## Abstract

This article studies the connection between risk taking and executive compensation in financial institutions. A model of shareholders, debtholders, depositors, and an executive demonstrates that (i) excess risk taking can be addressed by basing compensation on both stock price and the credit default swaps (CDS) spread, (ii) shareholders may not be able to commit to design such contracts, and (iii) they may not want to due to distortions from deposit insurance or unobservable tail risk. The advantage of using the CDS spread rather than deferred compensation or debt is due to the fact that it is a market price and reduces agency costs.

## 1. Introduction

It is well known that structuring chief executiver officer (CEO) incentives to maximize shareholder value in a levered firm tends to encourage excess risk taking. Indeed, the value of the stock for the levered firm is like the value of a call option and is increasing in the volatility (riskiness) of the assets held by the firm. This issue is particularly troublesome for banks. Although the average non-financial firm has about 35% debt,1 financial institutions have about 90% debt.2 At the same time, “banks can alter the risk composition of their assets more quickly than most nonfinancial industries, and … readily hide problems” (Levine, 2004, p. 4). Excess risk taking at financial institutions affects more than just creditors; it affects depositors, taxpayers, and potentially the financial system as a whole.

The academic literature has recognized this issue over the years and made several proposals to reduce risk taking, none of which have been adopted in the real world (we summarize these proposals below). Moreover, the link between compensation and risk remained strong throughout recent events, as evidenced by several recent empirical studies, which are listed in Figure 1.3 Along with a recent article by Edmans and Liu (2011),4 our goal is to revive the discussion on how to operationalize CEO incentive compensation toward reduced risky behavior by taking advantage of one of the most significant recent financial innovations: credit default swaps (CDS). Thus, we propose tying a CEO’s compensation in part to the financial firm’s CDS spread, which provides a market estimate of the default risk of the bank. In our proposed pay structure for bank CEOs, a high and increasing CDS spread would result in lower compensation, and vice-versa.5

Figure 1.

Recent results on the relationship between equity compensation and risk taking.

Figure 1.

Recent results on the relationship between equity compensation and risk taking.

Our model considers a fairly general formulation of the agency problem for banks where the bank CEO faces a multi-task problem: she chooses both the risk level of the bank and an effort level that increases mean returns. Although it may be in shareholders’ interests to commit to induce a CEO to take less risk as a way of lowering the cost of debt, we show that if the CEO’s actions are unobservable, she will be induced to undertake excessive risk in equilibrium. In addition, shareholders face a commitment problem, as they can always renegotiate the CEO compensation contract after the bank has issued its term debt. Also, as many commentators have pointed out, even if shareholders were able to commit to an incentive scheme, they would not want to due to the distortions in debt markets arising from deposit insurance and from unobservable tail risk.

A bank regulator, however, can align the CEO’s and social objectives in terms of risk choice by requiring that the CEO’s compensation be sufficiently exposed to the bank’s CDS spread. Alternatively, we show that better bank governance (or monitoring of the CEO) reduces the reliance on pay incentives, thus lowering both equity and CDS compensation. If labor market competition for CEOs increases, however, incentive pay must go up, increasing both types of variable compensation.

We further show why it is preferable to have CDS-based compensation rather than deferred cash or inside debt compensation to correct CEO risk-taking incentives. As in Holmstrom and Tirole (1993), it is preferable to build incentives around an ex-ante market measure of risk (the CDS spread) rather than just rely on ex-post performance pay. The added information contained in the CDS spread reduces agency costs for shareholders by reducing the overall risk exposure needed to discipline the CEO.

### 1.1 RELATED LITERATURE

We are by no means the first to consider risk-shifting by a CEO of a levered firm who is only compensated with equity. Several other papers, beginning with Jensen and Meckling (1976) have highlighted the risk-shifting problem for levered firms. Green (1984) proposes that firms issue warrants to eliminate excess risk-taking incentives. Hennessy and Tserlukevich (2008) show that callable bonds with call premia can discipline risk-shifting incentives and that convertible bonds can only discipline the most extreme forms of risk shifting. John and John (1993) suggest a default cost for the manager,6 whereas Brander and Poitevin (1992) propose a bonus contract (without equity compensation). John, Saunders, and Senbet (2000) specifically focus on banks and show that well-priced deposit insurance can possibly eliminate risk shifting. Most recently, Edmans and Liu (2011) demonstrate in a multi-task setting that giving the manager debt (either straight debt or deferred compensation) can solve the risk-shifting problem. We further this insight by demonstrating that using an instrument tied to market prices is cheaper, and analyze the incentives of banks to adopt this type of compensation structure. We explore the effects of governance and labor market competition on these components of the pay package. Bebchuk and Spamann (2010) also advocate linking pay to debt-like instruments.

In sum, our analysis advances this literature in the following main directions. First, Following Holmström and Tirole (1993), we rely on a market-based approach to correct risk-shifting incentives by exposing CEO compensation directly to the market’s assessment of the bank’s risk taking. Several recent papers suggest incorporating CDS spreads into other forms of regulation. Hart and Zingales (2010) suggest CDS spreads as a trigger for when financial firms should be asked to increase capital. Greenlaw et al. (2011) also propose CDS spreads as a macroprudential tool. Flannery, Houston, and Partnoy (2010) suggest replacing credit ratings with CDS spreads for both regulatory and private purposes.

Second, we show why optimal risk-taking incentives will not be implemented by shareholders: Shareholders suffer from a commitment problem in the model, which will be exacerbated by either the renegotiation of compensation contracts, deposit insurance, or the unobservability of tail risk to debt markets. Consequently, regulation may be necessary for implementation. Our analysis of deposit insurance highlights the richness of our model; we are able to incorporate shareholders, an executive, debtholders, depositors, and a regulator in a tractable framework.

Several recent empirical studies suggest that inside debt decreases risk taking. Wei and Yermack (2011) study non-financial firms and find that the revelation of more CEO inside debt compensation increases debt prices, decreases equity prices, and lowers the CDS spread for these firms.7Sundaram and Yermack (2007) find a significant positive effect of CEO inside debt compensation on the distance to default measure for a sample of 237 of the Fortune 500 firms from 1996 to 2002. Cassell et al. (2012) find a negative relationship between CEO inside debt compensation and the volatility of future firm stock returns, research and development expenditures, financial leverage, and a positive relationship with diversification and asset liquidity. Srivastav, Armitage, and Hagendorff (2013) find that CEOs at Troubled Asset Relief Program (TARP) banks with more CEO inside debt compensation are more likely to cut dividends and cut dividends by a larger amount. Bennett, Guntay, and Unal (2012) find that for Bank Holding Company CEO compensation there is a negative relationship between CEO inside debt compensation and default risk and a positive relationship with performance during the crisis.

The article is organized as follows: In Section 2, we write down the model. In Section 3, we analyze the optimal risk choice for a firm under concentrated ownership. Section 4 considers CEO risk choice under separation of ownership and control. Section 5 demonstrates that CDS-based compensation costs less than deferred compensation and debt. Section 6 characterizes optimal CDS-based compensation and why shareholders are unlikely to implement the first best. Section 7 discusses how CDS-based compensation could be implemented by regulators. Finally, Section 8 concludes. All proofs are in the Appendix.

## 2. The Model

We consider a bank that is run by a manager hired by shareholders under an incentive package designed to align the manager’s objectives with shareholders. The manager chooses: (i) the underlying riskiness of the bank’s investments and (ii) an effort level that affects the bank’s returns. We consider a classical incentive contract, where the manager receives a fixed wage and a payment that depends on the price of the bank’s stock, augmented by a payment that depends on the price of a CDS. In the benchmark model, we do not allow for deposit insurance, though in Section 6.2 we add deposit insurance to examine how the implicit subsidy in deposit insurance toward risk-taking affects the bank’s choice of risk.

### 2.1 INVESTMENT CHARACTERISTICS

#### 2.1.a Technology

The bank has access to an investment technology with the following characteristics. By investing an amount I the bank can get a gross return $x∼$ per unit invested, where $x∼$ can take the following values:

• a high return $x+eiΔ$ with probability q,

• a medium return x with probability $(1−2q)$, and

• a low return with probability q.

The low return can take two possible values:

• 1. $x−δ$ with probability γ, conditional on being in the low return state; the unconditional probability is $qγ$ and

• 2. 0 with probability $(1−γ).$

The variable q represents the risk of the investment, and will be a choice variable for the manager. This risk choice will be unobservable to the bond market. We restrict q to be in the interval $[0,½]$. An increase in q thus increases the likelihood of both the high and low return outcomes. Having more than two outcomes makes it possible for q to be a direct measure of risk, that is, the variance of the outcomes is strictly increasing in q for all $q∈[0,½]$.

