Bank Risk Dynamics Where Assets are Risky Debt Claims

Authors


  • The authors thank Dan Galai, Michel Crouhy, Zvi Wiener, Menachem Abudy, Beni Lauterbach and Linda Allen as well as the participants in the International Finance And Banking Society (IFABS) 2016 Conference and the International Risk Management Conference (IRMC) 2016 for useful comments. Raviv acknowledges the financial support of this study by the Israel Science Foundation (ISF) through grant number 969/15.

Abstract

The structural approach views firm's equity as a call option on the value of its assets, which motivates stockholders to increase risk. However, since bank assets are risky debt claims, bank equity resembles a subordinated debt. Using this assumption, and considering the strategic interaction between a bank and its debtor, we argue that risk shifting is limited to states in which the debtor is in financial distress. Furthermore, risk shifting increases with bankruptcy costs and decreases with bank capital. Thus, increasing a bank's capital affects stability, not only through the additional capital buffer, but also by affecting the risk shifting incentive.

1 Introduction

The structural approach for pricing corporate liabilities, developed by Merton (1974), views debt and equity as contingent claims on the firm's assets.1 In this framework, since the value of a firm's stock, which is equivalent to a call option, is positively related to the underlying asset's volatility, a stockholder aligned manager would be motivated to increase asset risk (Galai and Masulis, 1976; and Jensen and Meckling, 1976). However, rational creditors will consider these incentives when determining credit conditions. The financial literature suggests that banks are especially good at setting credit conditions and covenants, and monitoring their borrowers to limit risk shifting (Brealey et al., 1977; Campbell and Kracaw, 1980; Diamond, 1984; and Fama, 1985).

In contrast, empirical and theoretical work suggests that creditors' ability to limit risk shifting is more restricted in financial institutions as creditors are small and dispersed and bondholders may have explicit (deposit insurance) or implicit (too big to fail) guaranties. Moreover, banks are more complex and opaque with greater information asymmetries than non-financial firms; this makes bank asset risk hard to observe and assess while being easy to change and manipulate.2

In a recent paper, Nagel and Purnanandam (2015) claimed that, since bank's assets are risky loans, their value is capped.3 Therefore the value of the bank's stock is not equivalent to a call option on its asset as implied by the basic structural approach. Instead, a bank's assets are contingent claims on the value of their debtors' assets (see Figure 1). Thus, bank equity is economically equivalent to an ‘option-on-option' with a payoff function identical to that of subordinated debt, which can be replicated by a ‘bull spread' strategy (Black and Cox, 1976).4 Moreover, since bank assets are risky debt claims with limited upside, asset volatility depends on the debtor's asset value. Consequently, when assuming that bank assets follow a log-normal stochastic process with a constant volatility, the bank's probability of default and its costs of deposit insurance in bad times, when leverage is high, are underestimated.

Figure 1.

The value of the bank's asset, debt and equity at debt maturity The values in the figure refer to a bank with a single asset − a corporate loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with a face value of 60 that matures in one year.

Relying on the same assumption that bank assets are risky debt claims with limited upside (Dermine and Lajeri, 2001; Nagel and Purnanandam, 2015) and using the option pricing approach (Black and Scholes, 1973; Merton, 1974), we analyse the risk taking motivation of a bank's stockholder using a principal-agent model. We assume that the debtor – that is, the borrowing corporation – can shift the risk of its assets from its initial level with the consent of the creditor – the bank. The corporation and the bank are both managed by an equityholder aligned manager, whose goal is to maximise equity value. We determine if the stockholders are motivated to increase asset risk from its initial level and characterise the equilibrium level of asset risk. Moreover, we analyse the effect of risk shifting on the liabilities of the bank and the corporation, the cost of deposit insurance and the bank's probability of default.

We prove that, in contrast to the basic agent theory in which the stockholder is motivated to shift risk in both solvent and insolvent states (Jensen and Meckling, 1976), in our setting, the bank's stockholder is motivated to shift risk only when the debtor is in financial distress, that is, when the value of the debtor's assets is below the face value of the debtor's debt. Specifically, using a closed-form solution, we prove that risk shifting may occur only if the value of the debtor's assets is lower than the geometric mean of the face values of the bank's and debtor's debt.

As the value of the debtor's equity increases with asset risk, its stockholder would be motivated to increase asset risk in any state (Jensen and Meckling, 1976). In contrast, the creditor's risk taking motivation depends on the value of the debtor's assets. Here we distinguish between two cases. In the first case, where the value of the debtor's assets is above the risk shifting threshold, located between the face values of the debtor's debt and the creditor's debt, an increase in asset risk will decrease the value of the bank's equity. Therefore, an equityholder aligned bank manager is motivated to monitor the debtor tightly and restrict risk shifting by the debtor's stockholder. In equilibrium, risk shifting would not occur because, one of the stockholders – either of the debtor or of the creditor – would lose if the initial level of risk is changed.

In the second case, where the debtor is in financial distress and the value of the debtor's assets is below the threshold described above, the relationship between asset risk and the value of the bank's stock is hump-shaped. In this case, the bank's stockholder would willingly tolerate a shift in asset risk by the debtor, the corporation, if the shift increases its equity value. Therefore, in equilibrium the chosen level of asset risk is the level that maximizes the bank stockholder's position, that is the maximum level of the humped curve.

In our setting, the level of asset risk set in equilibrium depends on the bank's initial capital ratio. With all else equal, as the bank's equity layer increases, the stockholder's motivation to shift risk decreases. The result strengthens the argument for a higher capital adequacy in banks (Admati et al., 2013; Miles et al., 2013; and Turner, 2010), since the increase in a bank's capital not only decreases its probability of default and the cost of deposit insurance through the higher capital buffer, but also reduces the risk taking motivation of the bank's stockholders.5 Further, we show that an increase in leverage due to a negative shock to the value of debtor's assets, increases the probability of risk shifting. This can be explained intuitively, as the value of the corporation's assets decreases the payoff from an upside movement increases while the downside risk is constant.

According to the structural approach, in the event of firm failure equityholders surrender the firm to debtholders, who proceed to operate it.6 However, firm failure involves the costs of distress and bankruptcy, which may be substantial (Anderson and Sundaresan, 1996; and Sundaresan and Wang, 2014). Introducing these costs into our model increases risk shifting. First, the threshold of asset value under which equity value is hump-shaped with respect to asset risk is higher than the case with no bankruptcy costs, and equal to the face value of the debtor's debt discounted by the risk-free rate. Thus, with relatively high bankruptcy costs, a bank capital structure has no effect on the value of assets in which risk shifting might take place. Second, there is a positive relationship between bankruptcy costs and the asset risk chosen in equilibrium as long as the bankruptcy costs are lower than the bank's capital. However, a further increase in bankruptcy costs above the bank's capital, would not affect the level of asset risk chosen in equilibrium.

