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# Errors Found in Study Materials P2.T5. Market Risk (OLD thread)

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#### Part 2 Student

##### New Member
Meissner, Chapter 1, P21, the formula below seems erroneous: The square root sign should stop right before the very last "+" sign. In other words, "+PxPy" should be placed outside the square root sign. Could either Nicole or David please confirm this?

#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Part 2 Student Yes, our mistake, apologies. Thank you for noticing this! Cov(XY) = E(X*Y) - E(X)*E(Y) --> E(X)*E(Y) = Cov(XY) + E(X)*E(Y), where in the case of two binomials Cov(XY) = ρ(XY)*Std(X)*Std(Y) = ρ(XY)*sqrt[P_x*(1-P_x)]*sqrt[P_y*(1-P_y)] = ρ(XY)*sqrt[P_x*(1-P_x)]*[P_y*(1-P_y)] such that E(X)*E(Y) = ρ(XY)*sqrt[P_x*(1-P_x)]*[P_y*(1-P_y)] + E(X)*E(Y). cc @Nicole Seaman: it is exactly as @Part 2 Student says: the final "+PxPY" needs to come out from under the square root. Thanks!

#### emilioalzamora1

##### Well-Known Member
Hi,

most probably or rather certainly there is a formatting issue here with the formula for the sample std. deviation that you have copied.

Please see the original book by Meissner on page 6 where it says that the sqaure root spans over the whole term.

The sample standard deviation of asset 'x' equals the square root across the whole right-hand side of the equation over
'1' divided by 'n-1
' multiplied by squared difference between the return for a given period (t) and the mean return for the sample period.

The n-1 in the denomintaor ensures that we get an unbiased estimator of the true std. deviation.

My favourite book 'Econometric models' by Pindyck & Rubinfeld' gives an intuitive explanation (in case it is necessary) for why we need to divide by 'n-1' for the sample std. deviation:

Our sample is known to have 'n' data points, however, in order to compute the sample std. deviation we first need to compute the sample mean. This places a constraint on the 'n' data points that the 'n' observations sum to 'n' times the mean. This leaves 'n-1' unconstrained observations with which to compute the standard deviation.

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#### Nicole Seaman

##### Director of FRM Operations
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Hi David, I believe the below SD formula should not multiply by sqrt(n). Meissner material page 6.
View attachment 1044
Hello @rajeshtr

Thank you for pointing out this error. I've moved your thread here to the Study Notes Error thread, where you should post any errors that you find in the study notes. I will make sure to get this fixed as soon as possible.

Thank you,

Nicole

#### Part 2 Student

##### New Member
Hi Nicole / David,

In Tuckman, Chapter 7, Page 24, should the 2nd bullet point say $925.21 =$1,000/(1+5.25%/2)^3, instead of $925.81 =$1,000 / (1+5.25%/2)^3 ?

Thanks

#### Part 2 Student

##### New Member
Hi Nicole / David,

In Tuckman, Chapter 7, Page 26, should the first equation under "Similarly, on date 2, the state 2, 1, and 0 payoffs are, respectively" end with "=$5,000", instead of "=-$5,000"?

#### Part 2 Student

##### New Member
Hi Nicole / David,

In Tuckman, Chapter 7, Page 26, should the last equation on the page end with "= - $4,216.87", instead of "=$4,216.87"?

#### Part 2 Student

##### New Member
Hi Nicole / David,

In Tuckman, Chapter 8, Page 37, should the paragraph starting "Assume the current one-year spot rate is 10.0%..." end with"14%, 10% or 6%", instead of "14%, 12% or 6%"?

#### Nicole Seaman

##### Director of FRM Operations
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@David Harper CFA FRM

I've confirmed three of the four above errors need to be changed in the pdf. When you get a chance, can you confirm that the last error pointed out above by @Part 2 Student is an error? Thanks! Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
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@Nicole Seaman Yes, these are all mistakes  Re the last, @Part 2 Student is correct, the text should match the nodes under "Year 2" in the tree, so it should read "In this way, the one-year spot rate in the second year will evolve to either 14%, 10%% or 6% (see below)."

Thank you @Part 2 Student!

#### Part 2 Student

##### New Member
Thank you, Nicole and David, for your confirmation. It is always easier to hairsplit, but it was out of a sincere desire to help. I am sorry this has to come piecemeal, but the effort continues.

