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# Errors Found in Study Materials P1.T4. Valuation & Risk Models

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @JulioFRM are you sure, it looks okay to me (although note the unusually high Rf rate assumption of 12%): if Rf = 12.0% and T = 0.25, then p = (a-d)/(u-d) = [exp(.12*0.25) - 0.9]/(1.1-0.9) = 0.130455/0.2 = 0.65227267. Let me know ... Thanks!

#### JulioFRM

##### Member
oh yea it's right, I was putting a - in the 12%, thanks a lot.

#### QuantFFM

##### Member
Hi David,

this is from your Study Notes page 21 on Allen.
If K=100 how can there data points which are 165 periods ago?

I thought we have 250 data points and just look back at the 100 recent and these 100 recent data points we sort descending and do our computations on them.
So the oldest can be max. 100 periods ago?

Alphabet (Ticker: GOOG)
Exxon Mobil Corp (Ticker: XOM)
T 250
lambda, λ 0.98
K 100

Periods Hybrid (EXP)
Return Ago Weight Cumul
-5.46% 165 0.07% 0.07%
-5.06% 220 0.02% 0.10%
-3.87% 121 0.18% 0.28%
-3.58% 234 0.02% 0.29%
-3.51% 218 0.03% 0.32%
-2.94% 24 1.26% 1.58%
-2.88% 232 0.02% 1.60%
-2.66% 126 0.16% 1.76%
-2.64% 219 0.02% 1.79%
-2.41% 22 1.32% 3.11

Thank you a lot, Regards

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @QuantFFM Good catch, my mistake (apologies)! I just examined the XLS (it's in the learning sheet at https://www.bionicturtle.com/topic/learning-spreadsheet-allen-chapters-2-3/ ). The K = 100 is not used at all; it is just mistakenly displayed on the chart. Per the text ("we retrieved daily returns for the one year ending") the historical window is 250 days. And the 250 days is used to determine the hybrid weights. For example, the worst return of -5.46% is weighted according to (1-0.98)*0.98^(165-1)/(1-0.98^25) which is a product of the typical (1-λ)*λ^(n-1) = (1-0.98)*0.98^(165-1) and 1/(1-0.98^250) which is a slight upward multiplier that "trims" the total weight of 250 (truncated from an infinite series) days from 99.36% to a full 100.0% based on the 250 day window.

Re: "I thought we have 250 data points and just look back at the 100 recent and these 100 recent data points we sort descending and do our computations on them. "
No, my "uninvited" K = 100 is maybe confusing you. There is only a historical window (in this example, the recent T = 250 daily returns). They are sorted and weighted. And you can see from the weight example above, the only inputs to the weight are the actual day (and, if we are trimming up to 100.0% which is optional, then the 250 enters). I hope that explains, thank you again!

cc @Nicole Seaman I'll need to fix this

Note 04/02/19: This has been fixed in the PDF

Last edited by a moderator:

#### theapplecrispguy

##### New Member
Hi David. On Tuckman Chapter 4 -- page 74 of your notes, I cannot reconcile your spreadsheet with your description when you Compare and contrast DV01 and Duration for 2 1/8 bonds yielding 2.092%. Specifically your description in the 7th bullet on page 74 "divides by the price times one plus half the yield" and the Mac convexity description is similar. Your spreadsheet and hence the results on page 73 don't seem to include the "one plus half the yield" factor. Can you please explain. Thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @theapplecrispguy Sorry that's just a mistake in the text (cc @Nicole Seaman ) We've somehow inserted all those extra words uggh , it should be much simpler:

• [(0.75%/2 × 0.99925) + (0.75%/2 × 0.99648) + ((1.0 + 0.75%/2) × 0.99648)] * $100.00 = 1.00255 *$100.00 = $100.255 #### Maxim Rastorguev ##### Member Subscriber David, referring to this video 25:50 you went to realzed forward scenario in yields. As stated this meant that previous forwards now became spot. But when you calculated new price of the bond , it seemed you still treated rates as forward, not spot. Specificaly you discounted the final bond cash flow with both first rate and second rate (see picture). But if these are all spots, shouldnt we simply apply each rate for each cash flow? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @Maxim Rastorguev I mispoke (I do apologize) when I said "spot rates" plural, when I should have said singular spot rate. The Realized Forward scenario reflects the "realization" of current forward when we move forward in time. In the example shown: 1. On 5/30/10, for the bond that matures in 11/31/11, there is an initial forward 1.5 year forward rate curve give by {0.193%, 0.600%, 1.080%}. Using these forward rates, it is priced at$100.190. Please note at this time, the six month forward rate, F(0.5, 0.5) = 0.60% and the one-year forward rate, F(1.0, 0.5) = 1.080%.
2. Then we go forward in time six months, to 11/30/11 when the bond has only one year to mature. We can make different assumptions but the scenario assumption of Realized Forwards implies:
• The previous six-month forward rate of 0.60% becomes the six-month spot rate (note: this can also be called the six-month forward rate starting immediately)
• The previous one-year forward rate of 1.080% becomes the current six-month forward rate; hence the "realization" of the forward rates. The one-year bond (on 11/30/11) is priced with these two rates; I meant to say the 0.60% becomes the six-month spot rate, and the previous one-year forward becomes the six-month forward rate. They do not all become spot rates; rather, the pricing mechanics is exactly the same as in the initial step, we've just moved forward six months so there are only two cash flows to discount. I hope that's helpful, thanks!

#### Maxim Rastorguev

##### Member
Subscriber
Thank you, it is clear now!

#### Sixcarbs

##### Member
Subscriber
Amenc Chapter 4

It says under Sharpe and Jensen:

"Replacing this value of in the Jensen’s equation and dividing by σ(p) we get"

I think it should be "dividing by σ(m)."

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Sixcarbs Actually, it currently looks correct to me:

Given Jensen's E(Rp) - rf = α + β(p,m)* E(Rm - rf),
And assuming ρ(p,m) = 1.0 in a well-diversified portfolio s.t. β(p,m) = σ(p)/σ(m):
E(Rp) - rf = α + [σ(p)/σ(m)]*E(Rm - rf). Now divide by σ(p):
[E(Rp) - rf]/σ(p) = α/σ(p) + [σ(p)/σ(m)]*E(Rm - rf)/σ(p) = α/σ(p) + [σ(p)/σ(m)]*E(Rm - rf)/σ(p) =
[E(Rp) - rf]/σ(p) = α/σ(p) + E(Rm - rf)/σ(m). Thanks,

#### Sixcarbs

##### Member
Subscriber
@David Harper
Sorry, I am 1,000% wrong here. The notes are fine.