. The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. Panel A: Pooled OLS regressions Dependent variable: Book leverage i,t Loan growth volatility σ i (0.002) (0.002) Loan growth persistence ρ i (0.000) (0.000) Size control No Yes No Yes R N. Obs. 448, , , ,342 Panel B: With bank-fixed effects Dependent variable: Book leverage i,t ( Loans i,t,t 8 ) (0.001) (0.001) Loans i,t,t (0.000) (0.000) Loans i,t,t (0.000) (0.000) Size control No Yes No Yes No Yes Bank FE Yes Yes Yes Yes Yes Yes R N. Obs. 392, , , , , ,770 43

44 Table 7 Shocks and bank net hedging US data The table uses data on US banks to study the relationship between the structural properties of shocks to lending opportunities and banks net hedging. In Panel A, I regress net hedging on the volatility σ i and persistence ρ i of loan growth in logs. Net hedging for bank i at date t is defined in Equation (2). Denoting Loans i,t = log(loans i,t ) log(loans i,t 1 ), I define σ i and ρ i for any bank i respectively as the standard deviation and first-order autocorrelation of Loans it. In Panel B, I regress net hedging on time-varying measures of lending growth volatility. Denoting Loans i,t,t 8 = log(loans i,t ) log(loans i,t 8 ) (i.e., loan growth over 8 quarters) I use the squared value, the absolute and the signed value of Loans i,t,t 8 as measures of volatility. All regressions in Panel B include bank fixed effects. All regressions in Panel B include bank fixed effects. In both panels, control variables include a bank s size (log of total assets) and its ratio of book equity to total assets. I report regression estimates both in the sample of banks with non-zero net hedging, and in the sample including all banks with zero net hedging. Standard errors are in parentheses., and denote respectively statistical significance at the 10%, 5% and 1% levels. The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. Panel A: Pooled OLS regressions Dependent variable: Net hedging i,t Loan growth volatility σ i (0.018) (0.018) (0.001) Loan growth persistence ρ i (0.003) (0.003) (0.000) Controls No Yes Yes No Yes Yes Including zeros No No Yes No No Yes R N. Obs. 7,808 7, ,900 7,710 7, ,667 Panel B: With bank-fixed effects Dependent variable: Net hedging i,t ( Loans i,t,t 8 ) (0.010) (0.010) (0.000) Loans i,t,t (0.005) (0.005) (0.000) Loans i,t,t (0.003) (0.003) (0.000) Controls No Yes Yes No Yes Yes No Yes Yes Including zeros No No Yes No No Yes No No Yes Bank FE Yes Yes Yes Yes Yes Yes Yes Yes Yes R N. Obs. 7,363 7, ,240 7,363 7, ,240 7,363 7, ,240 44

45 Table 8 Financial constraints, financing needs, and net hedging US data The table uses data on US banks to study the relationship between financial constraints, financing needs and net hedging. Net hedging for bank i at date t is defined in Equation (2). In Panel A, I regress net hedging on four lagged measures of financial constraints: size (log of total assets), book equity over total assets, net income over total assets, and cash dividends over total assets. In Panel B, I test whether banks actively take pay-fixed swap positions in quarters when the increase leverage. The dependent variable is a dummy variable that takes value one if a bank takes a pay-fixed position in quarter t. A bank is considered to take a pay-fixed position in quarter t if net hedging increases in value, while gross hedging is also increasing. Increases in leverage are measures using the change in book leverage (i.e., 1 Book equity i,t /Total assets i,t ) between t and t + 1. Standard errors are in parentheses., and denote respectively statistical significance at the 10%, 5% and 1% levels. The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. Panel A: Financial constraints and net hedging Size i,t (0.001) (0.000) Book equity i,t Dependent variable: Net hedging i,t (0.019) (0.000) Net income i,t (0.062) (0.000) Cash dividends i,t (0.116) (0.003) Including zeros No Yes No Yes No Yes No Yes R N. Obs. 7, ,536 7, ,293 7, ,128 7, ,549 Panel B: Financing needs and net hedging Dependent variable: Dummy variable Take pay-fixed position at t Book leverage i,t (0.273) (0.276) (0.268) (0.271) Including zeros No No No No Bank FE No No Yes Yes Time FE No Yes No Yes R N. Obs. 7,935 7,935 7,935 7,935 45

46 Table 9 Net worth and bank hedging Simulated data This table uses data simulated from the model to study the relationship between bank net worth and hedging with interest rate derivatives. The decision to hedge is measured using a dummy variable equal to one if the bank uses non-zero swaps. The extent of hedging is measured by the absolute value of swaps taken, s t+1, normalized by a t+1. To study the decision to hedge, I estimate a probit model and a model with bank fixed-effects. To study the extent of hedging, I estimate a pooled OLS, a Tobit model and a model with bank fixed-effects. Panels A and B are respectively for γ = 0.89 and γ = 0. For each of these values, I solve for the policy function and simulate a panel of 1,000 banks. Each has a mean realization of the real factor z i drawn from a normal distribution with mean zero and standard deviation 0.1. Each bank is simulated for 300 periods in which it receives aggregate shocks { z t, r t }. The first 200 periods are dropped for each bank before the regression coefficients are estimated. Heteroskedasticity-consistent standard errors are in parentheses., and denote respectively statistical significance at the 10%, 5% and 1% levels. (1) (2) (3) (4) (5) Panel A: γ = 0.89 Decision to hedge i,t Extent of hedging i,t Probit Bank FE OLS Tobit Bank FE Net worth w i,t (0.0000) (0.0000) (0.0231) (0.0345) (0.0246) R N. Obs. 100, , , , ,000 Panel B: γ = 0 Decision to hedge i,t Extent of hedging i,t Probit Bank FE OLS Tobit Bank FE Net worth w i,t (0.0000) (0.0000) (0.0199) (0.0214) (0.0200) R N. Obs. 100, , , , ,000 46

47 Figure 1 Net hedging of US commercial banks This figure plots the distribution of net hedging for US commercial banks. Net hedging is defined in Equation (2). In each quarter, we represent the cross-sectional distribution of net hedging using a box plot. In each box plot, the horizontal dash represents the median and the diamond represents the mean. The whiskers represent the 5th and 95th percentiles. The gray rectangle represents the 25th and 75th percentiles. A positive (resp. negative) value of net hedging indicates a net pay-fixed (resp. pay-float) position at the bank level. The sample period is from 1997Q2 to 2013Q4. See Appendix A for details on the data. 47