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# effect of default probability on equity and mezzanine

#### southeuro

##### Member
hi all,

quick question.

I don't grasp the impact of increased default probability (holding correlation constant) has on these 2 tranches.

I'd think an increase would decrease the value of both, hence increase their value "at risk" not decrease.

appreciate any explanation. thank you.

#### afterworkguinness

##### Active Member
Hi,
(realizing this was asked nearly a year ago, but it may still be useful to people.) According to Malz "Increases in the default rate increase the bond losses and decrease the equity Internal Rate of Return." So you are correct, an increase in probability of default (PD) alone, with everything held held constant (notably default correction), value of all tranches will decrease.

Now, if we look a bit deeper and see what impact changing default correlation along side default probability has, we see the outcome isn't so intuitive.

Equity Tranche
As we said above, if we increase probability of default, we expect our value to go down. The question of how much loss in value is answered by looking at the combination of probability of default and default correlations. As probability of default rises, a rise in correlation helps ease the loss in value. The equity tranche likes high default correlation. For example let's say equity value given PD=.08 and correlation (p) = 0 (all credits are linearly independent) is $10,000. If we hold PD at .08 and increase default correlation to .3 we see that the value would be around 1,100 (made up numbers to show relationship). Now if we were to again keep PD the same and increase correlation lets say to .9 we would see that value is around$2,000.

Mezzanine Tranche
When PD is low, an increase in correlation increases the losses. Comparing high and low correlation when PD is high we see that high correlation and high PD result in smaller losses than low correlation and high PD. This is not intuitive, but empirical.

Sr. Tranche
This one is easy, no matter the value of PD, an increase in correlation means increased losses.

I've attached an except from the source text showing a visual relationship that makes this much easier to grasp.

Last edited:

#### vijayaraghavan sundararajan

##### Member
Subscriber
1 caveat about the above screenshots is that the graph for equity tranche is showing the value where for other 2 tranches its about losses. Simply put, in the graph of equity y axis is the value but for mezzanine and senior the y axis is loss.

#### Delo

##### Active Member
Subscriber
David's this video really helped to clear the concept of def prob and correlation

Default correlation in CDO or basket CDS

Subscriber

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thank you @Delo so glad it helped. Honestly, the simple Venn diagrams help me a lot when it comes to default correlations!

#### Kavita.bhangdia

##### Active Member
Hi Delo,
Please can you provide me the link to the above video.. I am not able to open it directly..

Thanks
Kavita

#### ShaktiRathore

##### Well-Known Member
Subscriber
hi
go to youtube https://www.youtube.com and type Default correlation in CDO or basket CDS in the search tab,the video shall appear.
thanks

#### Delo

##### Active Member
Subscriber
I understand the effect of Pd and Correlation changes it causes to value of tranche. But am unable to understand the impact on CVaR. See below.

#### [email protected]

##### New Member
@Delo did you find an answer for your above query for CVar?

#### vijayaraghavan sundararajan

##### Member
Subscriber
@Delo and @[email protected]:

Please refer the attached excel which shows the impact on increasing PD and correlation (while keeping either of it constant) on CVAR in various tranches of CDS.

Whilst I could solve for the impact in equity and senior, I found it bit manually cumbersome (read as excel) to simulate the effect of increasing correlation for mezzanine. In that case I would like to stress the fact that is empirical

#### Attachments

• CDS_CVAR.xlsx
22.3 KB · Views: 52

#### [email protected]

##### New Member
@vijayaraghavan sundararajan thanks for the excel, but dont understand the concept...!! I could read the same but the question is why it acts this way?

#### vijayaraghavan sundararajan

##### Member
Subscriber
@[email protected]

Sorry my friend. I know only the how and not the why of it..

I go with the assumption that it is the nature of the instrument and mezzanine exhibits more convexity than equity/senior.

Also the relevance of CVAR in this has to be much appreciated by linking it with the capital requirement under basel, so as to know the thumb rule of each instrument.

Perhaps I will be all ears when some one like @David Harper CFA FRM can explain the why of it?

#### [email protected]

##### New Member
I forgot all my concepts understanding this...its kinda tricky...!!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@[email protected] The impact of default correlation on CVaR can be understood in extremis if we imagine a really simple 3-bond, 3-tranche portfolio where, say, PD = 2.0% and CVaR confidence is 99.0%. By in extremis, I just mean let's imagine the default correlation, ρ, tending unrealistically toward perfect 1.0. The effect of perfect default correlation is to reduce a granular portfolio to a single bond; ie.., they all default together or none at all. At perfect default correlation, there is effectively one bond, and the 1.0% quantile (i.e., 99.0% CVaR) falls within the 2.0% loss tail: here the worst expected loss for each tranche is to be wiped out (they are all wiped out if all the bonds default). As Malz's CVaR = UL = 1% quantile - EL, perfect correlation is pushing the quantile for each tranche to maximum loss (or zero value).

This effect is illustrates in Malz' Chapter 9 Figures 9.2 to 9.4 (see my annotated 9.4 below where refers to the Senior Tranche) where the CVaR is visualized as the horizontal distance between the dotted/dashed line (i.e., 95 and 99 quantiles) and the solid line (mean). Higher correlation shifts the mean (EL) but is shifting the quantile toward total losses. This is from the following threads: