# Documentation

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# asrf

Asymptotic Single Risk Factor (ASRF) capital

## Syntax

``[capital,VaR] = asrf(PD,LGD,R)``
``[capital,VaR] = asrf(___,Name,Value)``

## Description

example

````[capital,VaR] = asrf(PD,LGD,R)` computes regulatory capital and value-at-risk using an ASRF model​.​```

example

````[capital,VaR] = asrf(___,Name,Value)` adds optional name-value pair arguments. ```

## Examples

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Load saved portfolio data.

```load CreditPortfolioData.mat ```

Compute asset correlation for corporate, sovereign, and bank exposures.

```R = 0.12 * (1-exp(-50*PD)) / (1-exp(-50)) +... 0.24 * (1 - (1-exp(-50*PD)) / (1-exp(-50))); ```

Compute the asymptotic single risk factor capital. By specifying the name-value pair argument for `EAD`, the `capital` is returned in terms of currency.

```capital = asrf(PD,LGD,R,'EAD',EAD); ```

Apply a maturity adjustment.

```b = (0.11852 - 0.05478 * log(PD)).^2; matAdj = (1 + (Maturity - 2.5) .* b) ./ (1 - 1.5 * b); adjustedCapital = capital .* matAdj; portfolioCapital = sum(adjustedCapital) ```
```portfolioCapital = 175.7865 ```

## Input Arguments

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Probability of default, specified as a `NumCounterparties`-by-`1` numeric vector with elements from `0` to `1`, representing the default probabilities for the counterparties.

Data Types: `double`

Loss given default, specified as a `NumCounterparties`-by-`1` numeric vector with elements from `0` to `1`, representing the fraction of exposure that is lost when a counterparty defaults. `LGD` is defined as (1 − Recovery). For example, an `LGD` of 0.6 implies a 40% recovery rate in the event of a default.

Data Types: `double`

Asset correlation, specified as a `NumCounterparties`-by-`1` numeric vector.

The asset correlations, `R`, have values from `0` to `1` and specify the correlation between assets in the same asset class.

### Note

The correlation between an asset value and the underlying single risk factor is `sqrt`(`R`). This value, `sqrt`(`R`), corresponds to the `Weights` input argument to the `creditDefaultCopula` and `creditMigrationCopula` classes for one-factor models.

Data Types: `double`

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `capital = asrf(PD,LGD,R,'EAD',EAD)`

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Exposure at default, specified as the comma-separated pair consisting of `'EAD'` and a `NumCounterparties`-by-`1` numeric vector of credit exposures.

If `EAD` is not specified, the default `EAD` is `1`, meaning that `capital` and `VaR` results are reported as a percentage of the counterparty's exposure. If `EAD` is specified, then `capital` and `VaR` are returned in units of currency.

Data Types: `double`

Value at risk level used when calculating the capital requirement, specified as the comma-separated pair consisting of `'VaRLevel'` and a decimal value between `0` and `1`.

Data Types: `double`

## Output Arguments

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Capital for each element in the portfolio, returned as a `NumCounterparties`-by-`1` vector. If the optional input `EAD` is specified, then `capital` is in units of currency. Otherwise, `capital` is reported as a percentage of each exposure.

Value-at-risk for each exposure, returned as a `NumCounterparties`-by-`1` vector. If the optional input `EAD` is specified, then `VaR` is in units of currency. Otherwise, `VaR` is reported as a percentage of each exposure.

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### ASRF Model Capital

In the ASRF model, capital is defined as the loss in excess of the expected loss (EL) at a high confidence level.

The formula for capital is

`capital = VaR - EL`

## Algorithms

The capital requirement formula for exposures is defined as

`$\begin{array}{l}VaR=EAD*LGD*\Phi \left(\frac{{\Phi }^{-1}\left(PD\right)-\sqrt{R}{\Phi }^{-1}\left(1-VaRLevel\right)}{\sqrt{1-R}}\right)\\ capital=VaR-EAD*LGD*PD\end{array}$`
where

`ɸ` is the normal CDF.

`ɸ`-1 is the inverse normal CDF.

`R` is asset correlation.

`EAD` is exposure at default.

`PD` is probability of default.

`LGD` is loss given default.

## References

[1] Gordy, M.B. "A risk-factor model foundation for ratings-based bank capital rule." Journal of Financial Intermediation. Vol. 12, pp. 199-232, 2003.