Systemic risk is a complex concept that requires knowledge of the size of single entities in a macro region and their interconnections. Indeed, the default of large and highly interconnected institutions may have a disastrous impact on the entire system. Therefore, we introduce a measure of systemic risk, first proposed by Huang et al (2009), that summarizes in one number the size, the probability of default and the degree of interconnectedness of an institution as part of larger system. This measure is the distress insurance price (DIP) and is an expected tail loss similar to a senior tranche of a collateralized debt obligation.

Let x be a tail threshold and L_{t}=\sum_{i=1}^{N}L_{i,t}w_{i,t} be the total loss of the portfolio as the weighted sum of the losses on each debt’s entity i at time t, L_{i,t}, with weights w_{i,t}=Debt_{i,t}/\sum_{i}Debt_{i,t}. Then, the T-year DIP (Distress Insurance Premium) at time t is

(1)   \begin{equation*} \mathbf{{DIP}}_{t}\left(T\right)=E^{\mathbb{Q}}[L_{t+T}\mathbf{\times\mathbbm{1}}\left\{ L_{t+T}\geq x\right\} ] \end{equation*}

where \mathbb{{Q}} indicates that the expectation is taken under the risk-neutral probability measure.

Using a risk neutral measure has the advantage of adjusting the actual systemic default risk by the market price of risk, that captures agents’ attitudes toward risk.

The expectation in equation (1) embeds the small probability of large losses. To estimate a the probability of the occurrence of a rare event, we follow Glasserman and Li (2005) and Grundke et al (2009) and employ a Bayesian technique: the Importance Sampling (IS). The IS approach twists the probability measure from which the loss paths are generated, such that important events become more likely. In other words, the twisting helps producing rare events even in a Normal-distributed world. For a complete presentation of the risk measure, we explain the main concepts behind the procedure.

The portfolio approach described here is one of the classical bottom-up approaches as it consists in piecing together information of the single entities, or subsystems, to give rise to a single or larger system. In our case, an entity is represented by the debt issued by a bank or a country.

Let us consider the following notation:

  • N: number of entities in the portfolio;
  • Y_{i}: default indicator (=1 if i-th entity defaults);
  • pd_{i}: marginal default probability of i-th entity;
  • ELGD_{i}: expected Loss Given Default of i-th entity;
  • L=ELGD_{1}Y_{1}+...+ELGD_{N}Y_{N}: Aggregate portfolio loss;
  • T: maturity of the portfolio.

Structural models a la Merton (1974) and Vasicek (1987) assume that a firm i defaults on its obligations the first time the asset return, R_{i,t}=\Delta\ln A_{i,t}, with asset value A, falls below a threshold, a_{i,t}\left(T\right) (defaulting in T years from time t). Let Y_{i,t}\left(T\right)=\mathbf{1}\left\{ R_{i,t} a_{i,t}\left(T\right)\right\} be our default indicator, the threshold is extracted by inverting the risk-neutral marginal default probability, pd_{i,t}\left(T\right), that is, a_{i,t}\left(T\right)=\Phi{{}^-}{{}^1}(pd_{i,t}\left(T\right)), with \Phi being the cumulative standard Normal distribution. The dependence among marginal default probabilities (or market interconnections) is captured by a Normal copula that is specified through a factor model, as in Vasicek (1987). Specifically,we specify a f-factor model for asset return as follows: , where the latter depends on f-global factors M_{t} and entity-specific idiosyncratic components Z_{i,t}, that is,

(2)   \begin{equation*} R_{i,t}=B_{i,t}M_{t}+\sqrt{1-B_{i,t}B_{i,t}^{\text{T}}}\mathbf{\cdot}Z_{i,t} \end{equation*}

where M_{t} is a set of f factors and Z_{i,t} is the firm idiosyncratic component, B_{i,t}=[\beta_{i,1,t},...,\beta_{i,F,t}] is the vector of loadings with \beta_{i,f,t}\in[-1,1] and \sum_{f=1}^{F}\beta_{i,f}{{}^2}\le1.

