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Expected shortfall

Thread starter #1
Hi David,

A bond with a face value of $10.0 million has a one-year probability of default (PD) of 1.0% and an expected recovery rate of 35.0%. What is the bond's one-year 99.0% expected shortfall (ES; aka, CVaR)?

a. $3.25 million
b. $6.5 million
c. $9.1 million
d. Not enough information: need the tail distribution

Your answer was B. $6.5 millionAs expected shortfall (ES) is the expected loss conditional on exceeding the VaR, and the VaR significance coincides with the PD, the ES is the expected (average) loss conditional on default, which is 1-recovery rate = 65% * $10 million = $6.5 million.

My question is, what if PD=2% and alpha is 1%, ES would be??? what is the relation between PD and the significance level?

Thanks
Imad
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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#3
Hi Imad,

Great question! ES is the conditional average loss; the average loss (conditional on) in the alpha% tail. In the case of alpha = 1%, the entire 1% tail is occupied by the default event, so the ES is the same $6.5 million!

What's unique here? We only have a single bond, so the Bernoulli distribution gives no loss for, in your case, 98% of the distribution, then the same loss for the 2% tail. Although this is a discrete Bernoulli, if we were to represent as a continuous CDF, it would be:



In the case of PD =2%, all X% ES have the same $6.5 million values for any X > 98%.

A test of understanding this might be a question like:
  • If the single bond PD = 2.0%, what is the 96.0% ES (assume LGD = 65%, same as above)?
 
Thread starter #4
Hi David,

I respond to your above question: for 96% confidence level, we are in the "no default" zone, as such Var is zero and ES is also zero.
We have a loss when the confidence level is 98% up to 99.9% because the bond will default by then, incurring a loss....am I right?

Thanks
Imad
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#5
Hi Imad,

You are correct about the VaR: if PD = 2%, then at 96%, we are in the no default (VaR is just the quantile).
However, ES is an average of the 4% tail, so it includes a zero from 96% to 98%, then the default from 98% to 100%, so the 96% ES would be (0*2% + 6.5*2%)/4% = $3.25 million
 
#7
Hi David

It means that PD is only used in calculating average (to reach ES) and it is recovery rate that matters to calculate loss. Can you please clear that in above case, if PD=3% then at 96% ES would be (0*1%+6.5*3%)/4%= $4.875 million?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#8
Yes @rana.nadeem I agree! In that scenario, the (unconditional) probability-weighted average loss of the worst 4.0% = (1% * 0 ) + (3% * $10.0 * 65%) = 0.1950 which is a conditional average of 0.1950/4% = $4.875 mm. Thanks,
 
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