Testing for Systemic Risk Using Stock Returns

Abstract

The literature proposes several stock return-based measures of systemic risk but does not include a classical hypothesis tests for detecting systemic risk. Using a joint null hypothesis of Gaussian returns and the absence of systemic risk, we develop a hypothesis test statistic to detect systemic risk in stock returns data. We apply our tests on conditional value-at-risk (CoVaR) and marginal expected shortfall (MES) estimates of the 50 largest US financial institutions using daily stock return data between 2006 and 2007. The CoVaR test identifies only one institution as systemically important while the MES test identifies 27 firms including some of the financial institutions that experienced distress in the past financial crisis. We perform a simulation analysis to assess the reliability of our proposed test statistics and find that our hypothesis tests have weak power, especially tests using CoVaR. We trace the power issue to the inherent variability of the nonparametric CoVaR and MES estimators that have been proposed in the literature. These estimators have large standard errors that increase as the tail dependence in stock returns strengthens.

Appendix

1.1 Parametric estimator for ∆CoVaR when returns are gaussian

In this section, we derive the CoVaR measure for the returns on a market portfolio, ${\tilde{R}}_M,$ conditional on a specific stock return, ${\tilde{R}}_j,$ equal to its 1 % VaR. The market portfolio conditional return distribution is given by,

$$ {\tilde{R}}_M\left|\left({\tilde{R}}_j={\mathrm{F}}^{-1}\left(.01,\ {\tilde{R}}_j\right)\right) \sim N\left[{\mu}_M+\rho \frac{\sigma_M}{\sigma_j}\left({\Phi}^{-1}\left(.01,\ {\tilde{R}}_j\right) - {\mu}_j\right),\ \left(1-{\rho}_{jM}^2\right){\sigma}_M^2\right], \right. $$

(A1)

where ${\mathrm{F}}^{-1}\left(.01,\ {\tilde{R}}_j\right)$ represents the inverse cumulative normal distribution of the unconditional return ${\tilde{R}}_j$ evaluated at the 0.01 cumulative probability. Using, ${\mathrm{F}}^{-1}\left(.01,\ {\tilde{R}}_j\right)={\mu}_j-2.32635\ {\sigma}_j$, the 1 % CoVaR for the market conditional on ${\tilde{R}}_j$ equal to its 1 % VaR is,

$$ CoVaR\left(\left.{\tilde{R}}_M\right|\left({\tilde{R}}_j={\mathrm{F}}^{-1}\left(.01,\ {\tilde{R}}_j\right)\right)\right)={\mu}_M-{\rho}_{jM}\frac{\sigma_M}{\sigma_j}\left(2.32635\ {\sigma}_j\right)-2.32635\ {\sigma}_M\sqrt{1-{\rho}_{jM}^2} $$

(A2)

Similarly, the return distribution for ${\tilde{R}}_M,$ conditional on ${\tilde{R}}_j$ equal to its median is,

$$ \left.{\tilde{R}}_M\right|\left({\tilde{R}}_j={\mathrm{F}}^{-1}\left(.50,\ {\tilde{R}}_j\right)\right) \sim N\left[{\mu}_M,\ \left(1-{\rho}_{jM}^2\right){\sigma}_M^2\right] $$

(A3)

Consequently, the CoVaR for the portfolio with ${\tilde{R}}_j$ evaluated at its median return is,

$$ CoVaR\left(\left.{\tilde{R}}_M\right|\left({\tilde{R}}_j={\mathrm{F}}^{-1}\left(.50,\ {\tilde{R}}_j\right)\right)\right)={\mu}_M-2.32635{\sigma}_M\sqrt{1-{\rho}_{jM}^2} $$

(A4)

Subtracting (A4) from (A2) gives the so-called contribution CoVaR measure, ∆CoVaR,

$$ \varDelta CoVaR\left(\left.{\tilde{R}}_M\right|\left({\tilde{R}}_j={\mathrm{F}}^{-1}\left(.01,\ {\tilde{R}}_j\right)\right)\right)=-2.32635\ {\rho}_{jM}{\sigma}_M $$

(A5)

1.2 Parametric estimator for MES when returns are gaussian

The marginal expected shortfall measure is the expected shortfall calculated from a conditional return distribution. The conditioning event is the return on a market portfolio, ${\tilde{R}}_M,$ less than or equal to its 5 % VaR value.

