Monetary Policy, Leverage, and Bank Risk-Taking

Transcription

1 Monetary Poliy, Leverage, and Bank Risk-Taking Giovanni Dell Ariia IMF and CEPR Lu Laeven IMF and CEPR Otober 200 Robert Maruez Boston University Abstrat We provide a theoretial foundation for the laim that prolonged periods of easy monetary onditions inrease bank risk taking. The net e et of a monetary poliy hange on bank monitoring (an inverse measure of risk taking) depends on the balane of three fores: interest rate pass-through, risk shifting, and leverage. When banks an adjust their apital strutures, a monetary easing leads to greater leverage and lower monitoring. However, if a bank s apital struture is xed, the balane depends on the degree of bank apitalization: when faing a poliy rate ut, well apitalized banks derease monitoring, while highly levered banks inrease it. Further, the balane of these e ets depends on the struture and ontestability of the banking industry, and is therefore likely to vary aross ountries and over time. The views expressed in this paper are those of the authors and do not neessarily represent those of the IMF. We thank Olivier Blanhard, Stijn Claessens, Gianni De Niolo, Hans Degryse, Kenihi Ueda, Fabian Valenia, and seminar partiipants at Boston University, Harvard Business Shool, Tilburg University, the Duth Central Bank, and the IMF for useful omments and disussions. Adess for orrespondene: Giovanni Dell Ariia, IMF, 700 9th Street NW, Washington, DC, USA. gdellariia@imf.org

2 Introdution The reent global nanial risis has brought the relationship between monetary poliy and bank risk taking to the forefront of the eonomi poliy debate. Many observers have blamed loose monetary poliy for the redit boom and the ensuing risis in the late 2000s, arguing that, in the run up to the risis, low interest rates and abundant liuidity led nanial intermediaries to take exessive risks by fueling asset pries and promoting leverage. The argument is that had monetary authorities raised interest rates earlier and more aggressively, the onseuenes of the bust would have been muh less severe. More reently, a related debate has been raging on whether ontinued exeptionally low interest rates are setting the stage for the next nanial risis. Fair or not, these laims have beome inreasingly popular both in aademia and in the business press. Surprisingly, however, the theoretial foundations for these laims have not been muh studied and hene are not well understood. Maroeonomi models have typially foused on the uantity rather than the uality of redit (e.g. the literature on the bank lending hannel) and have mostly abstrated from the notion of risk. Papers that onsider risk (e.g., nanial aelerator models in the spirit of Bernanke and Gertler, 989) explore primarily how hanges in the stane of monetary poliy a ets the riskiness of borrowers rather than the risk attitude of the banking system. In ontrast, exessive risk-taking by nanial intermediaries operating under limited liability and asymmetri information has been the fous of a large banking literature whih, however, has largely ignored monetary poliy. 2 This paper is an attempt to ll this gap. We develop a model of nanial intermediation where banks an engage in ostly monitoring to redue the redit risk in their loan portfolios. Monitoring e ort and the priing (i.e., interest rates) of bank assets and liabilities are endogenously determined and, in euilibrium, depend on a benhmark monetary poliy rate. We obtain three main ndings. First, for the ase where a bank s apital struture is xed exogenously, we nd that the e ets of monetary poliy hanges on bank monitoring and, hene, portfolio risk ritially depend on a bank s leverage: a monetary easing will lead highly apitalized banks to monitor less, while the opposite is true for poorly apitalized banks. See, for example, Rajan (200), Taylor (2009), or Borio and Zhu (2008). 2 Diamond and Rajan (2009) and Farhi and Tirole (2009) are reent exeptions, although these deal with the e ets of expetations of a maro bailout rather than the impliations of the monetary stane. Reviews of the older literature are in Boot and Greenbaum (993), Bhattaharya, Boot, and Thakor (998), and Carletti (2008).

3 We then endogenize banks apital strutures by allowing them to adjust their apital holdings in response to monetary poliy hanges. For this ase we nd that a ut in the poliy rate will lead banks to inrease their leverage. Re eting this inrease in leverage, our third main nding is that one leverage is allowed to be optimally hosen, a poliy rate ut will unambiguously lower bank monitoring and inrease risk taking. These results are onsistent with the evidene olleted by a growing empirial literature on the e ets of monetary poliy on risk-taking (see, for example, Maddaloni and Peyó, 200 and Ioannidou et al., 2009; Setion 2 gives a brief survey). A negative relationship between bank risk and the real poliy rate is also evident in data from the U.S. Terms of Business Lending Survey, as graphially illustrated in Figure??. In this gure, bank risk is measured using the weighted average internal risk rating assigned to loans by banks from the U.S. Terms of Business Lending Survey 3 and the real poliy rate is measured using the nominal federal funds rate adjusted for onsumer prie in ation. 4 Both variables are detrended by deduting their linear time trend and we use uarterly data from the seond uarter of 997 until the fourth uarter of Our model is based on two standard assumptions. First, banks are proteted by limited liability and hoose the degree to whih to monitor their borrowers or, euivalently, hoose the riskiness of their portfolios. Sine monitoring e ort is not observable, a bank s apital struture has a bearing on its risk-taking behavior. Seond, monetary poliy a ets the ost of a bank s liabilities through hanges in the risk-free rate. Under these two assumptions, we show that the balane of three oexisting fores - interest-rate pass-through, risk shifting, and leverage - determines how monetary poliy hanges a et a bank s risk taking. The rst is a pass-through e et that ats through the asset side of a bank s balane sheet. In our model, monetary easing redues the poliy rate, whih is then re eted in a redution of the interest rate on bank loans. This, in turn, redues the bank s gross return onditional on 3 The U.S. Terms of Business Lending Survey, whih is a uarterly survey on the terms of business lending of a strati ed sample of about 400 banks onduted by the U.S. Federal Reserve Bank. The survey asks partiipating banks about the terms of all ommerial and industrial loans issued during the rst full business week of the middle month in every uarter. The publily available version of this survey enompasses an aggregate version of the terms of business lending, disaggregated by type of banks. Loan risk ratings vary from to 5 and are inreasing in risk. We use the weighted average risk rating sore aggregate aross all partiipating banks as measure of bank risk. 4 The e etive federal funds rate is a volume-weighted average of rates on trades arranged by major brokers and alulated daily by the Federal Reserve Bank of New York using data provided by the brokers. As in ation rate we use the three-month average hange in the U.S. onsumer prie index. 2

