To account for credit risk, the investments in bonds are reduced by the portfolio loss
$$L_{t} = EAD_{t} \cdot LGD_{t} \cdot \tilde{L}_{t}$$
(8)
with exposure at default
\(EAD_{t}\), loss given default
\(LGD_{t}\), and normalized portfolio loss
\(\tilde{L}_{t}\).
4 The normalized portfolio loss is Vasicek distributed, i.e.,
$$\tilde{L}_{t} = \Phi \left( {\frac{{\sqrt {1 - \rho } \cdot \varepsilon_{t}^{L} - \Phi^{ - 1} \left( {PD_{t} } \right)}}{\sqrt \rho }} \right),$$
where
\(\Phi\) denotes the standard normal distribution,
\(PD_{t}\) denotes the default parameters,
\(\rho\) denotes the correlation parameter, and
\(\varepsilon_{t}^{L}\) denote independent standard normally distributed random variables.
5 The correlations between
\(\varepsilon_{t}^{S}\),
\(\varepsilon_{t}^{B}\), and
\(\varepsilon_{t}^{L}\) are denoted as
\(\rho_{SB}\),
\(\rho_{SL}\), and
\(\rho_{BL}\), respectively.
To model a financial shock in year
\(\tilde{t}\), we consider a price drop in combination with increased volatility for both investments
\(i \in \left\{ {S,B} \right\}\), i.e.,
$$\varepsilon_{t}^{i} = \left\{ {\begin{array}{*{20}c} { - \left| {\varepsilon_{t}^{i} } \right|\,\,\,{\text{if}}\,\,\,\,t = \tilde{t}} \\ {\,\varepsilon_{t}^{i} \,\,\,\,\,\,\,{\text{else}}\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.\,\,\,\,{\text{and}}\,\,\,\,\sigma_{i} = \left\{ {\begin{array}{*{20}c} {\sigma_{i} + \sigma_{i}^{shock} \,\,\,{\text{if}}\,\,\,\,t = \tilde{t}} \\ {\sigma_{i} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{else}}\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
(9)
for strictly positive constants
\(\sigma_{i}^{shock}\). Consequently, the asset prices in Eq. (
6) become
$$I_{t}^{i} = \left\{ {\begin{array}{*{20}c} {I_{t - 1}^{i} \cdot \exp \left( {\mu_{i} - \frac{{\left( {\sigma_{i} + \sigma_{i}^{shock} } \right)^{2} }}{2} - \left( {\sigma_{i} + \sigma_{i}^{shock} } \right) \cdot \left| {\varepsilon_{t}^{i} } \right|\,} \right)\,\,\,\,\,\,\,{\text{if}}\,\,\,\,t = \tilde{t}} \\ {I_{t - 1}^{i} \cdot \exp \left( {\mu_{i} - \frac{{\sigma_{i}^{2} }}{2}} \right) + \sigma_{i} \cdot \varepsilon_{t}^{i} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{else}}\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
and the returns in Eq. (
7) become
$$r_{t}^{i} = \left\{ {\begin{array}{*{20}c} {\mu_{i} - {{\left( {\sigma_{i} + \sigma_{i}^{shock} } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {\sigma_{i} + \sigma_{i}^{shock} } \right)^{2} } {\,2}}} \right. \kern-0pt} {\,2}} - \left( {\sigma_{i} + \sigma_{i}^{shock} } \right) \cdot \left| {\varepsilon_{t}^{i} } \right|\,\,\,\,\,\,\,\,{\text{if}}\,\,\,\,t = \tilde{t}} \\ {\mu_{i} - \sigma_{i}^{2} /2 + \sigma_{i} \cdot \varepsilon_{t}^{i} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{else}}{.}\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
To account for an increased credit risk in year
\(\tilde{t}\), we use two different default parameters, i.e.,
$$PD_{t} = \left\{ {\begin{array}{*{20}c} {PD^{shock} \,\,\,\,{\text{if}}\,\,\,\,t = \tilde{t}} \\ {PD\,\,\,\,\,\,\,\,\,\,\,\,{\text{else}}{.}\,\,\,\,\,\,\,\,\,\,} \\ \end{array} } \right.$$
(10)