Regulating Bank Portfolio Choice Under Asymmetric Information

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Regulating Bank Portfolio Choice Under Asymmetric Information Chris Anderson 2025-009 Please cite this paper as: Anderson, Chris (2025). “Regulating Bank Portfolio Choice Under Asymmetric Information,”FinanceandEconomicsDiscussionSeries2025-009. Washington: BoardofGovernors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2025.009. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Regulating Bank Portfolio Choice Under Asymmetric Information Chris Anderson∗ November 1, 2024 Abstract Regulating bank risk-taking is challenging since banks know more than regulators about the risks of their portfolios and can make adjustments to game regulations. To address this problem, I build a tractable model that incorporates this information asymmetry. The model is flexible enough to encompass many regulatory tools, although I focus on taxes. These taxes could also be interpreted as reflecting the shadow costs of other regulations, such as capital requirements. I showthatlinearrisk-sensitivetaxesshouldnot generallybesetmoreconservatively to address asymmetric information. I further show the efficacy of three regulatory tools: (1) not disclosing taxes to banks until after portfolio selection, (2) nonlinear taxes that respond to information contained in banks’ portfolio choice, and (3) taxes on banks’ realized profits that incentivize banks to reduce risk. ∗FederalReserveBoard. Email: christopher.s.anderson@frb.gov. IthankAndrewHawleyforexcellent research assistance. For helpful comments, I thank John Campbell, Chris Clayton, Sharjil Haque, Maximilian Jager, Charles Kahn (discussant), David McArthur, Marco Migueis, Matt Pritsker, Ben Ranish, Doriana Ruffino, Jeremy Stein, Nikola Tarashev, Cindy Vojtech, Ke Wang, and Yan Xue (discussant). I also thank the participants of the FMA Annual Meeting, IBEFA Summer Meeting, and the FDIC Bank Research Conference. The views in this paper are solely the author’s and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. 1

1 Introduction Modern banking regulation generally aims to be risk sensitive. For example, under the current Basel III framework, the primary determinants of capital requirements are riskbased measures that aim to assign higher requirements to riskier portfolios. Risk weights are a major input into these measures, which reflect regulators’ best guess of the risks of different exposures within banks’ portfolios. However, regulators typically know less about risk than banks, which can expose these frameworks to potential gaming. When regulators underestimate an asset’s risk, banks can take advantage by overweighting it in their portfolios. Therefore, a common heuristic among regulators is to be conservative on average, under the theory that such conservatism will counterbalance this gaming.1 However, it is unclear under which circumstances this heuristic should work or whether there are better ways of addressing banks’ information advantage. Furthermore, there has been surprisingly little theoretical work on this topic, despite its practical importance. In this paper, I build a tractable model to address the question of how to regulate banks’ portfolio choice in the presence of asymmetric information, which takes into account that banks will game the regulations ex-post. In this model, a bank selects its portfolio among a wide range of assets, which I interpret as reflecting investments in different granular subsectors (e.g., an asset might represent all loans to small biotechnology companies), subject to regulatory constraints. While these constraints can take many forms, for tractability, I focus on a regulator setting risk-sensitive taxes. These taxes could be interpreted as literal taxes or as shadow costs from quantity-based regulation, such as capital requirements. Each asset has a different level of risk, measured as its beta with respect to a single systematic risk factor, and profitability. Information asymmetry exists because the bank perfectly observes each asset’s risk and profitability, while the regulator only receives a noisy signal of risk. The bank and regulator share the objective of increasing profitability and decreasing risk, but bank risk-taking imposes a social externality that the bank does not internalize. Therefore, the regulator wants to reduce the riskiness of the bank’s portfolio, but also recognizes the social costs of the bank forgoing profitable investments. The major contribution of this paper is to develop a tractable and flexible framework that researchers can use to model regulation of bank portfolios under asymmetric information. The framework is rich enough to serve as a baseline for analyzing a wide range of regulatory tools and information asymmetries, but is still simple enough that most results can be solved by hand. I demonstrate how this framework can be used by applying it 1This observation comes from the author’s personal experiences working in banking supervision. 2

to analyzing three regulatory tools: (1) not disclosing taxes to banks until after portfolio selection, (2) nonlinear taxes that respond to information contained in banks’ portfolio choice, and (3) taxes on banks’ realized profits meant to align the incentives of the bank and the regulator. Before analyzing these tools, I begin by considering a familiar baseline: an assetspecific linear tax, which is similar to the linear risk weights used in most current capital regulation. Assuming a linear social cost to risk exposure, the regulator’s optimal policy is to set the tax exactly equal to the expected risk given their signals. Contrary to common intuition, the bank’s ex-post gaming of regulations does not mean the regulator should set taxes conservatively (i.e., higher than expected risk). The reason is that, in this environment, the costs from setting overly high taxes (i.e., dissuading socially optimal investments) cancel out the costs from setting overly low taxes (i.e., banks taking more risk than is socially optimal). Therefore, bank strategic behavior is not a rationale for conservatism by itself: nonlinearity of social costs or underweighting the benefits of profitable investments are necessary.2 I then consider how the three regulatory tools can address this asymmetric information problem. The first tool is strategic nondisclosure. Specifically, I consider a case in which the regulator sets a linear tax on exposure to each asset but does not reveal the magnitude of this tax to the bank until after it has selected its portfolio. Thus, the bank must then select its portfolio based on its best guess of these taxes. If the bank has no information about the regulator’s signals, then the bank guesses that taxes will be calibrated correctly on average, which achieves the regulator’s first-best outcome. However, even if the bank has imperfect information about the likely size of the tax, nondisclosure can still reduce the bank’s ability to game regulation. In practice, nondisclosure is used in stress tests since banks do not fully know the regulator’s model. The model supports maintaining opacity in stress testing since it reduces banks’ ability to take advantage of weaknesses in the stress testing models, similar to the conclusion in Leitner and Williams (2023). However, nondisclosure could also be applied more broadly, including limiting the disclosure of asset-specific risk weights. The second tool is a nonlinear tax on risk exposures. The bank investing more in an asset reveals information about its riskiness, which the regulator can incorporate by conditioning the tax rate on the bank’s investment. Specifically, the regulator optimally sets the marginal tax equal to the regulator’s estimate of risk, conditional on the bank’s investment in that asset (along with the regulator’s other information). This policy naturally leads to a nonlinear tax. Depending on the model’s parameters, this nonlinear 2While this paper takes a Bayesian perspective, accounting for model uncertainty by optimizing with respect to worst-case scenarios does not necessarily justify conservatism. Worst-case scenarios could include setting overly high taxes that prevent socially efficient investments. 3

tax could be increasing or decreasing. For example, if the bank invests more in riskier assets on average, then the regulator will optimally raise the marginal tax rate as the bank invests more. However, in certain cases, the bank might be investing more in an asset because it has lower risk, in which case a regulator would decrease marginal taxes. A nonlinear tax may also not be possible to implement if there is a sufficiently large divergence between the incentives of the bank and the regulator. The most direct policy implication here is that regulators should respond to banks’ portfolio choices, whether through formal policies, such as adjusting risk weights based on the concentration of banks’ investments, or informal ones, such as calibrating shocks in stress tests based on those same concentrations.3 The third tool is a tax on ex-post profits that aligns the bank’s risk-taking incentives with the regulator’s incentives. The tax reduces the expected return that banks receive for taking a given amount of risk, which worsens the bank’s after-tax risk–return tradeoff and effectively makes the bank more risk averse. If perfectly calibrated, this type of tax can perfectly align the incentives of the bank and the regulator. A flat tax on profits is not sufficient: Even though a flat tax reduces banks’ after-tax profits, it also reduces their after-tax portfolio risk in a similar way and thus does not change the risk–return tradeoff. One way to alter the risk–return tradeoff is to have a state-dependent tax that is higher during “good times” (i.e., when the stochastic discount factor (SDF) of banks’ investors is lower). Since banks’ profits are typically higher during booms (which reflect “good times”), a progressive tax on profits could approximate the outcome of a perfectly calibrated tax, similar to the progressive tax proposed by John et al. (1991), to align the incentives of banks’ shareholders. Such a policy could reduce risk-taking incentives at banks (or other financial institutions), even when regulators have little insight into those risks. While this paper is not the first to consider asymmetric information in the context of banking regulation (for example, see Giammarino et al. (1993), Chan et al. (1992), Wu and Zhao (2016), Perotti and Suarez (2018), and Leitner and Williams (2023)), the main high-level difference is its focus on portfolio choice, which considers that banks can invest in many assets and potentially game regulations by tilting their investments towards assets whose risks are underestimated by regulators. Therefore, this paper can address policy questions such as how regulators should specify asset-specific risk weights that take into account information asymmetries. Additionally, this paper considers a substantially different set of tools to address this problem. This paper is structured as follows. Section 2 lays out the model. Section 3 covers 3TheinformalapproachissimilartotheproposalinGreenwoodetal.(2017)toconsiderareasofhigh growth or profitability for banks when calibrating shocks used in stress tests. 4

regulation based on taxing expected risk exposure, including both linear and nonlinear taxes. Section 4 covers regulation based on nondisclosed taxes, which are not revealed to the bank until after it has selected its portfolio. Section 5 covers regulation based on taxing banks’ profits to reduce their effective risk aversion. Section 6 describes the policy applications. Section 7 concludes. 1.1 Related literature This paper relates most strongly to three broad strands of literature, including papers that study (1) information asymmetries between banks and regulators, (2) the effect of capital regulation on bank portfolio choice, and (3) the effect of taxes on banks. The most closely related papers address information asymmetries between banks and regulators. Perotti and Suarez (2018) consider optimal regulation when bank illiquidity imposes an externality, similar to how bank risk-taking imposes an externality in this current paper. They consider the use of both Pigovian taxes as well as quantity-based regulation, taking into account regulators’ uncertainty with respect to banks’ investment opportunities and gambling incentives, in the spirit of Weitzman (1974). One recent strandofliteratureconsidersmodelsecrecyinthecontextofstresstests, includingLeitner andWilliams(2023)andKimetal.(2024). Atahighlevel, thesepapersaddressquestions alongthelinesofhowstrictlytocalibratestresstestsoverallandwhethertorevealenough information about the models so that banks can determine whether they are passing on their own. Giammarino et al. (1993), Chan et al. (1992), and Wu and Zhao (2016) are other papers that address asymmetric information between banks and regulators across different contexts. The key differentiating factor of my paper is its focus on calibrating asset-specific taxes or risk weights, including how to limit banks’ ability to game those calibrations. The second strand of related literature addresses optimal capital regulation for banks, and particularly the impacts on portfolio choice. Kim and Santomero (1988) derive the optimal risk weights for a regulator with full information on asset riskiness with the goal of limiting failure probabilities below a certain level. Rochet (1992) similarly derives optimal regulatory policy, both in terms of asset risk weights for capital requirements and pricing of deposit insurance, for a regulator with full information. More recently, Glasserman and Kang (2014) consider the problem of a regulator choosing optimal risk weights, including a case in which the regulator does not know the mean return of each asset. The main difference is that this paper focuses on the problem of a regulator with limited information about the riskiness of individual assets, whereas the regulator in their paper has full knowledge of each asset’s riskiness. Greenwood et al. (2017) contains a simple model evaluating how various forms of bank capital regulation affect portfolio 5

