### Portfolio Credit Risk

This chapter is an extension of the discussion on credit risk to portfolios with several credit risky securities. The focus of this chapter is models measuring portfolio credit risk, employing a factor model whose key feature is a latent factor with normally distributed returns.

# Default Correlation

The following risk and return elements are considered when modeling a single credit risky position: the likelihood to default, LGD, the rating migration’s likelihood and severity, spread risks and the likelihood of restructuring the company’s debt for debts that are distressed.

In a portfolio of debts issued by various obligors, the possibility of encountering multiple defaults is driven by the additional concept of default correlation which is introduced to understand credit portfolio risk.

## Defining Default correlation

To understand default correlation, we consider the following simple framework: two companies, of default probabilities $${ \pi }_{ 1 }$$ and $${ \pi }_{ 2 }$$, over period $$\tau$$ and with a joint default probability $${ \pi }_{ 12 }$$. The likelihood of a default by at least one firm is computed as 1 less the first outcome’s likelihood or the last three outcomes’ sum of probabilities.

$$P\left[ Default\quad by\quad firm\quad 1\quad or\quad 2\quad or\quad both \right] ={ \pi }_{ 1 }+{ \pi }_{ 2 }-{ \pi }_{ 12 }$$

The two default processes that are Bernoulli-distributed have their expected values as:

$$E\left[ { X }_{ i } \right] ={ \pi }_{ i },\quad where\quad i=1,2$$

The joint defaults product mean is:

$$E\left[ { X }_{ 1 }{ X }_{ 2 } \right] ={ \pi }_{ 12 }$$

With variances:

$$E{ \left[ { X }_{ i } \right] }^{ 2 }-{ \left( E{ \left[ { X }_{ i } \right] } \right) }^{ 2 }={ \pi }_{ i }\left( 1-{ \pi }_{ i } \right) ,\quad where\quad i=1,2$$

The covariance is:

$$E\left[ { X }_{ 1 }{ X }_{ 2 } \right] -E\left[ { X }_{ 1 } \right] E\left[ { X }_{ 2 } \right] ={ \pi }_{ 12 }-{ \pi }_{ 1 }{ \pi }_{ 2 }$$

Finally, the default correlation is:

$${ \rho }_{ 12 }=\frac { { \pi }_{ 12 }-{ \pi }_{ 1 }{ \pi }_{ 2 } }{ \sqrt { { \pi }_{ 1 }\left( 1-{ \pi }_{ 1 } \right) } \sqrt { { \pi }_{ 2 }\left( 1-{ \pi }_{ 2 } \right) } } \quad \quad \quad \quad \quad \left( a \right)$$

The joint default probability can be determined by treating the default correlation as the primitive parameter and as the not joint default probability:

$${ \pi }_{ 12 }={ \rho }_{ 12 }\sqrt { { \pi }_{ 1 }\left( 1-{ \pi }_{ 1 } \right) } \sqrt { { \pi }_{ 2 }\left( 1-{ \pi }_{ 2 } \right) } +{ \pi }_{ 1 }{ \pi }_{ 2 }$$

More than one default joint probabilities and default correlations is contained in a portfolio with more than two credits.

## The Order of Magnitude of Default Correlation

For most debt-issuing firms, default is a relatively rare event. This means that by historical data of default, it is difficult to determine or approximate this parameter, and the magnitude of default correlation is small.

## Credit Portfolio Risk Measurement

Modeling default, default correlation, and loss given default should be modeled to determine the risk of the credit portfolio.

## The Effect of Granularity on Credit VaR

The term “granular” refers to increasing the number of credits in a portfolio such that the weight of each credit as a proportion of the total portfolio reduces. It can be shown that as a credit portfolio becomes more granular, the credit VaR decreases. However, with a low default probability, credit VaR is not impacted as much when the portfolio is broken down (becomes more granular).

# Default Distributions and Credit VaR with the Single-Factor Model

With a single factor model, default correlation can be varied via the beta of the credit to the market factor letting idiosyncratic risk play a role. Throughout this chapter, default correlation is allowed to take values between $$\left( 0,1 \right)$$.

## Conditional Default Distributions

A company’s $$i$$ ROA is:

$${ a }_{ i }={ \beta }_{ i }m+\sqrt { 1-{ \beta }_{ i }^{ 2 }{ \varepsilon }_{ i } } \quad \quad \quad \quad \quad i=1,2,\dots ,$$

Where $$i=1,2,\dots ,$$ is the number of firms, each with correlation $${ \beta }_{ i }$$ to the market factor and it’s standard deviation idiosyncratic risk $$\sqrt { 1-{ \beta }_{ i }^{ 2 } }$$, and its own idiosyncratic shock $${ \varepsilon }_{ i }$$.

