Question on T-Statistic

[agin_chackacherry]

Hi,

Page 33 of the study notes 2010 1 Foundation L1 mentions the value for the T-Statistic, but does not explain why this value is equal to Information ratio*sqrt(time period). It would be beneficial if any one can explain why this is so. The video tutorial skips this part of the calculation.

Thanks & Regards,
Agin

 

[David Harper CFA FRM]

Hi Agin,

That's from Amenc Chapter 4 where the derivation isn't given but please see this thread with Paul from last year:
http://forum.bionicturtle.com/viewreply/3096/

At that time, I was thinking:
IR = t-stat = alpha / StdDev (alpha) = alpha / SQRT [ Variance(alpha) ]
i.e., on the right we have same critical t/critical Z as we have for, say a regresssion coefficient, only now I have made explicit the variance in order to treat the time dimension
So over T years:
t-stat = (T*alpha) / SQRT[ T * Variance(alpha) ]
t-stat = (T*alpha) / ( SQRT[T] * StdDev(alpha) )
t-stat = alpha /StdDev(alpha) * T/SQRT[T]
t-stat = alpha /StdDev(alpha) * SQRT[T]

... so you'll note the key here is that alpha is an intercept and a t-stat is just a "test of the intercept."

Although it appears to me now that maybe a more intuitive derivation is:

t-stat [coefficient/standard error (coefficient)]
= t-stat [alpha / annualized standard error (alpha)]
.... where annualized standard error (alpha) = annualized tracking error = TE * SQRT(1/T), such that:

t-stat = [alpha / TE * SQRT(1/Y) ] = (alpha / TE) * SQRT(T) = IR * SQRT(T)
... I am glad you asked (!) because i find this a more satisfying interpretation: the t-stat is here a typical "critical t" (i.e., coefficient/standard error of coefficient) except that it "scales" the tracking error into an annualized tracking error, to product annualized/annualized ratio.

Thanks, David

 

[neveo]

Then I have a question about the practice question R4.P1.T1.404.1 where the the t-statistic of the supplied regression is given as the answer to the question "what is nearest to the residual-based IR?". Shouldn't the t-statistic from the regression in this problem be annualized to get the residual-based IR, since the regression is on monthly returns? In the following question, 404.2, where the returns are monthly, the monthly active/active ratio "IR" is annualized by multiplying by SQRT(12) in the provided answer.

Hi Agin,

That's from Amenc Chapter 4 where the derivation isn't given but please see this thread with Paul from last year:
http://forum.bionicturtle.com/viewreply/3096/

At that time, I was thinking:
IR = t-stat = alpha / StdDev (alpha) = alpha / SQRT [ Variance(alpha) ]
i.e., on the right we have same critical t/critical Z as we have for, say a regresssion coefficient, only now I have made explicit the variance in order to treat the time dimension
So over T years:
t-stat = (T*alpha) / SQRT[ T * Variance(alpha) ]
t-stat = (T*alpha) / ( SQRT[T] * StdDev(alpha) )
t-stat = alpha /StdDev(alpha) * T/SQRT[T]
t-stat = alpha /StdDev(alpha) * SQRT[T]

... so you'll note the key here is that alpha is an intercept and a t-stat is just a "test of the intercept."

Although it appears to me now that maybe a more intuitive derivation is:

t-stat [coefficient/standard error (coefficient)]
= t-stat [alpha / annualized standard error (alpha)]
.... where annualized standard error (alpha) = annualized tracking error = TE * SQRT(1/T), such that:

t-stat = [alpha / TE * SQRT(1/Y) ] = (alpha / TE) * SQRT(T) = IR * SQRT(T)
... I am glad you asked (!) because i find this a more satisfying interpretation: the t-stat is here a typical "critical t" (i.e., coefficient/standard error of coefficient) except that it "scales" the tracking error into an annualized tracking error, to product annualized/annualized ratio.

Thanks, David

R.

 

[David Harper CFA FRM]

Hi @neveo Great observation. The information ratio (in addition to having two numerator/denominator definitions) also can be defined by period (e.g., monthly as Bodie seems to do in Chapter) or annualized. However,

1. My question is, at best, imprecise for not explicitly stating; it would be okay if I clarified that it was looking for the "monthly IR." Further, as you show, it's inconsistent with 404.2
2. Regardless of FRM author variation, I do happen to think a default convention should be annualized IR (per Carl Bacon)

So, bottom line, I totally agree with you, thanks!

