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# Potential error in P2.T5. MR-9: 3.2.3. Kendall's tau and concordant/discordant pairs

#### David Harper CFA FRM

##### David Harper CFA FRM
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I wrote to Meissner (I don't see an errata yet for his new book, which is excellent) that I think the example for (3.2.3) Kendall's tau is incorrect. He shows five (x,y) pairs. Five (X,Y) sets implies 10 pairs (of X,Y pairs). 10 is the "triangle number" for 4 = (n-5), see http://en.wikipedia.org/wiki/Triangular_number; i.e., there are 10 pairs in the upper/lower matrix not including the diagonal. If you see the triangle, then n*(n-1)/2 will make sense! I love triangle numbers, I think I first encountered them in second grade and that's about the time I fell in love with math ...

My XLS is here https://www.dropbox.com/s/1f510upcpnp7ide/MR-9-kendalls-tau.xlsx?dl=0

He writes "The pairs {( 1,4),( 3,3)}, {( 2,5),( 3,3)}, {( 3,3 ),( 4,1)}, and {( 3,3),( 5,2)} are neither concordant nor discordant" but I think that's because the formulas in 3.2.3. are mis-stated. {(1,4),( 3,3)} is discordant because the directionality of X(i) and Y(i) differ: 1 < 3 = X1 < X2 but 4 > 3 = Y2 > Y1. I borrowed Carol Alexander's spreadsheet (apparently, it is easy to get this wrong!) http://www.carolalexander.org/phpBB/viewtopic.php?f=4&t=159 which employs the elegant math of the wikipedia entry http://en.wikipedia.org/wiki/Concordant_pair. This helped me grok concordant/discordant (it's a first appearance in the FRM).

Take {(1,4),( 3,3)}. The Excel formula is =SIGN(X2 - X1)*SIGN(Y2 - Y1); if that = 1, it's concordant because they are ranking the same direction. If it's = -1, they are discordant. How can they be neither? If either X2 = X1 or Y2 = Y1, such that the product is zero.

In this way, I get eight discordant pairs (of pairs) and a Kendall's tau of -0.60.

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#### David Harper CFA FRM

##### David Harper CFA FRM
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Gunter Meissner was kind enough to receive my feedback and reply immediately:

"Hi David,

Yes, you are correct. I misdefined the variables. As the spreadsheet says [i.e., he is referring to https://www.dropbox.com/s/1f510upcpnp7ide/MR-9-kendalls-tau.xlsx?dl=0 ]:

He (I ) mistakenly has: A pair is neither concordant nor discordant if x(t) = y(t) or x(t)* = y(t)*
But it should be: neither concordant nor discordant if x(t) = x(t)* or if y(t) = y(t)*

So in the example in the book, there are 2 concordant pairs, and 8 discordant pairs (and no non-concordant or non-discordant ones).

Hence the Kendall Thau Statistic is (2-8)/10 = -0.6.
Thanks for pointing this out!!!
Have a good weekend!
Gunter"

#### cbrach

##### New Member
Hi David,

I was wondering why the numbers in columns NC and ND do not sum to 2 and 8? The attached file contains a spreadsheet with my calculation of Kendall's Tau using your example data.

Best regards,
Connor

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#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @cbrach Right, thank you, yours looks more efficient at first glance. Mine double counts (because i didn't want to worry about stepping down the comparison; if you notice, mine are based on a blunt rectangular matrix). So, for example, with respect to the first two pairs--ie., (1,4) and (2,5) which are concordant because 2>1 and 5>4--your approach correctly scores +1 concordant pair (in your cell F3). Then you don't include the first pair in your next countif(.) function at F4. Mine is less efficient, you might notice my colored square double-counts this concordant pair in matrix-like fashion; i.e., 1.0s at L4 and K5, just where you'd expect duplicates in a triangle. But knowing that I am exactly double-counting, then I just divide by two in the final count. Yours looks correct at first blush! I think a lot about XLS design and because spreadsheet error is so commonplace, I definitely sometimes end up with a less efficient design because i get obsessed with robustness. Or, in this case, far less efficient because yours is much more intuitive! Thanks!

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#### cbrach

##### New Member
Greatly appreciate the quick response. Thanks again.

#### Ali Ehsan Abbas

##### New Member
There seems to be inconsistency on treatment of pairs neither concordant nor discordant.

My understanding is any pair (x,y) where x=y fits into the "neither" category.

Can someone or @David Harper CFA FRM please confirm.

Thanks.