Let $$m$$ and $${ \varepsilon }_{ i }$$ be uncorrelated standard normal variates. Then if $${ \varepsilon }_{ i }$$ are not correlated with each other:

$$m\sim N\left( 0,1 \right)$$

$${ \varepsilon }_{ i }=N\left( 0,1 \right) \quad \quad \quad i=1,2,\dots$$

$$Cov\left[ m,{ \varepsilon }_{ i } \right] =0\quad \quad \quad i=1,2,\dots$$

$$Cov\left[ { \varepsilon }_{ i },{ \varepsilon }_{ i } \right] =i\quad \quad \quad i,j=1,2,\dots$$

The pair of firms $$i$$ and $$j$$ has the correlation between the asset returns $${ \beta }_{ i } { \beta }_{ j }$$:

$$E\left[ { a }_{ i } \right] =0\quad \quad \quad \quad \quad \quad i=1,2,…$$

$$var\left[ { a }_{ i } \right] ={ \beta }_{ i }^{ 2 }+1-{ \beta }_{ i }^{ 2 }=1\quad \quad \quad \quad \quad \quad i=1,2,…$$

$$Cov\left[ { a }_{ i },{ a }_{ j } \right] =E\left[ \left( { \beta }_{ i }m+\sqrt { 1-{ \beta }_{ i }^{ 2 }{ \varepsilon }_{ i } } \right) \left( { \beta }_{ j }m+\sqrt { 1-{ \beta }_{ j }^{ 2 }j } \right) \right]$$

$$={ \beta }_{ i }{ \beta }_{ j },\quad \quad \quad \quad \quad i,j=1,2,\dots$$

Due to model assumptions, that only the relationship of companies’ returns correlates them to the market factor: If we suppose that $$m$$ takes on a particular value $$\bar { m }$$, then the distance to default-the asset return-increases or decreases, is given by:

$${ a }_{ i }-{ \beta }_{ i }\bar { m } +\sqrt { 1-{ \beta }_{ i }^{ 2 } } { \varepsilon }_{ i }\quad \quad \quad \quad i=1,2,\dots$$

Therefore, the conditional cumulative default probability function can be given as:

$$p\left( m \right) =\emptyset \left( \frac { { k }_{ i }-{ \beta }_{ i }{ m } }{ \sqrt { 1-{ \beta }_{ i }^{ 2 } } } \right) \quad \quad \quad \quad \quad i=1,2,\dots$$

Where $${ k }_{ i }$$ is the default threshold.

# Asset and Default Correlation

For any pair of credits $$i$$ and $$j$$, the cumulative return distribution has a correlation coefficient equal to $${ \beta }_{ i } { \beta }_{ j }$$, and is a bivariate standard normal in the single-factor model:

$$\begin{pmatrix} { a }_{ i } \\ { a }_{ j } \end{pmatrix}\sim N\left[ \begin{pmatrix} 0 \\ 0 \end{pmatrix}\begin{pmatrix} 1 & { \beta }_{ i }{ \beta }_{ j } \\ { \beta }_{ i } { \beta }_{ j } & 1 \end{pmatrix} \right]$$

The cumulative distribution function is $$\emptyset \begin{pmatrix} { a }_{ i } \\ { a }_{ j } \end{pmatrix}$$, and consequently the likelihood of a joint default being equal to the likelihood that the realized value falls in the region:

$$\left\{ -\infty \le { a }_{ i }\le { k }_{ i },-\infty \le { a }_{ j }\le { k }_{ j } \right\} :$$

$$\emptyset \begin{pmatrix} { k }_{ i } \\ { k }_{ j } \end{pmatrix}=P\left[ -\infty \le { a }_{ i }\le { k }_{ i },-\infty \le { a }_{ j }\le { k }_{ j } \right]$$

We substitute the $${ \pi }_{ ij }=\emptyset \begin{pmatrix} { k }_{ i } \\ { k }_{ j } \end{pmatrix}$$ equation labeled ($$a$$) of $${ \rho }_{ 12 }$$ to get the model’s default correlation. The linear correlation’s equation is:

$${ \rho }_{ ij= }\frac { \emptyset \begin{pmatrix} { k }_{ i } \\ { k }_{ j } \end{pmatrix}-{ \pi }_{ i }{ \pi }_{ j } }{ \sqrt { { \pi }_{ i }\left( 1-{ \pi }_{ i } \right) } \sqrt { { \pi }_{ j }\left( 1-{ \pi }_{ j } \right) } }$$

