Model-Based Measurement of Sector Concentration Risk in Credit Portfolios

Abstract

The focus of this chapter is on sector concentrations. This type of concentration risk can occur if there is more than one systematic risk factor that influences credit defaults. The main research questions that are analyzed in this chapter are:

• How can existing approaches for measuring sector concentration risk be modified and adjusted to be consistent with the Basel framework? Is the risk measure Value at Risk problematic when dealing with sector concentration risk?

• Which methods are capable of measuring concentration risk and how good do they perform in comparison? What are the advantages and disadvantages of these methods?

In order to deal with these questions, it is initially determined how a multi-factor model can be parameterized to obtain a capital requirement which is consistent with Basel II. Furthermore, the models of Pykhtin (Risk 17(3):85–90, 2004), Cespedes et al. (J Credit Risk, 2(3):57–85, 2006), and Düllmann (Measuring business sector concentration by an infection model. Discussion Paper, Series 2: Banking and Financial Studies, Deutsche Bundesbank, (3), 2006), which have been developed to approximate the risk in the presence of sector concentrations, are presented and modified. Then, the accuracy of these models concerning their ability to measure sector concentration risk is compared.

The main results of this section comply with Gürtler et al. (2010).

Appendix

5.1.1 Optimal Choice of the Single Correlation Factor

To relate $1/11 = 9.1\%$ to $\tilde{\bar{L}}$, it is assumed that the new systematic factor $\tilde{L}$ has a linear dependence to the original sector factors:⁶¹

$$ K \leq n $$

(5.73)

$$ \tilde{\bar{x}} = \sum\limits_{k = 1}^K {{b_k} \cdot {{\tilde{z}}_k},} $$

(5.74)

Condition (5.74) satisfies that the new systematic factor still has a variance of 1. In order to specify the correlation factors c _i and the coefficients b _k , it will be required that the loss ${\text{with}}\,\,\,\sum\limits_{k = 1}^K {b_k^2 = 1.}$ equals the conditional expectation of the “true” loss $\tilde{\bar{L}}$. This assures that the first element of the subsequently performed Taylor series expansion vanishes.⁶² To determine $b{E}(\tilde{L}|\tilde{\bar{x}})$, we first recall that the asset return of obligor i in sector s can be written as

$$ b{E}(\tilde{\bar{L}}|\tilde{\bar{x}}) $$

(5.75)

Now, each original sector factor ${\tilde{a}_{s,i}} = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\tilde{x}_s} + \sqrt {{1 - {\rho_{{\text{Intra}},i}}}} \cdot {\tilde{\xi }_i}.$ is decomposed into a part that is related to the single-factor ${\tilde{x}_s}$ and a part that is independent of this factor:

$$ \tilde{\bar{x}} $$

(5.76)

with ${\tilde{x}_s} = {\bar{\rho }_s} \cdot \tilde{\bar{x}} + \sqrt {{1 - {{\bar{\rho }}_s}^2}} \cdot {\tilde{\eta }_s}$. Using (5.2), (5.73), and the independence of ${\tilde{\eta }_s} \sim \mathcal{N}(0,1)$ if ${\tilde{z}_i},\;{\tilde{z}_j}$, the correlation parameter $i \ne j$ can be expressed as

$$ {\bar{\rho }_s} $$

(5.77)

Using this notation, the asset return (5.75) can now be written as

$$ \begin{array} {c} {{\bar{\rho }}_s} = {\text{Corr}}\left( {{{\tilde{x}}_s},\tilde{\bar{x}}} \right) = {\text{Corr}}\left( {\sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {{\tilde{z}}_k}}, \sum\limits_{k = 1}^K {{b_k} \cdot {{\tilde{z}}_k}} } \right) \\ = \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {b_k} \cdot b{V}\left( {{{\tilde{z}}_k}} \right)} = \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {b_k}.} \\ \end{array} $$

(5.78)

The independent standard normally distributed random variables $\begin{array} {c} {{\tilde{a}}_{s,i}} = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\tilde{x}}_s} + \sqrt {{1 - {\rho_{{\text{Intra}},i}}}} \cdot {{\tilde{\xi }}_i} \\ = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \left( {{{\bar{\rho }}_s} \cdot \tilde{\bar{x}} + \sqrt {{1 - {{\bar{\rho }}_s}^2}} \cdot {{\tilde{\eta }}_s}} \right) + \sqrt {{1 - {\rho_{{\text{Intra}},i}}}} \cdot {{\tilde{\xi }}_i} \\ = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_s} \cdot \tilde{\bar{x}} + \sqrt {{{\rho_{{\text{Intra}},i}} - {\rho_{{\text{Intra}},i}} \cdot {{\bar{\rho }}_s}^2}} \cdot {{\tilde{\eta }}_s} + \sqrt {{1 - {\rho_{{\text{Intra}},i}}}} \cdot {{\tilde{\xi }}_i}. \\ \end{array}$ and ${\tilde{\eta }_s}$ can be combined into a new standard normally distributed random variable ${\tilde{\xi }_i}$, leading to

$$ {\tilde{\zeta }_i} $$

(5.79)

with ${\tilde{a}_{s,i}} = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\bar{\rho }_i} \cdot \tilde{\bar{x}} + \sqrt {{1 - {{\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i}} \right)}^2}}} \cdot {\tilde{\zeta }_i},$ for each obligor i in sector s. Since the variable ${\bar{\rho }_i} = {\bar{\rho }_s}$ is independent of ${\tilde{\zeta }_i}$, we can use the known formula of the single-factor model for the conditional expectation

$$ \tilde{\bar{x}} $$

(5.80)

