# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

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Abstract

Using a simple model, this paper derives two results that provide guiding principles for hedging by, and capital regulation of, financial institutions. First, the adoption of value- maximizing hedging strategies by financial institutions minimizes the volatility of the regulatory capital ratio. The hedging incentives for financial institutions can therefore differ from those of nonfinancial firms, as it is commonly argued that nonfinancial firms have an incentive to reduce the volatility of cashflows. Second, asset substitution incentives for financial institutions are eliminated if and only if the regulatory capital ratio correlates perfectly with the market value of equity. This implies that, in order to eliminate asset substitution incentives for financial institutions, it is not sufficient for the capital requirements to be proportional to the systematic risks (the betas) of the assets. This result is consistent with the regulatory response to the global financial crisis, as the Basel III regulatory framework for banks focuses on aligning the regulatory definition of available capital with the market value of equity.

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Introduction

## Abstract

Using a simple model, I derive two results that provide guiding principles for hedging by, and capital regulation of, financial institutions. First, the adoption of value-maximizing hedging strategies by financial institutions minimizes the volatility of the regulatory capital ratio. The hedging incentives for financial institutions can therefore differ from those of nonfinancial firms, as it is commonly argued that nonfinancial firms have an incentive to reduce the volatility of cashflows. Second, asset substitution incentives for financial institutions are eliminated if and only if the regulatory capital ratio correlates perfectly with the market value of equity. This implies that, in order to eliminate asset substitution incentives for financial institutions, it is not sufficient for the capital requirements to be proportional to the systematic risks (the betas) of the assets. This result is consistent with the regulatory response to the global financial crisis, as the Basel III regulatory framework for banks focuses on aligning the regulatory definition of available capital with the market value of equity.

## 1 Introduction

Why do financial institutions apply regulatory hedging strategies that reduce the volatility of the regulatory capital ratio while it is commonly argued that nonfinancial firms have an incentive to apply cashflow hedging strategies that reduce the volatility of cashflows? For example, the British bank Barclays PLC and the Dutch bank ING specifically state in their annual reports that they apply regulatory hedging strategies.^{1}^{1} 1 On page 155 of its 2016 annual report, Barclays PLC states: “The Group’s capital ratio management strategy is to minimize the volatility of the capital ratios caused by foreign exchange rate movements.” On page 297 of its 2016 annual report, ING states: “As a result of the strategy to hedge the CET1 ratio a net foreign currency exposure exists.” In this paper, it is demonstrated that financial institutions have an incentive, first and foremost, to apply regulatory hedging strategies. As an implication of this result, it is shown that incentives for financial institutions to substitute low-risk assets for high-risk assets (asset substitution incentives) are eliminated if and only if the regulatory capital ratio correlates perfectly with the market value of equity.

This paper analyzes the impact of capital regulation on hedging by financial institutions. In so doing, two questions are explored. First, what are the value-maximizing hedging strategies for financial institutions with respect to a given regulatory capital ratio? Second, can capital regulation eliminate all asset substitution incentives for financial institutions? And, if so, how?

The answers to these questions flow from the insight that there is a positive relationship between the expected cost of a regulatory capital shortfall and the volatility of the regulatory capital ratio. I quantify this expected cost for a given regulatory capital ratio using option pricing techniques and, in the spirit of Froot et al (1993), by assuming an adjustment cost of equity (Donaldson 1961; Myers and Majluf 1984). While the literature both on hedging and on the relationship between capital regulation and bank risk taking is extensive, the relationship between capital regulation and hedging has received less attention.^{2}^{2} 2 Among others, Merton (1977), Kim and Santomero (1988), Keeley (1990), Rochet (1992) and Repullo (2004) contribute to moral hazard theory, which provides a justification for minimum capital requirements, as they reduce the possibilities for banks to extract value from the deposit insurance system. Capital buffer theory is a complementary strand of the literature and states that banks have an incentive to hold a capital buffer in excess of the minimum capital requirement as insurance against a costly breach of the minimum capital requirements (Marcus 1984; Milne and Whalley 2001). This paper explores the latter relation.

Froot et al (1993) show that firms want to match the supply and demand of funds internally rather than becoming dependent on external financing. Therefore, firms have an interest in hedging risks by reducing the volatility of cashflows, as this leads to a lower probability that a firm needs to attract external financing for its capital expenditure. Whereas Froot et al (1993) analyze hedging incentives for nonfinancial firms, this paper analyzes hedging incentives for financial institutions. Therefore, the analysis and model setup here differ from those in Froot et al (1993) in two crucial ways.

First, here we make the simplifying assumption that financial institutions only face an adjustment cost of equity and not an adjustment cost of debt. As opposed to nonfinancial firms, financial institutions are active in the wholesale funding market constantly, and they usually have direct access to debt financing. The intuition behind this assumption is that financial institutions can issue debt without an adjustment cost as long as they do not have a regulatory capital shortfall.

Second, since financial institutions are subject to capital regulation, this paper introduces a model in which capital, rather than cash, is the main variable. If financial institutions do not face an adjustment cost of debt as long as they do not have a regulatory capital shortfall, capital is a more binding constraint than cash.

The first result of the analysis is that financial institutions have an incentive, first and foremost, to apply regulatory hedging strategies. Using a linear factor model (see, for example, Chen and Ingersoll 1983; Grinblatt and Titman 1983), it is shown that, for a given regulatory capital ratio, value-maximizing hedging strategies by financial institutions minimize the volatility of the regulatory capital ratio. The hedging incentives for financial institutions can therefore differ from those of nonfinancial firms that have an incentive to reduce the volatility of cashflows, as derived by Froot et al (1993). A stylized real-life example is provided (Example 2.3) to illustrate that a hedging strategy which reduces the volatility of cashflows can lead to an increase in the volatility of the regulatory capital ratio and therefore to a higher expected cost of a regulatory capital shortfall.