In Tuckman, Chapter 8, Page 41, should the 3rd line of the paragraph starting "For example, we can add 20 basis points..." say "...the two-year discount factor is $0.826035", instead of "the two-year discount factor is$0.826025"?

#### Nicole Seaman

##### Director of FRM Operations
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Thank you, Nicole and David, for your confirmation. It is always easier to hairsplit, but it was out of a sincere desire to help. I am sorry this has to come piecemeal, but the effort continues.

In Tuckman, Chapter 8, Page 41, should the 3rd line of the paragraph starting "For example, we can add 20 basis points..." say "...the two-year discount factor is $0.826035", instead of "the two-year discount factor is$0.826025"?
@Part 2 Student

No apology necessary! That is the reason that we created these threads because we do not always spot the mistakes, and it helps us greatly when our members let us know when a mistake has been found. I've awarded you stars in this week's forum contest to show our appreciation for finding these mistakes. We strive to make sure that our materials are accurate, but unfortunately, we are only human and mistakes are inevitable. Thank you for continuing to point out any mistakes that you find. I've fixed all of the mistakes listed above, and they will be re-published in the next version of study notes (coming soon!). Nicole

#### Part 2 Student

##### New Member
Thank you, Nicole. Here's another:

In Tuckman, Chapter 9, Page 53, should the penultimate equation on the page be "dr = 2.0%*-0.2533*SQRT(1/12) = -0.14627%", instead of "dr = 3.0% + 2.0%*-0.2533*SQRT(1/12) = -0.14627%"?

#### Part 2 Student

##### New Member
Similarly, In Tuckman, Chapter 9, Page 54, should the penultimate equation in the "Rate change under Model 2" section be "dr = 1.0%*1/12 + 2.5%*0.80642*SQRT(1/12) = +0.6653%", instead of "dr = 4.0% + 1.0%*1/12 + 2.5%*0.80642*SQRT(1/12) = +0.6653%"?

#### Nicole Seaman

##### Director of FRM Operations
Staff member
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Thank you, Nicole. Here's another:

In Tuckman, Chapter 9, Page 53, should the penultimate equation on the page be "dr = 2.0%*-0.2533*SQRT(1/12) = -0.14627%", instead of "dr = 3.0% + 2.0%*-0.2533*SQRT(1/12) = -0.14627%"?

Similarly, In Tuckman, Chapter 9, Page 54, should the penultimate equation in the "Rate change under Model 2" section be "dr = 1.0%*1/12 + 2.5%*0.80642*SQRT(1/12) = +0.6653%", instead of "dr = 4.0% + 1.0%*1/12 + 2.5%*0.80642*SQRT(1/12) = +0.6653%"?

@David Harper CFA FRM AYC can you confirm the above two are errors please? Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
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@Part 2 Student Thank you so much for your attention to detail! Uggh too many errors here @Nicole Seaman Yes, @Part 2 Student is correct on both above errors!

#### Martin B

##### New Member
R37.P2.T5 , Page 5 -> definition of Monotonicity: R(L1) < R(L2) if L1 < L2

R35.P2.T5, Page 6 -> definition of Monotonicity: If X ≤ Y then rho(Y) ≤ rho(X)

I am wondering if this is correct or a error. #### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Martin B Yes, those are both faithful to their respective sources. This has been previously discussed; e.g., https://www.bionicturtle.com/forum/threads/coherent-risk-measure-monotonicity.8214 or https://www.bionicturtle.com/forum/threads/coherent-risk-measure.4723 ... basically, while R37 perhaps utilizes the more intuitive (and common) usage which holds that if the expected loss, L2, is greater than L1, its associated risk measure R(L2) should be greater than R(L1), whereas Dowd is actually strictly following the original authors of the coherent criteria (paper at http://trtl.bz/artzner-98-coherence) where Dowd makes gives the following interpretations: "Monotonicity means that a random cash flow or future value Y that is always greater than X should have a lower risk: this makes sense, because it means that less has to be added to Y than to X to make it acceptable, and the amount to be added is the risk measure;" while this has nevertheless been subject to some further debate over the years on this forum, I think the easiest resolution is to view this version as if the expected future value (Y) is greater than (X), then Y's risk measure should be less. (I will be honest, I am not convinced this is a proper interpretation of the original paper ....). Thanks!

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#### Martin B

##### New Member
Thank you very much @David Harper CFA FRM I completely missed the fact that here L means a loss while Dowd is working with expected future values. Now I understand where the difference comes from.

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