Bank default indicators with volatility clustering

Abstract

We estimate default measures for US banks using a model capable of handling volatility clustering like those observed during the Global Financial Crisis (GFC). In order to account for the time variation in volatility, we adapted a GARCH option pricing model which extends the seminal structural approach of default by Merton (J Finance 29(2):449, 1974) and calculated “distance to default” indicators that respond to heightened market developments. With its richer volatility dynamics, our results better reflect higher expected default probabilities precipitated by the GFC. The diagnostics show that the model generally outperforms standard models of default and offers relatively good indicators in assessing bank failures.

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Notes

  1. 1.

    There is overwhelming empirical evidence that GARCH models dominate the benchmark constant volatility Black–Scholes model achieving significant overall improvements in pricing performance. See Lehar et al. (2002), Hsieh and Ritchken (2005), Christoffersen et al. (2013) and Christoffersen et al. (2013).

  2. 2.

    As pointed out by Vassalou and Xing (2004) the theoretical distribution implied by the Merton model is the normal distribution. On the contrary, the KMV approach utilizes their own default database to derive an empirical distribution relating the distance-to-default to a default probability. In this regard, unlike the default probability calculated by KMV the probability measure in Eq. (4) may not correspond to the true probability of default in large samples.

  3. 3.

    \(N(-DD)\) is then the corresponding implied probability of default and sometimes called the expected default frequency (or EDF).

  4. 4.

    In the literature, there are several approaches [See Duan and Wang (2012) for their pros and cons].

  5. 5.

    We are very grateful to the anonymous referee for pointing out several points like this one.

  6. 6.

    The details of this classification of short term debt are elaborate and can be found in Harada et al. (2010).

  7. 7.

    Estimating implied default probabilities form CDS spreads is not a straight forward task as the process requires certain assumptions such as a value for the recovery rate. More importantly, this only yields risk-neutral probabilities unless we convert them into real world probabilities, which further requires an assumption on the value of risk aversion. Despite these differences we assume that the monotonic relationship between our default probabilities and CDS spreads is strong and therefore exploit the relationship.

  8. 8.

    We cannot rule out the possibility that the right-hand-side variables are endogenous as all variables could potentially capture some common economic conditions and hence the coefficients may reflect correlations.

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Acknowledgements

We thank the late Peter Christoffersen, Lynne Evans, Jens Hagendorff, Thomas Mazzoni and Aydin Ozkan for useful comments on an earlier version of the paper. In addition, we gratefully acknowledge very useful comments from an anonymous referee. Any remaining errors are our responsibility.

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Correspondence to Sel Dibooglu.

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Appendices

Appendix A: List of the sampled U.S. banks

See below table.

 Distressed: Distressed: Distressed:
Banks1=YesBanks1=YesBanks1=Yes
JPMORGAN CHASE BK NA FULTON BK COBIZ BK 
BANK OF AMER NA1SUSQUEHANNA BK PA FIRST BK 
CITIBANK NA1PROSPERITY BK WILSHIRE ST BK1
WACHOVIA BK NA1CITIZENS BUS BK FIRST RGNL BK1
WELLS FARGO BK NA SILICON VALLEY BK MAINSOURCE BK 
U S BK NA COLUMBUS B&TC1MACATAWA BK1
SUNTRUST BK NATIONAL PENN BK FIRST CMNTY BK NA 
NATIONAL CITY BK1CENTRAL PACIFIC BK1MERCANTILE BK MI1
REGIONS BK1NBT BK NA AMERIS BK1
STATE STREET B&TC AMCORE BK NA1COUNTY BK1
BRANCH BKG&TC COMMUNITY BK NA SOUTHSIDE BK 
PNC BK NA WESTAMERICA BK NEWBRIDGE BK1
CAPITAL ONE NA BANNER BK STERLING NB 
KEYBANK NA UNITED BK LAKE CITY BK 
MANUFACTURERS & TRADERS TC WESBANCO BK TRI CTY BK 
COMERICA BK HANMI BK1UNIVEST NB&TC 
FIFTH THIRD BK FRONTIER BK1FIRST BK 
NORTHERN TC CHEMICAL BK FIRST MRCH BK NA 
HUNTINGTON NB1BANCFIRST INTERVEST NB1
M&I MARSHALL & ILSLEY BK1BUSEY BK PEOPLES BK NA 
FIRST TENNESSEE BK NA MMPHS MIDWEST B&TC1CAMDEN NB 
COLONIAL BK NA1RENASANT BK FIDELITY BK1
ASSOCIATED BK NA1IBERIABANK CARDINAL BK 
ZIONS FIRST NB IMPERIAL CAP BK1CENTURY B&TC 
WEBSTER BK NA1FIRST CMNTY BK1STOCK YARDS B&TC 
TCF NB HANCOCK BK FIRST BK OF BEVERLY HILLS1
CITY NB PRIVATEBANK & TC FIRST UNITED B&TC 
COMMERCE BK NA S&T BK SUMMIT CMNTY BK 
BANK OF OK NA INTEGRA BK NA1PEAPACK GLADSTONE BK 
FROST NB FIRST FNCL BK NA WEST BK1
FIRST-CITIZENS B&TC REPUBLIC B&TC SIMMONS FIRST NB 
BANCO POPULAR SANDY SPRING BK BANK OF THE SIERRA 
BANCORPSOUTH BK COLUMBIA ST BK BANK OF KY 
 Distressed: Distressed: Distressed:
Banks1=YesBanks1=YesBanks1=Yes
VALLEY NB COMMUNITY TR BK INC CITIZENS & NORTHERN BK 
EAST WEST BK1BOSTON PRIVATE B&TC ROYAL BK AMERICA1
BANK OF HAWAII BANK OF THE OZARKS HAWTHORN BK 
FIRSTMERIT BK OLD SECOND NB1HEARTLAND B&TC 
CATHAY BK1CAPITAL CITY BK GERMAN AMERICAN BC 
INTERNATIONAL BK OF CMRC WASHINGTON TC BAYLAKE BK1
TRUSTMARK NB LAKELAND BK FIRST NB OF LONG ISLAND 
CORUS BK NA1CITY NB OF WV ALLIANCE BK1
UMPQUA BK VINEYARD BK NA1LORAIN NB 
UMB BK NA GREAT SOUTHERN BK COLUMBIA RIVER BK1
FIRST MIDWEST BK STILLWATER NB&TC REPUBLIC FIRST BK1
MB FNCL BK NA SEACOAST NB1NORTHRIM BK 
OLD NB BANK OF THE CASCADES1FIRST MID-IL B&T NA 