Finally, following the 2007–2009 financial crisis, federal regulators undertook a unique supervisory capital assessment programme under which large, complex bank-holding companies are expected to run stress tests to prove that they can manage real economic activity even under adverse economic conditions. By calibrating our model to typical market data, we demonstrate that not accounting for the possibility of risk shifting in bad economic states may lead to severe underestimation of the cost of deposit insurance. However, since risk shifting when bank assets are risky debt claims is limited to states in which the value of the debtor's assets is below the face value of the debtor's debt, the effect of risk shifting on a bank's probability of default is relatively minor.

1.1 Literature review

The literature regarding financial institutions' asset risk focuses on several major conflicts: the conflict between bank stockholders and depositors, which are usually represented by a benevolent regulator; the conflict between a bank and its debtors and the conflict between a bank's executives and its claimholders (stockholders and debtholders).

The conflict between equityholders and debtholders regarding the level of asset risk is described in the early work of Jensen and Meckling (1976) and Galai and Masulis (1976). A firm's equityholders are motivated to increase asset risk-since their payoff is a convex function of firm value due to the limited liability principle. Risk shifting increases equity value at the expense of creditors. Thus, by selecting riskier projects in a way that is not anticipated by creditors, equityholders can transfer wealth to their own benefit. Similarly, we assume that the stockholders of both the borrowing corporation and the lending bank are trying to maximise the value of their holding by choosing the level of asset risk without regard to the total value of the firm's assets. However, in our model, bank risk can only be increased by changing the asset risk of the debtor. Therefore, risk shifting from the initial level of asset risk can occur only with the consent of the stockholders of both – the debtor and the creditor. This assumption requires us to account for the strategic interaction between these stockholders.

The conflict between stockholders and bondholders in financial institutions appears to be mainly focused on the ability of the depositors and their representatives, the regulators, to monitor the level of a bank's asset risk (Acharya et al., 2016; Agoraki et al., 2011; Delis and Staikouras, 2011; Hilscher et al., 2015). However, over the last few decades, as the size and complexity of financial firms has increased, the regulators' ability to control and monitor bank's asset risk has deteriorated (Berger et al., 2000; Caprio and Levine, 2002; DeYoung et al., 2001; and Morgan, 2000). Moreover, the existence of deposit insurance motivated banks to take more risks and effectively shifted the agent problem from the bank's creditors to the regulators (Allen et al., 2015; Cooper and Ross, 2002; and Demirgu-Kunt and Detragiache, 1998).

The assumption that a bank's assets are risky debt claims limits the conflict between bank creditors and equityholders to states in which the bank's debtor is in financial distress, or more precisely, where the value of its assets is below the geometric mean of the face values of the debtor's and creditor's debts. This result holds true even though throughout most of our analysis we assume that a bank's debtholders and regulators are unable to monitor and restrict the bank's asset risk.

The conclusion that banks are motivated to shift asset risk in states of financial distress appears several times in the theoretical literature (Bruche and Llobet, 2014; and Diamond and Rajan, 2011). The idea that a stockholder's motivation to shift risk increases in states of financial distress is referred to as gambling for resurrection (Dewatripont et al., 1995; and Mishkin, 1992). When the value of a bank's equity is depleted, a bank may willingly take on large risks even if these risks are associated with low expected returns. If these gambles pay off, the bank may survive; if they do not, the bank would have failed anyway (Boyd and Hakenes, 2014). While the idea of risk shifting in financial distress appears in these papers as well, we consider the interaction that occurs between the stockholders of the bank and the debtor when bank assets are risky debt claims.

In this setting, we ignore possible deviations between the incentives of the bank's manager and claimholders, which can lead to risk shifting in solvent states, as described in a number of recent works, mostly written after the recent financial crisis. These works describe the positive effect of equity-based compensation on risk taking (Bhattacharyya and Purnanandam, 2011; and Cheng et al., 2010) and the negative effect of debt-like compensation, which is sometimes referred to as inside debt (Edmans and Liu, 2011; and Raviv and Sisli-Ciamarra, 2013).7

Finally, this paper is related to the debate that arose following the 2007–2009 financial crisis regarding the appropriate ways to regulate ‘too big to fail' financial institutions. First, the paper highlights the relationship between leverage and risk taking, strengthening the argument for a higher bank capital ratio (Admati et al., 2013; and Admati et al., 2011). Second, after the financial crisis, regulators around the world started conducting periodic stress tests for financial institutions. These are forward-looking assessments designed to determine if a bank would have adequate capital to withstand negative shocks in the future (Bayazitova and Shivdasani, 2012; Gofman, 2011; Goldstein and Sapra, 2014; Goldstein and Leitner, 2015; Greenlaw et al., 2012; and Peristiani et al., 2010). Our model contributes to this approach as we estimate a bank's resilience under the assumption that the bank's assets are risky debt claims.

The rest of the paper is organised as follows. Section 2 presents the liability structures of the corporation and bank and the valuation of the different claimholders' positions. Section 3 presents our equilibrium model for risk shifting, in which we also refer to the existence of bankruptcy costs. Section 4 includes a quantitative analysis of the effects of capital, bankruptcy costs and asset value on risk shifting. Section 5 concludes the paper.

2 Liability Structure and Valuation

This section describes the liability structures of both the bank and the corporation, defines the value of the claimholders' positions and defines the bank's probability of default and the cost of deposit insurance. We consider a single corporation and bank financed by equity and debt. Throughout the paper it is assumed that the bank and corporation are both managed by equity-aligned managers aiming to maximise the value of equity. Thus we refrain from an agency problem between the equityholder and manager, and focus on the agency problem between the equityholder and debtholder. It is also assumed that the managers of the bank and of the corporation are both fully informed, thus avoiding problems of information asymmetry.

2.1 The corporation's liability structure

The corporation is funded by equity with market value of SC and a single loan with face value of FC and market value of BC.8 The loan is a zero-coupon loan maturing at time T and the bank is the sole creditor. The value of the firm's assets, VC, follows a geometric Brownian motion according to the following equation:

display math(1)

where μ is the instantaneous expected return on the corporation's assets, σ is the volatility of the corporation's assets, and dW is a standard Wiener process. The event of default occurs at debt maturity, T, if the value of the asset, VT, is lower than the face value of the debt. If default occurs, the creditor takes over the firm without incurring any distress costs (this assumption is removed in section 3.3) and realises the residual assets of the firm, math formula. Otherwise, the debt is fully paid and the creditor, the bank, receives the entire face value of the debt, FC. The payoff to the corporation's debtholder can be expressed as math formula, or when rearranged:

display math(2)

As developed by Merton (1974), equation (2) is equivalent to the payoff of a risk-free debt minus a European put option. Therefore, the present value of the corporate debt is given by:

display math(3)

where r is the risk-free rate and math formula is the price of a European put option according to the Black and Scholes (1973) framework.