The manager can increase q at a cost c(q) per unit of investment. We will demonstrate below that the optimal amount of risk q is positive. We assume for simplicity that c(q) takes the quadratic form $c(q)=½αq2$, where α is a parameter that shifts the variable cost. In contrast to the standard principal-agent model, we take the cost c(q) to be a cost borne by the bank. A natural interpretation is that c(q) is the cost of originating assets with risk characteristics q.

The variable ei, where $i∈{L,H}$ is an observable but unverifiable effort choice by the manager that influences the high return. We set the high effort eH = 1 and the low effort $eL∈(0,1)$. Effort i has a utility cost Ki for the manager. The cost of high effort KH is $ψ>0$ and the cost of low effort KL is normalized to 0.

The parameter γ is outside the influence of the manager. We simply treat it as an exogenous probability. It represents the likelihood that the low return will not be a tail event that completely wipes out the bank.

#### 2.1.b The market

The bank raises funds through deposits and subordinated debt.8 For a total amount I of deposits and subordinated debt, it promises a return of $I(1+R)$. We assume that all lenders to the bank have an outside option of investing their money in an alternative that yields a safe gross return of $1$, say treasury bills. We exogenously fix I and assume that the bank is able to raise sufficient funds given that the expected rate of return R weakly exceeds the safe return. We later endogenize the amount of funds invested in the section on deposit insurance.9

A key constraint for the bank is that risk-neutral depositors obtain a return $1+R$ equal to or larger than the safe return 1. If the bank were to default only when investment returns are low, this constraint would be:10

(1)
$(1−q)(1+R)+qγ(x−δ)≥1.$

In equilibrium, the promised repayment R(q) is set so that this constraint binds:

(2)
$R(q)=1−qγ(x−δ)1−q−1.$

The most interesting case is the one where risk taking by the bank may lead to a default on its debt only when the low return occurs. For default to be restricted to the situation when the returns are low, two conditions must hold. First, when the low return occurs, the amount recovered is not sufficient to avoid default:

$1+R(q)>x−δ.$

The left-hand side (LHS) is the promised return for a unit of investment (given by Equation (2)) and the right-hand side (RHS) is the highest return possible in the low state per unit invested. Since R(q) is strictly increasing in q, we ensure that this inequality holds for all q, by setting q = 0. This gives us assumption A1a:

(A1a)
$1>x−δ.$

Second, we need to ensure that there is a sufficient return in the middle returns state to avoid default:

$x>1+R(q).$

Since R(q) is strictly increasing in q, we ensure that this inequality holds for all q, by setting $q=½$. This gives us assumption A1b:

(A1b)
$x>2−γ(x−δ).$

### 2.2 THE MANAGER

To simplify the algebra and notation, we assume that the discount rate is zero and all agents aside from the manager are risk-neutral. Given the manager’s compensation Wi, the manager’s utility over consumption is given by a constant absolute risk aversion (CARA) utility function:

$U(Wi)=1−e−aWi, i=H,L,$
and a > 0 is the coefficient of absolute risk aversion.

The manager faces two choices (i) the risk level of the bank q and (ii) the effort level ei. We restrict compensation to a classic linear incentive contract, where $w¯$ is the base pay, sE is the shares of equity, and sD is the loading on the CDSs of the bank. The price of equity is represented by $PEi$ and the CDS spread is represented by $PDi$, where the superscript i reflects the fact that these prices will depend on the manager’s effort choice $i∈{L,H}$:11

$Wi=w¯+sEPEi+sD(P¯−PDi).$

Since the CDS spread is increasing in the probability of default, it is judged relative to a high benchmark $P¯$ in order to align the manager’s incentives. The only requirement for this benchmark is that it be large enough to ensure that the manager receives a positive payment from the CDS compensation component.12 Paying the manager in equity- or CDS-based compensation costs the shareholders $1+z$ dollars, where z > 0 is a small transaction cost.

The manager’s decision to exert effort depends on her compensation and economic fundamentals. She will exert high effort if the following incentive constraint holds:

(3)
$U(w¯+sEPEH+sD(P¯−PDH))−ψ≥U(w¯+sEPEL+sD(P¯−PDL)).$

We make the following assumption about optimal effort choice:

(A2)
$PEH(qH∗)−WH(qH∗)>PEL(qL∗)−WL(qL∗),$
where we define $qi∗$ as the risk choice coming from the shareholders maximization problem when effort i is implemented. This assumption simply states that from the point of view of shareholders, it is optimal to have the manager undertake high effort.

Finally, we assume the manager has an outside option $U¯$ that reflects her opportunities managing other firms, so that the manager’s participation constraint given effort choice eH is:

(4)
$U(w¯+sEPEH+sD(P¯−PDH))−ψ≥U¯.$

### 2.3 TIMING

The timing of our model is as follows:

1. Incumbent shareholders hire a manager under a linear incentive contract $(w¯,sE,sD)$.

2. The manager chooses the bank’s risk q and effort ei.

3. The bank raises I to fund the asset from bondholders or depositors, with a promised return of $I(1+R(q))$.

4. The equity of the firm is priced at $PEi$ and the CDS spread at $PDi$. The manager gets paid Wi.

5. The returns on the asset $x∼$ are realized. Depositors and bondholders get paid first. If there are returns left the shareholders get the residual value.

### 2.4 FIRST-BEST

We begin by characterizing the first best outcome. The choice of q by a social planner maximizes the net expected return, which takes the following simple expression:

$maxq q(x+Δ)+(1−2q)x+qγ(x−δ)−12αq2.$

We assume there are no agency problems and that effort is high. It is immediate to see that the first best q is given by:

$qFB=Δ−x+γ(x−δ)α,$
when $Δ−x+γ(x−δ)>0$ and $qFB=0$ otherwise.

In other words, as long as there is upside there are gains to exposing the bank to some risk.

## 3. Ownership Concentration

We consider next the case where incumbent shareholders manage the firm with one voice. Shareholders choose q to maximize shareholder value net of the cost of debt. The cost of debt will reflect the market’s assessment of the risk the bank is taking. The debt market may or may not be able to observe the true risk q the bank is taking. Accordingly, we distinguish between two subcases. We first allow bond prices to reflect the perfectly observed risk q, and then we consider the case where q is not observed and where the debt market has rational expectations about the amount of risk chosen by the bank. In both, we will assume that there is no effort problem and high effort is implemented.

### 3.1 OBSERVABLE RISK

Given assumptions A1a and A1b, the optimization problem for shareholders when q is observable is:

$maxq q(x+Δ)+(1−2q)x−(1−q)(1+R)−12αq2$
subject to the constraint that risk-neutral depositors obtain a promised return R such that their expected return is equal to or larger than their safe return, given by Equation (1).

In equilibrium, the promised repayment R(q) is set so that this constraint binds, given by Equation (2). Substituting for R(q) in the shareholders’ maximization problem, we obtain the unconstrained problem:

$maxq q(x+Δ)+(1−2q)x−(1−qγ(x−δ))−12αq2.$

The first-order condition for the shareholders’ problem thus is:

(5)
$Δ−x+γ(x−δ)−αq=0,$

which gives the optimal choice:

$qo=Δ−x+γ(x−δ)α,$
when $Δ−x+γ(x−δ)>0$ and $qo=0$ otherwise.

As one would expect, the observability of q induces shareholders to limit their risk taking. Shareholders fully internalize the cost of greater risk taking, as it is immediately reflected in the cost of debt, so much so that the optimal solution for shareholders under observable q coincides exactly with the first-best solution: $qo=qFB$.13

### 3.2 UNOBSERVABLE RISK

Consider next the more realistic case where the choice of risk q is not observable to bondholders. In this case, the best bondholders can do is to form rational expectations about the bank’s optimal choice of q. This means, in particular, that if the bank changes its risk exposure at the margin, this change will not be reflected in the price of debt. As a result, the bank may be induced to take excessive risk when risk is unobservable.

With an expected risk level of $q^$, bondholders require a return of at least $R(q^)$, where

$R(q^)≥1−q^γ(x−δ)1−q^−1.$

The bank then chooses q knowing that it is unobservable and a change in q does not affect the cost of debt directly:

(6)
$maxq q(x+Δ)+(1−2q)x−(1−q)(1+R(q^))−12αq2.$

In the following proposition, we prove that shareholders choose an excessive amount of risk: $q^>qo$.

Proposition 1

If a rational expectations equilibrium exists, the amount of risk chosen ($q^$) is larger than the optimal amount (qo).

