- Email: [email protected]
Measuring systemic risk: A factor-augmented correlated default approach
J. Finan. Intermediation 21 (2012) 341–358
Contents lists available at SciVerse ScienceDirect
J. Finan. Intermediation j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j fi
Measuring systemic risk: A factor-augmented correlated default approach Sangwon Suh ⇑ School of Economics, Chung-Ang University, Republic of Korea
a r t i c l e
i n f o
Article history: Received 22 February 2011 Available online 31 October 2011 Keywords: Systemic risk Financial stability Correlated default approach Systemic risk contribution
a b s t r a c t In this paper, we extend existing correlated default models for measuring systemic risk by proposing a model that incorporates an observable common factor that features conditional heteroscedasticity. The addition of the common factor helps to effectively capture realistic time-varying characteristics in individual asset return volatility as well as return correlations. We apply the model for large US financial institutions. The common factor proves its importance in explaining asset return dynamics and measuring systemic risk. We also apply the model in the context of systemic risk contribution analysis and show its applicability. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The proper functioning of a financial system is crucial for economic stabilization and growth, and its malfunctioning can entail very high costs for the economy (see e.g., Hoggarth et al., 2002). Therefore, financial regulators have been alert to maintain financial stability by using various measures. In particular, they have paid more attention to the soundness of financial systems since the recent global financial crisis occurred during 2007–2009. The soundness of the financial system cannot be guaranteed by simply ensuring the soundness of individual financial institutions. The riskiness of the financial system as a whole (i.e., systemic risk) can be magnified through two channels: contagion and asset correlation. In the case of contagion, financial institutions are inter-dependent within the payment system or via direct loans. An initial financial institution gets into trouble and then other institutions may become distressed as a result. Early theoretical works on this type of systemic risk focus on lending between institutions (Rochet and Tirole, 1996) or the payment system (Cohen and Roberds, 1993). For ⇑ Fax: +82 2 813 5487. E-mail address: [email protected] 1042-9573/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfi.2011.10.003
342
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
empirical results, refer to, for example, Humphrey (1986), Angelini et al. (1996), and Furfine (2003). Recently, network models have been applied for this analysis, including Boss et al. (2004), Muller (2006), Nier et al. (2006), and Martinez-Jaramillo et al. (2010). Upper (2007) summarizes the research on financial contagion. Analysis from this contagion viewpoint requires detailed data on mutual exposure between institutions with various characteristics, thereby limiting its applicability. Furthermore, several works document evidence of small effects of the financial contagion channel; see, for example, Lang and Stulz (1992), Elsinger et al. (2006), Jorion and Zhang (2007, 2009), and Helwege (2010). In lieu of the aforementioned counterparty risk, we often observe in a financial crisis that a common factor simultaneously throws multiple financial institutions into financial distress. Under this viewpoint, the assets of financial institutions are correlated because of their exposure to such common factors. The common factors may have various forms, for example, belonging to the same industry (Helwege and Kleiman, 1997), exposure to a certain asset class (Fenn and Cole, 1994), information problems (Duffie and Lando, 2001; Yu, 2005), or macroeconomic factors (Helwege, 1996). Many empirical studies document strong effects of indirect inter-dependencies whereby markets also negatively respond to other institutions that are not directly related (but only indirectly related through a common factor) to a distressed institution. (See, for example, Crabbe, 1991; DeAngelo et al., 1994; Eichengreen and Mody, 2000; Fenn and Cole, 1994; and Jorion and Zhang, 2007, 2009.) De Nicolo and Kwast (2002) argue that financial consolidation raises the correlations of stock returns, and thereby the systemic risk as well. Nijskens and Wagner (2011) document that while institutions may have shed their individual credit risk through Credit Default Swaps (CDSs) or Collateralized Loan Obligations (CLOs), their correlations actually increase and the systemic risk becomes greater. In this paper, we follow the correlated default approach and propose a model in order to measure the systemic risk more accurately by using stock-market data. Balance-sheet information on financial institutions is available only on a relatively low-frequency (typically quarterly) basis and often with a significant time-lag while stock-market data provide timely, forward-looking information. CDS market data also deliver useful information about default probabilities but are available only for limited entities.1 This paper is closely related to Lehar (2005) and extends it in several aspects. Lehar (2005) employs Merton’s (1974) option pricing approach to estimate the default probability of individual financial institutions. We model the asset value of an individual financial institution as being exposed to an observable common factor as well as an unobservable individual factor. By introducing a common factor, we intend to effectively capture time-varying characteristics in asset correlations between financial institutions. This modeling feature contrasts with Lehar (2005) where asset correlations are basically constant over time and the time-varying property is obtained only through an exponentially-weighted moving average scheme. In addition, the common factor acts as a driving force of co-movement among the default probabilities of financial institutions. Moreover, the common factor is modeled to capture the volatility-clustering phenomenon in equity returns. We believe that these realistic modeling features help to measure the systemic risk more accurately. This paper also has an implication for systemic risk contribution analysis regarding which considerable literature has recently emerged. For example, Adrian and Brunnermeier (2010) propose a valueat-risk-based measure, called CoVaR, which quantifies how much an institution adds to the overall systemic risk. Acharya et al. (2010) devise an expected-shortfall-based measure by using the net equity returns of individual institutions conditional on the worst market outcome. They show that the measure has the ability to predict risks during a financial crisis. Both studies successfully estimate individual systemic risk contribution; however, they do not provide an overall systemic risk level. In fact, both loosely define systemic risk by using the worst market outcome event and do not explicitly define default events for individual institutions. Contrasting with them, this paper presents a model to permit the estimation of individual systemic risk contributions as well as the overall systemic risk level. We apply the newly proposed model for measuring the systemic risk of a group of 50 large US financial institutions for the period from January 1974 to March 2010. Our main findings are as follows. First, featured with time-varying volatility, the common factor significantly contributes to the
1
Huang et al. (2009) utilize CDS market data for measuring the systemic risk of a group of major financial institutions.
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
343
variation of the asset return. Second, the common factor also accounts for a considerable proportion of the correlations among asset returns. Furthermore, asset correlations become higher in a factored model than in an unfactored model during periods of financial distress. Third, the estimated systemic risk level shows the historically highest level during the recent financial crisis period. Without allowing correlation among individual factors, the estimated systemic risk exhibits a significantly lower level. However, without a common factor (but allowing correlations among individual factors) in the model, the systemic risk is estimated at a higher level during the financial crisis period. Fourth, the proposed model is also successful for analyzing systemic risk contributions. The systemic risk contributions measured with this model have the ability to predict risks during the crisis period. The measure also exhibits considerable stability during pre-crisis periods. The rest of the paper proceeds as follows. Section 2 explains the model, and Section 3 describes the data. Section 4 presents empirical results for the dynamics of the common factor, asset correlations, and measurements of systemic risk. A comparison with alternative model specifications is also provided. Section 5 applies the model for systemic risk contribution analysis. Section 6 concludes with discussions. 2. Methodology 2.1. Model Merton’s (1974) option pricing approach to regard equity as a call option has been widely accepted in the credit risk literature. We consider a discrete-time economy with N financial institutions (j = 1, . . . ,N). For institution j at time t, we denote by Vj,t and Dj,t the asset value and the matured debt amount, respectively. We define default as the event that Vj,t < Dj,t. We will infer the unobservable (market) asset values from observable market equity prices. In line with Merton (1974), equity will be considered as a call option which is written on the asset value with the strike price of the debt amount. Denoting by vj,t the log return of Vj,t, we model that under the physical measure P, the asset return depends upon a common factor xt with factor loading dj and also upon an unobservable individual factor wj,t. That is,
v j;t ¼ lj D þ dj ðxt rDÞ þ wj;t ;
ð1Þ
wj;t Nð0; nj Þ;
ð2Þ
where r denotes the annualized risk-free interest rate and D the length of time of one period. We choose the value-weighted, financial sector equity return as the observable common factor xt (log(Xt/Xt1)). Instead of specifying the common factor as a latent factor, fixing it in advance greatly facilitates parameter estimation because an observable common factor allows institution-by-institution estimation. In contrast, full information estimation by simultaneously using data on all financial institutions would be required for latent factor identification and estimation efficiency, which, however, is impractical for the case of many institutions in a system. For the dynamics of the common factor, we employ a GARCH-type model. Specifically, following Heston and Nandi (2000), the common factor is, under the physical measure P, modeled as:
pffiffiffiffiffi xt ¼ r D þ kht þ ht et ; pffiffiffiffiffiffiffiffiffi2 ht ¼ x þ a et1 c ht1 þ ght1 ;
ð3Þ ð4Þ
where et is a standard normal disturbance.2 This process is stationary with finite mean and variance if g + ac2 < 1 where g + ac2 p measures the degree of mean reversion. The annualized long-run volatility, deffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi noted by h, is defined as mðx þ aÞ=ð1 g ac2 Þ where m is the number of observations in a year. By allowing time-varying conditional volatility for the common factor, we intend to capture potential time2
We choose the first-order case for simplicity even though this GARCH-type model generalizes to higher-order cases.
344
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
varying features in the volatility and correlations of asset returns. Evidently, the conditional variance of an asset return becomes time-varying, i.e.,
Varðv j;t jut1 Þ r2j;t ¼ d2j ht þ nj :
ð5Þ
Now, equity is defined, under the risk-neutral measure Q, as a call option with the maturity of a prescribed multi-period:
Sj;t ¼ erDðTtÞ Et ½maxðV j;T Dj;T ; 0Þ;
ð6Þ
where Sj,t denotes the equity price of financial institution j at time t. Duan (1995) assumes that the risk-neutral measure Q satisfies the locally risk-neutral valuation relationship (LRNVR) that the expected return under the Q measure is the risk-free rate as usual, but the one-period ahead conditional variance of the return stays the same under the P and Q measures.3 Adopting the same assumption, Heston and Nandi (2000) show that under the Q measure, we have:
pffiffiffiffiffi 1 xt ¼ r D ht þ ht et ; 2 2 1 pffiffiffiffiffiffiffiffiffi ht1 þ ght1 : ht ¼ x þ a et1 c þ k þ 2
ð7Þ ð8Þ
Heston and Nandi (2000) derive the following moment-generating function for the future common factor:
h i f ð/Þ Et X /T ¼ X /t expðAðt; T; /Þ þ Bðt; T; /Þhtþ1 Þ;
ð9Þ
where the coefficients are recursively determined as follows:
AðT; T; /Þ ¼ 0;
ð10Þ
Aðt; T; /Þ ¼ Aðt þ 1; T; /Þ þ /rD þ Bðt þ 1; T; /Þc0
1 lnð1 2aBðt þ 1; T; /ÞÞ; 2
ð11Þ
BðT; T; /Þ ¼ 0;
ð12Þ
1 1=2ð/ cÞ2 Bðt; T; /Þ ¼ /ðk þ cÞ c2 þ gBðt þ 1; T; /Þ þ : 2 1 2aBðt þ 1; T; /Þ
ð13Þ
Utilizing these facts, we derive the moment-generating function for asset values. First, we note that under the Q measure,
log
V j;T XT ¼ ð1 dj Þr DðT tÞ þ dj log þ W Tj;t ; V j;t Xt
ð14Þ
where
W Tj;t wj;tþ1 þ . . . þ wj;T :
ð15Þ
Then, we can write: dj / ð1dj ÞrDðTtÞþ/W T j;t
V /j;T ¼ V /j;t X t
e
d/
X Tj :
ð16Þ
Therefore, we can derive the moment-generating function for asset values:
h i 2 d / g j ð/Þ Et V /j;T ¼ V /j;t X t j eð1dj ÞrDðTtÞþ/ nj ðTtÞ=2 f ðdj /Þ:
ð17Þ
From the assumption that equity is valued as a European call option, we have the equity valuation formula: 3
The LRNVR is valid under certain types of utility functions and consumption processes. Refer to Duan (1995) for details.