As the equity is the residual claim, its payoff at debt maturity is:

display math(4)

The value of the corporation's stock prior to debt maturity can be replicated by a European call option on the value of the corporation's assets, with a strike price equal to its face value of debt (Galai and Masulis, 1976):

display math(5)

where math formula is the price of a European call option as developed by Black and Scholes (1973).

2.2 The bank's liability structure

The bank is funded by equity with a market value of SB and zero-coupon debt with a face value of FB and market value of BB. It is assumed that the debtholders are many small depositors, who are unwilling or unable to monitor the bank manager's actions. Therefore, the bank's stockholder is the only claimholder who controls the level of the debtor's asset risk. As discussed in Marcus and Shaked (1984), due to the periodic frequency of supervisory audits, bank deposits are analogous to a debt claim with an effective maturity equal to the examination interval which we assume to be T.

As the bank has a single asset, the value of the bank's assets is equal to the value of the loan, math formula. Accordingly, the bank's assets payoff at maturity, math formula, is identical to the payoff of the corporation's debt, as expressed in equation (2). Therefore, the value of the bank's assets prior to debt maturity, VB, can be replicated by a long position in a risk-free debt and a short position in a European put option as described in equation (3). Since the bank is fully informed of the debtor's asset value and choice of risk, the bank can monitor the debtor. Consequently, the corporation can only shift risk from the initial level of asset risk set in the loan contract with the bank's consent (as explained in detail in section 3.1).

The payoff at maturity to the bank's debtholder is the minimum between the value of the bank's assets and the face value of the bank's debt, FB, and can be expressed as:

display math(6)

It is assumed that the bank is solvent at the time of debt issuance and at regulatory audits. If the bank is insolvent at the time of audit, the regulator will use the bank's residual assets to service the debt. This assumption requires the existence of a positive gap between the face value of the corporate debt, FC, and the face value of the bank's debt, FB. The corporate loan, which is the bank's single asset, is financed by both equity and debt. Therefore, the face value of the bank's loan is always lower than the total face value of the corporation. Under this assumption, equation (6) can be expressed as:

display math(7)

The payoff of the bank's debt can be replicated by a long position in a risk-free debt with a face value of FB and a short position in a European put option on the corporation's assets, with a strike price equal to the face value of the bank's debt. Therefore, the value of the bank's debt prior to debt maturity can be expressed as:

display math(8)

As the bank's equityholder is the residual claimant, its payoff at maturity is math formula. If the corporation is solvent at maturity, the equityholder receives the payoff math formula, which is the maximum payoff that the bank's equityholder can receive. This differs from the basic structural approach in which the equityholders' payoff is unbounded.

The bank equityholder's payoff at maturity can be rearranged and expressed as:

display math(9)

This payoff can be replicated by a long position in a call option, with a strike price equal to the face value of the bank's debt, FB, and by a short position in a call option, with a strike price equal to the face value of the corporation's debt, FC. Therefore, the value of the bank's equity prior to debt maturity is:

display math(10)

The above option position is known as a ‘bull spread' strategy; it was used to model junior debt by Black and Cox (1976). In our setting, the bank's equity acts similarly to a junior debt. Since the bank's asset is a single corporate loan and the corporation's single creditor is the bank, the bank's creditor has seniority in the event of the corporation's default. The bank's equityholder receives a positive residual only if the bank's creditor is fully paid. If the creditor is fully paid by the debtor, that is, if the bank's stockholder receives the amount of math formula, the corporation's stockholder receives a positive value − the residual assets of the corporation. Thus, the bank's equity is equivalent to a mezzanine debt in the corporation's capital structure. The position of each of the bank's claimholders and the value of the bank's assets at maturity are described in Figure 1.

2.3 Probability of default

In our setting the bank is in default if the value of the corporation's assets is below the face value of the bank's debt at debt maturity. As in Merton (1974), the bank's risk-neutral probability of default can be calculated using the following equation:

display math(11)

where math formula. The bank's probability of default decreases with the value of the corporation's assets as described by Figure 2. Equation (11) can be used to calculate the bank's probability of default following a negative shock to the value of the corporation's assets. This is equivalent to conducting a stress test for the bank under adverse scenarios. As discussed in section 3, when we take into account the possibility of risk shifting by using our equilibrium model, the probability of default may increase further.

Figure 2.

The bank default probability with asset risk that maximizes the position of the bank's equityholder The values in the figure refer to a bank with a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and deposits with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. The dotted line depicts the bank's probability of default when asset risk is constant and equal to the initial volatility of the corporate assets, 20%. The solid line is the bank's probability of default when the value of asset risk is the value that maximises the bank equityholder's position. The vertical dashed line indicates the threshold for risk shifting, V*.

2.4 The cost of deposit insurance

In the existence of deposit insurance, if the value of the bank's assets is below the face value of its deposit at maturity, the guarantor compensates the depositors with the difference between the two. The value of the deposit insurance equals the difference between the value of a risk-free debt with a notional amount equal to the face value of the secured deposit, and the value of the bank's risky debt. Thus, as discussed in Merton (1977) and Crouhy and Galai (1991), the insurance is equivalent to a long put option on the corporation's assets with a strike price equal to the face value of the bank's debt: math formula. The value of the deposit insurance per dollar of insured deposits is defined as:

display math(12)

The value of the deposit insurance increases with the corporation's asset risk, σ, and decreases with the corporation's asset value, VC (Figure 3). As shown in section 3, when taking into account the increase in asset volatility as a result of a decrease in the value of the debtor's assets, the increase in the value of deposit insurance may be substantial.

Figure 3.

The value of deposit insurance with asset risk that maximizes the bank equityholder's position The values in the figure refer to a bank with a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and deposits with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. The dotted line depicts the value of deposit insurance per dollar of deposits when the corporate asset risk is constant and equal to the initial asset risk of the debtor's assets, 20%. The solid line is the value of the deposit insurance per dollar of deposits when the value of asset risk is the value that maximises the bank equityholder's position. The vertical dashed line indicates the threshold for risk shifting, V*.

3 An Equilibrium Model For Risk Shifting

3.1 The framework of analysis

In our model the bank sets the face value of the corporate loan, FC, to account for the corporation's initial asset risk, math formula. When the loan is initiated, the bank's liabilities, the stock and the secured deposit, are fairly priced according to the corporation's asset risk and leverage.

It is assumed that the stockholders of both the corporation and the bank are trying to maximise the value their holdings without regard to the effect on the value of the bank or corporation assets. Thus, in the event of a risk shifting opportunity, which can increase the value of their holding at the expense of other claimholders – the bank depositor or the corporation debtholder, the stockholders would not reject it.

