Board of Governors of the Federal Reserve System

March 28, 2025

Outlining and Measuring the Benefits of Risk Sensitivity in Bank Capital Requirements

Marco Migueis¹

1. Introduction

Banks have incentives to operate with lower capital ratios than would be socially optimal due to deposit insurance and implicit government guarantees that socialize part of the costs of bank failures, particularly for the largest banks.² Given these incentives, regulatory capital requirements contribute to the safety and soundness of individual banks and to financial stability by setting minimum expectations for the amount of loss-absorbing equity that banks need to employ in their funding.³ However, requiring banks to fund themselves with more capital than they would otherwise choose also has costs. Increases in capital requirements can increase banks' cost of funding and, consequently, reduce banks' supply of credit, particularly as banks transition from low capital ratios to higher capital ratios.

In this note, I argue that risk sensitivity enhances the efficiency of capital requirements. A regulatory capital framework where requirements align with risk ensures that banks taking bigger risks fund themselves with more capital. Therefore, a risk-sensitive framework requires banks to internalize their risk-taking, mitigating the externalities their failure or distress may cause. From an alternative perspective, the more aligned capital requirements are with risk, the less costly it is to achieve a consistent standard of resilience across banks.

In addition, I put forward a conceptual framework for assessing the impact of risk sensitivity on the net benefits of capital requirements. Building upon frameworks that have been used to estimate the optimal level of bank capital, I find that risk sensitivity increases the net benefits of capital requirements and, under plausible assumptions, reduces the optimal level of capital.

The remainder of this note is organized as follows: Section 2 discusses the benefits of risk sensitivity in capital requirements; Section 3 discusses the main criticisms of risk-based capital requirements; Section 4 sketches a conceptual framework for measuring the effect of risk sensitivity on the net benefits of capital requirements; and Section 5 concludes and discusses potential directions of future research.

2. Benefits of risk sensitivity

In this section, I illustrate how risk sensitivity enhances the efficiency of capital requirements through two related arguments: (1) risk-sensitive capital requirements allow regulators to achieve a given level of conservatism across banks at the lowest cost, and (2) for a set amount of aggregate capital in the banking system, risk-sensitive capital requirements provide better targeting of loss absorbency capacity.

To demonstrate the benefits of risk-sensitive capital requirements, I consider two edge cases for how capital requirements are set: (1) a "leverage requirement," where requirements do not vary with risk and do not distinguish across assets; and (2) a "risk-based requirement," where requirements are perfectly correlated with the riskiness of specific assets. Also, I assume that banks can only invest in two assets: the "risky" asset and the "safe" asset. If the economy is hit by a shock of severity "x," the risky asset would be expected to lose 10 percent of its value while the safe asset would be expected to lose just 2 percent. In addition, I assume that banks are not constrained in the proportions they can invest in each asset, except that short selling is not allowed.

Lower cost for ensuring a given standard of conservatism
Capital regulation often aims to ensure that banks are resilient up to a certain standard of conservatism.⁴ As I show below, risk sensitivity lowers the cost of achieving a given standard of conservatism across banks. In illustrating this point, I will assume that regulators seek to ensure that all banks have sufficient capital to withstand an economic shock of severity "x."

Given possible bank portfolios, regulators would need to set a leverage requirement of 10 percent (equivalent to the loss experienced by the risky asset) to ensure that no bank would fail given an economic shock of severity "x."

Similar to risk-based capital requirements in the real world, I assume regulators in this hypothetical world would set a risk-based capital requirement that includes (1) a minimum capital ratio and (2) risk weights that apply to different assets in proportion to their risk. Given this structure of risk-based requirements and under the scenario considered, I assume the minimum risk-based capital ratio is set to 10 percent (similar to the leverage requirement) and the risk weights would be 100 percent for the risky asset and 20 percent for the safe asset.

Table 1 compares capital requirements to loss conditional on a shock of severity "x" for varying portfolio compositions under leverage and risk-based capital requirements.

