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

#### Elizabeth_Babalola

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#### David Harper CFA FRM

##### David Harper CFA FRM
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Hi @Elizabeth_Babalola Sorry but there should be an additional set of parenthesis in the first formula. It should read:

=(100*0.75%/2*0.99925)+(100*0.75%/2*0.99648)+((100+100*0.75%/2)*0.99135)

... The final cash flow, that includes the principal, is (100 + 100 * 0.75%/2) and that is multiplied by the discount factor; without the parens, my formula adds the $100 to the PV, rather than the discounted$100*0.99135. Apologies, we will add it to revisions .... Thank you for your awesome attention to detail!

Re: "Also the 0.875 cash flow/coupon was omitted from the solution."
Well these two (equivalent) formulas on page 7 are computing the model price of the 1.5 year bond referred to as the 3/4s (0.750% coupon) that expires on 11/30/2011. So it has three cash flows: in six months it pays a 0.750%/2 * $100.00 =$0.3750 coupon; in one year, another $0.3750 coupon, and finally, in 1.5 years$100.3750. Those three cash flows are discounted by multiplying by the respective discount factors: df(0.5) = 0.99925; df(1.0) = 0.99648; df(1.5) = 0.99135. The six-month bond, which we can refer to as "the 7/8s that matures in six months," and pays a 0.875%/2*100 = \$0.43750 s.a. coupon, has no role in pricing the 1.5 year bond. Now, in some applications, we might use the six-month bond to retrieve the six-month discount factor and then "bootstrap" that df(0.5) value to retrieve the 1.0 year discount factor, and finally use both df(0.5) and df(1.0) to infer the 1.5-year discount factor. Such an exercise would enforce, and in fact would be the very definition of presupposing, the Law of One Price because this law says there is only one discount function (set of discount factors) absent confounding factors (so basically, there is only one theoretical riskfree term structure/discount function).

But the point of this page per the label "Testing the Law of One Price" is to assume the previously derived discount function--i.e., df(0.5), df(1.0) and df(1.5)--already exists (i.e., "these bonds do not inform this discount function"). So we're basically assuming the discount function and using the same discount function to price all three bonds, which gets us the "model price" and then we compare that the the "market price" to determine which are "trading rich" or "trading cheap." Alternatively, we could forget the prior knowledge of the discount function, and use these three bonds to generate the implied set of discount factors. But then we wouldn't have a basis for determining "trade rich" versus "trade cheap." A bond trades rich/cheap when the observed (market) price varies from the model price (i.e., discounted per the discount function ) whose discount function is informed by other bonds. I hope that's interesting, maybe I even somehow addressed your second point, lol. Thanks,

#### Cassandra

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Good day, David.

On page 11 of chapter 2 study note you wrote:

"VaR estimated using the delta-normal model tends to overestimate the probability of high option value and underestimate the probability of low option value, which leads to overestimation (underestimation) of the VaR for a short (long) call position".

But it should actually be:
"VaR estimated using the delta-normal model tends to underestimate the probability of high option value and overestimate the probability of low option value, which leads to overestimation (underestimation) of the VaR for a long (short) call position".

Note for everyone:
Looking at the Call option graph in GARP text (Chapter 2 pg. 23), as stock prices move up the call option gains value faster if you draw a tangent line to this graph which is what a delta approximation is then you will have effectively, for high stock prices, underestimated the options value using the delta proxy. While for low stock prices you would have overestimated how low the option would be.

For a long position holder they would need to post more capital as we are overestimating how low the option value is when it is out of the money; which is the risk for a long call option holder (stock price decline). In essence long position holder is assuming things are worse than they actually are.

On the other hand the short call option holder is worried about the stock price going up (to become in the money) because he will lose if that happens but due to the delta underestimating how high the option value will actually be in this case he would thereby post less capital, as he is assuming things are better than they actually are and thus post less than he should.

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
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Hi @Cassandra Yes, I agree, that's an excellent analysis. Thank you, and sorry for any confusion. The numbers and visuals tell the truth, but I know why the words got distorted: there is a classic long-term problem interpreting the first clause: "VaR estimated using the delta-normal model tends to underestimate the probability of high option value and overestimate the probability of low option value." We probably mangled that because the interpretation can difficult to pin down; I will probably eliminate that phrase in favor of something concrete.

But, with respect to the second clause, I completely agree with you: ours is mistake and it should read "which leads to overestimation (underestimation) of the VaR for a long (short) call position". And, here your reference to capital posted is the best way to confirm! Thank you for spotting this ....

#### Cassandra

##### New Member
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Hi @Cassandra Yes, I agree, that's an excellent analysis. Thank you, and sorry for any confusion. The numbers and visuals tell the truth, but I know why the words got distorted: there is a classic long-term problem interpreting the first clause: "VaR estimated using the delta-normal model tends to underestimate the probability of high option value and overestimate the probability of low option value." We probably mangled that because the interpretation can difficult to pin down; I will probably eliminate that phrase in favor of something concrete.

But, with respect to the second clause, I completely agree with you: ours is mistake and it should read "which leads to overestimation (underestimation) of the VaR for a long (short) call position". And, here your reference to capital posted is the best way to confirm! Thank you for spotting this ....
Thanks David #### Erik

##### New Member
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On 903.1, I think the formula in the notes may have a typo.
Page 8:
$\left(1+\frac{f(t-0.5,t)}{2}\right)^{2t} = \frac{d(t-0.5)}{d(t)}$

The exponent on the left should reflect the change in time (0.5), not time value (t = 2.5) so that the exponent evaluates to 1.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @Erik Yes, you are absolutely correct, that is a typo. Thank you!