As a second result, it is derived that asset substitution incentives for financial institutions are eliminated totally if and only if the regulatory capital ratio correlates perfectly with the market value of equity of a financial institution. In a sense, this result complements the result of Rochet (1992), who shows that, in order to eliminate asset substitution incentives for banks that behave as competitive portfolio managers, minimum capital requirements should be based on theoretically correct risk weights. For computing theoretically correct risk weights, Rochet (1992) recommends making them proportional to the systematic risks (the betas) of the assets. Since, in the simple model of this paper, regulatory capital is a separate variable from the market value of equity and risk weights can vary through time, it is demonstrated that theoretically correct risk weights alone are not sufficient to eliminate asset substitution incentives. In addition, the regulatory definition of capital should be equal to the market value of equity, and the theoretically correct risk weights should be constant through time (eg, proportional to the “through-the-cycle” asset betas).

This second result is consistent with the regulatory response to the global financial crisis. The Basel III regulatory framework for banks focuses on aligning the regulatory definition of available capital with the market value of equity.^{3}^{3} 3 For example, under Basel III banks can no longer apply a prudential filter. Further, deductions are made from common equity under Basel III. Moreover, the second result is also consistent with the new regulatory framework for European insurance companies: Solvency II. In Solvency II, the regulatory definition of available capital is equal to equity on the market value balance sheet of an insurance company, and the minimum capital requirements are based on constant shocks.^{4}^{4} 4 Solvency II also makes several non-market-consistent assumptions, such as the ultimate forward rate.

## 2 The model

Consider a financial institution whose stock price ${({S}_{t})}_{t\ge 0}$ follows the process

$$\frac{\mathrm{d}{S}_{t}}{{S}_{t}}=\mu \mathrm{d}t+\sigma \mathrm{d}{W}_{1,t},\text{where}{S}_{0}=s\in {\mathbb{R}}_{\ge 0}\text{isfixed}.$$ | (2.1) |

Here, ${W}_{1,t}$ is a standard Brownian motion on a filtered probability space $(\mathrm{\Omega},\mathcal{F},{\{{\mathcal{F}}_{t}\}}_{t\ge 0},\mathbb{P})$ satisfying the usual conditions, $\mu $ is the drift and $\sigma >0$ is the standard deviation of the standard logarithmic returns of ${S}_{t}$. In the remainder of this paper, ${S}_{t}$ is referred to interchangeably as the stock price or the market value (of equity).

Regulatory capital, ${({A}_{t})}_{t\ge 0}$, is the regulator’s view on how much (equity) capital a financial institution has available, and, like ${({S}_{t})}_{t\ge 0}$, it is also assumed to be lognormally distributed. ${({A}_{t})}_{t\ge 0}$ is modeled by substituting ${\sigma}_{1}>0$ for $\sigma $ and ${\mu}_{1}$ for $\mu $ in (2.1):^{5}^{5} 5 Without loss of generality it can be assumed that the underlying stochastic process ${W}_{1,t}$ is different for ${({A}_{t})}_{t\ge 0}$. ${({A}_{t})}_{t\ge 0}$ is the regulatory definition of capital rather than the economic definition, where the economic definition of capital is simply the stock price.

$$\frac{\mathrm{d}{A}_{t}}{{A}_{t}}={\mu}_{1}\mathrm{d}t+{\sigma}_{1}\mathrm{d}{W}_{1,t},\text{where}{A}_{0}=a\in {\mathbb{R}}_{\ge 0}\text{isfixed}.$$ | (2.2) |

Regulatory risk capital is the amount of capital at risk according to the regulator (as a financial institution faces, among other risks, market, credit and insurance risks). Since regulatory risk capital is generally based on variables that can change – such as historical volatility, credit ratings or the value of a firm’s assets – changes in regulatory risk capital occur regardless of changes in regulation. In practice, regulatory risk capital need not be a continuous process, as the regulator uses proxies for the actual level of risk of a financial institution. As we do not want to introduce additional frictions when modeling regulatory risk capital, the simplifying assumption that regulatory risk capital ${({R}_{t})}_{t\ge 0}$ is also lognormally distributed is made:

$$\frac{\mathrm{d}{R}_{t}}{{R}_{t}}={\mu}_{2}\mathrm{d}t+{\sigma}_{2}\mathrm{d}{W}_{2,t},\text{where}{R}_{0}=r\in {\mathbb{R}}_{\ge 0}\text{isfixed}.$$ | (2.3) |

Suppose that the correlation between regulatory capital and regulatory risk capital is equal to $\rho $. More specifically, $E[\mathrm{d}{W}_{1,t}\mathrm{d}{W}_{2,t}]=\rho \mathrm{d}t$.

The regulatory capital ratio, ${F}_{t}={R}_{t}/{A}_{t}$, follows the process

$$\frac{\mathrm{d}{F}_{t}}{{F}_{t}}={\mu}_{\mathrm{com}}\mathrm{d}t+{\sigma}_{\mathrm{com}}\mathrm{d}{W}_{t},\text{where}{F}_{0}=f\in {\mathbb{R}}_{\ge 0}\text{isfixed},$$ | (2.4) |

and

${\sigma}_{\mathrm{com}}^{2}$ | $={\sigma}_{1}^{2}+{\sigma}_{2}^{2}-2\rho {\sigma}_{1}{\sigma}_{2},$ | (2.5) | ||

${\mu}_{\mathrm{com}}$ | $={\mu}_{2}-{\mu}_{1}.$ | (2.6) |

In practice, the regulatory capital ratio is generally defined as regulatory capital divided by regulatory risk capital. However, for the purposes of this paper, the regulatory capital ratio is defined as the reciprocal in order to make equations more intuitive.

###### Assumption 2.1.

Financial institutions are subject to minimum capital requirements which decree that regulatory capital is greater than or equal to regulatory risk capital (ie, ${F}_{t}\mathrm{\le}\mathrm{1}$). If regulatory capital is less than regulatory risk capital (ie, ${F}_{t}\mathrm{>}\mathrm{1}$), a financial institution is in regulatory default.

If a financial institution does not overcome a regulatory default, the regulator will force the institution to wind down. A financial institution can overcome a regulatory default by raising sufficient equity so that regulatory capital is again greater than or equal to regulatory risk capital. A financial institution can also improve (ie, lower) its regulatory capital ratio by substituting high-risk assets for low-risk assets (derisking). Derisking smoothly is generally a cost-efficient way for a financial institution to improve its regulatory capital ratio. For example, a costless form of derisking for banks is to not refinance loans. If, on the other hand, a financial institution is in regulatory default, it has to address this immediately, and rather than derisking smoothly it has to derisk abruptly, which can be costly. Hence, we assume that raising equity is less costly than derisking if a financial institution is in regulatory default.