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Ali Ehsan Abbas I moved your question to this thread. See above. I hope that helps, thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
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Thank you to @dlrm for sharing here https://www.bionicturtle.com/forum/...armans-and-kendalls-meissner.8220/#post-52499 this example of somebody replicating the mistake in Meissner's original text.

Here is the setup given:

But this set contains:
• 7 concordant pairs, and
• 3 discordant pairs, and
• Zero "neither" pairs. The set of pairs {(1,2), (5,5)} is concordant because 5>1 and 5>2. An example of a pair that is "neither concordant or discordant" is {(1,5), (2,5)} or also "neither" is {(5,2), (5,1)}.
• So the kendall's Tau is +0.40 per below (XLS is here https://www.dropbox.com/s/rth0hgje5h2u25o/0928-kendalls.xlsx?dl=0 )

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#### Karim_B

##### Active Member
Subscriber
Thank you to @dlrm for sharing here https://www.bionicturtle.com/forum/...armans-and-kendalls-meissner.8220/#post-52499 this example of somebody replicating the mistake in Meissner's original text.

Here is the setup given:

But this set contains:
• 7 concordant pairs, and
• 3 discordant pairs, and
• Zero "neither" pairs. The set of pairs {(1,2), (5,5)} is concordant because 5>1 and 5>2. An example of a pair that is "neither concordant or discordant" is {(1,5), (2,5)} or also "neither" is {(5,2), (5,1)}.
• So the kendall's Tau is +0.40 per below (XLS is here https://www.dropbox.com/s/rth0hgje5h2u25o/0928-kendalls.xlsx?dl=0 )
Hi @David Harper CFA FRM
This erroneous example came up again in the whatsapp group.

GARP is aware of the correct approach you mention above right?

Also is there an erratum link for Meissner's book mentioning it? I had a quick look online but didn't find one.

Thanks!
Karim

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Karim_B

Yes we definitely notified GARP (we notify GARP of all material errors, but candidly it sometimes seems to take time and repetition for some of them to permanently register ....)

Agreed, I just now checked the book's webpage at https://www.wiley.com/WileyCDA/Section/id-819643.html and I do not see any errata.

Thanks, David

#### Karim_B

##### Active Member
Subscriber
Thanks @David Harper CFA FRM
GARP confirmed they're aware of the Meissner error with Kendall's Tau.

From: Customer Cases <memberservices@garp.com>
Sent: Monday, April 23, 2018 9:12 PM
Subject: Re: Kendall's Tau calculation for FRM exam [ ref:_00D408wgM._5001W1GqxMJ:ref ]

Dear Karim,

Thank you. We are aware of the issue in the Meissner reading you have pointed out and consider it when constructing any questions that may appear on the FRM. "

Sincerely,
Denise Nikitopoulos

Member Services Team
Global Association of Risk Professionals (GARP)
111 Town Square Place, 14th Floor
Jersey City, NJ 07310
t: +1.201.719.7210
f: +1.201.222.5022
http://www.garp.org
Creating a culture of risk awareness®

--------------- Original Message ---------------
Sent: 4/21/2018 1:48 AM
To: memberservices@garp.com
Subject: Re: Kendall's Tau calculation for FRM exam [ ]

Thanks Denise
David said he's already flagged it to GARP, but since some of the other test prep providers are still using the incorrect formula I wanted to make sure the correction will be used in any FRM exam questions.

Karim
________________________________
From: Customer Cases <memberservices@garp.com>
Sent: Saturday, April 21, 2018 3:25 AM
Subject: RE: Kendall's Tau calculation for FRM exam [ ]

Dear Karim,

Thank you for bringing the matter to our attention. We will be reviewing it internally.

Sincerely,
Denise Nikitopoulos

Member Services Team
Global Association of Risk Professionals (GARP)
111 Town Square Place, 14th Floor
Jersey City, NJ 07310
t: +1.201.719.7210
f: +1.201.222.5022
http://www.garp.org
Creating a culture of risk awareness®
--------------- Original Message ---------------
Sent: 4/20/2018 1:47 PM
To: support@garp.com
Subject: Kendall's Tau calculation for FRM exam

Hi there
David Harper at Bionic Turtle pointed out an error in the Meissner reading regarding the calculation of Kendall's Tau for pairs that are neither concordant nor discordant.

This is from:
Gunter Meissner, Correlation Risk Modeling and Management (New York, NY: John Wiley & Sons, 2014).

