Accounting Information Quality, Capital Requirements, and Banks’ Risk Taking Carlos Corona, Lin Nan, and Gaoqing Zhang Carnegie Mellon University October 25, 2011

Abstract We study the implications of accounting information quality on banks’ risk-taking behavior. We show that the accounting information precision has a non-monotonic effect on banks’ risk-taking decisions. Surprisingly, when information precision is low, an improvement in precision actually induces banks to take more risk. In addition, at an intermediate level of information precision, the relation between accounting information precision and risk taking is also contingent on the stringency of capital requirement standards and the competitiveness of the banking industry. In particular, when either the capital requirement or the competitiveness of the industry is sufficiently high, increasing accounting information precision restrains risk taking. Moreover, when we consider a liquidation cost of assets sale to fulfill the capital requirement, the effect of the capital requirement on risk-taking behavior becomes more subtle. Specifically, for the capital requirement policy to be an effective tool in disciplining banks’ risk-taking behavior, either the banking industry should be highly competitive, or the accounting information should be sufficiently precise.

1

Introduction

Before the financial meltdown of 2008, neither regulators nor academia thoroughly examined the implications of accounting information quality in banks’ risk-taking behavior. The conventional wisdom of bank regulators focused mostly on the restraining effect of a higher capital requirement on banks’ risk-taking decisions. In fact, in the US, the Federal Deposit Insurance Corporation Improvement Act (FDICA), together with the Basel agreement, set the capital standards in the hope that requiring banks to 1

hold additional capital would restrain banks from taking too much leverage. In addition, regulators have also considered the competitiveness of the banking industry as a contributor to more aggressive risk taking and to banks’ failure. A previous official in the central bank stated that “in order to preserve the stability of the banking and financial industry, competition had to be restrained.” (Padoa-Schioppa, 2001, pg. 14). This view is in fact supported by some previous research studies such as Keeley (1990), Suarez (1994), and Matutes and Vives (1996). Nevertheless, more recently, accounting information quality has drawn considerable attention as a complementary regulatory tool to preserve financial stability. For instance, a 2010 report by the European Central Bank commented that “... the provision of more detailed information would help the market to assess the risks associated with asset-backed securities. ... it would unquestionably benefit all types of investors...” In this paper we examine the role of accounting information quality and its joint effect with other banking regulatory policies in restraining banks’ risk-taking behavior. Furthermore, we also examine how accounting information quality alters the effects that a capital requirement policy and the competitiveness of the banking industry have on banks’ risk-taking behavior. We examine a setting in which a bank, in competition with other banks in the deposits market, decides the level of risk of its loan investments. After the bank makes the risk decision, a public accounting signal provides information about the quality of the bank’s loan investments. This accounting information is used to monitor whether the bank meets the regulatory capital requirement. If the bank fails to meet the requirement based on the accounting information, it is forced to sell its risky assets to boost its capital ratio. Since the capital ratio requirement is usually calculated based on accounting information, the effectiveness of a capital requirement policy to deter banks’ risk-taking behavior should be examined jointly with the financial accounting information properties. In this paper, we examine this interaction assuming that banks improve their capital ratio through the sale of their risky assets. This is a frequently observed measure by banks to fulfill the capital requirement. During the 2008-2009 financial crisis, following huge write-downs and severe capital impairments, banks were often forced to sell a considerable amount of their risky assets (deleveraging) in the secondary market, even at a distressed or fire-sale price (Shleifer and Vishny,

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2011).12 In this paper we also want to examine an often neglected consequence of forced assets sales. Although the purpose of the sales is to increase banks’ capital ratio, ironically, it is widely observed that, after obtaining the cash proceeds from assets sales, banks immediately pay bonuses to top management and/or dividends to their shareholders. In the fall of 2008, the Washington Post noted that U.S. banks obtained $163 billion from the Troubled Asset Relief Program (TARP),3 and were on pace to pay more than half of that amount in dividends over three years. Also, the 17 largest banks that received bailout funds, including Goldman Sachs, Citigroup, Bank of America, and Wells Fargo, paid $1.6 billion in bonuses to their executives. Although regulators take measures to restrict this kind of “cash out” in bailout banks,4 it is difficult to completely forbid banks from spending the cash on bonuses and dividends. We incorporate this behavior in our model by assuming that part of the funds obtained by the assets sales are consumed by the decision makers in the bank. One can assume that the decision makers are either the current shareholders which distribute a dividend with part of the sale proceeds, or a manager that cares about the welfare of current shareholders but also uses part of those proceeds to pay his bonus. We show that the accounting information precision has a non-monotonic effect on banks’ risk-taking decisions, and that this effect is contingent on the stringency of the capital requirement standard and the competitiveness of the banking industry. We find that when accounting information is not precise, increasing accounting precision actually may encourage banks to take more risk. However, when accounting information is highly precise, increasing precision discourages risk taking. The capital requirement policy and the competitiveness of the banking industry interact with the accounting information only for intermediate levels of precision. In fact, both the capital requirement and the competitiveness strengthen the role of accounting 1 For instance, First Financial Network, an Oklahoma City-based loan sale advisor on behalf of FDIC, planned to sell 150 million dollars in loan participations from four failed banks in November, 2009. More recently, NP Paribas, one of the largest French banks, sold 96 billion dollars of assets to shore up capital and cut funding needs. 2 Sometimes the sale of risky assets may be in the form of securitization. As of the second quarter of 2008, the securitization market was of a size with an estimated outstanding of 10.24 trillion dollars in the United States and 2.25 trillion dollars in Europe. See ESF securitization report, 2008Q2. 3 A program of the United States government to purchase assets and equity from financial institutions to strengthen its financial sector. 4 For example, in 2009 the US government imposed a cap of $500,000 for top executives at companies that received large amounts of bailout money.

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information in disciplining risk taking. We also extend the model by considering a cost of assets sales. The main effects of accounting precision on risk-taking behavior are robust to the inclusion of such a cost in the model. In addition, we find that, when assets sales are costly, for the capital requirement policy to be an effective tool in disciplining banks’ risk-taking behavior, either the banking industry should be highly competitive, or the accounting information should be sufficiently precise. The paper is organized in the following way: Section 2 provides a literature review. In Section 3 we describe the main model and analyze the resulting equilibrium. In Section 4 we introduce a cost into the model, and in Section 5 we provide several robustness checks to our main results. Section 6 concludes the paper.

2

Literature Review

The extant literature has examined quite extensively the interaction between market competition and risk-taking behavior in the banking industry. However, the role of accounting disclosure in this interaction has often been neglected. Some studies along this line argue that a less competitive environment allows banks to enjoy higher rents that they are afraid to lose in case of failure. Therefore, lowering competition might improve economic efficiency by inducing banks to be more cautious in their risk-taking behavior to avoid failure. For example, Allen and Gale (2000) examine a model in which a bank selects from a set of projects with different risk levels in a Cournot competition in the deposit market. They show that the risk banks take and the probability of bank failure are strictly increasing in the number of banks competing in the market. Keeley (1990), Suarez (1994), and Matutes and Vives (1996) show that a less competitive market raises banks’ charter value,5 which increases the banks’ stake to lose in the case of failure, and leads to less aggressive risk-taking decisions. This argument is also shared by some banking regulators. However, there are some other studies that reach different conclusions. For example, Boyd and Nicolo (2004) argue that banks can be more aggressive in risk taking as the market becomes more concentrated. In particular, they consider an optimal contracting setting in which the risk level is privately selected by a borrower but can be indirectly induced by a bank through the menu of contracts offered. They show that a monopoly situation allows the bank to offer a more aggressive menu of 5

Charter value is the value that would be foregone if the bank closes.