Proof

See the Appendix. □

In fact, two possible solutions to the shareholders’ problem may exist, both of which satisfy the property that $q^>qo$. Only the smaller solution, however, is stable.14 Unobservable choices, although rationally expected, are thus riskier than observable choices, since the pricing of risk cannot respond to changes in unobservable risk choice. The bank’s shareholders are worse off with the riskier unobservable choice. This worse outcome is due to an inability of the bank’s shareholders to commit to a lower risk exposure. We will see in the next section that under separation of ownership and control the incentive contracting problem between the bank’s shareholders and the manager involves both the classical problem of aligning the shareholders’ and manager objectives and a commitment problem with respect to the bank’s bondholders. This joint contracting problem for a levered financial institution is an important conceptual difference with respect to the classical moral hazard incentive contracting problem of Mirrlees (1975) and Holmstrom (1979).

## 4. Separation of Ownership and Control and the Choice of Unobservable Risk

Suppose next that the bank CEO decides on the level of risk. The manager’s contract, as we stated before, is composed of three components: a fixed wage, a loading on equity, as well as a loading on the CDS spread. The equity part is standard and represents shares given to the manager as compensation. The CDS part is the innovation. In this section, we show that it can restore optimal risk-taking incentives. In Section 5, we discuss the advantages of using the CDS spread over other forms of debt and deferred compensation.

As in Holmstrom and Tirole (1993), we assume here that both equity and the CDSs are being traded by informed traders who observe signals that are perfectly correlated with the bank’s actual risk exposure q. Thus, although the bank’s actual risk choice is not observable ex-ante when the bank issues bonds it becomes effectively observable to analysts ex-post.15 In Section 6.3, we will analyze the case where tail risk is also unobservable to the market.

In order to analyze the optimal compensation contract we must first characterize equilibrium securities prices. The price of equity is given by the present discounted value of equity cash-flows, net of origination costs $c(q)$ and expected debt repayments $(1−q)(1+R(qT))$, where qT represents the risk level that bondholders believe the bank will implement through the compensation contract. Note that in the low return state the bank defaults and shareholders get nothing, so that the price of equity is given by:

(7)
$PEi=qi(x+eiΔ)+(1−2qi)x−(1−qi)(1+R(qiT))−12αqi2, i=H,L.$

Note that the price (and risk choice) depends on whether the manager exerted effort or not.

The CDS spread, in turn, given by:

(8)
$PDi=qi(1−γ(x−δ)1+R(qiT)), i=H,L$
where
$γ(x−δ)1+R(qiT)$
is the recovery rate of the investment.16 Note that the CDS spread does not directly depend on the effort choice, but can indirectly depend on effort through its feedback on the risk choice.

Note also the key simplification in our model that the risk choice qi is incorporated into the equity price and the CDS price through the valuations of informed traders. This implies that the risk-averse manager is not affected by risk with respect to these prices when her compensation is based on these prices (i.e., after the choice of qi and ei the variance: $var[PEi]=var[PDi]=0$.17

We begin by demonstrating that shareholders will choose a contract such that the manager’s participation and incentive constraints bind.

Proposition 2

In the shareholders’ optimal contract, the manager’s participation and effort-incentive constraint bind.

Proof

See the Appendix. $□$

When the manager’s compensation package only contains stock, so that $sD=0$, the manager’s and shareholders’ objectives with respect to the choice of risk q are perfectly aligned, so that the manager chooses the socially excessive level of risk $q^$. If instead, we allow the compensation package to be based also on CDS spreads, then first-best incentives can be restored with the proper loading factors on equity and the CDS spread. We show this in the following proposition:

Proposition 3

Assuming that the solution has the property $0, the first-best level of risk can be induced by setting $w¯$, $sE$, and $sD$such that:

• $sDsE=1+R(qo);$

• the participation constraint (4), and

• the effort constraint (3) bind.

Proof

See the Appendix. $□$

The number of shares to grant, the loading on the CDS component, and the fixed payment are jointly determined by the participation and effort constraints and the manager’s prescribed choice of risk. The ratio of the equity and CDS loadings should be set equal to the rate of return promised to bondholders at the first-best risk level. This relationship is directly derived from: (i) the manager’s optimal risk choice and (ii) by implementing the first-best level of risk.

Although the first-best risk level may be difficult to calculate, this provides a simple framework for thinking about how to balance incentives. Opening up the black box by substituting for qo provides further understanding:

(9)
$1+R(qo)=1−(Δ−x+γ(x−δ)α )γ(x−δ)1−Δ−x+γ(x−δ)α .$

The right-hand side (RHS) of this equation is increasing in the return on the safe investment (when the safe return is increased from 1). As the need to satisfy depositors with higher returns increases and depositors themselves are less sensitive to local change in risk, the manager will take on more risk. This is then reined in by pushing up the loading on the CDS portion of the manager’s contract. The term $(Δ−x+γ(x−δ))$ is the marginal return on a unit increase of risk, where the term $γ(x−δ)$ represents the expected default recovery amount. Assuming this is positive (otherwise $qo=0$), an increase in the marginal return increases this term. With higher returns to risk taking, the manager is controlled by exposing her more to the downside risk of default. Finally, the RHS of Equation (9) is decreasing in α, the marginal origination cost of q. If it is more costly to increase risk, there will be a reduced need for incentives to correct risk taking.

Although we cannot provide explicit solutions for $w¯$, $sE$, and $sD$, they are pinned down by the binding effort and participation constraints and the first-order condition of the manager’s maximization problem (written out in the Appendix as Equation (16)). We can thus study implicitly how they depend on the fundamental parameters of the environment. In the next proposition, we look at how they vary with the level of competition and the quality of bank governance.

Proposition 4

Implementing the first best level of risk, the optimal compensation package has the following properties:

1. Increased labor market competition (higher $U¯$) increases the amount of shares granted sEand the loading on the CDS component sD.

2. Stronger governance (lower ψ) decreases the amount of shares granted sEand the loading on the CDS component sD. It increases the fixed part of compensation $w¯$.

Proof

See the Appendix. $□$

Increased labor market competition for managers is reflected in a higher outside option $U¯$. An increase in the value of outside offers induces shareholders to offer the manager higher compensation and higher-powered compensation through increased weightings on equity and the CDS component. We also observe in the proof that the direction of change of the fixed part of compensation $w¯$ is ambiguous. Thus, raising compensation is cheaper to do through the equity and CDS components because it incentivizes effort and reduces the tightness of the effort constraint.

Following Holmstrom and Tirole (1997), we proxy for improved governance through a lower cost of exerting effort for the manger. Better governance thus makes it less likely that the manager will choose to enjoy her private benefit instead of increasing the profitability of the bank. Then, the lower the cost of high effort the lower the performance-based weightings in the manager’s compensation. As the total compensation remains the same (determined by the participation constraint), the fixed amount $w¯$ then goes up. These predictions are consistent with Fahlenbrach (2009), who finds a substitution between stronger governance (as measured by several variables relating to board quality, the ease of launching a proxy fight, and monitoring by outside large stakeholders) and pay for performance. Chung (2008) and Acharya and Volpin (2010) also find this substitution effect.18

## 5. Advantages of CDS-based Compensation

Our model demonstrates the efficacy of CDS-based compensation in reducing risk taking. It is, of course, possible that other instruments exposing the manager to default risk would reduce risk taking as well. One instrument often used in practice is deferred compensation. We now show that such deferred compensation, including inside debt itself, is less efficient than CDS-based compensation in controlling CEO risk taking.

Deferred compensation is a debt-like instrument since it is unsecured and therefore is vulnerable to the risk of default. To be able to model the effects of deferred compensation we slightly extend the model by allowing for two periods (t = 1, 2) of consumption for the manager. The first period is subsequent to market pricing of equity and CDS but before the realization of the state of nature. The second period follows the realization of the state. We continue to assume that there is no discounting so as not to artificially bias the results against deferred compensation.

Thus, consider deferred cash compensation C given in addition to (non-deferred) equity compensation. It is important to note that giving cash that can be consumed at period t = 1 would not affect risk taking at all. The key here is that it is deferred, so that the manager will lose some of it if there is a default. Deferred cash compensation could be in the form of actual compensation or pension benefits (i.e., inside debt). Assume that if the low return $(x−δ)$ occurs, the manager gets a return of ηC (where $η<1$ and represents the haircut on the manager’s claim), and when the low return of zero occurs, the manager gets nothing. Compensation for the manager then takes the form:

$WCi=w¯+sEPEi+(1−qi)Ci+qiγηCi, i=H,L.$

Since the cash compensation has two possible realizations this component of the pay package is risky. The expected utility for the manager under this package is then:

$(1−qi)U(w¯+sEPEi+Ci)+qiU(w¯+sEPEi+γηCi)−Ki, i=H,L.$

As we noted before, exposure to the price of equity and CDS is less risky (in fact it is riskless in the model, as market analysts are assumed to be able to perfectly infer qi and ei). We show in the next proposition that as result of this reduced risk, using CDS spread exposure is more efficient than using deferred compensation:

Proposition 5

As $z→0$, deferred cash compensation is more costly than CDS-based compensation.

