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# GARP.FRM.PQ.P2Important Difference in Information Ratio Formula (garp16-p2-72)

#### Liming

##### New Member
Dear David,
I’ve had some confusion, misunderstanding and doubts when doing 09 Level I Annotated Boot Camp. Appreciate your kind help on this!

I’ve noticed an important difference between you and FRM handbook with respect to calculating information ratio: in all your practice questions and study notes, the formula you used is such that: IR = Alpha/TE, however, the formula used by FRM handbook (as can be seen on page 388 of FRM 5th “Risk-Adjusted Performance Measurement”) is such that: IR = [mu(Rp) – MU(Rb)]/TE, where the numerator is excess return of portfolio versus benchmark return, not Alpha. And I think the excess return do not equal Alpha.

Thank you for your enlightenment and correction!
Cheers
Liming
16/11/09

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Liming,

if you search "alpha," you'll see lots of threads on this topic; e.g., http://www.bionicturtle.com/forum/viewthread/1242/
to repeat myself: I am using the assigned Grinold which is (IMO) the most technically accurate; e.g., IR = residual return / residual risk = alpha / StdDev(alpha)
...for good reasons including it is important to understand that ALPHA does NOT include BETA which is the mistake the other definition makes. The problem with (portfolio return - benchmark return) is simple: it's an active return not a residual return. It's not alpha because it includes beta!

the key reconciliation point is ratio consistency: the FRM handbook is not wrong if the denominator is the StdDev (numerator). If the TE is defined consistently, then it's an okay format of IR.

David

#### no_ming

##### Member
Hi Liming,

if you search "alpha," you'll see lots of threads on this topic; e.g., http://www.bionicturtle.com/forum/viewthread/1242/
to repeat myself: I am using the assigned Grinold which is (IMO) the most technically accurate; e.g., IR = residual return / residual risk = alpha / StdDev(alpha)
...for good reasons including it is important to understand that ALPHA does NOT include BETA which is the mistake the other definition makes. The problem with (portfolio return - benchmark return) is simple: it's an active return not a residual return. It's not alpha because it includes beta!

the key reconciliation point is ratio consistency: the FRM handbook is not wrong if the denominator is the StdDev (numerator). If the TE is defined consistently, then it's an okay format of IR.

David
Hi Mr. Harper, I have read lots of post about the confusion of the information ratio, I conclude the followings:

1. IR = Residual return (Jensen alpha)/Residual risk (Include effect of beta)
2. IR = Active return (alpha) /Tracking error (Not include effect if beta)

Do you agree?

Moreover, for the following questions (2016 Practice exam Q72), the SD of alpha is not directly equal to the residual risk and adjusted by the IC, it make me confused whether SD of residual return should be residual risk or residual risk*IC, can you help me? 72. An analyst regresses the returns of 100 stocks against the returns of a major market index. The resulting pool of 100 alphas has a residual risk of 18% and an information coefficient of 9%. If the alphas are normally distributed with a mean of 0%, roughly how many stocks have an alpha greater than 4% or less than -4%?

a. 5
b. 10
c. 20
d. 25

Explanation: The standard deviation (std) of the alphas = Residual Risk (volatility) x Information Coefficient (IC) = 0.20 * 0.10 = 0.02. So, 4% is twice the standard deviation of the alphas. The alphas follow normal distribution with mean 0, so about 5% of the alphas are out of the interval [-4%, 4%]. The total number of stocks is 100, so roughly there are 5 alphas that are out of the range. Last edited by a moderator:

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @no_ming

Yes, your two versions of IR are correct. You are correct except you should delete alpha = active return: alpha is a residual return. And you can delete "Jensen's" as Jensen's alpha is a special case of alpha (i.e, is itself technically the same alpha). In short: residual return = alpha, and active return is by itself. Our feedback over the years helped to inform the clarity of the ratio-consistency which you are illustrating; i.e., the denominator is the standard deviation of the numerator such that incorrect would be alpha/TE, if TE is active return, because that's residual/active. The two forms are active/active, or residual/residual, just as you show (on a very minor note: I consider "Jensen's alpha" per Amenc to be a special case of alpha under the single-factor CAPM where the benchmark is the market return or its proxy a broad index, but "alpha" is multi-factor). See https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ for old discussion on this which was input to GARP

Re: 72: earlier in the year, we also gave feedback on this question (see https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40716 ). Note the mathematical mistake. I don't like this question (more broadly, I think Grinold Chapter 14 is way too isolated), but technically per Grinold, they are just scaling the alpha. Alpha is the residual risk; ie, standard deviation of the residual return. So, I don't see a technical problem, it's just a scaling of the raw alphas ("residual risk of 18%) into a different "alpha scale" where, if the math error is corrected, 0.18 x 0.09 = 0.0162 as originally noted by @Kenji at https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40449 Thanks!

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#### no_ming

##### Member
Hi @no_ming

Yes, your two versions of IR are correct. Not only do I agree, but our feedback over the years helped to inform the clarity of the ratio-consistency which you are illustrating; i.e., the denominator is the standard deviation of the numerator such that incorrect would be alpha/TE, if TE is active return, because that's residual/active. The two forms are active/active, or residual/residual, just as you show (on a very minor note: I consider "Jensen's alpha" per Amenc to be a special case of alpha under the single-factor CAPM where the benchmark is the market return or its proxy a broad index, but "alpha" is multi-factor). See https://www.bionicturtle.com/forum/threads/information-ratio-definition.5554/ for old discussion on this which was input to GARP

Re: 72: earlier in the year, we also gave feedback on this question (see https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40716 ). Note the mathematical mistake. I don't like this question (more broadly, I think Grinold Chapter 14 is way too isolated), but technically per Grinold, they are just scaling the alpha. Alpha is the residual risk; ie, standard deviation of the residual return. So, I don't see a technical problem, it's just a scaling of the raw alphas ("residual risk of 18%) into a different "alpha scale" where, if the math error is corrected, 0.18 x 0.09 = 0.0162 as originally noted by @Kenji at https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40449 Thanks!
Hi, Mr. Harper, thanks for your reply, recall #2 above, you mentioned that "good reasons including it is important to understand that ALPHA does NOT include BETA which is the mistake the other definition makes. The problem with (portfolio return - benchmark return) is simple: it's an active return not a residual return. It's not alpha because it includes beta!"

It seems say that active return can not be called or rename as Alpha and only residual return can be called as Alpha, is that right?
Moreover, should the last sentence amend to It's not alpha because it DOES NOT include beta for the problem of active return?
Thanks #### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @no_ming Yes, that's correct. Sorry, you do have one mistake above. You should have instead the following; the following is correct:
1. IR = Residual return (alpha)/Residual risk (Include effect of beta)
2. 2. IR = Active return /Tracking error (aka, active risk; does not include effect if beta)
To say residual return or residual risk "includes the effect of beta" can be confusing. It is easy to illustrate with a simple example: imagine the riskfree-rate is 1.0% and the benchmark return is 4.0% such that the benchmark's excess return = 3.0%. Now imagine a portfolio with a beta of 0.80 returns 4.60%, such that the portfolio's excess return is 3.60%.
• The active return is the difference in excess returns = 3.60% - 3.00% = +0.60%; or just the difference in returns, 4.60% - 4.00% = +0.60%; i.e., active return is the portfolio's return relative to the benchmark
• The residual return = 3.60% - 3.0%*0.80 = 1.20%, or in this single-factor model, (jensen's) alpha = 4.60% - (3.00%*0.80) - 1.0% Rf = + 1.20%; i.e., residual return is the portfolio's excess return relative to (beta*benchmark excess return).
Alpha is residual return, it is not active return. Jensen's alpha is an alpha, it's just the special case of the single-factor CAPM (where there is only one beta). Tracking error connotes active risk, but some authors refer to residual risk also as tracking error. So tracking error needs careful definition. GARP knows all of this, this is based on Grinold and sussing out Amenc over literally years of usage. You can hopefully see that when referring to residual return, to say "it includes the effect of beta," is correct if we mean something like "the return is adjusted to discount the effect of beta." But it is confusing because residual risk could be interpreted to mean something like "the return excludes the contribution of beta." The math, however, is unambiguous. I hope this clarifies.