Under the assumption of bivariate normality, the conditional stock return is normally distributed, and consequently,

$$ E\left(\left.{\tilde{R}}_j\right|{\tilde{R}}_M={r}_M\right)={\mu}_j-\rho \frac{\sigma_j}{\sigma_M}\ {\mu}_M+\rho \frac{\sigma_j}{\sigma_M}{r}_M. $$

(A6)

Now, if ${\tilde{R}}_M$ is normally distributed with mean μ _M and standard deviation σ _M , then the expected value of the market return truncated above the value “b” is

$$ E\left(\left.{\tilde{R}}_M\right|{\tilde{R}}_M<b\right)={\mu}_M-{\sigma}_M\left[\frac{\phi \left(\frac{b-{\mu}_M}{\sigma_P}\right)}{\phi \left(\frac{b-{\mu}_M}{\sigma_M}\right)}\right], $$

(A7)

If b is the lower 5 % tail value, b = μ _M  − 1.645σ _M , the expected shortfall measure becomes:

$$ \begin{array}{c}\hfill E\left(\left.{\tilde{R}}_j\right|{\tilde{R}}_M<VaR\left({\tilde{R}}_M,\ 5\%\right)\right)={\mu}_j-\rho {\sigma}_j\left[\frac{\phi \left(-1.645\right)}{\Phi \left(-1.645\right)}\right]\hfill \\ {}\hfill ={\mu}_j-2.062839\ {\rho}_{jM}{\sigma}_M\hfill \end{array} $$

(A8)

1.3 Bivariate skew-t distribution

We use the definition of the multivariate skew-t definition given in Azzalini (2005). Let d be the dimension of the multivariate distribution. Let y be a (d × 1) random vector, and β and α (d × 1) vectors of constants. Ω is a (d × d) positive definite matrix. ν ∈ (0, ∞) be a scalar degrees of freedom parameter. The multivariate skew-t density y ~ f _T (y, β, Ω, α, ν) is defined as:

$$ {f}_T\left(y,\beta, \Omega, \alpha, \nu \right)=2{t}_d\left(y;\beta, \varOmega, \nu \right){T}_1\left({\alpha}^T{\omega}^{-1}\left(y-\beta \right){\left(\frac{\nu +d}{Q_y+\nu}\right)}^{0.5};\nu +d\right) $$

(20)

where,${t}_d\left(y;\beta, \varOmega, \nu \right)=\frac{\varGamma \left(0.5\left(\nu +d\right)\right)}{{\left|\varOmega \right|}^{0.5}{\left(\pi \nu \right)}^{d/2}\varGamma \left(0.5\nu \right){\left(1+{Q}_y/\nu \right)}^{\left(\nu +d\right)/2}}$is the density function of a d-dimensional student t random variate with ν degrees of freedom. T ₁ (x;ν + d) denotes the scalar t distribution with ν + d degrees of freedom; β is the location parameter that controls the distribution means; α is the parameter that controls the skewness of the distribution²³; and, Ω is a generalized covariance matrix.²⁴ The remaining parameters are, ω = diag(Ω)^0.5 and Q _y  = (y − β)^T Ω ^− 1(y − β).

The skew-t distribution nests the symmetric t-distribution, (α = 0 , 0 < ν < ∞), the skew-Gaussian distribution, (α ≠ 0, ν = ∞), and the symmetric Gaussian distribution, (α = 0, ν = ∞). The flexibility of the skew-t distribution is ideal for evaluating the behavior of the W _ΔCoVaR and W _MES test statistics under a number of alternative hypothesis that generate realistic representations of stock return data.