4

loan contracts to the borrower. This, in turn, reinforces the moral-hazard incentive of the borrower who responds by increasing the risk of failure. The related empirical evidence on this matter is mixed. Beck et al. (2003) study a panel data of 79 countries over 18 years and show that it is less likely to observe a bank crisis in a more concentrated market. Keeley (1990) and Dick (2006) provide similar empirical evidence. However, Jayaratne and Strahan (1998) find that following the deregulation in 1990s, the banking industry experienced a significant decline in loan losses. In addition, using a panel data of 21 industrialized countries over 10 years, Nicolo (2000) argues that a bank’s insolvency risk increases with its size. A second stream of related literature examines the relationship between capital requirements and banks’ risk-taking behavior. Buser et al (1981) and Dietrich and James (1983) provide insight on how raising capital requirements may restrict banks’ risk-taking behavior. Regulators seem to share this point of view and believe that a tightened capital requirement is an effective measure to restrain aggressive leverage taking. However, there are also studies indicating that the effect of a capital requirement on a bank’s risk-taking behavior is not monotonic. By considering a bank’s investment decision as a mean-variance portfolio-selection problem, an early study by Koehn and Santomero (1980) argues that a more stringent capital constraint may lead to a higher probability of bank failure if a higher capital requirement is imposed. Gennotte and Pyle (1991) obtain similar results by analyzing a model in which, with the presence of a deposit guarantee, raising the capital requirement can increase the probability of bank failure and lead to financial instability. Moreover, using data on the banking industry from 1984 to 1994, Calema and Rob (1999) quantitatively assess the relationship between bank capital and risk-taking with a structural estimation approach. They find that a tightened capital requirement may induce an ex-ante well-capitalized bank to take excessive risks. More recently, Laeven and Levine (2009) examine a database of 250 privately-owned banks across 48 countries and show that a more stringent capital regulation can encourage more excessive risk-taking behavior for banks with a sufficiently large owner. There are also several studies on the implication of accounting measurement for risk-taking behavior. For example, Li (2009) compares banks’ risk-taking behaviors under three different accounting regimes and finds that fair-value accounting may be less effective in controlling banks’ risk level when compared to other regimes. In addition, Burkhart and Strausz (2009) argue that more accounting transparency may exacerbate the assetsubstitution effect of debt and in turn lead to more excessive risk-taking 5

behaviors. Not many studies have been done on the interactions among capital standards, risk-taking and accounting rules. Among the few, Besanko and Kanatas (1996) study the effect of capital standards on bank safety in the presence of fair-value accounting rules. They assume that a bank satisfies the capital requirement by selling equities to outside investors and show that a more stringent capital requirement may raise the probability of bank failure. Different from our paper, the key factor driving their results comes from a dilution effect: increasing capital standards dilute insiders’ ownership which in turn reduces their incentives to exert effort in improving loan quality. More recently, Heaton, Lucas and McDonald (2010) show that the interaction between fair-value accounting rules and a fixed regulatory capital requirement can lead to an economic inefficiency that is absent under historical-cost accounting rules. A key difference between our paper and their study is that in their model, banks make risk-taking decisions after accounting disclosure and the arrival of new information affects only the equity price rather than real decisions (e.g. banks’ risk-taking decisions).

3 3.1

Model Setup

We examine a three-date setting in which there are  identical risk-neutral banks ( ≥ 2) competing in a market for deposits. At date 0, each bank decides on how much deposit to obtain and chooses the risk level at which it invests the deposit in a project (a loan). At date 1, an accounting signal, , is mechanically generated and observed publicly. The accounting signal is binary,  ∈ { }, where  stands for “good” and  stands for “bad.” In case of a bad signal, a bank has to sell some of its assets to fulfill the capital requirement. Finally, at date 2, the project’s outcome is realized. The time line of the model is shown below.

6

t=0

t=1

t=2

Each bank chooses  and  

An accounting signal   ∈ { } is realized for bank . If  the bank has to sell  portion of its assets to satisfy capital requirement.

Loan outcome realized.  ∈ { }

Time line. More specifically, at date 0, each bank  ∈ {1 2   } makes two decisions: the total amount of deposits,  , and the risk level at which it invests those deposits,  ∈ [0 1]. We make the simplifying assumption that a bank’s choices of  and  are private and not observed by outsiders.6 All  banks make both decisions simultaneously. The deposit P market is represented by an upward sloping inverse supply curve,  (   ), that yields the equilibrium deposit rate as a function of the aggregate bank deposit P amount,   . For simplicity, we assume that  has the following linear functional form: X X  ) =  +     ( 



with   0. That is, deposit amounts are perfect substitutes and increase the deposit rate. Since all deposits are fully insured by the Federal P Deposit Insurance Corporation (FDIC), the competitive deposit rate  (   ) depends on neither the individual nor the aggregate risk of all banks.7 To be consistent with the unobservability of  ,  needs to be unobservable to 6

In this model, we assume that the investment amount and the loan risk decision are not observable to the market. However, even if the market observes these decisions, as long as they are not perfectly observable, the market will not use them in the projection of future cash flows. This is because the bank does not have any relevant private information for the decisions and, therefore, the market can perfectly conjecture them without paying attention to any noisy signal about these decisions (Bagwell, 1995; Maggi, 1999). In addition, we have examined the case of an observable  , which seems to be a decision that is more likely to be observed. Although it is untractable analytically, in a numerical analysis we show that the observability of  does not bring any qualitative difference, and does not provide any additional insight (see Section 5). 7 Notice that the FDIC insurance is not a driving force for our main results but simplifies our analysis. Even if we do not have this insurance in the model and assume  depends on the market’s conjecture of total risk, it will only reduce banks’ return in -state but our results still remain.

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potential investors as well, otherwise the market would be able to conjecture  from  . 8 We assume that each bank  invests the total amount of its deposits in a bank loan that has an uncertain outcome,  . The loan outcome is characterized by a loan state,  ∈ { }. That is, the loan can end up either in a “high” state (), in which it yields a high outcome, or in a “low” state (), in which it yields a low outcome, which is normalized to zero. Neither the bank nor the outsiders observe the realized loan state and outcome until date 2. The risk level of the loan,  , affects the expected return from the loan in two ways. First, given the loan amount  , the higher the loan risk is, the higher is the return of the loan in the -state. In particular, in the -state the loan yields a cash flow of ( +  ) (where  represents the risk-free return), and in the -state the loan yields a zero outcome. Second, the probability that the loan ends up in the -state,  ( ), decreases with the risk of the loan. In particular, we assume that  ( ) follows a linear function  ( ) = 1 −  . These assumptions are consistent with the high-risk-high-returnPproperty of investments. At date P 2, bank  pays a deposit interest of  (   ) = ( +    ) only in the -state. Therefore, the outcome in the -state is equal to: X X  ) = [ − (  )] (1) ( +  ) − ( +  



In the -state, the bank defaults and repays the depositors to the limit of its on-balance assets and the rest is covered by the Federal Deposit Insurance program. That is, the return is zero. The outcome of the loan, therefore, can be characterized as follows: ½ P [ − (   )] if  =   = 0 if  = 

At date 1, an accounting signal  on the loan performance is generated and observed publicly. The quality of this accounting information is represented by an exogenous parameter, , which is the probability that the signal is correct. That is:  ( = | = ) =  ( = | = ) =  We assume that the accounting signal is informative, i.e., 8

1 2

   1.

As long as the observability of  is not perfect, the results of this basic model still hold. Nevertheless, we check the implications of an observable  , or equivalently an observable  , in Section 5, and find that the results are qualitatively the same.

8

Upon a bad accounting signal, the market value of a bank’s assets decreases. The bank then may fail to satisfy the capital-sufficiency examination by regulators. In the real world, the capital sufficiency requirement can be described by the constraint below:  ≥  -   where  represents the minimal capital ratio required by the regulators. -   is a weighted measure of the bank’s assets for regulatory purposes, which uses a larger weight for riskier assets and a lower weight for less risky assets (the weight on the risk-free asset such as cash is zero). According to the current accounting regulation, a bank’s assets are recognized at the lower of cost or fair value.9 Therefore, the assets impairment upon a bad signal decreases the bank’s assets value while the associated impairment loss reduces its equity value. These two effects jointly result in a lower capital ratio.10 . If the bank actually fails the capital examination, it is forced to take measures to satisfy the regulatory capital requirement. In particular, the bank can meet the capital requirement by selling a part of its risky assets for cash.11,12 Note that in our model, after  and  are selected by bank , the satisfaction of the capital requirement is solely determined by the realization of the accounting signal. We consider the nontrivial 9 Banks’ loans can be either held-for -investment (“HFI”) or held-for-sale (“HFS”). HFS loans are reported at the lower-of-cost-or-fair-value, with declines in fair value recognized in income (See SFAS No. 65; SFAS No. 5; SFAS No. 114; and SOP 03-3). Alternatively, banks may measure its loans at fair value regardless of whether they are HFI or HFS (See SFAS No. 159). In our model, we simply assume a bank’s loan assets are mark to market without distinguishing HFS and HFI. 10 For example, suppose there is a bank whose risky assets were 2 million dollars and its equity was recognized to be 1 million before a bad accounting signal, and the weight for risky assets is 100%. Its capital ratio therefore was 0.50. Upon a bad signal, assume the assets’ market value declines to 1.5 million. Then, the assets’ value is mark to market and the impairment loss reduces the equity book value to 0.5 million. The capital ratio after the accounting signal, therefore, declines to 0.33. 11 To illustrate how selling risky assets increases the capital ratio, suppose there is a bank whose risky assets were 2 million dollars and its equity was 1 million, and the weight for risky assets is 100%. Its capital ratio therefore was 0.50. Now suppose the bank sells 0.75 million dollars of its risky assets for cash. Since cash has zero weight in the calculation of risk-weighted assets, the new risk-weighted assets will be 1.25 million and the new capital requirement ratio becomes 0.80. 12 In practice, to meet capital requirement, an alternative to assets sales for banks is to issue new equities. We examine the case of equity issuance in the robustness checks section, Section 5 and find that, qualitatively, the mode with equity issuance yields the same results. Hence we will focus on the assets sales case in the main setting of this paper.