The mentioned condition $b{E}\left( {\tilde{L}|\tilde{\bar{x}}} \right) = \sum\limits_{i = 1}^n {{w_i} \cdot LG{D_i} \cdot \Phi \left[ {\frac{{{\Phi^{ - 1}}(P{D_i}) - \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i} \cdot {\Phi^{ - 1}}(\alpha )}}{{\sqrt {{1 - {{\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i}} \right)}^2}}} }}} \right]} .$ leads to

$$ \tilde{\bar{L}} = b{E}\left( {\tilde{L}|\tilde{\bar{x}}} \right) $$

(5.81)

using (5.9), (5.80), (5.77), and $\begin{array} {lll} \tilde{\bar{L}} &= b{E}\left( {\tilde{L}|\tilde{\bar{x}}} \right) \\ & \Leftrightarrow \Phi \left[ {\frac{{{\Phi^{ - 1}}(P{D_i}) - {c_i} \cdot \tilde{\bar{x}}}}{{\sqrt {{1 - {c_i}^2}} }}} \right] = \Phi \left[ {\frac{{{\Phi^{ - 1}}(P{D_i}) - \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i} \cdot {\Phi^{ - 1}}(\alpha )}}{{\sqrt {{1 - {{\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i}} \right)}^2}}} }}} \right] \\ & \Leftrightarrow {c_i} = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {{\bar{\rho }}_i} = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \sum\limits_{k = 1}^K {{\alpha_{i,k}} \cdot {b_k}}, \\ \end{array}$ for each obligor i in sector s. While ${\alpha_{i,k}} = {\alpha_{s,k}}$ and ${\rho_{{\text{Intra}},i}}$ are known, the coefficients ${\alpha_{i,k}}$ are unknown.

While (5.81) already satisfies that the first-order term of the Taylor series vanishes, the concrete choice of the parameter set {${b_k}$} is critical concerning the distance between the zeroth-order term ${b_k}$ and the unknown quantile ${q_\alpha }(\tilde{\bar{L}})$. Unfortunately, it is not obvious how this distance can be minimized. Thus, Pykhtin (2004) relies on the intuition that coefficients which maximize the (weighted) correlation between the single factor ${q_\alpha }(\tilde{L})$ and the sector factors {$\tilde{\bar{x}}$} should lead to good results. This leads to the following maximization problem:

$$ {\tilde{x}_s} $$

(5.82)

subject to

$$ \mathop {{\max }}\limits_{\left\{ {{b_k}} \right\}} \left( {\sum\limits_{i = 1}^n {{d_i} \cdot {{\bar{\rho }}_i}} } \right) = \mathop {{\max }}\limits_{\left\{ {{b_k}} \right\}} \left( {\sum\limits_{i = 1}^n {{d_i} \cdot \sum\limits_{k = 1}^K {{\alpha_{i,k}} \cdot {b_k}} } } \right) $$

(5.83)

The solution of this optimization problem is⁶³

$$ \sum\limits_{k = 1}^K {b_k^2 = 1.} $$

(5.84)

where the positive constant Lagrange multiplier ${b_k} = \sum\limits_{i = 1}^n {\frac{{{d_i} \cdot {\alpha_{ik}}}}{{2\tau }}},$ is chosen in a way that {$\tau$} satisfies the constraint. As a final step, the weighting factor d _i has to be chosen. After trying several specifications, Pykhtin (2004) uses

$$ {b_k} $$

(5.85)

which is the VaR formula in a single-factor model. The intuition behind this choice is that obligors with a high exposure in terms of VaR should have a large weight in the maximization problem whereas obligors with a small VaR should have a minor impact. Summing up, the correlation parameter c _i results from (5.81), where the coefficients b _k are determined by (5.83)–(5.85).

5.1.2 Conditional Correlation

The correlation conditional on ${d_i} = {w_i} \cdot LG{D_i} \cdot \Phi \left[ {\frac{{{\Phi^{ - 1}}(P{D_i}) + \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\Phi^{ - 1}}(\alpha )}}{{\sqrt {{1 - {\rho_{{\text{Intra}},i}}}} }}} \right],$ between the asset returns from (5.19) can be written as

$$ \tilde{\bar{x}} $$

(5.86)

using the independence of the factors $\begin{array} {c} \rho_{ij}^{\bar{x}} = {\text{Corr}}\left( {{{\tilde{a}}_{s,i}},{{\tilde{a}}_{t,j}}|\tilde{\bar{x}}} \right) \\ = \frac{{{\text{Cov}}\left( {\sum\limits_{k = 1}^K {\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\alpha_{s,k}} - {c_i} \cdot {b_k}} \right)} \cdot {{\tilde{z}}_k},\sum\limits_{k = 1}^K {\left( {\sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot {\alpha_{t,k}} - {c_j} \cdot {b_k}} \right)} \cdot {{\tilde{z}}_k}} \right)}}{{\sqrt {{b{V}\left( {{{\tilde{a}}_{s,i}}|\tilde{\bar{x}}} \right)}} \cdot \sqrt {{b{V}\left( {{{\tilde{a}}_{t,j}}|\tilde{\bar{x}}} \right)}} }} \\ = \frac{{\sum\limits_{k = 1}^K {\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\alpha_{s,k}} - {c_i} \cdot {b_k}} \right) \cdot \left( {\sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot {\alpha_{t,k}} - {c_j} \cdot {b_k}} \right)} }}{{\sqrt {{1 - {c_i}^2}} \cdot \sqrt {{1 - {c_j}^2}} }}, \\ \end{array}$. The numerator can be simplified using ${\tilde{z}_k}$ from (5.81) and $\sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {b_k}} = {{{c_i}} \mathord{\left/{\vphantom {{{c_i}} {\sqrt {{{\rho_{{\text{Intra}},i}}}} }}} \right.} {\sqrt {{{\rho_{{\text{Intra}},i}}}} }}$ from (5.74):

$$ \sum\limits_{k = 1}^K {b_k^2 = 1} $$

(5.87)

This leads to

$$ \begin{array} {lll} \sum\limits_{k = 1}^K {\left( {\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {\alpha_{s,k}} - {c_i} \cdot {b_k}} \right) \cdot \left( {\sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot {\alpha_{t,k}} - {c_j} \cdot {b_k}} \right)} \\ = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {\alpha_{t,k}}} - \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {c_j} \cdot \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {b_k}} \\ & - \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot {c_i} \cdot \sum\limits_{k = 1}^K {{\alpha_{t,k}} \cdot {b_k}} + {c_i} \cdot {c_j} \cdot \sum\limits_{k = 1}^K {{b_k}^2} \\ = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {\alpha_{t,k}}} - \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot {c_j} \cdot \frac{{{c_i}}}{{\sqrt {{{\rho_{{\text{Intra}},i}}}} }} \\ & - \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot {c_i} \cdot \frac{{{c_j}}}{{\sqrt {{{\rho_{{\text{Intra}},j}}}} }} + {c_i} \cdot {c_j} \\ = \sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {\alpha_{t,k}}} - {c_j} \cdot {c_i}. \\ \end{array} $$