###### Assumption 2.2.

If a financial institution is in regulatory default, it has to address this immediately by raising equity, which comes with an adjustment cost of equity.

There is a risk that regulatory risk capital exceeds regulatory capital. In this case, because of Assumption 2.2, the institution will raise equity. The amount of equity at time $t$, ${E}_{t}$, as a percentage of regulatory capital, ${A}_{t}$, that a financial institution needs to raise is equal to

$$\frac{{E}_{t}}{{A}_{t}}=\mathrm{max}[{F}_{t}-1,0].$$ | (2.7) |

The cost at time $t$, ${k}_{t}$, as a percentage of regulatory capital, ${A}_{t}$, of raising outside equity is equal to

$$\frac{{k}_{t}}{{A}_{t}}=f(\mathrm{max}[{F}_{t}-1,0]),$$ | (2.8) |

where $f(\cdot )$ is a convex function to account for the fact that there is a convex adjustment cost of equity (Myers and Majluf 1984).

Although a general convex function is conceptually straightforward to employ, it is analytically less tractable than its first-order approximation. For the sake of simplicity, I therefore assume that the cost of raising equity is a linear function of the amount that needs to be raised. This simplifying assumption makes the mathematics easier but does not affect the conclusions of this paper. With the aid of this simplifying assumption, the cost at time $t$, ${k}_{t}^{\prime}$, as a percentage of regulatory capital, ${A}_{t}$, of raising outside equity is equal to

$$\frac{{k}_{t}^{\prime}}{{A}_{t}}=c\mathrm{max}[{F}_{t}-1,0],$$ | (2.9) |

where $c$ is some constant.

### 2.1 The expected cost of a regulatory capital shortfall

For the sake of simplicity, the analyses in the remainder of this paper are conducted in the risk-neutral measure $\mathbb{Q}$, and $r$ is defined as the risk-free interest rate. The conclusions of this paper also hold in the physical measure $\mathbb{P}$. In this case, the risk-free interest rate $r$ should be replaced with the appropriate drift.

The expected cost at time $t$, ${k}_{t}^{\prime}$, as a percentage of regulatory capital, ${A}_{t}$, of a financial institution raising equity after one year is equal to a one-year call option on ${F}_{t}$ (Black and Scholes 1973). More specifically,

$$\frac{{k}_{t}^{\prime}}{{A}_{t}}=c[{F}_{t}N({d}_{1})-{\mathrm{e}}^{-r}N({d}_{2})],$$ | (2.10) |

where $N(\cdot )$ is the standard normal distribution and

${d}_{1}$ | $={\displaystyle \frac{\mathrm{ln}({F}_{t})+r+\frac{1}{2}{\sigma}_{\mathrm{com}}^{2}}{{\sigma}_{\mathrm{com}}}},$ | ||

${d}_{2}$ | $={d}_{1}-{\sigma}_{\mathrm{com}}.$ |

Equation (2.10) shows that the expected cost associated with the risk that regulatory risk capital exceeds regulatory capital is greater than or equal to zero. The only way for this expected cost to be zero is if the volatility of the regulatory capital ratio is zero. In practice, this is unlikely, unless regulatory capital and regulatory risk capital have the same volatility and the correlation between them is equal to one (see (2.5)). To illustrate that it is unlikely that the volatility of the regulatory capital ratio is zero, the online appendix to this paper gives a real-life example of a negative correlation between regulatory risk and regulatory capital. The lower the correlation between regulatory capital and regulatory risk capital, the higher the volatility of the regulatory capital ratio and, therefore, the higher the expected cost. In the remainder of this paper, the expected cost of a regulatory capital shortfall refers to the expected cost of a financial institution raising equity after one year because regulatory risk capital exceeds regulatory capital.

The fact that a financial institution is confronted with an expected cost of a regulatory capital shortfall as a result of capital regulation means that financial institutions need to manage this cost. Simply minimizing this expected cost of a regulatory capital shortfall allows us to derive implications for hedging by, and capital regulation for, financial institutions. In order to formally derive these implications, the next two subsections introduce a risk factor model for the regulatory capital ratio. Before that, a numerical example that illustrates the difference between cashflow hedging and hedging the regulatory capital ratio is given.

###### Example 2.3.

Consider a European bank with a US subsidiary. The total regulatory capital of this bank is €50 billion, and the regulatory risk capital is equal to €40 billion. This means that the European bank has a regulatory capital ratio of 80%. The US subsidiary has a regulatory capital position of USD15 billion and regulatory risk capital of USD7.5 billion. The regulatory capital of the US subsidiary is equal to the equity value of the US subsidiary. The expected future cashflows of the US subsidiary are equal to its equity value (USD15 billion). The €:USD exchange rate is equal to 1:1.5. Next, for this example, two hedging strategies are compared: first, the hedging strategy of minimizing the volatility of cashflows; and second, the hedging strategy of minimizing the volatility of the regulatory capital ratio.

If the European bank wants to protect its future cashflows against €:USD exchange rate movements, it has to sell USD15 billion and buy €10 billion. This hedge is different from the hedge where the European bank protects its regulatory capital ratio against €:USD exchange rate movements, which is derived below.

Since the regulatory capital ratio changes if either regulatory capital or regulatory risk capital changes, the European bank has to protect both against €:USD exchange rate movements if it wants to minimize the volatility of the regulatory capital ratio. Selling USD15 billion and buying €10 billion protects regulatory capital against exchange rate movements. In order to protect regulatory risk capital against €:USD exchange rate movements, the European bank has to buy USD9.375 billion (125% times USD7.5 billion of regulatory risk capital) and sell €6.25 billion. The net hedge is therefore that the European bank sells USD5.625 billion and buys €3.75 billion.^{6}^{6} 6 If the €:USD exchange rate moves to, for example, 1:1, the net hedging strategy bears a loss of €1.875 billion while the stake in the US subsidiary increases by €5 billion. Hence, in total, the regulatory capital position moves to €53.125 billion, thereby keeping the regulatory capital ratio constant at 80% as regulatory risk capital moves to €42.5 billion.

This example shows that a hedging strategy that minimizes the volatility of the regulatory capital ratio can be materially different from a hedging strategy that minimizes the volatility of cashflows.

