LVaR

[shanlane]

Hello,

Can LVaR be time scaled? If so, how?

It just seems strange because it is dealing with a spread and that the spread really wouldn't change as time goes on.

We could obviously scale the "regular" VaR and then add the LC, but If we are given a constant spread would the LC just stay at whatever percentage it is at even over a long holding period?

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

I think it's a very good point. I can't necessarily speak to current specialist practices (I've got to think quants/HFT are deep into this, in ways i can't even imagine...) but just in reference to our superficial (FRM) layer, we do NOT scale the liquidity cost (LC), we add it to the scaled VaR, on the implicit idea that, while even if it is random, it does not disperse like does return volatilty.

To me, this no-scaling make much more sense than scaling the LC: the spread incorporates its own time horizon, the 0.5 is the estimated liquidity cost to exit whether it requires five minutes or 3 days. If it's the liquidity cost to exit, I think it makes sense to add it similarly to various VaR horizons. This is why Dowd asserts, perhaps counter-intuitively (emph mine) "It is easy to show that the liquidity adjustment [as a ratio to VaR] (a) rises in proportion with the assumed spread, (b) falls as the confidence level increases, and (c) falls as the holding period increases." That is, it falls precisely because, under his model, an increasing horizon scales volatility/VaR but does not similarly scale the spread which is static (or roughly static).

So we (FRM) don't scale the LC, even under the random spread scenario, and I think no scaling makes much more sense than full-on square root rule (SRR) scaling. However, it would also make a lot of sense, to me, if the multiplier (k) in the random spread did slightly scale with the horizon, not quite with the square root of time, but just to incorporate some dispersion. But that's pure musing, I've not seen that... Thanks,

 

[shanlane]

That makes sense, except for the random spread scenario. If the volatility of the spread is, say, 1% per day, wouldn't this imply that over the course of one week the spread volatility would be sqrt(5) * 1% or am I looking at this incorrectly?

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon - It is tempting but the spread does not compound like returns compound. The use of SRR to compound a 1-day returns volatility into a 5-day returns volatility is based on rather narrow requirements that returns add (compound) and are i.i.d. But we don't really expect the spread to compound on itself. The random spread in Dowd is a "softer" idea: not that the spread compounds over time, but merely that the spread fluctuates around a constant mean. That's the superficial point.

To my previous reply, I am sure an argument can be made that the k multiplier ought to scale in some way with time ... to account for some "time decay" or time slippage in the spread ... but that is a far cry from justification of application of SQRT(x days) rule. Thanks,

 

[shanlane]

That makes perfect sense.

Thanks!

Shannon