## Credit Simulation Using Copulas

Predicting the credit losses for a counterparty depends on three main elements:

• Probability of default (`PD`)

• Exposure at default (`EAD`), the value of the instrument at some future time

• Loss given default (`LGD`), which is defined as 1 − Recovery

If these quantities are known at future time t, then the expected loss is `PD × EAD × LGD`. In this case, you can model the expected loss for a single counterparty by using a binomial distribution. The difficulty arises when you model a portfolio of these counterparties and you want to simulate them with some default correlation.

To simulate correlated defaults, the copula model associates each counterparty with a random variable, called a “latent” variable. These latent variables are correlated using some proxy for their credit worthiness, for example, their stock price. These latent variables are then mapped to default or nondefault outcomes such that the default occurs with probability `PD`.

This figure summarizes the copula simulation approach.

The random variable Ai associated to the ith counterparty falls in the default shaded region with probability `PD`i. If the simulated value falls in that region, it is interpreted as a default. The jth counterparty follows a similar pattern. If the Ai and Aj random variables are highly correlated, they tend to both have high values (no default), or both have low values (fall in the default region). Therefore, there is a default correlation.

### Factor Models

For M issuers, M(M − 1)/2 correlation parameters are required. For M = 1000, this is about half a million correlations. One practical variation of the approach is the one-factor model, which makes all the latent variables dependent on a single factor. This factor Z represents the underlying systemic credit quality in the economy. This model also includes a random idiosyncratic error.

`${A}_{i}={w}_{i}Z+\sqrt{1-{w}_{i}^{2}}{\epsilon }_{i}$`

This significantly reduces the input-data requirements, because now you need only the M sensitivities, that is, the weights `w`1,…,`w`M. If Z and εi are standard normal variables, then Ai is also a standard normal.

An extension of the one-factor model is a multifactor model.

`${A}_{i}={w}_{i1}{Z}_{1}+...+{w}_{iK}{Z}_{K}+{w}_{i\epsilon }^{}{\epsilon }_{i}$`

This model has several factors, each one associated with some underlying credit driver. For example, you can have factors for different regions or countries, or for different industries. Each latent variable is now a combination of several random variables plus the idiosyncratic error (epsilon) again.

When the latent variables Ai are normally distributed, there is a Gaussian copula. A common alternative is to let the latent variables follow a t distribution, which leads to a t copula. t copulas result in heavier tails than Gaussian copulas. Implied credit correlations are also larger with t copulas. Switching between these two copula approaches can provide important information on model risk.

### Supported Simulations

Risk Management Toolbox™ supports simulations for counterparty credit defaults and counterparty credit rating migrations.

#### Credit Default Simulation

The `creditDefaultCopula` object is used to simulate and analyze multifactor credit default simulations. These simulations assume that you calculated the main inputs to this model on your own. The main inputs to this model are:

• `PD` — Probability of default

• `EAD` — Exposure at default

• `LGD` — Loss given default (1 − Recovery)

• `Weights` — Factor and idiosyncratic weights

• `FactorCorrelation` — An optional factor correlation matrix for multifactor models

The `creditDefaultCopula` object enables you to simulate defaults using the multifactor copula and return the results as a distribution of losses on a portfolio and counterparty level. You can also use the `creditDefaultCopula` object to calculate several risk measures at the portfolio level and the risk contributions from individual obligors. The outputs of the `creditDefaultCopula` model and the associated functions are:

#### Credit Rating Migration Simulation

The `creditMigrationCopula` object enables you to simulate changes in credit rating for each counterparty.

The `creditMigrationCopula` object is used to simulate counterparty credit migrations. These simulations assume that you calculated the main inputs to this model on your own. The main inputs to this model are:

• `migrationValues` — Values of the counterparty positions for each credit rating.

• `ratings` — Current credit rating for each counterparty.

• `transitionMatrix` — Matrix of credit rating transition probabilities.

• `LGD` — Loss given default (1 − Recovery)

• `Weights` — Factor and idiosyncratic model weights

You can also use the `creditMigrationCopula` object to calculate several risk measures at the portfolio level and the risk contributions from individual obligors. The outputs of the `creditMigrationCopula` model and the associated functions are: