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unexpected vs expected losses

Hi David,

There seems to be an inconsistency in the way EC is calculated.

From your explanation, and the explanation in the reading, the unexpected loss is one sigma (or a multiple of sigmas) AWAY from the expected value of the portfolio, which should be the value of the portfolio minus the expected loss. Lets assume 95%, sigma of $30,000 and 2 sigmas. This would mean that if the expected loss is $5,000 and the unexpected loss is $60,000, then the total loss we would have to account for would be $65,000 (or $5,000 + 2*$30,000). $5,000 in loan loss reserve and an ADDITIONAL $60,000 for the unexpected loss. In your diagram (slide 27 of video 4e) you also say that the VaR is just this 2*sigma. According to the formula you have on slide 27, EC should be $60,000-$5,000=$55,000.

There is an inconsistency somewhere and I was hoping you could point out where it is.

Thanks in advance for your help.


David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Mike,

It's a sharp observation. This is now the same difference as relative versus absolute VaR but rather than market risk, we are now dealing in credit risk, such that the drift is the expected loss rather than the expected gain. Using your example:

EL = $5,000
UL = EC = 2*sigma = 2*$30,000 = $60,000; and now we have a choice w.r.t. VaR(alpha):

Relative VaR (i.e., relative to future expected mean) = $60,000
Absolute VaR (i.e., relative to current zero) = $5,000 EL + $60,000 = $65,000
please note: we can still use the same absolute VaR = -drift + sigma * deviate,
only now the drift is negative (!) so we have absolute CVaR = -(-EL) + sigma * deviate

(keep in mind that Ong defines UL as 1 sigma which is not consistent with us generally, we view UL = k*1 sigma; put another way, Ong's UL is the UL for a low confidence level).

My graphic probably should read, at the VaR, "EL + sigma*deviate", or "+sigma*deviate" ... i think that will reconcile your question ... but i only showed that "sigma*deviate" to emphasize that VaR and UL are multiples away from one standard deviation, if you follow.

The formula, EC(alpha) = VaR(alpha) - EL, is referring implicitly to an absolute VaR. In this way, I think, my formula and the graphic are consistent in depicting, as labelled, an absolute VaR(alpha). Why did i do that? Because that is how Basel II/III treats the IRB. In this way, the 2*sigma or k*sigma is actually not very significant: it merely wants to show that the quantile is far away from Ong's 1*sigma ... and here is the key point: that Ong's UL is but a special case of a relative VaR but at a quantile, implied by sigma = 1.0, that corresponds to a low confidence.

You probably want to know about how GARP/FRM treats this? Again, this specific issue (relative vs. absolute VaR in the credit and operational, not to mention market risk context) is one that i've requested they define for two years running (as you can see, the specific issue, like market risk, is that credit VaR can refer to either absolute or relative CVaR).

But don't let that distract from what is certain: as EL are priced in and covered by reserves, EC covers UL (not UL + EL).

I hope that helps, David
Hello David,

I need one help as i am calculating the VAR percentalies for below data to analyze the expected and unexpected loss for 3 level of confidence i.e 95%, 99.9% and 99%. The below are the fraud data for credit card business line.
January 2011--- 1000usd
february 2011... 2000 usd
april2011... 3000 usd
may2011-- 100 usd
june2011-- 56000 usd
july2011-- 500 usd
august 2011 1500 usd
september2011 1900 usd

Now i would like to know the following things :-

1) Mean for the above data and pls advise the formulae as well
2) what are the average rate of events per month
3) what is the expected loss per month
4) what will be the unexpected loss per month for confidence level 95% , 99%, and 99.9% (pls put the formulae also)

your prompt response would be highly appreciated.

Thanks & best Regards

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Ajay,

Two caveats:
  1. Your sample is very small (n = 8): the significance (or error associated with quantile/VaR estimates) probably overwhelm the answers. Put another way, it's possible the answers aren't significant; it is certain the errors of the estimates are high
  2. There are methodological assumptions; I could answer these questions a few ways
With that, I copied your data in this XLS http://db.tt/xaChwsqP
My superficial pass involves simply calculating average and quantile statistics; e.g., the 99% UL = 99% VaR (i.e., 99% percentile) - Average Loss (i.e., EL)

That's just illustrative, I hope that is a helpful start, thanks, David
Dear Mr Harper,

Would request you to please guide us where i can read the operational risk materials pertaining to Risk control self assessment for retail banking industry. And also the checklist of control testing for consumer banking retail .

Also would like to know do you have any VAR calculation sample file on excel for study.

Thanks & Best Regards
Ajay Jha

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Ajay,

1a. I'm not sure specifically, be we do affiliate sell KESDEE courses which can be found here @ http://www.bionicturtle.com/products
1b. And we have a new course developing with Pristine, see http://www.bionicturtle.com/news-an...ional-risk-credit-risk-training-with-pristine

2. We have several such learning XLS, these are available to paid FRM customers, see http://www.bionicturtle.com/how-to/spreadsheets/category/all
(next year we will be selling the XLS a la carte)

Thanks for your interest, David


New Member
Hi David!

I would like to know how one does the mathematical derivation of Unexpected Losses (UL), by definition this is the standard of deviation of Expected Losses (EL). We know that EL=PD*LGD*EaD, this means that UL^2 = E[EL^2] -(E[EL])^2, how does one continue from here?


New Member
Hi @David Harper CFA FRM , hope you had a good weekend like many of the candidates!

For Michael Ong's book, it seems that it has been uploaded in a landscape format. If so, would it be possible reload it in a portrait format please? The pages/layout seems quite strange at the moment.

Thanks a lot, Galaxy