 

[tosuhn]

Hi @David Harper,
For Question P1.T1.404.1, if the regression intercept is the alpha, what does the slope represent?

Also, how do I know when I should use residual based (ex ante) IR and active based (ex-post) IR?
Thanks in advance. Hope to hear from you soon.

 

[tosuhn]

Ohh btw there should be a typo in either the question or the answer for P1.T1.404.1. The intercept should be 0.171 instead of 0.0171.

 

[Alex_1]

Ohh btw there should be a typo in either the question or the answer for P1.T1.404.1. The intercept should be 0.171 instead of 0.0171.

Hi, have you looked here https://forum.bionicturtle.com/threads/p1-t1-404-tracking-error-information-ratio-and-sortino.7519/ ?
The alpha (or intercept) is 0.0171 in both the image and the answer and not 0.171

 

[Alex_1]

Hi @David Harper,
For Question P1.T1.404.1, if the regression intercept is the alpha, what does the slope represent?

Also, how do I know when I should use residual based (ex ante) IR and active based (ex-post) IR?
Thanks in advance. Hope to hear from you soon.

With regards to the ex-ante / ex-post IR, please refer to the Amenc notes, page 6

"Two definitions of information ratio

Currently (at least in the FRM) there are
two acceptable definitions of information ratio:

IR = Alpha / StdDev (alpha), but also acceptable because it is "ratio consistent:"

IR = (Active return) / StdDev (active return) = (Active return)/(Tracking error)

The first definition uses alpha which is also called the residual return, and this definition is implied by the assigned reading: "The information ratio, which is sometimes called the
appraisal ratio, is defined by the residual return of the portfolio compared with its residual risk. The residual return of a portfolio corresponds to the share of the return that is not explained by the benchmark."—Amenc page 114.

However, the second definition, which uses active return (i.e., the difference between the portfolio and the benchmark without any accounting for beta) is generally easier to compute.
As evidence, consider GARP’s 2012 Practice Exam Part 1, Question #3 [Notes by David Harper within square brackets]: "The information ratio may be calculated by either a comparison of the residual return to residual risk, or the excess return [i.e., active return] to tracking error [tends to refer to active risk; such that notice the ratio consistency]." Forum thread here @ https://forum.bionicturtle.com/threads/information-ratio-definition.5554/ "

In this case, regarding what represents the slope -> the slope in a regression is a measure of the average change in the dependent variable (y) given a change in the independent variable (x). This will be a learning outcome in part T2 (Quantitative Analysis), specifically in the part with regressions. I hope that helps.

 

[David Harper CFA FRM]

@tosuhn re: if the regression intercept is the alpha, what does the slope represent?

• Great question! The slope is arguably the point of the exercise; and while this concerns assigned Amenc, this is directly related to assigned Bodie (multifactor models). The model above is a single-index model, and M (or x) is a common factor. It is similar to the equity risk premium (ERP, aka MRP) in CAPM in the sense both are "common factors." The ERP is a broad theoretical common factor, whereas this one is a narrower (but likely correlating) market index. And it's coefficient can rightly be called a beta; so performance attributable here is similar to "hedge fund beta" (i.e., performance due simply to exposure to the factor). It's really at the center of all of Bodie Ch 10: the product of a (common) factor and a beta (aka, factor loading, sensitivity), where we hypothesize that some "beta performance" is due to sensitivity to a common factor.

Re: when I should use residual based (ex ante) IR and active based (ex-post) IR?

• Thank you @Alex! The fact that either is acceptable is why I wrote "residual-based information ratio (IR)?" which, I could have also written as "alpha-based information ratio (IR)?" Actually, I may suggest to GARP, alpha-based IR, I think that's better...

Re: there should be a typo in either the question or the answer for P1.T1.404.1.

• You are correct, there is a typo in the PDF (sorry!) that we have not yet reverted to the PDF, please see the source (where it has been corrected) https://forum.bionicturtle.com/threads/p1-t1-404-tracking-error-information-ratio-and-sortino.7519/
  i.e., IR = 0.0171/0.0180 = 0.950