Table: Capital requirements and risk of loss – Portfolio of $100

By aligning potential losses with required capital, the risk-based capital requirement would ensure that each bank's loss coverage meets the standard of conservatism sought by regulators while requiring the minimum amount of capital necessary in each case. Therefore, the risk-based capital requirement would ensure that banks face the minimum cost needed to achieve the standard of conservatism regulators seek. If this desired standard of conservatism reflects an optimal tradeoff between the costs and benefits of capital requirements, then the risk-based requirement maximizes net benefits.

Meanwhile, for banks that hold some portion of the safe asset, a leverage requirement would require capital in excess of the amount needed to meet the standard of conservatism. Such excess capital provides resilience to even bigger shocks and, therefore, has marginal benefits. Still, to the extent that the standard of conservatism is selected to optimize the net benefits of capital requirements, capital amounts above the selected standard would bring marginal costs larger than their marginal benefits.

Better alignment of capital with risk for a given aggregate level of capital
In the previous subsection, I assumed that regardless of employing leverage or risk-based capital requirements, regulators would aim to ensure that requirements would meet a certain standard of conservatism for all banks. However, setting a leverage requirement that achieves a reasonable standard of conservatism across banks with varying degrees of risk-taking would likely not be desirable in the real world. The riskiness of some financial assets (e.g., certain securitized products) can be so large relative to typical bank assets (e.g., lending to investment-grade businesses) that establishing a leverage requirement that appropriately covered risk for a bank focusing on high-risk assets would result in an excessively high requirement for banks that do not engage in risky activities. Instead of setting leverage requirements with the aim of ensuring sufficient loss absorbency for banks with high-risk portfolios, real-world regulators typically think of leverage requirements as a "back-stop" to risk-based requirements, which – by design – is supposed to be less strict than risk-based capital requirements under most situations.⁵

Returning to the scenario discussed so far, let us also assume that an equal number of banks are holding each of the five shares of the risky asset listed in Table 1 (0 percent, 25 percent, 50 percent, 75 percent, and 100 percent). Given this assumption, the average bank would have 50 percent of their portfolio invested in the risk asset and, thus, would have an expected loss of $6 under the economic shock of severity 'x.' Given this expected loss for the average bank, let us assume that regulators would set a leverage capital requirement of $6. Table 2 compares the two approaches to setting capital requirements in this revised scenario.

Table 2: Capital requirements and risk of loss – Portfolio of $100

Under this more realistic scenario, the leverage and risk-based requirements result in the same average capital requirement ($6). However, the resilience provided by the two types of requirements is quite distinct. Under the risk-based requirement, capital is aligned with risk and, thus, the requirement provides loss coverage at the same standard of conservatism across banks. Meanwhile, under the leverage requirement, banks with a large share of the risky asset are undercapitalized relative to a shock of severity "x" and banks with a large share of the safe asset maintain excess capital relative to such shock. Given this inefficient distribution of capital, bank failures are more likely under the leverage requirement than the risk-based requirement in this scenario, despite both requirements resulting in the same average amount of required capital across banks.

In this scenario, I assumed that the distribution of bank portfolios would be the same under the leverage or risk-based requirements. However, the leverage requirement would likely incentivize banks to direct their investment towards the risky asset. Suppose two banks have the same initial portfolio, one subject to the leverage requirement and the other to the risk-based requirement. From the same starting portfolio, the bank subject to the leverage requirement would likely move towards a riskier portfolio than the bank subject to the risk-based requirement because the relative funding cost of the risky asset would be higher for the bank subject to the risk-based requirement than for the bank subject to the leverage requirement. Therefore, the aggregate ratio of risk to capital across the banking industry would likely deteriorate over time under a leverage requirement relative to a risk-based one.

