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The following formulation is from Vasicek and refers to the cond. probability of the loss of a loan (equ. 3 in the reference): $$p(Y)=\Phi\left(\frac{\Phi^{-1}(p)-\sqrt{\rho}\,Y}{\sqrt{1-\rho}}\right).$$

Vasicek also states that the variables in his outlined asset value formula (equ. 1 in the reference) are jointly standard normal distributed with equal pairwise correlations $\rho$, leading to:

$$ X_i=\sqrt{\rho}\,Y+\sqrt{1-\rho}\,Z_i,$$

with $Y$ a portfolio commonn factor and $Z_i$ a company specific factor (equ. 2 in the reference).

Similar specifications are also used, for example, in connection with the risk-weighted assets of the Basel regulations or in the context of default correlations.

Why is it common or preferred to use a specification with the square root regarding the parameters (e.g., pairwise correlation) in factor models? Are there any specific benefits to using this specification?

Vasicek, O. A. (2002). The Distribution of Loan Portfolio Value, Risk 15(12), 160–162.

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  • The main motivation for the use of Vasicek single factor framework, is that the model produce analytically tractable formulas that are easy to implement and does not require any extensive numerical simulations. The model is able to reproduce the qualitative behavior of empirical credit loss distributions, namely fat tails and skewness [link]. This is especially true when you have many names in your portfolio that have a small exposure and are assumed to have a homogeneous pairwise correlation (also called Large Homogeneous Portfolio Approximation).

  • The model can be used whenever you want to model the credit loss distribution under a relatively simple framework. It can be viewed as the "Black-Scholes' equivalent" for credit portfolios or baskets. In terms of the Basel framework, the committee argues in the paper found here, on the reason for choosing the Single Factor Model:

  1. Regulatory requirements to the Basel credit risk model:

[...] The model specification was subject to an important restriction in order to fit supervisory needs:

The model should be portfolio invariant, i.e. the capital required for any given loan should only depend on the risk of that loan and must not depend on the portfolio it is added to. This characteristic has been deemed vital in order to make the new IRB framework applicable to a wider range of countries and institutions.

  1. Model specification:

[...] In the specification process of the Basel II model, it turned out that portfolio invariance of the capital requirements is a property with a strong influence on the structure of the portfolio model. It can be shown that essentially only so-called Asymptotic Single Risk Factor (ASRF) models are portfolio invariant (Gordy, 2003).

You can look further into (Gordy, 2003) if you want more details.

  • Lastly, the Large Homogeneous portfolio approximation can also be used to price CDO tranches under the same setup (see link for details).
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Why is there a square root in the formulation for the asset value $ X_i $ given the common factor $ Y $ and the idiosyncratic component $ Z_i $ below?

$$ X_i=\sqrt{\rho}\,Y+\sqrt{1-\rho}\,Z_i,$$

As you write it follows from the assumption of assets all having equal pairwise correlations $ \rho\ $ to each other and that all variables have unit variance. It's quite simple, trying with :$$ X_i=\beta\,Y+\gamma\,Z_i,$$ and using the assumed correlation : $$ \rho\ = cov(X_i, X_j)/\sqrt{Var(X_i)Var(X_j)}$$ further using independence between $ Z_i\ $and $ Z_j $ and $ Y $ gives us : $$ \rho\ = \beta^2\ Var(Y) /\sqrt{Var(X_i)Var(X_j)} = \beta^2\ $$

Hence the square root in the first term $ \beta\ = \sqrt{\rho} $ and taking the variance of $ X_i $ : $$ Var(X_i)= \beta^2 + \gamma^2 = 1 $$ now gives us the square root in the second term $ \gamma\ = \sqrt{1-\rho} $

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The question is not specific to Vasicek, the square root of correlation you see in the paper is a standard formula to generate correlated RVs. See for example this question

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