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
Sj;t erDðTtÞ Et ½maxðV j;T Dj;T ; 0Þ " i/ # " i/ # ! Z Z Dj;T g j ði/ þ 1Þ Dj;T g j ði/Þ 1 erDðTtÞ 1 1 1 1 d/ Dj;t d/ ; þ Re Re ¼ V j;t þ i/ i/ 2 2 p 0 p 0
345
ð18Þ
where g j ðÞ is obtained from gj() by replacing k with 1/2 and c with c⁄ (c + k + 1/2).4 2.2. Estimation Parameter estimation proceeds in two steps. First, we estimate the common factor parameters {x,a,g,c} in the system of (3) and (4) via the maximum likelihood method given the common factor observations. We make two assumptions for the estimation of the parameters related to the asset return process of individual institutions. Following Ronn and Verma (1986) and Lehar (2005), we assume that the maturity of the implied call option is one year. It may be difficult to determine the matured debt amount within the assumed maturity of 1 year. To avoid arbitrary choices of the matured debt amount, we also assume that the debt amount is a constant proportion of the total liability, that is, Dj,t = jjLj,t where Lj,t denotes the total liability and jj the constant proportion. Second, for one institution at a time via the maximum likelihood method, we estimate the parameters Hj = {lj, dj, nj, jj} for individual institution j0 s asset return. Given institution j’s equity price data Sj = [Sj,1 Sj,n]0 , total liability data Lj = [Lj,1 Lj,n]0 , and common factor data x = [x1 xn]0 , we apply the results of Duan (1994), Duan (2000) and derive the following log likelihood function.
log LðSj jx; Lj ; Hj Þ ¼
n n n X X n1 1X @Sj;t logð2pÞ log V j;t log r2j;t log 2 2 t¼2 @V j;t t¼2 t¼2
n 1X fv j;t ðr D þ dj ðxt r DÞÞg2 ; 2 t¼2 r2j;t
ð19Þ
" # Z 1 jj Lj;t erDðTtÞ i/ ði/ þ 1Þg j ði/ þ 1Þ @Sj;t 1 erDðTtÞ 1 jj Lj;t d/ ¼ þ Re @V j;t 2 p V j;t 0 pV j;t i/ Z 1 i/ Re½ jj Lj;t erDðTtÞ g j ði/Þd/:
ð20Þ
0
In the above, Vj,t and rj,t are the solutions to (18) and (5), and
vj,t is the log return of Vj,t.
2.3. Dynamics of individual factors The unobservable individual factor wj,t may contain not only uncorrelated idiosyncratic factors but also common factors other than the common factor xt. Therefore, the individual factors may be correlated with each other. In particular, we assume that the vector consisting of individual factors wt [w1,t wN,t]0 follows a multivariate normal distribution with time-varying covariance matrix; i.e.,
wt MVN½0; Xt ;
ð21Þ
where (j, k) element of Xt is njk,t. Then, we employ a diagonal-vech model for the dynamics of Xt. Specifically, the covariance njk,t at time t between the unobservable factors wj,t and wk,t is determined by:
njk;t ¼ cjk þ ajk wj;t1 wk;t1 þ bjk njk;t1 :
ð22Þ
Under this specification, the conditional correlation between the asset returns of institutions j and k also is time-varying, i.e.,
4
The debt is assumed to grow at the risk-free interest rate, following Lehar (2005).
346
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
qjk;t ¼
dj dk ht þ njk;t
rj;t rk;t
:
ð23Þ
This can be regarded as a generalized specification compared to Lehar (2005) who uses a simple exponentially weighted moving average method by applying a predetermined common decay factor for all institutions. b j for institution j For the estimating the time-varying covariance matrix Xt, we first use estimates H ^ j;t , which are deto construct unobservable time series {Vj,t} and {vj,t} and then obtain the residuals w fined as:
^ j;t v j;t ðl ^ j þ ^dj ðxt rDÞÞ: w
ð24Þ
We adopt the estimation algorithm proposed by Ledoit et al. (2003) that produces a positive semidefinite conditional covariance matrix.
2.4. Systemic risk measures Following Lehar (2005), we adopt two risk measures: (i) the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, f, (ND), and (ii) the probability that the proportion of the assets of defaulting financial institutions to the assets of all the financial institutions exceeds a prescribed threshold, f, (WA), over a fixed time-horizon. By using these two measures that allow a clear economic interpretation, we will illustrate the temporal trend of the overall systemic risk level. Obviously, these two measures are not exhaustive, and other measures may be devised according to the objectives of systemic risk analysis. Since our model specifies the dynamics pertaining to both individual institutions and their inter-dependence, it may permit the calculation of not only the above two measures but also a wider range of analyses of systemic risk.
2.5. Simulation Following Lehar (2005), we set 6 months as the time-horizon for measuring systemic risk. We employ Monte Carlo simulation in order to calculate the systemic risk measures because no analytical solution is available for those measures over a multi-period time horizon. We draw a standard normal random variate and then simulate a hypothetical future common factor according to (3). For simulating individual factors, we draw multivariate normal random variates as specified by (21). We perform 100,000 repetitions for simulation.
3. Data We choose the largest 50 US financial institutions (based on the book values of the total assets).5 Market capitalizations as well as equity prices are collected on a weekly basis from January 1973 until March 2010, and total assets and total liabilities (book values) are obtained quarterly and linearly interpolated. In order to be free from survivorship bias, we also include inactive financial institutions in the sample. 5 Lehar (2005) also analyzes data using a sample consisting of the largest 50 banks in the North American region. In this analysis, we choose sample financial institutions among commercial banks, saving banks, cooperative banks, real estate and mortgage banks, investment banks, and bank-holding and other holding companies.
347
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 Table 1 Estimation results for the common factor. Parameter
x
a
c
g
k
g + ac2
h
Estimate (p-value)
3.1e6 0.000
1.73e5 0.000
62.247 0.000
0.866 0.000
0.768 0.000
0.933
12.63%
Notes: The GARCH model for the common factor is described in (3), (4). (g + ac2) indicates the degree of mean reversion, and h the annualized long-run volatility.
4. Empirical results 4.1. Estimation of the common factor process We construct the common factor xt by calculating the financial sector value-weighted stock index return comprised of the sample financial institutions.6 For the sake of parsimony, we model the common factor to follow Heston and Nandi’s (2000) GARCH(1, 1) process. Table 1 shows the results of parameter estimation. All parameters are significantly estimated. The degree of mean reversion is estimated as 0.933, which is less than one. The annualized long-run volatility is estimated as 12.63%. Fig. 1 demonstrates the temporal trends of both the common factor and its conditional volatility. The conditional volatility sharply increased during several periods: 1974 (the period of the first oil shock and the ensuing stock market crash period), October 1987 (Black Monday), August 1990 (the first Gulf War period), August/September 1998 (the period of declaration of the Russian moratorium and the distress concerning Long Term Capital Management), and 2007/2008 (the global financial crisis period). 4.2. Estimation for individual institutions Given the estimation results for the common factor, we estimate for institution j (j = 1, . . . , N) the parameter vector Hj by maximizing the log likelihood function (19). The coefficient dj indicates the sensitivity of institution j’s asset return to the common factor as expressed in (1). All the ds are positively estimated with a mean of 0.48, ranging from 0.175 to 1.175. Fig. 2 demonstrates the empirical distribution of the d estimates. Most estimates of d are significantly less than one. We can explain this feature from the following facts: (i) Eq. (1) relates an individual asset return to not an overall asset return but the overall equity return and (ii) equity is a call option with the asset value as the underlying asset and therefore, equity and asset values are nonlinearly related. To clearly understand the reason, we approximate the change in the call option value (i.e., equity price) with the option delta, that is,
DS ffi C DV;
ð25Þ
where C denotes the call option delta.7 Therefore, using the approximation, we can write
vffi
DV 1 S DS 1 S 1 S ¼ ffi s¼ ðc þ bx þ Þ; V CV S CV CV
ð26Þ
where the last equality indicates the CAPM expression. Therefore d in (1) is approximated as
dffi
1 S b: CV
ð27Þ
The CAPM b distributes around one; however, two additional components affect d. Fig. 3 illustrates the equity price as the call option price S against the underlying asset value V and the strike price of K. We 6 Alternatively, we may use relevant, existing sector indices, for example, the S&P North American financial services sector index. However, we believe that a sector index that is directly constructed from our sample can better represent the overall movements of our sample than other broad indices. 7 We ignore the option theta for simplicity in the approximation.
348
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
0.01
0.6
0.009
0.4 Common factor (left scale)
0.008
0.2 0.007 0
0.006 0.005
-0.2
0.004
-0.4 Conditional volatility (right scale)
0.003
-0.6 0.002 -0.8
0.001
-1 1974
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
0 2009
Fig. 1. The common factor (i.e., weekly growth rate of the financial sector stock index) and its conditional volatility. The conditional volatility is estimated based on the results for GARCH(1, 1), as shown in Table 1.
20 18 16
Frequency
14 12 10 8 6 4 2 0 0
0.2
0.4
0.6
δ
0.8
1
1.2
1.4
Fig. 2. Histogram of ds. The coefficient d indicates the sensitivity of a financial institution’s asset return to the common factor as expressed in (1).
can show that the product of these two components is always less than unity, therefore the additional components lower the value of d. In Fig. 3, the tangent line passing through the tangent point S intersects the horizontal axis at V0 ; thus, we write the option delta as C = S/(V V0 ). Therefore, the product of the two components becomes S/(CV) = (V V0 )/V, which is always less than unity since 0 < V0 < V for V > 0. Following Acharya et al. (2010), we classify the sampled 50 financial institutions into four subgroups: depository institutions, other non-depository institutions, insurance companies, and brokerdealers companies. Table 2 presents the mean value of the d estimates for each subgroup. Among the subgroups, depositories and insurance companies exhibit relatively lower sensitivity to the common factor than other non-depository institutions and broker-dealers companies.
349
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
60
Equity price
S
45 degree line
Tangent Line
0
0
V' K
V
Asset value
100
Fig. 3. Equity price vs. asset value.
