# 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.

## 1 Answer

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.

• Thanks for the reply. If the unconditional probability is only use to parameterize the model then how do I interpret the sentence "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". It sounds like the unconditional* probability of a particular loss level is different from the unconditional probability of default. Is that a correct understanding? – AfterWorkGuinness Oct 13 '15 at 16:06
• The unconditional probability of default is for a particular asset. The unconditional probability of a particular loss level is for the entire portfolio. – user9403 Oct 13 '15 at 18:01
• Hi user9403. Thanks for your reply, and sorry for taking forever to look at it. The text says since asset returns are i.i.d, we can apply the law of large numbers and thus the loss level x(m) for the portfolio converges to a single credit. – AfterWorkGuinness Nov 19 '15 at 1:40
• After reading this thing over a ton of times, I think my source of confusion comes down to ambiguous wording by the author. He first presents it as a means to solve for a credit's "conditional cumulative probability of default" meaning the cumulative probability of default conditional on the market factor return m. He then says we can use it to solve for the unconditional probability of default. Unconditional PD is an alternate term for cumulative. He's saying the same thing a different way. I think – AfterWorkGuinness Nov 19 '15 at 1:44