4 Figure : U.S. bank risk and the real federal funds rate Risk of loans (detrended) Real Federal Funds Rate (detrended) (in %) its portfolio repaying, reduing the inentive for the bank to monitor. This e et is akin to the portfolio realloation e et present in portfolio hoie models. In these models, when monetary easing redues the real yield on safe assets, banks will typially inrease their demand for risky assets. 5 Seond, there is a standard risk-shifting e et that operates through the liability side of a bank s balane sheet. Monetary easing lower the osts of a bank s liabilities. Everything else eual, this inreases a bank s pro t when it sueeds and thus reates an inentive to limit risk taking in order to reap those gains. The extent of this e et, however, depends ritially on the degree of limited liability protetion a orded to the bank. 6 To see why, onsider a fully leveraged bank that is naned entirely through deposits/debt. Under limited liability, this bank will su er no losses in ase of failure. A poliy rate ut will inrease the bank s expeted net return on all assets by lowering the rate it has to pay on deposits. The bank an maximize this e et by reduing the risk of its portfolio, hoosing a safer portfolios for whih there is a higher probability the bank will have to repay depositors. In ontrast, for a bank fully funded by apital, the e et of a derease in the 5 The exeption would be banks with dereasing absolute risk aversion who, instead, would derease their holdings of risky assets (Fishburn and Porter, 976). 6 This is similar to what happens in models that study the e ets of ompetition for deposits on bank stability (Hellmann, Murdoh, and Stiglitz, 2000, Matutes and Vives, 2000, Cordella and Levy-Yeyati, 2003). 3

5 ost of its liabilities will, all other things eual, inrease the expeted net return uniformly aross portfolios and have little or no e et on the bank s risk hoies. When banks apital strutures are exogenously determined, the net e et of a monetary poliy hange on bank monitoring depends on the balane of these two e ets. This, in turn, depends on a bank s apital struture as well as the struture of the market in whih it operates. The risk-shifting e et is stronger the more bene ial is the limited liability protetion to the bank. This e et is therefore greatest for fully leveraged banks, and is lowest for banks with zero leverage who as a result have no limited liability protetion. In ontrast, the magnitude of the pass-through e et depends on how poliy rate hanges are re eted in hanges to lending rates. Thus, the magnitude of this e et depends on the market struture of the banking industry: it is minimal in the ase of a monopolist faing an inelasti demand funtion, when the pass-through onto the lending rate is zero; and it is maximal in the ase of perfet ompetition, when lending rates fully re et poliy rate hanges. It follows that the net e et of a monetary poliy hange may not be uniform aross times, banking systems or individual banks. Following a poliy rate ut, monitoring will derease when leverage is low and inrease when leverage is high. The position of this threshold level of leverage will, in turn, depend on the market struture of the banking industry. By ontrast, a third fore omes into play one we allow banks to optimally modify their apital struture in response to a monetary poliy hange. On the one hand, banks have an inentive to be levered sine holding apital is ostly. On the other hand, apital serves as a ommitment devie to limit risk taking and helps redue the ost of debt and deposits. Banks with limited liability tend to take exessive risk sine they do not internalize the losses they impose on depositors and bondholders. Bank apital redues this ageny problem: the more the bank has to lose in ase of failure, the more it will monitor its portfolio and invest more prudently. When investors annot observe a bank s monitoring but an only infer its euilibrium behavior, higher apital (i.e., lower leverage) will lower their expetations of a bank s risk-taking and, thus, redue the bank s ost of deposits and debt. Given that a poliy rate ut redues the ageny problem assoiated with limited liability, it follows that the bene t from holding apital will also be redued. In euilibrium, therefore, lower poliy rates will be assoiated with greater leverage. This result provides a simple miro-foundation for the empirial regularities doumented in reent papers, suh as in Aian and 4

6 Shin (2009). The addition of this leverage e et tilts the balane of the other two e ets: all else eual, more leverage means more risk taking. Our model s unambiguous predition when banks apital strutures are endogenous is onsistent with the laim that monetary easing leads to greater risk taking. Our ontribution to the existing literature is twofold. First, we provide a model that isolates the e et of monetary poliy hanges on bank risk tasking independently of other maroeonomi onsiderations related to asset values, liuidity provision, et. The model provides a theoretial foundation for some of the regularities reently doumented in the empirial literature, inluding the inverse relationship between monetary onditions and leverage, and the tendeny for banks to load up on risk during extended periods of loose monetary poliy. While our treatment of monetary poliy is obviously minimal (we take monetary poliy as exogenous and abstrat from other e ets linked to the maroeonomi yle), our paper an help bridge the gap between maroeonomi and banking models. Seond, our framework an help reonile the somewhat dihotomous preditions of two important strands of researh: the literature on the ight to uality and that on risk shifting linked to limited liability. The paper also ontributes to the ongoing poliy debate on whether maroprudential tools should omplement monetary poliy to safeguard maro nanial stability. We disuss this issue further in the onluding setion. The paper proeeds as follows: Setion 2 presents a brief survey of related theoretial and empirial work. Setion 3 introdues the model and examines the euilibrium when the bank apital struture is exogenous. Setion 4 solves the endogenous apital struture ase. Setion 5 examines the role of market struture, while Setions 6 and?? present some numerial examples and stylized fats. Setion 7 onludes. Proofs are mostly relegated to the appendix. 2 Related Literature Our paper is related to a well established literature studying the e ets of monetary poliy hanges on redit markets. The literature on nanial aelerators posits that monetary poliy tightening leads to more severe ageny problems by depressing borrowers net worth (see models in the spirit of Bernanke and Gertler, 989, and Bernanke et al., 996). The result is a ight to uality: rms more a eted by ageny problems will nd it harder to obtain external naning. However, this 5