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scenario in which a bank fails to meet the capital requirement only when a bad signal is generated. For simplicity, we assume that the portion of assets that need to be sold is a constant,  ∈ (0 1).13 Essentially, if the accounting signal realization is  the bank is forced to sell risky assets, keeping only a portion 1 −  of the loan assets originated at time 0. Otherwise, if the accounting signal realization is , the bank keeps all its assets.

3.2

Equilibrium

In this section, we define the equilibrium and explain the derivation of the main results. Definition 1 Equilibrium: We consider a symmetric Perfect Bayesian Equilibrium { ∗  ∗   } such that: • At date 0, each bank chooses the optimal {∗  ∗ } to maximize its expected future cash flow: max  ( | = ) + (1 −  )[   + (1 − ) ( | = )]

 

where  denotes the unconditional probability of a good accounting signal,14 and   are the bank’s proceeds from selling an  portion of its risky assets in the case of a bad signal. • The market price of the bank’s assets at date 1 upon the accounting signal,   is equal to the market’s updated expectation of the project future cash flow:  )  ∈ { }  = ( |     −

• In equilibrium, the market’s conjectures of a bank’s risk-taking and investing decisions equal the bank’s real decisions, i.e., (   ) = ( ∗  ∗ ). The model can be solved by backward induction. At date 2, if a bank gets a good accounting signal, the bank meets the capital requirement and there is 13 The effects of relaxing this assumption and allowing for an endogenous  is examined in the robustness checks section, Section 5. 14 The expression for this probability can be found in the appendix. Note that this probability is contingent on the amount of risk  .

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no need to sell assets. Hence, the assets market price information is irrelevant because the current shareholders keep their assets till the loan outcome is  ) = realized. The expected outcome given a  signal is ( |     − P  ( −  =1  )] , where  denotes the conditional probability of the -state given a good signal.15 With a bad signal, however, the bank is forced to sell a portion  of the risky assets for cash to shrink its riskadjusted assets. In that case, the asset market price (  ) equals the market’s conditional expectation of the project future cash flow:   =[

(1 − )(1 −  ) ]   (1 − )(1 −  ) +   

In this expression, the term in square brackets is the conditional probability of the -state given a bad signal, and the rest is the outcome in the -state. Note that the asset price   is only a function of the investors’ conjectures and, therefore, is ex ante independent of the banks’ actual choices. The expected payoff for the bank upon a bad signal is  ) =   ( |     −  +   [(1 − ) − 

 X

 ]

=1

where  is the conditional probability of the -state given a bad signal.16 Here we assume that after the bank gets cash proceeds from the sale of  portion of its risky loan, the bank won’t keep the cash idle and will consume some cash before date 2, which is represented by     with  ∈ (0 1] The use of the cash can be dividends to shareholders, compensation to management team, or other private managerial benefits.17 We also assume that once a capital-deficient bank ends up in the -state, the remaining cash proceeds are not sufficient to repay the depositors. However, by the virtue of limited liability, the bank is not liable to the outstanding balance. 15

The expression for this conditional probability can be found in the appendix. Note that this probability is contingent on the amount of risk  . 16 The expression for this conditional probability can be found in the appendix. Note that this probability is contingent on the amount of risk  . 17 As we mentioned in Section 1, it is difficult to forbid a bank from paying bonuses or dividends using the cash proceeds from the sale of its assets. This can be related to Jensen (1986)’s argument that there are agency costs of excessive free cash flows. Moreover, it is not efficient for a bank to keep all the cash proceeds from asset sale idle, because once the project/loan turns out to be in -state, the bank has enough cash flow from the loan project itself to pay off the depositors’ interest and it is not efficient to hold up the cash proceeds until date 2. Nevertheless, our main results hold even for a very small value of  As long as this benefit is non-zero, all the results remain.

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With these conditional expected payoffs for different signals, we can now specify the bank’s objective function. At date 0, each bank  chooses the deposit quantity  and the loan risk  to solve the following program:

max  [ ( −

 

 X

 )] +(1− ){

  + 

=1

 [(1−) −

 X

 ]}

=1

(2) Taking derivatives with respect to the two choice variables, we obtain two first-order conditions: 1 +  (− +  ) − 2 [ + (1 − )(1 − )] + [ (2  − 1)]  =0  − + 2  ( − )(1 −  ) = 0 1 +  − 

In equilibrium, (   ) = ( ∗  ∗ ) and (   ) = ( ∗  ∗ ) for all banks. Note that the aggregated deposit size for the rest of the banks in the market is − = ( −1)∗ . Solving the system of equations for the two choice variables, we obtain the expressions for the equilibrium optimal deposit amount and loan risk. The unique symmetric equilibrium is summarized in the following proposition: Proposition 1 There exists a unique symmetric equilibrium in which: (i) each bank makes the optimal decisions ( ∗  ∗ ) that satisfy p −2 − 22 − 4 1 3 ∗  and  = 2 1 1 − (1 − ) ∗   ∗ =  ( + 1) where, the coefficients (1  2  3 ) are defined as functions of (   ): 1 = (2  − 1)[(1 − ) − 1 − ( + 1)(1 − )(1 − )]

2 = −3 + 4 +  (2 − 3) + (1 − )2 + ( + 1)(1 − )(2 −  − 3 + 2)

3 = ( + 1)(1 − )(1 − (1 − ))

(ii) upon a bad accounting signal, the asset market price,  , is  =

(1 − )(1 −  ∗ ) 1 − (1 − ) ( ∗ )2 ( + 1) (1 − )(1 −  ∗ ) +  ∗ 12

The above proposition describes the expressions for the equilibrium choices of the deposit amount and loan risk amount, ( ∗  ∗ ) for each bank in the market. In addition, it shows the expression for the equilibrium selling price of the banks’ assets when the accounting signal is bad,  . The asset price when the signal is good is not relevant because the bank does not sell assets in that case. With Proposition 1, we are also able to examine how the equilibrium risk choice is affected by the quality of the accounting information.18 One would expect that an improvement in accounting information quality would restrain risk taking by the bank, and this is what we actually find when the accounting information is already very precise. However, we find that if the accounting precision is low, an improvement in the quality of accounting information may actually lead to more risk taking. ∗ That is,   can be positive. Figure 1 illustrates how a bank’s risk-taking strategy changes with the accounting information precision. There are four ∗ regions in the figure. In the far-left region,   is always positive, and in the ∗ far-right region,   is always negative. Capital requirement policy, competitiveness of the banking industry and  only play roles when the accounting information quality is of an intermediate level. ∗ With an intermediate quality of accounting information,    0 for a ∗ low capital requirement ratio while    0 for a high requirement ratio. The competitiveness,  , only has an impact in the two slim middle areas ∗ shown in Figure 1, where   is positive when  is small and negative if  is large. The effect of  is almost negligible, as it only plays a role in the ∗ sign of   in the middle-right region which is nearly invisible. In that tiny ∗ region,   is positive when  is small and negative when  is large. We summarize these results in the following proposition. Proposition 2 In the main setting, •

 ∗ 

 0 if  is sufficiently small;