Proof

See the Appendix. $□$

In the proof we begin by showing that the participation constraint binds in the deferred cash compensation case, and then show that deferred cash compensation is more costly due to risk aversion. Notice that this argument holds irrespective of what risk choice of the manager is implemented. We also note that the condition on z is sufficient, but not necessary; in fact deferred cash compensation may also be more costly when $z>>0$.

Proposition 5 discusses the differing costs of compensation. In order for shareholders to prefer one type of compensation over another, we must take into account the returns shareholders get as well as the cost. We examine this in the following corollary:

Corollary 1

As $z→0$, shareholders prefer CDS-based compensation to deferred cash compensation when both implement the same level of risk q.

Proof

See the Appendix. $□$

In Section 6, we give several reasons why optimal compensation may not be implemented. These reasons may also explain why we observe deferred cash and pension benefits as part of the compensation package in the real world and do not observe CDS-based compensation.

In principle, deferred cash compensation is very similar to compensation with a debt component, implying that CDS-based compensation should be better than debt as well. And given that debt is relatively illiquid it would be more costly to use than cash, so that CDS-based compensation dominates debt compensation even more.

One might imagine tying compensation to the price of debt rather giving the manager actual debt that they must hold. This would certainly be better for the arguments given above for CDS-based compensation. However, depending on the type of debt used, it may be difficult and/or costly to value the debt instrument at issuance and on an on-going basis. Individual debt issues vary by maturity, seniority, and specific covenants. Actual debt credit spreads may reflect liquidity and taxes (see Longstaff, Mithal, and Neis, 2005) and may lag CDS spreads (see Blanco, Brennan, and Marsh, 2005; Forte and Peña, 2009; Norden and Weber, 2009).

## 6. Optimal and Equilibrium CDS-based Compensation

Although it is in principle possible to make use of CDS price exposure to induce a levered bank’s manager to choose a socially optimal level of risk, it is far from obvious that a levered bank’s shareholders will want to align the manager’s risk-taking objectives. There are at least three reasons why we should not expect shareholders to offer socially optimal incentive contracts to their managers: renegotiation, deposit insurance, and unobservable tail risk.19 We explore these below.

### 6.1 RENEGOTIATION

As has been pointed out in the literature on the strategic role of incentive contracts (Caillaud, Jullien, and Picard, 1995; Katz, 1991; Persons, 1994), the optimal contract such that:

$sDsE=1+R(qo)$
may not have much commitment value if shareholders can (secretly) undo the contract once the bonds have been issued. If that is the case, shareholders will simply offer a new contract to the manager after the bond issue, inducing her to change the bank’s risk exposure away from qo.20 If that is possible, then the CDS-based contract will have no value and will therefore not be offered by shareholders. Although this issue is likely to be relevant in practice, we have not allowed for this possibility of renegotiation and revision of the bank’s risk choice in our model. One reason why we do not emphasize this problem is that disclosure of CEO compensation can to a large extent reduce the benefits of this strategic renegotiation. Still, it is worth emphasizing that some minimal form of regulation (such as mandatory disclosure) is required to make it worthwhile for shareholders to add this CDS exposure to CEO compensation contracts.

The next two reasons why shareholders may not offer the socially optimal contract to their manager are valid even if contracts can have full commitment power.

### 6.2 DEPOSIT INSURANCE

Suppose now, as is true in practice, that the bank funds its investments partly with deposits that are fully insured. Concretely, consider the liability structure where funds raised through debt issues are given by B and funds raised through insured deposits are given by L. The premium charged by the deposit insurance authority is π. Therefore, the total amount of funds other than equity capital available for investment is $B+L−π$.21 The premium $π$ set by the deposit insurance authority is set fairly and covers expected losses. It is given by:

$π=q(L−(B+L−π)γ(x−δ)),$
or
(10)
$π=q(L−(B+L)γ(x−δ))1−qγ(x−δ).$

We assume in this formulation that the amount recovered in default goes only to depositors, but is not enough to cover their obligations, which is equivalent to assuming that the premium is positive. Formally, it amounts to the following condition, which supersedes assumption A1a:22

(A3a)
$L−(B+L)γ(x−δ)>0.$

Assumption A1b, which assumes there will not be a default in the middle state for the main model continues to hold. Finally, we assume that $(B+L−π)$ is positive for all q:

(A4)
$B+L−12L>0.$

The expected return of debtholders depends on how much they will recover and is given by:

$(1−q)(1+R)≥1.$

The timing with deposit insurance is now:

1. Incumbent shareholders hire a manager under a linear incentive contract $(w¯,sE,sD)$, where $w¯$ is the base pay, sE is the shares of equity, and sD is the loading on the CDSs of the bank.

2. The deposit insurance authority sets π.

3. The manager chooses the bank’s risk q for the asset.

4. The bank raises B and L to fund the assets from debtholders and depositors, respectively, with a promised return $1+R$ and 1, respectively. It pays the deposit insurance premium π.

5. The equity of the firm is priced at PE and the CDS spread on the firm is priced at PD.

6. Returns $x∼$ are realized. Depositors and debtholders get paid first. In the case of bank failure, the recovery amount is distributed to depositors. The deposit insurance authority compensates depositors for any shortfall below 1.

Our first benchmark is the situation where q is perfectly observed by debtholders and the deposit insurance authority can set its fees after perfectly observing q.

Lemma 1

When q is observable to debtholders and the deposit insurance authority, there is a unique optimum qoDI. Furthermore,$qoDI.

Proof

See the Appendix. $□$

The bank takes less risk when there is deposit insurance because the deposit insurance premium increases with risk and therefore reduces the bank’s returns.

In our second benchmark, we maintain the observability of q to debtholders, but suppose that the deposit insurance authority set its premium in advance (i.e., the authority revises premia infrequently). The bank thus takes the premium π as fixed, but the deposit insurance authority rationally foresees the level q and sets the premium fairly. We then obtain the following result:

Lemma 2

When q is observable to debtholders but not to the deposit insurance authority the optimum $q2DI$satisfies the property $qoDI.

Proof

See the Appendix. $□$

The level of risk $q2DI$ is larger than qo and qoDI. When the bank is not sensitive to the cost it pays for deposit insurance it increases the level of risk q.

Now consider the case where q is unobservable to the deposit insurance authority and debtholders. Here, the bank takes both R(q) and the deposit insurance premium π as fixed.23 We then obtain:

Lemma 3

When q is unobservable to both debtholders and the deposit insurance authority the optimum $q^DI$satisfies the property $qoDI.

Proof

See the Appendix. $□$

As the proof highlights the optimum may not be unique. Any solution, however, has the property that it is larger than the first benchmark $qoDI$ and at least as large as the second benchmark $q2DI$. Thus, when q is unobservable shareholders prefer to increase the level of risk.

It is possible to incentivize the manager to implement qoDI by exposing the manager’s compensation to the CDS spread. The formula for the ratio of the share of CDS to the share of equity is now more complicated:

where
$Ω=qoDI(x+Δ)+(1−2qoDI)x+qoDIγ(x−δ)−12α(qoDI)2.$

Importantly, however, shareholders will not want to implement qoDI and implement the CDS contract. When the deposit insurance authority cannot observe risk directly it is only in the interest of shareholders to implement $q2DI$. Consequently, regulatory intervention is required in the presence of deposit insurance to be able to implement the socially desirable level of risk.

### 6.3 UNOBSERVABLE TAIL RISK

Although CDS spreads were fairly accurate as the crisis began in the second quarter of 2007, particularly for distinguishing risk across banks, they did not seem to reflect the underlying risks being taken leading up to the crisis.24 The CDS market therefore may not be as efficient as we assume in the text, particularly in terms of observing tail risk. We now look at the implications of this inefficiency for the model. We use the model of “local thinking” of Gennaioli and Shleifer (2010) to capture this effect.25

We assume that both bondholders and the CDS market are not aware of the lowest possible outcome (where the bank’s investment returns zero). The probabilities, from their point of view, for the remaining states are then conditional on the zero return state not occurring: the high return occurs with probability $q1−q+qγ$, the medium return occurs with probability $1−2q1−q+qγ$, and the low positive return occurs with probability $qγ1−q+qγ$. We maintain all other parts of the model. Bondholders still do not observe q but have “rational local expectations,” and the CDS market still observes q. The shareholders of the bank and manager are not assumed to be “local thinkers” and understand the risk being generated by the bank’s choices. All non-local thinkers, however, take into account that some agents think locally.

