Vasicek’s Single-Factor Model for Portfolio Credit Losses

# A little help with the Single Factor model for credit risk


# A little help with the Single Factor model for credit risk












I'm studying the "single factor model" in Malz text "Financial Risk Management - Models, History and Institutions". He only refers to it as such and gives it no proper name.

The model:

$a_{i} = \beta_{i}m+(\sqrt{1-\beta^2})\epsilon_{i}$

$\beta$ is the correlation of the firm to the state of the economy

$m$ is the state of the economy

I'm a bit confused here. The author first says we can use the model to convert an unconditional probability of default into one conditional on the state of the economy

and then further says "The unconditional probability of a particular loss level (the fraction of the portfolio that defaults) is equal to the probability that the market factor return that leads to that loss level is realized"

We find both probabilities in the same way:

$p(m) = \phi( \frac{k_{i}-\beta_{i}m}{\sqrt{1 - \beta^2}} )$

Where $k = \phi^{-1}(\pi)$

and $\pi$ = unconditional probability of default in the first usage and probability of realizing the market factor leading to observed the loss level in the second usage.

These sound opposite to me. In one usage we are finding a conditional PD and in another what is described as an unconditional.

## Answer by user9403 (score 4)

https://quant.stackexchange.com/a/21118

The name for the model is Vasicek's single factor model.

The model is very similar to CAPM: each asset has idiosyncratic and systemic risk with systemic risk driven by a single factor. Default occurs when an asset has a realization that is below some threshold. The level of this threshold doesn't matter; we can solve for it if we know the unconditional probability of default for the asset.

Much like CAPM, the idiosyncratic risk can be "diversified" away. In a large portfolio of homogenous assets the only "risk" (that is, variability around the expected loss) comes from the systemic variable. Without this variable the distribution converges to a Dirac-delta function around the expected loss. Hence the final output (the distribution of loss) from the model is solely dependent on the systemic variable.

To summarize, the unconditional default is used to parameterize the model, while the conditional is used to determine the output.