We assume that, some time after the contract is set, an exogenous shock to the value of the corporation's assets occurs, decreasing or increasing the value of the bank's assets. The change in asset value may change the sensitivity to asset risk of the bank's stock. The bank's stockholder might be willing to tolerate a change in asset risk by the debtor's stockholders. Therefore, the equilibrium solution for the decision variables and the value of the stockholders' positions are determined by a two-step backward induction. First, the corporation's stockholder chooses the level of asset risk, σ*, which maximises the value of her position, SC. This decision is made by considering the upper and lower bounds on asset risk set by the bank's stockholder: math formula. However, the domain of asset risk must contain the initial level of asset risk, math formula. Thus, the bank cannot force the corporation to change its asset risk from the initial level set in the terms of the contract. A shift in the level of asset risk may occur only if the two counterparts − the stockholders of the corporation and the bank − are both better off.

Second, after analysing the decision of the corporation's stockholder, the bank's stockholder chooses the lower and upper bounds on asset risk, math formula and math formula. In this domain, the value of bank stockholder's position, SB, is always greater than or equal to its initial value. Both math formula and math formula are positive and limited from above by the existing technologies governing the corporation's asset risk. If the stockholders of the bank and the corporation both chose a strategy and cannot both benefit from changing one strategy while the other remains unchanged, then the set of strategy choices and corresponding payoffs constitute a Nash equilibrium. We define the set of parameters and pay-offs in such an equilibrium as: math formula.

3.2 Risk shifting in equilibrium

3.2.1 The risk preference of the corporation's equityholder

The value of the corporation's stock, a call option on the value of its assets, increases with asset volatility, as illustrated in Figure 4. This result stems from the fact that the value of a call option increases with asset risk, as shown in Jensen and Meckling (1976). Thus, in all states of the world the corporation's stockholder would be motivated to increase asset risk from its initial level. Consequently, a lower bound on asset risk, math formula, set by the bank below the initial level of asset risk is always unbinding. Therefore, we ignore it throughout the rest of this paper. However, as discussed above, such a shift can occur only with the consent of the bank's stockholder.

Figure 4.

The value of the debtor's stock as a function of its asset risk The values in the figure refer to a corporation with a single debt instrument with a face value, FC, of 80 that matures in one year and where the risk-free rate is 1%. We analyse three scenarios where the corporation's asset value is 50, 70 and 100.

3.2.2 Bank equityholder risk preference

As demonstrated in Appendix A1, the sensitivity of the bank's equity value to the debtor's asset risk is divided into two segments defined by the following threshold of the corporation's asset value:

display math(13)

In the first case, when the value of the corporation's assets is above this threshold, V*, the bank equityholder's sensitivity to asset risk is negative (Figure 5). Hence, the value of the bank's stock decreases monotonically with asset risk. In this case, the bank stockholder would not tolerate any increase in asset risk by the corporation's stockholder, and the upper bound on asset risk is equal to the initial level of asset risk, math formula. Conversely, the position of the corporation's stockholder increases with asset risk. Therefore in equilibrium her choice would be to set asset risk at its maximum possible level, which is equal to the initial level of asset risk. The result of these conflicting interests is an equilibrium in which asset risk is equal to the initial level of asset risk, math formula.

Figure 5.

The value of bank equity as a function of the debtor's asset risk The values in the figure refer to a bank with a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. We plot the value of bank equity for three different corporate asset values: 100, 90 and 80. All of these values are above the threshold for risk shifting, V* which equals 67.2.

In the second, complementary case, in which the value of the assets is below the threshold, V*, the relationship between the value of the bank stock and asset risk is non-monotonic, and has a hump-shaped structure (Figure 6). The value of the bank's equity first increases with asset risk until it reaches its maximum value, then begins to decrease as asset risk increases further. The bank equityholder's position has a constrained maximum, math formula, which is defined in Appendix A1 as:

Figure 6.

The value of bank equity as a function of the debtor's asset risk The values in the figure refer to a bank with a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. We plot the value of bank equity for three different asset values − 67, 66, and 65 –all of which are below the threshold of risk shifting, V*, which is 67.2.

display math(14)
Figure 7.

The debtor's asset volatility that maximizes the position of the bank's equityholder The values in the figure refer to a bank whose assets consist of a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and deposits with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. The solid line is the asset risk of the debtor which maximises the position of the bank's equityholder, and the dotted line is the initial level of asset risk, which equals 20%. The vertical dashed line indicates the threshold for risk shifting, V*.

The level of the corporation's asset risk which maximises the position of the bank's equityholder for different asset values is presented in Figure 7.

Proposition 1. When the value of the corporation's assets is below the threshold level, V*, which is equal to the discounted geometric mean of the face values of the debtor's and creditor's debts, risk shifting to a higher level than the initial level of asset risk may occur in equilibrium. Thus, the condition is a necessary, but not sufficient condition for risk shifting.

Proof. When the value of the corporation's assets is below the threshold, V*, the value of the bank's equity is hump-shaped with respect to asset risk and the bank equityholder's position has a constrained maximum, math formula. When math formula, the bank's equityholder is better off by shifting to the lower level of asset risk, math formula. However, the corporation's equityholder would prefer not to reduce asset risk from its initial level because the value of her position increases with asset risk. Therefore, the equilibrium solution in such a case is the initial level of asset risk, math formula, and risk shifting would not occur. Conversely, if math formula the bank equityholder would tolerate an increase in the level of asset risk up to the point that maximises her position, math formula. Therefore, the bank would set the upper bound on asset risk at this level. Since the position of the corporation's equityholder increases with asset risk the result would be a shift in the level of risk from its initial level to the level of math formula.

Risk shifting only occurs under the following two conditions: (1) the value of assets is below some lower threshold, V*, and (2) at a given value of assets, the initial level of asset risk is below the level that maximises the position of the bank's stockholder, math formula. This is in contrast to the basic agent theory, developed by Jensen and Meckling (1976), in which risk shifting may occur in any state.

When the value of the corporation's assets is below the threshold, V*, the bank's equity value is maximised with a positive level of asset risk. However, risk shifting would occur only if asset value is below an additional threshold, V**. As proved in Appendix A1, when the value of the corporation's assets equals this threshold, the risk that maximises the position of the bank's equityholder is equal to the initial level of asset risk, math formula. We define V** as:

display math(15)

Note that V* is above V** whenever assets are risky. Also, both V* and V** depend on the geometric mean of the face values of the bank's debt and the corporate's debt. Since the face value of the bank's debt is lower than that of the corporation's debt, both thresholds are lower than the face value of the corporation's debt. Thus, when the value of the corporation's assets crosses these thresholds the corporation is already in financial distress.

Proposition 2. If the value of the corporation's assets is below the threshold V**, which is equal to the geometric mean of the face value of the debtor and creditor's debt discount by math formula), the asset risk would be shifted from its initial level, math formula, to the level that maximises the position of the bank's stockholder, math formula, as defined in equation  (14).

Proof. As shown in Appendix A1, when the corporation's asset value is below V** we find that math formula and, therefore, as proven in Proposition 1, the corporation's asset risk would increase to math formula which is higher than the initial risk.