(5.88)

5.1.3 Calculation of the Decomposed Variance

In order to determine the conditional variance, it is decomposed into the following terms:⁶⁴

$$ \rho_{ij}^{\bar{x}} = \frac{{\sqrt {{{\rho_{{\text{Intra}},i}}}} \cdot \sqrt {{{\rho_{{\text{Intra}},j}}}} \cdot \sum\limits_{k = 1}^K {{\alpha_{s,k}} \cdot {\alpha_{t,k}}} - {c_i} \cdot {c_j}}}{{\sqrt {{1 - {c_i}^2}} \cdot \sqrt {{1 - {c_j}^2}} }}. $$

(5.89)

For calculation of these terms, first the expressions (a) $b{V}\left( {\tilde{L}|\tilde{\bar{x}} = \bar{x}} \right) = b{V}\left[ {b{E}\left( {\tilde{L}|\left\{ {{{\tilde{z}}_k}} \right\}} \right)|\tilde{\bar{x}} = \bar{x}} \right] + b{E}\left[ {b{V}\left( {\tilde{L}|\left\{ {{{\tilde{z}}_k}} \right\}} \right)|\tilde{\bar{x}} = \bar{x}} \right]$, (b) $b{E}(\tilde{L}|\{ {\tilde{z}_k}\} )$, and (c) $b{E}({\tilde{L}^2}|\{ {\tilde{z}_k}\} )$ will be calculated. The conditional loss is given as

$$ b{V}(\tilde{L}|\{ {\tilde{z}_k}\} ) $$

(5.90)

and for stochastically independent LGDs this leads to

$$ \tilde{L}|\{ {\tilde{z}_k}\} = \sum\limits_i {{w_i}} \cdot \left( {{{\widetilde{{LGD}}}_i}|\{ {{\tilde{z}}_k}\} } \right) \cdot \left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right), $$

(5.91)

• (a) With $\tilde{L}|\{ {\tilde{z}_k}\} = \sum\limits_i {{w_i}} \cdot {\widetilde{{LGD}}_i} \cdot \left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right).$ and $b{E}\left( {{{\widetilde{{LGD}}}_i}} \right) = :ELG{D_i}$ we obtain:

$$ b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right) = :{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) $$

(5.92)

• (b) Consider that $b{E}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right) = \sum\limits_i {{w_i}} \cdot ELG{D_i} \cdot {p_i}\left( {\{ {{\tilde{z}}_k}\} } \right).$, $1_{\left\{ {{{\tilde{D}}_i}} \right\}}^2 = {1_{\left\{ {{{\tilde{D}}_i}} \right\}}}$, and

$$ b{E}\left( {{{\widetilde{{LGD}}}^2}} \right) = {b{E}^2}\left( {\widetilde{{LGD}}} \right) + b{V}\left( {\widetilde{{LGD}}} \right) = :ELG{D^2} + VLGD $$

(5.93)

as well as

$$ \begin{array} {c} b{E}\left( {LG{D_i}LG{D_j}} \right) = {\text{Cov}}\left( {LG{D_i},LG{D_j}} \right) + b{E}\left( {LG{D_i}} \right)b{E}\left( {LG{D_j}} \right) \\ = b{E}\left( {LG{D_i}} \right)b{E}\left( {LG{D_j}} \right) \\ = :ELG{D_i}ELG{D_j}, \\ \end{array} $$

(5.94)

Moreover, we have

$$ \begin{array} {c} b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right) = {\text{Cov}}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}},{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right) + b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right)b{E}\left( {{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right) \\ = b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right)b{E}\left( {{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right) \\ = {p_i}\left( {\{ {{\tilde{z}}_k}\} } \right){p_j}\left( {\{ {{\tilde{z}}_k}\} } \right). \\ \end{array} $$

(5.95)

$$ {\left( {\sum\limits_i {{x_i}} } \right)^2} = \sum\limits_i {\sum\limits_j {{x_i}{x_j} = \sum\limits_i {{x_i}^2} + \sum\limits_i {\sum\limits_{j \ne i} {{x_i}{x_j}} } } } $$

(5.96)

Thus, we obtain:

$$ \sum\limits_{j \ne i} {{x_i}{x_j}} = \sum\limits_j {{x_i}{x_j}} - {x_i}^2. $$

(5.97)

• (c) The conditional variance $\begin{array} {c} b{E}\left( {{{\tilde{L}}^2}|\{ {{\tilde{z}}_k}\} } \right) = b{E}\left[ {{{\sum\limits_i {\left( {{w_i}{{\widetilde{{LGD}}}_i}{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} \right)} }^2}|\{ {{\tilde{z}}_k}\} } \right] \\ = b{E}\left[ {\sum\limits_i {{w_i}^2} {{\widetilde{{LGD}}}_i}^21_{\left\{ {{{\tilde{D}}_i}} \right\}}^2|\{ {{\tilde{z}}_k}\} } \right] + b{E}\left[ {\sum\limits_i {\sum\limits_{j \ne i} {{w_i}{w_j}{{\widetilde{{LGD}}}_i}{{\widetilde{{LGD}}}_j}{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}|\{ {{\tilde{z}}_k}\} } } } \right] \\ = \sum\limits_i {{w_i}^2} b{E}\left( {LG{D_i}^2} \right){p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) \\ + \sum\limits_i {\sum\limits_{j \ne i} {{w_i}{w_j}b{E}\left( {LG{D_i}LG{D_j}} \right)b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}{1_{\left\{ {{{\tilde{D}}_j}} \right\}}}} \right)} } \\ = \sum\limits_i {{w_i}^2} \left( {ELG{D_i}^2 + VLG{D_i}} \right){p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) \\ + \sum\limits_i {\sum\limits_{j \ne i} {{w_i}{w_j}ELG{D_i}ELG{D_j}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right){p_j}\left( {\{ {{\tilde{z}}_k}\} } \right)} } \\ = \sum\limits_i {{w_i}^2} \left( {ELG{D_i}^2 + VLG{D_i}} \right){p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) - \sum\limits_i {{w_i}^2ELG{D_i}^2{p_i}^2\left( {\{ {{\tilde{z}}_k}\} } \right)} \\ + \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right){p_j}\left( {\{ {{\tilde{z}}_k}\} } \right)} } \\ = \sum\limits_i {{w_i}^2} \left( {ELG{D_i}^2\left[ {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) - {p_i}^2\left( {\{ {{\tilde{z}}_k}\} } \right)} \right] + VLG{D_i}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)} \right) \\ + {b{E}^2}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right). \\ \end{array}$ is equal to

$$ b{V}(\tilde{L}|\{ {\tilde{z}_k}\} ) $$

(5.98)

