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# Errors Found in Study Materials P1.T4. Valuation & Risk Models (OLD thread)

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#### Sixcarbs

##### Active Member
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Hi David,

I hope this is the right place for this. There is an error in the video on Hull Chapter 15. You say things correctly but the slide you are pointing at is incorrect.

It is in Hull's example 15.3

Slide says:

.17-(.2/2) = .15

.2 is the volatility, should be variance (.2)^2. Answer .15 is correct.

.17-[(.2)^2]/2= .15

Then below it slide says:

Standard deviation,
sqrt(.2/3)= 0.1155

Should be .2/sqrt(3)= 0.1155

Also, in Study notes, Hull Chapter 13, Binomial Trees. Page 6.

"The risk-neutral probability (= 0.6523) is found first."

I think it should be .5503, which it is in the spreadsheet below. I looked high and low for a way to come up with .6523 and could not calculate it or find a place for it.

I hope this helps.

Sixcarbs

#### tattoo

##### Member
Hi david.
In page 6 of R27-P1-T4, it says "The risk-neutral probability p= 0.6523". However, according to and I came up with the result of p is 0.5503 which is consistent with the result of the spreadsheet showed below. So where does "p= 0.6523" come from?
In addition, does the symbol "<" in the picture above(< probability of up jump) mean "less than"? If so, why is p less than probability of up jump? Many thanks.

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#### samuelfu9999

##### New Member
Hi David,

For 818.1 of the Practice Questions of R27 (P.93), the d1 of rate of
change of the option price with respect to the futures price should be ln[F(0)/K] + σ^2*T/2 )/ [σ*sqrt(T)].

For 818.3, it is missing in the pdf.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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@tattoo (cc: @Nicole Seaman )

1. You and @Sixcarbs are correct, thank you! There is a typo in the R27 Study Note, page 6, in the 3rd paragraph, instead of "(p = 0.6523)" it should read "(p = 0.5503)" to match the exhibit, as follows:
"The risk-neutral probability (p = 0.5503) is found first. The option prices at the final nodes are calculated as payoffs from the option. At the topmost node of time period two (T=0.5), the option value () is 3.2 (stock price of 24.2 minus strike price of 21). In the middle and lowermost nodes, the option is out of the money and so its value is zero (for and )."

2. In your binomial, it looks like you are discounting with exp[-(r-q)]; e.g., your 101.1160 = [(0.5126 * 189.3362) + (0.4874 * 10.000)] * exp[-(5.0% - 2.0%)*0.25] but you want [(0.5126 * 189.3362) + (0.4874 * 10.000)] * exp(-5.0%*0.25) = 100.6614; i.e., this is just a discounting function, you don't subtract the dividend yield here.

@samuelfu9999 Yes, totally agree thank you! (@Nicole Seaman it looks like 818.3 is missing ....)

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#### Nicole Seaman

##### Director of FRM Operations
Staff member
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Hi David,

For 818.3, it is missing in the pdf.
Hello @samuelfu9999

Thank you for pointing out that 818.3 is missing. I've fixed it in the PDF and will upload the corrected version to the study planner. In the future, when you find any errors in the practice questions, please comment directly in the PQ forum thread, as this thread is just for study note and video errors. The PQ forum threads are on the answer pages in the PDF documents.

@David Harper CFA FRM

Can you also let me know if samuelfu9999 is correct about 818.1? There is some discussion about this in the original thread, but I want to make sure I'm understanding it correctly before I change anything.

For 818.1 of the Practice Questions of R27 (P.93), the d1 of rate of
change of the option price with respect to the futures price should be ln[F(0)/K] + σ^2*T/2 )/ [σ*sqrt(T)].