•

 ∗ 

 0 if  is sufficiently large;

Observe that the equilibrium deposit size ∗ is linear in the equilibrium loan risk choice  ∗ . This is because an increase in risk level leads to a higher return in -state, but will not change the return in -state, as the return in -state is always zero. Therefore, the choice of the deposit size only depends on the cost-return trade-off in state  which is reflected in equation 1. Since in -state an increase in the loan risk increases the return from the loan without affecting the deposit rate, it shifts the optimal size upwards. Since ∗ decision is always linear in  ∗ and is relatively passive, we focus on the analysis of  ∗ decisions in our paper. 18

13

Figure 1: The sign of

 ∗ 

∗

• when  is of an intermediate value,    0 if  or  is sufficiently  ∗ small, and   0 if  or  is sufficiently large; ∗

•  only plays a role on the sign of   in rare cases, in which  ∗ if  is sufficiently small, and   0 otherwise

 ∗ 

0

Proposition 2 implies that for a given number of market participants  and a given capital requirement , the shape of the risk-taking decision  as a function of the accounting information precision  is single peaked, which is illustrated in Figure 2. When accounting information is highly precise, a bank will take less risk when investing in the project as the information becomes more precise, which is consistent with conventional wisdom. However, surprisingly, when the accounting information is noisy, a bank becomes more aggressive in risktaking as the precision of accounting information improves. To explain the

14

Figure 2:  ∗ peaks at intermediate  intuition behind this result, we rewrite a bank’s objective function as follows: 

 P



=1

] [ + (1 − )(1 − )](1 −  )[ −  + (1 − )(1 − ) | {z }

(3)

Assets-Holding Factor

+     [(2 |

− 1) + 1 −  ] {z }

Assets-Selling Factor

The first component in the bank’s objective function, the assets-holding factor, is the bank’s total expected return from its investment. It reflects a trade-off between the probability that the project P is in the -state, 1 −   and the expected return in the -state,  −  =1  . As a bank chooses a higher level of risk, the chance that its project will be in the -state is lowered, while the expected return once the project turns out to be successful is increased. This risk-return trade-off by itself determines a unique optimal level of risk, which is lower than the maximal level, 1. The second term in the bank’s objective function, the assets-selling factor, is the bank’s expected proceeds from the sale of the  portion of assets when a bad signal is realized. It is strictly increasing in  because a more aggressive risk-taking strategy makes a bad signal more likely, which in turn raises the likelihood of asset sales and hence the expected proceeds. Since the market price of the bank’s 15

assets upon a bad signal,    is independent of the bank’s actual decision of  , it gives the bank an incentive to take more risk. If the accounting information is noisy (that is,  is close to 12 ), the asset price upon a bad signal,    will be higher than that when accounting information is precise. A noisy accounting signal, hence, reinforces the assets-selling factor (the second term in Eq. 3). Further, as the assets-selling factor becomes larger, a bank tends to take more risk, which makes 2 − 1 more likely to be positive. When 2 − 1 is positive, the assets-selling factor increases with . That is, when  is small, as  increases, the assets-selling factor is strong and dominates other effects. That explains why we see a ∗ positive sign for   in the left region. On the other hand, if the accounting signal is very precise (that is,  is close to 1) the asset price upon a bad signal,   , will be low and the assets-selling factor weakened. Hence, when  is large, the assets-holding factor dominates. In addition, since the assets-holding factor is strictly increasing in the precision ([+(1−)(1−)] increases in ), as  increases, the bank becomes more reluctant to take risk as it tries to avoid the -state, even though taking less risk lowers the return in the -state. Therefore there is a negative association between the accounting information precision and risk taking when  is high. When the accounting information precision is of an intermediate value, the driving forces behind the relationship between a bank’s risk taking and the precision are more subtle. To understand them, it is useful to first examine how  ∗ changes with  and  In our model, as  or  increase, ∗ a bank becomes more aggressive in risk taking. That is,    0 and ∗ ∗  19 The intuition for   0 can be illustrated from (3). Increasing   0   raises the weight on the assets-selling factor (   ) and reduces that on the assets-holding factor (that is, [ + (1 − )(1 − )]), therefore it yields ∗ a higher level of risk in equilibrium. The intuition for    0 is that a harsher competition drives up the deposit interest, therefore, induces a bank to take more risky projects to maintain the margin. Now we are ready ∗ to explain how  or  affects the sign of    As  or  increases, a bank takes more risk. If the accounting information is precise, a more aggressive 19

The derivatives are computed as follows:  ∗ =− 

1 

2  ∗2 +  ∗ +  ∗ 2 1  + 2 ∗

3 

∗

 ∈ { } ∗

  In this manner, we can show that    0 and   0 The result,   0, is consistent with the findings of previous studies such as Koehn and Santomero (1980) and Gennotte ∗ and Pyle (1991). The other result,   0 is also consistent with predictions of previous  literature (Keely, 1990; Allen and Gale, 2000; Dick, 2006).

16

Figure 3: The effect of  on

 ∗ 

risk-taking decision reduces the asset price of the bank upon a bad signal. As a consequence, the assets-selling factor is weakened. Therefore, at an intermediate level of precision, when  or  is small, the bank takes more risk as precision improves; while when  or  is large, the bank takes less risk as precision improves. (The effect of  although negligible, can be explained in a similar way, as it is easy to verify that risk taking increases in .) It is interesting that with an intermediate level of accounting precision, changing the competitiveness of the banking industry and/or the capital requirement policy may switch the role of accounting information quality ∗ in banks’ risk-taking decisions. We illustrate the effect of  on   in Figure 320 . Specifically, when the banking industry becomes more competitive and/or the capital requirement policy is tightened, improving accounting information quality becomes more effective in discouraging banks from taking risk. Notice that in our model, the assets-selling factor is mainly driven from the cash benefit of assets sale. The proceeds from assets sale motivate banks to take more risk, as a higher risk level increases the probability of assets selling. In equilibrium, the market anticipates the banks’ incentive to take more risk and incorporates this into the asset price. Even if a bank had deviated by choosing a lower risk, it would not be compensated by a higher 20

The effect on  is similar.

17

asset price, as the asset price depends only on the conjectured risk level but is not contingent on the bank’s actual risk taking. In other words, a bank cannot help itself to take more risk.

4

Cost of Failure to Meet the Capital Requirement

In our main setting, there is no extra cost for a bank in the case of a bad accounting signal as the bank sells assets to fulfill the capital requirement. In reality, failure to meet the capital requirement induces many other costs. For example, a bank may incur a liquidity cost when selling assets in an illiquid market, which is well documented in the extant banking literature (Plantin, Sapra and Shin, 2008; Allen and Carletti, 2008). In this section, we introduce a cost of failure to meet the capital requirement. For convenience, we call this cost “liquidity cost.” In particular, we assume the cost is  =  where  is a constant. We assume  is proportional to  because the more assets a bank sells, the higher the liquidity cost is for the bank.21 Further, we assume there is an upper bound for  ,    .  is chosen so as to ensure banks’ return upon aPbad accounting signal is positive, namely,     +  [(1 − ) ( −  =1  ]  −   0. That is, the limited liability constraint is not violated when the accounting signal is bad. As in the main setting, at date 0, a bank chooses the optimal risk level ∗ maximizing its expected future cash flow: ∗ and investment  max  [ ( −

 

 X =1

 )] +(1− ){  (  − )+ [(1−)  −

Since we have already shown that the parameter  doesn’t play a significant role in the main setting, from now on, we assume  = 1 for simplicity and tractability. We find that there is a unique symmetric equilibrium in this case. We summarize the results in the following proposition. Proposition 3 With a liquidity cost, there exists a unique symmetric equilibrium in which ∗ = 1−(1−)  ∗  and  ∗ solves   3 +   2 +   +  = 0 and (i)  1 2 3 4    (+1) satisfies the second-order condition, where the coefficients (1  2  3  4 ) are 21

We have also numerically examined other cost functions such as a fixed cost, a cost which is proportional to investment  size  and a cost which is proportional to the expected cash flow ( = (1 −  )( −   =1  )  ), and we achieve similar results.

