P1.T4.VAR-HULL-DOWD_Ch2_TOPIC: Value-At-Risk (Var) Measure Of Risk

[gargi.adhikari]

In reference to: P1.T4.VAR-HULL-DOWD_Ch2_TOPIC: Value-At-Risk (Var) Measure Of Risk:

Hi All,
I am having some trouble connecting the dots between Quantile , Var & Confidence Level and how VAR = - Quantile.

Isn't VAR = some $ Amount ....??? ( say $5 Million at Risk with a 95 % of Confidence Level...? )
So having said that, how is VAR= - Quantile..? Given that, a Quantile is the Inverse of a CDF..?

I know I am probably missing an angle here for that AHA Moment on this topic... Please Help ....

 

[David Harper CFA FRM]

@gargi.adhikari It's a good question, I may not have the best way of approaching this (having thought about it too much, !) but, for me, key to the "aha" is the realization that the x-axis can any sequence of returns (%) or dollars ($). We need a probability distribution, but the x-axis which includes a characterization of the loss tail varies by situation. Dowd's approach is clinical: he mostly is using a standard normal variable, ~ N(0,1). So he is often either assuming arithmetic/geometric returns (%) or he is assuming P/L ($) are normal, in either case for the x-axis of a probability distribution. The really fundamental relationship here, assuming a normal distribution, is:

• quantile (q) = N^-1(probability); in Excel, eg, q = NORM.S.INV(p) = NORM.S.INV(0.95) = 1.645 quantile; the inverse is:
• probability (p) = N(q); e.g., p = NORM.S.DIST(z, true = cdf) = NORM.S.DIST(1.645) = 95.0% probability
• and by definition, p = N[N^1(p)]; e.g., there is a 5% probability we will exceed the 1.645 quantile which we retrieved as the quantile at 95%
• In Dowd's keep-it-simple-by-assuming-normal, VaR is typically given by (z) because he's assuming standard normal. So, z is the quantile.

So, given a probability distribution and whatever happens to be its x-axis (eg, %, $), VaR is just retrieving the value associated with a probability. As Dowd says, "VaR is simply [the negative of] the q(p) quantile ..." We do often work in percent returns (%), then multiply by price ($), but that's not greatly different than switching out the (translating) the x-axis from returns to P/L. Relative normal (%) VaR(α) = σ*z just multiplies by value to get dollars to get $VaR VaR(α) = P*σ*z, and that's just translating the x-axis from % to $ without changing the probability distribution.

I think it's downhill understand once you really grok both the essential requirements and flexibility of a probability distribution, esp analytical vs empirical. Further, consider a simulation (historical or monte carlo): a historical simulation is just a properly scaled histogram (e.g., worst 5% losses are in this dollar bin over here) and its x-axis can be dollar bins. I hope that's a start!

 

[gargi.adhikari]

@David Harper CFA FRM
Have a follow up question on the above...
You mentioned ..
"Relative normal (%) VaR(α) = σ*z just multiplies by value to get dollars to get $VaR VaR(α) = P*σ*z, and that's just translating the x-axis from % to $ without changing the probability distribution."
Now, given the fact that z= represents the point on the x-axis corresponding to a given certain area under the PDF that is, z= the standard normal variate for a given confidence level, I am a little fuzzy about how to go about converting this to get a $VAR...? I am with you on the σ*z...what does the "P" represent in the "$VaR VaR(α) = P*σ*z" though... ?

 

[ShaktiRathore]

Hi,
@gargi.adhikari i think David is referring to P as Portfolio value here therefore if σ*z is the portfolio Var in terms of % then the $ Var is simply multiply the σ*z% Var by Portfolio Value P to get the dollar loss.
$Var=%Var*($P)
thanks

 

[David Harper CFA FRM]

@gargi.adhikari Yes exactly what @ShaktiRathore says. The most basic VaR expression is, for example: if the daily volatility of normal returns are 1.0%, what is the 95% relative VaR? Answer: VaR(0.95) = σ*z = σ*1.645 = 1.0%*1.645 = 1.645%; i.e., the x axis is arithmetic returns. Or we can use dollars on the x-axis, effectively: if an asset has a price of $10.00 (P = $10) and its daily volatility of normal returns is 1.0%, what is the 95% relative VaR (relative = expected return is zero)? Answer: VaR(0.95) = P*σ*z = $10.00*1.0%*1.645 = $1.645. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM @ShaktiRathore Indebted to you guys! Thanks so very much.