While we show that a necessary condition for risk shifting is asset value below the lower threshold, V*, we did not relate this to the effect of leverage. Therefore, as in Merton (1974), we define the quasi leverage ratio as math formula. We can now express the lower threshold for risk shifting, which appears in equation (13), in terms of the bank's and corporation's leverage ratios. The value of the corporation's assets is equal to the lower threshold for risk shifting when the geometric mean between the bank and the corporation leverage ratio equals one: math formula. Thus, the bank's equityholder might be motivated to increase asset risk only if the corporation is in financial distress.

The expression for the level of asset risk that maximises the position of the bank's equityholder can be expressed in terms of leverage by rearranging equation (14):

display math(16)

The level of asset risk that maximises the position of the bank's stockholder increases with its leverage as well as the corporation's leverage. Since an increase in asset risk, increases the bank's probability of default and its costs of deposit insurance, our paper supports the advocates for higher capital ratio or lower leverage in banks. The decrease in leverage not only decreases a bank's probability of default due to the higher capital buffer, it also reduces events of risk shifting.

Thus far, we have assumed that by mutual agreement between the creditor and the debtor, asset risk can be shifted to any level. However, risk shifting is often limited by either the available technology or the regulator. Thus, it is interesting to study the equilibrium solution when the level of risk that maximises the position of the bank's stockholder cannot be reached. In this case, we try to ascertain if a bank's stockholder has incentive to shift risk locally, that is, above its initial level. If at the current level of asset risk the value of a bank's assets is below V**, the bank is motivated to shift asset risk.

Proposition 3. If the level of the corporation's risk is limited to the domain math formula, where math formula is lower than the level that maximises the position of the bank's stockholder, math formula, and if the value of the corporation's assets is lower than V**, then risk shifting would occur from its initial level of math formula, to the level of math formula.

Proof. As proven in Appendix A1, when math formula the equityholder's position increases with asset risk in the range math formula and decreases in the range math formula. Since math formula is lower than math formula the bank's equityholder would prefer math formula over the initial level of asset risk. Since the corporation's equityholder would like to increase asset risk as well, both will agree to increase asset risk to the maximum possible level math formula.

3.3 Risk shifting with bankruptcy costs

Our model accounts for bankruptcy costs by assuming that, if the corporation is in default at debt maturity, it's claimholders will bear costs of (1 − α) of its asset value, where math formula.9 Thus, the creditor's recovery rate is α and the value of the residual assets shrinks to αVC upon default. Following the convention in the financial literature (Anderson and Sundaresan, 1996; and Sundaresan and Wang, 2014), we assume that the recovery rate is constant and known upfront.

If at maturity the corporation's asset value is above its face value of debt, FC, the debtor is fully paid and receives the entire face value of the debt, FC. Otherwise, if the asset value is below FC, the creditor receives the residual assets of the corporation after bankruptcy costs, αVC. Therefore, the value of the bank's assets at maturity can be expressed as:

display math(17)

where 1{*} is an indicator function. The first term on the right-hand side is a long position of α units of the corporation's risky debt, as expressed in equation (2). The last term on the right-hand side is the payoff at maturity from a long position of (1 − α) units of a binary call option, with a strike price equal to the face value of the corporation's debt. A binary call option, also known as a digital option, pays out one unit of cash if the corporation's asset value is above the strike price at maturity.

The value of the bank's assets prior to debt maturity is given by:

display math(18)

where math formula is the price of a binary call option according to the Black and Scholes equations.10

The bank's claimholders’ have payoff functions which depends on the ratio between its the bank's book value capital ratio, defined as (FC − FB)/FC, and the costs of bankruptcy, (1 − α). We focus on the case where bankruptcy costs are greater than the bank's capital: (1 − α)> (FC − FB)/FC. This case can also be expressed as math formula. We focus on this case since it has a stronger effect on risk shifting.

Case 1: Bankruptcy costs are greater than the bank's capital ratio: math formula. In this case, the default event of the corporation leads automatically to a default event of the bank due to the large bankruptcy costs, which exceed the stake of the bank's stockholder in the corporation's debt (Figure 8). If the corporation is solvent at maturity, math formula, the bank's creditor is fully paid with the amount FB. If default occurs, math formula, the payoff of the bank's creditor is equal to the residual value of the bank's assets math formula. Therefore the value of its debt at maturity can be expressed as:

display math(19)
Figure 8.

The value of the bank's assets, debt and equity at debt maturity where the recovery rate is lower than the bank's face value of debt, αFC < FB The bank has a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. Bankruptcy costs are 30% of asset value.

The value of the bank's debt at any time prior to debt maturity can be expressed as:

display math(20)

As the residual claimant, the payoff of the bank's equityholder at maturity is math formula. This position can be replicated by a long position in math formula units of a binary call option, with a strike price equal to the face value of the corporation's debt. The value of the bank's stock can be expressed as:

display math(21)

In contrast to the debt, the value of the bank's equity does not depend on the cost of default, math formula. In the event of default, the bank's stockholder receives zero, that is, the stockholder is swapped out by its bondholder.11 When the value of the assets is below the threshold, V*, risk shifting to a higher level of asset risk may occur. The threshold, which is derived in Appendix A2, is expressed as follows:

display math(22)

In the first segment, above the threshold, the bank's equity value monotonically decreases with asset risk. Therefore, following the logic explained above for the case with no bankruptcy cost, the equilibrium asset risk is equal to the initial level. Below the threshold, the bank's equity value is hump-shaped with respect to asset risk, and the equityholder's position has a constrained maximum, math formula, which is defined in Appendix A2 as:

display math(23)

Proposition 4. In a model with bankruptcy costs which exceed the stake's of the bank's stockholder such that math formula, when the value of the corporation's assets is below the threshold level, V*, which is equal to the discounted value of the face value of the corporation's debt, risk shifting from the initial level of asset risk to a higher level may occur in equilibrium. Thus, the condition is necessary, but not sufficient, for risk shifting.

Proof. The proof follows the proof of Proposition 1.

A shift to a level of asset risk higher than the initial level, math formula, would occur only if the corporation's asset value is below the threshold, V**, which is derived in Appendix A2 as:

display math(24)

Proposition 5. In a model with bankruptcy costs which exceed the stake's of the bank's stockholder such that math formula, if the corporation's asset value is below V**, which is equal to the face value of the corporation's debt discounted by math formula, risk shifting will occur in equilibrium to the risk level of math formula, as defined in equation  (23).

Proof. The proof follows the proof of Proposition 2.

Based on our analysis, bankruptcy costs have several effects on risk taking. First, since the default event of the corporation leads to the bank's default, the value of the bank's debt and its leverage does not affect the equityholder's payoff or their decisions in equilibrium. Second, while in the model with no bankruptcy costs risk shifting occurs when the asset value is lower than the geometric mean of the face values of the bank's and debtor's debts, in the model with bankruptcy costs risk shifting would occur also for higher asset values, which are below the discounted value of the corporation's face value of debt.

