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# Example Tuckman 2011, page 156, Compute Key-Rate 01...

#### NoPasaran

##### New Member
Hello everyone

I watched David Harper's videos on Key Rate 01, but he uses spot rates, not par rates like in the example of Tuckman.
I have problems understanding that example, maybe someone is kind enough to enlighten me a little?

1.
Why are par rates used as key rates and not spot rates? Is that because par rates (or swap rates) is what we have available in the market and we derive the discount factors / spot rates / forward rates from these par rates?

2.
When we have key spot rates, as David has in his video, it is pretty clear how to value any bond when one of the key spot rates change.
Tuckman uses par rates as key rates. How do we go from change in a key (par) rate to valuing any bond?
Say we change a 5-year key par rate and adjust the adjacent par rates linearly (as in figure 5.1 on page 156), and then, given the term structure of all these par rates (key and non-key) we compute a spot rate curve with which we price the security at hand?
Is that the procedure, or did I not understand it correctly?

3.
On page 157 Tuckman writes: "The C-STRIP curve on that day was taken as the base pricing curve, with the key-rate shifts superimposed as appropriate". What is C-STRIP curve, is it spot curve derived from C-STRIPS? If yes, how do we superimpose the par key-rate shifts on it? Or is C-STRIP curve - par curve? It is very unclear what Tuckman means.

4.
Why is 30-year zero affected by changes in 2-year (or 5, or 10 for that matter) par key-rate? I mean, 30-year zero is discounted with 30-year spot rate, which is not affected by the change of, say, 2-year par key rate and the adjacent par rates (as per specification of linear effect in figure 5.1)? Or do we compute the value of 30-year zero in some convoluted way through par curve, that is, we shock the 2-year par rate, then from the whole par curve compute the discount factors and then use bond pricing formula 2.25 on page 78, where price of bond is related to the 30-year par rate (C(T), unchanged in our case), coupon (c, zero in our case) and annuity factor (A(T), which has changed due to changing discount factors), like P= 1+A(T)*(coupon-C(T))/2 ?

5.
Also, in table 5.2, we see change of 1bp in 30-year par rate, which changes the price or 30-year zero from 26.22311 to 26.10121. I understand what KR01 = 0.1219 means, it means changing 30-year par rate by 1bp changes zero's price by 0.1219 USD per 100 face value. But what does key-rate duration of 46.49 mean? Does it mean that if I change 30-year par rate by 1% (ONE PERCENT) the price of 30-year zero will change by 46.49%? I mean, in the formula 5.4 we divide the change in price by 0.01% and multiply by inverse of current bond price to come up with duration, so we use the price related to the 0.01% key-rate change and not to the 1% key-rate change. I am confused about what duration of 46.49 means in this example.

Hopefully there is someone around here who can help shed some light on these questions.

Thank you in advance for help!