Appendix B: Risk-neutralization of the HN-GARCH

This risk-neutral pricing framework relies on the following risk-neutralization of the processes in Eqs. (6) and (7):

$$\begin{aligned} \ln \, A_{t}= & {} \ln \, A_{t-1} + r -\tfrac{1}{2}h_t + \sqrt{h_t}\,z^*_t,\end{aligned}$$
(B.1)
$$\begin{aligned} h_t= & {} \beta _0+\beta _1\,h_{t-1}+\beta _2 (z^*_{t-1}-\gamma ^*\sqrt{h_{t-1}})^2, \end{aligned}$$
(B.2)

where \(\gamma ^*=\gamma +\lambda \) and \(z_t^*=z_t+\lambda \sqrt{h_t}\). Above, we simply transformed the GARCH process under the physical measure in equations (6) and (7) to a risk-neutral one \(\mathbf {Q}\) in which the underlying asset earns the risk-free rate: \(\mathbf {E}^{\mathbf {Q}}[e^{\ln (A_t/A_{t-1})}]=\exp {(r_t)}\). To this end, following Christoffersen et al. (2013), we use the stochastic discount factor:

$$\begin{aligned} \frac{\Lambda _t}{\Lambda _{t-1}}=\frac{e^{v_{t-1}z_t}}{\mathbf {E}^{\mathbf {P}}\left( e^{v_{t-1}z_t}\right) }=e^{e^{v_{t-1}z_t}-\tfrac{1}{2}v^2_{t-1}}. \end{aligned}$$

The no-arbitrage condition, \(\mathbf {E}^{\mathbf {Q}}[e^{\ln (A_t/A_{t-1})}]=e^{r_t}\), yields

$$\begin{aligned} \nonumber \mathbf {E}^{\mathbf {Q}}[e^{\ln (A_t/A_{t-1})}]= & {} \mathbf {E}^{\mathbf {P}}\Bigl [\frac{\Lambda _t}{\Lambda _{t-1}}e^{\ln (A_t/A_{t-1})}\Bigr ]\\\nonumber= & {} \mathbf {E}^{\mathbf {P}}\Bigl [e^{e^{v_{t-1}z_t}-\tfrac{1}{2}v^2_{t-1}}e^{r_t+(\lambda -\tfrac{1}{2})h_t+\sqrt{h_t}z_{t}}\Bigr ] \\= & {} e^{r_t+\lambda h_t+v_{t-1}\sqrt{h_t}z_{t}}=e^{r_t}, \end{aligned}$$
(B.3)

with the implication that \(v_{t-1}=-\lambda \) and hence \(z_t^*=z_t+\lambda \sqrt{h_t}\) under the risk-neutral-\(\mathbf {Q}\) measure.

Appendix C: The characteristic function

The characteristic function of the HN-GARCH model is represented by a set of difference equations:

$$\begin{aligned} f^*(t,T;i\phi )=A^\phi \exp (M_t + N_t h_{t+1}) \end{aligned}$$

with coefficients

$$\begin{aligned} M_t= & {} M_{t+1}+\phi r + N_{t+1}\beta _0-\frac{1}{2}\ln (1-2\beta _2N_{t+1})\end{aligned}$$
(C.1a)
$$\begin{aligned} N_t= & {} \phi \left( -\frac{1}{2}+\gamma ^*\right) --\frac{1}{2}\gamma ^{*2} +\frac{(\phi -\gamma ^*)^2}{2(1-2N_{t+1}\beta _2)} \beta _1N_{t+1} \end{aligned}$$
(C.1b)

Note that \(M_t\) and \(N_t\) are implicitly functions of T and \(\phi \). This system of difference equations can be solved backwards using the terminal condition \(M_T = N_T =0\).

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Kenc, T., Cevik, E.I. & Dibooglu, S. Bank default indicators with volatility clustering. Ann Finance (2020). https://doi.org/10.1007/s10436-020-00369-x

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Keywords

  • Default risk
  • Structural credit risk models
  • Contingent claims
  • GARCH option pricing
  • Bank defaults

JEL Classifications

  • G01
  • G21
  • G28