# asrf

Asymptotic Single Risk Factor (ASRF) capital

## Description

example

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

The ASRF model is useful because the Basel II documents propose this model as the standard for certain types of capital requirements. ASRF is not a Monte-Carlo model, so you can quickly compute the capital requirements for large credit portfolios. You can use the ASRF model to perform a quick sensitivity analysis and exploring "what-if" scenarios more easily than rerunning large simulations.

example

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

## Examples

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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.

b = (0.11852 - 0.05478 * log(PD)).^2;
matAdj = (1 + (Maturity - 2.5) .* b) ./ (1 - 1.5 * b);

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 Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

ϕ is the normal CDF.

ϕ-1 is the inverse normal CDF.

R is asset correlation.

PD is probability of default.

LGD is loss given default.

## References

[1] Basel Committee on Banking Supervision. "International Convergence of Capital Measurement and Capital Standards." June, 2006 (https://www.bis.org/publ/bcbs128.pdf).

[2] Basel Committee on Banking Supervision. "An Explanatory Note on the Basel II IRB Risk Weight Functions." July, 2005 (https://www.bis.org/bcbs/irbriskweight.pdf).

[3] Gordy, M.B. "A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules." Journal of Financial Intermediation. Vol. 12, pp. 199-232, 2003.

## Version History

Introduced in R2017b