Table 2 Estimation results for individual institutions.
Number Mean of ^ d ^ Mean of j
All
Depositories
Non-depository institutions
Insurance
Broker-dealers
50 0.480
33 0.427
8 0.631
2 0.363
7 0.589
0.166
0.090
0.322
0.100
0.361
Notes: The asset return model for an individual institution is described in (1). Here, j indicates a constant proportion of the matured debt amount relative to the total liabilities.
We assume that the debt amount matured within the maturity is a constant proportion of the total liability. Table 2 also reports the mean value of the proportion j. Depository institutions and insurance companies have a proportion of around 10% while other non-depository institutions and broker–dealers companies have a higher proportion, viz., more than 30% on average. To understand these empirical results, we may conjecture several factors that can affect the determination of the proportion j. As the maturity of the call option is assumed to be longer, the proportion j rises. As short-term debt accounts for a larger share of the total liabilities, the proportion j also increases, which may differentiate the proportion j across subgroups. Another factor may be related with funding stability. The matured debt may be paid by a liquidation of assets or funding other debts (via the receipt of deposits or the sale of debt instruments). As this repayment/funding cycle is regarded as less stable, the proportion j becomes larger too; therefore, the funding stability may also differentiate the proportion j across subgroups according to their business activities. The changes in an individual asset return are driven by two factors: the common factor and the individual factor. The variance of the asset return can be decomposed into two parts that are contributed by the two factors according to (5). Fig. 4 illustrates the empirical distribution of the average contribution of the common factor to the total variation over the sample period. The common factor proves its importance in explaining the variation in the asset return: on average, 19.2% (median of 18.0%) of the total variation in the asset return is attributable to variation in the common factor. 4.3. Correlations between asset returns Asset correlation is a major aspect that financial regulators should take into account regarding systemic risk. In particular, if the asset correlation is time-varying, then the asset correlation channel for
350
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
12
10
Frequency
8
6
4
2
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Contribution of common factor to the variation of asset return Fig. 4. Histogram of the average contribution of the common factor to the variation of the asset return.
1 0.9
total
0.8 0.7 0.6 0.5 0.4 0.3
common factor individual factor
0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 5. Correlations between the asset returns of individual institutions. The correlation coefficient between the asset returns of institutions j and k is obtained according to (23). This figure illustrates the median correlation of all possible bilateral correlation coefficients at each point in time. The median contributions by the common factor and individual factors are also illustrated.
systemic risk also becomes time-varying. Fig. 5 demonstrates the median correlation over all possible bilateral correlation coefficients between the asset returns of individual institutions. The correlation coefficient between the asset returns of institutions j and k is obtained according to (23). In addition, we decompose the correlation into two parts: the contribution by the common factor and that by the individual factor. The median correlation coefficient illustrates a high degree of positive correlation, fluctuating within a range of 0.40–0.88 with a mean of 0.61 (median of 0.59). The common factor contributes more to the asset correlation than the individual factors. On average, the common factor accounts for 0.44 while the individual factors accounts for 0.14 in the median asset correlation.
351
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 6. Systemic risk (ND): the benchmark factor-augmented model. Systemic risk is defined as the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, viz., f = 10% or 20%. The upper line corresponds to f = 10% and the lower one to f = 20%.
4.4. Systemic risk under the benchmark factor-augmented model We choose as our benchmark model the aforementioned factor-augmented model, use it for Monte Carlo simulation, and obtain two measurements of systemic risk. The default event of an individual institution is defined as the asset value’s being less than the debt level within the assumed time horizon, i.e., Vj,t+s < jjLj,t+s for s 6 0.5 years. We present two measures of systemic risk: (i) the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, f, (ND), and (ii) the probability that the proportion of the assets of defaulting financial institutions to the assets of all the financial institutions exceeds a prescribed threshold, f, (WA). The two measures are presented according to two systemic-risk thresholds, viz., f = 10% and 20%. The total number of sampled financial institutions varies from 15 to 50 during the sample period; therefore, the systemic-risk measure ND with f = 10% corresponds to the event that the number of defaulting institutions is more than one during an early period but more than five during a later period. Fig. 6 shows the systemic-risk measurements based on the number of distressed financial institutions (ND). Consistent with our intuition, as the threshold f becomes higher, the systemic risk measure decreases. The temporal trend in systemic risk indicates that a sharp increase in systemic risk occurred several times during the sample period: 1974, 1980, and 2008/2009 (the global financial crisis). The historically highest systemic risk level with the longest duration was recorded during the recent global financial crisis. Fig. 7 illustrates measurements of the systemic risk based on the weighted assets of defaulting financial institutions (WA). The WA measure also sharply increased in the same periods where the ND measure sharply increased; however, it also sharply increased at other times. This difference implies that the size effect on the measurement of systemic risk can be great. The historically highest systemic risk level with the longest duration under the WA measure was also recorded during the recent global financial crisis. Fig. 8 demonstrates the two systemic-risk measurements during the recent financial crisis period along with several specific dates: August 2007 (hedge fund failure and the start of crisis), March 2008 (acquisition of Bear Sterns by JPMorgan Chase), September 2008 (Lehman Brothers failure), March 2009 (stock market bottom). The systemic-risk measurements largely match up with those specific events; however, the pattern of the time variation depends on the choice of criterion. For example, the systemic-risk measure ND with a low threshold of 5% sensitively responds to a shock from an early
352
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 7. Systemic risk (WA): the benchmark factor-augmented model. Systemic risk is defined as the probability that the proportion of the assets of defaulting financial institutions to the assets of all financial institutions exceeds a prescribed threshold, viz., f = 10% or 20%. The upper line corresponds to f = 10% and the lower one to f = 20%.
1 Lehman Bros. fail
0.9 0.8
hedge funds fail Bear Sterns fail
0.7 0.6 0.5 0.4
WA (5%) WA (10%)
ND (5%)
0.3 0.2 ND (10%)
0.1 0 2007
2008
stock market bottom
2009
2010
Fig. 8. Systemic risk measures during the recent crisis period. Two systemic risk measures, ND and WA, under the benchmark factor-augmented model are displayed with two thresholds, viz., f = 5% or 10%. Several specific event dates are also demonstrated.
stage of the crisis and remains high until a late stage. In contrast, the ND with a relatively high threshold of 10% responds late to a shock but quickly subsides. Fig. 9 contrasts the systemic-risk measurements during the recent financial crisis period with several commonly used risk indicators: VIX (market volatility), the credit spread between Baa- and Aaarated yields (credit risk), and the difference between 3-month LIBOR and OIS (overnight index swap) (liquidity risk). Among the usual risk indicators, the VIX exhibits the most volatile movements; however, compared to the ND measure, it seems to subside too early because it greatly decreased even though the stock price stayed bottom. Compared to the ND measure, the credit risk indictor insensitively responded to shocks. The liquidity risk indicator too abruptly responded and then subsided too early.
353
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 1
2
Lehman Bros. fail
0.8
hedge funds fail
1.8
Bear Sterns fail
0.6
1.6 ND (10%)
0.4
1.4
ND (5%)
0.2
1.2 stock market bottom
1
0
-0.2
0.8
VIX
-0.4
Baa - Aaa
0.6
LIBOR - OIS
-0.6
0.4
-0.8
0.2
-1
0
2007
2008
2009
2010
Fig. 9. Systemic risk measures versus other risk indicators. The systemic risk measure ND (left scale) is displayed with two thresholds, viz., f = 5% or 10%. Commonly used risk indicators (right scale), VIX (market volatility, 102), Baa – Aaa (credit risk, 101), and LIBOR – OIS (liquidity risk, 101), are also demonstrated.
4.5. Systemic risk under alternative models We introduce a common factor in order to effectively capture time-varying characteristics in both volatilities and correlations between asset returns. To assess the effect of the common factor on the measurement of systemic risk, we calculate both the ND and the WA measures based on an unfactored model and compare the results with those of the benchmark model. The unfactored model does not contain a common factor, i.e.,
1
0.9
Unfactered but correlated model
0.8 0.7
Factered but uncorrelated model
0.6 0.5 0.4 0.3 0.2
Unfactered and uncorrelated model
0.1 0 -0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 10. Systemic risk (ND): alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008. This figure produces scatter plots of the ND measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model.
354
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 Unfactered but correlated model
0.8
Factered but uncorrelated model
0.7 0.6 0.5 0.4 0.3
Unfactered and uncorrelated model
0.2 0.1 0 -0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 11. Systemic risk (WA): alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008. This figure produces scatter plots of the WA measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model.
v j;t ¼ lj D þ wj;t :
ð28Þ
The other variation in model specification is whether or not to allow correlations between individual factors. We consider this variation in order to assess the effect of correlations between individual factors on the measurement of systemic risk. Therefore, we have four alternative model specifications: (i) (the benchmark) model with a common factor and correlated individual factors, (ii) a model without a common factor but with correlated individual factors, (iii) a model with a common factor but with uncorrelated individual factors, and (iv) a model without a common factor and with uncorrelated individual factors. We re-estimate the parameters of each alternative model specification. Fig. 10 demonstrates the effects of alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008 under the ND measure. In particular, Fig. 10 produces scatter plots of the ND measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model. Compared to the benchmark model, the two alternative models deliver much lower measurements of systemic risk: the model with a common factor but with uncorrelated individual factors (Specification iii), and the model without a common factor and with uncorrelated individual factors (Specification iv). This fact implies that correlations between individual factors can significantly affect the measurement of systemic risk. On the other hand, the other alternative model without a common factor but with correlated individual factors (Specification ii) yields higher measurements of systemic risk than the benchmark model.8 8 Our guess about the reason is as follows. The individual factors tend to have less persistent conditional variances than the common factor. The benchmark model differentiates the two factors, whereas the individual-factor-only model does not. Therefore, the persistence of the conditional variance of the unique factor under the individual-factor-only model may lie between those of the two factors under the benchmark model. During the recent financial crisis period, not only a large negative common shock hit all the financial institutions but negative individual shocks hit several unsound institutions as well. The individual-factoronly model does not distinguishes the two shocks and attributes more persistent conditional variance to negative individual shocks for unsound institutions than the benchmark model. This heightened and persistent conditional variance for fragile institutions might generate higher estimates of systemic risk.
355
Equity return: July 2007 to December 2008
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
0
1
2
3
4
5
6
7 -3
x 10
Equity return: July 2007 to December 2008
Δ ES: July 2006 to June 2007 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 -3
x 10
Δ ES: July 2006 to June 2007 Fig. 12. The ex-ante DESj5% measures versus the realized risks. This figure demonstrates the scatter plots of the ex-ante DESj5% measures against the realized risks. The upper panel contains all the sample institutions while the lower panel excludes two outlier institutions.
Fig. 11 demonstrates the effects of alternative model specifications against the benchmark model under the WA measure during the same period as in Fig. 10. Fig. 11 exhibits quantitatively different but qualitatively similar results as those in Fig. 10.