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#### no_ming

##### Member
Tank
Hi @no_ming Yes, that's correct. Sorry, you do have one mistake above. You should have instead the following; the following is correct:
1. IR = Residual return (alpha)/Residual risk (Include effect of beta)
2. 2. IR = Active return /Tracking error (aka, active risk; does not include effect if beta)
To say residual return or residual risk "includes the effect of beta" can be confusing. It is easy to illustrate with a simple example: imagine the riskfree-rate is 1.0% and the benchmark return is 4.0% such that the benchmark's excess return = 3.0%. Now imagine a portfolio with a beta of 0.80 returns 4.60%, such that the portfolio's excess return is 3.60%.
• The active return is the difference in excess returns = 3.60% - 3.00% = +0.60%; or just the difference in returns, 4.60% - 4.00% = +0.60%; i.e., active return is the portfolio's return relative to the benchmark
• The residual return = 3.60% - 3.0%*0.80 = 1.20%, or in this single-factor model, (jensen's) alpha = 4.60% - (3.00%*0.80) - 1.0% Rf = + 1.20%; i.e., residual return is the portfolio's excess return relative to (beta*benchmark excess return).
Alpha is residual return, it is not active return. Jensen's alpha is an alpha, it's just the special case of the single-factor CAPM (where there is only one beta). Tracking error connotes active risk, but some authors refer to residual risk also as tracking error. So tracking error needs careful definition. GARP knows all of this, this is based on Grinold and sussing out Amenc over literally years of usage. You can hopefully see that when referring to residual return, to say "it includes the effect of beta," is correct if we mean something like "the return is adjusted to discount the effect of beta." But it is confusing because residual risk could be interpreted to mean something like "the return excludes the contribution of beta." The math, however, is unambiguous. I hope this clarifies.
Very clear explanation & Many thanks, Mr. Harper, I got it~ #### farahm

##### Member
Hi Mr. Harper, I have read lots of post about the confusion of the information ratio, I conclude the followings:

1. IR = Residual return (Jensen alpha)/Residual risk (Include effect of beta)
2. IR = Active return (alpha) /Tracking error (Not include effect if beta)

Do you agree?

Moreover, for the following questions (2016 Practice exam Q72), the SD of alpha is not directly equal to the residual risk and adjusted by the IC, it make me confused whether SD of residual return should be residual risk or residual risk*IC, can you help me? View attachment 838
I don't understand how GARP is getting 5% of alphas from 0.02 std deviation of alphas? @David Harper CFA FRM

#### farahm

##### Member
Can anyone explain how to solve question 72 ? - Still don't understand how 5% was derived. Thank you

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@farahm Please note that, in May of this year, I both submitted a correction to this question (credit to Kenji who first spotted it https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40449) and further challenged it conceptually (based on its requiring more knowledge than found in the assignment; specifically, it requires additional Grinold, IMO). The answer to your specific question is likely simpler than you expect. Firstly, the question is utilizing Grinold's σ(refined α) = σ(α) * IC, based on σ(refined α) = σ(α) * IC * z-score but where score ~ N(0,1).

So the correct solution should read: the standard deviation (std) of the alphas = Residual Risk (volatility) * Information Coefficient (IC) = 0.18 * 0.09 = 0.0162 is the standard deviation of the alpha (versus 2.0%).

Then, to your question about the 5.0%, that's simply based on about two standard deviations including about 95% of the normal distribution: Pr[-1.96 < Z < 1.96] ~= 95%. In the incorrect version where σ(scaled α) = 2%, about 5% should be outside the +/- 2*2% interval; in the corrected version, about 5% should be outside the +/- 1.62%*2 interval. I still think the question has deeper problems, but I hope that helps.

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#### farahm

##### Member
Thank you David ! I agree with you, I am not a fan of this question - I'll take it as you explained it going in the exam as I dont have time to dig deeper on this one.