The scenario considered so far pitted a capital requirement with no differentiation relative to risk (the leverage requirement) against a capital requirement that perfectly aligns with risk (the risk-based requirement). Risk-based capital requirements are unlikely to align perfectly with risk in the real world. Still, the benefits of risk sensitivity discussed above – better alignment of capital with risk and lower cost of achieving a desired level of conservatism – would still apply when comparing more risk-sensitive to less risk-sensitive requirements.

3. Criticisms of risk-based capital requirements

Criticism of risk-based capital requirements falls into four main categories: (1) regulators cannot measure risk correctly; (2) outside pressures and other objectives may lead regulators not to adopt requirements that measure risk accurately; (3) attempts at introducing risk sensitivity redound into opportunities for regulatory arbitrage;⁶ and (4) a risk-sensitive requirement would be pro-cyclical.

The financial system is extremely complex and dynamic, and financial risk has multiple dimensions; therefore, establishing requirements that measure risk precisely over time and across its various dimensions is impossible.⁷ Still, certain financial metrics are consistently associated with risk (e.g., borrower leverage) and market participants – such as banks, bond analysts, or rating agencies – have expertise in identifying the relative riskiness of exposures.⁸ Regulators may not have access to all information banks have, but do have access to substantial information and can likely design requirements that correlate with risk.^9,10

A related concern is whether outside pressures and other governmental objectives can get in the way of regulators setting requirements that properly measure risk. The political economy literature suggests that regulated entities are incentivized to pressure regulators to set rules that favor them.¹¹ Banking industry advocacy for less strict regulatory rules can be observed in every rulemaking that U.S. federal bank regulators have issued, and the academic literature has documented examples of bank lobbying leading to the accumulation of risks.¹² Beyond the effects of lobbying, financial regulators may set regulatory requirements with other goals in mind in some cases, such as supporting financial inclusion or small businesses.^13,14 Balancing different goals is unavoidable in real-world policymaking; still, to the extent that regulators considering goals beyond bank resilience when setting capital requirements is seen as undesirable, statutory clarification of the role of bank regulators can be pursued.¹⁵

In some circumstances, complex regulatory requirements can increase opportunities for banks to engage in regulatory arbitrage.^16,17 No regulatory requirement can precisely capture the risks across all possible contingencies associated with financial products; therefore, banks are likely to explore situations where requirements underrate the risks of certain exposures. Relatedly, bank incentives likely bias modeling choices when capital requirements rely on bank internal models.¹⁸ But while complexity can open opportunities for regulatory arbitrage, bluntness does too. When requirements do not vary with risk, bank incentives point towards pursuing riskier activities. For example, before the 07-08 financial crisis, banks often reduced capital requirements by moving exposures off their balance sheet while retaining substantial risks.¹⁹ Given the potential for regulatory arbitrage under either overly simple or overly complex requirements, an efficient capital framework should aim to measure risk while being subject to frequent review to address arbitrage opportunities. Ultimately, regulatory arbitrage cannot be addressed through regulation alone. Continuous supervisory engagement is needed to monitor the risks resulting from new products and business strategies.

Risk sensitivity can result in the pro-cyclicality of capital requirements.²⁰ Upon an economic downturn, businesses and households often lose income and become more likely to default on their obligations to banks. Thus, bank exposures usually become riskier during economic downturns. Consequently, capital requirements that increase with risk would increase with economic downturns. While banks experience higher risk during economic downturns, pro-cyclicality in capital requirements is considered by most experts as undesirable. If banks cut their lending in response to higher requirements, such a reduction in lending can aggravate the economic slump.²¹ A reasonable balance between risk sensitivity and counter-cyclicality can be achieved by combining appropriate risk-weighting of bank exposures with a counter-cyclical level of overall requirement. Bank regulators have aimed to introduce counter-cyclicality in overall requirements through the "counter-cyclical capital buffer;"²² however, U.S. regulators have never used the counter-cyclical capital buffer since its introduction.

