When enough is not enough: bank capital and the Too-Big-To-Fail subsidy

Abstract

This paper takes a unique approach to study the relationship between bank capital and Too-Big-To-Fail (TBTF) during the Financial Crisis. A structural credit risk model is used to compute implied market value capital ratios which, when compared to traditional risk-based capital, illustrates the capital deficiency of large BHCs. As these BHCs’ implied capital deteriorated, their default probabilities spiked. The model is then used to solve for the amount of capital needed to reduce default probabilities. This amount is compared to the TARP capital infusions to quantify the TBTF subsidy which is associated with size and reliance on short-term volatile funding.

Appendix: Valuation of corporate securities in the compound option model

To value the securities within the compound option framework it is helpful to visualize the default process as a binomial tree. In Fig. 1 the solid nodes represent survival states, whereas the dashed nodes represent default states. The accompanying table shows the value of the corporate securities (equity, junior debt, and senior debt) at each node. Solving the expectations at node A results in the closed-form valuation equations below.

The value of the securities can be determined at node A (time $t=0$) by taking the discounted expected value of the future cash flows at each of the subsequent nodes. For equity and junior debt this results in nested expectations. Consequently, the assumption that Geometric Brownian Motion is the underlying stochastic process ensures that the solution is a function of the bivariate normal distribution, which is very manageable.

Now, for any time t, $0\le t<{{T}_{1}}$, evaluating the expectations at t gives the closed-form expressions for valuing all of the bank’s claims. Equity is a compound call on the bank’s assets in that shareholders have a claim to

$$\begin{aligned} \mathop {{\mathbb {E}}}\left[ {{e}^{-r{{\tau }_{1}}}}\,\max \left\{ {{E}_{{{T}_{1}}}}-{{F}_{1}},0 \right\} \left| {{V}_{t}} \right. \right] , \end{aligned}$$

(A.1)

where ${{E}_{{{T}_{1}}}}$ is equal to the value of a European call option at time ${T}_{1}$, which is not known at time t. Let ${{\tau }_{1}}=\left( {{T}_{1}}-t \right)$ be the time between t and the short-term senior debt maturity $T_{1}$ and${{\tau }_{2}}=\left( {{T}_{2}}-t \right)$ be the time between t and the long-term junior debt maturity $T_{2}$. The expectation given in Eq. (A.1) is equal to:³⁶

$$\begin{aligned} {{E}_{t}}={{V}_{t}}\,{{N}_{2}}\left( h_{1}^{+},h_{2}^{+};\,\rho \right) -{{F}_{2}}{{e}^{-r{{\tau }_{2}}}}\,{{N}_{2}}\left( h_{1}^{-},h_{2}^{-};\,\rho \right) -{{F}_{1}}{{e}^{-r{{\tau }_{1}}}}N\left( h_{1}^{-} \right) \end{aligned}$$

(A.2)

where $h_{1}^{+}=\frac{\ln \left( \frac{{{V}_{t}}}{{{\bar{V}}}} \right) +\left( r+\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{1}}}{\sigma \sqrt{{{\tau }_{1}}}}$,

$h_{1}^{-}=\frac{\ln \left( \frac{{{V}_{t}}}{{{\bar{V}}}} \right) +\left( r-\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{1}}}{\sigma \sqrt{{{\tau }_{1}}}}=h_{1}^{+}-\sigma \sqrt{{{\tau }_{1}}}$,

$h_{2}^{+}=\frac{\ln \left( \frac{{{V}_{t}}}{{{F}_{2}}} \right) +\left( r+\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{2}}}{\sigma \sqrt{{{\tau }_{2}}}}$,

$h_{2}^{-}=\frac{\ln \left( \frac{{{V}_{t}}}{{{F}_{2}}} \right) +\left( r-\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{2}}}{\sigma \sqrt{{{\tau }_{2}}}}=h_{2}^{+}-\sigma \sqrt{{{\tau }_{2}}}$, $N\left( \centerdot \right)$ denotes the cumulative standard normal distribution,

${{N}_{2}}\left( \centerdot \right)$ denotes the cumulative bivariate standard normal distribution, and correlation $\rho =\sqrt{{}^{{{\tau }_{1}}}\!\!\diagup \!\!{}_{{{\tau }_{2}}}\;}$which follows from the properties of Brownian Motion.

${\bar{V}}$ is the endogenous default boundary, derived in Sect. 3.4.