18

 X =1

 ]  }

defined as functions of (    ): 1 = −[1 − (1 − )](2  − 1){1 + ( + 1)[1 − (1 − )]}

2 = −[1 − (1 − )]{3 − 4  − 2 (1 − )2 −  [3  − 2 + (1 − )2 ]}

3 = ( + 1){−1 +  +  [(2 − (1 − ))(1 − )2 +   (1 + ) (1 − 2 )2 ]}

4 =    ( + 1)2 (1 − ) (2  − 1)

(ii)upon a bad accounting signal, the price of the bank’s assets,   is  =

(1 − )(1 − ∗ ) 1 − (1 − ) ( ∗ )2 ( + 1) (1 − )(1 − ∗ ) + ∗ 

A quick examination of the new objective function shows that the optimal level of risk decreases with the cost parameter  . This is because a bank pays the cost only when a bad accounting signal is realized and the probability of a bad signal is strictly increasing in the risk level ∗ . Corollary 1 A bank’s risk taking decreases in  ,

∗  

 0 ∗

Recall that in the main setting, we always have    0 When we introduce a cost and when the cost is sufficiently high ( is sufficiently large), an increase of capital requirement  may reduce the bank’s risk  ∗ level. That is, we may have   0. Figure 4 depicts how the sign of the  ∗ derivative  changes in the space of parameters (   ). In the shaded  ∗ space, the bank’s risk level still increases with  (   0). However, the bank becomes less aggressive in risk-taking as the capital requirement gets  ∗ tightened (   0) in the remaining space. The proposition below shows  ∗ that for a sufficiently large liquidity cost,  is always negative. Proposition 4 When   3, there exists a threshold such that always negative for  larger than that threshold. 22

∗  

is

 ∗

The threshold for   0 approaches zero when  goes to infinity. That is, in a sufficiently competitive market ( is large enough), even with a very small cost, raising the capital requirement always discourages banks from taking risks.23 Moreover, we find that when  approaches 1, the 22

Although we can only prove analytically that this result holds for   3, numerical analysis shows that the threshold also exists in the cases when  = 2 and  = 3. 23 ∗ Further numerical example shows that at =25, the cutoff  ∗ (   ) becomes so ∗  small that it is almost certain that   0.

19

Figure 4: The change of the sign of

∗  

in the space of parameters (   )

threshold also goes to zero. That is, in a world with a sufficiently informative accounting system, raising capital requirement can always discourage banks’ aggressive risk-taking behavior. In other words, our results suggest a “good” accounting system is essential for capital requirement policy to discipline banks from pursuing excessively risky project. For an economy with a poor-quality accounting information system( close to 12 ), raising capital requirement may in turn encourage more risk-taking behavior. Proposition 5 With a liquidity cost  , large and/or  is sufficiently large.

∗  

 0 when  is sufficiently

It is also interesting to examine how a bank’s risk-taking behavior changes  ∗ with the accounting information precision (  ) when we introduce the liquidity cost. Although this is not analytically tractable and we cannot provide closed-form solutions, we are able to show numerically that the derivative   moves continuously with  . As illustrated in Figure 5, as  moves from  ∗

0 to 0.0003, the region of   0 in ( ) changes smoothly.24 Further, we ∗ numerically compare the value of the derivative   in the main setting and ∗  

in the model with the cost, and find that introducing costs with realistic

24 Numerically, a cost of 0.0003 accounts for about 10% of each bank’s expected cash flow.

20

Figure 5: The change of the region of

∗  

 0 in (   )

values does not bring significant change to our results regarding   . This confirms the robustness of our results in Proposition 2 in the main setting.25 By introducing the cost, the competitiveness of the banking industry (which can be represented by  ) has a more pronounced effect on determining the sign of the derivative   than in the main setting. As illustrated in  ∗

Figure 6, as  increases, the region in which   0 gradually shrinks in the ( ) space. As  approaches infinity (perfect competition), the positive∗ ∗   26 This  region is negligible and it is almost always true that   0 implies that making the banking industry more competitive magnifies the role of accounting in restraining banks’ aggressive risk-taking behavior. 25

In addition, it implies that our previous result about ∗  

  

 

in the basic model is not

With a cost, is likely to be negative but we driven by the presence of a positive still observe similar results regarding how a bank’s risk-taking behavior changes with the accounting information precision  ∗  ∗ 26 ∗ More precisely, the positive-  region disappears when lim   12 (lim →∞  = ∗ 2 (2 lim  −1) (1−2 )−1 →∞

(1−2 )2

 →∞

) Since by lemma 1,

almost always true that

∗ 



 0

∗ lim   →∞

21

is close to 1 and well above

1 2,

it is

Figure 6: The region of

∗  

 0 gradually shrinks in the ( ) space

Proposition 6 With a liquidity cost, improving the competitiveness of the banking industry (increasing  ) makes the role of accounting information precision more pronounced in restraining banks’ risk-taking behavior.  ∗

To intuitively illustrate the results regarding  and bank’s objective function in a similar format as in (3): 

 P

∗   ,

we rewrite a



=1

] [ + (1 − )(1 − )](1 −  )[ −  + (1 − )(1 − ) | {z }

(4)

Assets-Holding Factor

+    {(2 |

− 1) + 1 −  } {z }

Assets-Selling Factor

−{(2 − 1) + 1 −  }  | {z } Cost Factor

The first two terms are the same as in the corresponding expression in the main setting. The third term is the bank’s expected cost for violating the capital requirement. It is strictly decreasing in the risk level  because a more aggressive risk-taking strategy makes a bad signal more likely, which 22

in turn raises the likelihood of capital insufficiency and hence the expected cost for the bank. Therefore, it reduces the bank’s incentive to take risk.  ∗ The intuition for   0 when  ,  and  are sufficiently large can be illustrated from (4). As in the main setting, the first two incentives jointly  ∗  ∗ yield   0. The third one, however, pushes  towards negativity. This is so because with a cost proportional to , an increase in  raises the penalty for taking risk, which in turn induces the bank to take a safer  ∗ investment strategy. Therefore, the sign of  is determined by weighting the third effect against the first two. Now consider how a change of  may affect the trade-off. If  is very large, on the one hand, a fierce competition increases the deposit interest rate, leading to a “negligible” bank profit level. On the other hand, by observing a large  , investors will rationally reduce their expectation of the loan performance, which yields a lower bank assets price. Overall, the strength of the first two factors will be considerably weakened by the increase of  . The risk-curbing incentive induced by the cost factor, however, is relatively immune to the competition shock.27 Therefore, as  increases and the first two factors become weaker, the risk-curbing incentive becomes  ∗ dominant and results in   0. The effect of a higher cost (a larger  ) or a more precise accounting information system (a higher ) is similar. Notice that as  increases, the cost factor becomes stronger. The other two factors, however, are not directly influenced by 28 Therefore, when  becomes sufficiently large, the risk-curbing incentive induced by the cost factor dominates and leads to ∗    0. Similarly, we can show that the cost factor is strictly increasing in , therefore for a sufficiently informative accounting system, the cost factor  ∗ outweighs the other two factors and makes  negative.  ∗ The intuition regarding the sign of  is similar. Recall that in the ∗ main setting, when  is sufficiently small (the left region in Figure 1),   ∗ is always positive. Moreover, the boundary of the positive-   region is independent of . In the case with a cost, however, the increase of  pushes  ∗ this boundary to the left and makes the region of positive  smaller. This result can be understood by explicitly considering how the inclusion of the additional cost factor may affect the trade-off in the main setting. 27

Note that the third factor indirectly depends on  through its dependence on  . Nevertheless, if we consider that explicitly, an increase of  raises the risk level  and will strengthen the cost factor. 28 Again, the first two factors indirectly depend on  through their dependence on  . But both the analytical and numerical results suggest that this indirect effect is secondary.

23

Specifically, when bank competition is very fierce (that is, when  is very  ∗ large), each bank will take a more aggressive competition strategy (   0), which makes (2 − 1) more likely to be positive. When (2 − 1) is positive,  ∗ the cost factor increases with  and makes   0 more likely. Further, as  increases, both a bank’s profit level and its asset price decrease, which considerably weakens the effects of the first two factors. That is, when  is sufficiently large, the cost factor dominates the other effects. As a  ∗  ∗ consequence,  is less likely to be positive and the positive-  region gradually shrinks as  increases, which is illustrated in Figure 6.