This leads to a new benchmark, where the level of risk chosen by shareholders would be:

$quo=Δ−x+γ(x−δ)+(1−γ)α .$

Not surprisingly, this is strictly larger than the original choice qo since the bondholders do not need to be compensated as much for the risk. When the manager is only compensated with equity she chooses the risk level $q^u$, where:

Proposition 6

If a rational expectations equilibrium exists, $q^>q^u>quo$.

Proof

See the Appendix. $□$

As in the main model, there is excessive risk taking as the interest rate the bondholders demand is not sensitive to small changes in risk. In the proof of Proposition 6, we show that if both $q^u$ and $q^$ exist and we apply the notion of stability that we used in Proposition 1, then $q^u. In other words, local thinking may lead to less risk ($q^u) than in the main model when bondholders do not observe q. The reason is that local thinking results in lower interest payments to bondholders, so that shareholders are less keen to avoid these states. From a social surplus perspective, it is thus better to have debtholders who think locally rather than fully rational debtholders. This logic, however, only goes so far. From the perspective of shareholders and the manager, local thinking in debt markets increases returns, but the increase in returns comes from two sources—an increase in actual expected returns and a transfer of expected returns from debtholders to shareholders. Debtholders expect (incorrectly) that their returns will be equal to 1, when in fact their expected returns will be less because of risk taking by the manager. So although returns are higher, debtholders are worse off in this scenario.

We now examine whether excess risk taking can be corrected by modifying the compensation package. Again, it is not in shareholders’ interest to implement such a correction. Even if they wanted to implement a correction, they would correct to $quo$ rather than qo. Regulatory intervention is required to achieve this reduction in risk taking.

The CDS market is affected by local thinking, and the spread is equal to:

(11)
$PDiu=qiγ1−qi+qiγ(1−x−δ1+Ru(qiT)), i=H,L,$
where $qiT$ is the risk level perceived by bondholders. The interest rate adjusted for local thinking is
(12)
$Rui(q)=(1−qi+qiγ)−qiγ(x−δ)1−qi−1, i=H,L.$

As the equity market is not affected by the low state, the price of shares is the same as in Equation (7) save for the fact that the local thinking interest rate $Rui(qi)$ is used.

We now solve for the compensation contract that implements the first-best risk level qo. Although it is conceivable that the social planner/regulator is also a local thinker, implementing qo remains the appropriate benchmark to see how compensation must be structured optimally. The following proposition shows that the weighting given to the CDS component relative to the equity component in the local thinking model may be more or less than that in the main model, depending on the parameters.

Proposition 7

A compensation contract under local thinking $(w¯u,sEu,sDu)$can implement the first best risk level qo. Such a contract may have a larger or smaller weighting on the CDS component relative to the equity component than the compensation contract $(w¯,sE,sD)$from Proposition 3, depending on the parameters.

Proof

See the Appendix. $□$

Even though bondholders and the CDS market ignore tail risk, including CDS in compensation contracts will improve incentives. Indeed, it is still the case that all-equity contracts cannot implement the first best and that contracts with some CDS-based compensation will implement the first best. There still needs to be a part of compensation that deals with default risk, even if it does not deal directly with tail risk. There is a tradeoff in determining how much to weight the CDS component. As local thinking leads to lower interest payments to bondholders, there is less risk taking (as shown in Proposition 6), making the CDS component less important. However, local thinking in the CDS market means that CDS spreads do not reflect risk sufficiently, so that they need to be overweighted. This implies that the parameters will determine which effect is larger in determining optimal compensation.

## 7. Regulatory Aspects of CDS-based Compensation

Our analysis suggests that regulatory intervention is needed to correct the bias in executive pay in favor of shareholder interests. The incentives provided induce managers to take excessive risks at the expense of financial stability. Bank regulators can enhance financial stability and increase expected social surplus by requiring that there be a debt component in bank CEO compensation; in particular, by requiring that bank CEOs be exposed to their own bank’s CDS spread. Although initial compensation regulations following the financial crisis did not focus on debt components for bank CEO compensation,26 more recently the issue of risk taking and bankers’ compensation has received renewed attention from bank regulators. Thus, the Federal Reserve, mandated by the Dodd–Frank act, recently released final rules27 on “Enhanced Prudential Standards,” which provided the Federal Reserve with the scope to regulate compensation and risk management at Bank Holding Companies28 in their annual bank examinations.29 Furthermore, Daniel Tarullo, the governor of the Federal Reserve System charged with prudential regulation, noted in remarks on June 9, 2014, that while compensation packages based on equity have “succeeded in better aligning the interests of shareholders and employees, they intensify the conflict between shareholder and regulatory interests” and concluded by saying that it is key to adopt measures that adjust “incentives so as to promote prudential objectives across the many risk decisions made within the firm.”30

The new regulatory framework in the USA permits compensation based on CDSs. We believe that the implementation of CDS-based compensation for bank CEOs, other executives, and traders making risk-taking decisions most compatible with current practice and regulations is to simply introduce a formula reducing the deferred bonuses set aside by the bank when the bank’s CDS spread increases.31 The higher the spread, indicating more risk, the bigger the cut in deferred bonuses. The undistributed bonuses could then also be added to bank reserves.

Nevertheless, given the reasons discussed in Section 6, boards (as representatives of shareholders) are unlikely to implement CDS-based compensation without persuasion. Therefore, incorporating it directly into regulation is important. Firms are also unlikely to introduce other forms of compensation if there is not a regulatory standard and guidance for that form of compensation.

Note that, by linking deferred compensation to changes in CDS spreads it is possible to partially correct for myopia in the CDS market. Of course, as we point out in Section 6.3, the precise formula and the weighting of CDS spreads in bankers’ compensation will no doubt require careful calibration. No matter how complex the determination of the exact weighting of CDS price risk exposure is in practice, it is safe to say based on our analysis that the optimal weight on CDS spreads is unlikely to be zero.

## 8. Conclusion

In this article, we propose using a recent innovation in financial markets, the CDS, to reduce risk taking of executives at highly levered financial firms. The CDS provides a market price for risk, which, when weighted correctly in a compensation contract that includes an equity component, can provide first-best risk incentives. We demonstrate that while in their interest, shareholders would be reluctant to adopt this type of contract due to a commitment problem that may be exacerbated by renegotiation, deposit insurance, and/or unobservable tail risk. Although other debt-like instruments are available, basing compensation on the CDS spread reduces agency costs and is thus cheaper for shareholders.

One side-benefit of our approach is that it creates a built-in stabilizer using compensation that would have been useful in the crisis. When banks’ performance deteriorates and their credit quality weakens (and they experience an increase in their CDS spread), the banks will be forced to conserve capital through the automatic adjustment of bonuses. Our approach is thus in a sense analogous to cutting dividends to protect the bank and its creditors. Although cutting dividends imposes a cost on all shareholders, our approach imposes a direct cost on risk-taking managers as well.

### Appendix A

#### A.1 PROOF OF PROPOSITION 1

Taking the first-order condition of the shareholders objective function (Equation (6)) gives:

$Δ−x+1+R(q^)−αq=0.$

In a rational expectations equilibrium, the choice of risk by the bank must be the same as depositors’ expectations, so that $q=q^$. This implies that the equilibrium choice of risk is determined by the intersection of the depositors’ and the banks optimal responses:

(A.1)
$Δ−x+1−q^γ(x−δ)1−q^=αq^.$

The solution is implicitly given by Equation (A.1). For a solution, we can show that $q^>qo.$ Indeed, recall that

$qo=Δ−x+γ(x−δ)α .$

The LHS of Equation (A.1) is $Δ−x+1$ when $q^=0$ and is increasing in $q^$ (given A1). We depict this in Figure 2, which demonstrates that $q^>qo$.

Figure 2.

Excess risk taking: $q^>qo$.

Figure 2.

Excess risk taking: $q^>qo$.

Using the quadratic formula on Equation (A.1), we find that the solution exists as long as

(A.2)
$(α+γ(x−δ)+Δ−x)2−4α(Δ−x+1)>0.$

There can be two solutions to Equation (A.1), as depicted in Figure 2. If we define $αq−(Δ−x)$as the price of the supply of risky debt and $1+R(q)$ as the price of the demand for risky debt, the smallest solution can be seen to be as stable. For this solution, if risk (q) were to be slightly larger, then the price of supply would be larger than the price that debtholders would be willing to accept, pushing risk lower. If risk (q) were to be slightly smaller, the price that debtholders would be willing to accept would be larger than the price at which risk is supplied, pushing risk higher.

