Model Risk confidence interval around Sample variance

[turtle2]

David,
Your FRM 2010  LEVEL II (QUESTIONS 1-20) page 18-19, Can you please explain,

SQRT(d.f./critical chi^2 @ 95%) * sample volatility < parameter volatility < SQRT(d.f./critical chi^2 @ 55%) * sample volatility
Where did you get critical chi^2 @ 55% ( i.e why 55%, is it a typo ? Is it supposed to be 5% ? ).

Are these critical chi^2 values need to be memorized like 1.645 and 1.96 for 95% and 99% confidence VaR calculations ?

Thanks.

Turtle2

 

[David Harper CFA FRM]

Hi Turtle2,

It's a double-typo (although the answer is still correct). First, note my intended 95% and 5% would be incorrect for a 95% confidence interval (two-tailed). Truly, an embarrassing error. I meant 97.5% and 2.5% because that gives 5% collectively in the tails. Sorry for the error and confusion; as I used Excel, the rest is actually correct (i.e., the chi-square values do reflect 2.5% and 97.5%):

So the corrected @ http://forum.bionicturtle.com/viewthread/2535/ ...
... Now reads (same answer):

"SQRT(d.f./critical chi^2 @ 97.5%) * sample volatility < parameter volatility < SQRT(d.f./critical chi^2 @ 2.5%) * sample volatility
SQRT(249/294.6) * 1% = 0.9194% < sigma < SQRT(249/207.2)*1% = 1.0963%

Such that 95% VaR confidence interval is given by:
SQRT(249/294.6) * 1% * 1.645 = 1.512% < VaR < SQRT(249/207.2)*1% * 1.645= 1.803%"

How did we get the sigma (the true population standard deviation) in the middle? By starting with our chi-square test statistic for a sample variance:

df*S^2/sigma^2, where df = n-1, S^2 = sample variance, sigma^2 = hypothesized pop variance.
And we know that for a 95% CI:
Chi^2 (2.5%,n-1) < df*S^2/sigma^2 < Chi^2 (97.5%,n-1), and solve for sigma:
For Chi^2 (2.5%,n-1) < df*S^2/sigma^2:
sigma^2 < df*S^2/Chi^2 (2.5%,n-1). Such that:
sigma < SQRT[df/Chi^2 (2.5%,n-1)]*S

Re: Are these critical chi^2 values need to be memorized like 1.645 and 2.33 for 95% and 99% confidence VaR calculations ?

Absolutely not, FRM will never expect this. Unlike the normal, each % has a different value for each d.f., so it is like the student's t: it is impossible to know all the critical values even just at 95%/99% CI.

Thanks, David