5

Robustness of Results

In this section, we provide several robustness checks for the main results. We examine the case of a perfectly observable deposit, or equivalently a perfectly observable deposit interest rate. We also examine a setting in which the portion of assets that need to be sold,  is endogenously decided by the capital requirement ratio. We further extend our model and study a setting in which the risk level, , is privately selected by a borrower but can be indirectly induced by the bank offering a menu of contracts. Lastly, we adapt our model to examine the scenario that, to meet capital requirement standards, a capital-deficient bank can also sell a portion of its equity interest (mostly via equity issuance in the public market) instead of selling risky assets. Observable Deposit Amount  In the main model we assume that the deposit quantity is not observable to external parties. Although the deposit size is reported in the financial statements, it would be hard to argue that it is perfectly observable to the market. The conclusions of our main model would be unchanged even if we allow for an imperfect observation of the deposit size because the bank has no private information at the time it makes the deposit size decision (see Bagwell, 1995).29 Nevertheless, in this section we examine the effect that a perfect signal about the deposit size would have on the equilibrium. The presence of a perfect signal about the deposit amount would change the analysis of the model in two ways. First, the market would not need to conjecture the deposit amount  to estimate future cash flows. Second, 29

Notice that this perfectly observable deposit case is equivalent to a case in which the deposit interest rate is perfectly observable.

24

the market would be able to use that information in its conjecture of the risk-taking decision as well. Therefore, the assets price would be specified as follows:   =[

(1 − )[1 −  ( )] ]  ( )  (1 − )[1 −  ( )] +  ( ) 

where,  ( ) is the conjecture the market makes of the bank’s loan risk decision using the perfect signal information about the deposit amount  . The first order conditions for both decisions are now:

1 +  (− +  ) − 2 [ + (1 − )(1 − )] + [(2  − 1)]  =0  − + 2  )(1 −  ) 1 +  −        +(1 −  )( + )=0    ( −

Notice that the first equation above is identical to the first order condition for the unobservable case. This is so because  is still unobservable and, therefore, the derivative with respect to  is unchanged. Nevertheless, the second expression has now an additional term that depends on the derivative of the conjectured risk with respect to the deposit amount. With these two first order conditions and the fact that in equilibrium the conjecture about the loan risk decision is true, we obtain a system of equations that implicitly determines the equilibrium bank deposit amount and loan risk decisions. This system of equations cannot be solved in closed form but we solve it numerically and obtain that, qualitatively, the results are the same as those in the main model. (The numerical results of all the robustness checks are illustrated in Figure 7.) Endogenized Asset Sale Amount  In the main setting, for tractability, we make the simplifying assumption that the portion of assets sold by a capital-deficient bank is fixed (as denoted by ) and is not directly related to the capital requirement ratio . In this section, we examine a bank competition model in which the portion of assets sold is endogenized, and we show that, numerically, the revised model yields similar results to our main model. 25

Consider the capital requirement constraint,  ≥    where  is the minimal capital ratio required. The denominator is riskweighted total assets on bank ’s balance sheet and  denotes the regulatory weight assigned to the risky loans. Hence to meet the capital requirement, the maximal amount of risky assets held by the bank, denoted by  , is  =

   

Accordingly, in the case of capital deficiency, the portion of risky assets that need to be sold is   = 1 −    Now note that the revised model is identical to our main model except that the fixed number  needs to be replaced with  30 , which is explicitly linked with the capital ratio . The revised model is not analytically tractable and yields no closed-form solutions. However, a numerical analysis indicates that the revised model provides similar conclusions to those of the main model. That is, the assumption about  in the main model is not a driving assumption of any of our results but helps to simplify our analysis. Risk-taking with a Borrower’s Moral Hazard Problem In our main model, we follow most previous studies in considering that a bank’s risktaking behavior is a result of an optimal portfolio selection. That is, a bank can freely select a project with the desirable risk characteristics,  , from a continuum of choices. Recently, Boyd and Nicolo (2004) examine an optimal contracting setting in which the risk level, , is privately selected by a borrower but can be indirectly induced by the bank through a menu of contracts offered. In this section, we incorporate the optimal contracting framework into our setup and show that it has no significant effect on our main conclusions. As in Boyd and Nicolo (2004), banks competeP in both loan and deposit markets. As a consequence, the demand curve  (   ) in the loan market 30

Note also that assuming  is observable to the investors is equivalent to assuming the deposit amount  is observable, which we have already addressed in the section above.

26

is downward sloping. In accordance with the main model, we assume a linear function for  : X X  (  ) =  −    



where  and  are both positive constants. We assume a linear function for  : X X  ) =  +     ( 



where  and  are both positive P constants. In this expression,  is positive, since the supply curve  (   ) in the deposit market is upward sloping. Given the loan rate  , borrowers optimally choose the risk level 0    1 to maximize max  ()( −  ) 

As in the main setting, a project indexed by  yields a return  with a probability  () and 0 otherwise. We also assume  () follows a linear function  () = 1 − . The first-order condition on  gives the induced risk level: P 1 +  (   )  = 2 Anticipating the borrowers’ optimal choice of , the bank chooses the investment size  to maximize max  [ ( −  )] + (1 −  ){   +   [(1 − )  −  ]}  P 1 +  (   )    = 2 where     P     are computed as in the main setting.    are functions of   as defined above. We numerically compare the results with those in our main setting. The two models yield very similar results except that in the revised one the risk level  is strictly decreasing in  , which is consistent with the results obtained by Boyd and Nicolo (2004). Therefore, our robustness analysis in this section suggests that most conclusions in our main model remain and they are not driven by the choice of a portfolio-selection framework. Equity Issuance To raise capital for a regulatory purpose, a bank can either sell a portion of its risky assets or issue new equity. Up until now, we have focused on assets sales. In this section, we show that, our setup can be 27

easily adapted to the case of equity issuance and we find that it only alters our results slightly. Consider the case that a capital-deficient bank sells a portion of its equity interest to raise new capital. Suppose that when a bad accounting signal is realized, the bank is forced to sell  portion of its equity. The expected proceeds to the current shareholders are: X  )]   + (1 − )  [ −  ( 

where the first term is the proceeds from issuing new equity and the second is the current shareholders’ share of the bank’s expected future cash flow. We assume investors offer a fair price for the bank’s equity given their conjectures: 

 = [

X (1 − )(1 −  ) ]( −   )    (1 − )(1 −  ) +  =1

When a good signal is realized, no asset sale is necessary and the bank receives (|     − ). Hence the bank’s total expected proceeds equal X  )]}  (|     − )+(1− ) {  +(1−)  [ −  ( 

Similar steps can be performed to solve the model with equity issuance and we find that, qualitatively, the results are the same as those in the main model. All the numerical results of the above robustness checks are summarized in the figure below. The left column of the panel depicts how the risk level  ∗ changes with  and the right column shows how the accounting effect  ∗  changes with respect to  or  (as the effects of  and  are similar, we only show the effect of one of them for each robustness check). In general, although most robustness checks can only be implemented numerically, they show that variations of the main setting do not bring significant change to our main results.

6

Conclusions

In this paper, we study the effect of accounting information quality on banks’ risk-taking behavior, and its interaction with the capital regulation and the 28

Figure 7: Numerics of robustness analysis. In the left column, the horizontal axis represents  and the vertical axis represents  ∗  In the right column, ∗ the horizontal axis represents  or  and the vertical axis represents    29

competitiveness of the banking industry. We examine a setting in which banks choose the risk level of their loan investments in a competitive environment. The banks are subject to capital requirement, which is measured using accounting information. In case of capital deficiency, a bank is forced to sell a portion of its risky assets to boost its capital ratio. We find that the accounting information precision has a non-monotonic effect on banks’ risk-taking decisions, and its effect is contingent on the tightness of the capital requirement and the competitiveness of the banking industry. We find that when accounting information is not precise, surprisingly, increasing accounting precision may encourage risk taking. However, when accounting information is highly precise, increasing precision may discourage risk taking. When the accounting precision is of an intermediate level, both the capital requirement and the competitiveness strengthen the disciplining role of accounting precision. We further extend the model by introducing a liquidity cost and find that, for the capital requirement policy to be an effective tool in disciplining banks’ risk-taking behavior, either the banking industry should be highly competitive, or the accounting information should be sufficiently precise. Our study sheds light on the interaction between different sets of policies in the banking industry, and provides implications for bank regulators. In particular, our findings suggest that different regulatory bodies should not consider their policies in isolation. In fact, the lack of coordination among regulators can lead to undesirable consequences. A recent Federal Reserve staff report criticized “the fractured nature of the global financial regulatory framework...and a collection of one-off, uncoordinated decisions by accounting and regulatory bodies...” (Pozsar et al., 2010). Our paper supports their arguments by providing a theoretical basis. The study of this paper rests on a set of simplifying assumptions that insure the tractability of the model. We numerically examine the consequence of relaxing each of these assumptions and confirm that our main findings are robust. However, a complete examination of all the possible combinations and variations of these assumptions is beyond the scope of a single paper.