• (d) Using the law of iterated expectation, we have

$$ \begin{array} {c} b{V}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right) = b{E}\left( {{{\tilde{L}}^2}|\{ {{\tilde{z}}_k}\} } \right) - {b{E}^2}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right) \\ = \sum\limits_i {{w_i}^2} \left( {ELG{D_i}^2\left[ {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) - {p_i}^2\left( {\{ {{\tilde{z}}_k}\} } \right)} \right] + VLG{D_i}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)} \right). \\ \end{array} $$

(5.99)

Thus, with (5.98) the expectation of the conditional variance can be written as

$$ {p_i}(\bar{x}) = b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\tilde{\bar{x}} = \bar{x}} \right) = b{E}\left[ {b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}|\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right] = b{E}\left[ {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right] $$

(5.100)

For independent idiosyncratic factors $\begin{array} {c} b{E}\left[ {b{V}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\tilde{\bar{x}} = \bar{x}} \right] = \sum\limits_i {{w_i}^2\left( {ELG{D_i}^2\left( {b{E}\left[ {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right] - b{E}\left[ {{p_i}^2\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right]} \right)} \right.} \\ \left. { + VLG{D_i}b{E}\left[ {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right]} \right) \\ = \sum\limits_i {{w_i}^2\left( {ELG{D_i}^2\left( {{p_i}(\bar{x}) - b{P}\left[ {\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}} = 1} \right) \wedge \left( {{1_{\left\{ {{{\tilde{D}}_i}^\prime } \right\}}} = 1} \right)|\bar{x}} \right]} \right)} \right.} \\ \left. { + VLG{D_i}{p_i}(\bar{x})} \right). \\ \end{array}$ and with

$$ {\tilde{\zeta }_i},{\tilde{\zeta }_i}^\prime \sim \mathcal{N}(0,1) $$

(5.101)

we get

$$ {p_i}(\bar{x}): = \Phi \left( {\frac{{{\Phi^{ - 1}}(P{D_i}) - {c_i} \cdot \bar{x}}}{{\sqrt {{1 - {c_i}^2}} }}} \right) \Leftrightarrow \frac{{{\Phi^{ - 1}}(PD) - {c_i} \cdot \bar{x}}}{{\sqrt {{1 - {c_i}^2}} }} = {\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right), $$

(5.102)

with the correlation conditional on $\begin{array} {c} b{P}\left[ {\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}} = 1} \right) \wedge \left( {{1_{\left\{ {{{\tilde{D}}_i}^\prime } \right\}}} = 1} \right)|\bar{x}} \right] \\ = b{P}\left[ {{c_i} \cdot \tilde{\bar{x}} + \sqrt {{1 - {c_i}^2}} \cdot {{\tilde{\zeta }}_i} \leq {\Phi^{ - 1}}(P{D_i}),{c_i} \cdot \tilde{x} + \sqrt {{1 - {c_i}^2}} \cdot {{\tilde{\zeta }}_i}^\prime \leq {\Phi^{ - 1}}(P{D_i})|\bar{x}} \right] \\ = b{P}\left[ {{{\tilde{\zeta }}_i} \leq \frac{{{\Phi^{ - 1}}(P{D_i}) - {c_i} \cdot \bar{x}}}{{\sqrt {{1 - {c_i}^2}} }},{{\tilde{\zeta }}_i}^\prime \leq \frac{{{\Phi^{ - 1}}(P{D_i}) - {c_i} \cdot \bar{x}}}{{\sqrt {{1 - {c_i}^2}} }}} \right] \\ = {\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right),{\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right),\rho_{ii}^{\bar{x}}} \right), \\ \end{array}$ of (5.20). Hence, (5.100) results in

$$ \bar{x} $$

(5.103)

• (e) Using (5.92), the variance of the conditional expectation can be expressed as

$$ \begin{array} {c} b{E}\left[ {b{V}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\tilde{\bar{x}} = \bar{x}} \right] = \sum\limits_i {{w_i}^2\left( {ELG{D_i}^2\left[ {{p_i}(\bar{x}) - {\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right),{\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right),\rho_{ii}^{\bar{x}}} \right)} \right]} \right.} \\ \left. { + VLG{D_i}{p_i}(\bar{x})} \right). \\ \end{array} $$

(5.104)

leading to

$$ \begin{array} {c} b{V}\left[ {b{E}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\tilde{\bar{x}} = x} \right] \\ = b{E}\left[ {{b{E}^2}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right] - {b{E}^2}\left[ {b{E}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right] \\ = b{E}\left( {{b{E}^2}\left[ {\sum\limits_i {{w_i}{{\widetilde{{LGD}}}_i}{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} |\{ {{\tilde{z}}_k}\} } \right]|\bar{x}} \right) - {b{E}^2}\left( {\sum\limits_i {{w_i}ELG{D_i}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} } \right) \\ = b{E}\left[ {{{\left( {\sum\limits_i {{w_i}ELG{D_i}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right)} } \right)}^2}|\bar{x}} \right] - {\left( {\sum\limits_i {{w_i}ELG{D_i}{p_i}(\bar{x})} } \right)^2}, \\ \end{array} $$

(5.105)