### 2.2 A risk factor model for the regulatory capital ratio

Consider an exogenous marketable risk factor (${\mathrm{mr}}_{t}$) that the financial institution is exposed to and that is assumed to be lognormally distributed:

$$\frac{{\mathrm{dmr}}_{t}}{{\mathrm{mr}}_{t}}={\mu}^{\mathrm{mr}}\mathrm{d}t+{\sigma}^{\mathrm{mr}}\mathrm{d}{W}_{t}^{\mathrm{mr}},\text{where}{\mathrm{mr}}_{0}=m\in \mathbb{R}\text{isfixed}.$$ | (2.11) |

A marketable risk factor has an observable market price (eg, interest rates) and can thus be hedged.

Further, I define ${\mathrm{or}}_{t}$ as all the risk factors other than ${\mathrm{mr}}_{t}$ that a financial institution is exposed to:

$$\frac{{\mathrm{dor}}_{t}}{{\mathrm{or}}_{t}}={\mu}^{\mathrm{or}}\mathrm{d}t+{\sigma}^{\mathrm{or}}\mathrm{d}{W}_{t}^{\mathrm{or}},\text{where}{\mathrm{or}}_{0}=o\in \mathbb{R}\text{isfixed}.$$ | (2.12) |

As commonly used in standard financial literature (see, for example, Chen and Ingersoll 1983; Grinblatt and Titman 1983), a (linear) factor model to determine the contribution of ${\mathrm{mr}}_{t}$ to movements of ${F}_{t}$ is introduced:

$\mathrm{ln}({F}_{t})$ | $=\mathrm{ln}({F}_{t}^{\mathrm{mr}})+\mathrm{ln}({F}_{t}^{\mathrm{or}})+{\epsilon}_{t},$ | (2.13) |

where

$\frac{\mathrm{d}{F}_{t}^{\mathrm{mr}}}{{F}_{t}^{\mathrm{mr}}}$ | $={\mu}_{\mathrm{com}}^{\mathrm{mr}}\mathrm{d}t+{\sigma}_{\mathrm{com}}^{\mathrm{mr}}\mathrm{d}{W}_{t}^{\mathrm{mr}},\text{where}{F}_{0}^{\mathrm{mr}}={f}^{\mathrm{mr}}\in {\mathbb{R}}_{\ge 0}\text{isfixed},$ | (2.14) | ||

$\frac{\mathrm{d}{F}_{t}^{\mathrm{or}}}{{F}_{t}^{\mathrm{or}}}$ | $={\mu}_{\mathrm{com}}^{\mathrm{or}}\mathrm{d}t+{\sigma}_{\mathrm{com}}^{\mathrm{or}}\mathrm{d}{W}_{t}^{\mathrm{or}},\text{where}{F}_{0}^{\mathrm{or}}={f}^{\mathrm{or}}\in {\mathbb{R}}_{\ge 0}\text{isfixed}.$ | (2.15) |

Hence, ${\sigma}_{\mathrm{com}}^{\mathrm{mr}}$ is a (deterministic) function of ${\sigma}^{\mathrm{mr}}$ and ${\sigma}_{\mathrm{com}}^{\mathrm{or}}$ is a (deterministic) function of ${\sigma}^{\mathrm{or}}$. ${\epsilon}_{t}$ is a normally distributed random variable error term with mean zero, which is independent of ${F}_{t}^{\mathrm{mr}}$ and ${F}_{t}^{\mathrm{or}}$. $\mathrm{d}{F}_{t}^{\mathrm{mr}}$ can be interpreted as the movements of ${F}_{t}$ caused by movements in ${\mathrm{mr}}_{t}$ while keeping all other risk factors unchanged. ${F}_{t}^{\mathrm{mr}}$ is referred to as the marketable risk factor of the regulatory capital ratio. $\mathrm{d}{F}_{t}^{\mathrm{or}}$ can be interpreted as the movements of ${F}_{t}$ caused by all risk factors other than ${\mathrm{mr}}_{t}$.

Since the marketable risk factor ${\mathrm{mr}}_{t}$ can correlate with other risk factors (eg, interest rates and prepayments can correlate), ${F}_{t}^{\mathrm{mr}}$ and ${F}_{t}^{\mathrm{or}}$ can correlate. It is assumed that the correlation between ${F}_{t}^{\mathrm{mr}}$ and ${F}_{t}^{\mathrm{or}}$ is equal to ${\rho}_{F}^{\mathrm{mo}}$ in the sense that

$$E[\mathrm{d}{W}_{t}^{\mathrm{mr}}\mathrm{d}{W}_{t}^{\mathrm{or}}]={\rho}_{F}^{\mathrm{mo}}\mathrm{d}t.$$ |

It can be derived that the correlation, ${\rho}_{F}^{\mathrm{mr}}$, between ${F}_{t}^{\mathrm{mr}}$ and ${F}_{t}$ is equal to^{7}^{7} 7 Equation (2.16) can be derived by solving the following two equations in one unknown (${\rho}_{F}^{\mathrm{mr}}$): ${({\sigma}_{\mathrm{com}})}^{2}={({\sigma}_{\mathrm{com}}^{\mathrm{or}})}^{2}+{({\sigma}_{\mathrm{com}}^{\mathrm{mr}})}^{2}+2{\sigma}_{\mathrm{com}}^{\mathrm{or}}{\sigma}_{\mathrm{com}}^{\mathrm{mr}}{\rho}_{F}^{\mathrm{mo}},{({\sigma}_{\mathrm{com}}^{\mathrm{or}})}^{2}={({\sigma}_{\mathrm{com}})}^{2}+{({\sigma}_{\mathrm{com}}^{\mathrm{mr}})}^{2}-2{\sigma}_{\mathrm{com}}{\sigma}_{\mathrm{com}}^{\mathrm{mr}}{\rho}_{F}^{\mathrm{mr}}.$

$${\rho}_{F}^{\mathrm{mr}}=\frac{{\sigma}_{\mathrm{com}}^{\mathrm{mr}}+{\rho}_{F}^{\mathrm{mo}}{\sigma}_{\mathrm{com}}^{\mathrm{or}}}{{\sigma}_{\mathrm{com}}}.$$ | (2.16) |

Suppose further that the correlation between ${S}_{t}$ and ${F}_{t}^{\mathrm{mr}}$ is equal to ${\rho}_{F,S}^{\mathrm{mr}}$. More specifically, $E[\mathrm{d}{W}_{1,t}\mathrm{d}{W}_{t}^{\mathrm{mr}}]={\rho}_{F,S}^{\mathrm{mr}}\mathrm{d}t$.