#### A.2 PROOF OF PROPOSITION 2

We assume the shareholders speak with one voice. The shareholders’ objective function is given by:

(A.3)
$PEH−(w¯+(1+z)[sEPEH+sD(P¯−PDH)])$
given that in A2 we assumed the shareholders prefer to implement high effort. The prices $PEH$ and $PDH$ are given by Equations (7) and (8), respectively. In the text, we assumed that both equity and CDS compensation would be more costly than cash, which is represented by z > 0. The variable $qiT$ represents the risk level that bondholders believe the bank will implement through the compensation contract for a given effort level $i∈{L,H}$. The shareholders must take this as fixed.

The shareholders face the participation constraint (Equation (4)) and the effort constraint (Equation (3)). We denote the Lagrange multipliers on those constraints $λP$ and λK, respectively. In addition, the manager maximizes his compensation by choosing q, giving us the following simplified first-order condition (given that $U′(W)>0$):

(A.4)
$eiΔ−x+(1+R(qiT))−αqi−sDsE(1−γ(x−δ)1+R(qiT))=0,i∈{L,H}.$

Notice that the second-order condition is negative. This choice represents additional constraints for the shareholders, with the multiplier $λMi$. We must examine this for both effort levels even though we have assumed the shareholders prefer to implement high effort. That is because we must define what happens off-the-equilibrium path, which will be important for implementing high effort using the effort constraint (Equation (3)).

The shareholders maximize Equation (A.3), choosing the components of the wage $w¯$, sE, and sD and taking into account the participation constraint, the effort constraint, and the manager’s risk choice (Equation (A.4)). The first-order conditions are, respectively:

Rewriting $FOCw¯$, we get:

$−1+λPU′(WH)+λK(U′(WH)−U′(WL))=0$
since $WH>WL$ (from the participation constraint) and $U′′<0$, this implies that $λP>0$ and the participation constraint binds.

Next, substituting $FOCw¯$ into $FOCsD$ and rewriting, we get:

(A.5)

Substituting $FOCw¯$ into $FOCsE$ and rewriting yields:

(A.6)

Taking Equation (A.5), multiplying it by $sDsE$, adding Equation (A.6) and rewriting using Equation (A.6) give us:

$λKU′(WL){sDsE(PDL−PDH)+(PEH−PEL)}=z(sDsE(P¯−PDH)+PEH).$

We can simplify further using the definitions of WH and WL:

$λKU′(WL)1sE{WH−WL}=z(sDsE(P¯−PDH)+PEH).$

Given that z > 0 and $WH>WL$, this implies that $λK>0$ and the effort constraint binds.

#### A.3 PROOF OF PROPOSITION 3

In a rational expectations equilibrium, bondholders have correct expectations about the choice of qH, so that $qHT=qH$. Using this equality, we can rewrite the first-order condition as follows:

(A.7)
$sDsE=Δ−x+(1+R(qHT))−αqHT1−γ(x−δ)1+R(qHT).$

Setting $qHT=qo$ and simplifying yields Equation (9) in the proposition.

#### A.4 PROOF OF PROPOSITION 4

We examine the solution where $sD=(1+R(qo))sE$. Note that $1+R(qo)$ is defined in the text as a function of parameters by Equation (9).

We start by introducing some additional notation:

(A.8)
$FK(ψ,U¯):=U(WH)−U(WL)−ψ.$

(A.9)
$FPC(ψ,U¯):=U(WH)−ψ−U¯.$

We suppress the arguments for FK and FPC and can then represent Equations (3) and (4) as binding by setting $FK=FPC=0$.

We apply the implicit function theorem, which, here, implies that

(A.10)
$dsE∗dy=−det[∂FK∂y∂FK∂w¯∗∂FPC∂y∂FPC∂w¯∗]det[∂FK∂sE∗∂FK∂w¯∗∂FPC∂sE∗∂FPC∂w¯∗] and dw¯∗dy=−det[∂FK∂sE∗∂FK∂y∂FPC∂sE∗∂FPC∂y]det[∂FK∂sE∗∂FK∂w¯∗∂FPC∂sE∗∂FPC∂w¯∗],$
where y is an arbitrary parameter. We begin by signing the denominator, which is the same in both expressions.

Note that we will use the envelope theorem for the subsequent calculations (and thus do not have to look at the effect on qi).

The following four expressions are the terms in the denominator:

It is obvious that $∂FPC∂w¯∗$ and $∂FPC∂sE∗$ are strictly positive. Since $WH>WL$ and $U′′<0, ∂FK∂w¯∗<0$. The sign of $∂FK∂sE∗$ depends on the parameters. Nevertheless, we can prove that the denominator is positive. The denominator is:

We can rewrite this expression by factoring out $1sE$ and simplifying:

$=U′(WH)U′(WL)1sE(WH−WL).$

Therefore, the denominator is positive.

We begin by finding the effect of $U¯$ on the compensation variables. Simply, $∂FK∂U¯=0$ and $∂FPC∂U¯=−1$. Therefore,

where the second expression has an ambiguous sign. The first expression, combined with the results above, gives us that $dsE∗dU¯>0$. Furthermore, we know that $dsD∗dU¯=(1+R(qo))dsE∗dU¯>0$.

Now we look at the effect of ψ on the compensation variables. Simply, $∂FK∂ψ=−1$ and $∂FPC∂ψ=−1$. Therefore,

$∂FK∂ψ∂FPC∂w¯∗−∂FK∂w¯∗∂FPC∂ψ=−U′(WL)<0∂FK∂sE∗∂FPC∂ψ−∂FK∂ψ∂FPC∂sE∗=U′(WL)(PEL+(1+R(qo))(P¯−PDL))>0.$

The first expression, combined with the results above, gives us that $dsE∗dψ>0$. Furthermore, we know that $dsD∗dψ=(1+R(qo))dsE∗dψ>0$. Finally, the second expression gives us $dw¯∗dψ<0$.

#### A.5 PROOF OF PROPOSITION 5

We will first prove that the participation constraint binds in the case where there is deferred cash compensation. As before, we assume it is optimal to implement high effort. The shareholders’ objective function is

(A.11)
$PEH−(w¯+(1+z)sEPEH+(1−qH)C+qHγηC).$

The participation constraint can be written as:

(A.12)
$(1−qH)U(w¯+sEPEH+C)+qHU(w¯+sEPEH+γηC)−ψ≥U¯.$

The effort constraint is now:

(A.13)

In addition, the manager maximizes his compensation by choosing q as before. For simplicity, we will define $W1i=w¯+sEPEi+C$ and $W2i=w¯+sEPEi+γηC$ for $i=H,L$. This gives us the manager’s risk choice:

(A.14)

Once again, we denote the Lagrange multipliers as $λP$ (participation constraint), $λK$ (effort), and $λMi$ (the manager’s risk choice). The first-order conditions are

$FOCC:0=−(1−qH+qHγη)+(λP+λK){(1−qH)U′(W1H)+qHγηU′(W2H)}−λK{(1−qL)U′(W1L)+qLγηU′(W2L)} −λMH{γηU′(W2H)−U′(W1H)+((1−qH)U′′(W1H) +qHγηU′′(W2H))(Δ−x+(1+R(qHT))−αqH)} −λML{γηU′(W2L)−U′(W1L)+((1−qL)U′′(W1L) +qLγηU′′(W2L))(Δ−x+(1+R(qLT))−αqL)}.$

Substituting $FOCw¯$ into $FOCsE$ and rewriting gives us:

(A.15)

Given that preferences are CARA, the following properties hold:

(A.16)
$U′(W)=a(1−U(W)).U′′(W)=−a2(1−U(W)).$

Substituting this into the second and third lines of Equation (A.15) and rewriting, we get:

(A.17)

Given that $U′(W)=a(1−U(W))$, it is straightforward to see the expression in the second and third lines in curly brackets is equal to the first-order condition for risk choice in Equation (A.14). Therefore, it is equal to zero. Furthermore, we set z = 0, which is one of the conditions for the proposition. This gives us:

$0=λK{(1−qL)U′(W1L)+qLU′(W2L)}.$

This implies that $λK=0$.

Using this fact to rewrite $FOCw¯$:

(A.18)

As above, the properties of the CARA function given in Equation (A.16) imply that the second line and third line of Equation (A.18) can be rewritten as equivalent to the first-order conditions for risk choice, making them equal to zero. This gives us:

$0=−1+λP{(1−qH)U′(W1H)+qHU′(W2H)}.$

This implies that $λP>0$ and the participation constraint binds.

Given that the participation constraint binds, we can now show that deferred cash compensation is more costly than CDS-based compensation.

