Malz single factor model

[saurabhpal49]

hi David,
Could you please explain the below mentioned sentence

Given m a realization of Ei less than or equal to Ki - Bim triggers default. As we let m vary from high to low values a smaller idisyncratic shock will suffice to trigger default

 

[David Harper CFA FRM]

Hi @saurabhpal49 Malz single-factor model is intermediate/advanced, I just want to "warn" you so that you do not expect to immediately grok it, as it's proven to be challenging . I copied below the latest version of my rendering of his example Malz 8.4. To grok the specific sentence you cite, IMO, is difficult without understanding the broader model (if you already get it, great!). Below I copied Malz example, which is specifically illustrated by my XLS further below.

To specifically answer your question, the meaning of "as we let m vary from high to low values a smaller idisyncratic shock will suffice to trigger default," is illustrated below: If you look at the market factor, m row, as it decreases from m = 0 to m = -2.33, notice the idiosyncratic (random) shock required to default (final row) is decreasing.

In this single factor model, the asset return is given by a(i) = β(i)*m + sqrt(1-β^2)*ϵ(i), where (m) is the single factor that represents the macro-economy and epsilon is the random standard normal shock. By specifying m = 1, the model simulations a downturn and, consequently a conditional distribution with a higher probability of default. I'll stop here because i don't want to overwhelm. I hope that's helpful, thanks!

Example 8.4 (Default Probability and Default Threshold) Suppose a firm has β(i) = 0.4 and k(i) = −2.33, it is a middling credit, but cyclical (relatively high βi). Its unconditional probability of default is Φ(−2.33) = 0.01. If we enter a modest economic downturn, with m = −1.0, the conditional asset return distribution is N(−0.4, sqrt[1 − 0.402]) or N(−0.4, 0.9165), and
the conditional default probability is found by computing the probability that this distribution takes on the value −2.33. That probability is 1.78 percent.

If we were in a stable economy with m = 0, we would need a shock of −2.33 standard deviations for the firm to die. But with the firm’s return already 0.4 in the hole because of an economy-wide recession, it takes only a 1.93 standard deviation additional shock to kill it.

Now suppose we have a more severe economic downturn, with m =−2.33. The firm’s conditional asset return distribution is N(−0.932, 0.9165) and the conditional default probability is 6.4 percent. A 0.93 standard deviation shock (ϵ(i) ≤ −0.93) will now trigger default [dharper: I disagree with this value]

This XLS is also here https://www.dropbox.com/s/7dnceu42dtpvzw7/P2.T6.Malz_Ch8_v2.xlsx?dl=0

 

[emilioalzamora1]

Hi @David Harper CFA FRM,

it's been a while since we have been touch. Hope you managed the last few months unscathed! Apparently GARP decided to follow the CFA Institute changing from paper-based to online exams. Interesting to see what has changed since the onset of the pandemic.

Would like to have your opinion about Malz' single factor model. I am just wondering about a few things:

• Any positive market factor (e..g. m = 0.2 or 1.0 or 2.0) implies a prospering economy (decreasing unemployment rate etc), right?
• The market factor can take on any value between approx. positive and negative 3, right? (z-score of the standard normal table)
• Given your example and assuming a beta of 0.4 and a negative market factor (-1.0) the unconditional PD changes from 1% to a conditional PD of 1.76%
  • If we then for example change the beta to 0.8 the conditional PD gets smaller (changes to 0.54%). This does not make sense because if the beta increases from 0.4 to 0.8 then the sensitivity to the negative market factor (-1.0) goes up and hence the firm is more heavily affected by the economy. This should imply that the conditional PD should increase as well but it does not.
  • If beta gets > approx. 0.4, then the conditional PD gets smaller the higher the beta becomes. Why? I have no explanation for that...

Any help/feedback is much appreciated.