Assuming that for all companies, the parameters are the same, i.e., $${ \beta }_{ i }=\beta ,{ k }_{ i }=k,\dots ,$$ then:

$$\emptyset \begin{pmatrix} { k } \\ { k } \end{pmatrix}=P\left[ -\infty \le { a }\le { k },-\infty \le { a }\le { k } \right]$$

And the default correlation for any pair of firms is:

$${ \rho }_{ = }\frac { \emptyset \begin{pmatrix} { k } \\ { k } \end{pmatrix}-{ \pi }^{ 2 } }{ \sqrt { { \pi }\left( 1-{ \pi } \right) } \sqrt { { \pi }\left( 1-{ \pi } \right) } }$$

# Credit VaR Using the Single-Factor Model

This section seeks to show how the single factor model is applied in the estimation of the credit $$VaR$$ of a granular homogeneous portfolio.

## Conditional Default Probability and Loss Level

The loss level $$x\left( m \right)$$, is the fraction of portfolio that defaults has its convergence to the conditional likelihood that a single credit defaults and is given by the following equation for each market factor level:

$$p\left( m \right) =\emptyset \left( \frac { k-\beta m }{ \sqrt { 1-{ \beta }^{ 2 } } } \right)$$

So we have that:

$$\begin{matrix} & lim & \\ N & \rightarrow & \infty \end{matrix}x\left( m \right) =p\left( m \right) \quad \quad \forall \quad m\quad \epsilon \quad R$$

The implication is that given the two parameters of the model, the likelihood of default and correlation, portfolio returns are market-factor driven.

## The Unconditional Default Probability and Loss Level

The unconditional default probability is equal to the likelihood that the market factor return leading to the loss is realized. The following steps give the process of computing the unconditional distribution:

• The level of loss should be treated as a random variable $$X$$, with realizations $$x$$.
• The market factor return and the loss level are related by:

$$x\left( m \right) =p\left( m \right) =\emptyset \left( \frac { k-\beta m }{ \sqrt { 1-{ \beta }^{ 2 } } } \right)$$

• The market factor return to a loss level given $$\bar { x }$$ is $$\bar { m }$$ and can be solved by:

$${ \emptyset }^{ -1 }\left( \bar { x } \right) = \frac { k-\beta \bar { m } }{ \sqrt { 1-{ \beta }^{ 2 } } }$$

Or:

$$\bar { m } =\frac { k-\sqrt { 1-{ \beta }^{ 2 } } { \emptyset }^{ -1 }\left( \bar { x } \right) }{ \beta }$$

• By assumption, the market factor is the standard normal:

$$P\left[ X\le \bar { x } \right] ={ \emptyset }\left( \bar { m } \right) =\emptyset \left( \frac { k-\sqrt { 1-{ \beta }^{ 2 } } { \emptyset }^{ -1 }\left( \bar { x } \right) }{ \beta } \right)$$

• To determine the probability distribution of $$X$$, this procedure is repeated.

# Practice Question

The conditional cumulative probability of default by a single credit is given by $$\emptyset \left( 3.254 \right) =0.1124$$.What is the default threshold if $${ \beta }_{ i }=0.74$$ and $$m=0.586$$ ?

1. 2.6223
2. 0.5092
3. 2.1887
4. 3.2154

The conditional cumulative default probability function can be given as:

$$p\left( m \right) =\emptyset \left( \frac { { k }_{ i }-{ \beta }_{ i }{ m } }{ \sqrt { 1-{ \beta }_{ i }^{ 2 } } } \right)$$

From the problem,we have: $${ \beta }_{ i }=0.74$$ and $$m=0.586$$. We have to solve for $${ k }_{ i }$$.

Therefore:

$$\emptyset \left( 2.56 \right) =\emptyset \left( \frac { { k }_{ i }-{ \beta }_{ i }{ m } }{ \sqrt { 1-{ \beta }_{ i }^{ 2 } } } \right) =0.1124$$

$$\Rightarrow \left( \frac { { k }_{ i }-0.74\times 0.586 }{ \sqrt { 1-{ 0.74 }^{ 2 } } } \right) =3.254$$

$$\Rightarrow { k }_{ i }-0.74\times 0.586=3.254\times \sqrt { 1-{ 0.74 }^{ 2 } }$$

$$\Rightarrow { k }_{ i }=3.254\times \sqrt { 1-{ 0.74 }^{ 2 } } +0.74\times 0.586$$

$$\Rightarrow { k }_{ i }=2.6223$$