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# R8.P1.T1.Amenc_Ch4_RISK_MGMT_Topic:INFORMATION_RATIO_RESIDUAL vs_ACTIVE

##### Active Member
In reference to R8.P1.T1.Amenc_Ch4_RISK_MGMT_Topic:INFORMATION_RATIO_RESIDUAL vs_ACTIVE:

The Residual Risk is the idiosyncratic Risk = any risk which is not the Systemic Risk
and that Total Risk = Systemic Risk + Residual Risk.

Having said that though, I am trying to get the standardized formula to use for calculating Info-Ratio when we use:-
a) Residual Return/ Residual Risk vs.
b) Active Return/ Active Risk

The study materials indicate just one formula for the Info-Ratio IR = E(Rp) - E(Rb) / Std-Dev( Rp - Rb)

Want to understand is whether the above stated formula is a) Residual Return/ Residual Risk or b) Active Return/ Active Risk... :-( ...?

Thanks much for all the help and insight on this topic ..

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari It's a great question! (although FYI if you had searched "information ratio" you would find this has been discussed many many times; also, you can use the tag = 'infomration ratio' https://www.bionicturtle.com/forum/tags/information-ratio/)

Please note that @Nicole Seaman recently published an updated R9. Amenc Study Note to the Study Planner (SP) at https://www.bionicturtle.com/topic/study-notes-amenc-chapter-4/
... in 2017, Amenc is R9 not R8
There are two valid definitions of information ratio
At least in the FRM, there are two acceptable definitions of information ratio:
• IR = alpha / σ(alpha)
• IR = active_return / σ(active_return) = active_return / tracking_error
These are both acceptable because they are ratio-consistent: the denominator is the standard deviation of the numerator. The first definition uses alpha, α, 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

However, the second definition (which uses active return; i.e., the difference between the portfolio and the benchmark without 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://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/
The new note also features my updated learning spreadsheet (which dynamic formula descriptors!) and you can see how I show the IR both ways: Here is the original thread on this issue https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ I hope that helps!

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##### Active Member
@David Harper CFA FRM Thank you so much for laying out the above in detail and for your patience- my bad - I just had the older BT notes open instead of the new updated one. The new notes clarifies the 2 separate formulas that I was looking for..

An example solidifies the concepts above.Am trying to find the entire example that the above is an excerpt of...I am not seeing in the notes somehow... :-(
Am trying to see how we get 3% as the Residual Error....seeing the example in it's entirety would help me a lot on this topic...

Again can't thank you enough for laying out the above- exactly what I was looking for !

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari I don't understand why you say you "Am trying to find the entire example that the above is an excerpt of" Were you able download the new note at https://www.bionicturtle.com/topic/study-notes-amenc-chapter-4/ ? Page 9 sets up the assumptions including "The residual risk (aka, non-systematic risk), σ(e), is simply assumed to be 3.0%" .

Then page 10 the displays the calculations (below) but the residual risk (aka, non-systematic risk) is simply an input per its yellow coloring. I always color inputs in yellow. The residual risk calculation would require a series-based analysis of simulated returns (as i did no page 12 for the Sortino) and i did not want to clutter this RAPMs with that here. I hope that helps! ##### Active Member
@David Harper CFA FRM Oh Gosh....completely missed seeing that ! I had already scrolled down and was looking all over the place but somehow overlooked this page ....please ignore ...silly me !! Thank you Thank you Thank you ....one last question on this ....would the Residual Risk always be given or is there a formula to calculate the Residual Risk...? I mean Residual Risk = is the Non-systemic Risk so common sense says there might not be a formula to quantify it as such...but wanted to double check with the Boss just to be sure ! #### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari re: the whole page, okay good i was a little concerned about the new note download or something, good to hear! Residual risk needs to be given unless there is a data series; from the data series, we can use the standard error of the regression (SER) so one formula is SER = sqrt(SSR/df) where SSR is sum of squared residuals. So, just as technically, alpha is the regression intercept, we'd want to either do a linear regression or take a standard deviation of the series of residuals (which should be close; actually there is more than one way to get residual risk, but I think SER is the most consistent for FRM!). You need data because the there really isn't an ex ante (before the performance) perspective like there is with active return and risk. We can say there is non zero expected active return and risk, but the expected alpha is zero. (Notice how my example contains a subtle cheat: the portfolio's expected return probably should be 10.8% such that expected jensen's alpha is zero). After the performance comes in, then we can measure the alpha with a regression. I hope that helps!

