Capital, reserve jargon

[shanlane]

Hello,

I am reading the Crouhy chapter now. It makes a lot of sense and reads really well. The only problem I have had so far is with the jargon. At one point (figure 14.12 on p 545) there is a graph showing "the poorer the quality of credit, the larger the Expected Loss ans attributed capital".

The idea is easy enough, but I thought EL was accounted for with reserves, not capital. Is this saying that when EL increases, the necessary capital increases? It seems like EL and capital would not have a direct relationship because capital is suppoed to cover unexpected losses, not EL. Is my thought process incorrect? My accounting is not terribly strong so I sometimes struggle with these concepts anyway, so that might be part of the problem.

Thanks,

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

I think, in an important sense, that you are correct. I hadn't noticed this Crouhy statement but IMO it is imprecise; and similarly the FRM would be very unlikely to frame it this way (for one thing, it requires reconciliation with the non-linear nature of UL). So, I think you've identified the more important prior concept: that reserves/provisions (i.e., accounts that show up as periodic expenses in the "business as usual") cover EL. Further, as our general format is absolute CVaR(confidence) = UL(confidence) + EL, where Crouhy uses WCL in place of absolute CVaR, to your point, there is not really a "necessary" rule that says higher EL --> higher UL.

"Capital" does suggest: not the reserves that are expensed and charged for EL, but the equity (or risk currency) that is is provisioned for unexpected losses (UL).

What he means is that, although it really requires assumption precision, they do tend to go up together. Softly, note that he's just drawn skewed distributions to reflect a lower quality credit (poorer; e.g., RR 5 is "poorer" than RR 3) that has distribution with both higher EL and greater dispersion (UL is a multiple of standard deviation, so we can view size of UL as dispersion).

More mathematically, for example, the Basel IRB function, without going into detail here, would increase the UL as PD increases, as the definition of UL is there a function of PD. As the Basel IRB is a representative distribution, Crouhy is correct as it applies to IRB that higher EL necessarily implies higher UL.

But that's just the model, and to your point, it's not a rule; it depends on our approach to characterizing EL and UL. It is similar to assuming a normal distribution (just to illustrate, right? a symmetrical distribution for credit risk would be a MISTAKE) where exposure = $100, EL (=PD*LGD) = 10% and volatility = 20%; although incorrect for the symmetry, we could say that CVaR is 3*sigma, such that absolute CVaR = $10 EL + $60 UL (i.e., 60 = 20% *3). Now increase EL to 20% and would could get: CVaR = $20 EL + $60 = $80 (i.e., to your point, it can't be wrong to maintain the UL @ $60) but more realistically if the EL increases then so does the standard deviation of the credit asset value, so more realistically probably something like CVaR = $20 + (3 * 25% * $100). So my read on this is that mathematically you are correct (more specifically, unless the assumptions are stated more precision, the UL, as a function of distributional sigma, is not necessarily an increasing function of EL, the mean) but that, in practice, it is probably true that, for a given exposure level, a higher PD will tend to correspond with greater variability in the EL.

I hope that helps, I don't pretend i have the last word on the interesting issue that you raise. It is really great how you are digging into the material!

 

[David Harper CFA FRM]

Sorry to append, it just occurred to me, among our common distributions are two that happen to illustrate his statement:

• If we used a Poisson distribution (which is totally weird for credit but it is skewed): as lambda = mean = variance, if we used lambda to calibrate EL, as variance would inform UL, then UL would scale directly with EL.
• More relevantly, Bernoulli PD: as the standard deviation [EDF] = SQRT[EDF*(1-EDF)], here too, the "unexpected loss" scales up with higher PD. For example, StdDev (PD = 1%) ~= 10% and StdDev (PD = 2%) ~= 14%. (although this only holds up to a certain point)

So, we can clearly specify distributions where UL increases with PD, over a selected domain, but we can find other that don't.

 

[shanlane]

Thank you for the extremely detailed explanation. If I may ask a VERY brief followup question: you mention "absolute CVaR" above and I do not believe I have seen that expression before. Is this kind of like absolute VaR where we need to include the drift (EL in this case, possibly)? If so, what is relative CVaR? Just the UL?

Thanks again and have a great weekend!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Bingo! Those are Jorion's terms (i think Crouhy uses elsewhere in his book, too). Absolute (C)VaR includes the drift, and therefore is the loss from current (zero); Relative VaR excludes the drift and is therefore the loss relative to the future expected value. It is exactly correct to say that relative CVaR = UL! (the confusing thing, of which there are many threads in here is that drift is positive for market risk and negative for op and credit risk).

Have a great weekend yourself!

 

[shanlane]

Thanks again.

Shannon