### 2.3 Changing the exposure to a marketable risk factor

Since a marketable risk factor has an observable and tradeable market price, financial institutions can change their exposure to a marketable risk factor.

###### Assumption 2.4.

A financial institution can change its exposure to a marketable risk factor by entering into a costless contract.

Before formalizing the implications of a change in exposure to a marketable risk factor, I first give a real-life example.

###### Example 2.5.

Consider a financial institution with regulatory capital of $\mathrm{\$}$1 billion and regulatory risk capital of $\mathrm{\$}$1 billion. The financial institution is exposed to interest rate risk and to several other risk factors such as credit risk and equity risk. Interest rate risk is uncorrelated with the other risk factors. Regardless of the level of interest rate, a downward shift in the ten-year interest rate level of 0.01% causes regulatory capital to decline by $\mathrm{\$}$1 million. The regulatory risk capital for interest rate risk is determined by the change in regulatory capital for a 1% parallel shift in interest rates and is therefore equal to $\mathrm{\$}$100 million.

From the above, it is clear that a 1% shift in the ten-year interest rate causes the regulatory capital ratio to change by 10%. The financial institution can reduce the sensitivity of its regulatory capital ratio to movements in the ten-year interest rate by entering into an interest rate swap contract. For example, if the financial institution enters into an interest rate swap contract where the financial institution makes a $\mathrm{\$}$1 million profit if interest rates decline by 0.01% (and a $\mathrm{\$}$1 million loss if interest rates increase by 0.01%), the regulatory capital ratio is no longer sensitive to movements in interest rates. This swap contract also improves the regulatory capital ratio from 100% to 90%.

Since ${\sigma}_{\mathrm{com}}^{\mathrm{mr}}$ is a (deterministic) function of ${\sigma}^{\mathrm{mr}}$, there is an ${\alpha}_{t}\in \mathbb{R}$ such that a change in exposure to ${\mathrm{mr}}_{t}$ changes the volatility of ${F}_{t}^{\mathrm{mr}}$ – following the dynamics of (2.14) – and the volatility of ${S}_{t}$ to

${\sigma}_{\mathrm{com}}^{*,\mathrm{mr}}$ | $=\sqrt{{[(1+{\alpha}_{t}){\sigma}_{\mathrm{com}}^{\mathrm{mr}}]}^{2}}=|(1+{\alpha}_{t})|{\sigma}_{\mathrm{com}}^{\mathrm{mr}}$ | (2.17) | ||

and | ||||

${\sigma}^{*}$ | $=\sqrt{{\sigma}^{2}+{\alpha}_{t}^{2}{[{\sigma}_{\mathrm{com}}^{\mathrm{mr}}]}^{2}+{\rho}_{F,S}^{\mathrm{mr}}\sigma {\alpha}_{t}{\sigma}_{\mathrm{com}}^{\mathrm{mr}}}.$ | (2.18) |

A change in exposure to the marketable risk factor ${\mathrm{mr}}_{t}$ can change the current value of ${R}_{t}$. A reduction in ${R}_{t}$ leads to a reduction in ${F}_{t}$, and it therefore leads to a reduction in the expected cost of a regulatory capital shortfall. However, a reduction in ${F}_{t}$ results in an opportunity cost since there is a cost associated with holding capital (Perold 2005). The institution can reduce this opportunity cost to zero by increasing risk elsewhere (or by paying out a dividend) so that the current value of ${F}_{t}$ is retained. I assume that a financial institution retains the current value of ${F}_{t}$ without raising or reducing capital. This assumption allows me to derive the hedging incentives for financial institutions for a given regulatory capital ratio.

###### Assumption 2.6.

If a financial institution changes its exposure to a marketable risk factor, ${\mathrm{mr}}_{t}$, it ensures that the current value of ${F}_{t}$ is retained without raising or reducing capital by changing the exposure to the other risks (${\mathrm{or}}_{t}$) proportionately so that ${\sigma}_{\mathrm{com}}^{\mathrm{or}}$ remains unchanged.

The numerical example in Section 5 illustrates how Assumption 2.6 works in practice. Assumption 2.6 is in the spirit of the capital buffer theory and is consistent with the common practice of financial institutions to target a specific regulatory capital ratio.^{8}^{8} 8 For example, on page 19 of its 2016 annual report, Barclays PLC states: “On this basis we currently expect our end-state CET1 ratio to be in a range of 12.3–12.8% and we remain confident in our capital trajectory.” In its 2016 annual report, JP Morgan Chase states: “The Firm continues to believe that over the next several years, it will operate with a Basel III CET1 capital ratio between 11% and 12.5%.” Empirical evidence from Heid et al (2004) and Jokipii and Milne (2011) supports the notion that banks hold a capital buffer in excess of the minimum capital requirement.

For the purposes of this paper, risk-weighted assets and theoretically correct risk weights are defined as follows. The definition for the theoretically correct risk weights is in the spirit of Rochet (1992).

###### Definition 2.7.

The risk-weighted assets of a financial institution are calculated as the sum of the market value of an asset multiplied by its regulatory determined risk weight. Regulatory risk capital is equal to the risk-weighted assets. Theoretically correct risk weights apply weights proportional to the systematic risk of an asset (systematic risk is measured by the market beta of an asset).

I also define asset substitution incentives.

###### Definition 2.8.

A financial institution has an asset substitution incentive if it can change its exposure to a marketable risk factor ${\mathrm{mr}}_{t}$ in such a way that both its current market value of equity and its risk profile increase. An increase in risk profile is defined as an increase in the absolute value of the covariance of the market value, ${S}_{t}$, with the marketable risk factor ${\mathrm{mr}}_{t}$.