4. Measuring the net benefits of a risk-based capital requirement

The economic literature includes several papers illustrating the benefits of applying risk-sensitive capital requirements to banks.²³ Also, several empirical studies have proposed frameworks to assess the costs and benefits of bank capital requirements to identify the range of optimal capital ratio requirements.²⁴ However, the literature lacks a methodology to integrate the degree of risk sensitivity in capital requirements with the estimation of optimal capital ratio requirements.²⁵ The remainder of this section sketches out an approach to fill this gap.

a) Baseline case with linear costs of capital

Studies like Basel Committee on Banking Supervision (2010b) and Firestone et al. (2019) consider the reduction of the likelihood of financial crises as the main macroeconomic benefit of higher bank capital. Better capitalized banks are less likely to fail and, therefore, less likely to provoke a financial crisis with sizeable economic costs. The empirical analysis provided in these papers suggests that increases in capital have a decreasing marginal benefit. When starting from low capital ratios, an increase in the ratios substantially reduces the probability of a financial crisis; while when starting from high capital ratios, an additional increase results in a much smaller marginal decrease of the likelihood of a financial crisis. Consistent with these findings, the analysis below assumes that the benefits of capital are a concave function of capital levels.

In addition, I assume that the societal benefits of bank capital requirements are proportional to bank asset size scaled by riskiness. All else equal, the failure of a larger bank disrupts more economic relations; therefore, the failure of a large bank generally results in higher social costs than the failure of a small bank.²⁶ In addition, the expected societal cost that can follow from a bank failure increases with the probability of failure and with the loss of value in case of failure; therefore, it is reasonable to assume that the benefits of capital are higher for riskier banks.²⁷

Given these assumptions and in the interest of mathematical tractability,²⁸ I assume the following functional form for the societal benefits of bank capital:

$$societal\ benefits\ of\ bank\ capital\ = ln(capital)\cdot risk\cdot assets$$

Given that debt generally enjoys a tax advantage, funding with higher own capital typically results in higher funding costs from the bank's perspective. These higher funding costs are passed to bank borrowers, increasing the cost of investment and decreasing economic growth. For simplicity, I will start by assuming that the societal costs of bank capital are linear.

$$societal\ costs\ of\ bank\ capital\ = c\cdot capital$$

Given these functional forms for the societal costs and benefits of bank capital, the optimal capital amount is given by the following expression:²⁹

$$\text{capital}^* = \frac{{risk\cdot assets}}{c}$$

Optimal capital is higher for riskier banks and banks with larger asset size. Also, the higher the societal cost of bank capital, the lower the optimal capital amount. Meanwhile, the optimal risk-adjusted capital ratio is constant:

$$\left(\frac{capital}{risk\cdot assets}\right)^* = \frac{1}{c}$$

To understand the benefits of capital requirements that account for risk, I compare the net benefits obtained when capital requirements vary with risk with the net benefits when capital requirements do not. To perform this comparison, I assume that "risk" is distributed according to a uniform distribution between 0 and 1.³⁰ In this scenario, if the capital requirement cannot vary with risk, the optimal capital requirement would be as follows:³¹

$$capital^{**} = \frac{assets}{2c}$$

Note that under the simple modeling assumptions considered so far, the average required amount of capital under a risk-based requirement would be the same as under the capital requirement that does not consider risk, as the expected value of "risk∙assets/c" is "assets/2c" when risk is distributed uniformly between 0 and 1.