#### bpdulog

##### Active Member
@farahm Please note that, in May of this year, I both submitted a correction to this question (credit to Kenji who first spotted it https://www.bionicturtle.com/forum/threads/2016-frm-part-ii-practice-exam-q-a.9350/#post-40449) and further challenged it conceptually (based on its requiring more knowledge than found in the assignment; specifically, it requires additional Grinold, IMO). The answer to your specific question is likely simpler than you expect. Firstly, the question is utilizing Grinold's α = σ(α) * IC, based on α = σ(α) * IC * z-score but where score ~ N(0,1).

So the correct solution should read: the standard deviation (std) of the alphas = Residual Risk (volatility) * Information Coefficient (IC) = 0.18 * 0.09 = 0.0162 is the standard deviation of the alpha (versus 2.0%).

Then, to your question about the 5.0%, that's simply based on about two standard deviations including about 95% of the normal distribution: Pr[-1.96 < Z < 1.96] ~= 95%. In the incorrect version where σ(scaled α) = 2%, about 5% should be outside the +/- 2*2% interval; in the corrected version, about 5% should be outside the +/- 1.62%*2 interval. I still think the question has deeper problems, but I hope that helps.
Just wanted to point out a similar question is int he 2017 practice exam, not sure if it was ever tested on the real exam

#### irwinchung

##### Member
Hi David,

On Grinold

standard deviation (std) of the alphas = Residual Risk (volatility) * Information Coefficient (IC)

Let's take a portfolio of 300 stocks.

What is Residual Risk (volatility)? Is is an average of each stock's tracking error? Sum of 300 tracking errors/300?

What is this the purpose of this equation? Really? What are we trying to predict?

Thanks!

#### emilioalzamora1

##### Well-Known Member
Can you please tell in which chapter this equation is mentioned? I have not yet seen anything like the equation you posted above.

I guess you are mixing things up here. The Information (IR) = alpha/residual risk

where residual risk = Tracking Error (TE)

The information coefficient (IC) is only part of the FLOAN (Fundamental Law of Active Management) having:

where IC = the manager's ability to predict the returns of the securities in the portfolio
breadth = number of securities in the portfolio

In principle, your example does not makes sense at all! Never heard about an average tracking error.

As I said, if you can show the equation in Grinold, I am happy to help.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@emilioalzamora1 @irwinchung is referring to Grinold's paragraph below (Scale the Alphas) which actually (unfortunately) has a corresponding practice question type; I say unfortunately because it really comes up without the scaffolding (proper context). As I mentioned above, this re-scaling of the alpha is developed in Grinold's chapter 10 where he develops a "forecasting rule of thumb:" Refined forecast = volatility * IC * (Z- score); i.e., Grinold formula 10.11. This is why Grinold says the alphas should have standard deviation, or "scale," of σ(refined α) = σ(α) * IC * Z-score, except score ~ N(0,1), so that σ(refined or forecast alpha) = σ(α)[aka, residual risk, volatility] * IC. So this is basically re-scaling a raw alpha into a refined alpha as a function of the manager's IC; e.g., Grinold (emphasis mine): "The forecasting rule of thumb [Eq. (10.11)] also shows the correct behavior in the limiting case of no forecasting skill. If the IC = 0, then the refined forecasts are all zero, as they should be in this case." I'm not convinced Q 63 is self-aware (realizes what it is asking for), is my current feedback to GARP. Thanks,

