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# Expected shortfall

#### Imad

##### Member
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 Subscriber 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: #### Roshan Ramdas

##### Active Member
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).

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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,

#### rana.nadeem

##### New Member
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,

Thanks David. Your explanations really help in understanding difficult concepts.