UL=SD(asset value)?

[ajsa]

Hi David,

The UL's definition is 1 SD of asset value (SD(V)). But UL is also the relative credit VAR and the number of SDs depend on the significance. Are they consistent?

Thanks.

 

[David Harper CFA FRM]

Hi asja,

IMO, it's another term GARP needs to standardize as a *practical* matter. To my knowledge, only the Ong reading has UL = 1 SD(V).

They are consistent if you accept that Ong's defintion is just a special case of the more general (and therefore, more accurate) credit VaR. In other words, using normality (just to illustrate, okay, the credit distribution isn't normal):

=NORMSDIST(1) = 84%
so, you can see, for any distribution there is a low confidence level that associates with the 1 SD
under normality, 84% confident VaR is the 1 SD. At this quantile, the defintions match.
(then 95% VaR could be re-expresses as a multiple of the 84% VaR!)

so more typically,
UL = WCL (confidence) - EL

...but you can see that if we lower the confidence, at some quantile on the distribution:
UL = WCL (lower confidence) - EL = 1 standard deviation

this view is supported by Ong's subsequent expression of EC as a multiple of UL
(where normally we say: EC = UL).
As in:
EC covers UL (normally) but if UL is only 1 standard deviation, then
EC covers Multiplier * Ong's 1 UL

Once again, if one gets comfortable with the notion that WCL, CVaR and UL are all calibrated merely as distributional quantiles, I think the apparent inconsistency is less trouble...and this leads to a more comfortable view in which:

UL is set by a quantile on the loss distribution (e.g, 95%),
and Ong has merely defined it by the low confidence that matches 1 SD

David

 

[ajsa]

Thanks David. so another definition (UL = AE * sqrt(EDF * VAR(LGD) + LGD^2 * VAR(EDF))) will still hold or not if we do not consider Ong's special case? In other word, is it derived from 1 SD or from the general definition?

Thanks.

 

[David Harper CFA FRM]

Hi asja - That's the derived value of (solution of) the one standard deviation, so not "derived from 1 SD" but instead "the solution to the question, what is the 1 SD?" So, it definitely does not generalize to other quantiles...it is sort of like the credit loss equivalent to the portfolio volatilty we calculate when we use, under mean variance, portfolio vol = SQRT[w1^2*sigma1^2 +w2^2*sigma2^2 + 2*w1*w2*2*Cov]...both solve for the 1 SD, then we may scale that output up, etc - David