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##### Active Member
@David Harper CFA FRM I have a follow up question on this topic:-
The CAPM Model states that :-
(Rp- Rf) = Alpha + Beta ( Rb - Rf) + Residual Error
So How is the Residual Error = the Tracking Error = Std-Dev( Rp-Rb) when Beta =1 Also, is the Alpha = the Active Return =( Rp - Rb) always ? ..... If Alpha = the Active Return, and Residual Error= the Tracking Error as indicated in some of the forums on BT, then:-

IR(Residual) = Alpha / Tracking Error which in turn = Active Return/ Tracking Error = IR( Active) !!    Last edited:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari Sorry, but those are not the definitions of alpha, active and tracking error that are above or in my notes. You seem to have them somwhat reversed from everything above. Did you see see the link above at https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ To recap one more time:
• IR can be active-based (perhaps more common) or residual-based (more technical and Grinold/Crouhy based)
• active IR = active return/active risk = active return/tracking error, because "tracking error" is common defined as active risk
• residual IR = residual return/residual risk = alpha/residual risk, because "alpha" technically is residual return
• It's okay to say IR = α/TE if we mean TE to refer to residual risk (although maybe unusual). This is because in both cases the denominators are standard deviations of their same numerator: active IR = active return/σ(active returns) and residual IR = α/σ(α) but we are really going to use α/σ(e)
• Given this, if the portfolio β = 1.0, then CAPM says E[R(p)] - Rf = E(α) + β*E[R(m) - Rf] = 0 + 1.0 * E[R(m) - Rf] = E[R(m) - Rf]; so beta of 1.0 says to expect an active return equal to the benchmark's excess return, but it's still expecting residual return, alpha, of zero. Active return "gets credit" for beta exposure, hence the saying that "some apparent alpha [if it is mis-measured as merely relative return] is "in truth actually beta in disquise" and why Grindold says we should use residual return
I ran a simulation for the learning spreadsheet (it's meant for Sortino, but I added a regression). See below. Please note that I simulated portfolio and benchmark returns over 20 days based on assumptions (in yellow) including a correlation assumption. From this we can observe:
• The ex-post active return = 3.19%; this is average excess of portfolio over benchmark. The active IR divides this by the tracking error, σ(P-B) = 19.02%, for an active IR of about 0.17. Again, active IR = avg(P-B)/σ(P-B)
• The (ex-post) residual return = 1.84% because it's the regression intercept. The residual IR divides this by SER = 19.323%, for a residual IR of about 0.095.
• Notice also we'd expect a β of ρ*σ(P)/σ(M) = 0.70*20%/10% = 1.40 but we have a small sample such that we get 1.30 (pretty close actually).
• Please consider two possible assumption changes:
• If I change the correlation to unrealistic, perfect 1.0 (100%), but leave σ(P) = 20% and σ(M) = 10%, then the plot would be a straight line and both alpha and SER would be zero, so residual IR would be zero (or undefined), but we would have a non-zero active IR (because beta would be 2.0 and there will be significant ouperformance versus the benchmark, but it will be entirely due to beta without any alpha!). In this case, residual IR = 0/0 but active IR = x/y.
• If I change correlation to 1.0 and portfolio volatility to 10%, then beta = 1.0 and then the ex post active return will be +4.0% but the σ(P-B) will be zero, so in this case of β=1.0, active IR = +4.0%/0. I hope that helps to clarify this! Last edited:

##### Active Member
@David Harper CFA FRM Thanks for much for laying this information in detail on the weekend - heartfelt gratitude...am digging into this right now to see what I've been missing ... :-(

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari sure, no problem, it actually gave me an excuse to add a worksheet to the Amenc learning XLS specifically to show both information ratios under a simulation (similar to how the notes simulates in order to illustrate the Sortino). And we will add this to the next update of the Amenc notes, I think it will be helpful because this is an extremely common point of confusion. I think this is looking pretty good (?), see below. Plust I couldn't resist jamming in a few key regression relationships! You only need to provide six input assumptions (in yellow) and then it simulates 36 months of returns and shows the IR both ways (highlit in light green and blue). All of the displayed formulas dynamically update After you dig in, let me know if you still have any question, please, because you are not the only one, I've lost count of how many IR threads we have, so it will be good to get some additional help into the Amenc note! ##### Active Member
@David Harper CFA FRM sorry to bother you over this - but if it's not too much trouble, could just this new spreadsheet be uploaded in Dropbox...?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari Can you wait until we upload this sheet as part of revised Amenc learning sheet at https://www.bionicturtle.com/topic/question-set-amenc-chapter-4-2/ , please? I would prefer to reserve this sheet for the paid members that help us keep the lights on ... we will post here when we've done that, it will be before the end of the week as I just need to make a few finishing touches i think. Thanks!

