calculate credit VaR with PD

[ajsa]

Hi David,

"A portfolio has two risky bonds. each bond is $500000. the monthly PD for each is 0.1682%. What is the best estimate of the monthly 99.9% credit VaR for this portfolio, assuming no default correlation and no recovery?
a) $841
b) $1682
c) $998318
d) $498318"

Answer is d. Could you please explain? I think c may make more sense, but cannot think straight


Thank you!

 

[hsuwang]

A portfolio has two risky bonds. each bond is $500000. the monthly PD for each is 0.1682%. What is the best estimate of the monthly 99.9% credit VaR for this portfolio, assuming no default correlation and no recovery?
a) $841
b) $1682
c) $998318
d) $498318​

Hello ajsa,

Here's how i approached this question, I'm not sure if I'm correct:
Since recovery=0, LGD=1, so EL=EAD*PD*1 (500,000*.1682*1=84,100)
99.9% alpha=3.09
84,100*3.09=259,869
assuming correlation=0, Portfolio VaR= sqrt(259,869^2+259,869^2)= 367,510

so... I guess the closest number you can get is D.

but... I thought if we want "credit VaR", we'll have to subtract EL from UL.. so 367,510(UL) - 84,100(EL)=283,410, which is even more "away" from the options we get..

But, since no default correlation probably doesn't mean no individual VaR correlation (not sure about this), if we assume VaR correlation = 1, then Portfolio VaR = 259,860 + 259,869 = 519,738
Credit VaR = 519,738 - 84,100 (EL) = 435,638. so maybe this looks more correct?

Please see if you agree with me and correct me if I'm wrong.
Thanks!

 

[David Harper CFA FRM]

Jack - Your try looks good except I don't think you can multiply the deviate (3.09) by the expected loss (which is a mean); you need to multiply the deviate by a dispersion (volatilty) metric. For that, you could try StdDev = SQRT(PD*(1-PD)) as PD is a bernouilli ... that was my first queston ("where is the volatility?"), but then i realized i could calculate the variance of a Bernouilli

...I think the question is flawed b/c it makes at least 2 assumptions
...but I think maybe it is looking for:

with 2 bonds you have 4 outcomes:
p(both survive) = 99.832%^2
p (both default) = 0.168%^2
p (one survive, one default) = 99.832%*0.168% = 0.1679....

so if you just visualize the default distribution in four pieces
up to 99.832% is both survive; e.g., 99% VaR = 0
at 99.832 up to 99.832% + 2*0.1682%*(1-0.1682%) = 99.99971% = one default
99.99971% to 100% = both default

...then notice the one default "transomes" or straddles the 99.9% VaR, so $500,000 looks good
then, notice to your point Jack, apparently they did deduct EL from that:

$500,000 - (EL = 500,000 * 0.1682% * 2) = 498,318

David

 

[hsuwang]

Hello David,
I'm a bit confused on the idea to multiply 3.09 by EL. I know it's incorrect to do it this way, but when looking at the ASRF IRB function, it seems like it is doing the same thing (scaling EL to make it a conditional EL, by incorporating 99.9% confidence), and then subtract EL from that. Please advice, I think I'm lost..

Thanks.

 

[David Harper CFA FRM]

Hi Jack,

The basel IRB is more complex (it's actually an application of a Gaussian copula) ... when you say "it seems like it is doing the same thing (scaling EL to make it a conditional EL, by incorporating 99.9% confidence), and then subtract EL from that" I think that is a FANTASTIC attempt to summarize IRB, I cannot do better in one sentence, but IRB is more sophisticated and different than simply "stretching" the PD with PD*deviate (higher confidence)

...just for example, to semi-support your point, i plugged the 1% PD into IRB. Assume 100% LGD, assume rho 20% correlation (with systemic factor), and I get UL = 13.55% ... and high sensitivity to rho ...you can see the IRB has "stretched" into a conditional EL that is significantly greater than the UL implied by 99.9% (i.e., 13.55% IRB conditional EL > 1% * NORMSINV(99.9%))...and this is due to the hidden complexity of the IRB: many truly outrageous assumptions have conspired to produce an elegant formula

...so i have two thought:
1. i think your "metaphorical" view of IRB is well stated; conditional EL is akin to a "stressed" (PD*LGD)
2. beyond that, until you want to deep dive the copula function, I would just treat the IRB as a "complex solution"

Hope that helps, David

 

[hsuwang]

ok, thanks David!