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.

Appendices

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

See below table.

	Distressed:		Distressed:		Distressed:
Banks	1=Yes	Banks	1=Yes	Banks	1=Yes
JPMORGAN CHASE BK NA		FULTON BK		COBIZ BK
BANK OF AMER NA	1	SUSQUEHANNA BK PA		FIRST BK
CITIBANK NA	1	PROSPERITY BK		WILSHIRE ST BK	1
WACHOVIA BK NA	1	CITIZENS BUS BK		FIRST RGNL BK	1
WELLS FARGO BK NA		SILICON VALLEY BK		MAINSOURCE BK
U S BK NA		COLUMBUS B&TC	1	MACATAWA BK	1
SUNTRUST BK		NATIONAL PENN BK		FIRST CMNTY BK NA
NATIONAL CITY BK	1	CENTRAL PACIFIC BK	1	MERCANTILE BK MI	1
REGIONS BK	1	NBT BK NA		AMERIS BK	1
STATE STREET B&TC		AMCORE BK NA	1	COUNTY BK	1
BRANCH BKG&TC		COMMUNITY BK NA		SOUTHSIDE BK
PNC BK NA		WESTAMERICA BK		NEWBRIDGE BK	1
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 BK	1	UNIVEST NB&TC
FIFTH THIRD BK		FRONTIER BK	1	FIRST BK
NORTHERN TC		CHEMICAL BK		FIRST MRCH BK NA
HUNTINGTON NB	1	BANCFIRST		INTERVEST NB	1
M&I MARSHALL & ILSLEY BK	1	BUSEY BK		PEOPLES BK NA
FIRST TENNESSEE BK NA MMPHS		MIDWEST B&TC	1	CAMDEN NB
COLONIAL BK NA	1	RENASANT BK		FIDELITY BK	1
ASSOCIATED BK NA	1	IBERIABANK		CARDINAL BK
ZIONS FIRST NB		IMPERIAL CAP BK	1	CENTURY B&TC
WEBSTER BK NA	1	FIRST CMNTY BK	1	STOCK YARDS B&TC
TCF NB		HANCOCK BK		FIRST BK OF BEVERLY HILLS	1
CITY NB		PRIVATEBANK & TC		FIRST UNITED B&TC
COMMERCE BK NA		S&T BK		SUMMIT CMNTY BK
BANK OF OK NA		INTEGRA BK NA	1	PEAPACK GLADSTONE BK
FROST NB		FIRST FNCL BK NA		WEST BK	1
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:
Banks	1=Yes	Banks	1=Yes	Banks	1=Yes
VALLEY NB		COMMUNITY TR BK INC		CITIZENS & NORTHERN BK
EAST WEST BK	1	BOSTON PRIVATE B&TC		ROYAL BK AMERICA	1
BANK OF HAWAII		BANK OF THE OZARKS		HAWTHORN BK
FIRSTMERIT BK		OLD SECOND NB	1	HEARTLAND B&TC
CATHAY BK	1	CAPITAL CITY BK		GERMAN AMERICAN BC
INTERNATIONAL BK OF CMRC		WASHINGTON TC		BAYLAKE BK	1
TRUSTMARK NB		LAKELAND BK		FIRST NB OF LONG ISLAND
CORUS BK NA	1	CITY NB OF WV		ALLIANCE BK	1
UMPQUA BK		VINEYARD BK NA	1	LORAIN NB
UMB BK NA		GREAT SOUTHERN BK		COLUMBIA RIVER BK	1
FIRST MIDWEST BK		STILLWATER NB&TC		REPUBLIC FIRST BK	1
MB FNCL BK NA		SEACOAST NB	1	NORTHRIM BK
OLD NB		BANK OF THE CASCADES	1	FIRST 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 $M_T = N_T =0$.