LVAR question

[crablegs]

Hello,

Came accrosss following GARP question from 2009:

You are holding 100 shares at USD 50. Daily historical mean and volatility of returns are 1% and 2%. Bid-ask spread daily historical volatility is 0.5% and 1%. Calculate daily LVAR at 99%

Answser:

VAR = 50 x 100 x (2.33 x 0.02 -0.01) = 183
LC = 50 x 100 x 0.5 x (2.33 x 0.01 x 0.005) = 71
LVAR = 254

• Why is it that we subtract 0.01 to the product of 2.33 x 0.02 in the VAR calculation?
• Why is it that we just don't use the historical mean for the spread?

Thank you!

 

[David Harper CFA FRM]

Hi Glen,

You capture two important points.

Given by answer, question should read "Bid-ask spread daily historical mean and volatility are 0.5% and 1%"

1. The question is imprecise, as written. "Relative VaR" would not subtract the 1% drift, but "absolute VaR" does. Relative VaR is loss relative to the future expect value; absolute VaR is loss relative to the initial value, and therefore mitigates the loss by any expected positive drift (the 1% offsets the downside of the 2% volatility). The answer given is best, only the question should specify "absolute VaR" or "VaR relative to the initial [current] value"
2. We could, again the question is a bit imprecise. Question should say "with exogenous [or stochastic or random] spread" but, as the question includes spread volatility, it's a hint to use the spread volatility. If we assume constant spread:

• LC = (50 * 100) * 0.5 * 0.005; i.e., one-half the spread, or if exogenous/random
• LC = (50 * 100) * [0.5 * (0.005 + 0.01*2.33)] = 71; i.e., if the spread widens to it's 99% worse

The answers, IMO, are exactly correct, only the question language is imprecise. I hope that explains, thanks,

 

[crablegs]

Hello Mr. Harper! Good to hear from you again. You helped me tremendously for Part I and I am sure glad to see you around for Part II again. Yes, the bid-ask spread data was for historical mean and volatility. Just got lazy typing.....

Thank you for the great explanation. This is a wake up call..... had frankly no clue about the distinction between absolute VaR and relative VaR.... will have to toss the dice on that one.

Thank you again for your time!

Glen

 

[David Harper CFA FRM]

Hi Glen, thanks! really glad to help ... absolute/relative is one of those topics that has dozens of forum posts accrued over the years, so i'd love to summarize in a "David's Notebook" post ASAP. Will try to do that...but I do prefer the way your absolute VaR is written above, it follows Dowd and is robust; i.e., LC adds. If we use (drift - vol*deviate), then adding LC is a common mistake. So best is:

(2.33 x 0.02 -0.01): absolute VaR% = -drift + volatility*deviate
Then liquidity cost is an addition to the VaR

you can't go wrong with this, because if you don't have drift (expected return), it "reduces" to relative VaR:
absolute VaR% [if drift = 0] = -0 + volatility*deviate = volatility*deviate = relative VaR

 

[ChadWOB]

Glen - great question. David thank you for the Relative and Absolute VaR explanations. I do not recalling seeing this in any of the textbooks.

May 1 - crunch time!

 

[southeuro]

Hi David,

here's a question out of Schweiser challenge problem set (similar to Glen's question but a bit more "twisted")

You are holding 100 shares at USD 30. Daily historical mean and volatility of returns are 2% and 3%. Bid-ask spread daily historical mean and volatility is 0.5% and 1%. Calculate daily LVAR at 99%, assuming the confidence parameter of the spread is equal to 2.58?

with your method, I answer:

VAR = 30 x 100 x (-2% + 2.33 x 3%) = 149.7 (same as their answer)

but I don't see how they calculate the LC...

namely, LC = 30 x 100 x 0.5 x (spread mean + 2.58 * 1%)

so my question: the absolute var you provide goes -drift (or mean) + vol * deviate whereas in this particular case they give mean + vol * deviate
How to deal with the change of sign? when to use which? am confused..

giving a separate confidence parameter is a mean way to get you tripped I think...

Many thanks!

 

[southeuro]

Hi David and/or others,

another rather easy (i think) question.. one from practice FRM.

Asset worth 1 million whose 95th percentile Var is 100,000 (using parametric method with normal distribution). Suppose the bid-ask spread has mean of .1 and st. deviation of .3 What's the 95th percentile liquidity adjusted Var assuming market risk Var and liquidity risk piece are uncorrelated?

answer is calculated as 100 * .5 * (.1 + .3 * 1.96)

Not sure at all why 1.96 is used instead of 1.645 -- aren't we supposed to use one-tail values for LVaR calculations?