Analogous to (5.102) and using the conditional correlation (5.20), this can be expressed as:

$$ \begin{array} {llll} b{V}\left[ {b{E}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\tilde{\bar{x}} = x} \right] \\ = b{E}\left[ {\sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) \cdot {p_j}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} } } \right] \\ & - \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}{p_i}(\bar{x}){p_j}(\bar{x})} } \\ = \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}b{E}\left( {{p_i}\left( {\{ {{\tilde{z}}_k}\} } \right) \cdot {p_j}\left( {\{ {{\tilde{z}}_k}\} } \right)|\bar{x}} \right)} } \\ & - \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}{p_i}(\bar{x}){p_j}(\bar{x})} } \\ = \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}\left[ {b{P}\left[ {\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}} = 1} \right) \wedge \left( {{1_{\left\{ {{{\tilde{D}}_j}} \right\}}} = 1} \right)|\bar{x}} \right] - {p_i}(\bar{x}){p_j}(\bar{x})} \right]} } . \\ \end{array} $$

(5.106)

5.1.4 Derivatives of the Decomposed Variance Terms

As both conditional variance terms are linear in the bivariate normal distribution, the derivative of the bivariate normal distribution will be calculated subsequently. Then, the derivatives of $\begin{array} {c} b{V}\left[ {b{E}\left( {\tilde{L}|\{ {{\tilde{z}}_k}\} } \right)|\tilde{\bar{x}} = \bar{x}} \right] = \sum\limits_i {\sum\limits_j {{w_i}{w_j}ELG{D_i}ELG{D_j}} } \\ \cdot \left[ {{\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}(\bar{x})} \right),{\Phi^{ - 1}}\left( {{p_j}(\bar{x})} \right),\rho_{ij}^{\bar{x}}} \right) - {p_i}(\bar{x}){p_j}(\bar{x})} \right]. \\ \end{array}$ and $\eta_{2,c}^\infty (\bar{x})$ will be computed.

Proposition

The derivative of the bivariate normal distribution can be written as:

$$ \eta_{2,c}^{\text{GA}}(\bar{x}) $$

(5.107)

Proof

Using the notation

$$ \begin{array} {c} \frac{d}{{dx}}{\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right),{\Phi^{ - 1}}\left( {{p_j}\left( {\bar{x}} \right)} \right),\rho_{ij}^{\bar{x}}} \right) = \frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}\Phi \left( {\frac{{{\Phi^{ - 1}}\left( {{p_j}\left( {\bar{x}} \right)} \right) - \rho_{ij}^{\bar{x}} \cdot {\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right)}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right) \\ + \frac{{d{p_j}\left( {\bar{x}} \right)}}{{d\bar{x}}}\Phi \left( {\frac{{{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right) - \rho_{ij}^{\bar{x}} \cdot {\Phi^{ - 1}}\left( {{p_j}\left( {\bar{x}} \right)} \right)}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right). \\ \end{array} $$

(5.108)

and the chain rule, we get

$$ \begin{array} {c}{*{20}{c}} {{y_i}(\bar{x}) = \frac{{{\Phi^{ - 1}}(P{D_i}) - {c_i} \cdot \bar{x}}}{{\sqrt {{1 - {c_i}^2}} }}, \,\,\,{y_j}(\bar{x}) = \frac{{{\Phi^{ - 1}}(P{D_j}) - {c_j} \cdot \bar{x}}}{{\sqrt {{1 - {c_j}^2}} }}} \\ \end{array}, $$

(5.109)

For calculation of term (II) and (IV), we rewrite the bivariate normal distribution according to Appendix 2.8.6 as

$$ \begin{array} {c} \frac{d}{{d\bar{x}}}{\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right),{\Phi^{ - 1}}\left( {{p_j}\left( {\bar{x}} \right)} \right),\rho_{ij}^{\bar{x}}} \right) \\ = \frac{d}{{d\bar{x}}}{\Phi_2}\left( {{y_i}\left( {\bar{x}} \right),{y_j}\left( {\bar{x}} \right),\rho_{ij}^{\bar{x}}} \right) \\ = \underbrace {\frac{{d{y_i}}}{{d\bar{x}}}}_{(I)}\underbrace {\frac{\partial }{{\partial {y_i}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right)}_{(II)} + \underbrace {\frac{{d{y_j}}}{{d\bar{x}}}}_{(III)}\underbrace {\frac{\partial }{{\partial {y_j}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right)}_{(IV)}. \\ \end{array} $$

(5.110)

Thus, we have

$$ {\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right) = \int\limits_{z = - \infty }^{{y_j}} {\varphi (z)\Phi \left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}} \cdot z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)dz} . $$

(5.111)

The term (*) is equivalent to

$$ \begin{array} {c} \frac{\partial }{{\partial {y_i}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right) = \frac{\partial }{{\partial {y_i}}}\int\limits_{z = - \infty }^{{y_j}} {\varphi (z)\Phi \left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}} \cdot z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)dz} \\ = \frac{1}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}\int\limits_{z = - \infty }^{{y_j}} {\varphi (z)\varphi \left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}} \cdot z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)dz} \\ = \frac{1}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}\int\limits_{z = - \infty }^{{y_j}} {\frac{1}{{2\pi }}\exp \left( { - \frac{1}{2}\underbrace {\left[ {{z^2} + {{\left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}} \cdot z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)}^2}} \right]}_{{(*)}}} \right)dz} . \\ \end{array} $$

(5.112)

Hence, (5.111) can be written as

$$ \begin{array} {c} {z^2} + {\left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}} \cdot z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)^2} = \frac{{\left( {1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}} \right){z^2} + {y_i}^2 - 2{y_i}\rho_{ij}^{\bar{x}}z + \left( {\rho_{ij}^{\bar{x}}} \right){z^2}}}{{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} \\ = \frac{{{z^2} - 2{y_i}\rho_{ij}^{\bar{x}}z + {y_i}^2}}{{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} \\ = \frac{{{z^2} - 2{y_i}\rho_{ij}^{\bar{x}}z + {y_i}^2 + {y_i}^2{{\left( {\rho_{ij}^{\bar{x}}} \right)}^2} - {y_i}^2{{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}}{{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} \\ = {\left( {\frac{{z - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)^2} + {y_i}^2. \\ \end{array} $$

(5.113)