The senior debt is to be paid at time $T_{1}$. If the market value of the assets at that time is greater than ${\bar{V}}$, then senior debt holders will be paid the amount $F_{1}$ in full. However, if the market value of the assets is less than or equal to ${\bar{V}}$ and less than the face value of the senior debt, then there is [total] default at time $T_{1}$. In this case, the assets will be liquidated and senior debt holders will only recover a portion of what they are owed; specifically they will recover the market value of the assets at the time of default, $T_{1}$. Thus, the value of the short-term senior debt at time t is given by the expectation

$$\begin{aligned} \mathop {{\mathbb {E}}}\left[ {{e}^{-r{{\tau }_{1}}}}\,\,\min \left\{ {{V}_{{{T}_{1}}}},{{F}_{1}} \right\} \left| {{V}_{t}} \right. \right] . \end{aligned}$$

(A.3)

It is relatively straightforward to show that the expectation given in Eq. (A.3) is equal to

$$\begin{aligned} {{S}_{t}}={{V}_{t}}\left[ 1-N\left( {{k}^{+}} \right) \right] +{{F}_{1}}\,{{e}^{-r{{\tau }_{1}}}}N\left( {{k}^{-}} \right) , \end{aligned}$$

(A.4)

where ${{k}^{+}}=\frac{\ln \left( \frac{{{V}_{t}}}{{{F}_{1}}} \right) +\left( r+\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{1}}}{\sigma \sqrt{{{\tau }_{1}}}}$ and ${{k}^{-}}=\frac{\ln \left( \frac{{{V}_{t}}}{{{F}_{1}}} \right) +\left( r-\frac{{{\sigma }^{2}}}{2} \right) {{\tau }_{1}}}{\sigma \sqrt{{{\tau }_{1}}}}={{k}^{+}}-\sigma \sqrt{{{\tau }_{1}}}$.

Then, finally, there is the bank’s junior debt which has the interesting property that when asset values are well above the default boundary they are a fixed claim and their prices behave like debt, but in times of distress they are a residual claim and their prices behave like equity. The model captures this characteristic of bank subordinated debentures well. The value of the long-term junior debt at time t is given by the expectation:

$$\begin{aligned} {\mathbb {E}}\left[ {{e}^{-r{{\tau }_{1}}}}\left( \left( {J}_{{T}_{1}}\,\cdot {\mathbb {I}}_{\left\{ {V}_{{T}_{1}}>{\bar{V}} \right\} } \right) +\left( \max \left\{ {V}_{{T}_{1}}-{{F}_{1}},0 \right\} \cdot {\mathbb {I}}_{\left\{ {V}_{{T}_{1}}\le {\bar{V}} \right\} } \right) \right) \left| {V}_{t} \right. \right] \end{aligned}$$

(A.5)

where ${\mathbb {I}}_{\left\{ V>{\bar{V}} \right\} }$ is the indicator function that equals 1 if $V>{\bar{V}}$ and 0 otherwise.

To obtain the closed-form solution for the long-term junior debt the expectation given in Eq. (A.5) can be evaluated directly or, since the expectation for short-term senior debt is considerably more straightforward, the fact that ${{V}_{t}}={{E}_{t}}+{{J}_{t}}+{{S}_{t}}$ can be used to solve for ${{J}_{t}}={{V}_{t}}-{{E}_{t}}-{{S}_{t}}$. Subtracting Eqs. (A.2) and (A.4) from $V_{t}$ gives:

$$\begin{aligned} {{J}_{t}}={{V}_{t}}\left[ N\left( {{k}^{+}}\right) -{{N}_{2}}\left( h_{1}^{+},\,h_{2}^{+};\,\rho \right) \right] +{{F}_{1}}{{e}^{-r{{\tau }_{1}}}}\left[ N\left( h_{1}^{-}\right) N\left( {{k}^{-}}\right) \right] +{{F}_{2}}{{e}^{-r{{\tau }_{2}}}}{{N}_{2}}\left( h_{1}^{-},h_{2}^{-};\,\rho \right) \end{aligned}$$

(A.6)

where all of the terms are as previously defined in Eqs. (A.2) and (A.4).

Although asset volatility is assumed to be constant, equity volatility is not constant but rather changes as a function of asset value and other parameters in the model. This was first shown by geske-1979. Equity, being a compound option on the underlying asset, has its own stochastic dynamics which can be specified using Ito’s Lemma. From the Ito expansion, the volatility term for equity is

$$\begin{aligned} {{\sigma }_{E}}=\frac{\partial E}{\partial V}\,\frac{V}{E}{{\sigma }_{V}}. \end{aligned}$$

(A.7)

The partial derivative $\frac{\partial E}{\partial V}$ can be found by differentiating Eq. (A.2) with respect to asset value and is known as the “equity delta”:

$$\begin{aligned} \frac{\partial E}{\partial V}={{N}_{2}}\left( h_{1}^{+},h_{2}^{+};\,\rho \right) . \end{aligned}$$

(A.8)

Plugging Eq. (A.8) into Eq. (A.7) gives

$$\begin{aligned} {{\sigma }_{E}}={{N}_{2}}\left( h_{1}^{+},h_{2}^{+};\,\rho \right) \,\frac{V}{E}{{\sigma }_{V}}. \end{aligned}$$

(A.9)

This equation says that, holding everything else constant, as more leverage is introduced the equity volatility becomes greater than the asset volatility. This leverage effect is consistent with standard corporate finance theory which says that leverage increases the riskiness of a firm’s equity as financial risk is compounded on top of the inherent total business risk (as proxied by ${{\sigma }_{V}}$).