choice, and the paper discusses how different types of regulations are subject to gaming by banks. While the broad topic of gaming regulations is closely related to the current paper, their model does not explicitly incorporate asymmetric information. The third strand of related literature discusses how tax policy affects bank behavior, which is particularly related to this paper’s results on taxing banks’ ex-post profits. John et al. (1991) is the most relevant, in that they propose setting a progressive tax to mitigate banks’ incentives to tax excessive risks. Shackelford et al. (2010) discuss various ways in which taxation may be used to address externalities in the financial sector. They note that information asymmetries make Pigouvian taxation to address these externalities difficult and broadly discuss how financial transactions taxes, taxes on bonuses, and levies on banks may partially mitigate those externalities. Empirically, Celerier et al. (2019) empirically demonstrate the impact of Belgium’s adoption of an equity subsidy, which allowed banks to deduct an estimate of the cost of equity from their taxes, on the composition of their portfolio. They show that the equity subsidy led banks to shift toward holding more loans rather than government bonds. And while not directly related to banks, this paper relates to a longstanding literature on the impact of taxation on risk-taking in general, such as in Domar and Musgrave (1944) and Stiglitz (1969). 2 The general framework This is a single-period model containing a bank and a regulator. There are many assets whose riskiness (which is reflected by each asset’s beta with respect to a common risk factor) and expected returns are drawn from known prior distributions. Although the framework is general, I think of a single asset in the model as mapping to an investment in some granular subcategory, such as “loans to small biotechnology companies.” The bank observes the betas and expected returns, whereas the regulator only receives noisy signals of each. The regulator can use these signals to establish regulatory constraints, which in general can take many forms, and then the bank selects its investments subject to those constraints. I typically assume that the bank’s starting equity is exogenous, although Section B in the appendix considers the case in which the bank endogenously selects its desired level of equity. Without loss of generality, I normalize the bank’s starting equity to one, so that all quantities can be interpreted as relative to the bank’s total capital. Large quantities of investments in risky assets correspond to high leverage since banks must issue debt (whether as deposits or other forms) to fund those positions. There is a continuum of risky assets indexed by i ∈ [0,1]. The bank selects q , its i 6

quantity of investment in asset i. The overall expected excess return to investing in asset i is a q − cq2. I assume that c > 0, so that there are diminishing returns to investment. i i 2 i q may be negative, which can be interpreted as the bank taking a short position.4 i Each asset is exposed to a single systematic factor F with an asset-specific β loading.5 i Without loss of generality, I set E[F] = 0 and Var(F) = 1. Thus, each asset i has a payoff X of the form6 i c X = a − q +β F. (1) i i i i 2 The bank’s objective is to maximize its market value, which is given by E [M(Π−T)], (2) B (cid:82)1 where Π = q X di is the bank’s pre-tax profit, T represents potential taxes paid to 0 i i the regulator, M is the investors’ stochastic discount factor (SDF), and E denotes an B expectation with respect to the bank’s information set. I assume that the investors’ SDF is7 M = 1−γF, (3) which implies that the bank maximizes E [(Π−T)]−γCov ((Π−T),F) (4) B B (cid:90) 1 c = (aq − q2 −γβ q )di−E [T]−γCov (T,F). (5) i 2 i i i B B 0 The γ term reflects the price of exposure to the systematic factor F, which the bank’s investors cannot diversify away. While the γ term could potentially reflect many forces, it is easiest to think of it as representing investors’ risk aversion. The bank internalizes 4Banks could either literally be taking a short position or achieving similar exposures through other means. For example, a bank might take a short position on a credit exposure through buying a credit default swap. 5The portfolio-invariant risk weights in Basel II and III can be justified by the assumption of a single systematic factor, as discussed in Gordy (2003). However, the assumption of a single factor is not as important in this paper’s framework. As will be explained in more detail shortly, risk in the context of thispaperisbasedoncovariancewiththeinvestors’SDFratherthanthevarianceofthebank’sportfolio. Under the SDF-based measure of risk, each asset’s risk contribution does not depend on the rest of the portfolio even if there are multiple systematic factors. 6For simplicity, I assume that there is no idiosyncratic risk. Since there is a continuum of assets, the idiosyncratic risk would be diversified away and would therefore not affect the model’s results. 7While the SDF can technically become negative for large values of F, this issue is not a concern for myapplication. ThisSDFresultsinapriceofγ perunitofexposuretothesystematicriskfactor,which leads to a more tractable model. 7

the price of risk exposure and only increases exposure to the systematic factor if justified by the profits. Here, I emphasize that risk is measured based on the bank’s portfolio beta rather than the variance. A negative beta portfolio is considered less risky than a zero-beta portfolio because it provides the bank’s investors with insurance by paying out during “bad times” when the marginal value of wealth is high. This linear price of risk exposure matches the approach taken by Froot and Stein (1998). The bank’s risk-taking imposes externalities η > 0 per unit of portfolio beta β = p (cid:82)1 q β di. This approach is a reduced-form way of reflecting a wide range of externalities, 0 i i including government bailouts and bankruptcy costs. The regulator’s objective is to maximize social welfare, which is (cid:20) (cid:90) 1 (cid:21) E MΠ−η q β di (6) R i i 0 (cid:20)(cid:90) 1 c (cid:21) =E (aq − q2 −(γ +η)β q )di , (7) R i 2 i i i 0 where E denotes an expectation with respect to the regulator’s information set, which R will be specified in greater detail shortly.8 The regulator’s objective is identical to the bank’s, except with higher effective risk aversion. The regulator is willing to allow the bank to take more risk if the associated profits are sufficiently high, but is more conservative in this tradeoff compared to the bank. The timing and information structure is as follows. First, the β loadings and a i i measures of investment profitability are drawn from a prior distribution known to both ˆ the bank and the regulator. Then, the regulator receives a noisy signal β for each asset. i Thesesignalsreflecttheregulator’simperfectinformationeachasset’sriskiness. Next, the regulatorusesthisinformationtodetermineregulatoryconstraintsonthebank. Thereare several types of regulatory constraints, but one example is the regulator selecting a tax k i thatthebankmustpayforeachunitofexposuretoasseti. Thiscostcouldbeinterpreted as either a direct cost, such as a premium for deposit insurance, or an indirect cost, such as the cost of higher capital requirements for a position. These payments represent a private cost to the bank but do not affect social surplus. Finally, the bank optimally selects its portfolio knowing the true β and a parameters, reflecting superior knowledge i i of assets’ riskiness, subject to paying any costs or following any constraints imposed by the regulator. 8Inprinciple,thebank’sinvestmentsmightalsoproducepositiveexternalitiesthattheregulatortakes into account. For simplicity, I’ll assume that there are no such externalities, which means that the bank iscapturingallsocialsurplus. However,ifpositiveexternalitieswereincludedinthemodel,theregulator would additionally include a subsidy that reflects these positive externalities. 8

In determining optimal regulation, the regulator faces a tradeoff. Since the regulator has a higher effective risk aversion than the bank, the regulator wants to reduce riskiness. But regulations meant to reduce riskiness might lead to socially inefficient investments, which reduce social welfare. The processes for each asset’s true profitability and beta are β = β ¯ +ub (8) i i a = a¯+χ(β −β ¯ )+ua, (9) i i i where ua and ub are mean-zero independent normal random variables with variances σ2 i i ua and σ2 , respectively. Therefore, the draws of a and β are independent across assets. ub i i χ is an important parameter that determines the strength of the relationship between each asset’s true risk and its baseline profitability. Since there is typically an equilibrium relationship between risk and return, it may be tempting to think that a should be i strongly positively related to β (i.e., that χ should be large). However, this relationship i is typically between risk and the marginal return. In this model, a only reflects the i initial marginal return starting from zero investment in an asset, whereas a −cq reflects i i the equilibrium marginal return. In equilibrium, investment in a highly profitable asset will drive down that asset’s marginal return until it exactly compensates for the asset’s risk. There is no similar logic for why the initial marginal return, a , should inherently i be strongly related to β . Therefore, I do not restrict the value of χ. i For each asset, the regulator receives a single signal ˆ β = β +e , (10) i i i where e is a mean-zero normal random variable with variance σ2. e are independent i e i across assets. Based on this signal structure, the regulator’s expectation of β conditional on obi ˆ serving β is i σ2 E[β |β ˆ ] = β ¯ + ub (β ˆ −β ¯ ) (11) i i σ2 +σ2 i ub e (cid:124) (cid:123)(cid:122) (cid:125) θ b based on formulas for Bayesian updating with normally-distributed random variables. Thissignalstructureisatractablerepresentationofregulatorshavinglessinformation about asset riskiness than banks. In practice, this information asymmetry could arise for many reasons. Banks could have better access to data, more staff, and a more specialized focus on particular products. This asymmetry could also potentially reflect a timing 9

effect, since risk weights may not be calibrated to the most recent market data. In that case, banks would have an information advantage from using more recent data. Since my focus is on how to address an information asymmetry, I assume that the signalstructureisexogenous. Inprinciple, aregulatormightalsoaddressthisinformation asymmetry by gathering additional data. Information acquisition could be modeled by ˆ allowing the regulator to pay a cost to improve the precision of the β signal. However, since my focus is not on optimal information acquisition and since including it would greatly complicate the model, I take the information structure as given. 2.1 No regulation To build intuition for the cases involving regulation that follow, I begin by considering the bank’s problem when there is no regulation. In this case, the bank solves (cid:90) 1 c max (aq − q2 −γβ q )di, (12) {qi} 0 i 2 i i i which quickly leads to the solution 1 q = (a −γβ ). (13) i i i c The bank’s investment in an asset increases in a and decreases in β . All else being equal, i i banks do not like to take risk and will only do so if they are sufficiently compensated for it. Common intuition is that banks will take advantage of weaknesses in regulation to increase their risk, but this intuition is only true if that risk is sufficiently compensated. The a − γβ term will appear in future results. I call this term the “initial riski i adjusted return.” In this model, the bank’s marginal return from investing in asset i is a −cq , so a represents the marginal return prior to making any investment (i.e., when q i i i i is zero). Since the bank dislikes beta exposure with a cost of γ, the −γβ term represents i a risk adjustment from the bank’s point of view. Whether banks invest more in high-beta assets depends on the relationship between each asset’s initial risk-adjusted return and its beta. Specifically, it depends on whether (cid:18) (cid:19) 1 Cov(q ,β ) = Cov (a −γβ ),β (14) i i i i i c 1 (cid:0) (cid:1) = Cov (χ−γ)ub +ua,ub (15) c i i i 1 = (χ−γ)σ2 (16) c ub 10

is positive or negative. This relationship will be positive when χ > γ, which means that higher betas are associated with higher initial risk-adjusted returns on average. Since χ could take on any value, this relationship could, in general, take either sign. Next, I consider what the portfolio beta and expected return look like in this case. The portfolio beta is (cid:90) 1 β = q β di. (17) p i i 0 Using the fact that draws of a and β are independent across assets and applying the i i law of large numbers implies that the expected portfolio beta is E[β ] = E[q β ] = E[q ]E[β ]+Cov(q ,β ). (18) p i i i i i i This familiar last term will determine how the portfolio beta compares to what would be expected ignoring the link between risk and portfolio choice. If this term is positive, the bank will adjust its portfolio to scale up risky investments. If this term is negative, it will adjust its portfolio to scale down risky investments. 2.2 The first-best solution Here I consider the first-best outcome, in which the regulator knows the true a and β i i parameters and directly selects investment quantities q to maximize social welfare. In i this case, the regulator maximizes (cid:90) 1(cid:16) c (cid:17) max a q − q2 −(γ +η)β q di, (19) {qi} 0 i i 2 i i i where there are no expectations due to the regulator’s perfect knowledge of all parameters. I will focus on understanding the behavior of the unconditional expectations of the portfolio beta, portfolio return, and social welfare. Proposition 1. In the first-best case, equilibrium investment in asset i is 1 q = (a −(γ +η)β ). (20) i i i c The equilibrium expected portfolio beta is 1 (cid:0) (cid:1) E[β ] = E[a β ]−(γ +η)E[β2] . (21) p c i i i 11