For solving the integral, we substitute $\begin{array} {c} \frac{\partial }{{\partial {y_i}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right) = \frac{1}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}\int\limits_{z = - \infty }^{{y_i}} {\frac{1}{{2\pi }}\exp \left( { - \frac{1}{2}\left[ {{y_i}^2 + {{\left( {\frac{{z - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)}^2}} \right]} \right)dz} \\ = \varphi \left( {{y_i}} \right)\int\limits_{z = - \infty }^{{y_j}} {\frac{1}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}\varphi \left( {\frac{{z - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)dz} . \\ \end{array}$, and thus $t: = \frac{{z - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{(\rho_{ij}^{\bar{x}})}^2}}} }}$. This leads to

$$ \frac{{dz}}{{dt}} = \sqrt {{1 - {{(\rho_{ij}^{\bar{x}})}^2}}} $$

(5.114)

Analogously, the term (IV) of (5.109) is equivalent to

$$ \begin{array} {c} \frac{\partial }{{\partial {y_i}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right) = \varphi \left( {{y_i}} \right)\int\limits_{t = - \infty }^{\frac{{{y_j} - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{(\rho_{ij}^{\bar{x}})}^2}}} }}} {\frac{1}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}\varphi (t)\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} dt} \\ = \varphi \left( {{y_i}} \right)\Phi \left( {\frac{{{y_j} - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right). \\ \end{array} $$

(5.115)

The derivatives (I) and (III) of (5.109) are given as

$$ \begin{array} {c} \frac{\partial }{{\partial {y_j}}}{\Phi_2}\left( {{y_i},{y_j},\rho_{ij}^{\bar{x}}} \right) = \frac{\partial }{{\partial {y_j}}}\int\limits_{z = - \infty }^{{y_i}} {(z)\Phi \left( {\frac{{{y_j} - \rho_{ij}^{\bar{x}}z}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right)dz} \\ = \varphi \left( {{y_j}} \right)\Phi \left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}}{y_j}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right). \\ \end{array} $$

(5.116)

Thus, inserting (5.114), (5.115), and (5.116) into (5.109), the derivative of the bivariate normal distribution finally results in

$$ \frac{{d{y_i}\left( {\bar{x}} \right)}}{{d\bar{x}}} = - \frac{{{c_i}}}{{\sqrt {{1 - {c_i}^2}} }} \,\,\,{\text{and}} \,\,\,\frac{{d{y_j}\left( {\bar{x}} \right)}}{{d\bar{x}}} = - \frac{{{c_j}}}{{\sqrt {{1 - {c_j}^2}} }}. $$

(5.117)

where the derivatives $\begin{array} {c} \frac{d}{{d\bar{x}}}{\Phi_2}\left( {{y_i}\left( {\bar{x}} \right),{y_j}\left( {\bar{x}} \right),\rho_{ij}^{\bar{x}}} \right) = - \frac{{{c_i}}}{{\sqrt {{1 - {c_i}^2}} }}\varphi \left( {{y_i}} \right)\Phi \left( {\frac{{{y_j} - \rho_{ij}^{\bar{x}}{y_i}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right) \\ - \frac{{{c_j}}}{{\sqrt {{1 - {c_j}^2}} }}\varphi \left( {{y_j}} \right)\Phi \left( {\frac{{{y_i} - \rho_{ij}^{\bar{x}}{y_j}}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right) \\ = \frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}\Phi \left( {\frac{{{\Phi^{ - 1}}\left[ {{p_j}\left( {\bar{x}} \right)} \right] - \rho_{ij}^{\bar{x}}{\Phi^{ - 1}}\left[ {{p_i}\left( {\bar{x}} \right)} \right]}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right) \\ + \frac{{d{p_j}\left( {\bar{x}} \right)}}{{d\bar{x}}}\Phi \left( {\frac{{{\Phi^{ - 1}}\left[ {{p_i}\left( {\bar{x}} \right)} \right] - \rho_{ij}^{\bar{x}}{\Phi^{ - 1}}\left[ {{p_j}\left( {\bar{x}} \right)} \right]}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right), \\ \end{array}$ and $\frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}$ are given by (5.16), which is equal to proposition (5.107).

As a next step, the derivatives of $\frac{{d{p_j}\left( {\bar{x}} \right)}}{{d\bar{x}}}$ and $\eta_{2,c}^\infty (\bar{x})$ will be calculated. With

$$ \eta_{2,c}^{\text{GA}}(\bar{x}) $$

(5.118)

we get

$$ \begin{array} {c} \eta_{2,c}^\infty \left( {\bar{x}} \right) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{w_i}{w_j}ELG{D_i}ELG{D_j}} } \\ \cdot \left[ {{\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right),{\Phi^{ - 1}}\left( {{p_j}\left( {\bar{x}} \right)} \right),\rho_{ij}^{\bar{x}}} \right) - {p_i}\left( {\bar{x}} \right){p_j}\left( {\bar{x}} \right)} \right], \\ \end{array} $$

(5.119)

Using the derivative of the bivariate normal distribution from (5.117) yields

$$ \begin{array} {c} \frac{{d\eta_{2,c}^\infty \left( {\bar{x}} \right)}}{{d\bar{x}}} = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{w_i}} } {w_j}ELG{D_i}ELG{D_j}\left[ {\frac{d}{{d\bar{x}}}{\Phi_2}\left( {{y_i}\left( {\bar{x}} \right),{y_j}\left( {\bar{x}} \right),\rho_{ij}^{\bar{x}}} \right) - \frac{d}{{d\bar{x}}}\left( {{p_i}\left( {\bar{x}} \right){p_j}\left( {\bar{x}} \right)} \right)} \right] \\ = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{w_i}} } {w_j}ELG{D_i}ELG{D_j} \\ \cdot \left[ {\frac{d}{{d\bar{x}}}{\Phi_2}\left( {{y_i}\left( {\bar{x}} \right),{y_j}\left( {\bar{x}} \right),\rho_{ij}^{\bar{x}}} \right) - \left( {\frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}{p_j}\left( {\bar{x}} \right) + \frac{{d{p_j}\left( {\bar{x}} \right)}}{{d\bar{x}}}{p_i}\left( {\bar{x}} \right)} \right)} \right]. \\ \end{array} $$