Scale the Alphas
Alphas have a natural structure, as we discussed in the forecasting rule of thumb in Chap. 10: α = volatility · IC · score. This structure includes a natural scale for the alphas. We expect the information coefficient (IC) and residual risk (volatility) for a set of alphas to be approximately constant, with the score having mean 0 and standard deviation 1 across the set. Hence the alphas should have mean 0 and standard deviation, or scale, of Std{α} ~ volatility · IC. An information coefficient of 0.05 and a typical residual risk of 30 percent would lead to an alpha scale of 1.5 percent. In this case, the mean alpha would be 0, with roughly two-thirds of the stocks having alphas between –1.5 percent and +1.5 percent and roughly 5 percent of the stocks having alphas larger than +3.0 percent or less than –3.0 percent. In Table 14.1, the original alphas have a standard deviation of 2.00 percent and the modified alphas have a standard deviation of 0.57 percent. This implies that the constraints in that example effectively shrank the IC by 62 percent, a significant reduction. There is value in noting this explicitly, rather than hiding it under a rug of optimizer constraints. The scale of the alphas will depend on the information coefficient of the manager. If the alphas input to portfolio construction do not have the proper scale, then rescale them. - Grinold Chapter 14, page 382
GARP 2017 P2.63
GARP 2017 P2. 63. An analyst regresses the returns of 300 stocks against the returns of a major market index. The resulting pool of 300 alphas has a residual risk of 15% and an information coefficient of 10%. If the alphas are normally distributed with a mean of 0%, roughly how many stocks have an alpha greater than 3.24% or less than -3.24%?
A. 5
B. 15
C. 30
D. 45

Explanation: The standard deviation (std) of the alphas = Residual Risk (volatility) x Information Coefficient (IC) = 0.15 * 0.10 = 0.0150. So, 3.24% is twice the standard deviation of the alphas. The alphas follow normal distribution with mean 0, so about 5% of the alphas are out of the interval [-3.24%, 3.24%]. The total number of stocks is 300, so roughly there are 15 alphas that are out of the range

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#### Galaxy

##### New Member
Subscriber
Hi David,

I was watching your video on "information ratio" (link as below), which was published on Nov 6, 2017. And I am trying to figure out how I can relate the content in that video to the readings in P2.T8 please.

To this end, can I check that the active information ratio that you mentioned in the video is the one described by Jorion (in Chapter 17 of his book "Value at Risk"), as "the expected tracking error divided by the volatility of the expected tracking error" (given that what he calls tracking error would be seen as "active return" by most practitioners.)

In contrast, the residual information ratio is the one described by Grinold & Kahn in Chapter 14 of their book.

Assuming I am on the right path so far, would it be right in suggesting that there is therefore a third version of information ratio, which would be "active return divided by tracking error" and this definition is the one most commonly used by practitioners.

Galaxy

#### emilioalzamora1

##### Well-Known Member
Hi @Galaxy,

first of all, the IR is defined as alpha (difference between portfolio return versus its benchmark) divided by the Tracking Error (which is the std. of the difference between the return of the managed portolfio versus its benchmark return).

For the exam: you only have to remember this basic (fundamental) idea described above.

Sometimes alpha is defined as the residual portfolio return (as in Amenc) where the denominator of the TE equation is sometimes defined as std deviation of the residual portfolio return. But this very rare or not applicable in practice. If you talk to fund managers etc. you will only hear alpha and TE.

In regression language, the std deviation of the residual portfolio return would then be the idiosyncratic term of the familiar porfolio std deviation equation:

sigma(p) = beta(p)*simga(m) + sigma(epsilon)

The difference described by David (active vs. residual IR) is simply explained by the fact that:

1.) Active IR (forget this term please - it does NOT exist in practice. It is called Informatio Ratio and nothing else) is a simple numerical exercise finding the difference between portfolio return and its benchmark and their respecticve std deviation.