The equilibrium expected portfolio return is 1 (cid:0) (cid:1) E[µ ] = E[a2]−(γ +η)2E[β2] . (22) p 2c i i The equilibrium social welfare is 1 E[µ −(γ +η)β ] = E[(a −(γ +η)β )2]. (23) p p i i 2c Proof. See Section D.1. The most interesting result is that social welfare depends on the expectation of a − i (γ +η)β squared. This term can be interpreted as the initial risk-adjusted return from i the regulator’s point of view. Whenever the initial risk-adjusted return differs from zero, the bank can make socially efficient investments. As these differences rise, the amount of socially efficient investments also rise, which increases social welfare. Even investments with negative initial risk-adjusted returns can create social value since banks can take short positions. The socially optimal investment in asset i matches the case without regulation, except with an effectively higher risk aversion of γ +η instead of γ. I also introduce a general result that is useful for calculating expected social welfare under various cases. Proposition 2. For all given random variables q representing a bank’s investment i choice, the expected social welfare can be expressed as c (cid:16) (cid:17) E[(qfb)2]−E[(q −qfb)2] , (24) 2 i i i where the first-best investment is 1 qfb = (a −(γ +η)β ). (25) i c i i Proof. See Section D.2. Intuitively, social welfare can be decomposed into one piece that depends on welfare under the first-best case and another piece that depends on deviations away from the first best. These deviations from the first best reflect the impact of socially inefficient investments. A particularly useful application of this result is comparing social welfare between two cases, for which it only becomes necessary to calculate the expected squared deviations from the first-best case. 12

3 Regulation through taxing estimated risk Here I consider how a regulator might use taxes on banks’ estimated risk to regulate their risk-taking. In interpreting the results, literal taxes are not required but instead something that imposes a direct or indirect cost to the bank. For example, risk-sensitive deposit insurance premiums would be a direct cost, while higher steady-state capital requirements would be an indirect cost. Section A discusses the relationship between settingtaxesandcapitalrequirementsinfurtherdetail. Additionally, thissectionassumes that the bank’s starting level of equity is exogenous. Section B.1 discusses an extension in which the bank selects its desired level of equity. 3.1 Linear tax Here the regulator picks a tax k that the bank must pay for each unit of asset i. The i proceeds of this tax do not have any social benefit, so they don’t enter the regulator’s objective function, but the bank optimizes conditional on the tax. The bank solves (cid:90) 1 c max (a q − q2 −γβ q −k q )di, (26) {qi} 0 i i 2 i i i i i taking k as given. The regulator solves i (cid:20)(cid:90) 1 c (cid:21) maxE (a q − q2 −(γ +η)β q )di , (27) {ki} R 0 i i 2 i i i taking into account how the choice of k affects the bank’s choice of q as well as infori i ˆ mation contained in the signal β . i Proposition 3. The bank optimally selects 1 q = (a −γβ −k ). (28) i i i i c The regulator optimally selects k = ηE [β ]. (29) i R i The equilibrium social welfare relative to the first-best is 1 − η2Var(β −E [β ]). (30) i R i 2c Proof. See Section D.3. 13

The regulator’s optimal policy is to set the tax equal to the expected risk given signals. At first glance this result may seem straightforward, but it runs counter to common intuitions. Financial regulation is often calibrated conservatively based on the argument that banks will take advantage of weaknesses in the regulatory framework. If the regulator underestimates an asset’s risk, then banks will overinvest in that asset. Calibrating regulations to be more conservative than the regulator’s expectations is then argued as necessary to prevent this type of gaming. This force exists in this model since banks will overinvest in assets if regulators underestimate their risk. But the optimal solution is not conservative because such conservatismiscostly. Ifregulatorsoverestimatetheriskofanasset,thenbankswillunderinvest and, thus, forgo socially-desirable investments. Due to the specification of the risk-taking externality, ηβ , as a linear function, the costs of overinvestment and underinvestment p are similar and, therefore, the regulator targets their expectation of optimal investment. Extensions that allow for a convex risk-taking externality, so that the extra costs of too much risk are substantially more than the benefits of reducing risk, can give rise to conservatism. Although even in these extensions, idiosyncratic uncertainty about the risks of individual assets is not sufficient to generate conservatism since idiosyncratic uncertainty diversifies across the portfolio. Instead, there also needs to be systematic uncertainty. Section C covers the topic in more detail. 3.1.1 Is a linear tax better than command? Here I consider whether a linear tax outperforms the command case, in which the regulator directly selects the bank’s investments. Intuitively, it would seem that the linear tax would always outperform since it controls for the expected externality while still allowing the bank to make use of its private information. However, for certain parameters, it is better for a regulator to select investments directly. In the command case, the regulator optimizes (cid:20)(cid:90) 1 c (cid:21) maxE (a q − q2 −γβ q −k q )di , (31) {qi} R 0 i i 2 i i i i i which leads to the solution 1 qcom = (E [a ]−(γ +η)E [β ]). (32) i c R i R i Proposition 4. Social welfare under the command case relative to the first-best is 1 ˜ − Var(a˜ −(γ +η)β ). (33) i i 2c 14

Social welfare under the command case is higher than in the linear tax case when ˜ ˜ 2(γ +η)Cov(a˜ ,β ) > Var(a˜ )+γ(γ +2η)Var(β ), (34) i i i i where x˜ = x−E [x] is the expectational error of the regulator. R Proof. See Section D.4. This expression illustrates that there are some circumstances under which the command case would be preferred to the linear tax. A necessary, but not sufficient, condition for command to be preferred to a linear tax ˜ is for Cov(a˜ ,β ) > 0. In that case, when the regulator underestimates the beta, the i i regulator also likely underestimates the expected return. The bank would then be more likely to invest more in assets for which the regulator has underestimated the risk. 3.2 Nonlinear tax Here the problem is similar to before, except that instead of picking a single linear tax governed by k , the regulator sets a nonlinear tax schedule k (q ). These nonlinear taxes i i i will implicitly make use of information contained in the bank’s choice of q . All else being i equal, banks prefer to increase q when expected returns are high (i.e., a is high) or risk i i is low (i.e., β is low), so the bank’s choice of q is a signal of the combination of these i i two parameters. Using the nonlinear tax is essentially a way for the regulator to set a ˆ tax conditioned not only on the signal β , but also on q . i i More specifically, the bank observes the choice of k (q) and then selects its portfolio i taking the tax function as given. The bank therefore maximizes over quantities (cid:90) 1 c max (a q − q2 −γβ q −k (q ))di, (35) {qi} 0 i i 2 i i i i i while the regulator maximizes over taxes (cid:20)(cid:90) 1 c (cid:21) max E (a q − q2 −(γ +η)β q )di , (36) {ki(qi)} 0 i i 2 i i i taking as given the bank’s optimal choice of q . Since adding a constant to taxes has no i effect on the bank’s choice, I select tax schedules that charge banks zero if they make no investments (i.e., k (0) = 0). i Proposition 5. The bank’s optimal choice of q satisfies i 1 q = (a −γβ −k(cid:48)(q )), (37) i c i i i i 15

and the regulator’s optimal tax k (q ) satisfies i i k(cid:48)(q ) = ηE[β |q ,β ˆ ] (38) i i i i i ˆ = ηE[β |a −γβ ,β ] (39) i i i i (cid:18) (cid:18) ¯(cid:19)(cid:19) (a −γβ )−a¯+γβ ¯ ˆ ¯ i i = η β +ω (β −β)+ω (40) 1 i 2 χ−γ = x+y(a −γβ ), (41) i i where 1/σ2 ω = e (42) 1 1/σ2 +1/σ2 +(χ−γ)2/σ2 ub e ua (χ−γ)2/σ2 ω = ua (43) 2 1/σ2 +1/σ2 +(χ−γ)2/σ2 ub e ua ˆ are the updating weights placed on the signals β and a −γβ respectively and i i i (cid:18) (cid:18) ¯(cid:19)(cid:19) −a¯+γβ ¯ ˆ ¯ x = η β +ω (β −β)+ω (44) 1 i 2 χ−γ η y = ω . (45) 2 χ−γ A nonlinear equilibrium only exists for y < 1, which can be characterized by χ−γ < 0 or χ−γ > ηω . In this equilibrium, the bank optimally invests 2 1 q = ((1−y)(a −γβ )−x) (46) i i i c and the regulator optimally sets a quadratic tax of the form (cid:18) (cid:19) x c y k (q ) = q + q2. (47) i i 1−y i 2 1−y i Otherwise, the regulator sets a linear tax. Proof. See Section D.6. In the case of a linear tax, the regulator sets the tax equal to their best guess of the beta. In the case of a nonlinear tax, the regulator sets the marginal tax equal to their best guess of the beta, taking into account the information contained in q . In this i environment, the regulator’s expectation of beta is linear in q , so the marginal tax is also i linear in q and the level of the tax is quadratic in q . i i The regulator observing q is equivalent to observing the risk-adjusted return a −γβ . i i i This equivalence holds because the risk-adjusted return is the only part of the bank’s 16

choice of q that the regulator does not know. Furthermore, the risk-adjusted return is a i noisy signal of β , which then affects the regulator’s desired marginal tax. i The nonlinear equilibrium only exists for χ−γ < 0 or χ−γ > ηω . Since ω ∈ [0,1], 2 2 χ−γ > η is a sufficient condition. These parameter restrictions correspond to the case in which equilibrium investment q increases as the risk-adjusted return a −γβ increases. i i i Otherwise, the regulator would want the bank’s investment to fall as the risk-adjusted return rises, which cannot be implemented with a nonlinear tax due to the divergence between the incentives of the bank and the regulator. When χ−γ < 0, assets with higher betas have lower risk-adjusted returns, on average, so banks invest less in riskier assets on average. Since the regulator prefers the bank to invest less in riskier assets as well, they are directionally aligned (although the bank and regulator differ on their desired investment magnitudes). In that case, the bank investing more indicates that an asset is less risky, so the regulator reduces the marginal tax rate as the bank invests more. For the other case, consider the sufficient condition χ − γ > η. In that case, assets with higher betas have such higher risk-adjusted returns, on average, that the regulator prefers the bank to invest more in risky assets. Since the bank prefers to invest more in risky assets as well, they are also directionally aligned. In that case, the bank investing more indicates that an asset is riskier, so the regulator raises the marginal tax rate as the bank invests more. The takeaway is that nonlinear taxes can allow the regulator to take advantage of additional information from the bank’s investment decisions, but only when both of their incentives are sufficiently aligned. Otherwise, there is not an equilibrium in which the bank reveals its information. For a straightforward extension of the model in which the prior distributions of a and i β vary across assets, such as by allowing asset-specific χ parameters, the regulator might i implement a nonlinear tax for specifically for those assets where incentives are sufficiently aligned and a linear tax for the rest. 4 Effects of nondisclosure Here I consider the policy of specifying taxes, similar to Section 3, but with the added twist that the regulator does not reveal these taxes to the bank until after they have made their portfolio choice. 17

