Lognormal VaR < normal VaR

[dla00]

Hi @David Harper CFA FRM

One doubt regarding the statement that is mentioned throughout Market Risk chapter "Lognormal VaR is always less than VaR" (e.g p12 of VitalSource notes, 707.3D). Can it be "less or equal"?

My question is the following: what would happen, if by any case, the coefficient of variation is the reciprocal of the cut-off quantile associated to the confidence level? e.g say z_{\alpha} = 1.645 and we have a coefficient of variation (meaning ratio sd/mean) equal to 1/1.645, it seems lognormal VaR will be equal to normal VaR, or am I missing something?

Here's an example of what I meant:

round(1 - math.exp(0.12 - 1.645*0.0729),5), round(-0.12 + 1.645*0.0729,5)
Out[119]: (-8e-05, -8e-05)

Thanks!

 

[David Harper CFA FRM]

HI @dla00 Yes, that's elegant because you are solving for μ - σ*z(α) = 0; i.e., μ = σ*z(α) --> μ/σ = z(α), or express via COV, 1/z(α) = σ/μ.
... and If we add the time dimension, sqrt(T), it applies at T = 1.0 year.

But, to me, that's one of the two pathological conditions: they are also equal as T converges to zero (ie., instantaneously). So, you are really solving for the incidental "break-even": when the VaR is breaching zero (from a loss expressed as a positive value) to a gain (expressed as a negative value), where, if we extend the timeline beyond 1.0 year, the negative value reflect profits but the LVaR will resume its status as "less than"! Even on the other side (gains) the LVaR is less. So, yes, I absolutely, mathematically agree that "≤" is correct ... I always concede the mathematical truth!
... but is it useful? Because the LVAR plot will always be "underneath" (dominated by) the VaR. Somewhere I did the math proof but can't find it, reference (for me at least)

• https://forum.bionicturtle.com/threads/lognormal-var.1597/post-59979
• https://forum.bionicturtle.com/threads/notes-from-my-review-of-garps-2017-part-2-practice-exam.10510
• https://forum.bionicturtle.com/thre...tations-of-value-at-risk-var-dowd-ch-2.13954/

P.S. I think it also might be the case (i.e., we can generalize your finding so to speak), that if we add the time dimension, SQRT(T), that we don't need the above COV condition: any combination ultimately has some T = x value where they hit the zero bound.