References [1] Allen, F., and E. Carletti. “Mark-to-Market Accounting and Liquidity Pricing.” Journal of Accounting and Economics 45 (2008): 358-378. [2] Allen, F., and D. Gale. Comparing Financial Systems, Cambridge, MA: MIT Press, 2000. 30

[3] Bagwell, K. “Commitment and Observability in Games.” Games and Economic Behavior, Vol. 8 (1995): 271-280. [4] Beck, T., A. Demirguc-Kunt, and R. Levine. “Bank Concentration and Crises.” Working paper, University of Minnesota, 2003. [5] Besanko, D., and G. Kanatas. “The Regulation of Bank Capital: Do Capital Standards Promote Bank Safety?” Journal of Financial Intermediation 5 (1996): 160—183. [6] Boyd, J., and G. Nicolo. “The Theory of Bank Risk Taking and Competition Revisited.” Journal of Finance 60 (2005): 1329-1343. [7] Burkhart, K., and Strausz. R. “Accounting Transparency and the Asset Substitution Problem.” Accounting Review 84 (2009): 689-712. [8] Buser, S., A. Chen, and E. Kane. “Federal Deposit Insurance, Regulatory Policy, and Optimal Bank Capital.” Journal of Finance 35 (1981): 51—60. [9] Calema, P., and R. Rob. “The Impact of Capital-Based Regulation on Bank Risk-Taking.” Journal of Financial Intermediation 8 (1999): 317-352. [10] Dick, A. “Nationwide Branching and its Impact on Market Structure, Quality and Bank Performance.” Journal of Business 79 (2006): 567-592. [11] Dietrich, J., and C. James. “Regulation and the Determination of Bank Capital Changes: A Note.” Journal of Finance 38 (1983): 1651— 1658. [12] Gennotte, G., and D. Pyle. “Capital Controls and Bank Risk.” Journal of Banking and Finance 15 (1991): 805-824. [13] Heaton, J., D. Lucas, and R. McDonald. “Is Mark-to-Market Accounting Destabilizing? Analysis and Implications for Policy.” Journal of Monetary Economics 57 (2010): 64-75. [14] Jayaratne, J., and P. Strahan. “Entry restrictions, Industry Evolution, and Dynamic Efficiency: Evidence from Commercial Banking.” Journal of Law and Economics 41 (1998): 239-275. [15] Jensen, M. C. “Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers.” The American Economic Review 76, No. 2, Papers and Proceedings of the Ninety-Eighth Annual Meeting of the American Economic Association (1986): 323-329.

31

[16] Keeley, M. “Deposit Insurance, Risk and Market Power in Banking.” American Economic Review 80 (1990): 1183-1200. [17] Koehn, M., and A. Santomero. “Regulation of Bank Capital and Portfolio Risk.” Journal of Finance 35 (1980): 1235-1244. [18] Laeven, L., and R. Levine. “Bank Governance, Regulation, and Risk Taking.” Journal of Financial Economics 93 (2009): 259-275. [19] Li, J. “Accounting for Banks, Capital Regulation and Risk-Taking.” Working paper, Carnegie Mellon University, 2009. [20] Maggi, G. “The Value of Commitment with Imperfect Observability and Private Information.” The RAND Journal of Economics 30 (1999): 555-574. [21] Matutes, C., and X. Vives. “Competition for Deposits, Fragility, and Insurance.” Journal of Financial Intermediation 5 (1996): 184-216. [22] De Nicolo, G. “Size, Charter Value and Risk in Banking: An International Perspective.” Working paper, International Finance Discussion Paper No. 689, 2000. [23] Padoa-Schioppa, T. “Bank Competition: A Changing Paradigm.” European Finance Review 5 (2001): 13-20. [24] Plantin, G., H. Sapra, and H. Shin. “Mark-to-Market: Panacea or Pandora’s Box?” Journal of Accounting Research 46 (2008): 435—460. [25] Pozsar, Z., T. Adrian, A. Ashcraft, and H. Boesky. “Shadow Banking.” Federal Reserve Bank of New York Staff Reports, No. 458, July 2010. [26] Shleifer, A., and R. Vishny. “Fire Sale in Finance and Macroeconomics.” Journal of Economic Perspectives 25 (2011): 29-48. [27] Suarez, J. “Closure Rules, Market Power, and Risk-taking in a Dynamic Model of Bank Behavior.” Working paper, London School of Economics/Financial Markets Group Paper No. 196, 1994.

Appendix I: Derivations of Banks’ Objective Function and Bayesian Probabilities Derivations of Banks’ Objective Function When a bad accounting signal is realized, the bank is forced to sell  portion of its loan. The expected proceeds from the assets sale will be X  )    + (1 − ) (|     − ) −   ( 

32

where the first term is the proceeds for assets sale, the second is the expected cash flow from the remaining loan held by the bank, and the third is the bank’s total expected deposit interest payment. We assume investors offer a fair price for the bank’s asset given their conjectures:   =

(1 − ) (1 −  )     (1 − ) (1 −  ) +    

Conditional on the bad accounting signal, the expected proceeds from the loan equals (|     − ) =

(1 − ) (1 −  )    (1 − ) (1 −  ) +  

When a good signal is realized, no assets sale is necessary and the bank receives (|     − ). Hence the bank’s total expected payoff, which is also its objective function, is X  )]  (|     − )+(1− ) [   +(1−) (|     − )−  ( 

Derivations of Bayesian ProbabilitiesDerivations of Bayesian Probabilities: The conditional probability of -state given a good signal is  =

 (1 −  ) ;  (1 −  ) + (1 − ) 

the conditional probability of -state given a bad signal is  =

(1 − ) (1 −  ) ; (1 − ) (1 −  ) +  

the probability that a good signal is realized is  =  (1 −  ) + (1 − )  ; and the probability that a bad signal is realized is  = 1 −  

33

Appendix II: Proofs Proof of Proposition 1 Proof. From the first-order conditions, we find that the investment size ∗ is proportional to the risk level  ∗ : ∗ =

1 − (1 − ) ∗    ( + 1)

Substituting this expression into the first-order condition of  ∗ gives the following quadratic equation: 1  ∗2 + 2  ∗ + 3 = 0

(5)

where the coefficients (1  2  3 ) are defined as functions of (  ): 1 = (2  − 1)[(1 − ) − 1 − ( + 1)(1 − )(1 − )]

2 = −3 + 4 +  (2 − 3) + (1 − )2 + ( + 1)(1 − )(2 −  − 3 + 2)

3 = ( + 1)(1 − )(1 − (1 − ))

A further analysis on the second-order condition shows that only the following solution is valid: p −2 − 22 − 4 1 3 ∗  =  (6) 2 1 Therefore, given our assumptions on (   ), there exists a unique equilibrium ( ∗  ∗ ). Proof of Proposition 2 Proof.  ∗  can be derived as follows:  ∗ =− 

1 

2 ∗ 3   +  2 1  ∗ + 2

 ∗2 +



First, the denominator 2 1  ∗ + 2 can be simplified as: q 22 − 4 1 3 

which is positive given that the equilibrium exists. Hence, the sign of  ∗  is solely determined by the numerator. We first examine the con∗2 + 2  ∗ + 3  0. With 1 dition for  ∗   0, which requires      a few algebra steps31 , this inequality can be reduced into a union of three conditions: 31

The steps are basic but tedious, which are available upon request.