7 says little about the riskiness of the marginal borrower that obtains naning beause monetary tightening inreases ageny problems aross the board, not just for rms that are intrinsially more a eted by ageny problems. Thakor (996) fouses on the uantity rather than the uality of redit. Yet, his model has impliations for bank risk taking. In Thakor (996), banks an invest in government seurities or extend loans to risky entrepreneurs. The impat of monetary poliy on the uantity of bank redit and thus on the riskiness of the bank portfolio depends on its relative e et on the bank intermediation margin on loans and seurities. While the impat on portfolio risk is not expliitly studied, if monetary easing were to redue the rate on seurities more than that on deposits, the opportunity ost of extending loans would fall and the portion of a bank s portfolio invested in loans would inrease; otherwise, the opposite would happen. Rajan (2005) identi es in the searh for yield a related, but di erent, mehanism through whih monetary poliy hanges may a et risk taking. He argues that nanial institutions may be indued to swith to riskier assets when a monetary poliy easing lowers the yield on their short-term assets relative to that on their long-term liabilities. This is a result of limited liability. If yields on safe assets remain low for a prolonged period, ontinued investment in safe assets will mean that a nanial institution will need to default on its long-term ommitments. A swith to riskier assets (and higher yields) may inrease the probability that it will be able to math its obligations. Dell Ariia and Maruez (2006a) nd that when banks fae an adverse seletion problem in seleting borrowers, monetary poliy easing may lead to a redit boom and lower lending standards. This is beause banks inentives to sreen out bad borrowers are redued when their ost of funds is lowered. More reently, Farhi and Tirole (2009) and Diamond and Rajan (2009) have examined the role of maro bailouts and olletive moral hazard on banks liuidity deisions. When banks expet a strong poliy response by the monetary authorities should a large negative shok our (a mehanism often referred to as the Greenspan put ), they will tend to take on exessive liuidity risk. This behavior, in turn, will inrease the likelihood that the entral bank will indeed respond to a shok by providing the neessary liuidity to the banking system. Unlike in this paper, their fous is on the reation funtion of the entral bank (the poliy regime) rather than on the poliy stane. Agur and Demertzis (200) present a redued form model of bank risk taking to fous on 6

8 how monetary poliymakers should balane the objetives of prie stability and nanial stability. Drees et al. (200) nd that the relationship between the poliy rate and risk taking depends on whether the primary soure of risk is the opaueness of a seurity or the idiosynrati risk of the underlying investment. Our paper also relates to a large theoretial literature examining the e ets of limited liability, leverage, and deposit rates on bank risk taking behavior. Several papers (for example, Matutes and Vives, 2000, Hellmann, Murdoh, and Stiglitz, 2000, Cordella and Levy-Yeyati, 2000, Repullo, 2004, and Boyd and De Niolo, 2005) have foused on how ompetition for deposits (i.e., higher deposit rates) exaerbates the ageny problem assoiated with limited liability and may ine iently inrease bank risk taking. 7 This e et is similar to the risk-shifting e et identi ed in this paper: more ompetition for deposits inreases the euilibrium deposit rate, ompressing intermediation margins and thus reduing a bank s inentives to invest in safe assets. The framework we use is based on Dell Ariia and Maruez (2006b) and Allen, Carletti, and Maruez (200). In partiular, the latter shows how banks may hoose to hold ostly apital to redue the premium demanded by depositors. They, however, ignore the e ets of monetary poliy and do not examine how leverage moves in response to poliy rate hanges. Our result that leverage is dereasing in the poliy rate is also related to that in Aian and Shin (2008). In their paper, leverage is limited by the moral hazard indued by the underlying risks in the environment. In our model, an inrease in the poliy rate exaerbates the ageny problem assoiated with limited liability, whih in turn leads to a redution in leverage. Finally, there is a small, but growing, empirial literature that links monetary poliy and bank risk taking. For example, Lown and Morgan (2006) show that redit standards in the U.S. tend to tighten following a monetary ontration. Similarly, Maddaloni and Peyó (200) nd that redit standards tend to loosen when overnight rates are lowered. Moreover, using Taylor rule residuals, they nd that holding rates low for prolonged periods of time softens lending standards even further. Similarly, Altunbas et al. (200) nd evidene that unusually low interest rates over an extended period of time ontributed to an inrease in banks risk-taking. Jimenez et al. (2008) and Ioannidou, Ongena, and Peyó (2009) use detailed information on borrower uality 7 Boyd and De Niolo (2005) also show that when moral hazard on the borrowers side is taken into aount, the result may be reversed. 7

