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# YouTubeT5-01: Lognormal Value at Risk

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Welcome to the first video in this new playlist that is devoted to Topic 5 in the FRM. Topic 5, Market Risk, is the first topic in Part 2. We will start here by comparing normal to lognormal VaR and, specifically, we are going to generalize to absolute VaR. Absolute VaR generalizes the relative VaR so it's the complete version of VaR. The key thing that we are going to do here is look at four different use cases so we can compare normal VaR to lognormal VaR in the single-period case. Normal is when we assume that the arithmetic returns are normally distributed and lognormal is when we assume that the geometric returns are normally distributed.

#### nictziak1

##### New Member
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Hi @David Harper CFA FRM
Very useful video & explanations - as always.
My question here is whether the lognormal VaR can also be applied to bond VaR calculation and provide a similar 'solution' to the problem of time-scaling. Would the relative 10-day VaR formula (so μ=0) then look like:

%VaR = 1-exp(-(σ(YTM)*δ*a)*sqrt(10)+(0.5*c*(a*σ(YTM))^2)*sqrt(10))

where
δ=modified duration
c= convexity
σ(YTM)= the sample standard deviation of the calculated ln(YTM t/ YTM t-1)

Much appreciated

#### frogs

##### New Member
Subscriber
Hi, @David Harper CFA FRM can you tell me around 17:15 in the video why you get the substitution for R' to be Mu - sigma(z)
It's very similar to normal but I can't see where you derived it.

EDIT: I kind of see how it is done. So a yes or no answer would suffice. If you want any kind of return (ie arithmetic, geometric) to be normally distributed you need Mu - sigma(z) factor?

Last edited:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @frogs Yes that is correct! This is all based in Dowd Chapter In both the normal versus lognormal VaR there is the same assumption that returns are normally distributed! The difference is due to whether arithmetic returns are normally distributed (what we call "normal VaR" or just "VaR") or whether geometric returns are normally distributed (what we call "lognormal VaR"). Both versions of VaR assume returns are normal, the differ in the definition of the return. Consequently, both versions define the critical return, r* = μ - σ*z, which is only breached with probability typically (significance = 1 - confidence) of 5.0% or 1.0%. In the above example where μ = +1.0% and σ = 5.0%, the critical value (at 95% confidence) is 1.0% - 5.0%*1.645 = -7.22%; if the return has a normal distribution, then only 5.0% of the time do we expect it to be worse than -7.22%. But the same critical return informs both the normal and lognormal VaR. Thanks,