4.1 Fully undisclosed linear tax In this case, the regulator specifies a linear tax, but the bank does not know the value of this tax until after it has selected its portfolio. One interpretation is that this scenario is similar to a stress test in which the bank does not know the regulator’s model. While not implemented in practice, another interpretation is that the bank selects its portfolio and the regulator only reveals the associated risk weights after the fact. In this situation, the bank solves (cid:90) 1 c max (a q − q2 −γβ q −E [k ]q )di, (48) {qi} 0 i i 2 i i i B i i where the bank’s information set does not include the regulator’s signals. Meanwhile, the regulator solves (cid:20)(cid:90) 1 c (cid:21) maxE (a q − q2 −(γ +η)β q )di , (49) {ki} R 0 i i 2 i i i taking into account the bank’s choice function. I solve for the set of solutions {q } and i {k } that jointly solve each optimization problem. i Proposition 6. One solution is for the bank to select 1 q = (a −(γ +η)β ) (50) i i i c and the regulator to select ˆ k = ηβ . (51) i i This solution achieves the first-best outcome for the regulator. Proof. See Section D.7. Tounderstandwhythisapproachachievesthefirst-bestoutcome, notethatthebank’s expectation of the tax is ˆ E [k ] = ηE [β ] = ηE [β +e ] = ηβ . (52) B i B i B i i i Intuitively, the bank expects that the regulator will set the tax correctly on average. Therefore, the bank invests as though the regulator had set all of the taxes correctly. ˆ Recall that β (cid:54)= E [β ], so the regulator is not setting the tax equal to their best i R i guess of the true beta. The rationale is that the regulator wants the tax to respond 18

one-for-one to changes in the true underlying beta. The regulator’s best guess of the true ¯ beta, E [β ], includes some regularization toward β that typically results in a response R i that is less than one-for-one. For practical application, the lesson here is that when regulators are not sharing ex-ante details about taxes, they should respond very strongly to any information they receive about an asset’s riskiness. Otherwise, banks will assume that regulators will not adjust taxes sufficiently in response to information. 4.2 Partially disclosed linear tax In practice, banks may have some sense of deficiencies in regulators’ models, even if they may not have full knowledge of those models. For example, banks may learn about some features of regulators’ stress testing models. In a dynamic setting, a bank may have learned some information from regulators’ prior actions. I model banks’ partial knowledge of regulators’ models by giving banks noisy signals of the regulator’s signals, which banks can use to guess the likely level of tax that regulators will specify. The set-up is exactly the same as before, except now banks receive a noisy signal of the regulator’s signal of the form ˆ s = β +w , (53) i i i where w are independent across assets and are normally distributed with mean zero and i variance σ2. w Proposition 7. If the bank receives noisy signals of the regulator’s signals, then a regulator’s choice of the optimal k must satisfy i E [E [k ]] = ηE [β ]. (54) R B i R i One solution satisfying this condition is for the regulator to set (cid:18) (cid:19) θ ¯ b ˆ ¯ k = η β + (β −β) , (55) i i θ +θ (1−θ ) b w b where σ2 θ = ub (56) b σ2 +σ2 ub e σ2 θ = w . (57) w σ2 +σ2 e w 19

ˆ θ is the weight that the regulator places on the signal β , and θ is the weight that the b i w bank places on the signal s . i Proof. See Section D.8. Intuitively, the condition for the optimality of k is that the regulator’s best guess of i the bank’s best guess of the regulator’s tax equals the regulator’s best guess of the risk externality imposed by the portfolio. Put another way, the regulator wants to calibrate the tax to give the bank the correct incentives on average. Existing results emerge as special cases of this framework. First, if the bank has a perfect signal of the regulator’s information, then σ = 0. This case is equivalent to when w theregulatorpre-announcestheloadings, inwhichcasetheresultisk = ηE [β ]. Second, i R i if the bank has no idea of the regulator’s information, which is modeled as σ → ∞, then w ˆ θ → 0, which matches the earlier result of the regulator setting k = ηβ . w i i In the intermediate case, as the bank’s signal of the regulator’s signal becomes noisier ˆ (i.e., the variance of w rises), the regulator places greater weight on β than they would i i in the case of optimal Bayesian updating, but does not place full weight on it as they would in the case of a fully undisclosed tax. 4.3 Nondisclosure with an aversion to idiosyncratic volatility A potential concern with nondisclosure of taxes is that it exposes banks to additional uncertainty over their future cash flows. In the baseline framework, banks are only averse to systematic risk, so this additional uncertainty does not pose any cost. But, in practice, factors such as capital market imperfections may make banks averse even to idiosyncratic volatility (Froot and Stein, 1998). Therefore, nondisclosure can pose costs to banks that the baseline framework does not capture. To incorporate these costs, I add a γ term reflecting banks’ aversion to idiosyncratic I volatility resulting from the uncertain taxes. One important question is whether the regulator considers idiosyncratic risk as imposing a social cost or not. To capture a wide range of possibilities, I assume that the regulator recognizes a social cost η associated I with the bank bearing idiosyncratic volatility. Since the regulator already recognizes indirect effects of volatility affecting banks’ portfolio choice, the η term reflects only I the direct costs of the bank bearing idiosyncratic volatility, even controlling for portfolio choice. For tractability reasons, I focus on the simpler case in which there is only one asset. In this case, the idiosyncratic risk only comes from the tax on the single asset. In the case with multiple assets, idiosyncratic risk will depend on the volatility of taxes across 20

many assets, in which case the correlation of the errors becomes important. To avoid introducing those complexities, I focus on one asset. Proposition 8. Suppose that banks are averse to the volatility of tax payments such that they maximize c max(a−γβ)q − q2 −E [kq]−γ σ (kq), (58) B I B q 2 where σ (kq) is the standard deviation of the taxes paid with respect to the bank’s infor- B mation set. Additionally, suppose that the regulator recognizes a social cost of η from the volatility I of tax payments, so that the regulator’s objective is to select a tax k to maximize c E [(a−γβ)q − q2 −ηβq −η σ (kq)]. (59) R I B 2 The regulator can achieve the first-best outcome by setting ˆ k = ηβ +(η −γ ) ησ sgn(q), (60) I I e (cid:124)(cid:123)(cid:122)(cid:125) σB(k) which results in overall taxes paid of ˆ kq = ηβq +(η −γ )ησ |q|. (61) I I e Proof. See Section D.9. ˆ The first piece of this expression, ηβ, is the same as in the baseline case without an ˆ aversion to idiosyncratic volatility. As before, the bank’s best guess of β is the true β, so nondisclosure by regulators forces the bank to use their information based on the true state of the world. The second piece of this expression relates to the idiosyncratic volatility. If the private and social costs of idiosyncratic volatility differ (i.e., to the extent that γ and η differ), I I the regulator should adjust the taxes to align the bank’s incentives. For example, if γ = η , so that the two are already aligned, there is no need to take action. I I But consider the case in which γ > 0 and η = 0, so that there is only a private cost I I to volatility. For positive q, the regulator should optimally reduce the size of the tax. The reason is that the volatility is already dissuading the bank from investing, so the tax does not need to be as high to achieve the optimal level of investment. In this case, the regulator can compensate for the costs imposed by higher uncertainty in the tax by reducing the average size of the tax. The idea generalizes: The regulator can compensate 21

for the costs imposed by higher regulatory uncertainty by reducing the average tightness of regulations. 5 Taxes on profits This section explores the potential of using taxes on ex-post bank profits to effectively reduce their risk aversion and, thus, influence their portfolio choice. The broad idea is that the regulator can calibrate a stochastic tax that the bank perceives as having a present value related to the riskiness of the bank’s investments. The regulator can then use taxes of this type to align the bank’s incentives with the regulator’s, even without any knowledge of risk or profitability of individual assets. 5.1 Achieving first best through a tax on expected profits In the generally infeasible case in which the regulator has perfect knowledge of the bank’s expected profits, it’s possible for the regulator to achieve a first-best outcome through an appropriately calibrated tax on them. The intuition is that reducing expected profits also reduces the profit per unit of risk the bank takes, which incentivizes the bank to reduce its risk and has an identical effect as reducing the bank’s risk aversion. Proposition 9. The regulator can achieve the first-best outcome by taxing a fraction η γ+η of the bank’s expected profits. Proof. In this case, the regulator imposes a tax of η (cid:90) 1 η c T = E [Π] = (a q − q2)di. (62) γ +η B γ +η i i 2 i 0 The bank’s objective is now to maximize (cid:90) 1(cid:18) γ (cid:16) c (cid:17) (cid:19) maxE [M(Π−T)] = max a q − q2 −γβ q di. (63) {qi} B {qi} 0 η +γ i i 2 i i i Multiplyingby(γ+η)/γ doesnotchangetheoptimalsolutionandyieldsthemaximization problem (cid:90) 1 c max (a q − q2 −(γ +η)β q )di, (64) {qi} 0 i i 2 i i i whichexactlymatchestheregulator’sobjectivefunctionandthereforeleadstoafirst-best outcome. 22