And learning objective:

Chapter 3. Statistical Correlation Models—Can We Apply Them to Finance? [MR–8]
• Assess the Pearson correlation approach, Spearman’s rank correlation, and Kendall’s ?, and evaluate their limitations
and usefulness in finance.

Here is the link to the BT forum which includes the correction:
https://www.bionicturtle.com/forum/...-concordant-discordant-pairs.8209/#post-33064<https://www.bionicturtle.com/forum/...-concordant-discordant-pairs.8209/#post-33064>

Quote from that post:
"Hi David,

Yes, you are correct. I misdefined the variables. As the spreadsheet says [i.e., he is referring to https://www.dropbox.com/s/1f510upcpnp7ide/MR-9-kendalls-tau.xlsx?dl=0 ]:

He (I [] ) mistakenly has: A pair is neither concordant nor discordant if x(t) = y(t) or x(t)* = y(t)*
But it should be: neither concordant nor discordant if x(t) = x(t)* or if y(t) = y(t)*

So in the example in the book, there are 2 concordant pairs, and 8 discordant pairs (and no non-concordant or non-discordant ones).

Hence the Kendall Thau Statistic is (2-8)/10 = -0.6.
Thanks for pointing this out!!!
Have a good weekend!
Gunter"

Could you please confirm that any FRM exam questions will take this correction into consideration since Meissner's original text still contains the error?

Thanks!
Karim

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @Karim_B Thank you for updating me on this, it's good to know they've incorporated (as they should have: we alerted them about this over three years ago! ... I don't think the book has issued an errata, to my knowledege)

#### Flashback

##### Active Member
Does anybody have some advanced approach how to quickly define concordants and disconcordants in Kendall? It's easy to make counting error and burn out a precious time.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Flashback I moved to my original post (where I identified a mistake in Meissner; I don't know if it has been fixed in the GARP readings yet?). Because I am visual, my preferred approach is illustrated below, see extract below; for me personally, this is relatively quick/easy (from WIR @ https://www.bionicturtle.com/forum/threads/week-in-risk-ending-august-28th.9813/post-44818 ). Also some discussions here:
I also wanted to share an update to my attempt to visualize the concept of concordant/discordant pairs. As mentioned above, and confirmed by the author himself, the Meissner text contains an error. For example, he writes that {(1,4), (3,3)} is neither, but the pair {(1,4), (3,3)} is actually discordant. Visually, concordance is easily revealed (see my new diagrams below): simply draw a line connecting the (Cartesian) points. If the slope is positive, the pairs are concordant. If the slope is negative, they are discordant. Easy, yes?

Or, equivalently, pairs are concordant if their relationship is captured in one of the green quadrants; they are discordant if their relationship falls into one of the red quadrants. In my diagrams, the X and Y axes are given by thick black lines. {(1,4), (4,7)} is concordant because (4,7) is "up and to the right" of (1,4). But {(1,4), (3,3)} is discordant because (3,3) is "down and to the right" of (1,4). {(1,4), (5,4)} is neither because they have a horizontal relationship; although not displayed, just for example, {(1,4), (1,7)} is also neither because they have a vertical relationship. I hope that helps solidify these concepts!

#### sparty

##### New Member
shouldn't this discussion fall under P2.T5? Unless of course the P1 syllabus has been updated and I've missed that update. Thanks.

Staff member
Subscriber
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#### Nicole Seaman

Staff member
Subscriber
HI @sparty Yes, that was a typo, it should refer to P2.T5 (edited above). Thank you for noticing! We do have an Error thread for each topic (e.g., , P2.T5 Errata at https://www.bionicturtle.com/forum/threads/errors-found-in-study-notes-p2-t5-market-risk.8758/) but these refer to our own errors, yet this is an error in the source Meissner. Although @Nicole Seaman maybe it does make sense to have a reference in P2.T5 ?
@David Harper CFA FRM

There is a comment from you regarding this in our T5 error notes thread here: https://www.bionicturtle.com/forum/...study-notes-p2-t5-market-risk.8758/post-44681. That post links to this thread.

Thanks!
Nicole

#### Jacques

##### New Member
It’s incredible that even Schweser blindly repeats such errors, including the Practice Exam. When we compare pairs (xt,yt) and (xt*,yt*) concordancy or discordancy is of course established by comparing each time xt with xt* and yt with yt*. The text explanation in Meissner is also wrong. Intuitively it must be wrong because if we had a series where for each t, xt = yt, Kendall tau should expected to be 1 (if all xt are different). In the Meissner text that would lead to Kendall tau of zero, which has to be wrong.