Mathematically, a financial institution has an asset substitution incentive if it can change its exposure to a marketable risk factor ${\mathrm{mr}}_{t}$ such that the new financial institution with stock price ${S}_{t}^{*}$ has the following properties:

${S}_{t}^{*}$ | $>{S}_{t},$ | ||

${\sigma}^{*}{\sigma}^{\mathrm{mr}}|{\rho}_{\mathrm{mr},{S}^{*}}|$ | $>\sigma {\sigma}^{\mathrm{mr}}|{\rho}_{\mathrm{mr},S}|,$ |

where ${\sigma}^{*}$ is the volatility of the value of the new financial institution and ${\rho}_{\mathrm{mr},{S}^{*}}$ is the correlation between the marketable risk factor and the value of the new financial institution.

The intuition behind Definition 2.8 is that an increase in risk profile only leads to an increase in the market value of equity if the financing rate (eg, the deposit rate for banks) does not increase proportionately to the increase in risk profile. Hence, a simultaneous increase in risk profile and the market value of equity implies a shift of value from the depositors or the policyholders to the stockholders.

The market value of a financial institution is modeled in (2.1) and includes the expected cost of a regulatory capital shortfall (modeled in (2.10)). This means that the market value excluding the expected cost of a regulatory shortfall, ${S}_{t}^{\prime}$, is equal to

$${S}_{t}^{\prime}={S}_{t}-{A}_{t}c[{F}_{t}N({d}_{1})-{\mathrm{e}}^{-r}N({d}_{2})].$$ | (2.19) |

${S}_{t}^{\prime}$ can therefore also be interpreted as the value of the financial institution in the fictitious case where the adjustment cost of equity is zero.

In order to distinguish hedging from regulatory arbitrage, I assume that, for a given regulatory capital ratio (${F}_{t}$), the current values of ${S}_{t}^{\prime}$ and ${A}_{t}$ are not affected by a change in exposure to the marketable risk factor ${\mathrm{mr}}_{t}$. This implies that, for a given regulatory capital ratio, asset substitution does not increase the value of stock excluding the expected cost of a regulatory capital shortfall. This assumption holds if capital requirements are proportional to the systematic risks (the betas) of the assets, as in this case financial institutions with the same regulatory capital ratio have the same beta.

###### Assumption 2.9.

The current values of ${S}_{t}^{\mathrm{\prime}}$ and ${A}_{t}$ are not affected by a change in exposure to the marketable risk factor ${\mathrm{mr}}_{t}$ or other risk factor ${\mathrm{or}}_{t}$ for a given regulatory capital ratio.

## 3 Hedging at financial institutions in the absence of asset substitution incentives for financial institutions

In this section, the hedging incentives for financial institutions are derived assuming an absence of asset substitution incentives for financial institutions. In the next section, we investigate what hedging strategy maximizes value for financial institutions taking into account the asset substitution incentives for financial institutions.

In the absence of asset substitution incentives for financial institutions, hedging for financial institutions is defined as follows (adapted from Smith and Stulz (1985) and in the spirit of Modigliani and Miller (1958)).

###### Definition 3.1.

Consider two institutions, $\mathrm{A}$ and $\mathrm{B}$, that differ from the institution with value ${S}_{t}$ only in their hedging policies. Institution $\mathrm{A}$ hedges more with respect to the marketable risk factor ${\mathrm{mr}}_{t}$ than institution $\mathrm{B}$ if the absolute value of the covariance of the market value of institution $\mathrm{A}$ with the marketable risk factor ${\mathrm{mr}}_{t}$ is less than that of institution $\mathrm{B}$.

Mathematically, if $\mathrm{A}$ and $\mathrm{B}$ differ only in their hedging policies, $\mathrm{A}$ hedges more with respect to the marketable risk factor ${\mathrm{mr}}_{t}$ than institution $\mathrm{B}$ if

${\sigma}^{\mathrm{A}}{\sigma}^{\mathrm{mr}}|{\rho}_{\mathrm{mr},{S}^{\mathrm{A}}}|$ | $$ |

where ${\sigma}^{\mathrm{A}}$ and ${\sigma}^{\mathrm{B}}$ are the volatilities of the value of institution $\mathrm{A}$ and institution $\mathrm{B}$, respectively; ${\sigma}^{\mathrm{mr}}$ is the volatility of the marketable risk factor; and ${\rho}_{\mathrm{mr},{S}^{\mathrm{A}}}$ and ${\rho}_{\mathrm{mr},{S}^{\mathrm{B}}}$ are the correlations between the marketable risk factor and the values of institution $\mathrm{A}$ and institution $\mathrm{B}$, respectively.

In order to formally derive the hedging incentives for financial institutions, I first define hedging incentives.

###### Definition 3.2.

In the absence of asset substitution incentives for financial institutions, a financial institution has an incentive to hedge marketable risk factor ${\mathrm{mr}}_{t}$ if there is a (hedging) strategy such that its current market value of equity increases and its risk profile decreases.

Mathematically, in the absence of asset substitution incentives for financial institutions, a financial institution has a hedging incentive if it can change its exposure to a marketable risk factor ${\mathrm{mr}}_{t}$ in such a way that the new financial institution with stock price ${S}_{t}^{*}$ has the following properties:

${S}_{t}^{*}$ | $>{S}_{t},$ | ||

${\sigma}^{*}{\sigma}^{\mathrm{mr}}|{\rho}_{\mathrm{mr},{S}^{*}}|$ | $$ |

where ${\sigma}^{*}$ is the volatility of the value of the new financial institution and ${\rho}_{\mathrm{mr},{S}^{*}}$ is the correlation between the marketable risk factor and the value of the new financial institution.

Since, in practice, financial institutions have asset substitution incentives, Definition 3.2 is a theoretical definition. In practice, financial institutions are therefore less concerned with the second property, as an increase in risk profile results in a shift of value from depositors or policyholders to stockholders (see Section 4).

The next proposition posits that a financial institution can increase its market value if it reduces the volatility of the regulatory capital ratio.

###### Proposition 3.3.

###### Proof.

See the online appendix. ∎

In the absence of asset substitution incentives, reducing the volatility of the regulatory capital ratio only qualifies as a hedging strategy if the second property of Definition 3.2 is also satisfied: namely, that the absolute covariance of the market value of the institution with the marketable risk factor is smaller than the absolute covariance without hedging. The following proposition gives, in the absence of asset substitution incentives, the conditions under which a financial institution has an incentive to hedge a marketable risk factor.