2. Residual IR involves regression analysis. Rarely done in practice!

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Right @emilioalzamora1 but help me out here, if you don't mind. We've worked hard to get GARP to be consistent (if nothing else so candidates know what to expect). As reflected finally in their in their practice question 2012.P1.3 (and wherever it recycles), the agreement is ratio-consistency by which I mean that any version of IR should be X/σ(X); ie., µ/σ(µ). The confusion arises because there are variations of µ and σ(µ). But under an agreement of ratio-consistency, we have only two information ratios that a candidate should expect:
• IR = (excess or active return)/σ(excess or active return); where the key feature here is that excess/active return is the simple (average) difference between the portfolio and benchmark return
• IR = (residual return)/σ(residual return); where the key feature here is that this a regression against the beta factors such that the residual is a "true" alpha intercept (uncorrelated to common factors)
From my perspective, the confusion arises almost entirely due to terminology:
• Jorion's (2007) terminology is obsolete for current purposes. His "tracking error" is what I prefer to call active return, but is sometimes called excess return (which i do not prefer because that typically means "in excess of riskfree"). His TEV is actually what we call active risk or "tracking error."
• A confusion is also engendered by IR = α/TE, which is extremely common notation. But for our purposes, it is not supposed to suggest (residual return)/(active risk) which is ratio inconsistent. It should be IR = α/σ(α). The problem here is that tracking error connotes "active risk."
• Finally, we do (e.g., Bodie, my XLS) see a "substitution" of σ(α) with σ(e); aks, standard error of the regression term. Because it makes a lot of statistical sense to use the SER (which is the volatility of unexplained factor, after all) as the risk associated with the alpha intercept. Consistent with your final point above about idiosyncratic term.
• I really hesitate to concede that "alpha" is anything other than the residual or regression intercept. We've worked hard to reinforce this definition (Tracking error not so much, clearly it has various definitions).
So setting aside the terminology differences, I think we want to stay with two ratio-consistent variations (if only for purposes of the FRM) that are confirmed by (eg) GARP's assumption given in the question "The information ratio may be calculated by either a comparison of the residual return to residual risk, or the excess return [dharper: aka, active return] to tracking error [dharper: aka, active risk]."
1. active IR = (active return)/TE, where TE = active risk
2. residual IR = α/σ(α) or α/SER
So no @Galaxy actually I don't understand your third variation? IMO, we just have your first two.

Please let me know what you think? I happen to be almost done with a memo going to GARP where I can mention this issue ... Thanks!

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#### emilioalzamora1

##### Well-Known Member
Hi @David Harper CFA FRM,

Sure. I agree whith your ratio-consitents explanations for the IR and for sure the outdated Jorion terminology have Tracking Error in the numerator. Jorion's VaR is still a superb book but with respect to the latter he got it completely wrong (or at least his terminology never made it into the real world).

I think it's a matter of the wording, however, I don't agree with your explanation on the original IR notation:

IR = α/TE

It is just written in a different form (not ratio conistent of course) but in practical language this is exactly what the IR denotes.

In the end it does not matter whether we write it in terms of

• residual return or active return / std deviation of the residual return or even active risk

OR

• in the format above: IR = α/TE

For me this is just a visual thing.

-------

Again, these expressions like active return or active risk simply exist in an isolated (GARP) world but in general I assume these terms cause confusions for candidates at later stages in their career as in practice these term are not applied across the industry.

Most of these notation problems arise due to the fact that various authors whose area of expertise is not necessarily performance measurement came up with some fancy new terms instead of the original language.

I have just found these CFA paper which also more or less refers to the original language (they call the numerator excess return relative to its benchmark):

https://www.cfapubs.org/doi/pdf/10.2469/ipmn.v2011.n1.7

In F. Lhabitant's book "Hedge Funds" it is the same:

The information ratio (IR) measures the annualized relative return (alpha) of a portfolio generated per unit of annualized relative risk (tracking error) fora portfolio that is managed against a benchmark. Or put differently, the IR compares the average differential return with its volatility. The latter is nothing more than the tracking error of the fund P with respect to the benchmark B. Algebraically: IR = R(p) - R(b) / TE.
In addition, Lhabitant writes: When the benchmark equals the risk-free rate, the information ratio equals the traditional Sharpe ratio.

Bill Sharpe in his old book "Investments" explains it in the following way:

..."dividing the ex-post alpha by an estimate of the ex-post unique (unsystematic) risk of the portfolio. This measure is known as the appraisal (information) ratio..."

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@emilioalzamora1 okay awesome thank you! And the CFA paper is helpful! And the F. Lhabitant's book, too. If I understand you, you are saying that not only is notationally IR = α/TE okay to use, it is probably the most common notation. But we just want to be mindful of what they each represent (as I don't think you are suggesting it should be residual_return/active_risk). Thanks again ...