Estimation of Systemic Shortfall Risk Measure using Stochastic Algorithms

TL;DR

Stochastic algorithms schemes are used in estimating MSRM and it is proved that the resulting estimators are consistent and asymptotically normal.

Abstract

Systemic risk measures were introduced to capture the global risk and the corresponding contagion effects that is generated by an interconnected system of financial institutions. To this purpose, two approaches were suggested. In the first one, systemic risk measures can be interpreted as the minimal amount of cash needed to secure a system after aggregating individual risks. In the second approach, systemic risk measures can be interpreted as the minimal amount of cash that secures a system by allocating capital to each single institution before aggregating individual risks. Although the theory behind these risk measures has been well investigated by several authors, the numerical part has been neglected so far. In this paper, we use stochastic algorithms schemes in estimating MSRM and prove that the resulting estimators are consistent and asymptotically normal. We also test numerically the performance of these algorithms on several examples.

Figures (13)

• Figure 1: Consistency of RM/PR estimators with for different values of $\rho$.
• Figure 2: Consistency of RM estimators with $c = 0.1$ for different values of $\rho$.
• Figure 3: Computational time (in seconds) as a function of the (average) absolute error (log log scale), for the RM algorithm ($\gamma=1$, $c=2$ and $\rho=0.5$).
• Figure 4: Convergence of the estimator $V_n$.
• Figure 5: Empirical cumulative density function of $m_n - m^*$.

Theorems & Definitions (35)

• Definition 1.1
• Example 1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 2.1: Almost sure convergence
• Remark 2.2
• Theorem 2.3: Asymptotic Normality