5. Systemic risk contribution Until now, we have proposed a model to specify the dynamics for individual institutions and their inter-dependence and use it to illustrate measurements of the overall systemic risk. However, aspects other than the estimation of the overall systemic risk level also have been investigated. Systemic risk contribution analysis can be such an example. Recently, Adrian and Brunnermeier (2010) proposed a measure for systemic risk, called CoVaR, which is the value at risk (VaR) of the financial system conditional on institutions’ being in distress. They define an institution’s contribution to systemic risk as the difference between the CoVaR conditional on the institution’s being in distress and the CoVaR at the median state of the institution. They also propose another measure with reverse direction, called exposure CoVaR, which measures the extent to which an individual institution is affected by systemic financial events. Acharya et al. (2010) present a simple model of systemic risk and show that each financial institution’s contribution to systemic risk can be measured as its systemic expected shortfall (SES), i.e., its propensity to be undercapitalized when the system as a whole is undercapitalized.9 9 We want to mention that both the previous studies do not explicitly define default events, which makes it difficult to measure an overall systemic risk level. In fact, both loosely define the systemic risk by using worst market outcome events. In this respect, the model proposed in this paper has an advantage of providing overall systemic risk measurements based on an explicit definition of default events.
356
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 -3
-3
x 10
1.8
6
Δ ES: July 2006 to June 2007
Δ ES: July 2006 to June 2007
7
5 4 3 2 1 1
2
3
4
5
6
x 10
7
x 10
Δ ES: July 2005 to June 2006
1.4 1.2 1 0.8 0.6
2
Δ ES: July 2007 to December 2008
4 2 0 -2 -4 1
2
3
4
5
6
0.8
1
1.2
1.4
7
x 10
Fig. 13. The stability of the outlier institutions.
DESj5%
1.8 -3 x 10
1.6
1.8 -3 x 10
-3
1.5
1
0.5
0 0.4
0.6
0.8
1
1.2
1.4
-3
Δ ES: July 2006 to June 2007
1.6
Δ ES: July 2005 to June 2006
-3
0
0.6
-3
6
Δ ES: July 2007 to December 2008
1.6
0.4 0.4
0 0
x 10
x 10
Δ ES: July 2006 to June 2007
measure. The left panels contain all the sample institutions while the right panels exclude
Motivated by these previous studies and in order to show the applicability of the model proposed in this paper for systemic risk contribution analysis, we modify the SES measure as follows:
# " # net assetj1 median worst E ¼E 1 C5% 1 C5% ; net assetj0 net assetj0 "
DESj5%
net assetj1
ð29Þ
where Cmedian and Cworst denote the set of events that the future common factor belongs to the middle 5% 5% 5% and lowest 5% regions in the distribution, respectively. The DESj5% measures institution j0 s systemic risk contribution by the difference between the expected value of the net asset (the asset value minus the matured debt) return conditional on the median state of the common factor and the expected value conditional on the worst outcome (say, 5%) of the common factor. The DESj5% measure is calculated via Monte Carlo simulation as in the calculation of the overall systemic risk measures. We want to note that the common factor also plays a crucial role as a conditioning variable in this systemic risk contribution analysis while the unfactored model has no such conditioning variable in which respect we may regard the unfactored model as less preferable to the factor-augmented benchmark model. Like Acharya et al. (2010), we first calculate the ex-ante DESj5% measure during the pre-crisis period of July 2006 to June 2007 and then examine whether this measure has any ability to predict the realized risk (defined by the losses in equity return) during the crisis period of July 2007 to December 2008. Fig. 12 demonstrates scatter plots of the ex-ante DESj5% measures against the realized risks. Consistent with Acharya et al. (2010), there exists a negative relationship between both ex-ante DESj5% measures and the realized risks, thus confirming the fact that the ex-ante DESj5% measure can predict the future risk. We also examine whether the ex-ante DESj5% measure is stable over time based on the closeness of the relationship between the DESj5% measures during the period of July 2006 to June 2007 and during the period of July 2005 to June 2006. Fig. 13 shows that the DESj5% measures during both pre-crisis
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
357
periods are positively and closely related. However, this positive relationship remains but becomes less tight during the crisis period as illustrated in the lower panels in Fig. 13. 6. Conclusions In this paper, we extend existing correlated default models for measuring systemic risk by proposing a model that incorporates a common factor and an individual factor. The common factor is observably constructed and featured with conditional heteroscedasticity. The addition of the common factor helps to effectively capture realistic time-varying characteristics in individual asset return volatilities as well as return correlations. The individual factors are correlated with each other and have timevarying conditional variances. We apply the model for a sample consisting of large US financial institutions. The common factor proves its importance in explaining asset return dynamics and measuring systemic risk. We also apply the model in the context of systemic risk contribution analysis and show its applicability. The analysis in this paper is extendable in several directions. We apply the model for the US case, but this method also can be applied for other regions or for international comparison. Moreover, the inclusion of the common factor also facilitates stress-test analysis. Hypothetical scenarios can be realistically constructed by making various assumptions on the common factor. We can also extend the model by adding more relevant common factors. It may be another extension to make the sensitivity to the common factor be time-varying. Acknowledgements I have benefited from helpful comments from two anonymous referees, Viral V. Acharya (Editor), Hyun Song Shin, and seminar participants at Hanyang University (ERICA Campus), Yeungnam University, 2010 Conference at The Korean Financial Engineering Society, Korea Institute of Finance, and the Bank of Korea. References Acharya, V.V., Pedersen, L.H., Philippon, T., Richardson, M., 2010. Measuring systemic risk. Working paper. Adrian, T., Brunnermeier, M.K., 2010. CoVaR. Federal Reserve Bank of New York Staff Reports No. 348. Angelini, P., Maresca, G., Russo, D., 1996. Systemic risk in the netting system. J. Bank. Finance 20, 853–868. Boss, M., Elsinger, H., Summer, M., Thurner, S., 2004. The network topology of the interbank market. Quant. Finance 4, 677–684. Cohen, H, Roberds, W., 1993. Towards the systematic measurement of systemic risk. Working paper 93-14, Federal Reserve Bank of Atlanta. Crabbe, L.E., 1991. Event risk: an analysis of losses to bond holders and ‘‘super poison put’’ bond covenants. J. Finance 46, 689– 706. De Nicolo, G., Kwast, M.L., 2002. Systemic risk and financial consolidation: are they related? J. Bank. Finance 26, 861–880. DeAngelo, H., DeAngelo, L., Gilson, S.C., 1994. The collapse of first executive corporation: junk bonds, adverse publicity, and the run on the bank phenomenon. J. Finance Econ. 36, 287–336. Duan, J.-C., 1994. Maximum likelihood estimation using the price data of the derivative contract. Math. Finance 4, 155–167. Duan, J.-C., 1995. The GARCH option pricing model. Math. Finance 5, 13–32. Duan, J.-C., 2000. Correction: maximum likelihood estimation using the price data of the derivative contract. Math. Finance 10, 461–462. Duffie, D., Lando, D., 2001. Term structure of credit spreads with incomplete accounting information. Econometrica 69, 633– 664. Eichengreen, B., Mody, A., 2000. What explains the changing spreads on emerging market dedt? In: Edwards, S. (Ed.), The Economics of International Capital Flows. University of Chicago Press. Elsinger, H., Lehar, A., Summer, M., 2006. Risk assessment for banking systems. Manage. Sci. 52, 1301–1314. Fenn, G.W., Cole, R.A., 1994. Announcements of asset-quality problems and contagion effects in the life insurance industry. J. Finance Econ. 35, 181–198. Furfine, C.H., 2003. Interbank exposures: quantifying the risk of contagion. J. Money, Credit, Banking 35, 111–128. Helwege, J., 1996. Determinants of S& L failure rates: estimates of a time-varying proportional hazard function. J. Finance Serv. Res. 7, 55–61. Helwege, J., 2010. Financial firm bankruptcy and systemic risk. J. Int. Finance Markets, Inst. Money 20, 1–12. Helwege, J., Kleiman, P., 1997. Understanding aggregate default rates of high-yield bonds. J. Fixed, Income 7, 55–61. Heston, S.L., Nandi, S., 2000. A closed-form GARCH option valuation model. Rev. Finance Stud. 13, 585–625. Hoggarth, G., Reis, R., Saporta, V., 2002. Costs of banking system instability: some empirical evidence. J. Bank. Finance 26, 825– 855.
358
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
Huang, X., Zhou, H., Zhu, H., 2009. A framework for assessing the systemic risk of major financial institutions. J. Bank. Finance 33, 2036–2049. Humphrey, D.B., 1986. Payments finality and risk of settlement failure. In: Saunders, Anthony, White, Lawrence (Eds.), Technology, and the Regulation of Financial Markets: Securities, Futures, and Banking. Lexington Books, Lexington, pp. 97– 120. Jorion, P., Zhang, G., 2007. Good and bad credit contagion: evidence from credit default swaps. J. Finance Econ. 84, 860–883. Jorion, P., Zhang, G., 2009. Credit contagion from counterparty risk. J. Finance 64, 2053–2087. Lang, L.H.P., Stulz, R.M., 1992. Contagion and competitive intra-industry effects of bankruptcy announcements. J. Finance Econ. 32, 45–60. Ledoit, O., Santa-Clara, P., Wolf, M., 2003. Flexible multivariate GARCH modeling with an application to international stock markets. Rev. Econ. Stat. 85, 735–747. Lehar, A., 2005. Measuring systemic risk: a risk management approach. J. Bank. Finance 29, 2577–2603. Martinez-Jaramillo, S., Perez, O.M., Embriz, F.A., Dey, F.L.G., 2010. Systemic risk, financial contagion and financial fragility. J. Econ. Dyn. Control 34, 2358–2374. Merton, R., 1974. On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29, 449–470. Muller, J., 2006. Interbank credit lines as a channel of contagion. J. Finance Serv. Res. 29, 37–60. Nier, E., Yang, J., Yorulmazer, T., Alentorn, A., 2006. Network models and financial stability. J. Econ. Dyn. Control 31, 2033–2060. Nijskens, R., Wagner, W., 2011. Credit risk transfer activities and systemic risk: how banks become less risky individually but posed greater risks to the financial system at the same time. J. Bank. Finance 35, 1391–1398. Rochet, J-C., Tirole, T., 1996. Interbank lending and systemic risk. J. Money, Credit, Banking 28, 733–762. Ronn, E.I., Verma, A.K., 1986. Pricing risk-adjusted deposit insurance: an option-based model. J. Financ. 41, 871–895. Upper, C., 2007. Using counterfactual simulations to assess the danger of contagion in interbank markets. BIS Working Papers 234, the Bank for International Settlements. Yu, F., 2005. Accounting transparency and the term structure of credit spreads. J. Finance Econ. 75, 53–84.