Thanks

 

[David Harper CFA FRM]

Hi @southeuro Yes, you are correct. It must be an old question: there is at least one old LVaR question which makes that mistake, but (I would search/find some of the discussion on this but it's actually years old....) GARP has long since corrected that and the LVaR spread, as you suggest, will indeed always be one-tailed. As it should, just like VaR, we are not concerned if the spread narrows, but only if it widens; there is only one direction that increases the liquidity risk. I hope that helps,

 

[southeuro]

thanks - was wrecking my brains over this..

also FYI, you mention somewhere that ES has no glaring weakness against the criteria of "not being intuitive", "not being coherent", and "not being stable". While I agree with the first 2, I think it's not stable as a measure. Here's something I found off the internet which makes sense to me as well on that:

"when the loss distribution is more fat-tailed, the ES estimates become more varied due to the large loss, and their estimation error becomes larger than the estimation error of VaR and vice versa. Thus ES varies more than VaR at low default rates if it is estimated with the same sample size."

source: http://www.ukessays.com/essays/finance/value-at-risk-versus-expected-shortfall-finance-essay.php

let me know your thoughts when you get a chance.

cheers,

 

[David Harper CFA FRM]

Hi @southeuro

do you know where I mention that ES is (effectively, by not being not!) stable? I just can't remember, I want to say that's based on a Basel document, but I connote "stability" firstly as a property of a distribution, not a measure like ES about a distribution (ES doesn't care if its distribution is stable or not), so i'm not exactly sure what your first refers to ...

Re: your quote: yes, I do agree with "when the loss distribution is more fat-tailed, the ES estimates become more varied due to the large loss." Dowd (long assigned) says that. I would not call this stability, I would call this estimate accuracy. (I actually think I could argue to call it precision, instead, but suffice for now accuracy or precision!). Realistically, we estimate an ES like we would estimate a VaR, so this quote refers (I think) to the standard error of the ES as an estimate. Larger sample leads to smaller standard error; and more accuracy/precision. So, I think your internet source is totally solid ... I'm not sure frankly what ES is stable exactly means, however (I could just be blanking, it sounds familiar ... but stability is a relevant distributional property so i am not sure ...). Thanks,

 

[southeuro]

hmm… came across something odd… would be great if you can help in the last stretch David.

100 shares with price $50. daily historical mean and volatility of stock is 1 and 2% respectively.
daily historical mean and volatility of spread is 0.5% and 1% respectively.
calculate LVAR at 99%?

in the calculation the var = 100 * 50 * (-0.01 + 2.33 * 0.02)
while the Lvar = 100 * 50 * 0.5 (+0.005 + 2.33 * 0.01)

why do we have the mean signs different for those?

Thanks & much appreciated

 

[David Harper CFA FRM]

Your second formula isn't exactly LVaR, it is liquidity cost (LC) = 100 * 50 * 0.5 (+0.005 + 2.33 * 0.01). I point that out to show why it has the same direction as volatility, not mean. These formulas are all expressed with losses as positive but if we just think about the natural direction:

drift is the stock expecting to go up a little
VaR is the downside expressed as some multiple of volatility; deviate*volatility
Liquidity cost is illiquidity making things even worse by lowering the exit price

so, naturally we have: + mean - volatility*deviate - LC, but to express loss in positive terms we switch this around to:
-(+ mean - volatility*deviate - LC) = -mean + volatility*deviate + LC

so

• this is the relative VaR(%): 2.33 * 0.02
• this is the absolute VaR(%): VaR(%) -0.01 + 2.33 * 0.02; worst loss mitigated by drift
• this is the liquidity cost (%): LC(%) = 0.5*(+0.005 + 2.33 * 0.01); i.e., the spread widened to its worst expected (narrow doesn't hurt us, what hurts us is the mean + widening!), but one-half (0.5) because we only need to exit not buy+exit
  
• The is the LVaR(%): -0.01 + (2.33 * 0.02) + (0.005 + 2.33 * 0.01); i.e., -drift + vol*deviate+LC = VaR + LC
  
• This is the LVaR($) 100*50*[(-0.01 + 2.33 * 0.02) + (0.005 + 2.33 * 0.01)]. I hope that explains why those answers are correct except your LVaR should be LC then LVaR = VaR+LC