Case 2: Bankruptcy costs are smaller than the bank's capital ratio, math formula. While in the previous case the corporation's default leads immediately to the default of the bank due to the high bankruptcy costs, in this case the relative low costs of bankruptcy may leave the stockholder with a residual value in the event of the corporation's default (see Figure 9). When the value of the bank's assets is greater than the ratio between the face value of the bank's debt and the recovery rate, math formula, the bank's creditor is fully paid math formula. Otherwise, the creditor receives the residual assets of the corporation, math formula. Therefore, the debtholder's payoff at maturity is:

display math(25)
Figure 9.

The value of the bank's asset, debt and equity at debt's maturity where the recovery rate is greater than the bank's face value of debt, αFC > FB The bank has a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with a face value of 60 and time to maturity of one year. The risk-free rate is 1%. Bankruptcy costs are 10% of asset value.

The payoff can be replicated by the following options strategy:

display math(26)

At maturity, the payoff of the bank's equityholder as the residual claimant is:

display math(27)

The equity payoff can be replicated by the following options:

display math(28)

As analysed in Appendix A3, the threshold, V*, where the relationship between the bank's equity value and the corporation's asset risk changes from monotonically decreasing to hump-shaped, is identical to that in the previous case and is given in equation (22). However, since the bank's stockholder has a positive payoff in the event of default, the level at which risk shifting occurs for a given level of initial asset risk, math formula, is now an increasing function of bankruptcy costs, and reaches its maximum when the costs are equal to the bank's capital. When bankruptcy costs increase further above the bank's capital, the threshold for risk shifting is constant, as there is no further effect on the value of the bank's equity.

4 Quantitative Analysis of Bank Risk Shifting

The model developed in the previous sections allows us to examine quantitatively the agency problem between the bank's equityholder and debtholder and its effect on the bank's choice of risk. In this section, we investigate numerically the equilibrium asset risk and its effects on the value of the bank's claims by calibrating our model to typical bank data. Moreover, we consider the effects of risk shifting on a bank's risk-neutral probability of default and its cost of deposit insurance. We first describe the choice of the base case parameters, then focus on the level of asset risk, which is set under different capital structures, asset values and bankruptcy costs.

In our base case, we consider a bank and a corporation with a single debt claim. The debt matures in one year, T = 1, following Marcus and Shaked (1984) and Ronn and Verma (1986). The one-year maturity is justified by the yearly frequency of regulatory audits. If an audit finds the market value of assets to be lower than the value of all liabilities, regulators have the ability to seize the bank. We consider an initial level of asset volatility, math formula, of 15%, similar to the asset risk of corporations that issued investment grade bonds (Huang and Huang, 2012). The face value of the debtor's debt is math formula, and that of the bank's debt math formula. We assume that all claims are initially fairly priced, which renders the bank's book-value debt-to-asset ratio 92%. Later, we decrease this ratio to 85%, in line with the approach of Admati et al. (2011). We assume no bankruptcy costs, 1 − α = 0. However, in section 4.3, we remove this assumption and set the bankruptcy costs at 15% of the corporation's asset value. This level is in the range of bankruptcy costs found by Bris et al. (2006). The annual risk-free rate, r, is set at 1%.

Given the recent debate over the necessity and desirability of bank's high leverage relative to that of non-financial corporations (Admati et al., 2011), we focus especially on the effects of leverage and the size of bankruptcy costs on the level of asset risk, and consequently, on the bank's costs of deposit insurance. We analyse both the effect of a bank's capital structure on risk shifting and the effect of an exogenous change in the value of the corporation's assets on the equilibrium level of asset risk.

Using the base case parameters, we find that, for any asset value below V* = 76, there is a positive level of asset risk, which maximizes the bank stockholder's position. Thus, risk shifting may occur only if the value of the debtor's assets is below this level (Proposition 1). When the debtor's initial level of asset risk is math formula, risk shifting occurs at any level of assets below V** = 75.1 (Proposition 2).

For example, when asset value decreases to the level of VC = 74, the value of the bank's assets is VB = 71.6 and the value of its equity and debt are SB = 2.5 and BB= 69, respectively (Table 1). The bank's default probability and cost of deposit insurance are 48.9% and 5.2%, respectively. However, in the case of this asset value, the position of the bank's equityholder is maximised at a risk level of math formula. Consequently, in equilibrium, asset risk would be shifted from its initial level of 15% to this level. When we account for risk shifting, the values of the bank's assets and debt decline to 69.3 and 66.7, respectively. However, in this case the stockholder is better off because the value of the stock increases to 2.6 due to risk shifting.

Table 1. Equilibrium outcomes
The base case parameters at the initial state 
FCFBσInitialrT       
8073.615%1%1       
Outcome variables
VCVBEBBBV*V**math formulamath formulaσ*PDBDIPD
Low asset value with no risk shifting
7471.62.5697675.122.9%15%15%48.9%5.2%
Low asset value with risk shifting
7469.32.666.77675.122.9%22.9%22.9%51.9%8.4%

The bank's probability of default increases to 51.9% and the cost of deposit insurance is 8.4% per unit of the debt's face value. These new levels reflect an increase of 6% in the risk-neutral probability of default and 60% in the value of deposit insurance per dollar of deposits. Since risk shifting occurs when the value of the debtor's assets is already below the face value of the debtor's debt, the bank's probability of default is moderately affected. However, the increase in asset risk affects the depositor's recovery rate even in states of default. Therefore, risk shifting substantially affects the cost of deposit insurance.

4.1 The effect of a bank's capital structure on risk shifting

Following the severe financial crisis of 2007–2009 and the European debt crisis, Admati et al. (2013), Miles et al. (2013) and Turner (2010) claimed that banks' fragility and high probability of default can be remedied by a substantial increase in banks' capital ratio. Thus, we analyse a case in which a bank's leverage is substantially lower than in our base case. The bank's debt-to-asset ratio is equal to the ratio between the face values of the bank's and debtor's debts. Since initially the two are fairly priced, the analysis captures the effect of a bank's capital ratio on risk-taking decisions.

We set the debt-to-asset ratio at 85% by changing the face value of the bank's debt to 68 instead of 73.6. In this case, for any asset value below V* = 73, there is an internal level of asset risk, which maximises the position of the bank's stockholder. This level is lower than 76, which was found when the debt-to-assets ratio was 92%. Moreover, assuming that the initial level of the debtor's asset risk is 15%, risk shifting occurs at any level of the debtor's assets below V** = 72.2. Consequently, in contrast to the base case, risk shifting would not occur when the asset value decreases to 74. In this case, the value of the bank's assets is 71.6, the value of equity is 5.9 and the value of its debt is 65.5. The bank's probability of default and its cost of deposit insurance are only 28.9% and 2.5%, respectively. The probability of default decreased due in part to the greater equity buffer and in part due to the lower motivation for risk shifting.

