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# Help [Bootstrapping]

#### laurel akin

##### New Member
I am so sorry to bother you but I really need help. Note: I am new to bonds, interest rates, forward rates .. etc.

In many examples ( for instance in Hull's book) I see something like this:
TTM Coupon Price
0.25 0 97.5
0.50 0 94.9
1.00 0 90.0
1.50 8 96.0
2.00 12 101.6

My question is ( i am a newbie): How do we know the price and the coupon rate of 1.5-year note? (As far as I know there is no 1.5-year bond in the US. ). Do we calculate this? (also 2.5, 3.5 .....)

I will be very grateful if somebody explains this to me.

#### Nicole Seaman

Staff member
Subscriber
I am so sorry to bother you but I really need help. Note: I am new to bonds, interest rates, forward rates .. etc.

In many examples ( for instance in Hull's book) I see something like this:
TTM Coupon Price
0.25 0 97.5
0.50 0 94.9
1.00 0 90.0
1.50 8 96.0
2.00 12 101.6

My question is ( i am a newbie): How do we know the price and the coupon rate of 1.5-year note? (As far as I know there is no 1.5-year bond in the US. ). Do we calculate this? (also 2.5, 3.5 .....)

I will be very grateful if somebody explains this to me.
Hello @laurel akin

While you are waiting for an answer from David or one of our other members, you may find it helpful to use the search function to see the other threads that have discussed these concepts. Our forum has 10 years of discussions covering all of the concepts in the FRM curriculum. If you type Bootstrapping into the search box, you will see that many different threads come up, and they may help to answer your questions. Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @laurel akin Don't be sorry to bother us, we have a forum to help (it gives me and Nicole a break from trying to translate Stulz writing into plain English ). As we are updating the Hull notes, I just recently revised this exhibit (I combined Table 4.3 and Table 4.4 below). Do you have Excel? If so, here is the sheet: https://www.dropbox.com/s/0eb3crugjr72pb4/0331-hull-4-3.xlsx?dl=0 So below I just transposed Hull's Table 4.3

Re: How do we know the price and the coupon rate of 1.5-year note? We are given the price and coupon as an assumption (you are correct there is no 1.5 year "on the run" T-note, T notes are issued with maturities of 2, 3, 5, 7, and 10 years; but a 2-year T-note issued six months ago, when it was on the run, today has a price--when it is off the run--and a term to maturity of 1.5 years). The given assumptions are in yellow (is my style).

This "bootstrap" method relies on the law of one price: each maturity has only one spot rate. So, we start by retrieving the 0.25 year spot rate which must be 10.127% per annum because $97.50*exp(R*0.25) =$100. By the time we get to solving for the 2.0-year spot rate (which is 10.808%) we have already solved for the spot rates at {0.25, .... 1.5}. So we just need the 2.0 rate that discounts the cash flows to the observed market (i.e., given assumption) price of $101.60. If you can open the Excel, I would recommend it. Let us know if you have further questions! Last edited: #### laurel akin ##### New Member I can't thank you enough. I really appreciate your reply. #### Flashback ##### Active Member If:$105.080 = ($5/2 x e - 0.01512/2) + ($100 + $5/2) x e -z2/2 x 2 What is a path to solve for z2? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Assuming the price (PV) is 105.08, the six month spot rate with continuous compounding is 0.01512 and the semi-annual coupon rate is 5.0% (paying$2.5 every six months) which is what your formula seems to reflect:
• if $105.08 = 2.5*exp(-0.01512*0.5) + 102.5*exp(-z*1.0), then: •$105.08 - 2.5*exp(-0.01512*0.5) = 102.5*exp(-z*1.0),
• [$105.08 - 2.5*exp(-0.01512*0.5)]/102.5 = exp(-z*1.0), and take ln(.) of both sides: • ln([$105.08 - 2.5*exp(-0.01512*0.5)]/102.5) = -z, or
• z = -ln([$105.08 - 2.5*exp(-0.01512*0.5)]/102.5) = -0.10%. Plugging this back in, price does work, so something may be wrong with assumptions #### Flashback ##### Active Member Fine! And Thank you on quick and clear explanations as always. I stuck with this yesterday. No, it isn’t wrong. I randomly changed the original numbers because of copyright. Doesn’t matter for plug in test. The path to solution was what I asked for. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @Flashback okay but please do not randomly change (scramble) the numbers. "Fair use" laws (https://en.wikipedia.org/wiki/Fair_use) allow you to quote the actual question. I do not want to support questions that both (i) aren't ours and (ii) are scrambled. Because it will be confusing. Plus, you will end up wasting my time (because I'll try to debug something when there wasn't really a problem). I am okay to support Schweser's questions, when I have time (I actually have a good relationship with them, or maybe I should say, I have a good opinion of them and their products I truly do) but please do us a favor and replicate them accurately with attribution. Last edited: #### Flashback ##### Active Member OK. I keep it to my knowledge. #### Flashback ##### Active Member I tried multiple times but couldn’t reach Schweser’s output. Now, it makes sense. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber @Flashback okay great. But i shared that link to show just one example of another forum visitor who plainly shared the source Schweser (and ended up even helping Schweser make a fix, which is a good thing as the highest priority is candidates' experience). Thanks, #### JamesVU2000 ##### Member 10.127% per annum because$97.50*exp(R*0.025) = $100? How do I back out r in this question? Thanks ! #### berrymucho ##### Member 10.127% per annum because$97.50*exp(R*0.025) = \$100? How do I back out r in this question? Thanks !
I think it's just a typo in post #3 above. The first maturity is 0.25 year (3 months), not 0.025. So ln(100/97.50)/0.25 = 10.127%.

#### JamesVU2000

##### Member
Thanks !
I think it's just a typo in post #3 above. The first maturity is 0.25 year (3 months), not 0.025. So ln(100/97.50)/0.25 = 10.127%.

#### San955

##### New Member
Hi @David Harper CFA FRM

I am trying to figure out when calculating the Discount Factor for 2 year whether I include the Discount Factor from 1 year if I have annual payments? Why would I or would not?

1 = (1+r_2year)*DF_2year +r_2year*DF_1year+ r_2year*DF_6_month

or should be

1 = (1 + r_2year)*DF_2year + r_2year*DF_1year

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