Analysis of the SRISK measure and its application to the Canadian banking and insurance industries

Abstract

In this paper, we analyse, modify, and apply one of the most widely used measures of systemic risk, SRISK, developed by Brownlees and Engle (in Rev Financ Stud 30:48–79, 2016). The measure is defined as the expected capital shortfall of a firm conditional on a prolonged market decline. We argue that segregated funds, also known as separate accounts in the US, should be excluded from actuarial liabilities when SRISK is calculated for insurance companies. We also demonstrate the importance of careful analysis of accounting standards when specifying the prudential capital ratio used in SRISK methodology. Based on the proposed adjustments to SRISK, we assess the systemic risk of the Canadian banking and insurance industries. It is shown that in its current implementation, the SRISK methodology substantially overestimates the systemic risk of Canadian insurance companies.

Appendices

Appendix A: GARCH-DCC model

Let $r_{it} = \log \left( {1 + R_{it} } \right)$ and $r_{mt} = \log \left( {1 + R_{mt} } \right)$ represent the logarithmic returns of the firm and the market, respectively. We assume that conditional on the information set ${\mathcal{F}}_{t - 1}$ available at time $t - 1$, the return pair has an unspecified distribution, ${\mathcal{D}}$, with zero mean and time varying covariance,

$$ \left. {\left[ {\begin{array}{*{20}c} {r_{it} } \\ {r_{mt} } \\ \end{array} } \right]} \right|{\mathcal{F}}_{t - 1} \sim{\mathcal{D}}\left( {0, \left[ {\begin{array}{*{20}c} {\sigma_{it}^{2} } & {\rho_{it} \sigma_{it} \sigma_{mt} } \\ {\rho_{it} \sigma_{it} \sigma_{mt} } & {\sigma_{mt}^{2} } \\ \end{array} } \right]} \right) $$

We use the GJR-GARCH volatility model and the standard DCC correlation model (Glosten et al. 1993). The GJR-GARCH model equations for the volatility dynamics are as follows:

$$ \begin{aligned} \sigma_{it}^{2} & = \omega_{Vi} + \alpha_{Vi} r_{it - 1}^{2} + \gamma_{Vi} r_{it - 1}^{2} I_{it - 1}^{ - } + \beta_{Vi} \sigma_{it - 1}^{2} , \\ \sigma_{mt}^{2} & = \omega_{Vm} + \alpha_{Vm} r_{mt - 1}^{2} + \gamma_{Vm} r_{mt - 1}^{2} I_{mt - 1}^{ - } + \beta_{Vm} \sigma_{mt - 1}^{2} , \\ \end{aligned} $$

with $I_{it}^{ - } = 1$ if $\left\{ {r_{it} < 0} \right\}$ and $I_{mt}^{ - } = 1$ if $\left\{ {r_{mt} < 0} \right\}$. The DCC specification models correlation through the volatility adjusted returns $\epsilon_{it} = r_{it} /\sigma_{it}$ and $\epsilon_{mt} = r_{mt} /\sigma_{mt}$

$$ Cor\left( {\begin{array}{*{20}c} {\epsilon_{it} } \\ {\epsilon_{mt} } \\ \end{array} } \right) = R_{t} = \left[ {\begin{array}{*{20}c} 1 & {\rho_{it} } \\ {\rho_{it} } & 1 \\ \end{array} } \right] = diag\left( {Q_{it} } \right)^{ - 1/2} Q_{it} diag\left( {Q_{it} } \right)^{ - 1/2} , $$

where $Q_{it}$ is the so-called pseudo correlation matrix. The DCC model then specifies the dynamics of the pseudo-correlation matrix $Q_{it}$ as

$$ Q_{it} = \left( {1 - \alpha_{Ci} - \beta_{Ci} } \right)S_{i} + \alpha_{Ci} \left[ {\begin{array}{*{20}c} {\epsilon_{it - 1} } \\ {\epsilon_{mt - 1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\epsilon_{it - 1} } \\ {\epsilon_{mt - 1} } \\ \end{array} } \right]^{'} + \beta_{Ci} Q_{it - 1} , $$

where $S_{i}$ is the unconditional correlation matrix of the firm and market adjusted returns.

Appendix B: LRMES estimation

The following bootstrap approach is used to evaluate LRMES (see Brownlees and Engle 2016).

• 1. Construct the GARCH-DCC standardized innovations

  $$ \epsilon_{mt} = \frac{{r_{mt} }}{{\sigma_{mt} }}\quad {\text{and}}\quad \xi_{it} = \left( {\frac{{r_{it} }}{{\sigma_{it} }} - \rho_{it} \frac{{r_{mt} }}{{\sigma_{mt} }}} \right)/\sqrt {1 - \rho_{it}^{2} } , $$

  for each $t = 1, \ldots ,T.$

• 2. Sample with replacement $S \times h$ pairs of standardized innovations $\left[ {\xi_{it} , \epsilon_{mt} } \right]^{\prime }$. Use these to construct $S$ pseudo samples of GARCH-DCC innovations from period $T + 1$ to period $T + h$, that is

  $$ \left[ {\begin{array}{*{20}c} {\xi_{iT + t}^{s} } \\ {\epsilon_{mT + t}^{s} } \\ \end{array} } \right]_{t = 1, \ldots , h} s = 1, \ldots , S. $$

• 3. Use the pseudo samples of GARCH-DCC innovations as inputs of the DCC and GARCH filters respectively using as initial conditions the last values of the conditional correlation $\rho_{iT}$ and variances $\sigma_{iT}^{2}$ and $\sigma_{mT}^{2}$. This step delivers $S$ pseudo samples of GARCH-DCC logarithmic returns from period $T + 1$ to period $T + h$ conditional on the realized process up to time $T$, that is

  $$ \left. {\left[ {\begin{array}{*{20}c} {r_{iT + t}^{s} } \\ {r_{mT + t}^{s} } \\ \end{array} } \right]_{t = 1, \ldots , h} } \right|{\mathcal{F}}_{T} s = 1, \ldots ,S. $$

• 4. Construct the multi-period arithmetic firm return of each pseudo sample

  $$ R_{iT + 1:T + h}^{s} = exp\left\{ {\mathop \sum \limits_{t = 1}^{h} r_{iT + t}^{s} } \right\} - 1 , $$

  and compute the multi-period arithmetic market return $R_{mT + 1:T + h}^{s}$ analogously.

• 5. Compute $LRMES$ as the Monte Carlo average of the simulated multi-period arithmetic returns conditional on the systemic event

  $$ LRMES_{iT} = - \frac{{\mathop \sum \nolimits_{s = 1}^{S} R_{iT + 1:T + h}^{s} I\left\{ {R_{mT + 1:T + h}^{s} < C} \right\}}}{{\mathop \sum \nolimits_{s = 1}^{S} I\left\{ {R_{mT + 1:T + h}^{s} < C} \right\}}}. $$