(5.120)

Comparing the terms on the right-hand side, it can be found that the first and second summand as well as the third and fourth summand only differ concerning the indices i and j. Due to the double sum, each combination of i and j occurs twice. Thus, (5.120) can be simplified to:⁶⁵

$$ \rho_{ij}^{\bar{x}} = \rho_{ji}^{\bar{x}} $$

(5.121)

Similarly, the derivative of

$$ \begin{array} {c} \frac{{d\eta_{2,c}^\infty \left( {\bar{x}} \right)}}{{d\bar{x}}} = 2 \cdot \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{w_i}} } {w_j}ELG{D_i}ELG{D_j}\frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}} \\ \cdot \left( {\Phi \left( {\frac{{{\Phi^{ - 1}}\left[ {{p_j}\left( {\bar{x}} \right)} \right] - \rho_{ij}^{\bar{x}}{\Phi^{ - 1}}\left[ {{p_i}\left( {\bar{x}} \right)} \right]}}{{\sqrt {{1 - {{\left( {\rho_{ij}^{\bar{x}}} \right)}^2}}} }}} \right) - {p_j}\left( {\bar{x}} \right)} \right). \\ \end{array} $$

(5.122)

is given as

$$ \begin{array} {lll} \eta_{2,c}^{\text{GA}}\left( {\bar{x}} \right) = \sum\limits_{i = 1}^n {{w_i}^2\left( {ELG{D_i}^2\left[ {{p_i}\left( {\bar{x}} \right) - {\Phi_2}\left( {{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right),{\Phi^{ - 1}}\left( {{p_i}\left( {\bar{x}} \right)} \right),\rho_{ii}^{\bar{x}}} \right)} \right]} \right.} \\ \left. { + VLG{D_i}{p_i}\left( {\bar{x}} \right)} \right) \\ \end{array} $$

(5.123)

Inserting the derivative of the bivariate normal distribution (5.117) finally leads to

$$ \frac{{d\eta_{2,c}^{\text{GA}}\left( {\bar{x}} \right)}}{{d\bar{x}}} = \sum\limits_{i = 1}^n {w_i^2} \left( {ELGD_i^2\left[ {\frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}} - \frac{d}{{d\bar{x}}}{\Phi_2}\left( {{y_i}\left( {\bar{x}} \right),{y_i}\left( {\bar{x}} \right),\rho_{ii}^{\bar{x}}} \right)} \right] + VLG{D_i}\frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}} \right). $$

(5.124)

5.1.5 Moment Matching in the BET-Model

5.1.5.1 Matching the First Moment

The expected loss of the original portfolio can be calculated as

$$\frac{{d\eta_{2,c}^{\text{GA}}\left( {\bar{x}} \right)}}{{d\bar{x}}} = \sum\limits_{i = 1}^n {w_i^2} \frac{{d{p_i}\left( {\bar{x}} \right)}}{{d\bar{x}}}\cdot \left( {ELGD_i^2\left[ {1 - 2\Phi \left( {\frac{{{\Phi^{ - 1}}\left[ {{p_i}\left( {\bar{x}} \right)} \right] - \rho_{ii}^{\bar{x}}{\Phi^{ - 1}}\left[ {{p_i}\left( {\bar{x}} \right)} \right]}}{{\sqrt {{1 - {{\left( {\rho_{ii}^{\bar{x}}} \right)}^2}}} }}} \right)} \right]} \right. \left. { + VLG{D_i}} \right).$$

(5.125)

and the expected loss of the hypothetical portfolio as

$$ b{E}\left( {{{\tilde{L}}^{\text{orig}}}} \right) = \sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot LGD \cdot b{E}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} \right)} } = \sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot LGD \cdot P{D_{s,i}}} } $$

(5.126)

with $\begin{array} {c} b{E}\left( {{{\tilde{L}}^{\text{hyp}}}} \right) = \sum\limits_{i = 1}^D {\frac{1}{D} \cdot LGD \cdot b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} \right)} = \frac{1}{D} \cdot LGD \cdot \sum\limits_{i = 1}^D {\bar{p}} \\ = \frac{1}{D} \cdot LGD \cdot D \cdot \bar{p} = LGD \cdot \bar{p}, \\ \end{array}$ for all i. Thus, matching the expectation for both portfolios leads to

$$ b{E}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} \right) = \bar{p} $$

(5.127)

5.1.5.2 Matching the Second Moment

For the original portfolio, the variance can be calculated as

$$ \begin{array} {lll} & b{E}\left( {{{\tilde{L}}^{\text{orig}}}} \right)\mathop { = }\limits^! b{E}\left( {{{\tilde{L}}^{\text{hyp}}}} \right) \\ \Leftrightarrow & \sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot LGD \cdot P{D_{s,i}}} } = LGD \cdot \bar{p} \\ \Leftrightarrow & \bar{p} = \sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot P{D_{s,i}}} } . \\ \end{array} $$

(5.128)

As the default variable is Bernoulli distributed, the variance terms equal

$$ \begin{array} {c} b{V}\left( {{{\tilde{L}}^{\text{orig}}}} \right) = b{V}\left( {\sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot LGD \cdot {1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} } } \right) \\ = LG{D^2} \cdot b{V}\left( {\sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot {1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} } } \right) \\ = LG{D^2} \cdot {\text{Cov}}\left( {\sum\limits_{s = 1}^S {\sum\limits_{i = 1}^{{n_s}} {{w_{s,i}} \cdot {1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} }, \sum\limits_{t = 1}^S {\sum\limits_{j = 1}^{{n_t}} {{w_{t,j}} \cdot {1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} } } \right) \\ = LG{D^2} \cdot \sum\limits_{s = 1}^S {\sum\limits_{t = 1}^S {\sum\limits_{i = 1}^{{n_s}} {\sum\limits_{j = 1}^{{n_t}} {{w_{s,i}} \cdot {w_{t,j}} \cdot {\text{Cov}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)} } } } \\ = LG{D^2} \cdot \sum\limits_{s = 1}^S {\sum\limits_{t = 1}^S {\sum\limits_{i = 1}^{{n_s}} {\sum\limits_{j = 1}^{{n_t}} {{w_{s,i}} \cdot {w_{t,j}} \cdot {\text{Corr}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) \cdot \sqrt {{b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} \right)}} \cdot \sqrt {{b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)}} .} } } } \\ \end{array} $$