I again emphasize that this outcome is generally not feasible since it relies on the regulator having perfect knowledge of a . However, it suggests that feasible strategies i that approximate a tax on expected returns might be a fruitful course of action. So I next examine feasible approaches that are in the same spirit. 5.2 Achieving the first best through a tax on ex-post profits It is possible to achieve the first best through an appropriately-calibrated tax on ex-post profits whose present value, from the bank’s point of view, equals a fraction of expected returns. Proposition 10. If the regulator sets an ex-post tax of 1 η T = Π, (65) M η +γ then portfolio choice will match the first-best case. In this equation, M and Π are random variables, while η and γ are constants. Proof. From Proposition 9, a tax that is set to be a fraction η/(η + γ) of the bank’s expectation of profits achieves the first best. The regulator can set an ex-post tax T with a present value equal to this tax by selecting T to satisfy η E [MT] = E [Π]. (66) B B η +γ Setting T as proposed leads to    η Π  η E M  = E [Π], (67) B B  η +γ M η +γ (cid:124) (cid:123)(cid:122) (cid:125) T so that the present value of the tax to the bank equals a tax on expected profits. From before, the regulator would ideally want to tax the bank’s expected returns. Unfortunately, a direct tax is infeasible because the regulator does not know asset-specific expected returns. However, the regulator can set a stochastic tax whose present value, from the bank’s perspective, equals some fraction of the portfolio’s expected return. State-dependent tax rates are required to achieve this result. Since M is high in “bad” times and low in “good” times, the taxes should be lower in “bad” times than in “good.” In practice, the taxes may not necessarily have to explicitly be state-dependent. For example, since bank profits are likely higher during good times than bad, setting a progressive tax rate on profits might achieve a similar outcome. Even a binary approach, 23

such as imposing an extra tax on profits above a particular threshold, may achieve an approximately similar outcome. If the aim is to tax a constant fraction of expected profits, it may be unclear why it isn’t sufficient to tax a constant fraction of realized profits. While this simpler flat tax reduces expected profits, it also reduces risk by the same amount. The overall risk–return tradeoffremainsunchangedalongwiththebank’sportfoliochoice. Themorecomplicated tax is necessary to alter the expected return without altering the risk. 6 Policy applications The model yields several broad insights for designing financial regulation that takes into account the information asymmetry between the regulator and the regulated. While the model focuses on specifying optimal taxes, these taxes could be interpreted as reflecting the shadow costs of capital requirements. Since capital requirements are a more common regulatory tool, I will focus on the model’s implications for setting risk weights as part of capital regulation. A first insight is that regulators can benefit by not disclosing information about risk weights to banks. Intuitively, if a bank does not know the regulator’s risk weights ahead of time, they will need to choose their investments based on their best guess of the risk weights. If a bank thinks that the regulator will set risk weights correctly, on average, then that bank will invest as though the regulator had set the risk weights correctly. Even if the bank has some information on the regulator’s likely risk weights, there is still benefit in disclosing as little information as possible to reduce possibilities to take advantage of misspecification. A potential concern is that not disclosing risk weights may create regulatory uncertainty that would dissuade socially valuable investment. However, reducing the average level of risk weights could counteract disincentive effects from uncertainty. The broad approach of not disclosing information to banks is already reflected in the spirit of current stress tests, in which banks provide information on their portfolios and do not know the full details of how regulators will calculate capital requirements. The model supports limiting disclosure of stress test models to limit banks from gaming those models. Furthermore, similar types of nondisclosure could apply to other parts of the regulatory framework. For example, in the Basel III capital framework, regulators publish prespecified risk weights for each asset. An alternative would be for regulators to ask banks to report data on exposures and then only tell them their aggregate capital requirements, without revealing any asset-specific risk weights. To limit learning, regulators could also 24

recalibrate risk weights more frequently. This approach would limit banks’ ability to take advantage of misspecified risk weights. Asecondinsightisthatregulatorscanaccountforinformationcontainedinthebank’s portfolio choice. The most direct application would be nonlinear risk weights that automatically change as banks concentrate investment in particular sectors. While the model is too stylized for the formulas to directly apply to policy settings, regulators can consider what information is revealed when a bank invests more in a particular type of asset. In some cases, concentration in an asset may indicate that the regulator has underestimated risk, and using a tool similar to a nonlinear risk weight may be a way of leaning back against that underestimation.9 But, in some cases, higher concentration may not be associated with higher risk. One approach to addressing this problem is to calibrate nonlinear risk weights based on the empirical relationship between concentration in particular types of assets and ex-post measures of risk, such as default rates on loans. However, this relationship would change if risk weights change, so multiple recalibrations may be required before converging to an equilibrium (if it exists at all). Practical application of such nonlinear formulas should also account for bank specialization. Some banks may specialize in particular lines of business and would therefore naturally have higher concentrations. Penalizing specialized banks is likely socially undesirable. For practical implementation, it would likely be preferable to use measures of concentration for the aggregate banking system rather than individual banks. An additional benefit of using measures based on the aggregate banking system is that individual banks may not internalize how their investments would change risk weights, which could reveal more information to regulators about the risks of individual assets. While formal rules, such as formulas for risk weights, could automatically respond to portfolio choice, a more informal supervisory process could also lead to a similar result. Supervisors could investigate the areas in which banks are investing much more (or less) than expected and consider adjusting risk weights for those areas. Risk weight adjustments could be direct, such as changing Basel III calibrations, or indirect, such as changing the shocks used in stress testing models as proposed by Greenwood et al. (2017). A third insight is that regulators could implement taxes on ex-post profits. Since the modelassumesthatbankshaveanexogenousamountofequityequaltoone, theempirical counterpartwouldbeataxwhoseratedependsonabank’sreturnonequity. Theintuition is that banks take risk to generate higher returns. Reducing expected returns through a tax thus reduces the bank’s incentives to take risk, which can effectively make them 9Sector-specificcountercyclicalcapitalbufferscanprovideasimilaroutcomeifadjustedappropriately. But if there are any difficulties in activating such buffers, an automatic nonlinear rule may be able to adjust more quickly. 25

more risk averse while still using their private information. However, it is important to take into account that taxes on ex-post profits affect both the after-tax risk and expected return, so a flat tax on profits would not be sufficient. In theory, an optimal solution is a state-dependent tax that is higher during “good times” (i.e., when investors’ SDF is lower). The level of this tax would rise with the magnitude of the externality that the bank’s risk-taking imposes. An explicitly statedependent tax could be calibrated based on measures of economic performance, such as unemployment or GDP growth. However, since bank profitability is likely to be highly correlated with economic performance, a progressive tax might also approximate the ideal outcome. For example, a higher marginal tax rate for banks with a return on equity above some predetermined threshold might discourage banks from excessive risk-taking by reducing the after-tax payoff of doing so. Additionally, a tax whose rate rises with a bank’s return on equity would incentivize banks to fund themselves with more equity since higher equity would mechanically reduce the return on equity. Empirically, Meiselman et al. (2018) show that high profitability for banks predicts higher tail risk for both the 2007-08 financial crisis as well as the 1980s savings and loan crisis. Given the empirical link between profits and risk, a well-calibrated tax on profits can effectively be viewed as a tax on risk, without requiring any knowledge on the regulator’s part of the riskiness of the bank’s portfolio. Onebenefitofataxonex-postprofitsisthatitcouldbequicklyappliedtotheshadow banking sector, even if regulators know little about its business model. In contrast, the regulatory framework for traditional banks may not be entirely appropriate for certain shadow banks, and it would likely take time to design a framework that appropriately reflects their activities. Even if the ultimate goal were to apply a more traditional regulatory framework to such institutions, a tax on profits could be useful as an interim measure while regulators prepare the longer-term framework. 7 Conclusion In this paper, I address the problem of how to regulate bank portfolio choice taking into account the asymmetry of information between banks and regulators. I construct a tractable model that explicitly accounts for this asymmetry. The model is flexible enough to allow for the consideration of a wide range of regulatory tools and information asymmetries, while also being simple enough to be easily solved by hand. Because of its flexibility and simplicity, this model can be adapted by researchers exploring related topics, both within the context of banking and elsewhere. I then use this model to explore the efficacy of several common regulatory tools and 26

to propose three less common tools: (1) not disclosing taxes to banks until after portfolio selection, (2) nonlinear taxes that respond to information contained in banks’ portfolio choice, and (3) state-dependent taxes on banks’ realized profits. While the model is intentionally stylized to aid in communicating intuitions, the broad takeaways could be applied in practice. Regulators could consciously not disclose information from banks to prevent them from gaming regulation, similar to how they already do so for stress tests. Nonlinear taxes (or risk weights) could automatically respond to banks concentrating their investments in a particular sector. And taxes on ex-post profits could incentivize banks to behave more risk averse, even if regulators know little about banks’ risks. Future work can extend the model along several dimensions. One extension is to focus on dynamic interactions, particularly in the case of information nondisclosure. Since banks can learn information about regulators’ models over time by observing outcomes, regulators may need to adjust their models or intentionally introduce noise to keep banks from learning too much. Another extension is to ease the single-factor assumption. While a single factor is an implicit assumption underlying much regulation (for example, Gordy (2003) discusses the importance of a single-factor assumption within capital regulation of the banking book), market risk must account for many correlated factors and hedges. Addressing this problem would further aid in regulating bank portfolio choice in the important, but more complicated, setting of banks’ trading books. References Beshears, J., Choi, J. J., Clayton, C., Harris, C., Laibson, D., and Madrian, B. C. (2020). Optimal Illiquidity. Working Paper. Celerier, C., Kick, T. K., and Ongena, S. (2019). Taxing Bank Leverage: The Effects on Bank Portfolio Allocation. SSRN Scholarly Paper ID 2829326, Social Science Research Network, Rochester, NY. Chan, Y.-S., Greenbaum, S. I., and Thakor, A. V. (1992). Is Fairly Priced Deposit Insurance Possible? The Journal of Finance, 47(1):227–245. Domar, E. D. and Musgrave, R. A. (1944). Proportional Income Taxation and Risk- Taking. The Quarterly Journal of Economics, 58(3):388–422. Froot, K. A. and Stein, J. C. (1998). Risk management, capital budgeting, and capital structurepolicyforfinancialinstitutions: Anintegratedapproach. Journal of Financial Economics, 47(1):55–82. 27

Giammarino, R. M., Lewis, T. R., and Sappington, D. E. M. (1993). An Incentive Approach to Banking Regulation. The Journal of Finance, 48(4):1523–1542. Glasserman, P. and Kang, W. (2014). Design of Risk Weights. Operations Research, 62(6):1204–1220. Gordy, M. B. (2003). A risk-factor model foundation for ratings-based bank capital rules. Journal of Financial Intermediation, 12(3):199–232. Greenwood, R., Hanson, S. G., Stein, J. C., and Sunderam, A. (2017). Strengthening and Streamlining Bank Capital Regulation. Brookings Papers on Economic Activity, 48(2 (Fall)):479–565. John, K., John, T. A., and Senbet, L. W. (1991). Risk-shifting incentives of depository institutions: Anewperspectiveonfederaldepositinsurancereform. Journal of Banking & Finance, 15(4):895–915. Kim, D. and Santomero, A. M. (1988). Risk in Banking and Capital Regulation. The Journal of Finance, 43(5):1219–1233. Kim, J. H. J., Kim, K., Liu, V., and Tanner, N. (2024). Optimal Communication in Banking Supervision. SSRN Scholarly Paper, Rochester, NY. Leitner, Y. and Williams, B. (2023). Model Secrecy and Stress Tests. The Journal of Finance, 78(2):1055–1095. Meiselman, B. S., Nagel, S., and Purnanandam, A. (2018). Judging Banks’ Risk by the Profits They Report. SSRN Scholarly Paper ID 3169730, Social Science Research Network, Rochester, NY. Perotti, E. and Suarez, J. (2018). A Pigovian Approach to Liquidity Regulation. 27th issue (November 2011) of the International Journal of Central Banking. Rochet, J.-C. (1992). Capital requirements and the behaviour of commercial banks. European Economic Review, 36(5):1137–1170. Shackelford, D. A., Shaviro, D. N., and Slemrod, J. (2010). Taxation and the financial sector. National Tax Journal, 63(4.1):781–806. Stiglitz, J. E. (1969). The Effects of Income, Wealth, and Capital Gains Taxation on Risk-Taking. The Quarterly Journal of Economics, 83(2):263–283. Weitzman, M. L. (1974). Prices vs. Quantities. The Review of Economic Studies, 41(4):477. 28