9 from redit registry databases for Europe and Bolivia. They nd a positive assoiation between low interest rates at loan origination and the probability of extending loans to borrowers with bad or no redit histories (i.e., risky borrowers). 3 A Simple Model of Bank Risk Taking Banks fae a negatively sloped demand funtion for loans, L(r L ), where r L is the gross interest rate the bank harges on loans. We assume for tratability that the demand funtion is linear, L = A br L. In setion 5, we examine the impat of alternative market strutures. 8 Loans are risky and a bank s portfolio needs to be monitored to inrease the probability of repayment. The bank is endowed with a monitoring tehnology, allowing the bank to exert monitoring e ort whih also represents the probability of loan repayment. This monitoring e ort entails a ost eual to 2 2 per dollar lent. 9 An alternative interpretation of this assumption is that banks have aess to a ontinuum of portfolios haraterized by a parameter 2 [0; ], with returns r L 2 and probability of suess.0 Banks fund themselves with two di erent types of liabilities. A portion k of a bank s liabilities represents a ost irrespetive of the bank s pro t, while a portion k is repaid only when the bank sueeds. Consistent with other existing models, k an represent the portion of bank assets naned with bank euity or apital. In this ase, k would be interpreted as the fration of the bank s portfolio naned by deposits. However, k an be also interpreted more generally as an inverse measure of the degree of limited liability protetion aorded to banks. For now, we treat k as exogenous. In Setion 4, we examine the ase where banks an adjust k in response to a hange in monetary poliy. For simpliity, we assume that the deposit rate is xed and eual to the poliy rate, r D = r. (We will relax this assumption later.) This is onsistent with the existene of deposit insurane, 8 The assumption of a downward sloping demand urve for loans is supported by broad empirial evidene (e.g., Den Haan, Sumner, and Yamashiro, 2007). More generally, the pass-through will depend on the ost struture of bank liabilities, inluding the proportion of retail versus wholesale deposits (Flannery, 982). Berlin and Mester (999) show that markups on loans derease as market rates inrease, implying that as market rates inrease, there is less than a one for one inrease in loan rates. 9 For a model in the same spirit but where banks hoose among portfolios with di erent risk/return harateristis, see Cordella and Levy-Yeyati (2003). 0 This latter interpretation orrespond to the lassi risk shifting problem between bondholders and shareholders, in that shareholder an hoose between investments that have a lower probability of suess, but that payo more onditional on suess. 8

10 for instane. Euity, however, is more ostly, with a yield r E = r +, with 0, whih is onsistent with an euity premium as a spread over the risk-free rate. Alternatively, the ost r E an be interpreted as the opportunity ost for shareholders of investing in the bank. We struture the model in two stages. For a xed poliy rate r, in stage banks hoose the interest rate to harge on loans, r L. In the seond stage, banks then hoose how muh to monitor their portfolio,. 3. Euilibrium when Leverage is Exogenous We solve the model by bakward indution, starting from the last stage. The bank s expeted pro t an be written as: = (r L r D ( k)) r E k 2 2 L(r L ); () whih re ets the fat that the bank s portfolio repays with probability. When the bank sueeds, it reeives a per-loan payment of r L and earns a return r L r D ( k) after repaying depositors. When it fails, it reeives no revenue, but, beause of limited liability, does not need to repay depositors. The term r E k represents the ost of euity to the bank or, euivalently, the opportunity ost of bank shareholders, whih is borne irrespetive of the bank s revenue. Taking the loan rate r L as given, the rst order ondition for bank monitoring an be written as whih (r L r D ( k)) r E k 2 2 L(r L ) = rl r D ( k) b = min ; : (2) Sine r D = r, we obtain immediately from (2) that the diret (i.e., for a given lending rate) e et of a poliy rate hike on bank monitoring 0. This is onsistent with most of the literature on the e ets of deposit ompetition on risk taking (see for example Hellmann et al., 2000). One way to interpret this result is that the short-term inentives banks with severe maturity mismathes have to monitor will be redued by an unexpeted inrease in the poliy rate. We assume that the premium on euity,, is independent of the poliy rate r. This is onsistent with our goal to isolate the e et of an exogenous hange in the stane of monetary poliy. However, from an asset priing perspetive these are likely to be orrelated through underlying ommon fators whih may ive the risk premium as well as the risk free rate. Our results ontinue to hold as long as the within period orrelation between and r is su iently di erent from (positive) one. 9

11 We an now solve the rst stage of the model where banks hoose the loan interest rate. Assuming that an interior solution exists, we substitute b into the expeted pro t funtion and obtain: 2 (b) = (r L r D ( k)) 2 2 r E k! L(r L ): (3) Maximizing (3) with respet to the loan rate yields the following rst order = L (r L ) r L r D ( k) (r L) (r L r D ( k)) 2 2 r E (r L) = 0: (4) From (4) we obtain our rst result. Proposition There exist a degree of apitalization, e k, suh that, for k < e k, bank monitoring dereases with the monetary poliy rate, db db > 0. < 0, while for k > e k it inreases with the poliy rate, The intuition behind this result is that a tightening of monetary poliy leads to an inrease in both the interest rate a bank harges on its loans and that it pays on its liabilities. The rst e et, whih re ets the pass-through of the poliy rate on loan rates, inreases the inentives to monitor. The seond e et, the risk-shifting e et, dereases monitoring inentives to the extent that it applies to liabilities that are repaid only in ase of suess. Indeed, from 2 is evident that an inrease in the ost of apital a ets the bank s monitoring e ort only through its e et on the lending rate. Thus, for a bank funded entirely through apital, the risk-shifting e et disappears. In ontrast, an inrease in the interest rate on deposits will also have a diret negative impat on b: In addition, a tightening of monetary poliy leads to a ompression of the intermediation margins, r L r D. Thus, for a bank entirely funded with deposits, the risk-shifting e et will dominate. In between the two extremes of full leverage or zero leverage, the bank s apital struture determines the net e et of a monetary poliy hange on risk taking. Banks with a higher leverage ratio will reat to a monetary poliy tightening by taking on more risk, while those with a lower leverage ratio will do the opposite. 2 It is straightforward to see that there always exist values of that guarantee an interior solution for. Later, we demonstrate numerially that an interior solution to the full model, where also bank leverage (k) is endogenous, exists. In other words, there is a wide range of parameter values for whih the rst order onditions haraterize the euilibrium. 0