(5.129)

and we obtain

$$ b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} \right) = P{D_{s,i}} \cdot \left( {1 - P{D_{s,i}}} \right)\quad {\text{and}}\quad b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) = P{D_{t,j}} \cdot \left( {1 - P{D_{t,j}}} \right) $$

(5.130)

Due to the independence of the default events in the hypothetical portfolio, the variance of this portfolio is

$$ \begin{array} {c} b{V}\left( {{{\tilde{L}}^{\text{orig}}}} \right) = LG{D^2} \cdot \sum\limits_{s = 1}^S {\sum\limits_{t = 1}^S {\sum\limits_{i = 1}^{{n_s}} {\sum\limits_{j = 1}^{{n_t}} {{w_{s,i}} \cdot {w_{t,j}} \cdot {\text{Corr}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)} } } } \\ \cdot \sqrt {{P{D_{s,i}}\left( {1 - P{D_{s,i}}} \right)P{D_{t,j}}\left( {1 - P{D_{t,j}}} \right)}} . \\ \end{array} $$

(5.131)

Matching the variance terms (5.130) and (5.131) leads to

$$ \begin{array} {c} b{V}\left( {{{\tilde{L}}^{\text{hyp}}}} \right) = b{V}\left( {\sum\limits_{i = 1}^D {\frac{1}{D} \cdot LGD \cdot {1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} } \right) = \frac{1}{{{D^2}}} \cdot LG{D^2} \cdot b{V}\left( {\sum\limits_{i = 1}^D {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} } \right) \\ = \frac{1}{{{D^2}}} \cdot LG{D^2} \cdot D \cdot b{V}\left( {{1_{\left\{ {{{\tilde{D}}_i}} \right\}}}} \right) = \frac{1}{D} \cdot LG{D^2} \cdot \bar{p} \cdot \left( {1 - \bar{p}} \right). \\ \end{array} $$

(5.132)

5.1.6 Interrelation of the Pairwise Default Correlation and the Asset Correlation

Using the standard calculus for the correlation and covariance as well as the variance of a Bernoulli distributed variable, the pairwise default correlation between borrower i in sector s and borrower j in sector t can be expressed as

$$ \begin{array} {lll} & b{V}\left( {{{\tilde{L}}^{\text{orig}}}} \right)\mathop { = }\limits^! b{V}\left( {{{\tilde{L}}^{\text{hyp}}}} \right) \\ \Leftrightarrow & D = \frac{{\bar{p} \cdot \left( {1 - \bar{p}} \right)}}{{\sum\limits_{s = 1}^S {\sum\limits_{t = 1}^S {\sum\limits_{i = 1}^{{n_s}} {\sum\limits_{j = 1}^{{n_t}} {{w_{s,i}} \cdot {w_{t,j}} \cdot {\text{Corr}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)} } } } \cdot \sqrt {{P{D_{s,i}}\left( {1 - P{D_{s,i}}} \right)P{D_{t,j}}\left( {1 - P{D_{t,j}}} \right)}} }}. \\ \end{array} $$

(5.133)

The expectation values of the individual default events equal $\begin{array} {c} {\text{Corr}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) = \frac{{{\text{Cov}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)}}{{\sqrt {{b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} \right)}} \cdot \sqrt {{b{V}\left( {{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)}} }} \\ = \frac{{{\text{Cov}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)}}{{\sqrt {{P{D_{s,i}} \cdot \left( {1 - P{D_{s,i}}} \right) \cdot P{D_{t,j}} \cdot \left( {1 - P{D_{t,j}}} \right)}} }} \\ = \frac{{b{E}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}} \cdot {1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) - b{E}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}}} \right) \cdot b{E}\left( {{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right)}}{{\sqrt {{P{D_{s,i}} \cdot \left( {1 - P{D_{s,i}}} \right) \cdot P{D_{t,j}} \cdot \left( {1 - P{D_{t,j}}} \right)}} }}. \\ \end{array}$ and $P{D_{s,i}}$. Similar to (5.102), assuming a normally distributed asset return, the expectation value of a simultaneous default can be written as

$$ P{D_{t,j}} $$

(5.134)

Thus, we get

$$ \begin{array} {c} b{E}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}} \cdot {1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) = b{P}\left[ {\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}} = 1} \right) \wedge \left( {{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}} = 1} \right)} \right] \\ = b{P}\left( {{{\tilde{a}}_{s,i}} \leq {\Phi^{ - 1}}(P{D_{s,i}}),{{\tilde{a}}_{t,j}} \leq {\Phi^{ - 1}}(P{D_{t,j}})} \right) \\ = {\Phi_2}\left( {{\Phi^{ - 1}}\left( {P{D_{s,i}}} \right),{\Phi^{ - 1}}\left( {P{D_{t,j}}} \right),{\text{Corr}}\left( {{{\tilde{a}}_{s,i}},{{\tilde{a}}_{t,j}}} \right)} \right). \\ \end{array} $$

(5.135)

5.1.7 Expected Number of Defaults in the Infectious Defaults Model

Due to the homogeneity of the portfolio and the stochastic independence of all indicator variables, the expected number of defaults is

$$ {\text{Corr}}\left( {{1_{\left\{ {{{\tilde{D}}_{s,i}}} \right\}}},{1_{\left\{ {{{\tilde{D}}_{t,j}}} \right\}}}} \right) = \frac{{{\Phi_2}\left( {{\Phi^{ - 1}}\left( {P{D_{s,i}}} \right),{\Phi^{ - 1}}\left( {P{D_{t,j}}} \right),{\text{Corr}}\left( {{{\tilde{a}}_{s,i}},{{\tilde{a}}_{t,j}}} \right)} \right) - P{D_{s,i}} \cdot P{D_{t,j}}}}{{\sqrt {{P{D_{s,i}}\left( {1 - P{D_{s,i}}} \right)P{D_{t,j}}\left( {1 - P{D_{t,j}}} \right)}} }} $$

(5.136)