If regulators impose a capital requirement that varies with risk, the net societal benefit of the capital requirement would be as follows:³²

$$\text{net societal benefit of risk based capital requirement} = \left( ln \left( \frac{\text{assets}}{c} \right) - \frac{3}{2} \right) \frac{\text{assets}}{2}$$

Meanwhile, if the capital requirement does not depend on risk, its net societal benefit would be as follows:³³

$$net\ societal\ benefit\ of\ nonrisk\ based\ capital\ requirement = \left( ln \left( \frac{assets}{2c} \right) -1 \right) \frac{assets}{2}$$

When comparing the two, the difference is always positive and equal to the following:

$$net\ societal\ benefit\ of\ risk\ based\ capital\ requirement \\ - net\ societal\ benefit\ of\ nonrisk\ based\ capital\ requirement \\ = \left( ln(2) - \frac{1}{2} \right) \frac{assets}{2}$$

So, by considering additional information relevant to the benefits of capital requirements, the risk-based capital requirement improves the net societal benefit of capital requirements.

b) Alternative assumptions: convex costs of capital and risk-reduction incentives

Consistent with Basel Committee on Banking Supervision (2010) and Firestone et al. (2019), the analysis up to this point assumed that the societal costs of higher bank capital requirements are linear. When considering required capital ratios in their historical (low) ranges and given a degree of substitutability between the banking system and other sources of funding for projects, the assumption of a linear societal cost of bank capital may be a reasonable approximation. But when considering required capital ratios outside of historical norms, the assumption of linearity is unlikely to hold. Beyond certain thresholds, higher capital requirements are likely to disrupt the liquidity and maturity transformation business model of banks and result in loss of funding for certain types of projects with high expected rates of return that rely on bank funding for screening reasons (e.g., small businesses, commercial real estate).³⁴ Therefore, the societal costs of capital requirements are likely to be convex, at least beyond a certain threshold.

In exploring the effect of risk sensitivity on capital requirements, let us consider convex societal costs of capital. In particular, I will assume that the societal cost of bank capital increases with the square of capital.

$$societal\ costs\ of\ bank\ capital’ = c\cdot capital^{2}$$

Under this assumption for the costs of bank capital, the optimal capital level – when capital can vary with risk – becomes as follows:³⁵

$$capital^{***} = \sqrt{\frac{risk\cdot assets}{2c}}$$

This means that if, as before, we assume "risk" to be a variable distributed uniformly between 0 and 1, the expected optimal capital becomes equal to the following:³⁶

$$E(capital^{***}) = \frac{\sqrt{2}}{3} \sqrt{\frac{assets}{c}}$$

Meanwhile, if we again assume that regulators cannot vary the required capital based on risk, then the optimal capital requirement becomes as follows:³⁷

$$capital^{****} = \frac{1}{2} \sqrt{\frac{assets}{c}} > \frac{\sqrt{2}}{3} \sqrt{\frac{assets}{c}}$$

So, when the societal costs of bank capital are proportional to the square of capital, the average optimal capital requirement is higher when regulators cannot differentiate the capital requirement relative to risk than when they can.³⁸ Therefore, the convexity of social costs of capital would provide additional support for the desirability of risk-based capital requirements, as they could be set optimally at lower average levels.

Similar to the leading papers from regulators on the estimation of optimal capital, the analysis so far assumes that the benefits and costs of capital requirements are a function of capital levels set by regulators and the risk of bank assets. And, like prior research, I have assumed that banks do not respond to capital requirements by rebalancing their portfolio. In particular, the analysis above does not include a mechanism for banks to reduce the risk of their portfolio when the capital requirements vary with risk. However, banks respond to incentives and, therefore, capital requirements that vary with risk incentivize banks to engage more in low-risk activities and less in high-risk activities.

At the moment, I have not developed a model that translates the incentives introduced by risk-sensitive capital requirements into a shift in the riskiness of bank assets. Still, it seems safe to assume that risk-sensitive capital requirements would induce an even bigger reduction in average optimal capital under such a model than when banks are assumed to not change the composition of their portfolio in reaction to capital requirements (as I have assumed in this note). So, bank incentives to avoid higher capital requirements provide another mechanism through which risk-based capital requirements can lead to lower average optimal capital requirements.

5. Conclusion and future work

Risk sensitivity can improve the efficiency of capital requirements. This note discusses two ways of conceptualizing this benefit: (1) risk-sensitive capital requirements make achieving a desired degree of conservatism in requirements cheaper, and (2) risk-sensitive capital requirements ensure that the aggregate pie of capital requirements is efficiently distributed across banks.

