Estimating Vasicek Asset Correlations Under Basel IRB Rules

# How do I estimate the factor sensitivity in a Vasicek Single Factor Model?


# How do I estimate the factor sensitivity in a Vasicek Single Factor Model?












I understand the formula of an asset return for an obligor i is given by the following: $$A_i = \sqrt{w_i}*Z + \sqrt{1-w_i}*\epsilon_i $$

My question is - How do I calculate $w_i$? I have the PD, LGD and EAD data for each obligor, as well as the covariance matrix for the asset class of each obligor (I have about 5,000 obligors with 3 asset classes with each obligor belonging to just one asset class).

From what I have read, $w_i$ is asset correlation. While I have the asset covariance matrix, I am unsure how to use it to derive the asset correlation (because each asset will have 2 covariances with the other two classes).

In another place, I read that it can be derived as asset correlation, $\rho$:

$$\rho = 0.12*(1-e^{-50*PD}/1-e^{-50}) + 0.24*[1-(1-e^{-50*PD}/1-e^{-50})]$$ (the values 0.12 and 0.24 have been chosen because $\rho$ can be within the chosen range of 12%-24%.

All this is very confusing and any help will be greatly appreciated.

## Answer by Pleb (score 2, accepted)

https://quant.stackexchange.com/a/68012

#### Calculating correlation under the Basel II accords:

Your second equation is almost in line with the international regulatory framework of the Basel II accords for calculating a banks minimum capital requirement, which is a framework telling you how to model PD, LGD and EAD following (at minimum) the foundational IRB approach. Any country that is a Basel committee member needs to satisfy these regulatory requirements.

Let us assume that your bank meet the requirements (at minimum) of the IRB approach, giving them the leeway to use their own PD estimates. Then the official document of the Basel II accords for minimum capital requirements (see above link) specify the correlation structure as:

$$\rho = A \cdot \frac{1-e^{-50\cdot PD}}{1-e^{-50}} + B \cdot \frac{1-(1-e^{-50\cdot PD})}{1-e^{-50\cdot PD}} \mathbin{\color{red}{-w^{SME}}},$$

where we observe that $\rho$ depends negatively on the obligors PD following from Vasicek model. Here, $w^{SME}$ is a firm-size adjustment term for small- and medium-sized entities (SME). Under IRB for corporate credits, banks are permitted to separately distinguish exposures of SMEs and larger borrowers. SMEs are defined as having annual corporate sales less than or equal to 50 million EUR. This adjustment term is given by:

$$w^{SME}=0.04 \cdot \left(1-\frac{S-5}{45}\right),$$

where $S$ is expressed as the total amount of annual sales in millions of euros, and needs to be between $5\leq S \leq 50$ million euros. Conclusively, SME's get a max reduction of $0.04$ in the correlation estimate compared to larger corporate exposures.

Be aware that for corporate, sovereign and bank exposures, $A$ and $B$ are set between $0.12$ and $0.24$, respectively (see pp. 63-64). As you wrote yourself, this implies that for low PD estimates, the correlation goes towards $0.24$ and vice versa. For high-volatility commercial real estate the correlation lies between $A=0.12$ and $B=0.3$ (pp. 65-66).

On the other hand, for retail exposures the support of the correlation is between $A=0.03$ and $B=0.16$ (pp. 77-78), with the exception of mortgages having a fixed correlation of $0.15$ (pp. 76-77). The stringent definitions of the different exposure types are provided at the beginning of the Basel II document.

I would recommend calculating the correlations using this approach, since a lot of countries need to satisfy the Basel II regulatory framework. Try and look up the central bank of your own country. They typically have practical (and shorter) papers for the calculation of different types of risk under regulatory frameworks.

I have provided my answer with page numbers, so you can do more in-depth research into the whopping 190 pages Basel II document. I hope my answer provides some insight.