## 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 timeLoss 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 *A*_{i} associated
to the *i*th 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 *j*th counterparty follows a similar
pattern. If the *A*_{i} and *A*_{j} 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 ε_{i} are
standard normal variables, then *A**i* is
also a standard normal.

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

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

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 *A**i* 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:

The full simulated distribution of portfolio losses across scenarios and the losses on each counterparty across scenarios. For more information, see

`creditDefaultCopula`

object properties and`simulate`

.Risk measures (

`VaR`

,`CVaR`

,`EL`

,`Std`

) with confidence intervals. See`portfolioRisk`

.Risk contributions per counterparty (for

`EL`

and`CVaR`

). See`riskContribution`

.Risk measures and associated confidence bands. See

`confidenceBands`

.Counterparty scenario details for individual losses for each counterparty. See

`getScenarios`

.

#### 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:

The full simulated distribution of portfolio values. For more information, see

`creditMigrationCopula`

object properties and`simulate`

.Risk measures (

`VaR`

,`CVaR`

,`EL`

,`Std`

) with confidence intervals. See`portfolioRisk`

.Risk contributions per counterparty (for

`EL`

and`CVaR`

). See`riskContribution`

.Risk measures and associated confidence bands. See

`confidenceBands`

.Counterparty scenario details for each counterparty. See

`getScenarios`

.

## See Also

`creditDefaultCopula`

| `creditMigrationCopula`

| `asrf`

## Related Examples

- creditDefaultCopula Simulation Workflow
- creditMigrationCopula Simulation Workflow
- Modeling Correlated Defaults with Copulas
- One-Factor Model Calibration