This note also provides a conceptual framework to assess the net benefits of risk sensitivity in setting capital requirements. Applying this conceptual framework to a specific policy assessment requires additional steps, three of which seem paramount: (1) how to integrate partial risk sensitivity into the benefits of requirements, (2) how to measure risk sensitivity, and (3) how to account for the impact of risk sensitivity on bank incentives.

To illustrate the benefits of risk sensitivity of capital requirements, this note compared two extreme cases: (a) no risk sensitivity and (b) perfect risk sensitivity. However, risk sensitivity is unlikely to be perfect in practical applications. Future work should seek to develop modeling approaches that incorporate partial risk sensitivity into calculating the net benefits of capital requirements.

A related challenge is measuring the degree of risk sensitivity of a capital requirement. The risk of exposures can be modeled based on relevant indicators. Still, given the uncertainty associated with risk in the real world, no measure can precisely capture risk across all circumstances. In practical applications that seek to compare approaches to set capital requirements, one possible methodology to estimate the relative net benefits of risk sensitivity is to (1) estimate how the approaches to set capital requirements correlate with an approach that is seen as more risk-sensitive; (2) assume that the more risk-sensitive approach perfectly represents risk; (3) model the net benefits of the alternative approaches under consideration (which requires addressing the challenge discussed in the previous paragraph); and (4) proxy for the difference in net benefits as the difference between the net benefits calculated under (3). For example, when comparing the net benefits of two standardized approaches to set capital requirements, the correlation of such approaches with bank-specific model-based measures of risk can be used to estimate the risk sensitivity of the two standardized approaches.

Lastly, an assessment of the net benefits of risk-sensitive capital requirements would ideally account for the impact on bank incentives. More risk-sensitive capital requirements are likely to incentivize banks to adopt less risky portfolios than risk-invariant capital requirements. Such behavioral reactions from banks should reduce the optimal amount of required capital under a risk-sensitive framework relative to a risk-invariant framework. Therefore, incorporating bank reactions is likely to increase the net benefits of a risk-sensitive capital framework relative to a risk-insensitive capital framework.

I am very interested in ideas for addressing these modeling challenges and open to collaborating.

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Appendix

Section 1
$$societal\ net\ benefits\ of\ bank\ capital=ln(capital)\cdot risk\cdot assets-c\cdot{capital}$$

Given that the societal net benefits of bank capital function is concave regarding the capital variable, the optimal level of capital is the amount for which the partial derivative of net benefits is equal to zero.

$$\frac{\partial societal\ net\ benefits}{\partial capital} = 0 \iff \frac{risk\cdot{assets}}{capital} - c = 0 \iff capital^{*} = \frac{risk\cdot{assets}}{c}$$

Section 2
The expected value of the societal net benefits of bank capital, when risk is a random variable with a uniform (0,1) distribution, is calculated as follows:

$$E (societal\ net\ benefits) = \int_{0}^1 (ln (capital)\cdot risk\cdot assets - c\cdot capital) drisk\\ = assets\cdot ln(capital) \int_{0}^{1} (risk)drisk - c\cdot capital \\ = \frac{assets}{2} ln(capital) - c\cdot capital$$

Given that this expected value function is concave regarding the capital variable, the optimal amount of capital when regulators cannot specify capital requirements as a function of risk is the amount for which the partial derivative of expected benefits is equal to zero.

$$\frac{\partial\ E(societal\ net\ benefits)}{\partial capital} = 0 \iff \frac{assets}{2capital} - c = 0 \iff capital^{**} = \frac{assets}{2c}$$

Section 3
Given the optimal amount of capital calculated in Section 1 of this appendix, the societal net benefits of bank capital are as follows:

$$societal\ net\ benefits\ of\ bank\ capital \\ = ln\left(\frac{risk\cdot assets}{c}\right)\cdot risk\cdot assets - c\cdot \frac{risk\cdot assets}{c} \\ = \bigg[ln\left(\frac{risk\cdot assets}{c}\right) -1 \bigg]\cdot risk\cdot assets$$

Assuming that risk is a random variable with a uniform (0,1) distribution, the expected societal net benefits of bank capital, conditional on this optimal capital amount derived in Section 1, are as follows:

$$E(societal\ net\ benefits) = \int_{0}^{1} \left(\bigg[ln\left(\frac{risk\cdot assets}{c}\right) -1 \bigg] risk\cdot assets \right)drisk \\ = \bigg[ \int_{0}^{1} (ln(risk)\cdot risk)drisk + \left( ln \left(\frac{assets}{c}\right) -1 \right) \int_{0}^{1} (risk)drisk\bigg]\cdot assets\\ \left[ \left[ \frac{risk^2 ln(risk)}{2} - \frac{risk^2}{4} \right]^{1}_{0} + \left( ln \left( \frac{assets}{c} \right) -1 \right) /2 \right]\cdot assets \\ = \left( ln \left( \frac{assets}{c} \right) - \frac{3}{2} \right)\cdot \frac{assets}{2}$$

Section 4
Given the optimal amount of capital calculated in Section 2 of this appendix, the expected societal net benefits of bank capital are as follows:

$$E(societal\ net\ benefits\ of\ bank\ capital) = \frac{assets}{2} ln \left(\frac{assets}{2c} \right) - c\cdot \frac{assets}{2c} \\ = \left( ln \left(\frac{assets}{2c} \right) -1 \right)\cdot \frac{assets}{2}$$

Section 5
In this scenario of costs, the societal net benefits of bank capital are as follows:

$$societal\ net\ benefits\ of\ bank\ capital=ln(capital)\cdot risk\cdot assets-c\cdot capital^2$$

Given that the societal net benefits of bank capital function is concave regarding the capital variable, the optimal level of capital is the amount for which the partial derivative of net benefits is equal to zero.

$$\frac{\partial societal\ net\ benefits}{\partial capital} = 0 \iff \frac{risk\cdot assets}{capital} - 2c\cdot capital = 0 \iff capital^{2} = \frac{risk\cdot assets}{2c} \\ \iff capital^{***} = \sqrt{\frac{risk\cdot assets}{2c}}$$

Section 6
Given the optimal level of capital calculated in Section 5 of this appendix and assuming that risk is a random variable with a uniform (0,1) distribution, the expected amount of optimal capital is calculated as follows:

$$E(capital^{***}) = \int_{0}^{1} (capital^{***}) drisk = \sqrt{\frac{assets}{2c}} \int_{0}^{1} (risk ^{\tfrac{1}{2}}) drisk = \frac{2}{3}\cdot \frac{1}{\sqrt{2}} \sqrt{\frac{assets}{c}} \\ = \frac{\sqrt{2}}{3} \sqrt{\frac{assets}{c}}$$

Section 7
The expected value of the societal net benefits of bank capital under this scenario of costs, when risk is a random variable with a uniform (0,1) distribution, is calculated as follows:

$$E(societal\ net\ benefits) = \int_{0}^{1} (ln(capital)\cdot risk\cdot assets - c\cdot capital^{2}) drisk \\ = \frac{assets}{2} ln(capital) - c\cdot capital^{2}$$

Given that this expected value function is concave regarding the capital variable, the optimal amount of capital when regulators cannot specify capital requirements as a function of risk is the amount for which the partial derivative of expected benefits is equal to zero.

$$\frac{\partial E (societal\ net\ benefits)}{\partial capital} = 0 \iff \frac{assets}{2capital} - 2c\cdot capital = 0 \iff capital^{2} = \frac{assets}{4c} \\ \iff capital^{****} = \frac{1}{2} \sqrt{\frac{assets}{c}}$$

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