The numerical example, which is also proved analytically in section 3.2, shows that, as the bank's equity layer increases, the stockholder's motivation to shift risk decreases (Figure 10). The result strengthens the argument for a higher capital adequacy in banks, since increasing a bank's capital not only affects its probability of default due to the higher capital buffer, but also reduces the stockholder's motivation to take risk.

Figure 10.

Optimal asset risk for different bank leverage ratios The values in the figure refer to a bank with a single loan with a face value of 80 and time to maturity of one year. The bank is financed with equity and a single bond with time to maturity of one year. In the first case, the face value of the bond is 73.6, representing a leverage ratio of 92% (dashed line). In the second case, the face value of the debt is 68, representing a leverage ratio of 85% (solid line). The risk-free rate is 1% and the debtor's asset risk, σ, is 15%.

4.2 The effect of severe market conditions on risk shifting

Studying the effect of an exogenous decline in the value of a debtor's assets on a bank's resilience is important for stress test analysis, where the solvency and the ability of a bank to function under severe market conditions are challenged. We focus on a scenario in which the initial value of the debtor's assets is VC = 121. Therefore, its risk-neutral probability of default is relatively low at 0.05% and the cost of deposit insurance is close to zero. In this case, the value of the bank's assets is 79.2 and the value of its equity and debt are 6.3 and 72.9, respectively. The debt is almost riskless since the yield spread over the risk-free rate is close to zero.

We introduce a severe decrease in the value of the debtor's assets by shifting it to 73 (a decrease of 2.6 standard deviations). The result would cause a decrease in the value of the bank's assets to 70.9. The value of the equity and debt are now 2.3 and 68.6, respectively.

This decrease in asset value affects the condition of the bank – the probability of default increases to 52.5% and the cost of deposit insurance is 5.8% per unit of debt. However, this scenario only partly describes the impact of the decrease in asset value because it ignores risk shifting. When the asset value is 73, the equilibrium level of asset risk is 28.2%. Consequently, the value of the bank's assets decreases to 67.2 and the value of debt equals 64.7. The value of the stock increases to 2.5 due to the positive effect of risk shifting. The higher level of asset risk moderately increases the bank's probability of default to 55.4%, and the cost of deposit insurance almost doubles to 11%.

4.3 The effect of bankruptcy costs

In the basic structural literature, the introduction of bankruptcy costs destroys asset value and decreases the bondholder's recovery rate. In the case of a bank, the introduction of bankruptcy costs also increases the cost of deposit insurance and thus the burden on taxpayers. In our model, the decline in the value of deposits, that is, the increase in the value of deposit insurance, is even more severe since the decrease in asset value encourages the bank's stockholders to shift asset risk to a higher level and thus further reduces the value of the deposits.

We illustrate this effect numerically through the case where the corporation's asset value is 74. Initially, with no possibility for risk shifting and no cost of default, the value of the bank's assets, debt, equity and deposit insurance per dollar are 71.6, 69, 2.5 and 5.2%, respectively (Table 2, Panel A, first row). The introduction of bankruptcy costs at 15% of the corporation's assets affects the bank's asset value through the deadweight costs of bankruptcy. The value of the bank's assets, debt and equity decrease to 64.4, 62.5 and 1.9, respectively, while the value of deposit insurance per dollar increases to 14.7% (Table 2, panel A, fourth row).

Table 2. Numerical analysis with bankruptcy cost
Panel A: The case of no risk shifting
Bankruptcy costBank assetsBank debtBank equityV*V**math formulaPDBDIPD
0%71.6692.57675.122.9%48.9%5.2%
4%69.667.62.179.277.533.8%59.7%7.2%
8%67.765.81.979.278.336.9%70.1%9.6%
15%64.462.51.979.278.336.9%70.1%14.7%
Panel B: The case of risk shifting
Bankruptcy costBank assetsBank debtBank equityPDBDIPD
0%66.163.62.554.9%12.6%
4%64.662.32.359.6%14.4%
8%62.2602.364.4%17.5%
15%59.757.42.364.4%21.3%

The effect of bankruptcy costs on a bank's claim is even more severe where we consider risk shifting. For the analysed case, since the level of asset risk that maximises the equityholder's position is math formula, we expect asset risk to increase in line with Proposition 5. When taking into account risk shifting, the value of the bank's assets and debt decrease to 59.7 and 57.4, respectively, while the value of the bank's equity and the value of deposit insurance per dollar increases to 2.3 and 21.3%, respectively (Table 2, panel B, fourth row). Thus, the cost of deposit insurance is more than four times that of the case with no bankruptcy costs, where 59% of the increase is due to the direct effect of the dead-weight costs of bankruptcy, and the rest is due to the effect of risk shifting.

5 Conclusion

We present a framework that takes into account that bank assets are risky debt claims with limited upside and analyse the risk-taking behaviour of banks by considering the strategic relationship between the debtor and creditor. We find that, in equilibrium, the level of asset risk may increase only when the debtor is in financial distress. This result contrasts with the basic agent theory (Jensen and Meckling, 1976), where the equityholder is motivated to increase risk in both solvent and insolvent states of the debtor.

We contribute to the debate over the optimal bank capital ratio by showing that bank stability increases when its capital ratio increases. While recent papers show that increasing bank capital decreases the events of costly default and the costs of deposit insurance (Admati et al., 2013; Miles et al., 2013; and Turner, 2010), we prove that a higher capital ratio can affect bank stability through a second channel, that is, its negative effect on the equityholders' risk-taking motivation. Moreover, we prove that the existence of bankruptcy costs increases the probability of risk shifting by extending the range of asset values for which risk shifting occurs, as well as by increasing the equilibrium asset risk, relative to the case with no bankruptcy cost.

By calibrating our model to typical market data, we show that not accounting for the possibility of risk shifting in bad economic states may lead to severe underestimation of the cost of deposit insurance. However, since risk shifting when a bank's assets are risky debt claims is limited to states in which the debtor is already in distress, the effect of risk shifting on a bank's probability of default is relatively minor.

Appendix A: Proofs

A1. Bank Equityholder's Position

The payoff of the bank's equityholder is equivalent to a portfolio of two call options on the value of the corporation's assets (equation (10)). To find the preferred asset risk of the bank's equityholder we calculate the derivative of the value of equity with respect to asset risk:

display math(A1)

where:

display math
display math

There is a constrained maximum for the bank equityholder's position with respect to asset risk in cases where the first derivative equals zero. There are two solutions for the equation when math formula or when a = b. By using the terms of the equation a = b we find the corporation's asset value, which maximises the equityholder's position as a function of asset risk:

display math(A2)

Based on Equation (A2), we define V** as the value of assets in which the equity value is maximised for a chosen level of asset risk σ. We note that the derivative changes its sign from positive to negative above the threshold, V**, meaning that the bank's equityholder would like to increase risk below that level and to decrease it above that level of assets.