Wu, H.-M. and Zhao, Y. (2016). Optimal Leverage Ratio and Capital Requirements with Limited Regulatory Power. Review of Finance, 20(6):2125–2150. A Relationship between taxes and capital requirements The aim of this section is to demonstrate how approaches based on taxes and capital requirements compare with each other. The exact relationship depends on the assumptions about capital. In several cases, there is an exact equivalence. In the most straightforward case, the supply of capital is elastic with constant cost relative to other funding sources of r . In c that case, a capital requirement of m per quantity held of asset i has the same effect on i the bank’s portfolio choice as an asset-specific tax of k = m ·r . i i c There is also an exact equivalence between the two approaches when the quantity of capital is exogenous and shocks to a and β are idiosyncratic (i.e., independent across i i assets). In that case, setting a capital requirement on a specific asset has the same effect on the bank’s portfolio choice as setting a tax equal to the shadow cost of the capital requirement. Proposition 11. Suppose that capital is exogenously set to 1 and the regulator sets a capital requirement of the form (cid:90) 1 m q di ≤ 1, (68) i i 0 where m indicates the capital requirement for asset i. If there are no aggregate shocks i to a and β (i.e., a and β are drawn independently across assets), then the Lagrange i i i i multiplier on the capital requirement λ is deterministic. The regulator can induce identical portfolio choice by the bank through setting a linear tax k according to i k = λm . (69) i i Proof. See Section D.10. However, the two approaches are not equivalent with systematic shocks. In that case, the Lagrange multiplier λ will be stochastic. The difference between capital requirements and taxes will essentially be a choice of price-based versus quantity-based regulation, the considerations of which are discussed most notably by Weitzman (1974). In the baseline 29

case in which there is a known linear social cost to bank risk-taking, a tax is likely the bettersolution. Butforparticularformsofnonlinearsocialcostfunctions, quantity-based regulation may be preferable. B Endogenous capital structure B.1 Linear taxes with endogenous capital structure The earlier analysis considers the case of linear taxes with an exogenous capital structure (i.e., banks begin with an exogenous amount of capital). I now consider the optimal policy when endogenizing banks’ choice of capital structure. The bank now selects the amount of equity with which it will fund itself. More equity reduces the social cost of bank risk taking for a given level of risk exposure. However, equity may potentially have both social and private costs relative to other sources of funding. Without delving into exactly why these costs exists, I suppose that the social cost of equity is r > 0 and the private cost of equity is r ≥ r > 0.10 These reflect the s p s change in the overall cost of capital from funding with an additional unit of equity, not the per-unit cost of equity. For example, if the Modigliani-Miller capital structure irrelevance theorem were to hold, then r = r = 0. I focus on the cases in which both r and r are positive s p s p since these reflect the interesting cases; if equity were socially costless, then the solution would be to hold sufficient equity such that there is no longer any social cost from bank risk-taking. The social cost of risk exposure is (cid:82)1 β q di η 0 i i . (70) e This social cost is exactly the same as before, except now the portfolio risk is scaled by the amount of equity e. I consider a case in which the regulator uses two tools. First, the regulator sets a tax that takes the form (cid:82)1 k q di 0 i i , (71) e 10If r = 0, so that there is no social cost of equity, then the solution is for the bank to fund itself s with as much equity as possible. For reasons of tractability, this model allows for potentially unlimited losses on the portfolio, including losses that are larger than the initial investment. Therefore to cover any potential loss would require equity levels approaching infinity. In a model that incorporated limited liability on investments, costless equity would imply that the bank should fund itself with 100% equity. 30

where k sets the magnitude of the asset-specific tax on asset i, but now it is also scaled i by the level of equity e since the social cost of risk also depends on the bank’s leverage. For this piece, the regulator solves for the optimal k . i Second, the regulator could also potentially subsidize equity at a linear rate of s. So the regulator solves for the optimal s as well. Proposition 12. If the bank solves (cid:90) 1(cid:18) c k q (cid:19) max (a −γβ )q − q2 − i i di−(r −s)e (72) {qi},e 0 i i i 2 i e p and the regulator solves (cid:20)(cid:90) 1(cid:18) c β q (cid:19) (cid:21) maxE (a −γβ )q − q2 −η i i di−r e , (73) {ki},s R 0 i i i 2 i e s taking into account the bank’s optimal choices of q and e, then the regulator optimally i sets k = ηE [β ] (74) i R i and s = r −r . (75) p s Proof. See Section D.11. The first part of this result – that the regulator sets the asset-specific tax k based i on the asset’s expected risk – is intuitively the same as the case with exogenous equity. The second part – that the regulator should set an equity subsidy – is new. Setting the subsidy like this reduces the bank’s marginal cost of equity from r to r , which aligns it p s with the social cost. The key difference when introducing endogenous capital structure is that tax policy also affects the bank’s incentives to fund itself with equity. When the regulator can align the private and social costs of equity, then the regulator only needs to worry about aligning asset-specific taxes with expected risks. However, the situation becomes more complicated when the regulator cannot introduce an equity subsidy and there are differences in the social and private costs of equity. In that case, asset-specific taxes can be used to influence the bank’s overall selected level of equity. For example, in typical specifications, setting k uniformly higher than ηE [β ] i R i can serve as a crude tax on leverage that encourages banks to fund themselves with more 31

equity. B.2 Tax on profits with endogenous capital structure Here I consider the regulator taxing ex-post profits, but allowing for the bank to select its level of equity endogenously. This section generalizes Section B.1 by allowing for a general function η(e) to characterize how equity affects the social cost of bank risk-taking instead of assuming that it takes the form η(e) = η/e. In the case with endogenous equity, the social objective is (cid:90) 1(cid:16) c (cid:17) a q − q2 −(γ +η(e))β q di−r e, (76) i i 2 i i i s 0 where η(e) is the regulator’s additional risk aversion, which now depends on the level of equity, and r is the social cost of equity funding. η(e) is decreasing in e, reflecting that s the social cost of bank risk taking declines as the bank becomes better capitalized since the bank is better able to absorb losses. The regulator can impose a tax T on the bank. Taking this tax into account, the bank solves (cid:90) 1(cid:16) c (cid:17) max a q − q2 −γβ q di−r e−E [MT], (77) e,{qi} 0 i i 2 i i i p B subject to e ≥ 0, where r is the bank’s private cost of equity funding. p Proposition 13. The regulator will perfectly align the bank’s incentives with its own by setting (cid:18) (cid:19) η(e) 1 T = −1 Π−(r −r )e. (78) p s γ M Proof. See Section D.12. (cid:82)1 The present value of the first term ultimately simplifies to η(e) β q di, so the bank 0 i i perceives it as a tax on risk. The second term is a subsidy for equity financing to align the private and social costs of equity. Since η(e) is decreasing in e, the magnitude of the tax on profits is also decreasing as e rises. The intuition is that better-capitalized banks will impose fewer externalities, so the magnitude of taxes on ex-post profits required to align their incentives is lower. 32

C Generalized social cost function with a linear tax My focus has been on a known linear social cost function of the form ηβ . However, there p may be nonlinearities in practice. For example, risk may have a small marginal social cost at low levels if banks are very unlikely to default. But at higher levels of risk, and with higher probabilities of defaulting, the marginal social cost may rise dramatically. Additionally, there may be uncertainty as to the magnitude of the social costs, even for a known level of risk. To address these concerns, I consider a generalized social cost function that allows for both nonlinearities and uncertainty over costs. Proposition 14. Suppose that the social cost to bank risk-taking is a function S(β ,η) p that is twice-differentiable, increasing in both arguments, and satisfies ∂2S(β ,η) ≥ 0. ∂β2 p p (cid:82)1 β = q β di is the bank’s portfolio beta and η is an exogenous random variable. Then p 0 i i the regulator’s optimal linear tax is (cid:20) (cid:21) ∂S k = E (β ,η)β (79) i R p i ∂β p (cid:20) (cid:21) (cid:18) (cid:19) ∂S ∂S = E (β ,η) E [β ]+Cov (β ,η),β . (80) R p R i R p i ∂β ∂β p p Proof. See Section D.5. One immediate observation is that if the β are independent across assets, then the i covariance term will be zero. In that case, the regulator’s solution is the same as in the case of the known linear social cost, except replacing the marginal social cost of risk with the expected marginal social cost, which is E [∂S (β ,η)]. R ∂βp p If there is a systematic shock to β , then the regulator should additionally consider i the covariance term. Assets whose riskiness is higher when the social cost of risk is high should receive a higher tax. D Additional details on proofs D.1 The first-best solution The first-order condition for equilibrium q is i a −cq −(γ +η)β = 0 i i i (81) 1 =⇒ q = (a −(γ +η)β ) i i i c 33

as desired. The second derivative, −c, is negative, indicating that this solution is a maximum. The expected portfolio beta is (cid:20)(cid:90) 1 (cid:21) E[β ] = E q β di p i i 0 (82) = E[q β ] i i 1 (cid:0) (cid:1) = E[a β ]−(γ +η)E[β2] . c i i i The expected portfolio return is (cid:20)(cid:90) 1 c (cid:21) E[µ ] =E (a q − q2)di p i i 2 i 0 c =E[a q ]− E[q2] i i 2 i (83) 1 c 1 = (E[a2]−(γ +η)E[a β ])− (E[a2]−2(γ +η)E[a β ]+(γ +η)2E[β2]) c i i i 2c2 i i i i 1 (cid:0) (cid:1) = E[a2]−(γ +η)2E[β2] . 2c i i Social welfare, represented by the regulator’s objective function, is 1 1 (cid:0) (cid:1) (cid:0) (cid:1) E[µ −(γ +η)β ] = E[a2]−(γ +η)2E[β2] − (γ +η) E[a β ]−(γ +η)E[β2] p p 2c i i c i i i (cid:18) (cid:19) 1 1 1 = E[a2]+ (γ +η)2E[β2]−(γ +η)E[a β ] c 2 i 2 i i i 1 = E[(a −(γ +η)β )2]. i i 2c (84) D.2 Expression of social welfare By definition, the regulator’s expected social welfare conditional on their information set is c E [a q − q2 −(γ +η)β q ] R i i 2 i i i (85) c =E [(a −(γ +η)β)q ]− E [q2]. R i i 2 R i 34