##### Active Member
@David Harper CFA FRM Completely Understand- NP at all Thanks so much
In the mean time, I got a hang of the above and also I think got a hang of how Residual Risk might be = Tracking Error only under the condition that Beta =1....

Having said that, the forum link ( https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ ) had let to some confusion on my part with the final conclusion stating that Residual Risk= Alpha/ Tracking Error ...

But then again reading through what you had explained at the very beginning of the forum and also this thread itself gave me the insight I was missing. The spreadsheet would additionally help solidify the concept. Infinite gratitude for the pains you took and your patience with my limited knowledge on this topic  Last edited:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari Sure thing, and thank you sincerely for your curiosity because this this is perennial issue for candidates and, not only can we improve the study note, but we continue to give GARP feedback (our feedback helped inform the ratio consistency that prevails in the exam methodology, or I should say, our relaying of, and pointing to, prior candidate feedback).

Re: I got a hang of the above and also I think got a hang of how Residual Error might be = Tracking Error only under the condition that Beta =1.
Awesome! IMO, that statement reflects an understanding. Other readers might be confused by all of this but I like to use a really simple single-period example: say Rf = 1.0% and the market's excess return = 4.0%. Consider the portfolio's gross return is 6.0% so its excess return = 5.0%, then we can say:
• The portfolio's active return = R(P) - R(B) = 5% - 4% = +1.0 regardless of beta b/c active return doesn't credit beta exposure; but if the portfolio's beta is 1.40 then:
• The portfolio's residual return = 5.0% - 1.4*4.0% = -0.60%. And, just as you suggest, if the portfolio's beta is 1.0, then
• The portfolio residual return = 5.0% - 1.0*4.0% = +1.0%; i.e., equal to its active return. And is a similar way, if beta = 1.0, my simulator will show active risk (aka, tracking error) does approximate residual risk (there is some further technical nuance due to the fact we are using SER but it's approximately similar!). So I think your statement is significant!
Re the above posted thread: IMO, he is right to say those statements are correct because "tracking error" seems to be employed slightly differently. However, Amenc and Grinold (and Bacon) do define Tracking Error as Active Risk. So, IMO, this is clearly the connotation of tracking error. But cabrown85 is just showing an alternative that is ratio-consistent with the numerator, which seems fine to me! Thanks!

##### Active Member
@David Harper CFA FRM The above example is indeed awesome ! - I have one little hitch in my understanding where you state the initial parameters:-
" say Rf = 1.0% and the market's excess return = 4.0%. Consider the portfolio's gross return is 6.0% so its excess return = 5.0%"
If the portfolio's gross return is 6.0%, would that not be the value of Rp= 6%. In the example we seem to have used (Rp- Rf)= (6%-1%) = the PF Excess Return as Rp...am sure am missing something in my understanding here...and there are so many nuances...want to make sure I understand the whole thing and not miss something.... :-(

Again cant thank you enough for your infinite patience on this thread...

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@gargi.adhikari Yes, sorry, I was being loose with gross vs excess returns. I think you are correct that if we don't qualify the terms, then "return" connotes gross return (more than excess), so R(p) = 6% is the implication. In my example, I meant to setup:
• R(M) = 5.0% and r(M) = 4.0%; where I might define R(.) as gross return and r(.) as excess return as "in excess of the risk-free" (borrowing from typical usage re: r=R-i where r is real and R is nominal and i is inflation). And I like capital "M" for market, similar to capital "B" for benchmark, then for the portfolio which i might denote with small 'p' because it's not universal then:
• For the portfolio, R(p) = 6.0% and r(p) = 5.0%.
• Then the active return is either R(p) - R(M) or r(p) - r(M).
But i do agree with you. Mine are are just definitions. If there is no context, then R(p) suggest the gross return (in the case, 6.0%) such that it's the burden of the writer to say "excess return of 5.0%" if net-of-risk-free is the intention. Thanks!

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##### Active Member
@David Harper CFA FRM Thanks so much for taking the time to clarify my ignorant doubts with so much patience. My apologies for being a pain on this topic. Infinite gratitude #### Nicole Seaman

Hi @gargi.adhikari Can you wait until we upload this sheet as part of revised Amenc learning sheet at https://www.bionicturtle.com/topic/question-set-amenc-chapter-4-2/ , please? I would prefer to reserve this sheet for the paid members that help us keep the lights on ... we will post here when we've done that, it will be before the end of the week as I just need to make a few finishing touches i think. Thanks!
I just wanted to let you know that the updated study notes and XLS have been published in the study planner 