By using the same equation where a = b we can now fix the level of assets to find the level of asset risk that maximises the value of the bank's stock:

display math(A3)

However, for both Equations A2 and A3 to hold – that is, for an internal solution to exist − the corporation's asset value must be below V* defined as: math formula.

A2. A Model with High Bankruptcy Costs

As discussed in section 3.3, when the costs of bankruptcy are high and greater than a bank's capital, the bank equityholder's position can be expressed as a portfolio of a digital call option (equation (21)). The price of a digital call option based on the Black and Scholes (1973) pricing equation is:

display math(A4)

where math formula and N(.) is the standard normal cumulative distribution.

To find the maximum value of the bank equityholder's position we first calculate the derivative of the position with respect to asset risk:

display math

There is a constrained maximum for the bank equityholder's position with respect to asset risk in cases where the value of the derivative is zero. Since the exponent of any number is positive and since we assume math formula, the derivative is equal zero when math formula. By rearranging the condition, we find the corporation's asset value, which maximises the equityholder's position as a function of asset risk:

display math(A5)

For a given value of assets, the risk that maximises the position of the bank's equityholder is:

display math(A6)

However, for both equations to hold – that is, for an internal solution to exist – the corporation's asset value must be below V* defined as: math formula. For any asset value below V* there is a positive asset value, math formula, which maximises the bank equityholder's position.

As discussed in section 3.3, when the costs of bankruptcy are high and greater than the bank's capital, the deadweight costs created at the corporation's default lead automatically to the bank's default. Therefore, the bank's risk-neutral probability of default can be expressed as:

display math(A7)

As discussed in section 2.3, in the case of a bank default, its assets, minus the deadweight costs of bankruptcy, are transferred to the bank's creditor and the depositors are compensated by the difference between the face value of deposits, FB, and the value of the bank's assets, math formula. Thus, the value of deposit insurance per dollar of debt can be expressed as:

display math(A8)

A3. A Model with Low Bankruptcy Costs

As discussed in section 3.3, in the model with low bankruptcy costs, the bank equityholder's position can be expressed as a portfolio of call options and a digital call option (equation (28)). To find the maximum value of the bank equityholder's position we calculate the derivative of the position with respect to asset risk:

display math(A9)

where: math formula, math formula and math formula.

There is a constrained maximum for the public position with respect to asset risk in cases where the value of the derivative is zero. Since the exponent of any number is positive, expressions a and b in Equation (A9) are positive. Moreover, since the value of d1(.) decreases with strike price, expression a is always smaller than expression b. Therefore if expression c is larger than one, the derivative is always negative and the bank's equity value decreases with risk. The derivative may equal zero when c is smaller than one. Rearranging the condition, c < 1, we find that there may be a constrained maximum when: math formula.

As discussed in section 3.3, in the model with the low cost of default, the bank defaults when the corporation's asset value minus the deadweight cost of default is below the face value of the bank's debt. Thus, the bank's risk-neutral probability of default can be expressed as:

display math(A10)

In the event of a bank default the depositor is compensated by the difference between the face value of deposits, FB, and the residual assets of the debtor, math formula. Thus, the value of deposit insurance per dollar of debt is:

display math(A11)

Notes

  1. 1

    A risky corporate bond is economically equivalent to a long position in a risk-free bond and a short position in a European put option on the firm's assets, with a strike price equal to the face value of its debt. Similarly, a firm's stock is economically equivalent to a call option on the value of the firm's assets with a strike price equal to the face value of its debt.

  2. 2

    Caprio and Levine (2002) discussed the corporate governance of banks and their opaqueness. Morgan (2000) found that bond analysts disagree more over bonds issued by banks than by non-financial firms suggesting that banks tend to be more opaque than non-financial firms.

  3. 3

    A similar analysis was suggested earlier by Dermine and Lajeri (2001).

  4. 4

    This strategy consists of a long position in a call option on the corporation's asset value, with a strike price equal to the face value of the bank's debt and a short position in a call option on the corporation's asset value, with a strike price equal to the face value of the debtor's debt.

  5. 5

    The costs of issuing more equity is a subject beyond the scope of our paper. De Nicolo et al. (2012) argued that even if capital requirements are initially beneficial, there is a point at which further increases become costly, reducing lending, efficiency and welfare.

  6. 6

    Merton (1974); Merton (1977) and Black and Cox (1976).

  7. 7

    In the financial literature there is contradicting evidence regarding the effect of executive compensation on risk taking (Murphy, 2013).

  8. 8

    To keep the notation as simple as possible, all variables without subscripts refer to time t.

  9. 9

    The cost of default generally refers to any cost that follows the event of default, which can destroy value. The costs of bankruptcy are the special costs related to formal legal processes. We use these two terms interchangeably as their effects on our results are identical.

  10. 10

    For the equation, see Appendix A2.

  11. 11

    The corporation's default may lead to a further reaction, where the bank's default creates additional bankruptcy costs. While such an event affects the value of debt, it has no effect on risk shifting or the robustness of our result.

Footnotes

  1. The table presents the set of parameters and pay-offs in equilibrium as well as the bank's probability of default and cost of deposit insurance for the case in which risk shifting is impossible and for the case in which risk shifting can take place. The base case parameters FC, FB, σ, r and T are the face values of the corporation and bank's debts, the corporation's asset risk, the risk-free rate and the time to maturity of the debt instruments.
  2. The positions in equilibrium VC, VB, EB and BB are the values of the corporation's asset and of the bank's assets, equity and debt. The thresholds for risk shifting are V*, the level of assets below which the position of the bank's stockholder is hump-shaped, as defined in equation (13), and V**, the threshold under which the bank equityholder would like to increase asset risk. The level of asset risk that maximises the bank equityholder's position, as defined in equation (14), is math formula, and the upper bound on corporate asset volatility set by the bank's equityholder and the equilibrium asset volatility as defined in section 3 are math formula and σ*, respectively. The bank's probability of default, as defined in equation (11), is PDB and the value of the bank's deposit insurance per dollar of insured deposits, as defined in equation (12), is DIPD.
  3. Panel A presents for different levels of bankruptcy cost, the value of the bank's liabilities, the threshold for risk shifting, V*, the threshold for risk shifting when initial asset risk equals 15%, V**, the level of asset risk in equilibrium, σ*, the bank's risk-neutral probability of default, PDB and its costs of deposit insurance, DIPD, for the base case parameters, which are presented in Table 1. Panel B reports the values of these parameters when taking into account equilibrium risk shifting. As the size of bankruptcy costs increases the threshold for risk shifting, V** increases, as well as the bank's probability of default. However, since the bank's capital is 8%, as the bankruptcy costs reach this level, a further increase in bankruptcy costs does not affect risk shifting or the bank's probability of default.

Ancillary