In both the CDS-based compensation case and the deferred cash compensation case, the participation constraint binds (Equations (4) and (A.12)). This implies that:

$U(WH)=(1−q)U(W1H)+qU(W2H),$
where WH is the total compensation in the CDS case and $W1H$ and $W2H$ are the total compensation in the two states of the deferred cash case. From concavity, we know that:
$U((1−q)W1H+qW2H)>(1−q)U(W1H)+qU(W2H).$

This implies that:

$WH<(1−q)W1H+qW2H.$

Given that z is equal to zero, the cost of compensation is thus smaller with CDS-based compensation than with deferred cash compensation. It is then straightforward to show that the shareholders are better off: if we implement the first-best level of risk qo and positive effort for both types of compensation, the price of equity is the same in both cases. Therefore, the shareholders’ objective function is larger when there is CDS-based compensation.

#### A.6 PROOF OF LEMMA 1

As in Section 3, where shareholders manage the firm with one voice, we set effort to be high and examine the shareholders’ maximization problem:

We can rewrite the problem as:

$maxq (B+L−P(q)){q(x+Δ)+(1−2q)x+qγ(x−δ)−12αq2}−(B+L).$

Not surprisingly, this expression represents total surplus. The difference with our previous model is that P is present and depends on q.

Taking the first-order condition of the objective function yields:

(A.19)

First note that $P′(q)=1q(1−qγ(x−δ))P(q)>0$. We will assume that $q(x+Δ)+(1−2q)x+qγ(x−δ)−12αq2>0$ for all $q∈[0,½]$, which says the project has positive NPV for all q. This then implies, from Equation (A.19), that for any solution, $Δ−x+γ(x−δ)−αq>0$.

Taking the derivative of the FOC, we get:

$−P′′(q){q(x+Δ)+(1−2q)x+qγ(x−δ)−12αq2}−2P′(q){Δ−x+γ(x−δ)−αq}−α(B+L−P(q)).$

The expression $P′′(q)=2qγ(x−δ)q2(1−qγ(x−δ))2P(q)>0$. Assumption A3 ensures that $B+L−P(q)>0$. As we noted above, the FOC implies that $Δ−x+γ(x−δ)−αqoDI>0$. Remember that qo from our original model satisfied the condition $Δ−x+γ(x−δ)−αqo=0$. This implies that $qoDI. Furthermore, it implies the objective function is strictly concave over the interval $[0,qo]$. It may not be concave over the interval $[qo,½]$, but as there is no inflection point in that interval, meaning that qoDI must be the global maximum.

#### A.7 PROOF OF LEMMA 2

As in Section 3, where shareholders manage the firm with one voice, we set effort to be high and examine the shareholders’ maximization problem:

Taking the first-order condition of the objective function and then setting $q=q2DI$ yields:

(A.20)
$0=(B+L−P(q2DI)){Δ−x−αq2DI}+L.$

Rewriting this equation, we get:

(A.21)
$B+L−P(q2DI)=Lαq2DI−(Δ−x).$

First, examine the LHS. At $q2DI=0$, it is equal to B + L. As we showed in Lemma 1, it is decreasing and concave, and Assumption A3 assures it is positive.

Next, examine the RHS and for now, take $Δ−xα<12$. When $q2DI<Δ−xα$, the RHS is negative, decreasing, concave, and approaching negative infinity as $q2DI$ approaches $Δ−xα−$. When $q2DI>Δ−xα$, the RHS is positive, decreasing, and concave (and approaching positive infinity as $q2DI$ approaches $Δ−xα+$). Therefore, the RHS may intersect the LHS when $q2DI>Δ−xα$ 0,1, or 2 times. If it does not intersect, that implies the solution is for the manager to choose the maximum $q2DI=½$ as the first-order condition in Equation (A.20) is positive holding fixed the deposit insurance authority expectations at $q2DI=½$. If it intersects twice, both intersections are rational expectations equilibria, but only the smaller one is stable (using the notion of stability that we use in Proposition 1).

If $Δ−xα>½$, the solution would be $q2DI=½$.

Now we prove that $qoDI. Clearly, this must be the case if $q2DI=½$. Otherwise, we can rewrite Equation (A.20) by adding and subtracting $γ(x−δ)(B+L−P(q2DI))$:

Simplifying, we get:

$0=(B+L−P(q2DI)){Δ−x+γ(x−δ)−αq2DI}+1qP(q2DI).$

This implies that $Δ−x+γ(x−δ)−αq2DI<0$, which proves our statement.

#### A.8 PROOF OF LEMMA 3

As in Section 3, where shareholders manage the firm with one voice, we set effort to be high and examine the shareholders’ maximization problem:

The first-order condition of the problem is (we also set $q=q^DI$):

(A.22)
$0=(B+L−P(q^DI)){Δ−x−αq^DI}+(B1−q^DI+L).$

We proceed with a similar approach to Lemma 2.

Rewriting the FOC, we get:

(A.23)
$B+L−P(q^DI)=(B1−q^DI+L)αq^DI−(Δ−x).$

First, examine the LHS. At $q2DI=0$, it is equal to B + L. As we showed in Lemma 2, it is decreasing and concave, and Assumption A3 assures it is positive.

Next, examine the RHS and take $Δ−xα<½$. When $q^DI<Δ−xα$, it is negative and approaching negative infinity as $q^DI$ approaches $Δ−xα−$. When $q^DI>Δ−xα$, it is positive (and coming from positive infinity as $q^DI$ approaches $Δ−xα+$). Furthermore, for any $q^DI$ such that $q^DI>Δ−xα$, the RHS is strictly larger than the RHS in Equation (A.21). Therefore, if there are zero intersections for Equation (A.21), there are zero for the current equation and $q^DI=½$. If there is one intersection for Equation (A.21), then the solution $q^DI$ is strictly larger than $q2DI$ and may be equal to $½$.32 If there are two intersections for Equation (A.21), then the solution $q^DI$ is strictly larger than the stable rational expectations equilibria $q2DI$.

If $Δ−xα>½$, the solution would be $q2DI=½$.

#### A.9 PROOF OF PROPOSITION 6

Bondholders must receive in expectation as much as the safe return given their beliefs:

$1+Ru(q^u)≥(1−q^u+q^uγ)−q^uγ(x−δ)1−q^u.$

The first-order condition for shareholders (or a manager who is compensated only in equity and a fixed component) assuming high effort has been implemented is:

$Δ−x+1+Ru(q^u)−αq=0.$

In a rational expectations equilibrium the choice of risk by the bank must be the same as depositors’ expectations, so that $q=q^u$. This implies that the equilibrium choice of risk is determined by the intersection of the depositors’ and the banks optimal responses:

(A.24)
$Δ−x+(1−q^u+q^uγ)−q^uγ(x−δ)1−q^u=αq^u.$

For a solution, we can show that $q^u>quo$. The LHS of Equation (A.24) is $Δ−x+1$ when $q^u=0$ and is increasing and convex in $q^u$ (given A1). We depict this in Figure 3, which demonstrates that $q^u>quo$. Furthermore, if we employ the same notion of stability as in Proposition 1 and assume existence both for Proposition 1 and here, the stable solution here $q^u$ is smaller than $q^$. This comparison can also be observed in Figure 3.

Figure 3.

Excess risk-taking with local thinking.

Figure 3.

Excess risk-taking with local thinking.

#### A.10 PROOF OF PROPOSITION 7

Assuming that high effort is optimal to implement, the equity price is:

$PEiu=qi(x+Δ)+(1−2qi)x−(1−q)(1+Ru(qiT))−12αqi2, i=H,L,$
where $qiT$ is the bondholders’ perceived risk choice.

The CDS spread is given by Equation (11).

The manager chooses the risk level qi to maximize her payoff:

$maxqiU(w¯u+sEuPEu(qi)+sDuPDu(qi)), i=H,L.$

This gives the following first-order condition:

$Δ−x+(1+Ru(qiT))−αqi−sDusEuγ(1−qi+qiγ)2(1−x−δ1+Ru(qiT))=0, i=H,L,$
where the second-order condition is negative.

In a rational expectations equilibrium, $qHT=qH$. Then implementing the risk choice qo gives us:

(A.25)
$sDusEu=Δ−x+(1+Ru(qo))−αqoγ(1−qo+qoγ)2(1−x−δ1+Ru(qo)).$

Consider setting $x−δ=0$. Given qo then equals $Δ−xα$, we find that:

$sDusEu=1+Ru(qo)γ(1−qo+qoγ)2=11−qo(1−qo+qoγ)3γ.$

The benchmark from the non-local thinking case (the main model) can then be written as $sDsE=1+R(qo)=11−qo$. Therefore, we analyze whether $(1−qo+qoγ)3γ$ is larger than, smaller than, or equal to 1.