34

• When

1 2

   23 , 0    1;

• When

2 3



• When

2 3



√ 2 2 , 0 √ 6−1 2 ,

   ();

b; — if  ()     (),    b and   b . — if  ()     (),    2

2

1−2  5−4 −4  and  () is the unique real root where  () = (1−) 2 ,  = 4(1−)2 of the following cubic polynomial with  as a parameter:

1 + 2  + 3 2 + 4 3 = 0 1 () = 4 2 + 4  − 5;

with

2 () = (8 − 14  + 10 2 − 16 3 + 12 4 ); 3 () = (−4 + 8  − 8 3 + 4 4 );

4 () = (1 − 4  + 6 2 − 4 3 + 4 )

In addition, b  is

[1 − (1 − )]2 [−5 + 4(1 − )2 + 4(1 + )] b  = 3(−2 + )(1 − )2 [22 + (1 − )2 − 1]

b is a function of (  ). and  ∗2 + 1 Now consider the condition for  ∗   0, which requires    2 ∗ 3   +   0. With similar steps, the inequality can be reduced into a union of three conditions: • When • When • When

√ 6−1 2 √ 3+1 4 2 3

   1, 0    1; 



√ 6−1 2 ,

 ()    1;

√ 6−1 2 ,

b; — if  ()     (),    b and   b — if  ()     (),    .

Figure 1 summarizes the conditions that determine the sign of  ∗  over the area ( ). There are four regions divided by three curves. The left curve is  (), the middle curve is  (), and the right curve is  (), 35

which are as defined above.  ∗  is always positive in the far-left region and always negative in the far-right region. In the middle-left region (the b and negative if    b, wider region),  ∗  is positive when    ∗ b where  is also defined as above. And in the middle-right region,   b and   b b and   b is positive when     and negative if    , where b  is also defined as above. Proof of Proposition 3 ∗ Proof. Our analysis shows that the first-order conditions of ∗ and  are: 1 +  (− +  ) − 2 [ + (1 − )(1 − )] + [ (2  − 1)](  −  ) = 0  − + 2  )(1 −  ) = 0 ( − 1+−

∗ ) and (     ) = ( ∗  ∗ ) for all banks. In equilibrium, (   ) = (∗       ∗ is From the first-order conditions, we have that the investment size  ∗ proportional to the risk level  : ∗  =

1 − (1 − ) ∗    ( + 1) 

Substituting this expression into the first-order condition of  ∗ gives us the following cubic equation: 1  ∗3 + 2  ∗2 + 3  ∗ + 4 = 0

(7)

where the coefficients (1  2  3  4 ) are defined as functions of (   ): 1 = −[1 − (1 − )](2  − 1){1 + ( + 1)[1 − (1 − )]}

2 = −[1 − (1 − )]{3 − 4  − 2 (1 − )2 −  [3  − 2 + (1 − )2 ]}

3 = ( + 1){−1 +  +  [(2 − (1 − ))(1 − )2 +   (1 + ) (1 − 2 )2 ]}

4 =    ( + 1)2 (1 − ) (2  − 1)

The cubic equation yields three real solutions. A further examination shows that there exists only one solution for ∗ listed below that satisfies the second-order condition, s.t., √ √ 2 2 (1 −  3) (1 +  3)(22 − 3 1 3 ) ∗ + +  = − 3 1 6 1 6 1 2 where 2 is a function of (1  2  3  4 ): r 3 1 (1 + 2 23 − 9 1 2 3 + 27 12 4 ) 2 = 2 q (2 23 − 9 1 2 3 + 27 12 4 )2 − 4 (22 − 3 1 3 )3  with 1 = 36

Proof of Proposition 4 Proof. Before the main proof, we first present the following two lemmas that will be used in deriving the results. Lemma 1 The optimal risk level ∗ 

+1 +5 .

Proof. To satisfy the second-order sufficient condition, the Hessian matrix evaluated at the optimal solution should be negative-definite with a positive determinant, such that: ¯ ¯ 2 ¯    2  ¯ [1 − (1 − )]2 (1 −  ) ( − +1 ) ¯  2  +5   ¯  0 ¯  2   2  ¯ = ¯ ¯   ( + 1) ( + 5) 2 





where  denotes a bank’s objective function. Since   1, the positive +1 determinant implies     +5 .

Lemma 2 For   3, satisfied:

∗  

is always negative if the following condition is    ∗ (∗    ) 

where  ∗ (∗    ) is a function of (    ) such that  ∗ (∗    ) =

∗ (1 − ){∗ 1 + ∗ 2 + 2 3 [∗2 (2 − 1) − ( + 1)(1 − )(1 − ∗ )]}  (1 + )2 (2  − 1)[(2  − 1) ∗ + 1 − ]

and 1  2  and 3 are, respectively, 1 = 5 − 4(1 − )2 − 6

2 = (2 − 1)[( + 1)∗ −  ]

3 = [1 − (1 − )

Proof. The first-order condition on  yields: ∗ = 

∗   ( + 1)

Substituting it into the first-order condition with respect to  along with ∗ ) and (     ) = ( ∗   ∗ ), the equilibrium conditions, (   ) = (∗       we get ∗ as an implicit function of :  (∗  ) = 0 37

Differentiate the equation with respect to  and by the implicit function  ∗ theorem, the derivative  can be derived as  (1 +  )2 (2  − 1)[(2  − 1) ∗ + 1 − ] [ −  ∗ (∗    )] ∗ =   1 ∗2 + 2 ∗ + 3 where  ∗ (∗   ) is defined in Proposition 4 and (1  2  3 ) are functions of (    ): 1 = −3 (2 − 1)[1 −  (1 − )][2 +  −  (1 − )]

2 = 2 3 {−3 + 4 + 2(1 − )2 +  [−2 + 3 + (1 − )2 ]}

3 = ( + 1) {1 −  −  [  ( + 1) (1 − 2 )2 + (1 + 3 )(1 − )2 ]}  ∗

The denominator of  is always negative when   3  along with our previous assumptions on (   ). Note that 1 is negative given 12    2 1 and 0    1. Further, it can be verified that +1 +5  − 2 1 . Now consider the denominator of

∗  

as a quadratic equation of ∗ :

(∗ ) = 1 ∗2 + 2 ∗ + 3  2 +1 ∗ ∗ Given 1  0 and +1 +5  − 2 1 , ( ) is strictly decreasing for   +5 (guaranteed by Lemma 1) and hence achieves its maximum at the point +1 ∗ = +1 +5 , the value of which (e.g. ( +5 )) is negative when   3. ∗ Therefore, the denominator, ( ), is always negative when   3.  ∗ To complete the proof, note that  is negative only when its numerator is positive, which yields    ∗ (∗    )  ∗

Now we will prove the existence of a threshold on  for   0. Recall  ∗ that by Lemma 2,  is negative if    ∗ (∗    ). The cutoff function  ∗ (∗    ) depends on ∗ and hence is also a function of  . To show that there exists a threshold b that is independent of ∗ , first we can show that  ∗ (∗    ) is strictly increasing in ∗ , which holds for   2. Meanwhile,  ∗ by Corollary 1,   0. Hence, according to the chain rule,  ∗ (∗    ) ∗  ∗ (∗    ) =  0  ∗   ∗

Now recall that  is always negative when    ∗ (∗    ). As  becomes larger, the LHS of the inequality increases while the RHS decreases 38

 ∗ ( ∗ )

 (  0), making it more likely for the equality to be satisfied.   ∗ Therefore, by continuity, there exists a threshold b such that  is negative when   b32 Proof of Proposition 5  ∗ Proof. Recall that according to Lemma 2,  is negative if    ∗ (∗    ). As  goes to infinity, the cutoff function  ∗ (∗    ) approaches zero.33

lim  ∗ (∗   ) = 0

 →∞

That is, the condition    ∗ (∗    ) always holds for any   0. Similarly, when  approaches 1, the cutoff function  ∗ (∗    ) also goes to zero.34 lim  ∗ (∗    ) = 0 →1

32 Since when  goes to infinity, the inequality is always satisfied. Hence by continuity, the inequality always holds for sufficiently large  . Numerically,  is around 0.00005 and always below the limited-liability upper bound  . 33 ∗ ∗ Note  ∗ (   ) also depends on  , which varies with . When  goes to infinity, ∗ lim →∞  = 1 by lemma 1 in the appendix. 34 ∗ ∗ Similar to the limit case when  goes to infinity,  ∗ (   ) also depends on  , ∗ which varies with . However, when  approaches 1, the limiting value lim→1  remains ∗ ∗  1) and hence will not affect lim→1  ∗ (   ). finite (0  

39