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

http://​www.​fdic.​gov/​news/​news/​press/​2008/​pr08100a.​html.

Since the risk-based capital ratios essentially use book values with re-weightings, this is analogous to when a stock price book-to-market ratio goes above 1 for a distressed firm (see Fama and French 1995; Griffin and Lemmon 2002).

See Basel Committee on Banking Supervision (2013) and Basel Committee on Banking Supervision (2014), respectively.

See, e.g., Bernanke (2009), Tarullo (2009) and Hanson et al. (2011). In fact, this is also addressed in Section 165 of the Dodd-Frank Act (US House of Representatives 2010).

For more detailed information on TARP and its constituent programs such as the CPP and TIP please see Wilson and Wu (2012), Egly and Mollick (2013), and Calomiris and Khan (2015).

See Merton (1977), Marcus and Shaked (1984), Ronn and Verma (1986, 1989), Pennacchi (1987a, b), Allen and Saunders (1993), and Cooperstein et al. (1995).

More precisely, they use a variation of the Ronn and Verma (1986) model where the strike price is a fraction of the outstanding liabilities’ book value.

I thank the referee for pointing out this connection.

If the model was indeed applied in an ex-ante capacity—for instance examining “what if” scenarios - it could easily be used for stress testing. For example, if regulators wanted to examine how much capital a bank would need if default probabilities were to drop to 5% or if asset values were to fall by 25%, the model could provide considerable insight despite that not being the focus of the present paper. Note that this would be slightly different from the macroprudential philosophy currently in favor, which examines the bank capital implications of factors such as interest rates, productivity growth, or unemployment rates. See Grundke and Pliszka (2018) for a proposed framework similar to the methodology in the present paper but from a macroeconomic perspective.

Interestingly, but only tangentially related to this paper, Wilson (2013) provides evidence of some negative outcomes of TARP for small banks. In that paper, the author documents the existence of “deadbeat banks” which are banks who received TARP support through the CPP but failed to pay dividends on those preferred shares.

E.g. Delianedis and Geske (2003), Leland (2004), Bharath and Shumway (2008).

He and Xiong (2012) present a very nice extension to the Leland and Toft (1996) model which accounts for rollover risk, or the risk that liquidity dries up in debt markets and the firm will not be able to roll-over their maturing debt as is assumed in the Leland (1994) and Leland and Toft (1996) framework. Interestingly, in the earlier unpublished working paper version, He and Xiong did extend the Leland and Toft (1996) model to include two classes of debt.

A referee pointed out that, since the TARP infusions were in the form of preferred stock, a structural credit risk model that differentiates preferred from common equity would be more ideal. The standard Geske compound option model discussed in this section portrays both preferred and common equity as the residual claimant. A detailed search of the structural credit risk literature was able to identify one paper, Finnerty (2008), that has nested equity claims similar to what the referee suggested. To incorporate such features into the current analysis is beyond the scope of the present paper and therefore is left to future research.

Adrian and Shin (2010) show that financial institutions tend to actively manage their leverage ratios in a procyclical manner.

The author thanks Mark Flannery for pointing out the fact that the potential for future capital infusions is, in itself, a compound option that is being evaluated by the market.

An interesting paper by Stanhouse and Stock (2016) also uses analytical techniques and stochastic mathematics to study bank capital. However, in their model, both the timing and the level of bank capital are endogenously solved within the context of a Brownian Motion process.

See pp. 551–552 in Geske (1977) for how debt subordination is treated in the compound option structural model although the recovery conditions were clarified and corrected in Geske and Johnson (1984)

The true market value of any bank’s assets is unobservable due to the opacity discussed in Flannery et al. (2004) and Flannery et al. (2013) where the latter article studies opaqueness during the financial crisis. The market values of the bank’s debt securities are either unobservable or hard to observe due to illiquidity. Therefore, equity is used to extract information about the unobservable values.

This is the same as the default condition in the Leland–Toft model; see Leland and Toft 1996, p. 994.

The derivations are available from the author upon request.

At this point, the model would collapse to the standard BSM structural model as described earlier in the Section.

For a derivation please see Geske 1977, 1979; Geske and Johnson 1984.

The model is used to solve for a market-implied asset value which could be used in the denominator of the MVC ratio, but the variations over time are almost identical so the simpler MVC ratio is used in the empirical analysis in this paper. It also reduces the potential impact for model error.

E.g. 60 day at-the-money puts, 30 day at-the-money calls, etc.

In symbols: \(\sigma _{E}=\frac{sd}{\sqrt{{1}\diagup {252}}}=sd\sqrt{252}\), where sd is the standard deviation of \(\ln \frac{{{P}_{t}}}{{{P}_{t-1}}}\) for each day over the previous three months. Three months is used to smooth out the noise that results from using a shorter window, such as one month.

Empirical evidence of this can be found in Fan et al. (2016) where the authors study the difference between the ex-post realized volatility and option implied volatility, referred to as the volatility risk premium, and it can clearly be seen that the gap is most pronounced through the crisis period.

Robustness checks were performed on the values for both \(T_{1}\) and \(T_{2}\), separately and together. It turns out that the results are insensitive to the value chosen for \(T_{2}\). However, the results are rather sensitive to the value chosen for \(T_{1}\), although the rank ordering of the capital needs remains the same. For instance, reducing the value of \(T_{1}\) to 0.5 (implying an average maturity of all short-term senior debt to be 6 months) causes default probabilities and the capital needs to fall to insignificant levels; but increasing \(T_{1}\) to 2 sometimes more than doubles the amount of capital needed to reduce default probabilities. To remain consistent with the most common KMV-type implementations in the literature as well as the traditional definition for “short-term liabilities” being those maturing within one year, results are reported and interpreted for \(T_{1}=1\).

This results in risk-neutral default probabilities. In order to compute the “true” probabilities under the physical measure, one would have to be able to accurately estimate the risk premium for each bank holding company over the course of the entire sample period. The cost of attempting to incorporate this into the model arguably outweighs the benefits, since it can be shown (mathematically) when the risk premium is non-negative, the risk-neutral default probability will serve as an upper bound on the physical default probability. This will result in more conservative estimates of default probabilities and capital needs.

In the interest of brevity, the other model inputs—\(F_{1}\), \(F_{2}\), \(T_{1}\), \(T_{2}\), and r—have been dropped from the expression.

See Crouhy et al. (2008) and Gorton (2009) for excellent accounts of the causes and extreme ramifications of the subprime crisis.

The leverage effect was first put forth by Black (1976) and Christie (1982).

See, for example, Kashyap et al. 2008; Squam Lake Working Group 2009b, a; Bernanke 2009; Tarullo 2009; Hanson et al. 2011; Tarullo 2012.

For a derivation please see Geske 1977, 1979; Geske and Johnson 1984.