###### Proposition 3.4.

In the absence of asset substitution incentives and if Assumptions 2.1–2.9 apply, for a given regulatory capital ratio a financial institution has an incentive to hedge if and only if the correlations of ${F}_{t}^{\mathrm{mr}}$ with both ${F}_{t}$ and ${S}_{t}$ have the same sign. In this case, there is a strategy that increases the value of the institution while lowering the risk profile of the institution.

###### Proof.

See the online appendix. ∎

The intuition behind Proposition 3.4 is as follows. Definition 3.2 specifies that a hedging strategy has to increase the market value of the institution while lowering the absolute value of the covariance of the market value, ${S}_{t}$, with the marketable risk factor ${\mathrm{mr}}_{t}$. Proposition 3.3 shows that a hedging strategy increases the market value of the financial institution if it reduces the volatility of the regulatory capital ratio. The only way for this hedging strategy to also reduce the absolute covariance of the market value, ${S}_{t}$, with the marketable risk factor ${\mathrm{mr}}_{t}$ is if the correlations of ${F}_{t}^{\mathrm{mr}}$ with both ${F}_{t}$ and ${S}_{t}$ have the same sign.

The next proposition shows that (in the absence of asset substitution incentives) if a financial institution has an incentive to hedge, there is one value-maximizing hedging strategy. This value-maximizing hedging strategy minimizes the volatility of the regulatory capital ratio under the constraint that the volatility of the value of the institution is also reduced. The next section discusses the fact that, given that financial institutions have asset substitution incentives, a value-maximizing hedging strategy for a marketable risk factor simply minimizes the volatility of the regulatory capital ratio. Before stating the next proposition, I first define the magnitude of a hedging strategy.

###### Definition 3.5.

###### Proposition 3.6.

In the absence of asset substitution incentives and if Assumptions 2.1–2.9 apply, if a financial institution has an incentive to hedge, there is one hedging strategy that maximizes the value of the institution for a given regulatory capital ratio. This hedging strategy minimizes the volatility of the regulatory capital ratio under the constraint that the volatility of the value of the institution is also reduced. The magnitude of this value-maximizing hedging strategy is equal to

${\alpha}_{t}=\mathrm{max}[-{\displaystyle \frac{2{\rho}_{F,S}^{\mathrm{mr}}\sigma}{{\sigma}_{\mathrm{com}}^{\mathrm{mr}}}},-{\displaystyle \frac{({\rho}_{F}^{\mathrm{mo}}{\sigma}_{\mathrm{com}}^{\mathrm{or}}+{\sigma}_{\mathrm{com}}^{\mathrm{mr}})}{{\sigma}_{\mathrm{com}}^{\mathrm{mr}}}}]\mathit{\hspace{1em}}\mathit{\text{if}}{\rho}_{F,S}^{\mathrm{mr}},{\rho}_{F}^{\mathrm{mr}}0,$ | (3.1) | ||

$$ | (3.2) |

###### Proof.

See the online appendix. ∎

## 4 Value-maximizing hedging strategies and asset substitution eliminating capital regulation

### 4.1 Value-maximizing hedging strategies at financial institutions

Recall Definition 2.8. Propositions 3.3, 3.4 and 3.6 assume that financial institutions do not have asset substitution incentives. Jensen and Meckling (1976) show, however, that leveraged firms have an incentive to shift value from debtholders to stockholders by investing in risky projects with a negative net present value. Moreover, deposit insurance causes a moral hazard problem in banking, resulting in asset substitution incentives for banks. (Life) insurance companies also have asset substitution incentives due to the long duration of (life) insurance policies.

If asset substitution incentives exist, the second property of Definition 3.2 is no longer relevant for financial institutions. Hence, if asset substitution incentives are taken into account, hedging at financial institutions, for a given regulatory capital ratio, focuses only on maximizing the current market value. Assumption 2.9 then implies that a value-maximizing hedging strategy for financial institutions minimizes the expected cost of a regulatory capital shortfall for a given regulatory capital ratio.

Thus, in view of (3.1) and (3.2), if asset substitution incentives exist, the magnitude (${\alpha}_{t}$) of a value-maximizing hedging strategy for financial institutions is equal to

$${\alpha}_{t}=-\frac{({\rho}_{F}^{\mathrm{mo}}{\sigma}_{\mathrm{com}}^{\mathrm{or}}+{\sigma}_{\mathrm{com}}^{\mathrm{mr}})}{{\sigma}_{\mathrm{com}}^{\mathrm{mr}}}.$$ | (4.1) |

If the correlation between the marketable risk factor and the other risk factors is equal to zero, a value-maximizing hedging strategy exactly immunizes the regulatory capital ratio against the marketable risk factor (ie, ${\alpha}_{t}=-1$). However, if the correlation between the marketable risk factor and the other risk factors is not equal to zero, a value-maximizing hedging strategy does not exactly immunize the regulatory capital ratio against the marketable risk factor (ie, ${\alpha}_{t}$ is different to $-1$). The reason for this is that the marketable risk factor diversifies against other risk factors. Hence, exactly immunizing the regulatory capital ratio against this marketable risk factor does not minimize the expected cost of a regulatory capital shortfall if the correlation between the marketable risk factor and the other risk factors is not zero.

In practice, diversification benefits are difficult to establish. Therefore, it is sensible to treat diversification and hedging strategies separately. In this case, a financial institution first determines in which areas it has a comparative advantage. In these areas, financial institutions want to have exposure and therefore do not hedge. In the areas where a financial institution does not have a comparative advantage, it can exactly immunize the regulatory capital ratio against the risk factors associated with these areas.

### 4.2 Asset substitution eliminating capital regulation

###### Proposition 4.1.

###### Proof.

See the online appendix. ∎

Proposition 4.1 implies that theoretically correct risk weights are not sufficient to eliminate asset substitution incentives for financial institutions. If risk weights are theoretically correct, a perfect correlation between the regulatory capital ratio and the market value of equity of a financial institution is only achieved if the regulatory definition of capital is also equal to the market value of equity and if the minimum capital requirements are based on constant theoretically correct risk weights (eg, “through-the-cycle” asset betas).

