# unexpected loss

Discussion in 'P1.T3. Financial Markets & Products (30%)' started by ajsa, Aug 22, 2009.

1. ### ajsaNew Member

Hi David,

You defined unexpected loss as the SD of asset value for bank loan. I wonder if SD(asset value)=SD(EL) given that I think asset value's mean has already factored in EL?

Also, since UL is a fraction of AE according to its fomula, I wonder if UL+EL=AE?

Thanks.

2. ### David Harper CFA FRMDavid Harper CFA FRM (test)

Hi ajsa -
Please note that particular definition of UL (i.e., UL = 1 SD) comes directly from the assigned Ong's Internal Credit Risk Models; I view it is a special case of the more general UL which is a multiple of (several multiples of) one standard deviation, as i tried to show in the tutorial. In my view, the best way to approach this is from the general (consistent with Basel II or assigned Crouhy): Credit VaR = EL + UL, where the Credit VaR and UL are a function of confidence and horizon....
...so Ong's UL is the special case where confidence is low such that UL is only one standard deviation...

re: "I wonder if SD(asset value)=SD(EL) given that I think asset value’s mean has already factored in EL?" In a manner of speaking, this is correct, but this language (while having technical justifaction) could easily mislead (IMO). The distribution here is not the EL per se but the future distribution of the credit (e.g., loan, bond, or more realistically: loan portfolio) asset. So, to use a metaphor to stock returns, we can think about stock returns and we talk about the first moment (expected return) and second moment (standard deviation or volatility) of returns. The analog here is not the EL per se, but the future value of the loan portfolio. The EL is the *mean* or expected value of the loss on the loan portfolio asset. The (Ong version of) UL is the dispersion (std deviation) of the (potential) loss distribution *around* the mean (EL). If you have Ong, see Fig 5.1...so, if we think about this in Gujarati's terms of a distribution that characterizes a random variable (and i think it's important to see this as a distribution):
the random variable is the future loss in credit asset (loan portfolio) and, as usual, we hope to approximate this risk with a loss distribution

EL = mean = first moment
UL = a quantile based on confidence = x * second moment
Ong's UL = the quantile associated with 1 standard deviation = 1*second moment (see how Ong's is just the special case of a generalized UL with infinite results as confidence/horizon vary?)
Credit VaR = UL + EL; e.g., Basel II Credit VaR= one-year 99.9% UL + EL

"Also, since UL is a fraction of AE according to its fomula, I wonder if UL+EL=AE?"
Good thinking, but not quite: UL + EL < AE simply because our confidence is less than 100%, the following is true:
EL + UL (only if confidence = 100%) = AE because the entire distribution (all of the exposure, AE) requires 100% confidence

rather if we are willing to abstract from Ong to the generalized UL, then:
EL + UL = credit VaR, or Ong's version would be EL + (multiplier)*UL = credit VaR, or
EL + economic capital = WCL; b/c Crouhy economic capital = unexpected loss and his WCL = credit VaR
...hope that helps, David