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# GARP.FRM.PQ.P1FMP

#### [email protected]

##### New Member
You are asked to find the price of the US Treasury note. The following table gives the prices of two out of three US Treasury notes for settlement on August 30, 2012. All three notes will mature exactly one year later on August 30, 2013. Assume annual coupon payments and that all three bonds have the same coupon payment date.
COUPON PRICE
5% 97.5
7% ?
8% 103.2
Approximately what would be the price of the 4 1/2 US Treasury note?
a. 99.64 Incorrect
b. 98.20 Incorrect
c. 98.64 Correct
d. 100.20 Incorrect

#### Deepak Chitnis

Subscriber
Hi @Namrata2001, I tried to solve the question but I am not 100% sure. first we need to find the value of 7% coupon treasury note. Like, 5%*X+8%*(1-X)=7%, after solving this X=0.33333 and 1-X=1-0.33333=0.66667. Then find the value, $97.50*0.33333+$103.2*0.66667=$101.29968. Then use the TVM function on calculator to find the value of 4 1/2 us treasury note, N=1, I/Y=4.5, PV=101.29968, PMT=7, CPT------->FV, then FV=98.85, it is not exact but I think @David Harper CFA FRM CIPM, can elaborate more. Thank you ! #### Dr. Jayanthi Sankaran ##### Well-Known Member Hi @Deepak Chitnis , I like your approach of creating a barbell portfolio of 5% coupon at$97.5 and 8% coupon at 103.2, to equate the bullet. Infact, this was what crossed my mind in approaching this problem. However, I was not very sure. However, the thing I don't understand in your TVM calculation for the 4.5% US T-note, is N = 1, I/Y = 4.5. As far as I know, 4.5% is the coupon rate and not the YTM. I could be mistaken Also, two more things I don't understand: PV = 101.29968 that you use as the price for the 4.5% US T-note and PMT = 7. It seems like there is a little muddle here in using the 7% rate and also the 4.5% rate. I could be wrong - please correct me if I am!

Thanks a tonne Jayanthi

#### Deepak Chitnis

Subscriber
Hi @Jayanthi Sankaran, I dont know the exact answer, it's just a try,$101.29968 it is just the value of 7% coupon bond which is derived as 5%*X+8%*(1-X)=7% after solving this X=0.33333 and 1-X=1-0.33333=0.66667. Then find the value,$97.50*0.33333+$103.2*0.66667=$101.29968 and $101.29968 is the just present value of 7% coupon bond. As I said earlier @David Harper CFA FRM CIPM will elaborate more. Hope that help. Thank you ! #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber This is a variation on a classic exam-type questions; for example https://www.bionicturtle.com/forum/threads/question-18-bonds-valuation.2110 and my own 309.3 @ https://www.bionicturtle.com/forum/threads/p1-t4-309-discount-factors-and-law-of-one-price.6848 Where the approach is exactly as @Deepak Chitnis describes (except with dollars): solving for X in 5x + 8(1-x) = 7, which is really X in 5x + 8(1-x) = 7*100% because you would expect that buying 100% of the par value of the$7 coupon would produce the same cash flow as allocating the same purchase price between the two other bonds. I mean, X solves for the two-bond portfolio that returns the same cash flows as owning 100% of par (i.e., 100% of the current price not 100) of the $7 coupon bond, which is simply a single cash flow of$107 in exactly one year. More simply even: an equivalent purchase either way--i.e., 100% of 7 coupon or allocated between the other two bonds--should return us $107 in one year. Using that approach, I too would get$101.30 for the answer. It is already a present value under this approach, so I don't see the reason for subsequent PV calculation (So i don't understand the given answer)

But last year hoangu90 identified a problem with this approach (see https://www.bionicturtle.com/forum/...factors-and-law-of-one-price.6848/#post-29277)
Namely, isn't the setup a violation of the law of one price:
• For the 5 coupon, the implied one year discount factor is 97.50/105 = 0.929, but
• For the 8 coupon, the implied one year discount factor is 103.20/108 = 0.956. Seems like a problem ...