Contents lists available at SciVerse ScienceDirect
J. Finan. Intermediation j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j fi
Measuring systemic risk: A factor-augmented correlated default approach Sangwon Suh ⇑ School of Economics, Chung-Ang University, Republic of Korea
a r t i c l e
i n f o
Article history: Received 22 February 2011 Available online 31 October 2011 Keywords: Systemic risk Financial stability Correlated default approach Systemic risk contribution
a b s t r a c t In this paper, we extend existing correlated default models for measuring systemic risk by proposing a model that incorporates an observable common factor that features conditional heteroscedasticity. The addition of the common factor helps to effectively capture realistic time-varying characteristics in individual asset return volatility as well as return correlations. We apply the model for large US financial institutions. The common factor proves its importance in explaining asset return dynamics and measuring systemic risk. We also apply the model in the context of systemic risk contribution analysis and show its applicability. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The proper functioning of a financial system is crucial for economic stabilization and growth, and its malfunctioning can entail very high costs for the economy (see e.g., Hoggarth et al., 2002). Therefore, financial regulators have been alert to maintain financial stability by using various measures. In particular, they have paid more attention to the soundness of financial systems since the recent global financial crisis occurred during 2007–2009. The soundness of the financial system cannot be guaranteed by simply ensuring the soundness of individual financial institutions. The riskiness of the financial system as a whole (i.e., systemic risk) can be magnified through two channels: contagion and asset correlation. In the case of contagion, financial institutions are inter-dependent within the payment system or via direct loans. An initial financial institution gets into trouble and then other institutions may become distressed as a result. Early theoretical works on this type of systemic risk focus on lending between institutions (Rochet and Tirole, 1996) or the payment system (Cohen and Roberds, 1993). For ⇑ Fax: +82 2 813 5487. E-mail address: [email protected] 1042-9573/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jfi.2011.10.003
342
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
empirical results, refer to, for example, Humphrey (1986), Angelini et al. (1996), and Furfine (2003). Recently, network models have been applied for this analysis, including Boss et al. (2004), Muller (2006), Nier et al. (2006), and Martinez-Jaramillo et al. (2010). Upper (2007) summarizes the research on financial contagion. Analysis from this contagion viewpoint requires detailed data on mutual exposure between institutions with various characteristics, thereby limiting its applicability. Furthermore, several works document evidence of small effects of the financial contagion channel; see, for example, Lang and Stulz (1992), Elsinger et al. (2006), Jorion and Zhang (2007, 2009), and Helwege (2010). In lieu of the aforementioned counterparty risk, we often observe in a financial crisis that a common factor simultaneously throws multiple financial institutions into financial distress. Under this viewpoint, the assets of financial institutions are correlated because of their exposure to such common factors. The common factors may have various forms, for example, belonging to the same industry (Helwege and Kleiman, 1997), exposure to a certain asset class (Fenn and Cole, 1994), information problems (Duffie and Lando, 2001; Yu, 2005), or macroeconomic factors (Helwege, 1996). Many empirical studies document strong effects of indirect inter-dependencies whereby markets also negatively respond to other institutions that are not directly related (but only indirectly related through a common factor) to a distressed institution. (See, for example, Crabbe, 1991; DeAngelo et al., 1994; Eichengreen and Mody, 2000; Fenn and Cole, 1994; and Jorion and Zhang, 2007, 2009.) De Nicolo and Kwast (2002) argue that financial consolidation raises the correlations of stock returns, and thereby the systemic risk as well. Nijskens and Wagner (2011) document that while institutions may have shed their individual credit risk through Credit Default Swaps (CDSs) or Collateralized Loan Obligations (CLOs), their correlations actually increase and the systemic risk becomes greater. In this paper, we follow the correlated default approach and propose a model in order to measure the systemic risk more accurately by using stock-market data. Balance-sheet information on financial institutions is available only on a relatively low-frequency (typically quarterly) basis and often with a significant time-lag while stock-market data provide timely, forward-looking information. CDS market data also deliver useful information about default probabilities but are available only for limited entities.1 This paper is closely related to Lehar (2005) and extends it in several aspects. Lehar (2005) employs Merton’s (1974) option pricing approach to estimate the default probability of individual financial institutions. We model the asset value of an individual financial institution as being exposed to an observable common factor as well as an unobservable individual factor. By introducing a common factor, we intend to effectively capture time-varying characteristics in asset correlations between financial institutions. This modeling feature contrasts with Lehar (2005) where asset correlations are basically constant over time and the time-varying property is obtained only through an exponentially-weighted moving average scheme. In addition, the common factor acts as a driving force of co-movement among the default probabilities of financial institutions. Moreover, the common factor is modeled to capture the volatility-clustering phenomenon in equity returns. We believe that these realistic modeling features help to measure the systemic risk more accurately. This paper also has an implication for systemic risk contribution analysis regarding which considerable literature has recently emerged. For example, Adrian and Brunnermeier (2010) propose a valueat-risk-based measure, called CoVaR, which quantifies how much an institution adds to the overall systemic risk. Acharya et al. (2010) devise an expected-shortfall-based measure by using the net equity returns of individual institutions conditional on the worst market outcome. They show that the measure has the ability to predict risks during a financial crisis. Both studies successfully estimate individual systemic risk contribution; however, they do not provide an overall systemic risk level. In fact, both loosely define systemic risk by using the worst market outcome event and do not explicitly define default events for individual institutions. Contrasting with them, this paper presents a model to permit the estimation of individual systemic risk contributions as well as the overall systemic risk level. We apply the newly proposed model for measuring the systemic risk of a group of 50 large US financial institutions for the period from January 1974 to March 2010. Our main findings are as follows. First, featured with time-varying volatility, the common factor significantly contributes to the
1
Huang et al. (2009) utilize CDS market data for measuring the systemic risk of a group of major financial institutions.
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
343
variation of the asset return. Second, the common factor also accounts for a considerable proportion of the correlations among asset returns. Furthermore, asset correlations become higher in a factored model than in an unfactored model during periods of financial distress. Third, the estimated systemic risk level shows the historically highest level during the recent financial crisis period. Without allowing correlation among individual factors, the estimated systemic risk exhibits a significantly lower level. However, without a common factor (but allowing correlations among individual factors) in the model, the systemic risk is estimated at a higher level during the financial crisis period. Fourth, the proposed model is also successful for analyzing systemic risk contributions. The systemic risk contributions measured with this model have the ability to predict risks during the crisis period. The measure also exhibits considerable stability during pre-crisis periods. The rest of the paper proceeds as follows. Section 2 explains the model, and Section 3 describes the data. Section 4 presents empirical results for the dynamics of the common factor, asset correlations, and measurements of systemic risk. A comparison with alternative model specifications is also provided. Section 5 applies the model for systemic risk contribution analysis. Section 6 concludes with discussions. 2. Methodology 2.1. Model Merton’s (1974) option pricing approach to regard equity as a call option has been widely accepted in the credit risk literature. We consider a discrete-time economy with N financial institutions (j = 1, . . . ,N). For institution j at time t, we denote by Vj,t and Dj,t the asset value and the matured debt amount, respectively. We define default as the event that Vj,t < Dj,t. We will infer the unobservable (market) asset values from observable market equity prices. In line with Merton (1974), equity will be considered as a call option which is written on the asset value with the strike price of the debt amount. Denoting by vj,t the log return of Vj,t, we model that under the physical measure P, the asset return depends upon a common factor xt with factor loading dj and also upon an unobservable individual factor wj,t. That is,
v j;t ¼ lj D þ dj ðxt rDÞ þ wj;t ;
ð1Þ
wj;t Nð0; nj Þ;
ð2Þ
where r denotes the annualized risk-free interest rate and D the length of time of one period. We choose the value-weighted, financial sector equity return as the observable common factor xt (log(Xt/Xt1)). Instead of specifying the common factor as a latent factor, fixing it in advance greatly facilitates parameter estimation because an observable common factor allows institution-by-institution estimation. In contrast, full information estimation by simultaneously using data on all financial institutions would be required for latent factor identification and estimation efficiency, which, however, is impractical for the case of many institutions in a system. For the dynamics of the common factor, we employ a GARCH-type model. Specifically, following Heston and Nandi (2000), the common factor is, under the physical measure P, modeled as:
pffiffiffiffiffi xt ¼ r D þ kht þ ht et ; pffiffiffiffiffiffiffiffiffi2 ht ¼ x þ a et1 c ht1 þ ght1 ;
ð3Þ ð4Þ
where et is a standard normal disturbance.2 This process is stationary with finite mean and variance if g + ac2 < 1 where g + ac2 p measures the degree of mean reversion. The annualized long-run volatility, deffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi noted by h, is defined as mðx þ aÞ=ð1 g ac2 Þ where m is the number of observations in a year. By allowing time-varying conditional volatility for the common factor, we intend to capture potential time2
We choose the first-order case for simplicity even though this GARCH-type model generalizes to higher-order cases.
344
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
varying features in the volatility and correlations of asset returns. Evidently, the conditional variance of an asset return becomes time-varying, i.e.,
Varðv j;t jut1 Þ r2j;t ¼ d2j ht þ nj :
ð5Þ
Now, equity is defined, under the risk-neutral measure Q, as a call option with the maturity of a prescribed multi-period:
Sj;t ¼ erDðTtÞ Et ½maxðV j;T Dj;T ; 0Þ;
ð6Þ
where Sj,t denotes the equity price of financial institution j at time t. Duan (1995) assumes that the risk-neutral measure Q satisfies the locally risk-neutral valuation relationship (LRNVR) that the expected return under the Q measure is the risk-free rate as usual, but the one-period ahead conditional variance of the return stays the same under the P and Q measures.3 Adopting the same assumption, Heston and Nandi (2000) show that under the Q measure, we have:
pffiffiffiffiffi 1 xt ¼ r D ht þ ht et ; 2 2 1 pffiffiffiffiffiffiffiffiffi ht1 þ ght1 : ht ¼ x þ a et1 c þ k þ 2
ð7Þ ð8Þ
Heston and Nandi (2000) derive the following moment-generating function for the future common factor:
h i f ð/Þ Et X /T ¼ X /t expðAðt; T; /Þ þ Bðt; T; /Þhtþ1 Þ;
ð9Þ
where the coefficients are recursively determined as follows:
AðT; T; /Þ ¼ 0;
ð10Þ
Aðt; T; /Þ ¼ Aðt þ 1; T; /Þ þ /rD þ Bðt þ 1; T; /Þc0
1 lnð1 2aBðt þ 1; T; /ÞÞ; 2
ð11Þ
BðT; T; /Þ ¼ 0;
ð12Þ
1 1=2ð/ cÞ2 Bðt; T; /Þ ¼ /ðk þ cÞ c2 þ gBðt þ 1; T; /Þ þ : 2 1 2aBðt þ 1; T; /Þ
ð13Þ
Utilizing these facts, we derive the moment-generating function for asset values. First, we note that under the Q measure,
log
V j;T XT ¼ ð1 dj Þr DðT tÞ þ dj log þ W Tj;t ; V j;t Xt
ð14Þ
where
W Tj;t wj;tþ1 þ . . . þ wj;T :
ð15Þ
Then, we can write: dj / ð1dj ÞrDðTtÞþ/W T j;t
V /j;T ¼ V /j;t X t
e
d/
X Tj :
ð16Þ
Therefore, we can derive the moment-generating function for asset values:
h i 2 d / g j ð/Þ Et V /j;T ¼ V /j;t X t j eð1dj ÞrDðTtÞþ/ nj ðTtÞ=2 f ðdj /Þ:
ð17Þ
From the assumption that equity is valued as a European call option, we have the equity valuation formula: 3
The LRNVR is valid under certain types of utility functions and consumption processes. Refer to Duan (1995) for details.