Simple algebra shows that, substituting equation (2) into the default indicator, the conditional default probability, conditional on the realization of the global factors, M_{t}=m_{t}, is given by

(3)   \begin{eqnarray*} PD_{i,t}(m_{t},T) & = & Pr\left(Y_{i,t}\left(T\right)=1|M_{t}=m_{t}\right)\nonumber \\  & = & Pr\left(R_{i,t}<a_{i,t}\left(T\right)|M_{t}=m_{t}\right)\nonumber \\  & = & Pr\left(B_{i}M_{t}+\sqrt{1-B_{i}B_{i}^{\text{T}}}\cdot Z_{i,t}<a_{i,t}\left(T\right)|M_{t}=m_{t}\right)\nonumber \\  & = & \text{\ensuremath{\Phi}}\left(\frac{a_{i,t}\left(T\right)-B_{i}m_{t}}{\sqrt{1-B_{i}B_{i}^{\text{T}}}}\right) \end{eqnarray*}

We employ the IS technique to estimate the probability of a loss greater than the threshold, or simply the tail probability, Pr(L\ge x). This procedure develops via two steps: In the first one, IS applies a twist to the original default probability when the simulated loss is not in the tail of the distribution. In other words, the initial marginal default probability at time t with maturity T of firm i, PD_{i,t}\left(T\right), is increased by a parameter \theta, such that the twisted probability is now equal to

    \[ PD_{i,t}(\theta,T)=\frac{PD_{i,t}\left(T\right)exp(\text{\ensuremath{\theta\times}}ELGD_{i})}{1+PD_{i,t}\left(T\right)(exp(\text{\ensuremath{\theta\times}}ELGD_{i})-1)} \]

The choice of \theta depends on whether the loss is in the tail or not. If L>x a tail loss is not rare, so we set \theta=0, that implies PD_{i,t}(\theta,T)=PD_{i,t}\left(T\right). If L<x a tail loss is rare, so \theta is optimally chosen to minimize the second moment of the estimator Pr(L\ge x). As shown in Glasserman and Li (2005), the optimal \theta shifts up the loss distribution so that its new mean is the threshold, E_{\theta}[L]=x.

The second step of the IS procedure deals with the simulations of the loss distribution. Differently from the plain Monte Carlo technique, the second step of the IS methodology consists in simulating the factors from a normal distribution with unit variance and an optimal mean for each factor f and time t, \mu_{f,t}^{*} .Finally, for each realization (simulation) of the common factor, the conditional risk-neutral loss distribution is simply

    \begin{eqnarray*} E^{Q}\left[L_{t+T}\mathbf{\times\mathbbm{1}}\left\{ L_{t+T}\geq x\right\} |M=m\right] & = & E^{Q}\left[L_{t+T}|L_{t+T}>x,M=m\right]\times\Pr\left\{ L_{t+T}>x|M=m\right\} \\  & = & \left[\sum_{i=1}^{N}Y_{i,t}\left(m,T\right)\times LGD_{i,t}\times w_{i,t}\right]\times\Pr\left\{ L_{t+T}>x|M=m\right\}  \end{eqnarray*}

where Y_{i,t}(m,T)\sim Bernulli\left(PD_{i,t}\left(m,T\right)\right).

The probability resulting from the two-step IS needs to be adjusted by the likelihood ratio that relates the original marginal probabilities to the twisted ones, the standard Normal distribution of the factors to the shifted one N(\mu,1) and keeps the probability in the range [0,1]. Therefore, the conditional expected total loss is

    \begin{eqnarray*} E^{Q}\left[L_{t+T}\mathbf{\times\mathbbm{1}}\left\{ L_{t+T}\geq x\right\} |M=m\right] & = & \widetilde{E}^{Q}[L_{t+T}\mathbf{\times\mathbbm{1}}\left\{ L_{t+T}\geq x\right\} exp\{-\theta(m_{t})L_{t+T}+\\  &  & +\psi(\theta(m_{t}),m_{t}))exp(-\mu_{t}^{*'}m_{t}+((\text{\ensuremath{\mu}}_{t}^{*'}\text{\ensuremath{\mu}}_{t}^{*})/2)\}|M=m] \end{eqnarray*}

where L_{t+T}=\sum_{i=1}^{N}Y_{i,t}\left(m,T\right)\times LGD_{i,t} and the second expectation is still risk-neutral but now under then new probability measure and adjusted by the likelihood ratio. Once again, the latter keeps the identity holding for the two expectations, E^{\mathbb{Q}} and \widetilde{E}^{\mathbb{Q}}. Averaging across all the realizations of the common factors, we get the unconditional expected total loss.


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