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#### [email protected]

##### New Member
what is the concept of 4 1/2 over here? n i am not able to calculate it

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @Namrata2001,

You need to calculate in the same fashion as @Deepak Chitnis and David have calculated above:

5X + 8(1-X) = 4.5
X = 1.66667
1 - X = -0.66667

In this situation, the long 5% coupon plus short 8% coupon replicate the 4.5% coupon. However, the price is:

1.6666*$97.5 + (-)(0.66667*$103.2) = $93.693 Thanks! Jayanthi #### Priyanka_Chandak23 ##### New Member @Jayanti Ma'am : 5X +8(1-X) = 4.5 gives us X=1.16667 and 1-X= 0.16667 This gives us , 1.16667*(97.50)-0.6667(103.20)=96.55 . Kindly help me with your calculation, I am not being able to understand the same. Also David suggests that it has we need to equate the coupon payments to 7 while you have equated them to 4.5. Kindly help. Thanks. #### Dr. Jayanthi Sankaran ##### Well-Known Member Hi Priyanka, 5X + 8(1-X) = 4.5 5X + 8 - 8X = 4.5 -3X = -8 + 4.5 = -3.5 X = 3.5/3 = 1.1667 1-X = 1 - 1.1667 = -0.1667 This gives us: 1.1667*(97.50) + (-0.1667)*(103.2) = 96.55 Yes, you are right - my mistake Instead of 1.1667, I wrongly typed in 1.6667. Thanks for pointing that out! What David is saying pertains to getting the price of the 7% coupon bond. The 7% coupon bond is equivalent to an X% long position in the 5% coupon bond and (1-X)% long position in the 8% coupon bond i.e. 5X + 8(1-X) = 7 (Here, he has chosen to use$ values instead of % for X - the results are the same!)
5X + 8 - 8X = 7
-3X = -1
X = 1/3 = 0.3333
1-X = 1-0.3333 = 0.6667
Hence, price of 7% coupon bond:
(0.3333*$97.5) + (0.6667*$103.2) = $101.30 Hope that helps! Last edited: #### Priyanka_Chandak23 ##### New Member @Jayanthi Ma'am : So, the answer to this question must pertain to the price of the coupon bond which is 101.30, right? Also, this question is flawed as pointed out by David but in an ideal situation we should equate the X% long position in the 5% coupon bond and (1-X)% long position in the 8% coupon bond as shown by you. Please confirm the same. Thanks a ton. #### Dr. Jayanthi Sankaran ##### Well-Known Member Hi Priyanka, There are two parts to this question: (1) Finding the price of the 7% coupon bond which is$101.30, and
(2) Finding the price of the 4.5% coupon bond which is $96.55* *Please note that you can get the$96.55 price for the 4.5% coupon bond whether you go,
*Long 5% coupon plus short 8% coupon or
*Long 5% coupon plus short 7% coupon

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I agree with @Dr. Jayanthi Sankaran 's approaches. I guess we have two versions of the problem: one solving for the price of a 4.5% coupon, one solving for the price of a 7.0% coupon. The unmentioned problem is that the question has a design flaw: it violates the "law of one price" which requires the one-year discount factor to be the same for all three bonds. Specifically, let's say the 5 bond is accurately priced; if so, the one year discount factor, df(1.0), is given by $97.50/105.00 = 0.92857. In this case, the price of the 8 coupon bond must be 0.92857 *$108.00 = $100.286. Then we can solve for either 7 or 4 /12 bond in the two scenarios below which are "internally consistent": •$5.00 coupon bond priced at $97.50 •$7.00 coupon bond priced at ???
• $8.00 coupon bond priced at$100.286
Or for that matter we can solve for:
• $5.00 coupon bond priced at$97.50
• $4.50 coupon bond priced at ??? •$8.00 coupon bond priced at $100.286 We can still solve per the intended method. For example, in the case of the 4 1/2 coupon, X = position in 5 coupon bond = 116.67% (see Dr Jayanthi's above), such that price of 4 1/2 bond =$97.036. The law of one price is here satisfied because 97.036/104.50 = 0.92857; ie, same discount factor. That's why this flaw was identified: you should have a quicker method to getting the price, per the shared discount factor, with 0.92857*104.50. But notice also that given this simple scenario we can also just use a simple ratio given by 104.50/105.00 * 97.500 = \$97.036 (both bonds are not really required). This question type is not well designed. BTW, GARP is aware, we reported this carefully at the time (2 years ago) when a customer pointed it out. I hope that helps!

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#### Priyanka_Chandak23

##### New Member
Thanks David and Dr.Jayanthi .
I think the main reason for my confusion was the flaw in the question with respect to the no parity of the discount factors across the bonds. But I think its solved now.