Using the fact that qfb = 1(a −(γ +η)β ) and expanding q = qfb +(q −ffb) leads to i c i i i i i i c cE [qfbq ]− E [(qfb +(q −qfb))2] R i i 2 R i i i c =cE [qfbq ]− E [(qfb)2 +2qfb(q −qfb)+(q −qfb)2] R i i 2 R i i i i i i (86) c c =cE [qfbq ]− E [(qfb)2]−cE [qfbq ]+cE [(qfb)2]− E [(q −qfb)2] R i i 2 R i R i i R i 2 R i i c = (E [(qfb)2]−E [(q −qfb)2]). 2 R i R i i Taking unconditional expectations leads to the desired result. D.3 Linear tax The first-order condition for the bank with respect to q is i a −cq −γβ −k = 0 (87) i i i i 1 =⇒ q = (a −γβ −k ) (88) i i i i c and this q is a maximum since the second derivative, −c, is negative. i The first-order condition for the regulator with respect to k , taking as given the i bank’s choice of q , is i (cid:20) (cid:21) ∂q i E (a −cq −(γ +η])β ) = 0 (89) R i i i ∂k i (cid:20) (cid:21) 1 E (k −ηβ )(− ) = 0 (90) R i i c k = ηE [β ]. (91) i R i Using the result from Proposition 2, the difference in social welfare compared to the first-best is c E[(q −qfb)2] (92) 2 i i c (cid:16)η(cid:17)2 = E[(−E [β ]+β )2]. (93) R i i 2 c ˜ I define β = β − E [β ], which reflects the regulator’s expectational error. I can then i i R i rewrite the expression as 1 = η2Var(β ˜ ). (94) i 2c 35

D.4 Command vs. linear tax Using the result from Proposition 2, the social welfare relative to the first-best is c − E[(qcom −qfb)2] (95) 2 i i c 1 =− E[(a˜ −(γ +η)β ˜ )2] (96) 2c2 i i 1 ˜ =− Var(a˜ −(γ +η)β ) (97) i i 2c Using results on the social welfare in the linear tax case from Proposition 3, the difference in social welfare between the command and the linear tax case is 1 (cid:16) (cid:17) − Var(a˜ −(γ +η)β ˜ )−η2Var(β ˜ ) . (98) i i i 2c This quantity is positive (indicating that the command solution provides higher social welfare) when η2Var(β ˜ ) > Var(a˜ −(γ +η)β ˜ ) (99) i i i =⇒ η2Var(β ˜ ) > Var(a˜ )+(γ +η)2Var(β ˜ )−2(γ +η)Cov(a˜ ,β ˜ ) (100) i i i i i ˜ ˜ =⇒ 2(γ +η)Cov(a˜ ,β ) > Var(a˜ )+γ(γ +2η)Var(β ) (101) i i i i D.5 Generalized social cost function and a linear tax The regulator’s goal is to maximize social welfare, which is given by (cid:20)(cid:90) 1(cid:16) c (cid:17) (cid:21) E aq − q2 −γβ q di−S(β ,η) , (102) R i 2 i i i p 0 (cid:82)1 where β = q β di is the portfolio beta. The regulator sets constraints subject to the p 0 i i bank’s choice of q . i First, consider the case in which a regulator sets an asset-specific linear tax k . Given i a linear tax, the bank maximizes c maxaq − q2 −γβ q −k q , (103) {qi} i 2 i i i i i which leads to the familiar optimal solution 1 q = (a −γβ −k ). (104) i i i i c Next, turn to the regulator’s problem of selecting optimal k taking the bank’s behavior i 36

as given. The regulator solves (cid:20)(cid:90) 1(cid:16) c (cid:17) (cid:21) maxE aq − q2 −γβ q di−S(β ,η) , (105) {ki} R 0 i 2 i i i p taking into account the effect on the bank’s choice of q . The regulator’s first-order i condition for k is i (cid:20)(cid:18) (cid:19) (cid:21) ∂S ∂q i E a −γβ −cq − β = 0, (106) R i i i i ∂β ∂k p i which leads to an optimal choice of (cid:20) (cid:21) ∂S k = E β . (107) i R i ∂β p Thetaxshouldbesettotheexpectedproductoftheassetbetamultipliedbythemarginal social cost. This term can be expanded to include a covariance as (cid:20) (cid:21) (cid:18) (cid:19) ∂S ∂S k = E E [β ]+Cov ,β . (108) i R R i i ∂β ∂β p p I then verify that the second-order condition holds with respect to k . The first derivative i with respect to k can be written as i (cid:20)(cid:18) (cid:19)(cid:18) (cid:19)(cid:21) ∂S 1 E k − (β ,η)β − , (109) R i p i ∂β c p which leads to an expression for the second derivative as (cid:20)(cid:18) 1∂2S (cid:19)(cid:18) 1 (cid:19)(cid:21) E 1+ (β ,η)β2 − , (110) R c ∂β2 p i c p which is negative since ∂2S(β ,η) ≥ 0. ∂β2 p p D.6 Nonlinear tax From the regulator’s point of view, observing q carries the same information as observing i a − γβ . The reason is that the bank’s first-order condition for selecting q , taking as i i i given the tax function k (q ), is i i cq +k(cid:48)(q ) = a −γβ . (111) i i i i i 37

Since q is sufficient for the regulator to pin down the left-hand side, then it is sufficient i to pin down the right-hand side. To simplify the problem, I take a mechanism design approach where the bank truthfully reports r = a − γβ to the regulator (the “r” stands for “risk-adjusted return”). i i i The regulator then specifies the quantity q (r ) and the tax k (q (r )) to maximize social i i i i i welfare subject to the bank’s incentive compatibility constraints. Collapsing the multidimensional types of a and β into a single a −γβ type is similar in spirit to the approach i i i i taken in Beshears et al. (2020). Since in this context the problem is separable across each asset i, for ease of readability I drop the i subscript in the rest of this section. Let r denote the true r and r(cid:48) the reported one. The change in the bank’s objective from modifying its report r(cid:48) is d c [rq(r(cid:48))− q(r(cid:48))2 −k(q(r(cid:48)))] = rq(cid:48)(r(cid:48))−cq(r(cid:48))q(cid:48)(r(cid:48))−k(cid:48)(q(r(cid:48)))q(cid:48)(r(cid:48)). (112) dr(cid:48) 2 The bank does not have an incentive to deviate when this derivative is zero at r(cid:48) = r, which is when (r−cq(r)−k(cid:48)(q(r))q(cid:48)(r) = 0. (113) The bank will optimally want to change its quantity invested as the risk-adjusted return r = a−γβ changes, so I expect that q(cid:48)(r) (cid:54)= 0 in this equilibrium. Therefore, I focus on satisfying the condition cq(r)+k(cid:48)(q(r)) = r. (114) Meanwhile, the regulator maximizes c E[(a−γβ −ηβ)q(r)− q(r)2|β ˆ ,r] (115) 2 c =(r−ηE[β|β ˆ ,r])q(r)− q(r)2 (116) 2 subject to the bank’s incentive compatibility constraint. I first substitute 1 q(r) = (r−k(cid:48)(q(r))) (117) c to obtain the regulator’s objective solely in terms of k(cid:48)(q(r)) as 1 c (cid:18) 1 (cid:19)2 (r−ηE[β|β ˆ ,r]) (r−k(cid:48)(q(r)))− (r−k(cid:48)(q(r))) . (118) c 2 c 38

I characterize the regulator’s solution in terms of k(cid:48)(q(r)) directly. Differentiating by k(cid:48)(q(r)) and rearranging yields the first-order condition k(cid:48)(q(r)) = ηE[β|β ˆ ,r]. (119) Substituting back into the bank’s incentive compatibility condition yields 1 (cid:16) (cid:17) ˆ q(r) = r−ηE[β|β,r] . (120) c ˆ Next, I explicitly characterize E[β|β,r]. Note that ¯ r = a−γβ = a¯+χu +u −γ(β +u ) (121) b a b ¯ = a¯−γβ +(χ−γ)u +u (122) b a so that r = a−γβ can be transformed to obtain a noisy signal of u as follows. b ¯ r−a¯+γβ u a = u + . (123) b χ−γ χ−γ The noise term is ua and it is independent of e, so that the error terms in this signal χ−γ ˆ and β are independent. Given the assumptions of normality, the expectation is i (cid:18) ¯(cid:19) r−a¯+γβ ˆ ¯ ˆ ¯ E[β|β,r] = β +ω (β −β)+ω , (124) 1 2 χ−γ where 1/σ2 ω = e (125) 1 1/σ2 +1/σ2 +(χ−γ)2/σ2 ub e ua (χ−γ)2/σ2 ω = ua . (126) 2 1/σ2 +1/σ2 +(χ−γ)2/σ2 ub e ua For notational simplicity, I will define x and y such that ˆ ηE[β|β,r] = x+yr, (127) where (cid:18) (cid:18) ¯(cid:19)(cid:19) −a¯+γβ ¯ ˆ ¯ x = η β +ω (β −β)+ω (128) 1 i 2 χ−γ η y = ω . (129) 2 χ−γ 39

Using this notation, straightforward algebra leads to an explicit expression of q(r) as 1 q(r) = ((1−y)r−x). (130) c I then solve for the k(q) function explicitly. I first solve for r in terms of q(r). Return to the first-order conditions for the bank and the regulator of k(cid:48)(q(r)) = r−cq(r) (131) k(cid:48)(q(r)) = ηE[β|β ˆ ,r] = x+yr. (132) Setting the right-hand sides of each equation equal to each other yields r−cq(r) = x+yr (133) x+cq(r) =⇒ r = (134) 1−y and then substitute in to the first equation to get x+cq(r) k(cid:48)(q(r)) = −cq(r) (135) 1−y (cid:18) (cid:19) x y = +c q(r). (136) 1−y 1−y Taking the antiderivative with respect to q(r) obtains (cid:18) (cid:19) x c y k(q(r)) = q(r)+ q(r)2, (137) 1−y 2 1−y where I omit the constant since k(0) = 0 by assumption. Now, I check the second-order conditions. The second derivative for the regulator’s objective function is −(1/c), so the regulator’s choice is a maximum. The second-order conditions for the bank are more complicated. First, I note that (cid:18) (cid:19) y k(cid:48)(cid:48)(q(r)) = c . (138) 1−y 40