Thank you,

Nicole

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @tattoo The final cash flow includes two components, the $100.00 par plus the coupon. The PV of the final cash flow is given by (100 + 100 × 0.75%/2) × 0.99135; i.e., the final (FV) cash flow is return of the$100.00 plus the coupon amount of 0.75%/2*100. However the formula nevertheless contains two typos (cc @Nicole Seaman this refers to R28 Study Notes page 6 ["Define the “law of one price”, explain it using an arbitrage argument, and describe how it can be applied to bond pricing"] such that the two indented (i.e., 2nd level) bullets should read:
• ($100.00 × 0.75%/2 × 0.99925) + ($100.00 × 0.75%/2 × 0.99648) + ($100.00 +$100.00 × 0.75%/2) × 0.99135 = $100.255, or • [(0.75%/2 × 0.99925) + (0.75%/2 × 0.99648) + ((1 + 0.75%/2) × 0.99135)] *$100.00 = 1.00255 * $100.00 =$100.255

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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@tattoo yea, it's confusing because the (t) has different roles: the exponent should be 2*Δt = 2*0.5 = 1.0, where Δt = 2.5 - 2.0; or the exponent can just be omitted altogether because it is just 1.0 in this case were we are retrieving a six-month forward rate under semi-annual compounding (our forward rate spans exactly one period). Although the logic isn't wrong. The second step is correct application if we understand the exponent, 2t, is really 2*Δt. I will modify this on later revision. Thanks,

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @pbhalava Yes, the use of L/P versus P/L should not change the VaR. In the pasted example above, the 95.0% 10-day absolute VaR = [-9.0%*(10/250) + 20.0%*sqrt(10/250)*1.645] * $100.00 =$6.22 regardless of L/P or P/L; in P/L, µ = 9.0% and in L/P, µ = -9.0% is the only input difference.

But as the text explains, the comparison above is between absolute VaR versus relative VaR: the relative VaR is 20.0%*sqrt(10/250)*1.645 * $100.00 =$6.58, which is greater (than absolute VaR) by exactly the drift because it omits the drift (and the relative VaR is just scaled volatility so it really doesn't care about L/P versus P/L). Relative VaR is the worst expected loss relative to the expected future value, and the expected future value is 9%*(10/250)*100 = $0.36 higher because that is the (positive) expected drift over the next ten days. The absolute VaR is less by the drift because it is the worst expected loss relative to today's position, so it's "giving credit" to the VaR for positive drift. The absolute VaR is the return-adjusted risk measure, is how I like to think about it; the relative VaR doesn't credit the drift (the return). So I don't see an error in the above, let me know if you still do? Thanks, #### pbhalava ##### New Member Hi @pbhalava Yes, the use of L/P versus P/L should not change the VaR. In the pasted example above, the 95.0% 10-day absolute VaR = [-9.0%*(10/250) + 20.0%*sqrt(10/250)*1.645] *$100.00 = $6.22 regardless of L/P or P/L; in P/L, µ = 9.0% and in L/P, µ = -9.0% is the only input difference. But as the text explains, the comparison above is between absolute VaR versus relative VaR: the relative VaR is 20.0%*sqrt(10/250)*1.645 *$100.00 = $6.58, which is greater (than absolute VaR) by exactly the drift because it omits the drift (and the relative VaR is just scaled volatility so it really doesn't care about L/P versus P/L). Relative VaR is the worst expected loss relative to the expected future value, and the expected future value is 9%*(10/250)*100 =$0.36 higher because that is the (positive) expected drift over the next ten days. The absolute VaR is less by the drift because it is the worst expected loss relative to today's position, so it's "giving credit" to the VaR for positive drift. The absolute VaR is the return-adjusted risk measure, is how I like to think about it; the relative VaR doesn't credit the drift (the return). So I don't see an error in the above, let me know if you still do? Thanks,

Thank you for the explanation. I thought, the values coming in the green highlighted portion should be the same as calculated in the paragraph above. But yes, I got the gist of it.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @Elizabeth_Babalola Yes, you are correct, it should be: ES(X) + ES(Y) ≥ ES(X + Y). Thank you for spotting these typos! @Nicole Seaman I have saved correct version (in ROOT) to T4-VRM-1-Ch1-FR-Measures-v3.01 (but this can await next batch; i.e., let's without publish until we have more to edit)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @Elizabeth_Babalola Yes, you are correct about both mistakes. Thank you for your attention to detail. We will fix on revision. In regard to page 32 of VRM-3, the red circled (see below) 0.2s should be 0.020, just as you suggest. And in regard to page 9 of VRM-4, yes the negative sign was mistakenly omitted. Thank you again. Status
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