12 It is worth noting that the results so far are obtained under the assumption that the priing of deposits is insensitive to risk (i.e., ), but does re et the underlying poliy rate r. This would be onsistent with the existene of deposit insurane, so that depositors are not onerned about being repaid by the bank, but nevertheless want to reeive a return that ompensates them for their opportunity ost, whih would be inorporated in the poliy rate r. 3 In what follows, we show that the result in Proposition is not iven by depositors insensitivity to risk, but rather by the bank s optimizing behavior given its desire to maximize its expeted return, whih inorporates not only the return onditional on suess but also the probability of suess. Assume now that depositors must be ompensated for the bank s expeted risk taking. Depositors annot diretly observe. However, from observing the apital ratio k they an infer the bank s euilibrium monitoring behavior, b. Given an opportunity ost of r, depositors will demand a promised repayment r D suh that r D E[jk] = r, or in other words r D = r E[jk]. The timing is as before, with the additional onstraint that depositors expetations about bank monitoring, E[jk], must in euilibrium be orret, so that E[jk] = b(r D jk). It is worth noting that this introdues an inentive for the bank to hold some apital. Euity is more expensive than deposits (or debt), but it allows the bank to ommit to a higher and thus redues the yield investors demand on instruments subjet to limited liability (i.e. debt or deposits). We exploit this aspet further in the next setion. We an now state the following result, whih parallels that in Proposition. Proposition 2 Suppose that depositors reuire ompensation for risk, so that r D = r E[jk]. Then there exist a degree of apitalization, e k, suh that, for k < e k, bank monitoring dereases with the monetary poliy rate, db 4 Endogenous Capital Struture < 0, while for k > e k it inreases with the poliy rate, db > 0. So far, we have assumed that the bank s degree of leverage or apitalization is exogenous. This setting ould apply, for instane, to the ase of individual banks that would optimally like to hoose a level of apital below some regulatory minimum. For suh banks, hanges in the poliy 3 Keeley (990) formally shows that when deposits are fully proteted by deposit insurane, the supply of deposits will not depend on bank risk.

13 rate would not be re eted in their apitalization deisions sine the regulatory onstraint would be binding. In this setion, we extend the model to allow for an endogenous apital struture and ontrast our results with those above for the ase of exogenous leverage. As apital struture will be endogenous, we adopt the framework introdued at the end of the previous setion and allow unseured investors to demand ompensation for the risk they expet to fae (in other words, we eliminate deposit insurane). 4 Spei ally, onsider the following extension to the model. At stage, banks hoose their desired apitalization ratio k. At stage 2, unseured investors observe the bank s hoie of k and set the interest rate they harge on the bank s liabilities. The last two stages are as before in that banks hoose the lending interest rate and then the extent of monitoring. 4. Euilibrium As before, we solve the model by bakward indution. The solutions for the last two stages are analogous to those in the previous setion. At stage three, unseured investors will demand a promised return of r D = r E[jk]. As we show below, this provides the bank with an inentive to hold some apital to redue the ost of borrowing. Formally, the objetive funtion is to maximize bank pro ts with respet to the apital ratio k: max = k b(br L r D ( k)) r E k 2 b2 L(br L ); subjet to r D = r E[jk] ; where b = b(r L ; k) is the euilibrium hoie of monitoring indued by the bank s hoie of the loan rate r L and apitalization ratio k, and br L = br L (k) is the optimal loan rate given k. In other words, the bank takes into aount the in uene of its hoie of k on its subseuent loan priing and monitoring deisions. The rst order ondition for k an be expressed as d dk L d dk = 0 4 In pratie, it may be more realisti to assume that some fration of bank liabilities are insured or insensitive to risk, while the remaining fration are uninsured so that their priing must re et the expeted amount of risk, suh as for subordinated debt. Allowing for these two kinds of liabilities in no way a ets our results, as we illustrate in Setion 6, where we inorporate both insured and ininsured deposits into our numerial examples. 2

14 sine the last two terms are zero from the envelope theorem. Substituting, this beomes d dk = (r L (r E r ) L(r L ) = 0; whih haraterizes the bank s optimal hoie of b k. As we show below in Proposition 3, b k is stritly positive for a broad range of parameter values. We an now use this to establish the following result. Proposition 3 Euilibrium bank leverage dereases with the monetary poliy rate: d b k > 0. The proposition establishes that, when an internal solution b k for the apitalization ratio exists, then b k will be inreasing in r. Put di erently, a low monetary poliy rate will indue banks to be more leveraged (i.e., to hold less apital). A poliy rate hike inreases the rate the bank has to pay on its debt liabilities and exaerbates the bank s ageny problem - note that at r = 0, a limit ase where the prinipal is not repaid at all, there is no moral hazard and b =. This e et is essentially the same as in the ight-to-uality literature (see for example Bernanke et al., 989). It follows that as the poliy rate inreases so does the bene t from holding apital, the only ommitment devie available to the bank to redue moral hazard. Put di erently, investors will allow banks to be more levered when the poliy rate is low relative to when it is high. A similar result is in Aian and Shin (2008), where leverage is a dereasing funtion of the moral hazard indued by the underlying risks in the environment, and evidene of this behavior is doumented in Aian and Shin (2009). The following result haraterizes banks loan priing deisions as a funtion of the monetary poliy rate, and will be useful in establishing the next main result. Lemma When bank leverage, the loan rate, and the level of monitoring are all optimally hosen with respet to the monetary poliy rate r, the optimal loan rate br L is inreasing in r : dbr L. The intuition for the lemma is straightforward: when the monetary poliy rate inreases, this raises the opportunity ost on all forms of naning. Conseuently, in euilibrium the rate that the bank harges on any loans also inreases. In other words, there is at least some pass through of the hanges in the bank s osts of funds onto the prie of (bank) redit, whih is re eted in a higher loan rate. 3