The bank’s second-order condition is satisfied when k(cid:48)(cid:48)(q(r))+c > 0 (139) (cid:18) (cid:19) y ⇐⇒ c +1 > 0 (140) 1−y ⇐⇒ y < 1. (141) Then, plugging in the value of y from earlier, I have η ω < 1, (142) 2 χ−γ which is equivalent to χ−γ < 0 or χ−γ > ηω . (143) 2 Note that ω is implicitly a function of χ−γ, but since ω ∈ [0,1], χ−γ > η is a sufficient 2 2 condition. Therefore, for this range of parameters, the bank’s solution is a maximum. D.7 Fully undisclosed linear tax To verify that these pair of choices are a solution, I first suppose that the regulator sets ˆ k = ηβ . The bank’s first-order condition for q leads to i i i 1 ˆ q = (a −γβ −ηE [β ]) (144) i i i B i c 1 = (a −γβ −ηE[β ¯ +ub +eb|ub]) (145) c i i i i i 1 = (a −γβ −η(β ¯ +ub)) (146) c i i i (cid:124) (cid:123)(cid:122) (cid:125) βi which is the desired solution. Since this quantity matches the first-best quantity from Proposition 1, it also maximizes the regulator’s objective function. Note that this solution is not unique. For all random variables x satisfying E [x ] = i B i ˆ β , the regulator will achieve first-best by setting k = ηx . While β is an obvious i i i i ˆ candidate, β with added noise or the fitted value from regressing β on aˆ are other i i i possibilities. 41

D.8 Partially disclosed linear tax First, I’ll establish the optimality condition for k . Starting with the bank’s problem, the i bank maximizes given knowledge over the distribution of k . The bank solves i (cid:90) 1 c max (a q − q2 −γβ q −E [k ]q )di, (147) {qi} 0 i i 2 i i i B i i which leads to the first-order condition 1 q = (a −γβ −E [k ]). (148) i i i B i c The second-order condition is satisfied since −c < 0, which ensures that the solution is a maximum. Given this choice of q for the bank, the regulator solves i (cid:20)(cid:90) 1 c (cid:21) maxE (a q − q2 −(γ +η)β q )di , (149) {ki} R 0 i i 2 i i i which leads to a first-order condition of    ∂q  E (a −cq −(γ +η)β ) i  = 0 (150) R i i i ∂k   i  (cid:124)(cid:123)(cid:122)(cid:125) =−(1/c) =⇒ E [E [k ]] = ηE [β ] (151) R B i R i as desired. Next, I show that the proposed solution satisfies this condition. I conjecture that the regulator sets a linear tax according to ¯ ˆ ¯ k = η(β +y(β −β)), (152) i i where y is an unknown constant. I then solve for the y that satisfies the previous firstorder condition. First, I begin by computing the left-hand side of the first-order condition, E [E [k ]]. R B i Start by taking the expectation of the tax with respect to the bank’s information set as ¯ ˆ ¯ E [k ] = η(β +yE [β −β]) (153) B i B i ¯ = η(β +y(u +E [e ])). (154) i B i 42

ˆ Recall that the bank receives a noisy signal s = β +w . Since the bank perfectly observes i i i β , then s −β = e +w is a noisy signal of e . Applying standard formulas for Bayesian i i i i i i updating with normally distributed variables yields σ2 E [e ] = e (s −β ) (155) B i σ2 +σ2 i i e w σ2 = e (e +w ). (156) σ2 +σ2 i i e w (cid:124) (cid:123)(cid:122) (cid:125) θw Thus, the bank’s expectation of the tax is ¯ E [k ] = η(β +y(u +θ (e +w ))). (157) B i i w i i Now take the regulator’s expectation of this quantity. E [w ] = 0, so the key pieces are R i E [e ] and E [u ]. Using the relationship R i R i ˆ β = β +e (158) i i i ˆ =⇒ β = E [β ]+E [e ] (159) i R i R i ¯ ˆ ¯ and recalling that E [β ] = β +θ (β −β) yields the relationships R i b i ˆ ¯ E [e ] = (1−θ )(β −β) (160) R i b i ˆ ¯ E [u ] = θ (β −β ). (161) R i b i i Substituting these leads to the expression ¯ E [E [k ]] = η(β +y(E [u ]+θ E [e ])) (162) R B i R i w R i ¯ ˆ ¯ = η(β +y(θ +θ (1−θ ))(β −β)). (163) b w b i Now I solve for a value of y that satisfies E [E [k ]] = ηE [β ]. y must ensure that R B i R i ¯ ˆ ¯ ¯ ˆ ¯ η(β +y(θ +θ (1−θ ))(β −β)) = η(β +θ (β −β)) (164) b w b i b i =⇒ y(θ +θ (1−θ )) = θ , (165) b w b b which leads to θ b y = (166) θ +θ (1−θ ) b w b as desired. 43

D.9 Nondisclosure with an aversion to idiosyncratic volatility If the regulator sets the tax in this manner, then the bank’s objective function transforms into the regulator’s, which immediately leads to the first best. Setting the two objectives equal, the bank and regulator have perfectly-aligned incentives when E [k]q +γ σ (k)|q| = ηβq +η σ (k)|q| (167) B I B I B ⇐⇒ E [k]+γ σ (k)sgn(q) = ηβ +η σ (k)sgn(q). (168) B I B I B ˆ Recall that β = β + e and that the bank knows β, but has no information about e. ˆ ˆ Therefore E [β] = β and σ (β) = σ . Using this fact yields B B e E [k] = (η −γ )ησ sgn(q)+ηβ (169) B I I e σ (k) = ησ . (170) B e Therefore it follows that the equation E [k]+γ σ (k)sgn(q) = ηβ +η σ (k)sgn(q) (171) B I B I B (η −γ )ησ sgn(q)+ηβ +γ ησ sgn(q) = ηβ +η ησ sgn(q) (172) I I e I e I e η ησ sgn(q)+ηβ = η ησ sgn(q)+ηβ (173) I e I e is satisfied and thus the incentives of the regulator and the bank are aligned. D.10 Equivalence between taxes and capital requirements with exogenous capital and idiosyncratic shocks If the regulator has set a capital requirement, then the bank solves (cid:90) 1 c max (a q − q2 −γβ q )di (174) {qi} 0 i i 2 i i i subject to the constraint that (cid:90) 1 m q di ≤ 1. (175) i i 0 Setting up the Lagrangian, the problem becomes (cid:90) 1 c (cid:18)(cid:90) 1 (cid:19) max (a q − q2 −γβ q )di−λ m q di−1 , (176) {qi} 0 i i 2 i i i 0 i i 44

which leads to the first-order condition for q of i a −cq −γβ −λm = 0 (177) i i i i 1 =⇒ q = (a −γβ −λm ). (178) i i i i c Compare this first-order condition to the case in which the regulator sets an asset-specific tax k , in which case the bank selects i 1 q = (a −γβ −k ). (179) i i i i c The regulator can replicate the outcome from the capital requirement by setting a tax k = λm . However, this replication is possible in this case because λ is deterministic, i i which I will show next. If the capital requirement is not binding, then λ = 0. Otherwise, multiplying the first-order condition for q by m and then integrating over i yields i i (cid:90) 1 1 (cid:90) 1 m q di = (m (a −γβ )−λm2)di = 1 (180) i i c i i i i 0 0 In the case in which there is only idiosyncratic uncertainty in a and β , the integral i i almost certainly equals the expectation, so that 1 (E[m (a −γβ )]−λE[m2]) = 1 (181) c i i i i E[m (a −γβ )]−c i i i =⇒ λ = , (182) E[m2] i which is deterministic. Intuitively, the idiosyncratic shocks to a and β diversify away i i in the aggregate so that the regulator understands how tightly the capital requirements will bind overall. D.11 Linear taxes with endogenous capital structure I will show that the proposed solution of s = r − r and k = ηE [β ] satisfies the p s i R i conditions for optimal s and k . i I start with the bank’s problem, taking s and k as given. The bank solves i (cid:90) 1(cid:18) c k q (cid:19) max (a −γβ )q − q2 − i i di−(r −s)e. (183) {qi},e 0 i i i 2 i e p 45

The first-order conditions yield (cid:18) (cid:19) 1 k i q = a −γβ − (184) i i i c e (cid:115) k p e = . (185) r −s p The second derivatives are Second of q : −c (186) i k p Second of e : −2 . (187) e3 (cid:113) At e = kp and s = r −r , the second derivative with respect to e is rp−s p s k 1 r p s e = −2 = −2 , (188) e2 e e which is negative since both r > 0 and e > 0. Thus, the second-order conditions for a s maximum are satisfied. Next, switch to the regulator’s problem. The regulator solves (cid:20)(cid:90) 1(cid:18) c β q (cid:19) (cid:21) maxE (a −γβ )q − q2 −η i i di−r e , (189) {ki},s R 0 i i i 2 i e s where q and e follow the bank’s strategy from before. The first-order conditions with i respect to k and s yield i (cid:34) (cid:90) 1(cid:18) β (cid:19) ∂q (cid:32) (cid:82)1 β q di (cid:33) ∂e (cid:35) E a −γβ −cq −η i i di+ η 0 i i −r = 0. (190) R i i i e ∂k e2 s ∂k 0 i i Substitute the expressions for cq and for e2 to obtain i (cid:34) (cid:90) 1(cid:18) k −ηβ (cid:19) ∂q (cid:32) (cid:82)1 β q di (cid:33) ∂e (cid:35) E i i i di+ η 0 i i (r −s)−r = 0. (191) R 0 e ∂k i (cid:82) 0 1 k i q i di p s ∂k i To verify that this first-order condition is satisfied for s = r − r and k = ηE [β ], p s i R i substitute in these values and apply the law of iterated expectations to obtain (cid:34) (cid:34) (cid:12) (cid:35)(cid:35) (cid:90) 1(cid:18) ηE [β ]−ηβ (cid:19) ∂q r (cid:90) 1 ∂e (cid:12) E E R i i i di+ s (ηβ −ηE [β ])q di(cid:12)e,{q },{β ˆ },{aˆ } = 0, R 0 e ∂k i (cid:82) 0 1 k i q i di 0 i R i i ∂k i (cid:12) (cid:12) i i i (192) 46

which follows since E [β − E [β]] = 0. Showing that the first-order condition with R i R respect to s is satisfied is identical, except that ∂qi and ∂e are replaced with ∂qi and ∂e. ∂ki ∂ki ∂s ∂s D.12 Taxes on profits with endogenous capital structure Social welfare equals the bank’s objective function when (cid:90) 1(cid:16) c (cid:17) (cid:90) 1(cid:16) c (cid:17) a q − q2 −(γ +η(e))β q di−r e = a q − q2 −γβ q di−r e−E [MT], i i 2 i i i s i i 2 i i i p B 0 0 (193) which simplifies to (cid:90) 1 E [MT] = η(e) β q di−(r −r )e. (194) B i i p s 0 The next step verifies that the specified T leads to this expression for E [MT]. Recall B that M = 1 − γF, where F is normalized such that E[F] = 0 and Var(F) = 1. The E [MT] term becomes B η(e) E[MT] = (E [Π]−E [MΠ])−E [M(r −r )e] (195) B B B p s γ η(e) = − Cov (M,Π)−(r −r )e (196) B p s γ η(e) (cid:18) (cid:90) 1 (cid:19) = − Cov 1−γF, β q F −(r −r )e (197) B i i p s γ 0 η(e) (cid:90) 1 = γ β q di−(r −r )e (198) i i p s γ 0 (cid:90) 1 = η(e) β q di−(r −r )e, (199) i i p s 0 as desired. This algebra uses the decomposition E[MΠ] = E[M]E[Π]+Cov(M,Π). 47