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
Sj;t erDðTtÞ Et ½maxðV j;T Dj;T ; 0Þ " i/ # " i/ # ! Z Z Dj;T g j ði/ þ 1Þ Dj;T g j ði/Þ 1 erDðTtÞ 1 1 1 1 d/ Dj;t d/ ; þ Re Re ¼ V j;t þ i/ i/ 2 2 p 0 p 0
345
ð18Þ
where g j ðÞ is obtained from gj() by replacing k with 1/2 and c with c⁄ (c + k + 1/2).4 2.2. Estimation Parameter estimation proceeds in two steps. First, we estimate the common factor parameters {x,a,g,c} in the system of (3) and (4) via the maximum likelihood method given the common factor observations. We make two assumptions for the estimation of the parameters related to the asset return process of individual institutions. Following Ronn and Verma (1986) and Lehar (2005), we assume that the maturity of the implied call option is one year. It may be difficult to determine the matured debt amount within the assumed maturity of 1 year. To avoid arbitrary choices of the matured debt amount, we also assume that the debt amount is a constant proportion of the total liability, that is, Dj,t = jjLj,t where Lj,t denotes the total liability and jj the constant proportion. Second, for one institution at a time via the maximum likelihood method, we estimate the parameters Hj = {lj, dj, nj, jj} for individual institution j0 s asset return. Given institution j’s equity price data Sj = [Sj,1 Sj,n]0 , total liability data Lj = [Lj,1 Lj,n]0 , and common factor data x = [x1 xn]0 , we apply the results of Duan (1994), Duan (2000) and derive the following log likelihood function.
log LðSj jx; Lj ; Hj Þ ¼
n n n X X n1 1X @Sj;t logð2pÞ log V j;t log r2j;t log 2 2 t¼2 @V j;t t¼2 t¼2
n 1X fv j;t ðr D þ dj ðxt r DÞÞg2 ; 2 t¼2 r2j;t
ð19Þ
" # Z 1 jj Lj;t erDðTtÞ i/ ði/ þ 1Þg j ði/ þ 1Þ @Sj;t 1 erDðTtÞ 1 jj Lj;t d/ ¼ þ Re @V j;t 2 p V j;t 0 pV j;t i/ Z 1 i/ Re½ jj Lj;t erDðTtÞ g j ði/Þd/:
ð20Þ
0
In the above, Vj,t and rj,t are the solutions to (18) and (5), and
vj,t is the log return of Vj,t.
2.3. Dynamics of individual factors The unobservable individual factor wj,t may contain not only uncorrelated idiosyncratic factors but also common factors other than the common factor xt. Therefore, the individual factors may be correlated with each other. In particular, we assume that the vector consisting of individual factors wt [w1,t wN,t]0 follows a multivariate normal distribution with time-varying covariance matrix; i.e.,
wt MVN½0; Xt ;
ð21Þ
where (j, k) element of Xt is njk,t. Then, we employ a diagonal-vech model for the dynamics of Xt. Specifically, the covariance njk,t at time t between the unobservable factors wj,t and wk,t is determined by:
njk;t ¼ cjk þ ajk wj;t1 wk;t1 þ bjk njk;t1 :
ð22Þ
Under this specification, the conditional correlation between the asset returns of institutions j and k also is time-varying, i.e.,
4
The debt is assumed to grow at the risk-free interest rate, following Lehar (2005).
346
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
qjk;t ¼
dj dk ht þ njk;t
rj;t rk;t
:
ð23Þ
This can be regarded as a generalized specification compared to Lehar (2005) who uses a simple exponentially weighted moving average method by applying a predetermined common decay factor for all institutions. b j for institution j For the estimating the time-varying covariance matrix Xt, we first use estimates H ^ j;t , which are deto construct unobservable time series {Vj,t} and {vj,t} and then obtain the residuals w fined as:
^ j;t v j;t ðl ^ j þ ^dj ðxt rDÞÞ: w
ð24Þ
We adopt the estimation algorithm proposed by Ledoit et al. (2003) that produces a positive semidefinite conditional covariance matrix.
2.4. Systemic risk measures Following Lehar (2005), we adopt two risk measures: (i) the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, f, (ND), and (ii) the probability that the proportion of the assets of defaulting financial institutions to the assets of all the financial institutions exceeds a prescribed threshold, f, (WA), over a fixed time-horizon. By using these two measures that allow a clear economic interpretation, we will illustrate the temporal trend of the overall systemic risk level. Obviously, these two measures are not exhaustive, and other measures may be devised according to the objectives of systemic risk analysis. Since our model specifies the dynamics pertaining to both individual institutions and their inter-dependence, it may permit the calculation of not only the above two measures but also a wider range of analyses of systemic risk.
2.5. Simulation Following Lehar (2005), we set 6 months as the time-horizon for measuring systemic risk. We employ Monte Carlo simulation in order to calculate the systemic risk measures because no analytical solution is available for those measures over a multi-period time horizon. We draw a standard normal random variate and then simulate a hypothetical future common factor according to (3). For simulating individual factors, we draw multivariate normal random variates as specified by (21). We perform 100,000 repetitions for simulation.
3. Data We choose the largest 50 US financial institutions (based on the book values of the total assets).5 Market capitalizations as well as equity prices are collected on a weekly basis from January 1973 until March 2010, and total assets and total liabilities (book values) are obtained quarterly and linearly interpolated. In order to be free from survivorship bias, we also include inactive financial institutions in the sample. 5 Lehar (2005) also analyzes data using a sample consisting of the largest 50 banks in the North American region. In this analysis, we choose sample financial institutions among commercial banks, saving banks, cooperative banks, real estate and mortgage banks, investment banks, and bank-holding and other holding companies.
347
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 Table 1 Estimation results for the common factor. Parameter
x
a
c
g
k
g + ac2
h
Estimate (p-value)
3.1e6 0.000
1.73e5 0.000
62.247 0.000
0.866 0.000
0.768 0.000
0.933
12.63%
Notes: The GARCH model for the common factor is described in (3), (4). (g + ac2) indicates the degree of mean reversion, and h the annualized long-run volatility.
4. Empirical results 4.1. Estimation of the common factor process We construct the common factor xt by calculating the financial sector value-weighted stock index return comprised of the sample financial institutions.6 For the sake of parsimony, we model the common factor to follow Heston and Nandi’s (2000) GARCH(1, 1) process. Table 1 shows the results of parameter estimation. All parameters are significantly estimated. The degree of mean reversion is estimated as 0.933, which is less than one. The annualized long-run volatility is estimated as 12.63%. Fig. 1 demonstrates the temporal trends of both the common factor and its conditional volatility. The conditional volatility sharply increased during several periods: 1974 (the period of the first oil shock and the ensuing stock market crash period), October 1987 (Black Monday), August 1990 (the first Gulf War period), August/September 1998 (the period of declaration of the Russian moratorium and the distress concerning Long Term Capital Management), and 2007/2008 (the global financial crisis period). 4.2. Estimation for individual institutions Given the estimation results for the common factor, we estimate for institution j (j = 1, . . . , N) the parameter vector Hj by maximizing the log likelihood function (19). The coefficient dj indicates the sensitivity of institution j’s asset return to the common factor as expressed in (1). All the ds are positively estimated with a mean of 0.48, ranging from 0.175 to 1.175. Fig. 2 demonstrates the empirical distribution of the d estimates. Most estimates of d are significantly less than one. We can explain this feature from the following facts: (i) Eq. (1) relates an individual asset return to not an overall asset return but the overall equity return and (ii) equity is a call option with the asset value as the underlying asset and therefore, equity and asset values are nonlinearly related. To clearly understand the reason, we approximate the change in the call option value (i.e., equity price) with the option delta, that is,
DS ffi C DV;
ð25Þ
where C denotes the call option delta.7 Therefore, using the approximation, we can write
vffi
DV 1 S DS 1 S 1 S ¼ ffi s¼ ðc þ bx þ Þ; V CV S CV CV
ð26Þ
where the last equality indicates the CAPM expression. Therefore d in (1) is approximated as
dffi
1 S b: CV
ð27Þ
The CAPM b distributes around one; however, two additional components affect d. Fig. 3 illustrates the equity price as the call option price S against the underlying asset value V and the strike price of K. We 6 Alternatively, we may use relevant, existing sector indices, for example, the S&P North American financial services sector index. However, we believe that a sector index that is directly constructed from our sample can better represent the overall movements of our sample than other broad indices. 7 We ignore the option theta for simplicity in the approximation.
348
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
0.01
0.6
0.009
0.4 Common factor (left scale)
0.008
0.2 0.007 0
0.006 0.005
-0.2
0.004
-0.4 Conditional volatility (right scale)
0.003
-0.6 0.002 -0.8
0.001
-1 1974
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
0 2009
Fig. 1. The common factor (i.e., weekly growth rate of the financial sector stock index) and its conditional volatility. The conditional volatility is estimated based on the results for GARCH(1, 1), as shown in Table 1.
20 18 16
Frequency
14 12 10 8 6 4 2 0 0
0.2
0.4
0.6
δ
0.8
1
1.2
1.4
Fig. 2. Histogram of ds. The coefficient d indicates the sensitivity of a financial institution’s asset return to the common factor as expressed in (1).
can show that the product of these two components is always less than unity, therefore the additional components lower the value of d. In Fig. 3, the tangent line passing through the tangent point S intersects the horizontal axis at V0 ; thus, we write the option delta as C = S/(V V0 ). Therefore, the product of the two components becomes S/(CV) = (V V0 )/V, which is always less than unity since 0 < V0 < V for V > 0. Following Acharya et al. (2010), we classify the sampled 50 financial institutions into four subgroups: depository institutions, other non-depository institutions, insurance companies, and brokerdealers companies. Table 2 presents the mean value of the d estimates for each subgroup. Among the subgroups, depositories and insurance companies exhibit relatively lower sensitivity to the common factor than other non-depository institutions and broker-dealers companies.
349
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
60
Equity price
S
45 degree line
Tangent Line
0
0
V' K
V
Asset value
100
Fig. 3. Equity price vs. asset value.
Table 2 Estimation results for individual institutions.
Number Mean of ^ d ^ Mean of j
All
Depositories
Non-depository institutions
Insurance
Broker-dealers
50 0.480
33 0.427
8 0.631
2 0.363
7 0.589
0.166
0.090
0.322
0.100
0.361
Notes: The asset return model for an individual institution is described in (1). Here, j indicates a constant proportion of the matured debt amount relative to the total liabilities.