15 We an now state our next main result: Proposition 4 When bank leverage is optimally hosen to maximize pro ts, monitoring will always inrease with the monetary poliy rate: db > 0. In ontrast to the result in Proposition, when bank leverage is endogenous we have that bank monitoring always inreases when the poliy rate r inreases. Relative to the ase where leverage is exogenous, here monetary poliy tightening a ets bank monitoring through the additional hannel of a derease in leverage, as per Proposition 3. Proposition 4 omplements this result along the dimension of bank monitoring, so that the aggregate e et of an inrease in the monetary poliy rate is for banks to be less levered and to take less risk (i.e., monitor more). Conversely, redutions in r that would aompany monetary easing should lead to more highly levered banks and redued monitoring e ort. It bears emphasizing that the lear ut e et of a hange in the monetary poliy rate arises only when banks are able to adjust their apital strutures (i.e., k) in response to hanges in r. Changes in bank leverage are, therefore, an important additional hannel through whih hanges in monetary poliy a et bank behavior. Moreover, Proposition 4 shows that the leverage e et an be su iently strong to overturn the diret e et on bank risk taking identi ed in Proposition 2 for the ase where leverage is exogenously given. At the same time, to the extent that some banks may be onstrained by regulation from adjusting their apital strutures (for instane, if their optimal apital holdings are below the minimum mandated by apital adeuay regulation), we may in pratie observe ross setional di erenes in banks reations to monetary poliy shoks. 5 Extension: The role of market struture This setion examines the e et of alternative assumptions on the struture of the loan market. We look at two diametrially opposed ases: First, a perfetly ompetitive redit market, where banks take the lending rate as given, whih is determined by market learing and a zero pro t ondition for the banks; and seond, a monopolist faing a loan demand funtion that is perfetly inelasti up to some xed loan rate R. This upper limit an be interpreted as either the maximum return on projets, or as the highest rate onsistent with borrowers satisfying their reservation utilities. 4

16 Under these two extreme strutures, we show that our results when leverage is endogenous ontinue to hold ualitatively. Spei ally, when the apital ratio k is endogenously determined, the leverage e et dominates and monetary easing will inrease bank risk taking. If banks are unable to adjust their apital strutures, however, the loan market struture does matter for how monetary poliy a ets risk taking. Intuitively, the pass-through of the monetary poliy rate on lending rates is higher the more ompetitive is the market. It follows that intermediation margins are less sensitive to monetary poliy hanges in more ompetitive markets. And this, in turn, results in a diminished risk shifting e et and onseuently a smaller region of leverage for whih monetary easing auses risk taking to derease. 5. The Perfet Competition Case Consider the following modi ation of our model to inorporate perfet ompetition. At stage, given a xed poliy rate, the lending rate is set ompetitively so that banks make zero expeted pro ts in euilibrium. At stage 2, banks hoose their desired leverage (or apitalization) ratio k. At stage 3, unseured investors observe the bank s hoie of k and set the interest rate they harge on the bank s liabilities, r D. And in the last stages, as before, banks hoose the extent of monitoring. Again, we solve the model by bakward indution. As for the ase where banks have market power analyzed in Setions 3 and 4, solving for the euilibrium monitoring and imposing r D = implies, as before r E[jk] b = r L + rl 2 4r ( k) : (5) 2 We rst onsider the ase where k is exogenous. For this ase, we impose a zero pro t ondition, b = L br L r k 2 b2 = 0; to obtain r L as a funtion of r and k. We an now state the following result. Proposition 5 In a perfetly ompetitive market, for a xed apitalization ratio k, bank monitoring inreases with the monetary poliy rate, db > 0, for k 2 (0; ], with db = 0 for k = 0. This result ontrasts with that obtained in Propositions and 2 for the ase where banks have market power. There, the e et of a hange in monetary poliy on risk taking depended on the 5

17 degree of bank apitalization, k, with dereased risk taking as the monetary poliy rate inreases for a su iently low level of k. Here db, the bank s response to hanges in monetary poliy in terms of monitoring, remains non-negative over its entire range, although it is still inreasing in k. This result stems from the fat that the pass-through of the poliy rate onto the loan rate is maximum in the ase of perfet ompetition, and must perfetly re et the inrease in the poliy rate. It follows that the pass-through e et dominates the risk-shifting e et, so that the region where db < 0 disappears. We next endogenize the apital ratio k, as in Setion 4. Banks maximize max = L br L r k k whih for the ase of perfet ompetition gives b k = r 2 L (r + ) r (r + 2) 2 : 2 b2 ; To obtain the lending rate, we impose the zero pro t ondition for banks: b = L b( b k)r L r b k 2 b2 ( b k) = 0: (6) r 2r (r +2) 2 3r +(r ) , From (6) we an solve for the euilibrium lending rate, apital, and monitoring as: br L = r b k = r +(r ) 2, and b =. From these we immediately obtain the following 3r +(r ) result. r 4(r +) 2 2(3r +(r ) ) Proposition 6 In a perfetly ompetitive market, euilibrium bank leverage dereases with the monetary poliy rate: d b k > 0. And, when bank leverage is optimally hosen to maximize pro ts, monitoring will always inrease with the monetary poliy rate: db > 0. This result extends Propositions 3 and 4 to the ase of perfet ompetition and establishes that even when redit markets are perfetly ompetitive, monetary easing in euilibrium lead banks to both hold less apital and take on more risk one one inorporates banks ability to adjust their optimal leverage ratios. 5.2 A Monopolist Faing Inelasti Demand Here, we assume as in the main part of the paper that banks an hoose the interest rate to harge, but also that there is a xed demand for loans, L, as long as the lending rate does not exeed a 6

