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# LVAR question

#### john.ophof

##### dra. Ing
Asset worth 1 million whose 95th VaR is 100.000. (normal assumption)
bid ask spread on the asset has a mean of USD 0.1 and a standard deviation of USD 0.3. What is the 95th percentile LVAR (VaR and liq risk are uncorrelated).

What is the liq var.

444000 is answere but I think it must be 344000.

In your examples you use 1.65 for the stand deviation around the mean but it is two sided so it should be 1.96

#### David Harper CFA FRM

Staff member
Subscriber
Hi John,

If you assume a two-tailed spread test, I agree with their $444,000 (443,000) But I do not agree with two-tailed test: our confidence concerns a spread widening (to one side of mean) not a narrowing (to the other side). Nobody's been able to explain to my why this should be a two tailed test? Prior forum threads here and here. Here is XLS David #### john.ophof ##### dra. Ing Indeed two tailed spread test I mean. Thanks for your help. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber ....i don't follow, sorry. My view is that it's all one-tailed here: one-tailed VaR plus one-tailed spread (and it's mean spread plus spread vol not mean minus spread vol)....please advise if you have better source/etc on why it should be a two tailed spread. Thanks, David #### john.ophof ##### dra. Ing The answer I got is: VaR = V*0.5 (mu-1.96sigma), I think it must be + 1.96 to get 344000 since the answer was 444000 for total VAR. so based on the 1.96 I deduct a two tailed spread around the mean spread (in stead of 1.65). #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Right, but the answer given is doubly incorrect so deductions from it may be flawed 1. it's not mu - sigma: if that is the case then mu of +10% and sigma of 5% (for argument's sake) would be zero. That's wrong. It should be mu + sigma (Culp's VaR generally is incorrect in regard to the +/- sign) 2. Per the threads above, logic suggests (to me) that the confidence is one-tailed (i.e., spread widen around mean is the concern, not spread narrowing) David #### [email protected] ##### New Member Hi David, Based on the same past year question: Asset worth 1 million whose 95th VaR is 100.000. (normal assumption) bid ask spread on the asset has a mean of USD 0.1 and a standard deviation of USD 0.3. What is the 95th percentile LVAR (VaR and liq risk are uncorrelated). For 2009 exam, we should only use LVaR = P[1-EXP(mu(R)-sigma(R) X normal deviate)+0.5 X spread] ------right? LVAR = 1,000,000[1-EXP(0.10-0.3X1.65)+0.5Xspread] and the "spread" is not given in this question, right? Your guidance, please. Regards Learning #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Learning - This is a great question: this is a good example of why the FRM can be especially difficult. You are absolutely correct (of course) that your LVaR is the "correct" because it follows the assignment (Dowd 14) However, especially as 2009 is "transition" year to full L2, please do expect this LVaR. ...if the question wants your type of LVaR, it will should say "lognormal LVaR" ...right? because this is an unusual VaR format relative to the Jorion/etc ...i am sure this sort of discrepancy will be ironed out in 2010 (in part, b/c i will give f/back) but for 2009, you just need to be flexible ... strictly speaking, you are correct, but, because the lognormal VaR is atypical, i would be ready for a "plain old" arithmetic LVaR (as above) ... depending on assumptions, they should be close David #### RomanS ##### New Member Hi everyone, apart from the question "one- or two-tailed" I was puzzled about the question itself wit respect to the information given. Sepecifically: why is the spread information given in USD-terms and not in %-terms? Best Roman #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Roman, I AGREE and, yikes, i think that is the 3rd problem with the question. I did not really see this before but the solution really assumes the 0.10 is a percentage spread! I am thinking, unless that is USD 0.10 spread for the portfolio, we need the average (mid) price. We don't have that, so it really is using 0.10 as a percentage. Wow, i did not think that question could get any worse....David #### RomanS ##### New Member ...yeah. When you think you are suprised, they will suprise you again... Best Roman #### mcarthur724 ##### Member Just want to clarify since I just came across a similar practice question last night and I had the same question in regards to whether to use 1.65 or 1.96 for the spread "z". I have seen it both ways in different questions. If this type of question appears on the exam Saturday would you suggest using 1.65? (That is what I gathered from the previous comments above. Just want to confirm. Thanks! Andrew #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Andrew, yes, it should be 1.65 for 95% on the spread (as with the VaR) as we are concerned only with the one side; the other side mitigates (reduces the mean). I am only aware of the incorrect usage in the GARP sample exam... thanks, David #### mcarthur724 ##### Member Thanks for clarifying David. By the way (as FYI), the current Schwesser Nov 2011 practice exam that they recommend taking the week before the exam uses 1.96, which is what confused me. Thanks again. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Sure. Well, I'd imagine they just replicated the error from GARP's sample question. FWIW, here is the source reading to which Jorion refers that validates the one-tail rationale: http://db.tt/hT1irIlp Thanks, David #### southeuro ##### Member Hi David, a small (and probably an easy to answer) question... hope you can help me with this one. A sample exam question at the back of GARP book (#2) calculates the liquidity cost as = (portfolio * .5 * spread/stock price) to get the answer, whereas Dodd gives that formula as portfolio * .5 * spread -- without dividing by the underlying's price. I think Dowd meant to write "percentage of spread" (hence justifying division by sotck price) rather than the spread itself, no? Many thanks again! Last edited: #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @southeuro Yes, that's definitely come up. Dowd defines spread as a percentage, such that it's units (%) are comparable to the drift (mu) and standard deviation (sigma); i.e., LVaR(%) = VaR(%) + 0.5*spread(%) and therefore LVaR($) = P* [VaR(%) + 0.5*spread(%)] ... which is just a direct way to the spread (%) which is more fundamentally = P(ask) - P(bid)/P(mid) and, in this way, the spread can be expressed in dollar terms simply as P(ask) - P(bid).

And GARP has asked for it both ways (why not? this is a classic instance of wanting to understand one concept but not get caught up in trivial formula differences). Here is from my 8.3, which is not atypical for an FRM question:
8.3. A $1 million portfolio in an asset has a daily mean return of zero and an annual volatility of 20.0%. The asset's bid price is$34 and the ask price is $35. What is the daily 95% confident liquidity-adjusted value at risk (LVaR) assuming a constant spread and lognormal VaR (geometric returns are normally distributed), in dollar terms? i.e., this question just gives the bid and ask, and expects you to realize the spread (%) = Spread = (35-34)/34.5 = 2.90%. I hope that helps! #### southeuro ##### Member yes it does! so I assume the answer would be LVAR = Var + LC LVAR = 1 mil * (1 - e^(.2)*(1.645)) + [1 + (.029/2[1 - e^(.2)(1.645))] then, correct? Thanks much! #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber We have the following (i didn't check, maybe they are the same): 8.3. B.$35,084
LVaR(%) = 2.0591% + 1.45% = 3.508%; LVaR($) =$35,084