We assume that the debt amount matured within the maturity is a constant proportion of the total liability. Table 2 also reports the mean value of the proportion j. Depository institutions and insurance companies have a proportion of around 10% while other non-depository institutions and broker–dealers companies have a higher proportion, viz., more than 30% on average. To understand these empirical results, we may conjecture several factors that can affect the determination of the proportion j. As the maturity of the call option is assumed to be longer, the proportion j rises. As short-term debt accounts for a larger share of the total liabilities, the proportion j also increases, which may differentiate the proportion j across subgroups. Another factor may be related with funding stability. The matured debt may be paid by a liquidation of assets or funding other debts (via the receipt of deposits or the sale of debt instruments). As this repayment/funding cycle is regarded as less stable, the proportion j becomes larger too; therefore, the funding stability may also differentiate the proportion j across subgroups according to their business activities. The changes in an individual asset return are driven by two factors: the common factor and the individual factor. The variance of the asset return can be decomposed into two parts that are contributed by the two factors according to (5). Fig. 4 illustrates the empirical distribution of the average contribution of the common factor to the total variation over the sample period. The common factor proves its importance in explaining the variation in the asset return: on average, 19.2% (median of 18.0%) of the total variation in the asset return is attributable to variation in the common factor. 4.3. Correlations between asset returns Asset correlation is a major aspect that financial regulators should take into account regarding systemic risk. In particular, if the asset correlation is time-varying, then the asset correlation channel for
350
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
12
10
Frequency
8
6
4
2
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Contribution of common factor to the variation of asset return Fig. 4. Histogram of the average contribution of the common factor to the variation of the asset return.
1 0.9
total
0.8 0.7 0.6 0.5 0.4 0.3
common factor individual factor
0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 5. Correlations between the asset returns of individual institutions. The correlation coefficient between the asset returns of institutions j and k is obtained according to (23). This figure illustrates the median correlation of all possible bilateral correlation coefficients at each point in time. The median contributions by the common factor and individual factors are also illustrated.
systemic risk also becomes time-varying. Fig. 5 demonstrates the median correlation over all possible bilateral correlation coefficients between the asset returns of individual institutions. The correlation coefficient between the asset returns of institutions j and k is obtained according to (23). In addition, we decompose the correlation into two parts: the contribution by the common factor and that by the individual factor. The median correlation coefficient illustrates a high degree of positive correlation, fluctuating within a range of 0.40–0.88 with a mean of 0.61 (median of 0.59). The common factor contributes more to the asset correlation than the individual factors. On average, the common factor accounts for 0.44 while the individual factors accounts for 0.14 in the median asset correlation.
351
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 6. Systemic risk (ND): the benchmark factor-augmented model. Systemic risk is defined as the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, viz., f = 10% or 20%. The upper line corresponds to f = 10% and the lower one to f = 20%.
4.4. Systemic risk under the benchmark factor-augmented model We choose as our benchmark model the aforementioned factor-augmented model, use it for Monte Carlo simulation, and obtain two measurements of systemic risk. The default event of an individual institution is defined as the asset value’s being less than the debt level within the assumed time horizon, i.e., Vj,t+s < jjLj,t+s for s 6 0.5 years. We present two measures of systemic risk: (i) the probability that the proportion of the number of defaulting financial institutions to the total number of financial institutions exceeds a prescribed threshold, f, (ND), and (ii) the probability that the proportion of the assets of defaulting financial institutions to the assets of all the financial institutions exceeds a prescribed threshold, f, (WA). The two measures are presented according to two systemic-risk thresholds, viz., f = 10% and 20%. The total number of sampled financial institutions varies from 15 to 50 during the sample period; therefore, the systemic-risk measure ND with f = 10% corresponds to the event that the number of defaulting institutions is more than one during an early period but more than five during a later period. Fig. 6 shows the systemic-risk measurements based on the number of distressed financial institutions (ND). Consistent with our intuition, as the threshold f becomes higher, the systemic risk measure decreases. The temporal trend in systemic risk indicates that a sharp increase in systemic risk occurred several times during the sample period: 1974, 1980, and 2008/2009 (the global financial crisis). The historically highest systemic risk level with the longest duration was recorded during the recent global financial crisis. Fig. 7 illustrates measurements of the systemic risk based on the weighted assets of defaulting financial institutions (WA). The WA measure also sharply increased in the same periods where the ND measure sharply increased; however, it also sharply increased at other times. This difference implies that the size effect on the measurement of systemic risk can be great. The historically highest systemic risk level with the longest duration under the WA measure was also recorded during the recent global financial crisis. Fig. 8 demonstrates the two systemic-risk measurements during the recent financial crisis period along with several specific dates: August 2007 (hedge fund failure and the start of crisis), March 2008 (acquisition of Bear Sterns by JPMorgan Chase), September 2008 (Lehman Brothers failure), March 2009 (stock market bottom). The systemic-risk measurements largely match up with those specific events; however, the pattern of the time variation depends on the choice of criterion. For example, the systemic-risk measure ND with a low threshold of 5% sensitively responds to a shock from an early
352
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
2010
Fig. 7. Systemic risk (WA): the benchmark factor-augmented model. Systemic risk is defined as the probability that the proportion of the assets of defaulting financial institutions to the assets of all financial institutions exceeds a prescribed threshold, viz., f = 10% or 20%. The upper line corresponds to f = 10% and the lower one to f = 20%.
1 Lehman Bros. fail
0.9 0.8
hedge funds fail Bear Sterns fail
0.7 0.6 0.5 0.4
WA (5%) WA (10%)
ND (5%)
0.3 0.2 ND (10%)
0.1 0 2007
2008
stock market bottom
2009
2010
Fig. 8. Systemic risk measures during the recent crisis period. Two systemic risk measures, ND and WA, under the benchmark factor-augmented model are displayed with two thresholds, viz., f = 5% or 10%. Several specific event dates are also demonstrated.
stage of the crisis and remains high until a late stage. In contrast, the ND with a relatively high threshold of 10% responds late to a shock but quickly subsides. Fig. 9 contrasts the systemic-risk measurements during the recent financial crisis period with several commonly used risk indicators: VIX (market volatility), the credit spread between Baa- and Aaarated yields (credit risk), and the difference between 3-month LIBOR and OIS (overnight index swap) (liquidity risk). Among the usual risk indicators, the VIX exhibits the most volatile movements; however, compared to the ND measure, it seems to subside too early because it greatly decreased even though the stock price stayed bottom. Compared to the ND measure, the credit risk indictor insensitively responded to shocks. The liquidity risk indicator too abruptly responded and then subsided too early.
353
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 1
2
Lehman Bros. fail
0.8
hedge funds fail
1.8
Bear Sterns fail
0.6
1.6 ND (10%)
0.4
1.4
ND (5%)
0.2
1.2 stock market bottom
1
0
-0.2
0.8
VIX
-0.4
Baa - Aaa
0.6
LIBOR - OIS
-0.6
0.4
-0.8
0.2
-1
0
2007
2008
2009
2010
Fig. 9. Systemic risk measures versus other risk indicators. The systemic risk measure ND (left scale) is displayed with two thresholds, viz., f = 5% or 10%. Commonly used risk indicators (right scale), VIX (market volatility, 102), Baa – Aaa (credit risk, 101), and LIBOR – OIS (liquidity risk, 101), are also demonstrated.
4.5. Systemic risk under alternative models We introduce a common factor in order to effectively capture time-varying characteristics in both volatilities and correlations between asset returns. To assess the effect of the common factor on the measurement of systemic risk, we calculate both the ND and the WA measures based on an unfactored model and compare the results with those of the benchmark model. The unfactored model does not contain a common factor, i.e.,
1
0.9
Unfactered but correlated model
0.8 0.7
Factered but uncorrelated model
0.6 0.5 0.4 0.3 0.2
Unfactered and uncorrelated model
0.1 0 -0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 10. Systemic risk (ND): alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008. This figure produces scatter plots of the ND measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model.
354
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
1 0.9 Unfactered but correlated model
0.8
Factered but uncorrelated model
0.7 0.6 0.5 0.4 0.3
Unfactered and uncorrelated model
0.2 0.1 0 -0.1 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 11. Systemic risk (WA): alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008. This figure produces scatter plots of the WA measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model.
v j;t ¼ lj D þ wj;t :
ð28Þ
The other variation in model specification is whether or not to allow correlations between individual factors. We consider this variation in order to assess the effect of correlations between individual factors on the measurement of systemic risk. Therefore, we have four alternative model specifications: (i) (the benchmark) model with a common factor and correlated individual factors, (ii) a model without a common factor but with correlated individual factors, (iii) a model with a common factor but with uncorrelated individual factors, and (iv) a model without a common factor and with uncorrelated individual factors. We re-estimate the parameters of each alternative model specification. Fig. 10 demonstrates the effects of alternative model specifications against the benchmark model during the financial crisis period of July 2007 to December 2008 under the ND measure. In particular, Fig. 10 produces scatter plots of the ND measures at each point in time under the benchmark model on the horizontal axis and under alternative models on the vertical axis. The point below (above) the 45 degree line indicates that the alternative model generates a lower (higher) level of systemic risk than the benchmark model. Compared to the benchmark model, the two alternative models deliver much lower measurements of systemic risk: the model with a common factor but with uncorrelated individual factors (Specification iii), and the model without a common factor and with uncorrelated individual factors (Specification iv). This fact implies that correlations between individual factors can significantly affect the measurement of systemic risk. On the other hand, the other alternative model without a common factor but with correlated individual factors (Specification ii) yields higher measurements of systemic risk than the benchmark model.8 8 Our guess about the reason is as follows. The individual factors tend to have less persistent conditional variances than the common factor. The benchmark model differentiates the two factors, whereas the individual-factor-only model does not. Therefore, the persistence of the conditional variance of the unique factor under the individual-factor-only model may lie between those of the two factors under the benchmark model. During the recent financial crisis period, not only a large negative common shock hit all the financial institutions but negative individual shocks hit several unsound institutions as well. The individual-factoronly model does not distinguishes the two shocks and attributes more persistent conditional variance to negative individual shocks for unsound institutions than the benchmark model. This heightened and persistent conditional variance for fragile institutions might generate higher estimates of systemic risk.
355
Equity return: July 2007 to December 2008
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2
0
1
2
3
4
5
6
7 -3
x 10
Equity return: July 2007 to December 2008
Δ ES: July 2006 to June 2007 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 -3
x 10
Δ ES: July 2006 to June 2007 Fig. 12. The ex-ante DESj5% measures versus the realized risks. This figure demonstrates the scatter plots of the ex-ante DESj5% measures against the realized risks. The upper panel contains all the sample institutions while the lower panel excludes two outlier institutions.
Fig. 11 demonstrates the effects of alternative model specifications against the benchmark model under the WA measure during the same period as in Fig. 10. Fig. 11 exhibits quantitatively different but qualitatively similar results as those in Fig. 10.