18 xed value of R. This setting an be interpreted as one where eah borrower has a unit demand for loans and R is the borrower s reservation loan rate. Demand beomes zero for r L > R. This eliminates any priing e ets on loan uantity and allows us to fous on a ase where the loan rate is not responsive to hanges in the ost of funding sine, given the xed, inelasti demand, it will always be set at the maximum value of br L = R. We an solve for b, imposing the ondition that r D = from whih we an state the following laim. r E[jk], and obtain b = R + p R 2 4r ( k) ; (7) 2 Claim For k 2 [0; ) xed, a monopolist bank faing a demand funtion that is perfetly inelasti db for r L R will always derease monitoring when the poliy rate is raised: < 0. For k =, k db = 0. k Proof: From 7 we an immediately write: db = k p R 2 4r ( k) < 0: This result stands in stark ontrast with what we obtained in Proposition 5 for the ase of perfet ompetition when leverage is exogenous. There, irrespetive of the level of leverage, risk taking was always dereasing in the poliy rate. Here, risk taking is always inreasing in the poliy rate. The di erene stems preisely from the extent to whih the bank passes onto the lending interest rate hanges in its osts stemming from hanges in the poliy rate. If demand is inelasti, the pass-through is zero as the lending rate is always held at its maximum, R, and thus annot adjust further when the monetary poliy rate hanges. Therefore, the impat of a hange in the poliy rate on monitoring, b, operates solely through the liability side of the bank balane sheet, reduing the bank s return in ase of suess and leading it to monitor less. Put di erently, there is only a risk-shifting e et. By ontrast, in the perfet ompetition ase the pass-through is at its maximum and the impat of a hange in r on the lending rate dominates the risk shifting e et. 7

19 This result holds in a more general setting. For example, in our main model it an be shown that the leverage threshold below whih a monetary poliy tightening leads to an inrease in risktaking is lower the atter is the loan demand funtion. Again, as demand beomes more elasti - whih an be interpreted as the market beoming more ompetitive - the interest rate pass-through inreases, making the net e et of a hange in the poliy rate on monitoring more positive. 5 To study the e et of a hange in monetary poliy when the monopolist bank an hoose the apitalization ratio k, we maximize bank pro ts with respet to k: max = L br r k k 2 b2 : This gives the rst order ondition r 2 + r R 2 p R 2 4r ( k) = 0; that has solution b k = R 2 (r + ) r (r + 2) 2 : (8) We an substitute the solution b k bak into the formula for b to obtain It is now immediate that Proposition 6 extends to this ase: d b k (r + ) b = R (r + 2) : (9) > 0 and db an adjust its target apital ratio in response to a hange in the monetary poliy rate. 6 A Numerial Example > 0 when the bank In this setion, we present some simple numerial simulations of the model. The purpose is twofold. First, we want to provide an intuitive graphial illustration of the e ets identi ed in this paper. Seond, sine most of our analysis relies on internal solutions for several of the hoie variables in the model, the example serves to demonstrate that there is a broad set of parameter values for whih suh solutions indeed exist. For the linear demand funtion L = A br L as above, we assume that A = 00 and b = 8. We also assume that 35 perent of the bank s liabilities onsist of insured deposits and the rest is 5 A formal proof for this result an be obtained on reuest from the authors. 8

20 Figure 2: Bank monitoring, b, as a funtion of the monetary poliy rate r for di erent values of bank apitalization, k ^ k=0.9 k=0.6 k=0.3 k=0.0 r* uninsured and therefore must be pried to re et its risk. This is to provide some realism to the numbers and also to over both ases onsidered in our analysis. Finally, we set the monitoring ost parameter = 9 and the euity premium,, to 6 perent. 6 Figure 2 illustrates Proposition. The euilibrium probability of loan repayment for di erent levels of k is plotted as a funtion of the poliy rate. The hart overs a broad range of real interest rate values (from negative 0 perent to positive 20 perent) enompassing the vast majority of realisti ases. From this piture it is easy to see how the response of a bank s risk taking to a hange in the monetary poliy rate depends on its apitalization. For low levels of k, bank monitoring b dereases with the poliy rate r, and the opposite happens at high levels. 7 When we allow the bank to hange is target leverage ratio, an additional e et emerges and the short-term ambiguity in the relationship between risk-taking and the poliy rate is resolved. As the poliy rate inreases, so does the ageny problem assoiated with limited liability. The bank s response is to derease its leverage ratio to limit the inrease in the interest rate it has to pay on its uninsured liabilities. Figure 3 desribes this relationship. The euilibrium leverage ratio is plotted against the real poliy interest rate. Note that, for illustrative purposes, the hart overs an extremely wide range of interest rates from minus 00 perent to plus 00 perent, whih are well beyond what typially ours in pratie. At extremely low values of the poliy rate (below minus 5 perent), the ageny problem is su iently small that the bank nds it optimal to be fully 6 An euity premium of 6 perent is onsistent with the historial average spread between U.S. stok returns and risk-free interest rate as reported in Mehra and Presott (985). 7 In our numerial example, the threshold value for k at whih the relationship between the poliy rate and bank risk taking reverses is about 0.55, whih is a fairly high apitalization ratio in pratie. 9