### 4.3 Derisking incentives and macroprudential regulation

The model in this paper can easily be extended to show that financial institutions have an incentive to derisk if they suffer losses, even if they continue to comply with the minimum capital requirements after the losses. Since banks are highly leveraged, if a bank suffers a loss, it needs to reduce its loan portfolio with a multiple of this loss in order to bring the regulatory capital ratio back to its original level. A capital conservation buffer allows the regulator to reduce the minimum capital requirement if banks suffer losses, thereby reducing banks’ incentives to derisk and thus reducing the potential negative knock-on effects on the real economy associated with bank derisking. In the online appendix, it is demonstrated that the model in this paper is also consistent with the introduction of a capital conservation buffer in the Basel III regulatory framework. This complements the results of Di Iasio (2013) and Repullo (2013) on macroprudential regulation.

## 5 A numerical example

The theoretical result of Proposition 3.6 and Section 4.1 is now illustrated with a numerical example.

Consider bank X with $\mathrm{\$}$10 million of capital and $\mathrm{\$}$90 million in deposits. Bank X therefore has $\mathrm{\$}$100 million of assets, of which $\mathrm{\$}$90 million are mortgages and $\mathrm{\$}$10 million is cash. The regulatory risk capital for credit risk is $\mathrm{\$}$4.5 million and the regulatory risk capital for interest rate risk is $\mathrm{\$}$0.5 million, so the regulatory capital ratio ${F}_{t}=0.5$ with volatility ${\sigma}_{\mathrm{com}}$. ${F}_{t}^{\mathrm{mr}}$ is the interest rate risk factor of the regulatory capital ratio and is equal to 1 with volatility ${\sigma}_{\mathrm{com}}^{\mathrm{mr}}$. Since regulatory risk capital for interest rate risk is equal to $\mathrm{\$}$0.5 million, ${F}_{t}^{\mathrm{mr}}=1$ implies that the regulatory capital for interest rate risk is $\mathrm{\$}$0.5 million. ${F}_{t}^{\mathrm{or}}$ is the credit risk factor of the regulatory capital ratio with a volatility ${\sigma}_{\mathrm{com}}^{\mathrm{or}}$. Since

$\mathrm{ln}({F}_{t})$ | $=\mathrm{ln}({F}_{t}^{\mathrm{mr}})+\mathrm{ln}({F}_{t}^{\mathrm{or}}),$ | (5.1) |

${F}_{t}^{\mathrm{or}}=0.5$. The correlation between interest rate risk and credit risk is zero, and the correlation between ${F}_{t}^{\mathrm{mr}}$ and ${F}_{t}^{\mathrm{or}}$ is also zero. If the ten-year interest rate declines by 1%, bank X loses $\mathrm{\$}$0.1 million, which translates into regulatory risk capital of $\mathrm{\$}$0.5 million. Bank X has no comparative advantage in trading interest rates and therefore wants to hedge its interest rate exposure while keeping its regulatory capital ratio unchanged. Following (4.1), bank X has an incentive to cancel out its interest rate risk. It can do so by entering into an interest rate swap contract where it pays the floating interest rate and receives the ten-year interest rate, where the notional is such that the market value of the swap increases by $\mathrm{\$}$0.1 million if interest rates decline by 1%. This transaction reduces the regulatory risk capital of bank X by $\mathrm{\$}$0.5 million. In order to keep its regulatory capital ratio unchanged, bank X therefore needs to increase its exposure to credit risk by investing the $\mathrm{\$}$10 million cash in mortgages, after which ${F}_{t}^{\mathrm{or}}={F}_{t}$ and ${\sigma}_{\mathrm{com}}^{\mathrm{or}}={\sigma}_{\mathrm{com}}$.

## 6 Conclusion

This paper uses option pricing techniques to derive the expected cost of a regulatory capital shortfall. Because financial institutions need to manage this expected cost, this has implications for hedging by, and capital regulation of, financial institutions. In a simple model, two results that provide guiding principles for hedging by, and capital regulation of, financial institutions are derived.

First, value-maximizing hedging strategies by financial institutions minimize the volatility of the regulatory capital ratio. A nonfinancial firm has an incentive to hedge if there is a strategy that increases its current market value and reduces its risk profile. Because of deposit insurance, depositors have less incentive to monitor banks. Therefore, deposit insurance causes an asset substitution problem in banking. Further, (life) insurance companies also have asset substitution incentives because of the long duration of their insurance policies. These asset substitution incentives encourage financial institutions to adopt hedging strategies that maximize the current market value of the institution for a given regulatory capital ratio regardless of the change in systematic risk. Financial institutions can maximize their current market value for a given regulatory capital ratio through minimizing the expected cost of a regulatory capital shortfall. In turn, this can be achieved by minimizing the volatility of the regulatory capital ratio. This result contrasts with the result that nonfinancial firms have an incentive to minimize the volatility of cashflows (Froot et al 1993). However, unlike the case for nonfinancial firms, cashflows are rarely a binding constraint for financial institutions, while minimum capital requirements are.

Second, in order to eliminate asset substitution incentives, capital regulation should be designed in such a way that the regulatory capital ratio correlates perfectly with the market value of equity of a financial institution. A practical way to achieve this is to equate regulatory capital with the market value of equity and to base minimum capital requirements on constant theoretically correct risk weights (eg, based on through-the-cycle asset betas).

It is hoped that this paper provides a more fundamental understanding of hedging by financial institutions. Moreover, policy implications for capital regulation are derived. There are several interesting areas for further research. The differences in hedging and risk management approaches between cooperative banks and commercial banks is one such area. The results of this paper flow from an adjustment cost of equity. Because adjustment costs of equity do not apply, or apply differently, to cooperative banks, one could find different results for hedging and risk management at cooperative banks. Altunbas et al (2007) study the relationship between capital and risk and find that this relationship differs between commercial and cooperative banks. Another interesting area for further research is how changes in capital regulation affect hedging at financial institutions. For example, in 2012 the Netherlands and Denmark introduced a non-market-consistent interest rate curve with which insurance companies can discount their liabilities. An interesting research question is whether the introduction of this non-market-consistent interest rate curve changed the hedging behavior of insurance companies in these countries.

## Declaration of interest

The views expressed in this paper are those of the author and do not necessarily reflect the position of the Dutch Authority for the Financial Markets. The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.

## Acknowledgements

The author thanks Ronald Bosman, Ruben Cox, Mark Mink, Bas Rooijmans and Edo Schets for useful comments, and is especially grateful to Roger Laeven and an anonymous referee for insightful and useful comments.

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