5. Systemic risk contribution Until now, we have proposed a model to specify the dynamics for individual institutions and their inter-dependence and use it to illustrate measurements of the overall systemic risk. However, aspects other than the estimation of the overall systemic risk level also have been investigated. Systemic risk contribution analysis can be such an example. Recently, Adrian and Brunnermeier (2010) proposed a measure for systemic risk, called CoVaR, which is the value at risk (VaR) of the financial system conditional on institutions’ being in distress. They define an institution’s contribution to systemic risk as the difference between the CoVaR conditional on the institution’s being in distress and the CoVaR at the median state of the institution. They also propose another measure with reverse direction, called exposure CoVaR, which measures the extent to which an individual institution is affected by systemic financial events. Acharya et al. (2010) present a simple model of systemic risk and show that each financial institution’s contribution to systemic risk can be measured as its systemic expected shortfall (SES), i.e., its propensity to be undercapitalized when the system as a whole is undercapitalized.9 9 We want to mention that both the previous studies do not explicitly define default events, which makes it difficult to measure an overall systemic risk level. In fact, both loosely define the systemic risk by using worst market outcome events. In this respect, the model proposed in this paper has an advantage of providing overall systemic risk measurements based on an explicit definition of default events.
356
S. Suh / J. Finan. Intermediation 21 (2012) 341–358 -3
-3
x 10
1.8
6
Δ ES: July 2006 to June 2007
Δ ES: July 2006 to June 2007
7
5 4 3 2 1 1
2
3
4
5
6
x 10
7
x 10
Δ ES: July 2005 to June 2006
1.4 1.2 1 0.8 0.6
2
Δ ES: July 2007 to December 2008
4 2 0 -2 -4 1
2
3
4
5
6
0.8
1
1.2
1.4
7
x 10
Fig. 13. The stability of the outlier institutions.
DESj5%
1.8 -3 x 10
1.6
1.8 -3 x 10
-3
1.5
1
0.5
0 0.4
0.6
0.8
1
1.2
1.4
-3
Δ ES: July 2006 to June 2007
1.6
Δ ES: July 2005 to June 2006
-3
0
0.6
-3
6
Δ ES: July 2007 to December 2008
1.6
0.4 0.4
0 0
x 10
x 10
Δ ES: July 2006 to June 2007
measure. The left panels contain all the sample institutions while the right panels exclude
Motivated by these previous studies and in order to show the applicability of the model proposed in this paper for systemic risk contribution analysis, we modify the SES measure as follows:
# " # net assetj1 median worst E ¼E 1 C5% 1 C5% ; net assetj0 net assetj0 "
DESj5%
net assetj1
ð29Þ
where Cmedian and Cworst denote the set of events that the future common factor belongs to the middle 5% 5% 5% and lowest 5% regions in the distribution, respectively. The DESj5% measures institution j0 s systemic risk contribution by the difference between the expected value of the net asset (the asset value minus the matured debt) return conditional on the median state of the common factor and the expected value conditional on the worst outcome (say, 5%) of the common factor. The DESj5% measure is calculated via Monte Carlo simulation as in the calculation of the overall systemic risk measures. We want to note that the common factor also plays a crucial role as a conditioning variable in this systemic risk contribution analysis while the unfactored model has no such conditioning variable in which respect we may regard the unfactored model as less preferable to the factor-augmented benchmark model. Like Acharya et al. (2010), we first calculate the ex-ante DESj5% measure during the pre-crisis period of July 2006 to June 2007 and then examine whether this measure has any ability to predict the realized risk (defined by the losses in equity return) during the crisis period of July 2007 to December 2008. Fig. 12 demonstrates scatter plots of the ex-ante DESj5% measures against the realized risks. Consistent with Acharya et al. (2010), there exists a negative relationship between both ex-ante DESj5% measures and the realized risks, thus confirming the fact that the ex-ante DESj5% measure can predict the future risk. We also examine whether the ex-ante DESj5% measure is stable over time based on the closeness of the relationship between the DESj5% measures during the period of July 2006 to June 2007 and during the period of July 2005 to June 2006. Fig. 13 shows that the DESj5% measures during both pre-crisis
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
357
periods are positively and closely related. However, this positive relationship remains but becomes less tight during the crisis period as illustrated in the lower panels in Fig. 13. 6. Conclusions In this paper, we extend existing correlated default models for measuring systemic risk by proposing a model that incorporates a common factor and an individual factor. The common factor is observably constructed and featured with conditional heteroscedasticity. The addition of the common factor helps to effectively capture realistic time-varying characteristics in individual asset return volatilities as well as return correlations. The individual factors are correlated with each other and have timevarying conditional variances. We apply the model for a sample consisting of large US financial institutions. The common factor proves its importance in explaining asset return dynamics and measuring systemic risk. We also apply the model in the context of systemic risk contribution analysis and show its applicability. The analysis in this paper is extendable in several directions. We apply the model for the US case, but this method also can be applied for other regions or for international comparison. Moreover, the inclusion of the common factor also facilitates stress-test analysis. Hypothetical scenarios can be realistically constructed by making various assumptions on the common factor. We can also extend the model by adding more relevant common factors. It may be another extension to make the sensitivity to the common factor be time-varying. Acknowledgements I have benefited from helpful comments from two anonymous referees, Viral V. Acharya (Editor), Hyun Song Shin, and seminar participants at Hanyang University (ERICA Campus), Yeungnam University, 2010 Conference at The Korean Financial Engineering Society, Korea Institute of Finance, and the Bank of Korea. References Acharya, V.V., Pedersen, L.H., Philippon, T., Richardson, M., 2010. Measuring systemic risk. Working paper. Adrian, T., Brunnermeier, M.K., 2010. CoVaR. Federal Reserve Bank of New York Staff Reports No. 348. Angelini, P., Maresca, G., Russo, D., 1996. Systemic risk in the netting system. J. Bank. Finance 20, 853–868. Boss, M., Elsinger, H., Summer, M., Thurner, S., 2004. The network topology of the interbank market. Quant. Finance 4, 677–684. Cohen, H, Roberds, W., 1993. Towards the systematic measurement of systemic risk. Working paper 93-14, Federal Reserve Bank of Atlanta. Crabbe, L.E., 1991. Event risk: an analysis of losses to bond holders and ‘‘super poison put’’ bond covenants. J. Finance 46, 689– 706. De Nicolo, G., Kwast, M.L., 2002. Systemic risk and financial consolidation: are they related? J. Bank. Finance 26, 861–880. DeAngelo, H., DeAngelo, L., Gilson, S.C., 1994. The collapse of first executive corporation: junk bonds, adverse publicity, and the run on the bank phenomenon. J. Finance Econ. 36, 287–336. Duan, J.-C., 1994. Maximum likelihood estimation using the price data of the derivative contract. Math. Finance 4, 155–167. Duan, J.-C., 1995. The GARCH option pricing model. Math. Finance 5, 13–32. Duan, J.-C., 2000. Correction: maximum likelihood estimation using the price data of the derivative contract. Math. Finance 10, 461–462. Duffie, D., Lando, D., 2001. Term structure of credit spreads with incomplete accounting information. Econometrica 69, 633– 664. Eichengreen, B., Mody, A., 2000. What explains the changing spreads on emerging market dedt? In: Edwards, S. (Ed.), The Economics of International Capital Flows. University of Chicago Press. Elsinger, H., Lehar, A., Summer, M., 2006. Risk assessment for banking systems. Manage. Sci. 52, 1301–1314. Fenn, G.W., Cole, R.A., 1994. Announcements of asset-quality problems and contagion effects in the life insurance industry. J. Finance Econ. 35, 181–198. Furfine, C.H., 2003. Interbank exposures: quantifying the risk of contagion. J. Money, Credit, Banking 35, 111–128. Helwege, J., 1996. Determinants of S& L failure rates: estimates of a time-varying proportional hazard function. J. Finance Serv. Res. 7, 55–61. Helwege, J., 2010. Financial firm bankruptcy and systemic risk. J. Int. Finance Markets, Inst. Money 20, 1–12. Helwege, J., Kleiman, P., 1997. Understanding aggregate default rates of high-yield bonds. J. Fixed, Income 7, 55–61. Heston, S.L., Nandi, S., 2000. A closed-form GARCH option valuation model. Rev. Finance Stud. 13, 585–625. Hoggarth, G., Reis, R., Saporta, V., 2002. Costs of banking system instability: some empirical evidence. J. Bank. Finance 26, 825– 855.
358
S. Suh / J. Finan. Intermediation 21 (2012) 341–358
Huang, X., Zhou, H., Zhu, H., 2009. A framework for assessing the systemic risk of major financial institutions. J. Bank. Finance 33, 2036–2049. Humphrey, D.B., 1986. Payments finality and risk of settlement failure. In: Saunders, Anthony, White, Lawrence (Eds.), Technology, and the Regulation of Financial Markets: Securities, Futures, and Banking. Lexington Books, Lexington, pp. 97– 120. Jorion, P., Zhang, G., 2007. Good and bad credit contagion: evidence from credit default swaps. J. Finance Econ. 84, 860–883. Jorion, P., Zhang, G., 2009. Credit contagion from counterparty risk. J. Finance 64, 2053–2087. Lang, L.H.P., Stulz, R.M., 1992. Contagion and competitive intra-industry effects of bankruptcy announcements. J. Finance Econ. 32, 45–60. Ledoit, O., Santa-Clara, P., Wolf, M., 2003. Flexible multivariate GARCH modeling with an application to international stock markets. Rev. Econ. Stat. 85, 735–747. Lehar, A., 2005. Measuring systemic risk: a risk management approach. J. Bank. Finance 29, 2577–2603. Martinez-Jaramillo, S., Perez, O.M., Embriz, F.A., Dey, F.L.G., 2010. Systemic risk, financial contagion and financial fragility. J. Econ. Dyn. Control 34, 2358–2374. Merton, R., 1974. On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29, 449–470. Muller, J., 2006. Interbank credit lines as a channel of contagion. J. Finance Serv. Res. 29, 37–60. Nier, E., Yang, J., Yorulmazer, T., Alentorn, A., 2006. Network models and financial stability. J. Econ. Dyn. Control 31, 2033–2060. Nijskens, R., Wagner, W., 2011. Credit risk transfer activities and systemic risk: how banks become less risky individually but posed greater risks to the financial system at the same time. J. Bank. Finance 35, 1391–1398. Rochet, J-C., Tirole, T., 1996. Interbank lending and systemic risk. J. Money, Credit, Banking 28, 733–762. Ronn, E.I., Verma, A.K., 1986. Pricing risk-adjusted deposit insurance: an option-based model. J. Financ. 41, 871–895. Upper, C., 2007. Using counterfactual simulations to assess the danger of contagion in interbank markets. BIS Working Papers 234, the Bank for International Settlements. Yu, F., 2005. Accounting transparency and the term structure of credit spreads. J. Finance Econ. 75, 53–84.