21 Figure 3: Optimal bank apitalization, b k, as a funtion of the real poliy interest rate r. k^ r* levered (more tehnially, k hits the zero-lower-bound orner solution). For more realisti ranges of the interest rate, the model admits an internal solution and bank apital k inreases with the poliy interest rate. However, the slope of this relationship is dereasing in the poliy rate. Eventually, the relationship beomes at one it hits its upper bound (this orresponds to b(k) = ; see below). Figure 4 illustrates the relationship between the bank s monitoring e ort/probability of repayment and the real poliy rate for the ase with endogenous leverage. For extremely low values of the real poliy rate (exatly the values for whih b k = 0), bank monitoring b is dereasing in the poliy rate. The intuition is straightforward. At these levels b k is in a orner (at zero) and does not move when the poliy rate hanges. It follows that the result related to a xed apital struture applies. And sine b k = 0, we obtain that db = db k=0 < 0. For the most realisti real poliy rate range between minus 0 perent and plus 20 perent, b admits an internal solution and is inreasing in r. Eventually, at a very high real interest rate (about 80 perent), b hits its upper bound, whih is exatly when the relationship between b k and r beomes at. 7 Disussion and Conlusions This paper provides a theoretial foundation for the laim that prolonged periods of easy monetary onditions inrease bank risk taking. In our model, the net e et of a monetary poliy hange on bank monitoring (an inverse measure of risk taking) depends on the balane of three fores: interest rate pass-through, risk shifting, and leverage. When banks an adjust their apital strutures, a monetary easing leads to greater leverage and lower monitoring. However, if a bank s apital 20

22 Figure 4: Euilibrium bank monitoring, b( b k), as a funtion of the real poliy interest rate. ^ r* struture is instead xed, the balane will depend on the degree of bank apitalization: when faing a poliy rate ut, well apitalized banks will derease monitoring, while highly levered banks will inrease it. Further, the balane of these e ets will depend on the struture and ontestability of the banking industry, and is therefore likely to vary aross ountries and over time. There are several potential extensions to our analysis that are useful to disuss. First, we model monetary poliy deisions as exogenous hanges in the real yield on safe assets. Of ourse, this is an approximation. In partiular, we abstrat from how entral banks respond to the eonomi yle and in ation pressures when hoosing their poliy stane. The next step should be to take into aount the role of the interation of the monetary poliy stane with the real yle in determining bank risk-taking. A promising avenue in this diretion may be to augment the model to examine how borrowers inentives hange over the yle. Another important simplifying assumption is that the ost of euity is independent from the bank s leverage. Yet, our results would ontinue to hold in a more omplex setting where the reuired return to euity is a inreasing in the degree of bank leverage. In this ase, it is straightforward to see that, everything else eual, euilibrium leverage would be lower than in our base model sine an inrease in apitalization would have the additional bene t of reduing euity osts. Also, leverage would ontinue to be dereasing in the poliy rate, although the exat shape of this relationship would depend on the funtional form assumed for the ost of euity as a funtion of leverage. A third simpli ation in the paper is that we fous on redit risk and abstrat from other 2

23 important aspets of the relationship between monetary poliy and risk taking, suh as liuidity risk. 8 While other frameworks may be better suited to study this issue (see, for example, Farhi and Tirole, 2009, and Stein, 200), our model ould be adapted to apture risks on the liability side of the bank s balane sheet. For instane, banks might hoose to nane themselves through expensive long-term debt instruments or heaper short-term deposits, whih, however, arry a greater liuidity risk. In that ontext, the trade-o for a bank would be between a wider intermediation margin and a greater risk of failure should a liuidity run ensue. Hene, dynamis similar to those in this paper ould be obtained. We leave all these extensions to future researh. The model has lear testable impliations. First, in situations where banks are relatively unonstrained in raising apital and an adjust their apital strutures, the model predits a negative relationship between the poliy rate (in real terms) and measures of bank risk. Seond, in situations where banks fae onstraints, suh as when their desired apital ratios are already below regulatory minimums for apital regulation, this negative relationship between the poliy rate and bank risk is less pronouned for poorly apitalized banks and in less ompetitive banking markets. Third, the model predits a negative relationship between the poliy rate and bank leverage. While we provide some simple empirial evidene in support of a negative relationship between poliy rate and bank risk, and poliy rate and leverage, we leave more rigorous empirial analysis of these relationships to future researh. The ndings in this paper bear on the debate about how to integrate maro-prudential regulation into the monetary poliy framework to meet the twin objetives of prie and nanial stability (see, for example, Blanhard et al., 2009). Whether a trade o between the two objetives emerges will depend on the type of shoks the eonomy is faing. For instane, no trade-o between prie and nanial stability may exist when an eonomy nears the peak of a yle, when banks tend to take the most risks and pries are under pressure. Under these onditions, monetary tightening will derease leverage and risk taking and, at the same time ontain prie pressures. In ontrast, a trade-o between the two objetives would emerge in an environment, suh as that in the runup to the urrent risis, with low in ation but exessive risk taking. Under these onditions, the poliy 8 A growing literature fouses on funding liuidity risk of banks and the adverse liuidity spirals that suh risk ould generate in the event of negative shoks (see Diamond and Rajan, 2008; Brunnermeier and Pedersen, 2009; and Aharya and Viswanathan, 200) and on the role of monetary poliy in altering bank fragility in the presene of liuidity risk (Aharya and Navi, 200; and Freixas et al., 200). 22