When γ = 0, $(1−qo+qoγ)3>γ$. When γ = 1, $(1−qo+qoγ)3=γ$. The slope of $(1−qo+qoγ)3$ is $3(1−qo+qoγ)2qo$ and its second derivative is $6(1−qo+qoγ)(qo)2$, which is positive. The slope of $γ$ is 1. Since $(1−qo+qoγ)3$ is increasing and convex, its intersection with γ at γ = 1 is either the only intersection or the second intersection. It is the only intersection if the slope of $(1−qo+qoγ)3$ at γ = 1 is less than or equal to 1, and is the second intersection if the slope of $(1−qo+qoγ)3$ at $γ=1$ is more than 1. If there is one intersection, then $sDusEu>sDsE$ for all $γ<1$. If there are two intersections, there exists some cutoff $γ⌣$, such that for $γ≤γ⌣, sDusEu>sDsE$, and for $γ>γ⌣, sDusEu.

Finally, the first derivative of $(1−qo+qoγ)3$ at γ = 1 is equal to $3qo$ or $3Δ−xα$ which may be larger or smaller than 1 depending on the parameters.

* We thank Douglas Diamond, Nicola Gennaioli, Frederic Malherbe, Matthew Osborne, Tarun Ramadorai, Nicolas Serrano-Velarde, Haluk Unal, Chenyang Wei, and participants at the AFA, EFA, FDIC CFR workshop, the GARP webcast, the ICFR conference on “The New Global Banking Regulations and the Cost of Intermediation,” and the Columbia Business School conference on “Governance, Executive Compensation and Excessive Risk in the Financial Services Industry” for helpful comments. We also thank the Editor (Franklin Allen) and two referees for helpful comments and suggestions. We are grateful to the FDIC CFR and GARP for financial support. Shapiro also thanks the Fundación Ramon Areces for support. The views expressed are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of New York or The Federal Reserve System.
1 In Panel (A) of Figure 1 of Graham, Leary, and Roberts (2014), the measure of total debt/capital in 2010 (their last year of data) is about 34%. Over the 2000–10 period, this fluctuates above 34%, but never below. Their data exclude financial firms, utilities, and railroads.
2 The Tier 1 Capital Ratio for Bank Holding Companies larger than \$500 bn in 2010 is about 11.5%, making the leverage ratio then about 88.5%. This is after a period of deleveraging, the leverage ratio for 2000–09 is above 90%. These numbers are similar for banks of different sizes. For details, see: http://www.newyorkfed.org/research/banking_research/quarterly_trends.html/QuarterlyTrends2013Q4.pdf
3 Other papers find this relationship holds in non-crisis times and for non-financial firms as well. For example, DeFusco, Johnson, and Zorn (1990) find that announcement of executive stock option plans cause firms’ stock prices to rise, their bond prices to fall, and equity volatility to rise, consistent with risk shifting.
4Edmans and Liu (2011) show that using inside debt (such as deferred compensation or pension benefits) will correct risk-taking incentives.
5 Although we focus on the compensation of bank CEOs, in practice the bonuses of all risk-takers can be linked to CDS spreads.
6John, Mehran, and Qian (2010) finds supporting evidence in the banking sector for the predictions on the determinants of the pay-performance ratio from John and John’s (1993) model.
7 In a previous version of this article, we take the Wei and Yermack (2011) approach and apply it to large financial institutions, finding that the disclosure of inside debt held by CEOs in large financial institutions decreased CDS spreads.
8 In this section, we will treat deposits and subordinated debt as equivalent. In Section 6.2 on deposit insurance, we model them separately.
9 In an earlier version of this article, we examined the role of leverage choice. That analysis is available upon request.
10 In an earlier version of the article, we assumed that if there was a default, only a fraction of the returns could be recovered. This does not affect the results, so we removed it to make the model more parsimonious.
11 The effort choice will affect the equity price directly through the value of the high return. The choice will affect the CDS spread indirectly, through the effect of effort on the risk choice.
12 Rather than directly sell a CDS to the manager, we envision the firm setting aside a pool of money that can be paid out to the manager according to the market price of the CDS. We discuss the implementation further in Section 7.
13 If default were costly in terms of lost resources, these would diverge, making a bank with debt more costly than an all-equity bank.
14 We define the notion of stability in the proof of the proposition.
15 In principle, this ex-post observability of q through the CDS and equity price could be incorporated into bond contracts ex-ante, but this is typically not done in practice (although step-up bonds incorporate credit ratings into the interest rate paid, as discussed in Manso, Strulovici, and Tchistyi (2010). We thank a referee for pointing this out). Accordingly, we shall not introduce this contractual contingency into bond contracts. Note that if we were to allow R to depend on the CDS spread we would introduce a potentially complex fixed-point problem for the equilibrium risk choice q.
16 More generally, if pi is the CDS spread and β is the discount factor, the no-arbitrage condition is
$pi(1+R(qiT))∑j=0nβj(1−qi)j=qi(1+R(qiT)−γ(x−δ))∑j=0nβj(1−q)j,$
where $1+R(qiT)$ is the face value and $γ(x−δ)$
is the expected amount recovered in default. In our basic model, n = 0, but one can observe that if periods were added, the CDS price would be the same.
17 Clearly, this is a simplification and we could add noise to these prices, which would affect the risk-averse manager. The key element here, which will be discussed in Section 5, is that due to the information aggregation in markets there will be less noise in market prices than in the outcomes. This is why basing compensation on CDS prices is better than basing it on debt.
18Mehran, Morrison, and Shapiro (2012) provide a review of the literature connecting corporate governance and risk taking by banks during the financial crisis.
19 There is a fourth more subtle reason: while the shareholders prefer qo when there is no manager and when a manager has stock-based compensation, they don’t prefer qo once there is compensation based on debt because their objective function has changed to incorporate the fact that the wage paid is based on the CDS spread as well.
20 What the equilibrium risk choice is depends on how the problem is formulated. One logic is that rational investors anticipate renegotiation and therefore pay no attention to the incentive contract in the first place. They act as if shareholders had no commitment power (as in Katz, 1991). Another logic is that they are naive and get fooled, in which case the final outcome is like the outcome in the section on naive bondholders.
21John, Saunders, and Senbet (2000) have a model of risk shifting in which deposit insurance restores first-best incentives for risk taking to shareholders (who then induce the manager to take the first-best risk choice). Our model differs along several dimensions. First, we posit that the money paid for deposit insurance up front comes from deposits and debt raised. Second, as the deposit insurance authority sets its premium first, the risk level is not responsive to the risk choice (although it will be accurate given rational expectations). Third, we allow for the return demanded by bondholders to depend on expectations of risk.
22 This condition implies that both the numerator and the denominator of the premium P in Equation (10) are positive (given that $q∈[0,½]$ and $rs∈(0,1)$).
23 Although the deposit insurance authority and the debtholders act at different times, the bank takes both of their actions as fixed. Nevertheless, we assume that both are rational and must be correct in equilibrium about the level of risk.
24 This can be observed, for example, in the graph of CDS spreads and the dispersion of CDS spreads in the chart on page 81 of Bank for International Settlements (2010). There, spreads are low and slightly decreasing, with low variance, from 2004 to the first quarter of 2007. We thank a referee for pointing to this evidence.
25Gennaioli, Shleifer, and Vishny (2012) use “local thinking” in the context of financial markets to explore the link between financial innovation and financial fragility.
26 The US Dodd–Frank bill was focused on say-on-pay, disclosure, and clawbacks. Although clawbacks could be used to replicate the incentive structure of debt-like compensation, the clawbacks in the Dodd–Frank bill are only to be used for accounting misstatements or malfeasance. In Europe, binding say-on-pay has become law in several countries and an European Union cap on variable pay for the financial services industry has recently been put in place.
27 The rule was finalized on March 27, 2014. It is available here: http://www.gpo.gov/fdsys/pkg/FR-2014-03-27/pdf/2014-05699.pdf
28 It should be noted that since the adoption of TARP and the subsequent passage of Dodd–Frank, all investment banking firms are in effect bank holding companies and are covered by the same laws and are subject to standard examinations and stress-tests. For a list of the largest US BHCs as of December 2013, see http://www.ffiec.gov/nicpubweb/nicweb/top50form.aspx
29 Making compensation and risk practices transparent to regulators (and internal auditors) requires substantial documentation (see the guidance on incentive compensation for further discussion: http://www.federalreserve.gov/newsevents/press/bcreg/20100621a.htm).
30 Tarullo (2014, p. 11–12).
31 In theory, there are other ways to implement CDS-based compensation, such as requiring that managers write a given amount of CDS (or buy swaps written by other insurers) for the duration of their employment contract.
32 There may be multiple solutions here, but all will be larger than $q2DI$.

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